Integrand size = 29, antiderivative size = 346 \[ \int \frac {(f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )^2}{\left (d+e x^m\right )^4} \, dx=-\frac {b^2 n^2 x^{1-m} (f x)^{-1+m}}{3 d^2 e m^3 \left (d+e x^m\right )}-\frac {b^2 n^2 x^{1-m} (f x)^{-1+m} \log (x)}{3 d^3 e m^2}+\frac {b n x^{1-m} (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )}{3 d e m^2 \left (d+e x^m\right )^2}-\frac {2 b n x (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )}{3 d^3 m^2 \left (d+e x^m\right )}-\frac {x^{1-m} (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )^2}{3 e m \left (d+e x^m\right )^3}-\frac {2 b n x^{1-m} (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {d x^{-m}}{e}\right )}{3 d^3 e m^2}+\frac {b^2 n^2 x^{1-m} (f x)^{-1+m} \log \left (d+e x^m\right )}{d^3 e m^3}+\frac {2 b^2 n^2 x^{1-m} (f x)^{-1+m} \operatorname {PolyLog}\left (2,-\frac {d x^{-m}}{e}\right )}{3 d^3 e m^3} \]
-1/3*b^2*n^2*x^(1-m)*(f*x)^(-1+m)/d^2/e/m^3/(d+e*x^m)-1/3*b^2*n^2*x^(1-m)* (f*x)^(-1+m)*ln(x)/d^3/e/m^2+1/3*b*n*x^(1-m)*(f*x)^(-1+m)*(a+b*ln(c*x^n))/ d/e/m^2/(d+e*x^m)^2-2/3*b*n*x*(f*x)^(-1+m)*(a+b*ln(c*x^n))/d^3/m^2/(d+e*x^ m)-1/3*x^(1-m)*(f*x)^(-1+m)*(a+b*ln(c*x^n))^2/e/m/(d+e*x^m)^3-2/3*b*n*x^(1 -m)*(f*x)^(-1+m)*(a+b*ln(c*x^n))*ln(1+d/e/(x^m))/d^3/e/m^2+b^2*n^2*x^(1-m) *(f*x)^(-1+m)*ln(d+e*x^m)/d^3/e/m^3+2/3*b^2*n^2*x^(1-m)*(f*x)^(-1+m)*polyl og(2,-d/e/(x^m))/d^3/e/m^3
Time = 0.60 (sec) , antiderivative size = 240, normalized size of antiderivative = 0.69 \[ \int \frac {(f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )^2}{\left (d+e x^m\right )^4} \, dx=\frac {x^{-m} (f x)^m \left (\frac {b m n \left (a+b \log \left (c x^n\right )\right )}{d \left (d+e x^m\right )^2}-\frac {m^2 \left (a+b \log \left (c x^n\right )\right )^2}{\left (d+e x^m\right )^3}+\frac {b n \left (2 a m-b n+2 b m \log \left (c x^n\right )\right )}{d^2 \left (d+e x^m\right )}-\frac {2 a b m n \log \left (d-d x^m\right )}{d^3}+\frac {3 b^2 n^2 \log \left (d-d x^m\right )}{d^3}+\frac {2 b^2 m n \left (n \log (x)-\log \left (c x^n\right )\right ) \log \left (d-d x^m\right )}{d^3}+\frac {2 b^2 n^2 \left (\frac {1}{2} m^2 \log ^2(x)+\left (-m \log (x)+\log \left (-\frac {e x^m}{d}\right )\right ) \log \left (d+e x^m\right )+\operatorname {PolyLog}\left (2,1+\frac {e x^m}{d}\right )\right )}{d^3}\right )}{3 e f m^3} \]
((f*x)^m*((b*m*n*(a + b*Log[c*x^n]))/(d*(d + e*x^m)^2) - (m^2*(a + b*Log[c *x^n])^2)/(d + e*x^m)^3 + (b*n*(2*a*m - b*n + 2*b*m*Log[c*x^n]))/(d^2*(d + e*x^m)) - (2*a*b*m*n*Log[d - d*x^m])/d^3 + (3*b^2*n^2*Log[d - d*x^m])/d^3 + (2*b^2*m*n*(n*Log[x] - Log[c*x^n])*Log[d - d*x^m])/d^3 + (2*b^2*n^2*((m ^2*Log[x]^2)/2 + (-(m*Log[x]) + Log[-((e*x^m)/d)])*Log[d + e*x^m] + PolyLo g[2, 1 + (e*x^m)/d]))/d^3))/(3*e*f*m^3*x^m)
Time = 1.44 (sec) , antiderivative size = 263, normalized size of antiderivative = 0.76, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.414, Rules used = {2777, 2776, 2791, 2776, 798, 54, 2009, 2791, 2773, 792, 2779, 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.
