Optimal. Leaf size=309 \[ \frac{2 (2 a-b) \sqrt{a+b} \cot (e+f x) \sqrt{\frac{b (1-\sec (e+f x))}{a+b}} \sqrt{-\frac{b (\sec (e+f x)+1)}{a-b}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right ),\frac{a+b}{a-b}\right )}{f}-\frac{2 (a-b) \sqrt{a+b} \cot (e+f x) \sqrt{\frac{b (1-\sec (e+f x))}{a+b}} \sqrt{-\frac{b (\sec (e+f x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{f}-\frac{2 a \sqrt{a+b} \cot (e+f x) \sqrt{\frac{b (1-\sec (e+f x))}{a+b}} \sqrt{-\frac{b (\sec (e+f x)+1)}{a-b}} \Pi \left (\frac{a+b}{a};\sin ^{-1}\left (\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{f} \]
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Rubi [A] time = 0.230189, antiderivative size = 309, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357, Rules used = {3781, 3921, 3784, 3832, 4004} \[ \frac{2 (2 a-b) \sqrt{a+b} \cot (e+f x) \sqrt{\frac{b (1-\sec (e+f x))}{a+b}} \sqrt{-\frac{b (\sec (e+f x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{f}-\frac{2 (a-b) \sqrt{a+b} \cot (e+f x) \sqrt{\frac{b (1-\sec (e+f x))}{a+b}} \sqrt{-\frac{b (\sec (e+f x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{f}-\frac{2 a \sqrt{a+b} \cot (e+f x) \sqrt{\frac{b (1-\sec (e+f x))}{a+b}} \sqrt{-\frac{b (\sec (e+f x)+1)}{a-b}} \Pi \left (\frac{a+b}{a};\sin ^{-1}\left (\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{f} \]
Antiderivative was successfully verified.
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Rule 3781
Rule 3921
Rule 3784
Rule 3832
Rule 4004
Rubi steps
\begin{align*} \int (a+b \sec (e+f x))^{3/2} \, dx &=b^2 \int \frac{\sec (e+f x) (1+\sec (e+f x))}{\sqrt{a+b \sec (e+f x)}} \, dx+\int \frac{a^2+(2 a-b) b \sec (e+f x)}{\sqrt{a+b \sec (e+f x)}} \, dx\\ &=-\frac{2 (a-b) \sqrt{a+b} \cot (e+f x) E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right ) \sqrt{\frac{b (1-\sec (e+f x))}{a+b}} \sqrt{-\frac{b (1+\sec (e+f x))}{a-b}}}{f}+a^2 \int \frac{1}{\sqrt{a+b \sec (e+f x)}} \, dx+((2 a-b) b) \int \frac{\sec (e+f x)}{\sqrt{a+b \sec (e+f x)}} \, dx\\ &=-\frac{2 (a-b) \sqrt{a+b} \cot (e+f x) E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right ) \sqrt{\frac{b (1-\sec (e+f x))}{a+b}} \sqrt{-\frac{b (1+\sec (e+f x))}{a-b}}}{f}+\frac{2 (2 a-b) \sqrt{a+b} \cot (e+f x) F\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right ) \sqrt{\frac{b (1-\sec (e+f x))}{a+b}} \sqrt{-\frac{b (1+\sec (e+f x))}{a-b}}}{f}-\frac{2 a \sqrt{a+b} \cot (e+f x) \Pi \left (\frac{a+b}{a};\sin ^{-1}\left (\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right ) \sqrt{\frac{b (1-\sec (e+f x))}{a+b}} \sqrt{-\frac{b (1+\sec (e+f x))}{a-b}}}{f}\\ \end{align*}
