Optimal. Leaf size=76 \[ a^6 \log (x)+3 a^5 b x^2+\frac {15}{4} a^4 b^2 x^4+\frac {10}{3} a^3 b^3 x^6+\frac {15}{8} a^2 b^4 x^8+\frac {3}{5} a b^5 x^{10}+\frac {b^6 x^{12}}{12} \]
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Rubi [A] time = 0.05, antiderivative size = 76, 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 {15}{8} a^2 b^4 x^8+\frac {10}{3} a^3 b^3 x^6+\frac {15}{4} a^4 b^2 x^4+3 a^5 b x^2+a^6 \log (x)+\frac {3}{5} a b^5 x^{10}+\frac {b^6 x^{12}}{12} \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 )^3}{x} \, dx &=\frac {\int \frac {\left (a b+b^2 x^2\right )^6}{x} \, dx}{b^6}\\ &=\frac {\operatorname {Subst}\left (\int \frac {\left (a b+b^2 x\right )^6}{x} \, dx,x,x^2\right )}{2 b^6}\\ &=\frac {\operatorname {Subst}\left (\int \left (6 a^5 b^7+\frac {a^6 b^6}{x}+15 a^4 b^8 x+20 a^3 b^9 x^2+15 a^2 b^{10} x^3+6 a b^{11} x^4+b^{12} x^5\right ) \, dx,x,x^2\right )}{2 b^6}\\ &=3 a^5 b x^2+\frac {15}{4} a^4 b^2 x^4+\frac {10}{3} a^3 b^3 x^6+\frac {15}{8} a^2 b^4 x^8+\frac {3}{5} a b^5 x^{10}+\frac {b^6 x^{12}}{12}+a^6 \log (x)\\ \end {align*}
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Mathematica [A] time = 0.00, size = 76, normalized size = 1.00 \begin {gather*} a^6 \log (x)+3 a^5 b x^2+\frac {15}{4} a^4 b^2 x^4+\frac {10}{3} a^3 b^3 x^6+\frac {15}{8} a^2 b^4 x^8+\frac {3}{5} a b^5 x^{10}+\frac {b^6 x^{12}}{12} \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 )^3}{x} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.79, size = 66, normalized size = 0.87 \begin {gather*} \frac {1}{12} \, b^{6} x^{12} + \frac {3}{5} \, a b^{5} x^{10} + \frac {15}{8} \, a^{2} b^{4} x^{8} + \frac {10}{3} \, a^{3} b^{3} x^{6} + \frac {15}{4} \, a^{4} b^{2} x^{4} + 3 \, a^{5} b x^{2} + a^{6} \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 = 69, normalized size = 0.91 \begin {gather*} \frac {1}{12} \, b^{6} x^{12} + \frac {3}{5} \, a b^{5} x^{10} + \frac {15}{8} \, a^{2} b^{4} x^{8} + \frac {10}{3} \, a^{3} b^{3} x^{6} + \frac {15}{4} \, a^{4} b^{2} x^{4} + 3 \, a^{5} b x^{2} + \frac {1}{2} \, a^{6} \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 = 67, normalized size = 0.88 \begin {gather*} \frac {b^{6} x^{12}}{12}+\frac {3 a \,b^{5} x^{10}}{5}+\frac {15 a^{2} b^{4} x^{8}}{8}+\frac {10 a^{3} b^{3} x^{6}}{3}+\frac {15 a^{4} b^{2} x^{4}}{4}+3 a^{5} b \,x^{2}+a^{6} \ln \relax (x ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.40, size = 69, normalized size = 0.91 \begin {gather*} \frac {1}{12} \, b^{6} x^{12} + \frac {3}{5} \, a b^{5} x^{10} + \frac {15}{8} \, a^{2} b^{4} x^{8} + \frac {10}{3} \, a^{3} b^{3} x^{6} + \frac {15}{4} \, a^{4} b^{2} x^{4} + 3 \, a^{5} b x^{2} + \frac {1}{2} \, a^{6} \log \left (x^{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 66, normalized size = 0.87 \begin {gather*} a^6\,\ln \relax (x)+\frac {b^6\,x^{12}}{12}+3\,a^5\,b\,x^2+\frac {3\,a\,b^5\,x^{10}}{5}+\frac {15\,a^4\,b^2\,x^4}{4}+\frac {10\,a^3\,b^3\,x^6}{3}+\frac {15\,a^2\,b^4\,x^8}{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.17, size = 76, normalized size = 1.00 \begin {gather*} a^{6} \log {\relax (x )} + 3 a^{5} b x^{2} + \frac {15 a^{4} b^{2} x^{4}}{4} + \frac {10 a^{3} b^{3} x^{6}}{3} + \frac {15 a^{2} b^{4} x^{8}}{8} + \frac {3 a b^{5} x^{10}}{5} + \frac {b^{6} x^{12}}{12} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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