Integrand size = 26, antiderivative size = 353 \[ \int (a+b x)^m (c+d x)^{-4-m} (e+f x)^2 \, dx=-\frac {(d e-c f) (a d f (3+m)-b (d e+c f (2+m))) (a+b x)^{1+m} (c+d x)^{-3-m}}{b d^2 (b c-a d) (3+m)}+\frac {\left (a^2 d^2 f^2 \left (6+5 m+m^2\right )-2 a b d f (3+m) (d e+c f (1+m))+b^2 \left (2 d^2 e^2+2 c d e f (1+m)+c^2 f^2 \left (2+3 m+m^2\right )\right )\right ) (a+b x)^{1+m} (c+d x)^{-2-m}}{b d^2 (b c-a d)^2 (2+m) (3+m)}+\frac {\left (a^2 d^2 f^2 \left (6+5 m+m^2\right )-2 a b d f (3+m) (d e+c f (1+m))+b^2 \left (2 d^2 e^2+2 c d e f (1+m)+c^2 f^2 \left (2+3 m+m^2\right )\right )\right ) (a+b x)^{1+m} (c+d x)^{-1-m}}{d^2 (b c-a d)^3 (1+m) (2+m) (3+m)}-\frac {f (a+b x)^{1+m} (c+d x)^{-3-m} (e+f x)}{b d} \]
-(-c*f+d*e)*(a*d*f*(3+m)-b*(d*e+c*f*(2+m)))*(b*x+a)^(1+m)*(d*x+c)^(-3-m)/b /d^2/(-a*d+b*c)/(3+m)+(a^2*d^2*f^2*(m^2+5*m+6)-2*a*b*d*f*(3+m)*(d*e+c*f*(1 +m))+b^2*(2*d^2*e^2+2*c*d*e*f*(1+m)+c^2*f^2*(m^2+3*m+2)))*(b*x+a)^(1+m)*(d *x+c)^(-2-m)/b/d^2/(-a*d+b*c)^2/(2+m)/(3+m)+(a^2*d^2*f^2*(m^2+5*m+6)-2*a*b *d*f*(3+m)*(d*e+c*f*(1+m))+b^2*(2*d^2*e^2+2*c*d*e*f*(1+m)+c^2*f^2*(m^2+3*m +2)))*(b*x+a)^(1+m)*(d*x+c)^(-1-m)/d^2/(-a*d+b*c)^3/(1+m)/(2+m)/(3+m)-f*(b *x+a)^(1+m)*(d*x+c)^(-3-m)*(f*x+e)/b/d
Time = 0.21 (sec) , antiderivative size = 286, normalized size of antiderivative = 0.81 \[ \int (a+b x)^m (c+d x)^{-4-m} (e+f x)^2 \, dx=-\frac {(a+b x)^{1+m} (c+d x)^{-3-m} \left (b^2 \left (2 d^2 e^2 x^2+2 c d e x (e (3+m)+f (1+m) x)+c^2 \left (e^2 \left (6+5 m+m^2\right )+2 e f \left (3+4 m+m^2\right ) x+f^2 \left (2+3 m+m^2\right ) x^2\right )\right )-2 a b \left (c^2 f (e (3+m)+f (1+m) x)+d^2 e x (e (1+m)+f (3+m) x)+c d \left (e^2 \left (3+4 m+m^2\right )+2 e f \left (5+4 m+m^2\right ) x+f^2 \left (3+4 m+m^2\right ) x^2\right )\right )+a^2 \left (2 c^2 f^2+2 c d f (e (1+m)+f (3+m) x)+d^2 \left (e^2 \left (2+3 m+m^2\right )+2 e f \left (3+4 m+m^2\right ) x+f^2 \left (6+5 m+m^2\right ) x^2\right )\right )\right )}{(-b c+a d)^3 (1+m) (2+m) (3+m)} \]
-(((a + b*x)^(1 + m)*(c + d*x)^(-3 - m)*(b^2*(2*d^2*e^2*x^2 + 2*c*d*e*x*(e *(3 + m) + f*(1 + m)*x) + c^2*(e^2*(6 + 5*m + m^2) + 2*e*f*(3 + 4*m + m^2) *x + f^2*(2 + 3*m + m^2)*x^2)) - 2*a*b*(c^2*f*(e*(3 + m) + f*(1 + m)*x) + d^2*e*x*(e*(1 + m) + f*(3 + m)*x) + c*d*(e^2*(3 + 4*m + m^2) + 2*e*f*(5 + 4*m + m^2)*x + f^2*(3 + 4*m + m^2)*x^2)) + a^2*(2*c^2*f^2 + 2*c*d*f*(e*(1 + m) + f*(3 + m)*x) + d^2*(e^2*(2 + 3*m + m^2) + 2*e*f*(3 + 4*m + m^2)*x + f^2*(6 + 5*m + m^2)*x^2))))/((-(b*c) + a*d)^3*(1 + m)*(2 + m)*(3 + m)))
Time = 0.43 (sec) , antiderivative size = 270, normalized size of antiderivative = 0.76, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {101, 25, 88, 55, 48}
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.
