Optimal. Leaf size=66 \[ \frac{5 a^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 b^{7/2}}-\frac{5 a x}{2 b^3}-\frac{x^5}{2 b \left (a+b x^2\right )}+\frac{5 x^3}{6 b^2} \]
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Rubi [A] time = 0.037731, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {28, 288, 302, 205} \[ \frac{5 a^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 b^{7/2}}-\frac{5 a x}{2 b^3}-\frac{x^5}{2 b \left (a+b x^2\right )}+\frac{5 x^3}{6 b^2} \]
Antiderivative was successfully verified.
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Rule 28
Rule 288
Rule 302
Rule 205
Rubi steps
\begin{align*} \int \frac{x^6}{a^2+2 a b x^2+b^2 x^4} \, dx &=b^2 \int \frac{x^6}{\left (a b+b^2 x^2\right )^2} \, dx\\ &=-\frac{x^5}{2 b \left (a+b x^2\right )}+\frac{5}{2} \int \frac{x^4}{a b+b^2 x^2} \, dx\\ &=-\frac{x^5}{2 b \left (a+b x^2\right )}+\frac{5}{2} \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\\ &=-\frac{5 a x}{2 b^3}+\frac{5 x^3}{6 b^2}-\frac{x^5}{2 b \left (a+b x^2\right )}+\frac{\left (5 a^2\right ) \int \frac{1}{a b+b^2 x^2} \, dx}{2 b^2}\\ &=-\frac{5 a x}{2 b^3}+\frac{5 x^3}{6 b^2}-\frac{x^5}{2 b \left (a+b x^2\right )}+\frac{5 a^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 b^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.0410447, size = 60, normalized size = 0.91 \[ \frac{x \left (-\frac{3 a^2}{a+b x^2}-12 a+2 b x^2\right )}{6 b^3}+\frac{5 a^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 b^{7/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.051, size = 57, normalized size = 0.9 \begin{align*}{\frac{{x}^{3}}{3\,{b}^{2}}}-2\,{\frac{ax}{{b}^{3}}}-{\frac{{a}^{2}x}{2\,{b}^{3} \left ( b{x}^{2}+a \right ) }}+{\frac{5\,{a}^{2}}{2\,{b}^{3}}\arctan \left ({bx{\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 = 1.71196, size = 348, normalized size = 5.27 \begin{align*} \left [\frac{4 \, b^{2} x^{5} - 20 \, a b x^{3} - 30 \, a^{2} x + 15 \,{\left (a b x^{2} + a^{2}\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{b x^{2} + 2 \, b x \sqrt{-\frac{a}{b}} - a}{b x^{2} + a}\right )}{12 \,{\left (b^{4} x^{2} + a b^{3}\right )}}, \frac{2 \, b^{2} x^{5} - 10 \, a b x^{3} - 15 \, a^{2} x + 15 \,{\left (a b x^{2} + a^{2}\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{b x \sqrt{\frac{a}{b}}}{a}\right )}{6 \,{\left (b^{4} x^{2} + a b^{3}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.4402, size = 107, normalized size = 1.62 \begin{align*} - \frac{a^{2} x}{2 a b^{3} + 2 b^{4} x^{2}} - \frac{2 a x}{b^{3}} - \frac{5 \sqrt{- \frac{a^{3}}{b^{7}}} \log{\left (x - \frac{b^{3} \sqrt{- \frac{a^{3}}{b^{7}}}}{a} \right )}}{4} + \frac{5 \sqrt{- \frac{a^{3}}{b^{7}}} \log{\left (x + \frac{b^{3} \sqrt{- \frac{a^{3}}{b^{7}}}}{a} \right )}}{4} + \frac{x^{3}}{3 b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13221, size = 82, normalized size = 1.24 \begin{align*} \frac{5 \, a^{2} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} b^{3}} - \frac{a^{2} x}{2 \,{\left (b x^{2} + a\right )} b^{3}} + \frac{b^{4} x^{3} - 6 \, a b^{3} x}{3 \, b^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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