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 = 0.10, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, 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*}
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Mathematica [C] time = 0.24, 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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fricas [F] time = 0.69, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (a^{2} \cos \left (f x + e\right )^{2} - a^{2}\right )} \sqrt {a \sin \left (f x + e\right )} \sqrt {b \tan \left (f x + e\right )}}{b \tan \left (f x + e\right )}, x\right ) \]
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 \[ \int \frac {\left (a \sin \left (f x + e\right )\right )^{\frac {5}{2}}}{\sqrt {b \tan \left (f x + e\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.56, size = 337, normalized size = 3.83 \[ \frac {2 \left (2 i \sqrt {\frac {1}{1+\cos \left (f x +e \right )}}\, \sqrt {\frac {\cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, i\right ) \sin \left (f x +e \right ) \cos \left (f x +e \right )-2 i \sin \left (f x +e \right ) \cos \left (f x +e \right ) \sqrt {\frac {1}{1+\cos \left (f x +e \right )}}\, \sqrt {\frac {\cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}\, \EllipticE \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, i\right )+2 i \sqrt {\frac {1}{1+\cos \left (f x +e \right )}}\, \sqrt {\frac {\cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, i\right ) \sin \left (f x +e \right )-2 i \sqrt {\frac {1}{1+\cos \left (f x +e \right )}}\, \sqrt {\frac {\cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}\, \EllipticE \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, i\right ) \sin \left (f x +e \right )+\cos ^{4}\left (f x +e \right )-3 \left (\cos ^{2}\left (f x +e \right )\right )+2 \cos \left (f x +e \right )\right ) \left (a \sin \left (f x +e \right )\right )^{\frac {5}{2}}}{5 f \sqrt {\frac {b \sin \left (f x +e \right )}{\cos \left (f x +e \right )}}\, \cos \left (f x +e \right ) \sin \left (f x +e \right )^{3}} \]
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 \[ \int \frac {\left (a \sin \left (f x + e\right )\right )^{\frac {5}{2}}}{\sqrt {b \tan \left (f x + e\right )}}\,{d x} \]
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 \[ \int \frac {{\left (a\,\sin \left (e+f\,x\right )\right )}^{5/2}}{\sqrt {b\,\mathrm {tan}\left (e+f\,x\right )}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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