Integrand size = 29, antiderivative size = 372 \[ \int (f x)^{-1+m} \left (d+e x^m\right )^3 \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\frac {2 b^2 d^3 n^2 x (f x)^{-1+m}}{m^3}+\frac {3 b^2 d^2 e n^2 x^{1+m} (f x)^{-1+m}}{4 m^3}+\frac {2 b^2 d e^2 n^2 x^{1+2 m} (f x)^{-1+m}}{9 m^3}+\frac {b^2 e^3 n^2 x^{1+3 m} (f x)^{-1+m}}{32 m^3}+\frac {b^2 d^4 n^2 x^{1-m} (f x)^{-1+m} \log ^2(x)}{4 e m}-\frac {2 b d^3 n x (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )}{m^2}-\frac {3 b d^2 e n x^{1+m} (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )}{2 m^2}-\frac {2 b d e^2 n x^{1+2 m} (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )}{3 m^2}-\frac {b e^3 n x^{1+3 m} (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )}{8 m^2}-\frac {b d^4 n x^{1-m} (f x)^{-1+m} \log (x) \left (a+b \log \left (c x^n\right )\right )}{2 e m}+\frac {x^{1-m} (f x)^{-1+m} \left (d+e x^m\right )^4 \left (a+b \log \left (c x^n\right )\right )^2}{4 e m} \]
2*b^2*d^3*n^2*x*(f*x)^(-1+m)/m^3+3/4*b^2*d^2*e*n^2*x^(1+m)*(f*x)^(-1+m)/m^ 3+2/9*b^2*d*e^2*n^2*x^(1+2*m)*(f*x)^(-1+m)/m^3+1/32*b^2*e^3*n^2*x^(1+3*m)* (f*x)^(-1+m)/m^3+1/4*b^2*d^4*n^2*x^(1-m)*(f*x)^(-1+m)*ln(x)^2/e/m-2*b*d^3* n*x*(f*x)^(-1+m)*(a+b*ln(c*x^n))/m^2-3/2*b*d^2*e*n*x^(1+m)*(f*x)^(-1+m)*(a +b*ln(c*x^n))/m^2-2/3*b*d*e^2*n*x^(1+2*m)*(f*x)^(-1+m)*(a+b*ln(c*x^n))/m^2 -1/8*b*e^3*n*x^(1+3*m)*(f*x)^(-1+m)*(a+b*ln(c*x^n))/m^2-1/2*b*d^4*n*x^(1-m )*(f*x)^(-1+m)*ln(x)*(a+b*ln(c*x^n))/e/m+1/4*x^(1-m)*(f*x)^(-1+m)*(d+e*x^m )^4*(a+b*ln(c*x^n))^2/e/m
Time = 0.19 (sec) , antiderivative size = 285, normalized size of antiderivative = 0.77 \[ \int (f x)^{-1+m} \left (d+e x^m\right )^3 \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\frac {(f x)^m \left (72 a^2 m^2 \left (4 d^3+6 d^2 e x^m+4 d e^2 x^{2 m}+e^3 x^{3 m}\right )-12 a b m n \left (48 d^3+36 d^2 e x^m+16 d e^2 x^{2 m}+3 e^3 x^{3 m}\right )+b^2 n^2 \left (576 d^3+216 d^2 e x^m+64 d e^2 x^{2 m}+9 e^3 x^{3 m}\right )+12 b m \left (12 a m \left (4 d^3+6 d^2 e x^m+4 d e^2 x^{2 m}+e^3 x^{3 m}\right )-b n \left (48 d^3+36 d^2 e x^m+16 d e^2 x^{2 m}+3 e^3 x^{3 m}\right )\right ) \log \left (c x^n\right )+72 b^2 m^2 \left (4 d^3+6 d^2 e x^m+4 d e^2 x^{2 m}+e^3 x^{3 m}\right ) \log ^2\left (c x^n\right )\right )}{288 f m^3} \]
((f*x)^m*(72*a^2*m^2*(4*d^3 + 6*d^2*e*x^m + 4*d*e^2*x^(2*m) + e^3*x^(3*m)) - 12*a*b*m*n*(48*d^3 + 36*d^2*e*x^m + 16*d*e^2*x^(2*m) + 3*e^3*x^(3*m)) + b^2*n^2*(576*d^3 + 216*d^2*e*x^m + 64*d*e^2*x^(2*m) + 9*e^3*x^(3*m)) + 12 *b*m*(12*a*m*(4*d^3 + 6*d^2*e*x^m + 4*d*e^2*x^(2*m) + e^3*x^(3*m)) - b*n*( 48*d^3 + 36*d^2*e*x^m + 16*d*e^2*x^(2*m) + 3*e^3*x^(3*m)))*Log[c*x^n] + 72 *b^2*m^2*(4*d^3 + 6*d^2*e*x^m + 4*d*e^2*x^(2*m) + e^3*x^(3*m))*Log[c*x^n]^ 2))/(288*f*m^3)
Time = 0.75 (sec) , antiderivative size = 252, normalized size of antiderivative = 0.68, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {2777, 2776, 2772, 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.
