\(\int \frac {\sqrt {c+d x}}{(a+b e^x)^2} \, dx\) [68]

Optimal result

Integrand size = 19, antiderivative size = 19 \[ \int \frac {\sqrt {c+d x}}{\left (a+b e^x\right )^2} \, dx=\text {Int}\left (\frac {\sqrt {c+d x}}{\left (a+b e^x\right )^2},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.03 (sec) , antiderivative size = 19, 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 = {} \[ \int \frac {\sqrt {c+d x}}{\left (a+b e^x\right )^2} \, dx=\int \frac {\sqrt {c+d x}}{\left (a+b e^x\right )^2} \, dx \]

[In]

Int[Sqrt[c + d*x]/(a + b*E^x)^2,x]

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 1.65 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.11 \[ \int \frac {\sqrt {c+d x}}{\left (a+b e^x\right )^2} \, dx=\int \frac {\sqrt {c+d x}}{\left (a+b e^x\right )^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A]

Not integrable

Time = 0.00 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.84

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.58 \[ \int \frac {\sqrt {c+d x}}{\left (a+b e^x\right )^2} \, dx=\int { \frac {\sqrt {d x + c}}{{\left (b e^{x} + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 5.03 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.89 \[ \int \frac {\sqrt {c+d x}}{\left (a+b e^x\right )^2} \, dx=\int \frac {\sqrt {c + d x}}{\left (a + b e^{x}\right )^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.29 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.95 \[ \int \frac {\sqrt {c+d x}}{\left (a+b e^x\right )^2} \, dx=\int { \frac {\sqrt {d x + c}}{{\left (b e^{x} + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.38 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.95 \[ \int \frac {\sqrt {c+d x}}{\left (a+b e^x\right )^2} \, dx=\int { \frac {\sqrt {d x + c}}{{\left (b e^{x} + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.95 \[ \int \frac {\sqrt {c+d x}}{\left (a+b e^x\right )^2} \, dx=\int \frac {\sqrt {c+d\,x}}{{\left (a+b\,{\mathrm {e}}^x\right )}^2} \,d x \]

[In]

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

[Out]

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