∫ 80+200x3+125x6+e2(−4−5x3+5x log(x)) ____________________________ 4+5x3 (−144+40x−160x3+50x4+25x6+(−160+40x+400x3−100x4) log(x)) ______________________________________________________________________________________________________________________________________________

3.88.98 \(\int \frac {80+200 x^3+125 x^6+e^{\frac {2 (-4-5 x^3+5 x \log (x))}{4+5 x^3}} (-144+40 x-160 x^3+50 x^4+25 x^6+(-160+40 x+400 x^3-100 x^4) \log (x))}{16+40 x^3+25 x^6} \, dx\)

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

________________________________________________________________________________________

Rubi [F] time = 21.71, 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 {80+200 x^3+125 x^6+e^{\frac {2 \left (-4-5 x^3+5 x \log (x)\right )}{4+5 x^3}} \left (-144+40 x-160 x^3+50 x^4+25 x^6+\left (-160+40 x+400 x^3-100 x^4\right ) \log (x)\right )}{16+40 x^3+25 x^6} \, dx \end {gather*}

Veriﬁcation is not applicable to the result.

[In]

Int[(80 + 200*x^3 + 125*x^6 + E^((2*(-4 - 5*x^3 + 5*x*Log[x]))/(4 + 5*x^3))*(-144 + 40*x - 160*x^3 + 50*x^4 +

25*x^6 + (-160 + 40*x + 400*x^3 - 100*x^4)*Log[x]))/(16 + 40*x^3 + 25*x^6),x]

[Out]

5*x + (9*(-1)^(1/3)*10^(2/3)*x^(1 + (10*x)/(4 + 5*x^3)))/((1 + (-1)^(1/3))^4*E^2*((-10)^(2/3) + 5*x)) - (10^(2

/3)*x^(1 + (10*x)/(4 + 5*x^3)))/(E^2*(10^(2/3) + 5*x)) - (10^(2/3)*x^(1 + (10*x)/(4 + 5*x^3)))/(E^2*(10^(2/3)

+ 5*(-1)^(2/3)*x)) + (5^(1/3)*x^((10*x)/(4 + 5*x^3))*(4 + 5*x^3))/(2^(2/3)*(1 + (-1)^(1/3))^4*E^2*((-10)^(2/3)

+ 5*x)) - ((-5)^(1/3)*x^((10*x)/(4 + 5*x^3))*(4 + 5*x^3))/(9*2^(2/3)*E^2*(10^(2/3) + 5*(-1)^(2/3)*x)) + (x^((

10*x)/(4 + 5*x^3))*(4 + 5*x^3))/(9*10^(1/3)*E^2*(2 + 10^(1/3)*x)) + (5*5^(1/3)*x^(6 + (10*x)/(4 + 5*x^3)))/(4*

2^(2/3)*(1 + (-1)^(1/3))^4*E^2*((-10)^(2/3) + 5*x)*(1 + (2*x)/(4 + 5*x^3))) - (5*(-5)^(1/3)*x^(6 + (10*x)/(4 +

5*x^3)))/(36*2^(2/3)*E^2*(10^(2/3) + 5*(-1)^(2/3)*x)*(1 + (2*x)/(4 + 5*x^3))) + (5^(2/3)*x^(6 + (10*x)/(4 + 5

*x^3)))/(36*2^(1/3)*E^2*(2 + 10^(1/3)*x)*(1 + (2*x)/(4 + 5*x^3))) - (10*10^(1/3)*x^(3 + (10*x)/(4 + 5*x^3)))/(

(1 + (-1)^(1/3))^4*E^2*((-10)^(2/3) + 5*x)*(1 + (5*x)/(4 + 5*x^3))) + (10*(-10)^(1/3)*x^(3 + (10*x)/(4 + 5*x^3

)))/(9*E^2*(10^(2/3) + 5*(-1)^(2/3)*x)*(1 + (5*x)/(4 + 5*x^3))) - (2*10^(2/3)*x^(3 + (10*x)/(4 + 5*x^3)))/(9*E

^2*(2 + 10^(1/3)*x)*(1 + (5*x)/(4 + 5*x^3))) + (10^(1/3)*x^((10*x)/(4 + 5*x^3))*(4 + 5*x^3)*(2 + (5*x)/(4 + 5*

x^3)))/((1 + (-1)^(1/3))^4*E^2*((-10)^(2/3) + 5*x)) + (10^(1/3)*x^((10*x)/(4 + 5*x^3))*(4 + 5*x^3)*(2 + (5*x)/

(4 + 5*x^3)))/(9*E^2*(10^(2/3) + 5*x)) - ((-10)^(1/3)*x^((10*x)/(4 + 5*x^3))*(4 + 5*x^3)*(2 + (5*x)/(4 + 5*x^3

)))/(9*E^2*(10^(2/3) + 5*(-1)^(2/3)*x)) - ((-1/5)^(1/3)*2^(2/3)*x^((10*x)/(4 + 5*x^3))*(4 + 5*x^3)*(3 + (5*x)/

