Integrand size = 25, antiderivative size = 153 \[ \int (f x)^m \left (d+e x^2\right )^2 \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {b d^2 n (f x)^{1+m}}{f (1+m)^2}-\frac {2 b d e n (f x)^{3+m}}{f^3 (3+m)^2}-\frac {b e^2 n (f x)^{5+m}}{f^5 (5+m)^2}+\frac {d^2 (f x)^{1+m} \left (a+b \log \left (c x^n\right )\right )}{f (1+m)}+\frac {2 d e (f x)^{3+m} \left (a+b \log \left (c x^n\right )\right )}{f^3 (3+m)}+\frac {e^2 (f x)^{5+m} \left (a+b \log \left (c x^n\right )\right )}{f^5 (5+m)} \]
-b*d^2*n*(f*x)^(1+m)/f/(1+m)^2-2*b*d*e*n*(f*x)^(3+m)/f^3/(3+m)^2-b*e^2*n*( f*x)^(5+m)/f^5/(5+m)^2+d^2*(f*x)^(1+m)*(a+b*ln(c*x^n))/f/(1+m)+2*d*e*(f*x) ^(3+m)*(a+b*ln(c*x^n))/f^3/(3+m)+e^2*(f*x)^(5+m)*(a+b*ln(c*x^n))/f^5/(5+m)
Time = 0.10 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.73 \[ \int (f x)^m \left (d+e x^2\right )^2 \left (a+b \log \left (c x^n\right )\right ) \, dx=x (f x)^m \left (-\frac {b d^2 n}{(1+m)^2}-\frac {2 b d e n x^2}{(3+m)^2}-\frac {b e^2 n x^4}{(5+m)^2}+\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{1+m}+\frac {2 d e x^2 \left (a+b \log \left (c x^n\right )\right )}{3+m}+\frac {e^2 x^4 \left (a+b \log \left (c x^n\right )\right )}{5+m}\right ) \]
x*(f*x)^m*(-((b*d^2*n)/(1 + m)^2) - (2*b*d*e*n*x^2)/(3 + m)^2 - (b*e^2*n*x ^4)/(5 + m)^2 + (d^2*(a + b*Log[c*x^n]))/(1 + m) + (2*d*e*x^2*(a + b*Log[c *x^n]))/(3 + m) + (e^2*x^4*(a + b*Log[c*x^n]))/(5 + m))
Time = 0.45 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.20, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {2792, 27, 1433, 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.
| \(\displaystyle \int \left (d+e x^2\right )^2 (f x)^m \left (a+b \log \left (c x^n\right )\right ) \, dx\) |
| \(\Big \downarrow \) 2792 |
| \(\displaystyle -b n \int \frac {(f x)^m \left (e^2 (m+1) (m+3) x^4+2 d e (m+1) (m+5) x^2+d^2 (m+3) (m+5)\right )}{m^3+9 m^2+23 m+15}dx+\frac {d^2 (f x)^{m+1} \left (a+b \log \left (c x^n\right )\right )}{f (m+1)}+\frac {2 d e (f x)^{m+3} \left (a+b \log \left (c x^n\right )\right )}{f^3 (m+3)}+\frac {e^2 (f x)^{m+5} \left (a+b \log \left (c x^n\right )\right )}{f^5 (m+5)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {b n \int (f x)^m \left (e^2 (m+1) (m+3) x^4+2 d e (m+1) (m+5) x^2+d^2 (m+3) (m+5)\right )dx}{m^3+9 m^2+23 m+15}+\frac {d^2 (f x)^{m+1} \left (a+b \log \left (c x^n\right )\right )}{f (m+1)}+\frac {2 d e (f x)^{m+3} \left (a+b \log \left (c x^n\right )\right )}{f^3 (m+3)}+\frac {e^2 (f x)^{m+5} \left (a+b \log \left (c x^n\right )\right )}{f^5 (m+5)}\) |
