\(\int \frac {\log (e (f (a+b x)^p (c+d x)^q)^r)}{x} \, dx\) [59]

Optimal result

Integrand size = 25, antiderivative size = 81 \[ \int \frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{x} \, dx=-p r \log (x) \log \left (1+\frac {b x}{a}\right )+\log (x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-q r \log (x) \log \left (1+\frac {d x}{c}\right )-p r \operatorname {PolyLog}\left (2,-\frac {b x}{a}\right )-q r \operatorname {PolyLog}\left (2,-\frac {d x}{c}\right ) \] Output:

-p*r*ln(x)*ln(1+b*x/a)+ln(x)*ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)-q*r*ln(x)*ln( 
1+d*x/c)-p*r*polylog(2,-b*x/a)-q*r*polylog(2,-d*x/c)
 

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.96 \[ \int \frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{x} \, dx=\log (x) \left (-p r \log \left (1+\frac {b x}{a}\right )+\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-q r \log \left (1+\frac {d x}{c}\right )\right )-p r \operatorname {PolyLog}\left (2,-\frac {b x}{a}\right )-q r \operatorname {PolyLog}\left (2,-\frac {d x}{c}\right ) \] Input:

Integrate[Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]/x,x]
 

Output:

Log[x]*(-(p*r*Log[1 + (b*x)/a]) + Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r] - q 
*r*Log[1 + (d*x)/c]) - p*r*PolyLog[2, -((b*x)/a)] - q*r*PolyLog[2, -((d*x) 
/c)]
 

Rubi [A] (verified)

Time = 0.52 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.15, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2980, 2754, 2838}

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.

\(\displaystyle \int \frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{x} \, dx\)

\(\Big \downarrow \) 2980

\(\displaystyle -b p r \int \frac {\log (x)}{a+b x}dx-d q r \int \frac {\log (x)}{c+d x}dx+\log (x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\)

\(\Big \downarrow \) 2754

\(\displaystyle -b p r \left (\frac {\log (x) \log \left (\frac {b x}{a}+1\right )}{b}-\frac {\int \frac {\log \left (\frac {b x}{a}+1\right )}{x}dx}{b}\right )-d q r \left (\frac {\log (x) \log \left (\frac {d x}{c}+1\right )}{d}-\frac {\int \frac {\log \left (\frac {d x}{c}+1\right )}{x}dx}{d}\right )+\log (x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\)

\(\Big \downarrow \) 2838

\(\displaystyle \log (x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-b p r \left (\frac {\operatorname {PolyLog}\left (2,-\frac {b x}{a}\right )}{b}+\frac {\log (x) \log \left (\frac {b x}{a}+1\right )}{b}\right )-d q r \left (\frac {\operatorname {PolyLog}\left (2,-\frac {d x}{c}\right )}{d}+\frac {\log (x) \log \left (\frac {d x}{c}+1\right )}{d}\right )\)

Input:

Int[Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]/x,x]
 

Output:

Log[x]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r] - b*p*r*((Log[x]*Log[1 + (b*x) 
/a])/b + PolyLog[2, -((b*x)/a)]/b) - d*q*r*((Log[x]*Log[1 + (d*x)/c])/d + 
PolyLog[2, -((d*x)/c)]/d)
 

Defintions of rubi rules used

rule 2754

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symb 
ol] :> Simp[Log[1 + e*(x/d)]*((a + b*Log[c*x^n])^p/e), x] - Simp[b*n*(p/e) 
  Int[Log[1 + e*(x/d)]*((a + b*Log[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, 
b, c, d, e, n}, x] && IGtQ[p, 0]
 

rule 2838

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 
, (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
 

rule 2980

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.)) 
^(r_.)]/((g_.) + (h_.)*(x_)), x_Symbol] :> Simp[Log[g + h*x]*(Log[e*(f*(a + 
 b*x)^p*(c + d*x)^q)^r]/h), x] + (-Simp[b*p*(r/h)   Int[Log[g + h*x]/(a + b 
*x), x], x] - Simp[d*q*(r/h)   Int[Log[g + h*x]/(c + d*x), x], x]) /; FreeQ 
[{a, b, c, d, e, f, g, h, p, q, r}, x] && NeQ[b*c - a*d, 0]
 

