\(\int (a+b x)^{-n} (c+d x) (e+f x)^{-5+n} \, dx\) [263]

Optimal result

Integrand size = 24, antiderivative size = 297 \[ \int (a+b x)^{-n} (c+d x) (e+f x)^{-5+n} \, dx=-\frac {(d e-c f) (a+b x)^{1-n} (e+f x)^{-4+n}}{f (b e-a f) (4-n)}+\frac {(3 b c f-a d f (4-n)+b d (e-e n)) (a+b x)^{1-n} (e+f x)^{-3+n}}{f (b e-a f)^2 (3-n) (4-n)}+\frac {2 b (3 b c f-a d f (4-n)+b d (e-e n)) (a+b x)^{1-n} (e+f x)^{-2+n}}{f (b e-a f)^3 (2-n) (3-n) (4-n)}+\frac {2 b^2 (3 b c f-a d f (4-n)+b d (e-e n)) (a+b x)^{1-n} (e+f x)^{-1+n}}{f (b e-a f)^4 (1-n) (2-n) (3-n) (4-n)} \] Output:

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

Mathematica [A] (verified)

Time = 0.28 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.50 \[ \int (a+b x)^{-n} (c+d x) (e+f x)^{-5+n} \, dx=\frac {(a+b x)^{1-n} (e+f x)^{-4+n} \left (-d e+c f-\frac {(3 b c f+a d f (-4+n)-b d e (-1+n)) (e+f x) \left ((b e-a f)^2 (-2+n) (-1+n)-2 b (b e-a f) (-1+n) (e+f x)+2 b^2 (e+f x)^2\right )}{(b e-a f)^3 (-3+n) (-2+n) (-1+n)}\right )}{f (-b e+a f) (-4+n)} \] Input:

Integrate[((c + d*x)*(e + f*x)^(-5 + n))/(a + b*x)^n,x]
 

Output:

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

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 241, normalized size of antiderivative = 0.81, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {88, 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.

\(\displaystyle \int (c+d x) (a+b x)^{-n} (e+f x)^{n-5} \, dx\)

\(\Big \downarrow \) 88

\(\displaystyle \frac {(-a d f (4-n)+3 b c f+b d e (1-n)) \int (a+b x)^{-n} (e+f x)^{n-4}dx}{f (4-n) (b e-a f)}-\frac {(a+b x)^{1-n} (d e-c f) (e+f x)^{n-4}}{f (4-n) (b e-a f)}\)

\(\Big \downarrow \) 55

\(\displaystyle \frac {(-a d f (4-n)+3 b c f+b d e (1-n)) \left (\frac {2 b \int (a+b x)^{-n} (e+f x)^{n-3}dx}{(3-n) (b e-a f)}+\frac {(a+b x)^{1-n} (e+f x)^{n-3}}{(3-n) (b e-a f)}\right )}{f (4-n) (b e-a f)}-\frac {(a+b x)^{1-n} (d e-c f) (e+f x)^{n-4}}{f (4-n) (b e-a f)}\)

\(\Big \downarrow \) 55

\(\displaystyle \frac {(-a d f (4-n)+3 b c f+b d e (1-n)) \left (\frac {2 b \left (\frac {b \int (a+b x)^{-n} (e+f x)^{n-2}dx}{(2-n) (b e-a f)}+\frac {(a+b x)^{1-n} (e+f x)^{n-2}}{(2-n) (b e-a f)}\right )}{(3-n) (b e-a f)}+\frac {(a+b x)^{1-n} (e+f x)^{n-3}}{(3-n) (b e-a f)}\right )}{f (4-n) (b e-a f)}-\frac {(a+b x)^{1-n} (d e-c f) (e+f x)^{n-4}}{f (4-n) (b e-a f)}\)

\(\Big \downarrow \) 48

\(\displaystyle \frac {\left (\frac {(a+b x)^{1-n} (e+f x)^{n-3}}{(3-n) (b e-a f)}+\frac {2 b \left (\frac {(a+b x)^{1-n} (e+f x)^{n-2}}{(2-n) (b e-a f)}+\frac {b (a+b x)^{1-n} (e+f x)^{n-1}}{(1-n) (2-n) (b e-a f)^2}\right )}{(3-n) (b e-a f)}\right ) (-a d f (4-n)+3 b c f+b d e (1-n))}{f (4-n) (b e-a f)}-\frac {(a+b x)^{1-n} (d e-c f) (e+f x)^{n-4}}{f (4-n) (b e-a f)}\)

Input:

Int[((c + d*x)*(e + f*x)^(-5 + n))/(a + b*x)^n,x]
 

Output:

-(((d*e - c*f)*(a + b*x)^(1 - n)*(e + f*x)^(-4 + n))/(f*(b*e - a*f)*(4 - n 
))) + ((3*b*c*f + b*d*e*(1 - n) - a*d*f*(4 - n))*(((a + b*x)^(1 - n)*(e + 
f*x)^(-3 + n))/((b*e - a*f)*(3 - n)) + (2*b*(((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))))/((b*e - a*f)*(3 - n))))/(f*(b*e - a*f)* 
(4 - 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. \(1187\) vs. \(2(297)=594\).

