\(\int \frac {d+e e^{h+i x}}{(a+b e^{h+i x}+c e^{2 h+2 i x}) (f+g x)} \, dx\) [576]

Optimal result

Integrand size = 44, antiderivative size = 44 \[ \int \frac {d+e e^{h+i x}}{\left (a+b e^{h+i x}+c e^{2 h+2 i x}\right ) (f+g x)} \, dx=d \text {Int}\left (\frac {1}{\left (a+b e^{h+i x}+c e^{2 h+2 i x}\right ) (f+g x)},x\right )+e \text {Int}\left (\frac {e^{h+i x}}{\left (a+b e^{h+i x}+c e^{2 h+2 i x}\right ) (f+g x)},x\right ) \]

[Out]

d*CannotIntegrate(1/(a+b*exp(i*x+h)+c*exp(2*i*x+2*h))/(g*x+f),x)+e*CannotIntegrate(exp(i*x+h)/(a+b*exp(i*x+h)+
c*exp(2*i*x+2*h))/(g*x+f),x)

Rubi [N/A]

Not integrable

Time = 0.65 (sec) , antiderivative size = 44, 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 = {} \[ \int \frac {d+e e^{h+i x}}{\left (a+b e^{h+i x}+c e^{2 h+2 i x}\right ) (f+g x)} \, dx=\int \frac {d+e e^{h+i x}}{\left (a+b e^{h+i x}+c e^{2 h+2 i x}\right ) (f+g x)} \, dx \]

[In]

Int[(d + e*E^(h + i*x))/((a + b*E^(h + i*x) + c*E^(2*h + 2*i*x))*(f + g*x)),x]

[Out]

d*Defer[Int][1/((a + b*E^(h + i*x) + c*E^(2*h + 2*i*x))*(f + g*x)), x] + e*Defer[Int][E^(h + i*x)/((a + b*E^(h
+ i*x) + c*E^(2*h + 2*i*x))*(f + g*x)), x]

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

Mathematica [N/A]

Not integrable

Time = 0.83 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.05 \[ \int \frac {d+e e^{h+i x}}{\left (a+b e^{h+i x}+c e^{2 h+2 i x}\right ) (f+g x)} \, dx=\int \frac {d+e e^{h+i x}}{\left (a+b e^{h+i x}+c e^{2 h+2 i x}\right ) (f+g x)} \, dx \]

[In]

Integrate[(d + e*E^(h + i*x))/((a + b*E^(h + i*x) + c*E^(2*h + 2*i*x))*(f + g*x)),x]

[Out]

Integrate[(d + e*E^(h + i*x))/((a + b*E^(h + i*x) + c*E^(2*h + 2*i*x))*(f + g*x)), x]

Maple [N/A]

Not integrable

Time = 0.05 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.93

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

[In]

int((d+e*exp(i*x+h))/(a+b*exp(i*x+h)+c*exp(2*i*x+2*h))/(g*x+f),x)

[Out]

int((d+e*exp(i*x+h))/(a+b*exp(i*x+h)+c*exp(2*i*x+2*h))/(g*x+f),x)

Fricas [N/A]

Not integrable

Time = 0.32 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.27 \[ \int \frac {d+e e^{h+i x}}{\left (a+b e^{h+i x}+c e^{2 h+2 i x}\right ) (f+g x)} \, dx=\int { \frac {e e^{\left (i x + h\right )} + d}{{\left (g x + f\right )} {\left (c e^{\left (2 \, i x + 2 \, h\right )} + b e^{\left (i x + h\right )} + a\right )}} \,d x } \]

[In]

integrate((d+e*exp(i*x+h))/(a+b*exp(i*x+h)+c*exp(2*i*x+2*h))/(g*x+f),x, algorithm="fricas")

[Out]

integral((e*e^(i*x + h) + d)/(a*g*x + a*f + (c*g*x + c*f)*e^(2*i*x + 2*h) + (b*g*x + b*f)*e^(i*x + h)), x)

Sympy [N/A]

Not integrable

Time = 54.98 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.95 \[ \int \frac {d+e e^{h+i x}}{\left (a+b e^{h+i x}+c e^{2 h+2 i x}\right ) (f+g x)} \, dx=\int \frac {d + e e^{h} e^{i x}}{\left (f + g x\right ) \left (a + b e^{h} e^{i x} + c e^{2 h} e^{2 i x}\right )}\, dx \]

[In]

integrate((d+e*exp(i*x+h))/(a+b*exp(i*x+h)+c*exp(2*i*x+2*h))/(g*x+f),x)

[Out]

Integral((d + e*exp(h)*exp(i*x))/((f + g*x)*(a + b*exp(h)*exp(i*x) + c*exp(2*h)*exp(2*i*x))), x)

Maxima [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.98 \[ \int \frac {d+e e^{h+i x}}{\left (a+b e^{h+i x}+c e^{2 h+2 i x}\right ) (f+g x)} \, dx=\int { \frac {e e^{\left (i x + h\right )} + d}{{\left (g x + f\right )} {\left (c e^{\left (2 \, i x + 2 \, h\right )} + b e^{\left (i x + h\right )} + a\right )}} \,d x } \]

[In]

integrate((d+e*exp(i*x+h))/(a+b*exp(i*x+h)+c*exp(2*i*x+2*h))/(g*x+f),x, algorithm="maxima")

[Out]

integrate((e*e^(i*x + h) + d)/((g*x + f)*(c*e^(2*i*x + 2*h) + b*e^(i*x + h) + a)), x)

Giac [N/A]

Not integrable

Time = 0.31 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.98 \[ \int \frac {d+e e^{h+i x}}{\left (a+b e^{h+i x}+c e^{2 h+2 i x}\right ) (f+g x)} \, dx=\int { \frac {e e^{\left (i x + h\right )} + d}{{\left (g x + f\right )} {\left (c e^{\left (2 \, i x + 2 \, h\right )} + b e^{\left (i x + h\right )} + a\right )}} \,d x } \]

[In]

integrate((d+e*exp(i*x+h))/(a+b*exp(i*x+h)+c*exp(2*i*x+2*h))/(g*x+f),x, algorithm="giac")

[Out]

integrate((e*e^(i*x + h) + d)/((g*x + f)*(c*e^(2*i*x + 2*h) + b*e^(i*x + h) + a)), x)

Mupad [N/A]

Not integrable

Time = 0.98 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.98 \[ \int \frac {d+e e^{h+i x}}{\left (a+b e^{h+i x}+c e^{2 h+2 i x}\right ) (f+g x)} \, dx=\int \frac {d+e\,{\mathrm {e}}^{h+i\,x}}{\left (f+g\,x\right )\,\left (a+b\,{\mathrm {e}}^{h+i\,x}+c\,{\mathrm {e}}^{2\,h+2\,i\,x}\right )} \,d x \]

[In]

int((d + e*exp(h + i*x))/((f + g*x)*(a + b*exp(h + i*x) + c*exp(2*h + 2*i*x))),x)

[Out]

int((d + e*exp(h + i*x))/((f + g*x)*(a + b*exp(h + i*x) + c*exp(2*h + 2*i*x))), x)