Optimal. Leaf size=100 \[ \frac{3 a^2 x^4}{2 b^5}-\frac{5 a^3 x^2}{b^6}+\frac{3 a^5}{b^7 \left (a+b x^2\right )}-\frac{a^6}{4 b^7 \left (a+b x^2\right )^2}+\frac{15 a^4 \log \left (a+b x^2\right )}{2 b^7}-\frac{a x^6}{2 b^4}+\frac{x^8}{8 b^3} \]
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Rubi [A] time = 0.0821957, antiderivative size = 100, 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} \[ \frac{3 a^2 x^4}{2 b^5}-\frac{5 a^3 x^2}{b^6}+\frac{3 a^5}{b^7 \left (a+b x^2\right )}-\frac{a^6}{4 b^7 \left (a+b x^2\right )^2}+\frac{15 a^4 \log \left (a+b x^2\right )}{2 b^7}-\frac{a x^6}{2 b^4}+\frac{x^8}{8 b^3} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^{13}}{\left (a+b x^2\right )^3} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^6}{(a+b x)^3} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (-\frac{10 a^3}{b^6}+\frac{6 a^2 x}{b^5}-\frac{3 a x^2}{b^4}+\frac{x^3}{b^3}+\frac{a^6}{b^6 (a+b x)^3}-\frac{6 a^5}{b^6 (a+b x)^2}+\frac{15 a^4}{b^6 (a+b x)}\right ) \, dx,x,x^2\right )\\ &=-\frac{5 a^3 x^2}{b^6}+\frac{3 a^2 x^4}{2 b^5}-\frac{a x^6}{2 b^4}+\frac{x^8}{8 b^3}-\frac{a^6}{4 b^7 \left (a+b x^2\right )^2}+\frac{3 a^5}{b^7 \left (a+b x^2\right )}+\frac{15 a^4 \log \left (a+b x^2\right )}{2 b^7}\\ \end{align*}
Mathematica [A] time = 0.0272045, size = 85, normalized size = 0.85 \[ \frac{12 a^2 b^2 x^4-40 a^3 b x^2+\frac{24 a^5}{a+b x^2}-\frac{2 a^6}{\left (a+b x^2\right )^2}+60 a^4 \log \left (a+b x^2\right )-4 a b^3 x^6+b^4 x^8}{8 b^7} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 91, normalized size = 0.9 \begin{align*} -5\,{\frac{{a}^{3}{x}^{2}}{{b}^{6}}}+{\frac{3\,{a}^{2}{x}^{4}}{2\,{b}^{5}}}-{\frac{a{x}^{6}}{2\,{b}^{4}}}+{\frac{{x}^{8}}{8\,{b}^{3}}}-{\frac{{a}^{6}}{4\,{b}^{7} \left ( b{x}^{2}+a \right ) ^{2}}}+3\,{\frac{{a}^{5}}{{b}^{7} \left ( b{x}^{2}+a \right ) }}+{\frac{15\,{a}^{4}\ln \left ( b{x}^{2}+a \right ) }{2\,{b}^{7}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.31735, size = 134, normalized size = 1.34 \begin{align*} \frac{12 \, a^{5} b x^{2} + 11 \, a^{6}}{4 \,{\left (b^{9} x^{4} + 2 \, a b^{8} x^{2} + a^{2} b^{7}\right )}} + \frac{15 \, a^{4} \log \left (b x^{2} + a\right )}{2 \, b^{7}} + \frac{b^{3} x^{8} - 4 \, a b^{2} x^{6} + 12 \, a^{2} b x^{4} - 40 \, a^{3} x^{2}}{8 \, b^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.20368, size = 261, normalized size = 2.61 \begin{align*} \frac{b^{6} x^{12} - 2 \, a b^{5} x^{10} + 5 \, a^{2} b^{4} x^{8} - 20 \, a^{3} b^{3} x^{6} - 68 \, a^{4} b^{2} x^{4} - 16 \, a^{5} b x^{2} + 22 \, a^{6} + 60 \,{\left (a^{4} b^{2} x^{4} + 2 \, a^{5} b x^{2} + a^{6}\right )} \log \left (b x^{2} + a\right )}{8 \,{\left (b^{9} x^{4} + 2 \, a b^{8} x^{2} + a^{2} b^{7}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.597757, size = 104, normalized size = 1.04 \begin{align*} \frac{15 a^{4} \log{\left (a + b x^{2} \right )}}{2 b^{7}} - \frac{5 a^{3} x^{2}}{b^{6}} + \frac{3 a^{2} x^{4}}{2 b^{5}} - \frac{a x^{6}}{2 b^{4}} + \frac{11 a^{6} + 12 a^{5} b x^{2}}{4 a^{2} b^{7} + 8 a b^{8} x^{2} + 4 b^{9} x^{4}} + \frac{x^{8}}{8 b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.81621, size = 138, normalized size = 1.38 \begin{align*} \frac{15 \, a^{4} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{7}} - \frac{45 \, a^{4} b^{2} x^{4} + 78 \, a^{5} b x^{2} + 34 \, a^{6}}{4 \,{\left (b x^{2} + a\right )}^{2} b^{7}} + \frac{b^{9} x^{8} - 4 \, a b^{8} x^{6} + 12 \, a^{2} b^{7} x^{4} - 40 \, a^{3} b^{6} x^{2}}{8 \, b^{12}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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