Optimal. Leaf size=24 \[ -\frac {a^2}{4 x^4}-\frac {a b}{x^2}+b^2 \log (x) \]
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Rubi [A] time = 0.01, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {266, 43} \[ -\frac {a^2}{4 x^4}-\frac {a b}{x^2}+b^2 \log (x) \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^2}{x^5} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(a+b x)^2}{x^3} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {a^2}{x^3}+\frac {2 a b}{x^2}+\frac {b^2}{x}\right ) \, dx,x,x^2\right )\\ &=-\frac {a^2}{4 x^4}-\frac {a b}{x^2}+b^2 \log (x)\\ \end {align*}
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Mathematica [A] time = 0.00, size = 24, normalized size = 1.00 \[ -\frac {a^2}{4 x^4}-\frac {a b}{x^2}+b^2 \log (x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.80, size = 28, normalized size = 1.17 \[ \frac {4 \, b^{2} x^{4} \log \relax (x) - 4 \, a b x^{2} - a^{2}}{4 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.14, size = 34, normalized size = 1.42 \[ \frac {1}{2} \, b^{2} \log \left (x^{2}\right ) - \frac {3 \, b^{2} x^{4} + 4 \, a b x^{2} + a^{2}}{4 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 23, normalized size = 0.96 \[ b^{2} \ln \relax (x )-\frac {a b}{x^{2}}-\frac {a^{2}}{4 x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.28, size = 26, normalized size = 1.08 \[ \frac {1}{2} \, b^{2} \log \left (x^{2}\right ) - \frac {4 \, a b x^{2} + a^{2}}{4 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 24, normalized size = 1.00 \[ b^2\,\ln \relax (x)-\frac {\frac {a^2}{4}+b\,a\,x^2}{x^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.17, size = 24, normalized size = 1.00 \[ b^{2} \log {\relax (x )} + \frac {- a^{2} - 4 a b x^{2}}{4 x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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