(4 + 5*x^3)))/((1 + (-1)^(1/3))^4*E^2*((-10)^(2/3) + 5*x)) + (2^(2/3)*x^((10*x)/(4 + 5*x^3))*(4 + 5*x^3)*(3 +

(5*x)/(4 + 5*x^3)))/(9*5^(1/3)*E^2*(10^(2/3) + 5*x)) + (2^(2/3)*x^((10*x)/(4 + 5*x^3))*(4 + 5*x^3)*(3 + (5*x)/

(4 + 5*x^3)))/(9*5^(1/3)*E^2*(10^(2/3) + 5*(-1)^(2/3)*x)) - (5*x^(5 + (10*x)/(4 + 5*x^3))*(3 + (5*x)/(4 + 5*x^

3)))/(36*E^2*((-10)^(2/3) + 5*x)*(1 + (2*x)/(4 + 5*x^3))*(2 + (5*x)/(4 + 5*x^3))) - (5*(-1)^(2/3)*x^(5 + (10*x

)/(4 + 5*x^3))*(3 + (5*x)/(4 + 5*x^3)))/(36*E^2*(10^(2/3) + 5*(-1)^(2/3)*x)*(1 + (2*x)/(4 + 5*x^3))*(2 + (5*x)

/(4 + 5*x^3))) - (5^(1/3)*x^(5 + (10*x)/(4 + 5*x^3))*(3 + (5*x)/(4 + 5*x^3)))/(18*2^(2/3)*E^2*(2 + 10^(1/3)*x)

*(1 + (2*x)/(4 + 5*x^3))*(2 + (5*x)/(4 + 5*x^3))) - (10^(1/3)*x^((10*x)/(4 + 5*x^3))*(4 + 5*x^3)*(2 + (5*x)/(4

+ 5*x^3)))/((1 + (-1)^(1/3))^4*E^2*((-10)^(2/3) + 5*x)*(1 + (10*x)/(4 + 5*x^3))) + ((-10)^(1/3)*x^((10*x)/(4

+ 5*x^3))*(4 + 5*x^3)*(2 + (5*x)/(4 + 5*x^3)))/(9*E^2*(10^(2/3) + 5*(-1)^(2/3)*x)*(1 + (10*x)/(4 + 5*x^3))) -

(2^(2/3)*x^((10*x)/(4 + 5*x^3))*(4 + 5*x^3)*(2 + (5*x)/(4 + 5*x^3)))/(9*5^(1/3)*E^2*(2 + 10^(1/3)*x)*(1 + (10*

x)/(4 + 5*x^3))) - (5*(-1/2)^(1/3)*5^(2/3)*x^(2 + (10*x)/(4 + 5*x^3))*(2 + (5*x)/(4 + 5*x^3)))/((1 + (-1)^(1/3

))^4*E^2*((-10)^(2/3) + 5*x)*(1 + (5*x)/(4 + 5*x^3))*(1 + (10*x)/(4 + 5*x^3))) + (5*5^(2/3)*x^(2 + (10*x)/(4 +

5*x^3))*(2 + (5*x)/(4 + 5*x^3)))/(9*2^(1/3)*E^2*(10^(2/3) + 5*(-1)^(2/3)*x)*(1 + (5*x)/(4 + 5*x^3))*(1 + (10*

x)/(4 + 5*x^3))) + (5*x^(2 + (10*x)/(4 + 5*x^3))*(2 + (5*x)/(4 + 5*x^3)))/(9*E^2*(2 + 10^(1/3)*x)*(1 + (5*x)/(

4 + 5*x^3))*(1 + (10*x)/(4 + 5*x^3))) + ((-1/5)^(1/3)*2^(2/3)*x^((10*x)/(4 + 5*x^3))*(4 + 5*x^3)*(3 + (5*x)/(4

+ 5*x^3)))/((1 + (-1)^(1/3))^4*E^2*((-10)^(2/3) + 5*x)*(1 + (10*x)/(4 + 5*x^3))) - (2^(2/3)*x^((10*x)/(4 + 5*

x^3))*(4 + 5*x^3)*(3 + (5*x)/(4 + 5*x^3)))/(9*5^(1/3)*E^2*(10^(2/3) + 5*(-1)^(2/3)*x)*(1 + (10*x)/(4 + 5*x^3))

) - (2*x^((10*x)/(4 + 5*x^3))*(4 + 5*x^3)*(3 + (5*x)/(4 + 5*x^3)))/(45*E^2*(2 + 10^(1/3)*x)*(1 + (10*x)/(4 + 5

*x^3))) + (5*x^(2 + (10*x)/(4 + 5*x^3))*(3 + (5*x)/(4 + 5*x^3)))/(9*E^2*((-10)^(2/3) + 5*x)*(1 + (5*x)/(4 + 5*

x^3))*(1 + (10*x)/(4 + 5*x^3))) + (5*(-1)^(2/3)*x^(2 + (10*x)/(4 + 5*x^3))*(3 + (5*x)/(4 + 5*x^3)))/(9*E^2*(10

