Optimal. Leaf size=44 \[ \frac {\left (\frac {b x^2}{2}+\frac {c x^3}{3}\right )^{n+1}}{n+1}+\frac {b x^2}{2}+\frac {c x^3}{3} \]
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Rubi [A] time = 0.01, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.032, Rules used = {1591} \[ \frac {\left (\frac {b x^2}{2}+\frac {c x^3}{3}\right )^{n+1}}{n+1}+\frac {b x^2}{2}+\frac {c x^3}{3} \]
Antiderivative was successfully verified.
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Rule 1591
Rubi steps
\begin {align*} \int \left (b x+c x^2\right ) \left (1+\left (\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,\frac {b x^2}{2}+\frac {c x^3}{3}\right )\\ &=\frac {b x^2}{2}+\frac {c x^3}{3}+\frac {\left (\frac {b x^2}{2}+\frac {c x^3}{3}\right )^{1+n}}{1+n}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 42, normalized size = 0.95 \[ \frac {x^2 (3 b+2 c x) \left (\left (\frac {b x^2}{2}+\frac {c x^3}{3}\right )^n+n+1\right )}{6 (n+1)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.64, size = 57, normalized size = 1.30 \[ \frac {2 \, {\left (c n + c\right )} x^{3} + 3 \, {\left (b n + b\right )} x^{2} + {\left (2 \, c x^{3} + 3 \, b x^{2}\right )} {\left (\frac {1}{3} \, c x^{3} + \frac {1}{2} \, b x^{2}\right )}^{n}}{6 \, {\left (n + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.39, size = 36, normalized size = 0.82 \[ \frac {1}{3} \, c x^{3} + \frac {1}{2} \, b x^{2} + \frac {{\left (\frac {1}{3} \, c x^{3} + \frac {1}{2} \, b x^{2}\right )}^{n + 1}}{n + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 37, normalized size = 0.84 \[ \frac {c \,x^{3}}{3}+\frac {b \,x^{2}}{2}+\frac {\left (\frac {1}{3} c \,x^{3}+\frac {1}{2} b \,x^{2}\right )^{n +1}}{n +1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.17, size = 71, normalized size = 1.61 \[ \frac {1}{3} \, c x^{3} + \frac {1}{2} \, b x^{2} + \frac {{\left (2 \, c x^{3} + 3 \, b x^{2}\right )} e^{\left (n \log \left (2 \, c x + 3 \, b\right ) + 2 \, n \log \relax (x)\right )}}{3^{n + 1} 2^{n + 1} n + 3^{n + 1} 2^{n + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.21, size = 37, normalized size = 0.84 \[ \frac {x^2\,\left (3\,b+2\,c\,x\right )\,\left (n+{\left (\frac {c\,x^3}{3}+\frac {b\,x^2}{2}\right )}^n+1\right )}{6\,\left (n+1\right )} \]
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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