\(\int \frac {-1800+2700 x-3148 x^2+2925 x^3-1350 x^4+225 x^5+e^{2 x^2} (-1800+2700 x-1350 x^2+225 x^3)+e^{x^2} (-2+7199 x-10796 x^2+5398 x^3-900 x^4)}{-1800 x^2+2700 x^3-1350 x^4+225 x^5+e^{2 x^2} (-1800+2700 x-1350 x^2+225 x^3)+e^{x^2} (3600 x-5400 x^2+2700 x^3-450 x^4)} \, dx\) [7547]

Optimal result

Integrand size = 146, antiderivative size = 34 \[ \int \frac {-1800+2700 x-3148 x^2+2925 x^3-1350 x^4+225 x^5+e^{2 x^2} \left (-1800+2700 x-1350 x^2+225 x^3\right )+e^{x^2} \left (-2+7199 x-10796 x^2+5398 x^3-900 x^4\right )}{-1800 x^2+2700 x^3-1350 x^4+225 x^5+e^{2 x^2} \left (-1800+2700 x-1350 x^2+225 x^3\right )+e^{x^2} \left (3600 x-5400 x^2+2700 x^3-450 x^4\right )} \, dx=x-\frac {x+\frac {4 x^2}{225 (-4+2 x)^2}}{x \left (-e^{x^2}+x\right )} \]

[Out]

x-(x+1/225*x^2/(-2+x)^2)/(-exp(x^2)+x)/x

Rubi [F]

\[ \int \frac {-1800+2700 x-3148 x^2+2925 x^3-1350 x^4+225 x^5+e^{2 x^2} \left (-1800+2700 x-1350 x^2+225 x^3\right )+e^{x^2} \left (-2+7199 x-10796 x^2+5398 x^3-900 x^4\right )}{-1800 x^2+2700 x^3-1350 x^4+225 x^5+e^{2 x^2} \left (-1800+2700 x-1350 x^2+225 x^3\right )+e^{x^2} \left (3600 x-5400 x^2+2700 x^3-450 x^4\right )} \, dx=\int \frac {-1800+2700 x-3148 x^2+2925 x^3-1350 x^4+225 x^5+e^{2 x^2} \left (-1800+2700 x-1350 x^2+225 x^3\right )+e^{x^2} \left (-2+7199 x-10796 x^2+5398 x^3-900 x^4\right )}{-1800 x^2+2700 x^3-1350 x^4+225 x^5+e^{2 x^2} \left (-1800+2700 x-1350 x^2+225 x^3\right )+e^{x^2} \left (3600 x-5400 x^2+2700 x^3-450 x^4\right )} \, dx \]

[In]

Int[(-1800 + 2700*x - 3148*x^2 + 2925*x^3 - 1350*x^4 + 225*x^5 + E^(2*x^2)*(-1800 + 2700*x - 1350*x^2 + 225*x^
3) + E^x^2*(-2 + 7199*x - 10796*x^2 + 5398*x^3 - 900*x^4))/(-1800*x^2 + 2700*x^3 - 1350*x^4 + 225*x^5 + E^(2*x
^2)*(-1800 + 2700*x - 1350*x^2 + 225*x^3) + E^x^2*(3600*x - 5400*x^2 + 2700*x^3 - 450*x^4)),x]

[Out]

