Integrand size = 19, antiderivative size = 185 \[ \int (a+b x)^n (c+d x)^{-5-n} \, dx=\frac {(a+b x)^{1+n} (c+d x)^{-4-n}}{(b c-a d) (4+n)}+\frac {3 b (a+b x)^{1+n} (c+d x)^{-3-n}}{(b c-a d)^2 (3+n) (4+n)}+\frac {6 b^2 (a+b x)^{1+n} (c+d x)^{-2-n}}{(b c-a d)^3 (2+n) (3+n) (4+n)}+\frac {6 b^3 (a+b x)^{1+n} (c+d x)^{-1-n}}{(b c-a d)^4 (1+n) (2+n) (3+n) (4+n)} \] Output:
(b*x+a)^(1+n)*(d*x+c)^(-4-n)/(-a*d+b*c)/(4+n)+3*b*(b*x+a)^(1+n)*(d*x+c)^(- 3-n)/(-a*d+b*c)^2/(3+n)/(4+n)+6*b^2*(b*x+a)^(1+n)*(d*x+c)^(-2-n)/(-a*d+b*c )^3/(2+n)/(3+n)/(4+n)+6*b^3*(b*x+a)^(1+n)*(d*x+c)^(-1-n)/(-a*d+b*c)^4/(1+n )/(2+n)/(3+n)/(4+n)
Time = 0.09 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.05 \[ \int (a+b x)^n (c+d x)^{-5-n} \, dx=\frac {(a+b x)^{1+n} (c+d x)^{-4-n} \left (-a^3 d^3 \left (6+11 n+6 n^2+n^3\right )+3 a^2 b d^2 \left (2+3 n+n^2\right ) (c (4+n)+d x)-3 a b^2 d (1+n) \left (c^2 \left (12+7 n+n^2\right )+2 c d (4+n) x+2 d^2 x^2\right )+b^3 \left (c^3 \left (24+26 n+9 n^2+n^3\right )+3 c^2 d \left (12+7 n+n^2\right ) x+6 c d^2 (4+n) x^2+6 d^3 x^3\right )\right )}{(b c-a d)^4 (1+n) (2+n) (3+n) (4+n)} \] Input:
Integrate[(a + b*x)^n*(c + d*x)^(-5 - n),x]
Output:
((a + b*x)^(1 + n)*(c + d*x)^(-4 - n)*(-(a^3*d^3*(6 + 11*n + 6*n^2 + n^3)) + 3*a^2*b*d^2*(2 + 3*n + n^2)*(c*(4 + n) + d*x) - 3*a*b^2*d*(1 + n)*(c^2* (12 + 7*n + n^2) + 2*c*d*(4 + n)*x + 2*d^2*x^2) + b^3*(c^3*(24 + 26*n + 9* n^2 + n^3) + 3*c^2*d*(12 + 7*n + n^2)*x + 6*c*d^2*(4 + n)*x^2 + 6*d^3*x^3) ))/((b*c - a*d)^4*(1 + n)*(2 + n)*(3 + n)*(4 + n))
Time = 0.24 (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.
