\(\int \frac {a+b \log (c (d (e+f x)^p)^q)}{g+h x} \, dx\) [545]

Optimal result

Integrand size = 26, antiderivative size = 68 \[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{g+h x} \, dx=\frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \log \left (\frac {f (g+h x)}{f g-e h}\right )}{h}+\frac {b p q \operatorname {PolyLog}\left (2,-\frac {h (e+f x)}{f g-e h}\right )}{h} \] Output:

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.99 \[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{g+h x} \, dx=\frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \log \left (\frac {f (g+h x)}{f g-e h}\right )}{h}+\frac {b p q \operatorname {PolyLog}\left (2,\frac {h (e+f x)}{-f g+e h}\right )}{h} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.70 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2895, 2841, 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.

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

\(\Big \downarrow \) 2895

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

\(\Big \downarrow \) 2841

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

\(\Big \downarrow \) 2840

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

\(\Big \downarrow \) 2838

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

Input:

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

Output:

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

Defintions of rubi rules used

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 2895

Int[((a_.) + Log[(c_.)*((d_.)*((e_.) + (f_.)*(x_))^(m_.))^(n_)]*(b_.))^(p_. 
)*(u_.), x_Symbol] :> Subst[Int[u*(a + b*Log[c*d^n*(e + f*x)^(m*n)])^p, x], 
 c*d^n*(e + f*x)^(m*n), c*(d*(e + f*x)^m)^n] /; FreeQ[{a, b, c, d, e, f, m, 
 n, p}, x] &&  !IntegerQ[n] &&  !(EqQ[d, 1] && EqQ[m, 1]) && IntegralFreeQ[ 
IntHide[u*(a + b*Log[c*d^n*(e + f*x)^(m*n)])^p, x]]
 

Maple [F]

\[\int \frac {a +b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )}{h x +g}d x\]

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{g+h x} \, dx=\frac {2 \left (\int \frac {\mathrm {log}\left (d^{q} \left (f x +e \right )^{p q} c \right )}{f h \,x^{2}+e h x +f g x +e g}d x \right ) b e h p q -2 \left (\int \frac {\mathrm {log}\left (d^{q} \left (f x +e \right )^{p q} c \right )}{f h \,x^{2}+e h x +f g x +e g}d x \right ) b f g p q +2 \,\mathrm {log}\left (h x +g \right ) a p q +\mathrm {log}\left (d^{q} \left (f x +e \right )^{p q} c \right )^{2} b}{2 h p q} \] Input:

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

Output:

(2*int(log(d**q*(e + f*x)**(p*q)*c)/(e*g + e*h*x + f*g*x + f*h*x**2),x)*b* 
e*h*p*q - 2*int(log(d**q*(e + f*x)**(p*q)*c)/(e*g + e*h*x + f*g*x + f*h*x* 
*2),x)*b*f*g*p*q + 2*log(g + h*x)*a*p*q + log(d**q*(e + f*x)**(p*q)*c)**2* 
b)/(2*h*p*q)