Integrand size = 27, antiderivative size = 171 \[ \int (f x)^{-1+m} \left (d+e x^m\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {b d^3 n x (f x)^{-1+m}}{m^2}-\frac {3 b d^2 e n x^{1+m} (f x)^{-1+m}}{4 m^2}-\frac {b d e^2 n x^{1+2 m} (f x)^{-1+m}}{3 m^2}-\frac {b e^3 n x^{1+3 m} (f x)^{-1+m}}{16 m^2}-\frac {b d^4 n x^{1-m} (f x)^{-1+m} \log (x)}{4 e m}+\frac {x^{1-m} (f x)^{-1+m} \left (d+e x^m\right )^4 \left (a+b \log \left (c x^n\right )\right )}{4 e m} \]
-b*d^3*n*x*(f*x)^(-1+m)/m^2-3/4*b*d^2*e*n*x^(1+m)*(f*x)^(-1+m)/m^2-1/3*b*d *e^2*n*x^(1+2*m)*(f*x)^(-1+m)/m^2-1/16*b*e^3*n*x^(1+3*m)*(f*x)^(-1+m)/m^2- 1/4*b*d^4*n*x^(1-m)*(f*x)^(-1+m)*ln(x)/e/m+1/4*x^(1-m)*(f*x)^(-1+m)*(d+e*x ^m)^4*(a+b*ln(c*x^n))/e/m
Time = 0.11 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.82 \[ \int (f x)^{-1+m} \left (d+e x^m\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {(f x)^m \left (12 a m \left (4 d^3+6 d^2 e x^m+4 d e^2 x^{2 m}+e^3 x^{3 m}\right )-b n \left (48 d^3+36 d^2 e x^m+16 d e^2 x^{2 m}+3 e^3 x^{3 m}\right )+12 b m \left (4 d^3+6 d^2 e x^m+4 d e^2 x^{2 m}+e^3 x^{3 m}\right ) \log \left (c x^n\right )\right )}{48 f m^2} \]
((f*x)^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)) + 12*b*m*(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]))/(48*f*m^2)
Time = 0.41 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.66, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {2777, 2776, 798, 49, 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.
| \(\displaystyle \int (f x)^{m-1} \left (d+e x^m\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx\) |
| \(\Big \downarrow \) 2777 |
| \(\displaystyle x^{1-m} (f x)^{m-1} \int x^{m-1} \left (e x^m+d\right )^3 \left (a+b \log \left (c x^n\right )\right )dx\) |
| \(\Big \downarrow \) 2776 |
| \(\displaystyle x^{1-m} (f x)^{m-1} \left (\frac {\left (d+e x^m\right )^4 \left (a+b \log \left (c x^n\right )\right )}{4 e m}-\frac {b n \int \frac {\left (e x^m+d\right )^4}{x}dx}{4 e m}\right )\) |
| \(\Big \downarrow \) 798 |
| \(\displaystyle x^{1-m} (f x)^{m-1} \left (\frac {\left (d+e x^m\right )^4 \left (a+b \log \left (c x^n\right )\right )}{4 e m}-\frac {b n \int x^{-m} \left (e x^m+d\right )^4dx^m}{4 e m^2}\right )\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle x^{1-m} (f x)^{m-1} \left (\frac {\left (d+e x^m\right )^4 \left (a+b \log \left (c x^n\right )\right )}{4 e m}-\frac {b n \int \left (d^4 x^{-m}+6 d^2 e^2 x^m+4 d e^3 x^{2 m}+e^4 x^{3 m}+4 d^3 e\right )dx^m}{4 e m^2}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle x^{1-m} (f x)^{m-1} \left (\frac {\left (d+e x^m\right )^4 \left (a+b \log \left (c x^n\right )\right )}{4 e m}-\frac {b n \left (d^4 \log \left (x^m\right )+4 d^3 e x^m+3 d^2 e^2 x^{2 m}+\frac {4}{3} d e^3 x^{3 m}+\frac {1}{4} e^4 x^{4 m}\right )}{4 e m^2}\right )\) |
