\(\int \frac {(a+b \log (c (d+e x)^n))^2}{f+g x} \, dx\) [226]

Optimal result

Integrand size = 24, antiderivative size = 111 \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f+g x} \, dx=\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e (f+g x)}{e f-d g}\right )}{g}+\frac {2 b n \left (a+b \log \left (c (d+e x)^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {g (d+e x)}{e f-d g}\right )}{g}-\frac {2 b^2 n^2 \operatorname {PolyLog}\left (3,-\frac {g (d+e x)}{e f-d g}\right )}{g} \] Output:

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

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.75 \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f+g x} \, dx=\frac {\left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2 \log (f+g x)+2 b n \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right ) \left (\log (d+e x) \log \left (\frac {e (f+g x)}{e f-d g}\right )+\operatorname {PolyLog}\left (2,\frac {g (d+e x)}{-e f+d g}\right )\right )+b^2 n^2 \left (\log ^2(d+e x) \log \left (\frac {e (f+g x)}{e f-d g}\right )+2 \log (d+e x) \operatorname {PolyLog}\left (2,\frac {g (d+e x)}{-e f+d g}\right )-2 \operatorname {PolyLog}\left (3,\frac {g (d+e x)}{-e f+d g}\right )\right )}{g} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.79 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.95, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2843, 2881, 2821, 7143}

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.

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

\(\Big \downarrow \) 2843

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

\(\Big \downarrow \) 2881

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

\(\Big \downarrow \) 2821

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

\(\Big \downarrow \) 7143

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 2821

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b 
_.))^(p_.))/(x_), x_Symbol] :> Simp[(-PolyLog[2, (-d)*f*x^m])*((a + b*Log[c 
*x^n])^p/m), x] + Simp[b*n*(p/m)   Int[PolyLog[2, (-d)*f*x^m]*((a + b*Log[c 
*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 
0] && EqQ[d*e, 1]
 

rule 2843

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)/((f_.) + (g_. 
)*(x_)), x_Symbol] :> Simp[Log[e*((f + g*x)/(e*f - d*g))]*((a + b*Log[c*(d 
+ e*x)^n])^p/g), x] - Simp[b*e*n*(p/g)   Int[Log[(e*(f + g*x))/(e*f - d*g)] 
*((a + b*Log[c*(d + e*x)^n])^(p - 1)/(d + e*x)), x], x] /; FreeQ[{a, b, c, 
d, e, f, g, n, p}, x] && NeQ[e*f - d*g, 0] && IGtQ[p, 1]
 

rule 2881

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + Log 
[(h_.)*((i_.) + (j_.)*(x_))^(m_.)]*(g_.))*((k_.) + (l_.)*(x_))^(r_.), x_Sym 
bol] :> Simp[1/e   Subst[Int[(k*(x/d))^r*(a + b*Log[c*x^n])^p*(f + g*Log[h* 
((e*i - d*j)/e + j*(x/e))^m]), x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, 
 f, g, h, i, j, k, l, n, p, r}, x] && EqQ[e*k - d*l, 0]
 

rule 7143

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S 
ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d 
, e, n, p}, x] && EqQ[b*d, a*e]
 

Maple [C] (warning: unable to verify)

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

Time = 1.37 (sec) , antiderivative size = 737, normalized size of antiderivative = 6.64

method	result	size
risch	\(\frac {b^{2} \ln \left (g \left (e x +d \right )-d g +e f \right ) \ln \left (e x +d \right )^{2} n^{2}}{g}-\frac {2 b^{2} \ln \left (g \left (e x +d \right )-d g +e f \right ) \ln \left (\left (e x +d \right )^{n}\right ) \ln \left (e x +d \right ) n}{g}+\frac {b^{2} \ln \left (g \left (e x +d \right )-d g +e f \right ) \ln \left (\left (e x +d \right )^{n}\right )^{2}}{g}+\frac {b^{2} n^{2} \ln \left (e x +d \right )^{2} \ln \left (1-\frac {g \left (e x +d \right )}{d g -e f}\right )}{g}+\frac {2 b^{2} n^{2} \ln \left (e x +d \right ) \operatorname {polylog}\left (2, \frac {g \left (e x +d \right )}{d g -e f}\right )}{g}-\frac {2 b^{2} n^{2} \operatorname {polylog}\left (3, \frac {g \left (e x +d \right )}{d g -e f}\right )}{g}-\frac {2 b^{2} n^{2} \operatorname {dilog}\left (\frac {g \left (e x +d \right )-d g +e f}{-d g +e f}\right ) \ln \left (e x +d \right )}{g}+\frac {2 b^{2} n \operatorname {dilog}\left (\frac {g \left (e x +d \right )-d g +e f}{-d g +e f}\right ) \ln \left (\left (e x +d \right )^{n}\right )}{g}-\frac {2 b^{2} n^{2} \ln \left (e x +d \right )^{2} \ln \left (\frac {g \left (e x +d \right )-d g +e f}{-d g +e f}\right )}{g}+\frac {2 b^{2} n \ln \left (e x +d \right ) \ln \left (\frac {g \left (e x +d \right )-d g +e f}{-d g +e f}\right ) \ln \left (\left (e x +d \right )^{n}\right )}{g}+\left (i b \pi \,\operatorname {csgn}\left (i \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}-i b \pi \,\operatorname {csgn}\left (i \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \right )-i b \pi \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3}+i b \pi \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2} \operatorname {csgn}\left (i c \right )+2 b \ln \left (c \right )+2 a \right ) b \left (\frac {\ln \left (\left (e x +d \right )^{n}\right ) \ln \left (g x +f \right )}{g}-\frac {n e \left (\frac {\operatorname {dilog}\left (\frac {\left (g x +f \right ) e +d g -e f}{d g -e f}\right )}{e}+\frac {\ln \left (g x +f \right ) \ln \left (\frac {\left (g x +f \right ) e +d g -e f}{d g -e f}\right )}{e}\right )}{g}\right )+\frac {{\left (i b \pi \,\operatorname {csgn}\left (i \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}-i b \pi \,\operatorname {csgn}\left (i \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \right )-i b \pi \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3}+i b \pi \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2} \operatorname {csgn}\left (i c \right )+2 b \ln \left (c \right )+2 a \right )}^{2} \ln \left (g x +f \right )}{4 g}\)	\(737\)

