3.45.80 \(\int \frac {(-x^3-x^4) \log (x)+(5 x^2+6 x^3+x^4+(5 x^2+6 x^3+x^4) \log (x)) \log (5+x)+(-500-600 x-100 x^2) \log ^2(5+x)+((5 x^2+11 x^3+2 x^4) \log (x) \log (5+x)+(500+1225 x+475 x^2+50 x^3) \log ^2(5+x)) \log (\frac {x^2 \log (x)+(100+25 x) \log (5+x)}{25 x \log (5+x)})}{(5 x^2+x^3) \log (x) \log (5+x)+(500+225 x+25 x^2) \log ^2(5+x)} \, dx\) [4480]

Optimal. Leaf size=28 \[ x (1+x) \log \left (\frac {4+x+\frac {x^2 \log (x)}{25 \log (5+x)}}{x}\right ) \]

[Out]

ln((4+1/25*x^2/ln(5+x)*ln(x)+x)/x)*(1+x)*x

________________________________________________________________________________________

Rubi [F]
time = 7.61, 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 {\left (-x^3-x^4\right ) \log (x)+\left (5 x^2+6 x^3+x^4+\left (5 x^2+6 x^3+x^4\right ) \log (x)\right ) \log (5+x)+\left (-500-600 x-100 x^2\right ) \log ^2(5+x)+\left (\left (5 x^2+11 x^3+2 x^4\right ) \log (x) \log (5+x)+\left (500+1225 x+475 x^2+50 x^3\right ) \log ^2(5+x)\right ) \log \left (\frac {x^2 \log (x)+(100+25 x) \log (5+x)}{25 x \log (5+x)}\right )}{\left (5 x^2+x^3\right ) \log (x) \log (5+x)+\left (500+225 x+25 x^2\right ) \log ^2(5+x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((-x^3 - x^4)*Log[x] + (5*x^2 + 6*x^3 + x^4 + (5*x^2 + 6*x^3 + x^4)*Log[x])*Log[5 + x] + (-500 - 600*x - 1
00*x^2)*Log[5 + x]^2 + ((5*x^2 + 11*x^3 + 2*x^4)*Log[x]*Log[5 + x] + (500 + 1225*x + 475*x^2 + 50*x^3)*Log[5 +
x]^2)*Log[(x^2*Log[x] + (100 + 25*x)*Log[5 + x])/(25*x*Log[5 + x])])/((5*x^2 + x^3)*Log[x]*Log[5 + x] + (500
+ 225*x + 25*x^2)*Log[5 + x]^2),x]

[Out]

