Integrand size = 26, antiderivative size = 205 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )}{x^4} \, dx=-\frac {2 b^2 e n^2 r}{81 x^3}-\frac {2 b e n (3 a+b n) r}{81 x^3}-\frac {e \left (9 a^2+6 a b n+2 b^2 n^2\right ) r}{81 x^3}-\frac {2 b^2 e n r \log \left (c x^n\right )}{27 x^3}-\frac {2 b e (3 a+b n) r \log \left (c x^n\right )}{27 x^3}-\frac {b^2 e r \log ^2\left (c x^n\right )}{9 x^3}-\frac {2 b^2 n^2 \left (d+e \log \left (f x^r\right )\right )}{27 x^3}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{9 x^3}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )}{3 x^3} \]
-2/81*b^2*e*n^2*r/x^3-2/81*b*e*n*(b*n+3*a)*r/x^3-1/81*e*(2*b^2*n^2+6*a*b*n +9*a^2)*r/x^3-2/27*b^2*e*n*r*ln(c*x^n)/x^3-2/27*b*e*(b*n+3*a)*r*ln(c*x^n)/ x^3-1/9*b^2*e*r*ln(c*x^n)^2/x^3-2/27*b^2*n^2*(d+e*ln(f*x^r))/x^3-2/9*b*n*( a+b*ln(c*x^n))*(d+e*ln(f*x^r))/x^3-1/3*(a+b*ln(c*x^n))^2*(d+e*ln(f*x^r))/x ^3
Time = 0.11 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.76 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )}{x^4} \, dx=-\frac {9 a^2 d+6 a b d n+2 b^2 d n^2+3 a^2 e r+4 a b e n r+2 b^2 e n^2 r+e \left (9 a^2+6 a b n+2 b^2 n^2\right ) \log \left (f x^r\right )+3 b^2 \log ^2\left (c x^n\right ) \left (3 d+e r+3 e \log \left (f x^r\right )\right )+2 b \log \left (c x^n\right ) \left (9 a d+3 b d n+3 a e r+2 b e n r+3 e (3 a+b n) \log \left (f x^r\right )\right )}{27 x^3} \]
-1/27*(9*a^2*d + 6*a*b*d*n + 2*b^2*d*n^2 + 3*a^2*e*r + 4*a*b*e*n*r + 2*b^2 *e*n^2*r + e*(9*a^2 + 6*a*b*n + 2*b^2*n^2)*Log[f*x^r] + 3*b^2*Log[c*x^n]^2 *(3*d + e*r + 3*e*Log[f*x^r]) + 2*b*Log[c*x^n]*(9*a*d + 3*b*d*n + 3*a*e*r + 2*b*e*n*r + 3*e*(3*a + b*n)*Log[f*x^r]))/x^3
Time = 0.46 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.95, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2813, 27, 2010, 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.