| \(\displaystyle \int (e+f x)^2 (a+b x)^m (c+d x)^{-m-4} \, dx\) |
| \(\Big \downarrow \) 101 |
| \(\displaystyle -\frac {\int -(a+b x)^m (c+d x)^{-m-4} \left ((b c-a d) (m+2) x f^2+a (c f-d e (m+3)) f+b e (d e+c f (m+1))\right )dx}{b d}-\frac {f (e+f x) (a+b x)^{m+1} (c+d x)^{-m-3}}{b d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int (a+b x)^m (c+d x)^{-m-4} \left ((b c-a d) (m+2) x f^2+a (c f-d e (m+3)) f+b e (d e+c f (m+1))\right )dx}{b d}-\frac {f (e+f x) (a+b x)^{m+1} (c+d x)^{-m-3}}{b d}\) |
| \(\Big \downarrow \) 88 |
| \(\displaystyle \frac {\frac {\left (2 b d (a f (c f-d e (m+3))+b e (c f (m+1)+d e))+f^2 (m+2) (b c-a d) (b c (m+1)-a d (m+3))\right ) \int (a+b x)^m (c+d x)^{-m-3}dx}{d (m+3) (b c-a d)}+\frac {(a+b x)^{m+1} (d e-c f) (c+d x)^{-m-3} (-a d f (m+3)+b c f (m+2)+b d e)}{d (m+3) (b c-a d)}}{b d}-\frac {f (e+f x) (a+b x)^{m+1} (c+d x)^{-m-3}}{b d}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle \frac {\frac {\left (2 b d (a f (c f-d e (m+3))+b e (c f (m+1)+d e))+f^2 (m+2) (b c-a d) (b c (m+1)-a d (m+3))\right ) \left (\frac {b \int (a+b x)^m (c+d x)^{-m-2}dx}{(m+2) (b c-a d)}+\frac {(a+b x)^{m+1} (c+d x)^{-m-2}}{(m+2) (b c-a d)}\right )}{d (m+3) (b c-a d)}+\frac {(a+b x)^{m+1} (d e-c f) (c+d x)^{-m-3} (-a d f (m+3)+b c f (m+2)+b d e)}{d (m+3) (b c-a d)}}{b d}-\frac {f (e+f x) (a+b x)^{m+1} (c+d x)^{-m-3}}{b d}\) |
| \(\Big \downarrow \) 48 |
| \(\displaystyle \frac {\frac {\left (\frac {(a+b x)^{m+1} (c+d x)^{-m-2}}{(m+2) (b c-a d)}+\frac {b (a+b x)^{m+1} (c+d x)^{-m-1}}{(m+1) (m+2) (b c-a d)^2}\right ) \left (2 b d (a f (c f-d e (m+3))+b e (c f (m+1)+d e))+f^2 (m+2) (b c-a d) (b c (m+1)-a d (m+3))\right )}{d (m+3) (b c-a d)}+\frac {(a+b x)^{m+1} (d e-c f) (c+d x)^{-m-3} (-a d f (m+3)+b c f (m+2)+b d e)}{d (m+3) (b c-a d)}}{b d}-\frac {f (e+f x) (a+b x)^{m+1} (c+d x)^{-m-3}}{b d}\) |
-((f*(a + b*x)^(1 + m)*(c + d*x)^(-3 - m)*(e + f*x))/(b*d)) + (((d*e - c*f )*(b*d*e + b*c*f*(2 + m) - a*d*f*(3 + m))*(a + b*x)^(1 + m)*(c + d*x)^(-3 - m))/(d*(b*c - a*d)*(3 + m)) + (((b*c - a*d)*f^2*(2 + m)*(b*c*(1 + m) - a *d*(3 + m)) + 2*b*d*(b*e*(d*e + c*f*(1 + m)) + a*f*(c*f - d*e*(3 + m))))*( ((a + b*x)^(1 + m)*(c + d*x)^(-2 - m))/((b*c - a*d)*(2 + m)) + (b*(a + b*x )^(1 + m)*(c + d*x)^(-1 - m))/((b*c - a*d)^2*(1 + m)*(2 + m))))/(d*(b*c - a*d)*(3 + m)))/(b*d)