x + (217*Defer[Int][(E^x^2 - x)^(-2), x])/225 - (2*Defer[Int][(E^x^2 - x)^(-1), x])/225 - (4*Defer[Int][1/((E^
x^2 - x)*(-2 + x)^3), x])/225 - (14*Defer[Int][1/((E^x^2 - x)^2*(-2 + x)^2), x])/225 - Defer[Int][1/((E^x^2 -
x)*(-2 + x)^2), x]/25 - (23*Defer[Int][1/((E^x^2 - x)^2*(-2 + x)), x])/225 - (8*Defer[Int][1/((E^x^2 - x)*(-2
+ x)), x])/225 - (2*Defer[Int][x/(E^x^2 - x)^2, x])/225 - 2*Defer[Int][x/(E^x^2 - x), x] - 2*Defer[Int][x^2/(E
^x^2 - x)^2, x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {1800-225 e^{2 x^2} (-2+x)^3-2700 x+3148 x^2-2925 x^3+1350 x^4-225 x^5-e^{x^2} \left (-2+7199 x-10796 x^2+5398 x^3-900 x^4\right )}{225 (2-x)^3 \left (e^{x^2}-x\right )^2} \, dx \\ & = \frac {1}{225} \int \frac {1800-225 e^{2 x^2} (-2+x)^3-2700 x+3148 x^2-2925 x^3+1350 x^4-225 x^5-e^{x^2} \left (-2+7199 x-10796 x^2+5398 x^3-900 x^4\right )}{(2-x)^3 \left (e^{x^2}-x\right )^2} \, dx \\ & = \frac {1}{225} \int \left (225-\frac {2-3599 x+5396 x^2-2698 x^3+450 x^4}{\left (e^{x^2}-x\right ) (-2+x)^3}-\frac {-900+899 x+1575 x^2-1798 x^3+450 x^4}{\left (e^{x^2}-x\right )^2 (-2+x)^2}\right ) \, dx \\ & = x-\frac {1}{225} \int \frac {2-3599 x+5396 x^2-2698 x^3+450 x^4}{\left (e^{x^2}-x\right ) (-2+x)^3} \, dx-\frac {1}{225} \int \frac {-900+899 x+1575 x^2-1798 x^3+450 x^4}{\left (e^{x^2}-x\right )^2 (-2+x)^2} \, dx \\ & = x-\frac {1}{225} \int \left (\frac {2}{e^{x^2}-x}+\frac {4}{\left (e^{x^2}-x\right ) (-2+x)^3}+\frac {9}{\left (e^{x^2}-x\right ) (-2+x)^2}+\frac {8}{\left (e^{x^2}-x\right ) (-2+x)}+\frac {450 x}{e^{x^2}-x}\right ) \, dx-\frac {1}{225} \int \left (-\frac {217}{\left (e^{x^2}-x\right )^2}+\frac {14}{\left (e^{x^2}-x\right )^2 (-2+x)^2}+\frac {23}{\left (e^{x^2}-x\right )^2 (-2+x)}+\frac {2 x}{\left (e^{x^2}-x\right )^2}+\frac {450 x^2}{\left (e^{x^2}-x\right )^2}\right ) \, dx \\ & = x-\frac {2}{225} \int \frac {1}{e^{x^2}-x} \, dx-\frac {2}{225} \int \frac {x}{\left (e^{x^2}-x\right )^2} \, dx-\frac {4}{225} \int \frac {1}{\left (e^{x^2}-x\right ) (-2+x)^3} \, dx-\frac {8}{225} \int \frac {1}{\left (e^{x^2}-x\right ) (-2+x)} \, dx-\frac {1}{25} \int \frac {1}{\left (e^{x^2}-x\right ) (-2+x)^2} \, dx-\frac {14}{225} \int \frac {1}{\left (e^{x^2}-x\right )^2 (-2+x)^2} \, dx-\frac {23}{225} \int \frac {1}{\left (e^{x^2}-x\right )^2 (-2+x)} \, dx+\frac {217}{225} \int \frac {1}{\left (e^{x^2}-x\right )^2} \, dx-2 \int \frac {x}{e^{x^2}-x} \, dx-2 \int \frac {x^2}{\left (e^{x^2}-x\right )^2} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 13.98 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.09 \[ \int \frac {-1800+2700 x-3148 x^2+2925 x^3-1350 x^4+225 x^5+e^{2 x^2} \left (-1800+2700 x-1350 x^2+225 x^3\right )+e^{x^2} \left (-2+7199 x-10796 x^2+5398 x^3-900 x^4\right )}{-1800 x^2+2700 x^3-1350 x^4+225 x^5+e^{2 x^2} \left (-1800+2700 x-1350 x^2+225 x^3\right )+e^{x^2} \left (3600 x-5400 x^2+2700 x^3-450 x^4\right )} \, dx=\frac {1}{225} \left (225 (-2+x)+\frac {900-899 x+225 x^2}{\left (e^{x^2}-x\right ) (-2+x)^2}\right ) \]

