Optimal. Leaf size=101 \[ -\frac {5 b^2 \log \left (a+b x^2\right )}{a^6}+\frac {10 b^2 \log (x)}{a^6}+\frac {3 b^2}{a^5 \left (a+b x^2\right )}+\frac {2 b}{a^5 x^2}+\frac {3 b^2}{4 a^4 \left (a+b x^2\right )^2}-\frac {1}{4 a^4 x^4}+\frac {b^2}{6 a^3 \left (a+b x^2\right )^3} \]
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Rubi [A] time = 0.10, antiderivative size = 101, 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, 44} \begin {gather*} \frac {3 b^2}{a^5 \left (a+b x^2\right )}+\frac {3 b^2}{4 a^4 \left (a+b x^2\right )^2}+\frac {b^2}{6 a^3 \left (a+b x^2\right )^3}-\frac {5 b^2 \log \left (a+b x^2\right )}{a^6}+\frac {10 b^2 \log (x)}{a^6}+\frac {2 b}{a^5 x^2}-\frac {1}{4 a^4 x^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 28
Rule 44
Rule 266
Rubi steps
\begin {align*} \int \frac {1}{x^5 \left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx &=b^4 \int \frac {1}{x^5 \left (a b+b^2 x^2\right )^4} \, dx\\ &=\frac {1}{2} b^4 \operatorname {Subst}\left (\int \frac {1}{x^3 \left (a b+b^2 x\right )^4} \, dx,x,x^2\right )\\ &=\frac {1}{2} b^4 \operatorname {Subst}\left (\int \left (\frac {1}{a^4 b^4 x^3}-\frac {4}{a^5 b^3 x^2}+\frac {10}{a^6 b^2 x}-\frac {1}{a^3 b (a+b x)^4}-\frac {3}{a^4 b (a+b x)^3}-\frac {6}{a^5 b (a+b x)^2}-\frac {10}{a^6 b (a+b x)}\right ) \, dx,x,x^2\right )\\ &=-\frac {1}{4 a^4 x^4}+\frac {2 b}{a^5 x^2}+\frac {b^2}{6 a^3 \left (a+b x^2\right )^3}+\frac {3 b^2}{4 a^4 \left (a+b x^2\right )^2}+\frac {3 b^2}{a^5 \left (a+b x^2\right )}+\frac {10 b^2 \log (x)}{a^6}-\frac {5 b^2 \log \left (a+b x^2\right )}{a^6}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 85, normalized size = 0.84 \begin {gather*} \frac {\frac {a \left (-3 a^4+15 a^3 b x^2+110 a^2 b^2 x^4+150 a b^3 x^6+60 b^4 x^8\right )}{x^4 \left (a+b x^2\right )^3}-60 b^2 \log \left (a+b x^2\right )+120 b^2 \log (x)}{12 a^6} \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 {1}{x^5 \left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 4.94, size = 178, normalized size = 1.76 \begin {gather*} \frac {60 \, a b^{4} x^{8} + 150 \, a^{2} b^{3} x^{6} + 110 \, a^{3} b^{2} x^{4} + 15 \, a^{4} b x^{2} - 3 \, a^{5} - 60 \, {\left (b^{5} x^{10} + 3 \, a b^{4} x^{8} + 3 \, a^{2} b^{3} x^{6} + a^{3} b^{2} x^{4}\right )} \log \left (b x^{2} + a\right ) + 120 \, {\left (b^{5} x^{10} + 3 \, a b^{4} x^{8} + 3 \, a^{2} b^{3} x^{6} + a^{3} b^{2} x^{4}\right )} \log \relax (x)}{12 \, {\left (a^{6} b^{3} x^{10} + 3 \, a^{7} b^{2} x^{8} + 3 \, a^{8} b x^{6} + a^{9} x^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 108, normalized size = 1.07 \begin {gather*} \frac {5 \, b^{2} \log \left (x^{2}\right )}{a^{6}} - \frac {5 \, b^{2} \log \left ({\left | b x^{2} + a \right |}\right )}{a^{6}} + \frac {110 \, b^{5} x^{6} + 366 \, a b^{4} x^{4} + 411 \, a^{2} b^{3} x^{2} + 157 \, a^{3} b^{2}}{12 \, {\left (b x^{2} + a\right )}^{3} a^{6}} - \frac {30 \, b^{2} x^{4} - 8 \, a b x^{2} + a^{2}}{4 \, a^{6} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 96, normalized size = 0.95 \begin {gather*} \frac {b^{2}}{6 \left (b \,x^{2}+a \right )^{3} a^{3}}+\frac {3 b^{2}}{4 \left (b \,x^{2}+a \right )^{2} a^{4}}+\frac {3 b^{2}}{\left (b \,x^{2}+a \right ) a^{5}}+\frac {10 b^{2} \ln \relax (x )}{a^{6}}-\frac {5 b^{2} \ln \left (b \,x^{2}+a \right )}{a^{6}}+\frac {2 b}{a^{5} x^{2}}-\frac {1}{4 a^{4} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.42, size = 114, normalized size = 1.13 \begin {gather*} \frac {60 \, b^{4} x^{8} + 150 \, a b^{3} x^{6} + 110 \, a^{2} b^{2} x^{4} + 15 \, a^{3} b x^{2} - 3 \, a^{4}}{12 \, {\left (a^{5} b^{3} x^{10} + 3 \, a^{6} b^{2} x^{8} + 3 \, a^{7} b x^{6} + a^{8} x^{4}\right )}} - \frac {5 \, b^{2} \log \left (b x^{2} + a\right )}{a^{6}} + \frac {5 \, b^{2} \log \left (x^{2}\right )}{a^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.66, size = 111, normalized size = 1.10 \begin {gather*} \frac {\frac {5\,b\,x^2}{4\,a^2}-\frac {1}{4\,a}+\frac {55\,b^2\,x^4}{6\,a^3}+\frac {25\,b^3\,x^6}{2\,a^4}+\frac {5\,b^4\,x^8}{a^5}}{a^3\,x^4+3\,a^2\,b\,x^6+3\,a\,b^2\,x^8+b^3\,x^{10}}-\frac {5\,b^2\,\ln \left (b\,x^2+a\right )}{a^6}+\frac {10\,b^2\,\ln \relax (x)}{a^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.72, size = 116, normalized size = 1.15 \begin {gather*} \frac {- 3 a^{4} + 15 a^{3} b x^{2} + 110 a^{2} b^{2} x^{4} + 150 a b^{3} x^{6} + 60 b^{4} x^{8}}{12 a^{8} x^{4} + 36 a^{7} b x^{6} + 36 a^{6} b^{2} x^{8} + 12 a^{5} b^{3} x^{10}} + \frac {10 b^{2} \log {\relax (x )}}{a^{6}} - \frac {5 b^{2} \log {\left (\frac {a}{b} + x^{2} \right )}}{a^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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