Optimal. Leaf size=70 \[ -\frac {\log \left (a+b x^2\right )}{2 a^4}+\frac {\log (x)}{a^4}+\frac {1}{2 a^3 \left (a+b x^2\right )}+\frac {1}{4 a^2 \left (a+b x^2\right )^2}+\frac {1}{6 a \left (a+b x^2\right )^3} \]
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Rubi [A] time = 0.08, antiderivative size = 70, 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} \[ \frac {1}{2 a^3 \left (a+b x^2\right )}+\frac {1}{4 a^2 \left (a+b x^2\right )^2}-\frac {\log \left (a+b x^2\right )}{2 a^4}+\frac {\log (x)}{a^4}+\frac {1}{6 a \left (a+b x^2\right )^3} \]
Antiderivative was successfully verified.
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Rule 28
Rule 44
Rule 266
Rubi steps
\begin {align*} \int \frac {1}{x \left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx &=b^4 \int \frac {1}{x \left (a b+b^2 x^2\right )^4} \, dx\\ &=\frac {1}{2} b^4 \operatorname {Subst}\left (\int \frac {1}{x \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}-\frac {1}{a b^3 (a+b x)^4}-\frac {1}{a^2 b^3 (a+b x)^3}-\frac {1}{a^3 b^3 (a+b x)^2}-\frac {1}{a^4 b^3 (a+b x)}\right ) \, dx,x,x^2\right )\\ &=\frac {1}{6 a \left (a+b x^2\right )^3}+\frac {1}{4 a^2 \left (a+b x^2\right )^2}+\frac {1}{2 a^3 \left (a+b x^2\right )}+\frac {\log (x)}{a^4}-\frac {\log \left (a+b x^2\right )}{2 a^4}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 54, normalized size = 0.77 \[ \frac {\frac {a \left (11 a^2+15 a b x^2+6 b^2 x^4\right )}{\left (a+b x^2\right )^3}-6 \log \left (a+b x^2\right )+12 \log (x)}{12 a^4} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.21, size = 134, normalized size = 1.91 \[ \frac {6 \, a b^{2} x^{4} + 15 \, a^{2} b x^{2} + 11 \, a^{3} - 6 \, {\left (b^{3} x^{6} + 3 \, a b^{2} x^{4} + 3 \, a^{2} b x^{2} + a^{3}\right )} \log \left (b x^{2} + a\right ) + 12 \, {\left (b^{3} x^{6} + 3 \, a b^{2} x^{4} + 3 \, a^{2} b x^{2} + a^{3}\right )} \log \relax (x)}{12 \, {\left (a^{4} b^{3} x^{6} + 3 \, a^{5} b^{2} x^{4} + 3 \, a^{6} b x^{2} + a^{7}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 70, normalized size = 1.00 \[ \frac {\log \left (x^{2}\right )}{2 \, a^{4}} - \frac {\log \left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{4}} + \frac {11 \, b^{3} x^{6} + 39 \, a b^{2} x^{4} + 48 \, a^{2} b x^{2} + 22 \, a^{3}}{12 \, {\left (b x^{2} + a\right )}^{3} a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 63, normalized size = 0.90 \[ \frac {1}{6 \left (b \,x^{2}+a \right )^{3} a}+\frac {1}{4 \left (b \,x^{2}+a \right )^{2} a^{2}}+\frac {1}{2 \left (b \,x^{2}+a \right ) a^{3}}+\frac {\ln \relax (x )}{a^{4}}-\frac {\ln \left (b \,x^{2}+a \right )}{2 a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.42, size = 82, normalized size = 1.17 \[ \frac {6 \, b^{2} x^{4} + 15 \, a b x^{2} + 11 \, a^{2}}{12 \, {\left (a^{3} b^{3} x^{6} + 3 \, a^{4} b^{2} x^{4} + 3 \, a^{5} b x^{2} + a^{6}\right )}} - \frac {\log \left (b x^{2} + a\right )}{2 \, a^{4}} + \frac {\log \left (x^{2}\right )}{2 \, a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.47, size = 78, normalized size = 1.11 \[ \frac {\ln \relax (x)}{a^4}+\frac {\frac {11}{12\,a}+\frac {5\,b\,x^2}{4\,a^2}+\frac {b^2\,x^4}{2\,a^3}}{a^3+3\,a^2\,b\,x^2+3\,a\,b^2\,x^4+b^3\,x^6}-\frac {\ln \left (b\,x^2+a\right )}{2\,a^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.56, size = 80, normalized size = 1.14 \[ \frac {11 a^{2} + 15 a b x^{2} + 6 b^{2} x^{4}}{12 a^{6} + 36 a^{5} b x^{2} + 36 a^{4} b^{2} x^{4} + 12 a^{3} b^{3} x^{6}} + \frac {\log {\relax (x )}}{a^{4}} - \frac {\log {\left (\frac {a}{b} + x^{2} \right )}}{2 a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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