3.81.57 \(\int \frac {e^{1+(4+e^x (-10+2 x)) \log (\frac {3 x}{2})+(4+e^x (-20+4 x)+e^{2 x} (25-10 x+x^2)) \log ^2(\frac {3 x}{2})} (4+e^x (-10+2 x)+(8+e^x (-40+2 x^2)+e^{2 x} (50-20 x+2 x^2)) \log (\frac {3 x}{2})+(e^x (-16 x+4 x^2)+e^{2 x} (40 x-18 x^2+2 x^3)) \log ^2(\frac {3 x}{2}))}{x} \, dx\)

Optimal. Leaf size=28 \[ e^5+e^{\left (-1+\left (-2+e^x (5-x)\right ) \log \left (\frac {3 x}{2}\right )\right )^2} \]

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Rubi [F]  time = 134.33, 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 (1+\left (4+e^x (-10+2 x)\right ) \log \left (\frac {3 x}{2}\right )+\left (4+e^x (-20+4 x)+e^{2 x} \left (25-10 x+x^2\right )\right ) \log ^2\left (\frac {3 x}{2}\right )\right ) \left (4+e^x (-10+2 x)+\left (8+e^x \left (-40+2 x^2\right )+e^{2 x} \left (50-20 x+2 x^2\right )\right ) \log \left (\frac {3 x}{2}\right )+\left (e^x \left (-16 x+4 x^2\right )+e^{2 x} \left (40 x-18 x^2+2 x^3\right )\right ) \log ^2\left (\frac {3 x}{2}\right )\right )}{x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(1 + (4 + E^x*(-10 + 2*x))*Log[(3*x)/2] + (4 + E^x*(-20 + 4*x) + E^(2*x)*(25 - 10*x + x^2))*Log[(3*x)/2
]^2)*(4 + E^x*(-10 + 2*x) + (8 + E^x*(-40 + 2*x^2) + E^(2*x)*(50 - 20*x + 2*x^2))*Log[(3*x)/2] + (E^x*(-16*x +
4*x^2) + E^(2*x)*(40*x - 18*x^2 + 2*x^3))*Log[(3*x)/2]^2))/x,x]

[Out]

