\(\int \frac {-15 x^2+e^{\frac {2-x^3+x^4}{x}} (10+10 x^3-15 x^4)}{12 x^2+3 e^{\frac {2 (2-x^3+x^4)}{x}} x^2-36 x^3+27 x^4+e^{\frac {2-x^3+x^4}{x}} (-12 x^2+18 x^3)} \, dx\) [8757]

Optimal result

Integrand size = 104, antiderivative size = 34 \[ \int \frac {-15 x^2+e^{\frac {2-x^3+x^4}{x}} \left (10+10 x^3-15 x^4\right )}{12 x^2+3 e^{\frac {2 \left (2-x^3+x^4\right )}{x}} x^2-36 x^3+27 x^4+e^{\frac {2-x^3+x^4}{x}} \left (-12 x^2+18 x^3\right )} \, dx=\frac {5}{\left (9+\frac {3 \left (-2+e^{\frac {2}{x}-x \left (x-x^2\right )}\right )}{x}\right ) x} \]

[Out]

5/(9+3*(exp(2/x-x*(-x^2+x))-2)/x)/x

Rubi [F]

\[ \int \frac {-15 x^2+e^{\frac {2-x^3+x^4}{x}} \left (10+10 x^3-15 x^4\right )}{12 x^2+3 e^{\frac {2 \left (2-x^3+x^4\right )}{x}} x^2-36 x^3+27 x^4+e^{\frac {2-x^3+x^4}{x}} \left (-12 x^2+18 x^3\right )} \, dx=\int \frac {-15 x^2+e^{\frac {2-x^3+x^4}{x}} \left (10+10 x^3-15 x^4\right )}{12 x^2+3 e^{\frac {2 \left (2-x^3+x^4\right )}{x}} x^2-36 x^3+27 x^4+e^{\frac {2-x^3+x^4}{x}} \left (-12 x^2+18 x^3\right )} \, dx \]

[In]

Int[(-15*x^2 + E^((2 - x^3 + x^4)/x)*(10 + 10*x^3 - 15*x^4))/(12*x^2 + 3*E^((2*(2 - x^3 + x^4))/x)*x^2 - 36*x^
3 + 27*x^4 + E^((2 - x^3 + x^4)/x)*(-12*x^2 + 18*x^3)),x]

[Out]

