Optimal. Leaf size=118 \[ -\frac {a^6}{10 b^7 \left (a+b x^2\right )^5}+\frac {3 a^5}{4 b^7 \left (a+b x^2\right )^4}-\frac {5 a^4}{2 b^7 \left (a+b x^2\right )^3}+\frac {5 a^3}{b^7 \left (a+b x^2\right )^2}-\frac {15 a^2}{2 b^7 \left (a+b x^2\right )}-\frac {3 a \log \left (a+b x^2\right )}{b^7}+\frac {x^2}{2 b^6} \]
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Rubi [A] time = 0.11, antiderivative size = 118, 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 {a^6}{10 b^7 \left (a+b x^2\right )^5}+\frac {3 a^5}{4 b^7 \left (a+b x^2\right )^4}-\frac {5 a^4}{2 b^7 \left (a+b x^2\right )^3}+\frac {5 a^3}{b^7 \left (a+b x^2\right )^2}-\frac {15 a^2}{2 b^7 \left (a+b x^2\right )}-\frac {3 a \log \left (a+b x^2\right )}{b^7}+\frac {x^2}{2 b^6} \end {gather*}
Antiderivative was successfully verified.
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Rule 28
Rule 43
Rule 266
Rubi steps
\begin {align*} \int \frac {x^{13}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx &=b^6 \int \frac {x^{13}}{\left (a b+b^2 x^2\right )^6} \, dx\\ &=\frac {1}{2} b^6 \operatorname {Subst}\left (\int \frac {x^6}{\left (a b+b^2 x\right )^6} \, dx,x,x^2\right )\\ &=\frac {1}{2} b^6 \operatorname {Subst}\left (\int \left (\frac {1}{b^{12}}+\frac {a^6}{b^{12} (a+b x)^6}-\frac {6 a^5}{b^{12} (a+b x)^5}+\frac {15 a^4}{b^{12} (a+b x)^4}-\frac {20 a^3}{b^{12} (a+b x)^3}+\frac {15 a^2}{b^{12} (a+b x)^2}-\frac {6 a}{b^{12} (a+b x)}\right ) \, dx,x,x^2\right )\\ &=\frac {x^2}{2 b^6}-\frac {a^6}{10 b^7 \left (a+b x^2\right )^5}+\frac {3 a^5}{4 b^7 \left (a+b x^2\right )^4}-\frac {5 a^4}{2 b^7 \left (a+b x^2\right )^3}+\frac {5 a^3}{b^7 \left (a+b x^2\right )^2}-\frac {15 a^2}{2 b^7 \left (a+b x^2\right )}-\frac {3 a \log \left (a+b x^2\right )}{b^7}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 101, normalized size = 0.86 \begin {gather*} -\frac {87 a^6+375 a^5 b x^2+600 a^4 b^2 x^4+400 a^3 b^3 x^6+50 a^2 b^4 x^8-50 a b^5 x^{10}+60 a \left (a+b x^2\right )^5 \log \left (a+b x^2\right )-10 b^6 x^{12}}{20 b^7 \left (a+b x^2\right )^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^{13}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.86, size = 190, normalized size = 1.61 \begin {gather*} \frac {10 \, b^{6} x^{12} + 50 \, a b^{5} x^{10} - 50 \, a^{2} b^{4} x^{8} - 400 \, a^{3} b^{3} x^{6} - 600 \, a^{4} b^{2} x^{4} - 375 \, a^{5} b x^{2} - 87 \, a^{6} - 60 \, {\left (a b^{5} x^{10} + 5 \, a^{2} b^{4} x^{8} + 10 \, a^{3} b^{3} x^{6} + 10 \, a^{4} b^{2} x^{4} + 5 \, a^{5} b x^{2} + a^{6}\right )} \log \left (b x^{2} + a\right )}{20 \, {\left (b^{12} x^{10} + 5 \, a b^{11} x^{8} + 10 \, a^{2} b^{10} x^{6} + 10 \, a^{3} b^{9} x^{4} + 5 \, a^{4} b^{8} x^{2} + a^{5} b^{7}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 95, normalized size = 0.81 \begin {gather*} \frac {x^{2}}{2 \, b^{6}} - \frac {3 \, a \log \left ({\left | b x^{2} + a \right |}\right )}{b^{7}} + \frac {137 \, a b^{5} x^{10} + 535 \, a^{2} b^{4} x^{8} + 870 \, a^{3} b^{3} x^{6} + 720 \, a^{4} b^{2} x^{4} + 300 \, a^{5} b x^{2} + 50 \, a^{6}}{20 \, {\left (b x^{2} + a\right )}^{5} b^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 109, normalized size = 0.92 \begin {gather*} -\frac {a^{6}}{10 \left (b \,x^{2}+a \right )^{5} b^{7}}+\frac {3 a^{5}}{4 \left (b \,x^{2}+a \right )^{4} b^{7}}-\frac {5 a^{4}}{2 \left (b \,x^{2}+a \right )^{3} b^{7}}+\frac {5 a^{3}}{\left (b \,x^{2}+a \right )^{2} b^{7}}+\frac {x^{2}}{2 b^{6}}-\frac {15 a^{2}}{2 \left (b \,x^{2}+a \right ) b^{7}}-\frac {3 a \ln \left (b \,x^{2}+a \right )}{b^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.47, size = 132, normalized size = 1.12 \begin {gather*} -\frac {150 \, a^{2} b^{4} x^{8} + 500 \, a^{3} b^{3} x^{6} + 650 \, a^{4} b^{2} x^{4} + 385 \, a^{5} b x^{2} + 87 \, a^{6}}{20 \, {\left (b^{12} x^{10} + 5 \, a b^{11} x^{8} + 10 \, a^{2} b^{10} x^{6} + 10 \, a^{3} b^{9} x^{4} + 5 \, a^{4} b^{8} x^{2} + a^{5} b^{7}\right )}} + \frac {x^{2}}{2 \, b^{6}} - \frac {3 \, a \log \left (b x^{2} + a\right )}{b^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.60, size = 132, normalized size = 1.12 \begin {gather*} \frac {x^2}{2\,b^6}-\frac {\frac {87\,a^6}{20\,b}+\frac {77\,a^5\,x^2}{4}+\frac {65\,a^4\,b\,x^4}{2}+25\,a^3\,b^2\,x^6+\frac {15\,a^2\,b^3\,x^8}{2}}{a^5\,b^6+5\,a^4\,b^7\,x^2+10\,a^3\,b^8\,x^4+10\,a^2\,b^9\,x^6+5\,a\,b^{10}\,x^8+b^{11}\,x^{10}}-\frac {3\,a\,\ln \left (b\,x^2+a\right )}{b^7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.97, size = 138, normalized size = 1.17 \begin {gather*} - \frac {3 a \log {\left (a + b x^{2} \right )}}{b^{7}} + \frac {- 87 a^{6} - 385 a^{5} b x^{2} - 650 a^{4} b^{2} x^{4} - 500 a^{3} b^{3} x^{6} - 150 a^{2} b^{4} x^{8}}{20 a^{5} b^{7} + 100 a^{4} b^{8} x^{2} + 200 a^{3} b^{9} x^{4} + 200 a^{2} b^{10} x^{6} + 100 a b^{11} x^{8} + 20 b^{12} x^{10}} + \frac {x^{2}}{2 b^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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