Optimal. Leaf size=76 \[ -\frac {a^6}{12 x^{12}}-\frac {3 a^5 b}{5 x^{10}}-\frac {15 a^4 b^2}{8 x^8}-\frac {10 a^3 b^3}{3 x^6}-\frac {15 a^2 b^4}{4 x^4}-\frac {3 a b^5}{x^2}+b^6 \log (x) \]
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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 a^4 b^2}{8 x^8}-\frac {10 a^3 b^3}{3 x^6}-\frac {15 a^2 b^4}{4 x^4}-\frac {3 a^5 b}{5 x^{10}}-\frac {a^6}{12 x^{12}}-\frac {3 a b^5}{x^2}+b^6 \log (x) \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^{13}} \, dx &=\frac {\int \frac {\left (a b+b^2 x^2\right )^6}{x^{13}} \, dx}{b^6}\\ &=\frac {\operatorname {Subst}\left (\int \frac {\left (a b+b^2 x\right )^6}{x^7} \, dx,x,x^2\right )}{2 b^6}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {a^6 b^6}{x^7}+\frac {6 a^5 b^7}{x^6}+\frac {15 a^4 b^8}{x^5}+\frac {20 a^3 b^9}{x^4}+\frac {15 a^2 b^{10}}{x^3}+\frac {6 a b^{11}}{x^2}+\frac {b^{12}}{x}\right ) \, dx,x,x^2\right )}{2 b^6}\\ &=-\frac {a^6}{12 x^{12}}-\frac {3 a^5 b}{5 x^{10}}-\frac {15 a^4 b^2}{8 x^8}-\frac {10 a^3 b^3}{3 x^6}-\frac {15 a^2 b^4}{4 x^4}-\frac {3 a b^5}{x^2}+b^6 \log (x)\\ \end {align*}
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Mathematica [A] time = 0.00, size = 76, normalized size = 1.00 \begin {gather*} -\frac {a^6}{12 x^{12}}-\frac {3 a^5 b}{5 x^{10}}-\frac {15 a^4 b^2}{8 x^8}-\frac {10 a^3 b^3}{3 x^6}-\frac {15 a^2 b^4}{4 x^4}-\frac {3 a b^5}{x^2}+b^6 \log (x) \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^{13}} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.77, size = 72, normalized size = 0.95 \begin {gather*} \frac {120 \, b^{6} x^{12} \log \relax (x) - 360 \, a b^{5} x^{10} - 450 \, a^{2} b^{4} x^{8} - 400 \, a^{3} b^{3} x^{6} - 225 \, a^{4} b^{2} x^{4} - 72 \, a^{5} b x^{2} - 10 \, a^{6}}{120 \, x^{12}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 80, normalized size = 1.05 \begin {gather*} \frac {1}{2} \, b^{6} \log \left (x^{2}\right ) - \frac {147 \, b^{6} x^{12} + 360 \, a b^{5} x^{10} + 450 \, a^{2} b^{4} x^{8} + 400 \, a^{3} b^{3} x^{6} + 225 \, a^{4} b^{2} x^{4} + 72 \, a^{5} b x^{2} + 10 \, a^{6}}{120 \, x^{12}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 67, normalized size = 0.88 \begin {gather*} b^{6} \ln \relax (x )-\frac {3 a \,b^{5}}{x^{2}}-\frac {15 a^{2} b^{4}}{4 x^{4}}-\frac {10 a^{3} b^{3}}{3 x^{6}}-\frac {15 a^{4} b^{2}}{8 x^{8}}-\frac {3 a^{5} b}{5 x^{10}}-\frac {a^{6}}{12 x^{12}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.40, size = 72, normalized size = 0.95 \begin {gather*} \frac {1}{2} \, b^{6} \log \left (x^{2}\right ) - \frac {360 \, a b^{5} x^{10} + 450 \, a^{2} b^{4} x^{8} + 400 \, a^{3} b^{3} x^{6} + 225 \, a^{4} b^{2} x^{4} + 72 \, a^{5} b x^{2} + 10 \, a^{6}}{120 \, x^{12}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 69, normalized size = 0.91 \begin {gather*} b^6\,\ln \relax (x)-\frac {\frac {a^6}{12}+\frac {3\,a^5\,b\,x^2}{5}+\frac {15\,a^4\,b^2\,x^4}{8}+\frac {10\,a^3\,b^3\,x^6}{3}+\frac {15\,a^2\,b^4\,x^8}{4}+3\,a\,b^5\,x^{10}}{x^{12}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.62, size = 73, normalized size = 0.96 \begin {gather*} b^{6} \log {\relax (x )} + \frac {- 10 a^{6} - 72 a^{5} b x^{2} - 225 a^{4} b^{2} x^{4} - 400 a^{3} b^{3} x^{6} - 450 a^{2} b^{4} x^{8} - 360 a b^{5} x^{10}}{120 x^{12}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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