\(\int \frac {(-384 x-768 x^2-384 x^3) \log (x)+(-384 x-1664 x^2-2304 x^3-1536 x^4-512 x^5) \log ^2(x)+(512 x^2+1536 x^3+1536 x^4+512 x^5) \log (x) \log (2 x+e^5 x)}{-27+(108 x+108 x^2) \log (2 x+e^5 x)+(-144 x^2-288 x^3-144 x^4) \log ^2(2 x+e^5 x)+(64 x^3+192 x^4+192 x^5+64 x^6) \log ^3(2 x+e^5 x)} \, dx\) [2052]

Optimal result

Integrand size = 169, antiderivative size = 31 \[ \int \frac {\left (-384 x-768 x^2-384 x^3\right ) \log (x)+\left (-384 x-1664 x^2-2304 x^3-1536 x^4-512 x^5\right ) \log ^2(x)+\left (512 x^2+1536 x^3+1536 x^4+512 x^5\right ) \log (x) \log \left (2 x+e^5 x\right )}{-27+\left (108 x+108 x^2\right ) \log \left (2 x+e^5 x\right )+\left (-144 x^2-288 x^3-144 x^4\right ) \log ^2\left (2 x+e^5 x\right )+\left (64 x^3+192 x^4+192 x^5+64 x^6\right ) \log ^3\left (2 x+e^5 x\right )} \, dx=\frac {4 \log ^2(x)}{\left (\frac {3}{x (4+4 x)}-\log \left (\left (2+e^5\right ) x\right )\right )^2} \]

[Out]

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

Rubi [F]

\[ \int \frac {\left (-384 x-768 x^2-384 x^3\right ) \log (x)+\left (-384 x-1664 x^2-2304 x^3-1536 x^4-512 x^5\right ) \log ^2(x)+\left (512 x^2+1536 x^3+1536 x^4+512 x^5\right ) \log (x) \log \left (2 x+e^5 x\right )}{-27+\left (108 x+108 x^2\right ) \log \left (2 x+e^5 x\right )+\left (-144 x^2-288 x^3-144 x^4\right ) \log ^2\left (2 x+e^5 x\right )+\left (64 x^3+192 x^4+192 x^5+64 x^6\right ) \log ^3\left (2 x+e^5 x\right )} \, dx=\int \frac {\left (-384 x-768 x^2-384 x^3\right ) \log (x)+\left (-384 x-1664 x^2-2304 x^3-1536 x^4-512 x^5\right ) \log ^2(x)+\left (512 x^2+1536 x^3+1536 x^4+512 x^5\right ) \log (x) \log \left (2 x+e^5 x\right )}{-27+\left (108 x+108 x^2\right ) \log \left (2 x+e^5 x\right )+\left (-144 x^2-288 x^3-144 x^4\right ) \log ^2\left (2 x+e^5 x\right )+\left (64 x^3+192 x^4+192 x^5+64 x^6\right ) \log ^3\left (2 x+e^5 x\right )} \, dx \]

[In]

Int[((-384*x - 768*x^2 - 384*x^3)*Log[x] + (-384*x - 1664*x^2 - 2304*x^3 - 1536*x^4 - 512*x^5)*Log[x]^2 + (512
*x^2 + 1536*x^3 + 1536*x^4 + 512*x^5)*Log[x]*Log[2*x + E^5*x])/(-27 + (108*x + 108*x^2)*Log[2*x + E^5*x] + (-1
44*x^2 - 288*x^3 - 144*x^4)*Log[2*x + E^5*x]^2 + (64*x^3 + 192*x^4 + 192*x^5 + 64*x^6)*Log[2*x + E^5*x]^3),x]

[Out]

