Optimal. Leaf size=83 \[ \frac {a^5}{2 b^6 \left (a+b x^2\right )}+\frac {5 a^4 \log \left (a+b x^2\right )}{2 b^6}-\frac {2 a^3 x^2}{b^5}+\frac {3 a^2 x^4}{4 b^4}-\frac {a x^6}{3 b^3}+\frac {x^8}{8 b^2} \]
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Rubi [A] time = 0.08, antiderivative size = 83, 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} \[ \frac {3 a^2 x^4}{4 b^4}-\frac {2 a^3 x^2}{b^5}+\frac {a^5}{2 b^6 \left (a+b x^2\right )}+\frac {5 a^4 \log \left (a+b x^2\right )}{2 b^6}-\frac {a x^6}{3 b^3}+\frac {x^8}{8 b^2} \]
Antiderivative was successfully verified.
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Rule 28
Rule 43
Rule 266
Rubi steps
\begin {align*} \int \frac {x^{11}}{a^2+2 a b x^2+b^2 x^4} \, dx &=b^2 \int \frac {x^{11}}{\left (a b+b^2 x^2\right )^2} \, dx\\ &=\frac {1}{2} b^2 \operatorname {Subst}\left (\int \frac {x^5}{\left (a b+b^2 x\right )^2} \, dx,x,x^2\right )\\ &=\frac {1}{2} b^2 \operatorname {Subst}\left (\int \left (-\frac {4 a^3}{b^7}+\frac {3 a^2 x}{b^6}-\frac {2 a x^2}{b^5}+\frac {x^3}{b^4}-\frac {a^5}{b^7 (a+b x)^2}+\frac {5 a^4}{b^7 (a+b x)}\right ) \, dx,x,x^2\right )\\ &=-\frac {2 a^3 x^2}{b^5}+\frac {3 a^2 x^4}{4 b^4}-\frac {a x^6}{3 b^3}+\frac {x^8}{8 b^2}+\frac {a^5}{2 b^6 \left (a+b x^2\right )}+\frac {5 a^4 \log \left (a+b x^2\right )}{2 b^6}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 72, normalized size = 0.87 \[ \frac {\frac {12 a^5}{a+b x^2}+60 a^4 \log \left (a+b x^2\right )-48 a^3 b x^2+18 a^2 b^2 x^4-8 a b^3 x^6+3 b^4 x^8}{24 b^6} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.04, size = 93, normalized size = 1.12 \[ \frac {3 \, b^{5} x^{10} - 5 \, a b^{4} x^{8} + 10 \, a^{2} b^{3} x^{6} - 30 \, a^{3} b^{2} x^{4} - 48 \, a^{4} b x^{2} + 12 \, a^{5} + 60 \, {\left (a^{4} b x^{2} + a^{5}\right )} \log \left (b x^{2} + a\right )}{24 \, {\left (b^{7} x^{2} + a b^{6}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 92, normalized size = 1.11 \[ \frac {5 \, a^{4} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{6}} - \frac {5 \, a^{4} b x^{2} + 4 \, a^{5}}{2 \, {\left (b x^{2} + a\right )} b^{6}} + \frac {3 \, b^{6} x^{8} - 8 \, a b^{5} x^{6} + 18 \, a^{2} b^{4} x^{4} - 48 \, a^{3} b^{3} x^{2}}{24 \, b^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 74, normalized size = 0.89 \[ \frac {x^{8}}{8 b^{2}}-\frac {a \,x^{6}}{3 b^{3}}+\frac {3 a^{2} x^{4}}{4 b^{4}}-\frac {2 a^{3} x^{2}}{b^{5}}+\frac {a^{5}}{2 \left (b \,x^{2}+a \right ) b^{6}}+\frac {5 a^{4} \ln \left (b \,x^{2}+a \right )}{2 b^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.37, size = 77, normalized size = 0.93 \[ \frac {a^{5}}{2 \, {\left (b^{7} x^{2} + a b^{6}\right )}} + \frac {5 \, a^{4} \log \left (b x^{2} + a\right )}{2 \, b^{6}} + \frac {3 \, b^{3} x^{8} - 8 \, a b^{2} x^{6} + 18 \, a^{2} b x^{4} - 48 \, a^{3} x^{2}}{24 \, b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.36, size = 79, normalized size = 0.95 \[ \frac {x^8}{8\,b^2}+\frac {a^5}{2\,b\,\left (b^6\,x^2+a\,b^5\right )}-\frac {a\,x^6}{3\,b^3}+\frac {5\,a^4\,\ln \left (b\,x^2+a\right )}{2\,b^6}+\frac {3\,a^2\,x^4}{4\,b^4}-\frac {2\,a^3\,x^2}{b^5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.32, size = 80, normalized size = 0.96 \[ \frac {a^{5}}{2 a b^{6} + 2 b^{7} x^{2}} + \frac {5 a^{4} \log {\left (a + b x^{2} \right )}}{2 b^{6}} - \frac {2 a^{3} x^{2}}{b^{5}} + \frac {3 a^{2} x^{4}}{4 b^{4}} - \frac {a x^{6}}{3 b^{3}} + \frac {x^{8}}{8 b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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