Optimal. Leaf size=22 \[ \frac{(e x)^{m+1} \left (a+b x^n\right )^{p+1}}{e} \]
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Rubi [A] time = 0.0148312, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.03, Rules used = {449} \[ \frac{(e x)^{m+1} \left (a+b x^n\right )^{p+1}}{e} \]
Antiderivative was successfully verified.
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Rule 449
Rubi steps
\begin{align*} \int (e x)^m \left (a+b x^n\right )^p \left (a (1+m)+b (1+m+n+n p) x^n\right ) \, dx &=\frac{(e x)^{1+m} \left (a+b x^n\right )^{1+p}}{e}\\ \end{align*}
Mathematica [C] time = 0.12954, size = 110, normalized size = 5. \[ x (e x)^m \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} \left (\frac{b x^n (m+n p+n+1) \, _2F_1\left (\frac{m+n+1}{n},-p;\frac{m+2 n+1}{n};-\frac{b x^n}{a}\right )}{m+n+1}+a \, _2F_1\left (\frac{m+1}{n},-p;\frac{m+n+1}{n};-\frac{b x^n}{a}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.07, size = 0, normalized size = 0. \begin{align*} \int \left ( ex \right ) ^{m} \left ( a+b{x}^{n} \right ) ^{p} \left ( a \left ( 1+m \right ) +b \left ( pn+m+n+1 \right ){x}^{n} \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.19691, size = 49, normalized size = 2.23 \begin{align*}{\left (a e^{m} x x^{m} + b e^{m} x e^{\left (m \log \left (x\right ) + n \log \left (x\right )\right )}\right )}{\left (b x^{n} + a\right )}^{p} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.03107, size = 107, normalized size = 4.86 \begin{align*}{\left (b x x^{n} e^{\left (m \log \left (e\right ) + m \log \left (x\right )\right )} + a x e^{\left (m \log \left (e\right ) + m \log \left (x\right )\right )}\right )}{\left (b x^{n} + a\right )}^{p} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 9.47734, size = 39, normalized size = 1.77 \begin{align*} a e^{m} x x^{m} \left (a + b x^{n}\right )^{p} + b e^{m} x x^{m} x^{n} \left (a + b x^{n}\right )^{p} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09913, size = 51, normalized size = 2.32 \begin{align*}{\left (b x^{n} + a\right )}^{p} b x x^{m} x^{n} e^{m} +{\left (b x^{n} + a\right )}^{p} a x x^{m} e^{m} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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