Integrand size = 26, antiderivative size = 494 \[ \int (a+b x)^m (c+d x)^{-5-m} (e+f x)^2 \, dx=-\frac {(d e-c f) (a d f (4+m)-b (2 d e+c f (2+m))) (a+b x)^{1+m} (c+d x)^{-4-m}}{2 b d^2 (b c-a d) (4+m)}+\frac {\left (a^2 d^2 f^2 \left (12+7 m+m^2\right )-2 a b d f (4+m) (2 d e+c f (1+m))+b^2 \left (6 d^2 e^2+4 c d e f (1+m)+c^2 f^2 \left (2+3 m+m^2\right )\right )\right ) (a+b x)^{1+m} (c+d x)^{-3-m}}{2 b d^2 (b c-a d)^2 (3+m) (4+m)}+\frac {\left (a^2 d^2 f^2 \left (12+7 m+m^2\right )-2 a b d f (4+m) (2 d e+c f (1+m))+b^2 \left (6 d^2 e^2+4 c d e f (1+m)+c^2 f^2 \left (2+3 m+m^2\right )\right )\right ) (a+b x)^{1+m} (c+d x)^{-2-m}}{d^2 (b c-a d)^3 (2+m) (3+m) (4+m)}+\frac {b \left (a^2 d^2 f^2 \left (12+7 m+m^2\right )-2 a b d f (4+m) (2 d e+c f (1+m))+b^2 \left (6 d^2 e^2+4 c d e f (1+m)+c^2 f^2 \left (2+3 m+m^2\right )\right )\right ) (a+b x)^{1+m} (c+d x)^{-1-m}}{d^2 (b c-a d)^4 (1+m) (2+m) (3+m) (4+m)}-\frac {f (a+b x)^{1+m} (c+d x)^{-4-m} (e+f x)}{2 b d} \]
-1/2*(-c*f+d*e)*(a*d*f*(4+m)-b*(2*d*e+c*f*(2+m)))*(b*x+a)^(1+m)*(d*x+c)^(- 4-m)/b/d^2/(-a*d+b*c)/(4+m)+1/2*(a^2*d^2*f^2*(m^2+7*m+12)-2*a*b*d*f*(4+m)* (2*d*e+c*f*(1+m))+b^2*(6*d^2*e^2+4*c*d*e*f*(1+m)+c^2*f^2*(m^2+3*m+2)))*(b* x+a)^(1+m)*(d*x+c)^(-3-m)/b/d^2/(-a*d+b*c)^2/(3+m)/(4+m)+(a^2*d^2*f^2*(m^2 +7*m+12)-2*a*b*d*f*(4+m)*(2*d*e+c*f*(1+m))+b^2*(6*d^2*e^2+4*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)/d^2/(-a*d+b*c)^3/(2+m)/ (3+m)/(4+m)+b*(a^2*d^2*f^2*(m^2+7*m+12)-2*a*b*d*f*(4+m)*(2*d*e+c*f*(1+m))+ b^2*(6*d^2*e^2+4*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)^4/(1+m)/(2+m)/(3+m)/(4+m)-1/2*f*(b*x+a)^(1+m)*(d*x+ c)^(-4-m)*(f*x+e)/b/d
Time = 0.45 (sec) , antiderivative size = 269, normalized size of antiderivative = 0.54 \[ \int (a+b x)^m (c+d x)^{-5-m} (e+f x)^2 \, dx=\frac {(a+b x)^{1+m} (c+d x)^{-4-m} \left (-\frac {(d e-c f) (2 b d e+b c f (2+m)-a d f (4+m))}{d (-b c+a d) (4+m)}-f (e+f x)+\frac {\left (a^2 d^2 f^2 \left (12+7 m+m^2\right )-2 a b d f (4+m) (2 d e+c f (1+m))+b^2 \left (6 d^2 e^2+4 c d e f (1+m)+c^2 f^2 \left (2+3 m+m^2\right )\right )\right ) (c+d x) \left (a^2 d^2 \left (2+3 m+m^2\right )-2 a b d (1+m) (c (3+m)+d x)+b^2 \left (c^2 \left (6+5 m+m^2\right )+2 c d (3+m) x+2 d^2 x^2\right )\right )}{d (b c-a d)^4 (1+m) (2+m) (3+m) (4+m)}\right )}{2 b d} \]
