Integrand size = 19, antiderivative size = 186 \[ \int (a+b x)^{-5-n} (c+d x)^n \, dx=-\frac {(a+b x)^{-4-n} (c+d x)^{1+n}}{(b c-a d) (4+n)}+\frac {3 d (a+b x)^{-3-n} (c+d x)^{1+n}}{(b c-a d)^2 (3+n) (4+n)}-\frac {6 d^2 (a+b x)^{-2-n} (c+d x)^{1+n}}{(b c-a d)^3 (2+n) (3+n) (4+n)}+\frac {6 d^3 (a+b x)^{-1-n} (c+d x)^{1+n}}{(b c-a d)^4 (1+n) (2+n) (3+n) (4+n)} \] Output:
-(b*x+a)^(-4-n)*(d*x+c)^(1+n)/(-a*d+b*c)/(4+n)+3*d*(b*x+a)^(-3-n)*(d*x+c)^ (1+n)/(-a*d+b*c)^2/(3+n)/(4+n)-6*d^2*(b*x+a)^(-2-n)*(d*x+c)^(1+n)/(-a*d+b* c)^3/(2+n)/(3+n)/(4+n)+6*d^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 \[ \int (a+b x)^{-5-n} (c+d x)^n \, dx=\frac {(a+b x)^{-4-n} (c+d x)^{1+n} \left (a^3 d^3 \left (24+26 n+9 n^2+n^3\right )-3 a^2 b d^2 \left (12+7 n+n^2\right ) (c+c n-d x)+3 a b^2 d (4+n) \left (c^2 \left (2+3 n+n^2\right )-2 c d (1+n) x+2 d^2 x^2\right )-b^3 \left (c^3 \left (6+11 n+6 n^2+n^3\right )-3 c^2 d \left (2+3 n+n^2\right ) x+6 c d^2 (1+n) x^2-6 d^3 x^3\right )\right )}{(b c-a d)^4 (1+n) (2+n) (3+n) (4+n)} \] Input:
Integrate[(a + b*x)^(-5 - n)*(c + d*x)^n,x]
Output:
((a + b*x)^(-4 - n)*(c + d*x)^(1 + n)*(a^3*d^3*(24 + 26*n + 9*n^2 + n^3) - 3*a^2*b*d^2*(12 + 7*n + n^2)*(c + c*n - d*x) + 3*a*b^2*d*(4 + n)*(c^2*(2 + 3*n + n^2) - 2*c*d*(1 + n)*x + 2*d^2*x^2) - b^3*(c^3*(6 + 11*n + 6*n^2 + n^3) - 3*c^2*d*(2 + 3*n + n^2)*x + 6*c*d^2*(1 + n)*x^2 - 6*d^3*x^3)))/((b *c - a*d)^4*(1 + n)*(2 + n)*(3 + n)*(4 + n))
Time = 0.27 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.03, 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.
| \(\displaystyle \int (a+b x)^{-n-5} (c+d x)^n \, dx\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle -\frac {3 d \int (a+b x)^{-n-4} (c+d x)^ndx}{(n+4) (b c-a d)}-\frac {(a+b x)^{-n-4} (c+d x)^{n+1}}{(n+4) (b c-a d)}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle -\frac {3 d \left (-\frac {2 d \int (a+b x)^{-n-3} (c+d x)^ndx}{(n+3) (b c-a d)}-\frac {(a+b x)^{-n-3} (c+d x)^{n+1}}{(n+3) (b c-a d)}\right )}{(n+4) (b c-a d)}-\frac {(a+b x)^{-n-4} (c+d x)^{n+1}}{(n+4) (b c-a d)}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle -\frac {3 d \left (-\frac {2 d \left (-\frac {d \int (a+b x)^{-n-2} (c+d x)^ndx}{(n+2) (b c-a d)}-\frac {(a+b x)^{-n-2} (c+d x)^{n+1}}{(n+2) (b c-a d)}\right )}{(n+3) (b c-a d)}-\frac {(a+b x)^{-n-3} (c+d x)^{n+1}}{(n+3) (b c-a d)}\right )}{(n+4) (b c-a d)}-\frac {(a+b x)^{-n-4} (c+d x)^{n+1}}{(n+4) (b c-a d)}\) |
| \(\Big \downarrow \) 48 |
| \(\displaystyle -\frac {(a+b x)^{-n-4} (c+d x)^{n+1}}{(n+4) (b c-a d)}-\frac {3 d \left (-\frac {(a+b x)^{-n-3} (c+d x)^{n+1}}{(n+3) (b c-a d)}-\frac {2 d \left (\frac {d (a+b x)^{-n-1} (c+d x)^{n+1}}{(n+1) (n+2) (b c-a d)^2}-\frac {(a+b x)^{-n-2} (c+d x)^{n+1}}{(n+2) (b c-a d)}\right )}{(n+3) (b c-a d)}\right )}{(n+4) (b c-a d)}\) |
Input:
Int[(a + b*x)^(-5 - n)*(c + d*x)^n,x]
Output:
-(((a + b*x)^(-4 - n)*(c + d*x)^(1 + n))/((b*c - a*d)*(4 + n))) - (3*d*(-( ((a + b*x)^(-3 - n)*(c + d*x)^(1 + n))/((b*c - a*d)*(3 + n))) - (2*d*(-((( a + b*x)^(-2 - n)*(c + d*x)^(1 + n))/((b*c - a*d)*(2 + n))) + (d*(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. \(660\) vs. \(2(186)=372\).
