3.4.50 \(\int \frac {(38+6 x) \log (\frac {-1875+475 x+47 x^2+x^3}{-5624+1425 x+141 x^2+3 x^3})}{421800-230603 x+15151 x^2+4302 x^3+207 x^4+3 x^5} \, dx\) [350]

3.4.50.1 Optimal result

Integrand size = 65, antiderivative size = 23 \begin {dmath*} \int \frac {(38+6 x) \log \left (\frac {-1875+475 x+47 x^2+x^3}{-5624+1425 x+141 x^2+3 x^3}\right )}{421800-230603 x+15151 x^2+4302 x^3+207 x^4+3 x^5} \, dx=\log ^2\left (\frac {1}{3+\frac {x}{(25+x)^2 \left (-3 x+x^2\right )}}\right ) \end {dmath*}

output

ln(1/(3+x/(x^2-3*x)/(x+25)^2))^2
 

3.4.50.2 Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.26 \begin {dmath*} \int \frac {(38+6 x) \log \left (\frac {-1875+475 x+47 x^2+x^3}{-5624+1425 x+141 x^2+3 x^3}\right )}{421800-230603 x+15151 x^2+4302 x^3+207 x^4+3 x^5} \, dx=\log ^2\left (\frac {(-3+x) (25+x)^2}{-5624+1425 x+141 x^2+3 x^3}\right ) \end {dmath*}

input

Integrate[((38 + 6*x)*Log[(-1875 + 475*x + 47*x^2 + x^3)/(-5624 + 1425*x + 
 141*x^2 + 3*x^3)])/(421800 - 230603*x + 15151*x^2 + 4302*x^3 + 207*x^4 + 
3*x^5),x]
 

output

Log[((-3 + x)*(25 + x)^2)/(-5624 + 1425*x + 141*x^2 + 3*x^3)]^2
 

3.4.50.3 Rubi [F]

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {(6 x+38) \log \left (\frac {x^3+47 x^2+475 x-1875}{3 x^3+141 x^2+1425 x-5624}\right )}{3 x^5+207 x^4+4302 x^3+15151 x^2-230603 x+421800} \, dx\)

\(\Big \downarrow \) 2463

\(\displaystyle \int \left (\frac {(6 x+38) \log \left (\frac {x^3+47 x^2+475 x-1875}{3 x^3+141 x^2+1425 x-5624}\right )}{28 (x-3)}-\frac {(6 x+38) \log \left (\frac {x^3+47 x^2+475 x-1875}{3 x^3+141 x^2+1425 x-5624}\right )}{28 (x+25)}-\frac {3 (x+25) (6 x+38) \log \left (\frac {x^3+47 x^2+475 x-1875}{3 x^3+141 x^2+1425 x-5624}\right )}{3 x^3+141 x^2+1425 x-5624}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {2 (3 x+19) \log \left (\frac {(x-3) (x+25)^2}{3 x^3+141 x^2+1425 x-5624}\right )}{(3-x) (x+25) \left (-3 x^3-141 x^2-1425 x+5624\right )}dx\)

\(\Big \downarrow \) 27

\(\displaystyle 2 \int \frac {(3 x+19) \log \left (\frac {(3-x) (x+25)^2}{-3 x^3-141 x^2-1425 x+5624}\right )}{(3-x) (x+25) \left (-3 x^3-141 x^2-1425 x+5624\right )}dx\)

\(\Big \downarrow \) 3008

\(\displaystyle 2 \int \left (\frac {\log \left (\frac {(3-x) (x+25)^2}{-3 x^3-141 x^2-1425 x+5624}\right )}{x-3}+\frac {2 \log \left (\frac {(3-x) (x+25)^2}{-3 x^3-141 x^2-1425 x+5624}\right )}{x+25}-\frac {3 \left (3 x^2+94 x+475\right ) \log \left (\frac {(3-x) (x+25)^2}{-3 x^3-141 x^2-1425 x+5624}\right )}{3 x^3+141 x^2+1425 x-5624}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle 2 \int \frac {(3 x+19) \log \left (\frac {(x-3) (x+25)^2}{3 x^3+141 x^2+1425 x-5624}\right )}{(3-x) (x+25) \left (-3 x^3-141 x^2-1425 x+5624\right )}dx\)