((a + b*x)^(1 + m)*(c + d*x)^(-4 - m)*(-(((d*e - c*f)*(2*b*d*e + b*c*f*(2 + m) - a*d*f*(4 + m)))/(d*(-(b*c) + a*d)*(4 + m))) - f*(e + f*x) + ((a^2*d ^2*f^2*(12 + 7*m + m^2) - 2*a*b*d*f*(4 + m)*(2*d*e + c*f*(1 + m)) + b^2*(6 *d^2*e^2 + 4*c*d*e*f*(1 + m) + c^2*f^2*(2 + 3*m + m^2)))*(c + d*x)*(a^2*d^ 2*(2 + 3*m + m^2) - 2*a*b*d*(1 + m)*(c*(3 + m) + d*x) + b^2*(c^2*(6 + 5*m + m^2) + 2*c*d*(3 + m)*x + 2*d^2*x^2)))/(d*(b*c - a*d)^4*(1 + m)*(2 + m)*( 3 + m)*(4 + m))))/(2*b*d)
Time = 0.48 (sec) , antiderivative size = 345, normalized size of antiderivative = 0.70, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {101, 25, 88, 55, 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.
| \(\displaystyle \int (e+f x)^2 (a+b x)^m (c+d x)^{-m-5} \, dx\) |
| \(\Big \downarrow \) 101 |
| \(\displaystyle -\frac {\int -(a+b x)^m (c+d x)^{-m-5} (b e (2 d e+c f (m+1))+a f (c f-d e (m+4))-f (a d f (m+3)-b (d e+c f (m+2))) x)dx}{2 b d}-\frac {f (e+f x) (a+b x)^{m+1} (c+d x)^{-m-4}}{2 b d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int (a+b x)^m (c+d x)^{-m-5} (b e (2 d e+c f (m+1))+a f (c f-d e (m+4))-f (a d f (m+3)-b (d e+c f (m+2))) x)dx}{2 b d}-\frac {f (e+f x) (a+b x)^{m+1} (c+d x)^{-m-4}}{2 b d}\) |
| \(\Big \downarrow \) 88 |
| \(\displaystyle \frac {\frac {\left (a^2 d^2 f^2 \left (m^2+7 m+12\right )-2 a b d f (m+4) (c f (m+1)+2 d e)+b^2 \left (c^2 f^2 \left (m^2+3 m+2\right )+4 c d e f (m+1)+6 d^2 e^2\right )\right ) \int (a+b x)^m (c+d x)^{-m-4}dx}{d (m+4) (b c-a d)}+\frac {(a+b x)^{m+1} (d e-c f) (c+d x)^{-m-4} (-a d f (m+4)+b c f (m+2)+2 b d e)}{d (m+4) (b c-a d)}}{2 b d}-\frac {f (e+f x) (a+b x)^{m+1} (c+d x)^{-m-4}}{2 b d}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle \frac {\frac {\left (a^2 d^2 f^2 \left (m^2+7 m+12\right )-2 a b d f (m+4) (c f (m+1)+2 d e)+b^2 \left (c^2 f^2 \left (m^2+3 m+2\right )+4 c d e f (m+1)+6 d^2 e^2\right )\right ) \left (\frac {2 b \int (a+b x)^m (c+d x)^{-m-3}dx}{(m+3) (b c-a d)}+\frac {(a+b