Maple [A] (verified)

Time = 2.91 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.27

method	result	size
parts	\(\ln \left (x \right ) \ln \left (e \left (f \left (b x +a \right )^{p} \left (d x +c \right )^{q}\right )^{r}\right )-\frac {r \left (\left (\frac {\operatorname {dilog}\left (\frac {b x +a}{a}\right )}{b}+\frac {\ln \left (x \right ) \ln \left (\frac {b x +a}{a}\right )}{b}\right ) b f p +\left (\frac {\operatorname {dilog}\left (\frac {d x +c}{c}\right )}{d}+\frac {\ln \left (x \right ) \ln \left (\frac {d x +c}{c}\right )}{d}\right ) d f q \right )}{f}\)	\(103\)

Input:

int(ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)/x,x,method=_RETURNVERBOSE)
 

Output:

ln(x)*ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)-r/f*((dilog((b*x+a)/a)/b+ln(x)*ln((b 
*x+a)/a)/b)*b*f*p+(dilog((d*x+c)/c)/d+ln(x)*ln((d*x+c)/c)/d)*d*f*q)
 

Fricas [F]

\[ \int \frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{x} \, dx=\int { \frac {\log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right )}{x} \,d x } \] Input:

integrate(log(e*(f*(b*x+a)^p*(d*x+c)^q)^r)/x,x, algorithm="fricas")
 

Output:

integral(log(((b*x + a)^p*(d*x + c)^q*f)^r*e)/x, x)
 

Sympy [F]

\[ \int \frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{x} \, dx=\int \frac {\log {\left (e \left (f \left (a + b x\right )^{p} \left (c + d x\right )^{q}\right )^{r} \right )}}{x}\, dx \] Input:

integrate(ln(e*(f*(b*x+a)**p*(d*x+c)**q)**r)/x,x)
 

Output:

Integral(log(e*(f*(a + b*x)**p*(c + d*x)**q)**r)/x, x)
 

Maxima [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.56 \[ \int \frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{x} \, dx=-\frac {{\left (f p \log \left (b x + a\right ) + f q \log \left (d x + c\right )\right )} r \log \left (x\right )}{f} + \log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right ) \log \left (x\right ) + \frac {{\left ({\left (\log \left (b x + a\right ) \log \left (-\frac {b x + a}{a} + 1\right ) + {\rm Li}_2\left (\frac {b x + a}{a}\right )\right )} f p + {\left (\log \left (d x + c\right ) \log \left (-\frac {d x + c}{c} + 1\right ) + {\rm Li}_2\left (\frac {d x + c}{c}\right )\right )} f q\right )} r}{f} \] Input:

integrate(log(e*(f*(b*x+a)^p*(d*x+c)^q)^r)/x,x, algorithm="maxima")
 

Output:

-(f*p*log(b*x + a) + f*q*log(d*x + c))*r*log(x)/f + log(((b*x + a)^p*(d*x 
+ c)^q*f)^r*e)*log(x) + ((log(b*x + a)*log(-(b*x + a)/a + 1) + dilog((b*x 
+ a)/a))*f*p + (log(d*x + c)*log(-(d*x + c)/c + 1) + dilog((d*x + c)/c))*f 
*q)*r/f
 

Giac [F]

\[ \int \frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{x} \, dx=\int { \frac {\log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right )}{x} \,d x } \] Input:

integrate(log(e*(f*(b*x+a)^p*(d*x+c)^q)^r)/x,x, algorithm="giac")
 

