Optimal. Leaf size=82 \[ \frac{15}{11} a^2 b^4 x^{11}+\frac{20}{9} a^3 b^3 x^9+\frac{15}{7} a^4 b^2 x^7+\frac{6}{5} a^5 b x^5+\frac{a^6 x^3}{3}+\frac{6}{13} a b^5 x^{13}+\frac{b^6 x^{15}}{15} \]
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Rubi [A] time = 0.0384399, 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} \[ \frac{15}{11} a^2 b^4 x^{11}+\frac{20}{9} a^3 b^3 x^9+\frac{15}{7} a^4 b^2 x^7+\frac{6}{5} a^5 b x^5+\frac{a^6 x^3}{3}+\frac{6}{13} a b^5 x^{13}+\frac{b^6 x^{15}}{15} \]
Antiderivative was successfully verified.
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Rule 28
Rule 270
Rubi steps
\begin{align*} \int x^2 \left (a^2+2 a b x^2+b^2 x^4\right )^3 \, dx &=\frac{\int x^2 \left (a b+b^2 x^2\right )^6 \, dx}{b^6}\\ &=\frac{\int \left (a^6 b^6 x^2+6 a^5 b^7 x^4+15 a^4 b^8 x^6+20 a^3 b^9 x^8+15 a^2 b^{10} x^{10}+6 a b^{11} x^{12}+b^{12} x^{14}\right ) \, dx}{b^6}\\ &=\frac{a^6 x^3}{3}+\frac{6}{5} a^5 b x^5+\frac{15}{7} a^4 b^2 x^7+\frac{20}{9} a^3 b^3 x^9+\frac{15}{11} a^2 b^4 x^{11}+\frac{6}{13} a b^5 x^{13}+\frac{b^6 x^{15}}{15}\\ \end{align*}
Mathematica [A] time = 0.0026024, size = 82, normalized size = 1. \[ \frac{15}{11} a^2 b^4 x^{11}+\frac{20}{9} a^3 b^3 x^9+\frac{15}{7} a^4 b^2 x^7+\frac{6}{5} a^5 b x^5+\frac{a^6 x^3}{3}+\frac{6}{13} a b^5 x^{13}+\frac{b^6 x^{15}}{15} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.041, size = 69, normalized size = 0.8 \begin{align*}{\frac{{a}^{6}{x}^{3}}{3}}+{\frac{6\,{a}^{5}b{x}^{5}}{5}}+{\frac{15\,{a}^{4}{b}^{2}{x}^{7}}{7}}+{\frac{20\,{a}^{3}{b}^{3}{x}^{9}}{9}}+{\frac{15\,{a}^{2}{b}^{4}{x}^{11}}{11}}+{\frac{6\,a{b}^{5}{x}^{13}}{13}}+{\frac{{b}^{6}{x}^{15}}{15}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.983176, size = 92, normalized size = 1.12 \begin{align*} \frac{1}{15} \, b^{6} x^{15} + \frac{6}{13} \, a b^{5} x^{13} + \frac{15}{11} \, a^{2} b^{4} x^{11} + \frac{20}{9} \, a^{3} b^{3} x^{9} + \frac{15}{7} \, a^{4} b^{2} x^{7} + \frac{6}{5} \, a^{5} b x^{5} + \frac{1}{3} \, a^{6} x^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.43171, size = 165, normalized size = 2.01 \begin{align*} \frac{1}{15} x^{15} b^{6} + \frac{6}{13} x^{13} b^{5} a + \frac{15}{11} x^{11} b^{4} a^{2} + \frac{20}{9} x^{9} b^{3} a^{3} + \frac{15}{7} x^{7} b^{2} a^{4} + \frac{6}{5} x^{5} b a^{5} + \frac{1}{3} x^{3} a^{6} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.078218, size = 80, normalized size = 0.98 \begin{align*} \frac{a^{6} x^{3}}{3} + \frac{6 a^{5} b x^{5}}{5} + \frac{15 a^{4} b^{2} x^{7}}{7} + \frac{20 a^{3} b^{3} x^{9}}{9} + \frac{15 a^{2} b^{4} x^{11}}{11} + \frac{6 a b^{5} x^{13}}{13} + \frac{b^{6} x^{15}}{15} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17251, size = 92, normalized size = 1.12 \begin{align*} \frac{1}{15} \, b^{6} x^{15} + \frac{6}{13} \, a b^{5} x^{13} + \frac{15}{11} \, a^{2} b^{4} x^{11} + \frac{20}{9} \, a^{3} b^{3} x^{9} + \frac{15}{7} \, a^{4} b^{2} x^{7} + \frac{6}{5} \, a^{5} b x^{5} + \frac{1}{3} \, a^{6} x^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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