4*Defer[Int][E^(1 + (2 + E^x*(-5 + x))*Log[(3*x)/2])^2/x, x] - 10*Defer[Int][E^(x + (1 + (2 + E^x*(-5 + x))*Lo
g[(3*x)/2])^2)/x, x] + 8*Defer[Int][(E^(1 + (2 + E^x*(-5 + x))*Log[(3*x)/2])^2*Log[(3*x)/2])/x, x] - 40*Defer[
Int][(E^(x + (1 + (2 + E^x*(-5 + x))*Log[(3*x)/2])^2)*Log[(3*x)/2])/x, x] + 50*Defer[Int][(E^(2*x + (1 + (2 +
E^x*(-5 + x))*Log[(3*x)/2])^2)*Log[(3*x)/2])/x, x] + 2*Defer[Int][E^(x + (1 + (2 + E^x*(-5 + x))*Log[(3*x)/2])
^2)*x*Log[(3*x)/2], x] + 2*Defer[Int][E^(2*x + (1 + (2 + E^x*(-5 + x))*Log[(3*x)/2])^2)*x*Log[(3*x)/2], x] + 4
*Defer[Subst][Defer[Int][E^(2*x + (1 + (2 + E^(2*x)*(-5 + 2*x))*Log[3*x])^2), x], x, x/2] - 40*Defer[Subst][De
fer[Int][E^(4*x + (1 + (2 + E^(2*x)*(-5 + 2*x))*Log[3*x])^2)*Log[3*x], x], x, x/2] - 32*Defer[Subst][Defer[Int
][E^(2*x + (1 + (2 + E^(2*x)*(-5 + 2*x))*Log[3*x])^2)*Log[3*x]^2, x], x, x/2] + 80*Defer[Subst][Defer[Int][E^(
4*x + (1 + (2 + E^(2*x)*(-5 + 2*x))*Log[3*x])^2)*Log[3*x]^2, x], x, x/2] + 16*Defer[Subst][Defer[Int][E^(2*x +
(1 + (2 + E^(2*x)*(-5 + 2*x))*Log[3*x])^2)*x*Log[3*x]^2, x], x, x/2] - 72*Defer[Subst][Defer[Int][E^(4*x + (1
+ (2 + E^(2*x)*(-5 + 2*x))*Log[3*x])^2)*x*Log[3*x]^2, x], x, x/2] + 16*Defer[Subst][Defer[Int][E^(4*x + (1 +
(2 + E^(2*x)*(-5 + 2*x))*Log[3*x])^2)*x^2*Log[3*x]^2, x], x, x/2]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\exp \left (\left (1+2 \log \left (\frac {3 x}{2}\right )-5 e^x \log \left (\frac {3 x}{2}\right )+e^x x \log \left (\frac {3 x}{2}\right )\right )^2\right ) \left (4+e^x (-10+2 x)+\left (8+e^x \left (-40+2 x^2\right )+e^{2 x} \left (50-20 x+2 x^2\right )\right ) \log \left (\frac {3 x}{2}\right )+\left (e^x \left (-16 x+4 x^2\right )+e^{2 x} \left (40 x-18 x^2+2 x^3\right )\right ) \log ^2\left (\frac {3 x}{2}\right )\right )}{x} \, dx\\ &=\int \frac {2 \exp \left (\left (1+\left (2+e^x (-5+x)\right ) \log \left (\frac {3 x}{2}\right )\right )^2\right ) \left (2+e^x (-5+x)+\left (4+e^{2 x} (-5+x)^2+e^x \left (-20+x^2\right )\right ) \log \left (\frac {3 x}{2}\right )+e^x \left (2+e^x (-5+x)\right ) (-4+x) x \log ^2\left (\frac {3 x}{2}\right )\right )}{x} \, dx\\ &=2 \int \frac {\exp \left (\left (1+\left (2+e^x (-5+x)\right ) \log \left (\frac {3 x}{2}\right )\right )^2\right ) \left (2+e^x (-5+x)+\left (4+e^{2 x} (-5+x)^2+e^x \left (-20+x^2\right )\right ) \log \left (\frac {3 x}{2}\right )+e^x \left (2+e^x (-5+x)\right ) (-4+x) x \log ^2\left (\frac {3 x}{2}\right )\right )}{x} \, dx\\ &=2 \int \left (\frac {2 \exp \left (\left (1+\left (2+e^x (-5+x)\right ) \log \left (\frac {3 x}{2}\right )\right )^2\right ) \left (1+2 \log \left (\frac {3 x}{2}\right )\right )}{x}+\frac {\exp \left (2 x+\left (1+\left (2+e^x (-5+x)\right ) \log \left (\frac {3 x}{2}\right )\right )^2\right ) (-5+x) \log \left (\frac {3 x}{2}\right ) \left (-5+x-4 x \log \left (\frac {3 x}{2}\right )+x^2 \log \left (\frac {3 x}{2}\right )\right )}{x}+\frac {\exp \left (x+\left (1+\left (2+e^x (-5+x)\right ) \log \left (\frac {3 x}{2}\right )\right )^2\right ) \left (-5+x-20 \log \left (\frac {3 x}{2}\right )+x^2 \log \left (\frac {3 x}{2}\right )-8 x \log ^2\left (\frac {3 x}{2}\right )+2 x^2 \log ^2\left (\frac {3 x}{2}\right )\right )}{x}\right ) \, dx\\ &=2 \int \frac {\exp \left (2 x+\left (1+\left (2+e^x (-5+x)\right ) \log \left (\frac {3 x}{2}\right )\right )^2\right ) (-5+x) \log \left (\frac {3 x}{2}\right ) \left (-5+x-4 x \log \left (\frac {3 x}{2}\right )+x^2 \log \left (\frac {3 x}{2}\right )\right )}{x} \, dx+2 \int \frac {\exp \left (x+\left (1+\left (2+e^x (-5+x)\right ) \log \left (\frac {3 x}{2}\right )\right )^2\right ) \left (-5+x-20 \log \left (\frac {3 x}{2}\right )+x^2 \log \left (\frac {3 x}{2}\right )-8 x \log ^2\left (\frac {3 x}{2}\right )+2 x^2 \log ^2\left (\frac {3 x}{2}\right )\right )}{x} \, dx+4 \int \frac {\exp \left (\left (1+\left (2+e^x (-5+x)\right ) \log \left (\frac {3 x}{2}\right )\right )^2\right ) \left (1+2 \log \left (\frac {3 x}{2}\right )\right )}{x} \, dx\\ &=2 \int \frac {\exp \left (x+\left (1+\left (2+e^x (-5+x)\right ) \log \left (\frac {3 x}{2}\right )\right )^2\right ) \left (-5+x+\left (-20+x^2\right ) \log \left (\frac {3 x}{2}\right )+2 (-4+x) x \log ^2\left (\frac {3 x}{2}\right )\right )}{x} \, dx+2 \int \left (\frac {\exp \left (2 x+\left (1+\left (2+e^x (-5+x)\right ) \log \left (\frac {3 x}{2}\right )\right )^2\right ) (-5+x)^2 \log \left (\frac {3 x}{2}\right )}{x}+\exp \left (2 x+\left (1+\left (2+e^x (-5+x)\right ) \log \left (\frac {3 x}{2}\right )\right )^2\right ) \left (20-9 x+x^2\right ) \log ^2\left (\frac {3 x}{2}\right )\right ) \, dx+4 \int \left (\frac {\exp \left (\left (1+\left (2+e^x (-5+x)\right ) \log \left (\frac {3 x}{2}\right )\right )^2\right )}{x}+\frac {2 \exp \left (\left (1+\left (2+e^x (-5+x)\right ) \log \left (\frac {3 x}{2}\right )\right )^2\right ) \log \left (\frac {3 x}{2}\right )}{x}\right ) \, dx\\ &=2 \int \frac {\exp \left (2 x+\left (1+\left (2+e^x (-5+x)\right ) \log \left (\frac {3 x}{2}\right )\right )^2\right ) (-5+x)^2 \log \left (\frac {3 x}{2}\right )}{x} \, dx+2 \int \exp \left (2 x+\left (1+\left (2+e^x (-5+x)\right ) \log \left (\frac {3 x}{2}\right )\right )^2\right ) \left (20-9 x+x^2\right ) \log ^2\left (\frac {3 x}{2}\right ) \, dx+2 \int \left (\frac {\exp \left (x+\left (1+\left (2+e^x (-5+x)\right ) \log \left (\frac {3 x}{2}\right )\right )^2\right ) (-5+x)}{x}+\frac {\exp \left (x+\left (1+\left (2+e^x (-5+x)\right ) \log \left (\frac {3 x}{2}\right )\right )^2\right ) \left (-20+x^2\right ) \log \left (\frac {3 x}{2}\right )}{x}+2 \exp \left (x+\left (1+\left (2+e^x (-5+x)\right ) \log \left (\frac {3 x}{2}\right )\right )^2\right ) (-4+x) \log ^2\left (\frac {3 x}{2}\right )\right ) \, dx+4 \int \frac {\exp \left (\left (1+\left (2+e^x (-5+x)\right ) \log \left (\frac {3 x}{2}\right )\right )^2\right )}{x} \, dx+8 \int \frac {\exp \left (\left (1+\left (2+e^x (-5+x)\right ) \log \left (\frac {3 x}{2}\right )\right )^2\right ) \log \left (\frac {3 x}{2}\right )}{x} \, dx\\ &=2 \int \frac {\exp \left (x+\left (1+\left (2+e^x (-5+x)\right ) \log \left (\frac {3 x}{2}\right )\right )^2\right ) (-5+x)}{x} \, dx+2 \int \frac {\exp \left (x+\left (1+\left (2+e^x (-5+x)\right ) \log \left (\frac {3 x}{2}\right )\right )^2\right ) \left (-20+x^2\right ) \log \left (\frac {3 x}{2}\right )}{x} \, dx+2 \int \left (-10 \exp \left (2 x+\left (1+\left (2+e^x (-5+x)\right ) \log \left (\frac {3 x}{2}\right )\right )^2\right ) \log \left (\frac {3 x}{2}\right )+\frac {25 \exp \left (2 x+\left (1+\left (2+e^x (-5+x)\right ) \log \left (\frac {3 x}{2}\right )\right )^2\right ) \log \left (\frac {3 x}{2}\right )}{x}+\exp \left (2 x+\left (1+\left (2+e^x (-5+x)\right ) \log \left (\frac {3 x}{2}\right )\right )^2\right ) x \log \left (\frac {3 x}{2}\right )\right ) \, dx+4 \int \frac {\exp \left (\left (1+\left (2+e^x (-5+x)\right ) \log \left (\frac {3 x}{2}\right )\right )^2\right )}{x} \, dx+4 \int \exp \left (x+\left (1+\left (2+e^x (-5+x)\right ) \log \left (\frac {3 x}{2}\right )\right )^2\right ) (-4+x) \log ^2\left (\frac {3 x}{2}\right ) \, dx+4 \operatorname {Subst}\left (\int \exp \left (4 x+\left (1+\left (2+e^{2 x} (-5+2 x)\right ) \log (3 x)\right )^2\right ) \left (20-18 x+4 x^2\right ) \log ^2(3 x) \, dx,x,\frac {x}{2}\right )+8 \int \frac {\exp \left (\left (1+\left (2+e^x (-5+x)\right ) \log \left (\frac {3 x}{2}\right )\right )^2\right ) \log \left (\frac {3 x}{2}\right )}{x} \, dx\\ &=2 \int \left (\exp \left (x+\left (1+\left (2+e^x (-5+x)\right ) \log \left (\frac {3 x}{2}\right )\right )^2\right )-\frac {5 \exp \left (x+\left (1+\left (2+e^x (-5+x)\right ) \log \left (\frac {3 x}{2}\right )\right )^2\right )}{x}\right ) \, dx+2 \int \exp \left (2 x+\left (1+\left (2+e^x (-5+x)\right ) \log \left (\frac {3 x}{2}\right )\right )^2\right ) x \log \left (\frac {3 x}{2}\right ) \, dx+2 \int \left (-\frac {20 \exp \left (x+\left (1+\left (2+e^x (-5+x)\right ) \log \left (\frac {3 x}{2}\right )\right )^2\right ) \log \left (\frac {3 x}{2}\right )}{x}+\exp \left (x+\left (1+\left (2+e^x (-5+x)\right ) \log \left (\frac {3 x}{2}\right )\right )^2\right ) x \log \left (\frac {3 x}{2}\right )\right ) \, dx+4 \int \frac {\exp \left (\left (1+\left (2+e^x (-5+x)\right ) \log \left (\frac {3 x}{2}\right )\right )^2\right )}{x} \, dx+4 \operatorname {Subst}\left (\int \left (20 \exp \left (4 x+\left (1+\left (2+e^{2 x} (-5+2 x)\right ) \log (3 x)\right )^2\right ) \log ^2(3 x)-18 \exp \left (4 x+\left (1+\left (2+e^{2 x} (-5+2 x)\right ) \log (3 x)\right )^2\right ) x \log ^2(3 x)+4 \exp \left (4 x+\left (1+\left (2+e^{2 x} (-5+2 x)\right ) \log (3 x)\right )^2\right ) x^2 \log ^2(3 x)\right ) \, dx,x,\frac {x}{2}\right )+8 \int \frac {\exp \left (\left (1+\left (2+e^x (-5+x)\right ) \log \left (\frac {3 x}{2}\right )\right )^2\right ) \log \left (\frac {3 x}{2}\right )}{x} \, dx+8 \operatorname {Subst}\left (\int \exp \left (2 x+\left (1+\left (2+e^{2 x} (-5+2 x)\right ) \log (3 x)\right )^2\right ) (-4+2 x) \log ^2(3 x) \, dx,x,\frac {x}{2}\right )-20 \int \exp \left (2 x+\left (1+\left (2+e^x (-5+x)\right ) \log \left (\frac {3 x}{2}\right )\right )^2\right ) \log \left (\frac {3 x}{2}\right ) \, dx+50 \int \frac {\exp \left (2 x+\left (1+\left (2+e^x (-5+x)\right ) \log \left (\frac {3 x}{2}\right )\right )^2\right ) \log \left (\frac {3 x}{2}\right )}{x} \, dx\\ &=2 \int \exp \left (x+\left (1+\left (2+e^x (-5+x)\right ) \log \left (\frac {3 x}{2}\right )\right )^2\right ) \, dx+2 \int \exp \left (x+\left (1+\left (2+e^x (-5+x)\right ) \log \left (\frac {3 x}{2}\right )\right )^2\right ) x \log \left (\frac {3 x}{2}\right ) \, dx+2 \int \exp \left (2 x+\left (1+\left (2+e^x (-5+x)\right ) \log \left (\frac {3 x}{2}\right )\right )^2\right ) x \log \left (\frac {3 x}{2}\right ) \, dx+4 \int \frac {\exp \left (\left (1+\left (2+e^x (-5+x)\right ) \log \left (\frac {3 x}{2}\right )\right )^2\right )}{x} \, dx+8 \int \frac {\exp \left (\left (1+\left (2+e^x (-5+x)\right ) \log \left (\frac {3 x}{2}\right )\right )^2\right ) \log \left (\frac {3 x}{2}\right )}{x} \, dx+8 \operatorname {Subst}\left (\int \left (-4 \exp \left (2 x+\left (1+\left (2+e^{2 x} (-5+2 x)\right ) \log (3 x)\right )^2\right ) \log ^2(3 x)+2 \exp \left (2 x+\left (1+\left (2+e^{2 x} (-5+2 x)\right ) \log (3 x)\right )^2\right ) x \log ^2(3 x)\right ) \, dx,x,\frac {x}{2}\right )-10 \int \frac {\exp \left (x+\left (1+\left (2+e^x (-5+x)\right ) \log \left (\frac {3 x}{2}\right )\right )^2\right )}{x} \, dx+16 \operatorname {Subst}\left (\int \exp \left (4 x+\left (1+\left (2+e^{2 x} (-5+2 x)\right ) \log (3 x)\right )^2\right ) x^2 \log ^2(3 x) \, dx,x,\frac {x}{2}\right )-40 \int \frac {\exp \left (x+\left (1+\left (2+e^x (-5+x)\right ) \log \left (\frac {3 x}{2}\right )\right )^2\right ) \log \left (\frac {3 x}{2}\right )}{x} \, dx-40 \operatorname {Subst}\left (\int \exp \left (4 x+\left (1+\left (2+e^{2 x} (-5+2 x)\right ) \log (3 x)\right )^2\right ) \log (3 x) \, dx,x,\frac {x}{2}\right )+50 \int \frac {\exp \left (2 x+\left (1+\left (2+e^x (-5+x)\right ) \log \left (\frac {3 x}{2}\right )\right )^2\right ) \log \left (\frac {3 x}{2}\right )}{x} \, dx-72 \operatorname {Subst}\left (\int \exp \left (4 x+\left (1+\left (2+e^{2 x} (-5+2 x)\right ) \log (3 x)\right )^2\right ) x \log ^2(3 x) \, dx,x,\frac {x}{2}\right )+80 \operatorname {Subst}\left (\int \exp \left (4 x+\left (1+\left (2+e^{2 x} (-5+2 x)\right ) \log (3 x)\right )^2\right ) \log ^2(3 x) \, dx,x,\frac {x}{2}\right )\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [F]  time = 99.89, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {e^{1+\left (4+e^x (-10+2 x)\right ) \log \left (\frac {3 x}{2}\right )+\left (4+e^x (-20+4 x)+e^{2 x} \left (25-10 x+x^2\right )\right ) \log ^2\left (\frac {3 x}{2}\right )} \left (4+e^x (-10+2 x)+\left (8+e^x \left (-40+2 x^2\right )+e^{2 x} \left (50-20 x+2 x^2\right )\right ) \log \left (\frac {3 x}{2}\right )+\left (e^x \left (-16 x+4 x^2\right )+e^{2 x} \left (40 x-18 x^2+2 x^3\right )\right ) \log ^2\left (\frac {3 x}{2}\right )\right )}{x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(E^(1 + (4 + E^x*(-10 + 2*x))*Log[(3*x)/2] + (4 + E^x*(-20 + 4*x) + E^(2*x)*(25 - 10*x + x^2))*Log[(
3*x)/2]^2)*(4 + E^x*(-10 + 2*x) + (8 + E^x*(-40 + 2*x^2) + E^(2*x)*(50 - 20*x + 2*x^2))*Log[(3*x)/2] + (E^x*(-
16*x + 4*x^2) + E^(2*x)*(40*x - 18*x^2 + 2*x^3))*Log[(3*x)/2]^2))/x,x]

