Integrand size = 29, antiderivative size = 298 \[ \int (f x)^{-1+m} \left (d+e x^m\right )^2 \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\frac {2 b^2 d^2 n^2 x (f x)^{-1+m}}{m^3}+\frac {b^2 d e n^2 x^{1+m} (f x)^{-1+m}}{2 m^3}+\frac {2 b^2 e^2 n^2 x^{1+2 m} (f x)^{-1+m}}{27 m^3}+\frac {b^2 d^3 n^2 x^{1-m} (f x)^{-1+m} \log ^2(x)}{3 e m}-\frac {2 b d^2 n x (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )}{m^2}-\frac {b d e n x^{1+m} (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )}{m^2}-\frac {2 b e^2 n x^{1+2 m} (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )}{9 m^2}-\frac {2 b d^3 n x^{1-m} (f x)^{-1+m} \log (x) \left (a+b \log \left (c x^n\right )\right )}{3 e m}+\frac {x^{1-m} (f x)^{-1+m} \left (d+e x^m\right )^3 \left (a+b \log \left (c x^n\right )\right )^2}{3 e m} \]
2*b^2*d^2*n^2*x*(f*x)^(-1+m)/m^3+1/2*b^2*d*e*n^2*x^(1+m)*(f*x)^(-1+m)/m^3+ 2/27*b^2*e^2*n^2*x^(1+2*m)*(f*x)^(-1+m)/m^3+1/3*b^2*d^3*n^2*x^(1-m)*(f*x)^ (-1+m)*ln(x)^2/e/m-2*b*d^2*n*x*(f*x)^(-1+m)*(a+b*ln(c*x^n))/m^2-b*d*e*n*x^ (1+m)*(f*x)^(-1+m)*(a+b*ln(c*x^n))/m^2-2/9*b*e^2*n*x^(1+2*m)*(f*x)^(-1+m)* (a+b*ln(c*x^n))/m^2-2/3*b*d^3*n*x^(1-m)*(f*x)^(-1+m)*ln(x)*(a+b*ln(c*x^n)) /e/m+1/3*x^(1-m)*(f*x)^(-1+m)*(d+e*x^m)^3*(a+b*ln(c*x^n))^2/e/m
Time = 0.15 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.69 \[ \int (f x)^{-1+m} \left (d+e x^m\right )^2 \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\frac {(f x)^m \left (18 a^2 m^2 \left (3 d^2+3 d e x^m+e^2 x^{2 m}\right )-6 a b m n \left (18 d^2+9 d e x^m+2 e^2 x^{2 m}\right )+b^2 n^2 \left (108 d^2+27 d e x^m+4 e^2 x^{2 m}\right )+6 b m \left (6 a m \left (3 d^2+3 d e x^m+e^2 x^{2 m}\right )-b n \left (18 d^2+9 d e x^m+2 e^2 x^{2 m}\right )\right ) \log \left (c x^n\right )+18 b^2 m^2 \left (3 d^2+3 d e x^m+e^2 x^{2 m}\right ) \log ^2\left (c x^n\right )\right )}{54 f m^3} \]
((f*x)^m*(18*a^2*m^2*(3*d^2 + 3*d*e*x^m + e^2*x^(2*m)) - 6*a*b*m*n*(18*d^2 + 9*d*e*x^m + 2*e^2*x^(2*m)) + b^2*n^2*(108*d^2 + 27*d*e*x^m + 4*e^2*x^(2 *m)) + 6*b*m*(6*a*m*(3*d^2 + 3*d*e*x^m + e^2*x^(2*m)) - b*n*(18*d^2 + 9*d* e*x^m + 2*e^2*x^(2*m)))*Log[c*x^n] + 18*b^2*m^2*(3*d^2 + 3*d*e*x^m + e^2*x ^(2*m))*Log[c*x^n]^2))/(54*f*m^3)
Time = 0.72 (sec) , antiderivative size = 212, normalized size of antiderivative = 0.71, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {2777, 2776, 2772, 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.
