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 = 0.01, antiderivative size = 50, normalized size of antiderivative = 1.00, 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*}
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Mathematica [A] time = 0.19, 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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fricas [A] time = 0.83, size = 72, normalized size = 1.44 \[ \frac {2 \, {\left (c n + c\right )} x^{3} + 3 \, {\left (b n + b\right )} x^{2} + {\left (2 \, c x^{3} + 3 \, b x^{2} + 6 \, a x\right )} {\left (\frac {1}{3} \, c x^{3} + \frac {1}{2} \, b x^{2} + a x\right )}^{n} + 6 \, {\left (a n + a\right )} x}{6 \, {\left (n + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.39, size = 42, normalized size = 0.84 \[ \frac {1}{3} \, c x^{3} + \frac {1}{2} \, b x^{2} + a x + \frac {{\left (\frac {1}{3} \, c x^{3} + \frac {1}{2} \, b x^{2} + a x\right )}^{n + 1}}{n + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 43, normalized size = 0.86 \[ \frac {c \,x^{3}}{3}+\frac {b \,x^{2}}{2}+a x +\frac {\left (\frac {1}{3} c \,x^{3}+\frac {1}{2} b \,x^{2}+a x \right )^{n +1}}{n +1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.22, size = 83, normalized size = 1.66 \[ \frac {1}{3} \, c x^{3} + \frac {1}{2} \, b x^{2} + a x + \frac {{\left (2 \, c x^{3} + 3 \, b x^{2} + 6 \, a x\right )} e^{\left (n \log \left (2 \, c x^{2} + 3 \, b x + 6 \, a\right ) + n \log \relax (x)\right )}}{3^{n + 1} 2^{n + 1} n + 3^{n + 1} 2^{n + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.17, size = 73, normalized size = 1.46 \[ a\,x+\left (\frac {3\,b\,x^2}{6\,n+6}+\frac {2\,c\,x^3}{6\,n+6}+\frac {6\,a\,x}{6\,n+6}\right )\,{\left (\frac {c\,x^3}{3}+\frac {b\,x^2}{2}+a\,x\right )}^n+\frac {b\,x^2}{2}+\frac {c\,x^3}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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