Optimal. Leaf size=49 \[ -\frac {a^4}{4 x^4}-\frac {2 a^3 b}{x^2}+6 a^2 b^2 \log (x)+2 a b^3 x^2+\frac {b^4 x^4}{4} \]
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Rubi [A] time = 0.04, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {28, 266, 43} \begin {gather*} 6 a^2 b^2 \log (x)-\frac {2 a^3 b}{x^2}-\frac {a^4}{4 x^4}+2 a b^3 x^2+\frac {b^4 x^4}{4} \end {gather*}
Antiderivative was successfully verified.
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Rule 28
Rule 43
Rule 266
Rubi steps
\begin {align*} \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^2}{x^5} \, dx &=\frac {\int \frac {\left (a b+b^2 x^2\right )^4}{x^5} \, dx}{b^4}\\ &=\frac {\operatorname {Subst}\left (\int \frac {\left (a b+b^2 x\right )^4}{x^3} \, dx,x,x^2\right )}{2 b^4}\\ &=\frac {\operatorname {Subst}\left (\int \left (4 a b^7+\frac {a^4 b^4}{x^3}+\frac {4 a^3 b^5}{x^2}+\frac {6 a^2 b^6}{x}+b^8 x\right ) \, dx,x,x^2\right )}{2 b^4}\\ &=-\frac {a^4}{4 x^4}-\frac {2 a^3 b}{x^2}+2 a b^3 x^2+\frac {b^4 x^4}{4}+6 a^2 b^2 \log (x)\\ \end {align*}
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Mathematica [A] time = 0.00, size = 49, normalized size = 1.00 \begin {gather*} -\frac {a^4}{4 x^4}-\frac {2 a^3 b}{x^2}+6 a^2 b^2 \log (x)+2 a b^3 x^2+\frac {b^4 x^4}{4} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^2}{x^5} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.81, size = 49, normalized size = 1.00 \begin {gather*} \frac {b^{4} x^{8} + 8 \, a b^{3} x^{6} + 24 \, a^{2} b^{2} x^{4} \log \relax (x) - 8 \, a^{3} b x^{2} - a^{4}}{4 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 59, normalized size = 1.20 \begin {gather*} \frac {1}{4} \, b^{4} x^{4} + 2 \, a b^{3} x^{2} + 3 \, a^{2} b^{2} \log \left (x^{2}\right ) - \frac {18 \, a^{2} b^{2} x^{4} + 8 \, a^{3} b x^{2} + a^{4}}{4 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 46, normalized size = 0.94 \begin {gather*} \frac {b^{4} x^{4}}{4}+2 a \,b^{3} x^{2}+6 a^{2} b^{2} \ln \relax (x )-\frac {2 a^{3} b}{x^{2}}-\frac {a^{4}}{4 x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.36, size = 48, normalized size = 0.98 \begin {gather*} \frac {1}{4} \, b^{4} x^{4} + 2 \, a b^{3} x^{2} + 3 \, a^{2} b^{2} \log \left (x^{2}\right ) - \frac {8 \, a^{3} b x^{2} + a^{4}}{4 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 48, normalized size = 0.98 \begin {gather*} \frac {b^4\,x^4}{4}-\frac {\frac {a^4}{4}+2\,b\,a^3\,x^2}{x^4}+2\,a\,b^3\,x^2+6\,a^2\,b^2\,\ln \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.22, size = 49, normalized size = 1.00 \begin {gather*} 6 a^{2} b^{2} \log {\relax (x )} + 2 a b^{3} x^{2} + \frac {b^{4} x^{4}}{4} + \frac {- a^{4} - 8 a^{3} b x^{2}}{4 x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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