Optimal. Leaf size=21 \[ x (-a-b x)^{-n} (a+b x)^n \]
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Rubi [A] time = 0.00, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {23, 8} \begin {gather*} x (-a-b x)^{-n} (a+b x)^n \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 23
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*}
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Mathematica [A] time = 0.00, size = 21, normalized size = 1.00 \begin {gather*} x (-a-b x)^{-n} (a+b x)^n \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.03, size = 25, normalized size = 1.19 \begin {gather*} \frac {(-a-b x)^{-n} (a+b x)^{n+1}}{b} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 2.63, size = 6, normalized size = 0.29 \begin {gather*} x \cos \left (\pi n\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.15, size = 1, normalized size = 0.05 \begin {gather*} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 26, normalized size = 1.24 \begin {gather*} x \,{\mathrm e}^{n \ln \left (b x +a \right )} {\mathrm e}^{-n \ln \left (-b x -a \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.99, size = 5, normalized size = 0.24 \begin {gather*} \left (-1\right )^{n} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.26, size = 21, normalized size = 1.00 \begin {gather*} \frac {x\,{\left (a+b\,x\right )}^n}{{\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 = 7.05, size = 44, normalized size = 2.10 \begin {gather*} \begin {cases} - \frac {a \left (- a - b x\right )^{- n} \left (a + b x\right )^{n}}{b} + x \left (- a - b x\right )^{- n} \left (a + b x\right )^{n} & \text {for}\: b \neq 0 \\a^{n} x \left (- a\right )^{- n} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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