Optimal. Leaf size=24 \[ -\frac {b^2}{4 x^4}-\frac {b c}{x^2}+c^2 \log (x) \]
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Rubi [A] time = 0.02, antiderivative size = 24, 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, 43} \[ -\frac {b^2}{4 x^4}-\frac {b c}{x^2}+c^2 \log (x) \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rule 1584
Rubi steps
\begin {align*} \int \frac {\left (b x^2+c x^4\right )^2}{x^9} \, dx &=\int \frac {\left (b+c x^2\right )^2}{x^5} \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(b+c x)^2}{x^3} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {b^2}{x^3}+\frac {2 b c}{x^2}+\frac {c^2}{x}\right ) \, dx,x,x^2\right )\\ &=-\frac {b^2}{4 x^4}-\frac {b c}{x^2}+c^2 \log (x)\\ \end {align*}
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Mathematica [A] time = 0.00, size = 24, normalized size = 1.00 \[ -\frac {b^2}{4 x^4}-\frac {b c}{x^2}+c^2 \log (x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.85, size = 28, normalized size = 1.17 \[ \frac {4 \, c^{2} x^{4} \log \relax (x) - 4 \, b c x^{2} - b^{2}}{4 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 34, normalized size = 1.42 \[ \frac {1}{2} \, c^{2} \log \left (x^{2}\right ) - \frac {3 \, c^{2} x^{4} + 4 \, b c x^{2} + b^{2}}{4 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 23, normalized size = 0.96 \[ c^{2} \ln \relax (x )-\frac {b c}{x^{2}}-\frac {b^{2}}{4 x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.32, size = 26, normalized size = 1.08 \[ \frac {1}{2} \, c^{2} \log \left (x^{2}\right ) - \frac {4 \, b c x^{2} + b^{2}}{4 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 24, normalized size = 1.00 \[ c^2\,\ln \relax (x)-\frac {\frac {b^2}{4}+c\,b\,x^2}{x^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.19, size = 24, normalized size = 1.00 \[ c^{2} \log {\relax (x )} + \frac {- b^{2} - 4 b c x^{2}}{4 x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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