Optimal. Leaf size=64 \[ \frac {2 \cos ^2(e+f x)^{7/12} (d \tan (e+f x))^{3/2} \, _2F_1\left (\frac {7}{12},\frac {3}{4};\frac {7}{4};\sin ^2(e+f x)\right )}{3 d f \sqrt [3]{b \sec (e+f x)}} \]
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Rubi [A] time = 0.05, antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {2617} \[ \frac {2 \cos ^2(e+f x)^{7/12} (d \tan (e+f x))^{3/2} \, _2F_1\left (\frac {7}{12},\frac {3}{4};\frac {7}{4};\sin ^2(e+f x)\right )}{3 d f \sqrt [3]{b \sec (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 2617
Rubi steps
\begin {align*} \int \frac {\sqrt {d \tan (e+f x)}}{\sqrt [3]{b \sec (e+f x)}} \, dx &=\frac {2 \cos ^2(e+f x)^{7/12} \, _2F_1\left (\frac {7}{12},\frac {3}{4};\frac {7}{4};\sin ^2(e+f x)\right ) (d \tan (e+f x))^{3/2}}{3 d f \sqrt [3]{b \sec (e+f x)}}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 62, normalized size = 0.97 \[ -\frac {3 d \sqrt [4]{-\tan ^2(e+f x)} \, _2F_1\left (-\frac {1}{6},\frac {1}{4};\frac {5}{6};\sec ^2(e+f x)\right )}{f \sqrt [3]{b \sec (e+f x)} \sqrt {d \tan (e+f x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.65, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\left (b \sec \left (f x + e\right )\right )^{\frac {2}{3}} \sqrt {d \tan \left (f x + e\right )}}{b \sec \left (f x + e\right )}, x\right ) \]
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 \[ \int \frac {\sqrt {d \tan \left (f x + e\right )}}{\left (b \sec \left (f x + e\right )\right )^{\frac {1}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.63, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {d \tan \left (f x +e \right )}}{\left (b \sec \left (f x +e \right )\right )^{\frac {1}{3}}}\, dx \]
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 \[ \int \frac {\sqrt {d \tan \left (f x + e\right )}}{\left (b \sec \left (f x + e\right )\right )^{\frac {1}{3}}}\,{d x} \]
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 \[ \int \frac {\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}}{{\left (\frac {b}{\cos \left (e+f\,x\right )}\right )}^{1/3}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {d \tan {\left (e + f x \right )}}}{\sqrt [3]{b \sec {\left (e + f x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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