| \(\displaystyle \int (f x)^{m-1} \left (d+e x^m\right )^3 \left (a+b \log \left (c x^n\right )\right )^2 \, dx\) |
| \(\Big \downarrow \) 2777 |
| \(\displaystyle x^{1-m} (f x)^{m-1} \int x^{m-1} \left (e x^m+d\right )^3 \left (a+b \log \left (c x^n\right )\right )^2dx\) |
| \(\Big \downarrow \) 2776 |
| \(\displaystyle x^{1-m} (f x)^{m-1} \left (\frac {\left (d+e x^m\right )^4 \left (a+b \log \left (c x^n\right )\right )^2}{4 e m}-\frac {b n \int \frac {\left (e x^m+d\right )^4 \left (a+b \log \left (c x^n\right )\right )}{x}dx}{2 e m}\right )\) |
| \(\Big \downarrow \) 2772 |
| \(\displaystyle x^{1-m} (f x)^{m-1} \left (\frac {\left (d+e x^m\right )^4 \left (a+b \log \left (c x^n\right )\right )^2}{4 e m}-\frac {b n \left (-b n \int \left (\frac {e \left (36 d^2 e x^m+16 d e^2 x^{2 m}+3 e^3 x^{3 m}+48 d^3\right ) x^{m-1}}{12 m}+\frac {d^4 \log (x)}{x}\right )dx+d^4 \log (x) \left (a+b \log \left (c x^n\right )\right )+\frac {4 d^3 e x^m \left (a+b \log \left (c x^n\right )\right )}{m}+\frac {3 d^2 e^2 x^{2 m} \left (a+b \log \left (c x^n\right )\right )}{m}+\frac {4 d e^3 x^{3 m} \left (a+b \log \left (c x^n\right )\right )}{3 m}+\frac {e^4 x^{4 m} \left (a+b \log \left (c x^n\right )\right )}{4 m}\right )}{2 e m}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle x^{1-m} (f x)^{m-1} \left (\frac {\left (d+e x^m\right )^4 \left (a+b \log \left (c x^n\right )\right )^2}{4 e m}-\frac {b n \left (d^4 \log (x) \left (a+b \log \left (c x^n\right )\right )+\frac {4 d^3 e x^m \left (a+b \log \left (c x^n\right )\right )}{m}+\frac {3 d^2 e^2 x^{2 m} \left (a+b \log \left (c x^n\right )\right )}{m}+\frac {4 d e^3 x^{3 m} \left (a+b \log \left (c x^n\right )\right )}{3 m}+\frac {e^4 x^{4 m} \left (a+b \log \left (c x^n\right )\right )}{4 m}-b n \left (\frac {1}{2} d^4 \log ^2(x)+\frac {4 d^3 e x^m}{m^2}+\frac {3 d^2 e^2 x^{2 m}}{2 m^2}+\frac {4 d e^3 x^{3 m}}{9 m^2}+\frac {e^4 x^{4 m}}{16 m^2}\right )\right )}{2 e m}\right )\) |
x^(1 - m)*(f*x)^(-1 + m)*(((d + e*x^m)^4*(a + b*Log[c*x^n])^2)/(4*e*m) - ( b*n*(-(b*n*((4*d^3*e*x^m)/m^2 + (3*d^2*e^2*x^(2*m))/(2*m^2) + (4*d*e^3*x^( 3*m))/(9*m^2) + (e^4*x^(4*m))/(16*m^2) + (d^4*Log[x]^2)/2)) + (4*d^3*e*x^m *(a + b*Log[c*x^n]))/m + (3*d^2*e^2*x^(2*m)*(a + b*Log[c*x^n]))/m + (4*d*e ^3*x^(3*m)*(a + b*Log[c*x^n]))/(3*m) + (e^4*x^(4*m)*(a + b*Log[c*x^n]))/(4 *m) + d^4*Log[x]*(a + b*Log[c*x^n])))/(2*e*m))
3.4.59.3.1 Defintions of rubi rules used
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 = 205.35 (sec) , antiderivative size = 617, normalized size of antiderivative = 1.66
| method | result | size |
