\(\int \frac {512 x^2-128 x^3+(1536 x^2-128 x^3) \log (x)+\frac {e^{5-x} (32 x^2+(160 x^2+64 x^3) \log (x))}{x}}{4096+\frac {e^{15-3 x}}{x^3}+\frac {e^{10-2 x} (48-12 x)}{x^2}-3072 x+768 x^2-64 x^3+\frac {e^{5-x} (768-384 x+48 x^2)}{x}} \, dx\) [7754]

Optimal result

Integrand size = 122, antiderivative size = 28 \[ \int \frac {512 x^2-128 x^3+\left (1536 x^2-128 x^3\right ) \log (x)+\frac {e^{5-x} \left (32 x^2+\left (160 x^2+64 x^3\right ) \log (x)\right )}{x}}{4096+\frac {e^{15-3 x}}{x^3}+\frac {e^{10-2 x} (48-12 x)}{x^2}-3072 x+768 x^2-64 x^3+\frac {e^{5-x} \left (768-384 x+48 x^2\right )}{x}} \, dx=\frac {2 x^3 \log (x)}{\left (4+\frac {e^{5-x}}{4 x}-x\right )^2} \]

[Out]

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

Rubi [F]

\[ \int \frac {512 x^2-128 x^3+\left (1536 x^2-128 x^3\right ) \log (x)+\frac {e^{5-x} \left (32 x^2+\left (160 x^2+64 x^3\right ) \log (x)\right )}{x}}{4096+\frac {e^{15-3 x}}{x^3}+\frac {e^{10-2 x} (48-12 x)}{x^2}-3072 x+768 x^2-64 x^3+\frac {e^{5-x} \left (768-384 x+48 x^2\right )}{x}} \, dx=\int \frac {512 x^2-128 x^3+\left (1536 x^2-128 x^3\right ) \log (x)+\frac {e^{5-x} \left (32 x^2+\left (160 x^2+64 x^3\right ) \log (x)\right )}{x}}{4096+\frac {e^{15-3 x}}{x^3}+\frac {e^{10-2 x} (48-12 x)}{x^2}-3072 x+768 x^2-64 x^3+\frac {e^{5-x} \left (768-384 x+48 x^2\right )}{x}} \, dx \]

[In]

Int[(512*x^2 - 128*x^3 + (1536*x^2 - 128*x^3)*Log[x] + (E^(5 - x)*(32*x^2 + (160*x^2 + 64*x^3)*Log[x]))/x)/(40
96 + E^(15 - 3*x)/x^3 + (E^(10 - 2*x)*(48 - 12*x))/x^2 - 3072*x + 768*x^2 - 64*x^3 + (E^(5 - x)*(768 - 384*x +
48*x^2))/x),x]

[Out]