-384*Defer[Int][(x*Log[x])/(-3 + 4*x*(1 + x)*Log[(2 + E^5)*x])^3, x] - 384*Defer[Int][(x^2*Log[x])/(-3 + 4*x*(
1 + x)*Log[(2 + E^5)*x])^3, x] - 128*(3 - 4*Log[2 + E^5])*Defer[Int][(x^2*Log[x])/(-3 + 4*x*(1 + x)*Log[(2 + E
^5)*x])^3, x] - 128*(3 - 4*Log[2 + E^5])*Defer[Int][(x^3*Log[x])/(-3 + 4*x*(1 + x)*Log[(2 + E^5)*x])^3, x] + 1
024*Log[2 + E^5]*Defer[Int][(x^3*Log[x])/(-3 + 4*x*(1 + x)*Log[(2 + E^5)*x])^3, x] + 1536*Log[2 + E^5]*Defer[I
nt][(x^4*Log[x])/(-3 + 4*x*(1 + x)*Log[(2 + E^5)*x])^3, x] + 512*Log[2 + E^5]*Defer[Int][(x^5*Log[x])/(-3 + 4*
x*(1 + x)*Log[(2 + E^5)*x])^3, x] - 384*Defer[Int][(x*Log[x]^2)/(-3 + 4*x*(1 + x)*Log[(2 + E^5)*x])^3, x] - 11
52*Defer[Int][(x^2*Log[x]^2)/(-3 + 4*x*(1 + x)*Log[(2 + E^5)*x])^3, x] - 768*Defer[Int][(x^3*Log[x]^2)/(-3 + 4
*x*(1 + x)*Log[(2 + E^5)*x])^3, x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {128 x (1+x) \log (x) \left (-\left ((1+x) \left (-3+4 x \log \left (2+e^5\right )+4 x^2 \log \left (2+e^5\right )\right )\right )+3 (1+2 x) \log (x)\right )}{\left (3-4 x (1+x) \log \left (\left (2+e^5\right ) x\right )\right )^3} \, dx \\ & = 128 \int \frac {x (1+x) \log (x) \left (-\left ((1+x) \left (-3+4 x \log \left (2+e^5\right )+4 x^2 \log \left (2+e^5\right )\right )\right )+3 (1+2 x) \log (x)\right )}{\left (3-4 x (1+x) \log \left (\left (2+e^5\right ) x\right )\right )^3} \, dx \\ & = 128 \int \left (\frac {x \log (x) \left (3+3 x \left (1-\frac {4}{3} \log \left (2+e^5\right )\right )-8 x^2 \log \left (2+e^5\right )-4 x^3 \log \left (2+e^5\right )+3 \log (x)+6 x \log (x)\right )}{\left (3-4 x \log \left (\left (2+e^5\right ) x\right )-4 x^2 \log \left (\left (2+e^5\right ) x\right )\right )^3}+\frac {x^2 \log (x) \left (3+3 x \left (1-\frac {4}{3} \log \left (2+e^5\right )\right )-8 x^2 \log \left (2+e^5\right )-4 x^3 \log \left (2+e^5\right )+3 \log (x)+6 x \log (x)\right )}{\left (3-4 x \log \left (\left (2+e^5\right ) x\right )-4 x^2 \log \left (\left (2+e^5\right ) x\right )\right )^3}\right ) \, dx \\ & = 128 \int \frac {x \log (x) \left (3+3 x \left (1-\frac {4}{3} \log \left (2+e^5\right )\right )-8 x^2 \log \left (2+e^5\right )-4 x^3 \log \left (2+e^5\right )+3 \log (x)+6 x \log (x)\right )}{\left (3-4 x \log \left (\left (2+e^5\right ) x\right )-4 x^2 \log \left (\left (2+e^5\right ) x\right )\right )^3} \, dx+128 \int \frac {x^2 \log (x) \left (3+3 x \left (1-\frac {4}{3} \log \left (2+e^5\right )\right )-8 x^2 \log \left (2+e^5\right )-4 x^3 \log \left (2+e^5\right )+3 \log (x)+6 x \log (x)\right )}{\left (3-4 x \log \left (\left (2+e^5\right ) x\right )-4 x^2 \log \left (\left (2+e^5\right ) x\right )\right )^3} \, dx \\ & = 128 \int \frac {x \log (x) \left (-\left ((1+x) \left (-3+4 x \log \left (2+e^5\right )+4 x^2 \log \left (2+e^5\right )\right )\right )+3 (1+2 x) \log (x)\right )}{\left (3-4 x (1+x) \log \left (\left (2+e^5\right ) x\right )\right )^3} \, dx+128 \int \frac {x^2 \log (x) \left (-\left ((1+x) \left (-3+4 x \log \left (2+e^5\right )+4 x^2 \log \left (2+e^5\right )\right )\right )+3 (1+2 x) \log (x)\right )}{\left (3-4 x (1+x) \log \left (\left (2+e^5\right ) x\right )\right )^3} \, dx \\ & = 128 \int \left (-\frac {3 x \log (x)}{\left (-3+4 x \log \left (\left (2+e^5\right ) x\right )+4 x^2 \log \left (\left (2+e^5\right ) x\right )\right )^3}-\frac {3 x^2 \left (1-\frac {4}{3} \log \left (2+e^5\right )\right ) \log (x)}{\left (-3+4 x \log \left (\left (2+e^5\right ) x\right )+4 x^2 \log \left (\left (2+e^5\right ) x\right )\right )^3}+\frac {8 x^3 \log \left (2+e^5\right ) \log (x)}{\left (-3+4 x \log \left (\left (2+e^5\right ) x\right )+4 x^2 \log \left (\left (2+e^5\right ) x\right )\right )^3}+\frac {4 x^4 \log \left (2+e^5\right ) \log (x)}{\left (-3+4 x \log \left (\left (2+e^5\right ) x\right )+4 x^2 \log \left (\left (2+e^5\right ) x\right )\right )^3}-\frac {3 x \log ^2(x)}{\left (-3+4 x \log \left (\left (2+e^5\right ) x\right )+4 x^2 \log \left (\left (2+e^5\right ) x\right )\right )^3}-\frac {6 x^2 \log ^2(x)}{\left (-3+4 x \log \left (\left (2+e^5\right ) x\right )+4 x^2 \log \left (\left (2+e^5\right ) x\right )\right )^3}\right ) \, dx+128 \int \left (-\frac {3 x^2 \log (x)}{\left (-3+4 x \log \left (\left (2+e^5\right ) x\right )+4 x^2 \log \left (\left (2+e^5\right ) x\right )\right )^3}-\frac {3 x^3 \left (1-\frac {4}{3} \log \left (2+e^5\right )\right ) \log (x)}{\left (-3+4 x \log \left (\left (2+e^5\right ) x\right )+4 x^2 \log \left (\left (2+e^5\right ) x\right )\right )^3}+\frac {8 x^4 \log \left (2+e^5\right ) \log (x)}{\left (-3+4 x \log \left (\left (2+e^5\right ) x\right )+4 x^2 \log \left (\left (2+e^5\right ) x\right )\right )^3}+\frac {4 x^5 \log \left (2+e^5\right ) \log (x)}{\left (-3+4 x \log \left (\left (2+e^5\right ) x\right )+4 x^2 \log \left (\left (2+e^5\right ) x\right )\right )^3}-\frac {3 x^2 \log ^2(x)}{\left (-3+4 x \log \left (\left (2+e^5\right ) x\right )+4 x^2 \log \left (\left (2+e^5\right ) x\right )\right )^3}-\frac {6 x^3 \log ^2(x)}{\left (-3+4 x \log \left (\left (2+e^5\right ) x\right )+4 x^2 \log \left (\left (2+e^5\right ) x\right )\right )^3}\right ) \, dx \\ & = -\left (384 \int \frac {x \log (x)}{\left (-3+4 x \log \left (\left (2+e^5\right ) x\right )+4 x^2 \log \left (\left (2+e^5\right ) x\right )\right )^3} \, dx\right )-384 \int \frac {x^2 \log (x)}{\left (-3+4 x \log \left (\left (2+e^5\right ) x\right )+4 x^2 \log \left (\left (2+e^5\right ) x\right )\right )^3} \, dx-384 \int \frac {x \log ^2(x)}{\left (-3+4 x \log \left (\left (2+e^5\right ) x\right )+4 x^2 \log \left (\left (2+e^5\right ) x\right )\right )^3} \, dx-384 \int \frac {x^2 \log ^2(x)}{\left (-3+4 x \log \left (\left (2+e^5\right ) x\right )+4 x^2 \log \left (\left (2+e^5\right ) x\right )\right )^3} \, dx-768 \int \frac {x^2 \log ^2(x)}{\left (-3+4 x \log \left (\left (2+e^5\right ) x\right )+4 x^2 \log \left (\left (2+e^5\right ) x\right )\right )^3} \, dx-768 \int \frac {x^3 \log ^2(x)}{\left (-3+4 x \log \left (\left (2+e^5\right ) x\right )+4 x^2 \log \left (\left (2+e^5\right ) x\right )\right )^3} \, dx-\left (128 \left (3-4 \log \left (2+e^5\right )\right )\right ) \int \frac {x^2 \log (x)}{\left (-3+4 x \log \left (\left (2+e^5\right ) x\right )+4 x^2 \log \left (\left (2+e^5\right ) x\right )\right )^3} \, dx-\left (128 \left (3-4 \log \left (2+e^5\right )\right )\right ) \int \frac {x^3 \log (x)}{\left (-3+4 x \log \left (\left (2+e^5\right ) x\right )+4 x^2 \log \left (\left (2+e^5\right ) x\right )\right )^3} \, dx+\left (512 \log \left (2+e^5\right )\right ) \int \frac {x^4 \log (x)}{\left (-3+4 x \log \left (\left (2+e^5\right ) x\right )+4 x^2 \log \left (\left (2+e^5\right ) x\right )\right )^3} \, dx+\left (512 \log \left (2+e^5\right )\right ) \int \frac {x^5 \log (x)}{\left (-3+4 x \log \left (\left (2+e^5\right ) x\right )+4 x^2 \log \left (\left (2+e^5\right ) x\right )\right )^3} \, dx+\left (1024 \log \left (2+e^5\right )\right ) \int \frac {x^3 \log (x)}{\left (-3+4 x \log \left (\left (2+e^5\right ) x\right )+4 x^2 \log \left (\left (2+e^5\right ) x\right )\right )^3} \, dx+\left (1024 \log \left (2+e^5\right )\right ) \int \frac {x^4 \log (x)}{\left (-3+4 x \log \left (\left (2+e^5\right ) x\right )+4 x^2 \log \left (\left (2+e^5\right ) x\right )\right )^3} \, dx \\ & = -\left (384 \int \frac {x \log (x)}{\left (-3+4 x (1+x) \log \left (\left (2+e^5\right ) x\right )\right )^3} \, dx\right )-384 \int \frac {x^2 \log (x)}{\left (-3+4 x (1+x) \log \left (\left (2+e^5\right ) x\right )\right )^3} \, dx-384 \int \frac {x \log ^2(x)}{\left (-3+4 x (1+x) \log \left (\left (2+e^5\right ) x\right )\right )^3} \, dx-384 \int \frac {x^2 \log ^2(x)}{\left (-3+4 x (1+x) \log \left (\left (2+e^5\right ) x\right )\right )^3} \, dx-768 \int \frac {x^2 \log ^2(x)}{\left (-3+4 x (1+x) \log \left (\left (2+e^5\right ) x\right )\right )^3} \, dx-768 \int \frac {x^3 \log ^2(x)}{\left (-3+4 x (1+x) \log \left (\left (2+e^5\right ) x\right )\right )^3} \, dx-\left (128 \left (3-4 \log \left (2+e^5\right )\right )\right ) \int \frac {x^2 \log (x)}{\left (-3+4 x (1+x) \log \left (\left (2+e^5\right ) x\right )\right )^3} \, dx-\left (128 \left (3-4 \log \left (2+e^5\right )\right )\right ) \int \frac {x^3 \log (x)}{\left (-3+4 x (1+x) \log \left (\left (2+e^5\right ) x\right )\right )^3} \, dx+\left (512 \log \left (2+e^5\right )\right ) \int \frac {x^4 \log (x)}{\left (-3+4 x (1+x) \log \left (\left (2+e^5\right ) x\right )\right )^3} \, dx+\left (512 \log \left (2+e^5\right )\right ) \int \frac {x^5 \log (x)}{\left (-3+4 x (1+x) \log \left (\left (2+e^5\right ) x\right )\right )^3} \, dx+\left (1024 \log \left (2+e^5\right )\right ) \int \frac {x^3 \log (x)}{\left (-3+4 x (1+x) \log \left (\left (2+e^5\right ) x\right )\right )^3} \, dx+\left (1024 \log \left (2+e^5\right )\right ) \int \frac {x^4 \log (x)}{\left (-3+4 x (1+x) \log \left (\left (2+e^5\right ) x\right )\right )^3} \, dx \\ \end{align*}

Mathematica [F(-1)]

Timed out. \[ \int \frac {\left (-384 x-768 x^2-384 x^3\right ) \log (x)+\left (-384 x-1664 x^2-2304 x^3-1536 x^4-512 x^5\right ) \log ^2(x)+\left (512 x^2+1536 x^3+1536 x^4+512 x^5\right ) \log (x) \log \left (2 x+e^5 x\right )}{-27+\left (108 x+108 x^2\right ) \log \left (2 x+e^5 x\right )+\left (-144 x^2-288 x^3-144 x^4\right ) \log ^2\left (2 x+e^5 x\right )+\left (64 x^3+192 x^4+192 x^5+64 x^6\right ) \log ^3\left (2 x+e^5 x\right )} \, dx=\text {\$Aborted} \]

[In]

Integrate[((-384*x - 768*x^2 - 384*x^3)*Log[x] + (-384*x - 1664*x^2 - 2304*x^3 - 1536*x^4 - 512*x^5)*Log[x]^2
+ (512*x^2 + 1536*x^3 + 1536*x^4 + 512*x^5)*Log[x]*Log[2*x + E^5*x])/(-27 + (108*x + 108*x^2)*Log[2*x + E^5*x]
+ (-144*x^2 - 288*x^3 - 144*x^4)*Log[2*x + E^5*x]^2 + (64*x^3 + 192*x^4 + 192*x^5 + 64*x^6)*Log[2*x + E^5*x]^
3),x]

[Out]

$Aborted

Maple [A] (verified)

Time = 2.59 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.06

method	result	size
risch	\(\frac {96 x^{2} \ln \left (x \right )+96 x \ln \left (x \right )-36}{\left (4 x^{2} \ln \left (x \right )+4 x \ln \left (x \right )-3\right )^{2}}\)	\(33\)
parallelrisch	\(\frac {24576 x^{4} \ln \left (x \right )^{2}+49152 x^{3} \ln \left (x \right )^{2}+24576 x^{2} \ln \left (x \right )^{2}}{6144 {\ln \left (x \left ({\mathrm e}^{5}+2\right )\right )}^{2} x^{4}+12288 {\ln \left (x \left ({\mathrm e}^{5}+2\right )\right )}^{2} x^{3}+6144 {\ln \left (x \left ({\mathrm e}^{5}+2\right )\right )}^{2} x^{2}-9216 \ln \left (x \left ({\mathrm e}^{5}+2\right )\right ) x^{2}-9216 \ln \left (x \left ({\mathrm e}^{5}+2\right )\right ) x +3456}\)	\(99\)
default	\(-\frac {128 \left (1+x \right ) x^{2} \left (x^{2}+2 x +1\right ) \ln \left (x \right )}{\left (4 x^{3}+8 x^{2}+10 x +3\right ) \left (4 x^{2} \ln \left (x \right )+4 x \ln \left (x \right )-3\right )}+\frac {64 x^{2} \ln \left (x \right ) \left (1+x \right ) \left (12 x^{4} \ln \left (x \right )+36 x^{3} \ln \left (x \right )+42 x^{2} \ln \left (x \right )+21 x \ln \left (x \right )-6 x^{2}+3 \ln \left (x \right )-12 x -6\right )}{\left (4 x^{2} \ln \left (x \right )+4 x \ln \left (x \right )-3\right )^{2} \left (4 x^{3}+8 x^{2}+10 x +3\right )}\)	\(136\)
parts	\(-\frac {128 \left (1+x \right ) x^{2} \left (x^{2}+2 x +1\right ) \ln \left (x \right )}{\left (4 x^{3}+8 x^{2}+10 x +3\right ) \left (4 x^{2} \ln \left (x \right )+4 x \ln \left (x \right )-3\right )}+\frac {64 x^{2} \ln \left (x \right ) \left (1+x \right ) \left (12 x^{4} \ln \left (x \right )+36 x^{3} \ln \left (x \right )+42 x^{2} \ln \left (x \right )+21 x \ln \left (x \right )-6 x^{2}+3 \ln \left (x \right )-12 x -6\right )}{\left (4 x^{2} \ln \left (x \right )+4 x \ln \left (x \right )-3\right )^{2} \left (4 x^{3}+8 x^{2}+10 x +3\right )}\)	\(136\)

[In]

int(((512*x^5+1536*x^4+1536*x^3+512*x^2)*ln(x)*ln(x*exp(5)+2*x)+(-512*x^5-1536*x^4-2304*x^3-1664*x^2-384*x)*ln
(x)^2+(-384*x^3-768*x^2-384*x)*ln(x))/((64*x^6+192*x^5+192*x^4+64*x^3)*ln(x*exp(5)+2*x)^3+(-144*x^4-288*x^3-14
4*x^2)*ln(x*exp(5)+2*x)^2+(108*x^2+108*x)*ln(x*exp(5)+2*x)-27),x,method=_RETURNVERBOSE)

[Out]

12*(8*x^2*ln(x)+8*x*ln(x)-3)/(4*x^2*ln(x)+4*x*ln(x)-3)^2

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 150 vs. \(2 (28) = 56\).

Time = 0.26 (sec) , antiderivative size = 150, normalized size of antiderivative = 4.84 \[ \int \frac {\left (-384 x-768 x^2-384 x^3\right ) \log (x)+\left (-384 x-1664 x^2-2304 x^3-1536 x^4-512 x^5\right ) \log ^2(x)+\left (512 x^2+1536 x^3+1536 x^4+512 x^5\right ) \log (x) \log \left (2 x+e^5 x\right )}{-27+\left (108 x+108 x^2\right ) \log \left (2 x+e^5 x\right )+\left (-144 x^2-288 x^3-144 x^4\right ) \log ^2\left (2 x+e^5 x\right )+\left (64 x^3+192 x^4+192 x^5+64 x^6\right ) \log ^3\left (2 x+e^5 x\right )} \, dx=-\frac {4 \, {\left (16 \, {\left (x^{4} + 2 \, x^{3} + x^{2}\right )} \log \left (e^{5} + 2\right )^{2} - 24 \, {\left (x^{2} + x\right )} \log \left (x\right ) - 8 \, {\left (3 \, x^{2} - 4 \, {\left (x^{4} + 2 \, x^{3} + x^{2}\right )} \log \left (x\right ) + 3 \, x\right )} \log \left (e^{5} + 2\right ) + 9\right )}}{16 \, {\left (x^{4} + 2 \, x^{3} + x^{2}\right )} \log \left (x\right )^{2} + 16 \, {\left (x^{4} + 2 \, x^{3} + x^{2}\right )} \log \left (e^{5} + 2\right )^{2} - 24 \, {\left (x^{2} + x\right )} \log \left (x\right ) - 8 \, {\left (3 \, x^{2} - 4 \, {\left (x^{4} + 2 \, x^{3} + x^{2}\right )} \log \left (x\right ) + 3 \, x\right )} \log \left (e^{5} + 2\right ) + 9} \]

[In]

integrate(((512*x^5+1536*x^4+1536*x^3+512*x^2)*log(x)*log(x*exp(5)+2*x)+(-512*x^5-1536*x^4-2304*x^3-1664*x^2-3
84*x)*log(x)^2+(-384*x^3-768*x^2-384*x)*log(x))/((64*x^6+192*x^5+192*x^4+64*x^3)*log(x*exp(5)+2*x)^3+(-144*x^4
-288*x^3-144*x^2)*log(x*exp(5)+2*x)^2+(108*x^2+108*x)*log(x*exp(5)+2*x)-27),x, algorithm="fricas")

[Out]

-4*(16*(x^4 + 2*x^3 + x^2)*log(e^5 + 2)^2 - 24*(x^2 + x)*log(x) - 8*(3*x^2 - 4*(x^4 + 2*x^3 + x^2)*log(x) + 3*
x)*log(e^5 + 2) + 9)/(16*(x^4 + 2*x^3 + x^2)*log(x)^2 + 16*(x^4 + 2*x^3 + x^2)*log(e^5 + 2)^2 - 24*(x^2 + x)*l
og(x) - 8*(3*x^2 - 4*(x^4 + 2*x^3 + x^2)*log(x) + 3*x)*log(e^5 + 2) + 9)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 243 vs. \(2 (24) = 48\).

Time = 0.39 (sec) , antiderivative size = 243, normalized size of antiderivative = 7.84 \[ \int \frac {\left (-384 x-768 x^2-384 x^3\right ) \log (x)+\left (-384 x-1664 x^2-2304 x^3-1536 x^4-512 x^5\right ) \log ^2(x)+\left (512 x^2+1536 x^3+1536 x^4+512 x^5\right ) \log (x) \log \left (2 x+e^5 x\right )}{-27+\left (108 x+108 x^2\right ) \log \left (2 x+e^5 x\right )+\left (-144 x^2-288 x^3-144 x^4\right ) \log ^2\left (2 x+e^5 x\right )+\left (64 x^3+192 x^4+192 x^5+64 x^6\right ) \log ^3\left (2 x+e^5 x\right )} \, dx=\frac {- 64 x^{4} \log {\left (2 + e^{5} \right )}^{2} - 128 x^{3} \log {\left (2 + e^{5} \right )}^{2} - 64 x^{2} \log {\left (2 + e^{5} \right )}^{2} + 96 x^{2} \log {\left (2 + e^{5} \right )} + 96 x \log {\left (2 + e^{5} \right )} + \left (- 128 x^{4} \log {\left (2 + e^{5} \right )} - 256 x^{3} \log {\left (2 + e^{5} \right )} - 128 x^{2} \log {\left (2 + e^{5} \right )} + 96 x^{2} + 96 x\right ) \log {\left (x \right )} - 36}{16 x^{4} \log {\left (2 + e^{5} \right )}^{2} + 32 x^{3} \log {\left (2 + e^{5} \right )}^{2} - 24 x^{2} \log {\left (2 + e^{5} \right )} + 16 x^{2} \log {\left (2 + e^{5} \right )}^{2} - 24 x \log {\left (2 + e^{5} \right )} + \left (16 x^{4} + 32 x^{3} + 16 x^{2}\right ) \log {\left (x \right )}^{2} + \left (32 x^{4} \log {\left (2 + e^{5} \right )} + 64 x^{3} \log {\left (2 + e^{5} \right )} - 24 x^{2} + 32 x^{2} \log {\left (2 + e^{5} \right )} - 24 x\right ) \log {\left (x \right )} + 9} \]

[In]

integrate(((512*x**5+1536*x**4+1536*x**3+512*x**2)*ln(x)*ln(x*exp(5)+2*x)+(-512*x**5-1536*x**4-2304*x**3-1664*
x**2-384*x)*ln(x)**2+(-384*x**3-768*x**2-384*x)*ln(x))/((64*x**6+192*x**5+192*x**4+64*x**3)*ln(x*exp(5)+2*x)**
3+(-144*x**4-288*x**3-144*x**2)*ln(x*exp(5)+2*x)**2+(108*x**2+108*x)*ln(x*exp(5)+2*x)-27),x)

[Out]

(-64*x**4*log(2 + exp(5))**2 - 128*x**3*log(2 + exp(5))**2 - 64*x**2*log(2 + exp(5))**2 + 96*x**2*log(2 + exp(
5)) + 96*x*log(2 + exp(5)) + (-128*x**4*log(2 + exp(5)) - 256*x**3*log(2 + exp(5)) - 128*x**2*log(2 + exp(5))
+ 96*x**2 + 96*x)*log(x) - 36)/(16*x**4*log(2 + exp(5))**2 + 32*x**3*log(2 + exp(5))**2 - 24*x**2*log(2 + exp(
5)) + 16*x**2*log(2 + exp(5))**2 - 24*x*log(2 + exp(5)) + (16*x**4 + 32*x**3 + 16*x**2)*log(x)**2 + (32*x**4*l
og(2 + exp(5)) + 64*x**3*log(2 + exp(5)) - 24*x**2 + 32*x**2*log(2 + exp(5)) - 24*x)*log(x) + 9)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 216 vs. \(2 (28) = 56\).

Time = 0.34 (sec) , antiderivative size = 216, normalized size of antiderivative = 6.97 \[ \int \frac {\left (-384 x-768 x^2-384 x^3\right ) \log (x)+\left (-384 x-1664 x^2-2304 x^3-1536 x^4-512 x^5\right ) \log ^2(x)+\left (512 x^2+1536 x^3+1536 x^4+512 x^5\right ) \log (x) \log \left (2 x+e^5 x\right )}{-27+\left (108 x+108 x^2\right ) \log \left (2 x+e^5 x\right )+\left (-144 x^2-288 x^3-144 x^4\right ) \log ^2\left (2 x+e^5 x\right )+\left (64 x^3+192 x^4+192 x^5+64 x^6\right ) \log ^3\left (2 x+e^5 x\right )} \, dx=-\frac {4 \, {\left (16 \, x^{4} \log \left (e^{5} + 2\right )^{2} + 32 \, x^{3} \log \left (e^{5} + 2\right )^{2} + 8 \, {\left (2 \, \log \left (e^{5} + 2\right )^{2} - 3 \, \log \left (e^{5} + 2\right )\right )} x^{2} + 8 \, {\left (4 \, x^{4} \log \left (e^{5} + 2\right ) + 8 \, x^{3} \log \left (e^{5} + 2\right ) + x^{2} {\left (4 \, \log \left (e^{5} + 2\right ) - 3\right )} - 3 \, x\right )} \log \left (x\right ) - 24 \, x \log \left (e^{5} + 2\right ) + 9\right )}}{16 \, x^{4} \log \left (e^{5} + 2\right )^{2} + 32 \, x^{3} \log \left (e^{5} + 2\right )^{2} + 8 \, {\left (2 \, \log \left (e^{5} + 2\right )^{2} - 3 \, \log \left (e^{5} + 2\right )\right )} x^{2} + 16 \, {\left (x^{4} + 2 \, x^{3} + x^{2}\right )} \log \left (x\right )^{2} + 8 \, {\left (4 \, x^{4} \log \left (e^{5} + 2\right ) + 8 \, x^{3} \log \left (e^{5} + 2\right ) + x^{2} {\left (4 \, \log \left (e^{5} + 2\right ) - 3\right )} - 3 \, x\right )} \log \left (x\right ) - 24 \, x \log \left (e^{5} + 2\right ) + 9} \]

[In]

integrate(((512*x^5+1536*x^4+1536*x^3+512*x^2)*log(x)*log(x*exp(5)+2*x)+(-512*x^5-1536*x^4-2304*x^3-1664*x^2-3
84*x)*log(x)^2+(-384*x^3-768*x^2-384*x)*log(x))/((64*x^6+192*x^5+192*x^4+64*x^3)*log(x*exp(5)+2*x)^3+(-144*x^4
-288*x^3-144*x^2)*log(x*exp(5)+2*x)^2+(108*x^2+108*x)*log(x*exp(5)+2*x)-27),x, algorithm="maxima")

[Out]

-4*(16*x^4*log(e^5 + 2)^2 + 32*x^3*log(e^5 + 2)^2 + 8*(2*log(e^5 + 2)^2 - 3*log(e^5 + 2))*x^2 + 8*(4*x^4*log(e
^5 + 2) + 8*x^3*log(e^5 + 2) + x^2*(4*log(e^5 + 2) - 3) - 3*x)*log(x) - 24*x*log(e^5 + 2) + 9)/(16*x^4*log(e^5
+ 2)^2 + 32*x^3*log(e^5 + 2)^2 + 8*(2*log(e^5 + 2)^2 - 3*log(e^5 + 2))*x^2 + 16*(x^4 + 2*x^3 + x^2)*log(x)^2
+ 8*(4*x^4*log(e^5 + 2) + 8*x^3*log(e^5 + 2) + x^2*(4*log(e^5 + 2) - 3) - 3*x)*log(x) - 24*x*log(e^5 + 2) + 9)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 239 vs. \(2 (28) = 56\).

Time = 0.30 (sec) , antiderivative size = 239, normalized size of antiderivative = 7.71 \[ \int \frac {\left (-384 x-768 x^2-384 x^3\right ) \log (x)+\left (-384 x-1664 x^2-2304 x^3-1536 x^4-512 x^5\right ) \log ^2(x)+\left (512 x^2+1536 x^3+1536 x^4+512 x^5\right ) \log (x) \log \left (2 x+e^5 x\right )}{-27+\left (108 x+108 x^2\right ) \log \left (2 x+e^5 x\right )+\left (-144 x^2-288 x^3-144 x^4\right ) \log ^2\left (2 x+e^5 x\right )+\left (64 x^3+192 x^4+192 x^5+64 x^6\right ) \log ^3\left (2 x+e^5 x\right )} \, dx=-\frac {4 \, {\left (32 \, x^{4} \log \left (x\right ) \log \left (e^{5} + 2\right ) + 16 \, x^{4} \log \left (e^{5} + 2\right )^{2} + 64 \, x^{3} \log \left (x\right ) \log \left (e^{5} + 2\right ) + 32 \, x^{3} \log \left (e^{5} + 2\right )^{2} + 32 \, x^{2} \log \left (x\right ) \log \left (e^{5} + 2\right ) + 16 \, x^{2} \log \left (e^{5} + 2\right )^{2} - 24 \, x^{2} \log \left (x\right ) - 24 \, x^{2} \log \left (e^{5} + 2\right ) - 24 \, x \log \left (x\right ) - 24 \, x \log \left (e^{5} + 2\right ) + 9\right )}}{16 \, x^{4} \log \left (x\right )^{2} + 32 \, x^{4} \log \left (x\right ) \log \left (e^{5} + 2\right ) + 16 \, x^{4} \log \left (e^{5} + 2\right )^{2} + 32 \, x^{3} \log \left (x\right )^{2} + 64 \, x^{3} \log \left (x\right ) \log \left (e^{5} + 2\right ) + 32 \, x^{3} \log \left (e^{5} + 2\right )^{2} + 16 \, x^{2} \log \left (x\right )^{2} + 32 \, x^{2} \log \left (x\right ) \log \left (e^{5} + 2\right ) + 16 \, x^{2} \log \left (e^{5} + 2\right )^{2} - 24 \, x^{2} \log \left (x\right ) - 24 \, x^{2} \log \left (e^{5} + 2\right ) - 24 \, x \log \left (x\right ) - 24 \, x \log \left (e^{5} + 2\right ) + 9} \]