| parallelrisch | \(-\frac {-288 b^{2} d^{3} \left (f x \right )^{m -1} \ln \left (c \,x^{n}\right )^{2} x \,m^{2}-72 x \,x^{3 m} \left (f x \right )^{m -1} a^{2} e^{3} m^{2}-9 x \,x^{3 m} \left (f x \right )^{m -1} b^{2} e^{3} n^{2}-432 b^{2} d^{2} e \left (f x \right )^{m -1} \ln \left (c \,x^{n}\right )^{2} x^{m} x \,m^{2}-864 x \,x^{m} \ln \left (c \,x^{n}\right ) \left (f x \right )^{m -1} a b \,d^{2} e \,m^{2}+432 x \,x^{m} \ln \left (c \,x^{n}\right ) \left (f x \right )^{m -1} b^{2} d^{2} e m n -576 x \,x^{2 m} \ln \left (c \,x^{n}\right ) \left (f x \right )^{m -1} a b d \,e^{2} m^{2}+192 x \,x^{2 m} \ln \left (c \,x^{n}\right ) \left (f x \right )^{m -1} b^{2} d \,e^{2} m n +432 x \,x^{m} \left (f x \right )^{m -1} a b \,d^{2} e m n +192 x \,x^{2 m} \left (f x \right )^{m -1} a b d \,e^{2} m n -288 x \left (f x \right )^{m -1} a^{2} d^{3} m^{2}-576 x \left (f x \right )^{m -1} b^{2} d^{3} n^{2}-216 x \,x^{m} \left (f x \right )^{m -1} b^{2} d^{2} e \,n^{2}-288 x \,x^{2 m} \left (f x \right )^{m -1} a^{2} d \,e^{2} m^{2}-64 x \,x^{2 m} \left (f x \right )^{m -1} b^{2} d \,e^{2} n^{2}-72 b^{2} e^{3} \left (f x \right )^{m -1} \ln \left (c \,x^{n}\right )^{2} x \,x^{3 m} m^{2}-576 x \ln \left (c \,x^{n}\right ) \left (f x \right )^{m -1} a b \,d^{3} m^{2}+576 x \ln \left (c \,x^{n}\right ) \left (f x \right )^{m -1} b^{2} d^{3} m n +576 x \left (f x \right )^{m -1} a b \,d^{3} m n -432 x \,x^{m} \left (f x \right )^{m -1} a^{2} d^{2} e \,m^{2}+36 x \,x^{3 m} \left (f x \right )^{m -1} a b \,e^{3} m n -144 x \,x^{3 m} \ln \left (c \,x^{n}\right ) \left (f x \right )^{m -1} a b \,e^{3} m^{2}+36 x \,x^{3 m} \ln \left (c \,x^{n}\right ) \left (f x \right )^{m -1} b^{2} e^{3} m n -288 b^{2} d \,e^{2} \left (f x \right )^{m -1} \ln \left (c \,x^{n}\right )^{2} x^{2 m} x \,m^{2}}{288 m^{3}}\) | \(617\) |
| risch | \(\text {Expression too large to display}\) | \(4156\) |
-1/288*(-288*b^2*d^3*(f*x)^(m-1)*ln(c*x^n)^2*x*m^2-72*x*(x^m)^3*(f*x)^(m-1 )*a^2*e^3*m^2-9*x*(x^m)^3*(f*x)^(m-1)*b^2*e^3*n^2-432*b^2*d^2*e*(f*x)^(m-1 )*ln(c*x^n)^2*x^m*x*m^2-864*x*x^m*ln(c*x^n)*(f*x)^(m-1)*a*b*d^2*e*m^2+432* x*x^m*ln(c*x^n)*(f*x)^(m-1)*b^2*d^2*e*m*n-576*x*(x^m)^2*ln(c*x^n)*(f*x)^(m -1)*a*b*d*e^2*m^2+192*x*(x^m)^2*ln(c*x^n)*(f*x)^(m-1)*b^2*d*e^2*m*n+432*x* x^m*(f*x)^(m-1)*a*b*d^2*e*m*n+192*x*(x^m)^2*(f*x)^(m-1)*a*b*d*e^2*m*n-288* x*(f*x)^(m-1)*a^2*d^3*m^2-576*x*(f*x)^(m-1)*b^2*d^3*n^2-216*x*x^m*(f*x)^(m -1)*b^2*d^2*e*n^2-288*x*(x^m)^2*(f*x)^(m-1)*a^2*d*e^2*m^2-64*x*(x^m)^2*(f* x)^(m-1)*b^2*d*e^2*n^2-72*b^2*e^3*(f*x)^(m-1)*ln(c*x^n)^2*x*(x^m)^3*m^2-57 6*x*ln(c*x^n)*(f*x)^(m-1)*a*b*d^3*m^2+576*x*ln(c*x^n)*(f*x)^(m-1)*b^2*d^3* m*n+576*x*(f*x)^(m-1)*a*b*d^3*m*n-432*x*x^m*(f*x)^(m-1)*a^2*d^2*e*m^2+36*x *(x^m)^3*(f*x)^(m-1)*a*b*e^3*m*n-144*x*(x^m)^3*ln(c*x^n)*(f*x)^(m-1)*a*b*e ^3*m^2+36*x*(x^m)^3*ln(c*x^n)*(f*x)^(m-1)*b^2*e^3*m*n-288*b^2*d*e^2*(f*x)^ (m-1)*ln(c*x^n)^2*(x^m)^2*x*m^2)/m^3
