3.35.94 \(\int \frac {-18 x^3+12 x^4-47 x^5+30 x^6-5 x^7+e^x (-9 x^5+6 x^6-x^7)+(-72 x^3+48 x^4-98 x^5+60 x^6-10 x^7+e^x (-18 x^5+21 x^6-8 x^7+x^8)) \log (x)+(30 x^5-6 x^6-6 x^7+x^8) \log ^2(x)}{36-24 x+184 x^2-120 x^3+245 x^4-150 x^5+25 x^6+e^{2 x} (9 x^4-6 x^5+x^6)+e^x (36 x^2-24 x^3+94 x^4-60 x^5+10 x^6)+(-60 x^2-16 x^3-126 x^4-44 x^5+60 x^6-10 x^7+e^x (-30 x^4-8 x^5+12 x^6-2 x^7)) \log (x)+(25 x^4+30 x^5-x^6-6 x^7+x^8) \log ^2(x)} \, dx\)

Optimal. Leaf size=33 \[ \frac {x^2}{\frac {5}{3-x}+x-\frac {5+e^x+\frac {2}{x^2}}{\log (x)}} \]

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Rubi [F]  time = 107.76, 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 {-18 x^3+12 x^4-47 x^5+30 x^6-5 x^7+e^x \left (-9 x^5+6 x^6-x^7\right )+\left (-72 x^3+48 x^4-98 x^5+60 x^6-10 x^7+e^x \left (-18 x^5+21 x^6-8 x^7+x^8\right )\right ) \log (x)+\left (30 x^5-6 x^6-6 x^7+x^8\right ) \log ^2(x)}{36-24 x+184 x^2-120 x^3+245 x^4-150 x^5+25 x^6+e^{2 x} \left (9 x^4-6 x^5+x^6\right )+e^x \left (36 x^2-24 x^3+94 x^4-60 x^5+10 x^6\right )+\left (-60 x^2-16 x^3-126 x^4-44 x^5+60 x^6-10 x^7+e^x \left (-30 x^4-8 x^5+12 x^6-2 x^7\right )\right ) \log (x)+\left (25 x^4+30 x^5-x^6-6 x^7+x^8\right ) \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-18*x^3 + 12*x^4 - 47*x^5 + 30*x^6 - 5*x^7 + E^x*(-9*x^5 + 6*x^6 - x^7) + (-72*x^3 + 48*x^4 - 98*x^5 + 60
*x^6 - 10*x^7 + E^x*(-18*x^5 + 21*x^6 - 8*x^7 + x^8))*Log[x] + (30*x^5 - 6*x^6 - 6*x^7 + x^8)*Log[x]^2)/(36 -
24*x + 184*x^2 - 120*x^3 + 245*x^4 - 150*x^5 + 25*x^6 + E^(2*x)*(9*x^4 - 6*x^5 + x^6) + E^x*(36*x^2 - 24*x^3 +
94*x^4 - 60*x^5 + 10*x^6) + (-60*x^2 - 16*x^3 - 126*x^4 - 44*x^5 + 60*x^6 - 10*x^7 + E^x*(-30*x^4 - 8*x^5 + 1
2*x^6 - 2*x^7))*Log[x] + (25*x^4 + 30*x^5 - x^6 - 6*x^7 + x^8)*Log[x]^2),x]

[Out]

