3.13 \(\int \frac{1}{(a+b e^{c+d x})^2 x} \, dx\)

Optimal. Leaf size=19 \[ \text{Unintegrable}\left (\frac{1}{x \left (a+b e^{c+d x}\right )^2},x\right ) \]

[Out]

Unintegrable[1/((a + b*E^(c + d*x))^2*x), x]

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Rubi [A]  time = 0.0465358, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{1}{\left (a+b e^{c+d x}\right )^2 x} \, dx \]

Verification is Not applicable to the result.

[In]

Int[1/((a + b*E^(c + d*x))^2*x),x]

[Out]

Defer[Int][1/((a + b*E^(c + d*x))^2*x), x]

Rubi steps

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

Mathematica [A]  time = 0.777473, size = 0, normalized size = 0. \[ \int \frac{1}{\left (a+b e^{c+d x}\right )^2 x} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[1/((a + b*E^(c + d*x))^2*x),x]

[Out]

Integrate[1/((a + b*E^(c + d*x))^2*x), x]

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Maple [A]  time = 0.1, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( a+b{{\rm e}^{dx+c}} \right ) ^{2}x}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b*exp(d*x+c))^2/x,x)

[Out]

int(1/(a+b*exp(d*x+c))^2/x,x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{a b d x e^{\left (d x + c\right )} + a^{2} d x} + \int \frac{d x + 1}{a b d x^{2} e^{\left (d x + c\right )} + a^{2} d x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/(a*b*d*x*e^(d*x + c) + a^2*d*x) + integrate((d*x + 1)/(a*b*d*x^2*e^(d*x + c) + a^2*d*x^2), x)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{b^{2} x e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a b x e^{\left (d x + c\right )} + a^{2} x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(1/(b^2*x*e^(2*d*x + 2*c) + 2*a*b*x*e^(d*x + c) + a^2*x), x)

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{a^{2} d x + a b d x e^{c + d x}} + \frac{\int \frac{d x}{a x^{2} + b x^{2} e^{c} e^{d x}}\, dx + \int \frac{1}{a x^{2} + b x^{2} e^{c} e^{d x}}\, dx}{a d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*exp(d*x+c))**2/x,x)

[Out]

1/(a**2*d*x + a*b*d*x*exp(c + d*x)) + (Integral(d*x/(a*x**2 + b*x**2*exp(c)*exp(d*x)), x) + Integral(1/(a*x**2
+ b*x**2*exp(c)*exp(d*x)), x))/(a*d)

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b e^{\left (d x + c\right )} + a\right )}^{2} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/((b*e^(d*x + c) + a)^2*x), x)