Integrand size = 21, antiderivative size = 143 \[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^2} \, dx=-\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{e (d+e x)}+\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{e^2}+\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{e^2}+\frac {2 b^2 n^2 \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{e^2}+\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{e^2}-\frac {2 b^2 n^2 \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )}{e^2} \] Output:
-x*(a+b*ln(c*x^n))^2/e/(e*x+d)+2*b*n*(a+b*ln(c*x^n))*ln(1+e*x/d)/e^2+(a+b* ln(c*x^n))^2*ln(1+e*x/d)/e^2+2*b^2*n^2*polylog(2,-e*x/d)/e^2+2*b*n*(a+b*ln (c*x^n))*polylog(2,-e*x/d)/e^2-2*b^2*n^2*polylog(3,-e*x/d)/e^2
Time = 0.16 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.99 \[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^2} \, dx=\frac {-\left (a+b \log \left (c x^n\right )\right )^2+\frac {d \left (a+b \log \left (c x^n\right )\right )^2}{d+e x}+2 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )+\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )+2 b^2 n^2 \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )+2 b n \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )-2 b^2 n^2 \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )}{e^2} \] Input:
Integrate[(x*(a + b*Log[c*x^n])^2)/(d + e*x)^2,x]
Output:
(-(a + b*Log[c*x^n])^2 + (d*(a + b*Log[c*x^n])^2)/(d + e*x) + 2*b*n*(a + b *Log[c*x^n])*Log[1 + (e*x)/d] + (a + b*Log[c*x^n])^2*Log[1 + (e*x)/d] + 2* b^2*n^2*PolyLog[2, -((e*x)/d)] + 2*b*n*(a + b*Log[c*x^n])*PolyLog[2, -((e* x)/d)] - 2*b^2*n^2*PolyLog[3, -((e*x)/d)])/e^2
Time = 0.43 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, 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.
| \(\displaystyle \int \frac {x \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^2} \, dx\) |
| \(\Big \downarrow \) 2795 |
| \(\displaystyle \int \left (\frac {\left (a+b \log \left (c x^n\right )\right )^2}{e (d+e x)}-\frac {d \left (a+b \log \left (c x^n\right )\right )^2}{e (d+e x)^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 b n \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^2}+\frac {2 b n \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{e^2}+\frac {\log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{e^2}-\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{e (d+e x)}+\frac {2 b^2 n^2 \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{e^2}-\frac {2 b^2 n^2 \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )}{e^2}\) |
Input:
Int[(x*(a + b*Log[c*x^n])^2)/(d + e*x)^2,x]
Output:
-((x*(a + b*Log[c*x^n])^2)/(e*(d + e*x))) + (2*b*n*(a + b*Log[c*x^n])*Log[ 1 + (e*x)/d])/e^2 + ((a + b*Log[c*x^n])^2*Log[1 + (e*x)/d])/e^2 + (2*b^2*n ^2*PolyLog[2, -((e*x)/d)])/e^2 + (2*b*n*(a + b*Log[c*x^n])*PolyLog[2, -((e *x)/d)])/e^2 - (2*b^2*n^2*PolyLog[3, -((e*x)/d)])/e^2
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]))
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.70 (sec) , antiderivative size = 609, normalized size of antiderivative = 4.26
| method | result | size |
