Optimal. Leaf size=97 \[ \frac{2 a^2 \sqrt{a \sec (c+d x)+a}}{d}-\frac{2 a^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right )}{d}+\frac{2 a (a \sec (c+d x)+a)^{3/2}}{3 d}+\frac{2 (a \sec (c+d x)+a)^{5/2}}{5 d} \]
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Rubi [A] time = 0.0650882, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {3880, 50, 63, 207} \[ \frac{2 a^2 \sqrt{a \sec (c+d x)+a}}{d}-\frac{2 a^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right )}{d}+\frac{2 a (a \sec (c+d x)+a)^{3/2}}{3 d}+\frac{2 (a \sec (c+d x)+a)^{5/2}}{5 d} \]
Antiderivative was successfully verified.
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Rule 3880
Rule 50
Rule 63
Rule 207
Rubi steps
\begin{align*} \int (a+a \sec (c+d x))^{5/2} \tan (c+d x) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(a+a x)^{5/2}}{x} \, dx,x,\sec (c+d x)\right )}{d}\\ &=\frac{2 (a+a \sec (c+d x))^{5/2}}{5 d}+\frac{a \operatorname{Subst}\left (\int \frac{(a+a x)^{3/2}}{x} \, dx,x,\sec (c+d x)\right )}{d}\\ &=\frac{2 a (a+a \sec (c+d x))^{3/2}}{3 d}+\frac{2 (a+a \sec (c+d x))^{5/2}}{5 d}+\frac{a^2 \operatorname{Subst}\left (\int \frac{\sqrt{a+a x}}{x} \, dx,x,\sec (c+d x)\right )}{d}\\ &=\frac{2 a^2 \sqrt{a+a \sec (c+d x)}}{d}+\frac{2 a (a+a \sec (c+d x))^{3/2}}{3 d}+\frac{2 (a+a \sec (c+d x))^{5/2}}{5 d}+\frac{a^3 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+a x}} \, dx,x,\sec (c+d x)\right )}{d}\\ &=\frac{2 a^2 \sqrt{a+a \sec (c+d x)}}{d}+\frac{2 a (a+a \sec (c+d x))^{3/2}}{3 d}+\frac{2 (a+a \sec (c+d x))^{5/2}}{5 d}+\frac{\left (2 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{-1+\frac{x^2}{a}} \, dx,x,\sqrt{a+a \sec (c+d x)}\right )}{d}\\ &=-\frac{2 a^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a+a \sec (c+d x)}}{\sqrt{a}}\right )}{d}+\frac{2 a^2 \sqrt{a+a \sec (c+d x)}}{d}+\frac{2 a (a+a \sec (c+d x))^{3/2}}{3 d}+\frac{2 (a+a \sec (c+d x))^{5/2}}{5 d}\\ \end{align*}
Mathematica [A] time = 0.20706, size = 82, normalized size = 0.85 \[ \frac{2 (a (\sec (c+d x)+1))^{5/2} \left (\sqrt{\sec (c+d x)+1} \left (3 \sec ^2(c+d x)+11 \sec (c+d x)+23\right )-15 \tanh ^{-1}\left (\sqrt{\sec (c+d x)+1}\right )\right )}{15 d (\sec (c+d x)+1)^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.029, size = 74, normalized size = 0.8 \begin{align*}{\frac{1}{d} \left ({\frac{2}{5} \left ( a+a\sec \left ( dx+c \right ) \right ) ^{{\frac{5}{2}}}}+{\frac{2\,a}{3} \left ( a+a\sec \left ( dx+c \right ) \right ) ^{{\frac{3}{2}}}}+2\,{a}^{2}\sqrt{a+a\sec \left ( dx+c \right ) }-2\,{a}^{5/2}{\it Artanh} \left ({\frac{\sqrt{a+a\sec \left ( dx+c \right ) }}{\sqrt{a}}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.98756, size = 729, normalized size = 7.52 \begin{align*} \left [\frac{15 \, a^{\frac{5}{2}} \cos \left (d x + c\right )^{2} \log \left (-8 \, a \cos \left (d x + c\right )^{2} + 4 \,{\left (2 \, \cos \left (d x + c\right )^{2} + \cos \left (d x + c\right )\right )} \sqrt{a} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} - 8 \, a \cos \left (d x + c\right ) - a\right ) + 4 \,{\left (23 \, a^{2} \cos \left (d x + c\right )^{2} + 11 \, a^{2} \cos \left (d x + c\right ) + 3 \, a^{2}\right )} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}}}{30 \, d \cos \left (d x + c\right )^{2}}, \frac{15 \, \sqrt{-a} a^{2} \arctan \left (\frac{2 \, \sqrt{-a} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{2 \, a \cos \left (d x + c\right ) + a}\right ) \cos \left (d x + c\right )^{2} + 2 \,{\left (23 \, a^{2} \cos \left (d x + c\right )^{2} + 11 \, a^{2} \cos \left (d x + c\right ) + 3 \, a^{2}\right )} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}}}{15 \, d \cos \left (d x + c\right )^{2}}\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 [A] time = 4.60844, size = 203, normalized size = 2.09 \begin{align*} \frac{\sqrt{2} a^{6}{\left (\frac{15 \, \sqrt{2} \arctan \left (\frac{\sqrt{2} \sqrt{-a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a}}{2 \, \sqrt{-a}}\right )}{\sqrt{-a} a^{3}} + \frac{2 \,{\left (15 \,{\left (a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - a\right )}^{2} - 10 \,{\left (a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - a\right )} a + 12 \, a^{2}\right )}}{{\left (a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - a\right )}^{2} \sqrt{-a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a} a^{3}}\right )} \mathrm{sgn}\left (\cos \left (d x + c\right )\right )}{15 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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