Optimal. Leaf size=84 \[ \frac{\sqrt{a+b x^2} (2 a B+A b)}{2 a}-\frac{(2 a B+A b) \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{2 \sqrt{a}}-\frac{A \left (a+b x^2\right )^{3/2}}{2 a x^2} \]
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Rubi [A] time = 0.0629452, antiderivative size = 84, 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 = {446, 78, 50, 63, 208} \[ \frac{\sqrt{a+b x^2} (2 a B+A b)}{2 a}-\frac{(2 a B+A b) \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{2 \sqrt{a}}-\frac{A \left (a+b x^2\right )^{3/2}}{2 a x^2} \]
Antiderivative was successfully verified.
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Rule 446
Rule 78
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\sqrt{a+b x^2} \left (A+B x^2\right )}{x^3} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{\sqrt{a+b x} (A+B x)}{x^2} \, dx,x,x^2\right )\\ &=-\frac{A \left (a+b x^2\right )^{3/2}}{2 a x^2}+\frac{(A b+2 a B) \operatorname{Subst}\left (\int \frac{\sqrt{a+b x}}{x} \, dx,x,x^2\right )}{4 a}\\ &=\frac{(A b+2 a B) \sqrt{a+b x^2}}{2 a}-\frac{A \left (a+b x^2\right )^{3/2}}{2 a x^2}+\frac{1}{4} (A b+2 a B) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,x^2\right )\\ &=\frac{(A b+2 a B) \sqrt{a+b x^2}}{2 a}-\frac{A \left (a+b x^2\right )^{3/2}}{2 a x^2}+\frac{(A b+2 a B) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x^2}\right )}{2 b}\\ &=\frac{(A b+2 a B) \sqrt{a+b x^2}}{2 a}-\frac{A \left (a+b x^2\right )^{3/2}}{2 a x^2}-\frac{(A b+2 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{2 \sqrt{a}}\\ \end{align*}
Mathematica [A] time = 0.0418051, size = 63, normalized size = 0.75 \[ \frac{1}{2} \left (\frac{\sqrt{a+b x^2} \left (2 B x^2-A\right )}{x^2}-\frac{(2 a B+A b) \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{\sqrt{a}}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 106, normalized size = 1.3 \begin{align*} -B\sqrt{a}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ) +B\sqrt{b{x}^{2}+a}-{\frac{A}{2\,a{x}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{Ab}{2}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){\frac{1}{\sqrt{a}}}}+{\frac{Ab}{2\,a}\sqrt{b{x}^{2}+a}} \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 = 1.6608, size = 332, normalized size = 3.95 \begin{align*} \left [\frac{{\left (2 \, B a + A b\right )} \sqrt{a} x^{2} \log \left (-\frac{b x^{2} - 2 \, \sqrt{b x^{2} + a} \sqrt{a} + 2 \, a}{x^{2}}\right ) + 2 \,{\left (2 \, B a x^{2} - A a\right )} \sqrt{b x^{2} + a}}{4 \, a x^{2}}, \frac{{\left (2 \, B a + A b\right )} \sqrt{-a} x^{2} \arctan \left (\frac{\sqrt{-a}}{\sqrt{b x^{2} + a}}\right ) +{\left (2 \, B a x^{2} - A a\right )} \sqrt{b x^{2} + a}}{2 \, a x^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 19.8911, size = 107, normalized size = 1.27 \begin{align*} - \frac{A \sqrt{b} \sqrt{\frac{a}{b x^{2}} + 1}}{2 x} - \frac{A b \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{2 \sqrt{a}} - B \sqrt{a} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )} + \frac{B a}{\sqrt{b} x \sqrt{\frac{a}{b x^{2}} + 1}} + \frac{B \sqrt{b} x}{\sqrt{\frac{a}{b x^{2}} + 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11272, size = 92, normalized size = 1.1 \begin{align*} \frac{2 \, \sqrt{b x^{2} + a} B b + \frac{{\left (2 \, B a b + A b^{2}\right )} \arctan \left (\frac{\sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} - \frac{\sqrt{b x^{2} + a} A b}{x^{2}}}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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