| \(\displaystyle \int (f x)^{m-1} \left (d+e x^m\right )^2 \left (a+b \log \left (c x^n\right )\right )^2 \, dx\) |
| \(\Big \downarrow \) 2777 |
| \(\displaystyle x^{1-m} (f x)^{m-1} \int x^{m-1} \left (e x^m+d\right )^2 \left (a+b \log \left (c x^n\right )\right )^2dx\) |
| \(\Big \downarrow \) 2776 |
| \(\displaystyle x^{1-m} (f x)^{m-1} \left (\frac {\left (d+e x^m\right )^3 \left (a+b \log \left (c x^n\right )\right )^2}{3 e m}-\frac {2 b n \int \frac {\left (e x^m+d\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x}dx}{3 e m}\right )\) |
| \(\Big \downarrow \) 2772 |
| \(\displaystyle x^{1-m} (f x)^{m-1} \left (\frac {\left (d+e x^m\right )^3 \left (a+b \log \left (c x^n\right )\right )^2}{3 e m}-\frac {2 b n \left (-b n \int \frac {e \left (9 d e x^m+2 e^2 x^{2 m}+18 d^2\right ) x^m+6 d^3 m \log (x)}{6 m x}dx+d^3 \log (x) \left (a+b \log \left (c x^n\right )\right )+\frac {3 d^2 e x^m \left (a+b \log \left (c x^n\right )\right )}{m}+\frac {3 d e^2 x^{2 m} \left (a+b \log \left (c x^n\right )\right )}{2 m}+\frac {e^3 x^{3 m} \left (a+b \log \left (c x^n\right )\right )}{3 m}\right )}{3 e m}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle x^{1-m} (f x)^{m-1} \left (\frac {\left (d+e x^m\right )^3 \left (a+b \log \left (c x^n\right )\right )^2}{3 e m}-\frac {2 b n \left (-\frac {b n \int \frac {e \left (9 d e x^m+2 e^2 x^{2 m}+18 d^2\right ) x^m+6 d^3 m \log (x)}{x}dx}{6 m}+d^3 \log (x) \left (a+b \log \left (c x^n\right )\right )+\frac {3 d^2 e x^m \left (a+b \log \left (c x^n\right )\right )}{m}+\frac {3 d e^2 x^{2 m} \left (a+b \log \left (c x^n\right )\right )}{2 m}+\frac {e^3 x^{3 m} \left (a+b \log \left (c x^n\right )\right )}{3 m}\right )}{3 e m}\right )\) |
| \(\Big \downarrow \) 2010 |
| \(\displaystyle x^{1-m} (f x)^{m-1} \left (\frac {\left (d+e x^m\right )^3 \left (a+b \log \left (c x^n\right )\right )^2}{3 e m}-\frac {2 b n \left (-\frac {b n \int \left (18 d^2 e x^{m-1}+9 d e^2 x^{2 m-1}+2 e^3 x^{3 m-1}+\frac {6 d^3 m \log (x)}{x}\right )dx}{6 m}+d^3 \log (x) \left (a+b \log \left (c x^n\right )\right )+\frac {3 d^2 e x^m \left (a+b \log \left (c x^n\right )\right )}{m}+\frac {3 d e^2 x^{2 m} \left (a+b \log \left (c x^n\right )\right )}{2 m}+\frac {e^3 x^{3 m} \left (a+b \log \left (c x^n\right )\right )}{3 m}\right )}{3 e m}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle x^{1-m} (f x)^{m-1} \left (\frac {\left (d+e x^m\right )^3 \left (a+b \log \left (c x^n\right )\right )^2}{3 e m}-\frac {2 b n \left (d^3 \log (x) \left (a+b \log \left (c x^n\right )\right )+\frac {3 d^2 e x^m \left (a+b \log \left (c x^n\right )\right )}{m}+\frac {3 d e^2 x^{2 m} \left (a+b \log \left (c x^n\right )\right )}{2 m}+\frac {e^3 x^{3 m} \left (a+b \log \left (c x^n\right )\right )}{3 m}-\frac {b n \left (3 d^3 m \log ^2(x)+\frac {18 d^2 e x^m}{m}+\frac {9 d e^2 x^{2 m}}{2 m}+\frac {2 e^3 x^{3 m}}{3 m}\right )}{6 m}\right )}{3 e m}\right )\) |
x^(1 - m)*(f*x)^(-1 + m)*(((d + e*x^m)^3*(a + b*Log[c*x^n])^2)/(3*e*m) - ( 2*b*n*(-1/6*(b*n*((18*d^2*e*x^m)/m + (9*d*e^2*x^(2*m))/(2*m) + (2*e^3*x^(3 *m))/(3*m) + 3*d^3*m*Log[x]^2))/m + (3*d^2*e*x^m*(a + b*Log[c*x^n]))/m + ( 3*d*e^2*x^(2*m)*(a + b*Log[c*x^n]))/(2*m) + (e^3*x^(3*m)*(a + b*Log[c*x^n] ))/(3*m) + d^3*Log[x]*(a + b*Log[c*x^n])))/(3*e*m))
3.4.60.3.1 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_.))*(x_)^(m_.)*((d_) + (e_.)*(x_)^(r_ .))^(q_.), x_Symbol] :> With[{u = IntHide[x^m*(d + e*x^r)^q, x]}, Simp[(a + b*Log[c*x^n]) u, x] - Simp[b*n Int[SimplifyIntegrand[u/x, x], x], x]] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[q, 0] && IntegerQ[m] && !(EqQ[q , 1] && EqQ[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])
Time = 50.86 (sec) , antiderivative size = 439, normalized size of antiderivative = 1.47
| method | result | size |
| parallelrisch | \(-\frac {-18 e^{2} b^{2} \left (f x \right )^{m -1} \ln \left (c \,x^{n}\right )^{2} x^{2 m} x \,m^{2}-36 x \,x^{2 m} \ln \left (c \,x^{n}\right ) \left (f x \right )^{m -1} a b \,e^{2} m^{2}+12 x \,x^{2 m} \ln \left (c \,x^{n}\right ) \left (f x \right )^{m -1} b^{2} e^{2} m n -54 b^{2} d e \left (f x \right )^{m -1} \ln \left (c \,x^{n}\right )^{2} x^{m} x \,m^{2}-18 x \,x^{2 m} \left (f x \right )^{m -1} a^{2} e^{2} m^{2}+12 x \,x^{2 m} \left (f x \right )^{m -1} a b \,e^{2} m n -4 x \,x^{2 m} \left (f x \right )^{m -1} b^{2} e^{2} n^{2}-108 x \,x^{m} \ln \left (c \,x^{n}\right ) \left (f x \right )^{m -1} a b d e \,m^{2}+54 x \,x^{m} \ln \left (c \,x^{n}\right ) \left (f x \right )^{m -1} b^{2} d e m n -54 b^{2} d^{2} \left (f x \right )^{m -1} \ln \left (c \,x^{n}\right )^{2} x \,m^{2}-54 x \,x^{m} \left (f x \right )^{m -1} a^{2} d e \,m^{2}+54 x \,x^{m} \left (f x \right )^{m -1} a b d e m n -27 x \,x^{m} \left (f x \right )^{m -1} b^{2} d e \,n^{2}-108 x \ln \left (c \,x^{n}\right ) \left (f x \right )^{m -1} a b \,d^{2} m^{2}+108 x \ln \left (c \,x^{n}\right ) \left (f x \right )^{m -1} b^{2} d^{2} m n -54 x \left (f x \right )^{m -1} a^{2} d^{2} m^{2}+108 x \left (f x \right )^{m -1} a b \,d^{2} m n -108 x \left (f x \right )^{m -1} b^{2} d^{2} n^{2}}{54 m^{3}}\) | \(439\) |
