3.8.56 \(\int (\frac {2 x}{\sqrt [3]{e^x+x}}+\frac {2 e^x x}{\sqrt [3]{e^x+x}}+3 (e^x+x)^{2/3}) \, dx\) [756]

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

[Out]

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

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Rubi [A]
time = 0.09, antiderivative size = 12, normalized size of antiderivative = 1.00, 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 = {2305, 2294} \begin {gather*} 3 x \left (x+e^x\right )^{2/3} \end {gather*}

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 2294

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]

Rule 2305

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]

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*}

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Mathematica [A]
time = 0.02, size = 12, normalized size = 1.00 \begin {gather*} 3 x \left (e^x+x\right )^{2/3} \end {gather*}

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 = 0.02, size = 10, normalized size = 0.83

method	result	size
risch	\(3 x \left ({\mathrm e}^{x}+x \right )^{\frac {2}{3}}\)	\(10\)

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,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.32, size = 16, normalized size = 1.33 \begin {gather*} \frac {3 \, {\left (x^{2} + x e^{x}\right )}}{{\left (x + e^{x}\right )}^{\frac {1}{3}}} \end {gather*}

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, algorithm="maxima")

[Out]

3*(x^2 + x*e^x)/(x + e^x)^(1/3)

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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

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, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (ha
s polynomial part)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 x e^{x} + 5 x + 3 e^{x}}{\sqrt [3]{x + e^{x}}}\, dx \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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, 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)

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Mupad [B]
time = 3.64, size = 9, normalized size = 0.75 \begin {gather*} 3\,x\,{\left (x+{\mathrm {e}}^x\right )}^{2/3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*x*(x + exp(x))^(2/3)

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