Optimal. Leaf size=48 \[ \frac{B \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{c^{3/2}}-\frac{A+B x}{c \sqrt{a+c x^2}} \]
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Rubi [A] time = 0.0183047, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {778, 217, 206} \[ \frac{B \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{c^{3/2}}-\frac{A+B x}{c \sqrt{a+c x^2}} \]
Antiderivative was successfully verified.
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Rule 778
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \frac{x (A+B x)}{\left (a+c x^2\right )^{3/2}} \, dx &=-\frac{A+B x}{c \sqrt{a+c x^2}}+\frac{B \int \frac{1}{\sqrt{a+c x^2}} \, dx}{c}\\ &=-\frac{A+B x}{c \sqrt{a+c x^2}}+\frac{B \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{a+c x^2}}\right )}{c}\\ &=-\frac{A+B x}{c \sqrt{a+c x^2}}+\frac{B \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{c^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0555763, size = 64, normalized size = 1.33 \[ \frac{\sqrt{a} B \sqrt{\frac{c x^2}{a}+1} \sinh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )-\sqrt{c} (A+B x)}{c^{3/2} \sqrt{a+c x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 54, normalized size = 1.1 \begin{align*} -{\frac{Bx}{c}{\frac{1}{\sqrt{c{x}^{2}+a}}}}+{B\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+a} \right ){c}^{-{\frac{3}{2}}}}-{\frac{A}{c}{\frac{1}{\sqrt{c{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 = 1.33938, size = 336, normalized size = 7. \begin{align*} \left [\frac{{\left (B c x^{2} + B a\right )} \sqrt{c} \log \left (-2 \, c x^{2} - 2 \, \sqrt{c x^{2} + a} \sqrt{c} x - a\right ) - 2 \,{\left (B c x + A c\right )} \sqrt{c x^{2} + a}}{2 \,{\left (c^{3} x^{2} + a c^{2}\right )}}, -\frac{{\left (B c x^{2} + B a\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{-c} x}{\sqrt{c x^{2} + a}}\right ) +{\left (B c x + A c\right )} \sqrt{c x^{2} + a}}{c^{3} x^{2} + a c^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 5.4927, size = 66, normalized size = 1.38 \begin{align*} A \left (\begin{cases} - \frac{1}{c \sqrt{a + c x^{2}}} & \text{for}\: c \neq 0 \\\frac{x^{2}}{2 a^{\frac{3}{2}}} & \text{otherwise} \end{cases}\right ) + B \left (\frac{\operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{c^{\frac{3}{2}}} - \frac{x}{\sqrt{a} c \sqrt{1 + \frac{c x^{2}}{a}}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.2318, size = 65, normalized size = 1.35 \begin{align*} -\frac{\frac{B x}{c} + \frac{A}{c}}{\sqrt{c x^{2} + a}} - \frac{B \log \left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right )}{c^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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