Optimal. Leaf size=27 \[ -\frac {a^2}{4 x^4}+2 a b \log (x)+\frac {b^2 x^4}{4} \]
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Rubi [A] time = 0.01, antiderivative size = 27, 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} \[ -\frac {a^2}{4 x^4}+2 a b \log (x)+\frac {b^2 x^4}{4} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int \frac {\left (a+b x^4\right )^2}{x^5} \, dx &=\frac {1}{4} \operatorname {Subst}\left (\int \frac {(a+b x)^2}{x^2} \, dx,x,x^4\right )\\ &=\frac {1}{4} \operatorname {Subst}\left (\int \left (b^2+\frac {a^2}{x^2}+\frac {2 a b}{x}\right ) \, dx,x,x^4\right )\\ &=-\frac {a^2}{4 x^4}+\frac {b^2 x^4}{4}+2 a b \log (x)\\ \end {align*}
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Mathematica [A] time = 0.00, size = 27, normalized size = 1.00 \[ -\frac {a^2}{4 x^4}+2 a b \log (x)+\frac {b^2 x^4}{4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.59, size = 27, normalized size = 1.00 \[ \frac {b^{2} x^{8} + 8 \, a b x^{4} \log \relax (x) - a^{2}}{4 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 33, normalized size = 1.22 \[ \frac {1}{4} \, b^{2} x^{4} + \frac {1}{2} \, a b \log \left (x^{4}\right ) - \frac {2 \, a b x^{4} + a^{2}}{4 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 24, normalized size = 0.89 \[ \frac {b^{2} x^{4}}{4}+2 a b \ln \relax (x )-\frac {a^{2}}{4 x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.28, size = 25, normalized size = 0.93 \[ \frac {1}{4} \, b^{2} x^{4} + \frac {1}{2} \, a b \log \left (x^{4}\right ) - \frac {a^{2}}{4 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 23, normalized size = 0.85 \[ \frac {b^2\,x^4}{4}-\frac {a^2}{4\,x^4}+2\,a\,b\,\ln \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.20, size = 24, normalized size = 0.89 \[ - \frac {a^{2}}{4 x^{4}} + 2 a b \log {\relax (x )} + \frac {b^{2} x^{4}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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