[In]

Integrate[(-1800 + 2700*x - 3148*x^2 + 2925*x^3 - 1350*x^4 + 225*x^5 + E^(2*x^2)*(-1800 + 2700*x - 1350*x^2 +
225*x^3) + E^x^2*(-2 + 7199*x - 10796*x^2 + 5398*x^3 - 900*x^4))/(-1800*x^2 + 2700*x^3 - 1350*x^4 + 225*x^5 +
E^(2*x^2)*(-1800 + 2700*x - 1350*x^2 + 225*x^3) + E^x^2*(3600*x - 5400*x^2 + 2700*x^3 - 450*x^4)),x]

[Out]

(225*(-2 + x) + (900 - 899*x + 225*x^2)/((E^x^2 - x)*(-2 + x)^2))/225

Maple [A] (verified)

Time = 0.12 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.03

method	result	size
risch	\(x -\frac {225 x^{2}-899 x +900}{225 \left (x^{2}-4 x +4\right ) \left (-{\mathrm e}^{x^{2}}+x \right )}\)	\(35\)
norman	\(\frac {-4+x^{4}-16 \,{\mathrm e}^{x^{2}}-13 x^{2}+\frac {4499 x}{225}+12 \,{\mathrm e}^{x^{2}} x -x^{3} {\mathrm e}^{x^{2}}}{\left (-{\mathrm e}^{x^{2}}+x \right ) \left (-2+x \right )^{2}}\)	\(52\)
parallelrisch	\(\frac {225 x^{4}-225 x^{3} {\mathrm e}^{x^{2}}-900-2925 x^{2}+2700 \,{\mathrm e}^{x^{2}} x +4499 x -3600 \,{\mathrm e}^{x^{2}}}{225 x^{3}-225 x^{2} {\mathrm e}^{x^{2}}-900 x^{2}+900 \,{\mathrm e}^{x^{2}} x +900 x -900 \,{\mathrm e}^{x^{2}}}\)	\(76\)

[In]

int(((225*x^3-1350*x^2+2700*x-1800)*exp(x^2)^2+(-900*x^4+5398*x^3-10796*x^2+7199*x-2)*exp(x^2)+225*x^5-1350*x^
4+2925*x^3-3148*x^2+2700*x-1800)/((225*x^3-1350*x^2+2700*x-1800)*exp(x^2)^2+(-450*x^4+2700*x^3-5400*x^2+3600*x
)*exp(x^2)+225*x^5-1350*x^4+2700*x^3-1800*x^2),x,method=_RETURNVERBOSE)

[Out]

x-1/225*(225*x^2-899*x+900)/(x^2-4*x+4)/(-exp(x^2)+x)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (30) = 60\).

Time = 0.27 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.00 \[ \int \frac {-1800+2700 x-3148 x^2+2925 x^3-1350 x^4+225 x^5+e^{2 x^2} \left (-1800+2700 x-1350 x^2+225 x^3\right )+e^{x^2} \left (-2+7199 x-10796 x^2+5398 x^3-900 x^4\right )}{-1800 x^2+2700 x^3-1350 x^4+225 x^5+e^{2 x^2} \left (-1800+2700 x-1350 x^2+225 x^3\right )+e^{x^2} \left (3600 x-5400 x^2+2700 x^3-450 x^4\right )} \, dx=\frac {225 \, x^{4} - 900 \, x^{3} + 675 \, x^{2} - 225 \, {\left (x^{3} - 4 \, x^{2} + 4 \, x\right )} e^{\left (x^{2}\right )} + 899 \, x - 900}{225 \, {\left (x^{3} - 4 \, x^{2} - {\left (x^{2} - 4 \, x + 4\right )} e^{\left (x^{2}\right )} + 4 \, x\right )}} \]

