3.755 \(\int \frac{5 x+e^x (3+2 x)}{\sqrt [3]{e^x+x}} \, dx\)

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 = 0.336884, 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]

Int[(5*x + E^x*(3 + 2*x))/(E^x + x)^(1/3),x]

[Out]

3*x*(E^x + x)^(2/3)

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

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 2261

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

Integrate[(5*x + E^x*(3 + 2*x))/(E^x + x)^(1/3),x]

[Out]

3*x*(E^x + x)^(2/3)

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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]

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

[Out]

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

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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]

integrate((5*x+exp(x)*(3+2*x))/(x+exp(x))^(1/3),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((5*x+exp(x)*(3+2*x))/(x+exp(x))^(1/3),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((5*x+exp(x)*(3+2*x))/(exp(x)+x)**(1/3),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{{\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]

integrate((5*x+exp(x)*(3+2*x))/(x+exp(x))^(1/3),x, algorithm="giac")

[Out]

integrate(((2*x + 3)*e^x + 5*x)/(x + e^x)^(1/3), x)