\(\int \frac {3+2 x^3+e^{x+17 x^2} (-1+x+34 x^2)}{9+e^{2 x+34 x^2}-18 x+9 x^2-6 x^3+6 x^4+x^6+e^{x+17 x^2} (-6+6 x+2 x^3)} \, dx\) [977]

Optimal result

Integrand size = 82, antiderivative size = 28 \[ \int \frac {3+2 x^3+e^{x+17 x^2} \left (-1+x+34 x^2\right )}{9+e^{2 x+34 x^2}-18 x+9 x^2-6 x^3+6 x^4+x^6+e^{x+17 x^2} \left (-6+6 x+2 x^3\right )} \, dx=\frac {x}{3-x \left (3+\frac {e^{x+17 x^2}+x^3}{x}\right )} \]

[Out]

x/(3-(3+(x^3+exp(x)*exp(17*x^2))/x)*x)

Rubi [F]

\[ \int \frac {3+2 x^3+e^{x+17 x^2} \left (-1+x+34 x^2\right )}{9+e^{2 x+34 x^2}-18 x+9 x^2-6 x^3+6 x^4+x^6+e^{x+17 x^2} \left (-6+6 x+2 x^3\right )} \, dx=\int \frac {3+2 x^3+e^{x+17 x^2} \left (-1+x+34 x^2\right )}{9+e^{2 x+34 x^2}-18 x+9 x^2-6 x^3+6 x^4+x^6+e^{x+17 x^2} \left (-6+6 x+2 x^3\right )} \, dx \]

[In]

Int[(3 + 2*x^3 + E^(x + 17*x^2)*(-1 + x + 34*x^2))/(9 + E^(2*x + 34*x^2) - 18*x + 9*x^2 - 6*x^3 + 6*x^4 + x^6
+ E^(x + 17*x^2)*(-6 + 6*x + 2*x^3)),x]

[Out]

6*Defer[Int][x/(-3 + E^(x + 17*x^2) + 3*x + x^3)^2, x] + 99*Defer[Int][x^2/(-3 + E^(x + 17*x^2) + 3*x + x^3)^2
, x] - 99*Defer[Int][x^3/(-3 + E^(x + 17*x^2) + 3*x + x^3)^2, x] - Defer[Int][x^4/(-3 + E^(x + 17*x^2) + 3*x +
x^3)^2, x] - 34*Defer[Int][x^5/(-3 + E^(x + 17*x^2) + 3*x + x^3)^2, x] - Defer[Int][(-3 + E^(x + 17*x^2) + 3*
x + x^3)^(-1), x] + Defer[Int][x/(-3 + E^(x + 17*x^2) + 3*x + x^3), x] + 34*Defer[Int][x^2/(-3 + E^(x + 17*x^2
) + 3*x + x^3), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {3+2 x^3+e^{x+17 x^2} \left (-1+x+34 x^2\right )}{\left (3-e^{x+17 x^2}-3 x-x^3\right )^2} \, dx \\ & = \int \left (\frac {-1+x+34 x^2}{-3+e^{x+17 x^2}+3 x+x^3}-\frac {x \left (-6-99 x+99 x^2+x^3+34 x^4\right )}{\left (-3+e^{x+17 x^2}+3 x+x^3\right )^2}\right ) \, dx \\ & = \int \frac {-1+x+34 x^2}{-3+e^{x+17 x^2}+3 x+x^3} \, dx-\int \frac {x \left (-6-99 x+99 x^2+x^3+34 x^4\right )}{\left (-3+e^{x+17 x^2}+3 x+x^3\right )^2} \, dx \\ & = -\int \left (-\frac {6 x}{\left (-3+e^{x+17 x^2}+3 x+x^3\right )^2}-\frac {99 x^2}{\left (-3+e^{x+17 x^2}+3 x+x^3\right )^2}+\frac {99 x^3}{\left (-3+e^{x+17 x^2}+3 x+x^3\right )^2}+\frac {x^4}{\left (-3+e^{x+17 x^2}+3 x+x^3\right )^2}+\frac {34 x^5}{\left (-3+e^{x+17 x^2}+3 x+x^3\right )^2}\right ) \, dx+\int \left (-\frac {1}{-3+e^{x+17 x^2}+3 x+x^3}+\frac {x}{-3+e^{x+17 x^2}+3 x+x^3}+\frac {34 x^2}{-3+e^{x+17 x^2}+3 x+x^3}\right ) \, dx \\ & = 6 \int \frac {x}{\left (-3+e^{x+17 x^2}+3 x+x^3\right )^2} \, dx-34 \int \frac {x^5}{\left (-3+e^{x+17 x^2}+3 x+x^3\right )^2} \, dx+34 \int \frac {x^2}{-3+e^{x+17 x^2}+3 x+x^3} \, dx+99 \int \frac {x^2}{\left (-3+e^{x+17 x^2}+3 x+x^3\right )^2} \, dx-99 \int \frac {x^3}{\left (-3+e^{x+17 x^2}+3 x+x^3\right )^2} \, dx-\int \frac {x^4}{\left (-3+e^{x+17 x^2}+3 x+x^3\right )^2} \, dx-\int \frac {1}{-3+e^{x+17 x^2}+3 x+x^3} \, dx+\int \frac {x}{-3+e^{x+17 x^2}+3 x+x^3} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 3.84 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.79 \[ \int \frac {3+2 x^3+e^{x+17 x^2} \left (-1+x+34 x^2\right )}{9+e^{2 x+34 x^2}-18 x+9 x^2-6 x^3+6 x^4+x^6+e^{x+17 x^2} \left (-6+6 x+2 x^3\right )} \, dx=-\frac {x}{-3+e^{x+17 x^2}+3 x+x^3} \]