[In]

integrate(((225*x^3-1350*x^2+2700*x-1800)*exp(x^2)^2+(-900*x^4+5398*x^3-10796*x^2+7199*x-2)*exp(x^2)+225*x^5-1
350*x^4+2925*x^3-3148*x^2+2700*x-1800)/((225*x^3-1350*x^2+2700*x-1800)*exp(x^2)^2+(-450*x^4+2700*x^3-5400*x^2+
3600*x)*exp(x^2)+225*x^5-1350*x^4+2700*x^3-1800*x^2),x, algorithm="fricas")

[Out]

1/225*(225*x^4 - 900*x^3 + 675*x^2 - 225*(x^3 - 4*x^2 + 4*x)*e^(x^2) + 899*x - 900)/(x^3 - 4*x^2 - (x^2 - 4*x
+ 4)*e^(x^2) + 4*x)

Sympy [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.15 \[ \int \frac {-1800+2700 x-3148 x^2+2925 x^3-1350 x^4+225 x^5+e^{2 x^2} \left (-1800+2700 x-1350 x^2+225 x^3\right )+e^{x^2} \left (-2+7199 x-10796 x^2+5398 x^3-900 x^4\right )}{-1800 x^2+2700 x^3-1350 x^4+225 x^5+e^{2 x^2} \left (-1800+2700 x-1350 x^2+225 x^3\right )+e^{x^2} \left (3600 x-5400 x^2+2700 x^3-450 x^4\right )} \, dx=x + \frac {225 x^{2} - 899 x + 900}{- 225 x^{3} + 900 x^{2} - 900 x + \left (225 x^{2} - 900 x + 900\right ) e^{x^{2}}} \]

[In]

integrate(((225*x**3-1350*x**2+2700*x-1800)*exp(x**2)**2+(-900*x**4+5398*x**3-10796*x**2+7199*x-2)*exp(x**2)+2
25*x**5-1350*x**4+2925*x**3-3148*x**2+2700*x-1800)/((225*x**3-1350*x**2+2700*x-1800)*exp(x**2)**2+(-450*x**4+2
700*x**3-5400*x**2+3600*x)*exp(x**2)+225*x**5-1350*x**4+2700*x**3-1800*x**2),x)

[Out]

x + (225*x**2 - 899*x + 900)/(-225*x**3 + 900*x**2 - 900*x + (225*x**2 - 900*x + 900)*exp(x**2))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (30) = 60\).

Time = 0.26 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.00 \[ \int \frac {-1800+2700 x-3148 x^2+2925 x^3-1350 x^4+225 x^5+e^{2 x^2} \left (-1800+2700 x-1350 x^2+225 x^3\right )+e^{x^2} \left (-2+7199 x-10796 x^2+5398 x^3-900 x^4\right )}{-1800 x^2+2700 x^3-1350 x^4+225 x^5+e^{2 x^2} \left (-1800+2700 x-1350 x^2+225 x^3\right )+e^{x^2} \left (3600 x-5400 x^2+2700 x^3-450 x^4\right )} \, dx=\frac {225 \, x^{4} - 900 \, x^{3} + 675 \, x^{2} - 225 \, {\left (x^{3} - 4 \, x^{2} + 4 \, x\right )} e^{\left (x^{2}\right )} + 899 \, x - 900}{225 \, {\left (x^{3} - 4 \, x^{2} - {\left (x^{2} - 4 \, x + 4\right )} e^{\left (x^{2}\right )} + 4 \, x\right )}} \]

[In]

integrate(((225*x^3-1350*x^2+2700*x-1800)*exp(x^2)^2+(-900*x^4+5398*x^3-10796*x^2+7199*x-2)*exp(x^2)+225*x^5-1
350*x^4+2925*x^3-3148*x^2+2700*x-1800)/((225*x^3-1350*x^2+2700*x-1800)*exp(x^2)^2+(-450*x^4+2700*x^3-5400*x^2+
3600*x)*exp(x^2)+225*x^5-1350*x^4+2700*x^3-1800*x^2),x, algorithm="maxima")

[Out]

1/225*(225*x^4 - 900*x^3 + 675*x^2 - 225*(x^3 - 4*x^2 + 4*x)*e^(x^2) + 899*x - 900)/(x^3 - 4*x^2 - (x^2 - 4*x
+ 4)*e^(x^2) + 4*x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 83 vs. \(2 (30) = 60\).

