Optimal. Leaf size=68 \[ -\frac{8 a^2 b \sqrt{a \sin (e+f x)}}{5 f \sqrt{b \tan (e+f x)}}-\frac{2 b (a \sin (e+f x))^{5/2}}{5 f \sqrt{b \tan (e+f x)}} \]
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Rubi [A] time = 0.0901038, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {2598, 2589} \[ -\frac{8 a^2 b \sqrt{a \sin (e+f x)}}{5 f \sqrt{b \tan (e+f x)}}-\frac{2 b (a \sin (e+f x))^{5/2}}{5 f \sqrt{b \tan (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 2598
Rule 2589
Rubi steps
\begin{align*} \int (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 \sqrt{b \tan (e+f x)}}+\frac{1}{5} \left (4 a^2\right ) \int \sqrt{a \sin (e+f x)} \sqrt{b \tan (e+f x)} \, dx\\ &=-\frac{8 a^2 b \sqrt{a \sin (e+f x)}}{5 f \sqrt{b \tan (e+f x)}}-\frac{2 b (a \sin (e+f x))^{5/2}}{5 f \sqrt{b \tan (e+f x)}}\\ \end{align*}
Mathematica [A] time = 0.186618, size = 51, normalized size = 0.75 \[ -\frac{a^2 \sqrt{a \sin (e+f x)} \sqrt{b \tan (e+f x)} (\sin (2 (e+f x))+8 \cot (e+f x))}{5 f} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.303, size = 493, normalized size = 7.3 \begin{align*} \text{result too large to display} \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}} \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 [A] time = 1.63554, size = 161, normalized size = 2.37 \begin{align*} \frac{2 \,{\left (a^{2} \cos \left (f x + e\right )^{3} - 5 \, a^{2} \cos \left (f x + e\right )\right )} \sqrt{a \sin \left (f x + e\right )} \sqrt{\frac{b \sin \left (f x + e\right )}{\cos \left (f x + e\right )}}}{5 \, f \sin \left (f x + e\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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