Optimal. Leaf size=49 \[ -\frac {a^4}{6 x^6}-\frac {a^3 b}{x^4}-\frac {3 a^2 b^2}{x^2}+4 a b^3 \log (x)+\frac {b^4 x^2}{2} \]
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Rubi [A] time = 0.03, 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} \[ -\frac {3 a^2 b^2}{x^2}-\frac {a^3 b}{x^4}-\frac {a^4}{6 x^6}+4 a b^3 \log (x)+\frac {b^4 x^2}{2} \]
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^7} \, dx &=\frac {\int \frac {\left (a b+b^2 x^2\right )^4}{x^7} \, dx}{b^4}\\ &=\frac {\operatorname {Subst}\left (\int \frac {\left (a b+b^2 x\right )^4}{x^4} \, dx,x,x^2\right )}{2 b^4}\\ &=\frac {\operatorname {Subst}\left (\int \left (b^8+\frac {a^4 b^4}{x^4}+\frac {4 a^3 b^5}{x^3}+\frac {6 a^2 b^6}{x^2}+\frac {4 a b^7}{x}\right ) \, dx,x,x^2\right )}{2 b^4}\\ &=-\frac {a^4}{6 x^6}-\frac {a^3 b}{x^4}-\frac {3 a^2 b^2}{x^2}+\frac {b^4 x^2}{2}+4 a b^3 \log (x)\\ \end {align*}
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Mathematica [A] time = 0.01, size = 49, normalized size = 1.00 \[ -\frac {a^4}{6 x^6}-\frac {a^3 b}{x^4}-\frac {3 a^2 b^2}{x^2}+4 a b^3 \log (x)+\frac {b^4 x^2}{2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.77, size = 50, normalized size = 1.02 \[ \frac {3 \, b^{4} x^{8} + 24 \, a b^{3} x^{6} \log \relax (x) - 18 \, a^{2} b^{2} x^{4} - 6 \, a^{3} b x^{2} - a^{4}}{6 \, x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 57, normalized size = 1.16 \[ \frac {1}{2} \, b^{4} x^{2} + 2 \, a b^{3} \log \left (x^{2}\right ) - \frac {22 \, a b^{3} x^{6} + 18 \, a^{2} b^{2} x^{4} + 6 \, a^{3} b x^{2} + a^{4}}{6 \, x^{6}} \]
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 \[ \frac {b^{4} x^{2}}{2}+4 a \,b^{3} \ln \relax (x )-\frac {3 a^{2} b^{2}}{x^{2}}-\frac {a^{3} b}{x^{4}}-\frac {a^{4}}{6 x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.31, size = 48, normalized size = 0.98 \[ \frac {1}{2} \, b^{4} x^{2} + 2 \, a b^{3} \log \left (x^{2}\right ) - \frac {18 \, a^{2} b^{2} x^{4} + 6 \, a^{3} b x^{2} + a^{4}}{6 \, x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 47, normalized size = 0.96 \[ \frac {b^4\,x^2}{2}-\frac {\frac {a^4}{6}+a^3\,b\,x^2+3\,a^2\,b^2\,x^4}{x^6}+4\,a\,b^3\,\ln \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.30, size = 49, normalized size = 1.00 \[ 4 a b^{3} \log {\relax (x )} + \frac {b^{4} x^{2}}{2} + \frac {- a^{4} - 6 a^{3} b x^{2} - 18 a^{2} b^{2} x^{4}}{6 x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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