-36*Defer[Int][(x^3*Log[x])/((-3 + x)*(2 + (5 + E^x)*x^2) + x^2*(5 + 3*x - x^2)*Log[x])^2, x] + 6*Defer[Int][(
x^4*Log[x])/((-3 + x)*(2 + (5 + E^x)*x^2) + x^2*(5 + 3*x - x^2)*Log[x])^2, x] - 7*Defer[Int][(x^5*Log[x])/((-3
+ x)*(2 + (5 + E^x)*x^2) + x^2*(5 + 3*x - x^2)*Log[x])^2, x] - 51*Defer[Int][(x^6*Log[x])/((-3 + x)*(2 + (5 +
E^x)*x^2) + x^2*(5 + 3*x - x^2)*Log[x])^2, x] + 36*Defer[Int][(x^7*Log[x])/((-3 + x)*(2 + (5 + E^x)*x^2) + x^
2*(5 + 3*x - x^2)*Log[x])^2, x] - 6*Defer[Int][(x^8*Log[x])/((-3 + x)*(2 + (5 + E^x)*x^2) + x^2*(5 + 3*x - x^2
)*Log[x])^2, x] + Defer[Int][(x^6*Log[x]^2)/((-3 + x)*(2 + (5 + E^x)*x^2) + x^2*(5 + 3*x - x^2)*Log[x])^2, x]
+ 10*Defer[Int][(x^7*Log[x]^2)/((-3 + x)*(2 + (5 + E^x)*x^2) + x^2*(5 + 3*x - x^2)*Log[x])^2, x] - 7*Defer[Int
][(x^8*Log[x]^2)/((-3 + x)*(2 + (5 + E^x)*x^2) + x^2*(5 + 3*x - x^2)*Log[x])^2, x] + Defer[Int][(x^9*Log[x]^2)
/((-3 + x)*(2 + (5 + E^x)*x^2) + x^2*(5 + 3*x - x^2)*Log[x])^2, x] - 3*Defer[Int][x^3/(-((-3 + x)*(2 + (5 + E^
x)*x^2)) + x^2*(-5 - 3*x + x^2)*Log[x]), x] + Defer[Int][x^4/(-((-3 + x)*(2 + (5 + E^x)*x^2)) + x^2*(-5 - 3*x
+ x^2)*Log[x]), x] - 6*Defer[Int][(x^3*Log[x])/(-((-3 + x)*(2 + (5 + E^x)*x^2)) + x^2*(-5 - 3*x + x^2)*Log[x])
, x] + 5*Defer[Int][(x^4*Log[x])/(-((-3 + x)*(2 + (5 + E^x)*x^2)) + x^2*(-5 - 3*x + x^2)*Log[x]), x] - Defer[I
nt][(x^5*Log[x])/(-((-3 + x)*(2 + (5 + E^x)*x^2)) + x^2*(-5 - 3*x + x^2)*Log[x]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {x^3 \left (-(-3+x)^2 \left (2+\left (5+e^x\right ) x^2\right )+(-3+x)^2 \left (-8-2 \left (5+e^x\right ) x^2+e^x x^3\right ) \log (x)+x^2 \left (30-6 x-6 x^2+x^3\right ) \log ^2(x)\right )}{\left ((-3+x) \left (2+\left (5+e^x\right ) x^2\right )+x^2 \left (5+3 x-x^2\right ) \log (x)\right )^2} \, dx\\ &=\int \left (-\frac {(-3+x) x^3 (-1-2 \log (x)+x \log (x))}{6-2 x+15 x^2+3 e^x x^2-5 x^3-e^x x^3-5 x^2 \log (x)-3 x^3 \log (x)+x^4 \log (x)}+\frac {x^3 \log (x) \left (-36+6 x-7 x^2-51 x^3+36 x^4-6 x^5+x^3 \log (x)+10 x^4 \log (x)-7 x^5 \log (x)+x^6 \log (x)\right )}{\left (6-2 x+15 x^2+3 e^x x^2-5 x^3-e^x x^3-5 x^2 \log (x)-3 x^3 \log (x)+x^4 \log (x)\right )^2}\right ) \, dx\\ &=-\int \frac {(-3+x) x^3 (-1-2 \log (x)+x \log (x))}{6-2 x+15 x^2+3 e^x x^2-5 x^3-e^x x^3-5 x^2 \log (x)-3 x^3 \log (x)+x^4 \log (x)} \, dx+\int \frac {x^3 \log (x) \left (-36+6 x-7 x^2-51 x^3+36 x^4-6 x^5+x^3 \log (x)+10 x^4 \log (x)-7 x^5 \log (x)+x^6 \log (x)\right )}{\left (6-2 x+15 x^2+3 e^x x^2-5 x^3-e^x x^3-5 x^2 \log (x)-3 x^3 \log (x)+x^4 \log (x)\right )^2} \, dx\\ &=-\int \frac {(3-x) x^3 (-1+(-2+x) \log (x))}{(-3+x) \left (2+\left (5+e^x\right ) x^2\right )-x^2 \left (-5-3 x+x^2\right ) \log (x)} \, dx+\int \left (-\frac {36 x^3 \log (x)}{\left (6-2 x+15 x^2+3 e^x x^2-5 x^3-e^x x^3-5 x^2 \log (x)-3 x^3 \log (x)+x^4 \log (x)\right )^2}+\frac {6 x^4 \log (x)}{\left (6-2 x+15 x^2+3 e^x x^2-5 x^3-e^x x^3-5 x^2 \log (x)-3 x^3 \log (x)+x^4 \log (x)\right )^2}-\frac {7 x^5 \log (x)}{\left (6-2 x+15 x^2+3 e^x x^2-5 x^3-e^x x^3-5 x^2 \log (x)-3 x^3 \log (x)+x^4 \log (x)\right )^2}-\frac {51 x^6 \log (x)}{\left (6-2 x+15 x^2+3 e^x x^2-5 x^3-e^x x^3-5 x^2 \log (x)-3 x^3 \log (x)+x^4 \log (x)\right )^2}+\frac {36 x^7 \log (x)}{\left (6-2 x+15 x^2+3 e^x x^2-5 x^3-e^x x^3-5 x^2 \log (x)-3 x^3 \log (x)+x^4 \log (x)\right )^2}-\frac {6 x^8 \log (x)}{\left (6-2 x+15 x^2+3 e^x x^2-5 x^3-e^x x^3-5 x^2 \log (x)-3 x^3 \log (x)+x^4 \log (x)\right )^2}+\frac {x^6 \log ^2(x)}{\left (6-2 x+15 x^2+3 e^x x^2-5 x^3-e^x x^3-5 x^2 \log (x)-3 x^3 \log (x)+x^4 \log (x)\right )^2}+\frac {10 x^7 \log ^2(x)}{\left (6-2 x+15 x^2+3 e^x x^2-5 x^3-e^x x^3-5 x^2 \log (x)-3 x^3 \log (x)+x^4 \log (x)\right )^2}-\frac {7 x^8 \log ^2(x)}{\left (6-2 x+15 x^2+3 e^x x^2-5 x^3-e^x x^3-5 x^2 \log (x)-3 x^3 \log (x)+x^4 \log (x)\right )^2}+\frac {x^9 \log ^2(x)}{\left (6-2 x+15 x^2+3 e^x x^2-5 x^3-e^x x^3-5 x^2 \log (x)-3 x^3 \log (x)+x^4 \log (x)\right )^2}\right ) \, dx\\ &=6 \int \frac {x^4 \log (x)}{\left (6-2 x+15 x^2+3 e^x x^2-5 x^3-e^x x^3-5 x^2 \log (x)-3 x^3 \log (x)+x^4 \log (x)\right )^2} \, dx-6 \int \frac {x^8 \log (x)}{\left (6-2 x+15 x^2+3 e^x x^2-5 x^3-e^x x^3-5 x^2 \log (x)-3 x^3 \log (x)+x^4 \log (x)\right )^2} \, dx-7 \int \frac {x^5 \log (x)}{\left (6-2 x+15 x^2+3 e^x x^2-5 x^3-e^x x^3-5 x^2 \log (x)-3 x^3 \log (x)+x^4 \log (x)\right )^2} \, dx-7 \int \frac {x^8 \log ^2(x)}{\left (6-2 x+15 x^2+3 e^x x^2-5 x^3-e^x x^3-5 x^2 \log (x)-3 x^3 \log (x)+x^4 \log (x)\right )^2} \, dx+10 \int \frac {x^7 \log ^2(x)}{\left (6-2 x+15 x^2+3 e^x x^2-5 x^3-e^x x^3-5 x^2 \log (x)-3 x^3 \log (x)+x^4 \log (x)\right )^2} \, dx-36 \int \frac {x^3 \log (x)}{\left (6-2 x+15 x^2+3 e^x x^2-5 x^3-e^x x^3-5 x^2 \log (x)-3 x^3 \log (x)+x^4 \log (x)\right )^2} \, dx+36 \int \frac {x^7 \log (x)}{\left (6-2 x+15 x^2+3 e^x x^2-5 x^3-e^x x^3-5 x^2 \log (x)-3 x^3 \log (x)+x^4 \log (x)\right )^2} \, dx-51 \int \frac {x^6 \log (x)}{\left (6-2 x+15 x^2+3 e^x x^2-5 x^3-e^x x^3-5 x^2 \log (x)-3 x^3 \log (x)+x^4 \log (x)\right )^2} \, dx+\int \frac {x^6 \log ^2(x)}{\left (6-2 x+15 x^2+3 e^x x^2-5 x^3-e^x x^3-5 x^2 \log (x)-3 x^3 \log (x)+x^4 \log (x)\right )^2} \, dx+\int \frac {x^9 \log ^2(x)}{\left (6-2 x+15 x^2+3 e^x x^2-5 x^3-e^x x^3-5 x^2 \log (x)-3 x^3 \log (x)+x^4 \log (x)\right )^2} \, dx-\int \left (-\frac {3 x^3 (-1-2 \log (x)+x \log (x))}{6-2 x+15 x^2+3 e^x x^2-5 x^3-e^x x^3-5 x^2 \log (x)-3 x^3 \log (x)+x^4 \log (x)}+\frac {x^4 (-1-2 \log (x)+x \log (x))}{6-2 x+15 x^2+3 e^x x^2-5 x^3-e^x x^3-5 x^2 \log (x)-3 x^3 \log (x)+x^4 \log (x)}\right ) \, dx\\ &=3 \int \frac {x^3 (-1-2 \log (x)+x \log (x))}{6-2 x+15 x^2+3 e^x x^2-5 x^3-e^x x^3-5 x^2 \log (x)-3 x^3 \log (x)+x^4 \log (x)} \, dx+6 \int \frac {x^4 \log (x)}{\left ((-3+x) \left (2+\left (5+e^x\right ) x^2\right )+x^2 \left (5+3 x-x^2\right ) \log (x)\right )^2} \, dx-6 \int \frac {x^8 \log (x)}{\left ((-3+x) \left (2+\left (5+e^x\right ) x^2\right )+x^2 \left (5+3 x-x^2\right ) \log (x)\right )^2} \, dx-7 \int \frac {x^5 \log (x)}{\left ((-3+x) \left (2+\left (5+e^x\right ) x^2\right )+x^2 \left (5+3 x-x^2\right ) \log (x)\right )^2} \, dx-7 \int \frac {x^8 \log ^2(x)}{\left ((-3+x) \left (2+\left (5+e^x\right ) x^2\right )+x^2 \left (5+3 x-x^2\right ) \log (x)\right )^2} \, dx+10 \int \frac {x^7 \log ^2(x)}{\left ((-3+x) \left (2+\left (5+e^x\right ) x^2\right )+x^2 \left (5+3 x-x^2\right ) \log (x)\right )^2} \, dx-36 \int \frac {x^3 \log (x)}{\left ((-3+x) \left (2+\left (5+e^x\right ) x^2\right )+x^2 \left (5+3 x-x^2\right ) \log (x)\right )^2} \, dx+36 \int \frac {x^7 \log (x)}{\left ((-3+x) \left (2+\left (5+e^x\right ) x^2\right )+x^2 \left (5+3 x-x^2\right ) \log (x)\right )^2} \, dx-51 \int \frac {x^6 \log (x)}{\left ((-3+x) \left (2+\left (5+e^x\right ) x^2\right )+x^2 \left (5+3 x-x^2\right ) \log (x)\right )^2} \, dx-\int \frac {x^4 (-1-2 \log (x)+x \log (x))}{6-2 x+15 x^2+3 e^x x^2-5 x^3-e^x x^3-5 x^2 \log (x)-3 x^3 \log (x)+x^4 \log (x)} \, dx+\int \frac {x^6 \log ^2(x)}{\left ((-3+x) \left (2+\left (5+e^x\right ) x^2\right )+x^2 \left (5+3 x-x^2\right ) \log (x)\right )^2} \, dx+\int \frac {x^9 \log ^2(x)}{\left ((-3+x) \left (2+\left (5+e^x\right ) x^2\right )+x^2 \left (5+3 x-x^2\right ) \log (x)\right )^2} \, dx\\ &=3 \int \frac {x^3 (1-(-2+x) \log (x))}{(-3+x) \left (2+\left (5+e^x\right ) x^2\right )-x^2 \left (-5-3 x+x^2\right ) \log (x)} \, dx+6 \int \frac {x^4 \log (x)}{\left ((-3+x) \left (2+\left (5+e^x\right ) x^2\right )+x^2 \left (5+3 x-x^2\right ) \log (x)\right )^2} \, dx-6 \int \frac {x^8 \log (x)}{\left ((-3+x) \left (2+\left (5+e^x\right ) x^2\right )+x^2 \left (5+3 x-x^2\right ) \log (x)\right )^2} \, dx-7 \int \frac {x^5 \log (x)}{\left ((-3+x) \left (2+\left (5+e^x\right ) x^2\right )+x^2 \left (5+3 x-x^2\right ) \log (x)\right )^2} \, dx-7 \int \frac {x^8 \log ^2(x)}{\left ((-3+x) \left (2+\left (5+e^x\right ) x^2\right )+x^2 \left (5+3 x-x^2\right ) \log (x)\right )^2} \, dx+10 \int \frac {x^7 \log ^2(x)}{\left ((-3+x) \left (2+\left (5+e^x\right ) x^2\right )+x^2 \left (5+3 x-x^2\right ) \log (x)\right )^2} \, dx-36 \int \frac {x^3 \log (x)}{\left ((-3+x) \left (2+\left (5+e^x\right ) x^2\right )+x^2 \left (5+3 x-x^2\right ) \log (x)\right )^2} \, dx+36 \int \frac {x^7 \log (x)}{\left ((-3+x) \left (2+\left (5+e^x\right ) x^2\right )+x^2 \left (5+3 x-x^2\right ) \log (x)\right )^2} \, dx-51 \int \frac {x^6 \log (x)}{\left ((-3+x) \left (2+\left (5+e^x\right ) x^2\right )+x^2 \left (5+3 x-x^2\right ) \log (x)\right )^2} \, dx+\int \frac {x^6 \log ^2(x)}{\left ((-3+x) \left (2+\left (5+e^x\right ) x^2\right )+x^2 \left (5+3 x-x^2\right ) \log (x)\right )^2} \, dx+\int \frac {x^9 \log ^2(x)}{\left ((-3+x) \left (2+\left (5+e^x\right ) x^2\right )+x^2 \left (5+3 x-x^2\right ) \log (x)\right )^2} \, dx-\int \frac {x^4 (1-(-2+x) \log (x))}{(-3+x) \left (2+\left (5+e^x\right ) x^2\right )-x^2 \left (-5-3 x+x^2\right ) \log (x)} \, dx\\ &=3 \int \left (-\frac {x^3}{6-2 x+15 x^2+3 e^x x^2-5 x^3-e^x x^3-5 x^2 \log (x)-3 x^3 \log (x)+x^4 \log (x)}-\frac {2 x^3 \log (x)}{6-2 x+15 x^2+3 e^x x^2-5 x^3-e^x x^3-5 x^2 \log (x)-3 x^3 \log (x)+x^4 \log (x)}+\frac {x^4 \log (x)}{6-2 x+15 x^2+3 e^x x^2-5 x^3-e^x x^3-5 x^2 \log (x)-3 x^3 \log (x)+x^4 \log (x)}\right ) \, dx+6 \int \frac {x^4 \log (x)}{\left ((-3+x) \left (2+\left (5+e^x\right ) x^2\right )+x^2 \left (5+3 x-x^2\right ) \log (x)\right )^2} \, dx-6 \int \frac {x^8 \log (x)}{\left ((-3+x) \left (2+\left (5+e^x\right ) x^2\right )+x^2 \left (5+3 x-x^2\right ) \log (x)\right )^2} \, dx-7 \int \frac {x^5 \log (x)}{\left ((-3+x) \left (2+\left (5+e^x\right ) x^2\right )+x^2 \left (5+3 x-x^2\right ) \log (x)\right )^2} \, dx-7 \int \frac {x^8 \log ^2(x)}{\left ((-3+x) \left (2+\left (5+e^x\right ) x^2\right )+x^2 \left (5+3 x-x^2\right ) \log (x)\right )^2} \, dx+10 \int \frac {x^7 \log ^2(x)}{\left ((-3+x) \left (2+\left (5+e^x\right ) x^2\right )+x^2 \left (5+3 x-x^2\right ) \log (x)\right )^2} \, dx-36 \int \frac {x^3 \log (x)}{\left ((-3+x) \left (2+\left (5+e^x\right ) x^2\right )+x^2 \left (5+3 x-x^2\right ) \log (x)\right )^2} \, dx+36 \int \frac {x^7 \log (x)}{\left ((-3+x) \left (2+\left (5+e^x\right ) x^2\right )+x^2 \left (5+3 x-x^2\right ) \log (x)\right )^2} \, dx-51 \int \frac {x^6 \log (x)}{\left ((-3+x) \left (2+\left (5+e^x\right ) x^2\right )+x^2 \left (5+3 x-x^2\right ) \log (x)\right )^2} \, dx+\int \frac {x^6 \log ^2(x)}{\left ((-3+x) \left (2+\left (5+e^x\right ) x^2\right )+x^2 \left (5+3 x-x^2\right ) \log (x)\right )^2} \, dx+\int \frac {x^9 \log ^2(x)}{\left ((-3+x) \left (2+\left (5+e^x\right ) x^2\right )+x^2 \left (5+3 x-x^2\right ) \log (x)\right )^2} \, dx-\int \left (-\frac {x^4}{6-2 x+15 x^2+3 e^x x^2-5 x^3-e^x x^3-5 x^2 \log (x)-3 x^3 \log (x)+x^4 \log (x)}-\frac {2 x^4 \log (x)}{6-2 x+15 x^2+3 e^x x^2-5 x^3-e^x x^3-5 x^2 \log (x)-3 x^3 \log (x)+x^4 \log (x)}+\frac {x^5 \log (x)}{6-2 x+15 x^2+3 e^x x^2-5 x^3-e^x x^3-5 x^2 \log (x)-3 x^3 \log (x)+x^4 \log (x)}\right ) \, dx\\ &=2 \int \frac {x^4 \log (x)}{6-2 x+15 x^2+3 e^x x^2-5 x^3-e^x x^3-5 x^2 \log (x)-3 x^3 \log (x)+x^4 \log (x)} \, dx-3 \int \frac {x^3}{6-2 x+15 x^2+3 e^x x^2-5 x^3-e^x x^3-5 x^2 \log (x)-3 x^3 \log (x)+x^4 \log (x)} \, dx+3 \int \frac {x^4 \log (x)}{6-2 x+15 x^2+3 e^x x^2-5 x^3-e^x x^3-5 x^2 \log (x)-3 x^3 \log (x)+x^4 \log (x)} \, dx-6 \int \frac {x^3 \log (x)}{6-2 x+15 x^2+3 e^x x^2-5 x^3-e^x x^3-5 x^2 \log (x)-3 x^3 \log (x)+x^4 \log (x)} \, dx+6 \int \frac {x^4 \log (x)}{\left ((-3+x) \left (2+\left (5+e^x\right ) x^2\right )+x^2 \left (5+3 x-x^2\right ) \log (x)\right )^2} \, dx-6 \int \frac {x^8 \log (x)}{\left ((-3+x) \left (2+\left (5+e^x\right ) x^2\right )+x^2 \left (5+3 x-x^2\right ) \log (x)\right )^2} \, dx-7 \int \frac {x^5 \log (x)}{\left ((-3+x) \left (2+\left (5+e^x\right ) x^2\right )+x^2 \left (5+3 x-x^2\right ) \log (x)\right )^2} \, dx-7 \int \frac {x^8 \log ^2(x)}{\left ((-3+x) \left (2+\left (5+e^x\right ) x^2\right )+x^2 \left (5+3 x-x^2\right ) \log (x)\right )^2} \, dx+10 \int \frac {x^7 \log ^2(x)}{\left ((-3+x) \left (2+\left (5+e^x\right ) x^2\right )+x^2 \left (5+3 x-x^2\right ) \log (x)\right )^2} \, dx-36 \int \frac {x^3 \log (x)}{\left ((-3+x) \left (2+\left (5+e^x\right ) x^2\right )+x^2 \left (5+3 x-x^2\right ) \log (x)\right )^2} \, dx+36 \int \frac {x^7 \log (x)}{\left ((-3+x) \left (2+\left (5+e^x\right ) x^2\right )+x^2 \left (5+3 x-x^2\right ) \log (x)\right )^2} \, dx-51 \int \frac {x^6 \log (x)}{\left ((-3+x) \left (2+\left (5+e^x\right ) x^2\right )+x^2 \left (5+3 x-x^2\right ) \log (x)\right )^2} \, dx+\int \frac {x^4}{6-2 x+15 x^2+3 e^x x^2-5 x^3-e^x x^3-5 x^2 \log (x)-3 x^3 \log (x)+x^4 \log (x)} \, dx-\int \frac {x^5 \log (x)}{6-2 x+15 x^2+3 e^x x^2-5 x^3-e^x x^3-5 x^2 \log (x)-3 x^3 \log (x)+x^4 \log (x)} \, dx+\int \frac {x^6 \log ^2(x)}{\left ((-3+x) \left (2+\left (5+e^x\right ) x^2\right )+x^2 \left (5+3 x-x^2\right ) \log (x)\right )^2} \, dx+\int \frac {x^9 \log ^2(x)}{\left ((-3+x) \left (2+\left (5+e^x\right ) x^2\right )+x^2 \left (5+3 x-x^2\right ) \log (x)\right )^2} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.16, size = 42, normalized size = 1.27 \begin {gather*} \frac {(-3+x) x^4 \log (x)}{-\left ((-3+x) \left (2+\left (5+e^x\right ) x^2\right )\right )+x^2 \left (-5-3 x+x^2\right ) \log (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-18*x^3 + 12*x^4 - 47*x^5 + 30*x^6 - 5*x^7 + E^x*(-9*x^5 + 6*x^6 - x^7) + (-72*x^3 + 48*x^4 - 98*x^
5 + 60*x^6 - 10*x^7 + E^x*(-18*x^5 + 21*x^6 - 8*x^7 + x^8))*Log[x] + (30*x^5 - 6*x^6 - 6*x^7 + x^8)*Log[x]^2)/
(36 - 24*x + 184*x^2 - 120*x^3 + 245*x^4 - 150*x^5 + 25*x^6 + E^(2*x)*(9*x^4 - 6*x^5 + x^6) + E^x*(36*x^2 - 24
*x^3 + 94*x^4 - 60*x^5 + 10*x^6) + (-60*x^2 - 16*x^3 - 126*x^4 - 44*x^5 + 60*x^6 - 10*x^7 + E^x*(-30*x^4 - 8*x
^5 + 12*x^6 - 2*x^7))*Log[x] + (25*x^4 + 30*x^5 - x^6 - 6*x^7 + x^8)*Log[x]^2),x]

