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 = 0.0315516, 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, Rules used = {2197} \[ \frac{e^{a x}}{a^2 (a x+1)} \]

Antiderivative was successfully verified.

[In]

Int[(E^(a*x)*x)/(1 + a*x)^2,x]

[Out]

E^(a*x)/(a^2*(1 + a*x))

Rule 2197

Int[(F_)^((c_.)*(v_))*(u_)^(m_.)*(w_), x_Symbol] :> With[{b = Coefficient[v, x, 1], d = Coefficient[u, x, 0],
e = Coefficient[u, x, 1], f = Coefficient[w, x, 0], g = Coefficient[w, x, 1]}, Simp[(g*u^(m + 1)*F^(c*v))/(b*c
*e*Log[F]), x] /; EqQ[e*g*(m + 1) - b*c*(e*f - d*g)*Log[F], 0]] /; FreeQ[{F, c, m}, x] && LinearQ[{u, v, w}, x
]

Rubi steps

\begin{align*} \int \frac{e^{a x} x}{(1+a x)^2} \, dx &=\frac{e^{a x}}{a^2 (1+a x)}\\ \end{align*}

Mathematica [A]  time = 0.045405, 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. \begin{align*}{\frac{{{\rm e}^{ax}}}{{a}^{2} \left ( ax+1 \right ) }} \end{align*}

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 = 0.950543, size = 22, normalized size = 1.38 \begin{align*} \frac{e^{\left (a x\right )}}{a^{3} x + a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(a*x)*x/(a*x+1)^2,x, algorithm="maxima")

[Out]

e^(a*x)/(a^3*x + a^2)

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Fricas [A]  time = 1.55689, size = 31, normalized size = 1.94 \begin{align*} \frac{e^{\left (a x\right )}}{a^{3} x + a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(a*x)*x/(a*x+1)^2,x, algorithm="fricas")

[Out]

e^(a*x)/(a^3*x + a^2)

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Sympy [A]  time = 0.116405, size = 12, normalized size = 0.75 \begin{align*} \frac{e^{a x}}{a^{3} x + a^{2}} \end{align*}

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 [A]  time = 1.10811, size = 38, normalized size = 2.38 \begin{align*} \frac{e^{\left (-{\left (a x + 1\right )}{\left (\frac{1}{a x + 1} - 1\right )}\right )}}{{\left (a x + 1\right )} a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^(-(a*x + 1)*(1/(a*x + 1) - 1))/((a*x + 1)*a^2)