\(\int \frac {45 x^2+e^{\frac {-5+x^3}{x^2}} (270+27 x^2+27 x^3)}{9 e^2 x^2-30 e x^3+25 x^4+9 e^{\frac {2 (-5+x^3)}{x^2}} x^4+e^{\frac {-5+x^3}{x^2}} (-18 e x^3+30 x^4)} \, dx\) [8696]

Optimal result

Integrand size = 94, antiderivative size = 28 \[ \int \frac {45 x^2+e^{\frac {-5+x^3}{x^2}} \left (270+27 x^2+27 x^3\right )}{9 e^2 x^2-30 e x^3+25 x^4+9 e^{\frac {2 \left (-5+x^3\right )}{x^2}} x^4+e^{\frac {-5+x^3}{x^2}} \left (-18 e x^3+30 x^4\right )} \, dx=\frac {3}{e-\left (\frac {5}{3}+e^{\frac {-\frac {5}{x}+x^2}{x}}\right ) x} \]

[Out]

3/(exp(1)-x*(5/3+exp((x^2-5/x)/x)))

Rubi [F]

\[ \int \frac {45 x^2+e^{\frac {-5+x^3}{x^2}} \left (270+27 x^2+27 x^3\right )}{9 e^2 x^2-30 e x^3+25 x^4+9 e^{\frac {2 \left (-5+x^3\right )}{x^2}} x^4+e^{\frac {-5+x^3}{x^2}} \left (-18 e x^3+30 x^4\right )} \, dx=\int \frac {45 x^2+e^{\frac {-5+x^3}{x^2}} \left (270+27 x^2+27 x^3\right )}{9 e^2 x^2-30 e x^3+25 x^4+9 e^{\frac {2 \left (-5+x^3\right )}{x^2}} x^4+e^{\frac {-5+x^3}{x^2}} \left (-18 e x^3+30 x^4\right )} \, dx \]

[In]

Int[(45*x^2 + E^((-5 + x^3)/x^2)*(270 + 27*x^2 + 27*x^3))/(9*E^2*x^2 - 30*E*x^3 + 25*x^4 + 9*E^((2*(-5 + x^3))
/x^2)*x^4 + E^((-5 + x^3)/x^2)*(-18*E*x^3 + 30*x^4)),x]

[Out]

