\(\int \frac {12 x+16 x^2+e^x (-2 x^2+2 x^3)}{9+48 x+64 x^2+e^{2 x} x^2+e^x (-6 x-16 x^2)} \, dx\) [2858]

Optimal result

Integrand size = 59, antiderivative size = 23 \[ \int \frac {12 x+16 x^2+e^x \left (-2 x^2+2 x^3\right )}{9+48 x+64 x^2+e^{2 x} x^2+e^x \left (-6 x-16 x^2\right )} \, dx=\frac {2 x^2}{x+\left (7-e^x+\frac {3}{x}\right ) x} \]

[Out]

2*x^2/(x*(3/x+7-exp(x))+x)

Rubi [F]

\[ \int \frac {12 x+16 x^2+e^x \left (-2 x^2+2 x^3\right )}{9+48 x+64 x^2+e^{2 x} x^2+e^x \left (-6 x-16 x^2\right )} \, dx=\int \frac {12 x+16 x^2+e^x \left (-2 x^2+2 x^3\right )}{9+48 x+64 x^2+e^{2 x} x^2+e^x \left (-6 x-16 x^2\right )} \, dx \]

[In]

Int[(12*x + 16*x^2 + E^x*(-2*x^2 + 2*x^3))/(9 + 48*x + 64*x^2 + E^(2*x)*x^2 + E^x*(-6*x - 16*x^2)),x]

[Out]

6*Defer[Int][x/(-3 - 8*x + E^x*x)^2, x] + 6*Defer[Int][x^2/(-3 - 8*x + E^x*x)^2, x] + 16*Defer[Int][x^3/(-3 -
8*x + E^x*x)^2, x] - 2*Defer[Int][x/(-3 - 8*x + E^x*x), x] + 2*Defer[Int][x^2/(-3 - 8*x + E^x*x), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {2 x \left (6-\left (-8+e^x\right ) x+e^x x^2\right )}{\left (3-\left (-8+e^x\right ) x\right )^2} \, dx \\ & = 2 \int \frac {x \left (6-\left (-8+e^x\right ) x+e^x x^2\right )}{\left (3-\left (-8+e^x\right ) x\right )^2} \, dx \\ & = 2 \int \left (\frac {(-1+x) x}{-3-8 x+e^x x}+\frac {x \left (3+3 x+8 x^2\right )}{\left (-3-8 x+e^x x\right )^2}\right ) \, dx \\ & = 2 \int \frac {(-1+x) x}{-3-8 x+e^x x} \, dx+2 \int \frac {x \left (3+3 x+8 x^2\right )}{\left (-3-8 x+e^x x\right )^2} \, dx \\ & = 2 \int \left (\frac {3 x}{\left (-3-8 x+e^x x\right )^2}+\frac {3 x^2}{\left (-3-8 x+e^x x\right )^2}+\frac {8 x^3}{\left (-3-8 x+e^x x\right )^2}\right ) \, dx+2 \int \left (-\frac {x}{-3-8 x+e^x x}+\frac {x^2}{-3-8 x+e^x x}\right ) \, dx \\ & = -\left (2 \int \frac {x}{-3-8 x+e^x x} \, dx\right )+2 \int \frac {x^2}{-3-8 x+e^x x} \, dx+6 \int \frac {x}{\left (-3-8 x+e^x x\right )^2} \, dx+6 \int \frac {x^2}{\left (-3-8 x+e^x x\right )^2} \, dx+16 \int \frac {x^3}{\left (-3-8 x+e^x x\right )^2} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 1.05 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.74 \[ \int \frac {12 x+16 x^2+e^x \left (-2 x^2+2 x^3\right )}{9+48 x+64 x^2+e^{2 x} x^2+e^x \left (-6 x-16 x^2\right )} \, dx=-\frac {2 x^2}{-3-8 x+e^x x} \]

[In]

Integrate[(12*x + 16*x^2 + E^x*(-2*x^2 + 2*x^3))/(9 + 48*x + 64*x^2 + E^(2*x)*x^2 + E^x*(-6*x - 16*x^2)),x]

