3.8.76 \(\int \frac {e^{e^{\frac {-5+x^2+x^3}{x}} (4 x-x^2)+8 e^{\frac {-5+x^2+x^3}{x}} x^3 \log (x)} (e^{\frac {-5+x^2+x^3}{x}} (20-x+2 x^2+15 x^3-2 x^4)+e^{\frac {-5+x^2+x^3}{x}} (40 x^2+24 x^3+8 x^4+16 x^5) \log (x))}{x} \, dx\)

Optimal. Leaf size=28 \[ e^{e^{-\frac {5}{x}+x+x^2} x \left (4-x+8 x^2 \log (x)\right )} \]

________________________________________________________________________________________

Rubi [F]  time = 26.80, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\exp \left (e^{\frac {-5+x^2+x^3}{x}} \left (4 x-x^2\right )+8 e^{\frac {-5+x^2+x^3}{x}} x^3 \log (x)\right ) \left (e^{\frac {-5+x^2+x^3}{x}} \left (20-x+2 x^2+15 x^3-2 x^4\right )+e^{\frac {-5+x^2+x^3}{x}} \left (40 x^2+24 x^3+8 x^4+16 x^5\right ) \log (x)\right )}{x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(E^((-5 + x^2 + x^3)/x)*(4*x - x^2) + 8*E^((-5 + x^2 + x^3)/x)*x^3*Log[x])*(E^((-5 + x^2 + x^3)/x)*(20
- x + 2*x^2 + 15*x^3 - 2*x^4) + E^((-5 + x^2 + x^3)/x)*(40*x^2 + 24*x^3 + 8*x^4 + 16*x^5)*Log[x]))/x,x]

[Out]