Time = 0.31 (sec) , antiderivative size = 592, normalized size of antiderivative = 1.59 \[ \int (f x)^{-1+m} \left (d+e x^m\right )^3 \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\frac {9 \, {\left (8 \, b^{2} e^{3} m^{2} n^{2} \log \left (x\right )^{2} + 8 \, b^{2} e^{3} m^{2} \log \left (c\right )^{2} + 8 \, a^{2} e^{3} m^{2} - 4 \, a b e^{3} m n + b^{2} e^{3} n^{2} + 4 \, {\left (4 \, a b e^{3} m^{2} - b^{2} e^{3} m n\right )} \log \left (c\right ) + 4 \, {\left (4 \, b^{2} e^{3} m^{2} n \log \left (c\right ) + 4 \, a b e^{3} m^{2} n - b^{2} e^{3} m n^{2}\right )} \log \left (x\right )\right )} f^{m - 1} x^{4 \, m} + 32 \, {\left (9 \, b^{2} d e^{2} m^{2} n^{2} \log \left (x\right )^{2} + 9 \, b^{2} d e^{2} m^{2} \log \left (c\right )^{2} + 9 \, a^{2} d e^{2} m^{2} - 6 \, a b d e^{2} m n + 2 \, b^{2} d e^{2} n^{2} + 6 \, {\left (3 \, a b d e^{2} m^{2} - b^{2} d e^{2} m n\right )} \log \left (c\right ) + 6 \, {\left (3 \, b^{2} d e^{2} m^{2} n \log \left (c\right ) + 3 \, a b d e^{2} m^{2} n - b^{2} d e^{2} m n^{2}\right )} \log \left (x\right )\right )} f^{m - 1} x^{3 \, m} + 216 \, {\left (2 \, b^{2} d^{2} e m^{2} n^{2} \log \left (x\right )^{2} + 2 \, b^{2} d^{2} e m^{2} \log \left (c\right )^{2} + 2 \, a^{2} d^{2} e m^{2} - 2 \, a b d^{2} e m n + b^{2} d^{2} e n^{2} + 2 \, {\left (2 \, a b d^{2} e m^{2} - b^{2} d^{2} e m n\right )} \log \left (c\right ) + 2 \, {\left (2 \, b^{2} d^{2} e m^{2} n \log \left (c\right ) + 2 \, a b d^{2} e m^{2} n - b^{2} d^{2} e m n^{2}\right )} \log \left (x\right )\right )} f^{m - 1} x^{2 \, m} + 288 \, {\left (b^{2} d^{3} m^{2} n^{2} \log \left (x\right )^{2} + b^{2} d^{3} m^{2} \log \left (c\right )^{2} + a^{2} d^{3} m^{2} - 2 \, a b d^{3} m n + 2 \, b^{2} d^{3} n^{2} + 2 \, {\left (a b d^{3} m^{2} - b^{2} d^{3} m n\right )} \log \left (c\right ) + 2 \, {\left (b^{2} d^{3} m^{2} n \log \left (c\right ) + a b d^{3} m^{2} n - b^{2} d^{3} m n^{2}\right )} \log \left (x\right )\right )} f^{m - 1} x^{m}}{288 \, m^{3}} \]
1/288*(9*(8*b^2*e^3*m^2*n^2*log(x)^2 + 8*b^2*e^3*m^2*log(c)^2 + 8*a^2*e^3* m^2 - 4*a*b*e^3*m*n + b^2*e^3*n^2 + 4*(4*a*b*e^3*m^2 - b^2*e^3*m*n)*log(c) + 4*(4*b^2*e^3*m^2*n*log(c) + 4*a*b*e^3*m^2*n - b^2*e^3*m*n^2)*log(x))*f^ (m - 1)*x^(4*m) + 32*(9*b^2*d*e^2*m^2*n^2*log(x)^2 + 9*b^2*d*e^2*m^2*log(c )^2 + 9*a^2*d*e^2*m^2 - 6*a*b*d*e^2*m*n + 2*b^2*d*e^2*n^2 + 6*(3*a*b*d*e^2 *m^2 - b^2*d*e^2*m*n)*log(c) + 6*(3*b^2*d*e^2*m^2*n*log(c) + 3*a*b*d*e^2*m ^2*n - b^2*d*e^2*m*n^2)*log(x))*f^(m - 1)*x^(3*m) + 216*(2*b^2*d^2*e*m^2*n ^2*log(x)^2 + 2*b^2*d^2*e*m^2*log(c)^2 + 2*a^2*d^2*e*m^2 - 2*a*b*d^2*e*m*n + b^2*d^2*e*n^2 + 2*(2*a*b*d^2*e*m^2 - b^2*d^2*e*m*n)*log(c) + 2*(2*b^2*d ^2*e*m^2*n*log(c) + 2*a*b*d^2*e*m^2*n - b^2*d^2*e*m*n^2)*log(x))*f^(m - 1) *x^(2*m) + 288*(b^2*d^3*m^2*n^2*log(x)^2 + b^2*d^3*m^2*log(c)^2 + a^2*d^3* m^2 - 2*a*b*d^3*m*n + 2*b^2*d^3*n^2 + 2*(a*b*d^3*m^2 - b^2*d^3*m*n)*log(c) + 2*(b^2*d^3*m^2*n*log(c) + a*b*d^3*m^2*n - b^2*d^3*m*n^2)*log(x))*f^(m - 1)*x^m)/m^3
Leaf count of result is larger than twice the leaf count of optimal. 760 vs. \(2 (364) = 728\).
