Optimal. Leaf size=56 \[ \frac{6}{7} a^2 b^2 x^7+\frac{4}{5} a^3 b x^5+\frac{a^4 x^3}{3}+\frac{4}{9} a b^3 x^9+\frac{b^4 x^{11}}{11} \]
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Rubi [A] time = 0.0288531, antiderivative size = 56, 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} \[ \frac{6}{7} a^2 b^2 x^7+\frac{4}{5} a^3 b x^5+\frac{a^4 x^3}{3}+\frac{4}{9} a b^3 x^9+\frac{b^4 x^{11}}{11} \]
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 )^2 \, dx &=\frac{\int x^2 \left (a b+b^2 x^2\right )^4 \, dx}{b^4}\\ &=\frac{\int \left (a^4 b^4 x^2+4 a^3 b^5 x^4+6 a^2 b^6 x^6+4 a b^7 x^8+b^8 x^{10}\right ) \, dx}{b^4}\\ &=\frac{a^4 x^3}{3}+\frac{4}{5} a^3 b x^5+\frac{6}{7} a^2 b^2 x^7+\frac{4}{9} a b^3 x^9+\frac{b^4 x^{11}}{11}\\ \end{align*}
Mathematica [A] time = 0.0021344, size = 56, normalized size = 1. \[ \frac{6}{7} a^2 b^2 x^7+\frac{4}{5} a^3 b x^5+\frac{a^4 x^3}{3}+\frac{4}{9} a b^3 x^9+\frac{b^4 x^{11}}{11} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.04, size = 47, normalized size = 0.8 \begin{align*}{\frac{{a}^{4}{x}^{3}}{3}}+{\frac{4\,{a}^{3}b{x}^{5}}{5}}+{\frac{6\,{a}^{2}{b}^{2}{x}^{7}}{7}}+{\frac{4\,a{b}^{3}{x}^{9}}{9}}+{\frac{{b}^{4}{x}^{11}}{11}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.989445, size = 62, normalized size = 1.11 \begin{align*} \frac{1}{11} \, b^{4} x^{11} + \frac{4}{9} \, a b^{3} x^{9} + \frac{6}{7} \, a^{2} b^{2} x^{7} + \frac{4}{5} \, a^{3} b x^{5} + \frac{1}{3} \, a^{4} x^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.51636, size = 107, normalized size = 1.91 \begin{align*} \frac{1}{11} x^{11} b^{4} + \frac{4}{9} x^{9} b^{3} a + \frac{6}{7} x^{7} b^{2} a^{2} + \frac{4}{5} x^{5} b a^{3} + \frac{1}{3} x^{3} a^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.070825, size = 53, normalized size = 0.95 \begin{align*} \frac{a^{4} x^{3}}{3} + \frac{4 a^{3} b x^{5}}{5} + \frac{6 a^{2} b^{2} x^{7}}{7} + \frac{4 a b^{3} x^{9}}{9} + \frac{b^{4} x^{11}}{11} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12835, size = 62, normalized size = 1.11 \begin{align*} \frac{1}{11} \, b^{4} x^{11} + \frac{4}{9} \, a b^{3} x^{9} + \frac{6}{7} \, a^{2} b^{2} x^{7} + \frac{4}{5} \, a^{3} b x^{5} + \frac{1}{3} \, a^{4} x^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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