Integrand size = 23, antiderivative size = 203 \[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^2} \, dx=-\frac {2 a b n x}{e^2}+\frac {2 b^2 n^2 x}{e^2}-\frac {2 b^2 n x \log \left (c x^n\right )}{e^2}+\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{e^2}+\frac {d x \left (a+b \log \left (c x^n\right )\right )^2}{e^2 (d+e x)}-\frac {2 b d n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{e^3}-\frac {2 d \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{e^3}-\frac {2 b^2 d n^2 \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{e^3}-\frac {4 b d n \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{e^3}+\frac {4 b^2 d n^2 \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )}{e^3} \] Output:
-2*a*b*n*x/e^2+2*b^2*n^2*x/e^2-2*b^2*n*x*ln(c*x^n)/e^2+x*(a+b*ln(c*x^n))^2 /e^2+d*x*(a+b*ln(c*x^n))^2/e^2/(e*x+d)-2*b*d*n*(a+b*ln(c*x^n))*ln(1+e*x/d) /e^3-2*d*(a+b*ln(c*x^n))^2*ln(1+e*x/d)/e^3-2*b^2*d*n^2*polylog(2,-e*x/d)/e ^3-4*b*d*n*(a+b*ln(c*x^n))*polylog(2,-e*x/d)/e^3+4*b^2*d*n^2*polylog(3,-e* x/d)/e^3
Time = 0.19 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.92 \[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^2} \, dx=\frac {d \left (a+b \log \left (c x^n\right )\right )^2+e x \left (a+b \log \left (c x^n\right )\right )^2-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )^2}{d+e x}-2 b e n x \left (a-b n+b \log \left (c x^n\right )\right )-2 b d n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )-2 d \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )-2 b^2 d n^2 \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )-4 b d n \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )+4 b^2 d n^2 \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )}{e^3} \] Input:
Integrate[(x^2*(a + b*Log[c*x^n])^2)/(d + e*x)^2,x]
Output:
(d*(a + b*Log[c*x^n])^2 + e*x*(a + b*Log[c*x^n])^2 - (d^2*(a + b*Log[c*x^n ])^2)/(d + e*x) - 2*b*e*n*x*(a - b*n + b*Log[c*x^n]) - 2*b*d*n*(a + b*Log[ c*x^n])*Log[1 + (e*x)/d] - 2*d*(a + b*Log[c*x^n])^2*Log[1 + (e*x)/d] - 2*b ^2*d*n^2*PolyLog[2, -((e*x)/d)] - 4*b*d*n*(a + b*Log[c*x^n])*PolyLog[2, -( (e*x)/d)] + 4*b^2*d*n^2*PolyLog[3, -((e*x)/d)])/e^3
Time = 0.50 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, 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.
| \(\displaystyle \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^2} \, dx\) |
| \(\Big \downarrow \) 2795 |
| \(\displaystyle \int \left (\frac {d^2 \left (a+b \log \left (c x^n\right )\right )^2}{e^2 (d+e x)^2}-\frac {2 d \left (a+b \log \left (c x^n\right )\right )^2}{e^2 (d+e x)}+\frac {\left (a+b \log \left (c x^n\right )\right )^2}{e^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {4 b d n \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^3}-\frac {2 b d n \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{e^3}-\frac {2 d \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{e^3}+\frac {d x \left (a+b \log \left (c x^n\right )\right )^2}{e^2 (d+e x)}+\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{e^2}-\frac {2 a b n x}{e^2}-\frac {2 b^2 n x \log \left (c x^n\right )}{e^2}-\frac {2 b^2 d n^2 \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{e^3}+\frac {4 b^2 d n^2 \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )}{e^3}+\frac {2 b^2 n^2 x}{e^2}\) |
Input:
Int[(x^2*(a + b*Log[c*x^n])^2)/(d + e*x)^2,x]
Output:
(-2*a*b*n*x)/e^2 + (2*b^2*n^2*x)/e^2 - (2*b^2*n*x*Log[c*x^n])/e^2 + (x*(a + b*Log[c*x^n])^2)/e^2 + (d*x*(a + b*Log[c*x^n])^2)/(e^2*(d + e*x)) - (2*b *d*n*(a + b*Log[c*x^n])*Log[1 + (e*x)/d])/e^3 - (2*d*(a + b*Log[c*x^n])^2* Log[1 + (e*x)/d])/e^3 - (2*b^2*d*n^2*PolyLog[2, -((e*x)/d)])/e^3 - (4*b*d* n*(a + b*Log[c*x^n])*PolyLog[2, -((e*x)/d)])/e^3 + (4*b^2*d*n^2*PolyLog[3, -((e*x)/d)])/e^3
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 = 0.83 (sec) , antiderivative size = 700, normalized size of antiderivative = 3.45
| method | result | size |
| risch | \(\frac {b^{2} \ln \left (x^{n}\right )^{2} x}{e^{2}}-\frac {b^{2} \ln \left (x^{n}\right )^{2} d^{2}}{e^{3} \left (e x +d \right )}-\frac {2 b^{2} \ln \left (x^{n}\right )^{2} d \ln \left (e x +d \right )}{e^{3}}+\frac {2 b^{2} n \ln \left (x \right ) \ln \left (x^{n}\right ) d}{e^{3}}-\frac {2 b^{2} n \ln \left (x^{n}\right ) d \ln \left (e x +d \right )}{e^{3}}-\frac {2 b^{2} n \ln \left (x^{n}\right ) x}{e^{2}}+\frac {2 b^{2} n^{2} x}{e^{2}}+\frac {2 b^{2} n^{2} \ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right ) d}{e^{3}}+\frac {2 b^{2} n^{2} \operatorname {dilog}\left (-\frac {e x}{d}\right ) d}{e^{3}}-\frac {b^{2} n^{2} d \ln \left (x \right )^{2}}{e^{3}}-\frac {4 b^{2} d \ln \left (x \right ) \ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right ) n^{2}}{e^{3}}-\frac {4 b^{2} d \ln \left (x \right ) \operatorname {dilog}\left (-\frac {e x}{d}\right ) n^{2}}{e^{3}}+\frac {4 b^{2} n d \ln \left (x^{n}\right ) \ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )}{e^{3}}+\frac {4 b^{2} n d \ln \left (x^{n}\right ) \operatorname {dilog}\left (-\frac {e x}{d}\right )}{e^{3}}+\frac {2 b^{2} d \,n^{2} \ln \left (e x +d \right ) \ln \left (x \right )^{2}}{e^{3}}-\frac {2 b^{2} d \,n^{2} \ln \left (x \right )^{2} \ln \left (1+\frac {e x}{d}\right )}{e^{3}}-\frac {4 b^{2} d \,n^{2} \ln \left (x \right ) \operatorname {polylog}\left (2, -\frac {e x}{d}\right )}{e^{3}}+\frac {4 b^{2} d \,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 {\ln \left (x^{n}\right ) x}{e^{2}}-\frac {\ln \left (x^{n}\right ) d^{2}}{e^{3} \left (e x +d \right )}-\frac {2 \ln \left (x^{n}\right ) d \ln \left (e x +d \right )}{e^{3}}-n \left (\frac {e x +d +d \ln \left (e x +d \right )-d \ln \left (e x \right )}{e^{3}}-\frac {2 d \left (\operatorname {dilog}\left (-\frac {e x}{d}\right )+\ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )\right )}{e^{3}}\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 {x}{e^{2}}-\frac {d^{2}}{e^{3} \left (e x +d \right )}-\frac {2 d \ln \left (e x +d \right )}{e^{3}}\right )}{4}\) | \(700\) |
