Optimal. Leaf size=65 \[ \frac{1}{5} x^5 (a C+A b)+\frac{1}{3} a A x^3+\frac{1}{6} x^6 (a D+b B)+\frac{1}{4} a B x^4+\frac{1}{7} b C x^7+\frac{1}{8} b D x^8 \]
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Rubi [A] time = 0.0764878, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.038, Rules used = {1802} \[ \frac{1}{5} x^5 (a C+A b)+\frac{1}{3} a A x^3+\frac{1}{6} x^6 (a D+b B)+\frac{1}{4} a B x^4+\frac{1}{7} b C x^7+\frac{1}{8} b D x^8 \]
Antiderivative was successfully verified.
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Rule 1802
Rubi steps
\begin{align*} \int x^2 \left (a+b x^2\right ) \left (A+B x+C x^2+D x^3\right ) \, dx &=\int \left (a A x^2+a B x^3+(A b+a C) x^4+(b B+a D) x^5+b C x^6+b D x^7\right ) \, dx\\ &=\frac{1}{3} a A x^3+\frac{1}{4} a B x^4+\frac{1}{5} (A b+a C) x^5+\frac{1}{6} (b B+a D) x^6+\frac{1}{7} b C x^7+\frac{1}{8} b D x^8\\ \end{align*}
Mathematica [A] time = 0.0155026, size = 65, normalized size = 1. \[ \frac{1}{5} x^5 (a C+A b)+\frac{1}{3} a A x^3+\frac{1}{6} x^6 (a D+b B)+\frac{1}{4} a B x^4+\frac{1}{7} b C x^7+\frac{1}{8} b D x^8 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 54, normalized size = 0.8 \begin{align*}{\frac{aA{x}^{3}}{3}}+{\frac{aB{x}^{4}}{4}}+{\frac{ \left ( Ab+aC \right ){x}^{5}}{5}}+{\frac{ \left ( Bb+aD \right ){x}^{6}}{6}}+{\frac{bC{x}^{7}}{7}}+{\frac{bD{x}^{8}}{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.05344, size = 72, normalized size = 1.11 \begin{align*} \frac{1}{8} \, D b x^{8} + \frac{1}{7} \, C b x^{7} + \frac{1}{6} \,{\left (D a + B b\right )} x^{6} + \frac{1}{4} \, B a x^{4} + \frac{1}{5} \,{\left (C a + A b\right )} x^{5} + \frac{1}{3} \, A a x^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.23049, size = 150, normalized size = 2.31 \begin{align*} \frac{1}{8} x^{8} b D + \frac{1}{7} x^{7} b C + \frac{1}{6} x^{6} a D + \frac{1}{6} x^{6} b B + \frac{1}{5} x^{5} a C + \frac{1}{5} x^{5} b A + \frac{1}{4} x^{4} a B + \frac{1}{3} x^{3} a A \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.06503, size = 60, normalized size = 0.92 \begin{align*} \frac{A a x^{3}}{3} + \frac{B a x^{4}}{4} + \frac{C b x^{7}}{7} + \frac{D b x^{8}}{8} + x^{6} \left (\frac{B b}{6} + \frac{D a}{6}\right ) + x^{5} \left (\frac{A b}{5} + \frac{C a}{5}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.53007, size = 77, normalized size = 1.18 \begin{align*} \frac{1}{8} \, D b x^{8} + \frac{1}{7} \, C b x^{7} + \frac{1}{6} \, D a x^{6} + \frac{1}{6} \, B b x^{6} + \frac{1}{5} \, C a x^{5} + \frac{1}{5} \, A b x^{5} + \frac{1}{4} \, B a x^{4} + \frac{1}{3} \, A a x^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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