Optimal. Leaf size=64 \[ -\frac{3 x}{8 a^2 \left (a x^2+b\right )}+\frac{3 \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{8 a^{5/2} \sqrt{b}}-\frac{x^3}{4 a \left (a x^2+b\right )^2} \]
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Rubi [A] time = 0.0200818, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {263, 288, 205} \[ -\frac{3 x}{8 a^2 \left (a x^2+b\right )}+\frac{3 \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{8 a^{5/2} \sqrt{b}}-\frac{x^3}{4 a \left (a x^2+b\right )^2} \]
Antiderivative was successfully verified.
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Rule 263
Rule 288
Rule 205
Rubi steps
\begin{align*} \int \frac{1}{\left (a+\frac{b}{x^2}\right )^3 x^2} \, dx &=\int \frac{x^4}{\left (b+a x^2\right )^3} \, dx\\ &=-\frac{x^3}{4 a \left (b+a x^2\right )^2}+\frac{3 \int \frac{x^2}{\left (b+a x^2\right )^2} \, dx}{4 a}\\ &=-\frac{x^3}{4 a \left (b+a x^2\right )^2}-\frac{3 x}{8 a^2 \left (b+a x^2\right )}+\frac{3 \int \frac{1}{b+a x^2} \, dx}{8 a^2}\\ &=-\frac{x^3}{4 a \left (b+a x^2\right )^2}-\frac{3 x}{8 a^2 \left (b+a x^2\right )}+\frac{3 \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{8 a^{5/2} \sqrt{b}}\\ \end{align*}
Mathematica [A] time = 0.0403315, size = 55, normalized size = 0.86 \[ \frac{3 \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{8 a^{5/2} \sqrt{b}}-\frac{5 a x^3+3 b x}{8 a^2 \left (a x^2+b\right )^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 47, normalized size = 0.7 \begin{align*}{\frac{1}{ \left ( a{x}^{2}+b \right ) ^{2}} \left ( -{\frac{5\,{x}^{3}}{8\,a}}-{\frac{3\,bx}{8\,{a}^{2}}} \right ) }+{\frac{3}{8\,{a}^{2}}\arctan \left ({ax{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \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 = 1.45685, size = 404, normalized size = 6.31 \begin{align*} \left [-\frac{10 \, a^{2} b x^{3} + 6 \, a b^{2} x + 3 \,{\left (a^{2} x^{4} + 2 \, a b x^{2} + b^{2}\right )} \sqrt{-a b} \log \left (\frac{a x^{2} - 2 \, \sqrt{-a b} x - b}{a x^{2} + b}\right )}{16 \,{\left (a^{5} b x^{4} + 2 \, a^{4} b^{2} x^{2} + a^{3} b^{3}\right )}}, -\frac{5 \, a^{2} b x^{3} + 3 \, a b^{2} x - 3 \,{\left (a^{2} x^{4} + 2 \, a b x^{2} + b^{2}\right )} \sqrt{a b} \arctan \left (\frac{\sqrt{a b} x}{b}\right )}{8 \,{\left (a^{5} b x^{4} + 2 \, a^{4} b^{2} x^{2} + a^{3} b^{3}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.638077, size = 109, normalized size = 1.7 \begin{align*} - \frac{3 \sqrt{- \frac{1}{a^{5} b}} \log{\left (- a^{2} b \sqrt{- \frac{1}{a^{5} b}} + x \right )}}{16} + \frac{3 \sqrt{- \frac{1}{a^{5} b}} \log{\left (a^{2} b \sqrt{- \frac{1}{a^{5} b}} + x \right )}}{16} - \frac{5 a x^{3} + 3 b x}{8 a^{4} x^{4} + 16 a^{3} b x^{2} + 8 a^{2} b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19668, size = 61, normalized size = 0.95 \begin{align*} \frac{3 \, \arctan \left (\frac{a x}{\sqrt{a b}}\right )}{8 \, \sqrt{a b} a^{2}} - \frac{5 \, a x^{3} + 3 \, b x}{8 \,{\left (a x^{2} + b\right )}^{2} a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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