Integrands of the form ''x''''m''(''a x'' + ''b'')''n''
Many of the following antiderivatives have a term of the form ln |ax + b|. Because this is undefined when x = −b / a, the most general form of the antiderivative replaces the constant of integration with a locally constant function. However, it is conventional to omit this from the notation. For example, is usually abbreviated as where C is to be understood as notation for a locally constant function of x. This convention will be adhered to in the following.
Integrands of the form ''x''''m'' / (''a x''2 + ''b x'' + ''c'')''n''
For
Integrands of the form ''x''''m'' (''a'' + ''b x''''n'')''p''
The resulting integrands are of the same form as the original integrand, so these reduction formulas can be repeatedly applied to drive the exponents m and p toward 0.
These reduction formulas can be used for integrands having integerand/orfractional exponents.
Integrands of the form (''A'' + ''B x'') (''a'' + ''b x'')''m'' (''c'' + ''d x'')''n'' (''e'' + ''f x'')''p''
The resulting integrands are of the same form as the original integrand, so these reduction formulas can be repeatedly applied to drive the exponents m, n and p toward 0.
These reduction formulas can be used for integrands having integer and/or fractional exponents.
Special cases of these reductions formulas can be used for integrands of the form by setting B to 0.
Integrands of the form ''x''''m'' (''A'' + ''B x''''n'') (''a'' + ''b x''''n'')''p'' (''c'' + ''d x''''n'')''q''
The resulting integrands are of the same form as the original integrand, so these reduction formulas can be repeatedly applied to drive the exponents m, p and q toward 0.
These reduction formulas can be used for integrands having integer and/or fractional exponents.
Special cases of these reductions formulas can be used for integrands of the form and by setting m and/or B to 0.
Integrands of the form (''d'' + ''e x'')''m'' (''a'' + ''b x'' + ''c x''2)''p'' when ''b''2 − 4 ''a c'' = 0
The resulting integrands are of the same form as the original integrand, so these reduction formulas can be repeatedly applied to drive the exponents m and p toward 0.
These reduction formulas can be used for integrands having integer and/or fractional exponents.
Special cases of these reductions formulas can be used for integrands of the form when by setting m to 0.
Integrands of the form (''d'' + ''e x'')''m'' (''A'' + ''B x'') (''a'' + ''b x'' + ''c x''2)''p''
The resulting integrands are of the same form as the original integrand, so these reduction formulas can be repeatedly applied to drive the exponents m and p toward 0.
These reduction formulas can be used for integrands having integer and/or fractional exponents.
Special cases of these reductions formulas can be used for integrands of the form and by setting m and/or B to 0.
Integrands of the form ''x''''m'' (''a'' + ''b x''''n'' + ''c x''2''n'')''p'' when ''b''2 − 4 ''a c'' = 0
The resulting integrands are of the same form as the original integrand, so these reduction formulas can be repeatedly applied to drive the exponents m and p toward 0.
These reduction formulas can be used for integrands having integer and/or fractional exponents.
Special cases of these reductions formulas can be used for integrands of the form when by setting m to 0.
Integrands of the form ''x''''m'' (''A'' + ''B x''''n'') (''a'' + ''b x''''n'' + ''c x''2''n'')''p''
The resulting integrands are of the same form as the original integrand, so these reduction formulas can be repeatedly applied to drive the exponents m and p toward 0.
These reduction formulas can be used for integrands having integer and/or fractional exponents.
Special cases of these reductions formulas can be used for integrands of the form and by setting m and/or B to 0.