x^(1 - m)*(f*x)^(-1 + m)*(-1/4*(b*n*(4*d^3*e*x^m + 3*d^2*e^2*x^(2*m) + (4* d*e^3*x^(3*m))/3 + (e^4*x^(4*m))/4 + d^4*Log[x^m]))/(e*m^2) + ((d + e*x^m) ^4*(a + b*Log[c*x^n]))/(4*e*m))
3.4.51.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
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_.))^(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 = 54.99 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.50
| method | result | size |
| parallelrisch | \(-\frac {-12 e^{3} b \left (f x \right )^{m -1} \ln \left (c \,x^{n}\right ) x \,x^{3 m} m -12 x \,x^{3 m} \left (f x \right )^{m -1} a \,e^{3} m +3 x \,x^{3 m} \left (f x \right )^{m -1} b \,e^{3} n -48 e^{2} d b \left (f x \right )^{m -1} \ln \left (c \,x^{n}\right ) x^{2 m} x m -48 x \,x^{2 m} \left (f x \right )^{m -1} a d \,e^{2} m +16 x \,x^{2 m} \left (f x \right )^{m -1} b d \,e^{2} n -72 e \,d^{2} b \left (f x \right )^{m -1} \ln \left (c \,x^{n}\right ) x^{m} x m -72 x \,x^{m} \left (f x \right )^{m -1} a \,d^{2} e m +36 x \,x^{m} \left (f x \right )^{m -1} b \,d^{2} e n -48 b \,d^{3} \left (f x \right )^{m -1} \ln \left (c \,x^{n}\right ) x m -48 x \left (f x \right )^{m -1} a \,d^{3} m +48 x \left (f x \right )^{m -1} b \,d^{3} n}{48 m^{2}}\) | \(256\) |
| risch | \(\frac {b \left (e^{3} x^{3 m}+4 d \,e^{2} x^{2 m}+6 d^{2} e \,x^{m}+4 d^{3}\right ) x \,{\mathrm e}^{\frac {\left (m -1\right ) \left (-i \pi \operatorname {csgn}\left (i f x \right )^{3}+i \pi \operatorname {csgn}\left (i f x \right )^{2} \operatorname {csgn}\left (i f \right )+i \pi \operatorname {csgn}\left (i f x \right )^{2} \operatorname {csgn}\left (i x \right )-i \pi \,\operatorname {csgn}\left (i f x \right ) \operatorname {csgn}\left (i f \right ) \operatorname {csgn}\left (i x \right )+2 \ln \left (x \right )+2 \ln \left (f \right )\right )}{2}} \ln \left (x^{n}\right )}{4 m}+\frac {\left (12 \ln \left (c \right ) b \,e^{3} x^{3 m} m -16 b d \,e^{2} n \,x^{2 m}+48 a d \,e^{2} x^{2 m} m +72 \ln \left (c \right ) b \,d^{2} e \,x^{m} m -6 i \pi b \,e^{3} \operatorname {csgn}\left (i c \,x^{n}\right )^{3} x^{3 m} m +24 i \pi b \,d^{3} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2} m +24 i \pi b \,d^{3} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2} m +48 \ln \left (c \right ) b d \,e^{2} x^{2 m} m -24 i \pi b \,d^{3} \operatorname {csgn}\left (i c \,x^{n}\right )^{3} m -24 i \pi b d \,e^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) x^{2 m} m -48 b \,d^{3} n +24 i \pi b d \,e^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2} x^{2 m} m -6 i \pi b \,e^{3} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) x^{3 m} m -3 b \,e^{3} n \,x^{3 m}+12 a \,e^{3} x^{3 m} m +24 i \pi b d \,e^{2} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2} x^{2 m} m +48 a \,d^{3} m +72 a \,d^{2} e \,x^{m} m +48 \ln \left (c \right ) b \,d^{3} m -36 i \pi b \,d^{2} e \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) x^{m} m -36 b \,d^{2} e n \,x^{m}+36 i \pi b \,d^{2} e \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2} x^{m} m +36 i \pi b \,d^{2} e \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2} x^{m} m -36 i \pi b \,d^{2} e \operatorname {csgn}\left (i c \,x^{n}\right )^{3} x^{m} m -24 i \pi b \,d^{3} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) m +6 i \pi b \,e^{3} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2} x^{3 m} m +6 i \pi b \,e^{3} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2} x^{3 m} m -24 i \pi b d \,e^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{3} x^{2 m} m \right ) x \,{\mathrm e}^{\frac {\left (m -1\right ) \left (-i \pi \operatorname {csgn}\left (i f x \right )^{3}+i \pi \operatorname {csgn}\left (i f x \right )^{2} \operatorname {csgn}\left (i f \right )+i \pi \operatorname {csgn}\left (i f x \right )^{2} \operatorname {csgn}\left (i x \right )-i \pi \,\operatorname {csgn}\left (i f x \right ) \operatorname {csgn}\left (i f \right ) \operatorname {csgn}\left (i x \right )+2 \ln \left (x \right )+2 \ln \left (f \right )\right )}{2}}}{48 m^{2}}\) | \(806\) |
-1/48*(-12*e^3*b*(f*x)^(m-1)*ln(c*x^n)*x*(x^m)^3*m-12*x*(x^m)^3*(f*x)^(m-1 )*a*e^3*m+3*x*(x^m)^3*(f*x)^(m-1)*b*e^3*n-48*e^2*d*b*(f*x)^(m-1)*ln(c*x^n) *(x^m)^2*x*m-48*x*(x^m)^2*(f*x)^(m-1)*a*d*e^2*m+16*x*(x^m)^2*(f*x)^(m-1)*b *d*e^2*n-72*e*d^2*b*(f*x)^(m-1)*ln(c*x^n)*x^m*x*m-72*x*x^m*(f*x)^(m-1)*a*d ^2*e*m+36*x*x^m*(f*x)^(m-1)*b*d^2*e*n-48*b*d^3*(f*x)^(m-1)*ln(c*x^n)*x*m-4 8*x*(f*x)^(m-1)*a*d^3*m+48*x*(f*x)^(m-1)*b*d^3*n)/m^2
Time = 0.29 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.13 \[ \int (f x)^{-1+m} \left (d+e x^m\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {3 \, {\left (4 \, b e^{3} m n \log \left (x\right ) + 4 \, b e^{3} m \log \left (c\right ) + 4 \, a e^{3} m - b e^{3} n\right )} f^{m - 1} x^{4 \, m} + 16 \, {\left (3 \, b d e^{2} m n \log \left (x\right ) + 3 \, b d e^{2} m \log \left (c\right ) + 3 \, a d e^{2} m - b d e^{2} n\right )} f^{m - 1} x^{3 \, m} + 36 \, {\left (2 \, b d^{2} e m n \log \left (x\right ) + 2 \, b d^{2} e m \log \left (c\right ) + 2 \, a d^{2} e m - b d^{2} e n\right )} f^{m - 1} x^{2 \, m} + 48 \, {\left (b d^{3} m n \log \left (x\right ) + b d^{3} m \log \left (c\right ) + a d^{3} m - b d^{3} n\right )} f^{m - 1} x^{m}}{48 \, m^{2}} \]
1/48*(3*(4*b*e^3*m*n*log(x) + 4*b*e^3*m*log(c) + 4*a*e^3*m - b*e^3*n)*f^(m - 1)*x^(4*m) + 16*(3*b*d*e^2*m*n*log(x) + 3*b*d*e^2*m*log(c) + 3*a*d*e^2* m - b*d*e^2*n)*f^(m - 1)*x^(3*m) + 36*(2*b*d^2*e*m*n*log(x) + 2*b*d^2*e*m* log(c) + 2*a*d^2*e*m - b*d^2*e*n)*f^(m - 1)*x^(2*m) + 48*(b*d^3*m*n*log(x) + b*d^3*m*log(c) + a*d^3*m - b*d^3*n)*f^(m - 1)*x^m)/m^2
Leaf count of result is larger than twice the leaf count of optimal. 333 vs. \(2 (160) = 320\).
