3.6.39 \(\int \frac {(a+b \log (c (d (e+f x)^p)^q))^3}{(g+h x) (i+j x)^2} \, dx\) [539]

3.6.39.1 Optimal result

Integrand size = 35, antiderivative size = 659 \[ \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{(g+h x) (i+j x)^2} \, dx=-\frac {j (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{(f i-e j) (h i-g j) (i+j x)}+\frac {h \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3 \log \left (\frac {f (g+h x)}{f g-e h}\right )}{(h i-g j)^2}+\frac {3 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \log \left (\frac {f (i+j x)}{f i-e j}\right )}{(f i-e j) (h i-g j)}-\frac {h \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3 \log \left (\frac {f (i+j x)}{f i-e j}\right )}{(h i-g j)^2}+\frac {3 b h p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \operatorname {PolyLog}\left (2,-\frac {h (e+f x)}{f g-e h}\right )}{(h i-g j)^2}+\frac {6 b^2 f p^2 q^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \operatorname {PolyLog}\left (2,-\frac {j (e+f x)}{f i-e j}\right )}{(f i-e j) (h i-g j)}-\frac {3 b h p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \operatorname {PolyLog}\left (2,-\frac {j (e+f x)}{f i-e j}\right )}{(h i-g j)^2}-\frac {6 b^2 h p^2 q^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \operatorname {PolyLog}\left (3,-\frac {h (e+f x)}{f g-e h}\right )}{(h i-g j)^2}-\frac {6 b^3 f p^3 q^3 \operatorname {PolyLog}\left (3,-\frac {j (e+f x)}{f i-e j}\right )}{(f i-e j) (h i-g j)}+\frac {6 b^2 h p^2 q^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \operatorname {PolyLog}\left (3,-\frac {j (e+f x)}{f i-e j}\right )}{(h i-g j)^2}+\frac {6 b^3 h p^3 q^3 \operatorname {PolyLog}\left (4,-\frac {h (e+f x)}{f g-e h}\right )}{(h i-g j)^2}-\frac {6 b^3 h p^3 q^3 \operatorname {PolyLog}\left (4,-\frac {j (e+f x)}{f i-e j}\right )}{(h i-g j)^2} \]

output

-j*(f*x+e)*(a+b*ln(c*(d*(f*x+e)^p)^q))^3/(-e*j+f*i)/(-g*j+h*i)/(j*x+i)+h*( 
a+b*ln(c*(d*(f*x+e)^p)^q))^3*ln(f*(h*x+g)/(-e*h+f*g))/(-g*j+h*i)^2+3*b*f*p 
*q*(a+b*ln(c*(d*(f*x+e)^p)^q))^2*ln(f*(j*x+i)/(-e*j+f*i))/(-e*j+f*i)/(-g*j 
+h*i)-h*(a+b*ln(c*(d*(f*x+e)^p)^q))^3*ln(f*(j*x+i)/(-e*j+f*i))/(-g*j+h*i)^ 
2+3*b*h*p*q*(a+b*ln(c*(d*(f*x+e)^p)^q))^2*polylog(2,-h*(f*x+e)/(-e*h+f*g)) 
/(-g*j+h*i)^2+6*b^2*f*p^2*q^2*(a+b*ln(c*(d*(f*x+e)^p)^q))*polylog(2,-j*(f* 
x+e)/(-e*j+f*i))/(-e*j+f*i)/(-g*j+h*i)-3*b*h*p*q*(a+b*ln(c*(d*(f*x+e)^p)^q 
))^2*polylog(2,-j*(f*x+e)/(-e*j+f*i))/(-g*j+h*i)^2-6*b^2*h*p^2*q^2*(a+b*ln 
(c*(d*(f*x+e)^p)^q))*polylog(3,-h*(f*x+e)/(-e*h+f*g))/(-g*j+h*i)^2-6*b^3*f 
*p^3*q^3*polylog(3,-j*(f*x+e)/(-e*j+f*i))/(-e*j+f*i)/(-g*j+h*i)+6*b^2*h*p^ 
2*q^2*(a+b*ln(c*(d*(f*x+e)^p)^q))*polylog(3,-j*(f*x+e)/(-e*j+f*i))/(-g*j+h 
*i)^2+6*b^3*h*p^3*q^3*polylog(4,-h*(f*x+e)/(-e*h+f*g))/(-g*j+h*i)^2-6*b^3* 
h*p^3*q^3*polylog(4,-j*(f*x+e)/(-e*j+f*i))/(-g*j+h*i)^2
 

