Optimal. Leaf size=34 \[ \frac{2 (b \tan (e+f x))^{5/2}}{5 b f (d \sec (e+f x))^{5/2}} \]
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Rubi [A] time = 0.0570212, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04, Rules used = {2605} \[ \frac{2 (b \tan (e+f x))^{5/2}}{5 b f (d \sec (e+f x))^{5/2}} \]
Antiderivative was successfully verified.
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Rule 2605
Rubi steps
\begin{align*} \int \frac{(b \tan (e+f x))^{3/2}}{(d \sec (e+f x))^{5/2}} \, dx &=\frac{2 (b \tan (e+f x))^{5/2}}{5 b f (d \sec (e+f x))^{5/2}}\\ \end{align*}
Mathematica [B] time = 1.33815, size = 141, normalized size = 4.15 \[ -\frac{b \sec ^{\frac{3}{2}}(e+f x) \sqrt{b \tan (e+f x)} \left (-\sqrt{\sec (e+f x)+1} \sec ^2\left (\frac{1}{2} (e+f x)\right )+\sqrt{\frac{1}{\cos (e+f x)+1}} \cos (3 (e+f x)) \sec ^{\frac{3}{2}}(e+f x)+\sqrt{\frac{1}{\cos (e+f x)+1}} \sqrt{\sec (e+f x)}\right )}{10 f \sqrt{\frac{1}{\cos (e+f x)+1}} (d \sec (e+f x))^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.151, size = 50, normalized size = 1.5 \begin{align*}{\frac{2\,\sin \left ( fx+e \right ) }{5\,f\cos \left ( fx+e \right ) } \left ({\frac{b\sin \left ( fx+e \right ) }{\cos \left ( fx+e \right ) }} \right ) ^{{\frac{3}{2}}} \left ({\frac{d}{\cos \left ( fx+e \right ) }} \right ) ^{-{\frac{5}{2}}}} \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{3}{2}}}{\left (d \sec \left (f x + e\right )\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.01878, size = 142, normalized size = 4.18 \begin{align*} -\frac{2 \,{\left (b \cos \left (f x + e\right )^{3} - b \cos \left (f x + e\right )\right )} \sqrt{\frac{b \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \sqrt{\frac{d}{\cos \left (f x + e\right )}}}{5 \, d^{3} f} \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{3}{2}}}{\left (d \sec \left (f x + e\right )\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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