[Out]

Integrate[(E^(1 + (4 + E^x*(-10 + 2*x))*Log[(3*x)/2] + (4 + E^x*(-20 + 4*x) + E^(2*x)*(25 - 10*x + x^2))*Log[(
3*x)/2]^2)*(4 + E^x*(-10 + 2*x) + (8 + E^x*(-40 + 2*x^2) + E^(2*x)*(50 - 20*x + 2*x^2))*Log[(3*x)/2] + (E^x*(-
16*x + 4*x^2) + E^(2*x)*(40*x - 18*x^2 + 2*x^3))*Log[(3*x)/2]^2))/x, x]

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fricas [B]  time = 0.66, size = 46, normalized size = 1.64 \begin {gather*} e^{\left ({\left ({\left (x^{2} - 10 \, x + 25\right )} e^{\left (2 \, x\right )} + 4 \, {\left (x - 5\right )} e^{x} + 4\right )} \log \left (\frac {3}{2} \, x\right )^{2} + 2 \, {\left ({\left (x - 5\right )} e^{x} + 2\right )} \log \left (\frac {3}{2} \, x\right ) + 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x^3-18*x^2+40*x)*exp(x)^2+(4*x^2-16*x)*exp(x))*log(3/2*x)^2+((2*x^2-20*x+50)*exp(x)^2+(2*x^2-40
)*exp(x)+8)*log(3/2*x)+(2*x-10)*exp(x)+4)*exp(((x^2-10*x+25)*exp(x)^2+(4*x-20)*exp(x)+4)*log(3/2*x)^2+((2*x-10
)*exp(x)+4)*log(3/2*x)+1)/x,x, algorithm="fricas")

[Out]

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

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giac [B]  time = 24.47, size = 94, normalized size = 3.36 \begin {gather*} e^{\left (x^{2} e^{\left (2 \, x\right )} \log \left (\frac {3}{2} \, x\right )^{2} - 10 \, x e^{\left (2 \, x\right )} \log \left (\frac {3}{2} \, x\right )^{2} + 4 \, x e^{x} \log \left (\frac {3}{2} \, x\right )^{2} + 2 \, x e^{x} \log \left (\frac {3}{2} \, x\right ) + 25 \, e^{\left (2 \, x\right )} \log \left (\frac {3}{2} \, x\right )^{2} - 20 \, e^{x} \log \left (\frac {3}{2} \, x\right )^{2} - 10 \, e^{x} \log \left (\frac {3}{2} \, x\right ) + 4 \, \log \left (\frac {3}{2} \, x\right )^{2} + 4 \, \log \left (\frac {3}{2} \, x\right ) + 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x^3-18*x^2+40*x)*exp(x)^2+(4*x^2-16*x)*exp(x))*log(3/2*x)^2+((2*x^2-20*x+50)*exp(x)^2+(2*x^2-40
)*exp(x)+8)*log(3/2*x)+(2*x-10)*exp(x)+4)*exp(((x^2-10*x+25)*exp(x)^2+(4*x-20)*exp(x)+4)*log(3/2*x)^2+((2*x-10
)*exp(x)+4)*log(3/2*x)+1)/x,x, algorithm="giac")

[Out]

e^(x^2*e^(2*x)*log(3/2*x)^2 - 10*x*e^(2*x)*log(3/2*x)^2 + 4*x*e^x*log(3/2*x)^2 + 2*x*e^x*log(3/2*x) + 25*e^(2*
x)*log(3/2*x)^2 - 20*e^x*log(3/2*x)^2 - 10*e^x*log(3/2*x) + 4*log(3/2*x)^2 + 4*log(3/2*x) + 1)

