Optimal. Leaf size=19 \[ \frac{x^n (a+b x)^{-n}}{a n} \]
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Rubi [A] time = 0.00262, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {37} \[ \frac{x^n (a+b x)^{-n}}{a n} \]
Antiderivative was successfully verified.
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Rule 37
Rubi steps
\begin{align*} \int x^{-1+n} (a+b x)^{-1-n} \, dx &=\frac{x^n (a+b x)^{-n}}{a n}\\ \end{align*}
Mathematica [A] time = 0.0047965, size = 19, normalized size = 1. \[ \frac{x^n (a+b x)^{-n}}{a n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 20, normalized size = 1.1 \begin{align*}{\frac{{x}^{n} \left ( bx+a \right ) ^{-n}}{an}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.13209, size = 30, normalized size = 1.58 \begin{align*} \frac{e^{\left (-n \log \left (b x + a\right ) + n \log \left (x\right )\right )}}{a n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.6104, size = 68, normalized size = 3.58 \begin{align*} \frac{{\left (b x^{2} + a x\right )}{\left (b x + a\right )}^{-n - 1} x^{n - 1}}{a 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 - 1} x^{n - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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