Optimal. Leaf size=44 \[ \frac{\left (\frac{b x^2}{2}+\frac{c x^3}{3}\right )^{n+1}}{n+1}+\frac{b x^2}{2}+\frac{c x^3}{3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0095982, antiderivative size = 44, normalized size of antiderivative = 1., 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.
[In]
[Out]
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*}
Mathematica [A] time = 0.0828407, 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.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.003, size = 37, normalized size = 0.8 \begin{align*}{\frac{b{x}^{2}}{2}}+{\frac{c{x}^{3}}{3}}+{\frac{1}{1+n} \left ({\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.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.70073, size = 96, normalized size = 2.18 \begin{align*} \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 \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.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.39462, size = 131, normalized size = 2.98 \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}\right )}{\left (\frac{1}{3} \, c x^{3} + \frac{1}{2} \, b x^{2}\right )}^{n}}{6 \,{\left (n + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.14951, size = 49, normalized size = 1.11 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]