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maple [F]  time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\left (\left (\left (2 x^{3}-18 x^{2}+40 x \right ) {\mathrm e}^{2 x}+\left (4 x^{2}-16 x \right ) {\mathrm e}^{x}\right ) \ln \left (\frac {3 x}{2}\right )^{2}+\left (\left (2 x^{2}-20 x +50\right ) {\mathrm e}^{2 x}+\left (2 x^{2}-40\right ) {\mathrm e}^{x}+8\right ) \ln \left (\frac {3 x}{2}\right )+\left (2 x -10\right ) {\mathrm e}^{x}+4\right ) {\mathrm e}^{\left (\left (x^{2}-10 x +25\right ) {\mathrm e}^{2 x}+\left (4 x -20\right ) {\mathrm e}^{x}+4\right ) \ln \left (\frac {3 x}{2}\right )^{2}+\left (\left (2 x -10\right ) {\mathrm e}^{x}+4\right ) \ln \left (\frac {3 x}{2}\right )+1}}{x}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((2*x^3-18*x^2+40*x)*exp(x)^2+(4*x^2-16*x)*exp(x))*ln(3/2*x)^2+((2*x^2-20*x+50)*exp(x)^2+(2*x^2-40)*exp(x
)+8)*ln(3/2*x)+(2*x-10)*exp(x)+4)*exp(((x^2-10*x+25)*exp(x)^2+(4*x-20)*exp(x)+4)*ln(3/2*x)^2+((2*x-10)*exp(x)+
4)*ln(3/2*x)+1)/x,x)

[Out]

int((((2*x^3-18*x^2+40*x)*exp(x)^2+(4*x^2-16*x)*exp(x))*ln(3/2*x)^2+((2*x^2-20*x+50)*exp(x)^2+(2*x^2-40)*exp(x
)+8)*ln(3/2*x)+(2*x-10)*exp(x)+4)*exp(((x^2-10*x+25)*exp(x)^2+(4*x-20)*exp(x)+4)*ln(3/2*x)^2+((2*x-10)*exp(x)+
4)*ln(3/2*x)+1)/x,x)

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maxima [B]  time = 2.72, size = 388, normalized size = 13.86 \begin {gather*} 81 \cdot 2^{-8 \, \log \relax (3) - 4} x^{4} e^{\left (x^{2} e^{\left (2 \, x\right )} \log \relax (3)^{2} - 2 \, x^{2} e^{\left (2 \, x\right )} \log \relax (3) \log \relax (2) + x^{2} e^{\left (2 \, x\right )} \log \relax (2)^{2} + 2 \, x^{2} e^{\left (2 \, x\right )} \log \relax (3) \log \relax (x) - 2 \, x^{2} e^{\left (2 \, x\right )} \log \relax (2) \log \relax (x) + x^{2} e^{\left (2 \, x\right )} \log \relax (x)^{2} - 10 \, x e^{\left (2 \, x\right )} \log \relax (3)^{2} + 4 \, x e^{x} \log \relax (3)^{2} + 20 \, x e^{\left (2 \, x\right )} \log \relax (3) \log \relax (2) - 8 \, x e^{x} \log \relax (3) \log \relax (2) - 10 \, x e^{\left (2 \, x\right )} \log \relax (2)^{2} + 4 \, x e^{x} \log \relax (2)^{2} - 20 \, x e^{\left (2 \, x\right )} \log \relax (3) \log \relax (x) + 8 \, x e^{x} \log \relax (3) \log \relax (x) + 20 \, x e^{\left (2 \, x\right )} \log \relax (2) \log \relax (x) - 8 \, x e^{x} \log \relax (2) \log \relax (x) - 10 \, x e^{\left (2 \, x\right )} \log \relax (x)^{2} + 4 \, x e^{x} \log \relax (x)^{2} + 2 \, x e^{x} \log \relax (3) + 25 \, e^{\left (2 \, x\right )} \log \relax (3)^{2} - 20 \, e^{x} \log \relax (3)^{2} - 2 \, x e^{x} \log \relax (2) - 50 \, e^{\left (2 \, x\right )} \log \relax (3) \log \relax (2) + 40 \, e^{x} \log \relax (3) \log \relax (2) + 25 \, e^{\left (2 \, x\right )} \log \relax (2)^{2} - 20 \, e^{x} \log \relax (2)^{2} + 2 \, x e^{x} \log \relax (x) + 50 \, e^{\left (2 \, x\right )} \log \relax (3) \log \relax (x) - 40 \, e^{x} \log \relax (3) \log \relax (x) - 50 \, e^{\left (2 \, x\right )} \log \relax (2) \log \relax (x) + 40 \, e^{x} \log \relax (2) \log \relax (x) + 25 \, e^{\left (2 \, x\right )} \log \relax (x)^{2} - 20 \, e^{x} \log \relax (x)^{2} - 10 \, e^{x} \log \relax (3) + 4 \, \log \relax (3)^{2} + 10 \, e^{x} \log \relax (2) + 4 \, \log \relax (2)^{2} - 10 \, e^{x} \log \relax (x) + 8 \, \log \relax (3) \log \relax (x) - 8 \, \log \relax (2) \log \relax (x) + 4 \, \log \relax (x)^{2} + 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x^3-18*x^2+40*x)*exp(x)^2+(4*x^2-16*x)*exp(x))*log(3/2*x)^2+((2*x^2-20*x+50)*exp(x)^2+(2*x^2-40
)*exp(x)+8)*log(3/2*x)+(2*x-10)*exp(x)+4)*exp(((x^2-10*x+25)*exp(x)^2+(4*x-20)*exp(x)+4)*log(3/2*x)^2+((2*x-10
)*exp(x)+4)*log(3/2*x)+1)/x,x, algorithm="maxima")