Time = 8.48 (sec) , antiderivative size = 333, normalized size of antiderivative = 1.95 \[ \int (f x)^{-1+m} \left (d+e x^m\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=\begin {cases} \frac {a d^{3} x \left (f x\right )^{m - 1}}{m} + \frac {3 a d^{2} e x x^{m} \left (f x\right )^{m - 1}}{2 m} + \frac {a d e^{2} x x^{2 m} \left (f x\right )^{m - 1}}{m} + \frac {a e^{3} x x^{3 m} \left (f x\right )^{m - 1}}{4 m} + \frac {b d^{3} x \left (f x\right )^{m - 1} \log {\left (c x^{n} \right )}}{m} - \frac {b d^{3} n x \left (f x\right )^{m - 1}}{m^{2}} + \frac {3 b d^{2} e x x^{m} \left (f x\right )^{m - 1} \log {\left (c x^{n} \right )}}{2 m} - \frac {3 b d^{2} e n x x^{m} \left (f x\right )^{m - 1}}{4 m^{2}} + \frac {b d e^{2} x x^{2 m} \left (f x\right )^{m - 1} \log {\left (c x^{n} \right )}}{m} - \frac {b d e^{2} n x x^{2 m} \left (f x\right )^{m - 1}}{3 m^{2}} + \frac {b e^{3} x x^{3 m} \left (f x\right )^{m - 1} \log {\left (c x^{n} \right )}}{4 m} - \frac {b e^{3} n x x^{3 m} \left (f x\right )^{m - 1}}{16 m^{2}} & \text {for}\: m \neq 0 \\\frac {\left (d + e\right )^{3} \left (\begin {cases} a \log {\left (x \right )} & \text {for}\: b = 0 \\- \left (- a - b \log {\left (c \right )}\right ) \log {\left (x \right )} & \text {for}\: n = 0 \\\frac {\left (- a - b \log {\left (c x^{n} \right )}\right )^{2}}{2 b n} & \text {otherwise} \end {cases}\right )}{f} & \text {otherwise} \end {cases} \]
Piecewise((a*d**3*x*(f*x)**(m - 1)/m + 3*a*d**2*e*x*x**m*(f*x)**(m - 1)/(2 *m) + a*d*e**2*x*x**(2*m)*(f*x)**(m - 1)/m + a*e**3*x*x**(3*m)*(f*x)**(m - 1)/(4*m) + b*d**3*x*(f*x)**(m - 1)*log(c*x**n)/m - b*d**3*n*x*(f*x)**(m - 1)/m**2 + 3*b*d**2*e*x*x**m*(f*x)**(m - 1)*log(c*x**n)/(2*m) - 3*b*d**2*e *n*x*x**m*(f*x)**(m - 1)/(4*m**2) + b*d*e**2*x*x**(2*m)*(f*x)**(m - 1)*log (c*x**n)/m - b*d*e**2*n*x*x**(2*m)*(f*x)**(m - 1)/(3*m**2) + b*e**3*x*x**( 3*m)*(f*x)**(m - 1)*log(c*x**n)/(4*m) - b*e**3*n*x*x**(3*m)*(f*x)**(m - 1) /(16*m**2), Ne(m, 0)), ((d + e)**3*Piecewise((a*log(x), Eq(b, 0)), (-(-a - b*log(c))*log(x), Eq(n, 0)), ((-a - b*log(c*x**n))**2/(2*b*n), True))/f, True))
Time = 0.21 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.48 \[ \int (f x)^{-1+m} \left (d+e x^m\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {b e^{3} f^{m - 1} x^{4 \, m} \log \left (c x^{n}\right )}{4 \, m} + \frac {b d e^{2} f^{m - 1} x^{3 \, m} \log \left (c x^{n}\right )}{m} + \frac {3 \, b d^{2} e f^{m - 1} x^{2 \, m} \log \left (c x^{n}\right )}{2 \, m} + \frac {a e^{3} f^{m - 1} x^{4 \, m}}{4 \, m} - \frac {b e^{3} f^{m - 1} n x^{4 \, m}}{16 \, m^{2}} + \frac {a d e^{2} f^{m - 1} x^{3 \, m}}{m} - \frac {b d e^{2} f^{m - 1} n x^{3 \, m}}{3 \, m^{2}} + \frac {3 \, a d^{2} e f^{m - 1} x^{2 \, m}}{2 \, m} - \frac {3 \, b d^{2} e f^{m - 1} n x^{2 \, m}}{4 \, m^{2}} - \frac {b d^{3} f^{m - 1} n x^{m}}{m^{2}} + \frac {\left (f x\right )^{m} b d^{3} \log \left (c x^{n}\right )}{f m} + \frac {\left (f x\right )^{m} a d^{3}}{f m} \]
1/4*b*e^3*f^(m - 1)*x^(4*m)*log(c*x^n)/m + b*d*e^2*f^(m - 1)*x^(3*m)*log(c *x^n)/m + 3/2*b*d^2*e*f^(m - 1)*x^(2*m)*log(c*x^n)/m + 1/4*a*e^3*f^(m - 1) *x^(4*m)/m - 1/16*b*e^3*f^(m - 1)*n*x^(4*m)/m^2 + a*d*e^2*f^(m - 1)*x^(3*m )/m - 1/3*b*d*e^2*f^(m - 1)*n*x^(3*m)/m^2 + 3/2*a*d^2*e*f^(m - 1)*x^(2*m)/ m - 3/4*b*d^2*e*f^(m - 1)*n*x^(2*m)/m^2 - b*d^3*f^(m - 1)*n*x^m/m^2 + (f*x )^m*b*d^3*log(c*x^n)/(f*m) + (f*x)^m*a*d^3/(f*m)
Leaf count of result is larger than twice the leaf count of optimal. 339 vs. \(2 (161) = 322\).
