Optimal. Leaf size=88 \[ \frac{4 a^2 E\left (\left .\frac{1}{2} (e+f x)\right |2\right ) \sqrt{a \sin (e+f x)}}{5 f \sqrt{\cos (e+f x)} \sqrt{b \tan (e+f x)}}-\frac{2 b (a \sin (e+f x))^{5/2}}{5 f (b \tan (e+f x))^{3/2}} \]
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Rubi [A] time = 0.104979, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {2598, 2601, 2639} \[ \frac{4 a^2 E\left (\left .\frac{1}{2} (e+f x)\right |2\right ) \sqrt{a \sin (e+f x)}}{5 f \sqrt{\cos (e+f x)} \sqrt{b \tan (e+f x)}}-\frac{2 b (a \sin (e+f x))^{5/2}}{5 f (b \tan (e+f x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 2598
Rule 2601
Rule 2639
Rubi steps
\begin{align*} \int \frac{(a \sin (e+f x))^{5/2}}{\sqrt{b \tan (e+f x)}} \, dx &=-\frac{2 b (a \sin (e+f x))^{5/2}}{5 f (b \tan (e+f x))^{3/2}}+\frac{1}{5} \left (2 a^2\right ) \int \frac{\sqrt{a \sin (e+f x)}}{\sqrt{b \tan (e+f x)}} \, dx\\ &=-\frac{2 b (a \sin (e+f x))^{5/2}}{5 f (b \tan (e+f x))^{3/2}}+\frac{\left (2 a^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{2 b (a \sin (e+f x))^{5/2}}{5 f (b \tan (e+f x))^{3/2}}+\frac{4 a^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)}}\\ \end{align*}
Mathematica [C] time = 0.233148, size = 87, normalized size = 0.99 \[ -\frac{a^2 \sin (2 (e+f x)) \sqrt{a \sin (e+f x)} \left (\cos ^2(e+f x)^{3/4}-\, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{3}{2};\sin ^2(e+f x)\right )\right )}{5 f \cos ^2(e+f x)^{3/4} \sqrt{b \tan (e+f x)}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.196, size = 337, normalized size = 3.8 \begin{align*}{\frac{2}{5\,f \left ( \sin \left ( fx+e \right ) \right ) ^{3}\cos \left ( fx+e \right ) } \left ( 2\,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 ) -2\,i\sin \left ( fx+e \right ) \cos \left ( fx+e \right ){\it EllipticE} \left ({\frac{i \left ( \cos \left ( fx+e \right ) -1 \right ) }{\sin \left ( fx+e \right ) }},i \right ) \sqrt{ \left ( \cos \left ( fx+e \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( fx+e \right ) }{\cos \left ( fx+e \right ) +1}}}+2\,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 ) -2\,i\sin \left ( fx+e \right ){\it EllipticE} \left ({\frac{i \left ( \cos \left ( fx+e \right ) -1 \right ) }{\sin \left ( fx+e \right ) }},i \right ) \sqrt{ \left ( \cos \left ( fx+e \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( fx+e \right ) }{\cos \left ( fx+e \right ) +1}}}+ \left ( \cos \left ( fx+e \right ) \right ) ^{4}-3\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}+2\,\cos \left ( fx+e \right ) \right ) \left ( a\sin \left ( fx+e \right ) \right ) ^{{\frac{5}{2}}}{\frac{1}{\sqrt{{\frac{b\sin \left ( fx+e \right ) }{\cos \left ( fx+e \right ) }}}}}} \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 \frac{\left (a \sin \left (f x + e\right )\right )^{\frac{5}{2}}}{\sqrt{b \tan \left (f x + e\right )}}\,{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 (-\frac{{\left (a^{2} \cos \left (f x + e\right )^{2} - a^{2}\right )} \sqrt{a \sin \left (f x + e\right )} \sqrt{b \tan \left (f x + e\right )}}{b \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a \sin \left (f x + e\right )\right )^{\frac{5}{2}}}{\sqrt{b \tan \left (f x + e\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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