27*Defer[Int][E^(1 + 10/x^2)/(3*E^(1 + 5/x^2) - 5*E^(5/x^2)*x - 3*E^x*x)^2, x] + 270*Defer[Int][E^(1 + 10/x^2)
/(x^3*(3*E^(1 + 5/x^2) - 5*E^(5/x^2)*x - 3*E^x*x)^2), x] + 27*Defer[Int][E^(1 + 10/x^2)/(x*(3*E^(1 + 5/x^2) -
5*E^(5/x^2)*x - 3*E^x*x)^2), x] - 9*Defer[Int][E^(5/x^2)/(3*E^(1 + 5/x^2) - 5*E^(5/x^2)*x - 3*E^x*x), x] - 450
*Defer[Int][E^(10/x^2)/(x^2*(-3*E^(1 + 5/x^2) + 5*E^(5/x^2)*x + 3*E^x*x)^2), x] - 45*Defer[Int][(E^(10/x^2)*x)
/(-3*E^(1 + 5/x^2) + 5*E^(5/x^2)*x + 3*E^x*x)^2, x] + 90*Defer[Int][E^(5/x^2)/(x^3*(-3*E^(1 + 5/x^2) + 5*E^(5/
x^2)*x + 3*E^x*x)), x] + 9*Defer[Int][E^(5/x^2)/(x*(-3*E^(1 + 5/x^2) + 5*E^(5/x^2)*x + 3*E^x*x)), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {9 e^{\frac {5}{x^2}} \left (5 e^{\frac {5}{x^2}} x^2+3 e^x \left (10+x^2+x^3\right )\right )}{x^2 \left (3 e^{1+\frac {5}{x^2}}-5 e^{\frac {5}{x^2}} x-3 e^x x\right )^2} \, dx \\ & = 9 \int \frac {e^{\frac {5}{x^2}} \left (5 e^{\frac {5}{x^2}} x^2+3 e^x \left (10+x^2+x^3\right )\right )}{x^2 \left (3 e^{1+\frac {5}{x^2}}-5 e^{\frac {5}{x^2}} x-3 e^x x\right )^2} \, dx \\ & = 9 \int \left (\frac {e^{\frac {5}{x^2}} \left (10+x^2+x^3\right )}{x^3 \left (-3 e^{1+\frac {5}{x^2}}+5 e^{\frac {5}{x^2}} x+3 e^x x\right )}-\frac {e^{\frac {10}{x^2}} \left (-30 e+50 x-3 e x^2-3 e x^3+5 x^4\right )}{x^3 \left (-3 e^{1+\frac {5}{x^2}}+5 e^{\frac {5}{x^2}} x+3 e^x x\right )^2}\right ) \, dx \\ & = 9 \int \frac {e^{\frac {5}{x^2}} \left (10+x^2+x^3\right )}{x^3 \left (-3 e^{1+\frac {5}{x^2}}+5 e^{\frac {5}{x^2}} x+3 e^x x\right )} \, dx-9 \int \frac {e^{\frac {10}{x^2}} \left (-30 e+50 x-3 e x^2-3 e x^3+5 x^4\right )}{x^3 \left (-3 e^{1+\frac {5}{x^2}}+5 e^{\frac {5}{x^2}} x+3 e^x x\right )^2} \, dx \\ & = -\left (9 \int \left (-\frac {3 e^{1+\frac {10}{x^2}}}{\left (3 e^{1+\frac {5}{x^2}}-5 e^{\frac {5}{x^2}} x-3 e^x x\right )^2}-\frac {30 e^{1+\frac {10}{x^2}}}{x^3 \left (3 e^{1+\frac {5}{x^2}}-5 e^{\frac {5}{x^2}} x-3 e^x x\right )^2}-\frac {3 e^{1+\frac {10}{x^2}}}{x \left (3 e^{1+\frac {5}{x^2}}-5 e^{\frac {5}{x^2}} x-3 e^x x\right )^2}+\frac {50 e^{\frac {10}{x^2}}}{x^2 \left (-3 e^{1+\frac {5}{x^2}}+5 e^{\frac {5}{x^2}} x+3 e^x x\right )^2}+\frac {5 e^{\frac {10}{x^2}} x}{\left (-3 e^{1+\frac {5}{x^2}}+5 e^{\frac {5}{x^2}} x+3 e^x x\right )^2}\right ) \, dx\right )+9 \int \left (-\frac {e^{\frac {5}{x^2}}}{3 e^{1+\frac {5}{x^2}}-5 e^{\frac {5}{x^2}} x-3 e^x x}+\frac {10 e^{\frac {5}{x^2}}}{x^3 \left (-3 e^{1+\frac {5}{x^2}}+5 e^{\frac {5}{x^2}} x+3 e^x x\right )}+\frac {e^{\frac {5}{x^2}}}{x \left (-3 e^{1+\frac {5}{x^2}}+5 e^{\frac {5}{x^2}} x+3 e^x x\right )}\right ) \, dx \\ & = -\left (9 \int \frac {e^{\frac {5}{x^2}}}{3 e^{1+\frac {5}{x^2}}-5 e^{\frac {5}{x^2}} x-3 e^x x} \, dx\right )+9 \int \frac {e^{\frac {5}{x^2}}}{x \left (-3 e^{1+\frac {5}{x^2}}+5 e^{\frac {5}{x^2}} x+3 e^x x\right )} \, dx+27 \int \frac {e^{1+\frac {10}{x^2}}}{\left (3 e^{1+\frac {5}{x^2}}-5 e^{\frac {5}{x^2}} x-3 e^x x\right )^2} \, dx+27 \int \frac {e^{1+\frac {10}{x^2}}}{x \left (3 e^{1+\frac {5}{x^2}}-5 e^{\frac {5}{x^2}} x-3 e^x x\right )^2} \, dx-45 \int \frac {e^{\frac {10}{x^2}} x}{\left (-3 e^{1+\frac {5}{x^2}}+5 e^{\frac {5}{x^2}} x+3 e^x x\right )^2} \, dx+90 \int \frac {e^{\frac {5}{x^2}}}{x^3 \left (-3 e^{1+\frac {5}{x^2}}+5 e^{\frac {5}{x^2}} x+3 e^x x\right )} \, dx+270 \int \frac {e^{1+\frac {10}{x^2}}}{x^3 \left (3 e^{1+\frac {5}{x^2}}-5 e^{\frac {5}{x^2}} x-3 e^x x\right )^2} \, dx-450 \int \frac {e^{\frac {10}{x^2}}}{x^2 \left (-3 e^{1+\frac {5}{x^2}}+5 e^{\frac {5}{x^2}} x+3 e^x x\right )^2} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 1.13 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.39 \[ \int \frac {45 x^2+e^{\frac {-5+x^3}{x^2}} \left (270+27 x^2+27 x^3\right )}{9 e^2 x^2-30 e x^3+25 x^4+9 e^{\frac {2 \left (-5+x^3\right )}{x^2}} x^4+e^{\frac {-5+x^3}{x^2}} \left (-18 e x^3+30 x^4\right )} \, dx=-\frac {9 e^{\frac {5}{x^2}}}{-3 e^{1+\frac {5}{x^2}}+5 e^{\frac {5}{x^2}} x+3 e^x x} \]