Input:
int(x^2*(a+b*ln(c*x^n))^2/(e*x+d)^2,x,method=_RETURNVERBOSE)
Output:
b^2*ln(x^n)^2/e^2*x-b^2*ln(x^n)^2/e^3*d^2/(e*x+d)-2*b^2*ln(x^n)^2/e^3*d*ln (e*x+d)+2*b^2*n/e^3*ln(x)*ln(x^n)*d-2*b^2*n*ln(x^n)/e^3*d*ln(e*x+d)-2*b^2* n*ln(x^n)/e^2*x+2*b^2*n^2*x/e^2+2*b^2/e^3*n^2*ln(e*x+d)*ln(-e*x/d)*d+2*b^2 /e^3*n^2*dilog(-e*x/d)*d-b^2/e^3*n^2*d*ln(x)^2-4*b^2/e^3*d*ln(x)*ln(e*x+d) *ln(-e*x/d)*n^2-4*b^2/e^3*d*ln(x)*dilog(-e*x/d)*n^2+4*b^2*n/e^3*d*ln(x^n)* ln(e*x+d)*ln(-e*x/d)+4*b^2*n/e^3*d*ln(x^n)*dilog(-e*x/d)+2*b^2/e^3*d*n^2*l n(e*x+d)*ln(x)^2-2*b^2/e^3*d*n^2*ln(x)^2*ln(1+e*x/d)-4*b^2/e^3*d*n^2*ln(x) *polylog(2,-e*x/d)+4*b^2*d*n^2*polylog(3,-e*x/d)/e^3+(I*Pi*b*csgn(I*x^n)*c sgn(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*x-ln (x^n)/e^3*d^2/(e*x+d)-2*ln(x^n)/e^3*d*ln(e*x+d)-n*(1/e^3*(e*x+d+d*ln(e*x+d )-d*ln(e*x))-2/e^3*d*(dilog(-e*x/d)+ln(e*x+d)*ln(-e*x/d))))+1/4*(I*Pi*b*cs gn(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*(x/e^2 -1/e^3*d^2/(e*x+d)-2/e^3*d*ln(e*x+d))
\[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^2} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} x^{2}}{{\left (e x + d\right )}^{2}} \,d x } \] Input:
integrate(x^2*(a+b*log(c*x^n))^2/(e*x+d)^2,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^2*x^2 + 2*d*e*x + d^2), x)
\[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^2} \, dx=\int \frac {x^{2} \left (a + b \log {\left (c x^{n} \right )}\right )^{2}}{\left (d + e x\right )^{2}}\, dx \] Input:
integrate(x**2*(a+b*ln(c*x**n))**2/(e*x+d)**2,x)
Output:
Integral(x**2*(a + b*log(c*x**n))**2/(d + e*x)**2, x)
\[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^2} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} x^{2}}{{\left (e x + d\right )}^{2}} \,d x } \] Input:
integrate(x^2*(a+b*log(c*x^n))^2/(e*x+d)^2,x, algorithm="maxima")
Output:
-a^2*(d^2/(e^4*x + d*e^3) - x/e^2 + 2*d*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^2*x^2 + 2*d*e*x + d^2), x)
\[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^2} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} x^{2}}{{\left (e x + d\right )}^{2}} \,d x } \] Input:
integrate(x^2*(a+b*log(c*x^n))^2/(e*x+d)^2,x, algorithm="giac")
Output:
integrate((b*log(c*x^n) + a)^2*x^2/(e*x + d)^2, x)
Timed out. \[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^2} \, dx=\int \frac {x^2\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2}{{\left (d+e\,x\right )}^2} \,d x \] Input:
int((x^2*(a + b*log(c*x^n))^2)/(d + e*x)^2,x)
Output:
int((x^2*(a + b*log(c*x^n))^2)/(d + e*x)^2, x)
\[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^2} \, dx=\frac {6 a^{2} d e n x -6 a b \,e^{2} n^{2} x^{2}+6 b^{2} d e \,n^{3} 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 ) a b \,d^{4} n -2 \mathrm {log}\left (x^{n} c \right )^{3} b^{2} d^{2}+3 \mathrm {log}\left (x^{n} c \right )^{2} b^{2} d e n x +6 \,\mathrm {log}\left (x^{n} c \right ) a b \,e^{2} n \,x^{2}+12 \,\mathrm {log}\left (x^{n} c \right ) b^{2} d e \,n^{2} x -6 a b d e \,n^{2} 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 ) a b \,d^{3} e n x -2 \mathrm {log}\left (x^{n} c \right )^{3} b^{2} d e x -6 \,\mathrm {log}\left (e x +d \right ) a^{2} d^{2} n -18 \,\mathrm {log}\left (e x +d \right ) b^{2} d^{2} n^{3}-6 \mathrm {log}\left (x^{n} c \right )^{2} a b d e x -18 \,\mathrm {log}\left (e x +d \right ) a b \,d^{2} n^{2}-6 \mathrm {log}\left (x^{n} c \right )^{2} a b \,d^{2}+3 a^{2} e^{2} n \,x^{2}+6 b^{2} e^{2} n^{3} x^{2}+6 \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^{4} n +18 \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^{4} n^{2}-18 \,\mathrm {log}\left (e x +d \right ) a b d e \,n^{2} x +24 \,\mathrm {log}\left (x^{n} c \right ) a b d e n x -6 \,\mathrm {log}\left (e x +d \right ) a^{2} d e n x -18 \,\mathrm {log}\left (e x +d \right ) b^{2} d e \,n^{3} x +6 \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} e n x +18 \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} e \,n^{2} x -9 \mathrm {log}\left (x^{n} c \right )^{2} b^{2} d^{2} n +3 \mathrm {log}\left (x^{n} c \right )^{2} b^{2} e^{2} n \,x^{2}-6 \,\mathrm {log}\left (x^{n} c \right ) b^{2} e^{2} n^{2} x^{2}}{3 e^{3} n \left (e x +d \right )} \] Input:
int(x^2*(a+b*log(c*x^n))^2/(e*x+d)^2,x)
Output:
(6*int(log(x**n*c)**2/(d**2*x + 2*d*e*x**2 + e**2*x**3),x)*b**2*d**4*n + 6 *int(log(x**n*c)**2/(d**2*x + 2*d*e*x**2 + e**2*x**3),x)*b**2*d**3*e*n*x + 12*int(log(x**n*c)/(d**2*x + 2*d*e*x**2 + e**2*x**3),x)*a*b*d**4*n + 12*i nt(log(x**n*c)/(d**2*x + 2*d*e*x**2 + e**2*x**3),x)*a*b*d**3*e*n*x + 18*in t(log(x**n*c)/(d**2*x + 2*d*e*x**2 + e**2*x**3),x)*b**2*d**4*n**2 + 18*int (log(x**n*c)/(d**2*x + 2*d*e*x**2 + e**2*x**3),x)*b**2*d**3*e*n**2*x - 6*l og(d + e*x)*a**2*d**2*n - 6*log(d + e*x)*a**2*d*e*n*x - 18*log(d + e*x)*a* b*d**2*n**2 - 18*log(d + e*x)*a*b*d*e*n**2*x - 18*log(d + e*x)*b**2*d**2*n **3 - 18*log(d + e*x)*b**2*d*e*n**3*x - 2*log(x**n*c)**3*b**2*d**2 - 2*log (x**n*c)**3*b**2*d*e*x - 6*log(x**n*c)**2*a*b*d**2 - 6*log(x**n*c)**2*a*b* d*e*x - 9*log(x**n*c)**2*b**2*d**2*n + 3*log(x**n*c)**2*b**2*d*e*n*x + 3*l og(x**n*c)**2*b**2*e**2*n*x**2 + 24*log(x**n*c)*a*b*d*e*n*x + 6*log(x**n*c )*a*b*e**2*n*x**2 + 12*log(x**n*c)*b**2*d*e*n**2*x - 6*log(x**n*c)*b**2*e* *2*n**2*x**2 + 6*a**2*d*e*n*x + 3*a**2*e**2*n*x**2 - 6*a*b*d*e*n**2*x - 6* a*b*e**2*n**2*x**2 + 6*b**2*d*e*n**3*x + 6*b**2*e**2*n**3*x**2)/(3*e**3*n* (d + e*x))