\(\Big \downarrow \) 3008

\(\displaystyle 2 \int \left (\frac {\log \left (\frac {(x-3) (x+25)^2}{3 x^3+141 x^2+1425 x-5624}\right )}{x-3}+\frac {2 \log \left (\frac {(x-3) (x+25)^2}{3 x^3+141 x^2+1425 x-5624}\right )}{x+25}-\frac {3 \left (3 x^2+94 x+475\right ) \log \left (\frac {(x-3) (x+25)^2}{3 x^3+141 x^2+1425 x-5624}\right )}{3 x^3+141 x^2+1425 x-5624}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle 2 \int \frac {(3 x+19) \log \left (\frac {(x-3) (x+25)^2}{3 x^3+141 x^2+1425 x-5624}\right )}{(3-x) (x+25) \left (-3 x^3-141 x^2-1425 x+5624\right )}dx\)

\(\Big \downarrow \) 3008

\(\displaystyle 2 \int \left (\frac {\log \left (\frac {(x-3) (x+25)^2}{3 x^3+141 x^2+1425 x-5624}\right )}{x-3}+\frac {2 \log \left (\frac {(x-3) (x+25)^2}{3 x^3+141 x^2+1425 x-5624}\right )}{x+25}-\frac {3 \left (3 x^2+94 x+475\right ) \log \left (\frac {(x-3) (x+25)^2}{3 x^3+141 x^2+1425 x-5624}\right )}{3 x^3+141 x^2+1425 x-5624}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle 2 \int \frac {(3 x+19) \log \left (\frac {(x-3) (x+25)^2}{3 x^3+141 x^2+1425 x-5624}\right )}{(3-x) (x+25) \left (-3 x^3-141 x^2-1425 x+5624\right )}dx\)

\(\Big \downarrow \) 3008

\(\displaystyle 2 \int \left (\frac {\log \left (\frac {(x-3) (x+25)^2}{3 x^3+141 x^2+1425 x-5624}\right )}{x-3}+\frac {2 \log \left (\frac {(x-3) (x+25)^2}{3 x^3+141 x^2+1425 x-5624}\right )}{x+25}-\frac {3 \left (3 x^2+94 x+475\right ) \log \left (\frac {(x-3) (x+25)^2}{3 x^3+141 x^2+1425 x-5624}\right )}{3 x^3+141 x^2+1425 x-5624}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle 2 \int \frac {(3 x+19) \log \left (\frac {(x-3) (x+25)^2}{3 x^3+141 x^2+1425 x-5624}\right )}{(3-x) (x+25) \left (-3 x^3-141 x^2-1425 x+5624\right )}dx\)

\(\Big \downarrow \) 3008

\(\displaystyle 2 \int \left (\frac {\log \left (\frac {(x-3) (x+25)^2}{3 x^3+141 x^2+1425 x-5624}\right )}{x-3}+\frac {2 \log \left (\frac {(x-3) (x+25)^2}{3 x^3+141 x^2+1425 x-5624}\right )}{x+25}-\frac {3 \left (3 x^2+94 x+475\right ) \log \left (\frac {(x-3) (x+25)^2}{3 x^3+141 x^2+1425 x-5624}\right )}{3 x^3+141 x^2+1425 x-5624}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle 2 \int \frac {(3 x+19) \log \left (\frac {(x-3) (x+25)^2}{3 x^3+141 x^2+1425 x-5624}\right )}{(3-x) (x+25) \left (-3 x^3-141 x^2-1425 x+5624\right )}dx\)

\(\Big \downarrow \) 3008

\(\displaystyle 2 \int \left (\frac {\log \left (\frac {(x-3) (x+25)^2}{3 x^3+141 x^2+1425 x-5624}\right )}{x-3}+\frac {2 \log \left (\frac {(x-3) (x+25)^2}{3 x^3+141 x^2+1425 x-5624}\right )}{x+25}-\frac {3 \left (3 x^2+94 x+475\right ) \log \left (\frac {(x-3) (x+25)^2}{3 x^3+141 x^2+1425 x-5624}\right )}{3 x^3+141 x^2+1425 x-5624}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle 2 \int \frac {(3 x+19) \log \left (\frac {(x-3) (x+25)^2}{3 x^3+141 x^2+1425 x-5624}\right )}{(3-x) (x+25) \left (-3 x^3-141 x^2-1425 x+5624\right )}dx\)

