Optimal. Leaf size=38 \[ \frac{x \left (a+b x^n\right )^{p+1} \, _2F_1\left (1,p+\frac{1}{n}+1;1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a} \]
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Rubi [A] time = 0.0111188, antiderivative size = 46, normalized size of antiderivative = 1.21, number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {246, 245} \[ x \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} \, _2F_1\left (\frac{1}{n},-p;1+\frac{1}{n};-\frac{b x^n}{a}\right ) \]
Antiderivative was successfully verified.
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Rule 246
Rule 245
Rubi steps
\begin{align*} \int \left (a+b x^n\right )^p \, dx &=\left (\left (a+b x^n\right )^p \left (1+\frac{b x^n}{a}\right )^{-p}\right ) \int \left (1+\frac{b x^n}{a}\right )^p \, dx\\ &=x \left (a+b x^n\right )^p \left (1+\frac{b x^n}{a}\right )^{-p} \, _2F_1\left (\frac{1}{n},-p;1+\frac{1}{n};-\frac{b x^n}{a}\right )\\ \end{align*}
Mathematica [A] time = 0.0054949, size = 46, normalized size = 1.21 \[ x \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} \, _2F_1\left (\frac{1}{n},-p;1+\frac{1}{n};-\frac{b x^n}{a}\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.069, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b{x}^{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 (b x^{n} + 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 (b x^{n} + a\right )}^{p}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 1.31222, size = 37, normalized size = 0.97 \begin{align*} \frac{a^{p} x \Gamma \left (\frac{1}{n}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{n}, - p \\ 1 + \frac{1}{n} \end{matrix}\middle |{\frac{b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (1 + \frac{1}{n}\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^{n} + a\right )}^{p}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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