Time = 38.32 (sec) , antiderivative size = 760, normalized size of antiderivative = 2.04 \[ \int (f x)^{-1+m} \left (d+e x^m\right )^3 \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\begin {cases} \frac {a^{2} d^{3} x \left (f x\right )^{m - 1}}{m} + \frac {3 a^{2} d^{2} e x x^{m} \left (f x\right )^{m - 1}}{2 m} + \frac {a^{2} d e^{2} x x^{2 m} \left (f x\right )^{m - 1}}{m} + \frac {a^{2} e^{3} x x^{3 m} \left (f x\right )^{m - 1}}{4 m} + \frac {2 a b d^{3} x \left (f x\right )^{m - 1} \log {\left (c x^{n} \right )}}{m} - \frac {2 a b d^{3} n x \left (f x\right )^{m - 1}}{m^{2}} + \frac {3 a b d^{2} e x x^{m} \left (f x\right )^{m - 1} \log {\left (c x^{n} \right )}}{m} - \frac {3 a b d^{2} e n x x^{m} \left (f x\right )^{m - 1}}{2 m^{2}} + \frac {2 a b d e^{2} x x^{2 m} \left (f x\right )^{m - 1} \log {\left (c x^{n} \right )}}{m} - \frac {2 a b d e^{2} n x x^{2 m} \left (f x\right )^{m - 1}}{3 m^{2}} + \frac {a b e^{3} x x^{3 m} \left (f x\right )^{m - 1} \log {\left (c x^{n} \right )}}{2 m} - \frac {a b e^{3} n x x^{3 m} \left (f x\right )^{m - 1}}{8 m^{2}} + \frac {b^{2} d^{3} x \left (f x\right )^{m - 1} \log {\left (c x^{n} \right )}^{2}}{m} - \frac {2 b^{2} d^{3} n x \left (f x\right )^{m - 1} \log {\left (c x^{n} \right )}}{m^{2}} + \frac {2 b^{2} d^{3} n^{2} x \left (f x\right )^{m - 1}}{m^{3}} + \frac {3 b^{2} d^{2} e x x^{m} \left (f x\right )^{m - 1} \log {\left (c x^{n} \right )}^{2}}{2 m} - \frac {3 b^{2} d^{2} e n x x^{m} \left (f x\right )^{m - 1} \log {\left (c x^{n} \right )}}{2 m^{2}} + \frac {3 b^{2} d^{2} e n^{2} x x^{m} \left (f x\right )^{m - 1}}{4 m^{3}} + \frac {b^{2} d e^{2} x x^{2 m} \left (f x\right )^{m - 1} \log {\left (c x^{n} \right )}^{2}}{m} - \frac {2 b^{2} d e^{2} n x x^{2 m} \left (f x\right )^{m - 1} \log {\left (c x^{n} \right )}}{3 m^{2}} + \frac {2 b^{2} d e^{2} n^{2} x x^{2 m} \left (f x\right )^{m - 1}}{9 m^{3}} + \frac {b^{2} e^{3} x x^{3 m} \left (f x\right )^{m - 1} \log {\left (c x^{n} \right )}^{2}}{4 m} - \frac {b^{2} e^{3} n x x^{3 m} \left (f x\right )^{m - 1} \log {\left (c x^{n} \right )}}{8 m^{2}} + \frac {b^{2} e^{3} n^{2} x x^{3 m} \left (f x\right )^{m - 1}}{32 m^{3}} & \text {for}\: m \neq 0 \\\frac {\left (d + e\right )^{3} \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**3*x*(f*x)**(m - 1)/m + 3*a**2*d**2*e*x*x**m*(f*x)**(m - 1)/(2*m) + a**2*d*e**2*x*x**(2*m)*(f*x)**(m - 1)/m + a**2*e**3*x*x**(3*m) *(f*x)**(m - 1)/(4*m) + 2*a*b*d**3*x*(f*x)**(m - 1)*log(c*x**n)/m - 2*a*b* d**3*n*x*(f*x)**(m - 1)/m**2 + 3*a*b*d**2*e*x*x**m*(f*x)**(m - 1)*log(c*x* *n)/m - 3*a*b*d**2*e*n*x*x**m*(f*x)**(m - 1)/(2*m**2) + 2*a*b*d*e**2*x*x** (2*m)*(f*x)**(m - 1)*log(c*x**n)/m - 2*a*b*d*e**2*n*x*x**(2*m)*(f*x)**(m - 1)/(3*m**2) + a*b*e**3*x*x**(3*m)*(f*x)**(m - 1)*log(c*x**n)/(2*m) - a*b* e**3*n*x*x**(3*m)*(f*x)**(m - 1)/(8*m**2) + b**2*d**3*x*(f*x)**(m - 1)*log (c*x**n)**2/m - 2*b**2*d**3*n*x*(f*x)**(m - 1)*log(c*x**n)/m**2 + 2*b**2*d **3*n**2*x*(f*x)**(m - 1)/m**3 + 3*b**2*d**2*e*x*x**m*(f*x)**(m - 1)*log(c *x**n)**2/(2*m) - 3*b**2*d**2*e*n*x*x**m*(f*x)**(m - 1)*log(c*x**n)/(2*m** 2) + 3*b**2*d**2*e*n**2*x*x**m*(f*x)**(m - 1)/(4*m**3) + b**2*d*e**2*x*x** (2*m)*(f*x)**(m - 1)*log(c*x**n)**2/m - 2*b**2*d*e**2*n*x*x**(2*m)*(f*x)** (m - 1)*log(c*x**n)/(3*m**2) + 2*b**2*d*e**2*n**2*x*x**(2*m)*(f*x)**(m - 1 )/(9*m**3) + b**2*e**3*x*x**(3*m)*(f*x)**(m - 1)*log(c*x**n)**2/(4*m) - b* *2*e**3*n*x*x**(3*m)*(f*x)**(m - 1)*log(c*x**n)/(8*m**2) + b**2*e**3*n**2* x*x**(3*m)*(f*x)**(m - 1)/(32*m**3), Ne(m, 0)), ((d + e)**3*Piecewise(((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 = 578, normalized size of antiderivative = 1.55 \[ \int (f x)^{-1+m} \left (d+e x^m\right )^3 \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\frac {b^{2} e^{3} f^{m - 1} x^{4 \, m} \log \left (c x^{n}\right )^{2}}{4 \, m} + \frac {b^{2} d e^{2} f^{m - 1} x^{3 \, m} \log \left (c x^{n}\right )^{2}}{m} + \frac {3 \, b^{2} d^{2} e f^{m - 1} x^{2 \, m} \log \left (c x^{n}\right )^{2}}{2 \, m} + \frac {a b e^{3} f^{m - 1} x^{4 \, m} \log \left (c x^{n}\right )}{2 \, m} + \frac {2 \, a b d e^{2} f^{m - 1} x^{3 \, m} \log \left (c x^{n}\right )}{m} + \frac {3 \, a b d^{2} e f^{m - 1} x^{2 \, m} \log \left (c x^{n}\right )}{m} - 2 \, {\left (\frac {f^{m - 1} n x^{m} \log \left (c x^{n}\right )}{m^{2}} - \frac {f^{m - 1} n^{2} x^{m}}{m^{3}}\right )} b^{2} d^{3} - \frac {3}{4} \, {\left (\frac {2 \, f^{m - 1} n x^{2 \, m} \log \left (c x^{n}\right )}{m^{2}} - \frac {f^{m - 1} n^{2} x^{2 \, m}}{m^{3}}\right )} b^{2} d^{2} e - \frac {2}{9} \, {\left (\frac {3 \, f^{m - 1} n x^{3 \, m} \log \left (c x^{n}\right )}{m^{2}} - \frac {f^{m - 1} n^{2} x^{3 \, m}}{m^{3}}\right )} b^{2} d e^{2} - \frac {1}{32} \, {\left (\frac {4 \, f^{m - 1} n x^{4 \, m} \log \left (c x^{n}\right )}{m^{2}} - \frac {f^{m - 1} n^{2} x^{4 \, m}}{m^{3}}\right )} b^{2} e^{3} + \frac {a^{2} e^{3} f^{m - 1} x^{4 \, m}}{4 \, m} - \frac {a b e^{3} f^{m - 1} n x^{4 \, m}}{8 \, m^{2}} + \frac {a^{2} d e^{2} f^{m - 1} x^{3 \, m}}{m} - \frac {2 \, a b d e^{2} f^{m - 1} n x^{3 \, m}}{3 \, m^{2}} + \frac {3 \, a^{2} d^{2} e f^{m - 1} x^{2 \, m}}{2 \, m} - \frac {3 \, a b d^{2} e f^{m - 1} n x^{2 \, m}}{2 \, m^{2}} - \frac {2 \, a b d^{3} f^{m - 1} n x^{m}}{m^{2}} + \frac {\left (f x\right )^{m} b^{2} d^{3} \log \left (c x^{n}\right )^{2}}{f m} + \frac {2 \, \left (f x\right )^{m} a b d^{3} \log \left (c x^{n}\right )}{f m} + \frac {\left (f x\right )^{m} a^{2} d^{3}}{f m} \]
1/4*b^2*e^3*f^(m - 1)*x^(4*m)*log(c*x^n)^2/m + b^2*d*e^2*f^(m - 1)*x^(3*m) *log(c*x^n)^2/m + 3/2*b^2*d^2*e*f^(m - 1)*x^(2*m)*log(c*x^n)^2/m + 1/2*a*b *e^3*f^(m - 1)*x^(4*m)*log(c*x^n)/m + 2*a*b*d*e^2*f^(m - 1)*x^(3*m)*log(c* x^n)/m + 3*a*b*d^2*e*f^(m - 1)*x^(2*m)*log(c*x^n)/m - 2*(f^(m - 1)*n*x^m*l og(c*x^n)/m^2 - f^(m - 1)*n^2*x^m/m^3)*b^2*d^3 - 3/4*(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^2*e - 2/9*(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*d*e^2 - 1/32*(4 *f^(m - 1)*n*x^(4*m)*log(c*x^n)/m^2 - f^(m - 1)*n^2*x^(4*m)/m^3)*b^2*e^3 + 1/4*a^2*e^3*f^(m - 1)*x^(4*m)/m - 1/8*a*b*e^3*f^(m - 1)*n*x^(4*m)/m^2 + a ^2*d*e^2*f^(m - 1)*x^(3*m)/m - 2/3*a*b*d*e^2*f^(m - 1)*n*x^(3*m)/m^2 + 3/2 *a^2*d^2*e*f^(m - 1)*x^(2*m)/m - 3/2*a*b*d^2*e*f^(m - 1)*n*x^(2*m)/m^2 - 2 *a*b*d^3*f^(m - 1)*n*x^m/m^2 + (f*x)^m*b^2*d^3*log(c*x^n)^2/(f*m) + 2*(f*x )^m*a*b*d^3*log(c*x^n)/(f*m) + (f*x)^m*a^2*d^3/(f*m)
Leaf count of result is larger than twice the leaf count of optimal. 995 vs. \(2 (354) = 708\).