Time = 0.43 (sec) , antiderivative size = 339, normalized size of antiderivative = 1.98 \[ \int (f x)^{-1+m} \left (d+e x^m\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {b e^{3} f^{m} n x^{4 \, m} \log \left (x\right )}{4 \, f m} + \frac {b d e^{2} f^{m} n x^{3 \, m} \log \left (x\right )}{f m} + \frac {3 \, b d^{2} e f^{m} n x^{2 \, m} \log \left (x\right )}{2 \, f m} + \frac {b d^{3} f^{m} n x^{m} \log \left (x\right )}{f m} + \frac {b e^{3} f^{m} x^{4 \, m} \log \left (c\right )}{4 \, f m} + \frac {b d e^{2} f^{m} x^{3 \, m} \log \left (c\right )}{f m} + \frac {3 \, b d^{2} e f^{m} x^{2 \, m} \log \left (c\right )}{2 \, f m} + \frac {b d^{3} f^{m} x^{m} \log \left (c\right )}{f m} + \frac {a e^{3} f^{m} x^{4 \, m}}{4 \, f m} - \frac {b e^{3} f^{m} n x^{4 \, m}}{16 \, f m^{2}} + \frac {a d e^{2} f^{m} x^{3 \, m}}{f m} - \frac {b d e^{2} f^{m} n x^{3 \, m}}{3 \, f m^{2}} + \frac {3 \, a d^{2} e f^{m} x^{2 \, m}}{2 \, f m} - \frac {3 \, b d^{2} e f^{m} n x^{2 \, m}}{4 \, f m^{2}} + \frac {a d^{3} f^{m} x^{m}}{f m} - \frac {b d^{3} f^{m} n x^{m}}{f m^{2}} \]
1/4*b*e^3*f^m*n*x^(4*m)*log(x)/(f*m) + b*d*e^2*f^m*n*x^(3*m)*log(x)/(f*m) + 3/2*b*d^2*e*f^m*n*x^(2*m)*log(x)/(f*m) + b*d^3*f^m*n*x^m*log(x)/(f*m) + 1/4*b*e^3*f^m*x^(4*m)*log(c)/(f*m) + b*d*e^2*f^m*x^(3*m)*log(c)/(f*m) + 3/ 2*b*d^2*e*f^m*x^(2*m)*log(c)/(f*m) + b*d^3*f^m*x^m*log(c)/(f*m) + 1/4*a*e^ 3*f^m*x^(4*m)/(f*m) - 1/16*b*e^3*f^m*n*x^(4*m)/(f*m^2) + a*d*e^2*f^m*x^(3* m)/(f*m) - 1/3*b*d*e^2*f^m*n*x^(3*m)/(f*m^2) + 3/2*a*d^2*e*f^m*x^(2*m)/(f* m) - 3/4*b*d^2*e*f^m*n*x^(2*m)/(f*m^2) + a*d^3*f^m*x^m/(f*m) - b*d^3*f^m*n *x^m/(f*m^2)
Timed out. \[ \int (f x)^{-1+m} \left (d+e x^m\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=\int {\left (f\,x\right )}^{m-1}\,{\left (d+e\,x^m\right )}^3\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \]