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3.97.26 \(\int \frac {x^{\frac {5 x^2}{15-8 x+6 x^2+(-8+2 x) \log (4)+\log ^2(4)}} (75 x-40 x^2+30 x^3+(-40 x+10 x^2) \log (4)+5 x \log ^2(4)+(150 x-40 x^2+(-80 x+10 x^2) \log (4)+10 x \log ^2(4)) \log (x))}{225-240 x+244 x^2-96 x^3+36 x^4+(-240+188 x-128 x^2+24 x^3) \log (4)+(94-48 x+16 x^2) \log ^2(4)+(-16+4 x) \log ^3(4)+\log ^4(4)} \, dx\)

Optimal. Leaf size=24 \[ x^{\frac {x}{x+\frac {-1+(-4+x+\log (4))^2}{5 x}}} \]

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Rubi [F]  time = 21.07, 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 {x^{\frac {5 x^2}{15-8 x+6 x^2+(-8+2 x) \log (4)+\log ^2(4)}} \left (75 x-40 x^2+30 x^3+\left (-40 x+10 x^2\right ) \log (4)+5 x \log ^2(4)+\left (150 x-40 x^2+\left (-80 x+10 x^2\right ) \log (4)+10 x \log ^2(4)\right ) \log (x)\right )}{225-240 x+244 x^2-96 x^3+36 x^4+\left (-240+188 x-128 x^2+24 x^3\right ) \log (4)+\left (94-48 x+16 x^2\right ) \log ^2(4)+(-16+4 x) \log ^3(4)+\log ^4(4)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(x^((5*x^2)/(15 - 8*x + 6*x^2 + (-8 + 2*x)*Log[4] + Log[4]^2))*(75*x - 40*x^2 + 30*x^3 + (-40*x + 10*x^2)*
Log[4] + 5*x*Log[4]^2 + (150*x - 40*x^2 + (-80*x + 10*x^2)*Log[4] + 10*x*Log[4]^2)*Log[x]))/(225 - 240*x + 244
*x^2 - 96*x^3 + 36*x^4 + (-240 + 188*x - 128*x^2 + 24*x^3)*Log[4] + (94 - 48*x + 16*x^2)*Log[4]^2 + (-16 + 4*x
)*Log[4]^3 + Log[4]^4),x]

[Out]

