Optimal. Leaf size=119 \[ -\frac{\left (a+b x^2\right )^{3/2} (3 a B+2 A b)}{3 a x}+\frac{b x \sqrt{a+b x^2} (3 a B+2 A b)}{2 a}+\frac{1}{2} \sqrt{b} (3 a B+2 A b) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )-\frac{A \left (a+b x^2\right )^{5/2}}{3 a x^3} \]
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Rubi [A] time = 0.0472946, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {453, 277, 195, 217, 206} \[ -\frac{\left (a+b x^2\right )^{3/2} (3 a B+2 A b)}{3 a x}+\frac{b x \sqrt{a+b x^2} (3 a B+2 A b)}{2 a}+\frac{1}{2} \sqrt{b} (3 a B+2 A b) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )-\frac{A \left (a+b x^2\right )^{5/2}}{3 a x^3} \]
Antiderivative was successfully verified.
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Rule 453
Rule 277
Rule 195
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x^4} \, dx &=-\frac{A \left (a+b x^2\right )^{5/2}}{3 a x^3}-\frac{(-2 A b-3 a B) \int \frac{\left (a+b x^2\right )^{3/2}}{x^2} \, dx}{3 a}\\ &=-\frac{(2 A b+3 a B) \left (a+b x^2\right )^{3/2}}{3 a x}-\frac{A \left (a+b x^2\right )^{5/2}}{3 a x^3}+\frac{(b (2 A b+3 a B)) \int \sqrt{a+b x^2} \, dx}{a}\\ &=\frac{b (2 A b+3 a B) x \sqrt{a+b x^2}}{2 a}-\frac{(2 A b+3 a B) \left (a+b x^2\right )^{3/2}}{3 a x}-\frac{A \left (a+b x^2\right )^{5/2}}{3 a x^3}+\frac{1}{2} (b (2 A b+3 a B)) \int \frac{1}{\sqrt{a+b x^2}} \, dx\\ &=\frac{b (2 A b+3 a B) x \sqrt{a+b x^2}}{2 a}-\frac{(2 A b+3 a B) \left (a+b x^2\right )^{3/2}}{3 a x}-\frac{A \left (a+b x^2\right )^{5/2}}{3 a x^3}+\frac{1}{2} (b (2 A b+3 a B)) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{x}{\sqrt{a+b x^2}}\right )\\ &=\frac{b (2 A b+3 a B) x \sqrt{a+b x^2}}{2 a}-\frac{(2 A b+3 a B) \left (a+b x^2\right )^{3/2}}{3 a x}-\frac{A \left (a+b x^2\right )^{5/2}}{3 a x^3}+\frac{1}{2} \sqrt{b} (2 A b+3 a B) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )\\ \end{align*}
Mathematica [C] time = 0.0767204, size = 83, normalized size = 0.7 \[ \frac{\sqrt{a+b x^2} (-3 a B-2 A b) \, _2F_1\left (-\frac{3}{2},-\frac{1}{2};\frac{1}{2};-\frac{b x^2}{a}\right )}{3 x \sqrt{\frac{b x^2}{a}+1}}-\frac{A \left (a+b x^2\right )^{5/2}}{3 a x^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 168, normalized size = 1.4 \begin{align*} -{\frac{A}{3\,a{x}^{3}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{2\,Ab}{3\,{a}^{2}x} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{2\,{b}^{2}Ax}{3\,{a}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{{b}^{2}Ax}{a}\sqrt{b{x}^{2}+a}}+A{b}^{{\frac{3}{2}}}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ) -{\frac{B}{ax} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{bBx}{a} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{3\,bBx}{2}\sqrt{b{x}^{2}+a}}+{\frac{3\,Ba}{2}\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 = 1.59618, size = 402, normalized size = 3.38 \begin{align*} \left [\frac{3 \,{\left (3 \, B a + 2 \, A b\right )} \sqrt{b} x^{3} \log \left (-2 \, b x^{2} - 2 \, \sqrt{b x^{2} + a} \sqrt{b} x - a\right ) + 2 \,{\left (3 \, B b x^{4} - 2 \,{\left (3 \, B a + 4 \, A b\right )} x^{2} - 2 \, A a\right )} \sqrt{b x^{2} + a}}{12 \, x^{3}}, -\frac{3 \,{\left (3 \, B a + 2 \, A b\right )} \sqrt{-b} x^{3} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right ) -{\left (3 \, B b x^{4} - 2 \,{\left (3 \, B a + 4 \, A b\right )} x^{2} - 2 \, A a\right )} \sqrt{b x^{2} + a}}{6 \, x^{3}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 5.48485, size = 202, normalized size = 1.7 \begin{align*} - \frac{A \sqrt{a} b}{x \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{A a \sqrt{b} \sqrt{\frac{a}{b x^{2}} + 1}}{3 x^{2}} - \frac{A b^{\frac{3}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{3} + A b^{\frac{3}{2}} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )} - \frac{A b^{2} x}{\sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{B a^{\frac{3}{2}}}{x \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{B \sqrt{a} b x \sqrt{1 + \frac{b x^{2}}{a}}}{2} - \frac{B \sqrt{a} b x}{\sqrt{1 + \frac{b x^{2}}{a}}} + \frac{3 B a \sqrt{b} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.14788, size = 279, normalized size = 2.34 \begin{align*} \frac{1}{2} \, \sqrt{b x^{2} + a} B b x - \frac{1}{4} \,{\left (3 \, B a \sqrt{b} + 2 \, A b^{\frac{3}{2}}\right )} \log \left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2}\right ) + \frac{2 \,{\left (3 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} B a^{2} \sqrt{b} + 6 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} A a b^{\frac{3}{2}} - 6 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} B a^{3} \sqrt{b} - 6 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} A a^{2} b^{\frac{3}{2}} + 3 \, B a^{4} \sqrt{b} + 4 \, A a^{3} b^{\frac{3}{2}}\right )}}{3 \,{\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} - a\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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