[In]

integrate(((512*x^5+1536*x^4+1536*x^3+512*x^2)*log(x)*log(x*exp(5)+2*x)+(-512*x^5-1536*x^4-2304*x^3-1664*x^2-3
84*x)*log(x)^2+(-384*x^3-768*x^2-384*x)*log(x))/((64*x^6+192*x^5+192*x^4+64*x^3)*log(x*exp(5)+2*x)^3+(-144*x^4
-288*x^3-144*x^2)*log(x*exp(5)+2*x)^2+(108*x^2+108*x)*log(x*exp(5)+2*x)-27),x, algorithm="giac")

[Out]

-4*(32*x^4*log(x)*log(e^5 + 2) + 16*x^4*log(e^5 + 2)^2 + 64*x^3*log(x)*log(e^5 + 2) + 32*x^3*log(e^5 + 2)^2 +
32*x^2*log(x)*log(e^5 + 2) + 16*x^2*log(e^5 + 2)^2 - 24*x^2*log(x) - 24*x^2*log(e^5 + 2) - 24*x*log(x) - 24*x*
log(e^5 + 2) + 9)/(16*x^4*log(x)^2 + 32*x^4*log(x)*log(e^5 + 2) + 16*x^4*log(e^5 + 2)^2 + 32*x^3*log(x)^2 + 64
*x^3*log(x)*log(e^5 + 2) + 32*x^3*log(e^5 + 2)^2 + 16*x^2*log(x)^2 + 32*x^2*log(x)*log(e^5 + 2) + 16*x^2*log(e
^5 + 2)^2 - 24*x^2*log(x) - 24*x^2*log(e^5 + 2) - 24*x*log(x) - 24*x*log(e^5 + 2) + 9)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (-384 x-768 x^2-384 x^3\right ) \log (x)+\left (-384 x-1664 x^2-2304 x^3-1536 x^4-512 x^5\right ) \log ^2(x)+\left (512 x^2+1536 x^3+1536 x^4+512 x^5\right ) \log (x) \log \left (2 x+e^5 x\right )}{-27+\left (108 x+108 x^2\right ) \log \left (2 x+e^5 x\right )+\left (-144 x^2-288 x^3-144 x^4\right ) \log ^2\left (2 x+e^5 x\right )+\left (64 x^3+192 x^4+192 x^5+64 x^6\right ) \log ^3\left (2 x+e^5 x\right )} \, dx=\int \frac {{\ln \left (x\right )}^2\,\left (512\,x^5+1536\,x^4+2304\,x^3+1664\,x^2+384\,x\right )+\ln \left (x\right )\,\left (384\,x^3+768\,x^2+384\,x\right )-\ln \left (2\,x+x\,{\mathrm {e}}^5\right )\,\ln \left (x\right )\,\left (512\,x^5+1536\,x^4+1536\,x^3+512\,x^2\right )}{\left (-64\,x^6-192\,x^5-192\,x^4-64\,x^3\right )\,{\ln \left (2\,x+x\,{\mathrm {e}}^5\right )}^3+\left (144\,x^4+288\,x^3+144\,x^2\right )\,{\ln \left (2\,x+x\,{\mathrm {e}}^5\right )}^2+\left (-108\,x^2-108\,x\right )\,\ln \left (2\,x+x\,{\mathrm {e}}^5\right )+27} \,d x \]

[In]

int((log(x)^2*(384*x + 1664*x^2 + 2304*x^3 + 1536*x^4 + 512*x^5) + log(x)*(384*x + 768*x^2 + 384*x^3) - log(2*
x + x*exp(5))*log(x)*(512*x^2 + 1536*x^3 + 1536*x^4 + 512*x^5))/(log(2*x + x*exp(5))^2*(144*x^2 + 288*x^3 + 14
4*x^4) - log(2*x + x*exp(5))*(108*x + 108*x^2) - log(2*x + x*exp(5))^3*(64*x^3 + 192*x^4 + 192*x^5 + 64*x^6) +
27),x)

[Out]

int((log(x)^2*(384*x + 1664*x^2 + 2304*x^3 + 1536*x^4 + 512*x^5) + log(x)*(384*x + 768*x^2 + 384*x^3) - log(2*
x + x*exp(5))*log(x)*(512*x^2 + 1536*x^3 + 1536*x^4 + 512*x^5))/(log(2*x + x*exp(5))^2*(144*x^2 + 288*x^3 + 14
4*x^4) - log(2*x + x*exp(5))*(108*x + 108*x^2) - log(2*x + x*exp(5))^3*(64*x^3 + 192*x^4 + 192*x^5 + 64*x^6) +
27), x)