Time = 0.62 (sec) , antiderivative size = 995, normalized size of antiderivative = 2.67 \[ \int (f x)^{-1+m} \left (d+e x^m\right )^3 \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\frac {b^{2} e^{3} f^{m} n^{2} x^{4 \, m} \log \left (x\right )^{2}}{4 \, f m} + \frac {b^{2} d e^{2} f^{m} n^{2} x^{3 \, m} \log \left (x\right )^{2}}{f m} + \frac {3 \, b^{2} d^{2} e f^{m} n^{2} x^{2 \, m} \log \left (x\right )^{2}}{2 \, f m} + \frac {b^{2} d^{3} f^{m} n^{2} x^{m} \log \left (x\right )^{2}}{f m} + \frac {b^{2} e^{3} f^{m} n x^{4 \, m} \log \left (c\right ) \log \left (x\right )}{2 \, f m} + \frac {2 \, b^{2} d e^{2} f^{m} n x^{3 \, m} \log \left (c\right ) \log \left (x\right )}{f m} + \frac {3 \, b^{2} d^{2} e f^{m} n x^{2 \, m} \log \left (c\right ) \log \left (x\right )}{f m} + \frac {2 \, b^{2} d^{3} f^{m} n x^{m} \log \left (c\right ) \log \left (x\right )}{f m} + \frac {b^{2} e^{3} f^{m} x^{4 \, m} \log \left (c\right )^{2}}{4 \, f m} + \frac {b^{2} d e^{2} f^{m} x^{3 \, m} \log \left (c\right )^{2}}{f m} + \frac {3 \, b^{2} d^{2} e f^{m} x^{2 \, m} \log \left (c\right )^{2}}{2 \, f m} + \frac {b^{2} d^{3} f^{m} x^{m} \log \left (c\right )^{2}}{f m} + \frac {a b e^{3} f^{m} n x^{4 \, m} \log \left (x\right )}{2 \, f m} - \frac {b^{2} e^{3} f^{m} n^{2} x^{4 \, m} \log \left (x\right )}{8 \, f m^{2}} + \frac {2 \, a b d e^{2} f^{m} n x^{3 \, m} \log \left (x\right )}{f m} - \frac {2 \, b^{2} d e^{2} f^{m} n^{2} x^{3 \, m} \log \left (x\right )}{3 \, f m^{2}} + \frac {3 \, a b d^{2} e f^{m} n x^{2 \, m} \log \left (x\right )}{f m} - \frac {3 \, b^{2} d^{2} e f^{m} n^{2} x^{2 \, m} \log \left (x\right )}{2 \, f m^{2}} + \frac {2 \, a b d^{3} f^{m} n x^{m} \log \left (x\right )}{f m} - \frac {2 \, b^{2} d^{3} f^{m} n^{2} x^{m} \log \left (x\right )}{f m^{2}} + \frac {a b e^{3} f^{m} x^{4 \, m} \log \left (c\right )}{2 \, f m} - \frac {b^{2} e^{3} f^{m} n x^{4 \, m} \log \left (c\right )}{8 \, f m^{2}} + \frac {2 \, a b d e^{2} f^{m} x^{3 \, m} \log \left (c\right )}{f m} - \frac {2 \, b^{2} d e^{2} f^{m} n x^{3 \, m} \log \left (c\right )}{3 \, f m^{2}} + \frac {3 \, a b d^{2} e f^{m} x^{2 \, m} \log \left (c\right )}{f m} - \frac {3 \, b^{2} d^{2} e f^{m} n x^{2 \, m} \log \left (c\right )}{2 \, f m^{2}} + \frac {2 \, a b d^{3} f^{m} x^{m} \log \left (c\right )}{f m} - \frac {2 \, b^{2} d^{3} f^{m} n x^{m} \log \left (c\right )}{f m^{2}} + \frac {a^{2} e^{3} f^{m} x^{4 \, m}}{4 \, f m} - \frac {a b e^{3} f^{m} n x^{4 \, m}}{8 \, f m^{2}} + \frac {b^{2} e^{3} f^{m} n^{2} x^{4 \, m}}{32 \, f m^{3}} + \frac {a^{2} d e^{2} f^{m} x^{3 \, m}}{f m} - \frac {2 \, a b d e^{2} f^{m} n x^{3 \, m}}{3 \, f m^{2}} + \frac {2 \, b^{2} d e^{2} f^{m} n^{2} x^{3 \, m}}{9 \, f m^{3}} + \frac {3 \, a^{2} d^{2} e f^{m} x^{2 \, m}}{2 \, f m} - \frac {3 \, a b d^{2} e f^{m} n x^{2 \, m}}{2 \, f m^{2}} + \frac {3 \, b^{2} d^{2} e f^{m} n^{2} x^{2 \, m}}{4 \, f m^{3}} + \frac {a^{2} d^{3} f^{m} x^{m}}{f m} - \frac {2 \, a b d^{3} f^{m} n x^{m}}{f m^{2}} + \frac {2 \, b^{2} d^{3} f^{m} n^{2} x^{m}}{f m^{3}} \]
1/4*b^2*e^3*f^m*n^2*x^(4*m)*log(x)^2/(f*m) + b^2*d*e^2*f^m*n^2*x^(3*m)*log (x)^2/(f*m) + 3/2*b^2*d^2*e*f^m*n^2*x^(2*m)*log(x)^2/(f*m) + b^2*d^3*f^m*n ^2*x^m*log(x)^2/(f*m) + 1/2*b^2*e^3*f^m*n*x^(4*m)*log(c)*log(x)/(f*m) + 2* b^2*d*e^2*f^m*n*x^(3*m)*log(c)*log(x)/(f*m) + 3*b^2*d^2*e*f^m*n*x^(2*m)*lo g(c)*log(x)/(f*m) + 2*b^2*d^3*f^m*n*x^m*log(c)*log(x)/(f*m) + 1/4*b^2*e^3* f^m*x^(4*m)*log(c)^2/(f*m) + b^2*d*e^2*f^m*x^(3*m)*log(c)^2/(f*m) + 3/2*b^ 2*d^2*e*f^m*x^(2*m)*log(c)^2/(f*m) + b^2*d^3*f^m*x^m*log(c)^2/(f*m) + 1/2* a*b*e^3*f^m*n*x^(4*m)*log(x)/(f*m) - 1/8*b^2*e^3*f^m*n^2*x^(4*m)*log(x)/(f *m^2) + 2*a*b*d*e^2*f^m*n*x^(3*m)*log(x)/(f*m) - 2/3*b^2*d*e^2*f^m*n^2*x^( 3*m)*log(x)/(f*m^2) + 3*a*b*d^2*e*f^m*n*x^(2*m)*log(x)/(f*m) - 3/2*b^2*d^2 *e*f^m*n^2*x^(2*m)*log(x)/(f*m^2) + 2*a*b*d^3*f^m*n*x^m*log(x)/(f*m) - 2*b ^2*d^3*f^m*n^2*x^m*log(x)/(f*m^2) + 1/2*a*b*e^3*f^m*x^(4*m)*log(c)/(f*m) - 1/8*b^2*e^3*f^m*n*x^(4*m)*log(c)/(f*m^2) + 2*a*b*d*e^2*f^m*x^(3*m)*log(c) /(f*m) - 2/3*b^2*d*e^2*f^m*n*x^(3*m)*log(c)/(f*m^2) + 3*a*b*d^2*e*f^m*x^(2 *m)*log(c)/(f*m) - 3/2*b^2*d^2*e*f^m*n*x^(2*m)*log(c)/(f*m^2) + 2*a*b*d^3* f^m*x^m*log(c)/(f*m) - 2*b^2*d^3*f^m*n*x^m*log(c)/(f*m^2) + 1/4*a^2*e^3*f^ m*x^(4*m)/(f*m) - 1/8*a*b*e^3*f^m*n*x^(4*m)/(f*m^2) + 1/32*b^2*e^3*f^m*n^2 *x^(4*m)/(f*m^3) + a^2*d*e^2*f^m*x^(3*m)/(f*m) - 2/3*a*b*d*e^2*f^m*n*x^(3* m)/(f*m^2) + 2/9*b^2*d*e^2*f^m*n^2*x^(3*m)/(f*m^3) + 3/2*a^2*d^2*e*f^m*x^( 2*m)/(f*m) - 3/2*a*b*d^2*e*f^m*n*x^(2*m)/(f*m^2) + 3/4*b^2*d^2*e*f^m*n^...
Timed out. \[ \int (f x)^{-1+m} \left (d+e x^m\right )^3 \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\int {\left (f\,x\right )}^{m-1}\,{\left (d+e\,x^m\right )}^3\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2 \,d x \]