\(\displaystyle \int \frac {(f x)^{m-1} \left (a+b \log \left (c x^n\right )\right )^2}{\left (d+e x^m\right )^4} \, dx\) |
\(\Big \downarrow \) 2777 |
\(\displaystyle x^{1-m} (f x)^{m-1} \int \frac {x^{m-1} \left (a+b \log \left (c x^n\right )\right )^2}{\left (e x^m+d\right )^4}dx\) |
\(\Big \downarrow \) 2776 |
\(\displaystyle x^{1-m} (f x)^{m-1} \left (\frac {2 b n \int \frac {a+b \log \left (c x^n\right )}{x \left (e x^m+d\right )^3}dx}{3 e m}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{3 e m \left (d+e x^m\right )^3}\right )\) |
\(\Big \downarrow \) 2791 |
\(\displaystyle x^{1-m} (f x)^{m-1} \left (\frac {2 b n \left (\frac {\int \frac {a+b \log \left (c x^n\right )}{x \left (e x^m+d\right )^2}dx}{d}-\frac {e \int \frac {x^{m-1} \left (a+b \log \left (c x^n\right )\right )}{\left (e x^m+d\right )^3}dx}{d}\right )}{3 e m}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{3 e m \left (d+e x^m\right )^3}\right )\) |
\(\Big \downarrow \) 2776 |
\(\displaystyle x^{1-m} (f x)^{m-1} \left (\frac {2 b n \left (\frac {\int \frac {a+b \log \left (c x^n\right )}{x \left (e x^m+d\right )^2}dx}{d}-\frac {e \left (\frac {b n \int \frac {1}{x \left (e x^m+d\right )^2}dx}{2 e m}-\frac {a+b \log \left (c x^n\right )}{2 e m \left (d+e x^m\right )^2}\right )}{d}\right )}{3 e m}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{3 e m \left (d+e x^m\right )^3}\right )\) |
\(\Big \downarrow \) 798 |
\(\displaystyle x^{1-m} (f x)^{m-1} \left (\frac {2 b n \left (\frac {\int \frac {a+b \log \left (c x^n\right )}{x \left (e x^m+d\right )^2}dx}{d}-\frac {e \left (\frac {b n \int \frac {x^{-m}}{\left (e x^m+d\right )^2}dx^m}{2 e m^2}-\frac {a+b \log \left (c x^n\right )}{2 e m \left (d+e x^m\right )^2}\right )}{d}\right )}{3 e m}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{3 e m \left (d+e x^m\right )^3}\right )\) |
\(\Big \downarrow \) 54 |
\(\displaystyle x^{1-m} (f x)^{m-1} \left (\frac {2 b n \left (\frac {\int \frac {a+b \log \left (c x^n\right )}{x \left (e x^m+d\right )^2}dx}{d}-\frac {e \left (\frac {b n \int \left (\frac {x^{-m}}{d^2}-\frac {e}{d^2 \left (e x^m+d\right )}-\frac {e}{d \left (e x^m+d\right )^2}\right )dx^m}{2 e m^2}-\frac {a+b \log \left (c x^n\right )}{2 e m \left (d+e x^m\right )^2}\right )}{d}\right )}{3 e m}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{3 e m \left (d+e x^m\right )^3}\right )\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle x^{1-m} (f x)^{m-1} \left (\frac {2 b n \left (\frac {\int \frac {a+b \log \left (c x^n\right )}{x \left (e x^m+d\right )^2}dx}{d}-\frac {e \left (\frac {b n \left (-\frac {\log \left (d+e x^m\right )}{d^2}+\frac {\log \left (x^m\right )}{d^2}+\frac {1}{d \left (d+e x^m\right )}\right )}{2 e m^2}-\frac {a+b \log \left (c x^n\right )}{2 e m \left (d+e x^m\right )^2}\right )}{d}\right )}{3 e m}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{3 e m \left (d+e x^m\right )^3}\right )\) |
\(\Big \downarrow \) 2791 |