[Out]

81*2^(-8*log(3) - 4)*x^4*e^(x^2*e^(2*x)*log(3)^2 - 2*x^2*e^(2*x)*log(3)*log(2) + x^2*e^(2*x)*log(2)^2 + 2*x^2*
e^(2*x)*log(3)*log(x) - 2*x^2*e^(2*x)*log(2)*log(x) + x^2*e^(2*x)*log(x)^2 - 10*x*e^(2*x)*log(3)^2 + 4*x*e^x*l
og(3)^2 + 20*x*e^(2*x)*log(3)*log(2) - 8*x*e^x*log(3)*log(2) - 10*x*e^(2*x)*log(2)^2 + 4*x*e^x*log(2)^2 - 20*x
*e^(2*x)*log(3)*log(x) + 8*x*e^x*log(3)*log(x) + 20*x*e^(2*x)*log(2)*log(x) - 8*x*e^x*log(2)*log(x) - 10*x*e^(
2*x)*log(x)^2 + 4*x*e^x*log(x)^2 + 2*x*e^x*log(3) + 25*e^(2*x)*log(3)^2 - 20*e^x*log(3)^2 - 2*x*e^x*log(2) - 5
0*e^(2*x)*log(3)*log(2) + 40*e^x*log(3)*log(2) + 25*e^(2*x)*log(2)^2 - 20*e^x*log(2)^2 + 2*x*e^x*log(x) + 50*e
^(2*x)*log(3)*log(x) - 40*e^x*log(3)*log(x) - 50*e^(2*x)*log(2)*log(x) + 40*e^x*log(2)*log(x) + 25*e^(2*x)*log
(x)^2 - 20*e^x*log(x)^2 - 10*e^x*log(3) + 4*log(3)^2 + 10*e^x*log(2) + 4*log(2)^2 - 10*e^x*log(x) + 8*log(3)*l
og(x) - 8*log(2)*log(x) + 4*log(x)^2 + 1)

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mupad [B]  time = 6.40, size = 429, normalized size = 15.32 \begin {gather*} \frac {81\,2^{40\,{\mathrm {e}}^x\,\ln \relax (3)}\,2^{10\,{\mathrm {e}}^x}\,2^{20\,x\,{\mathrm {e}}^{2\,x}\,\ln \relax (3)}\,3^{2\,x\,{\mathrm {e}}^x}\,x^{20\,x\,{\mathrm {e}}^{2\,x}\,\ln \relax (2)}\,x^{40\,{\mathrm {e}}^x\,\ln \relax (2)}\,x^{2\,x\,{\mathrm {e}}^x}\,x^{8\,\ln \relax (3)}\,x^{2\,x^2\,{\mathrm {e}}^{2\,x}\,\ln \relax (3)}\,x^4\,x^{50\,{\mathrm {e}}^{2\,x}\,\ln \relax (3)}\,x^{8\,x\,{\mathrm {e}}^x\,\ln \relax (3)}\,{\mathrm {e}}^{4\,{\ln \relax (x)}^2}\,{\mathrm {e}}^{x^2\,{\mathrm {e}}^{2\,x}\,{\ln \relax (x)}^2}\,{\mathrm {e}}^{-10\,x\,{\mathrm {e}}^{2\,x}\,{\ln \relax (2)}^2}\,{\mathrm {e}}^{-10\,x\,{\mathrm {e}}^{2\,x}\,{\ln \relax (3)}^2}\,{\mathrm {e}}^{25\,{\mathrm {e}}^{2\,x}\,{\ln \relax (x)}^2}\,{\mathrm {e}}^{4\,x\,{\mathrm {e}}^x\,{\ln \relax (x)}^2}\,\mathrm {e}\,{\mathrm {e}}^{-20\,{\mathrm {e}}^x\,{\ln \relax (2)}^2}\,{\mathrm {e}}^{-20\,{\mathrm {e}}^x\,{\ln \relax (3)}^2}\,{\mathrm {e}}^{4\,{\ln \relax (2)}^2}\,{\mathrm {e}}^{4\,{\ln \relax (3)}^2}\,{\mathrm {e}}^{x^2\,{\mathrm {e}}^{2\,x}\,{\ln \relax (2)}^2}\,{\mathrm {e}}^{x^2\,{\mathrm {e}}^{2\,x}\,{\ln \relax (3)}^2}\,{\mathrm {e}}^{-10\,x\,{\mathrm {e}}^{2\,x}\,{\ln \relax (x)}^2}\,{\mathrm {e}}^{25\,{\mathrm {e}}^{2\,x}\,{\ln \relax (2)}^2}\,{\mathrm {e}}^{25\,{\mathrm {e}}^{2\,x}\,{\ln \relax (3)}^2}\,{\mathrm {e}}^{4\,x\,{\mathrm {e}}^x\,{\ln \relax (2)}^2}\,{\mathrm {e}}^{4\,x\,{\mathrm {e}}^x\,{\ln \relax (3)}^2}\,{\mathrm {e}}^{-20\,{\mathrm {e}}^x\,{\ln \relax (x)}^2}}{16\,2^{2\,x\,{\mathrm {e}}^x}\,2^{8\,\ln \relax (3)}\,2^{2\,x^2\,{\mathrm {e}}^{2\,x}\,\ln \relax (3)}\,2^{50\,{\mathrm {e}}^{2\,x}\,\ln \relax (3)}\,2^{8\,x\,{\mathrm {e}}^x\,\ln \relax (3)}\,3^{10\,{\mathrm {e}}^x}\,x^{10\,{\mathrm {e}}^x}\,x^{20\,x\,{\mathrm {e}}^{2\,x}\,\ln \relax (3)}\,x^{40\,{\mathrm {e}}^x\,\ln \relax (3)}\,x^{8\,\ln \relax (2)}\,x^{2\,x^2\,{\mathrm {e}}^{2\,x}\,\ln \relax (2)}\,x^{50\,{\mathrm {e}}^{2\,x}\,\ln \relax (2)}\,x^{8\,x\,{\mathrm {e}}^x\,\ln \relax (2)}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(log((3*x)/2)*(exp(x)*(2*x - 10) + 4) + log((3*x)/2)^2*(exp(x)*(4*x - 20) + exp(2*x)*(x^2 - 10*x + 25)
+ 4) + 1)*(log((3*x)/2)^2*(exp(2*x)*(40*x - 18*x^2 + 2*x^3) - exp(x)*(16*x - 4*x^2)) + exp(x)*(2*x - 10) + lo
g((3*x)/2)*(exp(2*x)*(2*x^2 - 20*x + 50) + exp(x)*(2*x^2 - 40) + 8) + 4))/x,x)

[Out]

(81*2^(40*exp(x)*log(3))*2^(10*exp(x))*2^(20*x*exp(2*x)*log(3))*3^(2*x*exp(x))*x^(20*x*exp(2*x)*log(2))*x^(40*
exp(x)*log(2))*x^(2*x*exp(x))*x^(8*log(3))*x^(2*x^2*exp(2*x)*log(3))*x^4*x^(50*exp(2*x)*log(3))*x^(8*x*exp(x)*
log(3))*exp(4*log(x)^2)*exp(x^2*exp(2*x)*log(x)^2)*exp(-10*x*exp(2*x)*log(2)^2)*exp(-10*x*exp(2*x)*log(3)^2)*e
xp(25*exp(2*x)*log(x)^2)*exp(4*x*exp(x)*log(x)^2)*exp(1)*exp(-20*exp(x)*log(2)^2)*exp(-20*exp(x)*log(3)^2)*exp
(4*log(2)^2)*exp(4*log(3)^2)*exp(x^2*exp(2*x)*log(2)^2)*exp(x^2*exp(2*x)*log(3)^2)*exp(-10*x*exp(2*x)*log(x)^2
)*exp(25*exp(2*x)*log(2)^2)*exp(25*exp(2*x)*log(3)^2)*exp(4*x*exp(x)*log(2)^2)*exp(4*x*exp(x)*log(3)^2)*exp(-2
0*exp(x)*log(x)^2))/(16*2^(2*x*exp(x))*2^(8*log(3))*2^(2*x^2*exp(2*x)*log(3))*2^(50*exp(2*x)*log(3))*2^(8*x*ex
p(x)*log(3))*3^(10*exp(x))*x^(10*exp(x))*x^(20*x*exp(2*x)*log(3))*x^(40*exp(x)*log(3))*x^(8*log(2))*x^(2*x^2*e
xp(2*x)*log(2))*x^(50*exp(2*x)*log(2))*x^(8*x*exp(x)*log(2)))

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sympy [B]  time = 2.59, size = 51, normalized size = 1.82 \begin {gather*} e^{\left (\left (2 x - 10\right ) e^{x} + 4\right ) \log {\left (\frac {3 x}{2} \right )} + \left (\left (4 x - 20\right ) e^{x} + \left (x^{2} - 10 x + 25\right ) e^{2 x} + 4\right ) \log {\left (\frac {3 x}{2} \right )}^{2} + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x**3-18*x**2+40*x)*exp(x)**2+(4*x**2-16*x)*exp(x))*ln(3/2*x)**2+((2*x**2-20*x+50)*exp(x)**2+(2*
x**2-40)*exp(x)+8)*ln(3/2*x)+(2*x-10)*exp(x)+4)*exp(((x**2-10*x+25)*exp(x)**2+(4*x-20)*exp(x)+4)*ln(3/2*x)**2+
((2*x-10)*exp(x)+4)*ln(3/2*x)+1)/x,x)

[Out]

exp(((2*x - 10)*exp(x) + 4)*log(3*x/2) + ((4*x - 20)*exp(x) + (x**2 - 10*x + 25)*exp(2*x) + 4)*log(3*x/2)**2 +
1)

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