\(\int \frac {x (a+b \log (c x^n))^2}{(d+e x)^3} \, dx\) [109]

Optimal result

Integrand size = 21, antiderivative size = 112 \[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^3} \, dx=\frac {b n x \left (a+b \log \left (c x^n\right )\right )}{d e (d+e x)}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 d (d+e x)^2}-\frac {b n \left (a+b n+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{d e^2}-\frac {b^2 n^2 \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{d e^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.26 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.38 \[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^3} \, dx=\frac {-\frac {2 b n \left (a+b \log \left (c x^n\right )\right )}{d+e x}+\frac {\left (a+b \log \left (c x^n\right )\right )^2}{d}+\frac {d \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^2}-\frac {2 \left (a+b \log \left (c x^n\right )\right )^2}{d+e x}+\frac {2 b^2 n^2 (\log (x)-\log (d+e x))}{d}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{d}-\frac {2 b^2 n^2 \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{d}}{2 e^2} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.96, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2781, 2784, 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.

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

\(\Big \downarrow \) 2781

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

\(\Big \downarrow \) 2784

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

\(\Big \downarrow \) 2754

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

\(\Big \downarrow \) 2838

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

Input:

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

Output:

(x^2*(a + b*Log[c*x^n])^2)/(2*d*(d + e*x)^2) - (b*n*(-((x*(a + b*Log[c*x^n 
]))/(e*(d + e*x))) + (((a + b*n + b*Log[c*x^n])*Log[1 + (e*x)/d])/e + (b*n 
*PolyLog[2, -((e*x)/d)])/e)/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 2781

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

rule 2784

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)* 
(x_))^(q_.), x_Symbol] :> Simp[(f*x)^m*(d + e*x)^(q + 1)*((a + b*Log[c*x^n] 
)/(e*(q + 1))), x] - Simp[f/(e*(q + 1))   Int[(f*x)^(m - 1)*(d + e*x)^(q + 
1)*(a*m + b*n + b*m*Log[c*x^n]), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, 
x] && ILtQ[q, -1] && GtQ[m, 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 = 0.66 (sec) , antiderivative size = 484, normalized size of antiderivative = 4.32

method	result	size
risch	\(-\frac {b^{2} \ln \left (x^{n}\right )^{2}}{e^{2} \left (e x +d \right )}+\frac {b^{2} \ln \left (x^{n}\right )^{2} d}{2 e^{2} \left (e x +d \right )^{2}}-\frac {b^{2} n \ln \left (x^{n}\right )}{e^{2} \left (e x +d \right )}-\frac {b^{2} n \ln \left (x^{n}\right ) \ln \left (e x +d \right )}{e^{2} d}+\frac {b^{2} n \ln \left (x^{n}\right ) \ln \left (x \right )}{e^{2} d}-\frac {b^{2} n^{2} \ln \left (x \right )^{2}}{2 e^{2} d}-\frac {b^{2} n^{2} \ln \left (e x +d \right )}{e^{2} d}+\frac {b^{2} n^{2} \ln \left (x \right )}{e^{2} d}+\frac {b^{2} n^{2} \ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )}{e^{2} d}+\frac {b^{2} n^{2} \operatorname {dilog}\left (-\frac {e x}{d}\right )}{e^{2} 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^{2} \left (e x +d \right )}+\frac {\ln \left (x^{n}\right ) d}{2 e^{2} \left (e x +d \right )^{2}}-\frac {n \left (\frac {\ln \left (e x +d \right )}{d}+\frac {1}{e x +d}-\frac {\ln \left (x \right )}{d}\right )}{2 e^{2}}\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} \left (-\frac {1}{e^{2} \left (e x +d \right )}+\frac {d}{2 e^{2} \left (e x +d \right )^{2}}\right )}{4}\)	\(484\)

Input:

int(x*(a+b*ln(c*x^n))^2/(e*x+d)^3,x,method=_RETURNVERBOSE)
 

Output:

-b^2*ln(x^n)^2/e^2/(e*x+d)+1/2*b^2*ln(x^n)^2/e^2*d/(e*x+d)^2-b^2*n*ln(x^n) 
/e^2/(e*x+d)-b^2*n*ln(x^n)/e^2/d*ln(e*x+d)+b^2*n*ln(x^n)/e^2/d*ln(x)-1/2*b 
^2*n^2/e^2/d*ln(x)^2-b^2*n^2/e^2/d*ln(e*x+d)+b^2*n^2/e^2/d*ln(x)+b^2*n^2/e 
^2/d*ln(e*x+d)*ln(-e*x/d)+b^2*n^2/e^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)*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)/e^2/(e 
*x+d)+1/2*ln(x^n)/e^2*d/(e*x+d)^2-1/2*n/e^2*(1/d*ln(e*x+d)+1/(e*x+d)-1/d*l 
n(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)*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*(-1/e^2/(e*x+d)+1/2/e^2*d/(e*x+d)^2)
 

Fricas [F]

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

integrate(x*(a+b*log(c*x^n))^2/(e*x+d)^3,x, algorithm="fricas")
 