[In]

Integrate[(3 + 2*x^3 + E^(x + 17*x^2)*(-1 + x + 34*x^2))/(9 + E^(2*x + 34*x^2) - 18*x + 9*x^2 - 6*x^3 + 6*x^4
+ x^6 + E^(x + 17*x^2)*(-6 + 6*x + 2*x^3)),x]

[Out]

-(x/(-3 + E^(x + 17*x^2) + 3*x + x^3))

Maple [A] (verified)

Time = 0.12 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.79

method	result	size
risch	\(-\frac {x}{x^{3}+{\mathrm e}^{x \left (17 x +1\right )}+3 x -3}\)	\(22\)
parallelrisch	\(-\frac {x}{x^{3}+{\mathrm e}^{x} {\mathrm e}^{17 x^{2}}+3 x -3}\)	\(23\)

[In]

int(((34*x^2+x-1)*exp(x)*exp(17*x^2)+2*x^3+3)/(exp(x)^2*exp(17*x^2)^2+(2*x^3+6*x-6)*exp(x)*exp(17*x^2)+x^6+6*x
^4-6*x^3+9*x^2-18*x+9),x,method=_RETURNVERBOSE)

[Out]

-x/(x^3+exp(x*(17*x+1))+3*x-3)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.75 \[ \int \frac {3+2 x^3+e^{x+17 x^2} \left (-1+x+34 x^2\right )}{9+e^{2 x+34 x^2}-18 x+9 x^2-6 x^3+6 x^4+x^6+e^{x+17 x^2} \left (-6+6 x+2 x^3\right )} \, dx=-\frac {x}{x^{3} + 3 \, x + e^{\left (17 \, x^{2} + x\right )} - 3} \]

[In]

integrate(((34*x^2+x-1)*exp(x)*exp(17*x^2)+2*x^3+3)/(exp(x)^2*exp(17*x^2)^2+(2*x^3+6*x-6)*exp(x)*exp(17*x^2)+x
^6+6*x^4-6*x^3+9*x^2-18*x+9),x, algorithm="fricas")

[Out]

-x/(x^3 + 3*x + e^(17*x^2 + x) - 3)

Sympy [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.71 \[ \int \frac {3+2 x^3+e^{x+17 x^2} \left (-1+x+34 x^2\right )}{9+e^{2 x+34 x^2}-18 x+9 x^2-6 x^3+6 x^4+x^6+e^{x+17 x^2} \left (-6+6 x+2 x^3\right )} \, dx=- \frac {x}{x^{3} + 3 x + e^{x} e^{17 x^{2}} - 3} \]

[In]

integrate(((34*x**2+x-1)*exp(x)*exp(17*x**2)+2*x**3+3)/(exp(x)**2*exp(17*x**2)**2+(2*x**3+6*x-6)*exp(x)*exp(17
*x**2)+x**6+6*x**4-6*x**3+9*x**2-18*x+9),x)

[Out]

-x/(x**3 + 3*x + exp(x)*exp(17*x**2) - 3)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.75 \[ \int \frac {3+2 x^3+e^{x+17 x^2} \left (-1+x+34 x^2\right )}{9+e^{2 x+34 x^2}-18 x+9 x^2-6 x^3+6 x^4+x^6+e^{x+17 x^2} \left (-6+6 x+2 x^3\right )} \, dx=-\frac {x}{x^{3} + 3 \, x + e^{\left (17 \, x^{2} + x\right )} - 3} \]

[In]

integrate(((34*x^2+x-1)*exp(x)*exp(17*x^2)+2*x^3+3)/(exp(x)^2*exp(17*x^2)^2+(2*x^3+6*x-6)*exp(x)*exp(17*x^2)+x
^6+6*x^4-6*x^3+9*x^2-18*x+9),x, algorithm="maxima")

[Out]

-x/(x^3 + 3*x + e^(17*x^2 + x) - 3)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.75 \[ \int \frac {3+2 x^3+e^{x+17 x^2} \left (-1+x+34 x^2\right )}{9+e^{2 x+34 x^2}-18 x+9 x^2-6 x^3+6 x^4+x^6+e^{x+17 x^2} \left (-6+6 x+2 x^3\right )} \, dx=-\frac {x}{x^{3} + 3 \, x + e^{\left (17 \, x^{2} + x\right )} - 3} \]

[In]

integrate(((34*x^2+x-1)*exp(x)*exp(17*x^2)+2*x^3+3)/(exp(x)^2*exp(17*x^2)^2+(2*x^3+6*x-6)*exp(x)*exp(17*x^2)+x
^6+6*x^4-6*x^3+9*x^2-18*x+9),x, algorithm="giac")

[Out]

-x/(x^3 + 3*x + e^(17*x^2 + x) - 3)

Mupad [B] (verification not implemented)

Time = 9.14 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.75 \[ \int \frac {3+2 x^3+e^{x+17 x^2} \left (-1+x+34 x^2\right )}{9+e^{2 x+34 x^2}-18 x+9 x^2-6 x^3+6 x^4+x^6+e^{x+17 x^2} \left (-6+6 x+2 x^3\right )} \, dx=-\frac {x}{3\,x+{\mathrm {e}}^{17\,x^2+x}+x^3-3} \]

[In]

int((2*x^3 + exp(17*x^2)*exp(x)*(x + 34*x^2 - 1) + 3)/(exp(2*x)*exp(34*x^2) - 18*x + 9*x^2 - 6*x^3 + 6*x^4 + x
^6 + exp(17*x^2)*exp(x)*(6*x + 2*x^3 - 6) + 9),x)

[Out]

-x/(3*x + exp(x + 17*x^2) + x^3 - 3)