Output:

integrate(log(((b*x + a)^p*(d*x + c)^q*f)^r*e)/x, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{x} \, dx=\int \frac {\ln \left (e\,{\left (f\,{\left (a+b\,x\right )}^p\,{\left (c+d\,x\right )}^q\right )}^r\right )}{x} \,d x \] Input:

int(log(e*(f*(a + b*x)^p*(c + d*x)^q)^r)/x,x)
 

Output:

int(log(e*(f*(a + b*x)^p*(c + d*x)^q)^r)/x, x)
 

Reduce [F]

\[ \int \frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{x} \, dx =\text {Too large to display} \] Input:

int(log(e*(f*(b*x+a)^p*(d*x+c)^q)^r)/x,x)
                                                                                    
                                                                                    
 

Output:

(2*int(log(f**r*(c + d*x)**(q*r)*(a + b*x)**(p*r)*e)/(a**2*c*d*q*x + a**2* 
d**2*q*x**2 + a*b*c**2*p*x + a*b*c*d*p*x**2 + a*b*c*d*q*x**2 + a*b*d**2*q* 
x**3 + b**2*c**2*p*x**2 + b**2*c*d*p*x**3),x)*a**2*c*d*p*q*r + 2*int(log(f 
**r*(c + d*x)**(q*r)*(a + b*x)**(p*r)*e)/(a**2*c*d*q*x + a**2*d**2*q*x**2 
+ a*b*c**2*p*x + a*b*c*d*p*x**2 + a*b*c*d*q*x**2 + a*b*d**2*q*x**3 + b**2* 
c**2*p*x**2 + b**2*c*d*p*x**3),x)*a**2*c*d*q**2*r + 2*int(log(f**r*(c + d* 
x)**(q*r)*(a + b*x)**(p*r)*e)/(a**2*c*d*q*x + a**2*d**2*q*x**2 + a*b*c**2* 
p*x + a*b*c*d*p*x**2 + a*b*c*d*q*x**2 + a*b*d**2*q*x**3 + b**2*c**2*p*x**2 
 + b**2*c*d*p*x**3),x)*a*b*c**2*p**2*r + 2*int(log(f**r*(c + d*x)**(q*r)*( 
a + b*x)**(p*r)*e)/(a**2*c*d*q*x + a**2*d**2*q*x**2 + a*b*c**2*p*x + a*b*c 
*d*p*x**2 + a*b*c*d*q*x**2 + a*b*d**2*q*x**3 + b**2*c**2*p*x**2 + b**2*c*d 
*p*x**3),x)*a*b*c**2*p*q*r + 2*int(log(f**r*(c + d*x)**(q*r)*(a + b*x)**(p 
*r)*e)/(a*c*p + a*c*q + a*d*p*x + a*d*q*x + b*c*p*x + b*c*q*x + b*d*p*x**2 
 + b*d*q*x**2),x)*a*d*p**2*r + 2*int(log(f**r*(c + d*x)**(q*r)*(a + b*x)** 
(p*r)*e)/(a*c*p + a*c*q + a*d*p*x + a*d*q*x + b*c*p*x + b*c*q*x + b*d*p*x* 
*2 + b*d*q*x**2),x)*a*d*p*q*r + 2*int(log(f**r*(c + d*x)**(q*r)*(a + b*x)* 
*(p*r)*e)/(a*c*p + a*c*q + a*d*p*x + a*d*q*x + b*c*p*x + b*c*q*x + b*d*p*x 
**2 + b*d*q*x**2),x)*b*c*p*q*r + 2*int(log(f**r*(c + d*x)**(q*r)*(a + b*x) 
**(p*r)*e)/(a*c*p + a*c*q + a*d*p*x + a*d*q*x + b*c*p*x + b*c*q*x + b*d*p* 
x**2 + b*d*q*x**2),x)*b*c*q**2*r + log(f**r*(c + d*x)**(q*r)*(a + b*x)*...