3.756 \(\int (\frac{2 x}{\sqrt [3]{e^x+x}}+\frac{2 e^x x}{\sqrt [3]{e^x+x}}+3 (e^x+x)^{2/3}) \, dx\)

Optimal. Leaf size=12 \[ 3 x \left (x+e^x\right )^{2/3} \]

[Out]

3*x*(E^x + x)^(2/3)

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Rubi [A]  time = 0.143765, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.051, Rules used = {2273, 2262} \[ 3 x \left (x+e^x\right )^{2/3} \]

Antiderivative was successfully verified.

[In]

Int[(2*x)/(E^x + x)^(1/3) + (2*E^x*x)/(E^x + x)^(1/3) + 3*(E^x + x)^(2/3),x]

[Out]

3*x*(E^x + x)^(2/3)

Rule 2273

Int[(x_)^(m_.)*(E^(x_) + (x_)^(m_.))^(n_), x_Symbol] :> -Simp[(E^x + x^m)^(n + 1)/(n + 1), x] + (Dist[m, Int[x
^(m - 1)*(E^x + x^m)^n, x], x] + Int[(E^x + x^m)^(n + 1), x]) /; RationalQ[m, n] && GtQ[m, 0] && LtQ[n, 0] &&
NeQ[n, -1]

Rule 2262

Int[(F_)^((e_.)*((c_.) + (d_.)*(x_)))*(x_)^(m_.)*((b_.)*(F_)^((e_.)*((c_.) + (d_.)*(x_))) + (a_.)*(x_)^(n_.))^
(p_.), x_Symbol] :> Simp[(x^m*(a*x^n + b*F^(e*(c + d*x)))^(p + 1))/(b*d*e*(p + 1)*Log[F]), x] + (-Dist[m/(b*d*
e*(p + 1)*Log[F]), Int[x^(m - 1)*(a*x^n + b*F^(e*(c + d*x)))^(p + 1), x], x] - Dist[(a*n)/(b*d*e*Log[F]), Int[
x^(m + n - 1)*(a*x^n + b*F^(e*(c + d*x)))^p, x], x]) /; FreeQ[{F, a, b, c, d, e, m, n, p}, x] && NeQ[p, -1]

Rubi steps

\begin{align*} \int \left (\frac{2 x}{\sqrt [3]{e^x+x}}+\frac{2 e^x x}{\sqrt [3]{e^x+x}}+3 \left (e^x+x\right )^{2/3}\right ) \, dx &=2 \int \frac{x}{\sqrt [3]{e^x+x}} \, dx+2 \int \frac{e^x x}{\sqrt [3]{e^x+x}} \, dx+3 \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{1}{\sqrt [3]{e^x+x}} \, dx-2 \int \frac{x}{\sqrt [3]{e^x+x}} \, dx+2 \int \left (e^x+x\right )^{2/3} \, dx\\ &=3 x \left (e^x+x\right )^{2/3}\\ \end{align*}

Mathematica [A]  time = 0.0470589, size = 12, normalized size = 1. \[ 3 x \left (x+e^x\right )^{2/3} \]

Antiderivative was successfully verified.

[In]

Integrate[(2*x)/(E^x + x)^(1/3) + (2*E^x*x)/(E^x + x)^(1/3) + 3*(E^x + x)^(2/3),x]

[Out]

3*x*(E^x + x)^(2/3)

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Maple [A]  time = 0.043, 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]

int(2*x/(exp(x)+x)^(1/3)+2*exp(x)*x/(exp(x)+x)^(1/3)+3*(exp(x)+x)^(2/3),x)

[Out]

3*x*(exp(x)+x)^(2/3)

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Maxima [A]  time = 1.1051, 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]

integrate(2*x/(x+exp(x))^(1/3)+2*exp(x)*x/(x+exp(x))^(1/3)+3*(x+exp(x))^(2/3),x, algorithm="maxima")

[Out]

3*(x^2 + x*e^x)/(x + e^x)^(1/3)

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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]

integrate(2*x/(x+exp(x))^(1/3)+2*exp(x)*x/(x+exp(x))^(1/3)+3*(x+exp(x))^(2/3),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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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]

integrate(2*x/(exp(x)+x)**(1/3)+2*exp(x)*x/(exp(x)+x)**(1/3)+3*(exp(x)+x)**(2/3),x)

[Out]

Integral((2*x*exp(x) + 5*x + 3*exp(x))/(x + exp(x))**(1/3), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{2 \, x e^{x}}{{\left (x + e^{x}\right )}^{\frac{1}{3}}} + 3 \,{\left (x + e^{x}\right )}^{\frac{2}{3}} + \frac{2 \, x}{{\left (x + e^{x}\right )}^{\frac{1}{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2*x/(x+exp(x))^(1/3)+2*exp(x)*x/(x+exp(x))^(1/3)+3*(x+exp(x))^(2/3),x, algorithm="giac")

[Out]

integrate(2*x*e^x/(x + e^x)^(1/3) + 3*(x + e^x)^(2/3) + 2*x/(x + e^x)^(1/3), x)