\(\int \frac {(a+b \log (c x^n))^2}{(d+e x)^2} \, dx\) [103]

Optimal result

Integrand size = 20, antiderivative size = 77 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^2} \, dx=\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{d (d+e x)}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{d e}-\frac {2 b^2 n^2 \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{d e} \] Output:

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

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.05 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^2} \, dx=\frac {\left (a+b \log \left (c x^n\right )\right ) \left (a e x+b e x \log \left (c x^n\right )-2 b n (d+e x) \log \left (1+\frac {e x}{d}\right )\right )-2 b^2 n^2 (d+e x) \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{d e (d+e x)} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.92, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {2755, 2754, 2838}

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.

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

\(\Big \downarrow \) 2755

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

\(\Big \downarrow \) 2754

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

\(\Big \downarrow \) 2838

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

Input:

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

Output:

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

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 2755

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_))^2, x_Sy 
mbol] :> Simp[x*((a + b*Log[c*x^n])^p/(d*(d + e*x))), x] - Simp[b*n*(p/d) 
 Int[(a + b*Log[c*x^n])^(p - 1)/(d + e*x), x], x] /; FreeQ[{a, b, c, d, e, 
n, p}, x] && GtQ[p, 0]
 

rule 2838

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 
, (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
 

Maple [C] (warning: unable to verify)

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

Time = 0.51 (sec) , antiderivative size = 369, normalized size of antiderivative = 4.79

method	result	size
risch	\(-\frac {b^{2} \ln \left (x^{n}\right )^{2}}{e \left (e x +d \right )}-\frac {2 b^{2} n \ln \left (x^{n}\right ) \ln \left (e x +d \right )}{e d}+\frac {2 b^{2} n \ln \left (x^{n}\right ) \ln \left (x \right )}{e d}-\frac {b^{2} n^{2} \ln \left (x \right )^{2}}{e d}+\frac {2 b^{2} n^{2} \ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )}{e d}+\frac {2 b^{2} n^{2} \operatorname {dilog}\left (-\frac {e x}{d}\right )}{e d}+\left (i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )-i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )+2 b \ln \left (c \right )+2 a \right ) b \left (-\frac {\ln \left (x^{n}\right )}{e \left (e x +d \right )}+\frac {n \left (-\frac {\ln \left (e x +d \right )}{d}+\frac {\ln \left (x \right )}{d}\right )}{e}\right )-\frac {{\left (i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )-i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )+2 b \ln \left (c \right )+2 a \right )}^{2}}{4 \left (e x +d \right ) e}\)	\(369\)

Input:

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

Output:

-b^2*ln(x^n)^2/e/(e*x+d)-2*b^2/e*n*ln(x^n)/d*ln(e*x+d)+2*b^2/e*n*ln(x^n)/d 
*ln(x)-b^2/e*n^2/d*ln(x)^2+2*b^2/e*n^2/d*ln(e*x+d)*ln(-e*x/d)+2*b^2/e*n^2/ 
d*dilog(-e*x/d)+(I*Pi*b*csgn(I*x^n)*csgn(I*c*x^n)^2-I*Pi*b*csgn(I*x^n)*csg 
n(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)/e/(e*x+d)+1/e*n*(-1/d*ln(e*x+d)+1/d*ln(x)))- 
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)*c 
sgn(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/(e*x+d)/e
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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