| \(\displaystyle \int (a+b x)^n (c+d x)^{-n-5} \, dx\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle \frac {3 b \int (a+b x)^n (c+d x)^{-n-4}dx}{(n+4) (b c-a d)}+\frac {(a+b x)^{n+1} (c+d x)^{-n-4}}{(n+4) (b c-a d)}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle \frac {3 b \left (\frac {2 b \int (a+b x)^n (c+d x)^{-n-3}dx}{(n+3) (b c-a d)}+\frac {(a+b x)^{n+1} (c+d x)^{-n-3}}{(n+3) (b c-a d)}\right )}{(n+4) (b c-a d)}+\frac {(a+b x)^{n+1} (c+d x)^{-n-4}}{(n+4) (b c-a d)}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle \frac {3 b \left (\frac {2 b \left (\frac {b \int (a+b x)^n (c+d x)^{-n-2}dx}{(n+2) (b c-a d)}+\frac {(a+b x)^{n+1} (c+d x)^{-n-2}}{(n+2) (b c-a d)}\right )}{(n+3) (b c-a d)}+\frac {(a+b x)^{n+1} (c+d x)^{-n-3}}{(n+3) (b c-a d)}\right )}{(n+4) (b c-a d)}+\frac {(a+b x)^{n+1} (c+d x)^{-n-4}}{(n+4) (b c-a d)}\) |
| \(\Big \downarrow \) 48 |
| \(\displaystyle \frac {(a+b x)^{n+1} (c+d x)^{-n-4}}{(n+4) (b c-a d)}+\frac {3 b \left (\frac {(a+b x)^{n+1} (c+d x)^{-n-3}}{(n+3) (b c-a d)}+\frac {2 b \left (\frac {(a+b x)^{n+1} (c+d x)^{-n-2}}{(n+2) (b c-a d)}+\frac {b (a+b x)^{n+1} (c+d x)^{-n-1}}{(n+1) (n+2) (b c-a d)^2}\right )}{(n+3) (b c-a d)}\right )}{(n+4) (b c-a d)}\) |
Input:
Int[(a + b*x)^n*(c + d*x)^(-5 - n),x]
Output:
((a + b*x)^(1 + n)*(c + d*x)^(-4 - n))/((b*c - a*d)*(4 + n)) + (3*b*(((a + b*x)^(1 + n)*(c + d*x)^(-3 - n))/((b*c - a*d)*(3 + n)) + (2*b*(((a + b*x) ^(1 + n)*(c + d*x)^(-2 - n))/((b*c - a*d)*(2 + n)) + (b*(a + b*x)^(1 + n)* (c + d*x)^(-1 - n))/((b*c - a*d)^2*(1 + n)*(2 + n))))/((b*c - a*d)*(3 + n) )))/((b*c - a*d)*(4 + n))
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 = 0.38 (sec) , antiderivative size = 662, normalized size of antiderivative = 3.58
| method | result | size |
| gosper | \(-\frac {\left (b x +a \right )^{1+n} \left (x d +c \right )^{-4-n} \left (a^{3} d^{3} n^{3}-3 a^{2} b c \,d^{2} n^{3}-3 a^{2} b \,d^{3} n^{2} x +3 a \,b^{2} c^{2} d \,n^{3}+6 a \,b^{2} c \,d^{2} n^{2} x +6 a \,b^{2} d^{3} n \,x^{2}-b^{3} c^{3} n^{3}-3 b^{3} c^{2} d \,n^{2} x -6 b^{3} c \,d^{2} n \,x^{2}-6 d^{3} x^{3} b^{3}+6 a^{3} d^{3} n^{2}-21 a^{2} b c \,d^{2} n^{2}-9 a^{2} b \,d^{3} n x +24 a \,b^{2} c^{2} d \,n^{2}+30 a \,b^{2} c \,d^{2} n x +6 x^{2} a \,b^{2} d^{3}-9 b^{3} c^{3} n^{2}-21 b^{3} c^{2} d n x -24 x^{2} b^{3} c \,d^{2}+11 a^{3} d^{3} n -42 a^{2} b c \,d^{2} n -6 x \,a^{2} b \,d^{3}+57 a \,b^{2} c^{2} d n +24 x a \,b^{2} c \,d^{2}-26 b^{3} c^{3} n -36 x \,b^{3} c^{2} d +6 a^{3} d^{3}-24 a^{2} b c \,d^{2}+36 a \,b^{2} c^{2} d -24 b^{3} c^{3}\right )}{a^{4} d^{4} n^{4}-4 a^{3} b c \,d^{3} n^{4}+6 a^{2} b^{2} c^{2} d^{2} n^{4}-4 a \,b^{3} c^{3} d \,n^{4}+b^{4} c^{4} n^{4}+10 a^{4} d^{4} n^{3}-40 a^{3} b c \,d^{3} n^{3}+60 a^{2} b^{2} c^{2} d^{2} n^{3}-40 a \,b^{3} c^{3} d \,n^{3}+10 b^{4} c^{4} n^{3}+35 a^{4} d^{4} n^{2}-140 a^{3} b c \,d^{3} n^{2}+210 a^{2} b^{2} c^{2} d^{2} n^{2}-140 a \,b^{3} c^{3} d \,n^{2}+35 b^{4} c^{4} n^{2}+50 a^{4} d^{4} n -200 a^{3} b c \,d^{3} n +300 a^{2} b^{2} c^{2} d^{2} n -200 a \,b^{3} c^{3} d n +50 b^{4} c^{4} n +24 d^{4} a^{4}-96 a^{3} b c \,d^{3}+144 a^{2} b^{2} c^{2} d^{2}-96 a \,b^{3} c^{3} d +24 c^{4} b^{4}}\) | \(662\) |
| orering | \(-\frac {\left (b x +a \right ) \left (x d +c \right ) \left (a^{3} d^{3} n^{3}-3 a^{2} b c \,d^{2} n^{3}-3 a^{2} b \,d^{3} n^{2} x +3 a \,b^{2} c^{2} d \,n^{3}+6 a \,b^{2} c \,d^{2} n^{2} x +6 a \,b^{2} d^{3} n \,x^{2}-b^{3} c^{3} n^{3}-3 b^{3} c^{2} d \,n^{2} x -6 b^{3} c \,d^{2} n \,x^{2}-6 d^{3} x^{3} b^{3}+6 a^{3} d^{3} n^{2}-21 a^{2} b c \,d^{2} n^{2}-9 a^{2} b \,d^{3} n x +24 a \,b^{2} c^{2} d \,n^{2}+30 a \,b^{2} c \,d^{2} n x +6 x^{2} a \,b^{2} d^{3}-9 b^{3} c^{3} n^{2}-21 b^{3} c^{2} d n x -24 x^{2} b^{3} c \,d^{2}+11 a^{3} d^{3} n -42 a^{2} b c \,d^{2} n -6 x \,a^{2} b \,d^{3}+57 a \,b^{2} c^{2} d n +24 x a \,b^{2} c \,d^{2}-26 b^{3} c^{3} n -36 x \,b^{3} c^{2} d +6 a^{3} d^{3}-24 a^{2} b c \,d^{2}+36 a \,b^{2} c^{2} d -24 b^{3} c^{3}\right ) \left (b x +a \right )^{n} \left (x d +c \right )^{-5-n}}{a^{4} d^{4} n^{4}-4 a^{3} b c \,d^{3} n^{4}+6 a^{2} b^{2} c^{2} d^{2} n^{4}-4 a \,b^{3} c^{3} d \,n^{4}+b^{4} c^{4} n^{4}+10 a^{4} d^{4} n^{3}-40 a^{3} b c \,d^{3} n^{3}+60 a^{2} b^{2} c^{2} d^{2} n^{3}-40 a \,b^{3} c^{3} d \,n^{3}+10 b^{4} c^{4} n^{3}+35 a^{4} d^{4} n^{2}-140 a^{3} b c \,d^{3} n^{2}+210 a^{2} b^{2} c^{2} d^{2} n^{2}-140 a \,b^{3} c^{3} d \,n^{2}+35 b^{4} c^{4} n^{2}+50 a^{4} d^{4} n -200 a^{3} b c \,d^{3} n +300 a^{2} b^{2} c^{2} d^{2} n -200 a \,b^{3} c^{3} d n +50 b^{4} c^{4} n +24 d^{4} a^{4}-96 a^{3} b c \,d^{3}+144 a^{2} b^{2} c^{2} d^{2}-96 a \,b^{3} c^{3} d +24 c^{4} b^{4}}\) | \(670\) |
Input:
int((b*x+a)^n*(d*x+c)^(-5-n),x,method=_RETURNVERBOSE)
Output:
