Optimal. Leaf size=55 \[ \frac{b^2 x}{a^3}-\frac{b^{5/2} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{a^{7/2}}-\frac{b x^3}{3 a^2}+\frac{x^5}{5 a} \]
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Rubi [A] time = 0.0244078, antiderivative size = 55, 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} \[ \frac{b^2 x}{a^3}-\frac{b^{5/2} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{a^{7/2}}-\frac{b x^3}{3 a^2}+\frac{x^5}{5 a} \]
Antiderivative was successfully verified.
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Rule 263
Rule 302
Rule 205
Rubi steps
\begin{align*} \int \frac{x^4}{a+\frac{b}{x^2}} \, dx &=\int \frac{x^6}{b+a x^2} \, dx\\ &=\int \left (\frac{b^2}{a^3}-\frac{b x^2}{a^2}+\frac{x^4}{a}-\frac{b^3}{a^3 \left (b+a x^2\right )}\right ) \, dx\\ &=\frac{b^2 x}{a^3}-\frac{b x^3}{3 a^2}+\frac{x^5}{5 a}-\frac{b^3 \int \frac{1}{b+a x^2} \, dx}{a^3}\\ &=\frac{b^2 x}{a^3}-\frac{b x^3}{3 a^2}+\frac{x^5}{5 a}-\frac{b^{5/2} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{a^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.0227434, size = 55, normalized size = 1. \[ \frac{b^2 x}{a^3}-\frac{b^{5/2} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{a^{7/2}}-\frac{b x^3}{3 a^2}+\frac{x^5}{5 a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 49, normalized size = 0.9 \begin{align*}{\frac{{x}^{5}}{5\,a}}-{\frac{b{x}^{3}}{3\,{a}^{2}}}+{\frac{{b}^{2}x}{{a}^{3}}}-{\frac{{b}^{3}}{{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 = 1.48235, size = 278, normalized size = 5.05 \begin{align*} \left [\frac{6 \, a^{2} x^{5} - 10 \, a b x^{3} + 15 \, b^{2} \sqrt{-\frac{b}{a}} \log \left (\frac{a x^{2} - 2 \, a x \sqrt{-\frac{b}{a}} - b}{a x^{2} + b}\right ) + 30 \, b^{2} x}{30 \, a^{3}}, \frac{3 \, a^{2} x^{5} - 5 \, a b x^{3} - 15 \, b^{2} \sqrt{\frac{b}{a}} \arctan \left (\frac{a x \sqrt{\frac{b}{a}}}{b}\right ) + 15 \, b^{2} x}{15 \, a^{3}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.316712, size = 95, normalized size = 1.73 \begin{align*} \frac{\sqrt{- \frac{b^{5}}{a^{7}}} \log{\left (- \frac{a^{3} \sqrt{- \frac{b^{5}}{a^{7}}}}{b^{2}} + x \right )}}{2} - \frac{\sqrt{- \frac{b^{5}}{a^{7}}} \log{\left (\frac{a^{3} \sqrt{- \frac{b^{5}}{a^{7}}}}{b^{2}} + x \right )}}{2} + \frac{x^{5}}{5 a} - \frac{b x^{3}}{3 a^{2}} + \frac{b^{2} x}{a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.22603, size = 74, normalized size = 1.35 \begin{align*} -\frac{b^{3} \arctan \left (\frac{a x}{\sqrt{a b}}\right )}{\sqrt{a b} a^{3}} + \frac{3 \, a^{4} x^{5} - 5 \, a^{3} b x^{3} + 15 \, a^{2} b^{2} x}{15 \, a^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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