Optimal. Leaf size=150 \[ \frac {2 (e x)^{1+m} \, _2F_1\left (-\frac {3}{2},-\frac {2 i+2 i m+3 b d n}{4 b d n};-\frac {2 i+2 i m-b d n}{4 b d n};e^{2 i a d} \left (c x^n\right )^{2 i b d}\right ) \sin ^{\frac {3}{2}}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (2+2 m-3 i b d n) \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^{3/2}} \]
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Rubi [A]
time = 0.09, antiderivative size = 145, normalized size of antiderivative = 0.97, number of steps
used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {4581, 4579,
371} \begin {gather*} \frac {2 (e x)^{m+1} \, _2F_1\left (-\frac {3}{2},\frac {1}{4} \left (-\frac {2 i (m+1)}{b d n}-3\right );-\frac {2 i m-b d n+2 i}{4 b d n};e^{2 i a d} \left (c x^n\right )^{2 i b d}\right ) \sin ^{\frac {3}{2}}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (-3 i b d n+2 m+2) \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 371
Rule 4579
Rule 4581
Rubi steps
\begin {align*} \int (e x)^m \sin ^{\frac {3}{2}}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx &=\frac {\left ((e x)^{1+m} \left (c x^n\right )^{-\frac {1+m}{n}}\right ) \text {Subst}\left (\int x^{-1+\frac {1+m}{n}} \sin ^{\frac {3}{2}}(d (a+b \log (x))) \, dx,x,c x^n\right )}{e n}\\ &=\frac {\left ((e x)^{1+m} \left (c x^n\right )^{\frac {3 i b d}{2}-\frac {1+m}{n}} \sin ^{\frac {3}{2}}\left (d \left (a+b \log \left (c x^n\right )\right )\right )\right ) \text {Subst}\left (\int x^{-1-\frac {3 i b d}{2}+\frac {1+m}{n}} \left (1-e^{2 i a d} x^{2 i b d}\right )^{3/2} \, dx,x,c x^n\right )}{e n \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^{3/2}}\\ &=\frac {2 (e x)^{1+m} \, _2F_1\left (-\frac {3}{2},\frac {1}{4} \left (-3-\frac {2 i (1+m)}{b d n}\right );-\frac {2 i+2 i m-b d n}{4 b d n};e^{2 i a d} \left (c x^n\right )^{2 i b d}\right ) \sin ^{\frac {3}{2}}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (2+2 m-3 i b d n) \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 1.81, size = 256, normalized size = 1.71 \begin {gather*} \frac {2 (e x)^m \left (\frac {3 b^2 d^2 \sqrt {2-2 e^{2 i d \left (a+b \log \left (c x^n\right )\right )}} n^2 x \, _2F_1\left (\frac {1}{2},\frac {-2 i-2 i m+b d n}{4 b d n};-\frac {2 i+2 i m-5 b d n}{4 b d n};e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )}{\sqrt {-i e^{-i d \left (a+b \log \left (c x^n\right )\right )} \left (-1+e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )} (2+2 m+i b d n)}+x \sqrt {\sin \left (d \left (a+b \log \left (c x^n\right )\right )\right )} \left (-3 b d n \cos \left (d \left (a+b \log \left (c x^n\right )\right )\right )+2 (1+m) \sin \left (d \left (a+b \log \left (c x^n\right )\right )\right )\right )\right )}{4+8 m+4 m^2+9 b^2 d^2 n^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.09, size = 0, normalized size = 0.00 \[\int \left (e x \right )^{m} \left (\sin ^{\frac {3}{2}}\left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )\right )\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int {\sin \left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )}^{3/2}\,{\left (e\,x\right )}^m \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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