Optimal. Leaf size=66 \[ -\frac {3 c^2 \log \left (b+c x^2\right )}{2 b^4}+\frac {3 c^2 \log (x)}{b^4}+\frac {c^2}{2 b^3 \left (b+c x^2\right )}+\frac {c}{b^3 x^2}-\frac {1}{4 b^2 x^4} \]
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Rubi [A] time = 0.05, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {1584, 266, 44} \[ \frac {c^2}{2 b^3 \left (b+c x^2\right )}-\frac {3 c^2 \log \left (b+c x^2\right )}{2 b^4}+\frac {3 c^2 \log (x)}{b^4}+\frac {c}{b^3 x^2}-\frac {1}{4 b^2 x^4} \]
Antiderivative was successfully verified.
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Rule 44
Rule 266
Rule 1584
Rubi steps
\begin {align*} \int \frac {1}{x \left (b x^2+c x^4\right )^2} \, dx &=\int \frac {1}{x^5 \left (b+c x^2\right )^2} \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x^3 (b+c x)^2} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {1}{b^2 x^3}-\frac {2 c}{b^3 x^2}+\frac {3 c^2}{b^4 x}-\frac {c^3}{b^3 (b+c x)^2}-\frac {3 c^3}{b^4 (b+c x)}\right ) \, dx,x,x^2\right )\\ &=-\frac {1}{4 b^2 x^4}+\frac {c}{b^3 x^2}+\frac {c^2}{2 b^3 \left (b+c x^2\right )}+\frac {3 c^2 \log (x)}{b^4}-\frac {3 c^2 \log \left (b+c x^2\right )}{2 b^4}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 57, normalized size = 0.86 \[ \frac {-6 c^2 \log \left (b+c x^2\right )+b \left (\frac {2 c^2}{b+c x^2}-\frac {b}{x^4}+\frac {4 c}{x^2}\right )+12 c^2 \log (x)}{4 b^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.65, size = 90, normalized size = 1.36 \[ \frac {6 \, b c^{2} x^{4} + 3 \, b^{2} c x^{2} - b^{3} - 6 \, {\left (c^{3} x^{6} + b c^{2} x^{4}\right )} \log \left (c x^{2} + b\right ) + 12 \, {\left (c^{3} x^{6} + b c^{2} x^{4}\right )} \log \relax (x)}{4 \, {\left (b^{4} c x^{6} + b^{5} x^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 86, normalized size = 1.30 \[ \frac {3 \, c^{2} \log \left (x^{2}\right )}{2 \, b^{4}} - \frac {3 \, c^{2} \log \left ({\left | c x^{2} + b \right |}\right )}{2 \, b^{4}} + \frac {3 \, c^{3} x^{2} + 4 \, b c^{2}}{2 \, {\left (c x^{2} + b\right )} b^{4}} - \frac {9 \, c^{2} x^{4} - 4 \, b c x^{2} + b^{2}}{4 \, b^{4} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 61, normalized size = 0.92 \[ \frac {c^{2}}{2 \left (c \,x^{2}+b \right ) b^{3}}+\frac {3 c^{2} \ln \relax (x )}{b^{4}}-\frac {3 c^{2} \ln \left (c \,x^{2}+b \right )}{2 b^{4}}+\frac {c}{b^{3} x^{2}}-\frac {1}{4 b^{2} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.35, size = 70, normalized size = 1.06 \[ \frac {6 \, c^{2} x^{4} + 3 \, b c x^{2} - b^{2}}{4 \, {\left (b^{3} c x^{6} + b^{4} x^{4}\right )}} - \frac {3 \, c^{2} \log \left (c x^{2} + b\right )}{2 \, b^{4}} + \frac {3 \, c^{2} \log \left (x^{2}\right )}{2 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.17, size = 67, normalized size = 1.02 \[ \frac {\frac {3\,c\,x^2}{4\,b^2}-\frac {1}{4\,b}+\frac {3\,c^2\,x^4}{2\,b^3}}{c\,x^6+b\,x^4}-\frac {3\,c^2\,\ln \left (c\,x^2+b\right )}{2\,b^4}+\frac {3\,c^2\,\ln \relax (x)}{b^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.49, size = 68, normalized size = 1.03 \[ \frac {- b^{2} + 3 b c x^{2} + 6 c^{2} x^{4}}{4 b^{4} x^{4} + 4 b^{3} c x^{6}} + \frac {3 c^{2} \log {\relax (x )}}{b^{4}} - \frac {3 c^{2} \log {\left (\frac {b}{c} + x^{2} \right )}}{2 b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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