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

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0436893, 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 = {} \[ \int \frac{1}{\left (a+b e^{c+d x}\right )^2 x^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[1/((a + b*E^(c + d*x))^2*x^2),x]

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.613516, size = 0, normalized size = 0. \[ \int \frac{1}{\left (a+b e^{c+d x}\right )^2 x^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.103, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( a+b{{\rm e}^{dx+c}} \right ) ^{2}{x}^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{a b d x^{2} e^{\left (d x + c\right )} + a^{2} d x^{2}} + \int \frac{d x + 2}{a b d x^{3} e^{\left (d x + c\right )} + a^{2} d x^{3}}\,{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^2,x, algorithm="maxima")

[Out]

1/(a*b*d*x^2*e^(d*x + c) + a^2*d*x^2) + integrate((d*x + 2)/(a*b*d*x^3*e^(d*x + c) + a^2*d*x^3), x)

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{a^{2} d x^{2} + a b d x^{2} e^{c + d x}} + \frac{\int \frac{d x}{a x^{3} + b x^{3} e^{c} e^{d x}}\, dx + \int \frac{2}{a x^{3} + b x^{3} 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**2,x)

[Out]

1/(a**2*d*x**2 + a*b*d*x**2*exp(c + d*x)) + (Integral(d*x/(a*x**3 + b*x**3*exp(c)*exp(d*x)), x) + Integral(2/(
a*x**3 + b*x**3*exp(c)*exp(d*x)), x))/(a*d)

________________________________________________________________________________________

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^{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^2,x, algorithm="giac")

[Out]

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