| \(\Big \downarrow \) 1433 |
| \(\displaystyle -\frac {b n \int \left (d^2 (m+3) (m+5) (f x)^m+\frac {2 d e (m+1) (m+5) (f x)^{m+2}}{f^2}+\frac {e^2 (m+1) (m+3) (f x)^{m+4}}{f^4}\right )dx}{m^3+9 m^2+23 m+15}+\frac {d^2 (f x)^{m+1} \left (a+b \log \left (c x^n\right )\right )}{f (m+1)}+\frac {2 d e (f x)^{m+3} \left (a+b \log \left (c x^n\right )\right )}{f^3 (m+3)}+\frac {e^2 (f x)^{m+5} \left (a+b \log \left (c x^n\right )\right )}{f^5 (m+5)}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {d^2 (f x)^{m+1} \left (a+b \log \left (c x^n\right )\right )}{f (m+1)}+\frac {2 d e (f x)^{m+3} \left (a+b \log \left (c x^n\right )\right )}{f^3 (m+3)}+\frac {e^2 (f x)^{m+5} \left (a+b \log \left (c x^n\right )\right )}{f^5 (m+5)}-\frac {b n \left (\frac {d^2 (m+3) (m+5) (f x)^{m+1}}{f (m+1)}+\frac {2 d e (m+1) (m+5) (f x)^{m+3}}{f^3 (m+3)}+\frac {e^2 (m+1) (m+3) (f x)^{m+5}}{f^5 (m+5)}\right )}{m^3+9 m^2+23 m+15}\) |
-((b*n*((d^2*(3 + m)*(5 + m)*(f*x)^(1 + m))/(f*(1 + m)) + (2*d*e*(1 + m)*( 5 + m)*(f*x)^(3 + m))/(f^3*(3 + m)) + (e^2*(1 + m)*(3 + m)*(f*x)^(5 + m))/ (f^5*(5 + m))))/(15 + 23*m + 9*m^2 + m^3)) + (d^2*(f*x)^(1 + m)*(a + b*Log [c*x^n]))/(f*(1 + m)) + (2*d*e*(f*x)^(3 + m)*(a + b*Log[c*x^n]))/(f^3*(3 + m)) + (e^2*(f*x)^(5 + m)*(a + b*Log[c*x^n]))/(f^5*(5 + m))
3.4.19.3.1 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_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d*x)^m*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[p, 0] && (EqQ[p, 1] || !IntegerQ[(m + 1)/2])
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. \(915\) vs. \(2(153)=306\).
Time = 5.72 (sec) , antiderivative size = 916, normalized size of antiderivative = 5.99
| method | result | size |
| parallelrisch | \(-\frac {-225 x \left (f x \right )^{m} a \,d^{2}-150 x^{3} \left (f x \right )^{m} a d e -x^{5} \left (f x \right )^{m} a \,e^{2} m^{5}-13 x^{5} \left (f x \right )^{m} a \,e^{2} m^{4}-62 x^{5} \left (f x \right )^{m} a \,e^{2} m^{3}-134 x^{5} \left (f x \right )^{m} a \,e^{2} m^{2}-129 x^{5} \left (f x \right )^{m} a \,e^{2} m +9 x^{5} \left (f x \right )^{m} b \,e^{2} n -45 e^{2} b \ln \left (c \,x^{n}\right ) \left (f x \right )^{m} x^{5}-225 x \left (f x \right )^{m} \ln \left (c \,x^{n}\right ) b \,d^{2}-x \left (f x \right )^{m} a \,d^{2} m^{5}-17 