| risch | \(\frac {b^{2} \ln \left (x^{n}\right )^{2} d}{e^{2} \left (e x +d \right )}+\frac {b^{2} \ln \left (x^{n}\right )^{2} \ln \left (e x +d \right )}{e^{2}}+\frac {2 b^{2} \ln \left (x \right ) \ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right ) n^{2}}{e^{2}}+\frac {2 b^{2} \ln \left (x \right ) \operatorname {dilog}\left (-\frac {e x}{d}\right ) n^{2}}{e^{2}}-\frac {2 b^{2} n \ln \left (x^{n}\right ) \ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )}{e^{2}}-\frac {2 b^{2} n \ln \left (x^{n}\right ) \operatorname {dilog}\left (-\frac {e x}{d}\right )}{e^{2}}-\frac {b^{2} n^{2} \ln \left (e x +d \right ) \ln \left (x \right )^{2}}{e^{2}}+\frac {b^{2} n^{2} \ln \left (x \right )^{2} \ln \left (1+\frac {e x}{d}\right )}{e^{2}}+\frac {2 b^{2} n^{2} \ln \left (x \right ) \operatorname {polylog}\left (2, -\frac {e x}{d}\right )}{e^{2}}-\frac {2 b^{2} n^{2} \operatorname {polylog}\left (3, -\frac {e x}{d}\right )}{e^{2}}+\frac {2 b^{2} n \ln \left (x^{n}\right ) \ln \left (e x +d \right )}{e^{2}}-\frac {2 b^{2} n \ln \left (x^{n}\right ) \ln \left (x \right )}{e^{2}}+\frac {b^{2} n^{2} \ln \left (x \right )^{2}}{e^{2}}-\frac {2 b^{2} n^{2} \ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )}{e^{2}}-\frac {2 b^{2} n^{2} \operatorname {dilog}\left (-\frac {e x}{d}\right )}{e^{2}}+\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 ) d}{e^{2} \left (e x +d \right )}+\frac {\ln \left (x^{n}\right ) \ln \left (e x +d \right )}{e^{2}}-n \left (\frac {\operatorname {dilog}\left (-\frac {e x}{d}\right )+\ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )}{e^{2}}-\frac {\ln \left (e x +d \right )}{e^{2}}+\frac {\ln \left (e x \right )}{e^{2}}\right )\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 {d}{e^{2} \left (e x +d \right )}+\frac {\ln \left (e x +d \right )}{e^{2}}\right )}{4}\) | \(609\) |
Input:
int(x*(a+b*ln(c*x^n))^2/(e*x+d)^2,x,method=_RETURNVERBOSE)
Output:
b^2*ln(x^n)^2/e^2*d/(e*x+d)+b^2*ln(x^n)^2/e^2*ln(e*x+d)+2*b^2/e^2*ln(x)*ln (e*x+d)*ln(-e*x/d)*n^2+2*b^2/e^2*ln(x)*dilog(-e*x/d)*n^2-2*b^2*n/e^2*ln(x^ n)*ln(e*x+d)*ln(-e*x/d)-2*b^2*n/e^2*ln(x^n)*dilog(-e*x/d)-b^2/e^2*n^2*ln(e *x+d)*ln(x)^2+b^2/e^2*n^2*ln(x)^2*ln(1+e*x/d)+2*b^2/e^2*n^2*ln(x)*polylog( 2,-e*x/d)-2*b^2*n^2*polylog(3,-e*x/d)/e^2+2*b^2*n*ln(x^n)/e^2*ln(e*x+d)-2* b^2*n/e^2*ln(x^n)*ln(x)+b^2/e^2*n^2*ln(x)^2-2*b^2/e^2*n^2*ln(e*x+d)*ln(-e* x/d)-2*b^2/e^2*n^2*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*d/(e*x+d)+ln(x^n)/e^2*ln (e*x+d)-n*(1/e^2*(dilog(-e*x/d)+ln(e*x+d)*ln(-e*x/d))-1/e^2*ln(e*x+d)+1/e^ 2*ln(e*x)))+1/4*(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)^2*(1/e^2*d/(e*x+d)+1/e^2*ln(e*x+d))
\[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^2} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} x}{{\left (e x + d\right )}^{2}} \,d x } \] Input:
integrate(x*(a+b*log(c*x^n))^2/(e*x+d)^2,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^2*x^2 + 2*d* e*x + d^2), x)
\[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^2} \, dx=\int \frac {x \left (a + b \log {\left (c x^{n} \right )}\right )^{2}}{\left (d + e x\right )^{2}}\, dx \] Input:
integrate(x*(a+b*ln(c*x**n))**2/(e*x+d)**2,x)
Output:
Integral(x*(a + b*log(c*x**n))**2/(d + e*x)**2, x)
\[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^2} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} x}{{\left (e x + d\right )}^{2}} \,d x } \] Input:
integrate(x*(a+b*log(c*x^n))^2/(e*x+d)^2,x, algorithm="maxima")
Output:
a^2*(d/(e^3*x + d*e^2) + log(e*x + d)/e^2) + integrate((b^2*x*log(x^n)^2 + 2*(b^2*log(c) + a*b)*x*log(x^n) + (b^2*log(c)^2 + 2*a*b*log(c))*x)/(e^2*x ^2 + 2*d*e*x + d^2), x)
\[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^2} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} x}{{\left (e x + d\right )}^{2}} \,d x } \] Input:
integrate(x*(a+b*log(c*x^n))^2/(e*x+d)^2,x, algorithm="giac")
Output:
integrate((b*log(c*x^n) + a)^2*x/(e*x + d)^2, x)
Timed out. \[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^2} \, dx=\int \frac {x\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2}{{\left (d+e\,x\right )}^2} \,d x \] Input:
int((x*(a + b*log(c*x^n))^2)/(d + e*x)^2,x)
Output:
int((x*(a + b*log(c*x^n))^2)/(d + e*x)^2, x)
\[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^2} \, dx=\frac {-3 \left (\int \frac {\mathrm {log}\left (x^{n} c \right )^{2}}{e^{2} x^{3}+2 d e \,x^{2}+d^{2} x}d x \right ) b^{2} d^{3} n -3 \left (\int \frac {\mathrm {log}\left (x^{n} c \right )^{2}}{e^{2} x^{3}+2 d e \,x^{2}+d^{2} x}d x \right ) b^{2} d^{2} e n x -6 \left (\int \frac {\mathrm {log}\left (x^{n} c \right )}{e^{2} x^{3}+2 d e \,x^{2}+d^{2} x}d x \right ) a b \,d^{3} n -6 \left (\int \frac {\mathrm {log}\left (x^{n} c \right )}{e^{2} x^{3}+2 d e \,x^{2}+d^{2} x}d x \right ) a b \,d^{2} e n x -12 \left (\int \frac {\mathrm {log}\left (x^{n} c \right )}{e^{2} x^{3}+2 d e \,x^{2}+d^{2} x}d x \right ) b^{2} d^{3} n^{2}-12 \left (\int \frac {\mathrm {log}\left (x^{n} c \right )}{e^{2} x^{3}+2 d e \,x^{2}+d^{2} x}d x \right ) b^{2} d^{2} e \,n^{2} x +3 \,\mathrm {log}\left (e x +d \right ) a^{2} d n +3 \,\mathrm {log}\left (e x +d \right ) a^{2} e n x +12 \,\mathrm {log}\left (e x +d \right ) a b d \,n^{2}+12 \,\mathrm {log}\left (e x +d \right ) a b e \,n^{2} x +12 \,\mathrm {log}\left (e x +d \right ) b^{2} d \,n^{3}+12 \,\mathrm {log}\left (e x +d \right ) b^{2} e \,n^{3} x +\mathrm {log}\left (x^{n} c \right )^{3} b^{2} d +\mathrm {log}\left (x^{n} c \right )^{3} b^{2} e x +3 \mathrm {log}\left (x^{n} c \right )^{2} a b d +3 \mathrm {log}\left (x^{n} c \right )^{2} a b e x +6 \mathrm {log}\left (x^{n} c \right )^{2} b^{2} d n -12 \,\mathrm {log}\left (x^{n} c \right ) a b e n x -12 \,\mathrm {log}\left (x^{n} c \right ) b^{2} e \,n^{2} x -3 a^{2} e n x}{3 e^{2} n \left (e x +d \right )} \] Input:
int(x*(a+b*log(c*x^n))^2/(e*x+d)^2,x)
Output:
( - 3*int(log(x**n*c)**2/(d**2*x + 2*d*e*x**2 + e**2*x**3),x)*b**2*d**3*n - 3*int(log(x**n*c)**2/(d**2*x + 2*d*e*x**2 + e**2*x**3),x)*b**2*d**2*e*n* x - 6*int(log(x**n*c)/(d**2*x + 2*d*e*x**2 + e**2*x**3),x)*a*b*d**3*n - 6* int(log(x**n*c)/(d**2*x + 2*d*e*x**2 + e**2*x**3),x)*a*b*d**2*e*n*x - 12*i nt(log(x**n*c)/(d**2*x + 2*d*e*x**2 + e**2*x**3),x)*b**2*d**3*n**2 - 12*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**2*x + 3* log(d + e*x)*a**2*d*n + 3*log(d + e*x)*a**2*e*n*x + 12*log(d + e*x)*a*b*d* n**2 + 12*log(d + e*x)*a*b*e*n**2*x + 12*log(d + e*x)*b**2*d*n**3 + 12*log (d + e*x)*b**2*e*n**3*x + log(x**n*c)**3*b**2*d + log(x**n*c)**3*b**2*e*x + 3*log(x**n*c)**2*a*b*d + 3*log(x**n*c)**2*a*b*e*x + 6*log(x**n*c)**2*b** 2*d*n - 12*log(x**n*c)*a*b*e*n*x - 12*log(x**n*c)*b**2*e*n**2*x - 3*a**2*e *n*x)/(3*e**2*n*(d + e*x))