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 = 0.01, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.030, Rules used = {449} \begin {gather*} \frac {(e x)^{m+1} \left (a+b x^n\right )^{p+1}}{e} \end {gather*}
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*}
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Mathematica [C] time = 0.17, size = 110, normalized size = 5.00 \begin {gather*} 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 ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.26, size = 0, normalized size = 0.00 \begin {gather*} \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 \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.41, size = 40, normalized size = 1.82 \begin {gather*} {\left (b x x^{n} e^{\left (m \log \relax (e) + m \log \relax (x)\right )} + a x e^{\left (m \log \relax (e) + m \log \relax (x)\right )}\right )} {\left (b x^{n} + a\right )}^{p} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, size = 38, normalized size = 1.73 \begin {gather*} {\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.67, size = 0, normalized size = 0.00 \begin {gather*} \int \left (\left (n p +m +n +1\right ) b \,x^{n}+\left (m +1\right ) a \right ) \left (e x \right )^{m} \left (b \,x^{n}+a \right )^{p}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.95, size = 36, normalized size = 1.64 \begin {gather*} {\left (a e^{m} x x^{m} + b e^{m} x e^{\left (m \log \relax (x) + n \log \relax (x)\right )}\right )} {\left (b x^{n} + a\right )}^{p} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.94, size = 31, normalized size = 1.41 \begin {gather*} \left (a\,x\,{\left (e\,x\right )}^m+b\,x^{n+1}\,{\left (e\,x\right )}^m\right )\,{\left (a+b\,x^n\right )}^p \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 8.43, size = 39, normalized size = 1.77 \begin {gather*} 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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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