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

Optimal. Leaf size=16 \[ \frac{e^{a x}}{a^2 (a x+1)} \]

[Out]

E^(a*x)/(a^2*(1 + a*x))

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Rubi [A]  time = 0.0508024, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071 \[ \frac{e^{a x}}{a^2 (a x+1)} \]

Antiderivative was successfully verified.

[In]  Int[(E^(a*x)*x)/(1 + a*x)^2,x]

[Out]

E^(a*x)/(a^2*(1 + a*x))

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Rubi in Sympy [A]  time = 2.81589, size = 12, normalized size = 0.75 \[ \frac{e^{a x}}{a^{2} \left (a x + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

exp(a*x)/(a**2*(a*x + 1))

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Mathematica [A]  time = 0.0102228, size = 16, normalized size = 1. \[ \frac{e^{a x}}{a^2 (a x+1)} \]

Antiderivative was successfully verified.

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

[Out]

E^(a*x)/(a^2*(1 + a*x))

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Maple [A]  time = 0.003, size = 16, normalized size = 1. \[{\frac{{{\rm e}^{ax}}}{{a}^{2} \left ( ax+1 \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(exp(a*x)*x/(a*x+1)^2,x)

[Out]

exp(a*x)/a^2/(a*x+1)

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Maxima [A]  time = 1.3671, size = 22, normalized size = 1.38 \[ \frac{e^{\left (a x\right )}}{a^{3} x + a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

e^(a*x)/(a^3*x + a^2)

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Fricas [A]  time = 0.198744, size = 22, normalized size = 1.38 \[ \frac{e^{\left (a x\right )}}{a^{3} x + a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

e^(a*x)/(a^3*x + a^2)

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Sympy [A]  time = 0.124263, size = 12, normalized size = 0.75 \[ \frac{e^{a x}}{a^{3} x + a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

exp(a*x)/(a**3*x + a**2)

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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^(a*x)/(a*x + 1)^2,x, algorithm="giac")

[Out]

undef