Input:

int((a+b*ln(c*(e*x+d)^n))^2/(g*x+f),x,method=_RETURNVERBOSE)
 

Output:

b^2*ln(g*(e*x+d)-d*g+e*f)/g*ln(e*x+d)^2*n^2-2*b^2*ln(g*(e*x+d)-d*g+e*f)/g* 
ln((e*x+d)^n)*ln(e*x+d)*n+b^2*ln(g*(e*x+d)-d*g+e*f)/g*ln((e*x+d)^n)^2+b^2* 
n^2/g*ln(e*x+d)^2*ln(1-g*(e*x+d)/(d*g-e*f))+2*b^2*n^2/g*ln(e*x+d)*polylog( 
2,g*(e*x+d)/(d*g-e*f))-2*b^2*n^2/g*polylog(3,g*(e*x+d)/(d*g-e*f))-2*b^2*n^ 
2*dilog((g*(e*x+d)-d*g+e*f)/(-d*g+e*f))/g*ln(e*x+d)+2*b^2*n*dilog((g*(e*x+ 
d)-d*g+e*f)/(-d*g+e*f))/g*ln((e*x+d)^n)-2*b^2*n^2*ln(e*x+d)^2*ln((g*(e*x+d 
)-d*g+e*f)/(-d*g+e*f))/g+2*b^2*n*ln(e*x+d)*ln((g*(e*x+d)-d*g+e*f)/(-d*g+e* 
f))/g*ln((e*x+d)^n)+(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)*b*(ln((e*x+d)^n) 
*ln(g*x+f)/g-1/g*n*e*(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))+1/4*(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*ln(g*x+f)/g
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

Integral((a + b*log(c*(d + e*x)**n))**2/(f + g*x), x)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f+g x} \, dx=\frac {3 \left (\int \frac {\mathrm {log}\left (\left (e x +d \right )^{n} c \right )^{2}}{e g \,x^{2}+d g x +e f x +d f}d x \right ) b^{2} d g n -3 \left (\int \frac {\mathrm {log}\left (\left (e x +d \right )^{n} c \right )^{2}}{e g \,x^{2}+d g x +e f x +d f}d x \right ) b^{2} e f n +6 \left (\int \frac {\mathrm {log}\left (\left (e x +d \right )^{n} c \right )}{e g \,x^{2}+d g x +e f x +d f}d x \right ) a b d g n -6 \left (\int \frac {\mathrm {log}\left (\left (e x +d \right )^{n} c \right )}{e g \,x^{2}+d g x +e f x +d f}d x \right ) a b e f n +3 \,\mathrm {log}\left (g x +f \right ) a^{2} n +\mathrm {log}\left (\left (e x +d \right )^{n} c \right )^{3} b^{2}+3 \mathrm {log}\left (\left (e x +d \right )^{n} c \right )^{2} a b}{3 g n} \] Input:

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

Output:

(3*int(log((d + e*x)**n*c)**2/(d*f + d*g*x + e*f*x + e*g*x**2),x)*b**2*d*g 
*n - 3*int(log((d + e*x)**n*c)**2/(d*f + d*g*x + e*f*x + e*g*x**2),x)*b**2 
*e*f*n + 6*int(log((d + e*x)**n*c)/(d*f + d*g*x + e*f*x + e*g*x**2),x)*a*b 
*d*g*n - 6*int(log((d + e*x)**n*c)/(d*f + d*g*x + e*f*x + e*g*x**2),x)*a*b 
*e*f*n + 3*log(f + g*x)*a**2*n + log((d + e*x)**n*c)**3*b**2 + 3*log((d + 
e*x)**n*c)**2*a*b)/(3*g*n)