Optimal. Leaf size=21 \[ x (-a-b x)^{-n} (a+b x)^n \]
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Rubi [A] time = 0.0026255, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {23, 8} \[ x (-a-b x)^{-n} (a+b x)^n \]
Antiderivative was successfully verified.
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Rule 23
Rule 8
Rubi steps
\begin{align*} \int (-a-b x)^{-n} (a+b x)^n \, dx &=\left ((-a-b x)^{-n} (a+b x)^n\right ) \int 1 \, dx\\ &=x (-a-b x)^{-n} (a+b x)^n\\ \end{align*}
Mathematica [A] time = 0.0018964, size = 21, normalized size = 1. \[ x (-a-b x)^{-n} (a+b x)^n \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 26, normalized size = 1.2 \begin{align*}{\frac{x{{\rm e}^{n\ln \left ( bx+a \right ) }}}{{{\rm e}^{n\ln \left ( -bx-a \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.3917, size = 7, normalized size = 0.33 \begin{align*} \left (-1\right )^{n} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.71775, size = 18, normalized size = 0.86 \begin{align*} x \cos \left (\pi n\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 7.60881, size = 15, normalized size = 0.71 \begin{align*} x \left (- a - b x\right )^{- n} \left (a + b x\right )^{n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 3.27721, size = 1, normalized size = 0.05 \begin{align*} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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