Optimal. Leaf size=76 \[ \frac{3 A+2 B x}{3 a^2 \sqrt{a+b x^2}}-\frac{A \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{a^{5/2}}+\frac{A+B x}{3 a \left (a+b x^2\right )^{3/2}} \]
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Rubi [A] time = 0.0634318, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 6, 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} \[ \frac{3 A+2 B x}{3 a^2 \sqrt{a+b x^2}}-\frac{A \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{a^{5/2}}+\frac{A+B x}{3 a \left (a+b x^2\right )^{3/2}} \]
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 )^{5/2}} \, dx &=\frac{A+B x}{3 a \left (a+b x^2\right )^{3/2}}-\frac{\int \frac{-3 a A b-2 a b B x}{x \left (a+b x^2\right )^{3/2}} \, dx}{3 a^2 b}\\ &=\frac{A+B x}{3 a \left (a+b x^2\right )^{3/2}}+\frac{3 A+2 B x}{3 a^2 \sqrt{a+b x^2}}+\frac{\int \frac{3 a^2 A b^2}{x \sqrt{a+b x^2}} \, dx}{3 a^4 b^2}\\ &=\frac{A+B x}{3 a \left (a+b x^2\right )^{3/2}}+\frac{3 A+2 B x}{3 a^2 \sqrt{a+b x^2}}+\frac{A \int \frac{1}{x \sqrt{a+b x^2}} \, dx}{a^2}\\ &=\frac{A+B x}{3 a \left (a+b x^2\right )^{3/2}}+\frac{3 A+2 B x}{3 a^2 \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^2}\\ &=\frac{A+B x}{3 a \left (a+b x^2\right )^{3/2}}+\frac{3 A+2 B x}{3 a^2 \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^2 b}\\ &=\frac{A+B x}{3 a \left (a+b x^2\right )^{3/2}}+\frac{3 A+2 B x}{3 a^2 \sqrt{a+b x^2}}-\frac{A \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{a^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.0553323, size = 69, normalized size = 0.91 \[ \frac{a (4 A+3 B x)+b x^2 (3 A+2 B x)}{3 a^2 \left (a+b x^2\right )^{3/2}}-\frac{A \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{a^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 92, normalized size = 1.2 \begin{align*}{\frac{Bx}{3\,a} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}+{\frac{2\,Bx}{3\,{a}^{2}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}+{\frac{A}{3\,a} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}+{\frac{A}{{a}^{2}}{\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{5}{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 = 1.69197, size = 537, normalized size = 7.07 \begin{align*} \left [\frac{3 \,{\left (A b^{2} x^{4} + 2 \, A a b x^{2} + A a^{2}\right )} \sqrt{a} \log \left (-\frac{b x^{2} - 2 \, \sqrt{b x^{2} + a} \sqrt{a} + 2 \, a}{x^{2}}\right ) + 2 \,{\left (2 \, B a b x^{3} + 3 \, A a b x^{2} + 3 \, B a^{2} x + 4 \, A a^{2}\right )} \sqrt{b x^{2} + a}}{6 \,{\left (a^{3} b^{2} x^{4} + 2 \, a^{4} b x^{2} + a^{5}\right )}}, \frac{3 \,{\left (A b^{2} x^{4} + 2 \, A a b x^{2} + A a^{2}\right )} \sqrt{-a} \arctan \left (\frac{\sqrt{-a}}{\sqrt{b x^{2} + a}}\right ) +{\left (2 \, B a b x^{3} + 3 \, A a b x^{2} + 3 \, B a^{2} x + 4 \, A a^{2}\right )} \sqrt{b x^{2} + a}}{3 \,{\left (a^{3} b^{2} x^{4} + 2 \, a^{4} b x^{2} + a^{5}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 22.9917, size = 840, normalized size = 11.05 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19623, size = 111, normalized size = 1.46 \begin{align*} \frac{{\left ({\left (\frac{2 \, B b x}{a^{2}} + \frac{3 \, A b}{a^{2}}\right )} x + \frac{3 \, B}{a}\right )} x + \frac{4 \, A}{a}}{3 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}}} + \frac{2 \, A \arctan \left (-\frac{\sqrt{b} x - \sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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