Optimal. Leaf size=47 \[ \frac{x^{m+1} (a+b x)^n \left (\frac{b x}{a}+1\right )^{-n} \, _2F_1\left (m+1,-n;m+2;-\frac{b x}{a}\right )}{m+1} \]
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Rubi [A] time = 0.0111344, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {66, 64} \[ \frac{x^{m+1} (a+b x)^n \left (\frac{b x}{a}+1\right )^{-n} \, _2F_1\left (m+1,-n;m+2;-\frac{b x}{a}\right )}{m+1} \]
Antiderivative was successfully verified.
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Rule 66
Rule 64
Rubi steps
\begin{align*} \int x^m (a+b x)^n \, dx &=\left ((a+b x)^n \left (1+\frac{b x}{a}\right )^{-n}\right ) \int x^m \left (1+\frac{b x}{a}\right )^n \, dx\\ &=\frac{x^{1+m} (a+b x)^n \left (1+\frac{b x}{a}\right )^{-n} \, _2F_1\left (1+m,-n;2+m;-\frac{b x}{a}\right )}{1+m}\\ \end{align*}
Mathematica [A] time = 0.0096855, size = 47, normalized size = 1. \[ \frac{x^{m+1} (a+b x)^n \left (\frac{b x}{a}+1\right )^{-n} \, _2F_1\left (m+1,-n;m+2;-\frac{b x}{a}\right )}{m+1} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.054, size = 0, normalized size = 0. \begin{align*} \int{x}^{m} \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{\left (b x + a\right )}^{n} x^{m}\,{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 ({\left (b x + a\right )}^{n} x^{m}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 2.69871, size = 34, normalized size = 0.72 \begin{align*} \frac{a^{n} x x^{m} \Gamma \left (m + 1\right ){{}_{2}F_{1}\left (\begin{matrix} - n, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{b x e^{i \pi }}{a}} \right )}}{\Gamma \left (m + 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{\left (b x + a\right )}^{n} x^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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