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

Optimal. Leaf size=84 \[ d \text {Int}\left (\frac {1}{\left (a+b e^{h+i x}+c e^{2 h+2 i x}\right ) (f+g x)^2},x\right )+e \text {Int}\left (\frac {e^{h+i x}}{\left (a+b e^{h+i x}+c e^{2 h+2 i x}\right ) (f+g x)^2},x\right ) \]

[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)

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Rubi [A]
time = 0.62, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \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 \end {gather*}

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)^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*} \int \frac {d+e e^{h+577 x}}{\left (a+b e^{h+577 x}+c e^{2 h+1154 x}\right ) (f+g x)^2} \, dx &=\int \left (\frac {d}{\left (a+b e^{h+577 x}+c e^{2 h+1154 x}\right ) (f+g x)^2}+\frac {e e^{h+577 x}}{\left (a+b e^{h+577 x}+c e^{2 h+1154 x}\right ) (f+g x)^2}\right ) \, dx\\ &=d \int \frac {1}{\left (a+b e^{h+577 x}+c e^{2 h+1154 x}\right ) (f+g x)^2} \, dx+e \int \frac {e^{h+577 x}}{\left (a+b e^{h+577 x}+c e^{2 h+1154 x}\right ) (f+g x)^2} \, dx\\ \end {align*}

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Mathematica [A]
time = 7.41, size = 0, normalized size = 0.00 \begin {gather*} \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 \end {gather*}

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)^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]

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Maple [A]
time = 0.07, size = 0, normalized size = 0.00 \[\int \frac {d +e \,{\mathrm e}^{i x +h}}{\left (a +b \,{\mathrm e}^{i x +h}+c \,{\mathrm e}^{2 i x +2 h}\right ) \left (g x +f \right )^{2}}\, dx\]

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)^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)

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Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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)^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)

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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)^2,x, algorithm="fricas")

[Out]

integral((d*e^2 + e^(h + I*x + 3))/((a*g^2*x^2 + 2*a*f*g*x + a*f^2)*e^2 + (c*g^2*x^2 + 2*c*f*g*x + c*f^2)*e^(2
*h + 2*I*x + 2) + (b*g^2*x^2 + 2*b*f*g*x + b*f^2)*e^(h + I*x + 2)), x)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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)**2,x)

[Out]

Timed out

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Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)^2,x, algorithm="giac")

[Out]

integrate((d + e^(h + I*x + 1))/((g*x + f)^2*(c*e^(2*h + 2*I*x) + b*e^(h + I*x) + a)), x)

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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