\(\int \frac {27 e^{8 x}+e^{4 x} (64-256 x-144 x^2)}{9 e^{8 x}-24 e^{4 x} x+16 x^2} \, dx\) [8904]

Optimal result

Integrand size = 48, antiderivative size = 22 \[ \int \frac {27 e^{8 x}+e^{4 x} \left (64-256 x-144 x^2\right )}{9 e^{8 x}-24 e^{4 x} x+16 x^2} \, dx=\frac {x+4 (4+2 x)}{3-4 e^{-4 x} x} \]

[Out]

(9*x+16)/(3-4/exp(x)^4*x)

Rubi [F]

\[ \int \frac {27 e^{8 x}+e^{4 x} \left (64-256 x-144 x^2\right )}{9 e^{8 x}-24 e^{4 x} x+16 x^2} \, dx=\int \frac {27 e^{8 x}+e^{4 x} \left (64-256 x-144 x^2\right )}{9 e^{8 x}-24 e^{4 x} x+16 x^2} \, dx \]

[In]

Int[(27*E^(8*x) + E^(4*x)*(64 - 256*x - 144*x^2))/(9*E^(8*x) - 24*E^(4*x)*x + 16*x^2),x]

[Out]

-16/(3*(3*E^(4*x) - 4*x)) + (55*x)/(3*(3*E^(4*x) - 4*x)) + (12*x^2)/(3*E^(4*x) - 4*x) - (220*Defer[Int][x/(3*E
^(4*x) - 4*x)^2, x])/3 - 24*Defer[Int][x/(3*E^(4*x) - 4*x), x] - 48*Defer[Int][x^2/(3*E^(4*x) - 4*x)^2, x] + (
16*Defer[Subst][Defer[Int][(3*E^x - x)^(-2), x], x, 4*x])/3 - (55*Defer[Subst][Defer[Int][(3*E^x - x)^(-1), x]
, x, 4*x])/12 + (9*Defer[Subst][Defer[Int][E^x/(3*E^x - x), x], x, 4*x])/4

Rubi steps \begin{align*} \text {integral}& = \int \frac {e^{4 x} \left (64+27 e^{4 x}-256 x-144 x^2\right )}{\left (3 e^{4 x}-4 x\right )^2} \, dx \\ & = \int \left (\frac {9 e^{4 x}}{3 e^{4 x}-4 x}-\frac {4 e^{4 x} \left (-16+55 x+36 x^2\right )}{\left (3 e^{4 x}-4 x\right )^2}\right ) \, dx \\ & = -\left (4 \int \frac {e^{4 x} \left (-16+55 x+36 x^2\right )}{\left (3 e^{4 x}-4 x\right )^2} \, dx\right )+9 \int \frac {e^{4 x}}{3 e^{4 x}-4 x} \, dx \\ & = \frac {9}{4} \text {Subst}\left (\int \frac {e^x}{3 e^x-x} \, dx,x,4 x\right )-4 \int \left (-\frac {16 e^{4 x}}{\left (3 e^{4 x}-4 x\right )^2}+\frac {55 e^{4 x} x}{\left (3 e^{4 x}-4 x\right )^2}+\frac {36 e^{4 x} x^2}{\left (3 e^{4 x}-4 x\right )^2}\right ) \, dx \\ & = \frac {9}{4} \text {Subst}\left (\int \frac {e^x}{3 e^x-x} \, dx,x,4 x\right )+64 \int \frac {e^{4 x}}{\left (3 e^{4 x}-4 x\right )^2} \, dx-144 \int \frac {e^{4 x} x^2}{\left (3 e^{4 x}-4 x\right )^2} \, dx-220 \int \frac {e^{4 x} x}{\left (3 e^{4 x}-4 x\right )^2} \, dx \\ & = -\frac {16}{3 \left (3 e^{4 x}-4 x\right )}+\frac {55 x}{3 \left (3 e^{4 x}-4 x\right )}+\frac {12 x^2}{3 e^{4 x}-4 x}+\frac {9}{4} \text {Subst}\left (\int \frac {e^x}{3 e^x-x} \, dx,x,4 x\right )-\frac {55}{3} \int \frac {1}{3 e^{4 x}-4 x} \, dx+\frac {64}{3} \int \frac {1}{\left (3 e^{4 x}-4 x\right )^2} \, dx-24 \int \frac {x}{3 e^{4 x}-4 x} \, dx-48 \int \frac {x^2}{\left (3 e^{4 x}-4 x\right )^2} \, dx-\frac {220}{3} \int \frac {x}{\left (3 e^{4 x}-4 x\right )^2} \, dx \\ & = -\frac {16}{3 \left (3 e^{4 x}-4 x\right )}+\frac {55 x}{3 \left (3 e^{4 x}-4 x\right )}+\frac {12 x^2}{3 e^{4 x}-4 x}+\frac {9}{4} \text {Subst}\left (\int \frac {e^x}{3 e^x-x} \, dx,x,4 x\right )-\frac {55}{12} \text {Subst}\left (\int \frac {1}{3 e^x-x} \, dx,x,4 x\right )+\frac {16}{3} \text {Subst}\left (\int \frac {1}{\left (3 e^x-x\right )^2} \, dx,x,4 x\right )-24 \int \frac {x}{3 e^{4 x}-4 x} \, dx-48 \int \frac {x^2}{\left (3 e^{4 x}-4 x\right )^2} \, dx-\frac {220}{3} \int \frac {x}{\left (3 e^{4 x}-4 x\right )^2} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.51 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {27 e^{8 x}+e^{4 x} \left (64-256 x-144 x^2\right )}{9 e^{8 x}-24 e^{4 x} x+16 x^2} \, dx=\frac {\left (64+27 e^{4 x}\right ) x}{9 e^{4 x}-12 x} \]