| \(\displaystyle \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )}{x^4} \, dx\) |
| \(\Big \downarrow \) 2813 |
| \(\displaystyle -e r \int -\frac {9 a^2+6 b n a+2 b^2 n^2+9 b^2 \log ^2\left (c x^n\right )+6 b (3 a+b n) \log \left (c x^n\right )}{27 x^4}dx-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{9 x^3}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )}{3 x^3}-\frac {2 b^2 n^2 \left (d+e \log \left (f x^r\right )\right )}{27 x^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{27} e r \int \frac {9 a^2+6 b n a+2 b^2 n^2+9 b^2 \log ^2\left (c x^n\right )+6 b (3 a+b n) \log \left (c x^n\right )}{x^4}dx-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{9 x^3}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )}{3 x^3}-\frac {2 b^2 n^2 \left (d+e \log \left (f x^r\right )\right )}{27 x^3}\) |
| \(\Big \downarrow \) 2010 |
| \(\displaystyle \frac {1}{27} e r \int \left (\frac {9 b^2 \log ^2\left (c x^n\right )}{x^4}+\frac {6 b (3 a+b n) \log \left (c x^n\right )}{x^4}+\frac {9 a^2+6 b n a+2 b^2 n^2}{x^4}\right )dx-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{9 x^3}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )}{3 x^3}-\frac {2 b^2 n^2 \left (d+e \log \left (f x^r\right )\right )}{27 x^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{27} e r \left (-\frac {9 a^2+6 a b n+2 b^2 n^2}{3 x^3}-\frac {2 b (3 a+b n) \log \left (c x^n\right )}{x^3}-\frac {2 b n (3 a+b n)}{3 x^3}-\frac {3 b^2 \log ^2\left (c x^n\right )}{x^3}-\frac {2 b^2 n \log \left (c x^n\right )}{x^3}-\frac {2 b^2 n^2}{3 x^3}\right )-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{9 x^3}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )}{3 x^3}-\frac {2 b^2 n^2 \left (d+e \log \left (f x^r\right )\right )}{27 x^3}\) |
(e*r*((-2*b^2*n^2)/(3*x^3) - (2*b*n*(3*a + b*n))/(3*x^3) - (9*a^2 + 6*a*b* n + 2*b^2*n^2)/(3*x^3) - (2*b^2*n*Log[c*x^n])/x^3 - (2*b*(3*a + b*n)*Log[c *x^n])/x^3 - (3*b^2*Log[c*x^n]^2)/x^3))/27 - (2*b^2*n^2*(d + e*Log[f*x^r]) )/(27*x^3) - (2*b*n*(a + b*Log[c*x^n])*(d + e*Log[f*x^r]))/(9*x^3) - ((a + b*Log[c*x^n])^2*(d + e*Log[f*x^r]))/(3*x^3)
3.2.69.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x] , x] /; FreeQ[{c, m}, x] && SumQ[u] && !LinearQ[u, x] && !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.) + Log[(f_.)*(x_)^(r_ .)]*(e_.))*((g_.)*(x_))^(m_.), x_Symbol] :> With[{u = IntHide[(g*x)^m*(a + b*Log[c*x^n])^p, x]}, Simp[(d + e*Log[f*x^r]) u, x] - Simp[e*r Int[Simp lifyIntegrand[u/x, x], x], x]] /; FreeQ[{a, b, c, d, e, f, g, m, n, p, r}, x] && !(EqQ[p, 1] && EqQ[a, 0] && NeQ[d, 0])
