Integrand size = 19, antiderivative size = 185 \[ \int (a+b x)^m (c+d x)^{-5-m} \, dx=\frac {(a+b x)^{1+m} (c+d x)^{-4-m}}{(b c-a d) (4+m)}+\frac {3 b (a+b x)^{1+m} (c+d x)^{-3-m}}{(b c-a d)^2 (3+m) (4+m)}+\frac {6 b^2 (a+b x)^{1+m} (c+d x)^{-2-m}}{(b c-a d)^3 (2+m) (3+m) (4+m)}+\frac {6 b^3 (a+b x)^{1+m} (c+d x)^{-1-m}}{(b c-a d)^4 (1+m) (2+m) (3+m) (4+m)} \]
(b*x+a)^(1+m)*(d*x+c)^(-4-m)/(-a*d+b*c)/(4+m)+3*b*(b*x+a)^(1+m)*(d*x+c)^(- 3-m)/(-a*d+b*c)^2/(3+m)/(4+m)+6*b^2*(b*x+a)^(1+m)*(d*x+c)^(-2-m)/(-a*d+b*c )^3/(2+m)/(3+m)/(4+m)+6*b^3*(b*x+a)^(1+m)*(d*x+c)^(-1-m)/(-a*d+b*c)^4/(1+m )/(2+m)/(3+m)/(4+m)
Time = 0.09 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.05 \[ \int (a+b x)^m (c+d x)^{-5-m} \, dx=\frac {(a+b x)^{1+m} (c+d x)^{-4-m} \left (-a^3 d^3 \left (6+11 m+6 m^2+m^3\right )+3 a^2 b d^2 \left (2+3 m+m^2\right ) (c (4+m)+d x)-3 a b^2 d (1+m) \left (c^2 \left (12+7 m+m^2\right )+2 c d (4+m) x+2 d^2 x^2\right )+b^3 \left (c^3 \left (24+26 m+9 m^2+m^3\right )+3 c^2 d \left (12+7 m+m^2\right ) x+6 c d^2 (4+m) x^2+6 d^3 x^3\right )\right )}{(b c-a d)^4 (1+m) (2+m) (3+m) (4+m)} \]
((a + b*x)^(1 + m)*(c + d*x)^(-4 - m)*(-(a^3*d^3*(6 + 11*m + 6*m^2 + m^3)) + 3*a^2*b*d^2*(2 + 3*m + m^2)*(c*(4 + m) + d*x) - 3*a*b^2*d*(1 + m)*(c^2* (12 + 7*m + m^2) + 2*c*d*(4 + m)*x + 2*d^2*x^2) + b^3*(c^3*(24 + 26*m + 9* m^2 + m^3) + 3*c^2*d*(12 + 7*m + m^2)*x + 6*c*d^2*(4 + m)*x^2 + 6*d^3*x^3) ))/((b*c - a*d)^4*(1 + m)*(2 + m)*(3 + m)*(4 + m))
Time = 0.26 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.02, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {55, 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.
