Optimal. Leaf size=88 \[ \frac{b (b \tan (e+f x))^{3/2}}{f \sqrt{d \sec (e+f x)}}-\frac{3 b^2 E\left (\left .\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{b \tan (e+f x)}}{f \sqrt{\sin (e+f x)} \sqrt{d \sec (e+f x)}} \]
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Rubi [A] time = 0.112627, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {2611, 2616, 2640, 2639} \[ \frac{b (b \tan (e+f x))^{3/2}}{f \sqrt{d \sec (e+f x)}}-\frac{3 b^2 E\left (\left .\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{b \tan (e+f x)}}{f \sqrt{\sin (e+f x)} \sqrt{d \sec (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 2611
Rule 2616
Rule 2640
Rule 2639
Rubi steps
\begin{align*} \int \frac{(b \tan (e+f x))^{5/2}}{\sqrt{d \sec (e+f x)}} \, dx &=\frac{b (b \tan (e+f x))^{3/2}}{f \sqrt{d \sec (e+f x)}}-\frac{1}{2} \left (3 b^2\right ) \int \frac{\sqrt{b \tan (e+f x)}}{\sqrt{d \sec (e+f x)}} \, dx\\ &=\frac{b (b \tan (e+f x))^{3/2}}{f \sqrt{d \sec (e+f x)}}-\frac{\left (3 b^2 \sqrt{b \tan (e+f x)}\right ) \int \sqrt{b \sin (e+f x)} \, dx}{2 \sqrt{d \sec (e+f x)} \sqrt{b \sin (e+f x)}}\\ &=\frac{b (b \tan (e+f x))^{3/2}}{f \sqrt{d \sec (e+f x)}}-\frac{\left (3 b^2 \sqrt{b \tan (e+f x)}\right ) \int \sqrt{\sin (e+f x)} \, dx}{2 \sqrt{d \sec (e+f x)} \sqrt{\sin (e+f x)}}\\ &=-\frac{3 b^2 E\left (\left .\frac{1}{2} \left (e-\frac{\pi }{2}+f x\right )\right |2\right ) \sqrt{b \tan (e+f x)}}{f \sqrt{d \sec (e+f x)} \sqrt{\sin (e+f x)}}+\frac{b (b \tan (e+f x))^{3/2}}{f \sqrt{d \sec (e+f x)}}\\ \end{align*}
Mathematica [C] time = 0.710106, size = 74, normalized size = 0.84 \[ \frac{b^3 \left (3 \sqrt [4]{-\tan ^2(e+f x)} \, _2F_1\left (-\frac{1}{4},\frac{1}{4};\frac{3}{4};\sec ^2(e+f x)\right )+\tan ^2(e+f x)\right )}{f \sqrt{b \tan (e+f x)} \sqrt{d \sec (e+f x)}} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.223, size = 585, normalized size = 6.7 \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 \frac{\left (b \tan \left (f x + e\right )\right )^{\frac{5}{2}}}{\sqrt{d \sec \left (f x + e\right )}}\,{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 (\frac{\sqrt{d \sec \left (f x + e\right )} \sqrt{b \tan \left (f x + e\right )} b^{2} \tan \left (f x + e\right )^{2}}{d \sec \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (b \tan \left (f x + e\right )\right )^{\frac{5}{2}}}{\sqrt{d \sec \left (f x + e\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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