Optimal. Leaf size=36 \[ \frac{(a+b x)^{m+1} (c+d x)^{-m-1}}{(m+1) (b c-a d)} \]
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Rubi [A] time = 0.005813, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {37} \[ \frac{(a+b x)^{m+1} (c+d x)^{-m-1}}{(m+1) (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 37
Rubi steps
\begin{align*} \int (a+b x)^m (c+d x)^{-2-m} \, dx &=\frac{(a+b x)^{1+m} (c+d x)^{-1-m}}{(b c-a d) (1+m)}\\ \end{align*}
Mathematica [A] time = 0.0113719, size = 36, normalized size = 1. \[ \frac{(a+b x)^{m+1} (c+d x)^{-m-1}}{(m+1) (b c-a d)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 42, normalized size = 1.2 \begin{align*} -{\frac{ \left ( bx+a \right ) ^{1+m} \left ( dx+c \right ) ^{-1-m}}{adm-bcm+ad-bc}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m - 2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.58567, size = 124, normalized size = 3.44 \begin{align*} \frac{{\left (b d x^{2} + a c +{\left (b c + a d\right )} x\right )}{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m - 2}}{b c - a d +{\left (b c - a d\right )} m} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m - 2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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