\(\int \frac {a+b \log (c (d (e+f x)^p)^q)}{(g+h x) (i+j x)^2} \, dx\) [547]

Optimal result

Integrand size = 33, antiderivative size = 268 \[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{(g+h x) (i+j x)^2} \, dx=-\frac {b f p q \log (e+f x)}{(f i-e j) (h i-g j)}+\frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{(h i-g j) (i+j x)}+\frac {h \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \log \left (\frac {f (g+h x)}{f g-e h}\right )}{(h i-g j)^2}+\frac {b f p q \log (i+j x)}{(f i-e j) (h i-g j)}-\frac {h \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \log \left (\frac {f (i+j x)}{f i-e j}\right )}{(h i-g j)^2}+\frac {b h p q \operatorname {PolyLog}\left (2,-\frac {h (e+f x)}{f g-e h}\right )}{(h i-g j)^2}-\frac {b h p q \operatorname {PolyLog}\left (2,-\frac {j (e+f x)}{f i-e j}\right )}{(h i-g j)^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 225, normalized size of antiderivative = 0.84 \[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{(g+h x) (i+j x)^2} \, dx=\frac {\frac {a (h i-g j)}{i+j x}+\frac {b (h i-g j) \log \left (c \left (d (e+f x)^p\right )^q\right )}{i+j x}+h \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \log \left (\frac {f (g+h x)}{f g-e h}\right )-\frac {b f (h i-g j) p q (\log (e+f x)-\log (i+j x))}{f i-e j}-h \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \log \left (\frac {f (i+j x)}{f i-e j}\right )+b h p q \operatorname {PolyLog}\left (2,\frac {h (e+f x)}{-f g+e h}\right )-b h p q \operatorname {PolyLog}\left (2,\frac {j (e+f x)}{-f i+e j}\right )}{(h i-g j)^2} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 1.45 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2895, 2865, 2009}

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.

\(\displaystyle \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{(g+h x) (i+j x)^2} \, dx\)

\(\Big \downarrow \) 2895

\(\displaystyle \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{(g+h x) (i+j x)^2}dx\)

\(\Big \downarrow \) 2865

\(\displaystyle \int \left (\frac {h^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{(g+h x) (h i-g j)^2}-\frac {h j \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{(i+j x) (h i-g j)^2}-\frac {j \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{(i+j x)^2 (h i-g j)}\right )dx\)

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2865

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*(RFx_), x_Sy 
mbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, RFx, x]}, 
Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n}, x] && RationalFunctionQ[ 
RFx, x] && IntegerQ[p]
 

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]

\[\int \frac {a +b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )}{\left (h x +g \right ) \left (j x +i \right )^{2}}d x\]

Input:

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

Output:

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

Fricas [F]

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

integrate((a+b*log(c*(d*(f*x+e)^p)^q))/(h*x+g)/(j*x+i)^2,x, algorithm="fri 
cas")
 

Output:

integral((b*log(((f*x + e)^p*d)^q*c) + a)/(h*j^2*x^3 + g*i^2 + (2*h*i*j + 
g*j^2)*x^2 + (h*i^2 + 2*g*i*j)*x), x)
 

Sympy [F]

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

integrate((a+b*ln(c*(d*(f*x+e)**p)**q))/(h*x+g)/(j*x+i)**2,x)
 

Output:

Integral((a + b*log(c*(d*(e + f*x)**p)**q))/((g + h*x)*(i + j*x)**2), x)
 

Maxima [F]

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

integrate((a+b*log(c*(d*(f*x+e)^p)^q))/(h*x+g)/(j*x+i)^2,x, algorithm="max 
ima")
 

Output:

a*(h*log(h*x + g)/(h^2*i^2 - 2*g*h*i*j + g^2*j^2) - h*log(j*x + i)/(h^2*i^ 
2 - 2*g*h*i*j + g^2*j^2) + 1/(h*i^2 - g*i*j + (h*i*j - g*j^2)*x)) + b*inte 
grate((q*log(d) + log(((f*x + e)^p)^q) + log(c))/(h*j^2*x^3 + g*i^2 + (2*h 
*i*j + g*j^2)*x^2 + (h*i^2 + 2*g*i*j)*x), x)
 

Giac [F]

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

integrate((a+b*log(c*(d*(f*x+e)^p)^q))/(h*x+g)/(j*x+i)^2,x, algorithm="gia 
c")
 

Output:

integrate((b*log(((f*x + e)^p*d)^q*c) + a)/((h*x + g)*(j*x + i)^2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{(g+h x) (i+j x)^2} \, dx=\int \frac {a+b\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )}{\left (g+h\,x\right )\,{\left (i+j\,x\right )}^2} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{(g+h x) (i+j x)^2} \, dx=\left (\int \frac {\mathrm {log}\left (d^{q} \left (f x +e \right )^{p q} c \right )}{h \,j^{2} x^{3}+g \,j^{2} x^{2}+2 h i j \,x^{2}+2 g i j x -h x -g}d x \right ) b +\left (\int \frac {1}{h \,j^{2} x^{3}+g \,j^{2} x^{2}+2 h i j \,x^{2}+2 g i j x -h x -g}d x \right ) a \] Input:

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

Output:

int(log(d**q*(e + f*x)**(p*q)*c)/(2*g*i*j*x + g*j**2*x**2 - g + 2*h*i*j*x* 
*2 + h*j**2*x**3 - h*x),x)*b + int(1/(2*g*i*j*x + g*j**2*x**2 - g + 2*h*i* 
j*x**2 + h*j**2*x**3 - h*x),x)*a