\(\int (a+b x) (c+d x)^{-4+n} (e+f x)^{-n} \, dx\) [256]

Optimal result

Integrand size = 24, antiderivative size = 205 \[ \int (a+b x) (c+d x)^{-4+n} (e+f x)^{-n} \, dx=\frac {(b c-a d) (c+d x)^{-3+n} (e+f x)^{1-n}}{d (d e-c f) (3-n)}+\frac {(2 a d f+b c f (1-n)-b d e (3-n)) (c+d x)^{-2+n} (e+f x)^{1-n}}{d (d e-c f)^2 (2-n) (3-n)}-\frac {f (2 a d f+b c f (1-n)-b d e (3-n)) (c+d x)^{-1+n} (e+f x)^{1-n}}{d (d e-c f)^3 (1-n) (2-n) (3-n)} \] Output:

(-a*d+b*c)*(d*x+c)^(-3+n)*(f*x+e)^(1-n)/d/(-c*f+d*e)/(3-n)+(2*a*d*f+b*c*f* 
(1-n)-b*d*e*(3-n))*(d*x+c)^(-2+n)*(f*x+e)^(1-n)/d/(-c*f+d*e)^2/(2-n)/(3-n) 
-f*(2*a*d*f+b*c*f*(1-n)-b*d*e*(3-n))*(d*x+c)^(-1+n)*(f*x+e)^(1-n)/d/(-c*f+ 
d*e)^3/(1-n)/(2-n)/(3-n)
 

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.55 \[ \int (a+b x) (c+d x)^{-4+n} (e+f x)^{-n} \, dx=\frac {(c+d x)^{-3+n} (e+f x)^{1-n} \left (-b c+a d+\frac {(2 a d f+b d e (-3+n)-b c f (-1+n)) (c+d x) (-c f (-2+n)+d e (-1+n)+d f x)}{(d e-c f)^2 (-2+n) (-1+n)}\right )}{d (d e-c f) (-3+n)} \] Input:

Integrate[((a + b*x)*(c + d*x)^(-4 + n))/(e + f*x)^n,x]
 

Output:

((c + d*x)^(-3 + n)*(e + f*x)^(1 - n)*(-(b*c) + a*d + ((2*a*d*f + b*d*e*(- 
3 + n) - b*c*f*(-1 + n))*(c + d*x)*(-(c*f*(-2 + n)) + d*e*(-1 + n) + d*f*x 
))/((d*e - c*f)^2*(-2 + n)*(-1 + n))))/(d*(d*e - c*f)*(-3 + n))
 

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.89, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {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.

\(\displaystyle \int (a+b x) (c+d x)^{n-4} (e+f x)^{-n} \, dx\)

\(\Big \downarrow \) 88

\(\displaystyle \frac {(b c-a d) (c+d x)^{n-3} (e+f x)^{1-n}}{d (3-n) (d e-c f)}-\frac {(2 a d f+b c f (1-n)-b d e (3-n)) \int (c+d x)^{n-3} (e+f x)^{-n}dx}{d (3-n) (d e-c f)}\)

\(\Big \downarrow \) 55

\(\displaystyle \frac {(b c-a d) (c+d x)^{n-3} (e+f x)^{1-n}}{d (3-n) (d e-c f)}-\frac {(2 a d f+b c f (1-n)-b d e (3-n)) \left (-\frac {f \int (c+d x)^{n-2} (e+f x)^{-n}dx}{(2-n) (d e-c f)}-\frac {(c+d x)^{n-2} (e+f x)^{1-n}}{(2-n) (d e-c f)}\right )}{d (3-n) (d e-c f)}\)

\(\Big \downarrow \) 48

\(\displaystyle \frac {(b c-a d) (c+d x)^{n-3} (e+f x)^{1-n}}{d (3-n) (d e-c f)}-\frac {\left (\frac {f (c+d x)^{n-1} (e+f x)^{1-n}}{(1-n) (2-n) (d e-c f)^2}-\frac {(c+d x)^{n-2} (e+f x)^{1-n}}{(2-n) (d e-c f)}\right ) (2 a d f+b c f (1-n)-b d e (3-n))}{d (3-n) (d e-c f)}\)

Input:

Int[((a + b*x)*(c + d*x)^(-4 + n))/(e + f*x)^n,x]
 

Output:

