Optimal. Leaf size=84 \[ \frac {2 b \log \left (a+b x^2\right )}{a^5}-\frac {4 b \log (x)}{a^5}-\frac {3 b}{2 a^4 \left (a+b x^2\right )}-\frac {1}{2 a^4 x^2}-\frac {b}{2 a^3 \left (a+b x^2\right )^2}-\frac {b}{6 a^2 \left (a+b x^2\right )^3} \]
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Rubi [A] time = 0.09, antiderivative size = 84, 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^4 \left (a+b x^2\right )}-\frac {b}{2 a^3 \left (a+b x^2\right )^2}-\frac {b}{6 a^2 \left (a+b x^2\right )^3}+\frac {2 b \log \left (a+b x^2\right )}{a^5}-\frac {4 b \log (x)}{a^5}-\frac {1}{2 a^4 x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 28
Rule 44
Rule 266
Rubi steps
\begin {align*} \int \frac {1}{x^3 \left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx &=b^4 \int \frac {1}{x^3 \left (a b+b^2 x^2\right )^4} \, dx\\ &=\frac {1}{2} b^4 \operatorname {Subst}\left (\int \frac {1}{x^2 \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^2}-\frac {4}{a^5 b^3 x}+\frac {1}{a^2 b^2 (a+b x)^4}+\frac {2}{a^3 b^2 (a+b x)^3}+\frac {3}{a^4 b^2 (a+b x)^2}+\frac {4}{a^5 b^2 (a+b x)}\right ) \, dx,x,x^2\right )\\ &=-\frac {1}{2 a^4 x^2}-\frac {b}{6 a^2 \left (a+b x^2\right )^3}-\frac {b}{2 a^3 \left (a+b x^2\right )^2}-\frac {3 b}{2 a^4 \left (a+b x^2\right )}-\frac {4 b \log (x)}{a^5}+\frac {2 b \log \left (a+b x^2\right )}{a^5}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 70, normalized size = 0.83 \begin {gather*} -\frac {\frac {a \left (3 a^3+22 a^2 b x^2+30 a b^2 x^4+12 b^3 x^6\right )}{x^2 \left (a+b x^2\right )^3}-12 b \log \left (a+b x^2\right )+24 b \log (x)}{6 a^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 {1}{x^3 \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 [B] time = 0.92, size = 163, normalized size = 1.94 \begin {gather*} -\frac {12 \, a b^{3} x^{6} + 30 \, a^{2} b^{2} x^{4} + 22 \, a^{3} b x^{2} + 3 \, a^{4} - 12 \, {\left (b^{4} x^{8} + 3 \, a b^{3} x^{6} + 3 \, a^{2} b^{2} x^{4} + a^{3} b x^{2}\right )} \log \left (b x^{2} + a\right ) + 24 \, {\left (b^{4} x^{8} + 3 \, a b^{3} x^{6} + 3 \, a^{2} b^{2} x^{4} + a^{3} b x^{2}\right )} \log \relax (x)}{6 \, {\left (a^{5} b^{3} x^{8} + 3 \, a^{6} b^{2} x^{6} + 3 \, a^{7} b x^{4} + a^{8} x^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 93, normalized size = 1.11 \begin {gather*} -\frac {2 \, b \log \left (x^{2}\right )}{a^{5}} + \frac {2 \, b \log \left ({\left | b x^{2} + a \right |}\right )}{a^{5}} + \frac {4 \, b x^{2} - a}{2 \, a^{5} x^{2}} - \frac {22 \, b^{4} x^{6} + 75 \, a b^{3} x^{4} + 87 \, a^{2} b^{2} x^{2} + 35 \, a^{3} b}{6 \, {\left (b x^{2} + a\right )}^{3} a^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 77, normalized size = 0.92 \begin {gather*} -\frac {b}{6 \left (b \,x^{2}+a \right )^{3} a^{2}}-\frac {b}{2 \left (b \,x^{2}+a \right )^{2} a^{3}}-\frac {3 b}{2 \left (b \,x^{2}+a \right ) a^{4}}-\frac {4 b \ln \relax (x )}{a^{5}}+\frac {2 b \ln \left (b \,x^{2}+a \right )}{a^{5}}-\frac {1}{2 a^{4} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.40, size = 99, normalized size = 1.18 \begin {gather*} -\frac {12 \, b^{3} x^{6} + 30 \, a b^{2} x^{4} + 22 \, a^{2} b x^{2} + 3 \, a^{3}}{6 \, {\left (a^{4} b^{3} x^{8} + 3 \, a^{5} b^{2} x^{6} + 3 \, a^{6} b x^{4} + a^{7} x^{2}\right )}} + \frac {2 \, b \log \left (b x^{2} + a\right )}{a^{5}} - \frac {2 \, b \log \left (x^{2}\right )}{a^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.15, size = 97, normalized size = 1.15 \begin {gather*} \frac {2\,b\,\ln \left (b\,x^2+a\right )}{a^5}-\frac {\frac {1}{2\,a}+\frac {11\,b\,x^2}{3\,a^2}+\frac {5\,b^2\,x^4}{a^3}+\frac {2\,b^3\,x^6}{a^4}}{a^3\,x^2+3\,a^2\,b\,x^4+3\,a\,b^2\,x^6+b^3\,x^8}-\frac {4\,b\,\ln \relax (x)}{a^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.67, size = 102, normalized size = 1.21 \begin {gather*} \frac {- 3 a^{3} - 22 a^{2} b x^{2} - 30 a b^{2} x^{4} - 12 b^{3} x^{6}}{6 a^{7} x^{2} + 18 a^{6} b x^{4} + 18 a^{5} b^{2} x^{6} + 6 a^{4} b^{3} x^{8}} - \frac {4 b \log {\relax (x )}}{a^{5}} + \frac {2 b \log {\left (\frac {a}{b} + x^{2} \right )}}{a^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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