Optimal. Leaf size=149 \[ \frac{2 (e x)^{m+1} \sqrt{\sin \left (d \left (a+b \log \left (c x^n\right )\right )\right )} \text{Hypergeometric2F1}\left (-\frac{1}{2},-\frac{b d n+2 i m+2 i}{4 b d n},-\frac{-3 b d n+2 i m+2 i}{4 b d n},e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{e (-i b d n+2 m+2) \sqrt{1-e^{2 i a d} \left (c x^n\right )^{2 i b d}}} \]
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Rubi [A] time = 0.11187, antiderivative size = 145, normalized size of antiderivative = 0.97, number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, 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 4493
Rule 4491
Rule 364
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*}
Mathematica [B] time = 5.46119, size = 488, normalized size = 3.28 \[ 2 x (e x)^m \left (\frac{\sqrt{\sin \left (d \left (a+b \log \left (c x^n\right )\right )\right )} \sin \left (d \left (a+b \log \left (c x^n\right )-b n \log (x)\right )\right )}{2 (m+1) \sin \left (d \left (a+b \log \left (c x^n\right )-b n \log (x)\right )\right )+b d n \cos \left (d \left (a+b \log \left (c x^n\right )-b n \log (x)\right )\right )}-\frac{b d n x^{-i b d n} \sqrt{2-2 e^{2 i a d} \left (c x^n\right )^{2 i b d}} e^{i d \left (a+b \log \left (c x^n\right )-b n \log (x)\right )} \left ((b d n+2 i m+2 i) x^{2 i b d n} \text{Hypergeometric2F1}\left (\frac{1}{2},-\frac{i \left (\frac{3}{2} i b d n+m+1\right )}{2 b d n},-\frac{-7 b d n+2 i m+2 i}{4 b d n},e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )+(3 b d n-2 i m-2 i) \text{Hypergeometric2F1}\left (\frac{1}{2},-\frac{b d n+2 i m+2 i}{4 b d n},-\frac{-3 b d n+2 i m+2 i}{4 b d n},e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )\right )}{(-i b d n+2 m+2) (3 i b d n+2 m+2) \sqrt{-i e^{-i a d} \left (c x^n\right )^{-i b d} \left (-1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )} \left ((b d n-2 i m-2 i) e^{2 i d \left (a+b \log \left (c x^n\right )-b n \log (x)\right )}+b d n+2 i m+2 i\right )}\right ) \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.269, size = 0, normalized size = 0. \begin{align*} \int \left ( ex \right ) ^{m}\sqrt{\sin \left ( d \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) }\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (e x\right )^{m} \sqrt{\sin \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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