(-5*x^(1 + (5*x^2)/(15 - 8*x + 6*x^2 - 2*(4 - x)*Log[4] + Log[4]^2))*Hypergeometric2F1[1, 1 + (5*x^2)/(15 - 8*
x + 6*x^2 - 2*(4 - x)*Log[4] + Log[4]^2), 2 + (5*x^2)/(15 - 8*x + 6*x^2 - 2*(4 - x)*Log[4] + Log[4]^2), (6*x)/
(4 - Log[4] + I*Sqrt[74 - 40*Log[4] + 5*Log[4]^2])]*(4 - Log[4])*(2 + (5*x^2)/(15 - 8*x + 6*x^2 - 2*(4 - x)*Lo
g[4] + Log[4]^2)))/(2*(74 - 40*Log[4] + 5*Log[4]^2)*(1 + (5*x^2)/(15 - 8*x + 6*x^2 - 2*(4 - x)*Log[4] + Log[4]
^2))) - (5*x^(1 + (5*x^2)/(15 - 8*x + 6*x^2 - 2*(4 - x)*Log[4] + Log[4]^2))*Hypergeometric2F1[1, 1 + (5*x^2)/(
15 - 8*x + 6*x^2 - 2*(4 - x)*Log[4] + Log[4]^2), 2 + (5*x^2)/(15 - 8*x + 6*x^2 - 2*(4 - x)*Log[4] + Log[4]^2),
 (12*x)/(8 - (2*I)*Sqrt[74 - 40*Log[4] + 5*Log[4]^2] - Log[16])]*(4 - Log[4])*(2 + (5*x^2)/(15 - 8*x + 6*x^2 -
 2*(4 - x)*Log[4] + Log[4]^2)))/(2*(74 - 40*Log[4] + 5*Log[4]^2)*(1 + (5*x^2)/(15 - 8*x + 6*x^2 - 2*(4 - x)*Lo
g[4] + Log[4]^2))) - (45*x^(2 + (5*x^2)/(15 - 8*x + 6*x^2 - 2*(4 - x)*Log[4] + Log[4]^2))*Hypergeometric2F1[1,
 2 + (5*x^2)/(15 - 8*x + 6*x^2 - 2*(4 - x)*Log[4] + Log[4]^2), 3 + (5*x^2)/(15 - 8*x + 6*x^2 - 2*(4 - x)*Log[4
] + Log[4]^2), (6*x)/(4 - Log[4] + I*Sqrt[74 - 40*Log[4] + 5*Log[4]^2])]*(3 - Log[4])*(5 - Log[4]))/((74 - 40*
Log[4] + 5*Log[4]^2)^(3/2)*(2 + (5*x^2)/(15 - 8*x + 6*x^2 - 2*(4 - x)*Log[4] + Log[4]^2))*(4*I - I*Log[4] - Sq
rt[74 - 40*Log[4] + 5*Log[4]^2])) + (90*x^(3 + (5*x^2)/(15 - 8*x + 6*x^2 - 2*(4 - x)*Log[4] + Log[4]^2))*Hyper
geometric2F1[1, 3 + (5*x^2)/(15 - 8*x + 6*x^2 - 2*(4 - x)*Log[4] + Log[4]^2), 4 + (5*x^2)/(15 - 8*x + 6*x^2 -
2*(4 - x)*Log[4] + Log[4]^2), (6*x)/(4 - Log[4] + I*Sqrt[74 - 40*Log[4] + 5*Log[4]^2])]*(4 - Log[4]))/((74 - 4
0*Log[4] + 5*Log[4]^2)^(3/2)*(3 + (5*x^2)/(15 - 8*x + 6*x^2 - 2*(4 - x)*Log[4] + Log[4]^2))*(4*I - I*Log[4] -
Sqrt[74 - 40*Log[4] + 5*Log[4]^2])) - (270*x^(4 + (5*x^2)/(15 - 8*x + 6*x^2 - 2*(4 - x)*Log[4] + Log[4]^2))*Hy
pergeometric2F1[1, 4 + (5*x^2)/(15 - 8*x + 6*x^2 - 2*(4 - x)*Log[4] + Log[4]^2), 5 + (5*x^2)/(15 - 8*x + 6*x^2
 - 2*(4 - x)*Log[4] + Log[4]^2), (6*x)/(4 - Log[4] + I*Sqrt[74 - 40*Log[4] + 5*Log[4]^2])])/((74 - 40*Log[4] +
 5*Log[4]^2)^(3/2)*(4 + (5*x^2)/(15 - 8*x + 6*x^2 - 2*(4 - x)*Log[4] + Log[4]^2))*(4*I - I*Log[4] - Sqrt[74 -
40*Log[4] + 5*Log[4]^2])) + (15*x^(1 + (5*x^2)/(15 - 8*x + 6*x^2 - 2*(4 - x)*Log[4] + Log[4]^2))*Hypergeometri
c2F1[1, 1 + (5*x^2)/(15 - 8*x + 6*x^2 - 2*(4 - x)*Log[4] + Log[4]^2), 2 + (5*x^2)/(15 - 8*x + 6*x^2 - 2*(4 - x
)*Log[4] + Log[4]^2), (6*x)/(4 - Log[4] + I*Sqrt[74 - 40*Log[4] + 5*Log[4]^2])]*(3 - Log[4])*(5 - Log[4]))/(2*
(74 - 40*Log[4] + 5*Log[4]^2)*(4 - Log[4] + I*Sqrt[74 - 40*Log[4] + 5*Log[4]^2])) + (5*x^(1 + (5*x^2)/(15 - 8*
x + 6*x^2 - 2*(4 - x)*Log[4] + Log[4]^2))*Hypergeometric2F1[1, 1 + (5*x^2)/(15 - 8*x + 6*x^2 - 2*(4 - x)*Log[4
] + Log[4]^2), 2 + (5*x^2)/(15 - 8*x + 6*x^2 - 2*(4 - x)*Log[4] + Log[4]^2), (6*x)/(4 - Log[4] + I*Sqrt[74 - 4