\(\Big \downarrow \) 3008

\(\displaystyle 2 \int \left (\frac {\log \left (\frac {(x-3) (x+25)^2}{3 x^3+141 x^2+1425 x-5624}\right )}{x-3}+\frac {2 \log \left (\frac {(x-3) (x+25)^2}{3 x^3+141 x^2+1425 x-5624}\right )}{x+25}-\frac {3 \left (3 x^2+94 x+475\right ) \log \left (\frac {(x-3) (x+25)^2}{3 x^3+141 x^2+1425 x-5624}\right )}{3 x^3+141 x^2+1425 x-5624}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle 2 \int \frac {(3 x+19) \log \left (\frac {(x-3) (x+25)^2}{3 x^3+141 x^2+1425 x-5624}\right )}{(3-x) (x+25) \left (-3 x^3-141 x^2-1425 x+5624\right )}dx\)

\(\Big \downarrow \) 3008

\(\displaystyle 2 \int \left (\frac {\log \left (\frac {(x-3) (x+25)^2}{3 x^3+141 x^2+1425 x-5624}\right )}{x-3}+\frac {2 \log \left (\frac {(x-3) (x+25)^2}{3 x^3+141 x^2+1425 x-5624}\right )}{x+25}-\frac {3 \left (3 x^2+94 x+475\right ) \log \left (\frac {(x-3) (x+25)^2}{3 x^3+141 x^2+1425 x-5624}\right )}{3 x^3+141 x^2+1425 x-5624}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle 2 \int \frac {(3 x+19) \log \left (\frac {(x-3) (x+25)^2}{3 x^3+141 x^2+1425 x-5624}\right )}{(3-x) (x+25) \left (-3 x^3-141 x^2-1425 x+5624\right )}dx\)

\(\Big \downarrow \) 3008

\(\displaystyle 2 \int \left (\frac {\log \left (\frac {(x-3) (x+25)^2}{3 x^3+141 x^2+1425 x-5624}\right )}{x-3}+\frac {2 \log \left (\frac {(x-3) (x+25)^2}{3 x^3+141 x^2+1425 x-5624}\right )}{x+25}-\frac {3 \left (3 x^2+94 x+475\right ) \log \left (\frac {(x-3) (x+25)^2}{3 x^3+141 x^2+1425 x-5624}\right )}{3 x^3+141 x^2+1425 x-5624}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle 2 \int \frac {(3 x+19) \log \left (\frac {(x-3) (x+25)^2}{3 x^3+141 x^2+1425 x-5624}\right )}{(3-x) (x+25) \left (-3 x^3-141 x^2-1425 x+5624\right )}dx\)

\(\Big \downarrow \) 3008

\(\displaystyle 2 \int \left (\frac {\log \left (\frac {(x-3) (x+25)^2}{3 x^3+141 x^2+1425 x-5624}\right )}{x-3}+\frac {2 \log \left (\frac {(x-3) (x+25)^2}{3 x^3+141 x^2+1425 x-5624}\right )}{x+25}-\frac {3 \left (3 x^2+94 x+475\right ) \log \left (\frac {(x-3) (x+25)^2}{3 x^3+141 x^2+1425 x-5624}\right )}{3 x^3+141 x^2+1425 x-5624}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle 2 \int \frac {(3 x+19) \log \left (\frac {(x-3) (x+25)^2}{3 x^3+141 x^2+1425 x-5624}\right )}{(3-x) (x+25) \left (-3 x^3-141 x^2-1425 x+5624\right )}dx\)

\(\Big \downarrow \) 3008

\(\displaystyle 2 \int \left (\frac {\log \left (\frac {(x-3) (x+25)^2}{3 x^3+141 x^2+1425 x-5624}\right )}{x-3}+\frac {2 \log \left (\frac {(x-3) (x+25)^2}{3 x^3+141 x^2+1425 x-5624}\right )}{x+25}-\frac {3 \left (3 x^2+94 x+475\right ) \log \left (\frac {(x-3) (x+25)^2}{3 x^3+141 x^2+1425 x-5624}\right )}{3 x^3+141 x^2+1425 x-5624}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle 2 \int \frac {(3 x+19) \log \left (\frac {(x-3) (x+25)^2}{3 x^3+141 x^2+1425 x-5624}\right )}{(3-x) (x+25) \left (-3 x^3-141 x^2-1425 x+5624\right )}dx\)

\(\Big \downarrow \) 3008

\(\displaystyle 2 \int \left (\frac {\log \left (\frac {(x-3) (x+25)^2}{3 x^3+141 x^2+1425 x-5624}\right )}{x-3}+\frac {2 \log \left (\frac {(x-3) (x+25)^2}{3 x^3+141 x^2+1425 x-5624}\right )}{x+25}-\frac {3 \left (3 x^2+94 x+475\right ) \log \left (\frac {(x-3) (x+25)^2}{3 x^3+141 x^2+1425 x-5624}\right )}{3 x^3+141 x^2+1425 x-5624}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle 2 \int \frac {(3 x+19) \log \left (\frac {(x-3) (x+25)^2}{3 x^3+141 x^2+1425 x-5624}\right )}{(3-x) (x+25) \left (-3 x^3-141 x^2-1425 x+5624\right )}dx\)

