Optimal. Leaf size=95 \[ -\frac{3 A \sqrt{a+c x^2}}{2 a^2 x^2}+\frac{3 A c \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{2 a^{5/2}}-\frac{2 B \sqrt{a+c x^2}}{a^2 x}+\frac{A+B x}{a x^2 \sqrt{a+c x^2}} \]
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Rubi [A] time = 0.0795045, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {823, 835, 807, 266, 63, 208} \[ -\frac{3 A \sqrt{a+c x^2}}{2 a^2 x^2}+\frac{3 A c \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{2 a^{5/2}}-\frac{2 B \sqrt{a+c x^2}}{a^2 x}+\frac{A+B x}{a x^2 \sqrt{a+c x^2}} \]
Antiderivative was successfully verified.
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Rule 823
Rule 835
Rule 807
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{A+B x}{x^3 \left (a+c x^2\right )^{3/2}} \, dx &=\frac{A+B x}{a x^2 \sqrt{a+c x^2}}-\frac{\int \frac{-3 a A c-2 a B c x}{x^3 \sqrt{a+c x^2}} \, dx}{a^2 c}\\ &=\frac{A+B x}{a x^2 \sqrt{a+c x^2}}-\frac{3 A \sqrt{a+c x^2}}{2 a^2 x^2}+\frac{\int \frac{4 a^2 B c-3 a A c^2 x}{x^2 \sqrt{a+c x^2}} \, dx}{2 a^3 c}\\ &=\frac{A+B x}{a x^2 \sqrt{a+c x^2}}-\frac{3 A \sqrt{a+c x^2}}{2 a^2 x^2}-\frac{2 B \sqrt{a+c x^2}}{a^2 x}-\frac{(3 A c) \int \frac{1}{x \sqrt{a+c x^2}} \, dx}{2 a^2}\\ &=\frac{A+B x}{a x^2 \sqrt{a+c x^2}}-\frac{3 A \sqrt{a+c x^2}}{2 a^2 x^2}-\frac{2 B \sqrt{a+c x^2}}{a^2 x}-\frac{(3 A c) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+c x}} \, dx,x,x^2\right )}{4 a^2}\\ &=\frac{A+B x}{a x^2 \sqrt{a+c x^2}}-\frac{3 A \sqrt{a+c x^2}}{2 a^2 x^2}-\frac{2 B \sqrt{a+c x^2}}{a^2 x}-\frac{(3 A) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{c}+\frac{x^2}{c}} \, dx,x,\sqrt{a+c x^2}\right )}{2 a^2}\\ &=\frac{A+B x}{a x^2 \sqrt{a+c x^2}}-\frac{3 A \sqrt{a+c x^2}}{2 a^2 x^2}-\frac{2 B \sqrt{a+c x^2}}{a^2 x}+\frac{3 A c \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{2 a^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.159543, size = 75, normalized size = 0.79 \[ \frac{-\frac{a (A+2 B x)}{x^2}+3 A c \sqrt{\frac{c x^2}{a}+1} \tanh ^{-1}\left (\sqrt{\frac{c x^2}{a}+1}\right )-c (3 A+4 B x)}{2 a^2 \sqrt{a+c x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 101, normalized size = 1.1 \begin{align*} -{\frac{A}{2\,a{x}^{2}}{\frac{1}{\sqrt{c{x}^{2}+a}}}}-{\frac{3\,Ac}{2\,{a}^{2}}{\frac{1}{\sqrt{c{x}^{2}+a}}}}+{\frac{3\,Ac}{2}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{c{x}^{2}+a} \right ) } \right ){a}^{-{\frac{5}{2}}}}-{\frac{B}{ax}{\frac{1}{\sqrt{c{x}^{2}+a}}}}-2\,{\frac{Bcx}{{a}^{2}\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 = 1.35413, size = 474, normalized size = 4.99 \begin{align*} \left [\frac{3 \,{\left (A c^{2} x^{4} + A a c x^{2}\right )} \sqrt{a} \log \left (-\frac{c x^{2} + 2 \, \sqrt{c x^{2} + a} \sqrt{a} + 2 \, a}{x^{2}}\right ) - 2 \,{\left (4 \, B a c x^{3} + 3 \, A a c x^{2} + 2 \, B a^{2} x + A a^{2}\right )} \sqrt{c x^{2} + a}}{4 \,{\left (a^{3} c x^{4} + a^{4} x^{2}\right )}}, -\frac{3 \,{\left (A c^{2} x^{4} + A a c x^{2}\right )} \sqrt{-a} \arctan \left (\frac{\sqrt{-a}}{\sqrt{c x^{2} + a}}\right ) +{\left (4 \, B a c x^{3} + 3 \, A a c x^{2} + 2 \, B a^{2} x + A a^{2}\right )} \sqrt{c x^{2} + a}}{2 \,{\left (a^{3} c x^{4} + a^{4} x^{2}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 10.2129, size = 124, normalized size = 1.31 \begin{align*} A \left (- \frac{1}{2 a \sqrt{c} x^{3} \sqrt{\frac{a}{c x^{2}} + 1}} - \frac{3 \sqrt{c}}{2 a^{2} x \sqrt{\frac{a}{c x^{2}} + 1}} + \frac{3 c \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{c} x} \right )}}{2 a^{\frac{5}{2}}}\right ) + B \left (- \frac{1}{a \sqrt{c} x^{2} \sqrt{\frac{a}{c x^{2}} + 1}} - \frac{2 \sqrt{c}}{a^{2} \sqrt{\frac{a}{c x^{2}} + 1}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.27244, size = 231, normalized size = 2.43 \begin{align*} -\frac{\frac{B c x}{a^{2}} + \frac{A c}{a^{2}}}{\sqrt{c x^{2} + a}} - \frac{3 \, A c \arctan \left (-\frac{\sqrt{c} x - \sqrt{c x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{2}} + \frac{{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{3} A c + 2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} B a \sqrt{c} +{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )} A a c - 2 \, B a^{2} \sqrt{c}}{{\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} - a\right )}^{2} a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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