Mathematica [C] time = 6.11337, size = 882, normalized size = 2.85 \[ \frac{2 b \cos (e+f x) \sin (e+f x) (a+b \sec (e+f x))^{3/2}}{f (b+a \cos (e+f x))}+\frac{2 \left (-b^2 \sqrt{\frac{b-a}{a+b}} \tan ^5\left (\frac{1}{2} (e+f x)\right )+a b \sqrt{\frac{b-a}{a+b}} \tan ^5\left (\frac{1}{2} (e+f x)\right )-2 a b \sqrt{\frac{b-a}{a+b}} \tan ^3\left (\frac{1}{2} (e+f x)\right )+2 i a^2 \Pi \left (-\frac{a+b}{a-b};i \sinh ^{-1}\left (\sqrt{\frac{b-a}{a+b}} \tan \left (\frac{1}{2} (e+f x)\right )\right )|\frac{a+b}{a-b}\right ) \sqrt{1-\tan ^2\left (\frac{1}{2} (e+f x)\right )} \sqrt{\frac{-a \tan ^2\left (\frac{1}{2} (e+f x)\right )+b \tan ^2\left (\frac{1}{2} (e+f x)\right )+a+b}{a+b}} \tan ^2\left (\frac{1}{2} (e+f x)\right )+b^2 \sqrt{\frac{b-a}{a+b}} \tan \left (\frac{1}{2} (e+f x)\right )+a b \sqrt{\frac{b-a}{a+b}} \tan \left (\frac{1}{2} (e+f x)\right )-i (a-b) b E\left (i \sinh ^{-1}\left (\sqrt{\frac{b-a}{a+b}} \tan \left (\frac{1}{2} (e+f x)\right )\right )|\frac{a+b}{a-b}\right ) \sqrt{1-\tan ^2\left (\frac{1}{2} (e+f x)\right )} \left (\tan ^2\left (\frac{1}{2} (e+f x)\right )+1\right ) \sqrt{\frac{-a \tan ^2\left (\frac{1}{2} (e+f x)\right )+b \tan ^2\left (\frac{1}{2} (e+f x)\right )+a+b}{a+b}}-i (a-b)^2 \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{\frac{b-a}{a+b}} \tan \left (\frac{1}{2} (e+f x)\right )\right ),\frac{a+b}{a-b}\right ) \sqrt{1-\tan ^2\left (\frac{1}{2} (e+f x)\right )} \left (\tan ^2\left (\frac{1}{2} (e+f x)\right )+1\right ) \sqrt{\frac{-a \tan ^2\left (\frac{1}{2} (e+f x)\right )+b \tan ^2\left (\frac{1}{2} (e+f x)\right )+a+b}{a+b}}+2 i a^2 \Pi \left (-\frac{a+b}{a-b};i \sinh ^{-1}\left (\sqrt{\frac{b-a}{a+b}} \tan \left (\frac{1}{2} (e+f x)\right )\right )|\frac{a+b}{a-b}\right ) \sqrt{1-\tan ^2\left (\frac{1}{2} (e+f x)\right )} \sqrt{\frac{-a \tan ^2\left (\frac{1}{2} (e+f x)\right )+b \tan ^2\left (\frac{1}{2} (e+f x)\right )+a+b}{a+b}}\right ) (a+b \sec (e+f x))^{3/2}}{\sqrt{\frac{b-a}{a+b}} f (b+a \cos (e+f x))^{3/2} \sec ^{\frac{3}{2}}(e+f x) \sqrt{\frac{1}{1-\tan ^2\left (\frac{1}{2} (e+f x)\right )}} \left (\tan ^2\left (\frac{1}{2} (e+f x)\right )-1\right ) \left (\tan ^2\left (\frac{1}{2} (e+f x)\right )+1\right )^{3/2} \sqrt{\frac{-a \tan ^2\left (\frac{1}{2} (e+f x)\right )+b \tan ^2\left (\frac{1}{2} (e+f x)\right )+a+b}{\tan ^2\left (\frac{1}{2} (e+f x)\right )+1}}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.3, size = 1199, normalized size = 3.9 \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{\left (b \sec \left (f x + e\right ) + a\right )}^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [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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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \sec{\left (e + f x \right )}\right )^{\frac{3}{2}}\, dx \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 ) + a\right )}^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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