Optimal. Leaf size=13 \[ x^m \left (a+b x^n\right )^p \]
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Rubi [A] time = 0.0138087, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.032, Rules used = {449} \[ x^m \left (a+b x^n\right )^p \]
Antiderivative was successfully verified.
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Rule 449
Rubi steps
\begin{align*} \int x^{-1+m} \left (a+b x^n\right )^{-1+p} \left (a m+b (m+n p) x^n\right ) \, dx &=x^m \left (a+b x^n\right )^p\\ \end{align*}
Mathematica [C] time = 0.155663, size = 107, normalized size = 8.23 \[ \frac{x^m \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} \left (b x^n (m+n p) \, _2F_1\left (\frac{m+n}{n},1-p;\frac{m}{n}+2;-\frac{b x^n}{a}\right )+a (m+n) \, _2F_1\left (\frac{m}{n},1-p;\frac{m+n}{n};-\frac{b x^n}{a}\right )\right )}{a (m+n)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.075, size = 0, normalized size = 0. \begin{align*} \int{x}^{-1+m} \left ( a+b{x}^{n} \right ) ^{-1+p} \left ( am+b \left ( pn+m \right ){x}^{n} \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.15387, size = 22, normalized size = 1.69 \begin{align*} e^{\left (p \log \left (b x^{n} + a\right ) + m \log \left (x\right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.01659, size = 77, normalized size = 5.92 \begin{align*}{\left (b x x^{m - 1} x^{n} + a x x^{m - 1}\right )}{\left (b x^{n} + a\right )}^{p - 1} \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 [B] time = 1.60004, size = 95, normalized size = 7.31 \begin{align*} b x x^{n} e^{\left (p \log \left (b x^{n} + a\right ) + m \log \left (x\right ) - \log \left (b x^{n} + a\right ) - \log \left (x\right )\right )} + a x e^{\left (p \log \left (b x^{n} + a\right ) + m \log \left (x\right ) - \log \left (b x^{n} + a\right ) - \log \left (x\right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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