Optimal. Leaf size=45 \[ -\frac {a^2}{4 x^4}+\log (x) \left (2 a c+b^2\right )-\frac {a b}{x^2}+b c x^2+\frac {c^2 x^4}{4} \]
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Rubi [A] time = 0.04, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {1114, 698} \[ -\frac {a^2}{4 x^4}+\log (x) \left (2 a c+b^2\right )-\frac {a b}{x^2}+b c x^2+\frac {c^2 x^4}{4} \]
Antiderivative was successfully verified.
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Rule 698
Rule 1114
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2+c x^4\right )^2}{x^5} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {\left (a+b x+c x^2\right )^2}{x^3} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (2 b c+\frac {a^2}{x^3}+\frac {2 a b}{x^2}+\frac {b^2+2 a c}{x}+c^2 x\right ) \, dx,x,x^2\right )\\ &=-\frac {a^2}{4 x^4}-\frac {a b}{x^2}+b c x^2+\frac {c^2 x^4}{4}+\left (b^2+2 a c\right ) \log (x)\\ \end {align*}
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Mathematica [A] time = 0.02, size = 41, normalized size = 0.91 \[ \log (x) \left (2 a c+b^2\right )+\frac {\left (c x^4-a\right ) \left (a+4 b x^2+c x^4\right )}{4 x^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.86, size = 47, normalized size = 1.04 \[ \frac {c^{2} x^{8} + 4 \, b c x^{6} + 4 \, {\left (b^{2} + 2 \, a c\right )} x^{4} \log \relax (x) - 4 \, a b x^{2} - a^{2}}{4 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 60, normalized size = 1.33 \[ \frac {1}{4} \, c^{2} x^{4} + b c x^{2} + \frac {1}{2} \, {\left (b^{2} + 2 \, a c\right )} \log \left (x^{2}\right ) - \frac {3 \, b^{2} x^{4} + 6 \, a c x^{4} + 4 \, a b x^{2} + a^{2}}{4 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 43, normalized size = 0.96 \[ \frac {c^{2} x^{4}}{4}+b c \,x^{2}+2 a c \ln \relax (x )+b^{2} \ln \relax (x )-\frac {a b}{x^{2}}-\frac {a^{2}}{4 x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.34, size = 45, normalized size = 1.00 \[ \frac {1}{4} \, c^{2} x^{4} + b c x^{2} + \frac {1}{2} \, {\left (b^{2} + 2 \, a c\right )} \log \left (x^{2}\right ) - \frac {4 \, a b x^{2} + a^{2}}{4 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 43, normalized size = 0.96 \[ \ln \relax (x)\,\left (b^2+2\,a\,c\right )-\frac {\frac {a^2}{4}+b\,a\,x^2}{x^4}+\frac {c^2\,x^4}{4}+b\,c\,x^2 \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.37, size = 44, normalized size = 0.98 \[ b c x^{2} + \frac {c^{2} x^{4}}{4} + \left (2 a c + b^{2}\right ) \log {\relax (x )} + \frac {- a^{2} - 4 a b x^{2}}{4 x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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