Optimal. Leaf size=26 \[ \frac {1}{2} x^2 (-a-b x)^{-n} (a+b x)^n \]
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Rubi [A] time = 0.00, antiderivative size = 26, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {1}{2} x^2 (-a-b x)^{-n} (a+b x)^n \end {gather*}
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*}
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Mathematica [A] time = 0.00, size = 26, normalized size = 1.00 \begin {gather*} \frac {1}{2} x^2 (-a-b x)^{-n} (a+b x)^n \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.05, size = 34, normalized size = 1.31 \begin {gather*} \frac {(a-b x) (-a-b x)^{1-n} (a+b x)^n}{2 b^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.56, size = 9, normalized size = 0.35 \begin {gather*} \frac {1}{2} \, x^{2} \cos \left (\pi n\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.20, size = 5, normalized size = 0.19 \begin {gather*} \frac {1}{2} \, x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 25, normalized size = 0.96 \begin {gather*} \frac {x^{2} \left (-b x -a \right )^{-n} \left (b x +a \right )^{n}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.78, size = 8, normalized size = 0.31 \begin {gather*} \frac {1}{2} \, \left (-1\right )^{n} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.97, size = 24, normalized size = 0.92 \begin {gather*} \frac {x^2\,{\left (a+b\,x\right )}^n}{2\,{\left (-a-b\,x\right )}^n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 8.36, size = 19, normalized size = 0.73 \begin {gather*} \frac {x^{2} \left (- a - b x\right )^{- n} \left (a + b x\right )^{n}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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