Optimal. Leaf size=12 \[ 3 x \left (x+e^x\right )^{2/3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.336884, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {6742, 2273, 2261, 2262} \[ 3 x \left (x+e^x\right )^{2/3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6742
Rule 2273
Rule 2261
Rule 2262
Rubi steps
\begin{align*} \int \frac{5 x+e^x (3+2 x)}{\sqrt [3]{e^x+x}} \, dx &=\int \left (\frac{5 x}{\sqrt [3]{e^x+x}}+\frac{e^x (3+2 x)}{\sqrt [3]{e^x+x}}\right ) \, dx\\ &=5 \int \frac{x}{\sqrt [3]{e^x+x}} \, dx+\int \frac{e^x (3+2 x)}{\sqrt [3]{e^x+x}} \, dx\\ &=-\frac{15}{2} \left (e^x+x\right )^{2/3}+5 \int \frac{1}{\sqrt [3]{e^x+x}} \, dx+5 \int \left (e^x+x\right )^{2/3} \, dx+\int \left (\frac{3 e^x}{\sqrt [3]{e^x+x}}+\frac{2 e^x x}{\sqrt [3]{e^x+x}}\right ) \, dx\\ &=-\frac{15}{2} \left (e^x+x\right )^{2/3}+2 \int \frac{e^x x}{\sqrt [3]{e^x+x}} \, dx+3 \int \frac{e^x}{\sqrt [3]{e^x+x}} \, dx+5 \int \frac{1}{\sqrt [3]{e^x+x}} \, dx+5 \int \left (e^x+x\right )^{2/3} \, dx\\ &=-3 \left (e^x+x\right )^{2/3}+3 x \left (e^x+x\right )^{2/3}-2 \int \frac{x}{\sqrt [3]{e^x+x}} \, dx-3 \int \frac{1}{\sqrt [3]{e^x+x}} \, dx-3 \int \left (e^x+x\right )^{2/3} \, dx+5 \int \frac{1}{\sqrt [3]{e^x+x}} \, dx+5 \int \left (e^x+x\right )^{2/3} \, dx\\ &=3 x \left (e^x+x\right )^{2/3}-2 \int \frac{1}{\sqrt [3]{e^x+x}} \, dx-2 \int \left (e^x+x\right )^{2/3} \, dx-3 \int \frac{1}{\sqrt [3]{e^x+x}} \, dx-3 \int \left (e^x+x\right )^{2/3} \, dx+5 \int \frac{1}{\sqrt [3]{e^x+x}} \, dx+5 \int \left (e^x+x\right )^{2/3} \, dx\\ \end{align*}
Mathematica [A] time = 0.0917841, size = 12, normalized size = 1. \[ 3 x \left (x+e^x\right )^{2/3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.029, size = 10, normalized size = 0.8 \begin{align*} 3\,x \left ({{\rm e}^{x}}+x \right ) ^{2/3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.08398, size = 22, normalized size = 1.83 \begin{align*} \frac{3 \,{\left (x^{2} + x e^{x}\right )}}{{\left (x + e^{x}\right )}^{\frac{1}{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{2 x e^{x} + 5 x + 3 e^{x}}{\sqrt [3]{x + e^{x}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (2 \, x + 3\right )} e^{x} + 5 \, x}{{\left (x + e^{x}\right )}^{\frac{1}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]