^(2/3) + 5*(-1)^(2/3)*x)*(1 + (5*x)/(4 + 5*x^3))*(1 + (10*x)/(4 + 5*x^3))) + (10^(1/3)*x^(2 + (10*x)/(4 + 5*x^

3))*(3 + (5*x)/(4 + 5*x^3)))/(9*E^2*(2 + 10^(1/3)*x)*(1 + (5*x)/(4 + 5*x^3))*(1 + (10*x)/(4 + 5*x^3))) - (5^(1

/3)*x^((10*x)/(4 + 5*x^3))*(4 + 5*x^3)*(1 + (10*x)/(4 + 5*x^3)))/(2^(2/3)*(1 + (-1)^(1/3))^4*E^2*((-10)^(2/3)

+ 5*x)) - (5^(1/3)*x^((10*x)/(4 + 5*x^3))*(4 + 5*x^3)*(1 + (10*x)/(4 + 5*x^3)))/(9*2^(2/3)*E^2*(10^(2/3) + 5*x

)) + ((-5)^(1/3)*x^((10*x)/(4 + 5*x^3))*(4 + 5*x^3)*(1 + (10*x)/(4 + 5*x^3)))/(9*2^(2/3)*E^2*(10^(2/3) + 5*(-1

)^(2/3)*x)) + (25*5^(1/3)*x^(4 + (10*x)/(4 + 5*x^3)))/(2*2^(2/3)*(1 + (-1)^(1/3))^4*E^2*((-10)^(2/3) + 5*x)*(3

+ (10*x)/(4 + 5*x^3))) - (25*(-5)^(1/3)*x^(4 + (10*x)/(4 + 5*x^3)))/(18*2^(2/3)*E^2*(10^(2/3) + 5*(-1)^(2/3)*

x)*(3 + (10*x)/(4 + 5*x^3))) + (5*5^(2/3)*x^(4 + (10*x)/(4 + 5*x^3)))/(18*2^(1/3)*E^2*(2 + 10^(1/3)*x)*(3 + (1

0*x)/(4 + 5*x^3))) - (25*x^(3 + (10*x)/(4 + 5*x^3))*(2 + (5*x)/(4 + 5*x^3)))/(18*E^2*((-10)^(2/3) + 5*x)*(1 +

(5*x)/(4 + 5*x^3))*(3 + (10*x)/(4 + 5*x^3))) - (25*(-1)^(2/3)*x^(3 + (10*x)/(4 + 5*x^3))*(2 + (5*x)/(4 + 5*x^3

)))/(18*E^2*(10^(2/3) + 5*(-1)^(2/3)*x)*(1 + (5*x)/(4 + 5*x^3))*(3 + (10*x)/(4 + 5*x^3))) - (5*5^(1/3)*x^(3 +

(10*x)/(4 + 5*x^3))*(2 + (5*x)/(4 + 5*x^3)))/(9*2^(2/3)*E^2*(2 + 10^(1/3)*x)*(1 + (5*x)/(4 + 5*x^3))*(3 + (10*

x)/(4 + 5*x^3))) - (5*5^(1/3)*x^(3 + (10*x)/(4 + 5*x^3))*(3 + (5*x)/(4 + 5*x^3)))/(2^(2/3)*(1 + (-1)^(1/3))^4*

E^2*((-10)^(2/3) + 5*x)*(1 + (5*x)/(4 + 5*x^3))*(3 + (10*x)/(4 + 5*x^3))) + (5*(-5)^(1/3)*x^(3 + (10*x)/(4 + 5

*x^3))*(3 + (5*x)/(4 + 5*x^3)))/(9*2^(2/3)*E^2*(10^(2/3) + 5*(-1)^(2/3)*x)*(1 + (5*x)/(4 + 5*x^3))*(3 + (10*x)

/(4 + 5*x^3))) - (5^(2/3)*x^(3 + (10*x)/(4 + 5*x^3))*(3 + (5*x)/(4 + 5*x^3)))/(9*2^(1/3)*E^2*(2 + 10^(1/3)*x)*

(1 + (5*x)/(4 + 5*x^3))*(3 + (10*x)/(4 + 5*x^3))) - (5*(-1/2)^(1/3)*5^(2/3)*x^(4 + (10*x)/(4 + 5*x^3))*(3 + (5

*x)/(4 + 5*x^3)))/(2*(1 + (-1)^(1/3))^4*E^2*((-10)^(2/3) + 5*x)*(2 + (5*x)/(4 + 5*x^3))*(3 + (10*x)/(4 + 5*x^3

))) + (5*5^(2/3)*x^(4 + (10*x)/(4 + 5*x^3))*(3 + (5*x)/(4 + 5*x^3)))/(18*2^(1/3)*E^2*(10^(2/3) + 5*(-1)^(2/3)*

x)*(2 + (5*x)/(4 + 5*x^3))*(3 + (10*x)/(4 + 5*x^3))) + (5*x^(4 + (10*x)/(4 + 5*x^3))*(3 + (5*x)/(4 + 5*x^3)))/

(18*E^2*(2 + 10^(1/3)*x)*(2 + (5*x)/(4 + 5*x^3))*(3 + (10*x)/(4 + 5*x^3))) - (4*(-1/5)^(1/3)*2^(2/3)*x^((10*x)

/(4 + 5*x^3))*(4 + 5*x^3)*(3 + (10*x)/(4 + 5*x^3)))/((1 + (-1)^(1/3))^4*E^2*((-10)^(2/3) + 5*x)) + (4*2^(2/3)*

x^((10*x)/(4 + 5*x^3))*(4 + 5*x^3)*(3 + (10*x)/(4 + 5*x^3)))/(9*5^(1/3)*E^2*(10^(2/3) + 5*x)) + (4*2^(2/3)*x^(

(10*x)/(4 + 5*x^3))*(4 + 5*x^3)*(3 + (10*x)/(4 + 5*x^3)))/(9*5^(1/3)*E^2*(10^(2/3) + 5*(-1)^(2/3)*x)) + (4*(-1

/5)^(1/3)*2^(2/3)*x^((10*x)/(4 + 5*x^3))*(4 + 5*x^3)*(3 + (10*x)/(4 + 5*x^3)))/((1 + (-1)^(1/3))^4*E^2*((-10)^

(2/3) + 5*x)*(1 + (10*x)/(4 + 5*x^3))) - (4*2^(2/3)*x^((10*x)/(4 + 5*x^3))*(4 + 5*x^3)*(3 + (10*x)/(4 + 5*x^3)

))/(9*5^(1/3)*E^2*(10^(2/3) + 5*(-1)^(2/3)*x)*(1 + (10*x)/(4 + 5*x^3))) - (8*x^((10*x)/(4 + 5*x^3))*(4 + 5*x^3

)*(3 + (10*x)/(4 + 5*x^3)))/(45*E^2*(2 + 10^(1/3)*x)*(1 + (10*x)/(4 + 5*x^3))) + (20*x^(2 + (10*x)/(4 + 5*x^3)

)*(3 + (10*x)/(4 + 5*x^3)))/(9*E^2*((-10)^(2/3) + 5*x)*(1 + (5*x)/(4 + 5*x^3))*(1 + (10*x)/(4 + 5*x^3))) + (20

*(-1)^(2/3)*x^(2 + (10*x)/(4 + 5*x^3))*(3 + (10*x)/(4 + 5*x^3)))/(9*E^2*(10^(2/3) + 5*(-1)^(2/3)*x)*(1 + (5*x)

/(4 + 5*x^3))*(1 + (10*x)/(4 + 5*x^3))) + (4*10^(1/3)*x^(2 + (10*x)/(4 + 5*x^3))*(3 + (10*x)/(4 + 5*x^3)))/(9*

E^2*(2 + 10^(1/3)*x)*(1 + (5*x)/(4 + 5*x^3))*(1 + (10*x)/(4 + 5*x^3))) - ((-1/2)^(1/3)*5^(2/3)*x^(1 + (10*x)/(

4 + 5*x^3))*Hypergeometric2F1[1, 1 + (10*x)/(4 + 5*x^3), 2 + (10*x)/(4 + 5*x^3), ((-5)^(1/3)*x)/2^(2/3)])/((1

+ (-1)^(1/3))^4*E^2) - (18*(-1)^(5/6)*Sqrt[3]*x^(1 + (10*x)/(4 + 5*x^3))*Hypergeometric2F1[1, 1 + (10*x)/(4 +

5*x^3), 2 + (10*x)/(4 + 5*x^3), ((-5)^(1/3)*x)/2^(2/3)])/((1 + (-1)^(1/3))^5*E^2*(1 + (10*x)/(4 + 5*x^3))) + (

10*x^(2 + (10*x)/(4 + 5*x^3))*Hypergeometric2F1[1, 1 + (10*x)/(4 + 5*x^3), 2 + (10*x)/(4 + 5*x^3), ((-5)^(1/3)

*x)/2^(2/3)])/(E^2*(4 + 5*x^3)*(1 + (10*x)/(4 + 5*x^3))) + ((-1)^(1/3)*10^(2/3)*x^(1 + (10*x)/(4 + 5*x^3))*(2

+ (5*x)/(4 + 5*x^3))*Hypergeometric2F1[1, 1 + (10*x)/(4 + 5*x^3), 2 + (10*x)/(4 + 5*x^3), ((-5)^(1/3)*x)/2^(2/

3)])/((1 + (-1)^(1/3))^4*E^2*(1 + (10*x)/(4 + 5*x^3))) - (2*x^(1 + (10*x)/(4 + 5*x^3))*(3 + (5*x)/(4 + 5*x^3))

*Hypergeometric2F1[1, 1 + (10*x)/(4 + 5*x^3), 2 + (10*x)/(4 + 5*x^3), ((-5)^(1/3)*x)/2^(2/3)])/(9*E^2*(1 + (10

*x)/(4 + 5*x^3))) - (8*x^(1 + (10*x)/(4 + 5*x^3))*(3 + (10*x)/(4 + 5*x^3))*Hypergeometric2F1[1, 1 + (10*x)/(4

+ 5*x^3), 2 + (10*x)/(4 + 5*x^3), ((-5)^(1/3)*x)/2^(2/3)])/(9*E^2*(1 + (10*x)/(4 + 5*x^3))) + (5^(2/3)*x^(1 +

(10*x)/(4 + 5*x^3))*Hypergeometric2F1[1, 1 + (10*x)/(4 + 5*x^3), 2 + (10*x)/(4 + 5*x^3), -((5^(1/3)*x)/2^(2/3)

)])/(9*2^(1/3)*E^2) - (2*x^(1 + (10*x)/(4 + 5*x^3))*Hypergeometric2F1[1, 1 + (10*x)/(4 + 5*x^3), 2 + (10*x)/(4

+ 5*x^3), -((5^(1/3)*x)/2^(2/3))])/(E^2*(1 + (10*x)/(4 + 5*x^3))) + (10*x^(2 + (10*x)/(4 + 5*x^3))*Hypergeome

tric2F1[1, 1 + (10*x)/(4 + 5*x^3), 2 + (10*x)/(4 + 5*x^3), -((5^(1/3)*x)/2^(2/3))])/(E^2*(4 + 5*x^3)*(1 + (10*

x)/(4 + 5*x^3))) - (10^(2/3)*x^(1 + (10*x)/(4 + 5*x^3))*(2 + (5*x)/(4 + 5*x^3))*Hypergeometric2F1[1, 1 + (10*x

)/(4 + 5*x^3), 2 + (10*x)/(4 + 5*x^3), -((5^(1/3)*x)/2^(2/3))])/(9*E^2*(1 + (10*x)/(4 + 5*x^3))) - (2*x^(1 + (

10*x)/(4 + 5*x^3))*(3 + (5*x)/(4 + 5*x^3))*Hypergeometric2F1[1, 1 + (10*x)/(4 + 5*x^3), 2 + (10*x)/(4 + 5*x^3)

, -((5^(1/3)*x)/2^(2/3))])/(9*E^2*(1 + (10*x)/(4 + 5*x^3))) - (8*x^(1 + (10*x)/(4 + 5*x^3))*(3 + (10*x)/(4 + 5

*x^3))*Hypergeometric2F1[1, 1 + (10*x)/(4 + 5*x^3), 2 + (10*x)/(4 + 5*x^3), -((5^(1/3)*x)/2^(2/3))])/(9*E^2*(1

+ (10*x)/(4 + 5*x^3))) - ((-1/2)^(1/3)*5^(2/3)*x^(1 + (10*x)/(4 + 5*x^3))*Hypergeometric2F1[1, 1 + (10*x)/(4

+ 5*x^3), 2 + (10*x)/(4 + 5*x^3), -(((-1)^(2/3)*5^(1/3)*x)/2^(2/3))])/(9*E^2) + (10*x^(2 + (10*x)/(4 + 5*x^3))

*Hypergeometric2F1[1, 1 + (10*x)/(4 + 5*x^3), 2 + (10*x)/(4 + 5*x^3), -(((-1)^(2/3)*5^(1/3)*x)/2^(2/3))])/(E^2

*(4 + 5*x^3)*(1 + (10*x)/(4 + 5*x^3))) + ((-1)^(1/3)*10^(2/3)*x^(1 + (10*x)/(4 + 5*x^3))*(2 + (5*x)/(4 + 5*x^3

))*Hypergeometric2F1[1, 1 + (10*x)/(4 + 5*x^3), 2 + (10*x)/(4 + 5*x^3), -(((-1)^(2/3)*5^(1/3)*x)/2^(2/3))])/(9

*E^2*(1 + (10*x)/(4 + 5*x^3))) - (2*x^(1 + (10*x)/(4 + 5*x^3))*(3 + (5*x)/(4 + 5*x^3))*Hypergeometric2F1[1, 1

+ (10*x)/(4 + 5*x^3), 2 + (10*x)/(4 + 5*x^3), -(((-1)^(2/3)*5^(1/3)*x)/2^(2/3))])/(9*E^2*(1 + (10*x)/(4 + 5*x^

3))) - (8*x^(1 + (10*x)/(4 + 5*x^3))*(3 + (10*x)/(4 + 5*x^3))*Hypergeometric2F1[1, 1 + (10*x)/(4 + 5*x^3), 2 +

(10*x)/(4 + 5*x^3), -(((-1)^(2/3)*5^(1/3)*x)/2^(2/3))])/(9*E^2*(1 + (10*x)/(4 + 5*x^3))) - (2*x^(1 + (10*x)/(

4 + 5*x^3))*Hypergeometric2F1[1, 1 + (10*x)/(4 + 5*x^3), 2 + (10*x)/(4 + 5*x^3), (5^(1/3)*(1 - I*Sqrt[3])*x)/(

2*2^(2/3))])/(E^2*(1 + (10*x)/(4 + 5*x^3))) + (5*(-1)^(5/6)*Sqrt[3]*x^(2 + (10*x)/(4 + 5*x^3))*Hypergeometric2

F1[1, 2 + (10*x)/(4 + 5*x^3), 3 + (10*x)/(4 + 5*x^3), ((-5)^(1/3)*x)/2^(2/3)])/(2*(1 + (-1)^(1/3))^5*E^2*(1 +

(5*x)/(4 + 5*x^3))) + (5*x^(2 + (10*x)/(4 + 5*x^3))*Hypergeometric2F1[1, 2 + (10*x)/(4 + 5*x^3), 3 + (10*x)/(4

+ 5*x^3), -((5^(1/3)*x)/2^(2/3))])/(18*E^2*(1 + (5*x)/(4 + 5*x^3))) + (5*x^(2 + (10*x)/(4 + 5*x^3))*Hypergeom

etric2F1[1, 2 + (10*x)/(4 + 5*x^3), 3 + (10*x)/(4 + 5*x^3), (5^(1/3)*(1 - I*Sqrt[3])*x)/(2*2^(2/3))])/(18*E^2*

(1 + (5*x)/(4 + 5*x^3))) - (10*(-1)^(5/6)*Sqrt[3]*x^(4 + (10*x)/(4 + 5*x^3))*Hypergeometric2F1[1, 4 + (10*x)/(

4 + 5*x^3), 5 + (10*x)/(4 + 5*x^3), ((-5)^(1/3)*x)/2^(2/3)])/((1 + (-1)^(1/3))^5*E^2*(2 + (5*x)/(4 + 5*x^3)))

- (10*x^(4 + (10*x)/(4 + 5*x^3))*Hypergeometric2F1[1, 4 + (10*x)/(4 + 5*x^3), 5 + (10*x)/(4 + 5*x^3), -((5^(1/

3)*x)/2^(2/3))])/(9*E^2*(2 + (5*x)/(4 + 5*x^3))) - (10*x^(4 + (10*x)/(4 + 5*x^3))*Hypergeometric2F1[1, 4 + (10

*x)/(4 + 5*x^3), 5 + (10*x)/(4 + 5*x^3), (5^(1/3)*(1 - I*Sqrt[3])*x)/(2*2^(2/3))])/(9*E^2*(2 + (5*x)/(4 + 5*x^

3))) + (5*(-1)^(5/6)*Sqrt[3]*x^(5 + (10*x)/(4 + 5*x^3))*Hypergeometric2F1[1, 5 + (10*x)/(4 + 5*x^3), 6 + (10*x

)/(4 + 5*x^3), ((-5)^(1/3)*x)/2^(2/3)])/(4*(1 + (-1)^(1/3))^5*E^2*(1 + (2*x)/(4 + 5*x^3))) + (5*x^(5 + (10*x)/

(4 + 5*x^3))*Hypergeometric2F1[1, 5 + (10*x)/(4 + 5*x^3), 6 + (10*x)/(4 + 5*x^3), -((5^(1/3)*x)/2^(2/3))])/(36

*E^2*(1 + (2*x)/(4 + 5*x^3))) + (5*x^(5 + (10*x)/(4 + 5*x^3))*Hypergeometric2F1[1, 5 + (10*x)/(4 + 5*x^3), 6 +

(10*x)/(4 + 5*x^3), (5^(1/3)*(1 - I*Sqrt[3])*x)/(2*2^(2/3))])/(36*E^2*(1 + (2*x)/(4 + 5*x^3))) + (25*(-1)^(5/

6)*Sqrt[3]*x^(7 + (10*x)/(4 + 5*x^3))*Hypergeometric2F1[1, 7 + (10*x)/(4 + 5*x^3), 8 + (10*x)/(4 + 5*x^3), ((-

5)^(1/3)*x)/2^(2/3)])/(8*(1 + (-1)^(1/3))^5*E^2*(7 + (10*x)/(4 + 5*x^3))) + (25*x^(7 + (10*x)/(4 + 5*x^3))*Hyp

ergeometric2F1[1, 7 + (10*x)/(4 + 5*x^3), 8 + (10*x)/(4 + 5*x^3), -((5^(1/3)*x)/2^(2/3))])/(72*E^2*(7 + (10*x)

/(4 + 5*x^3))) + (25*x^(7 + (10*x)/(4 + 5*x^3))*Hypergeometric2F1[1, 7 + (10*x)/(4 + 5*x^3), 8 + (10*x)/(4 + 5