x)^{m+1} (c+d x)^{-m-3}}{(m+3) (b c-a d)}\right )}{d (m+4) (b c-a d)}+\frac {(a+b x)^{m+1} (d e-c f) (c+d x)^{-m-4} (-a d f (m+4)+b c f (m+2)+2 b d e)}{d (m+4) (b c-a d)}}{2 b d}-\frac {f (e+f x) (a+b x)^{m+1} (c+d x)^{-m-4}}{2 b d}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle \frac {\frac {\left (a^2 d^2 f^2 \left (m^2+7 m+12\right )-2 a b d f (m+4) (c f (m+1)+2 d e)+b^2 \left (c^2 f^2 \left (m^2+3 m+2\right )+4 c d e f (m+1)+6 d^2 e^2\right )\right ) \left (\frac {2 b \left (\frac {b \int (a+b x)^m (c+d x)^{-m-2}dx}{(m+2) (b c-a d)}+\frac {(a+b x)^{m+1} (c+d x)^{-m-2}}{(m+2) (b c-a d)}\right )}{(m+3) (b c-a d)}+\frac {(a+b x)^{m+1} (c+d x)^{-m-3}}{(m+3) (b c-a d)}\right )}{d (m+4) (b c-a d)}+\frac {(a+b x)^{m+1} (d e-c f) (c+d x)^{-m-4} (-a d f (m+4)+b c f (m+2)+2 b d e)}{d (m+4) (b c-a d)}}{2 b d}-\frac {f (e+f x) (a+b x)^{m+1} (c+d x)^{-m-4}}{2 b d}\) |
| \(\Big \downarrow \) 48 |
| \(\displaystyle \frac {\frac {\left (\frac {(a+b x)^{m+1} (c+d x)^{-m-3}}{(m+3) (b c-a d)}+\frac {2 b \left (\frac {(a+b x)^{m+1} (c+d x)^{-m-2}}{(m+2) (b c-a d)}+\frac {b (a+b x)^{m+1} (c+d x)^{-m-1}}{(m+1) (m+2) (b c-a d)^2}\right )}{(m+3) (b c-a d)}\right ) \left (a^2 d^2 f^2 \left (m^2+7 m+12\right )-2 a b d f (m+4) (c f (m+1)+2 d e)+b^2 \left (c^2 f^2 \left (m^2+3 m+2\right )+4 c d e f (m+1)+6 d^2 e^2\right )\right )}{d (m+4) (b c-a d)}+\frac {(a+b x)^{m+1} (d e-c f) (c+d x)^{-m-4} (-a d f (m+4)+b c f (m+2)+2 b d e)}{d (m+4) (b c-a d)}}{2 b d}-\frac {f (e+f x) (a+b x)^{m+1} (c+d x)^{-m-4}}{2 b d}\) |
-1/2*(f*(a + b*x)^(1 + m)*(c + d*x)^(-4 - m)*(e + f*x))/(b*d) + (((d*e - c *f)*(2*b*d*e + b*c*f*(2 + m) - a*d*f*(4 + m))*(a + b*x)^(1 + m)*(c + d*x)^ (-4 - m))/(d*(b*c - a*d)*(4 + m)) + ((a^2*d^2*f^2*(12 + 7*m + m^2) - 2*a*b *d*f*(4 + m)*(2*d*e + c*f*(1 + m)) + b^2*(6*d^2*e^2 + 4*c*d*e*f*(1 + m) + c^2*f^2*(2 + 3*m + m^2)))*(((a + b*x)^(1 + m)*(c + d*x)^(-3 - m))/((b*c - a*d)*(3 + m)) + (2*b*(((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))))/((b*c - a*d)*(3 + m))))/(d*(b*c - a*d)*(4 + m)))/(2*b*d)
3.32.7.3.1 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. \(1883\) vs. \(2(488)=976\).
