Optimal. Leaf size=57 \[ \frac{a^3}{2 b^4 \left (a+b x^2\right )}+\frac{3 a^2 \log \left (a+b x^2\right )}{2 b^4}-\frac{a x^2}{b^3}+\frac{x^4}{4 b^2} \]
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Rubi [A] time = 0.0524434, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {28, 266, 43} \[ \frac{a^3}{2 b^4 \left (a+b x^2\right )}+\frac{3 a^2 \log \left (a+b x^2\right )}{2 b^4}-\frac{a x^2}{b^3}+\frac{x^4}{4 b^2} \]
Antiderivative was successfully verified.
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Rule 28
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^7}{a^2+2 a b x^2+b^2 x^4} \, dx &=b^2 \int \frac{x^7}{\left (a b+b^2 x^2\right )^2} \, dx\\ &=\frac{1}{2} b^2 \operatorname{Subst}\left (\int \frac{x^3}{\left (a b+b^2 x\right )^2} \, dx,x,x^2\right )\\ &=\frac{1}{2} b^2 \operatorname{Subst}\left (\int \left (-\frac{2 a}{b^5}+\frac{x}{b^4}-\frac{a^3}{b^5 (a+b x)^2}+\frac{3 a^2}{b^5 (a+b x)}\right ) \, dx,x,x^2\right )\\ &=-\frac{a x^2}{b^3}+\frac{x^4}{4 b^2}+\frac{a^3}{2 b^4 \left (a+b x^2\right )}+\frac{3 a^2 \log \left (a+b x^2\right )}{2 b^4}\\ \end{align*}
Mathematica [A] time = 0.016188, size = 49, normalized size = 0.86 \[ \frac{\frac{2 a^3}{a+b x^2}+6 a^2 \log \left (a+b x^2\right )-4 a b x^2+b^2 x^4}{4 b^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.048, size = 52, normalized size = 0.9 \begin{align*} -{\frac{a{x}^{2}}{{b}^{3}}}+{\frac{{x}^{4}}{4\,{b}^{2}}}+{\frac{{a}^{3}}{2\,{b}^{4} \left ( b{x}^{2}+a \right ) }}+{\frac{3\,{a}^{2}\ln \left ( b{x}^{2}+a \right ) }{2\,{b}^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.961409, size = 73, normalized size = 1.28 \begin{align*} \frac{a^{3}}{2 \,{\left (b^{5} x^{2} + a b^{4}\right )}} + \frac{3 \, a^{2} \log \left (b x^{2} + a\right )}{2 \, b^{4}} + \frac{b x^{4} - 4 \, a x^{2}}{4 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.72323, size = 143, normalized size = 2.51 \begin{align*} \frac{b^{3} x^{6} - 3 \, a b^{2} x^{4} - 4 \, a^{2} b x^{2} + 2 \, a^{3} + 6 \,{\left (a^{2} b x^{2} + a^{3}\right )} \log \left (b x^{2} + a\right )}{4 \,{\left (b^{5} x^{2} + a b^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.403627, size = 53, normalized size = 0.93 \begin{align*} \frac{a^{3}}{2 a b^{4} + 2 b^{5} x^{2}} + \frac{3 a^{2} \log{\left (a + b x^{2} \right )}}{2 b^{4}} - \frac{a x^{2}}{b^{3}} + \frac{x^{4}}{4 b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12868, size = 90, normalized size = 1.58 \begin{align*} \frac{3 \, a^{2} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{4}} + \frac{b^{2} x^{4} - 4 \, a b x^{2}}{4 \, b^{4}} - \frac{3 \, a^{2} b x^{2} + 2 \, a^{3}}{2 \,{\left (b x^{2} + a\right )} b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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