\(\int (a+b x)^{-n} (c+d x) (e+f x)^{-4+n} \, dx\) [262]

Optimal result

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

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

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.88 \[ \int (a+b x)^{-n} (c+d x) (e+f x)^{-4+n} \, dx=\frac {(a+b x)^{1-n} (e+f x)^{-3+n} \left (a^2 f (-1+n) (-d e+c f (-2+n)+d f (-3+n) x)+b^2 \left (d e (-1+n) x (e (-3+n)-f x)+c \left (e^2 \left (6-5 n+n^2\right )-2 e f (-3+n) x+2 f^2 x^2\right )\right )+a b \left (2 c f (-1+n) (-e (-3+n)+f x)+d \left (e^2 (-3+n)-2 e f \left (5-4 n+n^2\right ) x+f^2 (-3+n) x^2\right )\right )\right )}{(-b e+a f)^3 (-3+n) (-2+n) (-1+n)} \] Input:

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

Output:

((a + b*x)^(1 - n)*(e + f*x)^(-3 + n)*(a^2*f*(-1 + n)*(-(d*e) + c*f*(-2 + 
n) + d*f*(-3 + n)*x) + b^2*(d*e*(-1 + n)*x*(e*(-3 + n) - f*x) + c*(e^2*(6 
- 5*n + n^2) - 2*e*f*(-3 + n)*x + 2*f^2*x^2)) + a*b*(2*c*f*(-1 + n)*(-(e*( 
-3 + n)) + f*x) + d*(e^2*(-3 + n) - 2*e*f*(5 - 4*n + n^2)*x + f^2*(-3 + n) 
*x^2))))/((-(b*e) + a*f)^3*(-3 + n)*(-2 + n)*(-1 + n))
 

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 182, 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.

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

\(\Big \downarrow \) 88

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

\(\Big \downarrow \) 55

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

\(\Big \downarrow \) 48

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

Input:

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

Output:

-(((d*e - c*f)*(a + b*x)^(1 - n)*(e + f*x)^(-3 + n))/(f*(b*e - a*f)*(3 - n 
))) + ((2*b*c*f + b*d*e*(1 - n) - a*d*f*(3 - n))*(((a + b*x)^(1 - n)*(e + 
f*x)^(-2 + n))/((b*e - a*f)*(2 - n)) + (b*(a + b*x)^(1 - n)*(e + f*x)^(-1 
+ n))/((b*e - a*f)^2*(1 - n)*(2 - n))))/(f*(b*e - a*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. \(504\) vs. \(2(205)=410\).

Time = 0.54 (sec) , antiderivative size = 505, normalized size of antiderivative = 2.46

method	result	size
gosper	\(\frac {\left (b x +a \right ) \left (f x +e \right )^{-3+n} \left (b x +a \right )^{-n} \left (a^{2} d \,f^{2} n^{2} x -2 a b d e f \,n^{2} x +a b d \,f^{2} n \,x^{2}+b^{2} d \,e^{2} n^{2} x -b^{2} d e f n \,x^{2}+a^{2} c \,f^{2} n^{2}-4 a^{2} d \,f^{2} n x -2 a b c e f \,n^{2}+2 a b c \,f^{2} n x +8 a b d e f n x -3 a b d \,f^{2} x^{2}+b^{2} c \,e^{2} n^{2}-2 b^{2} c e f n x +2 b^{2} c \,f^{2} x^{2}-4 b^{2} d \,e^{2} n x +b^{2} d e f \,x^{2}-3 a^{2} c \,f^{2} n -a^{2} d e f n +3 a^{2} d \,f^{2} x +8 a b c e f n -2 a b c \,f^{2} x +a b d \,e^{2} n -10 a b d e f x -5 b^{2} c \,e^{2} n +6 b^{2} c e f x +3 b^{2} d \,e^{2} x +2 a^{2} c \,f^{2}+a^{2} d e f -6 a b c e f -3 a b d \,e^{2}+6 b^{2} c \,e^{2}\right )}{a^{3} f^{3} n^{3}-3 a^{2} b e \,f^{2} n^{3}+3 a \,b^{2} e^{2} f \,n^{3}-b^{3} e^{3} n^{3}-6 a^{3} f^{3} n^{2}+18 a^{2} b e \,f^{2} n^{2}-18 a \,b^{2} e^{2} f \,n^{2}+6 b^{3} e^{3} n^{2}+11 a^{3} f^{3} n -33 a^{2} b e \,f^{2} n +33 a \,b^{2} e^{2} f n -11 b^{3} e^{3} n -6 a^{3} f^{3}+18 a^{2} b e \,f^{2}-18 a \,b^{2} e^{2} f +6 b^{3} e^{3}}\)	\(505\)
orering	\(\frac {\left (b x +a \right ) \left (f x +e \right ) \left (a^{2} d \,f^{2} n^{2} x -2 a b d e f \,n^{2} x +a b d \,f^{2} n \,x^{2}+b^{2} d \,e^{2} n^{2} x -b^{2} d e f n \,x^{2}+a^{2} c \,f^{2} n^{2}-4 a^{2} d \,f^{2} n x -2 a b c e f \,n^{2}+2 a b c \,f^{2} n x +8 a b d e f n x -3 a b d \,f^{2} x^{2}+b^{2} c \,e^{2} n^{2}-2 b^{2} c e f n x +2 b^{2} c \,f^{2} x^{2}-4 b^{2} d \,e^{2} n x +b^{2} d e f \,x^{2}-3 a^{2} c \,f^{2} n -a^{2} d e f n +3 a^{2} d \,f^{2} x +8 a b c e f n -2 a b c \,f^{2} x +a b d \,e^{2} n -10 a b d e f x -5 b^{2} c \,e^{2} n +6 b^{2} c e f x +3 b^{2} d \,e^{2} x +2 a^{2} c \,f^{2}+a^{2} d e f -6 a b c e f -3 a b d \,e^{2}+6 b^{2} c \,e^{2}\right ) \left (f x +e \right )^{-4+n} \left (b x +a \right )^{-n}}{a^{3} f^{3} n^{3}-3 a^{2} b e \,f^{2} n^{3}+3 a \,b^{2} e^{2} f \,n^{3}-b^{3} e^{3} n^{3}-6 a^{3} f^{3} n^{2}+18 a^{2} b e \,f^{2} n^{2}-18 a \,b^{2} e^{2} f \,n^{2}+6 b^{3} e^{3} n^{2}+11 a^{3} f^{3} n -33 a^{2} b e \,f^{2} n +33 a \,b^{2} e^{2} f n -11 b^{3} e^{3} n -6 a^{3} f^{3}+18 a^{2} b e \,f^{2}-18 a \,b^{2} e^{2} f +6 b^{3} e^{3}}\)	\(510\)
parallelrisch	\(\text {Expression too large to display}\)	\(1791\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 884 vs. \(2 (191) = 382\).