\(\Big \downarrow \) 3008

\(\displaystyle 2 \int \left (\frac {\log \left (\frac {(x-3) (x+25)^2}{3 x^3+141 x^2+1425 x-5624}\right )}{x-3}+\frac {2 \log \left (\frac {(x-3) (x+25)^2}{3 x^3+141 x^2+1425 x-5624}\right )}{x+25}-\frac {3 \left (3 x^2+94 x+475\right ) \log \left (\frac {(x-3) (x+25)^2}{3 x^3+141 x^2+1425 x-5624}\right )}{3 x^3+141 x^2+1425 x-5624}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle 2 \int \frac {(3 x+19) \log \left (\frac {(x-3) (x+25)^2}{3 x^3+141 x^2+1425 x-5624}\right )}{(3-x) (x+25) \left (-3 x^3-141 x^2-1425 x+5624\right )}dx\)

\(\Big \downarrow \) 3008

\(\displaystyle 2 \int \left (\frac {\log \left (\frac {(x-3) (x+25)^2}{3 x^3+141 x^2+1425 x-5624}\right )}{x-3}+\frac {2 \log \left (\frac {(x-3) (x+25)^2}{3 x^3+141 x^2+1425 x-5624}\right )}{x+25}-\frac {3 \left (3 x^2+94 x+475\right ) \log \left (\frac {(x-3) (x+25)^2}{3 x^3+141 x^2+1425 x-5624}\right )}{3 x^3+141 x^2+1425 x-5624}\right )dx\)

input

Int[((38 + 6*x)*Log[(-1875 + 475*x + 47*x^2 + x^3)/(-5624 + 1425*x + 141*x 
^2 + 3*x^3)])/(421800 - 230603*x + 15151*x^2 + 4302*x^3 + 207*x^4 + 3*x^5) 
,x]
 

output

$Aborted
 

3.4.50.3.1 Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2463

Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[ExpandIntegr 
and[u, Qx^p, x], x] /;  !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && Gt 
Q[Expon[Px, x], 2] &&  !BinomialQ[Px, x] &&  !TrinomialQ[Px, x] && ILtQ[p, 
0]
 

rule 3008

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*(RGx_), x_Symbol] :> With 
[{u = ExpandIntegrand[(a + b*Log[c*RFx^p])^n, RGx, x]}, Int[u, x] /; SumQ[u 
]] /; FreeQ[{a, b, c, p}, x] && RationalFunctionQ[RFx, x] && RationalFuncti 
onQ[RGx, x] && IGtQ[n, 0]
 

rule 7239

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl 
erIntegrandQ[v, u, x]]
 