-65536*Log[x]*Defer[Int][E^(2*x)/((-4 + x)*(E^5 - 4*E^x*(-4 + x)*x)^2), x] - 4096*Log[x]*Defer[Int][(E^(2*x)*x
)/(E^5 - 4*E^x*(-4 + x)*x)^2, x] - 1024*Log[x]*Defer[Int][(E^(2*x)*x^2)/(E^5 - 4*E^x*(-4 + x)*x)^2, x] - 256*L
og[x]*Defer[Int][(E^(2*x)*x^3)/(E^5 - 4*E^x*(-4 + x)*x)^2, x] + 32*Defer[Int][(E^(2*x)*x^4)/(E^5 - 4*E^x*(-4 +
x)*x)^2, x] + 32*Log[x]*Defer[Int][(E^(2*x)*x^4)/(E^5 - 4*E^x*(-4 + x)*x)^2, x] + 16384*Log[x]*Defer[Int][E^(
5 + 2*x)/(E^5 + 16*E^x*x - 4*E^x*x^2)^3, x] - 16384*Log[x]*Defer[Int][E^(2*x)/(E^5 + 16*E^x*x - 4*E^x*x^2)^2,
x] - 65536*Log[x]*Defer[Int][E^(5 + 2*x)/((-4 + x)*(-E^5 - 16*E^x*x + 4*E^x*x^2)^3), x] - 4096*Log[x]*Defer[In
t][(E^(5 + 2*x)*x)/(-E^5 - 16*E^x*x + 4*E^x*x^2)^3, x] - 1024*Log[x]*Defer[Int][(E^(5 + 2*x)*x^2)/(-E^5 - 16*E
^x*x + 4*E^x*x^2)^3, x] - 256*Log[x]*Defer[Int][(E^(5 + 2*x)*x^3)/(-E^5 - 16*E^x*x + 4*E^x*x^2)^3, x] - 128*Lo
g[x]*Defer[Int][(E^(5 + 2*x)*x^4)/(-E^5 - 16*E^x*x + 4*E^x*x^2)^3, x] - 64*Log[x]*Defer[Int][(E^(5 + 2*x)*x^5)
/(-E^5 - 16*E^x*x + 4*E^x*x^2)^3, x] - 16384*Defer[Int][Defer[Int][E^(5 + 2*x)/(E^5 - 4*E^x*(-4 + x)*x)^3, x]/
x, x] + 4096*Defer[Int][Defer[Int][-((E^(5 + 2*x)*x)/(E^5 - 4*E^x*(-4 + x)*x)^3), x]/x, x] + 1024*Defer[Int][D
efer[Int][-((E^(5 + 2*x)*x^2)/(E^5 - 4*E^x*(-4 + x)*x)^3), x]/x, x] + 256*Defer[Int][Defer[Int][-((E^(5 + 2*x)
*x^3)/(E^5 - 4*E^x*(-4 + x)*x)^3), x]/x, x] + 128*Defer[Int][Defer[Int][-((E^(5 + 2*x)*x^4)/(E^5 - 4*E^x*(-4 +
x)*x)^3), x]/x, x] + 64*Defer[Int][Defer[Int][-((E^(5 + 2*x)*x^5)/(E^5 - 4*E^x*(-4 + x)*x)^3), x]/x, x] + 163
84*Defer[Int][Defer[Int][E^(2*x)/(E^5 - 4*E^x*(-4 + x)*x)^2, x]/x, x] + 65536*Defer[Int][Defer[Int][E^(2*x)/((
-4 + x)*(E^5 - 4*E^x*(-4 + x)*x)^2), x]/x, x] + 4096*Defer[Int][Defer[Int][(E^(2*x)*x)/(E^5 - 4*E^x*(-4 + x)*x
)^2, x]/x, x] + 1024*Defer[Int][Defer[Int][(E^(2*x)*x^2)/(E^5 - 4*E^x*(-4 + x)*x)^2, x]/x, x] + 256*Defer[Int]
[Defer[Int][(E^(2*x)*x^3)/(E^5 - 4*E^x*(-4 + x)*x)^2, x]/x, x] - 32*Defer[Int][Defer[Int][(E^(2*x)*x^4)/(E^5 -
4*E^x*(-4 + x)*x)^2, x]/x, x] + 65536*Defer[Int][Defer[Int][E^(5 + 2*x)/((-4 + x)*(-E^5 + 4*E^x*(-4 + x)*x)^3
), x]/x, x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {32 e^{2 x} x^4 \left (e^5-4 e^x (-4+x) x-4 e^x (-12+x) x \log (x)+e^5 (5+2 x) \log (x)\right )}{\left (e^5-4 e^x (-4+x) x\right )^3} \, dx \\ & = 32 \int \frac {e^{2 x} x^4 \left (e^5-4 e^x (-4+x) x-4 e^x (-12+x) x \log (x)+e^5 (5+2 x) \log (x)\right )}{\left (e^5-4 e^x (-4+x) x\right )^3} \, dx \\ & = 32 \int \left (\frac {2 e^{5+2 x} x^4 \left (-4-2 x+x^2\right ) \log (x)}{(-4+x) \left (e^5+16 e^x x-4 e^x x^2\right )^3}+\frac {e^{2 x} x^4 (-4+x-12 \log (x)+x \log (x))}{(-4+x) \left (-e^5-16 e^x x+4 e^x x^2\right )^2}\right ) \, dx \\ & = 32 \int \frac {e^{2 x} x^4 (-4+x-12 \log (x)+x \log (x))}{(-4+x) \left (-e^5-16 e^x x+4 e^x x^2\right )^2} \, dx+64 \int \frac {e^{5+2 x} x^4 \left (-4-2 