-4*x - ExpIntegralEi[2*Log[5 + x]] + 12*Log[4 + x] - 20*Log[Log[5 + x]] + 9*LogIntegral[5 + x] + 100*Defer[Int
][(x^2*Log[x] + 100*Log[5 + x] + 25*x*Log[5 + x])^(-1), x] + 26*Defer[Int][x^2/(x^2*Log[x] + 100*Log[5 + x] +
25*x*Log[5 + x]), x] + Defer[Int][x^3/(x^2*Log[x] + 100*Log[5 + x] + 25*x*Log[5 + x]), x] - 500*Defer[Int][1/(
(5 + x)*(x^2*Log[x] + 100*Log[5 + x] + 25*x*Log[5 + x])), x] + 48*Defer[Int][Log[x]/(x^2*Log[x] + 100*Log[5 +
x] + 25*x*Log[5 + x]), x] - 12*Defer[Int][(x*Log[x])/(x^2*Log[x] + 100*Log[5 + x] + 25*x*Log[5 + x]), x] + 5*D
efer[Int][(x^2*Log[x])/(x^2*Log[x] + 100*Log[5 + x] + 25*x*Log[5 + x]), x] + Defer[Int][(x^3*Log[x])/(x^2*Log[
x] + 100*Log[5 + x] + 25*x*Log[5 + x]), x] - 192*Defer[Int][Log[x]/((4 + x)*(x^2*Log[x] + 100*Log[5 + x] + 25*
x*Log[5 + x])), x] + Defer[Int][Log[1 + 4/x + (x*Log[x])/(25*Log[5 + x])], x] + 2*Defer[Int][x*Log[1 + 4/x + (
x*Log[x])/(25*Log[5 + x])], x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-x^3 (1+x) \log (x)+x^2 \left (5+6 x+x^2\right ) (1+\log (x)) \log (5+x)-100 \left (5+6 x+x^2\right ) \log ^2(5+x)+\left (5+11 x+2 x^2\right ) \log (5+x) \left (x^2 \log (x)+25 (4+x) \log (5+x)\right ) \log \left (1+\frac {4}{x}+\frac {x \log (x)}{25 \log (5+x)}\right )}{(5+x) \log (5+x) \left (x^2 \log (x)+25 (4+x) \log (5+x)\right )} \, dx\\ &=\int \left (\frac {(1+x) \left (-x^3 \log (x)+5 x^2 \log (5+x)+x^3 \log (5+x)+5 x^2 \log (x) \log (5+x)+x^3 \log (x) \log (5+x)-500 \log ^2(5+x)-100 x \log ^2(5+x)\right )}{(5+x) \log (5+x) \left (x^2 \log (x)+100 \log (5+x)+25 x \log (5+x)\right )}+(1+2 x) \log \left (1+\frac {4}{x}+\frac {x \log (x)}{25 \log (5+x)}\right )\right ) \, dx\\ &=\int \frac {(1+x) \left (-x^3 \log (x)+5 x^2 \log (5+x)+x^3 \log (5+x)+5 x^2 \log (x) \log (5+x)+x^3 \log (x) \log (5+x)-500 \log ^2(5+x)-100 x \log ^2(5+x)\right )}{(5+x) \log (5+x) \left (x^2 \log (x)+100 \log (5+x)+25 x \log (5+x)\right )} \, dx+\int (1+2 x) \log \left (1+\frac {4}{x}+\frac {x \log (x)}{25 \log (5+x)}\right ) \, dx\\ &=\int \frac {(1+x) \left ((5+x) \left (x^2-100 \log (5+x)\right ) \log (5+x)+x^2 \log (x) (-x+(5+x) \log (5+x))\right )}{(5+x) \log (5+x) \left (x^2 \log (x)+25 (4+x) \log (5+x)\right )} \, dx+\int \left (\log \left (1+\frac {4}{x}+\frac {x \log (x)}{25 \log (5+x)}\right )+2 x \log \left (1+\frac {4}{x}+\frac {x \log (x)}{25 \log (5+x)}\right )\right ) \, dx\\ &=2 \int x \log \left (1+\frac {4}{x}+\frac {x \log (x)}{25 \log (5+x)}\right ) \, dx+\int \left (-\frac {4 (1+x)}{4+x}-\frac {x (1+x)}{(5+x) \log (5+x)}+\frac {25 x (1+x) (4+x)}{(5+x) \left (x^2 \log (x)+100 \log (5+x)+25 x \log (5+x)\right )}+\frac {x^2 (1+x) (4+x+8 \log (x)+x \log (x))}{(4+x) \left (x^2 \log (x)+100 \log (5+x)+25 x \log (5+x)\right )}\right ) \, dx+\int \log \left (1+\frac {4}{x}+\frac {x \log (x)}{25 \log (5+x)}\right ) \, dx\\ &=2 \int x \log \left (1+\frac {4}{x}+\frac {x \log (x)}{25 \log (5+x)}\right ) \, dx-4 \int \frac {1+x}{4+x} \, dx+25 \int \frac {x (1+x) (4+x)}{(5+x) \left (x^2 \log (x)+100 \log (5+x)+25 x \log (5+x)\right )} \, dx-\int \frac {x (1+x)}{(5+x) \log (5+x)} \, dx+\int \frac {x^2 (1+x) (4+x+8 \log (x)+x \log (x))}{(4+x) \left (x^2 \log (x)+100 \log (5+x)+25 x \log (5+x)\right )} \, dx+\int \log \left (1+\frac {4}{x}+\frac {x \log (x)}{25 \log (5+x)}\right ) \, dx\\ &=2 \int x \log \left (1+\frac {4}{x}+\frac {x \log (x)}{25 \log (5+x)}\right ) \, dx-4 \int \left (1-\frac {3}{4+x}\right ) \, dx+25 \int \left (\frac {4}{x^2 \log (x)+100 \log (5+x)+25 x \log (5+x)}+\frac {x^2}{x^2 \log (x)+100 \log (5+x)+25 x \log (5+x)}-\frac {20}{(5+x) \left (x^2 \log (x)+100 \log (5+x)+25 x \log (5+x)\right )}\right ) \, dx-\int \left (-\frac {4}{\log (5+x)}+\frac {x}{\log (5+x)}+\frac {20}{(5+x) \log (5+x)}\right ) \, dx+\int \left (\frac {12 (4+x+8 \log (x)+x \log (x))}{x^2 \log (x)+100 \log (5+x)+25 x \log (5+x)}-\frac {3 x (4+x+8 \log (x)+x \log (x))}{x^2 \log (x)+100 \log (5+x)+25 x \log (5+x)}+\frac {x^2 (4+x+8 \log (x)+x \log (x))}{x^2 \log (x)+100 \log (5+x)+25 x \log (5+x)}-\frac {48 (4+x+8 \log (x)+x \log (x))}{(4+x) \left (x^2 \log (x)+100 \log (5+x)+25 x \log (5+x)\right )}\right ) \, dx+\int \log \left (1+\frac {4}{x}+\frac {x \log (x)}{25 \log (5+x)}\right ) \, dx\\ &=-4 x+12 \log (4+x)+2 \int x \log \left (1+\frac {4}{x}+\frac {x \log (x)}{25 \log (5+x)}\right ) \, dx-3 \int \frac {x (4+x+8 \log (x)+x \log (x))}{x^2 \log (x)+100 \log (5+x)+25 x \log (5+x)} \, dx+4 \int \frac {1}{\log (5+x)} \, dx+12 \int \frac {4+x+8 \log (x)+x \log (x)}{x^2 \log (x)+100 \log (5+x)+25 x \log (5+x)} \, dx-20 \int \frac {1}{(5+x) \log (5+x)} \, dx+25 \int \frac {x^2}{x^2 \log (x)+100 \log (5+x)+25 x \log (5+x)} \, dx-48 \int \frac {4+x+8 \log (x)+x \log (x)}{(4+x) \left (x^2 \log (x)+100 \log (5+x)+25 x \log (5+x)\right )} \, dx+100 \int \frac {1}{x^2 \log (x)+100 \log (5+x)+25 x \log (5+x)} \, dx-500 \int \frac {1}{(5+x) \left (x^2 \log (x)+100 \log (5+x)+25 x \log (5+x)\right )} \, dx-\int \frac {x}{\log (5+x)} \, dx+\int \frac {x^2 (4+x+8 \log (x)+x \log (x))}{x^2 \log (x)+100 \log (5+x)+25 x \log (5+x)} \, dx+\int \log \left (1+\frac {4}{x}+\frac {x \log (x)}{25 \log (5+x)}\right ) \, dx\\ &=-4 x+12 \log (4+x)+2 \int x \log \left (1+\frac {4}{x}+\frac {x \log (x)}{25 \log (5+x)}\right ) \, dx-3 \int \left (\frac {4 x}{x^2 \log (x)+100 \log (5+x)+25 x \log (5+x)}+\frac {x^2}{x^2 \log (x)+100 \log (5+x)+25 x \log (5+x)}+\frac {8 x \log (x)}{x^2 \log (x)+100 \log (5+x)+25 x \log (5+x)}+\frac {x^2 \log (x)}{x^2 \log (x)+100 \log (5+x)+25 x \log (5+x)}\right ) \, dx+4 \text {Subst}\left (\int \frac {1}{\log (x)} \, dx,x,5+x\right )+12 \int \left (\frac {4}{x^2 \log (x)+100 \log (5+x)+25 x \log (5+x)}+\frac {x}{x^2 \log (x)+100 \log (5+x)+25 x \log (5+x)}+\frac {8 \log (x)}{x^2 \log (x)+100 \log (5+x)+25 x \log (5+x)}+\frac {x \log (x)}{x^2 \log (x)+100 \log (5+x)+25 x \log (5+x)}\right ) \, dx-20 \text {Subst}\left (\int \frac {1}{x \log (x)} \, dx,x,5+x\right )+25 \int \frac {x^2}{x^2 \log (x)+100 \log (5+x)+25 x \log (5+x)} \, dx-48 \int \left (\frac {4}{(4+x) \left (x^2 \log (x)+100 \log (5+x)+25 x \log (5+x)\right )}+\frac {x}{(4+x) \left (x^2 \log (x)+100 \log (5+x)+25 x \log (5+x)\right )}+\frac {8 \log (x)}{(4+x) \left (x^2 \log (x)+100 \log (5+x)+25 x \log (5+x)\right )}+\frac {x \log (x)}{(4+x) \left (x^2 \log (x)+100 \log (5+x)+25 x \log (5+x)\right )}\right ) \, dx+100 \int \frac {1}{x^2 \log (x)+100 \log (5+x)+25 x \log (5+x)} \, dx-500 \int \frac {1}{(5+x) \left (x^2 \log (x)+100 \log (5+x)+25 x \log (5+x)\right )} \, dx-\int \left (-\frac {5}{\log (5+x)}+\frac {5+x}{\log (5+x)}\right ) \, dx+\int \left (\frac {4 x^2}{x^2 \log (x)+100 \log (5+x)+25 x \log (5+x)}+\frac {x^3}{x^2 \log (x)+100 \log (5+x)+25 x \log (5+x)}+\frac {8 x^2 \log (x)}{x^2 \log (x)+100 \log (5+x)+25 x \log (5+x)}+\frac {x^3 \log (x)}{x^2 \log (x)+100 \log (5+x)+25 x \log (5+x)}\right ) \, dx+\int \log \left (1+\frac {4}{x}+\frac {x \log (x)}{25 \log (5+x)}\right ) \, dx\\ &=-4 x+12 \log (4+x)+4 \text {li}(5+x)+2 \int x \log \left (1+\frac {4}{x}+\frac {x \log (x)}{25 \log (5+x)}\right ) \, dx-3 \int \frac {x^2}{x^2 \log (x)+100 \log (5+x)+25 x \log (5+x)} \, dx-3 \int \frac {x^2 \log (x)}{x^2 \log (x)+100 \log (5+x)+25 x \log (5+x)} \, dx+4 \int \frac {x^2}{x^2 \log (x)+100 \log (5+x)+25 x \log (5+x)} \, dx+5 \int \frac {1}{\log (5+x)} \, dx+8 \int \frac {x^2 \log (x)}{x^2 \log (x)+100 \log (5+x)+25 x \log (5+x)} \, dx+12 \int \frac {x \log (x)}{x^2 \log (x)+100 \log (5+x)+25 x \log (5+x)} \, dx-20 \text {Subst}\left (\int \frac {1}{x} \, dx,x,\log (5+x)\right )-24 \int \frac {x \log (x)}{x^2 \log (x)+100 \log (5+x)+25 x \log (5+x)} \, dx+25 \int \frac {x^2}{x^2 \log (x)+100 \log (5+x)+25 x \log (5+x)} \, dx+48 \int \frac {1}{x^2 \log (x)+100 \log (5+x)+25 x \log (5+x)} \, dx-48 \int \frac {x}{(4+x) \left (x^2 \log (x)+100 \log (5+x)+25 x \log (5+x)\right )} \, dx-48 \int \frac {x \log (x)}{(4+x) \left (x^2 \log (x)+100 \log (5+x)+25 x \log (5+x)\right )} \, dx+96 \int \frac {\log (x)}{x^2 \log (x)+100 \log (5+x)+25 x \log (5+x)} \, dx+100 \int \frac {1}{x^2 \log (x)+100 \log (5+x)+25 x \log (5+x)} \, dx-192 \int \frac {1}{(4+x) \left (x^2 \log (x)+100 \log (5+x)+25 x \log (5+x)\right )} \, dx-384 \int \frac {\log (x)}{(4+x) \left (x^2 \log (x)+100 \log (5+x)+25 x \log (5+x)\right )} \, dx-500 \int \frac {1}{(5+x) \left (x^2 \log (x)+100 \log (5+x)+25 x \log (5+x)\right )} \, dx-\int \frac {5+x}{\log (5+x)} \, dx+\int \frac {x^3}{x^2 \log (x)+100 \log (5+x)+25 x \log (5+x)} \, dx+\int \frac {x^3 \log (x)}{x^2 \log (x)+100 \log (5+x)+25 x \log (5+x)} \, dx+\int \log \left (1+\frac {4}{x}+\frac {x \log (x)}{25 \log (5+x)}\right ) \, dx\\ &=-4 x+12 \log (4+x)-20 \log (\log (5+x))+4 \text {li}(5+x)+2 \int x \log \left (1+\frac {4}{x}+\frac {x \log (x)}{25 \log (5+x)}\right ) \, dx-3 \int \frac {x^2}{x^2 \log (x)+100 \log (5+x)+25 x \log (5+x)} \, dx-3 \int \frac {x^2 \log (x)}{x^2 \log (x)+100 \log (5+x)+25 x \log (5+x)} \, dx+4 \int \frac {x^2}{x^2 \log (x)+100 \log (5+x)+25 x \log (5+x)} \, dx+5 \text {Subst}\left (\int \frac {1}{\log (x)} \, dx,x,5+x\right )+8 \int \frac {x^2 \log (x)}{x^2 \log (x)+100 \log (5+x)+25 x \log (5+x)} \, dx+12 \int \frac {x \log (x)}{x^2 \log (x)+100 \log (5+x)+25 x \log (5+x)} \, dx-24 \int \frac {x \log (x)}{x^2 \log (x)+100 \log (5+x)+25 x \log (5+x)} \, dx+25 \int \frac {x^2}{x^2 \log (x)+100 \log (5+x)+25 x \log (5+x)} \, dx+48 \int \frac {1}{x^2 \log (x)+100 \log (5+x)+25 x \log (5+x)} \, dx-48 \int \left (\frac {1}{x^2 \log (x)+100 \log (5+x)+25 x \log (5+x)}-\frac {4}{(4+x) \left (x^2 \log (x)+100 \log (5+x)+25 x \log (5+x)\right )}\right ) \, dx-48 \int \left (\frac {\log (x)}{x^2 \log (x)+100 \log (5+x)+25 x \log (5+x)}-\frac {4 \log (x)}{(4+x) \left (x^2 \log (x)+100 \log (5+x)+25 x \log (5+x)\right )}\right ) \, dx+96 \int \frac {\log (x)}{x^2 \log (x)+100 \log (5+x)+25 x \log (5+x)} \, dx+100 \int \frac {1}{x^2 \log (x)+100 \log (5+x)+25 x \log (5+x)} \, dx-192 \int \frac {1}{(4+x) \left (x^2 \log (x)+100 \log (5+x)+25 x \log (5+x)\right )} \, dx-384 \int \frac {\log (x)}{(4+x) \left (x^2 \log (x)+100 \log (5+x)+25 x \log (5+x)\right )} \, dx-500 \int \frac {1}{(5+x) \left (x^2 \log (x)+100 \log (5+x)+25 x \log (5+x)\right )} \, dx+\int \frac {x^3}{x^2 \log (x)+100 \log (5+x)+25 x \log (5+x)} \, dx+\int \frac {x^3 \log (x)}{x^2 \log (x)+100 \log (5+x)+25 x \log (5+x)} \, dx+\int \log \left (1+\frac {4}{x}+\frac {x \log (x)}{25 \log (5+x)}\right ) \, dx-\text {Subst}\left (\int \frac {x}{\log (x)} \, dx,x,5+x\right )\\ &=-4 x+12 \log (4+x)-20 \log (\log (5+x))+9 \text {li}(5+x)+2 \int x \log \left (1+\frac {4}{x}+\frac {x \log (x)}{25 \log (5+x)}\right ) \, dx-3 \int \frac {x^2}{x^2 \log (x)+100 \log (5+x)+25 x \log (5+x)} \, dx-3 \int \frac {x^2 \log (x)}{x^2 \log (x)+100 \log (5+x)+25 x \log (5+x)} \, dx+4 \int \frac {x^2}{x^2 \log (x)+100 \log (5+x)+25 x \log (5+x)} \, dx+8 \int \frac {x^2 \log (x)}{x^2 \log (x)+100 \log (5+x)+25 x \log (5+x)} \, dx+12 \int \frac {x \log (x)}{x^2 \log (x)+100 \log (5+x)+25 x \log (5+x)} \, dx-24 \int \frac {x \log (x)}{x^2 \log (x)+100 \log (5+x)+25 x \log (5+x)} \, dx+25 \int \frac {x^2}{x^2 \log (x)+100 \log (5+x)+25 x \log (5+x)} \, dx-48 \int \frac {\log (x)}{x^2 \log (x)+100 \log (5+x)+25 x \log (5+x)} \, dx+96 \int \frac {\log (x)}{x^2 \log (x)+100 \log (5+x)+25 x \log (5+x)} \, dx+100 \int \frac {1}{x^2 \log (x)+100 \log (5+x)+25 x \log (5+x)} \, dx+192 \int \frac {\log (x)}{(4+x) \left (x^2 \log (x)+100 \log (5+x)+25 x \log (5+x)\right )} \, dx-384 \int \frac {\log (x)}{(4+x) \left (x^2 \log (x)+100 \log (5+x)+25 x \log (5+x)\right )} \, dx-500 \int \frac {1}{(5+x) \left (x^2 \log (x)+100 \log (5+x)+25 x \log (5+x)\right )} \, dx+\int \frac {x^3}{x^2 \log (x)+100 \log (5+x)+25 x \log (5+x)} \, dx+\int \frac {x^3 \log (x)}{x^2 \log (x)+100 \log (5+x)+25 x \log (5+x)} \, dx+\int \log \left (1+\frac {4}{x}+\frac {x \log (x)}{25 \log (5+x)}\right ) \, dx-\text {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,\log (5+x)\right )\\ &=-4 x-\text {Ei}(2 \log (5+x))+12 \log (4+x)-20 \log (\log (5+x))+9 \text {li}(5+x)+2 \int x \log \left (1+\frac {4}{x}+\frac {x \log (x)}{25 \log (5+x)}\right ) \, dx-3 \int \frac {x^2}{x^2 \log (x)+100 \log (5+x)+25 x \log (5+x)} \, dx-3 \int \frac {x^2 \log (x)}{x^2 \log (x)+100 \log (5+x)+25 x \log (5+x)} \, dx+4 \int \frac {x^2}{x^2 \log (x)+100 \log (5+x)+25 x \log (5+x)} \, dx+8 \int \frac {x^2 \log (x)}{x^2 \log (x)+100 \log (5+x)+25 x \log (5+x)} \, dx+12 \int \frac {x \log (x)}{x^2 \log (x)+100 \log (5+x)+25 x \log (5+x)} \, dx-24 \int \frac {x \log (x)}{x^2 \log (x)+100 \log (5+x)+25 x \log (5+x)} \, dx+25 \int \frac {x^2}{x^2 \log (x)+100 \log (5+x)+25 x \log (5+x)} \, dx-48 \int \frac {\log (x)}{x^2 \log (x)+100 \log (5+x)+25 x \log (5+x)} \, dx+96 \int \frac {\log (x)}{x^2 \log (x)+100 \log (5+x)+25 x \log (5+x)} \, dx+100 \int \frac {1}{x^2 \log (x)+100 \log (5+x)+25 x \log (5+x)} \, dx+192 \int \frac {\log (x)}{(4+x) \left (x^2 \log (x)+100 \log (5+x)+25 x \log (5+x)\right )} \, dx-384 \int \frac {\log (x)}{(4+x) \left (x^2 \log (x)+100 \log (5+x)+25 x \log (5+x)\right )} \, dx-500 \int \frac {1}{(5+x) \left (x^2 \log (x)+100 \log (5+x)+25 x \log (5+x)\right )} \, dx+\int \frac {x^3}{x^2 \log (x)+100 \log (5+x)+25 x \log (5+x)} \, dx+\int \frac {x^3 \log (x)}{x^2 \log (x)+100 \log (5+x)+25 x \log (5+x)} \, dx+\int \log \left (1+\frac {4}{x}+\frac {x \log (x)}{25 \log (5+x)}\right ) \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]
time = 0.10, size = 26, normalized size = 0.93 \begin {gather*} x (1+x) \log \left (1+\frac {4}{x}+\frac {x \log (x)}{25 \log (5+x)}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((-x^3 - x^4)*Log[x] + (5*x^2 + 6*x^3 + x^4 + (5*x^2 + 6*x^3 + x^4)*Log[x])*Log[5 + x] + (-500 - 600
*x - 100*x^2)*Log[5 + x]^2 + ((5*x^2 + 11*x^3 + 2*x^4)*Log[x]*Log[5 + x] + (500 + 1225*x + 475*x^2 + 50*x^3)*L
og[5 + x]^2)*Log[(x^2*Log[x] + (100 + 25*x)*Log[5 + x])/(25*x*Log[5 + x])])/((5*x^2 + x^3)*Log[x]*Log[5 + x] +
(500 + 225*x + 25*x^2)*Log[5 + x]^2),x]

