Optimal. Leaf size=35 \[ \frac {\left (a x+\frac {b x^2}{2}+c\right )^{n+1}}{n+1}+a x+\frac {b x^2}{2} \]
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Rubi [A] time = 0.01, antiderivative size = 35, normalized size of antiderivative = 1.00, 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 = {1591} \[ \frac {\left (a x+\frac {b x^2}{2}+c\right )^{n+1}}{n+1}+a x+\frac {b x^2}{2} \]
Antiderivative was successfully verified.
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Rule 1591
Rubi steps
\begin {align*} \int (a+b x) \left (1+\left (c+a x+\frac {b x^2}{2}\right )^n\right ) \, dx &=\operatorname {Subst}\left (\int \left (1+x^n\right ) \, dx,x,c+a x+\frac {b x^2}{2}\right )\\ &=a x+\frac {b x^2}{2}+\frac {\left (c+a x+\frac {b x^2}{2}\right )^{1+n}}{1+n}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 35, normalized size = 1.00 \[ \frac {\left (a x+\frac {b x^2}{2}+c\right )^{n+1}}{n+1}+a x+\frac {b x^2}{2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.95, size = 52, normalized size = 1.49 \[ \frac {{\left (b n + b\right )} x^{2} + {\left (b x^{2} + 2 \, a x + 2 \, c\right )} {\left (\frac {1}{2} \, b x^{2} + a x + c\right )}^{n} + 2 \, {\left (a n + a\right )} x}{2 \, {\left (n + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 32, normalized size = 0.91 \[ \frac {1}{2} \, b x^{2} + a x + c + \frac {{\left (\frac {1}{2} \, b x^{2} + a x + c\right )}^{n + 1}}{n + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 33, normalized size = 0.94 \[ \frac {b \,x^{2}}{2}+a x +c +\frac {\left (\frac {1}{2} b \,x^{2}+a x +c \right )^{n +1}}{n +1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.16, size = 54, normalized size = 1.54 \[ \frac {1}{2} \, b x^{2} + a x + \frac {{\left (b x^{2} + 2 \, a x + 2 \, c\right )} {\left (b x^{2} + 2 \, a x + 2 \, c\right )}^{n}}{2^{n + 1} n + 2^{n + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.11, size = 58, normalized size = 1.66 \[ a\,x+{\left (\frac {b\,x^2}{2}+a\,x+c\right )}^n\,\left (\frac {2\,c}{2\,n+2}+\frac {b\,x^2}{2\,n+2}+\frac {2\,a\,x}{2\,n+2}\right )+\frac {b\,x^2}{2} \]
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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