((b*c - a*d)*(c + d*x)^(-3 + n)*(e + f*x)^(1 - n))/(d*(d*e - c*f)*(3 - n)) 
 - ((2*a*d*f + b*c*f*(1 - n) - b*d*e*(3 - n))*(-(((c + d*x)^(-2 + n)*(e + 
f*x)^(1 - n))/((d*e - c*f)*(2 - n))) + (f*(c + d*x)^(-1 + n)*(e + f*x)^(1 
- n))/((d*e - c*f)^2*(1 - n)*(2 - n))))/(d*(d*e - c*f)*(3 - n))
 

Defintions of rubi rules used

rule 48

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]
 

rule 55

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])
 

rule 88

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(505\) vs. \(2(205)=410\).

Time = 0.54 (sec) , antiderivative size = 506, normalized size of antiderivative = 2.47

method	result	size
gosper	\(-\frac {\left (x d +c \right )^{-3+n} \left (f x +e \right ) \left (f x +e \right )^{-n} \left (b \,c^{2} f^{2} n^{2} x -2 b c d e f \,n^{2} x -b c d \,f^{2} n \,x^{2}+b \,d^{2} e^{2} n^{2} x +b \,d^{2} e f n \,x^{2}+a \,c^{2} f^{2} n^{2}-2 a c d e f \,n^{2}-2 a c d \,f^{2} n x +a \,d^{2} e^{2} n^{2}+2 a \,d^{2} e f n x +2 a \,d^{2} f^{2} x^{2}-4 b \,c^{2} f^{2} n x +8 b c d e f n x +b c d \,f^{2} x^{2}-4 b \,d^{2} e^{2} n x -3 b \,d^{2} e f \,x^{2}-5 a \,c^{2} f^{2} n +8 a c d e f n +6 a c d \,f^{2} x -3 a \,d^{2} e^{2} n -2 a \,d^{2} e f x +b \,c^{2} e f n +3 b \,c^{2} f^{2} x -b c d \,e^{2} n -10 b c d e f x +3 b \,d^{2} e^{2} x +6 a \,c^{2} f^{2}-6 a c d e f +2 a \,d^{2} e^{2}-3 b \,c^{2} e f +b c d \,e^{2}\right )}{c^{3} f^{3} n^{3}-3 c^{2} d e \,f^{2} n^{3}+3 c \,d^{2} e^{2} f \,n^{3}-d^{3} e^{3} n^{3}-6 c^{3} f^{3} n^{2}+18 c^{2} d e \,f^{2} n^{2}-18 c \,d^{2} e^{2} f \,n^{2}+6 d^{3} e^{3} n^{2}+11 c^{3} f^{3} n -33 c^{2} d e \,f^{2} n +33 c \,d^{2} e^{2} f n -11 d^{3} e^{3} n -6 c^{3} f^{3}+18 c^{2} d e \,f^{2}-18 c \,d^{2} e^{2} f +6 d^{3} e^{3}}\)	\(506\)
orering	\(-\frac {\left (f x +e \right ) \left (x d +c \right ) \left (b \,c^{2} f^{2} n^{2} x -2 b c d e f \,n^{2} x -b c d \,f^{2} n \,x^{2}+b \,d^{2} e^{2} n^{2} x +b \,d^{2} e f n \,x^{2}+a \,c^{2} f^{2} n^{2}-2 a c d e f \,n^{2}-2 a c d \,f^{2} n x +a \,d^{2} e^{2} n^{2}+2 a \,d^{2} e f n x +2 a \,d^{2} f^{2} x^{2}-4 b \,c^{2} f^{2} n x +8 b c d e f n x +b c d \,f^{2} x^{2}-4 b \,d^{2} e^{2} n x -3 b \,d^{2} e f \,x^{2}-5 a \,c^{2} f^{2} n +8 a c d e f n +6 a c d \,f^{2} x -3 a \,d^{2} e^{2} n -2 a \,d^{2} e f x +b \,c^{2} e f n +3 b \,c^{2} f^{2} x -b c d \,e^{2} n -10 b c d e f x +3 b \,d^{2} e^{2} x +6 a \,c^{2} f^{2}-6 a c d e f +2 a \,d^{2} e^{2}-3 b \,c^{2} e f +b c d \,e^{2}\right ) \left (x d +c \right )^{-4+n} \left (f x +e \right )^{-n}}{c^{3} f^{3} n^{3}-3 c^{2} d e \,f^{2} n^{3}+3 c \,d^{2} e^{2} f \,n^{3}-d^{3} e^{3} n^{3}-6 c^{3} f^{3} n^{2}+18 c^{2} d e \,f^{2} n^{2}-18 c \,d^{2} e^{2} f \,n^{2}+6 d^{3} e^{3} n^{2}+11 c^{3} f^{3} n -33 c^{2} d e \,f^{2} n +33 c \,d^{2} e^{2} f n -11 d^{3} e^{3} n -6 c^{3} f^{3}+18 c^{2} d e \,f^{2}-18 c \,d^{2} e^{2} f +6 d^{3} e^{3}}\)	\(511\)
parallelrisch	\(\text {Expression too large to display}\)	\(1801\)