x+x^2\right ) \log (x)}{(-4+x) \left (e^5+16 e^x x-4 e^x x^2\right )^3} \, dx \\ & = 32 \int \left (\frac {64 e^{2 x} (-4+x-12 \log (x)+x \log (x))}{\left (e^5+16 e^x x-4 e^x x^2\right )^2}+\frac {256 e^{2 x} (-4+x-12 \log (x)+x \log (x))}{(-4+x) \left (-e^5-16 e^x x+4 e^x x^2\right )^2}+\frac {16 e^{2 x} x (-4+x-12 \log (x)+x \log (x))}{\left (-e^5-16 e^x x+4 e^x x^2\right )^2}+\frac {4 e^{2 x} x^2 (-4+x-12 \log (x)+x \log (x))}{\left (-e^5-16 e^x x+4 e^x x^2\right )^2}+\frac {e^{2 x} x^3 (-4+x-12 \log (x)+x \log (x))}{\left (-e^5-16 e^x x+4 e^x x^2\right )^2}\right ) \, dx-64 \int \frac {256 \int \frac {e^{5+2 x}}{\left (e^5-4 e^x (-4+x) x\right )^3} \, dx-64 \int -\frac {e^{5+2 x} x}{\left (e^5-4 e^x (-4+x) x\right )^3} \, dx-16 \int -\frac {e^{5+2 x} x^2}{\left (e^5-4 e^x (-4+x) x\right )^3} \, dx-4 \int -\frac {e^{5+2 x} x^3}{\left (e^5-4 e^x (-4+x) x\right )^3} \, dx-2 \int -\frac {e^{5+2 x} x^4}{\left (e^5-4 e^x (-4+x) x\right )^3} \, dx-\int -\frac {e^{5+2 x} x^5}{\left (e^5-4 e^x (-4+x) x\right )^3} \, dx-1024 \int \frac {e^{5+2 x}}{(-4+x) \left (-e^5+4 e^x (-4+x) x\right )^3} \, dx}{x} \, dx-(64 \log (x)) \int \frac {e^{5+2 x} x^5}{\left (-e^5-16 e^x x+4 e^x x^2\right )^3} \, dx-(128 \log (x)) \int \frac {e^{5+2 x} x^4}{\left (-e^5-16 e^x x+4 e^x x^2\right )^3} \, dx-(256 \log (x)) \int \frac {e^{5+2 x} x^3}{\left (-e^5-16 e^x x+4 e^x x^2\right )^3} \, dx-(1024 \log (x)) \int \frac {e^{5+2 x} x^2}{\left (-e^5-16 e^x x+4 e^x x^2\right )^3} \, dx-(4096 \log (x)) \int \frac {e^{5+2 x} x}{\left (-e^5-16 e^x x+4 e^x x^2\right )^3} \, dx+(16384 \log (x)) \int \frac {e^{5+2 x}}{\left (e^5+16 e^x x-4 e^x x^2\right )^3} \, dx-(65536 \log (x)) \int \frac {e^{5+2 x}}{(-4+x) \left (-e^5-16 e^x x+4 e^x x^2\right )^3} \, dx \\ & = 32 \int \frac {e^{2 x} x^3 (-4+x-12 \log (x)+x \log (x))}{\left (-e^5-16 e^x x+4 e^x x^2\right )^2} \, dx-64 \int \left (\frac {256 \int \frac {e^{5+2 x}}{\left (e^5-4 e^x (-4+x) x\right )^3} \, dx+64 \int \frac {e^{5+2 x} x}{\left (e^5-4 e^x (-4+x) x\right )^3} \, dx+16 \int \frac {e^{5+2 x} x^2}{\left (e^5-4 e^x (-4+x) x\right )^3} \, dx+4 \int \frac {e^{5+2 x} x^3}{\left (e^5-4 e^x (-4+x) x\right )^3} \, dx+2 \int \frac {e^{5+2 x} x^4}{\left (e^5-4 e^x (-4+x) x\right )^3} \, dx+\int \frac {e^{5+2 x} x^5}{\left (e^5-4 e^x (-4+x) x\right )^3} \, dx}{x}-\frac {1024 \int \frac {e^{5+2 x}}{(-4+x) \left (-e^5+4 e^x (-4+x) x\right )^3} \, dx}{x}\right ) \, dx+128 \int \frac {e^{2 x} x^2 (-4+x-12 \log (x)+x \log (x))}{\left (-e^5-16 e^x x+4 e^x x^2\right )^2} \, dx+512 \int \frac {e^{2 x} x (-4+x-12 \log (x)+x \log (x))}{\left (-e^5-16 e^x x+4 e^x x^2\right )^2} \, dx+2048 \int \frac {e^{2 x} (-4+x-12 \log (x)+x \log (x))}{\left (e^5+16 e^x x-4 e^x x^2\right )^2} \, dx+8192 \int \frac {e^{2 x} (-4+x-12 \log (x)+x \log (x))}{(-4+x) \left (-e^5-16 e^x x+4 e^x x^2\right )^2} \, dx-(64 \log (x)) \int \frac {e^{5+2 x} x^5}{\left (-e^5-16 e^x x+4 e^x x^2\right )^3} \, dx-(128 \log (x)) \int \frac {e^{5+2 x} x^4}{\left (-e^5-16 e^x x+4 e^x x^2\right )^3} \, dx-(256 \log (x)) \int \frac {e^{5+2 x} x^3}{\left (-e^5-16 e^x x+4 e^x x^2\right )^3} \, dx-(1024 \log (x)) \int \frac {e^{5+2 x} x^2}{\left (-e^5-16 e^x x+4 e^x x^2\right )^3} \, dx-(4096 \log (x)) \int \frac {e^{5+2 x} x}{\left (-e^5-16 e^x x+4 e^x x^2\right )^3} \, dx+(16384 \log (x)) \int \frac {e^{5+2 x}}{\left (e^5+16 e^x x-4 e^x x^2\right )^3} \, dx-(65536 \log (x)) \int \frac {e^{5+2 x}}{(-4+x) \left (-e^5-16 e^x x+4 e^x x^2\right )^3} \, dx \\ & = 32 \int \frac {e^{2 x} x^3 (-4+x+(-12+x) \log (x))}{\left (e^5-4 e^x (-4+x) x\right )^2} \, dx-64 \int \frac {256 \int \frac {e^{5+2 x}}{\left (e^5-4 e^x (-4+x) x\right )^3} \, dx+64 \int \frac {e^{5+2 x} x}{\left (e^5-4 e^x (-4+x) x\right )^3} \, dx+16 \int \frac {e^{5+2 x} x^2}{\left (e^5-4 e^x (-4+x) x\right )^3} \, dx+4 \int \frac {e^{5+2 x} x^3}{\left (e^5-4 e^x (-4+x) x\right )^3} \, dx+2 \int \frac {e^{5+2 x} x^4}{\left (e^5-4 e^x (-4+x) x\right )^3} \, dx+\int \frac {e^{5+2 x} x^5}{\left (e^5-4 e^x (-4+x) x\right )^3} \, dx}{x} \, dx+128 \int \frac {e^{2 x} x^2 (-4+x+(-12+x) \log (x))}{\left (e^5-4 e^x (-4+x) x\right )^2} \, dx+512 \int \frac {e^{2 x} x (-4+x+(-12+x) \log (x))}{\left (e^5-4 e^x (-4+x) x\right )^2} \, dx+2048 \int \left (-\frac {4 e^{2 x}}{\left (e^5+16 e^x x-4 e^x x^2\right )^2}+\frac {e^{2 x} x}{\left (-e^5-16 e^x x+4 e^x x^2\right )^2}-\frac {12 e^{2 x} \log (x)}{\left (e^5+16 e^x x-4 e^x x^2\right )^2}+\frac {e^{2 x} x \log (x)}{\left (-e^5-16 e^x x+4 e^x x^2\right )^2}\right ) \, dx+8192 \int \left (-\frac {4 e^{2 x}}{(-4+x) \left (-e^5-16 e^x x+4 e^x x^2\right )^2}+\frac {e^{2 x} x}{(-4+x) \left (-e^5-16 e^x x+4 e^x x^2\right )^2}-\frac {12 e^{2 x} \log (x)}{(-4+x) \left (-e^5-16 e^x x+4 e^x x^2\right )^2}+\frac {e^{2 x} x \log (x)}{(-4+x) \left (-e^5-16 e^x x+4 e^x x^2\right )^2}\right ) \, dx+65536 \int \frac {\int \frac {e^{5+2 x}}{(-4+x) \left (-e^5+4 e^x (-4+x) x\right )^3} \, dx}{x} \, dx-(64 \log (x)) \int \frac {e^{5+2 x} x^5}{\left (-e^5-16 e^x x+4 e^x x^2\right )^3} \, dx-(128 \log (x)) \int \frac {e^{5+2 x} x^4}{\left (-e^5-16 e^x x+4 e^x x^2\right )^3} \, dx-(256 \log (x)) \int \frac {e^{5+2 x} x^3}{\left (-e^5-16 e^x x+4 e^x x^2\right )^3} \, dx-(1024 \log (x)) \int \frac {e^{5+2 x} x^2}{\left (-e^5-16 e^x x+4 e^x x^2\right )^3} \, dx-(4096 \log (x)) \int \frac {e^{5+2 x} x}{\left (-e^5-16 e^x x+4 e^x x^2\right )^3} \, dx+(16384 \log (x)) \int \frac {e^{5+2 x}}{\left (e^5+16 e^x x-4 e^x x^2\right )^3} \, dx-(65536 \log (x)) \int \frac {e^{5+2 x}}{(-4+x) \left (-e^5-16 e^x x+4 e^x x^2\right )^3} \, dx \\ & = 32 \int \left (-\frac {4 e^{2 x} x^3}{\left (-e^5-16 e^x x+4 e^x x^2\right )^2}+\frac {e^{2 x} x^4}{\left (-e^5-16 e^x x+4 e^x x^2\right )^2}-\frac {12 e^{2 x} x^3 \log (x)}{\left (-e^5-16 e^x x+4 e^x x^2\right )^2}+\frac {e^{2 x} x^4 \log (x)}{\left (-e^5-16 e^x x+4 e^x x^2\right )^2}\right ) \, dx-64 \int \left (\frac {2 \left (128 \int \frac {e^{5+2 x}}{\left (e^5-4 e^x (-4+x) x\right )^3} \, dx+32 \int \frac {e^{5+2 x} x}{\left (e^5-4 e^x (-4+x) x\right )^3} \, dx+8 \int \frac {e^{5+2 x} x^2}{\left (e^5-4 e^x (-4+x) x\right )^3} \, dx+2 \int \frac {e^{5+2 x} x^3}{\left (e^5-4 e^x (-4+x) x\right )^3} \, dx+\int \frac {e^{5+2 x} x^4}{\left (e^5-4 e^x (-4+x) x\right )^3} \, dx\right )}{x}+\frac {\int \frac {e^{5+2 x} x^5}{\left (e^5-4 e^x (-4+x) x\right )^3} \, dx}{x}\right ) \, dx+128 \int \left (-\frac {4 e^{2 x} x^2}{\left (-e^5-16 e^x x+4 e^x x^2\right )^2}+\frac {e^{2 x} x^3}{\left (-e^5-16 e^x x+4 e^x x^2\right )^2}-\frac {12 e^{2 x} x^2 \log (x)}{\left (-e^5-16 e^x x+4 e^x x^2\right )^2}+\frac {e^{2 x} x^3 \log (x)}{\left (-e^5-16 e^x x+4 e^x x^2\right )^2}\right ) \, dx+512 \int \left (-\frac {4 e^{2 x} x}{\left (-e^5-16 e^x x+4 e^x x^2\right )^2}+\frac {e^{2 x} x^2}{\left (-e^5-16 e^x x+4 e^x x^2\right )^2}-\frac {12 e^{2 x} x \log (x)}{\left (-e^5-16 e^x x+4 e^x x^2\right )^2}+\frac {e^{2 x} x^2 \log (x)}{\left (-e^5-16 e^x x+4 e^x x^2\right )^2}\right ) \, dx+2048 \int \frac {e^{2 x} x}{\left (-e^5-16 e^x x+4 e^x x^2\right )^2} \, dx+2048 \int \frac {e^{2 x} x \log (x)}{\left (-e^5-16 e^x x+4 e^x x^2\right )^2} \, dx-8192 \int \frac {e^{2 x}}{\left (e^5+16 e^x x-4 e^x x^2\right )^2} \, dx+8192 \int \frac {e^{2 x} x}{(-4+x) \left (-e^5-16 e^x x+4 e^x x^2\right )^2} \, dx+8192 \int \frac {e^{2 x} x \log (x)}{(-4+x) \left (-e^5-16 e^x x+4 e^x x^2\right )^2} \, dx-24576 \int \frac {e^{2 x} \log (x)}{\left (e^5+16 e^x x-4 e^x x^2\right )^2} \, dx-32768 \int \frac {e^{2 x}}{(-4+x) \left (-e^5-16 e^x x+4 e^x x^2\right )^2} \, dx+65536 \int \frac {\int \frac {e^{5+2 x}}{(-4+x) \left (-e^5+4 e^x (-4+x) x\right )^3} \, dx}{x} \, dx-98304 \int \frac {e^{2 x} \log (x)}{(-4+x) \left (-e^5-16 e^x x+4 e^x x^2\right )^2} \, dx-(64 \log (x)) \int \frac {e^{5+2 x} x^5}{\left (-e^5-16 e^x x+4 e^x x^2\right )^3} \, dx-(128 \log (x)) \int \frac {e^{5+2 x} x^4}{\left (-e^5-16 e^x x+4 e^x x^2\right )^3} \, dx-(256 \log (x)) \int \frac {e^{5+2 x} x^3}{\left (-e^5-16 e^x x+4 e^x x^2\right )^3} \, dx-(1024 \log (x)) \int \frac {e^{5+2 x} x^2}{\left (-e^5-16 e^x x+4 e^x x^2\right )^3} \, dx-(4096 \log (x)) \int \frac {e^{5+2 x} x}{\left (-e^5-16 e^x x+4 e^x x^2\right )^3} \, dx+(16384 \log (x)) \int \frac {e^{5+2 x}}{\left (e^5+16 e^x x-4 e^x x^2\right )^3} \, dx-(65536 \log (x)) \int \frac {e^{5+2 x}}{(-4+x) \left (-e^5-16 e^x x+4 e^x x^2\right )^3} \, dx \\ & = 32 \int \frac {e^{2 x} x^4}{\left (-e^5-16 e^x x+4 e^x x^2\right )^2} \, dx+32 \int \frac {e^{2 x} x^4 \log (x)}{\left (-e^5-16 e^x x+4 e^x x^2\right )^2} \, dx-64 \int \frac {\int \frac {e^{5+2 x} x^5}{\left (e^5-4 e^x (-4+x) x\right )^3} \, dx}{x} \, dx+128 \int \frac {e^{2 x} x^3 \log (x)}{\left (-e^5-16 e^x x+4 e^x x^2\right )^2} \, dx-128 \int \frac {128 \int \frac {e^{5+2 x}}{\left (e^5-4 e^x (-4+x) x\right )^3} \, dx+32 \int \frac {e^{5+2 x} x}{\left (e^5-4 e^x (-4+x) x\right )^3} \, dx+8 \int \frac {e^{5+2 x} x^2}{\left (e^5-4 e^x (-4+x) x\right )^3} \, dx+2 \int \frac {e^{5+2 x} x^3}{\left (e^5-4 e^x (-4+x) x\right )^3} \, dx+\int \frac {e^{5+2 x} x^4}{\left (e^5-4 e^x (-4+x) x\right )^3} \, dx}{x} \, dx-384 \int \frac {e^{2 x} x^3 \log (x)}{\left (-e^5-16 e^x x+4 e^x x^2\right )^2} \, dx+512 \int \frac {e^{2 x} x^2 \log (x)}{\left (-e^5-16 e^x x+4 e^x x^2\right )^2} \, dx-1536 \int \frac {e^{2 x} x^2 \log (x)}{\left (-e^5-16 e^x x+4 e^x x^2\right )^2} \, dx+2048 \int \frac {e^{2 x} x}{\left (e^5-4 e^x (-4+x) x\right )^2} \, dx-2048 \int \frac {e^{2 x} x}{\left (-e^5-16 e^x x+4 e^x x^2\right )^2} \, dx-2048 \int \frac {\int \frac {e^{2 x} x}{\left (e^5-4 e^x (-4+x) x\right )^2} \, dx}{x} \, dx-6144 \int \frac {e^{2 x} x \log (x)}{\left (-e^5-16 e^x x+4 e^x x^2\right )^2} \, dx+8192 \int \frac {e^{2 x} x}{(-4+x) \left (e^5-4 e^x (-4+x) x\right )^2} \, dx-8192 \int \frac {e^{2 x}}{\left (e^5+16 e^x x-4 e^x x^2\right )^2} \, dx-8192 \int \frac {\int \frac {e^{2 x}}{\left (e^5-4 e^x (-4+x) x\right )^2} \, dx+4 \int \frac {e^{2 x}}{(-4+x) \left (e^5-4 e^x (-4+x) x\right )^2} \, dx}{x} \, dx+24576 \int \frac {\int \frac {e^{2 x}}{\left (e^5-4 e^x (-4+x) x\right )^2} \, dx}{x} \, dx-32768 \int \frac {e^{2 x}}{(-4+x) \left (e^5-4 e^x (-4+x) x\right )^2} \, dx+65536 \int \frac {\int \frac {e^{5+2 x}}{(-4+x) \left (-e^5+4 e^x (-4+x) x\right )^3} \, dx}{x} \, dx+98304 \int \frac {\int \frac {e^{2 x}}{(-4+x) \left (e^5-4 e^x (-4+x) x\right )^2} \, dx}{x} \, dx-(64 \log (x)) \int \frac {e^{5+2 x} x^5}{\left (-e^5-16 e^x x+4 e^x x^2\right )^3} \, dx-(128 \log (x)) \int \frac {e^{5+2 x} x^4}{\left (-e^5-16 e^x x+4 e^x x^2\right )^3} \, dx-(256 \log (x)) \int \frac {e^{5+2 x} x^3}{\left (-e^5-16 e^x x+4 e^x x^2\right )^3} \, dx-(1024 \log (x)) \int \frac {e^{5+2 x} x^2}{\left (-e^5-16 e^x x+4 e^x x^2\right )^3} \, dx+(2048 \log (x)) \int \frac {e^{2 x} x}{\left (e^5-4 e^x (-4+x) x\right )^2} \, dx-(4096 \log (x)) \int \frac {e^{5+2 x} x}{\left (-e^5-16 e^x x+4 e^x x^2\right )^3} \, dx+(8192 \log (x)) \int \frac {e^{2 x}}{\left (e^5+16 e^x x-4 e^x x^2\right )^2} \, dx+(16384 \log (x)) \int \frac {e^{5+2 x}}{\left (e^5+16 e^x x-4 e^x x^2\right )^3} \, dx-(24576 \log (x)) \int \frac {e^{2 x}}{\left (e^5+16 e^x x-4 e^x x^2\right )^2} \, dx+(32768 \log (x)) \int \frac {e^{2 x}}{(-4+x) \left (e^5-4 e^x (-4+x) x\right )^2} \, dx-(65536 \log (x)) \int \frac {e^{5+2 x}}{(-4+x) \left (-e^5-16 e^x x+4 e^x x^2\right )^3} \, dx-(98304 \log (x)) \int \frac {e^{2 x}}{(-4+x) \left (e^5-4 e^x (-4+x) x\right )^2} \, dx \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [A] (verified)

Time = 5.11 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.96 \[ \int \frac {512 x^2-128 x^3+\left (1536 x^2-128 x^3\right ) \log (x)+\frac {e^{5-x} \left (32 x^2+\left (160 x^2+64 x^3\right ) \log (x)\right )}{x}}{4096+\frac {e^{15-3 x}}{x^3}+\frac {e^{10-2 x} (48-12 x)}{x^2}-3072 x+768 x^2-64 x^3+\frac {e^{5-x} \left (768-384 x+48 x^2\right )}{x}} \, dx=\frac {32 e^{2 x} x^5 \log (x)}{\left (e^5-4 e^x (-4+x) x\right )^2} \]

[In]

Integrate[(512*x^2 - 128*x^3 + (1536*x^2 - 128*x^3)*Log[x] + (E^(5 - x)*(32*x^2 + (160*x^2 + 64*x^3)*Log[x]))/
x)/(4096 + E^(15 - 3*x)/x^3 + (E^(10 - 2*x)*(48 - 12*x))/x^2 - 3072*x + 768*x^2 - 64*x^3 + (E^(5 - x)*(768 - 3
84*x + 48*x^2))/x),x]

[Out]

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

Maple [A] (verified)

Time = 0.25 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93

method	result	size
risch	\(\frac {32 x^{3} \ln \left (x \right )}{\left (4 x -\frac {{\mathrm e}^{5-x}}{x}-16\right )^{2}}\)	\(26\)
parallelrisch	\(\frac {32 x^{3} \ln \left (x \right )}{16 x^{2}-8 \,{\mathrm e}^{5-\ln \left (x \right )-x} x +\frac {{\mathrm e}^{-2 x +10}}{x^{2}}-128 x +32 \,{\mathrm e}^{5-\ln \left (x \right )-x}+256}\)	\(57\)

[In]

int((((64*x^3+160*x^2)*ln(x)+32*x^2)*exp(5-ln(x)-x)+(-128*x^3+1536*x^2)*ln(x)-128*x^3+512*x^2)/(exp(5-ln(x)-x)
^3+(-12*x+48)*exp(5-ln(x)-x)^2+(48*x^2-384*x+768)*exp(5-ln(x)-x)-64*x^3+768*x^2-3072*x+4096),x,method=_RETURNV
ERBOSE)

[Out]

32*x^3*ln(x)/(4*x-1/x*exp(5-x)-16)^2

Fricas [A] (verification not implemented)

none

Time = 0.40 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.57 \[ \int \frac {512 x^2-128 x^3+\left (1536 x^2-128 x^3\right ) \log (x)+\frac {e^{5-x} \left (32 x^2+\left (160 x^2+64 x^3\right ) \log (x)\right )}{x}}{4096+\frac {e^{15-3 x}}{x^3}+\frac {e^{10-2 x} (48-12 x)}{x^2}-3072 x+768 x^2-64 x^3+\frac {e^{5-x} \left (768-384 x+48 x^2\right )}{x}} \, dx=\frac {32 \, x^{3} \log \left (x\right )}{16 \, x^{2} - 8 \, {\left (x - 4\right )} e^{\left (-x - \log \left (x\right ) + 5\right )} - 128 \, x + e^{\left (-2 \, x - 2 \, \log \left (x\right ) + 10\right )} + 256} \]

[In]

integrate((((64*x^3+160*x^2)*log(x)+32*x^2)*exp(5-log(x)-x)+(-128*x^3+1536*x^2)*log(x)-128*x^3+512*x^2)/(exp(5
-log(x)-x)^3+(-12*x+48)*exp(5-log(x)-x)^2+(48*x^2-384*x+768)*exp(5-log(x)-x)-64*x^3+768*x^2-3072*x+4096),x, al
gorithm="fricas")

[Out]

32*x^3*log(x)/(16*x^2 - 8*(x - 4)*e^(-x - log(x) + 5) - 128*x + e^(-2*x - 2*log(x) + 10) + 256)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 42 vs. \(2 (20) = 40\).

Time = 0.12 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.50 \[ \int \frac {512 x^2-128 x^3+\left (1536 x^2-128 x^3\right ) \log (x)+\frac {e^{5-x} \left (32 x^2+\left (160 x^2+64 x^3\right ) \log (x)\right )}{x}}{4096+\frac {e^{15-3 x}}{x^3}+\frac {e^{10-2 x} (48-12 x)}{x^2}-3072 x+768 x^2-64 x^3+\frac {e^{5-x} \left (768-384 x+48 x^2\right )}{x}} \, dx=\frac {32 x^{5} \log {\left (x \right )}}{16 x^{4} - 128 x^{3} + 256 x^{2} + \left (- 8 x^{2} + 32 x\right ) e^{5 - x} + e^{10 - 2 x}} \]

[In]

integrate((((64*x**3+160*x**2)*ln(x)+32*x**2)*exp(5-ln(x)-x)+(-128*x**3+1536*x**2)*ln(x)-128*x**3+512*x**2)/(e
xp(5-ln(x)-x)**3+(-12*x+48)*exp(5-ln(x)-x)**2+(48*x**2-384*x+768)*exp(5-ln(x)-x)-64*x**3+768*x**2-3072*x+4096)
,x)

[Out]

32*x**5*log(x)/(16*x**4 - 128*x**3 + 256*x**2 + (-8*x**2 + 32*x)*exp(5 - x) + exp(10 - 2*x))