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

-a*b*n*(1/(e^3*x + d*e^2) + log(e*x + d)/(d*e^2) - log(x)/(d*e^2)) - 1/2*( 
(2*e*x + d)*log(x^n)^2/(e^4*x^2 + 2*d*e^3*x + d^2*e^2) - 2*integrate((e^2* 
x^2*log(c)^2 + (3*d*e*n*x + d^2*n + 2*(e^2*n + e^2*log(c))*x^2)*log(x^n))/ 
(e^5*x^4 + 3*d*e^4*x^3 + 3*d^2*e^3*x^2 + d^3*e^2*x), x))*b^2 - (2*e*x + d) 
*a*b*log(c*x^n)/(e^4*x^2 + 2*d*e^3*x + d^2*e^2) - 1/2*(2*e*x + d)*a^2/(e^4 
*x^2 + 2*d*e^3*x + d^2*e^2)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^3} \, dx=\frac {4 \left (\int \frac {\mathrm {log}\left (x^{n} c \right )}{e^{3} x^{4}+3 d \,e^{2} x^{3}+3 d^{2} e \,x^{2}+d^{3} x}d x \right ) b^{2} d^{5} n +8 \left (\int \frac {\mathrm {log}\left (x^{n} c \right )}{e^{3} x^{4}+3 d \,e^{2} x^{3}+3 d^{2} e \,x^{2}+d^{3} x}d x \right ) b^{2} d^{4} e n x +4 \left (\int \frac {\mathrm {log}\left (x^{n} c \right )}{e^{3} x^{4}+3 d \,e^{2} x^{3}+3 d^{2} e \,x^{2}+d^{3} x}d x \right ) b^{2} d^{3} e^{2} n \,x^{2}-4 \,\mathrm {log}\left (e x +d \right ) a b \,d^{2} n -8 \,\mathrm {log}\left (e x +d \right ) a b d e n x -4 \,\mathrm {log}\left (e x +d \right ) a b \,e^{2} n \,x^{2}-10 \,\mathrm {log}\left (e x +d \right ) b^{2} d^{2} n^{2}-20 \,\mathrm {log}\left (e x +d \right ) b^{2} d e \,n^{2} x -10 \,\mathrm {log}\left (e x +d \right ) b^{2} e^{2} n^{2} x^{2}-2 \mathrm {log}\left (x^{n} c \right )^{2} b^{2} d^{2}-4 \mathrm {log}\left (x^{n} c \right )^{2} b^{2} d e x +4 \,\mathrm {log}\left (x^{n} c \right ) a b \,e^{2} x^{2}-6 \,\mathrm {log}\left (x^{n} c \right ) b^{2} d^{2} n +4 \,\mathrm {log}\left (x^{n} c \right ) b^{2} e^{2} n \,x^{2}+6 \,\mathrm {log}\left (x \right ) b^{2} d^{2} n^{2}+12 \,\mathrm {log}\left (x \right ) b^{2} d e \,n^{2} x +6 \,\mathrm {log}\left (x \right ) b^{2} e^{2} n^{2} x^{2}+2 a^{2} e^{2} x^{2}-2 a b \,d^{2} n +2 a b \,e^{2} n \,x^{2}+b^{2} d^{2} n^{2}-b^{2} e^{2} n^{2} x^{2}}{4 d \,e^{2} \left (e^{2} x^{2}+2 d e x +d^{2}\right )} \] Input:

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

Output:

(4*int(log(x**n*c)/(d**3*x + 3*d**2*e*x**2 + 3*d*e**2*x**3 + e**3*x**4),x) 
*b**2*d**5*n + 8*int(log(x**n*c)/(d**3*x + 3*d**2*e*x**2 + 3*d*e**2*x**3 + 
 e**3*x**4),x)*b**2*d**4*e*n*x + 4*int(log(x**n*c)/(d**3*x + 3*d**2*e*x**2 
 + 3*d*e**2*x**3 + e**3*x**4),x)*b**2*d**3*e**2*n*x**2 - 4*log(d + e*x)*a* 
b*d**2*n - 8*log(d + e*x)*a*b*d*e*n*x - 4*log(d + e*x)*a*b*e**2*n*x**2 - 1 
0*log(d + e*x)*b**2*d**2*n**2 - 20*log(d + e*x)*b**2*d*e*n**2*x - 10*log(d 
 + e*x)*b**2*e**2*n**2*x**2 - 2*log(x**n*c)**2*b**2*d**2 - 4*log(x**n*c)** 
2*b**2*d*e*x + 4*log(x**n*c)*a*b*e**2*x**2 - 6*log(x**n*c)*b**2*d**2*n + 4 
*log(x**n*c)*b**2*e**2*n*x**2 + 6*log(x)*b**2*d**2*n**2 + 12*log(x)*b**2*d 
*e*n**2*x + 6*log(x)*b**2*e**2*n**2*x**2 + 2*a**2*e**2*x**2 - 2*a*b*d**2*n 
 + 2*a*b*e**2*n*x**2 + b**2*d**2*n**2 - b**2*e**2*n**2*x**2)/(4*d*e**2*(d* 
*2 + 2*d*e*x + e**2*x**2))