3.31.98.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp [(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{ a, b, c, d, m, n}, x] && EqQ[m + n + 2, 0] && NeQ[m, -1]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(S implify[m + n + 2]/((b*c - a*d)*(m + 1))) Int[(a + b*x)^Simplify[m + 1]*( c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && ILtQ[Simplify[m + n + 2], 0] && NeQ[m, -1] && !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[ c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (SumSimplerQ[m, 1] || !SumSimp lerQ[n, 1])
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^Simplify[p + 1], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && !RationalQ[p] && SumSimpl erQ[p, 1]
Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( p_), x_] :> Simp[b*(a + b*x)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 3))), x] + Simp[1/(d*f*(n + p + 3)) Int[(c + d*x)^n*(e + f*x)^p*Simp [a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f *(n + p + 4) - b*(d*e*(n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 3, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(740\) vs. \(2(353)=706\).
Time = 2.09 (sec) , antiderivative size = 741, normalized size of antiderivative = 2.10
| method | result | size |
| gosper | \(-\frac {\left (b x +a \right )^{1+m} \left (d x +c \right )^{-3-m} \left (a^{2} d^{2} f^{2} m^{2} x^{2}-2 a b c d \,f^{2} m^{2} x^{2}+b^{2} c^{2} f^{2} m^{2} x^{2}+2 a^{2} d^{2} e f \,m^{2} x +5 a^{2} d^{2} f^{2} m \,x^{2}-4 a b c d e f \,m^{2} x -8 a b c d \,f^{2} m \,x^{2}-2 a b \,d^{2} e f m \,x^{2}+2 b^{2} c^{2} e f \,m^{2} x +3 b^{2} c^{2} f^{2} m \,x^{2}+2 b^{2} c d e f m \,x^{2}+2 a^{2} c d \,f^{2} m x +a^{2} d^{2} e^{2} m^{2}+8 a^{2} d^{2} e f m x +6 a^{2} d^{2} f^{2} x^{2}-2 a b \,c^{2} f^{2} m x -2 a b c d \,e^{2} m^{2}-16 a b c d e f m x -6 a b c d \,f^{2} x^{2}-2 a b \,d^{2} e^{2} m x -6 a b \,d^{2} e f \,x^{2}+b^{2} c^{2} e^{2} m^{2}+8 b^{2} c^{2} e f m x +2 b^{2} c^{2} f^{2} x^{2}+2 b^{2} c d \,e^{2} m x +2 b^{2} c d e f \,x^{2}+2 b^{2} d^{2} e^{2} x^{2}+2 a^{2} c d e f m +6 a^{2} c d \,f^{2} x +3 a^{2} d^{2} e^{2} m +6 a^{2} d^{2} e f x -2 a b \,c^{2} e f m -2 a b \,c^{2} f^{2} x -8 a b c d \,e^{2} m -20 a b c d e f x -2 a b \,d^{2} e^{2} x +5 b^{2} c^{2} e^{2} m +6 b^{2} c^{2} e f x +6 b^{2} c d \,e^{2} x +2 a^{2} c^{2} f^{2}+2 a^{2} c d e f +2 a^{2} d^{2} e^{2}-6 a b \,c^{2} e f -6 a b c d \,e^{2}+6 b^{2} c^{2} e^{2}\right )}{a^{3} d^{3} m^{3}-3 a^{2} b c \,d^{2} m^{3}+3 a \,b^{2} c^{2} d \,m^{3}-b^{3} c^{3} m^{3}+6 a^{3} d^{3} m^{2}-18 a^{2} b c \,d^{2} m^{2}+18 a \,b^{2} c^{2} d \,m^{2}-6 b^{3} c^{3} m^{2}+11 a^{3} d^{3} m -33 a^{2} b c \,d^{2} m +33 a \,b^{2} c^{2} d m -11 b^{3} c^{3} m +6 a^{3} d^{3}-18 a^{2} b c \,d^{2}+18 a \,b^{2} c^{2} d -6 b^{3} c^{3}}\) | \(741\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(3645\) |
-(b*x+a)^(1+m)*(d*x+c)^(-3-m)/(a^3*d^3*m^3-3*a^2*b*c*d^2*m^3+3*a*b^2*c^2*d *m^3-b^3*c^3*m^3+6*a^3*d^3*m^2-18*a^2*b*c*d^2*m^2+18*a*b^2*c^2*d*m^2-6*b^3 *c^3*m^2+11*a^3*d^3*m-33*a^2*b*c*d^2*m+33*a*b^2*c^2*d*m-11*b^3*c^3*m+6*a^3 *d^3-18*a^2*b*c*d^2+18*a*b^2*c^2*d-6*b^3*c^3)*(a^2*d^2*f^2*m^2*x^2-2*a*b*c *d*f^2*m^2*x^2+b^2*c^2*f^2*m^2*x^2+2*a^2*d^2*e*f*m^2*x+5*a^2*d^2*f^2*m*x^2 -4*a*b*c*d*e*f*m^2*x-8*a*b*c*d*f^2*m*x^2-2*a*b*d^2*e*f*m*x^2+2*b^2*c^2*e*f *m^2*x+3*b^2*c^2*f^2*m*x^2+2*b^2*c*d*e*f*m*x^2+2*a^2*c*d*f^2*m*x+a^2*d^2*e ^2*m^2+8*a^2*d^2*e*f*m*x+6*a^2*d^2*f^2*x^2-2*a*b*c^2*f^2*m*x-2*a*b*c*d*e^2 *m^2-16*a*b*c*d*e*f*m*x-6*a*b*c*d*f^2*x^2-2*a*b*d^2*e^2*m*x-6*a*b*d^2*e*f* x^2+b^2*c^2*e^2*m^2+8*b^2*c^2*e*f*m*x+2*b^2*c^2*f^2*x^2+2*b^2*c*d*e^2*m*x+ 2*b^2*c*d*e*f*x^2+2*b^2*d^2*e^2*x^2+2*a^2*c*d*e*f*m+6*a^2*c*d*f^2*x+3*a^2* d^2*e^2*m+6*a^2*d^2*e*f*x-2*a*b*c^2*e*f*m-2*a*b*c^2*f^2*x-8*a*b*c*d*e^2*m- 20*a*b*c*d*e*f*x-2*a*b*d^2*e^2*x+5*b^2*c^2*e^2*m+6*b^2*c^2*e*f*x+6*b^2*c*d *e^2*x+2*a^2*c^2*f^2+2*a^2*c*d*e*f+2*a^2*d^2*e^2-6*a*b*c^2*e*f-6*a*b*c*d*e ^2+6*b^2*c^2*e^2)
Leaf count of result is larger than twice the leaf count of optimal. 1292 vs. \(2 (353) = 706\).
Time = 0.29 (sec) , antiderivative size = 1292, normalized size of antiderivative = 3.66 \[ \int (a+b x)^m (c+d x)^{-4-m} (e+f x)^2 \, dx=\text {Too large to display} \]