-Defer[Int][E^(-5/x + x - E^(-5/x + x + x^2)*(-4 + x)*x + x^2)*x^(8*E^(-5/x + x + x^2)*x^3), x] + 20*Defer[Int
][E^(-5/x + x - E^(-5/x + x + x^2)*(-4 + x)*x + x^2)*x^(-1 + 8*E^(-5/x + x + x^2)*x^3), x] + 2*Defer[Int][E^(-
5/x + x - E^(-5/x + x + x^2)*(-4 + x)*x + x^2)*x^(1 + 8*E^(-5/x + x + x^2)*x^3), x] + 40*Log[x]*Defer[Int][E^(
-5/x + x - E^(-5/x + x + x^2)*(-4 + x)*x + x^2)*x^(1 + 8*E^(-5/x + x + x^2)*x^3), x] + 15*Defer[Int][E^(-5/x +
x - E^(-5/x + x + x^2)*(-4 + x)*x + x^2)*x^(2 + 8*E^(-5/x + x + x^2)*x^3), x] + 24*Log[x]*Defer[Int][E^(-5/x
+ x - E^(-5/x + x + x^2)*(-4 + x)*x + x^2)*x^(2 + 8*E^(-5/x + x + x^2)*x^3), x] - 2*Defer[Int][E^(-5/x + x - E
^(-5/x + x + x^2)*(-4 + x)*x + x^2)*x^(3 + 8*E^(-5/x + x + x^2)*x^3), x] + 8*Log[x]*Defer[Int][E^(-5/x + x - E
^(-5/x + x + x^2)*(-4 + x)*x + x^2)*x^(3 + 8*E^(-5/x + x + x^2)*x^3), x] + 16*Log[x]*Defer[Int][E^(-5/x + x -
E^(-5/x + x + x^2)*(-4 + x)*x + x^2)*x^(4 + 8*E^(-5/x + x + x^2)*x^3), x] - 40*Defer[Int][Defer[Int][E^(-5/x +
x - E^(-5/x + x + x^2)*(-4 + x)*x + x^2)*x^(1 + 8*E^(-5/x + x + x^2)*x^3), x]/x, x] - 24*Defer[Int][Defer[Int
][E^(-5/x + x - E^(-5/x + x + x^2)*(-4 + x)*x + x^2)*x^(2 + 8*E^(-5/x + x + x^2)*x^3), x]/x, x] - 8*Defer[Int]
[Defer[Int][E^(-5/x + x - E^(-5/x + x + x^2)*(-4 + x)*x + x^2)*x^(3 + 8*E^(-5/x + x + x^2)*x^3), x]/x, x] - 16
*Defer[Int][Defer[Int][E^(-5/x + x - E^(-5/x + x + x^2)*(-4 + x)*x + x^2)*x^(4 + 8*E^(-5/x + x + x^2)*x^3), x]
/x, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int e^{-\frac {5}{x}+x-e^{-\frac {5}{x}+x+x^2} (-4+x) x+x^2} x^{-1+8 e^{-\frac {5}{x}+x+x^2} x^3} \left (20-x+2 x^2+15 x^3-2 x^4+8 x^2 \left (5+3 x+x^2+2 x^3\right ) \log (x)\right ) \, dx\\ &=\int \left (-e^{-\frac {5}{x}+x-e^{-\frac {5}{x}+x+x^2} (-4+x) x+x^2} x^{8 e^{-\frac {5}{x}+x+x^2} x^3}+20 e^{-\frac {5}{x}+x-e^{-\frac {5}{x}+x+x^2} (-4+x) x+x^2} x^{-1+8 e^{-\frac {5}{x}+x+x^2} x^3}+2 e^{-\frac {5}{x}+x-e^{-\frac {5}{x}+x+x^2} (-4+x) x+x^2} x^{1+8 e^{-\frac {5}{x}+x+x^2} x^3}+15 e^{-\frac {5}{x}+x-e^{-\frac {5}{x}+x+x^2} (-4+x) x+x^2} x^{2+8 e^{-\frac {5}{x}+x+x^2} x^3}-2 e^{-\frac {5}{x}+x-e^{-\frac {5}{x}+x+x^2} (-4+x) x+x^2} x^{3+8 e^{-\frac {5}{x}+x+x^2} x^3}+8 e^{-\frac {5}{x}+x-e^{-\frac {5}{x}+x+x^2} (-4+x) x+x^2} x^{1+8 e^{-\frac {5}{x}+x+x^2} x^3} \left (5+3 x+x^2+2 x^3\right ) \log (x)\right ) \, dx\\ &=2 \int e^{-\frac {5}{x}+x-e^{-\frac {5}{x}+x+x^2} (-4+x) x+x^2} x^{1+8 e^{-\frac {5}{x}+x+x^2} x^3} \, dx-2 \int e^{-\frac {5}{x}+x-e^{-\frac {5}{x}+x+x^2} (-4+x) x+x^2} x^{3+8 e^{-\frac {5}{x}+x+x^2} x^3} \, dx+8 \int e^{-\frac {5}{x}+x-e^{-\frac {5}{x}+x+x^2} (-4+x) x+x^2} x^{1+8 e^{-\frac {5}{x}+x+x^2} x^3} \left (5+3 x+x^2+2 x^3\right ) \log (x) \, dx+15 \int e^{-\frac {5}{x}+x-e^{-\frac {5}{x}+x+x^2} (-4+x) x+x^2} x^{2+8 e^{-\frac {5}{x}+x+x^2} x^3} \, dx+20 \int e^{-\frac {5}{x}+x-e^{-\frac {5}{x}+x+x^2} (-4+x) x+x^2} x^{-1+8 e^{-\frac {5}{x}+x+x^2} x^3} \, dx-\int e^{-\frac {5}{x}+x-e^{-\frac {5}{x}+x+x^2} (-4+x) x+x^2} x^{8 e^{-\frac {5}{x}+x+x^2} x^3} \, dx\\ &=2 \int e^{-\frac {5}{x}+x-e^{-\frac {5}{x}+x+x^2} (-4+x) x+x^2} x^{1+8 e^{-\frac {5}{x}+x+x^2} x^3} \, dx-2 \int e^{-\frac {5}{x}+x-e^{-\frac {5}{x}+x+x^2} (-4+x) x+x^2} x^{3+8 e^{-\frac {5}{x}+x+x^2} x^3} \, dx-8 \int \frac {5 \int e^{-\frac {5}{x}+x-e^{-\frac {5}{x}+x+x^2} (-4+x) x+x^2} x^{1+8 e^{-\frac {5}{x}+x+x^2} x^3} \, dx+3 \int e^{-\frac {5}{x}+x-e^{-\frac {5}{x}+x+x^2} (-4+x) x+x^2} x^{2+8 e^{-\frac {5}{x}+x+x^2} x^3} \, dx+\int e^{-\frac {5}{x}+x-e^{-\frac {5}{x}+x+x^2} (-4+x) x+x^2} x^{3+8 e^{-\frac {5}{x}+x+x^2} x^3} \, dx+2 \int e^{-\frac {5}{x}+x-e^{-\frac {5}{x}+x+x^2} (-4+x) x+x^2} x^{4+8 e^{-\frac {5}{x}+x+x^2} x^3} \, dx}{x} \, dx+15 \int e^{-\frac {5}{x}+x-e^{-\frac {5}{x}+x+x^2} (-4+x) x+x^2} x^{2+8 e^{-\frac {5}{x}+x+x^2} x^3} \, dx+20 \int e^{-\frac {5}{x}+x-e^{-\frac {5}{x}+x+x^2} (-4+x) x+x^2} x^{-1+8 e^{-\frac {5}{x}+x+x^2} x^3} \, dx+(8 \log (x)) \int e^{-\frac {5}{x}+x-e^{-\frac {5}{x}+x+x^2} (-4+x) x+x^2} x^{3+8 e^{-\frac {5}{x}+x+x^2} x^3} \, dx+(16 \log (x)) \int e^{-\frac {5}{x}+x-e^{-\frac {5}{x}+x+x^2} (-4+x) x+x^2} x^{4+8 e^{-\frac {5}{x}+x+x^2} x^3} \, dx+(24 \log (x)) \int e^{-\frac {5}{x}+x-e^{-\frac {5}{x}+x+x^2} (-4+x) x+x^2} x^{2+8 e^{-\frac {5}{x}+x+x^2} x^3} \, dx+(40 \log (x)) \int e^{-\frac {5}{x}+x-e^{-\frac {5}{x}+x+x^2} (-4+x) x+x^2} x^{1+8 e^{-\frac {5}{x}+x+x^2} x^3} \, dx-\int e^{-\frac {5}{x}+x-e^{-\frac {5}{x}+x+x^2} (-4+x) x+x^2} x^{8 e^{-\frac {5}{x}+x+x^2} x^3} \, dx\\ &=2 \int e^{-\frac {5}{x}+x-e^{-\frac {5}{x}+x+x^2} (-4+x) x+x^2} x^{1+8 e^{-\frac {5}{x}+x+x^2} x^3} \, dx-2 \int e^{-\frac {5}{x}+x-e^{-\frac {5}{x}+x+x^2} (-4+x) x+x^2} x^{3+8 e^{-\frac {5}{x}+x+x^2} x^3} \, dx-8 \int \left (\frac {5 \int e^{-\frac {5}{x}+x-e^{-\frac {5}{x}+x+x^2} (-4+x) x+x^2} x^{1+8 e^{-\frac {5}{x}+x+x^2} x^3} \, dx+3 \int e^{-\frac {5}{x}+x-e^{-\frac {5}{x}+x+x^2} (-4+x) x+x^2} x^{2+8 e^{-\frac {5}{x}+x+x^2} x^3} \, dx+\int e^{-\frac {5}{x}+x-e^{-\frac {5}{x}+x+x^2} (-4+x) x+x^2} x^{3+8 e^{-\frac {5}{x}+x+x^2} x^3} \, dx}{x}+\frac {2 \int e^{-\frac {5}{x}+x-e^{-\frac {5}{x}+x+x^2} (-4+x) x+x^2} x^{4+8 e^{-\frac {5}{x}+x+x^2} x^3} \, dx}{x}\right ) \, dx+15 \int e^{-\frac {5}{x}+x-e^{-\frac {5}{x}+x+x^2} (-4+x) x+x^2} x^{2+8 e^{-\frac {5}{x}+x+x^2} x^3} \, dx+20 \int e^{-\frac {5}{x}+x-e^{-\frac {5}{x}+x+x^2} (-4+x) x+x^2} x^{-1+8 e^{-\frac {5}{x}+x+x^2} x^3} \, dx+(8 \log (x)) \int e^{-\frac {5}{x}+x-e^{-\frac {5}{x}+x+x^2} (-4+x) x+x^2} x^{3+8 e^{-\frac {5}{x}+x+x^2} x^3} \, dx+(16 \log (x)) \int e^{-\frac {5}{x}+x-e^{-\frac {5}{x}+x+x^2} (-4+x) x+x^2} x^{4+8 