Time = 0.31 (sec) , antiderivative size = 83, normalized size of antiderivative = 2.44 \[ \int \frac {-1800+2700 x-3148 x^2+2925 x^3-1350 x^4+225 x^5+e^{2 x^2} \left (-1800+2700 x-1350 x^2+225 x^3\right )+e^{x^2} \left (-2+7199 x-10796 x^2+5398 x^3-900 x^4\right )}{-1800 x^2+2700 x^3-1350 x^4+225 x^5+e^{2 x^2} \left (-1800+2700 x-1350 x^2+225 x^3\right )+e^{x^2} \left (3600 x-5400 x^2+2700 x^3-450 x^4\right )} \, dx=\frac {225 \, x^{4} - 225 \, x^{3} e^{\left (x^{2}\right )} - 900 \, x^{3} + 900 \, x^{2} e^{\left (x^{2}\right )} + 675 \, x^{2} - 900 \, x e^{\left (x^{2}\right )} + 899 \, x - 900}{225 \, {\left (x^{3} - x^{2} e^{\left (x^{2}\right )} - 4 \, x^{2} + 4 \, x e^{\left (x^{2}\right )} + 4 \, x - 4 \, e^{\left (x^{2}\right )}\right )}} \]

[In]

integrate(((225*x^3-1350*x^2+2700*x-1800)*exp(x^2)^2+(-900*x^4+5398*x^3-10796*x^2+7199*x-2)*exp(x^2)+225*x^5-1
350*x^4+2925*x^3-3148*x^2+2700*x-1800)/((225*x^3-1350*x^2+2700*x-1800)*exp(x^2)^2+(-450*x^4+2700*x^3-5400*x^2+
3600*x)*exp(x^2)+225*x^5-1350*x^4+2700*x^3-1800*x^2),x, algorithm="giac")

[Out]

1/225*(225*x^4 - 225*x^3*e^(x^2) - 900*x^3 + 900*x^2*e^(x^2) + 675*x^2 - 900*x*e^(x^2) + 899*x - 900)/(x^3 - x
^2*e^(x^2) - 4*x^2 + 4*x*e^(x^2) + 4*x - 4*e^(x^2))

Mupad [B] (verification not implemented)

Time = 13.17 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.79 \[ \int \frac {-1800+2700 x-3148 x^2+2925 x^3-1350 x^4+225 x^5+e^{2 x^2} \left (-1800+2700 x-1350 x^2+225 x^3\right )+e^{x^2} \left (-2+7199 x-10796 x^2+5398 x^3-900 x^4\right )}{-1800 x^2+2700 x^3-1350 x^4+225 x^5+e^{2 x^2} \left (-1800+2700 x-1350 x^2+225 x^3\right )+e^{x^2} \left (3600 x-5400 x^2+2700 x^3-450 x^4\right )} \, dx=x-\frac {x^2-\frac {899\,x}{225}+4}{\left (x-{\mathrm {e}}^{x^2}\right )\,{\left (x-2\right )}^2} \]

[In]

int((2700*x - exp(x^2)*(10796*x^2 - 7199*x - 5398*x^3 + 900*x^4 + 2) + exp(2*x^2)*(2700*x - 1350*x^2 + 225*x^3
- 1800) - 3148*x^2 + 2925*x^3 - 1350*x^4 + 225*x^5 - 1800)/(exp(x^2)*(3600*x - 5400*x^2 + 2700*x^3 - 450*x^4)
+ exp(2*x^2)*(2700*x - 1350*x^2 + 225*x^3 - 1800) - 1800*x^2 + 2700*x^3 - 1350*x^4 + 225*x^5),x)

[Out]

x - (x^2 - (899*x)/225 + 4)/((x - exp(x^2))*(x - 2)^2)