[In]

Integrate[(27*E^(8*x) + E^(4*x)*(64 - 256*x - 144*x^2))/(9*E^(8*x) - 24*E^(4*x)*x + 16*x^2),x]

[Out]

((64 + 27*E^(4*x))*x)/(9*E^(4*x) - 12*x)

Maple [A] (verified)

Time = 0.12 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.14

method	result	size
risch	\(3 x -\frac {4 \left (9 x +16\right ) x}{3 \left (-3 \,{\mathrm e}^{4 x}+4 x \right )}\)	\(25\)
parallelrisch	\(-\frac {27 x \,{\mathrm e}^{4 x}+64 x}{3 \left (-3 \,{\mathrm e}^{4 x}+4 x \right )}\)	\(26\)
norman	\(\frac {-16 \,{\mathrm e}^{4 x}-9 x \,{\mathrm e}^{4 x}}{-3 \,{\mathrm e}^{4 x}+4 x}\)	\(28\)

[In]

int((27*exp(x)^8+(-144*x^2-256*x+64)*exp(x)^4)/(9*exp(x)^8-24*x*exp(x)^4+16*x^2),x,method=_RETURNVERBOSE)

[Out]

3*x-4/3*(9*x+16)*x/(-3*exp(4*x)+4*x)

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.14 \[ \int \frac {27 e^{8 x}+e^{4 x} \left (64-256 x-144 x^2\right )}{9 e^{8 x}-24 e^{4 x} x+16 x^2} \, dx=-\frac {27 \, x e^{\left (4 \, x\right )} + 64 \, x}{3 \, {\left (4 \, x - 3 \, e^{\left (4 \, x\right )}\right )}} \]

[In]

integrate((27*exp(x)^8+(-144*x^2-256*x+64)*exp(x)^4)/(9*exp(x)^8-24*x*exp(x)^4+16*x^2),x, algorithm="fricas")

[Out]

-1/3*(27*x*e^(4*x) + 64*x)/(4*x - 3*e^(4*x))

Sympy [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {27 e^{8 x}+e^{4 x} \left (64-256 x-144 x^2\right )}{9 e^{8 x}-24 e^{4 x} x+16 x^2} \, dx=3 x + \frac {36 x^{2} + 64 x}{- 12 x + 9 e^{4 x}} \]

[In]

integrate((27*exp(x)**8+(-144*x**2-256*x+64)*exp(x)**4)/(9*exp(x)**8-24*x*exp(x)**4+16*x**2),x)

[Out]

3*x + (36*x**2 + 64*x)/(-12*x + 9*exp(4*x))

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.14 \[ \int \frac {27 e^{8 x}+e^{4 x} \left (64-256 x-144 x^2\right )}{9 e^{8 x}-24 e^{4 x} x+16 x^2} \, dx=-\frac {27 \, x e^{\left (4 \, x\right )} + 64 \, x}{3 \, {\left (4 \, x - 3 \, e^{\left (4 \, x\right )}\right )}} \]

[In]

integrate((27*exp(x)^8+(-144*x^2-256*x+64)*exp(x)^4)/(9*exp(x)^8-24*x*exp(x)^4+16*x^2),x, algorithm="maxima")

[Out]

-1/3*(27*x*e^(4*x) + 64*x)/(4*x - 3*e^(4*x))

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.14 \[ \int \frac {27 e^{8 x}+e^{4 x} \left (64-256 x-144 x^2\right )}{9 e^{8 x}-24 e^{4 x} x+16 x^2} \, dx=-\frac {27 \, x e^{\left (4 \, x\right )} + 64 \, x}{3 \, {\left (4 \, x - 3 \, e^{\left (4 \, x\right )}\right )}} \]

[In]

integrate((27*exp(x)^8+(-144*x^2-256*x+64)*exp(x)^4)/(9*exp(x)^8-24*x*exp(x)^4+16*x^2),x, algorithm="giac")

[Out]

-1/3*(27*x*e^(4*x) + 64*x)/(4*x - 3*e^(4*x))

Mupad [B] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.05 \[ \int \frac {27 e^{8 x}+e^{4 x} \left (64-256 x-144 x^2\right )}{9 e^{8 x}-24 e^{4 x} x+16 x^2} \, dx=-\frac {x\,\left (27\,{\mathrm {e}}^{4\,x}+64\right )}{3\,\left (4\,x-3\,{\mathrm {e}}^{4\,x}\right )} \]

[In]

int((27*exp(8*x) - exp(4*x)*(256*x + 144*x^2 - 64))/(9*exp(8*x) - 24*x*exp(4*x) + 16*x^2),x)

[Out]

-(x*(27*exp(4*x) + 64))/(3*(4*x - 3*exp(4*x)))