(2*a^3*c^3*f^2 + (a*b^2*c^3 - 2*a^2*b*c^2*d + a^3*c*d^2)*e^2*m^2 + (2*b^3* d^3*e^2 + (b^3*c^2*d - 2*a*b^2*c*d^2 + a^2*b*d^3)*f^2*m^2 + 2*(b^3*c*d^2 - 3*a*b^2*d^3)*e*f + 2*(b^3*c^2*d - 3*a*b^2*c*d^2 + 3*a^2*b*d^3)*f^2 + (2*( b^3*c*d^2 - a*b^2*d^3)*e*f + (3*b^3*c^2*d - 8*a*b^2*c*d^2 + 5*a^2*b*d^3)*f ^2)*m)*x^4 + (8*b^3*c*d^2*e^2 + 8*(b^3*c^2*d - 3*a*b^2*c*d^2)*e*f + 2*(b^3 *c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 + 3*a^3*d^3)*f^2 + (2*(b^3*c^2*d - 2* a*b^2*c*d^2 + a^2*b*d^3)*e*f + (b^3*c^3 - a*b^2*c^2*d - a^2*b*c*d^2 + a^3* d^3)*f^2)*m^2 + (2*(b^3*c*d^2 - a*b^2*d^3)*e^2 + 2*(5*b^3*c^2*d - 8*a*b^2* c*d^2 + 3*a^2*b*d^3)*e*f + (3*b^3*c^3 - 7*a*b^2*c^2*d - a^2*b*c*d^2 + 5*a^ 3*d^3)*f^2)*m)*x^3 + 2*(3*a*b^2*c^3 - 3*a^2*b*c^2*d + a^3*c*d^2)*e^2 - 2*( 3*a^2*b*c^3 - a^3*c^2*d)*e*f + (12*b^3*c^2*d*e^2 + 12*a^3*c*d^2*f^2 + 6*(b ^3*c^3 - 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 + a^3*d^3)*e*f + ((b^3*c^2*d - 2*a* b^2*c*d^2 + a^2*b*d^3)*e^2 + 2*(b^3*c^3 - a*b^2*c^2*d - a^2*b*c*d^2 + a^3* d^3)*e*f + (a*b^2*c^3 - 2*a^2*b*c^2*d + a^3*c*d^2)*f^2)*m^2 + ((7*b^3*c^2* d - 8*a*b^2*c*d^2 + a^2*b*d^3)*e^2 + 8*(b^3*c^3 - a*b^2*c^2*d - a^2*b*c*d^ 2 + a^3*d^3)*e*f + (a*b^2*c^3 - 8*a^2*b*c^2*d + 7*a^3*c*d^2)*f^2)*m)*x^2 + ((5*a*b^2*c^3 - 8*a^2*b*c^2*d + 3*a^3*c*d^2)*e^2 - 2*(a^2*b*c^3 - a^3*c^2 *d)*e*f)*m + (8*a^3*c^2*d*f^2 + 2*(3*b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d ^2 + a^3*d^3)*e^2 - 8*(3*a^2*b*c^2*d - a^3*c*d^2)*e*f + ((b^3*c^3 - a*b^2* c^2*d - a^2*b*c*d^2 + a^3*d^3)*e^2 + 2*(a*b^2*c^3 - 2*a^2*b*c^2*d + a^3...
Exception generated. \[ \int (a+b x)^m (c+d x)^{-4-m} (e+f x)^2 \, dx=\text {Exception raised: HeuristicGCDFailed} \]
\[ \int (a+b x)^m (c+d x)^{-4-m} (e+f x)^2 \, dx=\int { {\left (f x + e\right )}^{2} {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m - 4} \,d x } \]
\[ \int (a+b x)^m (c+d x)^{-4-m} (e+f x)^2 \, dx=\int { {\left (f x + e\right )}^{2} {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m - 4} \,d x } \]
