\(\int \frac {48+720 x+108 x^2+42 x^3+6 x^4+(72+744 x+90 x^2+18 x^3) \log (x)+(36+54 x+18 x^2) \log ^2(x)+(6+6 x) \log ^3(x)}{8+12 x+6 x^2+x^3+(12+12 x+3 x^2) \log (x)+(6+3 x) \log ^2(x)+\log ^3(x)} \, dx\) [7082]

Optimal result

Integrand size = 106, antiderivative size = 21 \[ \int \frac {48+720 x+108 x^2+42 x^3+6 x^4+\left (72+744 x+90 x^2+18 x^3\right ) \log (x)+\left (36+54 x+18 x^2\right ) \log ^2(x)+(6+6 x) \log ^3(x)}{8+12 x+6 x^2+x^3+\left (12+12 x+3 x^2\right ) \log (x)+(6+3 x) \log ^2(x)+\log ^3(x)} \, dx=3 \left (25+x (2+x)+\frac {100 x^2}{(2+x+\log (x))^2}\right ) \]

[Out]

3*x*(2+x)+75+300*x^2/(x+ln(x)+2)^2

Rubi [F]

\[ \int \frac {48+720 x+108 x^2+42 x^3+6 x^4+\left (72+744 x+90 x^2+18 x^3\right ) \log (x)+\left (36+54 x+18 x^2\right ) \log ^2(x)+(6+6 x) \log ^3(x)}{8+12 x+6 x^2+x^3+\left (12+12 x+3 x^2\right ) \log (x)+(6+3 x) \log ^2(x)+\log ^3(x)} \, dx=\int \frac {48+720 x+108 x^2+42 x^3+6 x^4+\left (72+744 x+90 x^2+18 x^3\right ) \log (x)+\left (36+54 x+18 x^2\right ) \log ^2(x)+(6+6 x) \log ^3(x)}{8+12 x+6 x^2+x^3+\left (12+12 x+3 x^2\right ) \log (x)+(6+3 x) \log ^2(x)+\log ^3(x)} \, dx \]

[In]

Int[(48 + 720*x + 108*x^2 + 42*x^3 + 6*x^4 + (72 + 744*x + 90*x^2 + 18*x^3)*Log[x] + (36 + 54*x + 18*x^2)*Log[
x]^2 + (6 + 6*x)*Log[x]^3)/(8 + 12*x + 6*x^2 + x^3 + (12 + 12*x + 3*x^2)*Log[x] + (6 + 3*x)*Log[x]^2 + Log[x]^
3),x]

[Out]

6*x + 3*x^2 - 600*Defer[Int][x/(2 + x + Log[x])^3, x] - 600*Defer[Int][x^2/(2 + x + Log[x])^3, x] + 600*Defer[
Int][x/(2 + x + Log[x])^2, x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {6 \left (8+120 x+18 x^2+7 x^3+x^4+\left (12+124 x+15 x^2+3 x^3\right ) \log (x)+3 \left (2+3 x+x^2\right ) \log ^2(x)+(1+x) \log ^3(x)\right )}{(2+x+\log (x))^3} \, dx \\ & = 6 \int \frac {8+120 x+18 x^2+7 x^3+x^4+\left (12+124 x+15 x^2+3 x^3\right ) \log (x)+3 \left (2+3 x+x^2\right ) \log ^2(x)+(1+x) \log ^3(x)}{(2+x+\log (x))^3} \, dx \\ & = 6 \int \left (1+x-\frac {100 x (1+x)}{(2+x+\log (x))^3}+\frac {100 x}{(2+x+\log (x))^2}\right ) \, dx \\ & = 6 x+3 x^2-600 \int \frac {x (1+x)}{(2+x+\log (x))^3} \, dx+600 \int \frac {x}{(2+x+\log (x))^2} \, dx \\ & = 6 x+3 x^2+600 \int \frac {x}{(2+x+\log (x))^2} \, dx-600 \int \left (\frac {x}{(2+x+\log (x))^3}+\frac {x^2}{(2+x+\log (x))^3}\right ) \, dx \\ & = 6 x+3 x^2-600 \int \frac {x}{(2+x+\log (x))^3} \, dx-600 \int \frac {x^2}{(2+x+\log (x))^3} \, dx+600 \int \frac {x}{(2+x+\log (x))^2} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.76 \[ \int \frac {48+720 x+108 x^2+42 x^3+6 x^4+\left (72+744 x+90 x^2+18 x^3\right ) \log (x)+\left (36+54 x+18 x^2\right ) \log ^2(x)+(6+6 x) \log ^3(x)}{8+12 x+6 x^2+x^3+\left (12+12 x+3 x^2\right ) \log (x)+(6+3 x) \log ^2(x)+\log ^3(x)} \, dx=3 x \left (2+x+\frac {100 x}{(2+x+\log (x))^2}\right ) \]

