Optimal. Leaf size=53 \[ \frac{a^2 x^2}{2 b^3}-\frac{a^3 \log \left (a+b x^2\right )}{2 b^4}-\frac{a x^4}{4 b^2}+\frac{x^6}{6 b} \]
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Rubi [A] time = 0.0350458, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {266, 43} \[ \frac{a^2 x^2}{2 b^3}-\frac{a^3 \log \left (a+b x^2\right )}{2 b^4}-\frac{a x^4}{4 b^2}+\frac{x^6}{6 b} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^7}{a+b x^2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^3}{a+b x} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{a^2}{b^3}-\frac{a x}{b^2}+\frac{x^2}{b}-\frac{a^3}{b^3 (a+b x)}\right ) \, dx,x,x^2\right )\\ &=\frac{a^2 x^2}{2 b^3}-\frac{a x^4}{4 b^2}+\frac{x^6}{6 b}-\frac{a^3 \log \left (a+b x^2\right )}{2 b^4}\\ \end{align*}
Mathematica [A] time = 0.0053087, size = 53, normalized size = 1. \[ \frac{a^2 x^2}{2 b^3}-\frac{a^3 \log \left (a+b x^2\right )}{2 b^4}-\frac{a x^4}{4 b^2}+\frac{x^6}{6 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 46, normalized size = 0.9 \begin{align*}{\frac{{a}^{2}{x}^{2}}{2\,{b}^{3}}}-{\frac{a{x}^{4}}{4\,{b}^{2}}}+{\frac{{x}^{6}}{6\,b}}-{\frac{{a}^{3}\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 = 2.59227, size = 62, normalized size = 1.17 \begin{align*} -\frac{a^{3} \log \left (b x^{2} + a\right )}{2 \, b^{4}} + \frac{2 \, b^{2} x^{6} - 3 \, a b x^{4} + 6 \, a^{2} x^{2}}{12 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.2818, size = 99, normalized size = 1.87 \begin{align*} \frac{2 \, b^{3} x^{6} - 3 \, a b^{2} x^{4} + 6 \, a^{2} b x^{2} - 6 \, a^{3} \log \left (b x^{2} + a\right )}{12 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.305953, size = 44, normalized size = 0.83 \begin{align*} - \frac{a^{3} \log{\left (a + b x^{2} \right )}}{2 b^{4}} + \frac{a^{2} x^{2}}{2 b^{3}} - \frac{a x^{4}}{4 b^{2}} + \frac{x^{6}}{6 b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.38341, size = 63, normalized size = 1.19 \begin{align*} -\frac{a^{3} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{4}} + \frac{2 \, b^{2} x^{6} - 3 \, a b x^{4} + 6 \, a^{2} x^{2}}{12 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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