3.45 \(\int \frac{e^x x}{(1+x)^2} \, dx\)

Optimal. Leaf size=9 \[ \frac{e^x}{x+1} \]

[Out]

E^x/(1 + x)

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Rubi [A]  time = 0.0382296, antiderivative size = 9, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{e^x}{x+1} \]

Antiderivative was successfully verified.

[In]  Int[(E^x*x)/(1 + x)^2,x]

[Out]

E^x/(1 + x)

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Rubi in Sympy [A]  time = 2.0026, size = 5, normalized size = 0.56 \[ \frac{e^{x}}{x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

exp(x)/(x + 1)

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Mathematica [A]  time = 0.00537475, size = 9, normalized size = 1. \[ \frac{e^x}{x+1} \]

Antiderivative was successfully verified.

[In]  Integrate[(E^x*x)/(1 + x)^2,x]

[Out]

E^x/(1 + x)

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Maple [A]  time = 0.005, size = 9, normalized size = 1. \[{\frac{{{\rm e}^{x}}}{1+x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(exp(x)*x/(1+x)^2,x)

[Out]

exp(x)/(1+x)

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Maxima [A]  time = 1.34865, size = 11, normalized size = 1.22 \[ \frac{e^{x}}{x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

e^x/(x + 1)

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Fricas [A]  time = 0.202348, size = 11, normalized size = 1.22 \[ \frac{e^{x}}{x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

e^x/(x + 1)

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Sympy [A]  time = 0.076076, size = 5, normalized size = 0.56 \[ \frac{e^{x}}{x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(exp(x)*x/(1+x)**2,x)

[Out]

exp(x)/(x + 1)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \mathit{undef} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

undef