Time = 2.10 (sec) , antiderivative size = 1884, normalized size of antiderivative = 3.81
| method | result | size |
| gosper | \(\text {Expression too large to display}\) | \(1884\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(7366\) |
-(b*x+a)^(1+m)*(d*x+c)^(-4-m)/(a^4*d^4*m^4-4*a^3*b*c*d^3*m^4+6*a^2*b^2*c^2 *d^2*m^4-4*a*b^3*c^3*d*m^4+b^4*c^4*m^4+10*a^4*d^4*m^3-40*a^3*b*c*d^3*m^3+6 0*a^2*b^2*c^2*d^2*m^3-40*a*b^3*c^3*d*m^3+10*b^4*c^4*m^3+35*a^4*d^4*m^2-140 *a^3*b*c*d^3*m^2+210*a^2*b^2*c^2*d^2*m^2-140*a*b^3*c^3*d*m^2+35*b^4*c^4*m^ 2+50*a^4*d^4*m-200*a^3*b*c*d^3*m+300*a^2*b^2*c^2*d^2*m-200*a*b^3*c^3*d*m+5 0*b^4*c^4*m+24*a^4*d^4-96*a^3*b*c*d^3+144*a^2*b^2*c^2*d^2-96*a*b^3*c^3*d+2 4*b^4*c^4)*(a^3*d^3*f^2*m^3*x^2-3*a^2*b*c*d^2*f^2*m^3*x^2-a^2*b*d^3*f^2*m^ 2*x^3+3*a*b^2*c^2*d*f^2*m^3*x^2+2*a*b^2*c*d^2*f^2*m^2*x^3-b^3*c^3*f^2*m^3* x^2-b^3*c^2*d*f^2*m^2*x^3+2*a^3*d^3*e*f*m^3*x+8*a^3*d^3*f^2*m^2*x^2-6*a^2* b*c*d^2*e*f*m^3*x-23*a^2*b*c*d^2*f^2*m^2*x^2-4*a^2*b*d^3*e*f*m^2*x^2-7*a^2 *b*d^3*f^2*m*x^3+6*a*b^2*c^2*d*e*f*m^3*x+22*a*b^2*c^2*d*f^2*m^2*x^2+8*a*b^ 2*c*d^2*e*f*m^2*x^2+10*a*b^2*c*d^2*f^2*m*x^3+4*a*b^2*d^3*e*f*m*x^3-2*b^3*c ^3*e*f*m^3*x-7*b^3*c^3*f^2*m^2*x^2-4*b^3*c^2*d*e*f*m^2*x^2-3*b^3*c^2*d*f^2 *m*x^3-4*b^3*c*d^2*e*f*m*x^3+2*a^3*c*d^2*f^2*m^2*x+a^3*d^3*e^2*m^3+14*a^3* d^3*e*f*m^2*x+19*a^3*d^3*f^2*m*x^2-4*a^2*b*c^2*d*f^2*m^2*x-3*a^2*b*c*d^2*e ^2*m^3-44*a^2*b*c*d^2*e*f*m^2*x-58*a^2*b*c*d^2*f^2*m*x^2-3*a^2*b*d^3*e^2*m ^2*x-20*a^2*b*d^3*e*f*m*x^2-12*a^2*b*d^3*f^2*x^3+2*a*b^2*c^3*f^2*m^2*x+3*a *b^2*c^2*d*e^2*m^3+46*a*b^2*c^2*d*e*f*m^2*x+53*a*b^2*c^2*d*f^2*m*x^2+6*a*b ^2*c*d^2*e^2*m^2*x+40*a*b^2*c*d^2*e*f*m*x^2+8*a*b^2*c*d^2*f^2*x^3+6*a*b^2* d^3*e^2*m*x^2+16*a*b^2*d^3*e*f*x^3-b^3*c^3*e^2*m^3-16*b^3*c^3*e*f*m^2*x...
Leaf count of result is larger than twice the leaf count of optimal. 2662 vs. \(2 (488) = 976\).