-5*Defer[Int][E^(2*x^2)/(-2*E^x^2 + E^(2/x + x^3) + 3*E^x^2*x)^2, x] + (20*Defer[Int][E^(2*x^2)/(x^2*(-2*E^x^2
+ E^(2/x + x^3) + 3*E^x^2*x)^2), x])/3 - 10*Defer[Int][E^(2*x^2)/(x*(-2*E^x^2 + E^(2/x + x^3) + 3*E^x^2*x)^2)
, x] + (20*Defer[Int][(E^(2*x^2)*x)/(-2*E^x^2 + E^(2/x + x^3) + 3*E^x^2*x)^2, x])/3 - 20*Defer[Int][(E^(2*x^2)
*x^2)/(-2*E^x^2 + E^(2/x + x^3) + 3*E^x^2*x)^2, x] + 15*Defer[Int][(E^(2*x^2)*x^3)/(-2*E^x^2 + E^(2/x + x^3) +
3*E^x^2*x)^2, x] + (10*Defer[Int][E^x^2/(x^2*(-2*E^x^2 + E^(2/x + x^3) + 3*E^x^2*x)), x])/3 + (10*Defer[Int][
(E^x^2*x)/(-2*E^x^2 + E^(2/x + x^3) + 3*E^x^2*x), x])/3 - 5*Defer[Int][(E^x^2*x^2)/(-2*E^x^2 + E^(2/x + x^3) +
3*E^x^2*x), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {5 e^{x^2} \left (-3 e^{x^2} x^2-e^{\frac {2}{x}+x^3} \left (-2-2 x^3+3 x^4\right )\right )}{3 x^2 \left (e^{\frac {2}{x}+x^3}+e^{x^2} (-2+3 x)\right )^2} \, dx \\ & = \frac {5}{3} \int \frac {e^{x^2} \left (-3 e^{x^2} x^2-e^{\frac {2}{x}+x^3} \left (-2-2 x^3+3 x^4\right )\right )}{x^2 \left (e^{\frac {2}{x}+x^3}+e^{x^2} (-2+3 x)\right )^2} \, dx \\ & = \frac {5}{3} \int \left (-\frac {e^{x^2} \left (-2-2 x^3+3 x^4\right )}{x^2 \left (-2 e^{x^2}+e^{\frac {2}{x}+x^3}+3 e^{x^2} x\right )}+\frac {e^{2 x^2} \left (4-6 x-3 x^2+4 x^3-12 x^4+9 x^5\right )}{x^2 \left (-2 e^{x^2}+e^{\frac {2}{x}+x^3}+3 e^{x^2} x\right )^2}\right ) \, dx \\ & = -\left (\frac {5}{3} \int \frac {e^{x^2} \left (-2-2 x^3+3 x^4\right )}{x^2 \left (-2 e^{x^2}+e^{\frac {2}{x}+x^3}+3 e^{x^2} x\right )} \, dx\right )+\frac {5}{3} \int \frac {e^{2 x^2} \left (4-6 x-3 x^2+4 x^3-12 x^4+9 x^5\right )}{x^2 \left (-2 e^{x^2}+e^{\frac {2}{x}+x^3}+3 e^{x^2} x\right )^2} \, dx \\ & = \frac {5}{3} \int \left (-\frac {3 e^{2 x^2}}{\left (-2 e^{x^2}+e^{\frac {2}{x}+x^3}+3 e^{x^2} x\right )^2}+\frac {4 e^{2 x^2}}{x^2 \left (-2 e^{x^2}+e^{\frac {2}{x}+x^3}+3 e^{x^2} x\right )^2}-\frac {6 e^{2 x^2}}{x \left (-2 e^{x^2}+e^{\frac {2}{x}+x^3}+3 e^{x^2} x\right )^2}+\frac {4 e^{2 x^2} x}{\left (-2 e^{x^2}+e^{\frac {2}{x}+x^3}+3 e^{x^2} x\right )^2}-\frac {12 e^{2 x^2} x^2}{\left (-2 e^{x^2}+e^{\frac {2}{x}+x^3}+3 e^{x^2} x\right )^2}+\frac {9 e^{2 x^2} x^3}{\left (-2 e^{x^2}+e^{\frac {2}{x}+x^3}+3 e^{x^2} x\right )^2}\right ) \, dx-\frac {5}{3} \int \left (-\frac {2 e^{x^2}}{x^2 \left (-2 e^{x^2}+e^{\frac {2}{x}+x^3}+3 e^{x^2} x\right )}-\frac {2 e^{x^2} x}{-2 e^{x^2}+e^{\frac {2}{x}+x^3}+3 e^{x^2} x}+\frac {3 e^{x^2} x^2}{-2 e^{x^2}+e^{\frac {2}{x}+x^3}+3 e^{x^2} x}\right ) \, dx \\ & = \frac {10}{3} \int \frac {e^{x^2}}{x^2 \left (-2 e^{x^2}+e^{\frac {2}{x}+x^3}+3 e^{x^2} x\right )} \, dx+\frac {10}{3} \int \frac {e^{x^2} x}{-2 e^{x^2}+e^{\frac {2}{x}+x^3}+3 e^{x^2} x} \, dx-5 \int \frac {e^{2 x^2}}{\left (-2 e^{x^2}+e^{\frac {2}{x}+x^3}+3 e^{x^2} x\right )^2} \, dx-5 \int \frac {e^{x^2} x^2}{-2 e^{x^2}+e^{\frac {2}{x}+x^3}+3 e^{x^2} x} \, dx+\frac {20}{3} \int \frac {e^{2 x^2}}{x^2 \left (-2 e^{x^2}+e^{\frac {2}{x}+x^3}+3 e^{x^2} x\right )^2} \, dx+\frac {20}{3} \int \frac {e^{2 x^2} x}{\left (-2 e^{x^2}+e^{\frac {2}{x}+x^3}+3 e^{x^2} x\right )^2} \, dx-10 \int \frac {e^{2 x^2}}{x \left (-2 e^{x^2}+e^{\frac {2}{x}+x^3}+3 e^{x^2} x\right )^2} \, dx+15 \int \frac {e^{2 x^2} x^3}{\left (-2 e^{x^2}+e^{\frac {2}{x}+x^3}+3 e^{x^2} x\right )^2} \, dx-20 \int \frac {e^{2 x^2} x^2}{\left (-2 e^{x^2}+e^{\frac {2}{x}+x^3}+3 e^{x^2} x\right )^2} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 1.22 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00 \[ \int \frac {-15 x^2+e^{\frac {2-x^3+x^4}{x}} \left (10+10 x^3-15 x^4\right )}{12 x^2+3 e^{\frac {2 \left (2-x^3+x^4\right )}{x}} x^2-36 x^3+27 x^4+e^{\frac {2-x^3+x^4}{x}} \left (-12 x^2+18 x^3\right )} \, dx=\frac {5 e^{x^2}}{3 \left (e^{\frac {2}{x}+x^3}+e^{x^2} (-2+3 x)\right )} \]