[In]

Integrate[(45*x^2 + E^((-5 + x^3)/x^2)*(270 + 27*x^2 + 27*x^3))/(9*E^2*x^2 - 30*E*x^3 + 25*x^4 + 9*E^((2*(-5 +
x^3))/x^2)*x^4 + E^((-5 + x^3)/x^2)*(-18*E*x^3 + 30*x^4)),x]

[Out]

(-9*E^(5/x^2))/(-3*E^(1 + 5/x^2) + 5*E^(5/x^2)*x + 3*E^x*x)

Maple [A] (verified)

Time = 0.40 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93

method	result	size
norman	\(\frac {9}{-3 \,{\mathrm e}^{\frac {x^{3}-5}{x^{2}}} x +3 \,{\mathrm e}-5 x}\)	\(26\)
risch	\(\frac {9}{-3 \,{\mathrm e}^{\frac {x^{3}-5}{x^{2}}} x +3 \,{\mathrm e}-5 x}\)	\(26\)
parallelrisch	\(\frac {9}{-3 \,{\mathrm e}^{\frac {x^{3}-5}{x^{2}}} x +3 \,{\mathrm e}-5 x}\)	\(26\)

[In]

int(((27*x^3+27*x^2+270)*exp((x^3-5)/x^2)+45*x^2)/(9*x^4*exp((x^3-5)/x^2)^2+(-18*x^3*exp(1)+30*x^4)*exp((x^3-5
)/x^2)+9*x^2*exp(1)^2-30*x^3*exp(1)+25*x^4),x,method=_RETURNVERBOSE)

[Out]

9/(-3*exp((x^3-5)/x^2)*x+3*exp(1)-5*x)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.89 \[ \int \frac {45 x^2+e^{\frac {-5+x^3}{x^2}} \left (270+27 x^2+27 x^3\right )}{9 e^2 x^2-30 e x^3+25 x^4+9 e^{\frac {2 \left (-5+x^3\right )}{x^2}} x^4+e^{\frac {-5+x^3}{x^2}} \left (-18 e x^3+30 x^4\right )} \, dx=-\frac {9}{3 \, x e^{\left (\frac {x^{3} - 5}{x^{2}}\right )} + 5 \, x - 3 \, e} \]

[In]

integrate(((27*x^3+27*x^2+270)*exp((x^3-5)/x^2)+45*x^2)/(9*x^4*exp((x^3-5)/x^2)^2+(-18*x^3*exp(1)+30*x^4)*exp(
(x^3-5)/x^2)+9*x^2*exp(1)^2-30*x^3*exp(1)+25*x^4),x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.86 \[ \int \frac {45 x^2+e^{\frac {-5+x^3}{x^2}} \left (270+27 x^2+27 x^3\right )}{9 e^2 x^2-30 e x^3+25 x^4+9 e^{\frac {2 \left (-5+x^3\right )}{x^2}} x^4+e^{\frac {-5+x^3}{x^2}} \left (-18 e x^3+30 x^4\right )} \, dx=- \frac {9}{3 x e^{\frac {x^{3} - 5}{x^{2}}} + 5 x - 3 e} \]

