Optimal. Leaf size=52 \[ x \left (a+b (c x)^n\right )^p \left (\frac{b (c x)^n}{a}+1\right )^{-p} \, _2F_1\left (\frac{1}{n},-p;1+\frac{1}{n};-\frac{b (c x)^n}{a}\right ) \]
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Rubi [A] time = 0.0256385, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {247, 246, 245} \[ x \left (a+b (c x)^n\right )^p \left (\frac{b (c x)^n}{a}+1\right )^{-p} \, _2F_1\left (\frac{1}{n},-p;1+\frac{1}{n};-\frac{b (c x)^n}{a}\right ) \]
Antiderivative was successfully verified.
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Rule 247
Rule 246
Rule 245
Rubi steps
\begin{align*} \int \left (a+b (c x)^n\right )^p \, dx &=\frac{\operatorname{Subst}\left (\int \left (a+b x^n\right )^p \, dx,x,c x\right )}{c}\\ &=\frac{\left (\left (a+b (c x)^n\right )^p \left (1+\frac{b (c x)^n}{a}\right )^{-p}\right ) \operatorname{Subst}\left (\int \left (1+\frac{b x^n}{a}\right )^p \, dx,x,c x\right )}{c}\\ &=x \left (a+b (c x)^n\right )^p \left (1+\frac{b (c x)^n}{a}\right )^{-p} \, _2F_1\left (\frac{1}{n},-p;1+\frac{1}{n};-\frac{b (c x)^n}{a}\right )\\ \end{align*}
Mathematica [A] time = 0.0073823, size = 52, normalized size = 1. \[ x \left (a+b (c x)^n\right )^p \left (\frac{b (c x)^n}{a}+1\right )^{-p} \, _2F_1\left (\frac{1}{n},-p;1+\frac{1}{n};-\frac{b (c x)^n}{a}\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.081, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b \left ( cx \right ) ^{n} \right ) ^{p}\, 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 (\left (c x\right )^{n} b + a\right )}^{p}\,{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 (\left (c x\right )^{n} b + a\right )}^{p}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \left (c x\right )^{n}\right )^{p}\, dx \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 (\left (c x\right )^{n} b + a\right )}^{p}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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