Optimal. Leaf size=72 \[ a^2 x+\frac{2}{5} (64-a) x^5-2 (16-a) x^4+\frac{16}{3} (4-a) x^3+8 a x^2+\frac{x^9}{9}-x^8+\frac{32 x^7}{7}-\frac{40 x^6}{3} \]
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Rubi [A] time = 0.0297305, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {2061} \[ a^2 x+\frac{2}{5} (64-a) x^5-2 (16-a) x^4+\frac{16}{3} (4-a) x^3+8 a x^2+\frac{x^9}{9}-x^8+\frac{32 x^7}{7}-\frac{40 x^6}{3} \]
Antiderivative was successfully verified.
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Rule 2061
Rubi steps
\begin{align*} \int \left (a+8 x-8 x^2+4 x^3-x^4\right )^2 \, dx &=\int \left (a^2+16 a x+16 (4-a) x^2-8 (16-a) x^3+2 (64-a) x^4-80 x^5+32 x^6-8 x^7+x^8\right ) \, dx\\ &=a^2 x+8 a x^2+\frac{16}{3} (4-a) x^3-2 (16-a) x^4+\frac{2}{5} (64-a) x^5-\frac{40 x^6}{3}+\frac{32 x^7}{7}-x^8+\frac{x^9}{9}\\ \end{align*}
Mathematica [A] time = 0.0070781, size = 66, normalized size = 0.92 \[ a^2 x-\frac{2}{5} (a-64) x^5+2 (a-16) x^4-\frac{16}{3} (a-4) x^3+8 a x^2+\frac{x^9}{9}-x^8+\frac{32 x^7}{7}-\frac{40 x^6}{3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 63, normalized size = 0.9 \begin{align*}{\frac{{x}^{9}}{9}}-{x}^{8}+{\frac{32\,{x}^{7}}{7}}-{\frac{40\,{x}^{6}}{3}}+{\frac{ \left ( -2\,a+128 \right ){x}^{5}}{5}}+{\frac{ \left ( 8\,a-128 \right ){x}^{4}}{4}}+{\frac{ \left ( -16\,a+64 \right ){x}^{3}}{3}}+8\,a{x}^{2}+x{a}^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.08178, size = 88, normalized size = 1.22 \begin{align*} \frac{1}{9} \, x^{9} - x^{8} + \frac{32}{7} \, x^{7} - \frac{40}{3} \, x^{6} + \frac{128}{5} \, x^{5} - 32 \, x^{4} + a^{2} x + \frac{64}{3} \, x^{3} - \frac{2}{15} \,{\left (3 \, x^{5} - 15 \, x^{4} + 40 \, x^{3} - 60 \, x^{2}\right )} a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.33987, size = 165, normalized size = 2.29 \begin{align*} \frac{1}{9} x^{9} - x^{8} + \frac{32}{7} x^{7} - \frac{40}{3} x^{6} - \frac{2}{5} x^{5} a + \frac{128}{5} x^{5} + 2 x^{4} a - 32 x^{4} - \frac{16}{3} x^{3} a + \frac{64}{3} x^{3} + 8 x^{2} a + x a^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.068419, size = 65, normalized size = 0.9 \begin{align*} a^{2} x + 8 a x^{2} + \frac{x^{9}}{9} - x^{8} + \frac{32 x^{7}}{7} - \frac{40 x^{6}}{3} + x^{5} \left (\frac{128}{5} - \frac{2 a}{5}\right ) + x^{4} \left (2 a - 32\right ) + x^{3} \left (\frac{64}{3} - \frac{16 a}{3}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11347, size = 88, normalized size = 1.22 \begin{align*} \frac{1}{9} \, x^{9} - x^{8} + \frac{32}{7} \, x^{7} - \frac{2}{5} \, a x^{5} - \frac{40}{3} \, x^{6} + 2 \, a x^{4} + \frac{128}{5} \, x^{5} - \frac{16}{3} \, a x^{3} - 32 \, x^{4} + a^{2} x + 8 \, a x^{2} + \frac{64}{3} \, x^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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