Optimal. Leaf size=70 \[ \frac{2 b^2 \sqrt{\cos (e+f x)} F\left (\left .\frac{1}{2} (e+f x)\right |2\right ) \sqrt{b \sec (e+f x)}}{3 f}+\frac{2 b \sin (e+f x) (b \sec (e+f x))^{3/2}}{3 f} \]
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Rubi [A] time = 0.034645, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3768, 3771, 2641} \[ \frac{2 b^2 \sqrt{\cos (e+f x)} F\left (\left .\frac{1}{2} (e+f x)\right |2\right ) \sqrt{b \sec (e+f x)}}{3 f}+\frac{2 b \sin (e+f x) (b \sec (e+f x))^{3/2}}{3 f} \]
Antiderivative was successfully verified.
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Rule 3768
Rule 3771
Rule 2641
Rubi steps
\begin{align*} \int (b \sec (e+f x))^{5/2} \, dx &=\frac{2 b (b \sec (e+f x))^{3/2} \sin (e+f x)}{3 f}+\frac{1}{3} b^2 \int \sqrt{b \sec (e+f x)} \, dx\\ &=\frac{2 b (b \sec (e+f x))^{3/2} \sin (e+f x)}{3 f}+\frac{1}{3} \left (b^2 \sqrt{\cos (e+f x)} \sqrt{b \sec (e+f x)}\right ) \int \frac{1}{\sqrt{\cos (e+f x)}} \, dx\\ &=\frac{2 b^2 \sqrt{\cos (e+f x)} F\left (\left .\frac{1}{2} (e+f x)\right |2\right ) \sqrt{b \sec (e+f x)}}{3 f}+\frac{2 b (b \sec (e+f x))^{3/2} \sin (e+f x)}{3 f}\\ \end{align*}
Mathematica [A] time = 0.0753007, size = 51, normalized size = 0.73 \[ \frac{2 b^2 \sqrt{b \sec (e+f x)} \left (\tan (e+f x)+\sqrt{\cos (e+f x)} F\left (\left .\frac{1}{2} (e+f x)\right |2\right )\right )}{3 f} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.123, size = 128, normalized size = 1.8 \begin{align*} -{\frac{ \left ( -2+2\,\cos \left ( fx+e \right ) \right ) \cos \left ( fx+e \right ) \left ( \cos \left ( fx+e \right ) +1 \right ) ^{2}}{3\,f \left ( \sin \left ( fx+e \right ) \right ) ^{3}} \left ( i\cos \left ( fx+e \right ){\it EllipticF} \left ({\frac{i \left ( -1+\cos \left ( fx+e \right ) \right ) }{\sin \left ( fx+e \right ) }},i \right ) \sqrt{ \left ( \cos \left ( fx+e \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( fx+e \right ) }{\cos \left ( fx+e \right ) +1}}}\sin \left ( fx+e \right ) -\cos \left ( fx+e \right ) +1 \right ) \left ({\frac{b}{\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 \left (b \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sqrt{b \sec \left (f x + e\right )} b^{2} \sec \left (f x + e\right )^{2}, 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 \left (b \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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