Optimal. Leaf size=129 \[ \frac{x^3 \left (a+b x^2\right )}{3 b \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{a x \left (a+b x^2\right )}{b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{a^{3/2} \left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}} \]
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Rubi [A] time = 0.0460382, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {1112, 302, 205} \[ \frac{x^3 \left (a+b x^2\right )}{3 b \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{a x \left (a+b x^2\right )}{b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{a^{3/2} \left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}} \]
Antiderivative was successfully verified.
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Rule 1112
Rule 302
Rule 205
Rubi steps
\begin{align*} \int \frac{x^4}{\sqrt{a^2+2 a b x^2+b^2 x^4}} \, dx &=\frac{\left (a b+b^2 x^2\right ) \int \frac{x^4}{a b+b^2 x^2} \, dx}{\sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=\frac{\left (a b+b^2 x^2\right ) \int \left (-\frac{a}{b^3}+\frac{x^2}{b^2}+\frac{a^2}{b^2 \left (a b+b^2 x^2\right )}\right ) \, dx}{\sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac{a x \left (a+b x^2\right )}{b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{x^3 \left (a+b x^2\right )}{3 b \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (a^2 \left (a b+b^2 x^2\right )\right ) \int \frac{1}{a b+b^2 x^2} \, dx}{b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac{a x \left (a+b x^2\right )}{b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{x^3 \left (a+b x^2\right )}{3 b \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{a^{3/2} \left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ \end{align*}
Mathematica [A] time = 0.0250801, size = 66, normalized size = 0.51 \[ \frac{\left (a+b x^2\right ) \left (3 a^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )+\sqrt{b} x \left (b x^2-3 a\right )\right )}{3 b^{5/2} \sqrt{\left (a+b x^2\right )^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.217, size = 63, normalized size = 0.5 \begin{align*}{\frac{b{x}^{2}+a}{3\,{b}^{2}} \left ( \sqrt{ab}{x}^{3}b-3\,\sqrt{ab}xa+3\,{a}^{2}\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ) \right ){\frac{1}{\sqrt{ \left ( b{x}^{2}+a \right ) ^{2}}}}{\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 = 1.34042, size = 217, normalized size = 1.68 \begin{align*} \left [\frac{2 \, b x^{3} + 3 \, a \sqrt{-\frac{a}{b}} \log \left (\frac{b x^{2} + 2 \, b x \sqrt{-\frac{a}{b}} - a}{b x^{2} + a}\right ) - 6 \, a x}{6 \, b^{2}}, \frac{b x^{3} + 3 \, a \sqrt{\frac{a}{b}} \arctan \left (\frac{b x \sqrt{\frac{a}{b}}}{a}\right ) - 3 \, a x}{3 \, b^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.351204, size = 80, normalized size = 0.62 \begin{align*} - \frac{a x}{b^{2}} - \frac{\sqrt{- \frac{a^{3}}{b^{5}}} \log{\left (x - \frac{b^{2} \sqrt{- \frac{a^{3}}{b^{5}}}}{a} \right )}}{2} + \frac{\sqrt{- \frac{a^{3}}{b^{5}}} \log{\left (x + \frac{b^{2} \sqrt{- \frac{a^{3}}{b^{5}}}}{a} \right )}}{2} + \frac{x^{3}}{3 b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16783, size = 86, normalized size = 0.67 \begin{align*} \frac{a^{2} \arctan \left (\frac{b x}{\sqrt{a b}}\right ) \mathrm{sgn}\left (b x^{2} + a\right )}{\sqrt{a b} b^{2}} + \frac{b^{2} x^{3} \mathrm{sgn}\left (b x^{2} + a\right ) - 3 \, a b x \mathrm{sgn}\left (b x^{2} + a\right )}{3 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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