Optimal. Leaf size=47 \[ \frac{A+B x}{a \sqrt{a+b x^2}}-\frac{A \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{a^{3/2}} \]
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Rubi [A] time = 0.0380149, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {823, 12, 266, 63, 208} \[ \frac{A+B x}{a \sqrt{a+b x^2}}-\frac{A \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{a^{3/2}} \]
Antiderivative was successfully verified.
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Rule 823
Rule 12
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{A+B x}{x \left (a+b x^2\right )^{3/2}} \, dx &=\frac{A+B x}{a \sqrt{a+b x^2}}+\frac{\int \frac{a A b}{x \sqrt{a+b x^2}} \, dx}{a^2 b}\\ &=\frac{A+B x}{a \sqrt{a+b x^2}}+\frac{A \int \frac{1}{x \sqrt{a+b x^2}} \, dx}{a}\\ &=\frac{A+B x}{a \sqrt{a+b x^2}}+\frac{A \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,x^2\right )}{2 a}\\ &=\frac{A+B x}{a \sqrt{a+b x^2}}+\frac{A \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x^2}\right )}{a b}\\ &=\frac{A+B x}{a \sqrt{a+b x^2}}-\frac{A \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{a^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0272287, size = 47, normalized size = 1. \[ \frac{A+B x}{a \sqrt{a+b x^2}}-\frac{A \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{a^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 60, normalized size = 1.3 \begin{align*}{\frac{Bx}{a}{\frac{1}{\sqrt{b{x}^{2}+a}}}}+{\frac{A}{a}{\frac{1}{\sqrt{b{x}^{2}+a}}}}-{A\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{3}{2}}}} \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 = 1.84966, size = 338, normalized size = 7.19 \begin{align*} \left [\frac{{\left (A b x^{2} + A a\right )} \sqrt{a} \log \left (-\frac{b x^{2} - 2 \, \sqrt{b x^{2} + a} \sqrt{a} + 2 \, a}{x^{2}}\right ) + 2 \,{\left (B a x + A a\right )} \sqrt{b x^{2} + a}}{2 \,{\left (a^{2} b x^{2} + a^{3}\right )}}, \frac{{\left (A b x^{2} + A a\right )} \sqrt{-a} \arctan \left (\frac{\sqrt{-a}}{\sqrt{b x^{2} + a}}\right ) +{\left (B a x + A a\right )} \sqrt{b x^{2} + a}}{a^{2} b x^{2} + a^{3}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 5.90854, size = 206, normalized size = 4.38 \begin{align*} A \left (\frac{2 a^{3} \sqrt{1 + \frac{b x^{2}}{a}}}{2 a^{\frac{9}{2}} + 2 a^{\frac{7}{2}} b x^{2}} + \frac{a^{3} \log{\left (\frac{b x^{2}}{a} \right )}}{2 a^{\frac{9}{2}} + 2 a^{\frac{7}{2}} b x^{2}} - \frac{2 a^{3} \log{\left (\sqrt{1 + \frac{b x^{2}}{a}} + 1 \right )}}{2 a^{\frac{9}{2}} + 2 a^{\frac{7}{2}} b x^{2}} + \frac{a^{2} b x^{2} \log{\left (\frac{b x^{2}}{a} \right )}}{2 a^{\frac{9}{2}} + 2 a^{\frac{7}{2}} b x^{2}} - \frac{2 a^{2} b x^{2} \log{\left (\sqrt{1 + \frac{b x^{2}}{a}} + 1 \right )}}{2 a^{\frac{9}{2}} + 2 a^{\frac{7}{2}} b x^{2}}\right ) + \frac{B x}{a^{\frac{3}{2}} \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 = 1.189, size = 80, normalized size = 1.7 \begin{align*} \frac{\frac{B x}{a} + \frac{A}{a}}{\sqrt{b x^{2} + a}} + \frac{2 \, A \arctan \left (-\frac{\sqrt{b} x - \sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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