3.3.33 \(\int \frac {(a+b \log (c (d+e x)^n))^3}{(f+g x) (h+i x)^2} \, dx\) [233]

3.3.33.1 Optimal result

Integrand size = 31, antiderivative size = 602 \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{(f+g x) (h+i x)^2} \, dx=-\frac {i (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{(e h-d i) (g h-f i) (h+i x)}+\frac {g \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \log \left (\frac {e (f+g x)}{e f-d g}\right )}{(g h-f i)^2}+\frac {3 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e (h+i x)}{e h-d i}\right )}{(e h-d i) (g h-f i)}-\frac {g \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \log \left (\frac {e (h+i x)}{e h-d i}\right )}{(g h-f i)^2}+\frac {3 b g n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \operatorname {PolyLog}\left (2,-\frac {g (d+e x)}{e f-d g}\right )}{(g h-f i)^2}+\frac {6 b^2 e n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {i (d+e x)}{e h-d i}\right )}{(e h-d i) (g h-f i)}-\frac {3 b g n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \operatorname {PolyLog}\left (2,-\frac {i (d+e x)}{e h-d i}\right )}{(g h-f i)^2}-\frac {6 b^2 g n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \operatorname {PolyLog}\left (3,-\frac {g (d+e x)}{e f-d g}\right )}{(g h-f i)^2}-\frac {6 b^3 e n^3 \operatorname {PolyLog}\left (3,-\frac {i (d+e x)}{e h-d i}\right )}{(e h-d i) (g h-f i)}+\frac {6 b^2 g n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \operatorname {PolyLog}\left (3,-\frac {i (d+e x)}{e h-d i}\right )}{(g h-f i)^2}+\frac {6 b^3 g n^3 \operatorname {PolyLog}\left (4,-\frac {g (d+e x)}{e f-d g}\right )}{(g h-f i)^2}-\frac {6 b^3 g n^3 \operatorname {PolyLog}\left (4,-\frac {i (d+e x)}{e h-d i}\right )}{(g h-f i)^2} \]

output

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

3.3.33.2 Mathematica [A] (verified)

Time = 0.72 (sec) , antiderivative size = 1025, normalized size of antiderivative = 1.70 \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{(f+g x) (h+i x)^2} \, dx=\frac {(e h-d i) (g h-f i) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^3+g (e h-d i) (h+i x) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^3 \log (f+g x)-g (e h-d i) (h+i 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 ((g h-f i) (i (d+e x) \log (d+e x)-e (h+i x) \log (h+i x))-g (e h-d i) (h+i x) \left (\log (d+e x) \log \left (\frac {e (f+g x)}{e f-d g}\right )+\operatorname {PolyLog}\left (2,\frac {g (d+e x)}{-e f+d g}\right )\right )+g (e h-d i) (h+i x) \left (\log (d+e x) \log \left (\frac {e (h+i x)}{e h-d i}\right )+\operatorname {PolyLog}\left (2,\frac {i (d+e x)}{-e h+d i}\right )\right )\right )-3 b^2 n^2 \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right ) \left ((g h-f i) \left (\log (d+e x) \left (i (d+e x) \log (d+e x)-2 e (h+i x) \log \left (\frac {e (h+i x)}{e h-d i}\right )\right )-2 e (h+i x) \operatorname {PolyLog}\left (2,\frac {i (d+e x)}{-e h+d i}\right )\right )-g (e h-d i) (h+i x) \left (\log ^2(d+e x) \log \left (\frac {e (f+g x)}{e f-d g}\right )+2 \log (d+e x) \operatorname {PolyLog}\left (2,\frac {g (d+e x)}{-e f+d g}\right )-2 \operatorname {PolyLog}\left (3,\frac {g (d+e x)}{-e f+d g}\right )\right )+g (e h-d i) (h+i x) \left (\log ^2(d+e x) \log \left (\frac {e (h+i x)}{e h-d i}\right )+2 \log (d+e x) \operatorname {PolyLog}\left (2,\frac {i (d+e x)}{-e h+d i}\right )-2 \operatorname {PolyLog}\left (3,\frac {i (d+e x)}{-e h+d i}\right )\right )\right )-b^3 n^3 \left ((g h-f i) \left (\log ^2(d+e x) \left (i (d+e x) \log (d+e x)-3 e (h+i x) \log \left (\frac {e (h+i x)}{e h-d i}\right )\right )-6 e (h+i x) \log (d+e x) \operatorname {PolyLog}\left (2,\frac {i (d+e x)}{-e h+d i}\right )+6 e (h+i x) \operatorname {PolyLog}\left (3,\frac {i (d+e x)}{-e h+d i}\right )\right )-g (e h-d i) (h+i x) \left (\log ^3(d+e x) \log \left (\frac {e (f+g x)}{e f-d g}\right )+3 \log ^2(d+e x) \operatorname {PolyLog}\left (2,\frac {g (d+e x)}{-e f+d g}\right )-6 \log (d+e x) \operatorname {PolyLog}\left (3,\frac {g (d+e x)}{-e f+d g}\right )+6 \operatorname {PolyLog}\left (4,\frac {g (d+e x)}{-e f+d g}\right )\right )+g (e h-d i) (h+i x) \left (\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 {i (d+e x)}{-e h+d i}\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 {i (d+e x)}{-e h+d i}\right )\right )\right )}{(e h-d i) (g h-f i)^2 (h+i x)} \]

input

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

output

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

3.3.33.3 Rubi [A] (verified)

Time = 1.10 (sec) , antiderivative size = 602, 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.

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

\(\Big \downarrow \) 2865

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

\(\Big \downarrow \) 2009

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

input

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

output

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

3.3.33.3.1 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]
 

3.3.33.4 Maple [F]

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

input

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

output

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

3.3.33.5 Fricas [F]

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

input

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

3.3.33.6 Sympy [F(-1)]

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

input

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

output

Timed out
 

3.3.33.7 Maxima [F]

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

input

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

output

a^3*(g*log(g*x + f)/(g^2*h^2 - 2*f*g*h*i + f^2*i^2) - g*log(i*x + h)/(g^2* 
h^2 - 2*f*g*h*i + f^2*i^2) + 1/(g*h^2 - f*h*i + (g*h*i - f*i^2)*x)) + inte 
grate((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^2*x^3 + f*h^2 + (2*g*h*i + f*i 
^2)*x^2 + (g*h^2 + 2*f*h*i)*x), x)
 

3.3.33.8 Giac [F]

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

input

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

output

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

3.3.33.9 Mupad [F(-1)]

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

input

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

output

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