x \left (f x \right )^{m} a \,d^{2} m^{4}-110 x \left (f x \right )^{m} a \,d^{2} m^{3}-334 x \left (f x \right )^{m} a \,d^{2} m^{2}-465 x \left (f x \right )^{m} a \,d^{2} m +225 x \left (f x \right )^{m} b \,d^{2} n -2 x^{3} \left (f x \right )^{m} \ln \left (c \,x^{n}\right ) b d e \,m^{5}-30 x^{3} \left (f x \right )^{m} \ln \left (c \,x^{n}\right ) b d e \,m^{4}+2 x^{3} \left (f x \right )^{m} b d e \,m^{4} n -164 x^{3} \left (f x \right )^{m} \ln \left (c \,x^{n}\right ) b d e \,m^{3}+24 x^{3} \left (f x \right )^{m} b d e \,m^{3} n -396 x^{3} \left (f x \right )^{m} \ln \left (c \,x^{n}\right ) b d e \,m^{2}+92 x^{3} \left (f x \right )^{m} b d e \,m^{2} n -410 x^{3} \left (f x \right )^{m} \ln \left (c \,x^{n}\right ) b d e m +120 x^{3} \left (f x \right )^{m} b d e m n -x \left (f x \right )^{m} \ln \left (c \,x^{n}\right ) b \,d^{2} m^{5}-17 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 -110 x \left (f x \right )^{m} \ln \left (c \,x^{n}\right ) b \,d^{2} m^{3}+16 x \left (f x \right )^{m} b \,d^{2} m^{3} n -334 x \left (f x \right )^{m} \ln \left (c \,x^{n}\right ) b \,d^{2} m^{2}+94 x \left (f x \right )^{m} b \,d^{2} m^{2} n -465 x \left (f x \right )^{m} \ln \left (c \,x^{n}\right ) b \,d^{2} m +240 x \left (f x \right )^{m} b \,d^{2} m n -45 x^{5} \left (f x \right )^{m} a \,e^{2}-13 x^{5} \left (f x \right )^{m} \ln \left (c \,x^{n}\right ) b \,e^{2} m^{4}+x^{5} \left (f x \right )^{m} b \,e^{2} m^{4} n -62 x^{5} \left (f x \right )^{m} \ln \left (c \,x^{n}\right ) b \,e^{2} m^{3}+8 x^{5} \left (f x \right )^{m} b \,e^{2} m^{3} n -134 x^{5} \left (f x \right )^{m} \ln \left (c \,x^{n}\right ) b \,e^{2} m^{2}+22 x^{5} \left (f x \right )^{m} b \,e^{2} m^{2} n -150 b d e \ln \left (c \,x^{n}\right ) \left (f x \right )^{m} x^{3}-2 x^{3} \left (f x \right )^{m} a d e \,m^{5}-129 x^{5} \left (f x \right )^{m} \ln \left (c \,x^{n}\right ) b \,e^{2} m +24 x^{5} \left (f x \right )^{m} b \,e^{2} m n -30 x^{3} \left (f x \right )^{m} a d e \,m^{4}-164 x^{3} \left (f x \right )^{m} a d e \,m^{3}-396 x^{3} \left (f x \right )^{m} a d e \,m^{2}-410 x^{3} \left (f x \right )^{m} a d e m +50 x^{3} \left (f x \right )^{m} b d e n -x^{5} \left (f x \right )^{m} \ln \left (c \,x^{n}\right ) b \,e^{2} m^{5}}{\left (m^{2}+10 m +25\right ) \left (3+m \right )^{2} \left (1+m \right )^{2}}\) | \(916\) |
| risch | \(\text {Expression too large to display}\) | \(2724\) |
-(-225*x*(f*x)^m*a*d^2-150*x^3*(f*x)^m*a*d*e-x^5*(f*x)^m*a*e^2*m^5-13*x^5* (f*x)^m*a*e^2*m^4-62*x^5*(f*x)^m*a*e^2*m^3-134*x^5*(f*x)^m*a*e^2*m^2-129*x ^5*(f*x)^m*a*e^2*m+9*x^5*(f*x)^m*b*e^2*n-45*e^2*b*ln(c*x^n)*(f*x)^m*x^5-22 5*x*(f*x)^m*ln(c*x^n)*b*d^2-x*(f*x)^m*a*d^2*m^5-17*x*(f*x)^m*a*d^2*m^4-110 *x*(f*x)^m*a*d^2*m^3-334*x*(f*x)^m*a*d^2*m^2-465*x*(f*x)^m*a*d^2*m+225*x*( f*x)^m*b*d^2*n-2*x^3*(f*x)^m*ln(c*x^n)*b*d*e*m^5-30*x^3*(f*x)^m*ln(c*x^n)* b*d*e*m^4+2*x^3*(f*x)^m*b*d*e*m^4*n-164*x^3*(f*x)^m*ln(c*x^n)*b*d*e*m^3+24 *x^3*(f*x)^m*b*d*e*m^3*n-396*x^3*(f*x)^m*ln(c*x^n)*b*d*e*m^2+92*x^3*(f*x)^ m*b*d*e*m^2*n-410*x^3*(f*x)^m*ln(c*x^n)*b*d*e*m+120*x^3*(f*x)^m*b*d*e*m*n- x*(f*x)^m*ln(c*x^n)*b*d^2*m^5-17*x*(f*x)^m*ln(c*x^n)*b*d^2*m^4+x*(f*x)^m*b *d^2*m^4*n-110*x*(f*x)^m*ln(c*x^n)*b*d^2*m^3+16*x*(f*x)^m*b*d^2*m^3*n-334* x*(f*x)^m*ln(c*x^n)*b*d^2*m^2+94*x*(f*x)^m*b*d^2*m^2*n-465*x*(f*x)^m*ln(c* x^n)*b*d^2*m+240*x*(f*x)^m*b*d^2*m*n-45*x^5*(f*x)^m*a*e^2-13*x^5*(f*x)^m*l n(c*x^n)*b*e^2*m^4+x^5*(f*x)^m*b*e^2*m^4*n-62*x^5*(f*x)^m*ln(c*x^n)*b*e^2* m^3+8*x^5*(f*x)^m*b*e^2*m^3*n-134*x^5*(f*x)^m*ln(c*x^n)*b*e^2*m^2+22*x^5*( f*x)^m*b*e^2*m^2*n-150*b*d*e*ln(c*x^n)*(f*x)^m*x^3-2*x^3*(f*x)^m*a*d*e*m^5 -129*x^5*(f*x)^m*ln(c*x^n)*b*e^2*m+24*x^5*(f*x)^m*b*e^2*m*n-30*x^3*(f*x)^m *a*d*e*m^4-164*x^3*(f*x)^m*a*d*e*m^3-396*x^3*(f*x)^m*a*d*e*m^2-410*x^3*(f* x)^m*a*d*e*m+50*x^3*(f*x)^m*b*d*e*n-x^5*(f*x)^m*ln(c*x^n)*b*e^2*m^5)/(m^2+ 10*m+25)/(3+m)^2/(1+m)^2
Leaf count of result is larger than twice the leaf count of optimal. 633 vs. \(2 (153) = 306\).
Time = 0.30 (sec) , antiderivative size = 633, normalized size of antiderivative = 4.14 \[ \int (f x)^m \left (d+e x^2\right )^2 \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {{\left ({\left (a e^{2} m^{5} + 13 \, a e^{2} m^{4} + 62 \, a e^{2} m^{3} + 134 \, a e^{2} m^{2} + 129 \, a e^{2} m + 45 \, a e^{2} - {\left (b e^{2} m^{4} + 8 \, b e^{2} m^{3} + 22 \, b e^{2} m^{2} + 24 \, b e^{2} m + 9 \, b e^{2}\right )} n\right )} x^{5} + 2 \, {\left (a d e m^{5} + 15 \, a d e m^{4} + 82 \, a d e m^{3} + 198 \, a d e m^{2} + 205 \, a d e m + 75 \, a d e - {\left (b d e m^{4} + 12 \, b d e m^{3} + 46 \, b d e m^{2} + 60 \, b d e m + 25 \, b d e\right )} n\right )} x^{3} + {\left (a d^{2} m^{5} + 17 \, a d^{2} m^{4} + 110 \, a d^{2} m^{3} + 334 \, a d^{2} m^{2} + 465 \, a d^{2} m + 225 \, a d^{2} - {\left (b d^{2} m^{4} + 16 \, b d^{2} m^{3} + 94 \, b d^{2} m^{2} + 240 \, b d^{2} m + 225 \, b d^{2}\right )} n\right )} x + {\left ({\left (b e^{2} m^{5} + 13 \, b e^{2} m^{4} + 62 \, b e^{2} m^{3} + 134 \, b e^{2} m^{2} + 129 \, b e^{2} m + 45 \, b e^{2}\right )} x^{5} + 2 \, {\left (b d e m^{5} + 15 \, b d e m^{4} + 82 \, b d e m^{3} + 198 \, b d e m^{2} + 205 \, b d e m + 75 \, b d e\right )} x^{3} + {\left (b d^{2} m^{5} + 17 \, b d^{2} m^{4} + 110 \, b d^{2} m^{3} + 334 \, b d^{2} m^{2} + 465 \, b d^{2} m + 225 \, b d^{2}\right )} x\right )} \log \left (c\right ) + {\left ({\left (b e^{2} m^{5} + 13 \, b e^{2} m^{4} + 62 \, b e^{2} m^{3} + 134 \, b e^{2} m^{2} + 129 \, b e^{2} m + 45 \, b e^{2}\right )} n x^{5} + 2 \, {\left (b d e m^{5} + 15 \, b d e m^{4} + 82 \, b d e m^{3} + 198 \, b d e m^{2} + 205 \, b d e m + 75 \, b d e\right )} n x^{3} + {\left (b d^{2} m^{5} + 17 \, b d^{2} m^{4} + 110 \, b d^{2} m^{3} + 334 \, b d^{2} m^{2} + 465 \, b d^{2} m + 225 \, b d^{2}\right )} n x\right )} \log \left (x\right )\right )} e^{\left (m \log \left (f\right ) + m \log \left (x\right )\right )}}{m^{6} + 18 \, m^{5} + 127 \, m^{4} + 444 \, m^{3} + 799 \, m^{2} + 690 \, m + 225} \]
((a*e^2*m^5 + 13*a*e^2*m^4 + 62*a*e^2*m^3 + 134*a*e^2*m^2 + 129*a*e^2*m + 45*a*e^2 - (b*e^2*m^4 + 8*b*e^2*m^3 + 22*b*e^2*m^2 + 24*b*e^2*m + 9*b*e^2) *n)*x^5 + 2*(a*d*e*m^5 + 15*a*d*e*m^4 + 82*a*d*e*m^3 + 198*a*d*e*m^2 + 205 *a*d*e*m + 75*a*d*e - (b*d*e*m^4 + 12*b*d*e*m^3 + 46*b*d*e*m^2 + 60*b*d*e* m + 25*b*d*e)*n)*x^3 + (a*d^2*m^5 + 17*a*d^2*m^4 + 110*a*d^2*m^3 + 334*a*d ^2*m^2 + 465*a*d^2*m + 225*a*d^2 - (b*d^2*m^4 + 16*b*d^2*m^3 + 94*b*d^2*m^ 2 + 240*b*d^2*m + 225*b*d^2)*n)*x + ((b*e^2*m^5 + 13*b*e^2*m^4 + 62*b*e^2* m^3 + 134*b*e^2*m^2 + 129*b*e^2*m + 45*b*e^2)*x^5 + 2*(b*d*e*m^5 + 15*b*d* e*m^4 + 82*b*d*e*m^3 + 198*b*d*e*m^2 + 205*b*d*e*m + 75*b*d*e)*x^3 + (b*d^ 2*m^5 + 17*b*d^2*m^4 + 110*b*d^2*m^3 + 334*b*d^2*m^2 + 465*b*d^2*m + 225*b *d^2)*x)*log(c) + ((b*e^2*m^5 + 13*b*e^2*m^4 + 62*b*e^2*m^3 + 134*b*e^2*m^ 2 + 129*b*e^2*m + 45*b*e^2)*n*x^5 + 2*(b*d*e*m^5 + 15*b*d*e*m^4 + 82*b*d*e *m^3 + 198*b*d*e*m^2 + 205*b*d*e*m + 75*b*d*e)*n*x^3 + (b*d^2*m^5 + 17*b*d ^2*m^4 + 110*b*d^2*m^3 + 334*b*d^2*m^2 + 465*b*d^2*m + 225*b*d^2)*n*x)*log (x))*e^(m*log(f) + m*log(x))/(m^6 + 18*m^5 + 127*m^4 + 444*m^3 + 799*m^2 + 690*m + 225)
Leaf count of result is larger than twice the leaf count of optimal. 2820 vs. \(2 (146) = 292\).
Time = 5.45 (sec) , antiderivative size = 2820, normalized size of antiderivative = 18.43 \[ \int (f x)^m \left (d+e x^2\right )^2 \left (a+b \log \left (c x^n\right )\right ) \, dx=\text {Too large to display} \]
Piecewise(((-a*d**2/(4*x**4) - a*d*e/x**2 + a*e**2*log(x) + b*d**2*(-n/(16 *x**4) - log(c*x**n)/(4*x**4)) + 2*b*d*e*(-n/(4*x**2) - log(c*x**n)/(2*x** 2)) - b*e**2*Piecewise((-log(c)*log(x), Eq(n, 0)), (-log(c*x**n)**2/(2*n), True)))/f**5, Eq(m, -5)), ((-a*d**2/(2*x**2) + 2*a*d*e*log(c*x**n)/n + a* e**2*x**2/2 - b*d**2*n/(4*x**2) - b*d**2*log(c*x**n)/(2*x**2) + b*d*e*log( c*x**n)**2/n - b*e**2*n*x**2/4 + b*e**2*x**2*log(c*x**n)/2)/f**3, Eq(m, -3 )), ((a*d**2*log(c*x**n)/n + a*d*e*x**2 + a*e**2*x**4/4 + b*d**2*log(c*x** n)**2/(2*n) - b*d*e*n*x**2/2 + b*d*e*x**2*log(c*x**n) - b*e**2*n*x**4/16 + b*e**2*x**4*log(c*x**n)/4)/f, Eq(m, -1)), (a*d**2*m**5*x*(f*x)**m/(m**6 + 18*m**5 + 127*m**4 + 444*m**3 + 799*m**2 + 690*m + 225) + 17*a*d**2*m**4* x*(f*x)**m/(m**6 + 18*m**5 + 127*m**4 + 444*m**3 + 799*m**2 + 690*m + 225) + 110*a*d**2*m**3*x*(f*x)**m/(m**6 + 18*m**5 + 127*m**4 + 444*m**3 + 799* m**2 + 690*m + 225) + 334*a*d**2*m**2*x*(f*x)**m/(m**6 + 18*m**5 + 127*m** 4 + 444*m**3 + 799*m**2 + 690*m + 225) + 465*a*d**2*m*x*(f*x)**m/(m**6 + 1 8*m**5 + 127*m**4 + 444*m**3 + 799*m**2 + 690*m + 225) + 225*a*d**2*x*(f*x )**m/(m**6 + 18*m**5 + 127*m**4 + 444*m**3 + 799*m**2 + 690*m + 225) + 2*a *d*e*m**5*x**3*(f*x)**m/(m**6 + 18*m**5 + 127*m**4 + 444*m**3 + 799*m**2 + 690*m + 225) + 30*a*d*e*m**4*x**3*(f*x)**m/(m**6 + 18*m**5 + 127*m**4 + 4 44*m**3 + 799*m**2 + 690*m + 225) + 164*a*d*e*m**3*x**3*(f*x)**m/(m**6 + 1 8*m**5 + 127*m**4 + 444*m**3 + 799*m**2 + 690*m + 225) + 396*a*d*e*m**2...
Time = 0.20 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.27 \[ \int (f x)^m \left (d+e x^2\right )^2 \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {b e^{2} f^{m} x^{5} x^{m} \log \left (c x^{n}\right )}{m + 5} + \frac {a e^{2} f^{m} x^{5} x^{m}}{m + 5} - \frac {b e^{2} f^{m} n x^{5} x^{m}}{{\left (m + 5\right )}^{2}} + \frac {2 \, b d e f^{m} x^{3} x^{m} \log \left (c x^{n}\right )}{m + 3} + \frac {2 \, a d e f^{m} x^{3} x^{m}}{m + 3} - \frac {2 \, b d e f^{m} n x^{3} x^{m}}{{\left (m + 3\right )}^{2}} - \frac {b d^{2} f^{m} n x x^{m}}{{\left (m + 1\right )}^{2}} + \frac {\left (f x\right )^{m + 1} b d^{2} \log \left (c x^{n}\right )}{f {\left (m + 1\right )}} + \frac {\left (f x\right )^{m + 1} a d^{2}}{f {\left (m + 1\right )}} \]
b*e^2*f^m*x^5*x^m*log(c*x^n)/(m + 5) + a*e^2*f^m*x^5*x^m/(m + 5) - b*e^2*f ^m*n*x^5*x^m/(m + 5)^2 + 2*b*d*e*f^m*x^3*x^m*log(c*x^n)/(m + 3) + 2*a*d*e* f^m*x^3*x^m/(m + 3) - 2*b*d*e*f^m*n*x^3*x^m/(m + 3)^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. 396 vs. \(2 (153) = 306\).