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

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

Output:

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

Sympy [F(-2)]

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

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

Output:

Exception raised: HeuristicGCDFailed >> no luck
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [B] (verification not implemented)

Time = 1.66 (sec) , antiderivative size = 869, normalized size of antiderivative = 4.24 \[ \int (a+b x)^{-n} (c+d x) (e+f x)^{-4+n} \, dx=\frac {x\,{\left (e+f\,x\right )}^{n-4}\,\left (d\,a^3\,e\,f^2\,n^2-5\,d\,a^3\,e\,f^2\,n+4\,d\,a^3\,e\,f^2+c\,a^3\,f^3\,n^2-3\,c\,a^3\,f^3\,n+2\,c\,a^3\,f^3-2\,d\,a^2\,b\,e^2\,f\,n^2+8\,d\,a^2\,b\,e^2\,f\,n-12\,d\,a^2\,b\,e^2\,f-c\,a^2\,b\,e\,f^2\,n^2+7\,c\,a^2\,b\,e\,f^2\,n-6\,c\,a^2\,b\,e\,f^2+d\,a\,b^2\,e^3\,n^2-3\,d\,a\,b^2\,e^3\,n-c\,a\,b^2\,e^2\,f\,n^2+c\,a\,b^2\,e^2\,f\,n+6\,c\,a\,b^2\,e^2\,f+c\,b^3\,e^3\,n^2-5\,c\,b^3\,e^3\,n+6\,c\,b^3\,e^3\right )}{{\left (a\,f-b\,e\right )}^3\,{\left (a+b\,x\right )}^n\,\left (n^3-6\,n^2+11\,n-6\right )}+\frac {x^2\,{\left (e+f\,x\right )}^{n-4}\,\left (d\,a^3\,f^3\,n^2-4\,d\,a^3\,f^3\,n+3\,d\,a^3\,f^3-d\,a^2\,b\,e\,f^2\,n^2+4\,d\,a^2\,b\,e\,f^2\,n-9\,d\,a^2\,b\,e\,f^2+c\,a^2\,b\,f^3\,n^2-c\,a^2\,b\,f^3\,n-d\,a\,b^2\,e^2\,f\,n^2+4\,d\,a\,b^2\,e^2\,f\,n-9\,d\,a\,b^2\,e^2\,f-2\,c\,a\,b^2\,e\,f^2\,n^2+8\,c\,a\,b^2\,e\,f^2\,n+d\,b^3\,e^3\,n^2-4\,d\,b^3\,e^3\,n+3\,d\,b^3\,e^3+c\,b^3\,e^2\,f\,n^2-7\,c\,b^3\,e^2\,f\,n+12\,c\,b^3\,e^2\,f\right )}{{\left (a\,f-b\,e\right )}^3\,{\left (a+b\,x\right )}^n\,\left (n^3-6\,n^2+11\,n-6\right )}+\frac {a\,e\,{\left (e+f\,x\right )}^{n-4}\,\left (-d\,a^2\,e\,f\,n+d\,a^2\,e\,f+c\,a^2\,f^2\,n^2-3\,c\,a^2\,f^2\,n+2\,c\,a^2\,f^2+d\,a\,b\,e^2\,n-3\,d\,a\,b\,e^2-2\,c\,a\,b\,e\,f\,n^2+8\,c\,a\,b\,e\,f\,n-6\,c\,a\,b\,e\,f+c\,b^2\,e^2\,n^2-5\,c\,b^2\,e^2\,n+6\,c\,b^2\,e^2\right )}{{\left (a\,f-b\,e\right )}^3\,{\left (a+b\,x\right )}^n\,\left (n^3-6\,n^2+11\,n-6\right )}+\frac {b^2\,f^2\,x^4\,{\left (e+f\,x\right )}^{n-4}\,\left (2\,b\,c\,f-3\,a\,d\,f+b\,d\,e+a\,d\,f\,n-b\,d\,e\,n\right )}{{\left (a\,f-b\,e\right )}^3\,{\left (a+b\,x\right )}^n\,\left (n^3-6\,n^2+11\,n-6\right )}+\frac {b\,f\,x^3\,{\left (e+f\,x\right )}^{n-4}\,\left (4\,b\,e+a\,f\,n-b\,e\,n\right )\,\left (2\,b\,c\,f-3\,a\,d\,f+b\,d\,e+a\,d\,f\,n-b\,d\,e\,n\right )}{{\left (a\,f-b\,e\right )}^3\,{\left (a+b\,x\right )}^n\,\left (n^3-6\,n^2+11\,n-6\right )} \] Input:

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

Output:

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

Reduce [F]

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

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

Output:

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