\(\int \frac {(-3-\log (\frac {3}{x})) \sqrt [4]{4 x+x \log (\frac {3}{x})}}{50176 x+12544 x \log (\frac {3}{x})+\sqrt {4 x+x \log (\frac {3}{x})} (16 x+4 x \log (\frac {3}{x}))+\sqrt [4]{4 x+x \log (\frac {3}{x})} (1792 x+448 x \log (\frac {3}{x}))} \, dx\) [2996]

Optimal result

Integrand size = 102, antiderivative size = 20 \[ \int \frac {\left (-3-\log \left (\frac {3}{x}\right )\right ) \sqrt [4]{4 x+x \log \left (\frac {3}{x}\right )}}{50176 x+12544 x \log \left (\frac {3}{x}\right )+\sqrt {4 x+x \log \left (\frac {3}{x}\right )} \left (16 x+4 x \log \left (\frac {3}{x}\right )\right )+\sqrt [4]{4 x+x \log \left (\frac {3}{x}\right )} \left (1792 x+448 x \log \left (\frac {3}{x}\right )\right )} \, dx=2+\frac {1}{56+\sqrt [4]{x \left (4+\log \left (\frac {3}{x}\right )\right )}} \] Output:

2+1/(((ln(3/x)+4)*x)^(1/4)+56)
 

Mathematica [A] (verified)

Time = 3.54 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90 \[ \int \frac {\left (-3-\log \left (\frac {3}{x}\right )\right ) \sqrt [4]{4 x+x \log \left (\frac {3}{x}\right )}}{50176 x+12544 x \log \left (\frac {3}{x}\right )+\sqrt {4 x+x \log \left (\frac {3}{x}\right )} \left (16 x+4 x \log \left (\frac {3}{x}\right )\right )+\sqrt [4]{4 x+x \log \left (\frac {3}{x}\right )} \left (1792 x+448 x \log \left (\frac {3}{x}\right )\right )} \, dx=\frac {1}{56+\sqrt [4]{x \left (4+\log \left (\frac {3}{x}\right )\right )}} \] Input:

Integrate[((-3 - Log[3/x])*(4*x + x*Log[3/x])^(1/4))/(50176*x + 12544*x*Lo 
g[3/x] + Sqrt[4*x + x*Log[3/x]]*(16*x + 4*x*Log[3/x]) + (4*x + x*Log[3/x]) 
^(1/4)*(1792*x + 448*x*Log[3/x])),x]
 

Output:

(56 + (x*(4 + Log[3/x]))^(1/4))^(-1)
 

Rubi [A] (verified)

Time = 0.68 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.039, Rules used = {7239, 27, 25, 7237}

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.

\(\displaystyle \int \frac {\left (-\log \left (\frac {3}{x}\right )-3\right ) \sqrt [4]{4 x+x \log \left (\frac {3}{x}\right )}}{50176 x+12544 x \log \left (\frac {3}{x}\right )+\sqrt {4 x+x \log \left (\frac {3}{x}\right )} \left (16 x+4 x \log \left (\frac {3}{x}\right )\right )+\sqrt [4]{4 x+x \log \left (\frac {3}{x}\right )} \left (1792 x+448 x \log \left (\frac {3}{x}\right )\right )} \, dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-\log \left (\frac {3}{x}\right )-3}{4 \left (4 x+x \log \left (\frac {3}{x}\right )\right )^{3/4} \left (\sqrt [4]{x \left (\log \left (\frac {3}{x}\right )+4\right )}+56\right )^2}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{4} \int -\frac {\log \left (\frac {3}{x}\right )+3}{\left (\log \left (\frac {3}{x}\right ) x+4 x\right )^{3/4} \left (\sqrt [4]{x \left (\log \left (\frac {3}{x}\right )+4\right )}+56\right )^2}dx\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {1}{4} \int \frac {\log \left (\frac {3}{x}\right )+3}{\left (\log \left (\frac {3}{x}\right ) x+4 x\right )^{3/4} \left (\sqrt [4]{x \left (\log \left (\frac {3}{x}\right )+4\right )}+56\right )^2}dx\)

\(\Big \downarrow \) 7237

\(\displaystyle \frac {1}{\sqrt [4]{x \left (\log \left (\frac {3}{x}\right )+4\right )}+56}\)