[In]

Integrate[(48 + 720*x + 108*x^2 + 42*x^3 + 6*x^4 + (72 + 744*x + 90*x^2 + 18*x^3)*Log[x] + (36 + 54*x + 18*x^2
)*Log[x]^2 + (6 + 6*x)*Log[x]^3)/(8 + 12*x + 6*x^2 + x^3 + (12 + 12*x + 3*x^2)*Log[x] + (6 + 3*x)*Log[x]^2 + L
og[x]^3),x]

[Out]

3*x*(2 + x + (100*x)/(2 + x + Log[x])^2)

Maple [A] (verified)

Time = 0.86 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.05

method	result	size
risch	\(3 x^{2}+6 x +\frac {300 x^{2}}{\left (x +\ln \left (x \right )+2\right )^{2}}\)	\(22\)
default	\(\frac {24 x +6 x \ln \left (x \right )^{2}+24 x \ln \left (x \right )+3 x^{2} \ln \left (x \right )^{2}+3 x^{4}+18 x^{3}+336 x^{2}+6 x^{3} \ln \left (x \right )+24 x^{2} \ln \left (x \right )}{\left (x +\ln \left (x \right )+2\right )^{2}}\)	\(62\)
norman	\(\frac {-1344 \ln \left (x \right )-1320 x -336 \ln \left (x \right )^{2}-648 x \ln \left (x \right )+18 x^{3}+3 x^{4}+6 x \ln \left (x \right )^{2}+24 x^{2} \ln \left (x \right )+3 x^{2} \ln \left (x \right )^{2}+6 x^{3} \ln \left (x \right )-1344}{\left (x +\ln \left (x \right )+2\right )^{2}}\)	\(69\)
parallelrisch	\(\frac {24 x +6 x \ln \left (x \right )^{2}+24 x \ln \left (x \right )+3 x^{2} \ln \left (x \right )^{2}+3 x^{4}+18 x^{3}+336 x^{2}+6 x^{3} \ln \left (x \right )+24 x^{2} \ln \left (x \right )}{x^{2}+2 x \ln \left (x \right )+\ln \left (x \right )^{2}+4 x +4 \ln \left (x \right )+4}\)	\(79\)

[In]

int(((6+6*x)*ln(x)^3+(18*x^2+54*x+36)*ln(x)^2+(18*x^3+90*x^2+744*x+72)*ln(x)+6*x^4+42*x^3+108*x^2+720*x+48)/(l
n(x)^3+(6+3*x)*ln(x)^2+(3*x^2+12*x+12)*ln(x)+x^3+6*x^2+12*x+8),x,method=_RETURNVERBOSE)

[Out]

3*x^2+6*x+300*x^2/(x+ln(x)+2)^2

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (20) = 40\).

