Optimal. Leaf size=149 \[ \frac {2 (e x)^{m+1} \, _2F_1\left (-\frac {1}{2},-\frac {2 i m+b d n+2 i}{4 b d n};-\frac {2 i m-3 b d n+2 i}{4 b d n};e^{2 i a d} \left (c x^n\right )^{2 i b d}\right ) \sqrt {\sin \left (d \left (a+b \log \left (c x^n\right )\right )\right )}}{e (-i b d n+2 m+2) \sqrt {1-e^{2 i a d} \left (c x^n\right )^{2 i b d}}} \]
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Rubi [A] time = 0.11, 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} \[ \frac {2 (e x)^{m+1} \, _2F_1\left (-\frac {1}{2},\frac {1}{4} \left (-\frac {2 i (m+1)}{b d n}-1\right );-\frac {2 i m-3 b d n+2 i}{4 b d n};e^{2 i a d} \left (c x^n\right )^{2 i b d}\right ) \sqrt {\sin \left (d \left (a+b \log \left (c x^n\right )\right )\right )}}{e (-i b d n+2 m+2) \sqrt {1-e^{2 i a d} \left (c x^n\right )^{2 i b d}}} \]
Antiderivative was successfully verified.
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Rule 364
Rule 4491
Rule 4493
Rubi steps
\begin {align*} \int (e x)^m \sqrt {\sin \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}} \sqrt {\sin (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 {i b d}{2}-\frac {1+m}{n}} \sqrt {\sin \left (d \left (a+b \log \left (c x^n\right )\right )\right )}\right ) \operatorname {Subst}\left (\int x^{-1-\frac {i b d}{2}+\frac {1+m}{n}} \sqrt {1-e^{2 i a d} x^{2 i b d}} \, dx,x,c x^n\right )}{e n \sqrt {1-e^{2 i a d} \left (c x^n\right )^{2 i b d}}}\\ &=\frac {2 (e x)^{1+m} \, _2F_1\left (-\frac {1}{2},\frac {1}{4} \left (-1-\frac {2 i (1+m)}{b d n}\right );-\frac {2 i+2 i m-3 b d n}{4 b d n};e^{2 i a d} \left (c x^n\right )^{2 i b d}\right ) \sqrt {\sin \left (d \left (a+b \log \left (c x^n\right )\right )\right )}}{e (2+2 m-i b d n) \sqrt {1-e^{2 i a d} \left (c x^n\right )^{2 i b d}}}\\ \end {align*}
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Mathematica [B] time = 5.61, size = 488, normalized size = 3.28 \[ 2 x (e x)^m \left (\frac {\sqrt {\sin \left (d \left (a+b \log \left (c x^n\right )\right )\right )} \sin \left (d \left (a+b \log \left (c x^n\right )-b n \log (x)\right )\right )}{2 (m+1) \sin \left (d \left (a+b \log \left (c x^n\right )-b n \log (x)\right )\right )+b d n \cos \left (d \left (a+b \log \left (c x^n\right )-b n \log (x)\right )\right )}-\frac {b d n x^{-i b d n} \sqrt {2-2 e^{2 i a d} \left (c x^n\right )^{2 i b d}} e^{i d \left (a+b \log \left (c x^n\right )-b n \log (x)\right )} \left ((3 b d n-2 i m-2 i) \, _2F_1\left (\frac {1}{2},-\frac {2 i m+b d n+2 i}{4 b d n};-\frac {2 i m-3 b d n+2 i}{4 b d n};e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )+(b d n+2 i m+2 i) x^{2 i b d n} \, _2F_1\left (\frac {1}{2},-\frac {i \left (m+\frac {3}{2} i b d n+1\right )}{2 b d n};-\frac {2 i m-7 b d n+2 i}{4 b d n};e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )\right )}{(-i b d n+2 m+2) (3 i b d n+2 m+2) \sqrt {-i e^{-i a d} \left (c x^n\right )^{-i b d} \left (-1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )} \left ((b d n-2 i m-2 i) e^{2 i d \left (a+b \log \left (c x^n\right )-b n \log (x)\right )}+b d n+2 i m+2 i\right )}\right ) \]
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 \[ \int \left (e x\right )^{m} \sqrt {\sin \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.10, size = 0, normalized size = 0.00 \[ \int \left (e x \right )^{m} \left (\sqrt {\sin }\left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )\right )\, dx \]
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 \[ \int \left (e x\right )^{m} \sqrt {\sin \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )}\,{d x} \]
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 \[ \int \sqrt {\sin \left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )}\,{\left (e\,x\right )}^m \,d x \]
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 \[ \int \left (e x\right )^{m} \sqrt {\sin {\left (a d + b d \log {\left (c x^{n} \right )} \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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