Input:

Int[((-3 - Log[3/x])*(4*x + x*Log[3/x])^(1/4))/(50176*x + 12544*x*Log[3/x] 
 + Sqrt[4*x + x*Log[3/x]]*(16*x + 4*x*Log[3/x]) + (4*x + x*Log[3/x])^(1/4) 
*(1792*x + 448*x*Log[3/x])),x]
 

Output:

(56 + (x*(4 + Log[3/x]))^(1/4))^(-1)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 7237

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Si 
mp[q*(y^(m + 1)/(m + 1)), x] /;  !FalseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]
 

rule 7239

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

Maple [F]

\[\int \frac {\left (-\ln \left (\frac {3}{x}\right )-3\right ) \left (x \ln \left (\frac {3}{x}\right )+4 x \right )^{\frac {1}{4}}}{\left (4 x \ln \left (\frac {3}{x}\right )+16 x \right ) \sqrt {x \ln \left (\frac {3}{x}\right )+4 x}+\left (448 x \ln \left (\frac {3}{x}\right )+1792 x \right ) \left (x \ln \left (\frac {3}{x}\right )+4 x \right )^{\frac {1}{4}}+12544 x \ln \left (\frac {3}{x}\right )+50176 x}d x\]

Input:

int((-ln(3/x)-3)*(x*ln(3/x)+4*x)^(1/4)/((4*x*ln(3/x)+16*x)*(x*ln(3/x)+4*x) 
^(1/2)+(448*x*ln(3/x)+1792*x)*(x*ln(3/x)+4*x)^(1/4)+12544*x*ln(3/x)+50176* 
x),x)
 

Output:

int((-ln(3/x)-3)*(x*ln(3/x)+4*x)^(1/4)/((4*x*ln(3/x)+16*x)*(x*ln(3/x)+4*x) 
^(1/2)+(448*x*ln(3/x)+1792*x)*(x*ln(3/x)+4*x)^(1/4)+12544*x*ln(3/x)+50176* 
x),x)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 64 vs. \(2 (18) = 36\).

Time = 0.10 (sec) , antiderivative size = 64, normalized size of antiderivative = 3.20 \[ \int \frac {\left (-3-\log \left (\frac {3}{x}\right )\right ) \sqrt [4]{4 x+x \log \left (\frac {3}{x}\right )}}{50176 x+12544 x \log \left (\frac {3}{x}\right )+\sqrt {4 x+x \log \left (\frac {3}{x}\right )} \left (16 x+4 x \log \left (\frac {3}{x}\right )\right )+\sqrt [4]{4 x+x \log \left (\frac {3}{x}\right )} \left (1792 x+448 x \log \left (\frac {3}{x}\right )\right )} \, dx=\frac {{\left (x \log \left (\frac {3}{x}\right ) + 4 \, x\right )}^{\frac {3}{4}} - 56 \, \sqrt {x \log \left (\frac {3}{x}\right ) + 4 \, x} + 3136 \, {\left (x \log \left (\frac {3}{x}\right ) + 4 \, x\right )}^{\frac {1}{4}} - 175616}{x \log \left (\frac {3}{x}\right ) + 4 \, x - 9834496} \] Input:

integrate((-log(3/x)-3)*(x*log(3/x)+4*x)^(1/4)/((4*x*log(3/x)+16*x)*(x*log 
(3/x)+4*x)^(1/2)+(448*x*log(3/x)+1792*x)*(x*log(3/x)+4*x)^(1/4)+12544*x*lo 
g(3/x)+50176*x),x, algorithm="fricas")
 

Output:

((x*log(3/x) + 4*x)^(3/4) - 56*sqrt(x*log(3/x) + 4*x) + 3136*(x*log(3/x) + 
 4*x)^(1/4) - 175616)/(x*log(3/x) + 4*x - 9834496)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (-3-\log \left (\frac {3}{x}\right )\right ) \sqrt [4]{4 x+x \log \left (\frac {3}{x}\right )}}{50176 x+12544 x \log \left (\frac {3}{x}\right )+\sqrt {4 x+x \log \left (\frac {3}{x}\right )} \left (16 x+4 x \log \left (\frac {3}{x}\right )\right )+\sqrt [4]{4 x+x \log \left (\frac {3}{x}\right )} \left (1792 x+448 x \log \left (\frac {3}{x}\right )\right )} \, dx=\text {Timed out} \] Input:

