Optimal. Leaf size=37 \[ -\frac{(a+b x)^{-n-1} (c+d x)^{n+1}}{(n+1) (b c-a d)} \]
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Rubi [A] time = 0.0059839, antiderivative size = 37, 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} \[ -\frac{(a+b x)^{-n-1} (c+d x)^{n+1}}{(n+1) (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 37
Rubi steps
\begin{align*} \int (a+b x)^{-2-n} (c+d x)^n \, dx &=-\frac{(a+b x)^{-1-n} (c+d x)^{1+n}}{(b c-a d) (1+n)}\\ \end{align*}
Mathematica [A] time = 0.0127175, size = 38, normalized size = 1.03 \[ \frac{(a+b x)^{-n-1} (c+d x)^{n+1}}{(-n-1) (b c-a d)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 41, normalized size = 1.1 \begin{align*}{\frac{ \left ( bx+a \right ) ^{-1-n} \left ( dx+c \right ) ^{1+n}}{adn-bcn+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 )}^{-n - 2}{\left (d x + c\right )}^{n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.91859, size = 126, normalized size = 3.41 \begin{align*} -\frac{{\left (b d x^{2} + a c +{\left (b c + a d\right )} x\right )}{\left (b x + a\right )}^{-n - 2}{\left (d x + c\right )}^{n}}{b c - a d +{\left (b c - a d\right )} n} \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 )}^{-n - 2}{\left (d x + c\right )}^{n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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