3.6.39.2 Mathematica [A] (verified)

Time = 0.73 (sec) , antiderivative size = 1057, normalized size of antiderivative = 1.60 \[ \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{(g+h x) (i+j x)^2} \, dx=\frac {(f i-e j) (h i-g j) \left (a-b p q \log (e+f x)+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3+h (f i-e j) (i+j x) \left (a-b p q \log (e+f x)+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3 \log (g+h x)-h (f i-e j) (i+j x) \left (a-b p q \log (e+f x)+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3 \log (i+j x)-3 b p q \left (a-b p q \log (e+f x)+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \left ((h i-g j) (j (e+f x) \log (e+f x)-f (i+j x) \log (i+j x))-h (f i-e j) (i+j x) \left (\log (e+f x) \log \left (\frac {f (g+h x)}{f g-e h}\right )+\operatorname {PolyLog}\left (2,\frac {h (e+f x)}{-f g+e h}\right )\right )+h (f i-e j) (i+j x) \left (\log (e+f x) \log \left (\frac {f (i+j x)}{f i-e j}\right )+\operatorname {PolyLog}\left (2,\frac {j (e+f x)}{-f i+e j}\right )\right )\right )-3 b^2 p^2 q^2 \left (a-b p q \log (e+f x)+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \left ((h i-g j) \left (\log (e+f x) \left (j (e+f x) \log (e+f x)-2 f (i+j x) \log \left (\frac {f (i+j x)}{f i-e j}\right )\right )-2 f (i+j x) \operatorname {PolyLog}\left (2,\frac {j (e+f x)}{-f i+e j}\right )\right )-h (f i-e j) (i+j x) \left (\log ^2(e+f x) \log \left (\frac {f (g+h x)}{f g-e h}\right )+2 \log (e+f x) \operatorname {PolyLog}\left (2,\frac {h (e+f x)}{-f g+e h}\right )-2 \operatorname {PolyLog}\left (3,\frac {h (e+f x)}{-f g+e h}\right )\right )+h (f i-e j) (i+j x) \left (\log ^2(e+f x) \log \left (\frac {f (i+j x)}{f i-e j}\right )+2 \log (e+f x) \operatorname {PolyLog}\left (2,\frac {j (e+f x)}{-f i+e j}\right )-2 \operatorname {PolyLog}\left (3,\frac {j (e+f x)}{-f i+e j}\right )\right )\right )-b^3 p^3 q^3 \left ((h i-g j) \left (\log ^2(e+f x) \left (j (e+f x) \log (e+f x)-3 f (i+j x) \log \left (\frac {f (i+j x)}{f i-e j}\right )\right )-6 f (i+j x) \log (e+f x) \operatorname {PolyLog}\left (2,\frac {j (e+f x)}{-f i+e j}\right )+6 f (i+j x) \operatorname {PolyLog}\left (3,\frac {j (e+f x)}{-f i+e j}\right )\right )-h (f i-e j) (i+j x) \left (\log ^3(e+f x) \log \left (\frac {f (g+h x)}{f g-e h}\right )+3 \log ^2(e+f x) \operatorname {PolyLog}\left (2,\frac {h (e+f x)}{-f g+e h}\right )-6 \log (e+f x) \operatorname {PolyLog}\left (3,\frac {h (e+f x)}{-f g+e h}\right )+6 \operatorname {PolyLog}\left (4,\frac {h (e+f x)}{-f g+e h}\right )\right )+h (f i-e j) (i+j x) \left (\log ^3(e+f x) \log \left (\frac {f (i+j x)}{f i-e j}\right )+3 \log ^2(e+f x) \operatorname {PolyLog}\left (2,\frac {j (e+f x)}{-f i+e j}\right )-6 \log (e+f x) \operatorname {PolyLog}\left (3,\frac {j (e+f x)}{-f i+e j}\right )+6 \operatorname {PolyLog}\left (4,\frac {j (e+f x)}{-f i+e j}\right )\right )\right )}{(f i-e j) (h i-g j)^2 (i+j x)} \]

input

Integrate[(a + b*Log[c*(d*(e + f*x)^p)^q])^3/((g + h*x)*(i + j*x)^2),x]
 