-(b*x+a)^(1+n)*(d*x+c)^(-4-n)/(a^4*d^4*n^4-4*a^3*b*c*d^3*n^4+6*a^2*b^2*c^2 *d^2*n^4-4*a*b^3*c^3*d*n^4+b^4*c^4*n^4+10*a^4*d^4*n^3-40*a^3*b*c*d^3*n^3+6 0*a^2*b^2*c^2*d^2*n^3-40*a*b^3*c^3*d*n^3+10*b^4*c^4*n^3+35*a^4*d^4*n^2-140 *a^3*b*c*d^3*n^2+210*a^2*b^2*c^2*d^2*n^2-140*a*b^3*c^3*d*n^2+35*b^4*c^4*n^ 2+50*a^4*d^4*n-200*a^3*b*c*d^3*n+300*a^2*b^2*c^2*d^2*n-200*a*b^3*c^3*d*n+5 0*b^4*c^4*n+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*n^3-3*a^2*b*c*d^2*n^3-3*a^2*b*d^3*n^2*x+3*a*b^2*c^2*d* n^3+6*a*b^2*c*d^2*n^2*x+6*a*b^2*d^3*n*x^2-b^3*c^3*n^3-3*b^3*c^2*d*n^2*x-6* b^3*c*d^2*n*x^2-6*b^3*d^3*x^3+6*a^3*d^3*n^2-21*a^2*b*c*d^2*n^2-9*a^2*b*d^3 *n*x+24*a*b^2*c^2*d*n^2+30*a*b^2*c*d^2*n*x+6*a*b^2*d^3*x^2-9*b^3*c^3*n^2-2 1*b^3*c^2*d*n*x-24*b^3*c*d^2*x^2+11*a^3*d^3*n-42*a^2*b*c*d^2*n-6*a^2*b*d^3 *x+57*a*b^2*c^2*d*n+24*a*b^2*c*d^2*x-26*b^3*c^3*n-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 = 0.18 (sec) , antiderivative size = 954, normalized size of antiderivative = 5.16 \[ \int (a+b x)^n (c+d x)^{-5-n} \, dx =\text {Too large to display} \] Input:
integrate((b*x+a)^n*(d*x+c)^(-5-n),x, algorithm="fricas")
Output:
(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)*n)*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)*n^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)*n^2 + (9*b^4*c^2*d^2 - 10*a*b^3* c*d^3 + a^2*b^2*d^4)*n)*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)*n^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)*n^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)*n^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)*n)*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)*n + (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)*n^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)*n^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)*n)*x)*(b*x + a)^n*(d*x + c)^(-n - 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)*n^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)*n^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)*n^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)*n)
Exception generated. \[ \int (a+b x)^n (c+d x)^{-5-n} \, dx=\text {Exception raised: HeuristicGCDFailed} \] Input:
integrate((b*x+a)**n*(d*x+c)**(-5-n),x)
Output:
Exception raised: HeuristicGCDFailed >> no luck
\[ \int (a+b x)^n (c+d x)^{-5-n} \, dx=\int { {\left (b x + a\right )}^{n} {\left (d x + c\right )}^{-n - 5} \,d x } \] Input:
integrate((b*x+a)^n*(d*x+c)^(-5-n),x, algorithm="maxima")
Output:
integrate((b*x + a)^n*(d*x + c)^(-n - 5), x)
\[ \int (a+b x)^n (c+d x)^{-5-n} \, dx=\int { {\left (b x + a\right )}^{n} {\left (d x + c\right )}^{-n - 5} \,d x } \] Input:
integrate((b*x+a)^n*(d*x+c)^(-5-n),x, algorithm="giac")
Output:
integrate((b*x + a)^n*(d*x + c)^(-n - 5), x)
Time = 0.93 (sec) , antiderivative size = 945, normalized size of antiderivative = 5.11 \[ \int (a+b x)^n (c+d x)^{-5-n} \, dx=\frac {6\,b^4\,d^4\,x^5\,{\left (a+b\,x\right )}^n}{{\left (a\,d-b\,c\right )}^4\,{\left (c+d\,x\right )}^{n+5}\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}-\frac {a\,c\,{\left (a+b\,x\right )}^n\,\left (a^3\,d^3\,n^3+6\,a^3\,d^3\,n^2+11\,a^3\,d^3\,n+6\,a^3\,d^3-3\,a^2\,b\,c\,d^2\,n^3-21\,a^2\,b\,c\,d^2\,n^2-42\,a^2\,b\,c\,d^2\,n-24\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d\,n^3+24\,a\,b^2\,c^2\,d\,n^2+57\,a\,b^2\,c^2\,d\,n+36\,a\,b^2\,c^2\,d-b^3\,c^3\,n^3-9\,b^3\,c^3\,n^2-26\,b^3\,c^3\,n-24\,b^3\,c^3\right )}{{\left (a\,d-b\,c\right )}^4\,{\left (c+d\,x\right )}^{n+5}\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}-\frac {x\,{\left (a+b\,x\right )}^n\,\left (a^4\,d^4\,n^3+6\,a^4\,d^4\,n^2+11\,a^4\,d^4\,n+6\,a^4\,d^4-2\,a^3\,b\,c\,d^3\,n^3-18\,a^3\,b\,c\,d^3\,n^2-40\,a^3\,b\,c\,d^3\,n-24\,a^3\,b\,c\,d^3+9\,a^2\,b^2\,c^2\,d^2\,n^2+45\,a^2\,b^2\,c^2\,d^2\,n+36\,a^2\,b^2\,c^2\,d^2+2\,a\,b^3\,c^3\,d\,n^3+12\,a\,b^3\,c^3\,d\,n^2+10\,a\,b^3\,c^3\,d\,n-24\,a\,b^3\,c^3\,d-b^4\,c^4\,n^3-9\,b^4\,c^4\,n^2-26\,b^4\,c^4\,n-24\,b^4\,c^4\right )}{{\left (a\,d-b\,c\right )}^4\,{\left (c+d\,x\right )}^{n+5}\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}+\frac {3\,b^2\,d^2\,x^3\,{\left (a+b\,x\right )}^n\,\left (a^2\,d^2\,n^2+a^2\,d^2\,n-2\,a\,b\,c\,d\,n^2-10\,a\,b\,c\,d\,n+b^2\,c^2\,n^2+9\,b^2\,c^2\,n+20\,b^2\,c^2\right )}{{\left (a\,d-b\,c\right )}^4\,{\left (c+d\,x\right )}^{n+5}\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}+\frac {6\,b^3\,d^3\,x^4\,{\left (a+b\,x\right )}^n\,\left (5\,b\,c-a\,d\,n+b\,c\,n\right )}{{\left (a\,d-b\,c\right )}^4\,{\left (c+d\,x\right )}^{n+5}\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}+\frac {b\,d\,x^2\,{\left (a+b\,x\right )}^n\,\left (-a^3\,d^3\,n^3-3\,a^3\,d^3\,n^2-2\,a^3\,d^3\,n+3\,a^2\,b\,c\,d^2\,n^3+18\,a^2\,b\,c\,d^2\,n^2+15\,a^2\,b\,c\,d^2\,n-3\,a\,b^2\,c^2\,d\,n^3-27\,a\,b^2\,c^2\,d\,n^2-60\,a\,b^2\,c^2\,d\,n+b^3\,c^3\,n^3+12\,b^3\,c^3\,n^2+47\,b^3\,c^3\,n+60\,b^3\,c^3\right )}{{\left (a\,d-b\,c\right )}^4\,{\left (c+d\,x\right )}^{n+5}\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )} \] Input:
