\(\int \frac {(a+b \log (c x^n))^2}{d+e x} \, dx\) [95]

Optimal result

Integrand size = 20, antiderivative size = 72 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{d+e x} \, dx=\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{e}+\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{e}-\frac {2 b^2 n^2 \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )}{e} \] Output:

(a+b*ln(c*x^n))^2*ln(1+e*x/d)/e+2*b*n*(a+b*ln(c*x^n))*polylog(2,-e*x/d)/e- 
2*b^2*n^2*polylog(3,-e*x/d)/e
 

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.94 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{d+e x} \, dx=\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (\frac {d+e x}{d}\right )}{e}-\frac {2 b n \left (-\left (\left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )\right )+b n \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )\right )}{e} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.93, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {2754, 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.

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

\(\Big \downarrow \) 2754

\(\displaystyle \frac {\log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{e}-\frac {2 b n \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (\frac {e x}{d}+1\right )}{x}dx}{e}\)

\(\Big \downarrow \) 2821

\(\displaystyle \frac {\log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{e}-\frac {2 b n \left (b n \int \frac {\operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{x}dx-\operatorname {PolyLog}\left (2,-\frac {e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )\right )}{e}\)

\(\Big \downarrow \) 7143

\(\displaystyle \frac {\log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{e}-\frac {2 b n \left (b n \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )-\operatorname {PolyLog}\left (2,-\frac {e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )\right )}{e}\)

Input:

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

Output:

((a + b*Log[c*x^n])^2*Log[1 + (e*x)/d])/e - (2*b*n*(-((a + b*Log[c*x^n])*P 
olyLog[2, -((e*x)/d)]) + b*n*PolyLog[3, -((e*x)/d)]))/e
 

Defintions of rubi rules used

rule 2754

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

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 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 = 0.61 (sec) , antiderivative size = 445, normalized size of antiderivative = 6.18

method	result	size
risch	\(\frac {b^{2} \ln \left (x^{n}\right )^{2} \ln \left (e x +d \right )}{e}+\frac {2 b^{2} n^{2} \ln \left (x \right ) \ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )}{e}+\frac {2 b^{2} n^{2} \ln \left (x \right ) \operatorname {dilog}\left (-\frac {e x}{d}\right )}{e}-\frac {2 b^{2} n \ln \left (x^{n}\right ) \ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )}{e}-\frac {2 b^{2} n \ln \left (x^{n}\right ) \operatorname {dilog}\left (-\frac {e x}{d}\right )}{e}-\frac {b^{2} n^{2} \ln \left (e x +d \right ) \ln \left (x \right )^{2}}{e}+\frac {b^{2} n^{2} \ln \left (x \right )^{2} \ln \left (1+\frac {e x}{d}\right )}{e}+\frac {2 b^{2} n^{2} \ln \left (x \right ) \operatorname {polylog}\left (2, -\frac {e x}{d}\right )}{e}-\frac {2 b^{2} n^{2} \operatorname {polylog}\left (3, -\frac {e x}{d}\right )}{e}+\left (i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )-i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )+2 b \ln \left (c \right )+2 a \right ) b \left (\frac {\ln \left (x^{n}\right ) \ln \left (e x +d \right )}{e}-\frac {n \left (\operatorname {dilog}\left (-\frac {e x}{d}\right )+\ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )\right )}{e}\right )+\frac {{\left (i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )-i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )+2 b \ln \left (c \right )+2 a \right )}^{2} \ln \left (e x +d \right )}{4 e}\)	\(445\)

Input:

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

Output:

b^2*ln(x^n)^2*ln(e*x+d)/e+2*b^2/e*n^2*ln(x)*ln(e*x+d)*ln(-e*x/d)+2*b^2/e*n 
^2*ln(x)*dilog(-e*x/d)-2*b^2/e*n*ln(x^n)*ln(e*x+d)*ln(-e*x/d)-2*b^2/e*n*ln 
(x^n)*dilog(-e*x/d)-b^2/e*n^2*ln(e*x+d)*ln(x)^2+b^2/e*n^2*ln(x)^2*ln(1+e*x 
/d)+2*b^2/e*n^2*ln(x)*polylog(2,-e*x/d)-2*b^2*n^2*polylog(3,-e*x/d)/e+(I*P 
i*b*csgn(I*x^n)*csgn(I*c*x^n)^2-I*Pi*b*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c) 
-I*Pi*b*csgn(I*c*x^n)^3+I*Pi*b*csgn(I*c*x^n)^2*csgn(I*c)+2*b*ln(c)+2*a)*b* 
(ln(x^n)*ln(e*x+d)/e-1/e*n*(dilog(-e*x/d)+ln(e*x+d)*ln(-e*x/d)))+1/4*(I*Pi 
*b*csgn(I*x^n)*csgn(I*c*x^n)^2-I*Pi*b*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)- 
I*Pi*b*csgn(I*c*x^n)^3+I*Pi*b*csgn(I*c*x^n)^2*csgn(I*c)+2*b*ln(c)+2*a)^2*l 
n(e*x+d)/e
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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