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

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 = 0.50272, 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 \[ 3 x \left (x+e^x\right )^{2/3} \]

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)

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{5 x + \left (2 x + 3\right ) e^{x}}{\sqrt [3]{x + e^{x}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((5*x+exp(x)*(3+2*x))/(exp(x)+x)**(1/3),x)

[Out]

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

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Mathematica [A]  time = 0.0249667, 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 = 0.014, size = 10, normalized size = 0.8 \[ 3\,x \left ({{\rm e}^{x}}+x \right ) ^{2/3} \]

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 = 0.82782, size = 22, normalized size = 1.83 \[ \frac{3 \,{\left (x^{2} + x e^{x}\right )}}{{\left (x + e^{x}\right )}^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((2*x + 3)*e^x + 5*x)/(x + e^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. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((2*x + 3)*e^x + 5*x)/(x + e^x)^(1/3),x, algorithm="fricas")

[Out]

Exception raised: TypeError

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

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/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (2 \, x + 3\right )} e^{x} + 5 \, x}{{\left (x + e^{x}\right )}^{\frac{1}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((2*x + 3)*e^x + 5*x)/(x + e^x)^(1/3),x, algorithm="giac")

[Out]

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