\(\displaystyle x^{1-m} (f x)^{m-1} \left (\frac {2 b n \left (\frac {\frac {\int \frac {a+b \log \left (c x^n\right )}{x \left (e x^m+d\right )}dx}{d}-\frac {e \int \frac {x^{m-1} \left (a+b \log \left (c x^n\right )\right )}{\left (e x^m+d\right )^2}dx}{d}}{d}-\frac {e \left (\frac {b n \left (-\frac {\log \left (d+e x^m\right )}{d^2}+\frac {\log \left (x^m\right )}{d^2}+\frac {1}{d \left (d+e x^m\right )}\right )}{2 e m^2}-\frac {a+b \log \left (c x^n\right )}{2 e m \left (d+e x^m\right )^2}\right )}{d}\right )}{3 e m}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{3 e m \left (d+e x^m\right )^3}\right )\) |
\(\Big \downarrow \) 2773 |
\(\displaystyle x^{1-m} (f x)^{m-1} \left (\frac {2 b n \left (\frac {\frac {\int \frac {a+b \log \left (c x^n\right )}{x \left (e x^m+d\right )}dx}{d}-\frac {e \left (\frac {x^m \left (a+b \log \left (c x^n\right )\right )}{d m \left (d+e x^m\right )}-\frac {b n \int \frac {x^{m-1}}{e x^m+d}dx}{d m}\right )}{d}}{d}-\frac {e \left (\frac {b n \left (-\frac {\log \left (d+e x^m\right )}{d^2}+\frac {\log \left (x^m\right )}{d^2}+\frac {1}{d \left (d+e x^m\right )}\right )}{2 e m^2}-\frac {a+b \log \left (c x^n\right )}{2 e m \left (d+e x^m\right )^2}\right )}{d}\right )}{3 e m}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{3 e m \left (d+e x^m\right )^3}\right )\) |
\(\Big \downarrow \) 792 |
\(\displaystyle x^{1-m} (f x)^{m-1} \left (\frac {2 b n \left (\frac {\frac {\int \frac {a+b \log \left (c x^n\right )}{x \left (e x^m+d\right )}dx}{d}-\frac {e \left (\frac {x^m \left (a+b \log \left (c x^n\right )\right )}{d m \left (d+e x^m\right )}-\frac {b n \log \left (d+e x^m\right )}{d e m^2}\right )}{d}}{d}-\frac {e \left (\frac {b n \left (-\frac {\log \left (d+e x^m\right )}{d^2}+\frac {\log \left (x^m\right )}{d^2}+\frac {1}{d \left (d+e x^m\right )}\right )}{2 e m^2}-\frac {a+b \log \left (c x^n\right )}{2 e m \left (d+e x^m\right )^2}\right )}{d}\right )}{3 e m}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{3 e m \left (d+e x^m\right )^3}\right )\) |
\(\Big \downarrow \) 2779 |
\(\displaystyle x^{1-m} (f x)^{m-1} \left (\frac {2 b n \left (\frac {\frac {\frac {b n \int \frac {\log \left (\frac {d x^{-m}}{e}+1\right )}{x}dx}{d m}-\frac {\log \left (\frac {d x^{-m}}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d m}}{d}-\frac {e \left (\frac {x^m \left (a+b \log \left (c x^n\right )\right )}{d m \left (d+e x^m\right )}-\frac {b n \log \left (d+e x^m\right )}{d e m^2}\right )}{d}}{d}-\frac {e \left (\frac {b n \left (-\frac {\log \left (d+e x^m\right )}{d^2}+\frac {\log \left (x^m\right )}{d^2}+\frac {1}{d \left (d+e x^m\right )}\right )}{2 e m^2}-\frac {a+b \log \left (c x^n\right )}{2 e m \left (d+e x^m\right )^2}\right )}{d}\right )}{3 e m}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{3 e m \left (d+e x^m\right )^3}\right )\) |
\(\Big \downarrow \) 2838 |