e^{-\frac {5}{x}+x+x^2} x^3} \, dx+(24 \log (x)) \int e^{-\frac {5}{x}+x-e^{-\frac {5}{x}+x+x^2} (-4+x) x+x^2} x^{2+8 e^{-\frac {5}{x}+x+x^2} x^3} \, dx+(40 \log (x)) \int e^{-\frac {5}{x}+x-e^{-\frac {5}{x}+x+x^2} (-4+x) x+x^2} x^{1+8 e^{-\frac {5}{x}+x+x^2} x^3} \, dx-\int e^{-\frac {5}{x}+x-e^{-\frac {5}{x}+x+x^2} (-4+x) x+x^2} x^{8 e^{-\frac {5}{x}+x+x^2} x^3} \, dx\\ &=2 \int e^{-\frac {5}{x}+x-e^{-\frac {5}{x}+x+x^2} (-4+x) x+x^2} x^{1+8 e^{-\frac {5}{x}+x+x^2} x^3} \, dx-2 \int e^{-\frac {5}{x}+x-e^{-\frac {5}{x}+x+x^2} (-4+x) x+x^2} x^{3+8 e^{-\frac {5}{x}+x+x^2} x^3} \, dx-8 \int \frac {5 \int e^{-\frac {5}{x}+x-e^{-\frac {5}{x}+x+x^2} (-4+x) x+x^2} x^{1+8 e^{-\frac {5}{x}+x+x^2} x^3} \, dx+3 \int e^{-\frac {5}{x}+x-e^{-\frac {5}{x}+x+x^2} (-4+x) x+x^2} x^{2+8 e^{-\frac {5}{x}+x+x^2} x^3} \, dx+\int e^{-\frac {5}{x}+x-e^{-\frac {5}{x}+x+x^2} (-4+x) x+x^2} x^{3+8 e^{-\frac {5}{x}+x+x^2} x^3} \, dx}{x} \, dx+15 \int e^{-\frac {5}{x}+x-e^{-\frac {5}{x}+x+x^2} (-4+x) x+x^2} x^{2+8 e^{-\frac {5}{x}+x+x^2} x^3} \, dx-16 \int \frac {\int e^{-\frac {5}{x}+x-e^{-\frac {5}{x}+x+x^2} (-4+x) x+x^2} x^{4+8 e^{-\frac {5}{x}+x+x^2} x^3} \, dx}{x} \, dx+20 \int e^{-\frac {5}{x}+x-e^{-\frac {5}{x}+x+x^2} (-4+x) x+x^2} x^{-1+8 e^{-\frac {5}{x}+x+x^2} x^3} \, dx+(8 \log (x)) \int e^{-\frac {5}{x}+x-e^{-\frac {5}{x}+x+x^2} (-4+x) x+x^2} x^{3+8 e^{-\frac {5}{x}+x+x^2} x^3} \, dx+(16 \log (x)) \int e^{-\frac {5}{x}+x-e^{-\frac {5}{x}+x+x^2} (-4+x) x+x^2} x^{4+8 e^{-\frac {5}{x}+x+x^2} x^3} \, dx+(24 \log (x)) \int e^{-\frac {5}{x}+x-e^{-\frac {5}{x}+x+x^2} (-4+x) x+x^2} x^{2+8 e^{-\frac {5}{x}+x+x^2} x^3} \, dx+(40 \log (x)) \int e^{-\frac {5}{x}+x-e^{-\frac {5}{x}+x+x^2} (-4+x) x+x^2} x^{1+8 e^{-\frac {5}{x}+x+x^2} x^3} \, dx-\int e^{-\frac {5}{x}+x-e^{-\frac {5}{x}+x+x^2} (-4+x) x+x^2} x^{8 e^{-\frac {5}{x}+x+x^2} x^3} \, dx\\ &=2 \int e^{-\frac {5}{x}+x-e^{-\frac {5}{x}+x+x^2} (-4+x) x+x^2} x^{1+8 e^{-\frac {5}{x}+x+x^2} x^3} \, dx-2 \int e^{-\frac {5}{x}+x-e^{-\frac {5}{x}+x+x^2} (-4+x) x+x^2} x^{3+8 e^{-\frac {5}{x}+x+x^2} x^3} \, dx-8 \int \left (\frac {5 \int e^{-\frac {5}{x}+x-e^{-\frac {5}{x}+x+x^2} (-4+x) x+x^2} x^{1+8 e^{-\frac {5}{x}+x+x^2} x^3} \, dx+3 \int e^{-\frac {5}{x}+x-e^{-\frac {5}{x}+x+x^2} (-4+x) x+x^2} x^{2+8 e^{-\frac {5}{x}+x+x^2} x^3} \, dx}{x}+\frac {\int e^{-\frac {5}{x}+x-e^{-\frac {5}{x}+x+x^2} (-4+x) x+x^2} x^{3+8 e^{-\frac {5}{x}+x+x^2} x^3} \, dx}{x}\right ) \, dx+15 \int e^{-\frac {5}{x}+x-e^{-\frac {5}{x}+x+x^2} (-4+x) x+x^2} x^{2+8 e^{-\frac {5}{x}+x+x^2} x^3} \, dx-16 \int \frac {\int e^{-\frac {5}{x}+x-e^{-\frac {5}{x}+x+x^2} (-4+x) x+x^2} x^{4+8 e^{-\frac {5}{x}+x+x^2} x^3} \, dx}{x} \, dx+20 \int e^{-\frac {5}{x}+x-e^{-\frac {5}{x}+x+x^2} (-4+x) x+x^2} x^{-1+8 e^{-\frac {5}{x}+x+x^2} x^3} \, dx+(8 \log (x)) \int e^{-\frac {5}{x}+x-e^{-\frac {5}{x}+x+x^2} (-4+x) x+x^2} x^{3+8 e^{-\frac {5}{x}+x+x^2} x^3} \, dx+(16 \log (x)) \int e^{-\frac {5}{x}+x-e^{-\frac {5}{x}+x+x^2} (-4+x) x+x^2} x^{4+8 e^{-\frac {5}{x}+x+x^2} x^3} \, dx+(24 \log (x)) \int e^{-\frac {5}{x}+x-e^{-\frac {5}{x}+x+x^2} (-4+x) x+x^2} x^{2+8 e^{-\frac {5}{x}+x+x^2} x^3} \, dx+(40 \log (x)) \int e^{-\frac {5}{x}+x-e^{-\frac {5}{x}+x+x^2} (-4+x) x+x^2} x^{1+8 e^{-\frac {5}{x}+x+x^2} x^3} \, dx-\int e^{-\frac {5}{x}+x-e^{-\frac {5}{x}+x+x^2} (-4+x) x+x^2} x^{8 e^{-\frac {5}{x}+x+x^2} x^3} \, dx\\ &=2 \int e^{-\frac {5}{x}+x-e^{-\frac {5}{x}+x+x^2} (-4+x) x+x^2} x^{1+8 e^{-\frac {5}{x}+x+x^2} x^3} \, dx-2 \int e^{-\frac {5}{x}+x-e^{-\frac {5}{x}+x+x^2} (-4+x) x+x^2} x^{3+8 e^{-\frac {5}{x}+x+x^2} x^3} \, dx-8 \int \frac {5 \int e^{-\frac {5}{x}+x-e^{-\frac {5}{x}+x+x^2} (-4+x) x+x^2} x^{1+8 e^{-\frac {5}{x}+x+x^2} x^3} \, dx+3 \int e^{-\frac {5}{x}+x-e^{-\frac {5}{x}+x+x^2} (-4+x) x+x^2} x^{2+8 e^{-\frac {5}{x}+x+x^2} x^3} \, dx}{x} \, dx-8 \int \frac {\int e^{-\frac {5}{x}+x-e^{-\frac {5}{x}+x+x^2} (-4+x) x+x^2} x^{3+8 e^{-\frac {5}{x}+x+x^2} x^3} \, dx}{x} \, dx+15 \int e^{-\frac {5}{x}+x-e^{-\frac {5}{x}+x+x^2} (-4+x) x+x^2} x^{2+8 e^{-\frac {5}{x}+x+x^2} x^3} \, dx-16 \int \frac {\int e^{-\frac {5}{x}+x-e^{-\frac {5}{x}+x+x^2} (-4+x) x+x^2} x^{4+8 e^{-\frac {5}{x}+x+x^2} x^3} \, dx}{x} \, dx+20 \int e^{-\frac {5}{x}+x-e^{-\frac {5}{x}+x+x^2} (-4+x) x+x^2} x^{-1+8 e^{-\frac {5}{x}+x+x^2} x^3} \, dx+(8 \log (x)) \int e^{-\frac {5}{x}+x-e^{-\frac {5}{x}+x+x^2} (-4+x) x+x^2} x^{3+8 e^{-\frac {5}{x}+x+x^2} x^3} \, dx+(16 \log (x)) \int e^{-\frac {5}{x}+x-e^{-\frac {5}{x}+x+x^2} (-4+x) x+x^2} x^{4+8 e^{-\frac {5}{x}+x+x^2} x^3} \, dx+(24 \log (x)) \int e^{-\frac {5}{x}+x-e^{-\frac {5}{x}+x+x^2} (-4+x) x+x^2} x^{2+8 e^{-\frac {5}{x}+x+x^2} x^3} \, dx+(40 \log (x)) \int e^{-\frac {5}{x}+x-e^{-\frac {5}{x}+x+x^2} (-4+x) x+x^2} x^{1+8 e^{-\frac {5}{x}+x+x^2} x^3} \, dx-\int e^{-\frac {5}{x}+x-e^{-\frac {5}{x}+x+x^2} (-4+x) x+x^2} x^{8 e^{-\frac {5}{x}+x+x^2} x^3} \, dx\\ &=2 \int e^{-\frac {5}{x}+x-e^{-\frac {5}{x}+x+x^2} (-4+x) x+x^2} x^{1+8 e^{-\frac {5}{x}+x+x^2} x^3} \, dx-2 \int e^{-\frac {5}{x}+x-e^{-\frac {5}{x}+x+x^2} (-4+x) x+x^2} x^{3+8 e^{-\frac {5}{x}+x+x^2} x^3} \, dx-8 \int \left (\frac {5 \int e^{-\frac {5}{x}+x-e^{-\frac {5}{x}+x+x^2} (-4+x) x+x^2} x^{1+8 e^{-\frac {5}{x}+x+x^2} x^3} \, dx}{x}+\frac {3 \int e^{-\frac {5}{x}+x-e^{-\frac {5}{x}+x+x^2} (-4+x) x+x^2} x^{2+8 e^{-\frac {5}{x}+x+x^2} x^3} \, dx}{x}\right ) \, dx-8 \int \frac {\int e^{-\frac {5}{x}+x-e^{-\frac {5}{x}+x+x^2} (-4+x) x+x^2} x^{3+8 e^{-\frac {5}{x}+x+x^2} x^3} \, dx}{x} \, dx+15 \int e^{-\frac {5}{x}+x-e^{-\frac {5}{x}+x+x^2} (-4+x) x+x^2} x^{2+8 e^{-\frac {5}{x}+x+x^2} x^3} \, dx-16 \int \frac {\int e^{-\frac {5}{x}+x-e^{-\frac {5}{x}+x+x^2} (-4+x) x+x^2} x^{4+8 e^{-\frac {5}{x}+x+x^2} x^3} \, dx}{x} \, dx+20 \int e^{-\frac {5}{x}+x-e^{-\frac {5}{x}+x+x^2} (-4+x) x+x^2} x^{-1+8 e^{-\frac {5}{x}+x+x^2} x^3} \, dx+(8 \log (x)) \int e^{-\frac {5}{x}+x-e^{-\frac {5}{x}+x+x^2} (-4+x) x+x^2} x^{3+8 e^{-\frac {5}{x}+x+x^2} x^3} \, dx+(16 \log (x)) \int e^{-\frac {5}{x}+x-e^{-\frac {5}{x}+x+x^2} (-4+x) x+x^2} x^{4+8 e^{-\frac {5}{x}+x+x^2} x^3} \, dx+(24 \log (x)) \int e^{-\frac {5}{x}+x-e^{-\frac {5}{x}+x+x^2} (-4+x) x+x^2} x^{2+8 e^{-\frac {5}{x}+x+x^2} x^3} \, dx+(40 \log (x)) \int e^{-\frac {5}{x}+x-e^{-\frac {5}{x}+x+x^2} (-4+x) x+x^2} x^{1+8 e^{-\frac {5}{x}+x+x^2} x^3} \, dx-\int e^{-\frac {5}{x}+x-e^{-\frac {5}{x}+x+x^2} (-4+x) x+x^2} x^{8 e^{-\frac {5}{x}+x+x^2} x^3} \, dx\\ &=2 \int e^{-\frac {5}{x}+x-e^{-\frac {5}{x}+x+x^2} (-4+x) x+x^2} x^{1+8 e^{-\frac {5}{x}+x+x^2} x^3} \, dx-2 \int e^{-\frac {5}{x}+x-e^{-\frac {5}{x}+x+x^2} (-4+x) x+x^2} x^{3+8 e^{-\frac {5}{x}+x+x^2} x^3} \, dx-8 \int \frac {\int e^{-\frac {5}{x}+x-e^{-\frac {5}{x}+x+x^2} (-4+x) x+x^2} x^{3+8 e^{-\frac {5}{x}+x+x^2} x^3} \, dx}{x} \, dx+15 \int e^{-\frac {5}{x}+x-e^{-\frac {5}{x}+x+x^2} (-4+x) x+x^2} x^{2+8 e^{-\frac {5}{x}+x+x^2} x^3} \, dx-16 \int \frac {\int e^{-\frac {5}{x}+x-e^{-\frac {5}{x}+x+x^2} (-4+x) x+x^2} x^{4+8 e^{-\frac {5}{x}+x+x^2} x^3} \, dx}{x} \, dx+20 \int e^{-\frac {5}{x}+x-e^{-\frac {5}{x}+x+x^2} (-4+x) x+x^2} x^{-1+8 e^{-\frac {5}{x}+x+x^2} x^3} \, dx-24 \int \frac {\int e^{-\frac {5}{x}+x-e^{-\frac {5}{x}+x+x^2} (-4+x) x+x^2} x^{2+8 e^{-\frac {5}{x}+x+x^2} x^3} \, dx}{x} \, dx-40 \int \frac {\int e^{-\frac {5}{x}+x-e^{-\frac {5}{x}+x+x^2} (-4+x) x+x^2} x^{1+8 e^{-\frac {5}{x}+x+x^2} x^3} \, dx}{x} \, dx+(8 \log (x)) \int e^{-\frac {5}{x}+x-e^{-\frac {5}{x}+x+x^2} (-4+x) x+x^2} x^{3+8 e^{-\frac {5}{x}+x+x^2} x^3} \, dx+(16 \log (x)) \int e^{-\frac {5}{x}+x-e^{-\frac {5}{x}+x+x^2} (-4+x) x+x^2} x^{4+8 e^{-\frac {5}{x}+x+x^2} x^3} \, dx+(24 \log (x)) \int e^{-\frac {5}{x}+x-e^{-\frac {5}{x}+x+x^2} (-4+x) x+x^2} x^{2+8 e^{-\frac {5}{x}+x+x^2} x^3} \, dx+(40 \log (x)) \int e^{-\frac {5}{x}+x-e^{-\frac {5}{x}+x+x^2} (-4+x) x+x^2} x^{1+8 e^{-\frac {5}{x}+x+x^2} x^3} \, dx-\int e^{-\frac {5}{x}+x-e^{-\frac {5}{x}+x+x^2} (-4+x) x+x^2} x^{8 e^{-\frac {5}{x}+x+x^2} x^3} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.16, size = 40, normalized size = 1.43 \begin {gather*} e^{-e^{-\frac {5}{x}+x+x^2} (-4+x) x} x^{8 e^{-\frac {5}{x}+x+x^2} x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(E^((-5 + x^2 + x^3)/x)*(4*x - x^2) + 8*E^((-5 + x^2 + x^3)/x)*x^3*Log[x])*(E^((-5 + x^2 + x^3)/x
)*(20 - x + 2*x^2 + 15*x^3 - 2*x^4) + E^((-5 + x^2 + x^3)/x)*(40*x^2 + 24*x^3 + 8*x^4 + 16*x^5)*Log[x]))/x,x]