*x^3), (5^(1/3)*(1 - I*Sqrt[3])*x)/(2*2^(2/3))])/(72*E^2*(7 + (10*x)/(4 + 5*x^3))) - (20*2^(2/3)*Defer[Int][(x

^((10*x)/(4 + 5*x^3))*Log[x])/(-2^(2/3) + (-5)^(1/3)*x), x])/(3*E^2) + (5*2^(2/3)*Defer[Int][(x^(1 + (10*x)/(4

+ 5*x^3))*Log[x])/(-2^(2/3) + (-5)^(1/3)*x), x])/(3*E^2) - (20*2^(2/3)*Defer[Int][(x^((10*x)/(4 + 5*x^3))*Log

[x])/(-2^(2/3) - 5^(1/3)*x), x])/(3*E^2) + (5*2^(2/3)*Defer[Int][(x^(1 + (10*x)/(4 + 5*x^3))*Log[x])/(-2^(2/3)

- 5^(1/3)*x), x])/(3*E^2) - (20*2^(2/3)*Defer[Int][(x^((10*x)/(4 + 5*x^3))*Log[x])/(-2^(2/3) - (-1)^(2/3)*5^(

1/3)*x), x])/(3*E^2) + (5*2^(2/3)*Defer[Int][(x^(1 + (10*x)/(4 + 5*x^3))*Log[x])/(-2^(2/3) - (-1)^(2/3)*5^(1/3

)*x), x])/(3*E^2) - (480*Defer[Int][(x^((10*x)/(4 + 5*x^3))*Log[x])/(4 + 5*x^3)^2, x])/E^2 + (120*Defer[Int][(

x^(1 + (10*x)/(4 + 5*x^3))*Log[x])/(4 + 5*x^3)^2, x])/E^2

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=25 \int \frac {80+200 x^3+125 x^6+e^{\frac {2 \left (-4-5 x^3+5 x \log (x)\right )}{4+5 x^3}} \left (-144+40 x-160 x^3+50 x^4+25 x^6+\left (-160+40 x+400 x^3-100 x^4\right ) \log (x)\right )}{\left (20+25 x^3\right )^2} \, dx\\ &=25 \int \left (\frac {1}{5}+\frac {x^{\frac {10 x}{4+5 x^3}} \left (-144+40 x-160 x^3+50 x^4+25 x^6-160 \log (x)+40 x \log (x)+400 x^3 \log (x)-100 x^4 \log (x)\right )}{25 e^2 \left (4+5 x^3\right )^2}\right ) \, dx\\ &=5 x+\frac {\int \frac {x^{\frac {10 x}{4+5 x^3}} \left (-144+40 x-160 x^3+50 x^4+25 x^6-160 \log (x)+40 x \log (x)+400 x^3 \log (x)-100 x^4 \log (x)\right )}{\left (4+5 x^3\right )^2} \, dx}{e^2}\\ &=5 x+\frac {\int \left (-\frac {144 x^{\frac {10 x}{4+5 x^3}}}{\left (4+5 x^3\right )^2}+\frac {40 x^{1+\frac {10 x}{4+5 x^3}}}{\left (4+5 x^3\right )^2}-\frac {160 x^{3+\frac {10 x}{4+5 x^3}}}{\left (4+5 x^3\right )^2}+\frac {50 x^{4+\frac {10 x}{4+5 x^3}}}{\left (4+5 x^3\right )^2}+\frac {25 x^{6+\frac {10 x}{4+5 x^3}}}{\left (4+5 x^3\right )^2}-\frac {20 x^{\frac {10 x}{4+5 x^3}} \left (8-2 x-20 x^3+5 x^4\right ) \log (x)}{\left (4+5 x^3\right )^2}\right ) \, dx}{e^2}\\ &=5 x-\frac {20 \int \frac {x^{\frac {10 x}{4+5 x^3}} \left (8-2 x-20 x^3+5 x^4\right ) \log (x)}{\left (4+5 x^3\right )^2} \, dx}{e^2}+\frac {25 \int \frac {x^{6+\frac {10 x}{4+5 x^3}}}{\left (4+5 x^3\right )^2} \, dx}{e^2}+\frac {40 \int \frac {x^{1+\frac {10 x}{4+5 x^3}}}{\left (4+5 x^3\right )^2} \, dx}{e^2}+\frac {50 \int \frac {x^{4+\frac {10 x}{4+5 x^3}}}{\left (4+5 x^3\right )^2} \, dx}{e^2}-\frac {144 \int \frac {x^{\frac {10 x}{4+5 x^3}}}{\left (4+5 x^3\right )^2} \, dx}{e^2}-\frac {160 \int \frac {x^{3+\frac {10 x}{4+5 x^3}}}{\left (4+5 x^3\right )^2} \, dx}{e^2}\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A] time = 0.96, size = 25, normalized size = 0.89 \begin {gather*} 5 x+\frac {(-4+x) x^{\frac {10 x}{4+5 x^3}}}{e^2} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