Time = 8.13 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.31
| method | result | size |
| parallelrisch | \(-\frac {2 \ln \left (f \,x^{r}\right ) b^{2} e \,n^{5}+9 \ln \left (f \,x^{r}\right ) a^{2} e \,n^{3}+6 \ln \left (c \,x^{n}\right ) b^{2} d \,n^{4}+9 \ln \left (c \,x^{n}\right )^{2} b^{2} d \,n^{3}+2 b^{2} e \,n^{5} r +3 a^{2} e \,n^{3} r +6 a b d \,n^{4}+18 \ln \left (c \,x^{n}\right ) \ln \left (f \,x^{r}\right ) a b e \,n^{3}+6 \ln \left (c \,x^{n}\right ) a b e \,n^{3} r +4 a b e \,n^{4} r +2 b^{2} d \,n^{5}+9 a^{2} d \,n^{3}+9 e \,b^{2} \ln \left (f \,x^{r}\right ) \ln \left (c \,x^{n}\right )^{2} n^{3}+6 \ln \left (f \,x^{r}\right ) a b e \,n^{4}+6 \ln \left (c \,x^{n}\right ) \ln \left (f \,x^{r}\right ) b^{2} e \,n^{4}+4 \ln \left (c \,x^{n}\right ) b^{2} e \,n^{4} r +18 \ln \left (c \,x^{n}\right ) a b d \,n^{3}+3 \ln \left (c \,x^{n}\right )^{2} b^{2} e \,n^{3} r}{27 x^{3} n^{3}}\) | \(268\) |
| risch | \(\text {Expression too large to display}\) | \(8407\) |
-1/27/x^3*(2*ln(f*x^r)*b^2*e*n^5+9*ln(f*x^r)*a^2*e*n^3+6*ln(c*x^n)*b^2*d*n ^4+9*ln(c*x^n)^2*b^2*d*n^3+2*b^2*e*n^5*r+3*a^2*e*n^3*r+6*a*b*d*n^4+18*ln(c *x^n)*ln(f*x^r)*a*b*e*n^3+6*ln(c*x^n)*a*b*e*n^3*r+4*a*b*e*n^4*r+2*b^2*d*n^ 5+9*a^2*d*n^3+9*e*b^2*ln(f*x^r)*ln(c*x^n)^2*n^3+6*ln(f*x^r)*a*b*e*n^4+6*ln (c*x^n)*ln(f*x^r)*b^2*e*n^4+4*ln(c*x^n)*b^2*e*n^4*r+18*ln(c*x^n)*a*b*d*n^3 +3*ln(c*x^n)^2*b^2*e*n^3*r)/n^3
Time = 0.26 (sec) , antiderivative size = 329, normalized size of antiderivative = 1.60 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )}{x^4} \, dx=-\frac {9 \, b^{2} e n^{2} r \log \left (x\right )^{3} + 2 \, b^{2} d n^{2} + 6 \, a b d n + 9 \, a^{2} d + 3 \, {\left (b^{2} e r + 3 \, b^{2} d\right )} \log \left (c\right )^{2} + 9 \, {\left (2 \, b^{2} e n r \log \left (c\right ) + b^{2} e n^{2} \log \left (f\right ) + b^{2} d n^{2} + {\left (b^{2} e n^{2} + 2 \, a b e n\right )} r\right )} \log \left (x\right )^{2} + {\left (2 \, b^{2} e n^{2} + 4 \, a b e n + 3 \, a^{2} e\right )} r + 2 \, {\left (3 \, b^{2} d n + 9 \, a b d + {\left (2 \, b^{2} e n + 3 \, a b e\right )} r\right )} \log \left (c\right ) + {\left (2 \, b^{2} e n^{2} + 9 \, b^{2} e \log \left (c\right )^{2} + 6 \, a b e n + 9 \, a^{2} e + 6 \, {\left (b^{2} e n + 3 \, a b e\right )} \log \left (c\right )\right )} \log \left (f\right ) + 3 \, {\left (3 \, b^{2} e r \log \left (c\right )^{2} + 2 \, b^{2} d n^{2} + 6 \, a b d n + {\left (2 \, b^{2} e n^{2} + 4 \, a b e n + 3 \, a^{2} e\right )} r + 2 \, {\left (3 \, b^{2} d n + {\left (2 \, b^{2} e n + 3 \, a b e\right )} r\right )} \log \left (c\right ) + 2 \, {\left (b^{2} e n^{2} + 3 \, b^{2} e n \log \left (c\right ) + 3 \, a b e n\right )} \log \left (f\right )\right )} \log \left (x\right )}{27 \, x^{3}} \]
