Optimal. Leaf size=46 \[ \frac{1}{3} x^3 (a C+A c)+a A x+\frac{1}{2} a B x^2+\frac{1}{4} B c x^4+\frac{1}{5} c C x^5 \]
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Rubi [A] time = 0.028997, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {1657} \[ \frac{1}{3} x^3 (a C+A c)+a A x+\frac{1}{2} a B x^2+\frac{1}{4} B c x^4+\frac{1}{5} c C x^5 \]
Antiderivative was successfully verified.
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Rule 1657
Rubi steps
\begin{align*} \int \left (a+c x^2\right ) \left (A+B x+C x^2\right ) \, dx &=\int \left (a A+a B x+(A c+a C) x^2+B c x^3+c C x^4\right ) \, dx\\ &=a A x+\frac{1}{2} a B x^2+\frac{1}{3} (A c+a C) x^3+\frac{1}{4} B c x^4+\frac{1}{5} c C x^5\\ \end{align*}
Mathematica [A] time = 0.0124125, size = 46, normalized size = 1. \[ \frac{1}{3} x^3 (a C+A c)+a A x+\frac{1}{2} a B x^2+\frac{1}{4} B c x^4+\frac{1}{5} c C x^5 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.047, size = 39, normalized size = 0.9 \begin{align*} aAx+{\frac{aB{x}^{2}}{2}}+{\frac{ \left ( Ac+aC \right ){x}^{3}}{3}}+{\frac{Bc{x}^{4}}{4}}+{\frac{cC{x}^{5}}{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.994897, size = 51, normalized size = 1.11 \begin{align*} \frac{1}{5} \, C c x^{5} + \frac{1}{4} \, B c x^{4} + \frac{1}{2} \, B a x^{2} + \frac{1}{3} \,{\left (C a + A c\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 = 1.48862, size = 104, normalized size = 2.26 \begin{align*} \frac{1}{5} x^{5} c C + \frac{1}{4} x^{4} c B + \frac{1}{3} x^{3} a C + \frac{1}{3} x^{3} c 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 = 0.061914, size = 42, normalized size = 0.91 \begin{align*} A a x + \frac{B a x^{2}}{2} + \frac{B c x^{4}}{4} + \frac{C c x^{5}}{5} + x^{3} \left (\frac{A c}{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 = 1.1493, size = 54, normalized size = 1.17 \begin{align*} \frac{1}{5} \, C c x^{5} + \frac{1}{4} \, B c x^{4} + \frac{1}{3} \, C a x^{3} + \frac{1}{3} \, A c 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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