\(\int \frac {1}{(a+b e^{c+d x})^3 x} \, dx\) [21]

Optimal result

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

[Out]

Unintegrable(1/(a+b*exp(d*x+c))^3/x,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 = {} \[ \int \frac {1}{\left (a+b e^{c+d x}\right )^3 x} \, dx=\int \frac {1}{\left (a+b e^{c+d x}\right )^3 x} \, dx \]

[In]

Int[1/((a + b*E^(c + d*x))^3*x),x]

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 2.21 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.12 \[ \int \frac {1}{\left (a+b e^{c+d x}\right )^3 x} \, dx=\int \frac {1}{\left (a+b e^{c+d x}\right )^3 x} \, dx \]

[In]

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

[Out]

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

Maple [N/A]

Not integrable

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

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 53, normalized size of antiderivative = 3.12 \[ \int \frac {1}{\left (a+b e^{c+d x}\right )^3 x} \, dx=\int { \frac {1}{{\left (b e^{\left (d x + c\right )} + a\right )}^{3} x} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 2.22 (sec) , antiderivative size = 165, normalized size of antiderivative = 9.71 \[ \int \frac {1}{\left (a+b e^{c+d x}\right )^3 x} \, dx=\frac {3 a d x + a + \left (2 b d x + b\right ) e^{c + d x}}{2 a^{4} d^{2} x^{2} + 4 a^{3} b d^{2} x^{2} e^{c + d x} + 2 a^{2} b^{2} d^{2} x^{2} e^{2 c + 2 d x}} + \frac {\int \frac {3 d x}{a x^{3} + b x^{3} e^{c} e^{d x}}\, dx + \int \frac {2 d^{2} x^{2}}{a x^{3} + b x^{3} e^{c} e^{d x}}\, dx + \int \frac {2}{a x^{3} + b x^{3} e^{c} e^{d x}}\, dx}{2 a^{2} d^{2}} \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

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

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.32 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.06 \[ \int \frac {1}{\left (a+b e^{c+d x}\right )^3 x} \, dx=\int { \frac {1}{{\left (b e^{\left (d x + c\right )} + a\right )}^{3} x} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

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

[In]

int(1/(x*(a + b*exp(c + d*x))^3),x)

[Out]

int(1/(x*(a + b*exp(c + d*x))^3), x)