Optimal. Leaf size=126 \[ -\frac{24 a^2 b^2 E\left (\left .\frac{1}{2} (e+f x)\right |2\right ) \sqrt{a \sin (e+f x)}}{5 f \sqrt{\cos (e+f x)} \sqrt{b \tan (e+f x)}}+\frac{12 a^2 b \sqrt{a \sin (e+f x)} \sqrt{b \tan (e+f x)}}{5 f}-\frac{2 b (a \sin (e+f x))^{5/2} \sqrt{b \tan (e+f x)}}{5 f} \]
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Rubi [A] time = 0.169058, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {2598, 2594, 2601, 2639} \[ -\frac{24 a^2 b^2 E\left (\left .\frac{1}{2} (e+f x)\right |2\right ) \sqrt{a \sin (e+f x)}}{5 f \sqrt{\cos (e+f x)} \sqrt{b \tan (e+f x)}}+\frac{12 a^2 b \sqrt{a \sin (e+f x)} \sqrt{b \tan (e+f x)}}{5 f}-\frac{2 b (a \sin (e+f x))^{5/2} \sqrt{b \tan (e+f x)}}{5 f} \]
Antiderivative was successfully verified.
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Rule 2598
Rule 2594
Rule 2601
Rule 2639
Rubi steps
\begin{align*} \int (a \sin (e+f x))^{5/2} (b \tan (e+f x))^{3/2} \, dx &=-\frac{2 b (a \sin (e+f x))^{5/2} \sqrt{b \tan (e+f x)}}{5 f}+\frac{1}{5} \left (6 a^2\right ) \int \sqrt{a \sin (e+f x)} (b \tan (e+f x))^{3/2} \, dx\\ &=\frac{12 a^2 b \sqrt{a \sin (e+f x)} \sqrt{b \tan (e+f x)}}{5 f}-\frac{2 b (a \sin (e+f x))^{5/2} \sqrt{b \tan (e+f x)}}{5 f}-\frac{1}{5} \left (12 a^2 b^2\right ) \int \frac{\sqrt{a \sin (e+f x)}}{\sqrt{b \tan (e+f x)}} \, dx\\ &=\frac{12 a^2 b \sqrt{a \sin (e+f x)} \sqrt{b \tan (e+f x)}}{5 f}-\frac{2 b (a \sin (e+f x))^{5/2} \sqrt{b \tan (e+f x)}}{5 f}-\frac{\left (12 a^2 b^2 \sqrt{a \sin (e+f x)}\right ) \int \sqrt{\cos (e+f x)} \, dx}{5 \sqrt{\cos (e+f x)} \sqrt{b \tan (e+f x)}}\\ &=-\frac{24 a^2 b^2 E\left (\left .\frac{1}{2} (e+f x)\right |2\right ) \sqrt{a \sin (e+f x)}}{5 f \sqrt{\cos (e+f x)} \sqrt{b \tan (e+f x)}}+\frac{12 a^2 b \sqrt{a \sin (e+f x)} \sqrt{b \tan (e+f x)}}{5 f}-\frac{2 b (a \sin (e+f x))^{5/2} \sqrt{b \tan (e+f x)}}{5 f}\\ \end{align*}
Mathematica [C] time = 0.324921, size = 99, normalized size = 0.79 \[ \frac{a^2 b \sqrt{a \sin (e+f x)} \sqrt{b \tan (e+f x)} \left (\cos ^2(e+f x)^{3/4} (\cos (2 (e+f x))+11)-12 \cos ^2(e+f x) \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{3}{2};\sin ^2(e+f x)\right )\right )}{5 f \cos ^2(e+f x)^{3/4}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.217, size = 336, normalized size = 2.7 \begin{align*} -{\frac{2\,\cos \left ( fx+e \right ) }{5\,f \left ( \sin \left ( fx+e \right ) \right ) ^{5}} \left ( 12\,i\sqrt{ \left ( \cos \left ( fx+e \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( fx+e \right ) }{\cos \left ( fx+e \right ) +1}}}{\it EllipticF} \left ({\frac{i \left ( \cos \left ( fx+e \right ) -1 \right ) }{\sin \left ( fx+e \right ) }},i \right ) \sin \left ( fx+e \right ) \cos \left ( fx+e \right ) -12\,i\sin \left ( fx+e \right ) \cos \left ( fx+e \right ) \sqrt{ \left ( \cos \left ( fx+e \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( fx+e \right ) }{\cos \left ( fx+e \right ) +1}}}{\it EllipticE} \left ({\frac{i \left ( \cos \left ( fx+e \right ) -1 \right ) }{\sin \left ( fx+e \right ) }},i \right ) +12\,i\sqrt{ \left ( \cos \left ( fx+e \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( fx+e \right ) }{\cos \left ( fx+e \right ) +1}}}{\it EllipticF} \left ({\frac{i \left ( \cos \left ( fx+e \right ) -1 \right ) }{\sin \left ( fx+e \right ) }},i \right ) \sin \left ( fx+e \right ) -12\,i\sqrt{ \left ( \cos \left ( fx+e \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( fx+e \right ) }{\cos \left ( fx+e \right ) +1}}}{\it EllipticE} \left ({\frac{i \left ( \cos \left ( fx+e \right ) -1 \right ) }{\sin \left ( fx+e \right ) }},i \right ) \sin \left ( fx+e \right ) + \left ( \cos \left ( fx+e \right ) \right ) ^{4}-8\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}+12\,\cos \left ( fx+e \right ) -5 \right ) \left ( a\sin \left ( fx+e \right ) \right ) ^{{\frac{5}{2}}} \left ({\frac{b\sin \left ( fx+e \right ) }{\cos \left ( fx+e \right ) }} \right ) ^{{\frac{3}{2}}}} \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 (a \sin \left (f x + e\right )\right )^{\frac{5}{2}} \left (b \tan \left (f x + e\right )\right )^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-{\left (a^{2} b \cos \left (f x + e\right )^{2} - a^{2} b\right )} \sqrt{a \sin \left (f x + e\right )} \sqrt{b \tan \left (f x + e\right )} \tan \left (f x + e\right ), x\right ) \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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