Optimal. Leaf size=45 \[ \frac{x^{n+1} (a+b x)^{-n} \left (\frac{b x}{a}+1\right )^n \, _2F_1\left (n,n+1;n+2;-\frac{b x}{a}\right )}{n+1} \]
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Rubi [A] time = 0.0102049, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {66, 64} \[ \frac{x^{n+1} (a+b x)^{-n} \left (\frac{b x}{a}+1\right )^n \, _2F_1\left (n,n+1;n+2;-\frac{b x}{a}\right )}{n+1} \]
Antiderivative was successfully verified.
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Rule 66
Rule 64
Rubi steps
\begin{align*} \int x^n (a+b x)^{-n} \, dx &=\left ((a+b x)^{-n} \left (1+\frac{b x}{a}\right )^n\right ) \int x^n \left (1+\frac{b x}{a}\right )^{-n} \, dx\\ &=\frac{x^{1+n} (a+b x)^{-n} \left (1+\frac{b x}{a}\right )^n \, _2F_1\left (n,1+n;2+n;-\frac{b x}{a}\right )}{1+n}\\ \end{align*}
Mathematica [A] time = 0.0070472, size = 45, normalized size = 1. \[ \frac{x^{n+1} (a+b x)^{-n} \left (\frac{b x}{a}+1\right )^n \, _2F_1\left (n,n+1;n+2;-\frac{b x}{a}\right )}{n+1} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.048, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{n}}{ \left ( bx+a \right ) ^{n}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{n}}{{\left (b x + a\right )}^{n}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x^{n}}{{\left (b x + a\right )}^{n}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 41.13, size = 32, normalized size = 0.71 \begin{align*} \frac{a^{- n} x x^{n} \Gamma \left (n + 1\right ){{}_{2}F_{1}\left (\begin{matrix} n, n + 1 \\ n + 2 \end{matrix}\middle |{\frac{b x e^{i \pi }}{a}} \right )}}{\Gamma \left (n + 2\right )} \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 \frac{x^{n}}{{\left (b x + a\right )}^{n}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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