[Out]

x^(8*E^(-5/x + x + x^2)*x^3)/E^(E^(-5/x + x + x^2)*(-4 + x)*x)

________________________________________________________________________________________

fricas [A]  time = 0.71, size = 44, normalized size = 1.57 \begin {gather*} e^{\left (8 \, x^{3} e^{\left (\frac {x^{3} + x^{2} - 5}{x}\right )} \log \relax (x) - {\left (x^{2} - 4 \, x\right )} e^{\left (\frac {x^{3} + x^{2} - 5}{x}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((16*x^5+8*x^4+24*x^3+40*x^2)*exp((x^3+x^2-5)/x)*log(x)+(-2*x^4+15*x^3+2*x^2-x+20)*exp((x^3+x^2-5)/x
))*exp(8*x^3*exp((x^3+x^2-5)/x)*log(x)+(-x^2+4*x)*exp((x^3+x^2-5)/x))/x,x, algorithm="fricas")

[Out]

e^(8*x^3*e^((x^3 + x^2 - 5)/x)*log(x) - (x^2 - 4*x)*e^((x^3 + x^2 - 5)/x))

________________________________________________________________________________________

giac [A]  time = 0.62, size = 50, normalized size = 1.79 \begin {gather*} e^{\left (8 \, x^{3} e^{\left (x^{2} + x - \frac {5}{x}\right )} \log \relax (x) - x^{2} e^{\left (x^{2} + x - \frac {5}{x}\right )} + 4 \, x e^{\left (x^{2} + x - \frac {5}{x}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((16*x^5+8*x^4+24*x^3+40*x^2)*exp((x^3+x^2-5)/x)*log(x)+(-2*x^4+15*x^3+2*x^2-x+20)*exp((x^3+x^2-5)/x
))*exp(8*x^3*exp((x^3+x^2-5)/x)*log(x)+(-x^2+4*x)*exp((x^3+x^2-5)/x))/x,x, algorithm="giac")

