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

Optimal. Leaf size=17 \[ 3 \left (x+e^x\right )^{2/3} x+3 \log (x) \]

[Out]

3*x*(E^x + x)^(2/3) + 3*Log[x]

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Rubi [A]  time = 1.01534, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.093 \[ 3 \left (x+e^x\right )^{2/3} x+3 \log (x) \]

Antiderivative was successfully verified.

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

[Out]

3*x*(E^x + x)^(2/3) + 3*Log[x]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.123917, size = 17, normalized size = 1. \[ 3 \left (x+e^x\right )^{2/3} x+3 \log (x) \]

Antiderivative was successfully verified.

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

[Out]

3*x*(E^x + x)^(2/3) + 3*Log[x]

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Maple [F]  time = 0.024, size = 0, normalized size = 0. \[ \int{\frac{1}{x} \left ( 5\,{x}^{2}+3\,\sqrt [3]{{{\rm e}^{x}}+x}+{{\rm e}^{x}} \left ( 2\,{x}^{2}+3\,x \right ) \right ){\frac{1}{\sqrt [3]{{{\rm e}^{x}}+x}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.51708, size = 28, normalized size = 1.65 \[ \frac{3 \,{\left (x^{2} + x e^{x}\right )}}{{\left (x + e^{x}\right )}^{\frac{1}{3}}} + 3 \, \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*(x^2 + x*e^x)/(x + e^x)^(1/3) + 3*log(x)

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

[Out]

Exception raised: TypeError

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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