Time = 0.43 (sec) , antiderivative size = 2662, normalized size of antiderivative = 5.39 \[ \int (a+b x)^m (c+d x)^{-5-m} (e+f x)^2 \, dx=\text {Too large to display} \]
((a*b^3*c^4 - 3*a^2*b^2*c^3*d + 3*a^3*b*c^2*d^2 - a^4*c*d^3)*e^2*m^3 + (6* b^4*d^4*e^2 + (b^4*c^2*d^2 - 2*a*b^3*c*d^3 + a^2*b^2*d^4)*f^2*m^2 + 4*(b^4 *c*d^3 - 4*a*b^3*d^4)*e*f + 2*(b^4*c^2*d^2 - 4*a*b^3*c*d^3 + 6*a^2*b^2*d^4 )*f^2 + (4*(b^4*c*d^3 - a*b^3*d^4)*e*f + (3*b^4*c^2*d^2 - 10*a*b^3*c*d^3 + 7*a^2*b^2*d^4)*f^2)*m)*x^5 + (30*b^4*c*d^3*e^2 + (b^4*c^3*d - 3*a*b^3*c^2 *d^2 + 3*a^2*b^2*c*d^3 - a^3*b*d^4)*f^2*m^3 + 20*(b^4*c^2*d^2 - 4*a*b^3*c* d^3)*e*f + 10*(b^4*c^3*d - 4*a*b^3*c^2*d^2 + 6*a^2*b^2*c*d^3)*f^2 + (4*(b^ 4*c^2*d^2 - 2*a*b^3*c*d^3 + a^2*b^2*d^4)*e*f + (8*b^4*c^3*d - 23*a*b^3*c^2 *d^2 + 22*a^2*b^2*c*d^3 - 7*a^3*b*d^4)*f^2)*m^2 + (6*(b^4*c*d^3 - a*b^3*d^ 4)*e^2 + 8*(3*b^4*c^2*d^2 - 5*a*b^3*c*d^3 + 2*a^2*b^2*d^4)*e*f + (17*b^4*c ^3*d - 60*a*b^3*c^2*d^2 + 55*a^2*b^2*c*d^3 - 12*a^3*b*d^4)*f^2)*m)*x^4 + ( 60*b^4*c^2*d^2*e^2 + (2*(b^4*c^3*d - 3*a*b^3*c^2*d^2 + 3*a^2*b^2*c*d^3 - a ^3*b*d^4)*e*f + (b^4*c^4 - 2*a*b^3*c^3*d + 2*a^3*b*c*d^3 - a^4*d^4)*f^2)*m ^3 + 40*(b^4*c^3*d - 4*a*b^3*c^2*d^2)*e*f + 4*(2*b^4*c^4 - 8*a*b^3*c^3*d + 12*a^2*b^2*c^2*d^2 + 12*a^3*b*c*d^3 - 3*a^4*d^4)*f^2 + (3*(b^4*c^2*d^2 - 2*a*b^3*c*d^3 + a^2*b^2*d^4)*e^2 + 10*(2*b^4*c^3*d - 5*a*b^3*c^2*d^2 + 4*a ^2*b^2*c*d^3 - a^3*b*d^4)*e*f + (7*b^4*c^4 - 16*a*b^3*c^3*d + 3*a^2*b^2*c^ 2*d^2 + 14*a^3*b*c*d^3 - 8*a^4*d^4)*f^2)*m^2 + (3*(9*b^4*c^2*d^2 - 10*a*b^ 3*c*d^3 + a^2*b^2*d^4)*e^2 + 2*(29*b^4*c^3*d - 66*a*b^3*c^2*d^2 + 41*a^2*b ^2*c*d^3 - 4*a^3*b*d^4)*e*f + (14*b^4*c^4 - 46*a*b^3*c^3*d + 15*a^2*b^2...