output

((f*i - e*j)*(h*i - g*j)*(a - b*p*q*Log[e + f*x] + b*Log[c*(d*(e + f*x)^p) 
^q])^3 + h*(f*i - e*j)*(i + j*x)*(a - b*p*q*Log[e + f*x] + b*Log[c*(d*(e + 
 f*x)^p)^q])^3*Log[g + h*x] - h*(f*i - e*j)*(i + j*x)*(a - b*p*q*Log[e + f 
*x] + b*Log[c*(d*(e + f*x)^p)^q])^3*Log[i + j*x] - 3*b*p*q*(a - b*p*q*Log[ 
e + f*x] + b*Log[c*(d*(e + f*x)^p)^q])^2*((h*i - g*j)*(j*(e + f*x)*Log[e + 
 f*x] - f*(i + j*x)*Log[i + j*x]) - h*(f*i - e*j)*(i + j*x)*(Log[e + f*x]* 
Log[(f*(g + h*x))/(f*g - e*h)] + PolyLog[2, (h*(e + f*x))/(-(f*g) + e*h)]) 
 + h*(f*i - e*j)*(i + j*x)*(Log[e + f*x]*Log[(f*(i + j*x))/(f*i - e*j)] + 
PolyLog[2, (j*(e + f*x))/(-(f*i) + e*j)])) - 3*b^2*p^2*q^2*(a - b*p*q*Log[ 
e + f*x] + b*Log[c*(d*(e + f*x)^p)^q])*((h*i - g*j)*(Log[e + f*x]*(j*(e + 
f*x)*Log[e + f*x] - 2*f*(i + j*x)*Log[(f*(i + j*x))/(f*i - e*j)]) - 2*f*(i 
 + j*x)*PolyLog[2, (j*(e + f*x))/(-(f*i) + e*j)]) - h*(f*i - e*j)*(i + j*x 
)*(Log[e + f*x]^2*Log[(f*(g + h*x))/(f*g - e*h)] + 2*Log[e + f*x]*PolyLog[ 
2, (h*(e + f*x))/(-(f*g) + e*h)] - 2*PolyLog[3, (h*(e + f*x))/(-(f*g) + e* 
h)]) + h*(f*i - e*j)*(i + j*x)*(Log[e + f*x]^2*Log[(f*(i + j*x))/(f*i - e* 
j)] + 2*Log[e + f*x]*PolyLog[2, (j*(e + f*x))/(-(f*i) + e*j)] - 2*PolyLog[ 
3, (j*(e + f*x))/(-(f*i) + e*j)])) - b^3*p^3*q^3*((h*i - g*j)*(Log[e + f*x 
]^2*(j*(e + f*x)*Log[e + f*x] - 3*f*(i + j*x)*Log[(f*(i + j*x))/(f*i - e*j 
)]) - 6*f*(i + j*x)*Log[e + f*x]*PolyLog[2, (j*(e + f*x))/(-(f*i) + e*j)] 
+ 6*f*(i + j*x)*PolyLog[3, (j*(e + f*x))/(-(f*i) + e*j)]) - h*(f*i - e*...
 

3.6.39.3 Rubi [A] (verified)

Time = 2.04 (sec) , antiderivative size = 659, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, 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.

\(\displaystyle \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{(g+h x) (i+j x)^2} \, dx\)

\(\Big \downarrow \) 2895

\(\displaystyle \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{(g+h x) (i+j x)^2}dx\)

\(\Big \downarrow \) 2865

\(\displaystyle \int \left (\frac {h^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{(g+h x) (h i-g j)^2}-\frac {h j \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{(i+j x) (h i-g j)^2}-\frac {j \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{(i+j x)^2 (h i-g j)}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {6 b^2 f p^2 q^2 \operatorname {PolyLog}\left (2,-\frac {j (e+f x)}{f i-e j}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{(f i-e j) (h i-g j)}-\frac {6 b^2 h p^2 q^2 \operatorname {PolyLog}\left (3,-\frac {h (e+f x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{(h i-g j)^2}+\frac {6 b^2 h p^2 q^2 \operatorname {PolyLog}\left (3,-\frac {j (e+f x)}{f i-e j}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{(h i-g j)^2}+\frac {3 b h p q \operatorname {PolyLog}\left (2,-\frac {h (e+f x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{(h i-g j)^2}-\frac {3 b h p q \operatorname {PolyLog}\left (2,-\frac {j (e+f x)}{f i-e j}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{(h i-g j)^2}+\frac {3 b f p q \log \left (\frac {f (i+j x)}{f i-e j}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{(f i-e j) (h i-g j)}-\frac {j (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{(i+j x) (f i-e j) (h i-g j)}+\frac {h \log \left (\frac {f (g+h x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{(h i-g j)^2}-\frac {h \log \left (\frac {f (i+j x)}{f i-e j}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{(h i-g j)^2}-\frac {6 b^3 f p^3 q^3 \operatorname {PolyLog}\left (3,-\frac {j (e+f x)}{f i-e j}\right )}{(f i-e j) (h i-g j)}+\frac {6 b^3 h p^3 q^3 \operatorname {PolyLog}\left (4,-\frac {h (e+f x)}{f g-e h}\right )}{(h i-g j)^2}-\frac {6 b^3 h p^3 q^3 \operatorname {PolyLog}\left (4,-\frac {j (e+f x)}{f i-e j}\right )}{(h i-g j)^2}\)

input

Int[(a + b*Log[c*(d*(e + f*x)^p)^q])^3/((g + h*x)*(i + j*x)^2),x]
 

output

-((j*(e + f*x)*(a + b*Log[c*(d*(e + f*x)^p)^q])^3)/((f*i - e*j)*(h*i - g*j 
)*(i + j*x))) + (h*(a + b*Log[c*(d*(e + f*x)^p)^q])^3*Log[(f*(g + h*x))/(f 
*g - e*h)])/(h*i - g*j)^2 + (3*b*f*p*q*(a + b*Log[c*(d*(e + f*x)^p)^q])^2* 
Log[(f*(i + j*x))/(f*i - e*j)])/((f*i - e*j)*(h*i - g*j)) - (h*(a + b*Log[ 
c*(d*(e + f*x)^p)^q])^3*Log[(f*(i + j*x))/(f*i - e*j)])/(h*i - g*j)^2 + (3 
*b*h*p*q*(a + b*Log[c*(d*(e + f*x)^p)^q])^2*PolyLog[2, -((h*(e + f*x))/(f* 
g - e*h))])/(h*i - g*j)^2 + (6*b^2*f*p^2*q^2*(a + b*Log[c*(d*(e + f*x)^p)^ 
q])*PolyLog[2, -((j*(e + f*x))/(f*i - e*j))])/((f*i - e*j)*(h*i - g*j)) - 
(3*b*h*p*q*(a + b*Log[c*(d*(e + f*x)^p)^q])^2*PolyLog[2, -((j*(e + f*x))/( 
f*i - e*j))])/(h*i - g*j)^2 - (6*b^2*h*p^2*q^2*(a + b*Log[c*(d*(e + f*x)^p 
)^q])*PolyLog[3, -((h*(e + f*x))/(f*g - e*h))])/(h*i - g*j)^2 - (6*b^3*f*p 
^3*q^3*PolyLog[3, -((j*(e + f*x))/(f*i - e*j))])/((f*i - e*j)*(h*i - g*j)) 
 + (6*b^2*h*p^2*q^2*(a + b*Log[c*(d*(e + f*x)^p)^q])*PolyLog[3, -((j*(e + 
f*x))/(f*i - e*j))])/(h*i - g*j)^2 + (6*b^3*h*p^3*q^3*PolyLog[4, -((h*(e + 
 f*x))/(f*g - e*h))])/(h*i - g*j)^2 - (6*b^3*h*p^3*q^3*PolyLog[4, -((j*(e 
+ f*x))/(f*i - e*j))])/(h*i - g*j)^2
 

3.6.39.3.1 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]]
 

3.6.39.4 Maple [F]

\[\int \frac {{\left (a +b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )\right )}^{3}}{\left (h x +g \right ) \left (j x +i \right )^{2}}d x\]

input

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

output

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

3.6.39.5 Fricas [F]

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

input

integrate((a+b*log(c*(d*(f*x+e)^p)^q))^3/(h*x+g)/(j*x+i)^2,x, algorithm="f 
ricas")
 

output

integral((b^3*log(((f*x + e)^p*d)^q*c)^3 + 3*a*b^2*log(((f*x + e)^p*d)^q*c 
)^2 + 3*a^2*b*log(((f*x + e)^p*d)^q*c) + a^3)/(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)
 

3.6.39.6 Sympy [F]

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

input

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

output

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

3.6.39.7 Maxima [F]

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

input

integrate((a+b*log(c*(d*(f*x+e)^p)^q))^3/(h*x+g)/(j*x+i)^2,x, algorithm="m 
axima")
 

output

a^3*(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)) + inte 
grate((b^3*log(((f*x + e)^p)^q)^3 + 3*(q*log(d) + log(c))*a^2*b + 3*(q^2*l 
og(d)^2 + 2*q*log(c)*log(d) + log(c)^2)*a*b^2 + (q^3*log(d)^3 + 3*q^2*log( 
c)*log(d)^2 + 3*q*log(c)^2*log(d) + log(c)^3)*b^3 + 3*((q*log(d) + log(c)) 
*b^3 + a*b^2)*log(((f*x + e)^p)^q)^2 + 3*(2*(q*log(d) + log(c))*a*b^2 + (q 
^2*log(d)^2 + 2*q*log(c)*log(d) + log(c)^2)*b^3 + a^2*b)*log(((f*x + e)^p) 
^q))/(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)
 

3.6.39.8 Giac [F]

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

input

integrate((a+b*log(c*(d*(f*x+e)^p)^q))^3/(h*x+g)/(j*x+i)^2,x, algorithm="g 
iac")
 

output

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

3.6.39.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{(g+h x) (i+j x)^2} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )\right )}^3}{\left (g+h\,x\right )\,{\left (i+j\,x\right )}^2} \,d x \]

input

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

output

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