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

Optimal. Leaf size=82 \[ d \text{CannotIntegrate}\left (\frac{1}{(f+g x) \left (a+b e^{h+i x}+c e^{2 h+2 i x}\right )},x\right )+e \text{CannotIntegrate}\left (\frac{e^{h+i x}}{(f+g x) \left (a+b e^{h+i x}+c e^{2 h+2 i x}\right )},x\right ) \]

[Out]

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

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Rubi [A]  time = 1.01787, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{d+e e^{h+i x}}{\left (a+b e^{h+i x}+c e^{2 h+2 i x}\right ) (f+g x)} \, dx \]

Verification is Not applicable to the result.

[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*} \int \frac{d+e e^{h+576 x}}{\left (a+b e^{h+576 x}+c e^{2 h+1152 x}\right ) (f+g x)} \, dx &=\int \left (\frac{d}{\left (a+b e^{h+576 x}+c e^{2 h+1152 x}\right ) (f+g x)}+\frac{e e^{h+576 x}}{\left (a+b e^{h+576 x}+c e^{2 h+1152 x}\right ) (f+g x)}\right ) \, dx\\ &=d \int \frac{1}{\left (a+b e^{h+576 x}+c e^{2 h+1152 x}\right ) (f+g x)} \, dx+e \int \frac{e^{h+576 x}}{\left (a+b e^{h+576 x}+c e^{2 h+1152 x}\right ) (f+g x)} \, dx\\ \end{align*}

Mathematica [A]  time = 0.412382, size = 0, normalized size = 0. \[ \int \frac{d+e e^{h+i x}}{\left (a+b e^{h+i x}+c e^{2 h+2 i x}\right ) (f+g x)} \, dx \]

Verification is Not applicable to the result.

[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]

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Maple [A]  time = 0.156, size = 0, normalized size = 0. \begin{align*} \int{\frac{d+e{{\rm e}^{ix+h}}}{ \left ( a+b{{\rm e}^{ix+h}}+c{{\rm e}^{2\,ix+2\,h}} \right ) \left ( gx+f \right ) }}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{e e^{\left (i x + h\right )} + d}{a g x + a f +{\left (c g x + c f\right )} e^{\left (2 \, i x + 2 \, h\right )} +{\left (b g x + b f\right )} e^{\left (i x + h\right )}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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]

Timed out

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)