3.4.50.4 Maple [A] (verified)

Time = 1.27 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.52

method	result	size
norman	\(\ln \left (\frac {x^{3}+47 x^{2}+475 x -1875}{3 x^{3}+141 x^{2}+1425 x -5624}\right )^{2}\)	\(35\)
parts	\(-2 \ln \left (\frac {x^{3}+47 x^{2}+475 x -1875}{3 x^{3}+141 x^{2}+1425 x -5624}\right ) \ln \left (3 x^{3}+141 x^{2}+1425 x -5624\right )+4 \ln \left (x +25\right ) \ln \left (\frac {x^{3}+47 x^{2}+475 x -1875}{3 x^{3}+141 x^{2}+1425 x -5624}\right )+2 \ln \left (-3+x \right ) \ln \left (\frac {x^{3}+47 x^{2}+475 x -1875}{3 x^{3}+141 x^{2}+1425 x -5624}\right )-\ln \left (-3+x \right )^{2}-4 \ln \left (-3+x \right ) \ln \left (\frac {25}{28}+\frac {x}{28}\right )+2 \ln \left (-3+x \right ) \ln \left (3 x^{3}+141 x^{2}+1425 x -5624\right )+2 \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (3 \textit {\_Z}^{3}+141 \textit {\_Z}^{2}+1425 \textit {\_Z} -5624\right )}{\sum }\left (-\ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (3 x^{3}+141 x^{2}+1425 x -5624\right )+\ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\frac {\underline {\hspace {1.25 ex}}\alpha +47-\sqrt {-3 \underline {\hspace {1.25 ex}}\alpha ^{2}-94 \underline {\hspace {1.25 ex}}\alpha +309}+2 x}{3 \underline {\hspace {1.25 ex}}\alpha +47-\sqrt {-3 \underline {\hspace {1.25 ex}}\alpha ^{2}-94 \underline {\hspace {1.25 ex}}\alpha +309}}\right )+\ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\frac {\underline {\hspace {1.25 ex}}\alpha +47+\sqrt {-3 \underline {\hspace {1.25 ex}}\alpha ^{2}-94 \underline {\hspace {1.25 ex}}\alpha +309}+2 x}{3 \underline {\hspace {1.25 ex}}\alpha +47+\sqrt {-3 \underline {\hspace {1.25 ex}}\alpha ^{2}-94 \underline {\hspace {1.25 ex}}\alpha +309}}\right )+\operatorname {dilog}\left (\frac {\underline {\hspace {1.25 ex}}\alpha +47-\sqrt {-3 \underline {\hspace {1.25 ex}}\alpha ^{2}-94 \underline {\hspace {1.25 ex}}\alpha +309}+2 x}{3 \underline {\hspace {1.25 ex}}\alpha +47-\sqrt {-3 \underline {\hspace {1.25 ex}}\alpha ^{2}-94 \underline {\hspace {1.25 ex}}\alpha +309}}\right )+\operatorname {dilog}\left (\frac {\underline {\hspace {1.25 ex}}\alpha +47+\sqrt {-3 \underline {\hspace {1.25 ex}}\alpha ^{2}-94 \underline {\hspace {1.25 ex}}\alpha +309}+2 x}{3 \underline {\hspace {1.25 ex}}\alpha +47+\sqrt {-3 \underline {\hspace {1.25 ex}}\alpha ^{2}-94 \underline {\hspace {1.25 ex}}\alpha +309}}\right )+\frac {\ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right )^{2}}{2}\right )\right )+4 \ln \left (x +25\right ) \ln \left (3 x^{3}+141 x^{2}+1425 x -5624\right )-4 \ln \left (x +25\right )^{2}-4 \left (\ln \left (x +25\right )-\ln \left (\frac {25}{28}+\frac {x}{28}\right )\right ) \ln \left (-\frac {x}{28}+\frac {3}{28}\right )\)	\(455\)
default	\(\text {Expression too large to display}\)	\(1016\)
risch	\(\text {Expression too large to display}\)	\(6661\)

input

int((6*x+38)*ln((x^3+47*x^2+475*x-1875)/(3*x^3+141*x^2+1425*x-5624))/(3*x^ 
5+207*x^4+4302*x^3+15151*x^2-230603*x+421800),x,method=_RETURNVERBOSE)
 

output

ln((x^3+47*x^2+475*x-1875)/(3*x^3+141*x^2+1425*x-5624))^2
 

3.4.50.5 Fricas [A] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.48 \begin {dmath*} \int \frac {(38+6 x) \log \left (\frac {-1875+475 x+47 x^2+x^3}{-5624+1425 x+141 x^2+3 x^3}\right )}{421800-230603 x+15151 x^2+4302 x^3+207 x^4+3 x^5} \, dx=\log \left (\frac {x^{3} + 47 \, x^{2} + 475 \, x - 1875}{3 \, x^{3} + 141 \, x^{2} + 1425 \, x - 5624}\right )^{2} \end {dmath*}

input

integrate((6*x+38)*log((x^3+47*x^2+475*x-1875)/(3*x^3+141*x^2+1425*x-5624) 
)/(3*x^5+207*x^4+4302*x^3+15151*x^2-230603*x+421800),x, algorithm=\
 

output

log((x^3 + 47*x^2 + 475*x - 1875)/(3*x^3 + 141*x^2 + 1425*x - 5624))^2
 

3.4.50.6 Sympy [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.35 \begin {dmath*} \int \frac {(38+6 x) \log \left (\frac {-1875+475 x+47 x^2+x^3}{-5624+1425 x+141 x^2+3 x^3}\right )}{421800-230603 x+15151 x^2+4302 x^3+207 x^4+3 x^5} \, dx=\log {\left (\frac {x^{3} + 47 x^{2} + 475 x - 1875}{3 x^{3} + 141 x^{2} + 1425 x - 5624} \right )}^{2} \end {dmath*}

input

integrate((6*x+38)*ln((x**3+47*x**2+475*x-1875)/(3*x**3+141*x**2+1425*x-56 
24))/(3*x**5+207*x**4+4302*x**3+15151*x**2-230603*x+421800),x)
 

output

log((x**3 + 47*x**2 + 475*x - 1875)/(3*x**3 + 141*x**2 + 1425*x - 5624))** 
2
 

3.4.50.7 Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 139 vs. \(2 (23) = 46\).