Time = 0.29 (sec) , antiderivative size = 68, normalized size of antiderivative = 3.24 \[ \int \frac {48+720 x+108 x^2+42 x^3+6 x^4+\left (72+744 x+90 x^2+18 x^3\right ) \log (x)+\left (36+54 x+18 x^2\right ) \log ^2(x)+(6+6 x) \log ^3(x)}{8+12 x+6 x^2+x^3+\left (12+12 x+3 x^2\right ) \log (x)+(6+3 x) \log ^2(x)+\log ^3(x)} \, dx=\frac {3 \, {\left (x^{4} + 6 \, x^{3} + {\left (x^{2} + 2 \, x\right )} \log \left (x\right )^{2} + 112 \, x^{2} + 2 \, {\left (x^{3} + 4 \, x^{2} + 4 \, x\right )} \log \left (x\right ) + 8 \, x\right )}}{x^{2} + 2 \, {\left (x + 2\right )} \log \left (x\right ) + \log \left (x\right )^{2} + 4 \, x + 4} \]

[In]

integrate(((6+6*x)*log(x)^3+(18*x^2+54*x+36)*log(x)^2+(18*x^3+90*x^2+744*x+72)*log(x)+6*x^4+42*x^3+108*x^2+720
*x+48)/(log(x)^3+(6+3*x)*log(x)^2+(3*x^2+12*x+12)*log(x)+x^3+6*x^2+12*x+8),x, algorithm="fricas")

[Out]

3*(x^4 + 6*x^3 + (x^2 + 2*x)*log(x)^2 + 112*x^2 + 2*(x^3 + 4*x^2 + 4*x)*log(x) + 8*x)/(x^2 + 2*(x + 2)*log(x)
+ log(x)^2 + 4*x + 4)

Sympy [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.62 \[ \int \frac {48+720 x+108 x^2+42 x^3+6 x^4+\left (72+744 x+90 x^2+18 x^3\right ) \log (x)+\left (36+54 x+18 x^2\right ) \log ^2(x)+(6+6 x) \log ^3(x)}{8+12 x+6 x^2+x^3+\left (12+12 x+3 x^2\right ) \log (x)+(6+3 x) \log ^2(x)+\log ^3(x)} \, dx=3 x^{2} + \frac {300 x^{2}}{x^{2} + 4 x + \left (2 x + 4\right ) \log {\left (x \right )} + \log {\left (x \right )}^{2} + 4} + 6 x \]

[In]

integrate(((6+6*x)*ln(x)**3+(18*x**2+54*x+36)*ln(x)**2+(18*x**3+90*x**2+744*x+72)*ln(x)+6*x**4+42*x**3+108*x**
2+720*x+48)/(ln(x)**3+(6+3*x)*ln(x)**2+(3*x**2+12*x+12)*ln(x)+x**3+6*x**2+12*x+8),x)

[Out]

3*x**2 + 300*x**2/(x**2 + 4*x + (2*x + 4)*log(x) + log(x)**2 + 4) + 6*x

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (20) = 40\).

Time = 0.22 (sec) , antiderivative size = 68, normalized size of antiderivative = 3.24 \[ \int \frac {48+720 x+108 x^2+42 x^3+6 x^4+\left (72+744 x+90 x^2+18 x^3\right ) \log (x)+\left (36+54 x+18 x^2\right ) \log ^2(x)+(6+6 x) \log ^3(x)}{8+12 x+6 x^2+x^3+\left (12+12 x+3 x^2\right ) \log (x)+(6+3 x) \log ^2(x)+\log ^3(x)} \, dx=\frac {3 \, {\left (x^{4} + 6 \, x^{3} + {\left (x^{2} + 2 \, x\right )} \log \left (x\right )^{2} + 112 \, x^{2} + 2 \, {\left (x^{3} + 4 \, x^{2} + 4 \, x\right )} \log \left (x\right ) + 8 \, x\right )}}{x^{2} + 2 \, {\left (x + 2\right )} \log \left (x\right ) + \log \left (x\right )^{2} + 4 \, x + 4} \]

[In]

integrate(((6+6*x)*log(x)^3+(18*x^2+54*x+36)*log(x)^2+(18*x^3+90*x^2+744*x+72)*log(x)+6*x^4+42*x^3+108*x^2+720
*x+48)/(log(x)^3+(6+3*x)*log(x)^2+(3*x^2+12*x+12)*log(x)+x^3+6*x^2+12*x+8),x, algorithm="maxima")

[Out]

3*(x^4 + 6*x^3 + (x^2 + 2*x)*log(x)^2 + 112*x^2 + 2*(x^3 + 4*x^2 + 4*x)*log(x) + 8*x)/(x^2 + 2*(x + 2)*log(x)
+ log(x)^2 + 4*x + 4)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 59 vs. \(2 (20) = 40\).