[Out]

e^(8*x^3*e^(x^2 + x - 5/x)*log(x) - x^2*e^(x^2 + x - 5/x) + 4*x*e^(x^2 + x - 5/x))

________________________________________________________________________________________

maple [A]  time = 0.04, size = 42, normalized size = 1.50

method	result	size
risch	\(x^{8 \,{\mathrm e}^{\frac {x^{3}+x^{2}-5}{x}} x^{3}} {\mathrm e}^{-\left (x -4\right ) x \,{\mathrm e}^{\frac {x^{3}+x^{2}-5}{x}}}\)	\(42\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((16*x^5+8*x^4+24*x^3+40*x^2)*exp((x^3+x^2-5)/x)*ln(x)+(-2*x^4+15*x^3+2*x^2-x+20)*exp((x^3+x^2-5)/x))*exp(
8*x^3*exp((x^3+x^2-5)/x)*ln(x)+(-x^2+4*x)*exp((x^3+x^2-5)/x))/x,x,method=_RETURNVERBOSE)

[Out]

x^(8*exp((x^3+x^2-5)/x)*x^3)*exp(-(x-4)*x*exp((x^3+x^2-5)/x))

________________________________________________________________________________________

maxima [A]  time = 1.21, size = 50, normalized size = 1.79 \begin {gather*} e^{\left (8 \, x^{3} e^{\left (x^{2} + x - \frac {5}{x}\right )} \log \relax (x) - x^{2} e^{\left (x^{2} + x - \frac {5}{x}\right )} + 4 \, x e^{\left (x^{2} + x - \frac {5}{x}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((16*x^5+8*x^4+24*x^3+40*x^2)*exp((x^3+x^2-5)/x)*log(x)+(-2*x^4+15*x^3+2*x^2-x+20)*exp((x^3+x^2-5)/x
))*exp(8*x^3*exp((x^3+x^2-5)/x)*log(x)+(-x^2+4*x)*exp((x^3+x^2-5)/x))/x,x, algorithm="maxima")

[Out]

e^(8*x^3*e^(x^2 + x - 5/x)*log(x) - x^2*e^(x^2 + x - 5/x) + 4*x*e^(x^2 + x - 5/x))

________________________________________________________________________________________

mupad [B]  time = 0.85, size = 54, normalized size = 1.93 \begin {gather*} x^{8\,x^3\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^{-\frac {5}{x}}\,{\mathrm {e}}^x}\,{\mathrm {e}}^{-x^2\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^{-\frac {5}{x}}\,{\mathrm {e}}^x}\,{\mathrm {e}}^{4\,x\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^{-\frac {5}{x}}\,{\mathrm {e}}^x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(exp((x^2 + x^3 - 5)/x)*(4*x - x^2) + 8*x^3*exp((x^2 + x^3 - 5)/x)*log(x))*(exp((x^2 + x^3 - 5)/x)*(2*
x^2 - x + 15*x^3 - 2*x^4 + 20) + exp((x^2 + x^3 - 5)/x)*log(x)*(40*x^2 + 24*x^3 + 8*x^4 + 16*x^5)))/x,x)

[Out]

x^(8*x^3*exp(x^2)*exp(-5/x)*exp(x))*exp(-x^2*exp(x^2)*exp(-5/x)*exp(x))*exp(4*x*exp(x^2)*exp(-5/x)*exp(x))

________________________________________________________________________________________

sympy [A]  time = 1.05, size = 39, normalized size = 1.39 \begin {gather*} e^{8 x^{3} e^{\frac {x^{3} + x^{2} - 5}{x}} \log {\relax (x )} + \left (- x^{2} + 4 x\right ) e^{\frac {x^{3} + x^{2} - 5}{x}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((16*x**5+8*x**4+24*x**3+40*x**2)*exp((x**3+x**2-5)/x)*ln(x)+(-2*x**4+15*x**3+2*x**2-x+20)*exp((x**3
+x**2-5)/x))*exp(8*x**3*exp((x**3+x**2-5)/x)*ln(x)+(-x**2+4*x)*exp((x**3+x**2-5)/x))/x,x)

[Out]

exp(8*x**3*exp((x**3 + x**2 - 5)/x)*log(x) + (-x**2 + 4*x)*exp((x**3 + x**2 - 5)/x))

________________________________________________________________________________________