\(\int \frac {x^2 (a+b \log (c x^n))^2}{(d+e x)^3} \, dx\) [108]

Optimal result

Integrand size = 23, antiderivative size = 232 \[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^3} \, dx=-\frac {b n x \left (a+b \log \left (c x^n\right )\right )}{e^2 (d+e x)}+\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 e^3}-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 e^3 (d+e x)^2}-\frac {2 x \left (a+b \log \left (c x^n\right )\right )^2}{e^2 (d+e x)}+\frac {b^2 n^2 \log (d+e x)}{e^3}+\frac {3 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{e^3}+\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{e^3}+\frac {3 b^2 n^2 \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{e^3}+\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{e^3}-\frac {2 b^2 n^2 \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )}{e^3} \] Output:

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

Mathematica [A] (verified)

Time = 0.28 (sec) , antiderivative size = 212, normalized size of antiderivative = 0.91 \[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^3} \, dx=\frac {\frac {2 b d n \left (a+b \log \left (c x^n\right )\right )}{d+e x}-3 \left (a+b \log \left (c x^n\right )\right )^2-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^2}+\frac {4 d \left (a+b \log \left (c x^n\right )\right )^2}{d+e x}-2 b^2 n^2 (\log (x)-\log (d+e x))+6 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )+2 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )+6 b^2 n^2 \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )+4 b n \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )-4 b^2 n^2 \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )}{2 e^3} \] Input:

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

Output:

((2*b*d*n*(a + b*Log[c*x^n]))/(d + e*x) - 3*(a + b*Log[c*x^n])^2 - (d^2*(a 
 + b*Log[c*x^n])^2)/(d + e*x)^2 + (4*d*(a + b*Log[c*x^n])^2)/(d + e*x) - 2 
*b^2*n^2*(Log[x] - Log[d + e*x]) + 6*b*n*(a + b*Log[c*x^n])*Log[1 + (e*x)/ 
d] + 2*(a + b*Log[c*x^n])^2*Log[1 + (e*x)/d] + 6*b^2*n^2*PolyLog[2, -((e*x 
)/d)] + 4*b*n*(a + b*Log[c*x^n])*PolyLog[2, -((e*x)/d)] - 4*b^2*n^2*PolyLo 
g[3, -((e*x)/d)])/(2*e^3)
 

Rubi [A] (verified)

Time = 0.69 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.13, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2795, 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.

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

\(\Big \downarrow \) 2795

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {d^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 e^3 (d+e x)^2}+\frac {2 b n \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^3}-\frac {b n \log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{e^3}+\frac {4 b n \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{e^3}+\frac {\log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{e^3}-\frac {b n x \left (a+b \log \left (c x^n\right )\right )}{e^2 (d+e x)}-\frac {2 x \left (a+b \log \left (c x^n\right )\right )^2}{e^2 (d+e x)}+\frac {b^2 n^2 \operatorname {PolyLog}\left (2,-\frac {d}{e x}\right )}{e^3}+\frac {4 b^2 n^2 \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{e^3}-\frac {2 b^2 n^2 \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )}{e^3}+\frac {b^2 n^2 \log (d+e x)}{e^3}\)

Input:

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

Output:

-((b*n*x*(a + b*Log[c*x^n]))/(e^2*(d + e*x))) - (b*n*Log[1 + d/(e*x)]*(a + 
 b*Log[c*x^n]))/e^3 - (d^2*(a + b*Log[c*x^n])^2)/(2*e^3*(d + e*x)^2) - (2* 
x*(a + b*Log[c*x^n])^2)/(e^2*(d + e*x)) + (b^2*n^2*Log[d + e*x])/e^3 + (4* 
b*n*(a + b*Log[c*x^n])*Log[1 + (e*x)/d])/e^3 + ((a + b*Log[c*x^n])^2*Log[1 
 + (e*x)/d])/e^3 + (b^2*n^2*PolyLog[2, -(d/(e*x))])/e^3 + (4*b^2*n^2*PolyL 
og[2, -((e*x)/d)])/e^3 + (2*b*n*(a + b*Log[c*x^n])*PolyLog[2, -((e*x)/d)]) 
/e^3 - (2*b^2*n^2*PolyLog[3, -((e*x)/d)])/e^3
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2795

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + 
(e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[ 
c*x^n])^p, (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b 
, c, d, e, f, m, n, p, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IGtQ[p, 0 
] && IntegerQ[m] && IntegerQ[r]))
 

Maple [C] (warning: unable to verify)