Input:

int((b*x+a)*(d*x+c)^(-4+n)/((f*x+e)^n),x,method=_RETURNVERBOSE)
 

Output:

-(d*x+c)^(-3+n)*(f*x+e)/((f*x+e)^n)/(c^3*f^3*n^3-3*c^2*d*e*f^2*n^3+3*c*d^2 
*e^2*f*n^3-d^3*e^3*n^3-6*c^3*f^3*n^2+18*c^2*d*e*f^2*n^2-18*c*d^2*e^2*f*n^2 
+6*d^3*e^3*n^2+11*c^3*f^3*n-33*c^2*d*e*f^2*n+33*c*d^2*e^2*f*n-11*d^3*e^3*n 
-6*c^3*f^3+18*c^2*d*e*f^2-18*c*d^2*e^2*f+6*d^3*e^3)*(b*c^2*f^2*n^2*x-2*b*c 
*d*e*f*n^2*x-b*c*d*f^2*n*x^2+b*d^2*e^2*n^2*x+b*d^2*e*f*n*x^2+a*c^2*f^2*n^2 
-2*a*c*d*e*f*n^2-2*a*c*d*f^2*n*x+a*d^2*e^2*n^2+2*a*d^2*e*f*n*x+2*a*d^2*f^2 
*x^2-4*b*c^2*f^2*n*x+8*b*c*d*e*f*n*x+b*c*d*f^2*x^2-4*b*d^2*e^2*n*x-3*b*d^2 
*e*f*x^2-5*a*c^2*f^2*n+8*a*c*d*e*f*n+6*a*c*d*f^2*x-3*a*d^2*e^2*n-2*a*d^2*e 
*f*x+b*c^2*e*f*n+3*b*c^2*f^2*x-b*c*d*e^2*n-10*b*c*d*e*f*x+3*b*d^2*e^2*x+6* 
a*c^2*f^2-6*a*c*d*e*f+2*a*d^2*e^2-3*b*c^2*e*f+b*c*d*e^2)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 884 vs. \(2 (187) = 374\).

Time = 0.13 (sec) , antiderivative size = 884, normalized size of antiderivative = 4.31 \[ \int (a+b x) (c+d x)^{-4+n} (e+f x)^{-n} \, dx =\text {Too large to display} \] Input:

integrate((b*x+a)*(d*x+c)^(-4+n)/((f*x+e)^n),x, algorithm="fricas")
 

Output:

-(6*a*c^3*e*f^2 - (3*b*d^3*e*f^2 - (b*c*d^2 + 2*a*d^3)*f^3 - (b*d^3*e*f^2 
- b*c*d^2*f^3)*n)*x^4 + (b*c^2*d + 2*a*c*d^2)*e^3 - 3*(b*c^3 + 2*a*c^2*d)* 
e^2*f - (12*b*c*d^2*e*f^2 - 4*(b*c^2*d + 2*a*c*d^2)*f^3 - (b*d^3*e^2*f - 2 
*b*c*d^2*e*f^2 + b*c^2*d*f^3)*n^2 + (3*b*d^3*e^2*f - 2*(4*b*c*d^2 + a*d^3) 
*e*f^2 + (5*b*c^2*d + 2*a*c*d^2)*f^3)*n)*x^3 + (a*c*d^2*e^3 - 2*a*c^2*d*e^ 
2*f + a*c^3*e*f^2)*n^2 + (3*b*d^3*e^3 - 9*b*c*d^2*e^2*f - 9*b*c^2*d*e*f^2 
+ 3*(b*c^3 + 4*a*c^2*d)*f^3 + (b*d^3*e^3 - (b*c*d^2 - a*d^3)*e^2*f - (b*c^ 
2*d + 2*a*c*d^2)*e*f^2 + (b*c^3 + a*c^2*d)*f^3)*n^2 - (4*b*d^3*e^3 - (4*b* 
c*d^2 - a*d^3)*e^2*f - 4*(b*c^2*d + 2*a*c*d^2)*e*f^2 + (4*b*c^3 + 7*a*c^2* 
d)*f^3)*n)*x^2 - (5*a*c^3*e*f^2 + (b*c^2*d + 3*a*c*d^2)*e^3 - (b*c^3 + 8*a 
*c^2*d)*e^2*f)*n + (6*a*c^2*d*e*f^2 + 6*a*c^3*f^3 + 2*(2*b*c*d^2 + a*d^3)* 
e^3 - 6*(2*b*c^2*d + a*c*d^2)*e^2*f + (a*c^3*f^3 + (b*c*d^2 + a*d^3)*e^3 - 
 (2*b*c^2*d + a*c*d^2)*e^2*f + (b*c^3 - a*c^2*d)*e*f^2)*n^2 - (5*a*c^3*f^3 
 + (5*b*c*d^2 + 3*a*d^3)*e^3 - (8*b*c^2*d + 7*a*c*d^2)*e^2*f + (3*b*c^3 - 
a*c^2*d)*e*f^2)*n)*x)*(d*x + c)^(n - 4)/((6*d^3*e^3 - 18*c*d^2*e^2*f + 18* 
c^2*d*e*f^2 - 6*c^3*f^3 - (d^3*e^3 - 3*c*d^2*e^2*f + 3*c^2*d*e*f^2 - c^3*f 
^3)*n^3 + 6*(d^3*e^3 - 3*c*d^2*e^2*f + 3*c^2*d*e*f^2 - c^3*f^3)*n^2 - 11*( 
d^3*e^3 - 3*c*d^2*e^2*f + 3*c^2*d*e*f^2 - c^3*f^3)*n)*(f*x + e)^n)
 

Sympy [F(-2)]

Exception generated. \[ \int (a+b x) (c+d x)^{-4+n} (e+f x)^{-n} \, dx=\text {Exception raised: HeuristicGCDFailed} \] Input:

integrate((b*x+a)*(d*x+c)**(-4+n)/((f*x+e)**n),x)
 

Output:

Exception raised: HeuristicGCDFailed >> no luck
 

Maxima [F]

\[ \int (a+b x) (c+d x)^{-4+n} (e+f x)^{-n} \, dx=\int { \frac {{\left (b x + a\right )} {\left (d x + c\right )}^{n - 4}}{{\left (f x + e\right )}^{n}} \,d x } \] Input:

integrate((b*x+a)*(d*x+c)^(-4+n)/((f*x+e)^n),x, algorithm="maxima")
 

Output:

integrate((b*x + a)*(d*x + c)^(n - 4)/(f*x + e)^n, x)
 

Giac [F]

\[ \int (a+b x) (c+d x)^{-4+n} (e+f x)^{-n} \, dx=\int { \frac {{\left (b x + a\right )} {\left (d x + c\right )}^{n - 4}}{{\left (f x + e\right )}^{n}} \,d x } \] Input:

integrate((b*x+a)*(d*x+c)^(-4+n)/((f*x+e)^n),x, algorithm="giac")
 

Output:

integrate((b*x + a)*(d*x + c)^(n - 4)/(f*x + e)^n, x)
 

Mupad [B] (verification not implemented)

Time = 1.61 (sec) , antiderivative size = 874, normalized size of antiderivative = 4.26 \[ \int (a+b x) (c+d x)^{-4+n} (e+f x)^{-n} \, dx=-\frac {x\,{\left (c+d\,x\right )}^{n-4}\,\left (b\,c^3\,e\,f^2\,n^2-3\,b\,c^3\,e\,f^2\,n+a\,c^3\,f^3\,n^2-5\,a\,c^3\,f^3\,n+6\,a\,c^3\,f^3-2\,b\,c^2\,d\,e^2\,f\,n^2+8\,b\,c^2\,d\,e^2\,f\,n-12\,b\,c^2\,d\,e^2\,f-a\,c^2\,d\,e\,f^2\,n^2+a\,c^2\,d\,e\,f^2\,n+6\,a\,c^2\,d\,e\,f^2+b\,c\,d^2\,e^3\,n^2-5\,b\,c\,d^2\,e^3\,n+4\,b\,c\,d^2\,e^3-a\,c\,d^2\,e^2\,f\,n^2+7\,a\,c\,d^2\,e^2\,f\,n-6\,a\,c\,d^2\,e^2\,f+a\,d^3\,e^3\,n^2-3\,a\,d^3\,e^3\,n+2\,a\,d^3\,e^3\right )}{{\left (e+f\,x\right )}^n\,{\left (c\,f-d\,e\right )}^3\,\left (n^3-6\,n^2+11\,n-6\right )}-\frac {x^2\,{\left (c+d\,x\right )}^{n-4}\,\left (b\,c^3\,f^3\,n^2-4\,b\,c^3\,f^3\,n+3\,b\,c^3\,f^3-b\,c^2\,d\,e\,f^2\,n^2+4\,b\,c^2\,d\,e\,f^2\,n-9\,b\,c^2\,d\,e\,f^2+a\,c^2\,d\,f^3\,n^2-7\,a\,c^2\,d\,f^3\,n+12\,a\,c^2\,d\,f^3-b\,c\,d^2\,e^2\,f\,n^2+4\,b\,c\,d^2\,e^2\,f\,n-9\,b\,c\,d^2\,e^2\,f-2\,a\,c\,d^2\,e\,f^2\,n^2+8\,a\,c\,d^2\,e\,f^2\,n+b\,d^3\,e^3\,n^2-4\,b\,d^3\,e^3\,n+3\,b\,d^3\,e^3+a\,d^3\,e^2\,f\,n^2-a\,d^3\,e^2\,f\,n\right )}{{\left (e+f\,x\right )}^n\,{\left (c\,f-d\,e\right )}^3\,\left (n^3-6\,n^2+11\,n-6\right )}-\frac {c\,e\,{\left (c+d\,x\right )}^{n-4}\,\left (b\,c^2\,e\,f\,n-3\,b\,c^2\,e\,f+a\,c^2\,f^2\,n^2-5\,a\,c^2\,f^2\,n+6\,a\,c^2\,f^2-b\,c\,d\,e^2\,n+b\,c\,d\,e^2-2\,a\,c\,d\,e\,f\,n^2+8\,a\,c\,d\,e\,f\,n-6\,a\,c\,d\,e\,f+a\,d^2\,e^2\,n^2-3\,a\,d^2\,e^2\,n+2\,a\,d^2\,e^2\right )}{{\left (e+f\,x\right )}^n\,{\left (c\,f-d\,e\right )}^3\,\left (n^3-6\,n^2+11\,n-6\right )}-\frac {d^2\,f^2\,x^4\,{\left (c+d\,x\right )}^{n-4}\,\left (2\,a\,d\,f+b\,c\,f-3\,b\,d\,e-b\,c\,f\,n+b\,d\,e\,n\right )}{{\left (e+f\,x\right )}^n\,{\left (c\,f-d\,e\right )}^3\,\left (n^3-6\,n^2+11\,n-6\right )}-\frac {d\,f\,x^3\,{\left (c+d\,x\right )}^{n-4}\,\left (4\,c\,f-c\,f\,n+d\,e\,n\right )\,\left (2\,a\,d\,f+b\,c\,f-3\,b\,d\,e-b\,c\,f\,n+b\,d\,e\,n\right )}{{\left (e+f\,x\right )}^n\,{\left (c\,f-d\,e\right )}^3\,\left (n^3-6\,n^2+11\,n-6\right )} \] Input:

int(((a + b*x)*(c + d*x)^(n - 4))/(e + f*x)^n,x)
 

Output:

- (x*(c + d*x)^(n - 4)*(6*a*c^3*f^3 + 2*a*d^3*e^3 + a*c^3*f^3*n^2 + a*d^3* 
e^3*n^2 + 4*b*c*d^2*e^3 - 5*a*c^3*f^3*n - 3*a*d^3*e^3*n - 6*a*c*d^2*e^2*f 
+ 6*a*c^2*d*e*f^2 - 12*b*c^2*d*e^2*f - 5*b*c*d^2*e^3*n - 3*b*c^3*e*f^2*n + 
 b*c*d^2*e^3*n^2 + b*c^3*e*f^2*n^2 + 7*a*c*d^2*e^2*f*n + a*c^2*d*e*f^2*n + 
 8*b*c^2*d*e^2*f*n - a*c*d^2*e^2*f*n^2 - a*c^2*d*e*f^2*n^2 - 2*b*c^2*d*e^2 
*f*n^2))/((e + f*x)^n*(c*f - d*e)^3*(11*n - 6*n^2 + n^3 - 6)) - (x^2*(c + 
d*x)^(n - 4)*(3*b*c^3*f^3 + 3*b*d^3*e^3 + b*c^3*f^3*n^2 + b*d^3*e^3*n^2 + 
12*a*c^2*d*f^3 - 4*b*c^3*f^3*n - 4*b*d^3*e^3*n - 9*b*c*d^2*e^2*f - 9*b*c^2 
*d*e*f^2 - 7*a*c^2*d*f^3*n - a*d^3*e^2*f*n + a*c^2*d*f^3*n^2 + a*d^3*e^2*f 
*n^2 + 8*a*c*d^2*e*f^2*n + 4*b*c*d^2*e^2*f*n + 4*b*c^2*d*e*f^2*n - 2*a*c*d 
^2*e*f^2*n^2 - b*c*d^2*e^2*f*n^2 - b*c^2*d*e*f^2*n^2))/((e + f*x)^n*(c*f - 
 d*e)^3*(11*n - 6*n^2 + n^3 - 6)) - (c*e*(c + d*x)^(n - 4)*(6*a*c^2*f^2 + 
2*a*d^2*e^2 + a*c^2*f^2*n^2 + a*d^2*e^2*n^2 + b*c*d*e^2 - 3*b*c^2*e*f - 5* 
a*c^2*f^2*n - 3*a*d^2*e^2*n - 6*a*c*d*e*f - b*c*d*e^2*n + b*c^2*e*f*n - 2* 
a*c*d*e*f*n^2 + 8*a*c*d*e*f*n))/((e + f*x)^n*(c*f - d*e)^3*(11*n - 6*n^2 + 
 n^3 - 6)) - (d^2*f^2*x^4*(c + d*x)^(n - 4)*(2*a*d*f + b*c*f - 3*b*d*e - b 
*c*f*n + b*d*e*n))/((e + f*x)^n*(c*f - d*e)^3*(11*n - 6*n^2 + n^3 - 6)) - 
(d*f*x^3*(c + d*x)^(n - 4)*(4*c*f - c*f*n + d*e*n)*(2*a*d*f + b*c*f - 3*b* 
d*e - b*c*f*n + b*d*e*n))/((e + f*x)^n*(c*f - d*e)^3*(11*n - 6*n^2 + n^3 - 
 6))
 

Reduce [F]

\[ \int (a+b x) (c+d x)^{-4+n} (e+f x)^{-n} \, dx=\left (\int \frac {\left (d x +c \right )^{n}}{\left (f x +e \right )^{n} c^{4}+4 \left (f x +e \right )^{n} c^{3} d x +6 \left (f x +e \right )^{n} c^{2} d^{2} x^{2}+4 \left (f x +e \right )^{n} c \,d^{3} x^{3}+\left (f x +e \right )^{n} d^{4} x^{4}}d x \right ) a +\left (\int \frac {\left (d x +c \right )^{n} x}{\left (f x +e \right )^{n} c^{4}+4 \left (f x +e \right )^{n} c^{3} d x +6 \left (f x +e \right )^{n} c^{2} d^{2} x^{2}+4 \left (f x +e \right )^{n} c \,d^{3} x^{3}+\left (f x +e \right )^{n} d^{4} x^{4}}d x \right ) b \] Input:

int((b*x+a)*(d*x+c)^(-4+n)/((f*x+e)^n),x)
 

Output:

int((c + d*x)**n/((e + f*x)**n*c**4 + 4*(e + f*x)**n*c**3*d*x + 6*(e + f*x 
)**n*c**2*d**2*x**2 + 4*(e + f*x)**n*c*d**3*x**3 + (e + f*x)**n*d**4*x**4) 
,x)*a + int(((c + d*x)**n*x)/((e + f*x)**n*c**4 + 4*(e + f*x)**n*c**3*d*x 
+ 6*(e + f*x)**n*c**2*d**2*x**2 + 4*(e + f*x)**n*c*d**3*x**3 + (e + f*x)** 
n*d**4*x**4),x)*b