Optimal. Leaf size=69 \[ -\frac{3 b}{2 a^2 \sqrt{a+b x^2}}+\frac{3 b \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{2 a^{5/2}}-\frac{1}{2 a x^2 \sqrt{a+b x^2}} \]
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Rubi [A] time = 0.0390216, antiderivative size = 68, normalized size of antiderivative = 0.99, number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {266, 51, 63, 208} \[ -\frac{3 \sqrt{a+b x^2}}{2 a^2 x^2}+\frac{3 b \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{2 a^{5/2}}+\frac{1}{a x^2 \sqrt{a+b x^2}} \]
Antiderivative was successfully verified.
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Rule 266
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{x^3 \left (a+b x^2\right )^{3/2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x^2 (a+b x)^{3/2}} \, dx,x,x^2\right )\\ &=\frac{1}{a x^2 \sqrt{a+b x^2}}+\frac{3 \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{a+b x}} \, dx,x,x^2\right )}{2 a}\\ &=\frac{1}{a x^2 \sqrt{a+b x^2}}-\frac{3 \sqrt{a+b x^2}}{2 a^2 x^2}-\frac{(3 b) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,x^2\right )}{4 a^2}\\ &=\frac{1}{a x^2 \sqrt{a+b x^2}}-\frac{3 \sqrt{a+b x^2}}{2 a^2 x^2}-\frac{3 \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x^2}\right )}{2 a^2}\\ &=\frac{1}{a x^2 \sqrt{a+b x^2}}-\frac{3 \sqrt{a+b x^2}}{2 a^2 x^2}+\frac{3 b \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{2 a^{5/2}}\\ \end{align*}
Mathematica [C] time = 0.007585, size = 35, normalized size = 0.51 \[ -\frac{b \, _2F_1\left (-\frac{1}{2},2;\frac{1}{2};\frac{b x^2}{a}+1\right )}{a^2 \sqrt{a+b x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 63, normalized size = 0.9 \begin{align*} -{\frac{1}{2\,a{x}^{2}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}-{\frac{3\,b}{2\,{a}^{2}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}+{\frac{3\,b}{2}\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 = 1.35542, size = 382, normalized size = 5.54 \begin{align*} \left [\frac{3 \,{\left (b^{2} x^{4} + a b x^{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 (3 \, a b x^{2} + a^{2}\right )} \sqrt{b x^{2} + a}}{4 \,{\left (a^{3} b x^{4} + a^{4} x^{2}\right )}}, -\frac{3 \,{\left (b^{2} x^{4} + a b x^{2}\right )} \sqrt{-a} \arctan \left (\frac{\sqrt{-a}}{\sqrt{b x^{2} + a}}\right ) +{\left (3 \, a b x^{2} + a^{2}\right )} \sqrt{b x^{2} + a}}{2 \,{\left (a^{3} b 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 = 3.19302, size = 73, normalized size = 1.06 \begin{align*} - \frac{1}{2 a \sqrt{b} x^{3} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{3 \sqrt{b}}{2 a^{2} x \sqrt{\frac{a}{b x^{2}} + 1}} + \frac{3 b \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{2 a^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.72575, size = 89, normalized size = 1.29 \begin{align*} -\frac{1}{2} \, b{\left (\frac{3 \, \arctan \left (\frac{\sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{2}} + \frac{3 \, b x^{2} + a}{{\left ({\left (b x^{2} + a\right )}^{\frac{3}{2}} - \sqrt{b x^{2} + a} a\right )} a^{2}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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