Time = 0.38 (sec) , antiderivative size = 661, normalized size of antiderivative = 3.55
| method | result | size |
| gosper | \(\frac {\left (b x +a \right )^{-4-n} \left (x d +c \right )^{1+n} \left (a^{3} d^{3} n^{3}-3 a^{2} b c \,d^{2} n^{3}+3 a^{2} b \,d^{3} n^{2} x +3 a \,b^{2} c^{2} d \,n^{3}-6 a \,b^{2} c \,d^{2} n^{2} x +6 a \,b^{2} d^{3} n \,x^{2}-b^{3} c^{3} n^{3}+3 b^{3} c^{2} d \,n^{2} x -6 b^{3} c \,d^{2} n \,x^{2}+6 d^{3} x^{3} b^{3}+9 a^{3} d^{3} n^{2}-24 a^{2} b c \,d^{2} n^{2}+21 a^{2} b \,d^{3} n x +21 a \,b^{2} c^{2} d \,n^{2}-30 a \,b^{2} c \,d^{2} n x +24 x^{2} a \,b^{2} d^{3}-6 b^{3} c^{3} n^{2}+9 b^{3} c^{2} d n x -6 x^{2} b^{3} c \,d^{2}+26 a^{3} d^{3} n -57 a^{2} b c \,d^{2} n +36 x \,a^{2} b \,d^{3}+42 a \,b^{2} c^{2} d n -24 x a \,b^{2} c \,d^{2}-11 b^{3} c^{3} n +6 x \,b^{3} c^{2} d +24 a^{3} d^{3}-36 a^{2} b c \,d^{2}+24 a \,b^{2} c^{2} d -6 b^{3} c^{3}\right )}{a^{4} d^{4} n^{4}-4 a^{3} b c \,d^{3} n^{4}+6 a^{2} b^{2} c^{2} d^{2} n^{4}-4 a \,b^{3} c^{3} d \,n^{4}+b^{4} c^{4} n^{4}+10 a^{4} d^{4} n^{3}-40 a^{3} b c \,d^{3} n^{3}+60 a^{2} b^{2} c^{2} d^{2} n^{3}-40 a \,b^{3} c^{3} d \,n^{3}+10 b^{4} c^{4} n^{3}+35 a^{4} d^{4} n^{2}-140 a^{3} b c \,d^{3} n^{2}+210 a^{2} b^{2} c^{2} d^{2} n^{2}-140 a \,b^{3} c^{3} d \,n^{2}+35 b^{4} c^{4} n^{2}+50 a^{4} d^{4} n -200 a^{3} b c \,d^{3} n +300 a^{2} b^{2} c^{2} d^{2} n -200 a \,b^{3} c^{3} d n +50 b^{4} c^{4} n +24 d^{4} a^{4}-96 a^{3} b c \,d^{3}+144 a^{2} b^{2} c^{2} d^{2}-96 a \,b^{3} c^{3} d +24 c^{4} b^{4}}\) | \(661\) |
| orering | \(\frac {\left (b x +a \right ) \left (x d +c \right ) \left (a^{3} d^{3} n^{3}-3 a^{2} b c \,d^{2} n^{3}+3 a^{2} b \,d^{3} n^{2} x +3 a \,b^{2} c^{2} d \,n^{3}-6 a \,b^{2} c \,d^{2} n^{2} x +6 a \,b^{2} d^{3} n \,x^{2}-b^{3} c^{3} n^{3}+3 b^{3} c^{2} d \,n^{2} x -6 b^{3} c \,d^{2} n \,x^{2}+6 d^{3} x^{3} b^{3}+9 a^{3} d^{3} n^{2}-24 a^{2} b c \,d^{2} n^{2}+21 a^{2} b \,d^{3} n x +21 a \,b^{2} c^{2} d \,n^{2}-30 a \,b^{2} c \,d^{2} n x +24 x^{2} a \,b^{2} d^{3}-6 b^{3} c^{3} n^{2}+9 b^{3} c^{2} d n x -6 x^{2} b^{3} c \,d^{2}+26 a^{3} d^{3} n -57 a^{2} b c \,d^{2} n +36 x \,a^{2} b \,d^{3}+42 a \,b^{2} c^{2} d n -24 x a \,b^{2} c \,d^{2}-11 b^{3} c^{3} n +6 x \,b^{3} c^{2} d +24 a^{3} d^{3}-36 a^{2} b c \,d^{2}+24 a \,b^{2} c^{2} d -6 b^{3} c^{3}\right ) \left (b x +a \right )^{-5-n} \left (x d +c \right )^{n}}{a^{4} d^{4} n^{4}-4 a^{3} b c \,d^{3} n^{4}+6 a^{2} b^{2} c^{2} d^{2} n^{4}-4 a \,b^{3} c^{3} d \,n^{4}+b^{4} c^{4} n^{4}+10 a^{4} d^{4} n^{3}-40 a^{3} b c \,d^{3} n^{3}+60 a^{2} b^{2} c^{2} d^{2} n^{3}-40 a \,b^{3} c^{3} d \,n^{3}+10 b^{4} c^{4} n^{3}+35 a^{4} d^{4} n^{2}-140 a^{3} b c \,d^{3} n^{2}+210 a^{2} b^{2} c^{2} d^{2} n^{2}-140 a \,b^{3} c^{3} d \,n^{2}+35 b^{4} c^{4} n^{2}+50 a^{4} d^{4} n -200 a^{3} b c \,d^{3} n +300 a^{2} b^{2} c^{2} d^{2} n -200 a \,b^{3} c^{3} d n +50 b^{4} c^{4} n +24 d^{4} a^{4}-96 a^{3} b c \,d^{3}+144 a^{2} b^{2} c^{2} d^{2}-96 a \,b^{3} c^{3} d +24 c^{4} b^{4}}\) | \(669\) |
Input:
int((b*x+a)^(-5-n)*(d*x+c)^n,x,method=_RETURNVERBOSE)
Output:
(b*x+a)^(-4-n)*(d*x+c)^(1+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*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)*(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+9*a^3*d^3*n^2-24*a^2*b*c*d^2*n^2+21*a^2*b*d^3 *n*x+21*a*b^2*c^2*d*n^2-30*a*b^2*c*d^2*n*x+24*a*b^2*d^3*x^2-6*b^3*c^3*n^2+ 9*b^3*c^2*d*n*x-6*b^3*c*d^2*x^2+26*a^3*d^3*n-57*a^2*b*c*d^2*n+36*a^2*b*d^3 *x+42*a*b^2*c^2*d*n-24*a*b^2*c*d^2*x-11*b^3*c^3*n+6*b^3*c^2*d*x+24*a^3*d^3 -36*a^2*b*c*d^2+24*a*b^2*c^2*d-6*b^3*c^3)
Leaf count of result is larger than twice the leaf count of optimal. 959 vs. \(2 (186) = 372\).
Time = 0.12 (sec) , antiderivative size = 959, normalized size of antiderivative = 5.16 \[ \int (a+b x)^{-5-n} (c+d x)^n \, dx =\text {Too large to display} \] Input:
integrate((b*x+a)^(-5-n)*(d*x+c)^n,x, algorithm="fricas")
Output:
(6*b^4*d^4*x^5 - 6*a*b^3*c^4 + 24*a^2*b^2*c^3*d - 36*a^3*b*c^2*d^2 + 24*a^ 4*c*d^3 + 6*(5*a*b^3*d^4 - (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*a^2*b^2*d^4 + (b ^4*c^2*d^2 - 2*a*b^3*c*d^3 + a^2*b^2*d^4)*n^2 + (b^4*c^2*d^2 - 10*a*b^3*c* d^3 + 9*a^2*b^2*d^4)*n)*x^3 - 3*(2*a*b^3*c^4 - 7*a^2*b^2*c^3*d + 8*a^3*b*c ^2*d^2 - 3*a^4*c*d^3)*n^2 + (60*a^3*b*d^4 - (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*(b^4*c^3*d - 6*a*b^3*c^2*d^2 + 9*a^2 *b^2*c*d^3 - 4*a^3*b*d^4)*n^2 - (2*b^4*c^3*d - 15*a*b^3*c^2*d^2 + 60*a^2*b ^2*c*d^3 - 47*a^3*b*d^4)*n)*x^2 - (11*a*b^3*c^4 - 42*a^2*b^2*c^3*d + 57*a^ 3*b*c^2*d^2 - 26*a^4*c*d^3)*n - (6*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 - 24*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*(2*b^4*c^4 - 6*a*b^3*c^3*d + 3*a^2*b^2*c^2*d^2 + 4*a^3*b*c*d^3 - 3*a^4*d^4)*n^2 + (11*b^4*c^4 - 40*a*b^3*c^3*d + 45*a^2*b^2 *c^2*d^2 + 10*a^3*b*c*d^3 - 26*a^4*d^4)*n)*x)*(b*x + a)^(-n - 5)*(d*x + c) ^n/(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. \[ \int (a+b x)^{-5-n} (c+d x)^n \, dx=\text {Exception raised: HeuristicGCDFailed} \] Input:
integrate((b*x+a)**(-5-n)*(d*x+c)**n,x)
Output:
Exception raised: HeuristicGCDFailed >> no luck
\[ \int (a+b x)^{-5-n} (c+d x)^n \, dx=\int { {\left (b x + a\right )}^{-n - 5} {\left (d x + c\right )}^{n} \,d x } \] Input:
integrate((b*x+a)^(-5-n)*(d*x+c)^n,x, algorithm="maxima")
Output:
integrate((b*x + a)^(-n - 5)*(d*x + c)^n, x)
\[ \int (a+b x)^{-5-n} (c+d x)^n \, dx=\int { {\left (b x + a\right )}^{-n - 5} {\left (d x + c\right )}^{n} \,d x } \] Input:
integrate((b*x+a)^(-5-n)*(d*x+c)^n,x, algorithm="giac")
Output:
integrate((b*x + a)^(-n - 5)*(d*x + c)^n, x)
Time = 0.94 (sec) , antiderivative size = 944, normalized size of antiderivative = 5.08 \[ \int (a+b x)^{-5-n} (c+d x)^n \, dx=\frac {a\,c\,{\left (c+d\,x\right )}^n\,\left (a^3\,d^3\,n^3+9\,a^3\,d^3\,n^2+26\,a^3\,d^3\,n+24\,a^3\,d^3-3\,a^2\,b\,c\,d^2\,n^3-24\,a^2\,b\,c\,d^2\,n^2-57\,a^2\,b\,c\,d^2\,n-36\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d\,n^3+21\,a\,b^2\,c^2\,d\,n^2+42\,a\,b^2\,c^2\,d\,n+24\,a\,b^2\,c^2\,d-b^3\,c^3\,n^3-6\,b^3\,c^3\,n^2-11\,b^3\,c^3\,n-6\,b^3\,c^3\right )}{{\left (a\,d-b\,c\right )}^4\,{\left (a+b\,x\right )}^{n+5}\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}-\frac {x\,{\left (c+d\,x\right )}^n\,\left (-a^4\,d^4\,n^3-9\,a^4\,d^4\,n^2-26\,a^4\,d^4\,n-24\,a^4\,d^4+2\,a^3\,b\,c\,d^3\,n^3+12\,a^3\,b\,c\,d^3\,n^2+10\,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^3-18\,a\,b^3\,c^3\,d\,n^2-40\,a\,b^3\,c^3\,d\,n-24\,a\,b^3\,c^3\,d+b^4\,c^4\,n^3+6\,b^4\,c^4\,n^2+11\,b^4\,c^4\,n+6\,b^4\,c^4\right )}{{\left (a\,d-b\,c\right )}^4\,{\left (a+b\,x\right )}^{n+5}\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}+\frac {6\,b^4\,d^4\,x^5\,{\left (c+d\,x\right )}^n}{{\left (a\,d-b\,c\right )}^4\,{\left (a+b\,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 (c+d\,x\right )}^n\,\left (a^2\,d^2\,n^2+9\,a^2\,d^2\,n+20\,a^2\,d^2-2\,a\,b\,c\,d\,n^2-10\,a\,b\,c\,d\,n+b^2\,c^2\,n^2+b^2\,c^2\,n\right )}{{\left (a\,d-b\,c\right )}^4\,{\left (a+b\,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 (c+d\,x\right )}^n\,\left (5\,a\,d+a\,d\,n-b\,c\,n\right )}{{\left (a\,d-b\,c\right )}^4\,{\left (a+b\,x\right )}^{n+5}\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}+\frac {b\,d\,x^2\,{\left (c+d\,x\right )}^n\,\left (a^3\,d^3\,n^3+12\,a^3\,d^3\,n^2+47\,a^3\,d^3\,n+60\,a^3\,d^3-3\,a^2\,b\,c\,d^2\,n^3-27\,a^2\,b\,c\,d^2\,n^2-60\,a^2\,b\,c\,d^2\,n+3\,a\,b^2\,c^2\,d\,n^3+18\,a\,b^2\,c^2\,d\,n^2+15\,a\,b^2\,c^2\,d\,n-b^3\,c^3\,n^3-3\,b^3\,c^3\,n^2-2\,b^3\,c^3\,n\right )}{{\left (a\,d-b\,c\right )}^4\,{\left (a+b\,x\right )}^{n+5}\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )} \] Input:
int((c + d*x)^n/(a + b*x)^(n + 5),x)
Output:
(a*c*(c + d*x)^n*(24*a^3*d^3 - 6*b^3*c^3 + 26*a^3*d^3*n - 11*b^3*c^3*n + 9 *a^3*d^3*n^2 - 6*b^3*c^3*n^2 + a^3*d^3*n^3 - b^3*c^3*n^3 + 24*a*b^2*c^2*d - 36*a^2*b*c*d^2 + 42*a*b^2*c^2*d*n - 57*a^2*b*c*d^2*n + 21*a*b^2*c^2*d*n^ 2 - 24*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*(a + b*x)^(n + 5)*(50*n + 35*n^2 + 10*n^3 + n^4 + 24)) - (x*(c + d*x )^n*(6*b^4*c^4 - 24*a^4*d^4 - 26*a^4*d^4*n + 11*b^4*c^4*n - 9*a^4*d^4*n^2 + 6*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 - 40*a*b^3*c^3*d*n + 10*a^3*b*c*d^3*n + 9*a^2*b^2 *c^2*d^2*n^2 - 18*a*b^3*c^3*d*n^2 + 12*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*(a + b*x)^(n + 5)*(50*n + 35*n^2 + 10*n^3 + n^4 + 24)) + (6*b^4*d^4*x^5*(c + d*x)^n)/(( a*d - b*c)^4*(a + b*x)^(n + 5)*(50*n + 35*n^2 + 10*n^3 + n^4 + 24)) + (3*b ^2*d^2*x^3*(c + d*x)^n*(20*a^2*d^2 + 9*a^2*d^2*n + 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*(a + b*x)^( n + 5)*(50*n + 35*n^2 + 10*n^3 + n^4 + 24)) + (6*b^3*d^3*x^4*(c + d*x)^n*( 5*a*d + a*d*n - b*c*n))/((a*d - b*c)^4*(a + b*x)^(n + 5)*(50*n + 35*n^2 + 10*n^3 + n^4 + 24)) + (b*d*x^2*(c + d*x)^n*(60*a^3*d^3 + 47*a^3*d^3*n - 2* b^3*c^3*n + 12*a^3*d^3*n^2 - 3*b^3*c^3*n^2 + a^3*d^3*n^3 - b^3*c^3*n^3 + 1 5*a*b^2*c^2*d*n - 60*a^2*b*c*d^2*n + 18*a*b^2*c^2*d*n^2 - 27*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*(a + b*x)^(...
\[ \int (a+b x)^{-5-n} (c+d x)^n \, dx=\int \frac {\left (d x +c \right )^{n}}{\left (b x +a \right )^{n} a^{5}+5 \left (b x +a \right )^{n} a^{4} b x +10 \left (b x +a \right )^{n} a^{3} b^{2} x^{2}+10 \left (b x +a \right )^{n} a^{2} b^{3} x^{3}+5 \left (b x +a \right )^{n} a \,b^{4} x^{4}+\left (b x +a \right )^{n} b^{5} x^{5}}d x \] Input:
int((b*x+a)^(-5-n)*(d*x+c)^n,x)
Output:
int((c + d*x)**n/((a + b*x)**n*a**5 + 5*(a + b*x)**n*a**4*b*x + 10*(a + b* x)**n*a**3*b**2*x**2 + 10*(a + b*x)**n*a**2*b**3*x**3 + 5*(a + b*x)**n*a*b **4*x**4 + (a + b*x)**n*b**5*x**5),x)