Optimal. Leaf size=66 \[ \frac{5 b^{3/2} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{2 a^{7/2}}-\frac{5 b x}{2 a^3}+\frac{5 x^3}{6 a^2}-\frac{x^5}{2 a \left (a x^2+b\right )} \]
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Rubi [A] time = 0.0261688, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {263, 288, 302, 205} \[ \frac{5 b^{3/2} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{2 a^{7/2}}-\frac{5 b x}{2 a^3}+\frac{5 x^3}{6 a^2}-\frac{x^5}{2 a \left (a x^2+b\right )} \]
Antiderivative was successfully verified.
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Rule 263
Rule 288
Rule 302
Rule 205
Rubi steps
\begin{align*} \int \frac{x^2}{\left (a+\frac{b}{x^2}\right )^2} \, dx &=\int \frac{x^6}{\left (b+a x^2\right )^2} \, dx\\ &=-\frac{x^5}{2 a \left (b+a x^2\right )}+\frac{5 \int \frac{x^4}{b+a x^2} \, dx}{2 a}\\ &=-\frac{x^5}{2 a \left (b+a x^2\right )}+\frac{5 \int \left (-\frac{b}{a^2}+\frac{x^2}{a}+\frac{b^2}{a^2 \left (b+a x^2\right )}\right ) \, dx}{2 a}\\ &=-\frac{5 b x}{2 a^3}+\frac{5 x^3}{6 a^2}-\frac{x^5}{2 a \left (b+a x^2\right )}+\frac{\left (5 b^2\right ) \int \frac{1}{b+a x^2} \, dx}{2 a^3}\\ &=-\frac{5 b x}{2 a^3}+\frac{5 x^3}{6 a^2}-\frac{x^5}{2 a \left (b+a x^2\right )}+\frac{5 b^{3/2} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{2 a^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.0410853, size = 60, normalized size = 0.91 \[ \frac{x \left (-\frac{3 b^2}{a x^2+b}+2 a x^2-12 b\right )}{6 a^3}+\frac{5 b^{3/2} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{2 a^{7/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 57, normalized size = 0.9 \begin{align*}{\frac{{x}^{3}}{3\,{a}^{2}}}-2\,{\frac{bx}{{a}^{3}}}-{\frac{{b}^{2}x}{2\,{a}^{3} \left ( a{x}^{2}+b \right ) }}+{\frac{5\,{b}^{2}}{2\,{a}^{3}}\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 = 1.46385, size = 348, normalized size = 5.27 \begin{align*} \left [\frac{4 \, a^{2} x^{5} - 20 \, a b x^{3} - 30 \, b^{2} x + 15 \,{\left (a b x^{2} + b^{2}\right )} \sqrt{-\frac{b}{a}} \log \left (\frac{a x^{2} + 2 \, a x \sqrt{-\frac{b}{a}} - b}{a x^{2} + b}\right )}{12 \,{\left (a^{4} x^{2} + a^{3} b\right )}}, \frac{2 \, a^{2} x^{5} - 10 \, a b x^{3} - 15 \, b^{2} x + 15 \,{\left (a b x^{2} + b^{2}\right )} \sqrt{\frac{b}{a}} \arctan \left (\frac{a x \sqrt{\frac{b}{a}}}{b}\right )}{6 \,{\left (a^{4} x^{2} + a^{3} b\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.585575, size = 107, normalized size = 1.62 \begin{align*} - \frac{b^{2} x}{2 a^{4} x^{2} + 2 a^{3} b} - \frac{5 \sqrt{- \frac{b^{3}}{a^{7}}} \log{\left (- \frac{a^{3} \sqrt{- \frac{b^{3}}{a^{7}}}}{b} + x \right )}}{4} + \frac{5 \sqrt{- \frac{b^{3}}{a^{7}}} \log{\left (\frac{a^{3} \sqrt{- \frac{b^{3}}{a^{7}}}}{b} + x \right )}}{4} + \frac{x^{3}}{3 a^{2}} - \frac{2 b x}{a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15197, size = 82, normalized size = 1.24 \begin{align*} \frac{5 \, b^{2} \arctan \left (\frac{a x}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} a^{3}} - \frac{b^{2} x}{2 \,{\left (a x^{2} + b\right )} a^{3}} + \frac{a^{4} x^{3} - 6 \, a^{3} b x}{3 \, a^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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