\(\displaystyle x^{1-m} (f x)^{m-1} \left (\frac {2 b n \left (\frac {\frac {\frac {b n \operatorname {PolyLog}\left (2,-\frac {d x^{-m}}{e}\right )}{d m^2}-\frac {\log \left (\frac {d x^{-m}}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d m}}{d}-\frac {e \left (\frac {x^m \left (a+b \log \left (c x^n\right )\right )}{d m \left (d+e x^m\right )}-\frac {b n \log \left (d+e x^m\right )}{d e m^2}\right )}{d}}{d}-\frac {e \left (\frac {b n \left (-\frac {\log \left (d+e x^m\right )}{d^2}+\frac {\log \left (x^m\right )}{d^2}+\frac {1}{d \left (d+e x^m\right )}\right )}{2 e m^2}-\frac {a+b \log \left (c x^n\right )}{2 e m \left (d+e x^m\right )^2}\right )}{d}\right )}{3 e m}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{3 e m \left (d+e x^m\right )^3}\right )\) |
x^(1 - m)*(f*x)^(-1 + m)*(-1/3*(a + b*Log[c*x^n])^2/(e*m*(d + e*x^m)^3) + (2*b*n*(-((e*(-1/2*(a + b*Log[c*x^n])/(e*m*(d + e*x^m)^2) + (b*n*(1/(d*(d + e*x^m)) + Log[x^m]/d^2 - Log[d + e*x^m]/d^2))/(2*e*m^2)))/d) + (-((e*((x ^m*(a + b*Log[c*x^n]))/(d*m*(d + e*x^m)) - (b*n*Log[d + e*x^m])/(d*e*m^2)) )/d) + (-(((a + b*Log[c*x^n])*Log[1 + d/(e*x^m)])/(d*m)) + (b*n*PolyLog[2, -(d/(e*x^m))])/(d*m^2))/d)/d))/(3*e*m))
3.4.66.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[E xpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[m, 0] && IntegerQ[n] && !(IGtQ[n, 0] && LtQ[m + n + 2, 0])
Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveConten t[a + b*x^n, x]]/(b*n), x] /; FreeQ[{a, b, m, n}, x] && EqQ[m, n - 1]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[1/n Subst [Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)* (x_)^(r_.))^(q_), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^r)^(q + 1)*((a + b*Log[c*x^n])/(d*f*(m + 1))), x] - Simp[b*(n/(d*(m + 1))) Int[(f*x)^m*(d + e*x^r)^(q + 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, n, q, r}, x] && Eq Q[m + r*(q + 1) + 1, 0] && NeQ[m, -1]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_))^(q_.), x_Symbol] :> Simp[f^m*(d + e*x^r)^(q + 1)*((a + b*L og[c*x^n])^p/(e*r*(q + 1))), x] - Simp[b*f^m*n*(p/(e*r*(q + 1))) Int[(d + e*x^r)^(q + 1)*((a + b*Log[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d , e, f, m, n, q, r}, x] && EqQ[m, r - 1] && IGtQ[p, 0] && (IntegerQ[m] || G tQ[f, 0]) && NeQ[r, n] && NeQ[q, -1]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_)*(x_))^(m_.)*((d_) + ( e_.)*(x_)^(r_))^(q_.), x_Symbol] :> Simp[(f*x)^m/x^m Int[x^m*(d + e*x^r)^ q*(a + b*Log[c*x^n])^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, q, r}, x] && EqQ[m, r - 1] && IGtQ[p, 0] && !(IntegerQ[m] || GtQ[f, 0])
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_)^(r _.))), x_Symbol] :> Simp[(-Log[1 + d/(e*x^r)])*((a + b*Log[c*x^n])^p/(d*r)) , x] + Simp[b*n*(p/(d*r)) Int[Log[1 + d/(e*x^r)]*((a + b*Log[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[p, 0]
Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_)^(r_.))^ (q_))/(x_), x_Symbol] :> Simp[1/d Int[(d + e*x^r)^(q + 1)*((a + b*Log[c*x ^n])^p/x), x], x] - Simp[e/d Int[x^(r - 1)*(d + e*x^r)^q*(a + b*Log[c*x^n ])^p, x], x] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[p, 0] && ILtQ[q, -1 ]
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]
\[\int \frac {\left (f x \right )^{m -1} {\left (a +b \ln \left (c \,x^{n}\right )\right )}^{2}}{\left (d +e \,x^{m}\right )^{4}}d x\]
Leaf count of result is larger than twice the leaf count of optimal. 810 vs. \(2 (331) = 662\).
