\(\int \frac {e^{\frac {e^{-\frac {20-9 x^2-5 x^3}{9 x+5 x^2}} (1+2 e^{\frac {20-9 x^2-5 x^3}{9 x+5 x^2}} x^2)}{x}-\frac {20-9 x^2-5 x^3}{9 x+5 x^2}} (180+119 x-9 x^2+65 x^3+25 x^4+e^{\frac {20-9 x^2-5 x^3}{9 x+5 x^2}} (162 x^3+180 x^4+50 x^5))}{81 x^3+90 x^4+25 x^5} \, dx\) [10307]

Optimal result

Integrand size = 174, antiderivative size = 28 \[ \int \frac {e^{\frac {e^{-\frac {20-9 x^2-5 x^3}{9 x+5 x^2}} \left (1+2 e^{\frac {20-9 x^2-5 x^3}{9 x+5 x^2}} x^2\right )}{x}-\frac {20-9 x^2-5 x^3}{9 x+5 x^2}} \left (180+119 x-9 x^2+65 x^3+25 x^4+e^{\frac {20-9 x^2-5 x^3}{9 x+5 x^2}} \left (162 x^3+180 x^4+50 x^5\right )\right )}{81 x^3+90 x^4+25 x^5} \, dx=e^{\frac {e^{x-\frac {5}{x+\frac {5}{4} x (1+x)}}}{x}+2 x} \]

[Out]

exp(1/x/exp(5/(5/4*(1+x)*x+x)-x)+2*x)

Rubi [F]

\[ \int \frac {e^{\frac {e^{-\frac {20-9 x^2-5 x^3}{9 x+5 x^2}} \left (1+2 e^{\frac {20-9 x^2-5 x^3}{9 x+5 x^2}} x^2\right )}{x}-\frac {20-9 x^2-5 x^3}{9 x+5 x^2}} \left (180+119 x-9 x^2+65 x^3+25 x^4+e^{\frac {20-9 x^2-5 x^3}{9 x+5 x^2}} \left (162 x^3+180 x^4+50 x^5\right )\right )}{81 x^3+90 x^4+25 x^5} \, dx=\int \frac {\exp \left (\frac {e^{-\frac {20-9 x^2-5 x^3}{9 x+5 x^2}} \left (1+2 e^{\frac {20-9 x^2-5 x^3}{9 x+5 x^2}} x^2\right )}{x}-\frac {20-9 x^2-5 x^3}{9 x+5 x^2}\right ) \left (180+119 x-9 x^2+65 x^3+25 x^4+e^{\frac {20-9 x^2-5 x^3}{9 x+5 x^2}} \left (162 x^3+180 x^4+50 x^5\right )\right )}{81 x^3+90 x^4+25 x^5} \, dx \]

[In]

Int[(E^((1 + 2*E^((20 - 9*x^2 - 5*x^3)/(9*x + 5*x^2))*x^2)/(E^((20 - 9*x^2 - 5*x^3)/(9*x + 5*x^2))*x) - (20 -
9*x^2 - 5*x^3)/(9*x + 5*x^2))*(180 + 119*x - 9*x^2 + 65*x^3 + 25*x^4 + E^((20 - 9*x^2 - 5*x^3)/(9*x + 5*x^2))*
(162*x^3 + 180*x^4 + 50*x^5)))/(81*x^3 + 90*x^4 + 25*x^5),x]

[Out]

2*Defer[Int][E^(E^(x - 20/(9*x + 5*x^2))/x + 2*x), x] + (20*Defer[Int][E^(E^(x - 20/(9*x + 5*x^2))/x - (20 - 2
7*x^2 - 15*x^3)/(x*(9 + 5*x)))/x^3, x])/9 - Defer[Int][E^(E^(x - 20/(9*x + 5*x^2))/x - (20 - 27*x^2 - 15*x^3)/
(x*(9 + 5*x)))/x^2, x] + (229*Defer[Int][E^(E^(x - 20/(9*x + 5*x^2))/x - (20 - 27*x^2 - 15*x^3)/(x*(9 + 5*x)))
/x, x])/729 + (2500*Defer[Int][E^(E^(x - 20/(9*x + 5*x^2))/x - (20 - 27*x^2 - 15*x^3)/(x*(9 + 5*x)))/(9 + 5*x)
^2, x])/81 + (2500*Defer[Int][E^(E^(x - 20/(9*x + 5*x^2))/x - (20 - 27*x^2 - 15*x^3)/(x*(9 + 5*x)))/(9 + 5*x),
x])/729

Rubi steps \begin{align*} \text {integral}& = \int \frac {\exp \left (\frac {e^{-\frac {20-9 x^2-5 x^3}{9 x+5 x^2}} \left (1+2 e^{\frac {20-9 x^2-5 x^3}{9 x+5 x^2}} x^2\right )}{x}-\frac {20-9 x^2-5 x^3}{9 x+5 x^2}\right ) \left (180+119 x-9 x^2+65 x^3+25 x^4+e^{\frac {20-9 x^2-5 x^3}{9 x+5 x^2}} \left (162 x^3+180 x^4+50 x^5\right )\right )}{x^3 \left (81+90 x+25 x^2\right )} \, dx \\ & = \int \frac {\exp \left (\frac {e^{-\frac {20-9 x^2-5 x^3}{9 x+5 x^2}} \left (1+2 e^{\frac {20-9 x^2-5 x^3}{9 x+5 x^2}} x^2\right )}{x}-\frac {20-9 x^2-5 x^3}{9 x+5 x^2}\right ) \left (180+119 x-9 x^2+65 x^3+25 x^4+e^{\frac {20-9 x^2-5 x^3}{9 x+5 x^2}} \left (162 x^3+180 x^4+50 x^5\right )\right )}{x^3 (9+5 x)^2} \, dx \\ & = \int \frac {\exp \left (\frac {e^{x-\frac {20}{9 x+5 x^2}}}{x}-\frac {20-27 x^2-15 x^3}{x (9+5 x)}\right ) \left (180+119 x-9 x^2+65 x^3+25 x^4+2 e^{\frac {20-9 x^2-5 x^3}{9 x+5 x^2}} x^3 (9+5 x)^2\right )}{x^3 (9+5 x)^2} \, dx \\ & = \int \left (2 \exp \left (\frac {e^{x-\frac {20}{9 x+5 x^2}}}{x}-\frac {20-27 x^2-15 x^3}{x (9+5 x)}+\frac {20-9 x^2-5 x^3}{x (9+5 x)}\right )+\frac {65 \exp \left (\frac {e^{x-\frac {20}{9 x+5 x^2}}}{x}-\frac {20-27 x^2-15 x^3}{x (9+5 x)}\right )}{(9+5 x)^2}+\frac {180 \exp \left (\frac {e^{x-\frac {20}{9 x+5 x^2}}}{x}-\frac {20-27 x^2-15 x^3}{x (9+5 x)}\right )}{x^3 (9+5 x)^2}+\frac {119 \exp \left (\frac {e^{x-\frac {20}{9 x+5 x^2}}}{x}-\frac {20-27 x^2-15 x^3}{x (9+5 x)}\right )}{x^2 (9+5 x)^2}-\frac {9 \exp \left (\frac {e^{x-\frac {20}{9 x+5 x^2}}}{x}-\frac {20-27 x^2-15 x^3}{x (9+5 x)}\right )}{x (9+5 x)^2}+\frac {25 \exp \left (\frac {e^{x-\frac {20}{9 x+5 x^2}}}{x}-\frac {20-27 x^2-15 x^3}{x (9+5 x)}\right ) x}{(9+5 x)^2}\right ) \, dx \\ & = 2 \int \exp \left (\frac {e^{x-\frac {20}{9 x+5 x^2}}}{x}-\frac {20-27 x^2-15 x^3}{x (9+5 x)}+\frac {20-9 x^2-5 x^3}{x (9+5 x)}\right ) \, dx-9 \int \frac {\exp \left (\frac {e^{x-\frac {20}{9 x+5 x^2}}}{x}-\frac {20-27 x^2-15 x^3}{x (9+5 x)}\right )}{x (9+5 x)^2} \, dx+25 \int \frac {\exp \left (\frac {e^{x-\frac {20}{9 x+5 x^2}}}{x}-\frac {20-27 x^2-15 x^3}{x (9+5 x)}\right ) x}{(9+5 x)^2} \, dx+65 \int \frac {\exp \left (\frac {e^{x-\frac {20}{9 x+5 x^2}}}{x}-\frac {20-27 x^2-15 x^3}{x (9+5 x)}\right )}{(9+5 x)^2} \, dx+119 \int \frac {\exp \left (\frac {e^{x-\frac {20}{9 x+5 x^2}}}{x}-\frac {20-27 x^2-15 x^3}{x (9+5 x)}\right )}{x^2 (9+5 x)^2} \, dx+180 \int \frac {\exp \left (\frac {e^{x-\frac {20}{9 x+5 x^2}}}{x}-\frac {20-27 x^2-15 x^3}{x (9+5 x)}\right )}{x^3 (9+5 x)^2} \, dx \\ & = 2 \int e^{\frac {e^{x-\frac {20}{9 x+5 x^2}}}{x}+2 x} \, dx-9 \int \left (\frac {e^{\frac {e^{x-\frac {20}{9 x+5 x^2}}}{x}-\frac {20-27 x^2-15 x^3}{x (9+5 x)}}}{81 x}-\frac {5 e^{\frac {e^{x-\frac {20}{9 x+5 x^2}}}{x}-\frac {20-27 x^2-15 x^3}{x (9+5 x)}}}{9 (9+5 x)^2}-\frac {5 e^{\frac {e^{x-\frac {20}{9 x+5 x^2}}}{x}-\frac {20-27 x^2-15 x^3}{x (9+5 x)}}}{81 (9+5 x)}\right ) \, dx+25 \int \left (-\frac {9 e^{\frac {e^{x-\frac {20}{9 x+5 x^2}}}{x}-\frac {20-27 x^2-15 x^3}{x (9+5 x)}}}{5 (9+5 x)^2}+\frac {e^{\frac {e^{x-\frac {20}{9 x+5 x^2}}}{x}-\frac {20-27 x^2-15 x^3}{x (9+5 x)}}}{5 (9+5 x)}\right ) \, dx+65 \int \frac {e^{\frac {e^{x-\frac {20}{9 x+5 x^2}}}{x}-\frac {20-27 x^2-15 x^3}{x (9+5 x)}}}{(9+5 x)^2} \, dx+119 \int \left (\frac {e^{\frac {e^{x-\frac {20}{9 x+5 x^2}}}{x}-\frac {20-27 x^2-15 x^3}{x (9+5 x)}}}{81 x^2}-\frac {10 e^{\frac {e^{x-\frac {20}{9 x+5 x^2}}}{x}-\frac {20-27 x^2-15 x^3}{x (9+5 x)}}}{729 x}+\frac {25 e^{\frac {e^{x-\frac {20}{9 x+5 x^2}}}{x}-\frac {20-27 x^2-15 x^3}{x (9+5 x)}}}{81 (9+5 x)^2}+\frac {50 e^{\frac {e^{x-\frac {20}{9 x+5 x^2}}}{x}-\frac {20-27 x^2-15 x^3}{x (9+5 x)}}}{729 (9+5 x)}\right ) \, dx+180 \int \left (\frac {e^{\frac {e^{x-\frac {20}{9 x+5 x^2}}}{x}-\frac {20-27 x^2-15 x^3}{x (9+5 x)}}}{81 x^3}-\frac {10 e^{\frac {e^{x-\frac {20}{9 x+5 x^2}}}{x}-\frac {20-27 x^2-15 x^3}{x (9+5 x)}}}{729 x^2}+\frac {25 e^{\frac {e^{x-\frac {20}{9 x+5 x^2}}}{x}-\frac {20-27 x^2-15 x^3}{x (9+5 x)}}}{2187 x}-\frac {125 e^{\frac {e^{x-\frac {20}{9 x+5 x^2}}}{x}-\frac {20-27 x^2-15 x^3}{x (9+5 x)}}}{729 (9+5 x)^2}-\frac {125 e^{\frac {e^{x-\frac {20}{9 x+5 x^2}}}{x}-\frac {20-27 x^2-15 x^3}{x (9+5 x)}}}{2187 (9+5 x)}\right ) \, dx \\ & = -\left (\frac {1}{9} \int \frac {e^{\frac {e^{x-\frac {20}{9 x+5 x^2}}}{x}-\frac {20-27 x^2-15 x^3}{x (9+5 x)}}}{x} \, dx\right )+\frac {5}{9} \int \frac {e^{\frac {e^{x-\frac {20}{9 x+5 x^2}}}{x}-\frac {20-27 x^2-15 x^3}{x (9+5 x)}}}{9+5 x} \, dx+\frac {119}{81} \int \frac {e^{\frac {e^{x-\frac {20}{9 x+5 x^2}}}{x}-\frac {20-27 x^2-15 x^3}{x (9+5 x)}}}{x^2} \, dx-\frac {1190}{729} \int \frac {e^{\frac {e^{x-\frac {20}{9 x+5 x^2}}}{x}-\frac {20-27 x^2-15 x^3}{x (9+5 x)}}}{x} \, dx+2 \int e^{\frac {e^{x-\frac {20}{9 x+5 x^2}}}{x}+2 x} \, dx+\frac {500}{243} \int \frac {e^{\frac {e^{x-\frac {20}{9 x+5 x^2}}}{x}-\frac {20-27 x^2-15 x^3}{x (9+5 x)}}}{x} \, dx+\frac {20}{9} \int \frac {e^{\frac {e^{x-\frac {20}{9 x+5 x^2}}}{x}-\frac {20-27 x^2-15 x^3}{x (9+5 x)}}}{x^3} \, dx-\frac {200}{81} \int \frac {e^{\frac {e^{x-\frac {20}{9 x+5 x^2}}}{x}-\frac {20-27 x^2-15 x^3}{x (9+5 x)}}}{x^2} \, dx+5 \int \frac {e^{\frac {e^{x-\frac {20}{9 x+5 x^2}}}{x}-\frac {20-27 x^2-15 x^3}{x (9+5 x)}}}{(9+5 x)^2} \, dx+5 \int \frac {e^{\frac {e^{x-\frac {20}{9 x+5 x^2}}}{x}-\frac {20-27 x^2-15 x^3}{x (9+5 x)}}}{9+5 x} \, dx+\frac {5950}{729} \int \frac {e^{\frac {e^{x-\frac {20}{9 x+5 x^2}}}{x}-\frac {20-27 x^2-15 x^3}{x (9+5 x)}}}{9+5 x} \, dx-\frac {2500}{243} \int \frac {e^{\frac {e^{x-\frac {20}{9 x+5 x^2}}}{x}-\frac {20-27 x^2-15 x^3}{x (9+5 x)}}}{9+5 x} \, dx-\frac {2500}{81} \int \frac {e^{\frac {e^{x-\frac {20}{9 x+5 x^2}}}{x}-\frac {20-27 x^2-15 x^3}{x (9+5 x)}}}{(9+5 x)^2} \, dx+\frac {2975}{81} \int \frac {e^{\frac {e^{x-\frac {20}{9 x+5 x^2}}}{x}-\frac {20-27 x^2-15 x^3}{x (9+5 x)}}}{(9+5 x)^2} \, dx-45 \int \frac {e^{\frac {e^{x-\frac {20}{9 x+5 x^2}}}{x}-\frac {20-27 x^2-15 x^3}{x (9+5 x)}}}{(9+5 x)^2} \, dx+65 \int \frac {e^{\frac {e^{x-\frac {20}{9 x+5 x^2}}}{x}-\frac {20-27 x^2-15 x^3}{x (9+5 x)}}}{(9+5 x)^2} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.14 \[ \int \frac {e^{\frac {e^{-\frac {20-9 x^2-5 x^3}{9 x+5 x^2}} \left (1+2 e^{\frac {20-9 x^2-5 x^3}{9 x+5 x^2}} x^2\right )}{x}-\frac {20-9 x^2-5 x^3}{9 x+5 x^2}} \left (180+119 x-9 x^2+65 x^3+25 x^4+e^{\frac {20-9 x^2-5 x^3}{9 x+5 x^2}} \left (162 x^3+180 x^4+50 x^5\right )\right )}{81 x^3+90 x^4+25 x^5} \, dx=e^{\frac {e^{-\frac {20}{9 x}+x+\frac {100}{9 (9+5 x)}}}{x}+2 x} \]

[In]

Integrate[(E^((1 + 2*E^((20 - 9*x^2 - 5*x^3)/(9*x + 5*x^2))*x^2)/(E^((20 - 9*x^2 - 5*x^3)/(9*x + 5*x^2))*x) -
(20 - 9*x^2 - 5*x^3)/(9*x + 5*x^2))*(180 + 119*x - 9*x^2 + 65*x^3 + 25*x^4 + E^((20 - 9*x^2 - 5*x^3)/(9*x + 5*
x^2))*(162*x^3 + 180*x^4 + 50*x^5)))/(81*x^3 + 90*x^4 + 25*x^5),x]

[Out]

E^(E^(-20/(9*x) + x + 100/(9*(9 + 5*x)))/x + 2*x)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(61\) vs. \(2(28)=56\).

Time = 1.41 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.21

method	result	size
risch	\({\mathrm e}^{\frac {\left (2 x^{2} {\mathrm e}^{-\frac {5 x^{3}+9 x^{2}-20}{x \left (5 x +9\right )}}+1\right ) {\mathrm e}^{\frac {5 x^{3}+9 x^{2}-20}{x \left (5 x +9\right )}}}{x}}\)	\(62\)
parallelrisch	\(\frac {22500 x^{2} {\mathrm e}^{\frac {\left (2 x^{2} {\mathrm e}^{-\frac {5 x^{3}+9 x^{2}-20}{x \left (5 x +9\right )}}+1\right ) {\mathrm e}^{\frac {5 x^{3}+9 x^{2}-20}{x \left (5 x +9\right )}}}{x}}+40500 x \,{\mathrm e}^{\frac {\left (2 x^{2} {\mathrm e}^{-\frac {5 x^{3}+9 x^{2}-20}{x \left (5 x +9\right )}}+1\right ) {\mathrm e}^{\frac {5 x^{3}+9 x^{2}-20}{x \left (5 x +9\right )}}}{x}}}{4500 x \left (5 x +9\right )}\)	\(150\)
norman	\(\frac {\left (9 x^{2} {\mathrm e}^{\frac {-5 x^{3}-9 x^{2}+20}{5 x^{2}+9 x}} {\mathrm e}^{\frac {\left (2 x^{2} {\mathrm e}^{\frac {-5 x^{3}-9 x^{2}+20}{5 x^{2}+9 x}}+1\right ) {\mathrm e}^{-\frac {-5 x^{3}-9 x^{2}+20}{5 x^{2}+9 x}}}{x}}+5 x^{3} {\mathrm e}^{\frac {-5 x^{3}-9 x^{2}+20}{5 x^{2}+9 x}} {\mathrm e}^{\frac {\left (2 x^{2} {\mathrm e}^{\frac {-5 x^{3}-9 x^{2}+20}{5 x^{2}+9 x}}+1\right ) {\mathrm e}^{-\frac {-5 x^{3}-9 x^{2}+20}{5 x^{2}+9 x}}}{x}}\right ) {\mathrm e}^{-\frac {-5 x^{3}-9 x^{2}+20}{5 x^{2}+9 x}}}{x^{2} \left (5 x +9\right )}\)	\(228\)

[In]

int(((50*x^5+180*x^4+162*x^3)*exp((-5*x^3-9*x^2+20)/(5*x^2+9*x))+25*x^4+65*x^3-9*x^2+119*x+180)*exp((2*x^2*exp
((-5*x^3-9*x^2+20)/(5*x^2+9*x))+1)/x/exp((-5*x^3-9*x^2+20)/(5*x^2+9*x)))/(25*x^5+90*x^4+81*x^3)/exp((-5*x^3-9*
x^2+20)/(5*x^2+9*x)),x,method=_RETURNVERBOSE)

[Out]

exp((2*x^2*exp(-(5*x^3+9*x^2-20)/x/(5*x+9))+1)/x*exp((5*x^3+9*x^2-20)/x/(5*x+9)))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 82 vs. \(2 (26) = 52\).

Time = 0.27 (sec) , antiderivative size = 82, normalized size of antiderivative = 2.93 \[ \int \frac {e^{\frac {e^{-\frac {20-9 x^2-5 x^3}{9 x+5 x^2}} \left (1+2 e^{\frac {20-9 x^2-5 x^3}{9 x+5 x^2}} x^2\right )}{x}-\frac {20-9 x^2-5 x^3}{9 x+5 x^2}} \left (180+119 x-9 x^2+65 x^3+25 x^4+e^{\frac {20-9 x^2-5 x^3}{9 x+5 x^2}} \left (162 x^3+180 x^4+50 x^5\right )\right )}{81 x^3+90 x^4+25 x^5} \, dx=e^{\left (\frac {15 \, x^{3} + 27 \, x^{2} + {\left (5 \, x + 9\right )} e^{\left (\frac {5 \, x^{3} + 9 \, x^{2} - 20}{5 \, x^{2} + 9 \, x}\right )} - 20}{5 \, x^{2} + 9 \, x} - \frac {5 \, x^{3} + 9 \, x^{2} - 20}{5 \, x^{2} + 9 \, x}\right )} \]

[In]

integrate(((50*x^5+180*x^4+162*x^3)*exp((-5*x^3-9*x^2+20)/(5*x^2+9*x))+25*x^4+65*x^3-9*x^2+119*x+180)*exp((2*x
^2*exp((-5*x^3-9*x^2+20)/(5*x^2+9*x))+1)/x/exp((-5*x^3-9*x^2+20)/(5*x^2+9*x)))/(25*x^5+90*x^4+81*x^3)/exp((-5*
x^3-9*x^2+20)/(5*x^2+9*x)),x, algorithm="fricas")

[Out]

e^((15*x^3 + 27*x^2 + (5*x + 9)*e^((5*x^3 + 9*x^2 - 20)/(5*x^2 + 9*x)) - 20)/(5*x^2 + 9*x) - (5*x^3 + 9*x^2 -
20)/(5*x^2 + 9*x))

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (22) = 44\).

Time = 0.43 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.89 \[ \int \frac {e^{\frac {e^{-\frac {20-9 x^2-5 x^3}{9 x+5 x^2}} \left (1+2 e^{\frac {20-9 x^2-5 x^3}{9 x+5 x^2}} x^2\right )}{x}-\frac {20-9 x^2-5 x^3}{9 x+5 x^2}} \left (180+119 x-9 x^2+65 x^3+25 x^4+e^{\frac {20-9 x^2-5 x^3}{9 x+5 x^2}} \left (162 x^3+180 x^4+50 x^5\right )\right )}{81 x^3+90 x^4+25 x^5} \, dx=e^{\frac {\left (2 x^{2} e^{\frac {- 5 x^{3} - 9 x^{2} + 20}{5 x^{2} + 9 x}} + 1\right ) e^{- \frac {- 5 x^{3} - 9 x^{2} + 20}{5 x^{2} + 9 x}}}{x}} \]

[In]

integrate(((50*x**5+180*x**4+162*x**3)*exp((-5*x**3-9*x**2+20)/(5*x**2+9*x))+25*x**4+65*x**3-9*x**2+119*x+180)
*exp((2*x**2*exp((-5*x**3-9*x**2+20)/(5*x**2+9*x))+1)/x/exp((-5*x**3-9*x**2+20)/(5*x**2+9*x)))/(25*x**5+90*x**
4+81*x**3)/exp((-5*x**3-9*x**2+20)/(5*x**2+9*x)),x)

[Out]

exp((2*x**2*exp((-5*x**3 - 9*x**2 + 20)/(5*x**2 + 9*x)) + 1)*exp(-(-5*x**3 - 9*x**2 + 20)/(5*x**2 + 9*x))/x)

Maxima [A] (verification not implemented)

none

Time = 0.48 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93 \[ \int \frac {e^{\frac {e^{-\frac {20-9 x^2-5 x^3}{9 x+5 x^2}} \left (1+2 e^{\frac {20-9 x^2-5 x^3}{9 x+5 x^2}} x^2\right )}{x}-\frac {20-9 x^2-5 x^3}{9 x+5 x^2}} \left (180+119 x-9 x^2+65 x^3+25 x^4+e^{\frac {20-9 x^2-5 x^3}{9 x+5 x^2}} \left (162 x^3+180 x^4+50 x^5\right )\right )}{81 x^3+90 x^4+25 x^5} \, dx=e^{\left (2 \, x + \frac {e^{\left (x + \frac {100}{9 \, {\left (5 \, x + 9\right )}} - \frac {20}{9 \, x}\right )}}{x}\right )} \]

[In]

integrate(((50*x^5+180*x^4+162*x^3)*exp((-5*x^3-9*x^2+20)/(5*x^2+9*x))+25*x^4+65*x^3-9*x^2+119*x+180)*exp((2*x
^2*exp((-5*x^3-9*x^2+20)/(5*x^2+9*x))+1)/x/exp((-5*x^3-9*x^2+20)/(5*x^2+9*x)))/(25*x^5+90*x^4+81*x^3)/exp((-5*
x^3-9*x^2+20)/(5*x^2+9*x)),x, algorithm="maxima")

[Out]

e^(2*x + e^(x + 100/9/(5*x + 9) - 20/9/x)/x)

Giac [F]

\[ \int \frac {e^{\frac {e^{-\frac {20-9 x^2-5 x^3}{9 x+5 x^2}} \left (1+2 e^{\frac {20-9 x^2-5 x^3}{9 x+5 x^2}} x^2\right )}{x}-\frac {20-9 x^2-5 x^3}{9 x+5 x^2}} \left (180+119 x-9 x^2+65 x^3+25 x^4+e^{\frac {20-9 x^2-5 x^3}{9 x+5 x^2}} \left (162 x^3+180 x^4+50 x^5\right )\right )}{81 x^3+90 x^4+25 x^5} \, dx=\int { \frac {{\left (25 \, x^{4} + 65 \, x^{3} - 9 \, x^{2} + 2 \, {\left (25 \, x^{5} + 90 \, x^{4} + 81 \, x^{3}\right )} e^{\left (-\frac {5 \, x^{3} + 9 \, x^{2} - 20}{5 \, x^{2} + 9 \, x}\right )} + 119 \, x + 180\right )} e^{\left (\frac {{\left (2 \, x^{2} e^{\left (-\frac {5 \, x^{3} + 9 \, x^{2} - 20}{5 \, x^{2} + 9 \, x}\right )} + 1\right )} e^{\left (\frac {5 \, x^{3} + 9 \, x^{2} - 20}{5 \, x^{2} + 9 \, x}\right )}}{x} + \frac {5 \, x^{3} + 9 \, x^{2} - 20}{5 \, x^{2} + 9 \, x}\right )}}{25 \, x^{5} + 90 \, x^{4} + 81 \, x^{3}} \,d x } \]

[In]

integrate(((50*x^5+180*x^4+162*x^3)*exp((-5*x^3-9*x^2+20)/(5*x^2+9*x))+25*x^4+65*x^3-9*x^2+119*x+180)*exp((2*x
^2*exp((-5*x^3-9*x^2+20)/(5*x^2+9*x))+1)/x/exp((-5*x^3-9*x^2+20)/(5*x^2+9*x)))/(25*x^5+90*x^4+81*x^3)/exp((-5*
x^3-9*x^2+20)/(5*x^2+9*x)),x, algorithm="giac")

[Out]

integrate((25*x^4 + 65*x^3 - 9*x^2 + 2*(25*x^5 + 90*x^4 + 81*x^3)*e^(-(5*x^3 + 9*x^2 - 20)/(5*x^2 + 9*x)) + 11
9*x + 180)*e^((2*x^2*e^(-(5*x^3 + 9*x^2 - 20)/(5*x^2 + 9*x)) + 1)*e^((5*x^3 + 9*x^2 - 20)/(5*x^2 + 9*x))/x + (
5*x^3 + 9*x^2 - 20)/(5*x^2 + 9*x))/(25*x^5 + 90*x^4 + 81*x^3), x)

Mupad [B] (verification not implemented)

Time = 19.23 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.71 \[ \int \frac {e^{\frac {e^{-\frac {20-9 x^2-5 x^3}{9 x+5 x^2}} \left (1+2 e^{\frac {20-9 x^2-5 x^3}{9 x+5 x^2}} x^2\right )}{x}-\frac {20-9 x^2-5 x^3}{9 x+5 x^2}} \left (180+119 x-9 x^2+65 x^3+25 x^4+e^{\frac {20-9 x^2-5 x^3}{9 x+5 x^2}} \left (162 x^3+180 x^4+50 x^5\right )\right )}{81 x^3+90 x^4+25 x^5} \, dx={\mathrm {e}}^{\frac {{\mathrm {e}}^{\frac {5\,x^2}{5\,x+9}}\,{\mathrm {e}}^{-\frac {20}{5\,x^2+9\,x}}\,{\mathrm {e}}^{\frac {9\,x}{5\,x+9}}}{x}}\,{\mathrm {e}}^{2\,x} \]

[In]

int((exp((9*x^2 + 5*x^3 - 20)/(9*x + 5*x^2))*exp((exp((9*x^2 + 5*x^3 - 20)/(9*x + 5*x^2))*(2*x^2*exp(-(9*x^2 +
5*x^3 - 20)/(9*x + 5*x^2)) + 1))/x)*(119*x + exp(-(9*x^2 + 5*x^3 - 20)/(9*x + 5*x^2))*(162*x^3 + 180*x^4 + 50
*x^5) - 9*x^2 + 65*x^3 + 25*x^4 + 180))/(81*x^3 + 90*x^4 + 25*x^5),x)

[Out]

exp((exp((5*x^2)/(5*x + 9))*exp(-20/(9*x + 5*x^2))*exp((9*x)/(5*x + 9)))/x)*exp(2*x)