Integrand size = 23, antiderivative size = 153 \[ \int (f x)^m (d+e x)^2 \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {b d^2 n (f x)^{1+m}}{f (1+m)^2}-\frac {2 b d e n (f x)^{2+m}}{f^2 (2+m)^2}-\frac {b e^2 n (f x)^{3+m}}{f^3 (3+m)^2}+\frac {d^2 (f x)^{1+m} \left (a+b \log \left (c x^n\right )\right )}{f (1+m)}+\frac {2 d e (f x)^{2+m} \left (a+b \log \left (c x^n\right )\right )}{f^2 (2+m)}+\frac {e^2 (f x)^{3+m} \left (a+b \log \left (c x^n\right )\right )}{f^3 (3+m)} \]
-b*d^2*n*(f*x)^(1+m)/f/(1+m)^2-2*b*d*e*n*(f*x)^(2+m)/f^2/(2+m)^2-b*e^2*n*( f*x)^(3+m)/f^3/(3+m)^2+d^2*(f*x)^(1+m)*(a+b*ln(c*x^n))/f/(1+m)+2*d*e*(f*x) ^(2+m)*(a+b*ln(c*x^n))/f^2/(2+m)+e^2*(f*x)^(3+m)*(a+b*ln(c*x^n))/f^3/(3+m)
Time = 0.10 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.71 \[ \int (f x)^m (d+e x)^2 \left (a+b \log \left (c x^n\right )\right ) \, dx=x (f x)^m \left (-\frac {b d^2 n}{(1+m)^2}-\frac {2 b d e n x}{(2+m)^2}-\frac {b e^2 n x^2}{(3+m)^2}+\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{1+m}+\frac {2 d e x \left (a+b \log \left (c x^n\right )\right )}{2+m}+\frac {e^2 x^2 \left (a+b \log \left (c x^n\right )\right )}{3+m}\right ) \]
x*(f*x)^m*(-((b*d^2*n)/(1 + m)^2) - (2*b*d*e*n*x)/(2 + m)^2 - (b*e^2*n*x^2 )/(3 + m)^2 + (d^2*(a + b*Log[c*x^n]))/(1 + m) + (2*d*e*x*(a + b*Log[c*x^n ]))/(2 + m) + (e^2*x^2*(a + b*Log[c*x^n]))/(3 + m))
Time = 0.44 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.20, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {2792, 27, 1140, 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.
| \(\displaystyle \int (d+e x)^2 (f x)^m \left (a+b \log \left (c x^n\right )\right ) \, dx\) |
| \(\Big \downarrow \) 2792 |
| \(\displaystyle -b n \int \frac {(f x)^m \left ((m+2) (m+3) d^2+2 e (m+1) (m+3) x d+e^2 (m+1) (m+2) x^2\right )}{m^3+6 m^2+11 m+6}dx+\frac {d^2 (f x)^{m+1} \left (a+b \log \left (c x^n\right )\right )}{f (m+1)}+\frac {2 d e (f x)^{m+2} \left (a+b \log \left (c x^n\right )\right )}{f^2 (m+2)}+\frac {e^2 (f x)^{m+3} \left (a+b \log \left (c x^n\right )\right )}{f^3 (m+3)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {b n \int (f x)^m \left ((m+2) (m+3) d^2+2 e (m+1) (m+3) x d+e^2 (m+1) (m+2) x^2\right )dx}{m^3+6 m^2+11 m+6}+\frac {d^2 (f x)^{m+1} \left (a+b \log \left (c x^n\right )\right )}{f (m+1)}+\frac {2 d e (f x)^{m+2} \left (a+b \log \left (c x^n\right )\right )}{f^2 (m+2)}+\frac {e^2 (f x)^{m+3} \left (a+b \log \left (c x^n\right )\right )}{f^3 (m+3)}\) |