int((a + b*x)^n/(c + d*x)^(n + 5),x)
Output:
(6*b^4*d^4*x^5*(a + b*x)^n)/((a*d - b*c)^4*(c + d*x)^(n + 5)*(50*n + 35*n^ 2 + 10*n^3 + n^4 + 24)) - (a*c*(a + b*x)^n*(6*a^3*d^3 - 24*b^3*c^3 + 11*a^ 3*d^3*n - 26*b^3*c^3*n + 6*a^3*d^3*n^2 - 9*b^3*c^3*n^2 + a^3*d^3*n^3 - b^3 *c^3*n^3 + 36*a*b^2*c^2*d - 24*a^2*b*c*d^2 + 57*a*b^2*c^2*d*n - 42*a^2*b*c *d^2*n + 24*a*b^2*c^2*d*n^2 - 21*a^2*b*c*d^2*n^2 + 3*a*b^2*c^2*d*n^3 - 3*a ^2*b*c*d^2*n^3))/((a*d - b*c)^4*(c + d*x)^(n + 5)*(50*n + 35*n^2 + 10*n^3 + n^4 + 24)) - (x*(a + b*x)^n*(6*a^4*d^4 - 24*b^4*c^4 + 11*a^4*d^4*n - 26* b^4*c^4*n + 6*a^4*d^4*n^2 - 9*b^4*c^4*n^2 + a^4*d^4*n^3 - b^4*c^4*n^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*n - 40 *a^3*b*c*d^3*n + 9*a^2*b^2*c^2*d^2*n^2 + 12*a*b^3*c^3*d*n^2 - 18*a^3*b*c*d ^3*n^2 + 2*a*b^3*c^3*d*n^3 - 2*a^3*b*c*d^3*n^3 + 45*a^2*b^2*c^2*d^2*n))/(( a*d - b*c)^4*(c + d*x)^(n + 5)*(50*n + 35*n^2 + 10*n^3 + n^4 + 24)) + (3*b ^2*d^2*x^3*(a + b*x)^n*(20*b^2*c^2 + a^2*d^2*n + 9*b^2*c^2*n + a^2*d^2*n^2 + b^2*c^2*n^2 - 10*a*b*c*d*n - 2*a*b*c*d*n^2))/((a*d - b*c)^4*(c + d*x)^( n + 5)*(50*n + 35*n^2 + 10*n^3 + n^4 + 24)) + (6*b^3*d^3*x^4*(a + b*x)^n*( 5*b*c - a*d*n + b*c*n))/((a*d - b*c)^4*(c + d*x)^(n + 5)*(50*n + 35*n^2 + 10*n^3 + n^4 + 24)) + (b*d*x^2*(a + b*x)^n*(60*b^3*c^3 - 2*a^3*d^3*n + 47* b^3*c^3*n - 3*a^3*d^3*n^2 + 12*b^3*c^3*n^2 - a^3*d^3*n^3 + b^3*c^3*n^3 - 6 0*a*b^2*c^2*d*n + 15*a^2*b*c*d^2*n - 27*a*b^2*c^2*d*n^2 + 18*a^2*b*c*d^2*n ^2 - 3*a*b^2*c^2*d*n^3 + 3*a^2*b*c*d^2*n^3))/((a*d - b*c)^4*(c + d*x)^(...
\[ \int (a+b x)^n (c+d x)^{-5-n} \, dx=\int \frac {\left (b x +a \right )^{n}}{\left (d x +c \right )^{n} c^{5}+5 \left (d x +c \right )^{n} c^{4} d x +10 \left (d x +c \right )^{n} c^{3} d^{2} x^{2}+10 \left (d x +c \right )^{n} c^{2} d^{3} x^{3}+5 \left (d x +c \right )^{n} c \,d^{4} x^{4}+\left (d x +c \right )^{n} d^{5} x^{5}}d x \] Input:
int((b*x+a)^n*(d*x+c)^(-5-n),x)
Output:
int((a + b*x)**n/((c + d*x)**n*c**5 + 5*(c + d*x)**n*c**4*d*x + 10*(c + d* x)**n*c**3*d**2*x**2 + 10*(c + d*x)**n*c**2*d**3*x**3 + 5*(c + d*x)**n*c*d **4*x**4 + (c + d*x)**n*d**5*x**5),x)