Optimal. Leaf size=64 \[ \frac {6 \cos ^2(e+f x)^{2/3} \, _2F_1\left (\frac {2}{3},\frac {17}{12};\frac {29}{12};\sin ^2(e+f x)\right ) (b \sin (e+f x))^{3/2} (d \tan (e+f x))^{4/3}}{17 d f} \]
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Rubi [A]
time = 0.06, 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 = {2682, 2657}
\begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 2657
Rule 2682
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 = 10.53, size = 72, normalized size = 1.12 \begin {gather*} \frac {6 \cos (e+f x) \, _2F_1\left (\frac {17}{12},\frac {7}{4};\frac {29}{12};-\tan ^2(e+f x)\right ) \sec ^2(e+f x)^{7/4} (b \sin (e+f x))^{5/2} \sqrt [3]{d \tan (e+f x)}}{17 b f} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.20, 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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
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 \begin {gather*} \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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