0*Log[4] + 5*Log[4]^2])]*(3 + (5*x^2)/(15 - 8*x + 6*x^2 - 2*(4 - x)*Log[4] + Log[4]^2))*(4 - Log[4] + I*Sqrt[7
4 - 40*Log[4] + 5*Log[4]^2]))/(4*(74 - 40*Log[4] + 5*Log[4]^2)*(1 + (5*x^2)/(15 - 8*x + 6*x^2 - 2*(4 - x)*Log[
4] + Log[4]^2))) + (3*x^(-1 + (5*x^2)/(15 - 8*x + 6*x^2 - 2*(4 - x)*Log[4] + Log[4]^2))*(3 - Log[4])*(5 - Log[
4])*(15 - 8*x + 6*x^2 - 2*(4 - x)*Log[4] + Log[4]^2))/(2*(74 - 40*Log[4] + 5*Log[4]^2)*(4 - 6*x - Log[4] + I*S
qrt[74 - 40*Log[4] + 5*Log[4]^2])) - (15*x^(2 + (5*x^2)/(15 - 8*x + 6*x^2 - 2*(4 - x)*Log[4] + Log[4]^2))*(4 -
 Log[4]))/((74 - 40*Log[4] + 5*Log[4]^2)*(1 + (5*x^2)/(15 - 8*x + 6*x^2 - 2*(4 - x)*Log[4] + Log[4]^2))*(4 - 6
*x - Log[4] + I*Sqrt[74 - 40*Log[4] + 5*Log[4]^2])) - (3*x^(-1 + (5*x^2)/(15 - 8*x + 6*x^2 - 2*(4 - x)*Log[4]
+ Log[4]^2))*(3 - Log[4])*(5 - Log[4])*(15 - 8*x + 6*x^2 - 2*(4 - x)*Log[4] + Log[4]^2)*(1 + (5*x^2)/(15 - 8*x
 + 6*x^2 - 2*(4 - x)*Log[4] + Log[4]^2)))/(2*(74 - 40*Log[4] + 5*Log[4]^2)*(4 - 6*x - Log[4] + I*Sqrt[74 - 40*
Log[4] + 5*Log[4]^2])) + (45*x^(3 + (5*x^2)/(15 - 8*x + 6*x^2 - 2*(4 - x)*Log[4] + Log[4]^2)))/((74 - 40*Log[4
] + 5*Log[4]^2)*(2 + (5*x^2)/(15 - 8*x + 6*x^2 - 2*(4 - x)*Log[4] + Log[4]^2))*(4 - 6*x - Log[4] + I*Sqrt[74 -
 40*Log[4] + 5*Log[4]^2])) + (x^(-1 + (5*x^2)/(15 - 8*x + 6*x^2 - 2*(4 - x)*Log[4] + Log[4]^2))*(4 - Log[4])*(
15 - 8*x + 6*x^2 - 2*(4 - x)*Log[4] + Log[4]^2)*(2 + (5*x^2)/(15 - 8*x + 6*x^2 - 2*(4 - x)*Log[4] + Log[4]^2))
*(4 - Log[4] + I*Sqrt[74 - 40*Log[4] + 5*Log[4]^2]))/(2*(74 - 40*Log[4] + 5*Log[4]^2)*(4 - 6*x - Log[4] + I*Sq
rt[74 - 40*Log[4] + 5*Log[4]^2])) - (x^(-1 + (5*x^2)/(15 - 8*x + 6*x^2 - 2*(4 - x)*Log[4] + Log[4]^2))*(4 - Lo
g[4])*(15 - 8*x + 6*x^2 - 2*(4 - x)*Log[4] + Log[4]^2)*(2 + (5*x^2)/(15 - 8*x + 6*x^2 - 2*(4 - x)*Log[4] + Log
[4]^2))*(4 - Log[4] + I*Sqrt[74 - 40*Log[4] + 5*Log[4]^2]))/(2*(74 - 40*Log[4] + 5*Log[4]^2)*(1 + (5*x^2)/(15
- 8*x + 6*x^2 - 2*(4 - x)*Log[4] + Log[4]^2))*(4 - 6*x - Log[4] + I*Sqrt[74 - 40*Log[4] + 5*Log[4]^2])) + (15*
x^(2 + (5*x^2)/(15 - 8*x + 6*x^2 - 2*(4 - x)*Log[4] + Log[4]^2))*(3 + (5*x^2)/(15 - 8*x + 6*x^2 - 2*(4 - x)*Lo
g[4] + Log[4]^2))*(4 - Log[4] + I*Sqrt[74 - 40*Log[4] + 5*Log[4]^2]))/(2*(74 - 40*Log[4] + 5*Log[4]^2)*(1 + (5
*x^2)/(15 - 8*x + 6*x^2 - 2*(4 - x)*Log[4] + Log[4]^2))*(2 + (5*x^2)/(15 - 8*x + 6*x^2 - 2*(4 - x)*Log[4] + Lo
g[4]^2))*(4 - 6*x - Log[4] + I*Sqrt[74 - 40*Log[4] + 5*Log[4]^2])) - (x^(-1 + (5*x^2)/(15 - 8*x + 6*x^2 - 2*(4
 - x)*Log[4] + Log[4]^2))*(15 - 8*x + 6*x^2 - 2*(4 - x)*Log[4] + Log[4]^2)*(3 + (5*x^2)/(15 - 8*x + 6*x^2 - 2*
(4 - x)*Log[4] + Log[4]^2))*(4 - Log[4] + I*Sqrt[74 - 40*Log[4] + 5*Log[4]^2])^2)/(4*(74 - 40*Log[4] + 5*Log[4
]^2)*(4 - 6*x - Log[4] + I*Sqrt[74 - 40*Log[4] + 5*Log[4]^2])) + (x^(-1 + (5*x^2)/(15 - 8*x + 6*x^2 - 2*(4 - x
)*Log[4] + Log[4]^2))*(15 - 8*x + 6*x^2 - 2*(4 - x)*Log[4] + Log[4]^2)*(3 + (5*x^2)/(15 - 8*x + 6*x^2 - 2*(4 -
 x)*Log[4] + Log[4]^2))*(4 - Log[4] + I*Sqrt[74 - 40*Log[4] + 5*Log[4]^2])^2)/(4*(74 - 40*Log[4] + 5*Log[4]^2)