| \(\Big \downarrow \) 1140 |
| \(\displaystyle -\frac {b n \int \left (d^2 (m+2) (m+3) (f x)^m+\frac {2 d e (m+1) (m+3) (f x)^{m+1}}{f}+\frac {e^2 (m+1) (m+2) (f x)^{m+2}}{f^2}\right )dx}{m^3+6 m^2+11 m+6}+\frac {d^2 (f x)^{m+1} \left (a+b \log \left (c x^n\right )\right )}{f (m+1)}+\frac {2 d e (f x)^{m+2} \left (a+b \log \left (c x^n\right )\right )}{f^2 (m+2)}+\frac {e^2 (f x)^{m+3} \left (a+b \log \left (c x^n\right )\right )}{f^3 (m+3)}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {d^2 (f x)^{m+1} \left (a+b \log \left (c x^n\right )\right )}{f (m+1)}+\frac {2 d e (f x)^{m+2} \left (a+b \log \left (c x^n\right )\right )}{f^2 (m+2)}+\frac {e^2 (f x)^{m+3} \left (a+b \log \left (c x^n\right )\right )}{f^3 (m+3)}-\frac {b n \left (\frac {d^2 (m+2) (m+3) (f x)^{m+1}}{f (m+1)}+\frac {2 d e (m+1) (m+3) (f x)^{m+2}}{f^2 (m+2)}+\frac {e^2 (m+1) (m+2) (f x)^{m+3}}{f^3 (m+3)}\right )}{m^3+6 m^2+11 m+6}\) |
-((b*n*((d^2*(2 + m)*(3 + m)*(f*x)^(1 + m))/(f*(1 + m)) + (2*d*e*(1 + m)*( 3 + m)*(f*x)^(2 + m))/(f^2*(2 + m)) + (e^2*(1 + m)*(2 + m)*(f*x)^(3 + m))/ (f^3*(3 + m))))/(6 + 11*m + 6*m^2 + m^3)) + (d^2*(f*x)^(1 + m)*(a + b*Log[ c*x^n]))/(f*(1 + m)) + (2*d*e*(f*x)^(2 + m)*(a + b*Log[c*x^n]))/(f^2*(2 + m)) + (e^2*(f*x)^(3 + m)*(a + b*Log[c*x^n]))/(f^3*(3 + m))
3.2.63.3.1 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[((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x _Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[p, 0]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)* (x_)^(r_.))^(q_.), x_Symbol] :> With[{u = IntHide[(f*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] /; ((EqQ[r, 1] || EqQ[r, 2]) && IntegerQ[m] && IntegerQ[q - 1/2] ) || InverseFunctionFreeQ[u, x]] /; FreeQ[{a, b, c, d, e, f, m, n, q, r}, x ] && IntegerQ[2*q] && ((IntegerQ[m] && IntegerQ[r]) || IGtQ[q, 0])
Leaf count of result is larger than twice the leaf count of optimal. \(910\) vs. \(2(153)=306\).
Time = 1.05 (sec) , antiderivative size = 911, normalized size of antiderivative = 5.95
| method | result | size |
| parallelrisch | \(-\frac {-12 x^{3} \left (f x \right )^{m} a \,e^{2}-36 x \left (f x \right )^{m} a \,d^{2}-36 x \left (f x \right )^{m} \ln \left (c \,x^{n}\right ) b \,d^{2}-12 x^{3} \left (f x \right )^{m} \ln \left (c \,x^{n}\right ) b \,e^{2}-x^{3} \left (f x \right )^{m} a \,e^{2} m^{5}-9 x^{3} \left (f x \right )^{m} a \,e^{2} m^{4}-31 x^{3} \left (f x \right )^{m} a \,e^{2} m^{3}-x \left (f x \right )^{m} a \,d^{2} m^{5}-51 x^{3} \left (f x \right )^{m} a \,e^{2} m^{2}-11 x \left (f x \right )^{m} a \,d^{2} m^{4}-40 x^{3} \left (f x \right )^{m} a \,e^{2} m +4 x^{3} \left (f x \right )^{m} b \,e^{2} n -47 x \left (f x \right )^{m} a \,d^{2} m^{3}-97 x \left (f x \right )^{m} a \,d^{2} m^{2}-36 x^{2} \left (f x \right )^{m} a d e -96 x \left (f x \right )^{m} a \,d^{2} m +36 x \left (f x \right )^{m} b \,d^{2} n -x^{3} \left (f x \right )^{m} \ln \left (c \,x^{n}\right ) b \,e^{2} m^{5}-36 x^{2} \left (f x \right )^{m} \ln \left (c \,x^{n}\right ) b d e -9 x^{3} \left (f x \right )^{m} \ln \left (c \,x^{n}\right ) b \,e^{2} m^{4}+x^{3} \left (f x \right )^{m} b \,e^{2} m^{4} n -31 x^{3} \left (f x \right )^{m} \ln \left (c \,x^{n}\right ) b \,e^{2} m^{3}+6 x^{3} \left (f x \right )^{m} b \,e^{2} m^{3} n -2 x^{2} \left (f x \right )^{m} a d e \,m^{5}-x \left (f x \right )^{m} \ln \left (c \,x^{n}\right ) b \,d^{2} m^{5}-51 x^{3} \left (f x \right )^{m} \ln \left (c \,x^{n}\right ) b \,e^{2} m^{2}+13 x^{3} \left (f x \right )^{m} b \,e^{2} m^{2} n -20 x^{2} \left (f x \right )^{m} a d e \,m^{4}-11 x \left (f x \right )^{m} \ln \left (c \,x^{n}\right ) b \,d^{2} m^{4}+x \left (f x \right )^{m} b \,d^{2} m^{4} n -40 x^{3} \left (f x \right )^{m} \ln \left (c \,x^{n}\right ) b \,e^{2} m +12 x^{3} \left (f x \right )^{m} b \,e^{2} m n -76 x^{2} \left (f x \right )^{m} a d e \,m^{3}-47 x \left (f x \right )^{m} \ln \left (c \,x^{n}\right ) b \,d^{2} m^{3}+10 x \left (f x \right )^{m} b \,d^{2} m^{3} n -136 x^{2} \left (f x \right )^{m} a d e \,m^{2}-97 x \left (f x \right )^{m} \ln \left (c \,x^{n}\right ) b \,d^{2} m^{2}+37 x \left (f x \right )^{m} b \,d^{2} m^{2} n -114 x^{2} \left (f x \right )^{m} a d e m +18 x^{2} \left (f x \right )^{m} b d e n -96 x \left (f x \right )^{m} \ln \left (c \,x^{n}\right ) b \,d^{2} m +60 x \left (f x \right )^{m} b \,d^{2} m n -2 x^{2} \left (f x \right )^{m} \ln \left (c \,x^{n}\right ) b d e \,m^{5}-20 x^{2} \left (f x \right )^{m} \ln \left (c \,x^{n}\right ) b d e \,m^{4}+2 x^{2} \left (f x \right )^{m} b d e \,m^{4} n -76 x^{2} \left (f x \right )^{m} \ln \left (c \,x^{n}\right ) b d e \,m^{3}+16 x^{2} \left (f x \right )^{m} b d e \,m^{3} n -136 x^{2} \left (f x \right )^{m} \ln \left (c \,x^{n}\right ) b d e \,m^{2}+44 x^{2} \left (f x \right )^{m} b d e \,m^{2} n -114 x^{2} \left (f x \right )^{m} \ln \left (c \,x^{n}\right ) b d e m +48 x^{2} \left (f x \right )^{m} b d e m n}{\left (3+m \right )^{2} \left (1+m \right )^{2} \left (2+m \right )^{2}}\) | \(911\) |
