3.1.53 \(\int \frac {\log (e (f (a+b x)^p (c+d x)^q)^r)}{g k+h k x} \, dx\) [53]

3.1.53.1 Optimal result

Integrand size = 32, antiderivative size = 172 \[ \int \frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{g k+h k x} \, dx=-\frac {p r \log \left (-\frac {h (a+b x)}{b g-a h}\right ) \log (g k+h k x)}{h k}-\frac {q r \log \left (-\frac {h (c+d x)}{d g-c h}\right ) \log (g k+h k x)}{h k}+\frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \log (g k+h k x)}{h k}-\frac {p r \operatorname {PolyLog}\left (2,\frac {b (g+h x)}{b g-a h}\right )}{h k}-\frac {q r \operatorname {PolyLog}\left (2,\frac {d (g+h x)}{d g-c h}\right )}{h k} \]

output

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

3.1.53.2 Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.97 \[ \int \frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{g k+h k x} \, dx=\frac {-p r \log (a+b x) \log (g+h x)-q r \log (c+d x) \log (g+h x)+\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \log (g+h x)+p r \log (a+b x) \log \left (\frac {b (g+h x)}{b g-a h}\right )+q r \log (c+d x) \log \left (\frac {d (g+h x)}{d g-c h}\right )+p r \operatorname {PolyLog}\left (2,\frac {h (a+b x)}{-b g+a h}\right )+q r \operatorname {PolyLog}\left (2,\frac {h (c+d x)}{-d g+c h}\right )}{h k} \]

input

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

output

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

3.1.53.3 Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {2980, 2841, 27, 2840, 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.

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

\(\Big \downarrow \) 2980

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

\(\Big \downarrow \) 2841

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2840

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

\(\Big \downarrow \) 2838

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

input

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

output

(Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]*Log[g*k + h*k*x])/(h*k) - (b*p*r*((L 
og[-((h*(a + b*x))/(b*g - a*h))]*Log[g*k + h*k*x])/b + PolyLog[2, (b*(g + 
h*x))/(b*g - a*h)]/b))/(h*k) - (d*q*r*((Log[-((h*(c + d*x))/(d*g - c*h))]* 
Log[g*k + h*k*x])/d + PolyLog[2, (d*(g + h*x))/(d*g - c*h)]/d))/(h*k)
 

3.1.53.3.1 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 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 2840

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/((f_.) + (g_.)*(x_)), x_ 
Symbol] :> Simp[1/g   Subst[Int[(a + b*Log[1 + c*e*(x/g)])/x, x], x, f + g* 
x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[g + c 
*(e*f - d*g), 0]
 

rule 2841

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

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]
 

3.1.53.4 Maple [A] (verified)

Time = 51.53 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.12

method	result	size
parts	\(\frac {\ln \left (e \left (f \left (b x +a \right )^{p} \left (d x +c \right )^{q}\right )^{r}\right ) \ln \left (h x +g \right )}{k h}-\frac {r \left (b p h \left (\frac {\operatorname {dilog}\left (\frac {\left (h x +g \right ) b +a h -b g}{a h -b g}\right )}{b}+\frac {\ln \left (h x +g \right ) \ln \left (\frac {\left (h x +g \right ) b +a h -b g}{a h -b g}\right )}{b}\right )+d q h \left (\frac {\operatorname {dilog}\left (\frac {d \left (h x +g \right )+c h -d g}{c h -d g}\right )}{d}+\frac {\ln \left (h x +g \right ) \ln \left (\frac {d \left (h x +g \right )+c h -d g}{c h -d g}\right )}{d}\right )\right )}{k \,h^{2}}\)	\(192\)

input

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

output

ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)/k*ln(h*x+g)/h-1/k/h^2*r*(b*p*h*(dilog(((h* 
x+g)*b+a*h-b*g)/(a*h-b*g))/b+ln(h*x+g)*ln(((h*x+g)*b+a*h-b*g)/(a*h-b*g))/b 
)+d*q*h*(dilog((d*(h*x+g)+c*h-d*g)/(c*h-d*g))/d+ln(h*x+g)*ln((d*(h*x+g)+c* 
h-d*g)/(c*h-d*g))/d))
 

3.1.53.5 Fricas [F]

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

input

integrate(log(e*(f*(b*x+a)^p*(d*x+c)^q)^r)/(h*k*x+g*k),x, algorithm="frica 
s")
 

output

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

3.1.53.6 Sympy [F(-1)]

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

input

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

output

Timed out
 

3.1.53.7 Maxima [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.19 \[ \int \frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{g k+h k x} \, dx=\frac {{\left (\frac {{\left (\log \left (b x + a\right ) \log \left (\frac {b h x + a h}{b g - a h} + 1\right ) + {\rm Li}_2\left (-\frac {b h x + a h}{b g - a h}\right )\right )} f p}{h k} + \frac {{\left (\log \left (d x + c\right ) \log \left (\frac {d h x + c h}{d g - c h} + 1\right ) + {\rm Li}_2\left (-\frac {d h x + c h}{d g - c h}\right )\right )} f q}{h k}\right )} r}{f} - \frac {{\left (f p \log \left (b x + a\right ) + f q \log \left (d x + c\right )\right )} r \log \left (h k x + g k\right )}{f h k} + \frac {\log \left (h k x + g k\right ) \log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right )}{h k} \]

input

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

output

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

3.1.53.8 Giac [F]

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

input

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

output

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

3.1.53.9 Mupad [F(-1)]

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

input

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

output

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