*(1 + (5*x^2)/(15 - 8*x + 6*x^2 - 2*(4 - x)*Log[4] + Log[4]^2))*(4 - 6*x - Log[4] + I*Sqrt[74 - 40*Log[4] + 5*
Log[4]^2])) + (90*x^(2 + (5*x^2)/(15 - 8*x + 6*x^2 - 2*(4 - x)*Log[4] + Log[4]^2))*Hypergeometric2F1[1, 2 + (5
*x^2)/(15 - 8*x + 6*x^2 - 2*(4 - x)*Log[4] + Log[4]^2), 3 + (5*x^2)/(15 - 8*x + 6*x^2 - 2*(4 - x)*Log[4] + Log
[4]^2), (12*x)/(8 - (2*I)*Sqrt[74 - 40*Log[4] + 5*Log[4]^2] - Log[16])]*(3 - Log[4])*(5 - Log[4]))/((74 - 40*L
og[4] + 5*Log[4]^2)^(3/2)*(2 + (5*x^2)/(15 - 8*x + 6*x^2 - 2*(4 - x)*Log[4] + Log[4]^2))*(2*Sqrt[74 - 40*Log[4
] + 5*Log[4]^2] + I*(8 - Log[16]))) - (180*x^(3 + (5*x^2)/(15 - 8*x + 6*x^2 - 2*(4 - x)*Log[4] + Log[4]^2))*Hy
pergeometric2F1[1, 3 + (5*x^2)/(15 - 8*x + 6*x^2 - 2*(4 - x)*Log[4] + Log[4]^2), 4 + (5*x^2)/(15 - 8*x + 6*x^2
 - 2*(4 - x)*Log[4] + Log[4]^2), (12*x)/(8 - (2*I)*Sqrt[74 - 40*Log[4] + 5*Log[4]^2] - Log[16])]*(4 - Log[4]))
/((74 - 40*Log[4] + 5*Log[4]^2)^(3/2)*(3 + (5*x^2)/(15 - 8*x + 6*x^2 - 2*(4 - x)*Log[4] + Log[4]^2))*(2*Sqrt[7
4 - 40*Log[4] + 5*Log[4]^2] + I*(8 - Log[16]))) + (540*x^(4 + (5*x^2)/(15 - 8*x + 6*x^2 - 2*(4 - x)*Log[4] + L
og[4]^2))*Hypergeometric2F1[1, 4 + (5*x^2)/(15 - 8*x + 6*x^2 - 2*(4 - x)*Log[4] + Log[4]^2), 5 + (5*x^2)/(15 -
 8*x + 6*x^2 - 2*(4 - x)*Log[4] + Log[4]^2), (12*x)/(8 - (2*I)*Sqrt[74 - 40*Log[4] + 5*Log[4]^2] - Log[16])])/
((74 - 40*Log[4] + 5*Log[4]^2)^(3/2)*(4 + (5*x^2)/(15 - 8*x + 6*x^2 - 2*(4 - x)*Log[4] + Log[4]^2))*(2*Sqrt[74
 - 40*Log[4] + 5*Log[4]^2] + I*(8 - Log[16]))) + (15*x^(1 + (5*x^2)/(15 - 8*x + 6*x^2 - 2*(4 - x)*Log[4] + Log
[4]^2))*Hypergeometric2F1[1, 1 + (5*x^2)/(15 - 8*x + 6*x^2 - 2*(4 - x)*Log[4] + Log[4]^2), 2 + (5*x^2)/(15 - 8
*x + 6*x^2 - 2*(4 - x)*Log[4] + Log[4]^2), (12*x)/(8 - (2*I)*Sqrt[74 - 40*Log[4] + 5*Log[4]^2] - Log[16])]*(3
- Log[4])*(5 - Log[4]))/((74 - 40*Log[4] + 5*Log[4]^2)*(8 - (2*I)*Sqrt[74 - 40*Log[4] + 5*Log[4]^2] - Log[16])
) + (5*x^(1 + (5*x^2)/(15 - 8*x + 6*x^2 - 2*(4 - x)*Log[4] + Log[4]^2))*Hypergeometric2F1[1, 1 + (5*x^2)/(15 -
 8*x + 6*x^2 - 2*(4 - x)*Log[4] + Log[4]^2), 2 + (5*x^2)/(15 - 8*x + 6*x^2 - 2*(4 - x)*Log[4] + Log[4]^2), (12
*x)/(8 - (2*I)*Sqrt[74 - 40*Log[4] + 5*Log[4]^2] - Log[16])]*(3 + (5*x^2)/(15 - 8*x + 6*x^2 - 2*(4 - x)*Log[4]
 + Log[4]^2))*(8 - (2*I)*Sqrt[74 - 40*Log[4] + 5*Log[4]^2] - Log[16]))/(8*(74 - 40*Log[4] + 5*Log[4]^2)*(1 + (
5*x^2)/(15 - 8*x + 6*x^2 - 2*(4 - x)*Log[4] + Log[4]^2))) + (3*x^(-1 + (5*x^2)/(15 - 8*x + 6*x^2 - 2*(4 - x)*L
og[4] + Log[4]^2))*(3 - Log[4])*(5 - Log[4])*(15 - 8*x + 6*x^2 - 2*(4 - x)*Log[4] + Log[4]^2))/((74 - 40*Log[4
] + 5*Log[4]^2)*(8 - 12*x - (2*I)*Sqrt[74 - 40*Log[4] + 5*Log[4]^2] - Log[16])) - (30*x^(2 + (5*x^2)/(15 - 8*x
 + 6*x^2 - 2*(4 - x)*Log[4] + Log[4]^2))*(4 - Log[4]))/((74 - 40*Log[4] + 5*Log[4]^2)*(1 + (5*x^2)/(15 - 8*x +
 6*x^2 - 2*(4 - x)*Log[4] + Log[4]^2))*(8 - 12*x - (2*I)*Sqrt[74 - 40*Log[4] + 5*Log[4]^2] - Log[16])) - (3*x^