[In]

Integrate[(-15*x^2 + E^((2 - x^3 + x^4)/x)*(10 + 10*x^3 - 15*x^4))/(12*x^2 + 3*E^((2*(2 - x^3 + x^4))/x)*x^2 -
36*x^3 + 27*x^4 + E^((2 - x^3 + x^4)/x)*(-12*x^2 + 18*x^3)),x]

[Out]

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

Maple [A] (verified)

Time = 0.23 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.74

method	result	size
norman	\(\frac {5}{3 \left (-2+{\mathrm e}^{\frac {x^{4}-x^{3}+2}{x}}+3 x \right )}\)	\(25\)
risch	\(\frac {5}{3 \left (-2+{\mathrm e}^{\frac {x^{4}-x^{3}+2}{x}}+3 x \right )}\)	\(25\)
parallelrisch	\(\frac {5}{3 \left (-2+{\mathrm e}^{\frac {x^{4}-x^{3}+2}{x}}+3 x \right )}\)	\(25\)

[In]

int(((-15*x^4+10*x^3+10)*exp((x^4-x^3+2)/x)-15*x^2)/(3*x^2*exp((x^4-x^3+2)/x)^2+(18*x^3-12*x^2)*exp((x^4-x^3+2
)/x)+27*x^4-36*x^3+12*x^2),x,method=_RETURNVERBOSE)

[Out]

5/3/(-2+exp((x^4-x^3+2)/x)+3*x)

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.71 \[ \int \frac {-15 x^2+e^{\frac {2-x^3+x^4}{x}} \left (10+10 x^3-15 x^4\right )}{12 x^2+3 e^{\frac {2 \left (2-x^3+x^4\right )}{x}} x^2-36 x^3+27 x^4+e^{\frac {2-x^3+x^4}{x}} \left (-12 x^2+18 x^3\right )} \, dx=\frac {5}{3 \, {\left (3 \, x + e^{\left (\frac {x^{4} - x^{3} + 2}{x}\right )} - 2\right )}} \]

[In]

integrate(((-15*x^4+10*x^3+10)*exp((x^4-x^3+2)/x)-15*x^2)/(3*x^2*exp((x^4-x^3+2)/x)^2+(18*x^3-12*x^2)*exp((x^4
-x^3+2)/x)+27*x^4-36*x^3+12*x^2),x, algorithm="fricas")

[Out]

5/3/(3*x + e^((x^4 - x^3 + 2)/x) - 2)

