Optimal. Leaf size=73 \[ -\frac {a^6}{8 x^8}-\frac {a^5 b}{x^6}-\frac {15 a^4 b^2}{4 x^4}-\frac {10 a^3 b^3}{x^2}+15 a^2 b^4 \log (x)+3 a b^5 x^2+\frac {b^6 x^4}{4} \]
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Rubi [A] time = 0.05, antiderivative size = 73, 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}{4 x^4}-\frac {10 a^3 b^3}{x^2}+15 a^2 b^4 \log (x)-\frac {a^5 b}{x^6}-\frac {a^6}{8 x^8}+3 a b^5 x^2+\frac {b^6 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 )^3}{x^9} \, dx &=\frac {\int \frac {\left (a b+b^2 x^2\right )^6}{x^9} \, dx}{b^6}\\ &=\frac {\operatorname {Subst}\left (\int \frac {\left (a b+b^2 x\right )^6}{x^5} \, dx,x,x^2\right )}{2 b^6}\\ &=\frac {\operatorname {Subst}\left (\int \left (6 a b^{11}+\frac {a^6 b^6}{x^5}+\frac {6 a^5 b^7}{x^4}+\frac {15 a^4 b^8}{x^3}+\frac {20 a^3 b^9}{x^2}+\frac {15 a^2 b^{10}}{x}+b^{12} x\right ) \, dx,x,x^2\right )}{2 b^6}\\ &=-\frac {a^6}{8 x^8}-\frac {a^5 b}{x^6}-\frac {15 a^4 b^2}{4 x^4}-\frac {10 a^3 b^3}{x^2}+3 a b^5 x^2+\frac {b^6 x^4}{4}+15 a^2 b^4 \log (x)\\ \end {align*}
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Mathematica [A] time = 0.01, size = 73, normalized size = 1.00 \begin {gather*} -\frac {a^6}{8 x^8}-\frac {a^5 b}{x^6}-\frac {15 a^4 b^2}{4 x^4}-\frac {10 a^3 b^3}{x^2}+15 a^2 b^4 \log (x)+3 a b^5 x^2+\frac {b^6 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 )^3}{x^9} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.78, size = 72, normalized size = 0.99 \begin {gather*} \frac {2 \, b^{6} x^{12} + 24 \, a b^{5} x^{10} + 120 \, a^{2} b^{4} x^{8} \log \relax (x) - 80 \, a^{3} b^{3} x^{6} - 30 \, a^{4} b^{2} x^{4} - 8 \, a^{5} b x^{2} - a^{6}}{8 \, x^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 81, normalized size = 1.11 \begin {gather*} \frac {1}{4} \, b^{6} x^{4} + 3 \, a b^{5} x^{2} + \frac {15}{2} \, a^{2} b^{4} \log \left (x^{2}\right ) - \frac {125 \, a^{2} b^{4} x^{8} + 80 \, a^{3} b^{3} x^{6} + 30 \, a^{4} b^{2} x^{4} + 8 \, a^{5} b x^{2} + a^{6}}{8 \, x^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 68, normalized size = 0.93 \begin {gather*} \frac {b^{6} x^{4}}{4}+3 a \,b^{5} x^{2}+15 a^{2} b^{4} \ln \relax (x )-\frac {10 a^{3} b^{3}}{x^{2}}-\frac {15 a^{4} b^{2}}{4 x^{4}}-\frac {a^{5} b}{x^{6}}-\frac {a^{6}}{8 x^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.31, size = 70, normalized size = 0.96 \begin {gather*} \frac {1}{4} \, b^{6} x^{4} + 3 \, a b^{5} x^{2} + \frac {15}{2} \, a^{2} b^{4} \log \left (x^{2}\right ) - \frac {80 \, a^{3} b^{3} x^{6} + 30 \, a^{4} b^{2} x^{4} + 8 \, a^{5} b x^{2} + a^{6}}{8 \, x^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 69, normalized size = 0.95 \begin {gather*} \frac {b^6\,x^4}{4}-\frac {\frac {a^6}{8}+a^5\,b\,x^2+\frac {15\,a^4\,b^2\,x^4}{4}+10\,a^3\,b^3\,x^6}{x^8}+3\,a\,b^5\,x^2+15\,a^2\,b^4\,\ln \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.42, size = 73, normalized size = 1.00 \begin {gather*} 15 a^{2} b^{4} \log {\relax (x )} + 3 a b^{5} x^{2} + \frac {b^{6} x^{4}}{4} + \frac {- a^{6} - 8 a^{5} b x^{2} - 30 a^{4} b^{2} x^{4} - 80 a^{3} b^{3} x^{6}}{8 x^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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