| risch | \(\text {Expression too large to display}\) | \(2636\) |
-(-12*x^3*(f*x)^m*a*e^2-36*x*(f*x)^m*a*d^2-36*x*(f*x)^m*ln(c*x^n)*b*d^2-12 *x^3*(f*x)^m*ln(c*x^n)*b*e^2-x^3*(f*x)^m*a*e^2*m^5-9*x^3*(f*x)^m*a*e^2*m^4 -31*x^3*(f*x)^m*a*e^2*m^3-x*(f*x)^m*a*d^2*m^5-51*x^3*(f*x)^m*a*e^2*m^2-11* x*(f*x)^m*a*d^2*m^4-40*x^3*(f*x)^m*a*e^2*m+4*x^3*(f*x)^m*b*e^2*n-47*x*(f*x )^m*a*d^2*m^3-97*x*(f*x)^m*a*d^2*m^2-36*x^2*(f*x)^m*a*d*e-96*x*(f*x)^m*a*d ^2*m+36*x*(f*x)^m*b*d^2*n-x^3*(f*x)^m*ln(c*x^n)*b*e^2*m^5-36*x^2*(f*x)^m*l n(c*x^n)*b*d*e-9*x^3*(f*x)^m*ln(c*x^n)*b*e^2*m^4+x^3*(f*x)^m*b*e^2*m^4*n-3 1*x^3*(f*x)^m*ln(c*x^n)*b*e^2*m^3+6*x^3*(f*x)^m*b*e^2*m^3*n-2*x^2*(f*x)^m* a*d*e*m^5-x*(f*x)^m*ln(c*x^n)*b*d^2*m^5-51*x^3*(f*x)^m*ln(c*x^n)*b*e^2*m^2 +13*x^3*(f*x)^m*b*e^2*m^2*n-20*x^2*(f*x)^m*a*d*e*m^4-11*x*(f*x)^m*ln(c*x^n )*b*d^2*m^4+x*(f*x)^m*b*d^2*m^4*n-40*x^3*(f*x)^m*ln(c*x^n)*b*e^2*m+12*x^3* (f*x)^m*b*e^2*m*n-76*x^2*(f*x)^m*a*d*e*m^3-47*x*(f*x)^m*ln(c*x^n)*b*d^2*m^ 3+10*x*(f*x)^m*b*d^2*m^3*n-136*x^2*(f*x)^m*a*d*e*m^2-97*x*(f*x)^m*ln(c*x^n )*b*d^2*m^2+37*x*(f*x)^m*b*d^2*m^2*n-114*x^2*(f*x)^m*a*d*e*m+18*x^2*(f*x)^ m*b*d*e*n-96*x*(f*x)^m*ln(c*x^n)*b*d^2*m+60*x*(f*x)^m*b*d^2*m*n-2*x^2*(f*x )^m*ln(c*x^n)*b*d*e*m^5-20*x^2*(f*x)^m*ln(c*x^n)*b*d*e*m^4+2*x^2*(f*x)^m*b *d*e*m^4*n-76*x^2*(f*x)^m*ln(c*x^n)*b*d*e*m^3+16*x^2*(f*x)^m*b*d*e*m^3*n-1 36*x^2*(f*x)^m*ln(c*x^n)*b*d*e*m^2+44*x^2*(f*x)^m*b*d*e*m^2*n-114*x^2*(f*x )^m*ln(c*x^n)*b*d*e*m+48*x^2*(f*x)^m*b*d*e*m*n)/(3+m)^2/(1+m)^2/(2+m)^2
Leaf count of result is larger than twice the leaf count of optimal. 633 vs. \(2 (153) = 306\).
Time = 0.32 (sec) , antiderivative size = 633, normalized size of antiderivative = 4.14 \[ \int (f x)^m (d+e x)^2 \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {{\left ({\left (a e^{2} m^{5} + 9 \, a e^{2} m^{4} + 31 \, a e^{2} m^{3} + 51 \, a e^{2} m^{2} + 40 \, a e^{2} m + 12 \, a e^{2} - {\left (b e^{2} m^{4} + 6 \, b e^{2} m^{3} + 13 \, b e^{2} m^{2} + 12 \, b e^{2} m + 4 \, b e^{2}\right )} n\right )} x^{3} + 2 \, {\left (a d e m^{5} + 10 \, a d e m^{4} + 38 \, a d e m^{3} + 68 \, a d e m^{2} + 57 \, a d e m + 18 \, a d e - {\left (b d e m^{4} + 8 \, b d e m^{3} + 22 \, b d e m^{2} + 24 \, b d e m + 9 \, b d e\right )} n\right )} x^{2} + {\left (a d^{2} m^{5} + 11 \, a d^{2} m^{4} + 47 \, a