Time = 0.31 (sec) , antiderivative size = 810, normalized size of antiderivative = 2.34 \[ \int \frac {(f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )^2}{\left (d+e x^m\right )^4} \, dx=\frac {{\left (b^{2} e^{3} m^{2} n^{2} \log \left (x\right )^{2} + {\left (2 \, b^{2} e^{3} m^{2} n \log \left (c\right ) + 2 \, a b e^{3} m^{2} n - 3 \, b^{2} e^{3} m n^{2}\right )} \log \left (x\right )\right )} f^{m - 1} x^{3 \, m} + {\left (3 \, b^{2} d e^{2} m^{2} n^{2} \log \left (x\right )^{2} + 2 \, b^{2} d e^{2} m n \log \left (c\right ) + 2 \, a b d e^{2} m n - b^{2} d e^{2} n^{2} + {\left (6 \, b^{2} d e^{2} m^{2} n \log \left (c\right ) + 6 \, a b d e^{2} m^{2} n - 7 \, b^{2} d e^{2} m n^{2}\right )} \log \left (x\right )\right )} f^{m - 1} x^{2 \, m} + {\left (3 \, b^{2} d^{2} e m^{2} n^{2} \log \left (x\right )^{2} + 5 \, b^{2} d^{2} e m n \log \left (c\right ) + 5 \, a b d^{2} e m n - 2 \, b^{2} d^{2} e n^{2} + 2 \, {\left (3 \, b^{2} d^{2} e m^{2} n \log \left (c\right ) + 3 \, a b d^{2} e m^{2} n - 2 \, b^{2} d^{2} e m n^{2}\right )} \log \left (x\right )\right )} f^{m - 1} x^{m} - {\left (b^{2} d^{3} m^{2} \log \left (c\right )^{2} + a^{2} d^{3} m^{2} - 3 \, a b d^{3} m n + b^{2} d^{3} n^{2} + {\left (2 \, a b d^{3} m^{2} - 3 \, b^{2} d^{3} m n\right )} \log \left (c\right )\right )} f^{m - 1} - 2 \, {\left (b^{2} e^{3} f^{m - 1} n^{2} x^{3 \, m} + 3 \, b^{2} d e^{2} f^{m - 1} n^{2} x^{2 \, m} + 3 \, b^{2} d^{2} e f^{m - 1} n^{2} x^{m} + b^{2} d^{3} f^{m - 1} n^{2}\right )} {\rm Li}_2\left (-\frac {e x^{m} + d}{d} + 1\right ) - {\left ({\left (2 \, b^{2} e^{3} m n \log \left (c\right ) + 2 \, a b e^{3} m n - 3 \, b^{2} e^{3} n^{2}\right )} f^{m - 1} x^{3 \, m} + 3 \, {\left (2 \, b^{2} d e^{2} m n \log \left (c\right ) + 2 \, a b d e^{2} m n - 3 \, b^{2} d e^{2} n^{2}\right )} f^{m - 1} x^{2 \, m} + 3 \, {\left (2 \, b^{2} d^{2} e m n \log \left (c\right ) + 2 \, a b d^{2} e m n - 3 \, b^{2} d^{2} e n^{2}\right )} f^{m - 1} x^{m} + {\left (2 \, b^{2} d^{3} m n \log \left (c\right ) + 2 \, a b d^{3} m n - 3 \, b^{2} d^{3} n^{2}\right )} f^{m - 1}\right )} \log \left (e x^{m} + d\right ) - 2 \, {\left (b^{2} e^{3} f^{m - 1} m n^{2} x^{3 \, m} \log \left (x\right ) + 3 \, b^{2} d e^{2} f^{m - 1} m n^{2} x^{2 \, m} \log \left (x\right ) + 3 \, b^{2} d^{2} e f^{m - 1} m n^{2} x^{m} \log \left (x\right ) + b^{2} d^{3} f^{m - 1} m n^{2} \log \left (x\right )\right )} \log \left (\frac {e x^{m} + d}{d}\right )}{3 \, {\left (d^{3} e^{4} m^{3} x^{3 \, m} + 3 \, d^{4} e^{3} m^{3} x^{2 \, m} + 3 \, d^{5} e^{2} m^{3} x^{m} + d^{6} e m^{3}\right )}} \]
