Optimal. Leaf size=88 \[ \frac{4 a^2 \sqrt{\cos (e+f x)} F\left (\left .\frac{1}{2} (e+f x)\right |2\right ) \sqrt{b \tan (e+f x)}}{3 f \sqrt{a \sin (e+f x)}}-\frac{2 b (a \sin (e+f x))^{3/2}}{3 f \sqrt{b \tan (e+f x)}} \]
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Rubi [A] time = 0.105774, 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, 2641} \[ \frac{4 a^2 \sqrt{\cos (e+f x)} F\left (\left .\frac{1}{2} (e+f x)\right |2\right ) \sqrt{b \tan (e+f x)}}{3 f \sqrt{a \sin (e+f x)}}-\frac{2 b (a \sin (e+f x))^{3/2}}{3 f \sqrt{b \tan (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 2598
Rule 2601
Rule 2641
Rubi steps
\begin{align*} \int (a \sin (e+f x))^{3/2} \sqrt{b \tan (e+f x)} \, dx &=-\frac{2 b (a \sin (e+f x))^{3/2}}{3 f \sqrt{b \tan (e+f x)}}+\frac{1}{3} \left (2 a^2\right ) \int \frac{\sqrt{b \tan (e+f x)}}{\sqrt{a \sin (e+f x)}} \, dx\\ &=-\frac{2 b (a \sin (e+f x))^{3/2}}{3 f \sqrt{b \tan (e+f x)}}+\frac{\left (2 a^2 \sqrt{\cos (e+f x)} \sqrt{b \tan (e+f x)}\right ) \int \frac{1}{\sqrt{\cos (e+f x)}} \, dx}{3 \sqrt{a \sin (e+f x)}}\\ &=-\frac{2 b (a \sin (e+f x))^{3/2}}{3 f \sqrt{b \tan (e+f x)}}+\frac{4 a^2 \sqrt{\cos (e+f x)} F\left (\left .\frac{1}{2} (e+f x)\right |2\right ) \sqrt{b \tan (e+f x)}}{3 f \sqrt{a \sin (e+f x)}}\\ \end{align*}
Mathematica [A] time = 0.312267, size = 80, normalized size = 0.91 \[ -\frac{2 a b \sqrt{a \sin (e+f x)} \left (\sin (e+f x) \sqrt [4]{\cos ^2(e+f x)}-2 F\left (\left .\frac{1}{2} \sin ^{-1}(\sin (e+f x))\right |2\right )\right )}{3 f \sqrt [4]{\cos ^2(e+f x)} \sqrt{b \tan (e+f x)}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.199, size = 131, normalized size = 1.5 \begin{align*} -{\frac{2}{3\,f \left ( \cos \left ( fx+e \right ) -1 \right ) \sin \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 ) + \left ( \cos \left ( fx+e \right ) \right ) ^{2}-\cos \left ( fx+e \right ) \right ) \left ( a\sin \left ( fx+e \right ) \right ) ^{{\frac{3}{2}}}\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 \left (a \sin \left (f x + e\right )\right )^{\frac{3}{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 (\sqrt{a \sin \left (f x + e\right )} \sqrt{b \tan \left (f x + e\right )} a \sin \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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