integrate((-ln(3/x)-3)*(x*ln(3/x)+4*x)**(1/4)/((4*x*ln(3/x)+16*x)*(x*ln(3/ 
x)+4*x)**(1/2)+(448*x*ln(3/x)+1792*x)*(x*ln(3/x)+4*x)**(1/4)+12544*x*ln(3/ 
x)+50176*x),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\left (-3-\log \left (\frac {3}{x}\right )\right ) \sqrt [4]{4 x+x \log \left (\frac {3}{x}\right )}}{50176 x+12544 x \log \left (\frac {3}{x}\right )+\sqrt {4 x+x \log \left (\frac {3}{x}\right )} \left (16 x+4 x \log \left (\frac {3}{x}\right )\right )+\sqrt [4]{4 x+x \log \left (\frac {3}{x}\right )} \left (1792 x+448 x \log \left (\frac {3}{x}\right )\right )} \, dx=\int { -\frac {{\left (x \log \left (\frac {3}{x}\right ) + 4 \, x\right )}^{\frac {1}{4}} {\left (\log \left (\frac {3}{x}\right ) + 3\right )}}{4 \, {\left (3136 \, x \log \left (\frac {3}{x}\right ) + {\left (x \log \left (\frac {3}{x}\right ) + 4 \, x\right )}^{\frac {3}{2}} + 112 \, {\left (x \log \left (\frac {3}{x}\right ) + 4 \, x\right )}^{\frac {5}{4}} + 12544 \, x\right )}} \,d x } \] Input:

integrate((-log(3/x)-3)*(x*log(3/x)+4*x)^(1/4)/((4*x*log(3/x)+16*x)*(x*log 
(3/x)+4*x)^(1/2)+(448*x*log(3/x)+1792*x)*(x*log(3/x)+4*x)^(1/4)+12544*x*lo 
g(3/x)+50176*x),x, algorithm="maxima")
 

Output:

-1/4*integrate((x*log(3/x) + 4*x)^(1/4)*(log(3/x) + 3)/(3136*x*log(3/x) + 
(x*log(3/x) + 4*x)^(3/2) + 112*(x*log(3/x) + 4*x)^(5/4) + 12544*x), x)
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90 \[ \int \frac {\left (-3-\log \left (\frac {3}{x}\right )\right ) \sqrt [4]{4 x+x \log \left (\frac {3}{x}\right )}}{50176 x+12544 x \log \left (\frac {3}{x}\right )+\sqrt {4 x+x \log \left (\frac {3}{x}\right )} \left (16 x+4 x \log \left (\frac {3}{x}\right )\right )+\sqrt [4]{4 x+x \log \left (\frac {3}{x}\right )} \left (1792 x+448 x \log \left (\frac {3}{x}\right )\right )} \, dx=\frac {1}{{\left (x \log \left (\frac {3}{x}\right ) + 4 \, x\right )}^{\frac {1}{4}} + 56} \] Input:

integrate((-log(3/x)-3)*(x*log(3/x)+4*x)^(1/4)/((4*x*log(3/x)+16*x)*(x*log 
(3/x)+4*x)^(1/2)+(448*x*log(3/x)+1792*x)*(x*log(3/x)+4*x)^(1/4)+12544*x*lo 
g(3/x)+50176*x),x, algorithm="giac")
 

Output:

1/((x*log(3/x) + 4*x)^(1/4) + 56)
 

Mupad [B] (verification not implemented)

Time = 3.29 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90 \[ \int \frac {\left (-3-\log \left (\frac {3}{x}\right )\right ) \sqrt [4]{4 x+x \log \left (\frac {3}{x}\right )}}{50176 x+12544 x \log \left (\frac {3}{x}\right )+\sqrt {4 x+x \log \left (\frac {3}{x}\right )} \left (16 x+4 x \log \left (\frac {3}{x}\right )\right )+\sqrt [4]{4 x+x \log \left (\frac {3}{x}\right )} \left (1792 x+448 x \log \left (\frac {3}{x}\right )\right )} \, dx=\frac {1}{{\left (4\,x+x\,\ln \left (\frac {3}{x}\right )\right )}^{1/4}+56} \] Input:

int(-((log(3/x) + 3)*(4*x + x*log(3/x))^(1/4))/(50176*x + (4*x + x*log(3/x 
))^(1/2)*(16*x + 4*x*log(3/x)) + (4*x + x*log(3/x))^(1/4)*(1792*x + 448*x* 
log(3/x)) + 12544*x*log(3/x)),x)
 