1/3*((b^2*e^3*m^2*n^2*log(x)^2 + (2*b^2*e^3*m^2*n*log(c) + 2*a*b*e^3*m^2*n - 3*b^2*e^3*m*n^2)*log(x))*f^(m - 1)*x^(3*m) + (3*b^2*d*e^2*m^2*n^2*log(x )^2 + 2*b^2*d*e^2*m*n*log(c) + 2*a*b*d*e^2*m*n - b^2*d*e^2*n^2 + (6*b^2*d* e^2*m^2*n*log(c) + 6*a*b*d*e^2*m^2*n - 7*b^2*d*e^2*m*n^2)*log(x))*f^(m - 1 )*x^(2*m) + (3*b^2*d^2*e*m^2*n^2*log(x)^2 + 5*b^2*d^2*e*m*n*log(c) + 5*a*b *d^2*e*m*n - 2*b^2*d^2*e*n^2 + 2*(3*b^2*d^2*e*m^2*n*log(c) + 3*a*b*d^2*e*m ^2*n - 2*b^2*d^2*e*m*n^2)*log(x))*f^(m - 1)*x^m - (b^2*d^3*m^2*log(c)^2 + a^2*d^3*m^2 - 3*a*b*d^3*m*n + b^2*d^3*n^2 + (2*a*b*d^3*m^2 - 3*b^2*d^3*m*n )*log(c))*f^(m - 1) - 2*(b^2*e^3*f^(m - 1)*n^2*x^(3*m) + 3*b^2*d*e^2*f^(m - 1)*n^2*x^(2*m) + 3*b^2*d^2*e*f^(m - 1)*n^2*x^m + b^2*d^3*f^(m - 1)*n^2)* dilog(-(e*x^m + d)/d + 1) - ((2*b^2*e^3*m*n*log(c) + 2*a*b*e^3*m*n - 3*b^2 *e^3*n^2)*f^(m - 1)*x^(3*m) + 3*(2*b^2*d*e^2*m*n*log(c) + 2*a*b*d*e^2*m*n - 3*b^2*d*e^2*n^2)*f^(m - 1)*x^(2*m) + 3*(2*b^2*d^2*e*m*n*log(c) + 2*a*b*d ^2*e*m*n - 3*b^2*d^2*e*n^2)*f^(m - 1)*x^m + (2*b^2*d^3*m*n*log(c) + 2*a*b* d^3*m*n - 3*b^2*d^3*n^2)*f^(m - 1))*log(e*x^m + d) - 2*(b^2*e^3*f^(m - 1)* m*n^2*x^(3*m)*log(x) + 3*b^2*d*e^2*f^(m - 1)*m*n^2*x^(2*m)*log(x) + 3*b^2* d^2*e*f^(m - 1)*m*n^2*x^m*log(x) + b^2*d^3*f^(m - 1)*m*n^2*log(x))*log((e* x^m + d)/d))/(d^3*e^4*m^3*x^(3*m) + 3*d^4*e^3*m^3*x^(2*m) + 3*d^5*e^2*m^3* x^m + d^6*e*m^3)
Timed out. \[ \int \frac {(f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )^2}{\left (d+e x^m\right )^4} \, dx=\text {Timed out} \]
\[ \int \frac {(f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )^2}{\left (d+e x^m\right )^4} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} \left (f x\right )^{m - 1}}{{\left (e x^{m} + d\right )}^{4}} \,d x } \]
1/3*a*b*f^m*n*((2*e*x^m + 3*d)/((d^2*e^3*f*m*x^(2*m) + 2*d^3*e^2*f*m*x^m + d^4*e*f*m)*m) + 2*log(x)/(d^3*e*f*m) - 2*log(e*x^m + d)/(d^3*e*f*m^2)) - 1/3*(f^m*log(x^n)^2/(e^4*f*m*x^(3*m) + 3*d*e^3*f*m*x^(2*m) + 3*d^2*e^2*f*m *x^m + d^3*e*f*m) - 3*integrate(1/3*(3*e*f^m*m*x^m*log(c)^2 + 2*(d*f^m*n + (3*e*f^m*m*log(c) + e*f^m*n)*x^m)*log(x^n))/(e^5*f*m*x*x^(4*m) + 4*d*e^4* f*m*x*x^(3*m) + 6*d^2*e^3*f*m*x*x^(2*m) + 4*d^3*e^2*f*m*x*x^m + d^4*e*f*m* x), x))*b^2 - 2/3*a*b*f^m*log(c*x^n)/(e^4*f*m*x^(3*m) + 3*d*e^3*f*m*x^(2*m ) + 3*d^2*e^2*f*m*x^m + d^3*e*f*m) - 1/3*a^2*f^m/(e^4*f*m*x^(3*m) + 3*d*e^ 3*f*m*x^(2*m) + 3*d^2*e^2*f*m*x^m + d^3*e*f*m)
\[ \int \frac {(f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )^2}{\left (d+e x^m\right )^4} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} \left (f x\right )^{m - 1}}{{\left (e x^{m} + d\right )}^{4}} \,d x } \]
Timed out. \[ \int \frac {(f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )^2}{\left (d+e x^m\right )^4} \, dx=\int \frac {{\left (f\,x\right )}^{m-1}\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2}{{\left (d+e\,x^m\right )}^4} \,d x \]