[Out]

((-3 + x)*x^4*Log[x])/(-((-3 + x)*(2 + (5 + E^x)*x^2)) + x^2*(-5 - 3*x + x^2)*Log[x])

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fricas [A]  time = 0.52, size = 60, normalized size = 1.82 \begin {gather*} -\frac {{\left (x^{5} - 3 \, x^{4}\right )} \log \relax (x)}{5 \, x^{3} - 15 \, x^{2} + {\left (x^{3} - 3 \, x^{2}\right )} e^{x} - {\left (x^{4} - 3 \, x^{3} - 5 \, x^{2}\right )} \log \relax (x) + 2 \, x - 6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^8-6*x^7-6*x^6+30*x^5)*log(x)^2+((x^8-8*x^7+21*x^6-18*x^5)*exp(x)-10*x^7+60*x^6-98*x^5+48*x^4-72*
x^3)*log(x)+(-x^7+6*x^6-9*x^5)*exp(x)-5*x^7+30*x^6-47*x^5+12*x^4-18*x^3)/((x^8-6*x^7-x^6+30*x^5+25*x^4)*log(x)
^2+((-2*x^7+12*x^6-8*x^5-30*x^4)*exp(x)-10*x^7+60*x^6-44*x^5-126*x^4-16*x^3-60*x^2)*log(x)+(x^6-6*x^5+9*x^4)*e
xp(x)^2+(10*x^6-60*x^5+94*x^4-24*x^3+36*x^2)*exp(x)+25*x^6-150*x^5+245*x^4-120*x^3+184*x^2-24*x+36),x, algorit
hm="fricas")

[Out]

-(x^5 - 3*x^4)*log(x)/(5*x^3 - 15*x^2 + (x^3 - 3*x^2)*e^x - (x^4 - 3*x^3 - 5*x^2)*log(x) + 2*x - 6)

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giac [B]  time = 0.44, size = 108, normalized size = 3.27 \begin {gather*} \frac {x^{5} \log \relax (x) - 2 \, x^{4} \log \relax (x) - x^{3} e^{x} - 3 \, x^{3} \log \relax (x) - 5 \, x^{3} + 3 \, x^{2} e^{x} - 5 \, x^{2} \log \relax (x) + 15 \, x^{2} - 2 \, x + 6}{x^{4} \log \relax (x) - x^{3} e^{x} - 3 \, x^{3} \log \relax (x) - 5 \, x^{3} + 3 \, x^{2} e^{x} - 5 \, x^{2} \log \relax (x) + 15 \, x^{2} - 2 \, x + 6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^8-6*x^7-6*x^6+30*x^5)*log(x)^2+((x^8-8*x^7+21*x^6-18*x^5)*exp(x)-10*x^7+60*x^6-98*x^5+48*x^4-72*
x^3)*log(x)+(-x^7+6*x^6-9*x^5)*exp(x)-5*x^7+30*x^6-47*x^5+12*x^4-18*x^3)/((x^8-6*x^7-x^6+30*x^5+25*x^4)*log(x)
^2+((-2*x^7+12*x^6-8*x^5-30*x^4)*exp(x)-10*x^7+60*x^6-44*x^5-126*x^4-16*x^3-60*x^2)*log(x)+(x^6-6*x^5+9*x^4)*e
xp(x)^2+(10*x^6-60*x^5+94*x^4-24*x^3+36*x^2)*exp(x)+25*x^6-150*x^5+245*x^4-120*x^3+184*x^2-24*x+36),x, algorit
hm="giac")

[Out]

(x^5*log(x) - 2*x^4*log(x) - x^3*e^x - 3*x^3*log(x) - 5*x^3 + 3*x^2*e^x - 5*x^2*log(x) + 15*x^2 - 2*x + 6)/(x^
4*log(x) - x^3*e^x - 3*x^3*log(x) - 5*x^3 + 3*x^2*e^x - 5*x^2*log(x) + 15*x^2 - 2*x + 6)

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maple [B]  time = 0.05, size = 124, normalized size = 3.76

method	result	size
risch	\(\frac {x^{2} \left (x -3\right )}{x^{2}-3 x -5}+\frac {x^{2} \left ({\mathrm e}^{x} x^{4}+5 x^{4}-6 \,{\mathrm e}^{x} x^{3}-30 x^{3}+9 \,{\mathrm e}^{x} x^{2}+47 x^{2}-12 x +18\right )}{\left (x^{2}-3 x -5\right ) \left (x^{4} \ln \relax (x )-3 x^{3} \ln \relax (x )-{\mathrm e}^{x} x^{3}-5 x^{2} \ln \relax (x )+3 \,{\mathrm e}^{x} x^{2}-5 x^{3}+15 x^{2}-2 x +6\right )}\)	\(124\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^8-6*x^7-6*x^6+30*x^5)*ln(x)^2+((x^8-8*x^7+21*x^6-18*x^5)*exp(x)-10*x^7+60*x^6-98*x^5+48*x^4-72*x^3)*ln
(x)+(-x^7+6*x^6-9*x^5)*exp(x)-5*x^7+30*x^6-47*x^5+12*x^4-18*x^3)/((x^8-6*x^7-x^6+30*x^5+25*x^4)*ln(x)^2+((-2*x
^7+12*x^6-8*x^5-30*x^4)*exp(x)-10*x^7+60*x^6-44*x^5-126*x^4-16*x^3-60*x^2)*ln(x)+(x^6-6*x^5+9*x^4)*exp(x)^2+(1
0*x^6-60*x^5+94*x^4-24*x^3+36*x^2)*exp(x)+25*x^6-150*x^5+245*x^4-120*x^3+184*x^2-24*x+36),x,method=_RETURNVERB
OSE)

[Out]

x^2*(x-3)/(x^2-3*x-5)+x^2*(exp(x)*x^4+5*x^4-6*exp(x)*x^3-30*x^3+9*exp(x)*x^2+47*x^2-12*x+18)/(x^2-3*x-5)/(x^4*
ln(x)-3*x^3*ln(x)-exp(x)*x^3-5*x^2*ln(x)+3*exp(x)*x^2-5*x^3+15*x^2-2*x+6)

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maxima [A]  time = 0.81, size = 60, normalized size = 1.82 \begin {gather*} -\frac {{\left (x^{5} - 3 \, x^{4}\right )} \log \relax (x)}{5 \, x^{3} - 15 \, x^{2} + {\left (x^{3} - 3 \, x^{2}\right )} e^{x} - {\left (x^{4} - 3 \, x^{3} - 5 \, x^{2}\right )} \log \relax (x) + 2 \, x - 6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^8-6*x^7-6*x^6+30*x^5)*log(x)^2+((x^8-8*x^7+21*x^6-18*x^5)*exp(x)-10*x^7+60*x^6-98*x^5+48*x^4-72*
x^3)*log(x)+(-x^7+6*x^6-9*x^5)*exp(x)-5*x^7+30*x^6-47*x^5+12*x^4-18*x^3)/((x^8-6*x^7-x^6+30*x^5+25*x^4)*log(x)
^2+((-2*x^7+12*x^6-8*x^5-30*x^4)*exp(x)-10*x^7+60*x^6-44*x^5-126*x^4-16*x^3-60*x^2)*log(x)+(x^6-6*x^5+9*x^4)*e
xp(x)^2+(10*x^6-60*x^5+94*x^4-24*x^3+36*x^2)*exp(x)+25*x^6-150*x^5+245*x^4-120*x^3+184*x^2-24*x+36),x, algorit
hm="maxima")

[Out]

-(x^5 - 3*x^4)*log(x)/(5*x^3 - 15*x^2 + (x^3 - 3*x^2)*e^x - (x^4 - 3*x^3 - 5*x^2)*log(x) + 2*x - 6)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} -\int \frac {\ln \relax (x)\,\left ({\mathrm {e}}^x\,\left (-x^8+8\,x^7-21\,x^6+18\,x^5\right )+72\,x^3-48\,x^4+98\,x^5-60\,x^6+10\,x^7\right )-{\ln \relax (x)}^2\,\left (x^8-6\,x^7-6\,x^6+30\,x^5\right )+18\,x^3-12\,x^4+47\,x^5-30\,x^6+5\,x^7+{\mathrm {e}}^x\,\left (x^7-6\,x^6+9\,x^5\right )}{{\mathrm {e}}^{2\,x}\,\left (x^6-6\,x^5+9\,x^4\right )-24\,x-\ln \relax (x)\,\left ({\mathrm {e}}^x\,\left (2\,x^7-12\,x^6+8\,x^5+30\,x^4\right )+60\,x^2+16\,x^3+126\,x^4+44\,x^5-60\,x^6+10\,x^7\right )+{\mathrm {e}}^x\,\left (10\,x^6-60\,x^5+94\,x^4-24\,x^3+36\,x^2\right )+{\ln \relax (x)}^2\,\left (x^8-6\,x^7-x^6+30\,x^5+25\,x^4\right )+184\,x^2-120\,x^3+245\,x^4-150\,x^5+25\,x^6+36} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(x)*(exp(x)*(18*x^5 - 21*x^6 + 8*x^7 - x^8) + 72*x^3 - 48*x^4 + 98*x^5 - 60*x^6 + 10*x^7) - log(x)^2*
(30*x^5 - 6*x^6 - 6*x^7 + x^8) + 18*x^3 - 12*x^4 + 47*x^5 - 30*x^6 + 5*x^7 + exp(x)*(9*x^5 - 6*x^6 + x^7))/(ex
p(2*x)*(9*x^4 - 6*x^5 + x^6) - 24*x - log(x)*(exp(x)*(30*x^4 + 8*x^5 - 12*x^6 + 2*x^7) + 60*x^2 + 16*x^3 + 126
*x^4 + 44*x^5 - 60*x^6 + 10*x^7) + exp(x)*(36*x^2 - 24*x^3 + 94*x^4 - 60*x^5 + 10*x^6) + log(x)^2*(25*x^4 + 30
*x^5 - x^6 - 6*x^7 + x^8) + 184*x^2 - 120*x^3 + 245*x^4 - 150*x^5 + 25*x^6 + 36),x)

[Out]

-int((log(x)*(exp(x)*(18*x^5 - 21*x^6 + 8*x^7 - x^8) + 72*x^3 - 48*x^4 + 98*x^5 - 60*x^6 + 10*x^7) - log(x)^2*
(30*x^5 - 6*x^6 - 6*x^7 + x^8) + 18*x^3 - 12*x^4 + 47*x^5 - 30*x^6 + 5*x^7 + exp(x)*(9*x^5 - 6*x^6 + x^7))/(ex
p(2*x)*(9*x^4 - 6*x^5 + x^6) - 24*x - log(x)*(exp(x)*(30*x^4 + 8*x^5 - 12*x^6 + 2*x^7) + 60*x^2 + 16*x^3 + 126
*x^4 + 44*x^5 - 60*x^6 + 10*x^7) + exp(x)*(36*x^2 - 24*x^3 + 94*x^4 - 60*x^5 + 10*x^6) + log(x)^2*(25*x^4 + 30
*x^5 - x^6 - 6*x^7 + x^8) + 184*x^2 - 120*x^3 + 245*x^4 - 150*x^5 + 25*x^6 + 36), x)

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sympy [B]  time = 0.89, size = 65, normalized size = 1.97 \begin {gather*} \frac {- x^{5} \log {\relax (x )} + 3 x^{4} \log {\relax (x )}}{- x^{4} \log {\relax (x )} + 3 x^{3} \log {\relax (x )} + 5 x^{3} + 5 x^{2} \log {\relax (x )} - 15 x^{2} + 2 x + \left (x^{3} - 3 x^{2}\right ) e^{x} - 6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x**8-6*x**7-6*x**6+30*x**5)*ln(x)**2+((x**8-8*x**7+21*x**6-18*x**5)*exp(x)-10*x**7+60*x**6-98*x**5
+48*x**4-72*x**3)*ln(x)+(-x**7+6*x**6-9*x**5)*exp(x)-5*x**7+30*x**6-47*x**5+12*x**4-18*x**3)/((x**8-6*x**7-x**
6+30*x**5+25*x**4)*ln(x)**2+((-2*x**7+12*x**6-8*x**5-30*x**4)*exp(x)-10*x**7+60*x**6-44*x**5-126*x**4-16*x**3-
60*x**2)*ln(x)+(x**6-6*x**5+9*x**4)*exp(x)**2+(10*x**6-60*x**5+94*x**4-24*x**3+36*x**2)*exp(x)+25*x**6-150*x**
5+245*x**4-120*x**3+184*x**2-24*x+36),x)

[Out]

(-x**5*log(x) + 3*x**4*log(x))/(-x**4*log(x) + 3*x**3*log(x) + 5*x**3 + 5*x**2*log(x) - 15*x**2 + 2*x + (x**3
- 3*x**2)*exp(x) - 6)

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