Optimal. Leaf size=26 \[ \frac{1}{2} x^2 (-a-b x)^{-n} (a+b x)^n \]
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Rubi [A] time = 0.0030901, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {23, 30} \[ \frac{1}{2} x^2 (-a-b x)^{-n} (a+b x)^n \]
Antiderivative was successfully verified.
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Rule 23
Rule 30
Rubi steps
\begin{align*} \int x (-a-b x)^{-n} (a+b x)^n \, dx &=\left ((-a-b x)^{-n} (a+b x)^n\right ) \int x \, dx\\ &=\frac{1}{2} x^2 (-a-b x)^{-n} (a+b x)^n\\ \end{align*}
Mathematica [A] time = 0.0027887, size = 26, normalized size = 1. \[ \frac{1}{2} x^2 (-a-b x)^{-n} (a+b x)^n \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 25, normalized size = 1. \begin{align*}{\frac{{x}^{2} \left ( bx+a \right ) ^{n}}{2\, \left ( -bx-a \right ) ^{n}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.18332, size = 11, normalized size = 0.42 \begin{align*} \frac{1}{2} \, \left (-1\right )^{n} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.80333, size = 26, normalized size = 1. \begin{align*} \frac{1}{2} \, x^{2} \cos \left (\pi n\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 20.1702, size = 19, normalized size = 0.73 \begin{align*} \frac{x^{2} \left (- a - b x\right )^{- n} \left (a + b x\right )^{n}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.40948, size = 7, normalized size = 0.27 \begin{align*} \frac{1}{2} \, x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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