| \(\displaystyle \int (a+b x)^m (c+d x)^{-m-5} \, dx\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle \frac {3 b \int (a+b x)^m (c+d x)^{-m-4}dx}{(m+4) (b c-a d)}+\frac {(a+b x)^{m+1} (c+d x)^{-m-4}}{(m+4) (b c-a d)}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle \frac {3 b \left (\frac {2 b \int (a+b x)^m (c+d x)^{-m-3}dx}{(m+3) (b c-a d)}+\frac {(a+b x)^{m+1} (c+d x)^{-m-3}}{(m+3) (b c-a d)}\right )}{(m+4) (b c-a d)}+\frac {(a+b x)^{m+1} (c+d x)^{-m-4}}{(m+4) (b c-a d)}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle \frac {3 b \left (\frac {2 b \left (\frac {b \int (a+b x)^m (c+d x)^{-m-2}dx}{(m+2) (b c-a d)}+\frac {(a+b x)^{m+1} (c+d x)^{-m-2}}{(m+2) (b c-a d)}\right )}{(m+3) (b c-a d)}+\frac {(a+b x)^{m+1} (c+d x)^{-m-3}}{(m+3) (b c-a d)}\right )}{(m+4) (b c-a d)}+\frac {(a+b x)^{m+1} (c+d x)^{-m-4}}{(m+4) (b c-a d)}\) |
| \(\Big \downarrow \) 48 |
| \(\displaystyle \frac {(a+b x)^{m+1} (c+d x)^{-m-4}}{(m+4) (b c-a d)}+\frac {3 b \left (\frac {(a+b x)^{m+1} (c+d x)^{-m-3}}{(m+3) (b c-a d)}+\frac {2 b \left (\frac {(a+b x)^{m+1} (c+d x)^{-m-2}}{(m+2) (b c-a d)}+\frac {b (a+b x)^{m+1} (c+d x)^{-m-1}}{(m+1) (m+2) (b c-a d)^2}\right )}{(m+3) (b c-a d)}\right )}{(m+4) (b c-a d)}\) |
((a + b*x)^(1 + m)*(c + d*x)^(-4 - m))/((b*c - a*d)*(4 + m)) + (3*b*(((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) )))/((b*c - a*d)*(4 + m))
3.32.9.3.1 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])
Leaf count of result is larger than twice the leaf count of optimal. \(661\) vs. \(2(185)=370\).
Time = 1.65 (sec) , antiderivative size = 662, normalized size of antiderivative = 3.58
| method | result | size |
| gosper | \(-\frac {\left (b x +a \right )^{1+m} \left (d x +c \right )^{-4-m} \left (a^{3} d^{3} m^{3}-3 a^{2} b c \,d^{2} m^{3}-3 a^{2} b \,d^{3} m^{2} x +3 a \,b^{2} c^{2} d \,m^{3}+6 a \,b^{2} c \,d^{2} m^{2} x +6 a \,b^{2} d^{3} m \,x^{2}-b^{3} c^{3} m^{3}-3 b^{3} c^{2} d \,m^{2} x -6 b^{3} c \,d^{2} m \,x^{2}-6 d^{3} x^{3} b^{3}+6 a^{3} d^{3} m^{2}-21 a^{2} b c \,d^{2} m^{2}-9 a^{2} b \,d^{3} m x +24 a \,b^{2} c^{2} d \,m^{2}+30 a \,b^{2} c \,d^{2} m x +6 x^{2} a \,b^{2} d^{3}-9 b^{3} c^{3} m^{2}-21 b^{3} c^{2} d m x -24 x^{2} b^{3} c \,d^{2}+11 a^{3} d^{3} m -42 a^{2} b c \,d^{2} m -6 x \,a^{2} b \,d^{3}+57 a \,b^{2} c^{2} d m +24 x a \,b^{2} c \,d^{2}-26 b^{3} c^{3} m -36 x \,b^{3} c^{2} d +6 a^{3} d^{3}-24 a^{2} b c \,d^{2}+36 a \,b^{2} c^{2} d -24 b^{3} c^{3}\right )}{a^{4} d^{4} m^{4}-4 a^{3} b c \,d^{3} m^{4}+6 a^{2} b^{2} c^{2} d^{2} m^{4}-4 a \,b^{3} c^{3} d \,m^{4}+b^{4} c^{4} m^{4}+10 a^{4} d^{4} m^{3}-40 a^{3} b c \,d^{3} m^{3}+60 a^{2} b^{2} c^{2} d^{2} m^{3}-40 a \,b^{3} c^{3} d \,m^{3}+10 b^{4} c^{4} m^{3}+35 a^{4} d^{4} m^{2}-140 a^{3} b c \,d^{3} m^{2}+210 a^{2} b^{2} c^{2} d^{2} m^{2}-140 a \,b^{3} c^{3} d \,m^{2}+35 b^{4} c^{4} m^{2}+50 a^{4} d^{4} m -200 a^{3} b c \,d^{3} m +300 a^{2} b^{2} c^{2} d^{2} m -200 a \,b^{3} c^{3} d m +50 b^{4} c^{4} m +24 a^{4} d^{4}-96 a^{3} b c \,d^{3}+144 a^{2} b^{2} c^{2} d^{2}-96 a \,b^{3} c^{3} d +24 b^{4} c^{4}}\) | \(662\) |
-(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*m^3-3*a^2*b*c*d^2*m^3-3*a^2*b*d^3*m^2*x+3*a*b^2*c^2*d* m^3+6*a*b^2*c*d^2*m^2*x+6*a*b^2*d^3*m*x^2-b^3*c^3*m^3-3*b^3*c^2*d*m^2*x-6* b^3*c*d^2*m*x^2-6*b^3*d^3*x^3+6*a^3*d^3*m^2-21*a^2*b*c*d^2*m^2-9*a^2*b*d^3 *m*x+24*a*b^2*c^2*d*m^2+30*a*b^2*c*d^2*m*x+6*a*b^2*d^3*x^2-9*b^3*c^3*m^2-2 1*b^3*c^2*d*m*x-24*b^3*c*d^2*x^2+11*a^3*d^3*m-42*a^2*b*c*d^2*m-6*a^2*b*d^3 *x+57*a*b^2*c^2*d*m+24*a*b^2*c*d^2*x-26*b^3*c^3*m-36*b^3*c^2*d*x+6*a^3*d^3 -24*a^2*b*c*d^2+36*a*b^2*c^2*d-24*b^3*c^3)
Leaf count of result is larger than twice the leaf count of optimal. 954 vs. \(2 (185) = 370\).
Time = 0.27 (sec) , antiderivative size = 954, normalized size of antiderivative = 5.16 \[ \int (a+b x)^m (c+d x)^{-5-m} \, dx=\frac {{\left (6 \, b^{4} d^{4} x^{5} + 24 \, a b^{3} c^{4} - 36 \, a^{2} b^{2} c^{3} d + 24 \, a^{3} b c^{2} d^{2} - 6 \, a^{4} c d^{3} + 6 \, {\left (5 \, b^{4} c d^{3} + {\left (b^{4} c d^{3} - a b^{3} d^{4}\right )} m\right )} x^{4} + {\left (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}\right )} m^{3} + 3 \, {\left (20 \, b^{4} c^{2} d^{2} + {\left (b^{4} c^{2} d^{2} - 2 \, a b^{3} c d^{3} + a^{2} b^{2} d^{4}\right )} m^{2} + {\left (9 \, b^{4} c^{2} d^{2} - 10 \, a b^{3} c d^{3} + a^{2} b^{2} d^{4}\right )} m\right )} x^{3} + 3 \, {\left (3 \, a b^{3} c^{4} - 8 \, a^{2} b^{2} c^{3} d + 7 \, a^{3} b c^{2} d^{2} - 2 \, a^{4} c d^{3}\right )} m^{2} + {\left (60 \, b^{4} c^{3} d + {\left (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}\right )} m^{3} + 