[Out]

x*(1 + x)*Log[1 + 4/x + (x*Log[x])/(25*Log[5 + x])]

________________________________________________________________________________________

Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 7.74, size = 979, normalized size = 34.96

method	result	size
risch	\(\text {Expression too large to display}\)	\(979\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((50*x^3+475*x^2+1225*x+500)*ln(5+x)^2+(2*x^4+11*x^3+5*x^2)*ln(x)*ln(5+x))*ln(1/25*((25*x+100)*ln(5+x)+x^
2*ln(x))/x/ln(5+x))+(-100*x^2-600*x-500)*ln(5+x)^2+((x^4+6*x^3+5*x^2)*ln(x)+x^4+6*x^3+5*x^2)*ln(5+x)+(-x^4-x^3
)*ln(x))/((25*x^2+225*x+500)*ln(5+x)^2+(x^3+5*x^2)*ln(x)*ln(5+x)),x,method=_RETURNVERBOSE)

[Out]

-2*x*ln(5)-x*ln(x)-2*x^2*ln(5)-x^2*ln(x)-1/2*I*Pi*x^2*csgn(I/ln(5+x))*csgn(I*(x^2*ln(x)+25*x*ln(5+x)+100*ln(5+
x)))*csgn(I/ln(5+x)*(x^2*ln(x)+25*x*ln(5+x)+100*ln(5+x)))-1/2*I*Pi*x^2*csgn(I/ln(5+x)*(x^2*ln(x)+25*x*ln(5+x)+
100*ln(5+x)))^3-1/2*I*Pi*x*csgn(I/ln(5+x)*(x^2*ln(x)+25*x*ln(5+x)+100*ln(5+x)))^3-1/2*I*Pi*x^2*csgn(I/x/ln(5+x
)*(x^2*ln(x)+25*x*ln(5+x)+100*ln(5+x)))^3-1/2*I*Pi*x*csgn(I/ln(5+x))*csgn(I*(x^2*ln(x)+25*x*ln(5+x)+100*ln(5+x
)))*csgn(I/ln(5+x)*(x^2*ln(x)+25*x*ln(5+x)+100*ln(5+x)))+1/2*I*Pi*x^2*csgn(I/x/ln(5+x)*(x^2*ln(x)+25*x*ln(5+x)
+100*ln(5+x)))^2*csgn(I/x)+1/2*I*Pi*x*csgn(I/ln(5+x)*(x^2*ln(x)+25*x*ln(5+x)+100*ln(5+x)))*csgn(I/x/ln(5+x)*(x
^2*ln(x)+25*x*ln(5+x)+100*ln(5+x)))^2+1/2*I*Pi*x^2*csgn(I/ln(5+x)*(x^2*ln(x)+25*x*ln(5+x)+100*ln(5+x)))*csgn(I
/x/ln(5+x)*(x^2*ln(x)+25*x*ln(5+x)+100*ln(5+x)))^2+1/2*I*Pi*x*csgn(I*(x^2*ln(x)+25*x*ln(5+x)+100*ln(5+x)))*csg
n(I/ln(5+x)*(x^2*ln(x)+25*x*ln(5+x)+100*ln(5+x)))^2+1/2*I*Pi*x^2*csgn(I*(x^2*ln(x)+25*x*ln(5+x)+100*ln(5+x)))*
csgn(I/ln(5+x)*(x^2*ln(x)+25*x*ln(5+x)+100*ln(5+x)))^2-1/2*I*Pi*x*csgn(I/ln(5+x)*(x^2*ln(x)+25*x*ln(5+x)+100*l
n(5+x)))*csgn(I/x/ln(5+x)*(x^2*ln(x)+25*x*ln(5+x)+100*ln(5+x)))*csgn(I/x)-1/2*I*Pi*x*csgn(I/x/ln(5+x)*(x^2*ln(
x)+25*x*ln(5+x)+100*ln(5+x)))^3+1/2*I*Pi*x*csgn(I/ln(5+x))*csgn(I/ln(5+x)*(x^2*ln(x)+25*x*ln(5+x)+100*ln(5+x))
)^2+1/2*I*Pi*x^2*csgn(I/ln(5+x))*csgn(I/ln(5+x)*(x^2*ln(x)+25*x*ln(5+x)+100*ln(5+x)))^2-x*ln(ln(5+x))-x^2*ln(l
n(5+x))+(x^2+x)*ln(x^2*ln(x)+25*x*ln(5+x)+100*ln(5+x))+1/2*I*Pi*x*csgn(I/x/ln(5+x)*(x^2*ln(x)+25*x*ln(5+x)+100
*ln(5+x)))^2*csgn(I/x)-1/2*I*Pi*x^2*csgn(I/ln(5+x)*(x^2*ln(x)+25*x*ln(5+x)+100*ln(5+x)))*csgn(I/x/ln(5+x)*(x^2
*ln(x)+25*x*ln(5+x)+100*ln(5+x)))*csgn(I/x)

________________________________________________________________________________________

Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (28) = 56\).
time = 0.57, size = 61, normalized size = 2.18 \begin {gather*} -2 \, x^{2} \log \left (5\right ) - 2 \, x \log \left (5\right ) + {\left (x^{2} + x\right )} \log \left (x^{2} \log \left (x\right ) + 25 \, x \log \left (x + 5\right ) + 100 \, \log \left (x + 5\right )\right ) - {\left (x^{2} + x\right )} \log \left (x\right ) - {\left (x^{2} + x\right )} \log \left (\log \left (x + 5\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((50*x^3+475*x^2+1225*x+500)*log(5+x)^2+(2*x^4+11*x^3+5*x^2)*log(x)*log(5+x))*log(1/25*((25*x+100)*
log(5+x)+x^2*log(x))/x/log(5+x))+(-100*x^2-600*x-500)*log(5+x)^2+((x^4+6*x^3+5*x^2)*log(x)+x^4+6*x^3+5*x^2)*lo
g(5+x)+(-x^4-x^3)*log(x))/((25*x^2+225*x+500)*log(5+x)^2+(x^3+5*x^2)*log(x)*log(5+x)),x, algorithm="maxima")

