\(\int \frac {(a+b \log (c (d+e x)^n))^3}{(f+g x) (h+i x)} \, dx\) [232]

Optimal result

Integrand size = 31, antiderivative size = 372 \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{(f+g x) (h+i x)} \, dx=\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3 \log \left (\frac {e (f+g x)}{e f-d g}\right )}{g h-f i}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3 \log \left (\frac {e (h+i x)}{e h-d i}\right )}{g h-f i}+\frac {3 b n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \operatorname {PolyLog}\left (2,-\frac {g (d+e x)}{e f-d g}\right )}{g h-f i}-\frac {3 b n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \operatorname {PolyLog}\left (2,-\frac {i (d+e x)}{e h-d i}\right )}{g h-f i}-\frac {6 b^2 n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \operatorname {PolyLog}\left (3,-\frac {g (d+e x)}{e f-d g}\right )}{g h-f i}+\frac {6 b^2 n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \operatorname {PolyLog}\left (3,-\frac {i (d+e x)}{e h-d i}\right )}{g h-f i}+\frac {6 b^3 n^3 \operatorname {PolyLog}\left (4,-\frac {g (d+e x)}{e f-d g}\right )}{g h-f i}-\frac {6 b^3 n^3 \operatorname {PolyLog}\left (4,-\frac {i (d+e x)}{e h-d i}\right )}{g h-f i} \] Output:

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

Mathematica [A] (verified)

Time = 0.45 (sec) , antiderivative size = 599, normalized size of antiderivative = 1.61 \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{(f+g x) (h+i x)} \, dx=\frac {\left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^3 \log (f+g x)-\left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^3 \log (h+i x)+3 b n \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2 \left (\log (d+e x) \left (\log \left (\frac {e (f+g x)}{e f-d g}\right )-\log \left (\frac {e (h+i x)}{e h-d i}\right )\right )+\operatorname {PolyLog}\left (2,\frac {g (d+e x)}{-e f+d g}\right )-\operatorname {PolyLog}\left (2,\frac {i (d+e x)}{-e h+d i}\right )\right )+6 b^2 n^2 \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right ) \left (\frac {1}{2} \log ^2(d+e x) \log \left (\frac {e (f+g x)}{e f-d g}\right )-\frac {1}{2} \log ^2(d+e x) \log \left (\frac {e (h+i x)}{e h-d i}\right )+\log (d+e x) \operatorname {PolyLog}\left (2,\frac {g (d+e x)}{-e f+d g}\right )-\log (d+e x) \operatorname {PolyLog}\left (2,\frac {i (d+e x)}{-e h+d i}\right )-\operatorname {PolyLog}\left (3,\frac {g (d+e x)}{-e f+d g}\right )+\operatorname {PolyLog}\left (3,\frac {i (d+e x)}{-e h+d i}\right )\right )+b^3 n^3 \left (\log ^3(d+e x) \log \left (\frac {e (f+g x)}{e f-d g}\right )-\log ^3(d+e x) \log \left (\frac {e (h+i x)}{e h-d i}\right )+3 \log ^2(d+e x) \operatorname {PolyLog}\left (2,\frac {g (d+e x)}{-e f+d g}\right )-3 \log ^2(d+e x) \operatorname {PolyLog}\left (2,\frac {i (d+e x)}{-e h+d i}\right )-6 \log (d+e x) \operatorname {PolyLog}\left (3,\frac {g (d+e x)}{-e f+d g}\right )+6 \log (d+e x) \operatorname {PolyLog}\left (3,\frac {i (d+e x)}{-e h+d i}\right )+6 \operatorname {PolyLog}\left (4,\frac {g (d+e x)}{-e f+d g}\right )-6 \operatorname {PolyLog}\left (4,\frac {i (d+e x)}{-e h+d i}\right )\right )}{g h-f i} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 1.44 (sec) , antiderivative size = 372, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {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.

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

\(\Big \downarrow \) 2865

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

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]
 

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 13.37 (sec) , antiderivative size = 2696, normalized size of antiderivative = 7.25

method	result	size
risch	\(\text {Expression too large to display}\)	\(2696\)

Input:

int((a+b*ln(c*(e*x+d)^n))^3/(g*x+f)/(i*x+h),x,method=_RETURNVERBOSE)
 

Output:

3/4*(I*b*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2-I*b*Pi*csgn(I*(e*x+d)^ 
n)*csgn(I*c*(e*x+d)^n)*csgn(I*c)-I*b*Pi*csgn(I*c*(e*x+d)^n)^3+I*b*Pi*csgn( 
I*c*(e*x+d)^n)^2*csgn(I*c)+2*b*ln(c)+2*a)^2*b*(ln((e*x+d)^n)/(f*i-g*h)*ln( 
i*x+h)-ln((e*x+d)^n)/(f*i-g*h)*ln(g*x+f)-e*n*(1/(f*i-g*h)*(dilog(((i*x+h)* 
e+d*i-e*h)/(d*i-e*h))/e+ln(i*x+h)*ln(((i*x+h)*e+d*i-e*h)/(d*i-e*h))/e)-1/( 
f*i-g*h)*(dilog(((g*x+f)*e+d*g-e*f)/(d*g-e*f))/e+ln(g*x+f)*ln(((g*x+f)*e+d 
*g-e*f)/(d*g-e*f))/e)))+1/8*(I*b*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^ 
2-I*b*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)*csgn(I*c)-I*b*Pi*csgn(I*c*( 
e*x+d)^n)^3+I*b*Pi*csgn(I*c*(e*x+d)^n)^2*csgn(I*c)+2*b*ln(c)+2*a)^3*(1/(f* 
i-g*h)*ln(i*x+h)-1/(f*i-g*h)*ln(g*x+f))-6*b^3*n^2/(f*i-g*h)*polylog(3,-i*( 
e*x+d)/(-d*i+e*h))*ln((e*x+d)^n)+6*b^3*n^2/(f*i-g*h)*polylog(3,-g*(e*x+d)/ 
(-d*g+e*f))*ln((e*x+d)^n)+3*b^3*n^3/(f*i-g*h)*ln(e*x+d)^2*polylog(2,-g*(e* 
x+d)/(-d*g+e*f))-b^3/(f*i-g*h)*ln((e*x+d)*i-d*i+e*h)*ln(e*x+d)^3*n^3+b^3/( 
f*i-g*h)*ln(g*(e*x+d)-d*g+e*f)*ln(e*x+d)^3*n^3+3*b^3*n^3/(f*i-g*h)*dilog(( 
(e*x+d)*i-d*i+e*h)/(-d*i+e*h))*ln(e*x+d)^2+3*b^3*n/(f*i-g*h)*dilog(((e*x+d 
)*i-d*i+e*h)/(-d*i+e*h))*ln((e*x+d)^n)^2+3*b^3*n^3/(f*i-g*h)*ln(e*x+d)^3*l 
n(((e*x+d)*i-d*i+e*h)/(-d*i+e*h))-3*b^3*n^3/(f*i-g*h)*dilog((g*(e*x+d)-d*g 
+e*f)/(-d*g+e*f))*ln(e*x+d)^2-3*b^3*n/(f*i-g*h)*dilog((g*(e*x+d)-d*g+e*f)/ 
(-d*g+e*f))*ln((e*x+d)^n)^2-3*b^3*n^3/(f*i-g*h)*ln(e*x+d)^3*ln((g*(e*x+d)- 
d*g+e*f)/(-d*g+e*f))-2*b^3*n^3/(f*i-g*h)*ln(e*x+d)^3*ln(1+i*(e*x+d)/(-d...
 

Fricas [F]

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

integrate((a+b*log(c*(e*x+d)^n))^3/(g*x+f)/(i*x+h),x, algorithm="fricas")
 

Output:

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

Sympy [F]

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

integrate((a+b*ln(c*(e*x+d)**n))**3/(g*x+f)/(i*x+h),x)
 

Output:

Integral((a + b*log(c*(d + e*x)**n))**3/((f + g*x)*(h + i*x)), x)
 

Maxima [F]

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

integrate((a+b*log(c*(e*x+d)^n))^3/(g*x+f)/(i*x+h),x, algorithm="maxima")
 

Output:

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

Giac [F]

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

integrate((a+b*log(c*(e*x+d)^n))^3/(g*x+f)/(i*x+h),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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