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

Time = 0.80 (sec) , antiderivative size = 738, normalized size of antiderivative = 3.18

method	result	size
risch	\(\frac {2 b^{2} \ln \left (x^{n}\right )^{2} d}{e^{3} \left (e x +d \right )}+\frac {b^{2} \ln \left (x^{n}\right )^{2} \ln \left (e x +d \right )}{e^{3}}-\frac {b^{2} \ln \left (x^{n}\right )^{2} d^{2}}{2 e^{3} \left (e x +d \right )^{2}}+\frac {b^{2} n \ln \left (x^{n}\right ) d}{e^{3} \left (e x +d \right )}+\frac {3 b^{2} n \ln \left (x^{n}\right ) \ln \left (e x +d \right )}{e^{3}}-\frac {3 b^{2} n \ln \left (x^{n}\right ) \ln \left (x \right )}{e^{3}}+\frac {b^{2} n^{2} \ln \left (e x +d \right )}{e^{3}}-\frac {b^{2} n^{2} \ln \left (x \right )}{e^{3}}+\frac {3 b^{2} n^{2} \ln \left (x \right )^{2}}{2 e^{3}}-\frac {3 b^{2} n^{2} \ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )}{e^{3}}-\frac {3 b^{2} n^{2} \operatorname {dilog}\left (-\frac {e x}{d}\right )}{e^{3}}+\frac {2 b^{2} \ln \left (x \right ) \ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right ) n^{2}}{e^{3}}+\frac {2 b^{2} \ln \left (x \right ) \operatorname {dilog}\left (-\frac {e x}{d}\right ) n^{2}}{e^{3}}-\frac {2 b^{2} n \ln \left (x^{n}\right ) \ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )}{e^{3}}-\frac {2 b^{2} n \ln \left (x^{n}\right ) \operatorname {dilog}\left (-\frac {e x}{d}\right )}{e^{3}}-\frac {b^{2} n^{2} \ln \left (e x +d \right ) \ln \left (x \right )^{2}}{e^{3}}+\frac {b^{2} n^{2} \ln \left (x \right )^{2} \ln \left (1+\frac {e x}{d}\right )}{e^{3}}+\frac {2 b^{2} n^{2} \ln \left (x \right ) \operatorname {polylog}\left (2, -\frac {e x}{d}\right )}{e^{3}}-\frac {2 b^{2} n^{2} \operatorname {polylog}\left (3, -\frac {e x}{d}\right )}{e^{3}}+\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 {2 \ln \left (x^{n}\right ) d}{e^{3} \left (e x +d \right )}+\frac {\ln \left (x^{n}\right ) \ln \left (e x +d \right )}{e^{3}}-\frac {\ln \left (x^{n}\right ) d^{2}}{2 e^{3} \left (e x +d \right )^{2}}-\frac {n \left (-\frac {d}{e^{3} \left (e x +d \right )}-\frac {3 \ln \left (e x +d \right )}{e^{3}}+\frac {3 \ln \left (e x \right )}{e^{3}}+\frac {2 \ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )}{e^{3}}+\frac {2 \operatorname {dilog}\left (-\frac {e x}{d}\right )}{e^{3}}\right )}{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 {2 d}{e^{3} \left (e x +d \right )}+\frac {\ln \left (e x +d \right )}{e^{3}}-\frac {d^{2}}{2 e^{3} \left (e x +d \right )^{2}}\right )}{4}\)	\(738\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

( - 6*int(log(x**n*c)**2/(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 - 12*int(log(x**n*c)**2/(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 - 6*int(log(x**n*c)**2/(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 - 12*i 
nt(log(x**n*c)/(d**3*x + 3*d**2*e*x**2 + 3*d*e**2*x**3 + e**3*x**4),x)*a*b 
*d**5*n - 24*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)*a*b*d**4*e*n*x - 12*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)*a*b*d**3*e**2*n*x**2 - 36*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**2 - 72*i 
nt(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**2*x - 36*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**2*x**2 + 6*log(d + e*x)*a**2*d**2*n + 
 12*log(d + e*x)*a**2*d*e*n*x + 6*log(d + e*x)*a**2*e**2*n*x**2 + 36*log(d 
 + e*x)*a*b*d**2*n**2 + 72*log(d + e*x)*a*b*d*e*n**2*x + 36*log(d + e*x)*a 
*b*e**2*n**2*x**2 + 54*log(d + e*x)*b**2*d**2*n**3 + 108*log(d + e*x)*b**2 
*d*e*n**3*x + 54*log(d + e*x)*b**2*e**2*n**3*x**2 + 2*log(x**n*c)**3*b**2* 
d**2 + 4*log(x**n*c)**3*b**2*d*e*x + 2*log(x**n*c)**3*b**2*e**2*x**2 + 6*l 
og(x**n*c)**2*a*b*d**2 + 12*log(x**n*c)**2*a*b*d*e*x + 6*log(x**n*c)**2*a* 
b*e**2*x**2 + 18*log(x**n*c)**2*b**2*d**2*n + 18*log(x**n*c)**2*b**2*d*e*n 
*x + 18*log(x**n*c)*a*b*d**2*n - 18*log(x**n*c)*a*b*e**2*n*x**2 + 36*lo...