\(\int \frac {6340+144 e^{2 x}-640 x+16 x^2+e^x (-1911+87 x)}{6400+144 e^{2 x}-640 x+16 x^2+e^x (-1920+96 x)} \, dx\) [3862]

Optimal result

Integrand size = 55, antiderivative size = 21 \[ \int \frac {6340+144 e^{2 x}-640 x+16 x^2+e^x (-1911+87 x)}{6400+144 e^{2 x}-640 x+16 x^2+e^x (-1920+96 x)} \, dx=x+\frac {x}{16 \left (-5+e^x+\frac {1}{3} (-5+x)\right )} \]

[Out]

x+1/16*x/(1/3*x-20/3+exp(x))

Rubi [F]

\[ \int \frac {6340+144 e^{2 x}-640 x+16 x^2+e^x (-1911+87 x)}{6400+144 e^{2 x}-640 x+16 x^2+e^x (-1920+96 x)} \, dx=\int \frac {6340+144 e^{2 x}-640 x+16 x^2+e^x (-1911+87 x)}{6400+144 e^{2 x}-640 x+16 x^2+e^x (-1920+96 x)} \, dx \]

[In]

Int[(6340 + 144*E^(2*x) - 640*x + 16*x^2 + E^x*(-1911 + 87*x))/(6400 + 144*E^(2*x) - 640*x + 16*x^2 + E^x*(-19
20 + 96*x)),x]

[Out]

x - (63*Defer[Int][x/(-20 + 3*E^x + x)^2, x])/16 + (3*Defer[Int][x^2/(-20 + 3*E^x + x)^2, x])/16 + (3*Defer[In
t][(-20 + 3*E^x + x)^(-1), x])/16 - (3*Defer[Int][x/(-20 + 3*E^x + x), x])/16

Rubi steps \begin{align*} \text {integral}& = \int \frac {6340+144 e^{2 x}-640 x+16 x^2+e^x (-1911+87 x)}{16 \left (20-3 e^x-x\right )^2} \, dx \\ & = \frac {1}{16} \int \frac {6340+144 e^{2 x}-640 x+16 x^2+e^x (-1911+87 x)}{\left (20-3 e^x-x\right )^2} \, dx \\ & = \frac {1}{16} \int \left (16+\frac {3 (-21+x) x}{\left (-20+3 e^x+x\right )^2}-\frac {3 (-1+x)}{-20+3 e^x+x}\right ) \, dx \\ & = x+\frac {3}{16} \int \frac {(-21+x) x}{\left (-20+3 e^x+x\right )^2} \, dx-\frac {3}{16} \int \frac {-1+x}{-20+3 e^x+x} \, dx \\ & = x+\frac {3}{16} \int \left (-\frac {21 x}{\left (-20+3 e^x+x\right )^2}+\frac {x^2}{\left (-20+3 e^x+x\right )^2}\right ) \, dx-\frac {3}{16} \int \left (-\frac {1}{-20+3 e^x+x}+\frac {x}{-20+3 e^x+x}\right ) \, dx \\ & = x+\frac {3}{16} \int \frac {x^2}{\left (-20+3 e^x+x\right )^2} \, dx+\frac {3}{16} \int \frac {1}{-20+3 e^x+x} \, dx-\frac {3}{16} \int \frac {x}{-20+3 e^x+x} \, dx-\frac {63}{16} \int \frac {x}{\left (-20+3 e^x+x\right )^2} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.87 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \frac {6340+144 e^{2 x}-640 x+16 x^2+e^x (-1911+87 x)}{6400+144 e^{2 x}-640 x+16 x^2+e^x (-1920+96 x)} \, dx=\frac {1}{16} \left (16 x+\frac {3 x}{-20+3 e^x+x}\right ) \]

[In]

Integrate[(6340 + 144*E^(2*x) - 640*x + 16*x^2 + E^x*(-1911 + 87*x))/(6400 + 144*E^(2*x) - 640*x + 16*x^2 + E^
x*(-1920 + 96*x)),x]

[Out]

(16*x + (3*x)/(-20 + 3*E^x + x))/16

Maple [A] (verified)