Output:

1/((4*x + x*log(3/x))^(1/4) + 56)
 

Reduce [F]

\[ \int \frac {\left (-3-\log \left (\frac {3}{x}\right )\right ) \sqrt [4]{4 x+x \log \left (\frac {3}{x}\right )}}{50176 x+12544 x \log \left (\frac {3}{x}\right )+\sqrt {4 x+x \log \left (\frac {3}{x}\right )} \left (16 x+4 x \log \left (\frac {3}{x}\right )\right )+\sqrt [4]{4 x+x \log \left (\frac {3}{x}\right )} \left (1792 x+448 x \log \left (\frac {3}{x}\right )\right )} \, dx=-\frac {\left (\int \frac {x^{\frac {1}{4}} \left (\mathrm {log}\left (\frac {3}{x}\right )+4\right )^{\frac {1}{4}} \mathrm {log}\left (\frac {3}{x}\right )}{112 x^{\frac {5}{4}} \left (\mathrm {log}\left (\frac {3}{x}\right )+4\right )^{\frac {1}{4}} \mathrm {log}\left (\frac {3}{x}\right )+448 x^{\frac {5}{4}} \left (\mathrm {log}\left (\frac {3}{x}\right )+4\right )^{\frac {1}{4}}+\sqrt {x}\, \sqrt {\mathrm {log}\left (\frac {3}{x}\right )+4}\, \mathrm {log}\left (\frac {3}{x}\right ) x +4 \sqrt {x}\, \sqrt {\mathrm {log}\left (\frac {3}{x}\right )+4}\, x +3136 \,\mathrm {log}\left (\frac {3}{x}\right ) x +12544 x}d x \right )}{4}-\frac {3 \left (\int \frac {x^{\frac {1}{4}} \left (\mathrm {log}\left (\frac {3}{x}\right )+4\right )^{\frac {1}{4}}}{112 x^{\frac {5}{4}} \left (\mathrm {log}\left (\frac {3}{x}\right )+4\right )^{\frac {1}{4}} \mathrm {log}\left (\frac {3}{x}\right )+448 x^{\frac {5}{4}} \left (\mathrm {log}\left (\frac {3}{x}\right )+4\right )^{\frac {1}{4}}+\sqrt {x}\, \sqrt {\mathrm {log}\left (\frac {3}{x}\right )+4}\, \mathrm {log}\left (\frac {3}{x}\right ) x +4 \sqrt {x}\, \sqrt {\mathrm {log}\left (\frac {3}{x}\right )+4}\, x +3136 \,\mathrm {log}\left (\frac {3}{x}\right ) x +12544 x}d x \right )}{4} \] Input:

int((-log(3/x)-3)*(x*log(3/x)+4*x)^(1/4)/((4*x*log(3/x)+16*x)*(x*log(3/x)+ 
4*x)^(1/2)+(448*x*log(3/x)+1792*x)*(x*log(3/x)+4*x)^(1/4)+12544*x*log(3/x) 
+50176*x),x)
 

Output:

( - int((x**(1/4)*(log(3/x) + 4)**(1/4)*log(3/x))/(112*x**(1/4)*(log(3/x) 
+ 4)**(1/4)*log(3/x)*x + 448*x**(1/4)*(log(3/x) + 4)**(1/4)*x + sqrt(x)*sq 
rt(log(3/x) + 4)*log(3/x)*x + 4*sqrt(x)*sqrt(log(3/x) + 4)*x + 3136*log(3/ 
x)*x + 12544*x),x) - 3*int((x**(1/4)*(log(3/x) + 4)**(1/4))/(112*x**(1/4)* 
(log(3/x) + 4)**(1/4)*log(3/x)*x + 448*x**(1/4)*(log(3/x) + 4)**(1/4)*x + 
sqrt(x)*sqrt(log(3/x) + 4)*log(3/x)*x + 4*sqrt(x)*sqrt(log(3/x) + 4)*x + 3 
136*log(3/x)*x + 12544*x),x))/4