Optimal. Leaf size=50 \[ \frac{\left (a x+\frac{b x^2}{2}+\frac{c x^3}{3}\right )^{n+1}}{n+1}+a x+\frac{b x^2}{2}+\frac{c x^3}{3} \]
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Rubi [A] time = 0.0091307, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.029, Rules used = {1591} \[ \frac{\left (a x+\frac{b x^2}{2}+\frac{c x^3}{3}\right )^{n+1}}{n+1}+a x+\frac{b x^2}{2}+\frac{c x^3}{3} \]
Antiderivative was successfully verified.
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Rule 1591
Rubi steps
\begin{align*} \int \left (a+b x+c x^2\right ) \left (1+\left (a x+\frac{b x^2}{2}+\frac{c x^3}{3}\right )^n\right ) \, dx &=\operatorname{Subst}\left (\int \left (1+x^n\right ) \, dx,x,a x+\frac{b x^2}{2}+\frac{c x^3}{3}\right )\\ &=a x+\frac{b x^2}{2}+\frac{c x^3}{3}+\frac{\left (a x+\frac{b x^2}{2}+\frac{c x^3}{3}\right )^{1+n}}{1+n}\\ \end{align*}
Mathematica [A] time = 0.187863, size = 49, normalized size = 0.98 \[ \frac{x (6 a+x (3 b+2 c x)) \left (\left (a x+\frac{b x^2}{2}+\frac{c x^3}{3}\right )^n+n+1\right )}{6 (n+1)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 43, normalized size = 0.9 \begin{align*} ax+{\frac{b{x}^{2}}{2}}+{\frac{c{x}^{3}}{3}}+{\frac{1}{1+n} \left ( ax+{\frac{b{x}^{2}}{2}}+{\frac{c{x}^{3}}{3}} \right ) ^{1+n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.7359, size = 112, normalized size = 2.24 \begin{align*} \frac{1}{3} \, c x^{3} + \frac{1}{2} \, b x^{2} + a x + \frac{{\left (2 \, c x^{3} + 3 \, b x^{2} + 6 \, a x\right )} e^{\left (n \log \left (2 \, c x^{2} + 3 \, b x + 6 \, a\right ) + n \log \left (x\right )\right )}}{3^{n + 1} 2^{n + 1} n + 3^{n + 1} 2^{n + 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.37342, size = 171, normalized size = 3.42 \begin{align*} \frac{2 \,{\left (c n + c\right )} x^{3} + 3 \,{\left (b n + b\right )} x^{2} +{\left (2 \, c x^{3} + 3 \, b x^{2} + 6 \, a x\right )}{\left (\frac{1}{3} \, c x^{3} + \frac{1}{2} \, b x^{2} + a x\right )}^{n} + 6 \,{\left (a n + a\right )} x}{6 \,{\left (n + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19638, size = 57, normalized size = 1.14 \begin{align*} \frac{1}{3} \, c x^{3} + \frac{1}{2} \, b x^{2} + a x + \frac{{\left (\frac{1}{3} \, c x^{3} + \frac{1}{2} \, b x^{2} + a x\right )}^{n + 1}}{n + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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