Time = 0.25 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.71

method	result	size
risch	\(x +\frac {3 x}{16 \left (-20+3 \,{\mathrm e}^{x}+x \right )}\)	\(15\)
norman	\(\frac {x^{2}+\frac {951 \,{\mathrm e}^{x}}{16}+3 \,{\mathrm e}^{x} x -\frac {1585}{4}}{-20+3 \,{\mathrm e}^{x}+x}\)	\(25\)
parallelrisch	\(\frac {48 x^{2}+144 \,{\mathrm e}^{x} x -951 x}{-960+144 \,{\mathrm e}^{x}+48 x}\)	\(26\)

[In]

int((144*exp(x)^2+(87*x-1911)*exp(x)+16*x^2-640*x+6340)/(144*exp(x)^2+(96*x-1920)*exp(x)+16*x^2-640*x+6400),x,
method=_RETURNVERBOSE)

[Out]

x+3/16*x/(-20+3*exp(x)+x)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.19 \[ \int \frac {6340+144 e^{2 x}-640 x+16 x^2+e^x (-1911+87 x)}{6400+144 e^{2 x}-640 x+16 x^2+e^x (-1920+96 x)} \, dx=\frac {16 \, x^{2} + 48 \, x e^{x} - 317 \, x}{16 \, {\left (x + 3 \, e^{x} - 20\right )}} \]

[In]

integrate((144*exp(x)^2+(87*x-1911)*exp(x)+16*x^2-640*x+6340)/(144*exp(x)^2+(96*x-1920)*exp(x)+16*x^2-640*x+64
00),x, algorithm="fricas")

[Out]

1/16*(16*x^2 + 48*x*e^x - 317*x)/(x + 3*e^x - 20)

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.67 \[ \int \frac {6340+144 e^{2 x}-640 x+16 x^2+e^x (-1911+87 x)}{6400+144 e^{2 x}-640 x+16 x^2+e^x (-1920+96 x)} \, dx=x + \frac {3 x}{16 x + 48 e^{x} - 320} \]

[In]

integrate((144*exp(x)**2+(87*x-1911)*exp(x)+16*x**2-640*x+6340)/(144*exp(x)**2+(96*x-1920)*exp(x)+16*x**2-640*
x+6400),x)

[Out]

x + 3*x/(16*x + 48*exp(x) - 320)

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.19 \[ \int \frac {6340+144 e^{2 x}-640 x+16 x^2+e^x (-1911+87 x)}{6400+144 e^{2 x}-640 x+16 x^2+e^x (-1920+96 x)} \, dx=\frac {16 \, x^{2} + 48 \, x e^{x} - 317 \, x}{16 \, {\left (x + 3 \, e^{x} - 20\right )}} \]

[In]

integrate((144*exp(x)^2+(87*x-1911)*exp(x)+16*x^2-640*x+6340)/(144*exp(x)^2+(96*x-1920)*exp(x)+16*x^2-640*x+64
00),x, algorithm="maxima")

[Out]

1/16*(16*x^2 + 48*x*e^x - 317*x)/(x + 3*e^x - 20)

Giac [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.19 \[ \int \frac {6340+144 e^{2 x}-640 x+16 x^2+e^x (-1911+87 x)}{6400+144 e^{2 x}-640 x+16 x^2+e^x (-1920+96 x)} \, dx=\frac {16 \, x^{2} + 48 \, x e^{x} - 317 \, x}{16 \, {\left (x + 3 \, e^{x} - 20\right )}} \]

[In]

integrate((144*exp(x)^2+(87*x-1911)*exp(x)+16*x^2-640*x+6340)/(144*exp(x)^2+(96*x-1920)*exp(x)+16*x^2-640*x+64
00),x, algorithm="giac")

[Out]

1/16*(16*x^2 + 48*x*e^x - 317*x)/(x + 3*e^x - 20)

Mupad [B] (verification not implemented)

Time = 9.50 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.76 \[ \int \frac {6340+144 e^{2 x}-640 x+16 x^2+e^x (-1911+87 x)}{6400+144 e^{2 x}-640 x+16 x^2+e^x (-1920+96 x)} \, dx=x+\frac {3\,x}{16\,x+48\,{\mathrm {e}}^x-320} \]

[In]

int((144*exp(2*x) - 640*x + exp(x)*(87*x - 1911) + 16*x^2 + 6340)/(144*exp(2*x) - 640*x + exp(x)*(96*x - 1920)
+ 16*x^2 + 6400),x)

[Out]

x + (3*x)/(16*x + 48*exp(x) - 320)