[In]

integrate(((27*x**3+27*x**2+270)*exp((x**3-5)/x**2)+45*x**2)/(9*x**4*exp((x**3-5)/x**2)**2+(-18*x**3*exp(1)+30
*x**4)*exp((x**3-5)/x**2)+9*x**2*exp(1)**2-30*x**3*exp(1)+25*x**4),x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.11 \[ \int \frac {45 x^2+e^{\frac {-5+x^3}{x^2}} \left (270+27 x^2+27 x^3\right )}{9 e^2 x^2-30 e x^3+25 x^4+9 e^{\frac {2 \left (-5+x^3\right )}{x^2}} x^4+e^{\frac {-5+x^3}{x^2}} \left (-18 e x^3+30 x^4\right )} \, dx=-\frac {9 \, e^{\left (\frac {5}{x^{2}}\right )}}{3 \, x e^{x} + {\left (5 \, x - 3 \, e\right )} e^{\left (\frac {5}{x^{2}}\right )}} \]

[In]

integrate(((27*x^3+27*x^2+270)*exp((x^3-5)/x^2)+45*x^2)/(9*x^4*exp((x^3-5)/x^2)^2+(-18*x^3*exp(1)+30*x^4)*exp(
(x^3-5)/x^2)+9*x^2*exp(1)^2-30*x^3*exp(1)+25*x^4),x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.89 \[ \int \frac {45 x^2+e^{\frac {-5+x^3}{x^2}} \left (270+27 x^2+27 x^3\right )}{9 e^2 x^2-30 e x^3+25 x^4+9 e^{\frac {2 \left (-5+x^3\right )}{x^2}} x^4+e^{\frac {-5+x^3}{x^2}} \left (-18 e x^3+30 x^4\right )} \, dx=-\frac {9}{3 \, x e^{\left (\frac {x^{3} - 5}{x^{2}}\right )} + 5 \, x - 3 \, e} \]

[In]

integrate(((27*x^3+27*x^2+270)*exp((x^3-5)/x^2)+45*x^2)/(9*x^4*exp((x^3-5)/x^2)^2+(-18*x^3*exp(1)+30*x^4)*exp(
(x^3-5)/x^2)+9*x^2*exp(1)^2-30*x^3*exp(1)+25*x^4),x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {45 x^2+e^{\frac {-5+x^3}{x^2}} \left (270+27 x^2+27 x^3\right )}{9 e^2 x^2-30 e x^3+25 x^4+9 e^{\frac {2 \left (-5+x^3\right )}{x^2}} x^4+e^{\frac {-5+x^3}{x^2}} \left (-18 e x^3+30 x^4\right )} \, dx=\int \frac {45\,x^2+{\mathrm {e}}^{\frac {x^3-5}{x^2}}\,\left (27\,x^3+27\,x^2+270\right )}{9\,x^4\,{\mathrm {e}}^{\frac {2\,\left (x^3-5\right )}{x^2}}-{\mathrm {e}}^{\frac {x^3-5}{x^2}}\,\left (18\,x^3\,\mathrm {e}-30\,x^4\right )+9\,x^2\,{\mathrm {e}}^2-30\,x^3\,\mathrm {e}+25\,x^4} \,d x \]

[In]

int((45*x^2 + exp((x^3 - 5)/x^2)*(27*x^2 + 27*x^3 + 270))/(9*x^4*exp((2*(x^3 - 5))/x^2) - exp((x^3 - 5)/x^2)*(
18*x^3*exp(1) - 30*x^4) + 9*x^2*exp(2) - 30*x^3*exp(1) + 25*x^4),x)

[Out]

int((45*x^2 + exp((x^3 - 5)/x^2)*(27*x^2 + 27*x^3 + 270))/(9*x^4*exp((2*(x^3 - 5))/x^2) - exp((x^3 - 5)/x^2)*(
18*x^3*exp(1) - 30*x^4) + 9*x^2*exp(2) - 30*x^3*exp(1) + 25*x^4), x)