Time = 0.29 (sec) , antiderivative size = 59, normalized size of antiderivative = 2.81 \[ \int \frac {48+720 x+108 x^2+42 x^3+6 x^4+\left (72+744 x+90 x^2+18 x^3\right ) \log (x)+\left (36+54 x+18 x^2\right ) \log ^2(x)+(6+6 x) \log ^3(x)}{8+12 x+6 x^2+x^3+\left (12+12 x+3 x^2\right ) \log (x)+(6+3 x) \log ^2(x)+\log ^3(x)} \, dx=3 \, x^{2} + 6 \, x + \frac {300 \, {\left (x^{3} + x^{2}\right )}}{x^{3} + 2 \, x^{2} \log \left (x\right ) + x \log \left (x\right )^{2} + 5 \, x^{2} + 6 \, x \log \left (x\right ) + \log \left (x\right )^{2} + 8 \, x + 4 \, \log \left (x\right ) + 4} \]

[In]

integrate(((6+6*x)*log(x)^3+(18*x^2+54*x+36)*log(x)^2+(18*x^3+90*x^2+744*x+72)*log(x)+6*x^4+42*x^3+108*x^2+720
*x+48)/(log(x)^3+(6+3*x)*log(x)^2+(3*x^2+12*x+12)*log(x)+x^3+6*x^2+12*x+8),x, algorithm="giac")

[Out]

3*x^2 + 6*x + 300*(x^3 + x^2)/(x^3 + 2*x^2*log(x) + x*log(x)^2 + 5*x^2 + 6*x*log(x) + log(x)^2 + 8*x + 4*log(x
) + 4)

Mupad [B] (verification not implemented)

Time = 13.07 (sec) , antiderivative size = 68, normalized size of antiderivative = 3.24 \[ \int \frac {48+720 x+108 x^2+42 x^3+6 x^4+\left (72+744 x+90 x^2+18 x^3\right ) \log (x)+\left (36+54 x+18 x^2\right ) \log ^2(x)+(6+6 x) \log ^3(x)}{8+12 x+6 x^2+x^3+\left (12+12 x+3 x^2\right ) \log (x)+(6+3 x) \log ^2(x)+\log ^3(x)} \, dx=3\,x\,\left (x+2\right )+\frac {3\,x\,\left (x^3+6\,x^2+112\,x+8\right )-3\,x\,{\left (x+2\right )}^3+\ln \left (x\right )\,\left (3\,x\,\left (2\,x^2+8\,x+8\right )-3\,x\,\left (2\,x+4\right )\,\left (x+2\right )\right )}{{\left (x+\ln \left (x\right )+2\right )}^2} \]

[In]

int((720*x + log(x)^2*(54*x + 18*x^2 + 36) + 108*x^2 + 42*x^3 + 6*x^4 + log(x)^3*(6*x + 6) + log(x)*(744*x + 9
0*x^2 + 18*x^3 + 72) + 48)/(12*x + log(x)^3 + log(x)*(12*x + 3*x^2 + 12) + 6*x^2 + x^3 + log(x)^2*(3*x + 6) +
8),x)

[Out]

3*x*(x + 2) + (3*x*(112*x + 6*x^2 + x^3 + 8) - 3*x*(x + 2)^3 + log(x)*(3*x*(8*x + 2*x^2 + 8) - 3*x*(2*x + 4)*(
x + 2)))/(x + log(x) + 2)^2