Optimal. Leaf size=82 \[ \frac{15}{17} a^2 b^4 x^{17}+\frac{4}{3} a^3 b^3 x^{15}+\frac{15}{13} a^4 b^2 x^{13}+\frac{6}{11} a^5 b x^{11}+\frac{a^6 x^9}{9}+\frac{6}{19} a b^5 x^{19}+\frac{b^6 x^{21}}{21} \]
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Rubi [A] time = 0.0456947, 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} \[ \frac{15}{17} a^2 b^4 x^{17}+\frac{4}{3} a^3 b^3 x^{15}+\frac{15}{13} a^4 b^2 x^{13}+\frac{6}{11} a^5 b x^{11}+\frac{a^6 x^9}{9}+\frac{6}{19} a b^5 x^{19}+\frac{b^6 x^{21}}{21} \]
Antiderivative was successfully verified.
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Rule 28
Rule 270
Rubi steps
\begin{align*} \int x^8 \left (a^2+2 a b x^2+b^2 x^4\right )^3 \, dx &=\frac{\int x^8 \left (a b+b^2 x^2\right )^6 \, dx}{b^6}\\ &=\frac{\int \left (a^6 b^6 x^8+6 a^5 b^7 x^{10}+15 a^4 b^8 x^{12}+20 a^3 b^9 x^{14}+15 a^2 b^{10} x^{16}+6 a b^{11} x^{18}+b^{12} x^{20}\right ) \, dx}{b^6}\\ &=\frac{a^6 x^9}{9}+\frac{6}{11} a^5 b x^{11}+\frac{15}{13} a^4 b^2 x^{13}+\frac{4}{3} a^3 b^3 x^{15}+\frac{15}{17} a^2 b^4 x^{17}+\frac{6}{19} a b^5 x^{19}+\frac{b^6 x^{21}}{21}\\ \end{align*}
Mathematica [A] time = 0.0031235, size = 82, normalized size = 1. \[ \frac{15}{17} a^2 b^4 x^{17}+\frac{4}{3} a^3 b^3 x^{15}+\frac{15}{13} a^4 b^2 x^{13}+\frac{6}{11} a^5 b x^{11}+\frac{a^6 x^9}{9}+\frac{6}{19} a b^5 x^{19}+\frac{b^6 x^{21}}{21} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.042, size = 69, normalized size = 0.8 \begin{align*}{\frac{{a}^{6}{x}^{9}}{9}}+{\frac{6\,{a}^{5}b{x}^{11}}{11}}+{\frac{15\,{a}^{4}{b}^{2}{x}^{13}}{13}}+{\frac{4\,{a}^{3}{b}^{3}{x}^{15}}{3}}+{\frac{15\,{a}^{2}{b}^{4}{x}^{17}}{17}}+{\frac{6\,a{b}^{5}{x}^{19}}{19}}+{\frac{{b}^{6}{x}^{21}}{21}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.980238, size = 92, normalized size = 1.12 \begin{align*} \frac{1}{21} \, b^{6} x^{21} + \frac{6}{19} \, a b^{5} x^{19} + \frac{15}{17} \, a^{2} b^{4} x^{17} + \frac{4}{3} \, a^{3} b^{3} x^{15} + \frac{15}{13} \, a^{4} b^{2} x^{13} + \frac{6}{11} \, a^{5} b x^{11} + \frac{1}{9} \, a^{6} x^{9} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.72178, size = 170, normalized size = 2.07 \begin{align*} \frac{1}{21} x^{21} b^{6} + \frac{6}{19} x^{19} b^{5} a + \frac{15}{17} x^{17} b^{4} a^{2} + \frac{4}{3} x^{15} b^{3} a^{3} + \frac{15}{13} x^{13} b^{2} a^{4} + \frac{6}{11} x^{11} b a^{5} + \frac{1}{9} x^{9} a^{6} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.082118, size = 80, normalized size = 0.98 \begin{align*} \frac{a^{6} x^{9}}{9} + \frac{6 a^{5} b x^{11}}{11} + \frac{15 a^{4} b^{2} x^{13}}{13} + \frac{4 a^{3} b^{3} x^{15}}{3} + \frac{15 a^{2} b^{4} x^{17}}{17} + \frac{6 a b^{5} x^{19}}{19} + \frac{b^{6} x^{21}}{21} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17663, size = 92, normalized size = 1.12 \begin{align*} \frac{1}{21} \, b^{6} x^{21} + \frac{6}{19} \, a b^{5} x^{19} + \frac{15}{17} \, a^{2} b^{4} x^{17} + \frac{4}{3} \, a^{3} b^{3} x^{15} + \frac{15}{13} \, a^{4} b^{2} x^{13} + \frac{6}{11} \, a^{5} b x^{11} + \frac{1}{9} \, a^{6} x^{9} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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