3 \, {\left (4 \, b^{4} c^{3} d - 9 \, a b^{3} c^{2} d^{2} + 6 \, a^{2} b^{2} c d^{3} - a^{3} b d^{4}\right )} m^{2} + {\left (47 \, b^{4} c^{3} d - 60 \, a b^{3} c^{2} d^{2} + 15 \, a^{2} b^{2} c d^{3} - 2 \, a^{3} b d^{4}\right )} m\right )} x^{2} + {\left (26 \, a b^{3} c^{4} - 57 \, a^{2} b^{2} c^{3} d + 42 \, a^{3} b c^{2} d^{2} - 11 \, a^{4} c d^{3}\right )} m + {\left (24 \, b^{4} c^{4} + 24 \, a b^{3} c^{3} d - 36 \, a^{2} b^{2} c^{2} d^{2} + 24 \, a^{3} b c d^{3} - 6 \, a^{4} d^{4} + {\left (b^{4} c^{4} - 2 \, a b^{3} c^{3} d + 2 \, a^{3} b c d^{3} - a^{4} d^{4}\right )} m^{3} + 3 \, {\left (3 \, b^{4} c^{4} - 4 \, a b^{3} c^{3} d - 3 \, a^{2} b^{2} c^{2} d^{2} + 6 \, a^{3} b c d^{3} - 2 \, a^{4} d^{4}\right )} m^{2} + {\left (26 \, b^{4} c^{4} - 10 \, a b^{3} c^{3} d - 45 \, a^{2} b^{2} c^{2} d^{2} + 40 \, a^{3} b c d^{3} - 11 \, a^{4} d^{4}\right )} m\right )} x\right )} {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m - 5}}{24 \, b^{4} c^{4} - 96 \, a b^{3} c^{3} d + 144 \, a^{2} b^{2} c^{2} d^{2} - 96 \, a^{3} b c d^{3} + 24 \, a^{4} d^{4} + {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} m^{4} + 10 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} m^{3} + 35 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} m^{2} + 50 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} m} \]
(6*b^4*d^4*x^5 + 24*a*b^3*c^4 - 36*a^2*b^2*c^3*d + 24*a^3*b*c^2*d^2 - 6*a^ 4*c*d^3 + 6*(5*b^4*c*d^3 + (b^4*c*d^3 - a*b^3*d^4)*m)*x^4 + (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)*m^3 + 3*(20*b^4*c^2*d^2 + (b ^4*c^2*d^2 - 2*a*b^3*c*d^3 + a^2*b^2*d^4)*m^2 + (9*b^4*c^2*d^2 - 10*a*b^3* c*d^3 + a^2*b^2*d^4)*m)*x^3 + 3*(3*a*b^3*c^4 - 8*a^2*b^2*c^3*d + 7*a^3*b*c ^2*d^2 - 2*a^4*c*d^3)*m^2 + (60*b^4*c^3*d + (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)*m^3 + 3*(4*b^4*c^3*d - 9*a*b^3*c^2*d^2 + 6*a ^2*b^2*c*d^3 - a^3*b*d^4)*m^2 + (47*b^4*c^3*d - 60*a*b^3*c^2*d^2 + 15*a^2* b^2*c*d^3 - 2*a^3*b*d^4)*m)*x^2 + (26*a*b^3*c^4 - 57*a^2*b^2*c^3*d + 42*a^ 3*b*c^2*d^2 - 11*a^4*c*d^3)*m + (24*b^4*c^4 + 24*a*b^3*c^3*d - 36*a^2*b^2* c^2*d^2 + 24*a^3*b*c*d^3 - 6*a^4*d^4 + (b^4*c^4 - 2*a*b^3*c^3*d + 2*a^3*b* c*d^3 - a^4*d^4)*m^3 + 3*(3*b^4*c^4 - 4*a*b^3*c^3*d - 3*a^2*b^2*c^2*d^2 + 6*a^3*b*c*d^3 - 2*a^4*d^4)*m^2 + (26*b^4*c^4 - 10*a*b^3*c^3*d - 45*a^2*b^2 *c^2*d^2 + 40*a^3*b*c*d^3 - 11*a^4*d^4)*m)*x)*(b*x + a)^m*(d*x + c)^(-m - 5)/(24*b^4*c^4 - 96*a*b^3*c^3*d + 144*a^2*b^2*c^2*d^2 - 96*a^3*b*c*d^3 + 2 4*a^4*d^4 + (b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*m^4 + 10*(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b* c*d^3 + a^4*d^4)*m^3 + 35*(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4 *a^3*b*c*d^3 + a^4*d^4)*m^2 + 50*(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2* d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*m)
Exception generated. \[ \int (a+b x)^m (c+d x)^{-5-m} \, dx=\text {Exception raised: HeuristicGCDFailed} \]
\[ \int (a+b x)^m (c+d x)^{-5-m} \, dx=\int { {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m - 5} \,d x } \]
\[ \int (a+b x)^m (c+d x)^{-5-m} \, dx=\int { {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m - 5} \,d x } \]
Time = 4.27 (sec) , antiderivative size = 945, normalized size of antiderivative = 5.11 \[ \int (a+b x)^m (c+d x)^{-5-m} \, dx=\frac {6\,b^4\,d^4\,x^5\,{\left (a+b\,x\right )}^m}{{\left (a\,d-b\,c\right )}^4\,{\left (c+d\,x\right )}^{m+5}\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )}-\frac {x\,{\left (a+b\,x\right )}^m\,\left (a^4\,d^4\,m^3+6\,a^4\,d^4\,m^2+11\,a^4\,d^4\,m+6\,a^4\,d^4-2\,a^3\,b\,c\,d^3\,m^3-18\,a^3\,b\,c\,d^3\,m^2-40\,a^3\,b\,c\,d^3\,m-24\,a^3\,b\,c\,d^3+9\,a^2\,b^2\,c^2\,d^2\,m^2+45\,a^2\,b^2\,c^2\,d^2\,m+36\,a^2\,b^2\,c^2\,d^2+2\,a\,b^3\,c^3\,d\,m^3+12\,a\,b^3\,c^3\,d\,m^2+10\,a\,b^3\,c^3\,d\,m-24\,a\,b^3\,c^3\,d-b^4\,c^4\,m^3-9\,b^4\,c^4\,m^2-26\,b^4\,c^4\,m-24\,b^4\,c^4\right )}{{\left (a\,d-b\,c\right )}^4\,{\left (c+d\,x\right )}^{m+5}\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )}-\frac {a\,c\,{\left (a+b\,x\right )}^m\,\left (a^3\,d^3\,m^3+6\,a^3\,d^3\,m^2+11\,a^3\,d^3\,m+6\,a^3\,d^3-3\,a^2\,b\,c\,d^2\,m^3-21\,a^2\,b\,c\,d^2\,m^2-42\,a^2\,b\,c\,d^2\,m-24\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d\,m^3+24\,a\,b^2\,c^2\,d\,m^2+57\,a\,b^2\,c^2\,d\,m+36\,a\,b^2\,c^2\,d-b^3\,c^3\,m^3-9\,b^3\,c^3\,m^2-26\,b^3\,c^3\,m-24\,b^3\,c^3\right )}{{\left (a\,d-b\,c\right )}^4\,{\left (c+d\,x\right )}^{m+5}\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )}+\frac {3\,b^2\,d^2\,x^3\,{\left (a+b\,x\right )}^m\,\left (a^2\,d^2\,m^2+a^2\,d^2\,m-2\,a\,b\,c\,d\,m^2-10\,a\,b\,c\,d\,m+b^2\,c^2\,m^2+9\,b^2\,c^2\,m+20\,b^2\,c^2\right )}{{\left (a\,d-b\,c\right )}^4\,{\left (c+d\,x\right )}^{m+5}\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )}+\frac {6\,b^3\,d^3\,x^4\,{\left (a+b\,x\right )}^m\,\left (5\,b\,c-a\,d\,m+b\,c\,m\right )}{{\left (a\,d-b\,c\right )}^4\,{\left (c+d\,x\right )}^{m+5}\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )}+\frac {b\,d\,x^2\,{\left (a+b\,x\right )}^m\,\left (-a^3\,d^3\,m^3-3\,a^3\,d^3\,m^2-2\,a^3\,d^3\,m+3\,a^2\,b\,c\,d^2\,m^3+18\,a^2\,b\,c\,d^2\,m^2+15\,a^2\,b\,c\,d^2\,m-3\,a\,b^2\,c^2\,d\,m^3-27\,a\,b^2\,c^2\,d\,m^2-60\,a\,b^2\,c^2\,d\,m+b^3\,c^3\,m^3+12\,b^3\,c^3\,m^2+47\,b^3\,c^3\,m+60\,b^3\,c^3\right )}{{\left (a\,d-b\,c\right )}^4\,{\left (c+d\,x\right )}^{m+5}\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )} \]
(6*b^4*d^4*x^5*(a + b*x)^m)/((a*d - b*c)^4*(c + d*x)^(m + 5)*(50*m + 35*m^ 2 + 10*m^3 + m^4 + 24)) - (x*(a + b*x)^m*(6*a^4*d^4 - 24*b^4*c^4 + 11*a^4* d^4*m - 26*b^4*c^4*m + 6*a^4*d^4*m^2 - 9*b^4*c^4*m^2 + a^4*d^4*m^3 - b^4*c ^4*m^3 + 36*a^2*b^2*c^2*d^2 - 24*a*b^3*c^3*d - 24*a^3*b*c*d^3 + 10*a*b^3*c ^3*d*m - 40*a^3*b*c*d^3*m + 9*a^2*b^2*c^2*d^2*m^2 + 12*a*b^3*c^3*d*m^2 - 1 8*a^3*b*c*d^3*m^2 + 2*a*b^3*c^3*d*m^3 - 2*a^3*b*c*d^3*m^3 + 45*a^2*b^2*c^2 *d^2*m))/((a*d - b*c)^4*(c + d*x)^(m + 5)*(50*m + 35*m^2 + 10*m^3 + m^4 + 24)) - (a*c*(a + b*x)^m*(6*a^3*d^3 - 24*b^3*c^3 + 11*a^3*d^3*m - 26*b^3*c^ 3*m + 6*a^3*d^3*m^2 - 9*b^3*c^3*m^2 + a^3*d^3*m^3 - b^3*c^3*m^3 + 36*a*b^2 *c^2*d - 24*a^2*b*c*d^2 + 57*a*b^2*c^2*d*m - 42*a^2*b*c*d^2*m + 24*a*b^2*c ^2*d*m^2 - 21*a^2*b*c*d^2*m^2 + 3*a*b^2*c^2*d*m^3 - 3*a^2*b*c*d^2*m^3))/(( a*d - b*c)^4*(c + d*x)^(m + 5)*(50*m + 35*m^2 + 10*m^3 + m^4 + 24)) + (3*b ^2*d^2*x^3*(a + b*x)^m*(20*b^2*c^2 + a^2*d^2*m + 9*b^2*c^2*m + a^2*d^2*m^2 + b^2*c^2*m^2 - 10*a*b*c*d*m - 2*a*b*c*d*m^2))/((a*d - b*c)^4*(c + d*x)^( m + 5)*(50*m + 35*m^2 + 10*m^3 + m^4 + 24)) + (6*b^3*d^3*x^4*(a + b*x)^m*( 5*b*c - a*d*m + b*c*m))/((a*d - b*c)^4*(c + d*x)^(m + 5)*(50*m + 35*m^2 + 10*m^3 + m^4 + 24)) + (b*d*x^2*(a + b*x)^m*(60*b^3*c^3 - 2*a^3*d^3*m + 47* b^3*c^3*m - 3*a^3*d^3*m^2 + 12*b^3*c^3*m^2 - a^3*d^3*m^3 + b^3*c^3*m^3 - 6 0*a*b^2*c^2*d*m + 15*a^2*b*c*d^2*m - 27*a*b^2*c^2*d*m^2 + 18*a^2*b*c*d^2*m ^2 - 3*a*b^2*c^2*d*m^3 + 3*a^2*b*c*d^2*m^3))/((a*d - b*c)^4*(c + d*x)^(...