Sympy [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.56 \[ \int \frac {-15 x^2+e^{\frac {2-x^3+x^4}{x}} \left (10+10 x^3-15 x^4\right )}{12 x^2+3 e^{\frac {2 \left (2-x^3+x^4\right )}{x}} x^2-36 x^3+27 x^4+e^{\frac {2-x^3+x^4}{x}} \left (-12 x^2+18 x^3\right )} \, dx=\frac {5}{9 x + 3 e^{\frac {x^{4} - x^{3} + 2}{x}} - 6} \]

[In]

integrate(((-15*x**4+10*x**3+10)*exp((x**4-x**3+2)/x)-15*x**2)/(3*x**2*exp((x**4-x**3+2)/x)**2+(18*x**3-12*x**
2)*exp((x**4-x**3+2)/x)+27*x**4-36*x**3+12*x**2),x)

[Out]

5/(9*x + 3*exp((x**4 - x**3 + 2)/x) - 6)

Maxima [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.85 \[ \int \frac {-15 x^2+e^{\frac {2-x^3+x^4}{x}} \left (10+10 x^3-15 x^4\right )}{12 x^2+3 e^{\frac {2 \left (2-x^3+x^4\right )}{x}} x^2-36 x^3+27 x^4+e^{\frac {2-x^3+x^4}{x}} \left (-12 x^2+18 x^3\right )} \, dx=\frac {5 \, e^{\left (x^{2}\right )}}{3 \, {\left ({\left (3 \, x - 2\right )} e^{\left (x^{2}\right )} + e^{\left (x^{3} + \frac {2}{x}\right )}\right )}} \]

[In]

integrate(((-15*x^4+10*x^3+10)*exp((x^4-x^3+2)/x)-15*x^2)/(3*x^2*exp((x^4-x^3+2)/x)^2+(18*x^3-12*x^2)*exp((x^4
-x^3+2)/x)+27*x^4-36*x^3+12*x^2),x, algorithm="maxima")

[Out]

5/3*e^(x^2)/((3*x - 2)*e^(x^2) + e^(x^3 + 2/x))

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.71 \[ \int \frac {-15 x^2+e^{\frac {2-x^3+x^4}{x}} \left (10+10 x^3-15 x^4\right )}{12 x^2+3 e^{\frac {2 \left (2-x^3+x^4\right )}{x}} x^2-36 x^3+27 x^4+e^{\frac {2-x^3+x^4}{x}} \left (-12 x^2+18 x^3\right )} \, dx=\frac {5}{3 \, {\left (3 \, x + e^{\left (\frac {x^{4} - x^{3} + 2}{x}\right )} - 2\right )}} \]

[In]

integrate(((-15*x^4+10*x^3+10)*exp((x^4-x^3+2)/x)-15*x^2)/(3*x^2*exp((x^4-x^3+2)/x)^2+(18*x^3-12*x^2)*exp((x^4
-x^3+2)/x)+27*x^4-36*x^3+12*x^2),x, algorithm="giac")

[Out]

5/3/(3*x + e^((x^4 - x^3 + 2)/x) - 2)

Mupad [B] (verification not implemented)

Time = 13.63 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.79 \[ \int \frac {-15 x^2+e^{\frac {2-x^3+x^4}{x}} \left (10+10 x^3-15 x^4\right )}{12 x^2+3 e^{\frac {2 \left (2-x^3+x^4\right )}{x}} x^2-36 x^3+27 x^4+e^{\frac {2-x^3+x^4}{x}} \left (-12 x^2+18 x^3\right )} \, dx=\frac {5}{3\,\left (3\,x+{\mathrm {e}}^{x^3}\,{\mathrm {e}}^{2/x}\,{\mathrm {e}}^{-x^2}-2\right )} \]

[In]

int((exp((x^4 - x^3 + 2)/x)*(10*x^3 - 15*x^4 + 10) - 15*x^2)/(3*x^2*exp((2*(x^4 - x^3 + 2))/x) - exp((x^4 - x^
3 + 2)/x)*(12*x^2 - 18*x^3) + 12*x^2 - 36*x^3 + 27*x^4),x)

[Out]

5/(3*(3*x + exp(x^3)*exp(2/x)*exp(-x^2) - 2))