Optimal. Leaf size=60 \[ \frac{1}{3} x^3 (a C+A b)+a A x+\frac{1}{4} x^4 (a D+b B)+\frac{1}{2} a B x^2+\frac{1}{5} b C x^5+\frac{1}{6} b D x^6 \]
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Rubi [A] time = 0.0411648, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043, Rules used = {1810} \[ \frac{1}{3} x^3 (a C+A b)+a A x+\frac{1}{4} x^4 (a D+b B)+\frac{1}{2} a B x^2+\frac{1}{5} b C x^5+\frac{1}{6} b D x^6 \]
Antiderivative was successfully verified.
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Rule 1810
Rubi steps
\begin{align*} \int \left (a+b x^2\right ) \left (A+B x+C x^2+D x^3\right ) \, dx &=\int \left (a A+a B x+(A b+a C) x^2+(b B+a D) x^3+b C x^4+b D x^5\right ) \, dx\\ &=a A x+\frac{1}{2} a B x^2+\frac{1}{3} (A b+a C) x^3+\frac{1}{4} (b B+a D) x^4+\frac{1}{5} b C x^5+\frac{1}{6} b D x^6\\ \end{align*}
Mathematica [A] time = 0.0089797, size = 60, normalized size = 1. \[ \frac{1}{3} x^3 (a C+A b)+a A x+\frac{1}{4} x^4 (a D+b B)+\frac{1}{2} a B x^2+\frac{1}{5} b C x^5+\frac{1}{6} b D x^6 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 51, normalized size = 0.9 \begin{align*} aAx+{\frac{Ba{x}^{2}}{2}}+{\frac{ \left ( Ab+aC \right ){x}^{3}}{3}}+{\frac{ \left ( Bb+aD \right ){x}^{4}}{4}}+{\frac{bC{x}^{5}}{5}}+{\frac{bD{x}^{6}}{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.03893, size = 68, normalized size = 1.13 \begin{align*} \frac{1}{6} \, D b x^{6} + \frac{1}{5} \, C b x^{5} + \frac{1}{4} \,{\left (D a + B b\right )} x^{4} + \frac{1}{2} \, B a x^{2} + \frac{1}{3} \,{\left (C a + A b\right )} x^{3} + A a x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.28853, size = 142, normalized size = 2.37 \begin{align*} \frac{1}{6} x^{6} b D + \frac{1}{5} x^{5} b C + \frac{1}{4} x^{4} a D + \frac{1}{4} x^{4} b B + \frac{1}{3} x^{3} a C + \frac{1}{3} x^{3} b A + \frac{1}{2} x^{2} a B + x a A \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.062735, size = 56, normalized size = 0.93 \begin{align*} A a x + \frac{B a x^{2}}{2} + \frac{C b x^{5}}{5} + \frac{D b x^{6}}{6} + x^{4} \left (\frac{B b}{4} + \frac{D a}{4}\right ) + x^{3} \left (\frac{A b}{3} + \frac{C a}{3}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15165, size = 73, normalized size = 1.22 \begin{align*} \frac{1}{6} \, D b x^{6} + \frac{1}{5} \, C b x^{5} + \frac{1}{4} \, D a x^{4} + \frac{1}{4} \, B b x^{4} + \frac{1}{3} \, C a x^{3} + \frac{1}{3} \, A b x^{3} + \frac{1}{2} \, B a x^{2} + A a x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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