(-1 + (5*x^2)/(15 - 8*x + 6*x^2 - 2*(4 - x)*Log[4] + Log[4]^2))*(3 - Log[4])*(5 - Log[4])*(15 - 8*x + 6*x^2 -
2*(4 - x)*Log[4] + Log[4]^2)*(1 + (5*x^2)/(15 - 8*x + 6*x^2 - 2*(4 - x)*Log[4] + Log[4]^2)))/((74 - 40*Log[4]
+ 5*Log[4]^2)*(8 - 12*x - (2*I)*Sqrt[74 - 40*Log[4] + 5*Log[4]^2] - Log[16])) + (90*x^(3 + (5*x^2)/(15 - 8*x +
 6*x^2 - 2*(4 - x)*Log[4] + Log[4]^2)))/((74 - 40*Log[4] + 5*Log[4]^2)*(2 + (5*x^2)/(15 - 8*x + 6*x^2 - 2*(4 -
 x)*Log[4] + Log[4]^2))*(8 - 12*x - (2*I)*Sqrt[74 - 40*Log[4] + 5*Log[4]^2] - Log[16])) + (x^(-1 + (5*x^2)/(15
 - 8*x + 6*x^2 - 2*(4 - x)*Log[4] + Log[4]^2))*(15 - 8*x + 6*x^2 - 2*(4 - x)*Log[4] + Log[4]^2)*(3 + (5*x^2)/(
15 - 8*x + 6*x^2 - 2*(4 - x)*Log[4] + Log[4]^2))*(2*Sqrt[74 - 40*Log[4] + 5*Log[4]^2] + I*(8 - Log[16]))^2)/(8
*(74 - 40*Log[4] + 5*Log[4]^2)*(8 - 12*x - (2*I)*Sqrt[74 - 40*Log[4] + 5*Log[4]^2] - Log[16])) - (x^(-1 + (5*x
^2)/(15 - 8*x + 6*x^2 - 2*(4 - x)*Log[4] + Log[4]^2))*(15 - 8*x + 6*x^2 - 2*(4 - x)*Log[4] + Log[4]^2)*(3 + (5
*x^2)/(15 - 8*x + 6*x^2 - 2*(4 - x)*Log[4] + Log[4]^2))*(2*Sqrt[74 - 40*Log[4] + 5*Log[4]^2] + I*(8 - Log[16])
)^2)/(8*(74 - 40*Log[4] + 5*Log[4]^2)*(1 + (5*x^2)/(15 - 8*x + 6*x^2 - 2*(4 - x)*Log[4] + Log[4]^2))*(8 - 12*x
 - (2*I)*Sqrt[74 - 40*Log[4] + 5*Log[4]^2] - Log[16])) + (x^(-1 + (5*x^2)/(15 - 8*x + 6*x^2 - 2*(4 - x)*Log[4]
 + Log[4]^2))*(4 - Log[4])*(15 - 8*x + 6*x^2 - 2*(4 - x)*Log[4] + Log[4]^2)*(2 + (5*x^2)/(15 - 8*x + 6*x^2 - 2
*(4 - x)*Log[4] + Log[4]^2))*(8 - (2*I)*Sqrt[74 - 40*Log[4] + 5*Log[4]^2] - Log[16]))/(2*(74 - 40*Log[4] + 5*L
og[4]^2)*(8 - 12*x - (2*I)*Sqrt[74 - 40*Log[4] + 5*Log[4]^2] - Log[16])) - (x^(-1 + (5*x^2)/(15 - 8*x + 6*x^2
- 2*(4 - x)*Log[4] + Log[4]^2))*(4 - Log[4])*(15 - 8*x + 6*x^2 - 2*(4 - x)*Log[4] + Log[4]^2)*(2 + (5*x^2)/(15
 - 8*x + 6*x^2 - 2*(4 - x)*Log[4] + Log[4]^2))*(8 - (2*I)*Sqrt[74 - 40*Log[4] + 5*Log[4]^2] - Log[16]))/(2*(74
 - 40*Log[4] + 5*Log[4]^2)*(1 + (5*x^2)/(15 - 8*x + 6*x^2 - 2*(4 - x)*Log[4] + Log[4]^2))*(8 - 12*x - (2*I)*Sq
rt[74 - 40*Log[4] + 5*Log[4]^2] - Log[16])) + (15*x^(2 + (5*x^2)/(15 - 8*x + 6*x^2 - 2*(4 - x)*Log[4] + Log[4]
^2))*(3 + (5*x^2)/(15 - 8*x + 6*x^2 - 2*(4 - x)*Log[4] + Log[4]^2))*(8 - (2*I)*Sqrt[74 - 40*Log[4] + 5*Log[4]^
2] - Log[16]))/(2*(74 - 40*Log[4] + 5*Log[4]^2)*(1 + (5*x^2)/(15 - 8*x + 6*x^2 - 2*(4 - x)*Log[4] + Log[4]^2))
*(2 + (5*x^2)/(15 - 8*x + 6*x^2 - 2*(4 - x)*Log[4] + Log[4]^2))*(8 - 12*x - (2*I)*Sqrt[74 - 40*Log[4] + 5*Log[
4]^2] - Log[16])) + 10*(3 - Log[4])*(5 - Log[4])*Defer[Int][(x^(1 + (5*x^2)/(15 - 8*x + 6*x^2 + 2*(-4 + x)*Log
[4] + Log[4]^2))*Log[x])/(6*x^2 - 2*x*(4 - Log[4]) + (3 - Log[4])*(5 - Log[4]))^2, x] - 10*(4 - Log[4])*Defer[
Int][(x^(2 + (5*x^2)/(15 - 8*x + 6*x^2 + 2*(-4 + x)*Log[4] + Log[4]^2))*Log[x])/(6*x^2 - 2*x*(4 - Log[4]) + (3
