\(\int \frac {1}{(a+b e^{c+d x})^3 x^2} \, dx\) [22]

Optimal result

Integrand size = 17, antiderivative size = 17 \[ \int \frac {1}{\left (a+b e^{c+d x}\right )^3 x^2} \, dx=\text {Int}\left (\frac {1}{\left (a+b e^{c+d x}\right )^3 x^2},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.03 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {1}{\left (a+b e^{c+d x}\right )^3 x^2} \, dx=\int \frac {1}{\left (a+b e^{c+d x}\right )^3 x^2} \, dx \]

[In]

Int[1/((a + b*E^(c + d*x))^3*x^2),x]

[Out]

Defer[Int][1/((a + b*E^(c + d*x))^3*x^2), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{\left (a+b e^{c+d x}\right )^3 x^2} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 1.17 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.12 \[ \int \frac {1}{\left (a+b e^{c+d x}\right )^3 x^2} \, dx=\int \frac {1}{\left (a+b e^{c+d x}\right )^3 x^2} \, dx \]

[In]

Integrate[1/((a + b*E^(c + d*x))^3*x^2),x]

[Out]

Integrate[1/((a + b*E^(c + d*x))^3*x^2), x]

Maple [N/A]

Not integrable

Time = 0.16 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.94

\[\int \frac {1}{\left (a +b \,{\mathrm e}^{d x +c}\right )^{3} x^{2}}d x\]

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 61, normalized size of antiderivative = 3.59 \[ \int \frac {1}{\left (a+b e^{c+d x}\right )^3 x^2} \, dx=\int { \frac {1}{{\left (b e^{\left (d x + c\right )} + a\right )}^{3} x^{2}} \,d x } \]

[In]

integrate(1/(a+b*exp(d*x+c))^3/x^2,x, algorithm="fricas")

[Out]

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

Sympy [N/A]

Not integrable

Time = 2.08 (sec) , antiderivative size = 165, normalized size of antiderivative = 9.71 \[ \int \frac {1}{\left (a+b e^{c+d x}\right )^3 x^2} \, dx=\frac {3 a d x + 2 a + \left (2 b d x + 2 b\right ) e^{c + d x}}{2 a^{4} d^{2} x^{3} + 4 a^{3} b d^{2} x^{3} e^{c + d x} + 2 a^{2} b^{2} d^{2} x^{3} e^{2 c + 2 d x}} + \frac {\int \frac {3 d x}{a x^{4} + b x^{4} e^{c} e^{d x}}\, dx + \int \frac {d^{2} x^{2}}{a x^{4} + b x^{4} e^{c} e^{d x}}\, dx + \int \frac {3}{a x^{4} + b x^{4} e^{c} e^{d x}}\, dx}{a^{2} d^{2}} \]

[In]

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

[Out]

(3*a*d*x + 2*a + (2*b*d*x + 2*b)*exp(c + d*x))/(2*a**4*d**2*x**3 + 4*a**3*b*d**2*x**3*exp(c + d*x) + 2*a**2*b*
*2*d**2*x**3*exp(2*c + 2*d*x)) + (Integral(3*d*x/(a*x**4 + b*x**4*exp(c)*exp(d*x)), x) + Integral(d**2*x**2/(a
*x**4 + b*x**4*exp(c)*exp(d*x)), x) + Integral(3/(a*x**4 + b*x**4*exp(c)*exp(d*x)), x))/(a**2*d**2)

Maxima [N/A]

Not integrable

Time = 0.23 (sec) , antiderivative size = 128, normalized size of antiderivative = 7.53 \[ \int \frac {1}{\left (a+b e^{c+d x}\right )^3 x^2} \, dx=\int { \frac {1}{{\left (b e^{\left (d x + c\right )} + a\right )}^{3} x^{2}} \,d x } \]

[In]

integrate(1/(a+b*exp(d*x+c))^3/x^2,x, algorithm="maxima")

[Out]

1/2*(3*a*d*x + 2*(b*d*x*e^c + b*e^c)*e^(d*x) + 2*a)/(a^2*b^2*d^2*x^3*e^(2*d*x + 2*c) + 2*a^3*b*d^2*x^3*e^(d*x
+ c) + a^4*d^2*x^3) + integrate((d^2*x^2 + 3*d*x + 3)/(a^2*b*d^2*x^4*e^(d*x + c) + a^3*d^2*x^4), x)

Giac [N/A]

Not integrable

Time = 0.37 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.06 \[ \int \frac {1}{\left (a+b e^{c+d x}\right )^3 x^2} \, dx=\int { \frac {1}{{\left (b e^{\left (d x + c\right )} + a\right )}^{3} x^{2}} \,d x } \]

[In]

integrate(1/(a+b*exp(d*x+c))^3/x^2,x, algorithm="giac")

[Out]

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

Mupad [N/A]

Not integrable

Time = 0.16 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.06 \[ \int \frac {1}{\left (a+b e^{c+d x}\right )^3 x^2} \, dx=\int \frac {1}{x^2\,{\left (a+b\,{\mathrm {e}}^{c+d\,x}\right )}^3} \,d x \]

[In]

int(1/(x^2*(a + b*exp(c + d*x))^3),x)

[Out]

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