[Out]

(-2*x^2)/(-3 - 8*x + E^x*x)

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.74

method	result	size
norman	\(-\frac {2 x^{2}}{{\mathrm e}^{x} x -8 x -3}\)	\(17\)
risch	\(-\frac {2 x^{2}}{{\mathrm e}^{x} x -8 x -3}\)	\(17\)
parallelrisch	\(-\frac {2 x^{2}}{{\mathrm e}^{x} x -8 x -3}\)	\(17\)

[In]

int(((2*x^3-2*x^2)*exp(x)+16*x^2+12*x)/(exp(x)^2*x^2+(-16*x^2-6*x)*exp(x)+64*x^2+48*x+9),x,method=_RETURNVERBO
SE)

[Out]

-2*x^2/(exp(x)*x-8*x-3)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.70 \[ \int \frac {12 x+16 x^2+e^x \left (-2 x^2+2 x^3\right )}{9+48 x+64 x^2+e^{2 x} x^2+e^x \left (-6 x-16 x^2\right )} \, dx=-\frac {2 \, x^{2}}{x e^{x} - 8 \, x - 3} \]

[In]

integrate(((2*x^3-2*x^2)*exp(x)+16*x^2+12*x)/(exp(x)^2*x^2+(-16*x^2-6*x)*exp(x)+64*x^2+48*x+9),x, algorithm="f
ricas")

[Out]

-2*x^2/(x*e^x - 8*x - 3)

Sympy [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.65 \[ \int \frac {12 x+16 x^2+e^x \left (-2 x^2+2 x^3\right )}{9+48 x+64 x^2+e^{2 x} x^2+e^x \left (-6 x-16 x^2\right )} \, dx=- \frac {2 x^{2}}{x e^{x} - 8 x - 3} \]

[In]

integrate(((2*x**3-2*x**2)*exp(x)+16*x**2+12*x)/(exp(x)**2*x**2+(-16*x**2-6*x)*exp(x)+64*x**2+48*x+9),x)

[Out]

-2*x**2/(x*exp(x) - 8*x - 3)

Maxima [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.70 \[ \int \frac {12 x+16 x^2+e^x \left (-2 x^2+2 x^3\right )}{9+48 x+64 x^2+e^{2 x} x^2+e^x \left (-6 x-16 x^2\right )} \, dx=-\frac {2 \, x^{2}}{x e^{x} - 8 \, x - 3} \]

[In]

integrate(((2*x^3-2*x^2)*exp(x)+16*x^2+12*x)/(exp(x)^2*x^2+(-16*x^2-6*x)*exp(x)+64*x^2+48*x+9),x, algorithm="m
axima")

[Out]

-2*x^2/(x*e^x - 8*x - 3)

Giac [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.70 \[ \int \frac {12 x+16 x^2+e^x \left (-2 x^2+2 x^3\right )}{9+48 x+64 x^2+e^{2 x} x^2+e^x \left (-6 x-16 x^2\right )} \, dx=-\frac {2 \, x^{2}}{x e^{x} - 8 \, x - 3} \]

[In]

integrate(((2*x^3-2*x^2)*exp(x)+16*x^2+12*x)/(exp(x)^2*x^2+(-16*x^2-6*x)*exp(x)+64*x^2+48*x+9),x, algorithm="g
iac")

[Out]

-2*x^2/(x*e^x - 8*x - 3)

Mupad [B] (verification not implemented)

Time = 9.16 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.74 \[ \int \frac {12 x+16 x^2+e^x \left (-2 x^2+2 x^3\right )}{9+48 x+64 x^2+e^{2 x} x^2+e^x \left (-6 x-16 x^2\right )} \, dx=\frac {2\,x^2}{8\,x-x\,{\mathrm {e}}^x+3} \]

[In]

int((12*x - exp(x)*(2*x^2 - 2*x^3) + 16*x^2)/(48*x + x^2*exp(2*x) - exp(x)*(6*x + 16*x^2) + 64*x^2 + 9),x)

[Out]

(2*x^2)/(8*x - x*exp(x) + 3)