 - Log[4])*(5 - Log[4]))^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {x^{\frac {5 x^2}{15-8 x+6 x^2+(-8+2 x) \log (4)+\log ^2(4)}} \left (-40 x^2+30 x^3+\left (-40 x+10 x^2\right ) \log (4)+x \left (75+5 \log ^2(4)\right )+\left (150 x-40 x^2+\left (-80 x+10 x^2\right ) \log (4)+10 x \log ^2(4)\right ) \log (x)\right )}{225-240 x+244 x^2-96 x^3+36 x^4+\left (-240+188 x-128 x^2+24 x^3\right ) \log (4)+\left (94-48 x+16 x^2\right ) \log ^2(4)+(-16+4 x) \log ^3(4)+\log ^4(4)} \, dx\\ &=\int \frac {5 x^{1+\frac {5 x^2}{15-8 x+6 x^2+2 (-4+x) \log (4)+\log ^2(4)}} \left (6 x^2+2 x (-4+\log (4))+15 \left (1+\frac {2}{15} (-4+\log (2)) \log (4)\right )+2 \left (15+x (-4+\log (4))-8 \log (4)+\log ^2(4)\right ) \log (x)\right )}{\left (6 x^2-2 x (4-\log (4))+(3-\log (4)) (5-\log (4))\right )^2} \, dx\\ &=5 \int \frac {x^{1+\frac {5 x^2}{15-8 x+6 x^2+2 (-4+x) \log (4)+\log ^2(4)}} \left (6 x^2+2 x (-4+\log (4))+15 \left (1+\frac {2}{15} (-4+\log (2)) \log (4)\right )+2 \left (15+x (-4+\log (4))-8 \log (4)+\log ^2(4)\right ) \log (x)\right )}{\left (6 x^2-2 x (4-\log (4))+(3-\log (4)) (5-\log (4))\right )^2} \, dx\\ &=5 \int \left (\frac {6 x^{3+\frac {5 x^2}{15-8 x+6 x^2+2 (-4+x) \log (4)+\log ^2(4)}}}{\left (6 x^2-2 x (4-\log (4))+(3-\log (4)) (5-\log (4))\right )^2}+\frac {x^{1+\frac {5 x^2}{15-8 x+6 x^2+2 (-4+x) \log (4)+\log ^2(4)}} (3-\log (4)) (5-\log (4))}{\left (6 x^2-2 x (4-\log (4))+(3-\log (4)) (5-\log (4))\right )^2}+\frac {2 x^{2+\frac {5 x^2}{15-8 x+6 x^2+2 (-4+x) \log (4)+\log ^2(4)}} (-4+\log (4))}{\left (6 x^2-2 x (4-\log (4))+(3-\log (4)) (5-\log (4))\right )^2}+\frac {2 x^{1+\frac {5 x^2}{15-8 x+6 x^2+2 (-4+x) \log (4)+\log ^2(4)}} (-x (4-\log (4))+(3-\log (4)) (5-\log (4))) \log (x)}{\left (6 x^2-2 x (4-\log (4))+(3-\log (4)) (5-\log (4))\right )^2}\right ) \, dx\\ &=10 \int \frac {x^{1+\frac {5 x^2}{15-8 x+6 x^2+2 (-4+x) \log (4)+\log ^2(4)}} (-x (4-\log (4))+(3-\log (4)) (5-\log (4))) \log (x)}{\left (6 x^2-2 x (4-\log (4))+(3-\log (4)) (5-\log (4))\right )^2} \, dx+30 \int \frac {x^{3+\frac {5 x^2}{15-8 x+6 x^2+2 (-4+x) \log (4)+\log ^2(4)}}}{\left (6 x^2-2 x (4-\log (4))+(3-\log (4)) (5-\log (4))\right )^2} \, dx-(10 (4-\log (4))) \int \frac {x^{2+\frac {5 x^2}{15-8 x+6 x^2+2 (-4+x) \log (4)+\log ^2(4)}}}{\left (6 x^2-2 x (4-\log (4))+(3-\log (4)) (5-\log (4))\right )^2} \, dx+(5 (3-\log (4)) (5-\log (4))) \int \frac {x^{1+\frac {5 x^2}{15-8 x+6 x^2+2 (-4+x) \log (4)+\log ^2(4)}}}{\left (6 x^2-2 x (4-\log (4))+(3-\log (4)) (5-\log (4))\right )^2} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.10, size = 32, normalized size = 1.33 \begin {gather*} x^{\frac {5 x^2}{15-8 x+6 x^2-8 \log (4)+2 x \log (4)+\log ^2(4)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^((5*x^2)/(15 - 8*x + 6*x^2 + (-8 + 2*x)*Log[4] + Log[4]^2))*(75*x - 40*x^2 + 30*x^3 + (-40*x + 10
*x^2)*Log[4] + 5*x*Log[4]^2 + (150*x - 40*x^2 + (-80*x + 10*x^2)*Log[4] + 10*x*Log[4]^2)*Log[x]))/(225 - 240*x
 + 244*x^2 - 96*x^3 + 36*x^4 + (-240 + 188*x - 128*x^2 + 24*x^3)*Log[4] + (94 - 48*x + 16*x^2)*Log[4]^2 + (-16
 + 4*x)*Log[4]^3 + Log[4]^4),x]

[Out]

x^((5*x^2)/(15 - 8*x + 6*x^2 - 8*Log[4] + 2*x*Log[4] + Log[4]^2))

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fricas [A]  time = 0.62, size = 32, normalized size = 1.33 \begin {gather*} x^{\frac {5 \, x^{2}}{6 \, x^{2} + 4 \, {\left (x - 4\right )} \log \relax (2) + 4 \, \log \relax (2)^{2} - 8 \, x + 15}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((40*x*log(2)^2+2*(10*x^2-80*x)*log(2)-40*x^2+150*x)*log(x)+20*x*log(2)^2+2*(10*x^2-40*x)*log(2)+30*
x^3-40*x^2+75*x)*exp(5*x^2*log(x)/(4*log(2)^2+2*(2*x-8)*log(2)+6*x^2-8*x+15))/(16*log(2)^4+8*(4*x-16)*log(2)^3
+4*(16*x^2-48*x+94)*log(2)^2+2*(24*x^3-128*x^2+188*x-240)*log(2)+36*x^4-96*x^3+244*x^2-240*x+225),x, algorithm
="fricas")

[Out]

x^(5*x^2/(6*x^2 + 4*(x - 4)*log(2) + 4*log(2)^2 - 8*x + 15))

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((40*x*log(2)^2+2*(10*x^2-80*x)*log(2)-40*x^2+150*x)*log(x)+20*x*log(2)^2+2*(10*x^2-40*x)*log(2)+30*
x^3-40*x^2+75*x)*exp(5*x^2*log(x)/(4*log(2)^2+2*(2*x-8)*log(2)+6*x^2-8*x+15))/(16*log(2)^4+8*(4*x-16)*log(2)^3
+4*(16*x^2-48*x+94)*log(2)^2+2*(24*x^3-128*x^2+188*x-240)*log(2)+36*x^4-96*x^3+244*x^2-240*x+225),x, algorithm
="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Evaluation time: 4.27Unable to divide, perhaps due to rounding error%%%{20565489540003765682176000000000000
000000000,[

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maple [A]  time = 0.54, size = 35, normalized size = 1.46

method result size
risch \(x^{\frac {5 x^{2}}{4 \ln \relax (2)^{2}+4 x \ln \relax (2)+6 x^{2}-16 \ln \relax (2)-8 x +15}}\) \(35\)
norman \(\frac {\left (4 \ln \relax (2)^{2}-16 \ln \relax (2)+15\right ) {\mathrm e}^{\frac {5 x^{2} \ln \relax (x )}{4 \ln \relax (2)^{2}+2 \left (2 x -8\right ) \ln \relax (2)+6 x^{2}-8 x +15}}+\left (4 \ln \relax (2)-8\right ) x \,{\mathrm e}^{\frac {5 x^{2} \ln \relax (x )}{4 \ln \relax (2)^{2}+2 \left (2 x -8\right ) \ln \relax (2)+6 x^{2}-8 x +15}}+6 x^{2} {\mathrm e}^{\frac {5 x^{2} \ln \relax (x )}{4 \ln \relax (2)^{2}+2 \left (2 x -8\right ) \ln \relax (2)+6 x^{2}-8 x +15}}}{4 \ln \relax (2)^{2}+4 x \ln \relax (2)+6 x^{2}-16 \ln \relax (2)-8 x +15}\) \(161\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((40*x*ln(2)^2+2*(10*x^2-80*x)*ln(2)-40*x^2+150*x)*ln(x)+20*x*ln(2)^2+2*(10*x^2-40*x)*ln(2)+30*x^3-40*x^2+
75*x)*exp(5*x^2*ln(x)/(4*ln(2)^2+2*(2*x-8)*ln(2)+6*x^2-8*x+15))/(16*ln(2)^4+8*(4*x-16)*ln(2)^3+4*(16*x^2-48*x+
94)*ln(2)^2+2*(24*x^3-128*x^2+188*x-240)*ln(2)+36*x^4-96*x^3+244*x^2-240*x+225),x,method=_RETURNVERBOSE)

