Optimal. Leaf size=109 \[ \frac {a^5}{10 b^6 \left (a+b x^2\right )^5}-\frac {5 a^4}{8 b^6 \left (a+b x^2\right )^4}+\frac {5 a^3}{3 b^6 \left (a+b x^2\right )^3}-\frac {5 a^2}{2 b^6 \left (a+b x^2\right )^2}+\frac {5 a}{2 b^6 \left (a+b x^2\right )}+\frac {\log \left (a+b x^2\right )}{2 b^6} \]
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Rubi [A] time = 0.10, antiderivative size = 109, 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} \[ \frac {a^5}{10 b^6 \left (a+b x^2\right )^5}-\frac {5 a^4}{8 b^6 \left (a+b x^2\right )^4}+\frac {5 a^3}{3 b^6 \left (a+b x^2\right )^3}-\frac {5 a^2}{2 b^6 \left (a+b x^2\right )^2}+\frac {5 a}{2 b^6 \left (a+b x^2\right )}+\frac {\log \left (a+b x^2\right )}{2 b^6} \]
Antiderivative was successfully verified.
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Rule 28
Rule 43
Rule 266
Rubi steps
\begin {align*} \int \frac {x^{11}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx &=b^6 \int \frac {x^{11}}{\left (a b+b^2 x^2\right )^6} \, dx\\ &=\frac {1}{2} b^6 \operatorname {Subst}\left (\int \frac {x^5}{\left (a b+b^2 x\right )^6} \, dx,x,x^2\right )\\ &=\frac {1}{2} b^6 \operatorname {Subst}\left (\int \left (-\frac {a^5}{b^{11} (a+b x)^6}+\frac {5 a^4}{b^{11} (a+b x)^5}-\frac {10 a^3}{b^{11} (a+b x)^4}+\frac {10 a^2}{b^{11} (a+b x)^3}-\frac {5 a}{b^{11} (a+b x)^2}+\frac {1}{b^{11} (a+b x)}\right ) \, dx,x,x^2\right )\\ &=\frac {a^5}{10 b^6 \left (a+b x^2\right )^5}-\frac {5 a^4}{8 b^6 \left (a+b x^2\right )^4}+\frac {5 a^3}{3 b^6 \left (a+b x^2\right )^3}-\frac {5 a^2}{2 b^6 \left (a+b x^2\right )^2}+\frac {5 a}{2 b^6 \left (a+b x^2\right )}+\frac {\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 = 0.66 \[ \frac {\frac {a \left (137 a^4+625 a^3 b x^2+1100 a^2 b^2 x^4+900 a b^3 x^6+300 b^4 x^8\right )}{\left (a+b x^2\right )^5}+60 \log \left (a+b x^2\right )}{120 b^6} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.79, size = 168, normalized size = 1.54 \[ \frac {300 \, a b^{4} x^{8} + 900 \, a^{2} b^{3} x^{6} + 1100 \, a^{3} b^{2} x^{4} + 625 \, a^{4} b x^{2} + 137 \, a^{5} + 60 \, {\left (b^{5} x^{10} + 5 \, a b^{4} x^{8} + 10 \, a^{2} b^{3} x^{6} + 10 \, a^{3} b^{2} x^{4} + 5 \, a^{4} b x^{2} + a^{5}\right )} \log \left (b x^{2} + a\right )}{120 \, {\left (b^{11} x^{10} + 5 \, a b^{10} x^{8} + 10 \, a^{2} b^{9} x^{6} + 10 \, a^{3} b^{8} x^{4} + 5 \, a^{4} b^{7} x^{2} + a^{5} b^{6}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 75, normalized size = 0.69 \[ \frac {\log \left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{6}} - \frac {137 \, b^{4} x^{10} + 385 \, a b^{3} x^{8} + 470 \, a^{2} b^{2} x^{6} + 270 \, a^{3} b x^{4} + 60 \, a^{4} x^{2}}{120 \, {\left (b x^{2} + a\right )}^{5} b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 98, normalized size = 0.90 \[ \frac {a^{5}}{10 \left (b \,x^{2}+a \right )^{5} b^{6}}-\frac {5 a^{4}}{8 \left (b \,x^{2}+a \right )^{4} b^{6}}+\frac {5 a^{3}}{3 \left (b \,x^{2}+a \right )^{3} b^{6}}-\frac {5 a^{2}}{2 \left (b \,x^{2}+a \right )^{2} b^{6}}+\frac {5 a}{2 \left (b \,x^{2}+a \right ) b^{6}}+\frac {\ln \left (b \,x^{2}+a \right )}{2 b^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.42, size = 121, normalized size = 1.11 \[ \frac {300 \, a b^{4} x^{8} + 900 \, a^{2} b^{3} x^{6} + 1100 \, a^{3} b^{2} x^{4} + 625 \, a^{4} b x^{2} + 137 \, a^{5}}{120 \, {\left (b^{11} x^{10} + 5 \, a b^{10} x^{8} + 10 \, a^{2} b^{9} x^{6} + 10 \, a^{3} b^{8} x^{4} + 5 \, a^{4} b^{7} x^{2} + a^{5} b^{6}\right )}} + \frac {\log \left (b x^{2} + a\right )}{2 \, b^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.37, size = 119, normalized size = 1.09 \[ \frac {\frac {137\,a^5}{120\,b^6}+\frac {5\,a\,x^8}{2\,b^2}+\frac {15\,a^2\,x^6}{2\,b^3}+\frac {55\,a^3\,x^4}{6\,b^4}+\frac {125\,a^4\,x^2}{24\,b^5}}{a^5+5\,a^4\,b\,x^2+10\,a^3\,b^2\,x^4+10\,a^2\,b^3\,x^6+5\,a\,b^4\,x^8+b^5\,x^{10}}+\frac {\ln \left (b\,x^2+a\right )}{2\,b^6} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.81, size = 124, normalized size = 1.14 \[ \frac {137 a^{5} + 625 a^{4} b x^{2} + 1100 a^{3} b^{2} x^{4} + 900 a^{2} b^{3} x^{6} + 300 a b^{4} x^{8}}{120 a^{5} b^{6} + 600 a^{4} b^{7} x^{2} + 1200 a^{3} b^{8} x^{4} + 1200 a^{2} b^{9} x^{6} + 600 a b^{10} x^{8} + 120 b^{11} x^{10}} + \frac {\log {\left (a + b x^{2} \right )}}{2 b^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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