Time = 0.37 (sec) , antiderivative size = 396, normalized size of antiderivative = 2.59 \[ \int (f x)^m \left (d+e x^2\right )^2 \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {b e^{2} f^{4} f^{m} x^{5} x^{m} \log \left (c\right )}{f^{4} m + 5 \, f^{4}} + \frac {a e^{2} f^{4} f^{m} x^{5} x^{m}}{f^{4} m + 5 \, f^{4}} + \frac {b e^{2} f^{m} m n x^{5} x^{m} \log \left (x\right )}{m^{2} + 10 \, m + 25} + \frac {5 \, b e^{2} f^{m} n x^{5} x^{m} \log \left (x\right )}{m^{2} + 10 \, m + 25} - \frac {b e^{2} f^{m} n x^{5} x^{m}}{m^{2} + 10 \, m + 25} + \frac {2 \, b d e f^{2} f^{m} x^{3} x^{m} \log \left (c\right )}{f^{2} m + 3 \, f^{2}} + \frac {2 \, b d e f^{m} m n x^{3} x^{m} \log \left (x\right )}{m^{2} + 6 \, m + 9} + \frac {2 \, a d e f^{2} f^{m} x^{3} x^{m}}{f^{2} m + 3 \, f^{2}} + \frac {6 \, b d e f^{m} n x^{3} x^{m} \log \left (x\right )}{m^{2} + 6 \, m + 9} - \frac {2 \, b d e f^{m} n x^{3} x^{m}}{m^{2} + 6 \, m + 9} + \frac {b d^{2} f^{m} m n x x^{m} \log \left (x\right )}{m^{2} + 2 \, m + 1} + \frac {b d^{2} f^{m} n x x^{m} \log \left (x\right )}{m^{2} + 2 \, m + 1} - \frac {b d^{2} f^{m} n x x^{m}}{m^{2} + 2 \, m + 1} + \frac {\left (f x\right )^{m} b d^{2} x \log \left (c\right )}{m + 1} + \frac {\left (f x\right )^{m} a d^{2} x}{m + 1} \]
b*e^2*f^4*f^m*x^5*x^m*log(c)/(f^4*m + 5*f^4) + a*e^2*f^4*f^m*x^5*x^m/(f^4* m + 5*f^4) + b*e^2*f^m*m*n*x^5*x^m*log(x)/(m^2 + 10*m + 25) + 5*b*e^2*f^m* n*x^5*x^m*log(x)/(m^2 + 10*m + 25) - b*e^2*f^m*n*x^5*x^m/(m^2 + 10*m + 25) + 2*b*d*e*f^2*f^m*x^3*x^m*log(c)/(f^2*m + 3*f^2) + 2*b*d*e*f^m*m*n*x^3*x^ m*log(x)/(m^2 + 6*m + 9) + 2*a*d*e*f^2*f^m*x^3*x^m/(f^2*m + 3*f^2) + 6*b*d *e*f^m*n*x^3*x^m*log(x)/(m^2 + 6*m + 9) - 2*b*d*e*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) + 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) + (f*x)^m*b*d^2*x* log(c)/(m + 1) + (f*x)^m*a*d^2*x/(m + 1)
Timed out. \[ \int (f x)^m \left (d+e x^2\right )^2 \left (a+b \log \left (c x^n\right )\right ) \, dx=\int {\left (f\,x\right )}^m\,{\left (e\,x^2+d\right )}^2\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \]