Optimal. Leaf size=141 \[ -\frac{5}{2} a^{3/2} A b \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )+\frac{15}{8} a^2 \sqrt{b} B \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )-\frac{\left (a+b x^2\right )^{5/2} (2 A-B x)}{4 x^2}-\frac{5 \left (a+b x^2\right )^{3/2} (3 a B-2 A b x)}{12 x}+\frac{5}{8} a b \sqrt{a+b x^2} (4 A+3 B x) \]
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Rubi [A] time = 0.117082, antiderivative size = 141, 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} \[ -\frac{5}{2} a^{3/2} A b \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )+\frac{15}{8} a^2 \sqrt{b} B \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )-\frac{\left (a+b x^2\right )^{5/2} (2 A-B x)}{4 x^2}-\frac{5 \left (a+b x^2\right )^{3/2} (3 a B-2 A b x)}{12 x}+\frac{5}{8} a b \sqrt{a+b x^2} (4 A+3 B x) \]
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+b x^2\right )^{5/2}}{x^3} \, dx &=-\frac{(2 A-B x) \left (a+b x^2\right )^{5/2}}{4 x^2}-\frac{5}{16} \int \frac{(-4 a B-8 A b x) \left (a+b x^2\right )^{3/2}}{x^2} \, dx\\ &=-\frac{5 (3 a B-2 A b x) \left (a+b x^2\right )^{3/2}}{12 x}-\frac{(2 A-B x) \left (a+b x^2\right )^{5/2}}{4 x^2}+\frac{5}{32} \int \frac{(16 a A b+24 a b B x) \sqrt{a+b x^2}}{x} \, dx\\ &=\frac{5}{8} a b (4 A+3 B x) \sqrt{a+b x^2}-\frac{5 (3 a B-2 A b x) \left (a+b x^2\right )^{3/2}}{12 x}-\frac{(2 A-B x) \left (a+b x^2\right )^{5/2}}{4 x^2}+\frac{5 \int \frac{32 a^2 A b^2+24 a^2 b^2 B x}{x \sqrt{a+b x^2}} \, dx}{64 b}\\ &=\frac{5}{8} a b (4 A+3 B x) \sqrt{a+b x^2}-\frac{5 (3 a B-2 A b x) \left (a+b x^2\right )^{3/2}}{12 x}-\frac{(2 A-B x) \left (a+b x^2\right )^{5/2}}{4 x^2}+\frac{1}{2} \left (5 a^2 A b\right ) \int \frac{1}{x \sqrt{a+b x^2}} \, dx+\frac{1}{8} \left (15 a^2 b B\right ) \int \frac{1}{\sqrt{a+b x^2}} \, dx\\ &=\frac{5}{8} a b (4 A+3 B x) \sqrt{a+b x^2}-\frac{5 (3 a B-2 A b x) \left (a+b x^2\right )^{3/2}}{12 x}-\frac{(2 A-B x) \left (a+b x^2\right )^{5/2}}{4 x^2}+\frac{1}{4} \left (5 a^2 A b\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,x^2\right )+\frac{1}{8} \left (15 a^2 b B\right ) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{x}{\sqrt{a+b x^2}}\right )\\ &=\frac{5}{8} a b (4 A+3 B x) \sqrt{a+b x^2}-\frac{5 (3 a B-2 A b x) \left (a+b x^2\right )^{3/2}}{12 x}-\frac{(2 A-B x) \left (a+b x^2\right )^{5/2}}{4 x^2}+\frac{15}{8} a^2 \sqrt{b} B \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )+\frac{1}{2} \left (5 a^2 A\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x^2}\right )\\ &=\frac{5}{8} a b (4 A+3 B x) \sqrt{a+b x^2}-\frac{5 (3 a B-2 A b x) \left (a+b x^2\right )^{3/2}}{12 x}-\frac{(2 A-B x) \left (a+b x^2\right )^{5/2}}{4 x^2}+\frac{15}{8} a^2 \sqrt{b} B \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )-\frac{5}{2} a^{3/2} A b \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )\\ \end{align*}
Mathematica [C] time = 0.0259002, size = 92, normalized size = 0.65 \[ \frac{A b \left (a+b x^2\right )^{7/2} \, _2F_1\left (2,\frac{7}{2};\frac{9}{2};\frac{b x^2}{a}+1\right )}{7 a^2}-\frac{a^2 B \sqrt{a+b x^2} \, _2F_1\left (-\frac{5}{2},-\frac{1}{2};\frac{1}{2};-\frac{b x^2}{a}\right )}{x \sqrt{\frac{b x^2}{a}+1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 181, normalized size = 1.3 \begin{align*} -{\frac{A}{2\,a{x}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{Ab}{2\,a} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{5\,Ab}{6} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{5\,Ab}{2}{a}^{{\frac{3}{2}}}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ) }+{\frac{5\,Aab}{2}\sqrt{b{x}^{2}+a}}-{\frac{B}{ax} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{bBx}{a} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{5\,bBx}{4} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{15\,Bbax}{8}\sqrt{b{x}^{2}+a}}+{\frac{15\,B{a}^{2}}{8}\sqrt{b}\ln \left ( x\sqrt{b}+\sqrt{b{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 = 1.70185, size = 1316, normalized size = 9.33 \begin{align*} \left [\frac{45 \, B a^{2} \sqrt{b} x^{2} \log \left (-2 \, b x^{2} - 2 \, \sqrt{b x^{2} + a} \sqrt{b} x - a\right ) + 60 \, A a^{\frac{3}{2}} b x^{2} \log \left (-\frac{b x^{2} - 2 \, \sqrt{b x^{2} + a} \sqrt{a} + 2 \, a}{x^{2}}\right ) + 2 \,{\left (6 \, B b^{2} x^{5} + 8 \, A b^{2} x^{4} + 27 \, B a b x^{3} + 56 \, A a b x^{2} - 24 \, B a^{2} x - 12 \, A a^{2}\right )} \sqrt{b x^{2} + a}}{48 \, x^{2}}, -\frac{45 \, B a^{2} \sqrt{-b} x^{2} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right ) - 30 \, A a^{\frac{3}{2}} b x^{2} \log \left (-\frac{b x^{2} - 2 \, \sqrt{b x^{2} + a} \sqrt{a} + 2 \, a}{x^{2}}\right ) -{\left (6 \, B b^{2} x^{5} + 8 \, A b^{2} x^{4} + 27 \, B a b x^{3} + 56 \, A a b x^{2} - 24 \, B a^{2} x - 12 \, A a^{2}\right )} \sqrt{b x^{2} + a}}{24 \, x^{2}}, \frac{120 \, A \sqrt{-a} a b x^{2} \arctan \left (\frac{\sqrt{-a}}{\sqrt{b x^{2} + a}}\right ) + 45 \, B a^{2} \sqrt{b} x^{2} \log \left (-2 \, b x^{2} - 2 \, \sqrt{b x^{2} + a} \sqrt{b} x - a\right ) + 2 \,{\left (6 \, B b^{2} x^{5} + 8 \, A b^{2} x^{4} + 27 \, B a b x^{3} + 56 \, A a b x^{2} - 24 \, B a^{2} x - 12 \, A a^{2}\right )} \sqrt{b x^{2} + a}}{48 \, x^{2}}, -\frac{45 \, B a^{2} \sqrt{-b} x^{2} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right ) - 60 \, A \sqrt{-a} a b x^{2} \arctan \left (\frac{\sqrt{-a}}{\sqrt{b x^{2} + a}}\right ) -{\left (6 \, B b^{2} x^{5} + 8 \, A b^{2} x^{4} + 27 \, B a b x^{3} + 56 \, A a b x^{2} - 24 \, B a^{2} x - 12 \, A a^{2}\right )} \sqrt{b x^{2} + a}}{24 \, x^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 11.3184, size = 279, normalized size = 1.98 \begin{align*} - \frac{5 A a^{\frac{3}{2}} b \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{2} - \frac{A a^{2} \sqrt{b} \sqrt{\frac{a}{b x^{2}} + 1}}{2 x} + \frac{2 A a^{2} \sqrt{b}}{x \sqrt{\frac{a}{b x^{2}} + 1}} + \frac{2 A a b^{\frac{3}{2}} x}{\sqrt{\frac{a}{b x^{2}} + 1}} + A b^{2} \left (\begin{cases} \frac{\sqrt{a} x^{2}}{2} & \text{for}\: b = 0 \\\frac{\left (a + b x^{2}\right )^{\frac{3}{2}}}{3 b} & \text{otherwise} \end{cases}\right ) - \frac{B a^{\frac{5}{2}}}{x \sqrt{1 + \frac{b x^{2}}{a}}} + B a^{\frac{3}{2}} b x \sqrt{1 + \frac{b x^{2}}{a}} - \frac{7 B a^{\frac{3}{2}} b x}{8 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{3 B \sqrt{a} b^{2} x^{3}}{8 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{15 B a^{2} \sqrt{b} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{8} + \frac{B b^{3} x^{5}}{4 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18631, size = 296, normalized size = 2.1 \begin{align*} \frac{5 \, A a^{2} b \arctan \left (-\frac{\sqrt{b} x - \sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} - \frac{15}{8} \, B a^{2} \sqrt{b} \log \left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right ) + \frac{1}{24} \,{\left (56 \, A a b +{\left (27 \, B a b + 2 \,{\left (3 \, B b^{2} x + 4 \, A b^{2}\right )} x\right )} x\right )} \sqrt{b x^{2} + a} + \frac{{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{3} A a^{2} b + 2 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} B a^{3} \sqrt{b} +{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )} A a^{3} b - 2 \, B a^{4} \sqrt{b}}{{\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} - a\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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