Optimal. Leaf size=42 \[ \frac{b^{3/2} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{a^{5/2}}-\frac{b x}{a^2}+\frac{x^3}{3 a} \]
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Rubi [A] time = 0.0193837, antiderivative size = 42, 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, 302, 205} \[ \frac{b^{3/2} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{a^{5/2}}-\frac{b x}{a^2}+\frac{x^3}{3 a} \]
Antiderivative was successfully verified.
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Rule 263
Rule 302
Rule 205
Rubi steps
\begin{align*} \int \frac{x^2}{a+\frac{b}{x^2}} \, dx &=\int \frac{x^4}{b+a x^2} \, dx\\ &=\int \left (-\frac{b}{a^2}+\frac{x^2}{a}+\frac{b^2}{a^2 \left (b+a x^2\right )}\right ) \, dx\\ &=-\frac{b x}{a^2}+\frac{x^3}{3 a}+\frac{b^2 \int \frac{1}{b+a x^2} \, dx}{a^2}\\ &=-\frac{b x}{a^2}+\frac{x^3}{3 a}+\frac{b^{3/2} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{a^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.0188225, size = 42, normalized size = 1. \[ \frac{b^{3/2} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{a^{5/2}}-\frac{b x}{a^2}+\frac{x^3}{3 a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 38, normalized size = 0.9 \begin{align*}{\frac{{x}^{3}}{3\,a}}-{\frac{bx}{{a}^{2}}}+{\frac{{b}^{2}}{{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 = 1.44814, size = 217, normalized size = 5.17 \begin{align*} \left [\frac{2 \, a x^{3} + 3 \, b \sqrt{-\frac{b}{a}} \log \left (\frac{a x^{2} + 2 \, a x \sqrt{-\frac{b}{a}} - b}{a x^{2} + b}\right ) - 6 \, b x}{6 \, a^{2}}, \frac{a x^{3} + 3 \, b \sqrt{\frac{b}{a}} \arctan \left (\frac{a x \sqrt{\frac{b}{a}}}{b}\right ) - 3 \, b x}{3 \, a^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.31167, size = 80, normalized size = 1.9 \begin{align*} - \frac{\sqrt{- \frac{b^{3}}{a^{5}}} \log{\left (- \frac{a^{2} \sqrt{- \frac{b^{3}}{a^{5}}}}{b} + x \right )}}{2} + \frac{\sqrt{- \frac{b^{3}}{a^{5}}} \log{\left (\frac{a^{2} \sqrt{- \frac{b^{3}}{a^{5}}}}{b} + x \right )}}{2} + \frac{x^{3}}{3 a} - \frac{b x}{a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18616, size = 54, normalized size = 1.29 \begin{align*} \frac{b^{2} \arctan \left (\frac{a x}{\sqrt{a b}}\right )}{\sqrt{a b} a^{2}} + \frac{a^{2} x^{3} - 3 \, a b x}{3 \, a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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