Optimal. Leaf size=134 \[ \frac{i b e^{i c} \left (-i d x^3\right )^{\frac{1}{3} (-m-1)} (e x)^{m+1} \text{Gamma}\left (\frac{m+1}{3},-i d x^3\right )}{6 e}-\frac{i b e^{-i c} \left (i d x^3\right )^{\frac{1}{3} (-m-1)} (e x)^{m+1} \text{Gamma}\left (\frac{m+1}{3},i d x^3\right )}{6 e}+\frac{a (e x)^{m+1}}{e (m+1)} \]
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Rubi [A] time = 0.0995589, antiderivative size = 134, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {14, 3389, 2218} \[ \frac{i b e^{i c} \left (-i d x^3\right )^{\frac{1}{3} (-m-1)} (e x)^{m+1} \text{Gamma}\left (\frac{m+1}{3},-i d x^3\right )}{6 e}-\frac{i b e^{-i c} \left (i d x^3\right )^{\frac{1}{3} (-m-1)} (e x)^{m+1} \text{Gamma}\left (\frac{m+1}{3},i d x^3\right )}{6 e}+\frac{a (e x)^{m+1}}{e (m+1)} \]
Antiderivative was successfully verified.
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Rule 14
Rule 3389
Rule 2218
Rubi steps
\begin{align*} \int (e x)^m \left (a+b \sin \left (c+d x^3\right )\right ) \, dx &=\int \left (a (e x)^m+b (e x)^m \sin \left (c+d x^3\right )\right ) \, dx\\ &=\frac{a (e x)^{1+m}}{e (1+m)}+b \int (e x)^m \sin \left (c+d x^3\right ) \, dx\\ &=\frac{a (e x)^{1+m}}{e (1+m)}+\frac{1}{2} (i b) \int e^{-i c-i d x^3} (e x)^m \, dx-\frac{1}{2} (i b) \int e^{i c+i d x^3} (e x)^m \, dx\\ &=\frac{a (e x)^{1+m}}{e (1+m)}+\frac{i b e^{i c} (e x)^{1+m} \left (-i d x^3\right )^{\frac{1}{3} (-1-m)} \Gamma \left (\frac{1+m}{3},-i d x^3\right )}{6 e}-\frac{i b e^{-i c} (e x)^{1+m} \left (i d x^3\right )^{\frac{1}{3} (-1-m)} \Gamma \left (\frac{1+m}{3},i d x^3\right )}{6 e}\\ \end{align*}
Mathematica [A] time = 1.60107, size = 149, normalized size = 1.11 \[ \frac{x \left (d^2 x^6\right )^{\frac{1}{3} (-m-1)} (e x)^m \left (-i b (m+1) (\cos (c)-i \sin (c)) \left (-i d x^3\right )^{\frac{m+1}{3}} \text{Gamma}\left (\frac{m+1}{3},i d x^3\right )+i b (m+1) (\cos (c)+i \sin (c)) \left (i d x^3\right )^{\frac{m+1}{3}} \text{Gamma}\left (\frac{m+1}{3},-i d x^3\right )+6 a \left (d^2 x^6\right )^{\frac{m+1}{3}}\right )}{6 (m+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.124, size = 0, normalized size = 0. \begin{align*} \int \left ( ex \right ) ^{m} \left ( a+b\sin \left ( d{x}^{3}+c \right ) \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.74211, size = 274, normalized size = 2.04 \begin{align*} \frac{6 \, \left (e x\right )^{m} a d x -{\left (b e^{2} m + b e^{2}\right )} e^{\left (-\frac{1}{3} \,{\left (m - 2\right )} \log \left (\frac{i \, d}{e^{3}}\right ) - i \, c\right )} \Gamma \left (\frac{1}{3} \, m + \frac{1}{3}, i \, d x^{3}\right ) -{\left (b e^{2} m + b e^{2}\right )} e^{\left (-\frac{1}{3} \,{\left (m - 2\right )} \log \left (-\frac{i \, d}{e^{3}}\right ) + i \, c\right )} \Gamma \left (\frac{1}{3} \, m + \frac{1}{3}, -i \, d x^{3}\right )}{6 \,{\left (d m + d\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 (e x\right )^{m} \left (a + b \sin{\left (c + d x^{3} \right )}\right )\, 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 (b \sin \left (d x^{3} + c\right ) + a\right )} \left (e x\right )^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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