Optimal. Leaf size=70 \[ -\frac {b^4}{2 a^5 \left (a x^2+b\right )}-\frac {2 b^3 \log \left (a x^2+b\right )}{a^5}+\frac {3 b^2 x^2}{2 a^4}-\frac {b x^4}{2 a^3}+\frac {x^6}{6 a^2} \]
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Rubi [A] time = 0.05, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {263, 266, 43} \[ \frac {3 b^2 x^2}{2 a^4}-\frac {b^4}{2 a^5 \left (a x^2+b\right )}-\frac {2 b^3 \log \left (a x^2+b\right )}{a^5}-\frac {b x^4}{2 a^3}+\frac {x^6}{6 a^2} \]
Antiderivative was successfully verified.
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Rule 43
Rule 263
Rule 266
Rubi steps
\begin {align*} \int \frac {x^5}{\left (a+\frac {b}{x^2}\right )^2} \, dx &=\int \frac {x^9}{\left (b+a x^2\right )^2} \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^4}{(b+a x)^2} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {3 b^2}{a^4}-\frac {2 b x}{a^3}+\frac {x^2}{a^2}+\frac {b^4}{a^4 (b+a x)^2}-\frac {4 b^3}{a^4 (b+a x)}\right ) \, dx,x,x^2\right )\\ &=\frac {3 b^2 x^2}{2 a^4}-\frac {b x^4}{2 a^3}+\frac {x^6}{6 a^2}-\frac {b^4}{2 a^5 \left (b+a x^2\right )}-\frac {2 b^3 \log \left (b+a x^2\right )}{a^5}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 60, normalized size = 0.86 \[ \frac {a^3 x^6-3 a^2 b x^4-\frac {3 b^4}{a x^2+b}-12 b^3 \log \left (a x^2+b\right )+9 a b^2 x^2}{6 a^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.94, size = 81, normalized size = 1.16 \[ \frac {a^{4} x^{8} - 2 \, a^{3} b x^{6} + 6 \, a^{2} b^{2} x^{4} + 9 \, a b^{3} x^{2} - 3 \, b^{4} - 12 \, {\left (a b^{3} x^{2} + b^{4}\right )} \log \left (a x^{2} + b\right )}{6 \, {\left (a^{6} x^{2} + a^{5} b\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 80, normalized size = 1.14 \[ -\frac {2 \, b^{3} \log \left ({\left | a x^{2} + b \right |}\right )}{a^{5}} + \frac {a^{4} x^{6} - 3 \, a^{3} b x^{4} + 9 \, a^{2} b^{2} x^{2}}{6 \, a^{6}} + \frac {4 \, a b^{3} x^{2} + 3 \, b^{4}}{2 \, {\left (a x^{2} + b\right )} a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 63, normalized size = 0.90 \[ \frac {x^{6}}{6 a^{2}}-\frac {b \,x^{4}}{2 a^{3}}+\frac {3 b^{2} x^{2}}{2 a^{4}}-\frac {b^{4}}{2 \left (a \,x^{2}+b \right ) a^{5}}-\frac {2 b^{3} \ln \left (a \,x^{2}+b \right )}{a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.88, size = 65, normalized size = 0.93 \[ -\frac {b^{4}}{2 \, {\left (a^{6} x^{2} + a^{5} b\right )}} - \frac {2 \, b^{3} \log \left (a x^{2} + b\right )}{a^{5}} + \frac {a^{2} x^{6} - 3 \, a b x^{4} + 9 \, b^{2} x^{2}}{6 \, a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 68, normalized size = 0.97 \[ \frac {x^6}{6\,a^2}-\frac {b^4}{2\,a\,\left (a^5\,x^2+b\,a^4\right )}-\frac {b\,x^4}{2\,a^3}-\frac {2\,b^3\,\ln \left (a\,x^2+b\right )}{a^5}+\frac {3\,b^2\,x^2}{2\,a^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.29, size = 66, normalized size = 0.94 \[ - \frac {b^{4}}{2 a^{6} x^{2} + 2 a^{5} b} + \frac {x^{6}}{6 a^{2}} - \frac {b x^{4}}{2 a^{3}} + \frac {3 b^{2} x^{2}}{2 a^{4}} - \frac {2 b^{3} \log {\left (a x^{2} + b \right )}}{a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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