| risch | \(\text {Expression too large to display}\) | \(3038\) |
-1/54*(-18*e^2*b^2*(f*x)^(m-1)*ln(c*x^n)^2*(x^m)^2*x*m^2-36*x*(x^m)^2*ln(c *x^n)*(f*x)^(m-1)*a*b*e^2*m^2+12*x*(x^m)^2*ln(c*x^n)*(f*x)^(m-1)*b^2*e^2*m *n-54*b^2*d*e*(f*x)^(m-1)*ln(c*x^n)^2*x^m*x*m^2-18*x*(x^m)^2*(f*x)^(m-1)*a ^2*e^2*m^2+12*x*(x^m)^2*(f*x)^(m-1)*a*b*e^2*m*n-4*x*(x^m)^2*(f*x)^(m-1)*b^ 2*e^2*n^2-108*x*x^m*ln(c*x^n)*(f*x)^(m-1)*a*b*d*e*m^2+54*x*x^m*ln(c*x^n)*( f*x)^(m-1)*b^2*d*e*m*n-54*b^2*d^2*(f*x)^(m-1)*ln(c*x^n)^2*x*m^2-54*x*x^m*( f*x)^(m-1)*a^2*d*e*m^2+54*x*x^m*(f*x)^(m-1)*a*b*d*e*m*n-27*x*x^m*(f*x)^(m- 1)*b^2*d*e*n^2-108*x*ln(c*x^n)*(f*x)^(m-1)*a*b*d^2*m^2+108*x*ln(c*x^n)*(f* x)^(m-1)*b^2*d^2*m*n-54*x*(f*x)^(m-1)*a^2*d^2*m^2+108*x*(f*x)^(m-1)*a*b*d^ 2*m*n-108*x*(f*x)^(m-1)*b^2*d^2*n^2)/m^3
Time = 0.31 (sec) , antiderivative size = 419, normalized size of antiderivative = 1.41 \[ \int (f x)^{-1+m} \left (d+e x^m\right )^2 \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\frac {2 \, {\left (9 \, b^{2} e^{2} m^{2} n^{2} \log \left (x\right )^{2} + 9 \, b^{2} e^{2} m^{2} \log \left (c\right )^{2} + 9 \, a^{2} e^{2} m^{2} - 6 \, a b e^{2} m n + 2 \, b^{2} e^{2} n^{2} + 6 \, {\left (3 \, a b e^{2} m^{2} - b^{2} e^{2} m n\right )} \log \left (c\right ) + 6 \, {\left (3 \, b^{2} e^{2} m^{2} n \log \left (c\right ) + 3 \, a b e^{2} m^{2} n - b^{2} e^{2} m n^{2}\right )} \log \left (x\right )\right )} f^{m - 1} x^{3 \, m} + 27 \, {\left (2 \, b^{2} d e m^{2} n^{2} \log \left (x\right )^{2} + 2 \, b^{2} d e m^{2} \log \left (c\right )^{2} + 2 \, a^{2} d e m^{2} - 2 \, a b d e m n + b^{2} d e n^{2} + 2 \, {\left (2 \, a b d e m^{2} - b^{2} d e m n\right )} \log \left (c\right ) + 2 \, {\left (2 \, b^{2} d e m^{2} n \log \left (c\right ) + 2 \, a b d e m^{2} n - b^{2} d e m n^{2}\right )} \log \left (x\right )\right )} f^{m - 1} x^{2 \, m} + 54 \, {\left (b^{2} d^{2} m^{2} n^{2} \log \left (x\right )^{2} + b^{2} d^{2} m^{2} \log \left (c\right )^{2} + a^{2} d^{2} m^{2} - 2 \, a b d^{2} m n + 2 \, b^{2} d^{2} n^{2} + 2 \, {\left (a b d^{2} m^{2} - b^{2} d^{2} m n\right )} \log \left (c\right ) + 2 \, {\left (b^{2} d^{2} m^{2} n \log \left (c\right ) + a b d^{2} m^{2} n - b^{2} d^{2} m n^{2}\right )} \log \left (x\right )\right )} f^{m - 1} x^{m}}{54 \, m^{3}} \]
