Optimal. Leaf size=82 \[ -\frac{15 a^4 b^2}{13 x^{13}}-\frac{20 a^3 b^3}{11 x^{11}}-\frac{5 a^2 b^4}{3 x^9}-\frac{2 a^5 b}{5 x^{15}}-\frac{a^6}{17 x^{17}}-\frac{6 a b^5}{7 x^7}-\frac{b^6}{5 x^5} \]
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Rubi [A] time = 0.0377724, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {28, 270} \[ -\frac{15 a^4 b^2}{13 x^{13}}-\frac{20 a^3 b^3}{11 x^{11}}-\frac{5 a^2 b^4}{3 x^9}-\frac{2 a^5 b}{5 x^{15}}-\frac{a^6}{17 x^{17}}-\frac{6 a b^5}{7 x^7}-\frac{b^6}{5 x^5} \]
Antiderivative was successfully verified.
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Rule 28
Rule 270
Rubi steps
\begin{align*} \int \frac{\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^{18}} \, dx &=\frac{\int \frac{\left (a b+b^2 x^2\right )^6}{x^{18}} \, dx}{b^6}\\ &=\frac{\int \left (\frac{a^6 b^6}{x^{18}}+\frac{6 a^5 b^7}{x^{16}}+\frac{15 a^4 b^8}{x^{14}}+\frac{20 a^3 b^9}{x^{12}}+\frac{15 a^2 b^{10}}{x^{10}}+\frac{6 a b^{11}}{x^8}+\frac{b^{12}}{x^6}\right ) \, dx}{b^6}\\ &=-\frac{a^6}{17 x^{17}}-\frac{2 a^5 b}{5 x^{15}}-\frac{15 a^4 b^2}{13 x^{13}}-\frac{20 a^3 b^3}{11 x^{11}}-\frac{5 a^2 b^4}{3 x^9}-\frac{6 a b^5}{7 x^7}-\frac{b^6}{5 x^5}\\ \end{align*}
Mathematica [A] time = 0.0068018, size = 82, normalized size = 1. \[ -\frac{15 a^4 b^2}{13 x^{13}}-\frac{20 a^3 b^3}{11 x^{11}}-\frac{5 a^2 b^4}{3 x^9}-\frac{2 a^5 b}{5 x^{15}}-\frac{a^6}{17 x^{17}}-\frac{6 a b^5}{7 x^7}-\frac{b^6}{5 x^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.049, size = 69, normalized size = 0.8 \begin{align*} -{\frac{{a}^{6}}{17\,{x}^{17}}}-{\frac{2\,{a}^{5}b}{5\,{x}^{15}}}-{\frac{15\,{a}^{4}{b}^{2}}{13\,{x}^{13}}}-{\frac{20\,{a}^{3}{b}^{3}}{11\,{x}^{11}}}-{\frac{5\,{a}^{2}{b}^{4}}{3\,{x}^{9}}}-{\frac{6\,a{b}^{5}}{7\,{x}^{7}}}-{\frac{{b}^{6}}{5\,{x}^{5}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.999607, size = 95, normalized size = 1.16 \begin{align*} -\frac{51051 \, b^{6} x^{12} + 218790 \, a b^{5} x^{10} + 425425 \, a^{2} b^{4} x^{8} + 464100 \, a^{3} b^{3} x^{6} + 294525 \, a^{4} b^{2} x^{4} + 102102 \, a^{5} b x^{2} + 15015 \, a^{6}}{255255 \, x^{17}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.58771, size = 198, normalized size = 2.41 \begin{align*} -\frac{51051 \, b^{6} x^{12} + 218790 \, a b^{5} x^{10} + 425425 \, a^{2} b^{4} x^{8} + 464100 \, a^{3} b^{3} x^{6} + 294525 \, a^{4} b^{2} x^{4} + 102102 \, a^{5} b x^{2} + 15015 \, a^{6}}{255255 \, x^{17}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.793198, size = 75, normalized size = 0.91 \begin{align*} - \frac{15015 a^{6} + 102102 a^{5} b x^{2} + 294525 a^{4} b^{2} x^{4} + 464100 a^{3} b^{3} x^{6} + 425425 a^{2} b^{4} x^{8} + 218790 a b^{5} x^{10} + 51051 b^{6} x^{12}}{255255 x^{17}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11932, size = 95, normalized size = 1.16 \begin{align*} -\frac{51051 \, b^{6} x^{12} + 218790 \, a b^{5} x^{10} + 425425 \, a^{2} b^{4} x^{8} + 464100 \, a^{3} b^{3} x^{6} + 294525 \, a^{4} b^{2} x^{4} + 102102 \, a^{5} b x^{2} + 15015 \, a^{6}}{255255 \, x^{17}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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