Optimal. Leaf size=79 \[ \frac{2 \cos ^2(e+f x)^{5/4} (b \tan (e+f x))^{5/2} (a \sin (e+f x))^m \, _2F_1\left (\frac{5}{4},\frac{1}{4} (2 m+5);\frac{1}{4} (2 m+9);\sin ^2(e+f x)\right )}{b f (2 m+5)} \]
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Rubi [A] time = 0.115703, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {2602, 2577} \[ \frac{2 \cos ^2(e+f x)^{5/4} (b \tan (e+f x))^{5/2} (a \sin (e+f x))^m \, _2F_1\left (\frac{5}{4},\frac{1}{4} (2 m+5);\frac{1}{4} (2 m+9);\sin ^2(e+f x)\right )}{b f (2 m+5)} \]
Antiderivative was successfully verified.
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Rule 2602
Rule 2577
Rubi steps
\begin{align*} \int (a \sin (e+f x))^m (b \tan (e+f x))^{3/2} \, dx &=\frac{\left (a \cos ^{\frac{5}{2}}(e+f x) (b \tan (e+f x))^{5/2}\right ) \int \frac{(a \sin (e+f x))^{\frac{3}{2}+m}}{\cos ^{\frac{3}{2}}(e+f x)} \, dx}{b (a \sin (e+f x))^{5/2}}\\ &=\frac{2 \cos ^2(e+f x)^{5/4} \, _2F_1\left (\frac{5}{4},\frac{1}{4} (5+2 m);\frac{1}{4} (9+2 m);\sin ^2(e+f x)\right ) (a \sin (e+f x))^m (b \tan (e+f x))^{5/2}}{b f (5+2 m)}\\ \end{align*}
Mathematica [A] time = 8.10383, size = 87, normalized size = 1.1 \[ \frac{2 (b \tan (e+f x))^{5/2} \sec ^2(e+f x)^{m/2} (a \sin (e+f x))^m \, _2F_1\left (\frac{m+2}{2},\frac{1}{4} (2 m+5);\frac{1}{4} (2 m+9);-\tan ^2(e+f x)\right )}{b f (2 m+5)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.14, size = 0, normalized size = 0. \begin{align*} \int \left ( a\sin \left ( fx+e \right ) \right ) ^{m} \left ( b\tan \left ( fx+e \right ) \right ) ^{{\frac{3}{2}}}\, dx \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 (b \tan \left (f x + e\right )\right )^{\frac{3}{2}} \left (a \sin \left (f x + e\right )\right )^{m}\,{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{b \tan \left (f x + e\right )} \left (a \sin \left (f x + e\right )\right )^{m} 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(-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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