Optimal. Leaf size=150 \[ \frac {2 (e x)^{m+1} \, _2F_1\left (-\frac {3}{2},-\frac {2 i m+3 b d n+2 i}{4 b d n};-\frac {2 i m-b d n+2 i}{4 b d n};e^{2 i a d} \left (c x^n\right )^{2 i b d}\right ) \sin ^{\frac {3}{2}}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (-3 i b d n+2 m+2) \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^{3/2}} \]
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Rubi [A] time = 0.13, 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 = {4493, 4491, 364} \[ \frac {2 (e x)^{m+1} \, _2F_1\left (-\frac {3}{2},\frac {1}{4} \left (-\frac {2 i (m+1)}{b d n}-3\right );-\frac {2 i m-b d n+2 i}{4 b d n};e^{2 i a d} \left (c x^n\right )^{2 i b d}\right ) \sin ^{\frac {3}{2}}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (-3 i b d n+2 m+2) \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 364
Rule 4491
Rule 4493
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 ) \operatorname {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 ) \operatorname {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 = 2.04, size = 235, normalized size = 1.57 \[ \frac {2 (e x)^m \left (x (i b d n+2 m+2) \sin \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \left (2 (m+1) \sin \left (d \left (a+b \log \left (c x^n\right )\right )\right )-3 b d n \cos \left (d \left (a+b \log \left (c x^n\right )\right )\right )\right )-3 b^2 d^2 n^2 x \left (-1+e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right ) \, _2F_1\left (1,-\frac {2 i m-3 b d n+2 i}{4 b d n};-\frac {2 i m-5 b d n+2 i}{4 b d n};e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )\right )}{(i b d n+2 m+2) (-3 i b d n+2 m+2) (3 i b d n+2 m+2) \sqrt {\sin \left (d \left (a+b \log \left (c x^n\right )\right )\right )}} \]
Warning: Unable to verify antiderivative.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
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 \[ \int \left (e x\right )^{m} \sin \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.35, 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 \[ \int \left (e x\right )^{m} \sin \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )^{\frac {3}{2}}\,{d x} \]
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 \[ \int {\sin \left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )}^{3/2}\,{\left (e\,x\right )}^m \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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