Integrate[(80 + 200*x^3 + 125*x^6 + E^((2*(-4 - 5*x^3 + 5*x*Log[x]))/(4 + 5*x^3))*(-144 + 40*x - 160*x^3 + 50*

x^4 + 25*x^6 + (-160 + 40*x + 400*x^3 - 100*x^4)*Log[x]))/(16 + 40*x^3 + 25*x^6),x]

[Out]

5*x + ((-4 + x)*x^((10*x)/(4 + 5*x^3)))/E^2

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fricas [A] time = 0.48, size = 32, normalized size = 1.14 \begin {gather*} {\left (x - 4\right )} e^{\left (-\frac {2 \, {\left (5 \, x^{3} - 5 \, x \log \relax (x) + 4\right )}}{5 \, x^{3} + 4}\right )} + 5 \, x \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-100*x^4+400*x^3+40*x-160)*log(x)+25*x^6+50*x^4-160*x^3+40*x-144)*exp((5*x*log(x)-5*x^3-4)/(5*x^3

+4))^2+125*x^6+200*x^3+80)/(25*x^6+40*x^3+16),x, algorithm="fricas")

[Out]

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

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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {125 \, x^{6} + 200 \, x^{3} + {\left (25 \, x^{6} + 50 \, x^{4} - 160 \, x^{3} - 20 \, {\left (5 \, x^{4} - 20 \, x^{3} - 2 \, x + 8\right )} \log \relax (x) + 40 \, x - 144\right )} e^{\left (-\frac {2 \, {\left (5 \, x^{3} - 5 \, x \log \relax (x) + 4\right )}}{5 \, x^{3} + 4}\right )} + 80}{25 \, x^{6} + 40 \, x^{3} + 16}\,{d x} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-100*x^4+400*x^3+40*x-160)*log(x)+25*x^6+50*x^4-160*x^3+40*x-144)*exp((5*x*log(x)-5*x^3-4)/(5*x^3

+4))^2+125*x^6+200*x^3+80)/(25*x^6+40*x^3+16),x, algorithm="giac")

[Out]

integrate((125*x^6 + 200*x^3 + (25*x^6 + 50*x^4 - 160*x^3 - 20*(5*x^4 - 20*x^3 - 2*x + 8)*log(x) + 40*x - 144)

*e^(-2*(5*x^3 - 5*x*log(x) + 4)/(5*x^3 + 4)) + 80)/(25*x^6 + 40*x^3 + 16), x)

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maple [A] time = 0.10, size = 27, normalized size = 0.96


method	result	size

risch	\(5 x +\left (x -4\right ) x^{\frac {10 x}{5 x^{3}+4}} {\mathrm e}^{-2}\)	\(27\)

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((((-100*x^4+400*x^3+40*x-160)*ln(x)+25*x^6+50*x^4-160*x^3+40*x-144)*exp((5*x*ln(x)-5*x^3-4)/(5*x^3+4))^2+1

25*x^6+200*x^3+80)/(25*x^6+40*x^3+16),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A] time = 0.57, size = 25, normalized size = 0.89 \begin {gather*} {\left (x - 4\right )} e^{\left (\frac {10 \, x \log \relax (x)}{5 \, x^{3} + 4} - 2\right )} + 5 \, x \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-100*x^4+400*x^3+40*x-160)*log(x)+25*x^6+50*x^4-160*x^3+40*x-144)*exp((5*x*log(x)-5*x^3-4)/(5*x^3

+4))^2+125*x^6+200*x^3+80)/(25*x^6+40*x^3+16),x, algorithm="maxima")

[Out]

(x - 4)*e^(10*x*log(x)/(5*x^3 + 4) - 2) + 5*x

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mupad [B] time = 5.57, size = 49, normalized size = 1.75 \begin {gather*} 5\,x+x^{\frac {10\,x}{5\,x^3+4}}\,{\mathrm {e}}^{-\frac {10\,x^3}{5\,x^3+4}-\frac {8}{5\,x^3+4}}\,\left (x-4\right ) \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-(2*(5*x^3 - 5*x*log(x) + 4))/(5*x^3 + 4))*(40*x - 160*x^3 + 50*x^4 + 25*x^6 + log(x)*(40*x + 400*x^3

- 100*x^4 - 160) - 144) + 200*x^3 + 125*x^6 + 80)/(40*x^3 + 25*x^6 + 16),x)

[Out]

5*x + x^((10*x)/(5*x^3 + 4))*exp(- (10*x^3)/(5*x^3 + 4) - 8/(5*x^3 + 4))*(x - 4)

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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-100*x**4+400*x**3+40*x-160)*ln(x)+25*x**6+50*x**4-160*x**3+40*x-144)*exp((5*x*ln(x)-5*x**3-4)/(5

*x**3+4))**2+125*x**6+200*x**3+80)/(25*x**6+40*x**3+16),x)

[Out]

Timed out

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