Optimal. Leaf size=70 \[ -\frac {a^4}{2 b^5 \left (a+b x^2\right )}-\frac {2 a^3 \log \left (a+b x^2\right )}{b^5}+\frac {3 a^2 x^2}{2 b^4}-\frac {a x^4}{2 b^3}+\frac {x^6}{6 b^2} \]
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Rubi [A] time = 0.06, antiderivative size = 70, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {3 a^2 x^2}{2 b^4}-\frac {a^4}{2 b^5 \left (a+b x^2\right )}-\frac {2 a^3 \log \left (a+b x^2\right )}{b^5}-\frac {a x^4}{2 b^3}+\frac {x^6}{6 b^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 28
Rule 43
Rule 266
Rubi steps
\begin {align*} \int \frac {x^9}{a^2+2 a b x^2+b^2 x^4} \, dx &=b^2 \int \frac {x^9}{\left (a b+b^2 x^2\right )^2} \, dx\\ &=\frac {1}{2} b^2 \operatorname {Subst}\left (\int \frac {x^4}{\left (a b+b^2 x\right )^2} \, dx,x,x^2\right )\\ &=\frac {1}{2} b^2 \operatorname {Subst}\left (\int \left (\frac {3 a^2}{b^6}-\frac {2 a x}{b^5}+\frac {x^2}{b^4}+\frac {a^4}{b^6 (a+b x)^2}-\frac {4 a^3}{b^6 (a+b x)}\right ) \, dx,x,x^2\right )\\ &=\frac {3 a^2 x^2}{2 b^4}-\frac {a x^4}{2 b^3}+\frac {x^6}{6 b^2}-\frac {a^4}{2 b^5 \left (a+b x^2\right )}-\frac {2 a^3 \log \left (a+b x^2\right )}{b^5}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 60, normalized size = 0.86 \begin {gather*} \frac {-\frac {3 a^4}{a+b x^2}-12 a^3 \log \left (a+b x^2\right )+9 a^2 b x^2-3 a b^2 x^4+b^3 x^6}{6 b^5} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^9}{a^2+2 a b x^2+b^2 x^4} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.59, size = 81, normalized size = 1.16 \begin {gather*} \frac {b^{4} x^{8} - 2 \, a b^{3} x^{6} + 6 \, a^{2} b^{2} x^{4} + 9 \, a^{3} b x^{2} - 3 \, a^{4} - 12 \, {\left (a^{3} b x^{2} + a^{4}\right )} \log \left (b x^{2} + a\right )}{6 \, {\left (b^{6} x^{2} + a b^{5}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 80, normalized size = 1.14 \begin {gather*} -\frac {2 \, a^{3} \log \left ({\left | b x^{2} + a \right |}\right )}{b^{5}} + \frac {b^{4} x^{6} - 3 \, a b^{3} x^{4} + 9 \, a^{2} b^{2} x^{2}}{6 \, b^{6}} + \frac {4 \, a^{3} b x^{2} + 3 \, a^{4}}{2 \, {\left (b x^{2} + a\right )} b^{5}} \end {gather*}
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 \begin {gather*} \frac {x^{6}}{6 b^{2}}-\frac {a \,x^{4}}{2 b^{3}}+\frac {3 a^{2} x^{2}}{2 b^{4}}-\frac {a^{4}}{2 \left (b \,x^{2}+a \right ) b^{5}}-\frac {2 a^{3} \ln \left (b \,x^{2}+a \right )}{b^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.35, size = 65, normalized size = 0.93 \begin {gather*} -\frac {a^{4}}{2 \, {\left (b^{6} x^{2} + a b^{5}\right )}} - \frac {2 \, a^{3} \log \left (b x^{2} + a\right )}{b^{5}} + \frac {b^{2} x^{6} - 3 \, a b x^{4} + 9 \, a^{2} x^{2}}{6 \, b^{4}} \end {gather*}
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 \begin {gather*} \frac {x^6}{6\,b^2}-\frac {a^4}{2\,b\,\left (b^5\,x^2+a\,b^4\right )}-\frac {a\,x^4}{2\,b^3}-\frac {2\,a^3\,\ln \left (b\,x^2+a\right )}{b^5}+\frac {3\,a^2\,x^2}{2\,b^4} \end {gather*}
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 \begin {gather*} - \frac {a^{4}}{2 a b^{5} + 2 b^{6} x^{2}} - \frac {2 a^{3} \log {\left (a + b x^{2} \right )}}{b^{5}} + \frac {3 a^{2} x^{2}}{2 b^{4}} - \frac {a x^{4}}{2 b^{3}} + \frac {x^{6}}{6 b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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