-1/27*(9*b^2*e*n^2*r*log(x)^3 + 2*b^2*d*n^2 + 6*a*b*d*n + 9*a^2*d + 3*(b^2 *e*r + 3*b^2*d)*log(c)^2 + 9*(2*b^2*e*n*r*log(c) + b^2*e*n^2*log(f) + b^2* d*n^2 + (b^2*e*n^2 + 2*a*b*e*n)*r)*log(x)^2 + (2*b^2*e*n^2 + 4*a*b*e*n + 3 *a^2*e)*r + 2*(3*b^2*d*n + 9*a*b*d + (2*b^2*e*n + 3*a*b*e)*r)*log(c) + (2* b^2*e*n^2 + 9*b^2*e*log(c)^2 + 6*a*b*e*n + 9*a^2*e + 6*(b^2*e*n + 3*a*b*e) *log(c))*log(f) + 3*(3*b^2*e*r*log(c)^2 + 2*b^2*d*n^2 + 6*a*b*d*n + (2*b^2 *e*n^2 + 4*a*b*e*n + 3*a^2*e)*r + 2*(3*b^2*d*n + (2*b^2*e*n + 3*a*b*e)*r)* log(c) + 2*(b^2*e*n^2 + 3*b^2*e*n*log(c) + 3*a*b*e*n)*log(f))*log(x))/x^3
Time = 1.78 (sec) , antiderivative size = 342, normalized size of antiderivative = 1.67 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )}{x^4} \, dx=- \frac {a^{2} d}{3 x^{3}} - \frac {a^{2} e r}{9 x^{3}} - \frac {a^{2} e \log {\left (f x^{r} \right )}}{3 x^{3}} - \frac {2 a b d n}{9 x^{3}} - \frac {2 a b d \log {\left (c x^{n} \right )}}{3 x^{3}} - \frac {4 a b e n r}{27 x^{3}} - \frac {2 a b e n \log {\left (f x^{r} \right )}}{9 x^{3}} - \frac {2 a b e r \log {\left (c x^{n} \right )}}{9 x^{3}} - \frac {2 a b e \log {\left (c x^{n} \right )} \log {\left (f x^{r} \right )}}{3 x^{3}} - \frac {2 b^{2} d n^{2}}{27 x^{3}} - \frac {2 b^{2} d n \log {\left (c x^{n} \right )}}{9 x^{3}} - \frac {b^{2} d \log {\left (c x^{n} \right )}^{2}}{3 x^{3}} - \frac {2 b^{2} e n^{2} r}{27 x^{3}} - \frac {2 b^{2} e n^{2} \log {\left (f x^{r} \right )}}{27 x^{3}} - \frac {4 b^{2} e n r \log {\left (c x^{n} \right )}}{27 x^{3}} - \frac {2 b^{2} e n \log {\left (c x^{n} \right )} \log {\left (f x^{r} \right )}}{9 x^{3}} - \frac {b^{2} e r \log {\left (c x^{n} \right )}^{2}}{9 x^{3}} - \frac {b^{2} e \log {\left (c x^{n} \right )}^{2} \log {\left (f x^{r} \right )}}{3 x^{3}} \]
-a**2*d/(3*x**3) - a**2*e*r/(9*x**3) - a**2*e*log(f*x**r)/(3*x**3) - 2*a*b *d*n/(9*x**3) - 2*a*b*d*log(c*x**n)/(3*x**3) - 4*a*b*e*n*r/(27*x**3) - 2*a *b*e*n*log(f*x**r)/(9*x**3) - 2*a*b*e*r*log(c*x**n)/(9*x**3) - 2*a*b*e*log (c*x**n)*log(f*x**r)/(3*x**3) - 2*b**2*d*n**2/(27*x**3) - 2*b**2*d*n*log(c *x**n)/(9*x**3) - b**2*d*log(c*x**n)**2/(3*x**3) - 2*b**2*e*n**2*r/(27*x** 3) - 2*b**2*e*n**2*log(f*x**r)/(27*x**3) - 4*b**2*e*n*r*log(c*x**n)/(27*x* *3) - 2*b**2*e*n*log(c*x**n)*log(f*x**r)/(9*x**3) - b**2*e*r*log(c*x**n)** 2/(9*x**3) - b**2*e*log(c*x**n)**2*log(f*x**r)/(3*x**3)
Time = 0.21 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.12 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )}{x^4} \, dx=-\frac {1}{9} \, b^{2} e {\left (\frac {r}{x^{3}} + \frac {3 \, \log \left (f x^{r}\right )}{x^{3}}\right )} \log \left (c x^{n}\right )^{2} - \frac {2}{9} \, a b e {\left (\frac {r}{x^{3}} + \frac {3 \, \log \left (f x^{r}\right )}{x^{3}}\right )} \log \left (c x^{n}\right ) - \frac {2}{27} \, b^{2} e {\left (\frac {{\left (r \log \left (x\right ) + r + \log \left (f\right )\right )} n^{2}}{x^{3}} + \frac {n {\left (2 \, r + 3 \, \log \left (f\right ) + 3 \, \log \left (x^{r}\right )\right )} \log \left (c x^{n}\right )}{x^{3}}\right )} - \frac {2}{27} \, b^{2} d {\left (\frac {n^{2}}{x^{3}} + \frac {3 \, n \log \left (c x^{n}\right )}{x^{3}}\right )} - \frac {2 \, a b e n {\left (2 \, r + 3 \, \log \left (f\right ) + 3 \, \log \left (x^{r}\right )\right )}}{27 \, x^{3}} - \frac {b^{2} d \log \left (c x^{n}\right )^{2}}{3 \, x^{3}} - \frac {2 \, a b d n}{9 \, x^{3}} - \frac {a^{2} e r}{9 \, x^{3}} - \frac {2 \, a b d \log \left (c x^{n}\right )}{3 \, x^{3}} - \frac {a^{2} e \log \left (f x^{r}\right )}{3 \, x^{3}} - \frac {a^{2} d}{3 \, x^{3}} \]
-1/9*b^2*e*(r/x^3 + 3*log(f*x^r)/x^3)*log(c*x^n)^2 - 2/9*a*b*e*(r/x^3 + 3* log(f*x^r)/x^3)*log(c*x^n) - 2/27*b^2*e*((r*log(x) + r + log(f))*n^2/x^3 + n*(2*r + 3*log(f) + 3*log(x^r))*log(c*x^n)/x^3) - 2/27*b^2*d*(n^2/x^3 + 3 *n*log(c*x^n)/x^3) - 2/27*a*b*e*n*(2*r + 3*log(f) + 3*log(x^r))/x^3 - 1/3* b^2*d*log(c*x^n)^2/x^3 - 2/9*a*b*d*n/x^3 - 1/9*a^2*e*r/x^3 - 2/3*a*b*d*log (c*x^n)/x^3 - 1/3*a^2*e*log(f*x^r)/x^3 - 1/3*a^2*d/x^3
Time = 0.30 (sec) , antiderivative size = 352, normalized size of antiderivative = 1.72 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )}{x^4} \, dx=-\frac {b^{2} e n^{2} r \log \left (x\right )^{3}}{3 \, x^{3}} - \frac {{\left (b^{2} e n^{2} r + 2 \, b^{2} e n r \log \left (c\right ) + b^{2} e n^{2} \log \left (f\right ) + b^{2} d n^{2} + 2 \, a b e n r\right )} \log \left (x\right )^{2}}{3 \, x^{3}} - \frac {{\left (2 \, b^{2} e n^{2} r + 4 \, b^{2} e n r \log \left (c\right ) + 3 \, b^{2} e r \log \left (c\right )^{2} + 2 \, b^{2} e n^{2} \log \left (f\right ) + 6 \, b^{2} e n \log \left (c\right ) \log \left (f\right ) + 2 \, b^{2} d n^{2} + 4 \, a b e n r + 6 \, b^{2} d n \log \left (c\right ) + 6 \, a b e r \log \left (c\right ) + 6 \, a b e n \log \left (f\right ) + 6 \, a b d n + 3 \, a^{2} e r\right )} \log \left (x\right )}{9 \, x^{3}} - \frac {2 \, b^{2} e n^{2} r + 4 \, b^{2} e n r \log \left (c\right ) + 3 \, b^{2} e r \log \left (c\right )^{2} + 2 \, b^{2} e n^{2} \log \left (f\right ) + 6 \, b^{2} e n \log \left (c\right ) \log \left (f\right ) + 9 \, b^{2} e \log \left (c\right )^{2} \log \left (f\right ) + 2 \, b^{2} d n^{2} + 4 \, a b e n r + 6 \, b^{2} d n \log \left (c\right ) + 6 \, a b e r \log \left (c\right ) + 9 \, b^{2} d \log \left (c\right )^{2} + 6 \, a b e n \log \left (f\right ) + 18 \, a b e \log \left (c\right ) \log \left (f\right ) + 6 \, a b d n + 3 \, a^{2} e r + 18 \, a b d \log \left (c\right ) + 9 \, a^{2} e \log \left (f\right ) + 9 \, a^{2} d}{27 \, x^{3}} \]
-1/3*b^2*e*n^2*r*log(x)^3/x^3 - 1/3*(b^2*e*n^2*r + 2*b^2*e*n*r*log(c) + b^ 2*e*n^2*log(f) + b^2*d*n^2 + 2*a*b*e*n*r)*log(x)^2/x^3 - 1/9*(2*b^2*e*n^2* r + 4*b^2*e*n*r*log(c) + 3*b^2*e*r*log(c)^2 + 2*b^2*e*n^2*log(f) + 6*b^2*e *n*log(c)*log(f) + 2*b^2*d*n^2 + 4*a*b*e*n*r + 6*b^2*d*n*log(c) + 6*a*b*e* r*log(c) + 6*a*b*e*n*log(f) + 6*a*b*d*n + 3*a^2*e*r)*log(x)/x^3 - 1/27*(2* b^2*e*n^2*r + 4*b^2*e*n*r*log(c) + 3*b^2*e*r*log(c)^2 + 2*b^2*e*n^2*log(f) + 6*b^2*e*n*log(c)*log(f) + 9*b^2*e*log(c)^2*log(f) + 2*b^2*d*n^2 + 4*a*b *e*n*r + 6*b^2*d*n*log(c) + 6*a*b*e*r*log(c) + 9*b^2*d*log(c)^2 + 6*a*b*e* n*log(f) + 18*a*b*e*log(c)*log(f) + 6*a*b*d*n + 3*a^2*e*r + 18*a*b*d*log(c ) + 9*a^2*e*log(f) + 9*a^2*d)/x^3
Time = 0.85 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.93 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )}{x^4} \, dx=-\ln \left (f\,x^r\right )\,\left (\ln \left (c\,x^n\right )\,\left (\frac {2\,a\,b\,e}{3\,x^3}+\frac {2\,b^2\,e\,n}{9\,x^3}\right )+\frac {a^2\,e}{3\,x^3}+\frac {2\,b^2\,e\,n^2}{27\,x^3}+\frac {b^2\,e\,{\ln \left (c\,x^n\right )}^2}{3\,x^3}+\frac {2\,a\,b\,e\,n}{9\,x^3}\right )-\frac {\frac {a^2\,d}{3}+\frac {2\,b^2\,d\,n^2}{27}+\frac {a^2\,e\,r}{9}+\frac {2\,b^2\,e\,n^2\,r}{27}+\frac {2\,a\,b\,d\,n}{9}+\frac {4\,a\,b\,e\,n\,r}{27}}{x^3}-\frac {b^2\,{\ln \left (c\,x^n\right )}^2\,\left (3\,d+e\,r\right )}{9\,x^3}-\frac {2\,b\,\ln \left (c\,x^n\right )\,\left (9\,a\,d+3\,b\,d\,n+3\,a\,e\,r+2\,b\,e\,n\,r\right )}{27\,x^3} \]
- log(f*x^r)*(log(c*x^n)*((2*a*b*e)/(3*x^3) + (2*b^2*e*n)/(9*x^3)) + (a^2* e)/(3*x^3) + (2*b^2*e*n^2)/(27*x^3) + (b^2*e*log(c*x^n)^2)/(3*x^3) + (2*a* b*e*n)/(9*x^3)) - ((a^2*d)/3 + (2*b^2*d*n^2)/27 + (a^2*e*r)/9 + (2*b^2*e*n ^2*r)/27 + (2*a*b*d*n)/9 + (4*a*b*e*n*r)/27)/x^3 - (b^2*log(c*x^n)^2*(3*d + e*r))/(9*x^3) - (2*b*log(c*x^n)*(9*a*d + 3*b*d*n + 3*a*e*r + 2*b*e*n*r)) /(27*x^3)