Optimal. Leaf size=64 \[ \frac {6 \cos ^2(e+f x)^{2/3} (b \sin (e+f x))^{3/2} (d \tan (e+f x))^{4/3} \, _2F_1\left (\frac {2}{3},\frac {17}{12};\frac {29}{12};\sin ^2(e+f x)\right )}{17 d f} \]
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Rubi [A] time = 0.09, antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {2602, 2577} \[ \frac {6 \cos ^2(e+f x)^{2/3} (b \sin (e+f x))^{3/2} (d \tan (e+f x))^{4/3} \, _2F_1\left (\frac {2}{3},\frac {17}{12};\frac {29}{12};\sin ^2(e+f x)\right )}{17 d f} \]
Antiderivative was successfully verified.
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Rule 2577
Rule 2602
Rubi steps
\begin {align*} \int (b \sin (e+f x))^{3/2} \sqrt [3]{d \tan (e+f x)} \, dx &=\frac {\left (b \cos ^{\frac {4}{3}}(e+f x) (d \tan (e+f x))^{4/3}\right ) \int \frac {(b \sin (e+f x))^{11/6}}{\sqrt [3]{\cos (e+f x)}} \, dx}{d (b \sin (e+f x))^{4/3}}\\ &=\frac {6 \cos ^2(e+f x)^{2/3} \, _2F_1\left (\frac {2}{3},\frac {17}{12};\frac {29}{12};\sin ^2(e+f x)\right ) (b \sin (e+f x))^{3/2} (d \tan (e+f x))^{4/3}}{17 d f}\\ \end {align*}
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Mathematica [A] time = 0.50, size = 72, normalized size = 1.12 \[ \frac {6 \cos (e+f x) \sec ^2(e+f x)^{7/4} (b \sin (e+f x))^{5/2} \sqrt [3]{d \tan (e+f x)} \, _2F_1\left (\frac {17}{12},\frac {7}{4};\frac {29}{12};-\tan ^2(e+f x)\right )}{17 b f} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {b \sin \left (f x + e\right )} \left (d \tan \left (f x + e\right )\right )^{\frac {1}{3}} b \sin \left (f x + e\right ), x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [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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maple [F] time = 0.43, size = 0, normalized size = 0.00 \[ \int \left (b \sin \left (f x +e \right )\right )^{\frac {3}{2}} \left (d \tan \left (f x +e \right )\right )^{\frac {1}{3}}\, dx \]
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 \[ \int \left (b \sin \left (f x + e\right )\right )^{\frac {3}{2}} \left (d \tan \left (f x + e\right )\right )^{\frac {1}{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int {\left (b\,\sin \left (e+f\,x\right )\right )}^{3/2}\,{\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{1/3} \,d x \]
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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