Optimal. Leaf size=50 \[ a^4 \log (x)+2 a^3 b x^2+\frac {3}{2} a^2 b^2 x^4+\frac {2}{3} a b^3 x^6+\frac {b^4 x^8}{8} \]
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Rubi [A] time = 0.03, antiderivative size = 50, 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*} \frac {3}{2} a^2 b^2 x^4+2 a^3 b x^2+a^4 \log (x)+\frac {2}{3} a b^3 x^6+\frac {b^4 x^8}{8} \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} \, dx &=\frac {\int \frac {\left (a b+b^2 x^2\right )^4}{x} \, dx}{b^4}\\ &=\frac {\operatorname {Subst}\left (\int \frac {\left (a b+b^2 x\right )^4}{x} \, dx,x,x^2\right )}{2 b^4}\\ &=\frac {\operatorname {Subst}\left (\int \left (4 a^3 b^5+\frac {a^4 b^4}{x}+6 a^2 b^6 x+4 a b^7 x^2+b^8 x^3\right ) \, dx,x,x^2\right )}{2 b^4}\\ &=2 a^3 b x^2+\frac {3}{2} a^2 b^2 x^4+\frac {2}{3} a b^3 x^6+\frac {b^4 x^8}{8}+a^4 \log (x)\\ \end {align*}
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Mathematica [A] time = 0.00, size = 50, normalized size = 1.00 \begin {gather*} a^4 \log (x)+2 a^3 b x^2+\frac {3}{2} a^2 b^2 x^4+\frac {2}{3} a b^3 x^6+\frac {b^4 x^8}{8} \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} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.77, size = 44, normalized size = 0.88 \begin {gather*} \frac {1}{8} \, b^{4} x^{8} + \frac {2}{3} \, a b^{3} x^{6} + \frac {3}{2} \, a^{2} b^{2} x^{4} + 2 \, a^{3} b x^{2} + a^{4} \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 47, normalized size = 0.94 \begin {gather*} \frac {1}{8} \, b^{4} x^{8} + \frac {2}{3} \, a b^{3} x^{6} + \frac {3}{2} \, a^{2} b^{2} x^{4} + 2 \, a^{3} b x^{2} + \frac {1}{2} \, a^{4} \log \left (x^{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 45, normalized size = 0.90 \begin {gather*} \frac {b^{4} x^{8}}{8}+\frac {2 a \,b^{3} x^{6}}{3}+\frac {3 a^{2} b^{2} x^{4}}{2}+2 a^{3} b \,x^{2}+a^{4} \ln \relax (x ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.34, size = 47, normalized size = 0.94 \begin {gather*} \frac {1}{8} \, b^{4} x^{8} + \frac {2}{3} \, a b^{3} x^{6} + \frac {3}{2} \, a^{2} b^{2} x^{4} + 2 \, a^{3} b x^{2} + \frac {1}{2} \, a^{4} \log \left (x^{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 44, normalized size = 0.88 \begin {gather*} a^4\,\ln \relax (x)+\frac {b^4\,x^8}{8}+2\,a^3\,b\,x^2+\frac {2\,a\,b^3\,x^6}{3}+\frac {3\,a^2\,b^2\,x^4}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.13, size = 49, normalized size = 0.98 \begin {gather*} a^{4} \log {\relax (x )} + 2 a^{3} b x^{2} + \frac {3 a^{2} b^{2} x^{4}}{2} + \frac {2 a b^{3} x^{6}}{3} + \frac {b^{4} x^{8}}{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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