Time = 4.31 (sec) , antiderivative size = 1485, normalized size of antiderivative = 4.21 \[ \int (a+b x)^m (c+d x)^{-4-m} (e+f x)^2 \, dx=-\frac {x^3\,{\left (a+b\,x\right )}^m\,\left (a^3\,d^3\,f^2\,m^2+5\,a^3\,d^3\,f^2\,m+6\,a^3\,d^3\,f^2-a^2\,b\,c\,d^2\,f^2\,m^2-a^2\,b\,c\,d^2\,f^2\,m+6\,a^2\,b\,c\,d^2\,f^2+2\,a^2\,b\,d^3\,e\,f\,m^2+6\,a^2\,b\,d^3\,e\,f\,m-a\,b^2\,c^2\,d\,f^2\,m^2-7\,a\,b^2\,c^2\,d\,f^2\,m-6\,a\,b^2\,c^2\,d\,f^2-4\,a\,b^2\,c\,d^2\,e\,f\,m^2-16\,a\,b^2\,c\,d^2\,e\,f\,m-24\,a\,b^2\,c\,d^2\,e\,f-2\,a\,b^2\,d^3\,e^2\,m+b^3\,c^3\,f^2\,m^2+3\,b^3\,c^3\,f^2\,m+2\,b^3\,c^3\,f^2+2\,b^3\,c^2\,d\,e\,f\,m^2+10\,b^3\,c^2\,d\,e\,f\,m+8\,b^3\,c^2\,d\,e\,f+2\,b^3\,c\,d^2\,e^2\,m+8\,b^3\,c\,d^2\,e^2\right )}{{\left (a\,d-b\,c\right )}^3\,{\left (c+d\,x\right )}^{m+4}\,\left (m^3+6\,m^2+11\,m+6\right )}-\frac {x^2\,{\left (a+b\,x\right )}^m\,\left (a^3\,c\,d^2\,f^2\,m^2+7\,a^3\,c\,d^2\,f^2\,m+12\,a^3\,c\,d^2\,f^2+2\,a^3\,d^3\,e\,f\,m^2+8\,a^3\,d^3\,e\,f\,m+6\,a^3\,d^3\,e\,f-2\,a^2\,b\,c^2\,d\,f^2\,m^2-8\,a^2\,b\,c^2\,d\,f^2\,m-2\,a^2\,b\,c\,d^2\,e\,f\,m^2-8\,a^2\,b\,c\,d^2\,e\,f\,m-18\,a^2\,b\,c\,d^2\,e\,f+a^2\,b\,d^3\,e^2\,m^2+a^2\,b\,d^3\,e^2\,m+a\,b^2\,c^3\,f^2\,m^2+a\,b^2\,c^3\,f^2\,m-2\,a\,b^2\,c^2\,d\,e\,f\,m^2-8\,a\,b^2\,c^2\,d\,e\,f\,m-18\,a\,b^2\,c^2\,d\,e\,f-2\,a\,b^2\,c\,d^2\,e^2\,m^2-8\,a\,b^2\,c\,d^2\,e^2\,m+2\,b^3\,c^3\,e\,f\,m^2+8\,b^3\,c^3\,e\,f\,m+6\,b^3\,c^3\,e\,f+b^3\,c^2\,d\,e^2\,m^2+7\,b^3\,c^2\,d\,e^2\,m+12\,b^3\,c^2\,d\,e^2\right )}{{\left (a\,d-b\,c\right )}^3\,{\left (c+d\,x\right )}^{m+4}\,\left (m^3+6\,m^2+11\,m+6\right )}-\frac {x\,{\left (a+b\,x\right )}^m\,\left (2\,a^3\,c^2\,d\,f^2\,m+8\,a^3\,c^2\,d\,f^2+2\,a^3\,c\,d^2\,e\,f\,m^2+10\,a^3\,c\,d^2\,e\,f\,m+8\,a^3\,c\,d^2\,e\,f+a^3\,d^3\,e^2\,m^2+3\,a^3\,d^3\,e^2\,m+2\,a^3\,d^3\,e^2-2\,a^2\,b\,c^3\,f^2\,m-4\,a^2\,b\,c^2\,d\,e\,f\,m^2-16\,a^2\,b\,c^2\,d\,e\,f\,m-24\,a^2\,b\,c^2\,d\,e\,f-a^2\,b\,c\,d^2\,e^2\,m^2-7\,a^2\,b\,c\,d^2\,e^2\,m-6\,a^2\,b\,c\,d^2\,e^2+2\,a\,b^2\,c^3\,e\,f\,m^2+6\,a\,b^2\,c^3\,e\,f\,m-a\,b^2\,c^2\,d\,e^2\,m^2-a\,b^2\,c^2\,d\,e^2\,m+6\,a\,b^2\,c^2\,d\,e^2+b^3\,c^3\,e^2\,m^2+5\,b^3\,c^3\,e^2\,m+6\,b^3\,c^3\,e^2\right )}{{\left (a\,d-b\,c\right )}^3\,{\left (c+d\,x\right )}^{m+4}\,\left (m^3+6\,m^2+11\,m+6\right )}-\frac {a\,c\,{\left (a+b\,x\right )}^m\,\left (2\,a^2\,c^2\,f^2+2\,a^2\,c\,d\,e\,f\,m+2\,a^2\,c\,d\,e\,f+a^2\,d^2\,e^2\,m^2+3\,a^2\,d^2\,e^2\,m+2\,a^2\,d^2\,e^2-2\,a\,b\,c^2\,e\,f\,m-6\,a\,b\,c^2\,e\,f-2\,a\,b\,c\,d\,e^2\,m^2-8\,a\,b\,c\,d\,e^2\,m-6\,a\,b\,c\,d\,e^2+b^2\,c^2\,e^2\,m^2+5\,b^2\,c^2\,e^2\,m+6\,b^2\,c^2\,e^2\right )}{{\left (a\,d-b\,c\right )}^3\,{\left (c+d\,x\right )}^{m+4}\,\left (m^3+6\,m^2+11\,m+6\right )}-\frac {b\,d\,x^4\,{\left (a+b\,x\right )}^m\,\left (a^2\,d^2\,f^2\,m^2+5\,a^2\,d^2\,f^2\,m+6\,a^2\,d^2\,f^2-2\,a\,b\,c\,d\,f^2\,m^2-8\,a\,b\,c\,d\,f^2\,m-6\,a\,b\,c\,d\,f^2-2\,a\,b\,d^2\,e\,f\,m-6\,a\,b\,d^2\,e\,f+b^2\,c^2\,f^2\,m^2+3\,b^2\,c^2\,f^2\,m+2\,b^2\,c^2\,f^2+2\,b^2\,c\,d\,e\,f\,m+2\,b^2\,c\,d\,e\,f+2\,b^2\,d^2\,e^2\right )}{{\left (a\,d-b\,c\right )}^3\,{\left (c+d\,x\right )}^{m+4}\,\left (m^3+6\,m^2+11\,m+6\right )} \]
- (x^3*(a + b*x)^m*(6*a^3*d^3*f^2 + 2*b^3*c^3*f^2 + 8*b^3*c*d^2*e^2 + 5*a^ 3*d^3*f^2*m + 3*b^3*c^3*f^2*m + a^3*d^3*f^2*m^2 + b^3*c^3*f^2*m^2 + 8*b^3* c^2*d*e*f - 6*a*b^2*c^2*d*f^2 + 6*a^2*b*c*d^2*f^2 - 2*a*b^2*d^3*e^2*m + 2* b^3*c*d^2*e^2*m - a*b^2*c^2*d*f^2*m^2 - a^2*b*c*d^2*f^2*m^2 - 24*a*b^2*c*d ^2*e*f + 6*a^2*b*d^3*e*f*m + 10*b^3*c^2*d*e*f*m - 7*a*b^2*c^2*d*f^2*m - a^ 2*b*c*d^2*f^2*m + 2*a^2*b*d^3*e*f*m^2 + 2*b^3*c^2*d*e*f*m^2 - 4*a*b^2*c*d^ 2*e*f*m^2 - 16*a*b^2*c*d^2*e*f*m))/((a*d - b*c)^3*(c + d*x)^(m + 4)*(11*m + 6*m^2 + m^3 + 6)) - (x^2*(a + b*x)^m*(12*a^3*c*d^2*f^2 + 12*b^3*c^2*d*e^ 2 + 6*a^3*d^3*e*f + 6*b^3*c^3*e*f + 8*a^3*d^3*e*f*m + 8*b^3*c^3*e*f*m + a* b^2*c^3*f^2*m + a^2*b*d^3*e^2*m + 7*a^3*c*d^2*f^2*m + 7*b^3*c^2*d*e^2*m + 2*a^3*d^3*e*f*m^2 + 2*b^3*c^3*e*f*m^2 + a*b^2*c^3*f^2*m^2 + a^2*b*d^3*e^2* m^2 + a^3*c*d^2*f^2*m^2 + b^3*c^2*d*e^2*m^2 - 2*a*b^2*c*d^2*e^2*m^2 - 2*a^ 2*b*c^2*d*f^2*m^2 - 18*a*b^2*c^2*d*e*f - 18*a^2*b*c*d^2*e*f - 8*a*b^2*c*d^ 2*e^2*m - 8*a^2*b*c^2*d*f^2*m - 2*a*b^2*c^2*d*e*f*m^2 - 2*a^2*b*c*d^2*e*f* m^2 - 8*a*b^2*c^2*d*e*f*m - 8*a^2*b*c*d^2*e*f*m))/((a*d - b*c)^3*(c + d*x) ^(m + 4)*(11*m + 6*m^2 + m^3 + 6)) - (x*(a + b*x)^m*(2*a^3*d^3*e^2 + 6*b^3 *c^3*e^2 + 8*a^3*c^2*d*f^2 + 3*a^3*d^3*e^2*m + 5*b^3*c^3*e^2*m + a^3*d^3*e ^2*m^2 + b^3*c^3*e^2*m^2 + 8*a^3*c*d^2*e*f + 6*a*b^2*c^2*d*e^2 - 6*a^2*b*c *d^2*e^2 - 2*a^2*b*c^3*f^2*m + 2*a^3*c^2*d*f^2*m - a*b^2*c^2*d*e^2*m^2 - a ^2*b*c*d^2*e^2*m^2 - 24*a^2*b*c^2*d*e*f + 6*a*b^2*c^3*e*f*m + 10*a^3*c*...