Time = 0.24 (sec) , antiderivative size = 139, normalized size of antiderivative = 6.04 \begin {dmath*} \int \frac {(38+6 x) \log \left (\frac {-1875+475 x+47 x^2+x^3}{-5624+1425 x+141 x^2+3 x^3}\right )}{421800-230603 x+15151 x^2+4302 x^3+207 x^4+3 x^5} \, dx=2 \, {\left (2 \, \log \left (x + 25\right ) + \log \left (x - 3\right )\right )} \log \left (3 \, x^{3} + 141 \, x^{2} + 1425 \, x - 5624\right ) - \log \left (3 \, x^{3} + 141 \, x^{2} + 1425 \, x - 5624\right )^{2} - 4 \, \log \left (x + 25\right )^{2} - 4 \, \log \left (x + 25\right ) \log \left (x - 3\right ) - \log \left (x - 3\right )^{2} - 2 \, {\left (\log \left (3 \, x^{3} + 141 \, x^{2} + 1425 \, x - 5624\right ) - 2 \, \log \left (x + 25\right ) - \log \left (x - 3\right )\right )} \log \left (\frac {x^{3} + 47 \, x^{2} + 475 \, x - 1875}{3 \, x^{3} + 141 \, x^{2} + 1425 \, x - 5624}\right ) \end {dmath*}

input

integrate((6*x+38)*log((x^3+47*x^2+475*x-1875)/(3*x^3+141*x^2+1425*x-5624) 
)/(3*x^5+207*x^4+4302*x^3+15151*x^2-230603*x+421800),x, algorithm=\
 

output

2*(2*log(x + 25) + log(x - 3))*log(3*x^3 + 141*x^2 + 1425*x - 5624) - log( 
3*x^3 + 141*x^2 + 1425*x - 5624)^2 - 4*log(x + 25)^2 - 4*log(x + 25)*log(x 
 - 3) - log(x - 3)^2 - 2*(log(3*x^3 + 141*x^2 + 1425*x - 5624) - 2*log(x + 
 25) - log(x - 3))*log((x^3 + 47*x^2 + 475*x - 1875)/(3*x^3 + 141*x^2 + 14 
25*x - 5624))
 

3.4.50.8 Giac [A] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.48 \begin {dmath*} \int \frac {(38+6 x) \log \left (\frac {-1875+475 x+47 x^2+x^3}{-5624+1425 x+141 x^2+3 x^3}\right )}{421800-230603 x+15151 x^2+4302 x^3+207 x^4+3 x^5} \, dx=\log \left (\frac {x^{3} + 47 \, x^{2} + 475 \, x - 1875}{3 \, x^{3} + 141 \, x^{2} + 1425 \, x - 5624}\right )^{2} \end {dmath*}

input

integrate((6*x+38)*log((x^3+47*x^2+475*x-1875)/(3*x^3+141*x^2+1425*x-5624) 
)/(3*x^5+207*x^4+4302*x^3+15151*x^2-230603*x+421800),x, algorithm=\
 

output

log((x^3 + 47*x^2 + 475*x - 1875)/(3*x^3 + 141*x^2 + 1425*x - 5624))^2
 

3.4.50.9 Mupad [B] (verification not implemented)

Time = 13.89 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.04 \begin {dmath*} \int \frac {(38+6 x) \log \left (\frac {-1875+475 x+47 x^2+x^3}{-5624+1425 x+141 x^2+3 x^3}\right )}{421800-230603 x+15151 x^2+4302 x^3+207 x^4+3 x^5} \, dx={\ln \left (\frac {1}{3}-\frac {1}{3\,\left (3\,x^3+141\,x^2+1425\,x-5624\right )}\right )}^2 \end {dmath*}

input

int((log((475*x + 47*x^2 + x^3 - 1875)/(1425*x + 141*x^2 + 3*x^3 - 5624))* 
(6*x + 38))/(15151*x^2 - 230603*x + 4302*x^3 + 207*x^4 + 3*x^5 + 421800),x 
)
 

output

log(1/3 - 1/(3*(1425*x + 141*x^2 + 3*x^3 - 5624)))^2