[Out]

x^(5*x^2/(4*ln(2)^2+4*x*ln(2)+6*x^2-16*ln(2)-8*x+15))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} 5 \, \int \frac {{\left (6 \, x^{3} + 4 \, x \log \relax (2)^{2} - 8 \, x^{2} + 4 \, {\left (x^{2} - 4 \, x\right )} \log \relax (2) + 2 \, {\left (4 \, x \log \relax (2)^{2} - 4 \, x^{2} + 2 \, {\left (x^{2} - 8 \, x\right )} \log \relax (2) + 15 \, x\right )} \log \relax (x) + 15 \, x\right )} x^{\frac {5 \, x^{2}}{6 \, x^{2} + 4 \, {\left (x - 4\right )} \log \relax (2) + 4 \, \log \relax (2)^{2} - 8 \, x + 15}}}{36 \, x^{4} + 32 \, {\left (x - 4\right )} \log \relax (2)^{3} + 16 \, \log \relax (2)^{4} - 96 \, x^{3} + 8 \, {\left (8 \, x^{2} - 24 \, x + 47\right )} \log \relax (2)^{2} + 244 \, x^{2} + 8 \, {\left (6 \, x^{3} - 32 \, x^{2} + 47 \, x - 60\right )} \log \relax (2) - 240 \, x + 225}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((40*x*log(2)^2+2*(10*x^2-80*x)*log(2)-40*x^2+150*x)*log(x)+20*x*log(2)^2+2*(10*x^2-40*x)*log(2)+30*
x^3-40*x^2+75*x)*exp(5*x^2*log(x)/(4*log(2)^2+2*(2*x-8)*log(2)+6*x^2-8*x+15))/(16*log(2)^4+8*(4*x-16)*log(2)^3
+4*(16*x^2-48*x+94)*log(2)^2+2*(24*x^3-128*x^2+188*x-240)*log(2)+36*x^4-96*x^3+244*x^2-240*x+225),x, algorithm
="maxima")

[Out]

5*integrate((6*x^3 + 4*x*log(2)^2 - 8*x^2 + 4*(x^2 - 4*x)*log(2) + 2*(4*x*log(2)^2 - 4*x^2 + 2*(x^2 - 8*x)*log
(2) + 15*x)*log(x) + 15*x)*x^(5*x^2/(6*x^2 + 4*(x - 4)*log(2) + 4*log(2)^2 - 8*x + 15))/(36*x^4 + 32*(x - 4)*l
og(2)^3 + 16*log(2)^4 - 96*x^3 + 8*(8*x^2 - 24*x + 47)*log(2)^2 + 244*x^2 + 8*(6*x^3 - 32*x^2 + 47*x - 60)*log
(2) - 240*x + 225), x)

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mupad [B]  time = 6.92, size = 35, normalized size = 1.46 \begin {gather*} {\mathrm {e}}^{\frac {5\,x^2\,\ln \relax (x)}{4\,x\,\ln \relax (2)-16\,\ln \relax (2)-8\,x+4\,{\ln \relax (2)}^2+6\,x^2+15}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp((5*x^2*log(x))/(2*log(2)*(2*x - 8) - 8*x + 4*log(2)^2 + 6*x^2 + 15))*(75*x - 2*log(2)*(40*x - 10*x^2)
 + log(x)*(150*x - 2*log(2)*(80*x - 10*x^2) + 40*x*log(2)^2 - 40*x^2) + 20*x*log(2)^2 - 40*x^2 + 30*x^3))/(8*l
og(2)^3*(4*x - 16) - 240*x + 2*log(2)*(188*x - 128*x^2 + 24*x^3 - 240) + 4*log(2)^2*(16*x^2 - 48*x + 94) + 16*
log(2)^4 + 244*x^2 - 96*x^3 + 36*x^4 + 225),x)

[Out]

exp((5*x^2*log(x))/(4*x*log(2) - 16*log(2) - 8*x + 4*log(2)^2 + 6*x^2 + 15))

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sympy [A]  time = 0.99, size = 34, normalized size = 1.42 \begin {gather*} e^{\frac {5 x^{2} \log {\relax (x )}}{6 x^{2} - 8 x + \left (4 x - 16\right ) \log {\relax (2 )} + 4 \log {\relax (2 )}^{2} + 15}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((40*x*ln(2)**2+2*(10*x**2-80*x)*ln(2)-40*x**2+150*x)*ln(x)+20*x*ln(2)**2+2*(10*x**2-40*x)*ln(2)+30*
x**3-40*x**2+75*x)*exp(5*x**2*ln(x)/(4*ln(2)**2+2*(2*x-8)*ln(2)+6*x**2-8*x+15))/(16*ln(2)**4+8*(4*x-16)*ln(2)*
*3+4*(16*x**2-48*x+94)*ln(2)**2+2*(24*x**3-128*x**2+188*x-240)*ln(2)+36*x**4-96*x**3+244*x**2-240*x+225),x)

[Out]

exp(5*x**2*log(x)/(6*x**2 - 8*x + (4*x - 16)*log(2) + 4*log(2)**2 + 15))

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