d^{2} m^{3} + 97 \, a d^{2} m^{2} + 96 \, a d^{2} m + 36 \, a d^{2} - {\left (b d^{2} m^{4} + 10 \, b d^{2} m^{3} + 37 \, b d^{2} m^{2} + 60 \, b d^{2} m + 36 \, b d^{2}\right )} n\right )} x + {\left ({\left (b e^{2} m^{5} + 9 \, b e^{2} m^{4} + 31 \, b e^{2} m^{3} + 51 \, b e^{2} m^{2} + 40 \, b e^{2} m + 12 \, b e^{2}\right )} x^{3} + 2 \, {\left (b d e m^{5} + 10 \, b d e m^{4} + 38 \, b d e m^{3} + 68 \, b d e m^{2} + 57 \, b d e m + 18 \, b d e\right )} x^{2} + {\left (b d^{2} m^{5} + 11 \, b d^{2} m^{4} + 47 \, b d^{2} m^{3} + 97 \, b d^{2} m^{2} + 96 \, b d^{2} m + 36 \, b d^{2}\right )} x\right )} \log \left (c\right ) + {\left ({\left (b e^{2} m^{5} + 9 \, b e^{2} m^{4} + 31 \, b e^{2} m^{3} + 51 \, b e^{2} m^{2} + 40 \, b e^{2} m + 12 \, b e^{2}\right )} n x^{3} + 2 \, {\left (b d e m^{5} + 10 \, b d e m^{4} + 38 \, b d e m^{3} + 68 \, b d e m^{2} + 57 \, b d e m + 18 \, b d e\right )} n x^{2} + {\left (b d^{2} m^{5} + 11 \, b d^{2} m^{4} + 47 \, b d^{2} m^{3} + 97 \, b d^{2} m^{2} + 96 \, b d^{2} m + 36 \, b d^{2}\right )} n x\right )} \log \left (x\right )\right )} e^{\left (m \log \left (f\right ) + m \log \left (x\right )\right )}}{m^{6} + 12 \, m^{5} + 58 \, m^{4} + 144 \, m^{3} + 193 \, m^{2} + 132 \, m + 36} \]
((a*e^2*m^5 + 9*a*e^2*m^4 + 31*a*e^2*m^3 + 51*a*e^2*m^2 + 40*a*e^2*m + 12* a*e^2 - (b*e^2*m^4 + 6*b*e^2*m^3 + 13*b*e^2*m^2 + 12*b*e^2*m + 4*b*e^2)*n) *x^3 + 2*(a*d*e*m^5 + 10*a*d*e*m^4 + 38*a*d*e*m^3 + 68*a*d*e*m^2 + 57*a*d* e*m + 18*a*d*e - (b*d*e*m^4 + 8*b*d*e*m^3 + 22*b*d*e*m^2 + 24*b*d*e*m + 9* b*d*e)*n)*x^2 + (a*d^2*m^5 + 11*a*d^2*m^4 + 47*a*d^2*m^3 + 97*a*d^2*m^2 + 96*a*d^2*m + 36*a*d^2 - (b*d^2*m^4 + 10*b*d^2*m^3 + 37*b*d^2*m^2 + 60*b*d^ 2*m + 36*b*d^2)*n)*x + ((b*e^2*m^5 + 9*b*e^2*m^4 + 31*b*e^2*m^3 + 51*b*e^2 *m^2 + 40*b*e^2*m + 12*b*e^2)*x^3 + 2*(b*d*e*m^5 + 10*b*d*e*m^4 + 38*b*d*e *m^3 + 68*b*d*e*m^2 + 57*b*d*e*m + 18*b*d*e)*x^2 + (b*d^2*m^5 + 11*b*d^2*m ^4 + 47*b*d^2*m^3 + 97*b*d^2*m^2 + 96*b*d^2*m + 36*b*d^2)*x)*log(c) + ((b* e^2*m^5 + 9*b*e^2*m^4 + 31*b*e^2*m^3 + 51*b*e^2*m^2 + 40*b*e^2*m + 12*b*e^ 2)*n*x^3 + 2*(b*d*e*m^5 + 10*b*d*e*m^4 + 38*b*d*e*m^3 + 68*b*d*e*m^2 + 57* b*d*e*m + 18*b*d*e)*n*x^2 + (b*d^2*m^5 + 11*b*d^2*m^4 + 47*b*d^2*m^3 + 97* b*d^2*m^2 + 96*b*d^2*m + 36*b*d^2)*n*x)*log(x))*e^(m*log(f) + m*log(x))/(m ^6 + 12*m^5 + 58*m^4 + 144*m^3 + 193*m^2 + 132*m + 36)
Leaf count of result is larger than twice the leaf count of optimal. 2791 vs. \(2 (146) = 292\).
Time = 3.51 (sec) , antiderivative size = 2791, normalized size of antiderivative = 18.24 \[ \int (f x)^m (d+e x)^2 \left (a+b \log \left (c x^n\right )\right ) \, dx=\text {Too large to display} \]
Piecewise(((-a*d**2/(2*x**2) - 2*a*d*e/x + a*e**2*log(x) + b*d**2*(-n/(4*x **2) - log(c*x**n)/(2*x**2)) + 2*b*d*e*(-n/x - log(c*x**n)/x) - b*e**2*Pie cewise((-log(c)*log(x), Eq(n, 0)), (-log(c*x**n)**2/(2*n), True)))/f**3, E q(m, -3)), ((-a*d**2/x + 2*a*d*e*log(c*x**n)/n + a*e**2*x - b*d**2*n/x - b *d**2*log(c*x**n)/x + b*d*e*log(c*x**n)**2/n - b*e**2*n*x + b*e**2*x*log(c *x**n))/f**2, Eq(m, -2)), ((a*d**2*log(c*x**n)/n + 2*a*d*e*x + a*e**2*x**2 /2 + b*d**2*log(c*x**n)**2/(2*n) - 2*b*d*e*n*x + 2*b*d*e*x*log(c*x**n) - b *e**2*n*x**2/4 + b*e**2*x**2*log(c*x**n)/2)/f, Eq(m, -1)), (a*d**2*m**5*x* (f*x)**m/(m**6 + 12*m**5 + 58*m**4 + 144*m**3 + 193*m**2 + 132*m + 36) + 1 1*a*d**2*m**4*x*(f*x)**m/(m**6 + 12*m**5 + 58*m**4 + 144*m**3 + 193*m**2 + 132*m + 36) + 47*a*d**2*m**3*x*(f*x)**m/(m**6 + 12*m**5 + 58*m**4 + 144*m **3 + 193*m**2 + 132*m + 36) + 97*a*d**2*m**2*x*(f*x)**m/(m**6 + 12*m**5 + 58*m**4 + 144*m**3 + 193*m**2 + 132*m + 36) + 96*a*d**2*m*x*(f*x)**m/(m** 6 + 12*m**5 + 58*m**4 + 144*m**3 + 193*m**2 + 132*m + 36) + 36*a*d**2*x*(f *x)**m/(m**6 + 12*m**5 + 58*m**4 + 144*m**3 + 193*m**2 + 132*m + 36) + 2*a *d*e*m**5*x**2*(f*x)**m/(m**6 + 12*m**5 + 58*m**4 + 144*m**3 + 193*m**2 + 132*m + 36) + 20*a*d*e*m**4*x**2*(f*x)**m/(m**6 + 12*m**5 + 58*m**4 + 144* m**3 + 193*m**2 + 132*m + 36) + 76*a*d*e*m**3*x**2*(f*x)**m/(m**6 + 12*m** 5 + 58*m**4 + 144*m**3 + 193*m**2 + 132*m + 36) + 136*a*d*e*m**2*x**2*(f*x )**m/(m**6 + 12*m**5 + 58*m**4 + 144*m**3 + 193*m**2 + 132*m + 36) + 11...
Time = 0.24 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.27 \[ \int (f x)^m (d+e x)^2 \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {b e^{2} f^{m} x^{3} x^{m} \log \left (c x^{n}\right )}{m + 3} + \frac {a e^{2} f^{m} x^{3} x^{m}}{m + 3} - \frac {b e^{2} f^{m} n x^{3} x^{m}}{{\left (m + 3\right )}^{2}} + \frac {2 \, b d e f^{m} x^{2} x^{m} \log \left (c x^{n}\right )}{m + 2} + \frac {2 \, a d e f^{m} x^{2} x^{m}}{m + 2} - \frac {2 \, b d e f^{m} n x^{2} x^{m}}{{\left (m + 2\right )}^{2}} - \frac {b d^{2} f^{m} n x x^{m}}{{\left (m + 1\right )}^{2}} + \frac {\left (f x\right )^{m + 1} b d^{2} \log \left (c x^{n}\right )}{f {\left (m + 1\right )}} + \frac {\left (f x\right )^{m + 1} a d^{2}}{f {\left (m + 1\right )}} \]
b*e^2*f^m*x^3*x^m*log(c*x^n)/(m + 3) + a*e^2*f^m*x^3*x^m/(m + 3) - b*e^2*f ^m*n*x^3*x^m/(m + 3)^2 + 2*b*d*e*f^m*x^2*x^m*log(c*x^n)/(m + 2) + 2*a*d*e* f^m*x^2*x^m/(m + 2) - 2*b*d*e*f^m*n*x^2*x^m/(m + 2)^2 - b*d^2*f^m*n*x*x^m/ (m + 1)^2 + (f*x)^(m + 1)*b*d^2*log(c*x^n)/(f*(m + 1)) + (f*x)^(m + 1)*a*d ^2/(f*(m + 1))
Leaf count of result is larger than twice the leaf count of optimal. 374 vs. \(2 (153) = 306\).
Time = 0.34 (sec) , antiderivative size = 374, normalized size of antiderivative = 2.44 \[ \int (f x)^m (d+e x)^2 \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {b e^{2} f^{2} f^{m} x^{3} x^{m} \log \left (c\right )}{f^{2} m + 3 \, f^{2}} + \frac {b e^{2} f^{m} m n x^{3} x^{m} \log \left (x\right )}{m^{2} + 6 \, m + 9} + \frac {a e^{2} f^{2} f^{m} x^{3} x^{m}}{f^{2} m + 3 \, f^{2}} + \frac {2 \, b d e f^{m} m n x^{2} x^{m} \log \left (x\right )}{m^{2} + 4 \, m + 4} + \frac {3 \, b e^{2} f^{m} n x^{3} x^{m} \log \left (x\right )}{m^{2} + 6 \, m + 9} - \frac {b e^{2} f^{m} n x^{3} x^{m}}{m^{2} + 6 \, m + 9} + \frac {b d^{2} f^{m} m n x x^{m} \log \left (x\right )}{m^{2} + 2 \, m + 1} + \frac {4 \, b d e f^{m} n x^{2} x^{m} \log \left (x\right )}{m^{2} + 4 \, m + 4} - \frac {2 \, b d e f^{m} n x^{2} x^{m}}{m^{2} + 4 \, m + 4} + \frac {2 \, b d e f^{m} x^{2} x^{m} \log \left (c\right )}{m + 2} + \frac {b d^{2} f^{m} n x x^{m} \log \left (x\right )}{m^{2} + 2 \, m + 1} - \frac {b d^{2} f^{m} n x x^{m}}{m^{2} + 2 \, m + 1} + \frac {2 \, a d e f^{m} x^{2} x^{m}}{m + 2} + \frac {\left (f x\right )^{m} b d^{2} x \log \left (c\right )}{m + 1} + \frac {\left (f x\right )^{m} a d^{2} x}{m + 1} \]
b*e^2*f^2*f^m*x^3*x^m*log(c)/(f^2*m + 3*f^2) + b*e^2*f^m*m*n*x^3*x^m*log(x )/(m^2 + 6*m + 9) + a*e^2*f^2*f^m*x^3*x^m/(f^2*m + 3*f^2) + 2*b*d*e*f^m*m* n*x^2*x^m*log(x)/(m^2 + 4*m + 4) + 3*b*e^2*f^m*n*x^3*x^m*log(x)/(m^2 + 6*m + 9) - b*e^2*f^m*n*x^3*x^m/(m^2 + 6*m + 9) + b*d^2*f^m*m*n*x*x^m*log(x)/( m^2 + 2*m + 1) + 4*b*d*e*f^m*n*x^2*x^m*log(x)/(m^2 + 4*m + 4) - 2*b*d*e*f^ m*n*x^2*x^m/(m^2 + 4*m + 4) + 2*b*d*e*f^m*x^2*x^m*log(c)/(m + 2) + b*d^2*f ^m*n*x*x^m*log(x)/(m^2 + 2*m + 1) - b*d^2*f^m*n*x*x^m/(m^2 + 2*m + 1) + 2* a*d*e*f^m*x^2*x^m/(m + 2) + (f*x)^m*b*d^2*x*log(c)/(m + 1) + (f*x)^m*a*d^2 *x/(m + 1)
Timed out. \[ \int (f x)^m (d+e x)^2 \left (a+b \log \left (c x^n\right )\right ) \, dx=\int {\left (f\,x\right )}^m\,\left (a+b\,\ln \left (c\,x^n\right )\right )\,{\left (d+e\,x\right )}^2 \,d x \]