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.86 \[ \int \frac {512 x^2-128 x^3+\left (1536 x^2-128 x^3\right ) \log (x)+\frac {e^{5-x} \left (32 x^2+\left (160 x^2+64 x^3\right ) \log (x)\right )}{x}}{4096+\frac {e^{15-3 x}}{x^3}+\frac {e^{10-2 x} (48-12 x)}{x^2}-3072 x+768 x^2-64 x^3+\frac {e^{5-x} \left (768-384 x+48 x^2\right )}{x}} \, dx=\frac {32 \, x^{5} e^{\left (2 \, x\right )} \log \left (x\right )}{16 \, {\left (x^{4} - 8 \, x^{3} + 16 \, x^{2}\right )} e^{\left (2 \, x\right )} - 8 \, {\left (x^{2} e^{5} - 4 \, x e^{5}\right )} e^{x} + e^{10}} \]

[In]

integrate((((64*x^3+160*x^2)*log(x)+32*x^2)*exp(5-log(x)-x)+(-128*x^3+1536*x^2)*log(x)-128*x^3+512*x^2)/(exp(5
-log(x)-x)^3+(-12*x+48)*exp(5-log(x)-x)^2+(48*x^2-384*x+768)*exp(5-log(x)-x)-64*x^3+768*x^2-3072*x+4096),x, al
gorithm="maxima")

[Out]

32*x^5*e^(2*x)*log(x)/(16*(x^4 - 8*x^3 + 16*x^2)*e^(2*x) - 8*(x^2*e^5 - 4*x*e^5)*e^x + e^10)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.82 \[ \int \frac {512 x^2-128 x^3+\left (1536 x^2-128 x^3\right ) \log (x)+\frac {e^{5-x} \left (32 x^2+\left (160 x^2+64 x^3\right ) \log (x)\right )}{x}}{4096+\frac {e^{15-3 x}}{x^3}+\frac {e^{10-2 x} (48-12 x)}{x^2}-3072 x+768 x^2-64 x^3+\frac {e^{5-x} \left (768-384 x+48 x^2\right )}{x}} \, dx=\frac {32 \, x^{5} \log \left (x\right )}{16 \, x^{4} - 128 \, x^{3} - 8 \, x^{2} e^{\left (-x + 5\right )} + 256 \, x^{2} + 32 \, x e^{\left (-x + 5\right )} + e^{\left (-2 \, x + 10\right )}} \]

[In]

integrate((((64*x^3+160*x^2)*log(x)+32*x^2)*exp(5-log(x)-x)+(-128*x^3+1536*x^2)*log(x)-128*x^3+512*x^2)/(exp(5
-log(x)-x)^3+(-12*x+48)*exp(5-log(x)-x)^2+(48*x^2-384*x+768)*exp(5-log(x)-x)-64*x^3+768*x^2-3072*x+4096),x, al
gorithm="giac")

[Out]

32*x^5*log(x)/(16*x^4 - 128*x^3 - 8*x^2*e^(-x + 5) + 256*x^2 + 32*x*e^(-x + 5) + e^(-2*x + 10))

Mupad [F(-1)]

Timed out. \[ \int \frac {512 x^2-128 x^3+\left (1536 x^2-128 x^3\right ) \log (x)+\frac {e^{5-x} \left (32 x^2+\left (160 x^2+64 x^3\right ) \log (x)\right )}{x}}{4096+\frac {e^{15-3 x}}{x^3}+\frac {e^{10-2 x} (48-12 x)}{x^2}-3072 x+768 x^2-64 x^3+\frac {e^{5-x} \left (768-384 x+48 x^2\right )}{x}} \, dx=\int \frac {\ln \left (x\right )\,\left (1536\,x^2-128\,x^3\right )+512\,x^2-128\,x^3+{\mathrm {e}}^{5-\ln \left (x\right )-x}\,\left (\ln \left (x\right )\,\left (64\,x^3+160\,x^2\right )+32\,x^2\right )}{{\mathrm {e}}^{15-3\,\ln \left (x\right )-3\,x}-3072\,x-{\mathrm {e}}^{10-2\,\ln \left (x\right )-2\,x}\,\left (12\,x-48\right )+{\mathrm {e}}^{5-\ln \left (x\right )-x}\,\left (48\,x^2-384\,x+768\right )+768\,x^2-64\,x^3+4096} \,d x \]

[In]

int((log(x)*(1536*x^2 - 128*x^3) + 512*x^2 - 128*x^3 + exp(5 - log(x) - x)*(log(x)*(160*x^2 + 64*x^3) + 32*x^2
))/(exp(15 - 3*log(x) - 3*x) - 3072*x - exp(10 - 2*log(x) - 2*x)*(12*x - 48) + exp(5 - log(x) - x)*(48*x^2 - 3
84*x + 768) + 768*x^2 - 64*x^3 + 4096),x)

[Out]

int((log(x)*(1536*x^2 - 128*x^3) + 512*x^2 - 128*x^3 + exp(5 - log(x) - x)*(log(x)*(160*x^2 + 64*x^3) + 32*x^2
))/(exp(15 - 3*log(x) - 3*x) - 3072*x - exp(10 - 2*log(x) - 2*x)*(12*x - 48) + exp(5 - log(x) - x)*(48*x^2 - 3
84*x + 768) + 768*x^2 - 64*x^3 + 4096), x)