Exception generated. \[ \int (a+b x)^m (c+d x)^{-5-m} (e+f x)^2 \, dx=\text {Exception raised: HeuristicGCDFailed} \]
\[ \int (a+b x)^m (c+d x)^{-5-m} (e+f x)^2 \, dx=\int { {\left (f x + e\right )}^{2} {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m - 5} \,d x } \]
\[ \int (a+b x)^m (c+d x)^{-5-m} (e+f x)^2 \, dx=\int { {\left (f x + e\right )}^{2} {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m - 5} \,d x } \]
Time = 6.27 (sec) , antiderivative size = 2932, normalized size of antiderivative = 5.94 \[ \int (a+b x)^m (c+d x)^{-5-m} (e+f x)^2 \, dx=\text {Too large to display} \]
(x^3*(a + b*x)^m*(8*b^4*c^4*f^2 - 12*a^4*d^4*f^2 - 19*a^4*d^4*f^2*m + 14*b ^4*c^4*f^2*m + 60*b^4*c^2*d^2*e^2 - 8*a^4*d^4*f^2*m^2 + 7*b^4*c^4*f^2*m^2 - a^4*d^4*f^2*m^3 + b^4*c^4*f^2*m^3 + 40*b^4*c^3*d*e*f + 48*a^2*b^2*c^2*d^ 2*f^2 + 3*a^2*b^2*d^4*e^2*m^2 + 3*b^4*c^2*d^2*e^2*m^2 - 32*a*b^3*c^3*d*f^2 + 48*a^3*b*c*d^3*f^2 + 3*a^2*b^2*d^4*e^2*m + 27*b^4*c^2*d^2*e^2*m - 6*a*b ^3*c*d^3*e^2*m^2 - 16*a*b^3*c^3*d*f^2*m^2 + 14*a^3*b*c*d^3*f^2*m^2 - 2*a*b ^3*c^3*d*f^2*m^3 + 2*a^3*b*c*d^3*f^2*m^3 - 8*a^3*b*d^4*e*f*m + 58*b^4*c^3* d*e*f*m + 15*a^2*b^2*c^2*d^2*f^2*m - 160*a*b^3*c^2*d^2*e*f - 30*a*b^3*c*d^ 3*e^2*m - 46*a*b^3*c^3*d*f^2*m + 36*a^3*b*c*d^3*f^2*m - 10*a^3*b*d^4*e*f*m ^2 - 2*a^3*b*d^4*e*f*m^3 + 20*b^4*c^3*d*e*f*m^2 + 2*b^4*c^3*d*e*f*m^3 + 3* a^2*b^2*c^2*d^2*f^2*m^2 - 132*a*b^3*c^2*d^2*e*f*m + 82*a^2*b^2*c*d^3*e*f*m - 50*a*b^3*c^2*d^2*e*f*m^2 + 40*a^2*b^2*c*d^3*e*f*m^2 - 6*a*b^3*c^2*d^2*e *f*m^3 + 6*a^2*b^2*c*d^3*e*f*m^3))/((a*d - b*c)^4*(c + d*x)^(m + 5)*(50*m + 35*m^2 + 10*m^3 + m^4 + 24)) - ((a + b*x)^m*(6*a^4*c*d^3*e^2 - 8*a^3*b*c ^4*f^2 - 24*a*b^3*c^4*e^2 + 2*a^4*c^3*d*f^2 + 24*a^2*b^2*c^4*e*f + 4*a^4*c ^2*d^2*e*f - 26*a*b^3*c^4*e^2*m - 2*a^3*b*c^4*f^2*m + 11*a^4*c*d^3*e^2*m + 2*a^4*c^3*d*f^2*m + 36*a^2*b^2*c^3*d*e^2 - 24*a^3*b*c^2*d^2*e^2 - 9*a*b^3 *c^4*e^2*m^2 - a*b^3*c^4*e^2*m^3 + 6*a^4*c*d^3*e^2*m^2 + a^4*c*d^3*e^2*m^3 + 57*a^2*b^2*c^3*d*e^2*m - 42*a^3*b*c^2*d^2*e^2*m + 2*a^2*b^2*c^4*e*f*m^2 + 2*a^4*c^2*d^2*e*f*m^2 - 16*a^3*b*c^3*d*e*f + 24*a^2*b^2*c^3*d*e^2*m^...