Optimal. Leaf size=134 \[ \frac {a (e x)^{m+1}}{e (m+1)}+\frac {i b e^{i c} \left (-i d x^2\right )^{\frac {1}{2} (-m-1)} (e x)^{m+1} \Gamma \left (\frac {m+1}{2},-i d x^2\right )}{4 e}-\frac {i b e^{-i c} \left (i d x^2\right )^{\frac {1}{2} (-m-1)} (e x)^{m+1} \Gamma \left (\frac {m+1}{2},i d x^2\right )}{4 e} \]
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Rubi [A] time = 0.12, antiderivative size = 134, normalized size of antiderivative = 1.00, 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} \[ \frac {i b e^{i c} \left (-i d x^2\right )^{\frac {1}{2} (-m-1)} (e x)^{m+1} \text {Gamma}\left (\frac {m+1}{2},-i d x^2\right )}{4 e}-\frac {i b e^{-i c} \left (i d x^2\right )^{\frac {1}{2} (-m-1)} (e x)^{m+1} \text {Gamma}\left (\frac {m+1}{2},i d x^2\right )}{4 e}+\frac {a (e x)^{m+1}}{e (m+1)} \]
Antiderivative was successfully verified.
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Rule 14
Rule 2218
Rule 3389
Rubi steps
\begin {align*} \int (e x)^m \left (a+b \sin \left (c+d x^2\right )\right ) \, dx &=\int \left (a (e x)^m+b (e x)^m \sin \left (c+d x^2\right )\right ) \, dx\\ &=\frac {a (e x)^{1+m}}{e (1+m)}+b \int (e x)^m \sin \left (c+d x^2\right ) \, dx\\ &=\frac {a (e x)^{1+m}}{e (1+m)}+\frac {1}{2} (i b) \int e^{-i c-i d x^2} (e x)^m \, dx-\frac {1}{2} (i b) \int e^{i c+i d x^2} (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^2\right )^{\frac {1}{2} (-1-m)} \Gamma \left (\frac {1+m}{2},-i d x^2\right )}{4 e}-\frac {i b e^{-i c} (e x)^{1+m} \left (i d x^2\right )^{\frac {1}{2} (-1-m)} \Gamma \left (\frac {1+m}{2},i d x^2\right )}{4 e}\\ \end {align*}
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Mathematica [A] time = 1.51, size = 149, normalized size = 1.11 \[ \frac {x \left (d^2 x^4\right )^{\frac {1}{2} (-m-1)} (e x)^m \left (4 a \left (d^2 x^4\right )^{\frac {m+1}{2}}-i b (m+1) (\cos (c)-i \sin (c)) \left (-i d x^2\right )^{\frac {m+1}{2}} \Gamma \left (\frac {m+1}{2},i d x^2\right )+i b (m+1) (\cos (c)+i \sin (c)) \left (i d x^2\right )^{\frac {m+1}{2}} \Gamma \left (\frac {m+1}{2},-i d x^2\right )\right )}{4 (m+1)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.78, size = 98, normalized size = 0.73 \[ \frac {4 \, \left (e x\right )^{m} a d x - {\left (b e m + b e\right )} e^{\left (-\frac {1}{2} \, {\left (m - 1\right )} \log \left (\frac {i \, d}{e^{2}}\right ) - i \, c\right )} \Gamma \left (\frac {1}{2} \, m + \frac {1}{2}, i \, d x^{2}\right ) - {\left (b e m + b e\right )} e^{\left (-\frac {1}{2} \, {\left (m - 1\right )} \log \left (-\frac {i \, d}{e^{2}}\right ) + i \, c\right )} \Gamma \left (\frac {1}{2} \, m + \frac {1}{2}, -i \, d x^{2}\right )}{4 \, {\left (d m + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \sin \left (d x^{2} + c\right ) + a\right )} \left (e x\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.26, size = 0, normalized size = 0.00 \[ \int \left (e x \right )^{m} \left (a +b \sin \left (d \,x^{2}+c \right )\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ b e^{m} \int x^{m} \sin \left (d x^{2} + c\right )\,{d x} + \frac {\left (e x\right )^{m + 1} a}{e {\left (m + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (e\,x\right )}^m\,\left (a+b\,\sin \left (d\,x^2+c\right )\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (e x\right )^{m} \left (a + b \sin {\left (c + d x^{2} \right )}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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