1/54*(2*(9*b^2*e^2*m^2*n^2*log(x)^2 + 9*b^2*e^2*m^2*log(c)^2 + 9*a^2*e^2*m ^2 - 6*a*b*e^2*m*n + 2*b^2*e^2*n^2 + 6*(3*a*b*e^2*m^2 - b^2*e^2*m*n)*log(c ) + 6*(3*b^2*e^2*m^2*n*log(c) + 3*a*b*e^2*m^2*n - b^2*e^2*m*n^2)*log(x))*f ^(m - 1)*x^(3*m) + 27*(2*b^2*d*e*m^2*n^2*log(x)^2 + 2*b^2*d*e*m^2*log(c)^2 + 2*a^2*d*e*m^2 - 2*a*b*d*e*m*n + b^2*d*e*n^2 + 2*(2*a*b*d*e*m^2 - b^2*d* e*m*n)*log(c) + 2*(2*b^2*d*e*m^2*n*log(c) + 2*a*b*d*e*m^2*n - b^2*d*e*m*n^ 2)*log(x))*f^(m - 1)*x^(2*m) + 54*(b^2*d^2*m^2*n^2*log(x)^2 + b^2*d^2*m^2* log(c)^2 + a^2*d^2*m^2 - 2*a*b*d^2*m*n + 2*b^2*d^2*n^2 + 2*(a*b*d^2*m^2 - b^2*d^2*m*n)*log(c) + 2*(b^2*d^2*m^2*n*log(c) + a*b*d^2*m^2*n - b^2*d^2*m* n^2)*log(x))*f^(m - 1)*x^m)/m^3
Time = 12.60 (sec) , antiderivative size = 552, normalized size of antiderivative = 1.85 \[ \int (f x)^{-1+m} \left (d+e x^m\right )^2 \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\begin {cases} \frac {a^{2} d^{2} x \left (f x\right )^{m - 1}}{m} + \frac {a^{2} d e x x^{m} \left (f x\right )^{m - 1}}{m} + \frac {a^{2} e^{2} x x^{2 m} \left (f x\right )^{m - 1}}{3 m} + \frac {2 a b d^{2} x \left (f x\right )^{m - 1} \log {\left (c x^{n} \right )}}{m} - \frac {2 a b d^{2} n x \left (f x\right )^{m - 1}}{m^{2}} + \frac {2 a b d e x x^{m} \left (f x\right )^{m - 1} \log {\left (c x^{n} \right )}}{m} - \frac {a b d e n x x^{m} \left (f x\right )^{m - 1}}{m^{2}} + \frac {2 a b e^{2} x x^{2 m} \left (f x\right )^{m - 1} \log {\left (c x^{n} \right )}}{3 m} - \frac {2 a b e^{2} n x x^{2 m} \left (f x\right )^{m - 1}}{9 m^{2}} + \frac {b^{2} d^{2} x \left (f x\right )^{m - 1} \log {\left (c x^{n} \right )}^{2}}{m} - \frac {2 b^{2} d^{2} n x \left (f x\right )^{m - 1} \log {\left (c x^{n} \right )}}{m^{2}} + \frac {2 b^{2} d^{2} n^{2} x \left (f x\right )^{m - 1}}{m^{3}} + \frac {b^{2} d e x x^{m} \left (f x\right )^{m - 1} \log {\left (c x^{n} \right )}^{2}}{m} - \frac {b^{2} d e n x x^{m} \left (f x\right )^{m - 1} \log {\left (c x^{n} \right )}}{m^{2}} + \frac {b^{2} d e n^{2} x x^{m} \left (f x\right )^{m - 1}}{2 m^{3}} + \frac {b^{2} e^{2} x x^{2 m} \left (f x\right )^{m - 1} \log {\left (c x^{n} \right )}^{2}}{3 m} - \frac {2 b^{2} e^{2} n x x^{2 m} \left (f x\right )^{m - 1} \log {\left (c x^{n} \right )}}{9 m^{2}} + \frac {2 b^{2} e^{2} n^{2} x x^{2 m} \left (f x\right )^{m - 1}}{27 m^{3}} & \text {for}\: m \neq 0 \\\frac {\left (d + e\right )^{2} \left (\begin {cases} \frac {a^{2} \log {\left (c x^{n} \right )} + a b \log {\left (c x^{n} \right )}^{2} + \frac {b^{2} \log {\left (c x^{n} \right )}^{3}}{3}}{n} & \text {for}\: n \neq 0 \\\left (a^{2} + 2 a b \log {\left (c \right )} + b^{2} \log {\left (c \right )}^{2}\right ) \log {\left (x \right )} & \text {otherwise} \end {cases}\right )}{f} & \text {otherwise} \end {cases} \]
Piecewise((a**2*d**2*x*(f*x)**(m - 1)/m + a**2*d*e*x*x**m*(f*x)**(m - 1)/m + a**2*e**2*x*x**(2*m)*(f*x)**(m - 1)/(3*m) + 2*a*b*d**2*x*(f*x)**(m - 1) *log(c*x**n)/m - 2*a*b*d**2*n*x*(f*x)**(m - 1)/m**2 + 2*a*b*d*e*x*x**m*(f* x)**(m - 1)*log(c*x**n)/m - a*b*d*e*n*x*x**m*(f*x)**(m - 1)/m**2 + 2*a*b*e **2*x*x**(2*m)*(f*x)**(m - 1)*log(c*x**n)/(3*m) - 2*a*b*e**2*n*x*x**(2*m)* (f*x)**(m - 1)/(9*m**2) + b**2*d**2*x*(f*x)**(m - 1)*log(c*x**n)**2/m - 2* b**2*d**2*n*x*(f*x)**(m - 1)*log(c*x**n)/m**2 + 2*b**2*d**2*n**2*x*(f*x)** (m - 1)/m**3 + b**2*d*e*x*x**m*(f*x)**(m - 1)*log(c*x**n)**2/m - b**2*d*e* n*x*x**m*(f*x)**(m - 1)*log(c*x**n)/m**2 + b**2*d*e*n**2*x*x**m*(f*x)**(m - 1)/(2*m**3) + b**2*e**2*x*x**(2*m)*(f*x)**(m - 1)*log(c*x**n)**2/(3*m) - 2*b**2*e**2*n*x*x**(2*m)*(f*x)**(m - 1)*log(c*x**n)/(9*m**2) + 2*b**2*e** 2*n**2*x*x**(2*m)*(f*x)**(m - 1)/(27*m**3), Ne(m, 0)), ((d + e)**2*Piecewi se(((a**2*log(c*x**n) + a*b*log(c*x**n)**2 + b**2*log(c*x**n)**3/3)/n, Ne( n, 0)), ((a**2 + 2*a*b*log(c) + b**2*log(c)**2)*log(x), True))/f, True))
Time = 0.23 (sec) , antiderivative size = 417, normalized size of antiderivative = 1.40 \[ \int (f x)^{-1+m} \left (d+e x^m\right )^2 \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\frac {b^{2} e^{2} f^{m - 1} x^{3 \, m} \log \left (c x^{n}\right )^{2}}{3 \, m} + \frac {b^{2} d e f^{m - 1} x^{2 \, m} \log \left (c x^{n}\right )^{2}}{m} + \frac {2 \, a b e^{2} f^{m - 1} x^{3 \, m} \log \left (c x^{n}\right )}{3 \, m} + \frac {2 \, a b d e f^{m - 1} x^{2 \, m} \log \left (c x^{n}\right )}{m} - 2 \, {\left (\frac {f^{m - 1} n x^{m} \log \left (c x^{n}\right )}{m^{2}} - \frac {f^{m - 1} n^{2} x^{m}}{m^{3}}\right )} b^{2} d^{2} - \frac {1}{2} \, {\left (\frac {2 \, f^{m - 1} n x^{2 \, m} \log \left (c x^{n}\right )}{m^{2}} - \frac {f^{m - 1} n^{2} x^{2 \, m}}{m^{3}}\right )} b^{2} d e - \frac {2}{27} \, {\left (\frac {3 \, f^{m - 1} n x^{3 \, m} \log \left (c x^{n}\right )}{m^{2}} - \frac {f^{m - 1} n^{2} x^{3 \, m}}{m^{3}}\right )} b^{2} e^{2} + \frac {a^{2} e^{2} f^{m - 1} x^{3 \, m}}{3 \, m} - \frac {2 \, a b e^{2} f^{m - 1} n x^{3 \, m}}{9 \, m^{2}} + \frac {a^{2} d e f^{m - 1} x^{2 \, m}}{m} - \frac {a b d e f^{m - 1} n x^{2 \, m}}{m^{2}} - \frac {2 \, a b d^{2} f^{m - 1} n x^{m}}{m^{2}} + \frac {\left (f x\right )^{m} b^{2} d^{2} \log \left (c x^{n}\right )^{2}}{f m} + \frac {2 \, \left (f x\right )^{m} a b d^{2} \log \left (c x^{n}\right )}{f m} + \frac {\left (f x\right )^{m} a^{2} d^{2}}{f m} \]
1/3*b^2*e^2*f^(m - 1)*x^(3*m)*log(c*x^n)^2/m + b^2*d*e*f^(m - 1)*x^(2*m)*l og(c*x^n)^2/m + 2/3*a*b*e^2*f^(m - 1)*x^(3*m)*log(c*x^n)/m + 2*a*b*d*e*f^( m - 1)*x^(2*m)*log(c*x^n)/m - 2*(f^(m - 1)*n*x^m*log(c*x^n)/m^2 - f^(m - 1 )*n^2*x^m/m^3)*b^2*d^2 - 1/2*(2*f^(m - 1)*n*x^(2*m)*log(c*x^n)/m^2 - f^(m - 1)*n^2*x^(2*m)/m^3)*b^2*d*e - 2/27*(3*f^(m - 1)*n*x^(3*m)*log(c*x^n)/m^2 - f^(m - 1)*n^2*x^(3*m)/m^3)*b^2*e^2 + 1/3*a^2*e^2*f^(m - 1)*x^(3*m)/m - 2/9*a*b*e^2*f^(m - 1)*n*x^(3*m)/m^2 + a^2*d*e*f^(m - 1)*x^(2*m)/m - a*b*d* e*f^(m - 1)*n*x^(2*m)/m^2 - 2*a*b*d^2*f^(m - 1)*n*x^m/m^2 + (f*x)^m*b^2*d^ 2*log(c*x^n)^2/(f*m) + 2*(f*x)^m*a*b*d^2*log(c*x^n)/(f*m) + (f*x)^m*a^2*d^ 2/(f*m)
Leaf count of result is larger than twice the leaf count of optimal. 715 vs. \(2 (286) = 572\).
Time = 0.52 (sec) , antiderivative size = 715, normalized size of antiderivative = 2.40 \[ \int (f x)^{-1+m} \left (d+e x^m\right )^2 \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\frac {b^{2} e^{2} f^{m} n^{2} x^{3 \, m} \log \left (x\right )^{2}}{3 \, f m} + \frac {b^{2} d e f^{m} n^{2} x^{2 \, m} \log \left (x\right )^{2}}{f m} + \frac {b^{2} d^{2} f^{m} n^{2} x^{m} \log \left (x\right )^{2}}{f m} + \frac {2 \, b^{2} e^{2} f^{m} n x^{3 \, m} \log \left (c\right ) \log \left (x\right )}{3 \, f m} + \frac {2 \, b^{2} d e f^{m} n x^{2 \, m} \log \left (c\right ) \log \left (x\right )}{f m} + \frac {2 \, b^{2} d^{2} f^{m} n x^{m} \log \left (c\right ) \log \left (x\right )}{f m} + \frac {b^{2} e^{2} f^{m} x^{3 \, m} \log \left (c\right )^{2}}{3 \, f m} + \frac {b^{2} d e f^{m} x^{2 \, m} \log \left (c\right )^{2}}{f m} + \frac {b^{2} d^{2} f^{m} x^{m} \log \left (c\right )^{2}}{f m} + \frac {2 \, a b e^{2} f^{m} n x^{3 \, m} \log \left (x\right )}{3 \, f m} - \frac {2 \, b^{2} e^{2} f^{m} n^{2} x^{3 \, m} \log \left (x\right )}{9 \, f m^{2}} + \frac {2 \, a b d e f^{m} n x^{2 \, m} \log \left (x\right )}{f m} - \frac {b^{2} d e f^{m} n^{2} x^{2 \, m} \log \left (x\right )}{f m^{2}} + \frac {2 \, a b d^{2} f^{m} n x^{m} \log \left (x\right )}{f m} - \frac {2 \, b^{2} d^{2} f^{m} n^{2} x^{m} \log \left (x\right )}{f m^{2}} + \frac {2 \, a b e^{2} f^{m} x^{3 \, m} \log \left (c\right )}{3 \, f m} - \frac {2 \, b^{2} e^{2} f^{m} n x^{3 \, m} \log \left (c\right )}{9 \, f m^{2}} + \frac {2 \, a b d e f^{m} x^{2 \, m} \log \left (c\right )}{f m} - \frac {b^{2} d e f^{m} n x^{2 \, m} \log \left (c\right )}{f m^{2}} + \frac {2 \, a b d^{2} f^{m} x^{m} \log \left (c\right )}{f m} - \frac {2 \, b^{2} d^{2} f^{m} n x^{m} \log \left (c\right )}{f m^{2}} + \frac {a^{2} e^{2} f^{m} x^{3 \, m}}{3 \, f m} - \frac {2 \, a b e^{2} f^{m} n x^{3 \, m}}{9 \, f m^{2}} + \frac {2 \, b^{2} e^{2} f^{m} n^{2} x^{3 \, m}}{27 \, f m^{3}} + \frac {a^{2} d e f^{m} x^{2 \, m}}{f m} - \frac {a b d e f^{m} n x^{2 \, m}}{f m^{2}} + \frac {b^{2} d e f^{m} n^{2} x^{2 \, m}}{2 \, f m^{3}} + \frac {a^{2} d^{2} f^{m} x^{m}}{f m} - \frac {2 \, a b d^{2} f^{m} n x^{m}}{f m^{2}} + \frac {2 \, b^{2} d^{2} f^{m} n^{2} x^{m}}{f m^{3}} \]
1/3*b^2*e^2*f^m*n^2*x^(3*m)*log(x)^2/(f*m) + b^2*d*e*f^m*n^2*x^(2*m)*log(x )^2/(f*m) + b^2*d^2*f^m*n^2*x^m*log(x)^2/(f*m) + 2/3*b^2*e^2*f^m*n*x^(3*m) *log(c)*log(x)/(f*m) + 2*b^2*d*e*f^m*n*x^(2*m)*log(c)*log(x)/(f*m) + 2*b^2 *d^2*f^m*n*x^m*log(c)*log(x)/(f*m) + 1/3*b^2*e^2*f^m*x^(3*m)*log(c)^2/(f*m ) + b^2*d*e*f^m*x^(2*m)*log(c)^2/(f*m) + b^2*d^2*f^m*x^m*log(c)^2/(f*m) + 2/3*a*b*e^2*f^m*n*x^(3*m)*log(x)/(f*m) - 2/9*b^2*e^2*f^m*n^2*x^(3*m)*log(x )/(f*m^2) + 2*a*b*d*e*f^m*n*x^(2*m)*log(x)/(f*m) - b^2*d*e*f^m*n^2*x^(2*m) *log(x)/(f*m^2) + 2*a*b*d^2*f^m*n*x^m*log(x)/(f*m) - 2*b^2*d^2*f^m*n^2*x^m *log(x)/(f*m^2) + 2/3*a*b*e^2*f^m*x^(3*m)*log(c)/(f*m) - 2/9*b^2*e^2*f^m*n *x^(3*m)*log(c)/(f*m^2) + 2*a*b*d*e*f^m*x^(2*m)*log(c)/(f*m) - b^2*d*e*f^m *n*x^(2*m)*log(c)/(f*m^2) + 2*a*b*d^2*f^m*x^m*log(c)/(f*m) - 2*b^2*d^2*f^m *n*x^m*log(c)/(f*m^2) + 1/3*a^2*e^2*f^m*x^(3*m)/(f*m) - 2/9*a*b*e^2*f^m*n* x^(3*m)/(f*m^2) + 2/27*b^2*e^2*f^m*n^2*x^(3*m)/(f*m^3) + a^2*d*e*f^m*x^(2* m)/(f*m) - a*b*d*e*f^m*n*x^(2*m)/(f*m^2) + 1/2*b^2*d*e*f^m*n^2*x^(2*m)/(f* m^3) + a^2*d^2*f^m*x^m/(f*m) - 2*a*b*d^2*f^m*n*x^m/(f*m^2) + 2*b^2*d^2*f^m *n^2*x^m/(f*m^3)
Timed out. \[ \int (f x)^{-1+m} \left (d+e x^m\right )^2 \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\int {\left (f\,x\right )}^{m-1}\,{\left (d+e\,x^m\right )}^2\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2 \,d x \]