Optimal. Leaf size=82 \[ -\frac{15 a^4 b^2}{11 x^{11}}-\frac{20 a^3 b^3}{9 x^9}-\frac{15 a^2 b^4}{7 x^7}-\frac{6 a^5 b}{13 x^{13}}-\frac{a^6}{15 x^{15}}-\frac{6 a b^5}{5 x^5}-\frac{b^6}{3 x^3} \]
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Rubi [A] time = 0.0383924, 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} \[ -\frac{15 a^4 b^2}{11 x^{11}}-\frac{20 a^3 b^3}{9 x^9}-\frac{15 a^2 b^4}{7 x^7}-\frac{6 a^5 b}{13 x^{13}}-\frac{a^6}{15 x^{15}}-\frac{6 a b^5}{5 x^5}-\frac{b^6}{3 x^3} \]
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^{16}} \, dx &=\frac{\int \frac{\left (a b+b^2 x^2\right )^6}{x^{16}} \, dx}{b^6}\\ &=\frac{\int \left (\frac{a^6 b^6}{x^{16}}+\frac{6 a^5 b^7}{x^{14}}+\frac{15 a^4 b^8}{x^{12}}+\frac{20 a^3 b^9}{x^{10}}+\frac{15 a^2 b^{10}}{x^8}+\frac{6 a b^{11}}{x^6}+\frac{b^{12}}{x^4}\right ) \, dx}{b^6}\\ &=-\frac{a^6}{15 x^{15}}-\frac{6 a^5 b}{13 x^{13}}-\frac{15 a^4 b^2}{11 x^{11}}-\frac{20 a^3 b^3}{9 x^9}-\frac{15 a^2 b^4}{7 x^7}-\frac{6 a b^5}{5 x^5}-\frac{b^6}{3 x^3}\\ \end{align*}
Mathematica [A] time = 0.0095794, size = 82, normalized size = 1. \[ -\frac{15 a^4 b^2}{11 x^{11}}-\frac{20 a^3 b^3}{9 x^9}-\frac{15 a^2 b^4}{7 x^7}-\frac{6 a^5 b}{13 x^{13}}-\frac{a^6}{15 x^{15}}-\frac{6 a b^5}{5 x^5}-\frac{b^6}{3 x^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.05, size = 69, normalized size = 0.8 \begin{align*} -{\frac{{a}^{6}}{15\,{x}^{15}}}-{\frac{6\,{a}^{5}b}{13\,{x}^{13}}}-{\frac{15\,{a}^{4}{b}^{2}}{11\,{x}^{11}}}-{\frac{20\,{a}^{3}{b}^{3}}{9\,{x}^{9}}}-{\frac{15\,{a}^{2}{b}^{4}}{7\,{x}^{7}}}-{\frac{6\,a{b}^{5}}{5\,{x}^{5}}}-{\frac{{b}^{6}}{3\,{x}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01108, size = 95, normalized size = 1.16 \begin{align*} -\frac{15015 \, b^{6} x^{12} + 54054 \, a b^{5} x^{10} + 96525 \, a^{2} b^{4} x^{8} + 100100 \, a^{3} b^{3} x^{6} + 61425 \, a^{4} b^{2} x^{4} + 20790 \, a^{5} b x^{2} + 3003 \, a^{6}}{45045 \, x^{15}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.6635, size = 190, normalized size = 2.32 \begin{align*} -\frac{15015 \, b^{6} x^{12} + 54054 \, a b^{5} x^{10} + 96525 \, a^{2} b^{4} x^{8} + 100100 \, a^{3} b^{3} x^{6} + 61425 \, a^{4} b^{2} x^{4} + 20790 \, a^{5} b x^{2} + 3003 \, a^{6}}{45045 \, x^{15}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.750857, size = 75, normalized size = 0.91 \begin{align*} - \frac{3003 a^{6} + 20790 a^{5} b x^{2} + 61425 a^{4} b^{2} x^{4} + 100100 a^{3} b^{3} x^{6} + 96525 a^{2} b^{4} x^{8} + 54054 a b^{5} x^{10} + 15015 b^{6} x^{12}}{45045 x^{15}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18698, size = 95, normalized size = 1.16 \begin{align*} -\frac{15015 \, b^{6} x^{12} + 54054 \, a b^{5} x^{10} + 96525 \, a^{2} b^{4} x^{8} + 100100 \, a^{3} b^{3} x^{6} + 61425 \, a^{4} b^{2} x^{4} + 20790 \, a^{5} b x^{2} + 3003 \, a^{6}}{45045 \, x^{15}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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