\(\int \frac {d+e e^{h+i x}}{(a+b e^{h+i x}+c e^{2 h+2 i x}) (f+g x)^2} \, dx\) [577]

Optimal result

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

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.56 (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 = {} \[ \int \frac {d+e e^{h+i x}}{\left (a+b e^{h+i x}+c e^{2 h+2 i x}\right ) (f+g x)^2} \, dx=\int \frac {d+e e^{h+i x}}{\left (a+b e^{h+i x}+c e^{2 h+2 i x}\right ) (f+g x)^2} \, dx \]

[In]

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

[Out]

d*Defer[Int][1/((a + b*E^(h + i*x) + c*E^(2*h + 2*i*x))*(f + g*x)^2), x] + e*Defer[Int][E^(h + i*x)/((a + b*E^
(h + i*x) + c*E^(2*h + 2*i*x))*(f + g*x)^2), 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)^2}+\frac {e e^{h+i x}}{\left (a+b e^{h+i x}+c e^{2 h+2 i x}\right ) (f+g x)^2}\right ) \, dx \\ & = d \int \frac {1}{\left (a+b e^{h+i x}+c e^{2 h+2 i x}\right ) (f+g x)^2} \, 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)^2} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 7.31 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.05 \[ \int \frac {d+e e^{h+i x}}{\left (a+b e^{h+i x}+c e^{2 h+2 i x}\right ) (f+g x)^2} \, dx=\int \frac {d+e e^{h+i x}}{\left (a+b e^{h+i x}+c e^{2 h+2 i x}\right ) (f+g x)^2} \, dx \]

[In]

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

Maple [N/A]

Not integrable

Time = 0.07 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.93

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

[In]

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

Fricas [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 92, normalized size of antiderivative = 2.09 \[ \int \frac {d+e e^{h+i x}}{\left (a+b e^{h+i x}+c e^{2 h+2 i x}\right ) (f+g x)^2} \, dx=\int { \frac {e e^{\left (i x + h\right )} + d}{{\left (g x + f\right )}^{2} {\left (c e^{\left (2 \, i x + 2 \, h\right )} + b e^{\left (i x + h\right )} + a\right )}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 0.00 (sec) , antiderivative size = 1, normalized size of antiderivative = 0.02 \[ \int \frac {d+e e^{h+i x}}{\left (a+b e^{h+i x}+c e^{2 h+2 i x}\right ) (f+g x)^2} \, dx=\int \frac {d + e e^{h + i x}}{\left (f + g x\right )^{2} \left (a + b e^{h + i x} + c e^{2 h + 2 i x}\right )} \, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.98 \[ \int \frac {d+e e^{h+i x}}{\left (a+b e^{h+i x}+c e^{2 h+2 i x}\right ) (f+g x)^2} \, dx=\int { \frac {e e^{\left (i x + h\right )} + d}{{\left (g x + f\right )}^{2} {\left (c e^{\left (2 \, i x + 2 \, h\right )} + b e^{\left (i x + h\right )} + a\right )}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.61 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.98 \[ \int \frac {d+e e^{h+i x}}{\left (a+b e^{h+i x}+c e^{2 h+2 i x}\right ) (f+g x)^2} \, dx=\int { \frac {e e^{\left (i x + h\right )} + d}{{\left (g x + f\right )}^{2} {\left (c e^{\left (2 \, i x + 2 \, h\right )} + b e^{\left (i x + h\right )} + a\right )}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 0.76 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.98 \[ \int \frac {d+e e^{h+i x}}{\left (a+b e^{h+i x}+c e^{2 h+2 i x}\right ) (f+g x)^2} \, dx=\int \frac {d+e\,{\mathrm {e}}^{h+i\,x}}{{\left (f+g\,x\right )}^2\,\left (a+b\,{\mathrm {e}}^{h+i\,x}+c\,{\mathrm {e}}^{2\,h+2\,i\,x}\right )} \,d x \]

[In]

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