Optimal. Leaf size=136 \[ \frac{15}{8} a^2 A \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )+a^{5/2} (-B) \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )+\frac{1}{8} a \sqrt{a+c x^2} (8 a B+15 A c x)-\frac{\left (a+c x^2\right )^{5/2} (5 A-B x)}{5 x}+\frac{1}{12} \left (a+c x^2\right )^{3/2} (4 a B+15 A c x) \]
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Rubi [A] time = 0.120381, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {813, 815, 844, 217, 206, 266, 63, 208} \[ \frac{15}{8} a^2 A \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )+a^{5/2} (-B) \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )+\frac{1}{8} a \sqrt{a+c x^2} (8 a B+15 A c x)-\frac{\left (a+c x^2\right )^{5/2} (5 A-B x)}{5 x}+\frac{1}{12} \left (a+c x^2\right )^{3/2} (4 a B+15 A c x) \]
Antiderivative was successfully verified.
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Rule 813
Rule 815
Rule 844
Rule 217
Rule 206
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{(A+B x) \left (a+c x^2\right )^{5/2}}{x^2} \, dx &=-\frac{(5 A-B x) \left (a+c x^2\right )^{5/2}}{5 x}-\frac{1}{2} \int \frac{(-2 a B-10 A c x) \left (a+c x^2\right )^{3/2}}{x} \, dx\\ &=\frac{1}{12} (4 a B+15 A c x) \left (a+c x^2\right )^{3/2}-\frac{(5 A-B x) \left (a+c x^2\right )^{5/2}}{5 x}-\frac{\int \frac{\left (-8 a^2 B c-30 a A c^2 x\right ) \sqrt{a+c x^2}}{x} \, dx}{8 c}\\ &=\frac{1}{8} a (8 a B+15 A c x) \sqrt{a+c x^2}+\frac{1}{12} (4 a B+15 A c x) \left (a+c x^2\right )^{3/2}-\frac{(5 A-B x) \left (a+c x^2\right )^{5/2}}{5 x}-\frac{\int \frac{-16 a^3 B c^2-30 a^2 A c^3 x}{x \sqrt{a+c x^2}} \, dx}{16 c^2}\\ &=\frac{1}{8} a (8 a B+15 A c x) \sqrt{a+c x^2}+\frac{1}{12} (4 a B+15 A c x) \left (a+c x^2\right )^{3/2}-\frac{(5 A-B x) \left (a+c x^2\right )^{5/2}}{5 x}+\left (a^3 B\right ) \int \frac{1}{x \sqrt{a+c x^2}} \, dx+\frac{1}{8} \left (15 a^2 A c\right ) \int \frac{1}{\sqrt{a+c x^2}} \, dx\\ &=\frac{1}{8} a (8 a B+15 A c x) \sqrt{a+c x^2}+\frac{1}{12} (4 a B+15 A c x) \left (a+c x^2\right )^{3/2}-\frac{(5 A-B x) \left (a+c x^2\right )^{5/2}}{5 x}+\frac{1}{2} \left (a^3 B\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+c x}} \, dx,x,x^2\right )+\frac{1}{8} \left (15 a^2 A c\right ) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{a+c x^2}}\right )\\ &=\frac{1}{8} a (8 a B+15 A c x) \sqrt{a+c x^2}+\frac{1}{12} (4 a B+15 A c x) \left (a+c x^2\right )^{3/2}-\frac{(5 A-B x) \left (a+c x^2\right )^{5/2}}{5 x}+\frac{15}{8} a^2 A \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )+\frac{\left (a^3 B\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{c}+\frac{x^2}{c}} \, dx,x,\sqrt{a+c x^2}\right )}{c}\\ &=\frac{1}{8} a (8 a B+15 A c x) \sqrt{a+c x^2}+\frac{1}{12} (4 a B+15 A c x) \left (a+c x^2\right )^{3/2}-\frac{(5 A-B x) \left (a+c x^2\right )^{5/2}}{5 x}+\frac{15}{8} a^2 A \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )-a^{5/2} B \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )\\ \end{align*}
Mathematica [C] time = 0.218287, size = 117, normalized size = 0.86 \[ -\frac{a^3 A \sqrt{\frac{c x^2}{a}+1} \, _2F_1\left (-\frac{5}{2},-\frac{1}{2};\frac{1}{2};-\frac{c x^2}{a}\right )}{x \sqrt{a+c x^2}}+\frac{1}{15} B \sqrt{a+c x^2} \left (23 a^2+11 a c x^2+3 c^2 x^4\right )-a^{5/2} B \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 158, normalized size = 1.2 \begin{align*}{\frac{B}{5} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{aB}{3} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-B{a}^{{\frac{5}{2}}}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{c{x}^{2}+a} \right ) } \right ) +B\sqrt{c{x}^{2}+a}{a}^{2}-{\frac{A}{ax} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{Acx}{a} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{5\,Acx}{4} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{15\,aAcx}{8}\sqrt{c{x}^{2}+a}}+{\frac{15\,A{a}^{2}}{8}\sqrt{c}\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+a} \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 = 1.77564, size = 1319, normalized size = 9.7 \begin{align*} \left [\frac{225 \, A a^{2} \sqrt{c} x \log \left (-2 \, c x^{2} - 2 \, \sqrt{c x^{2} + a} \sqrt{c} x - a\right ) + 120 \, B a^{\frac{5}{2}} x \log \left (-\frac{c x^{2} - 2 \, \sqrt{c x^{2} + a} \sqrt{a} + 2 \, a}{x^{2}}\right ) + 2 \,{\left (24 \, B c^{2} x^{5} + 30 \, A c^{2} x^{4} + 88 \, B a c x^{3} + 135 \, A a c x^{2} + 184 \, B a^{2} x - 120 \, A a^{2}\right )} \sqrt{c x^{2} + a}}{240 \, x}, -\frac{225 \, A a^{2} \sqrt{-c} x \arctan \left (\frac{\sqrt{-c} x}{\sqrt{c x^{2} + a}}\right ) - 60 \, B a^{\frac{5}{2}} x \log \left (-\frac{c x^{2} - 2 \, \sqrt{c x^{2} + a} \sqrt{a} + 2 \, a}{x^{2}}\right ) -{\left (24 \, B c^{2} x^{5} + 30 \, A c^{2} x^{4} + 88 \, B a c x^{3} + 135 \, A a c x^{2} + 184 \, B a^{2} x - 120 \, A a^{2}\right )} \sqrt{c x^{2} + a}}{120 \, x}, \frac{240 \, B \sqrt{-a} a^{2} x \arctan \left (\frac{\sqrt{-a}}{\sqrt{c x^{2} + a}}\right ) + 225 \, A a^{2} \sqrt{c} x \log \left (-2 \, c x^{2} - 2 \, \sqrt{c x^{2} + a} \sqrt{c} x - a\right ) + 2 \,{\left (24 \, B c^{2} x^{5} + 30 \, A c^{2} x^{4} + 88 \, B a c x^{3} + 135 \, A a c x^{2} + 184 \, B a^{2} x - 120 \, A a^{2}\right )} \sqrt{c x^{2} + a}}{240 \, x}, -\frac{225 \, A a^{2} \sqrt{-c} x \arctan \left (\frac{\sqrt{-c} x}{\sqrt{c x^{2} + a}}\right ) - 120 \, B \sqrt{-a} a^{2} x \arctan \left (\frac{\sqrt{-a}}{\sqrt{c x^{2} + a}}\right ) -{\left (24 \, B c^{2} x^{5} + 30 \, A c^{2} x^{4} + 88 \, B a c x^{3} + 135 \, A a c x^{2} + 184 \, B a^{2} x - 120 \, A a^{2}\right )} \sqrt{c x^{2} + a}}{120 \, x}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 14.8022, size = 318, normalized size = 2.34 \begin{align*} - \frac{A a^{\frac{5}{2}}}{x \sqrt{1 + \frac{c x^{2}}{a}}} + A a^{\frac{3}{2}} c x \sqrt{1 + \frac{c x^{2}}{a}} - \frac{7 A a^{\frac{3}{2}} c x}{8 \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{3 A \sqrt{a} c^{2} x^{3}}{8 \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{15 A a^{2} \sqrt{c} \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{8} + \frac{A c^{3} x^{5}}{4 \sqrt{a} \sqrt{1 + \frac{c x^{2}}{a}}} - B a^{\frac{5}{2}} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{c} x} \right )} + \frac{B a^{3}}{\sqrt{c} x \sqrt{\frac{a}{c x^{2}} + 1}} + \frac{B a^{2} \sqrt{c} x}{\sqrt{\frac{a}{c x^{2}} + 1}} + 2 B a c \left (\begin{cases} \frac{\sqrt{a} x^{2}}{2} & \text{for}\: c = 0 \\\frac{\left (a + c x^{2}\right )^{\frac{3}{2}}}{3 c} & \text{otherwise} \end{cases}\right ) + B c^{2} \left (\begin{cases} - \frac{2 a^{2} \sqrt{a + c x^{2}}}{15 c^{2}} + \frac{a x^{2} \sqrt{a + c x^{2}}}{15 c} + \frac{x^{4} \sqrt{a + c x^{2}}}{5} & \text{for}\: c \neq 0 \\\frac{\sqrt{a} x^{4}}{4} & \text{otherwise} \end{cases}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15919, size = 203, normalized size = 1.49 \begin{align*} \frac{2 \, B a^{3} \arctan \left (-\frac{\sqrt{c} x - \sqrt{c x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} - \frac{15}{8} \, A a^{2} \sqrt{c} \log \left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right ) + \frac{2 \, A a^{3} \sqrt{c}}{{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} - a} + \frac{1}{120} \,{\left (184 \, B a^{2} +{\left (135 \, A a c + 2 \,{\left (44 \, B a c + 3 \,{\left (4 \, B c^{2} x + 5 \, A c^{2}\right )} x\right )} x\right )} x\right )} \sqrt{c x^{2} + a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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