[Out]

-2*x^2*log(5) - 2*x*log(5) + (x^2 + x)*log(x^2*log(x) + 25*x*log(x + 5) + 100*log(x + 5)) - (x^2 + x)*log(x) -
(x^2 + x)*log(log(x + 5))

________________________________________________________________________________________

Fricas [A]
time = 0.36, size = 34, normalized size = 1.21 \begin {gather*} {\left (x^{2} + x\right )} \log \left (\frac {x^{2} \log \left (x\right ) + 25 \, {\left (x + 4\right )} \log \left (x + 5\right )}{25 \, x \log \left (x + 5\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((50*x^3+475*x^2+1225*x+500)*log(5+x)^2+(2*x^4+11*x^3+5*x^2)*log(x)*log(5+x))*log(1/25*((25*x+100)*
log(5+x)+x^2*log(x))/x/log(5+x))+(-100*x^2-600*x-500)*log(5+x)^2+((x^4+6*x^3+5*x^2)*log(x)+x^4+6*x^3+5*x^2)*lo
g(5+x)+(-x^4-x^3)*log(x))/((25*x^2+225*x+500)*log(5+x)^2+(x^3+5*x^2)*log(x)*log(5+x)),x, algorithm="fricas")

[Out]

(x^2 + x)*log(1/25*(x^2*log(x) + 25*(x + 4)*log(x + 5))/(x*log(x + 5)))

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((50*x**3+475*x**2+1225*x+500)*ln(5+x)**2+(2*x**4+11*x**3+5*x**2)*ln(x)*ln(5+x))*ln(1/25*((25*x+100
)*ln(5+x)+x**2*ln(x))/x/ln(5+x))+(-100*x**2-600*x-500)*ln(5+x)**2+((x**4+6*x**3+5*x**2)*ln(x)+x**4+6*x**3+5*x*
*2)*ln(5+x)+(-x**4-x**3)*ln(x))/((25*x**2+225*x+500)*ln(5+x)**2+(x**3+5*x**2)*ln(x)*ln(5+x)),x)

[Out]

Exception raised: PolynomialError >> 1/(75*x**3 + 975*x**2 + 4200*x + 6000) contains an element of the set of
generators.

________________________________________________________________________________________

Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((50*x^3+475*x^2+1225*x+500)*log(5+x)^2+(2*x^4+11*x^3+5*x^2)*log(x)*log(5+x))*log(1/25*((25*x+100)*
log(5+x)+x^2*log(x))/x/log(5+x))+(-100*x^2-600*x-500)*log(5+x)^2+((x^4+6*x^3+5*x^2)*log(x)+x^4+6*x^3+5*x^2)*lo
g(5+x)+(-x^4-x^3)*log(x))/((25*x^2+225*x+500)*log(5+x)^2+(x^3+5*x^2)*log(x)*log(5+x)),x, algorithm="giac")

[Out]

Timed out

________________________________________________________________________________________

Mupad [B]
time = 3.84, size = 63, normalized size = 2.25 \begin {gather*} \frac {\ln \left (\frac {\frac {x^2\,\ln \left (x\right )}{25}+\frac {\ln \left (x+5\right )\,\left (25\,x+100\right )}{25}}{x\,\ln \left (x+5\right )}\right )\,\left (x^5+10\,x^4+29\,x^3+20\,x^2\right )}{x\,\left (x+4\right )\,\left (x+5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x + 5)*(5*x^2 + 6*x^3 + x^4 + log(x)*(5*x^2 + 6*x^3 + x^4)) - log(x)*(x^3 + x^4) - log(x + 5)^2*(600*
x + 100*x^2 + 500) + log(((x^2*log(x))/25 + (log(x + 5)*(25*x + 100))/25)/(x*log(x + 5)))*(log(x + 5)^2*(1225*
x + 475*x^2 + 50*x^3 + 500) + log(x + 5)*log(x)*(5*x^2 + 11*x^3 + 2*x^4)))/(log(x + 5)^2*(225*x + 25*x^2 + 500
) + log(x + 5)*log(x)*(5*x^2 + x^3)),x)

[Out]

(log(((x^2*log(x))/25 + (log(x + 5)*(25*x + 100))/25)/(x*log(x + 5)))*(20*x^2 + 29*x^3 + 10*x^4 + x^5))/(x*(x
+ 4)*(x + 5))

________________________________________________________________________________________