Time = 0.56 (sec) , antiderivative size = 1188, normalized size of antiderivative = 4.00

method	result	size
gosper	\(\text {Expression too large to display}\)	\(1188\)
orering	\(\text {Expression too large to display}\)	\(1193\)
parallelrisch	\(\text {Expression too large to display}\)	\(3499\)

Input:

int((d*x+c)*(f*x+e)^(-5+n)/((b*x+a)^n),x,method=_RETURNVERBOSE)
 

Output:

(b*x+a)*(f*x+e)^(-4+n)/((b*x+a)^n)/(a^4*f^4*n^4-4*a^3*b*e*f^3*n^4+6*a^2*b^ 
2*e^2*f^2*n^4-4*a*b^3*e^3*f*n^4+b^4*e^4*n^4-10*a^4*f^4*n^3+40*a^3*b*e*f^3* 
n^3-60*a^2*b^2*e^2*f^2*n^3+40*a*b^3*e^3*f*n^3-10*b^4*e^4*n^3+35*a^4*f^4*n^ 
2-140*a^3*b*e*f^3*n^2+210*a^2*b^2*e^2*f^2*n^2-140*a*b^3*e^3*f*n^2+35*b^4*e 
^4*n^2-50*a^4*f^4*n+200*a^3*b*e*f^3*n-300*a^2*b^2*e^2*f^2*n+200*a*b^3*e^3* 
f*n-50*b^4*e^4*n+24*a^4*f^4-96*a^3*b*e*f^3+144*a^2*b^2*e^2*f^2-96*a*b^3*e^ 
3*f+24*b^4*e^4)*(a^3*d*f^3*n^3*x-3*a^2*b*d*e*f^2*n^3*x+2*a^2*b*d*f^3*n^2*x 
^2+3*a*b^2*d*e^2*f*n^3*x-4*a*b^2*d*e*f^2*n^2*x^2+2*a*b^2*d*f^3*n*x^3-b^3*d 
*e^3*n^3*x+2*b^3*d*e^2*f*n^2*x^2-2*b^3*d*e*f^2*n*x^3+a^3*c*f^3*n^3-7*a^3*d 
*f^3*n^2*x-3*a^2*b*c*e*f^2*n^3+3*a^2*b*c*f^3*n^2*x+22*a^2*b*d*e*f^2*n^2*x- 
10*a^2*b*d*f^3*n*x^2+3*a*b^2*c*e^2*f*n^3-6*a*b^2*c*e*f^2*n^2*x+6*a*b^2*c*f 
^3*n*x^2-23*a*b^2*d*e^2*f*n^2*x+20*a*b^2*d*e*f^2*n*x^2-8*a*b^2*d*f^3*x^3-b 
^3*c*e^3*n^3+3*b^3*c*e^2*f*n^2*x-6*b^3*c*e*f^2*n*x^2+6*b^3*c*f^3*x^3+8*b^3 
*d*e^3*n^2*x-10*b^3*d*e^2*f*n*x^2+2*b^3*d*e*f^2*x^3-6*a^3*c*f^3*n^2-a^3*d* 
e*f^2*n^2+14*a^3*d*f^3*n*x+21*a^2*b*c*e*f^2*n^2-9*a^2*b*c*f^3*n*x+2*a^2*b* 
d*e^2*f*n^2-53*a^2*b*d*e*f^2*n*x+8*a^2*b*d*f^3*x^2-24*a*b^2*c*e^2*f*n^2+30 
*a*b^2*c*e*f^2*n*x-6*a*b^2*c*f^3*x^2-a*b^2*d*e^3*n^2+58*a*b^2*d*e^2*f*n*x- 
34*a*b^2*d*e*f^2*x^2+9*b^3*c*e^3*n^2-21*b^3*c*e^2*f*n*x+24*b^3*c*e*f^2*x^2 
-19*b^3*d*e^3*n*x+8*b^3*d*e^2*f*x^2+11*a^3*c*f^3*n+3*a^3*d*e*f^2*n-8*a^3*d 
*f^3*x-42*a^2*b*c*e*f^2*n+6*a^2*b*c*f^3*x-10*a^2*b*d*e^2*f*n+34*a^2*b*d...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1740 vs. \(2 (273) = 546\).

Time = 0.18 (sec) , antiderivative size = 1740, normalized size of antiderivative = 5.86 \[ \int (a+b x)^{-n} (c+d x) (e+f x)^{-5+n} \, dx=\text {Too large to display} \] Input:

integrate((d*x+c)*(f*x+e)^(-5+n)/((b*x+a)^n),x, algorithm="fricas")
 

Output:

-(6*a^4*c*e*f^3 - 2*(b^4*d*e*f^3 + (3*b^4*c - 4*a*b^3*d)*f^4 - (b^4*d*e*f^ 
3 - a*b^3*d*f^4)*n)*x^5 - 12*(2*a*b^3*c - a^2*b^2*d)*e^4 + 4*(9*a^2*b^2*c 
- 2*a^3*b*d)*e^3*f - 2*(12*a^3*b*c - a^4*d)*e^2*f^2 - 2*(5*b^4*d*e^2*f^2 + 
 5*(3*b^4*c - 4*a*b^3*d)*e*f^3 + (b^4*d*e^2*f^2 - 2*a*b^3*d*e*f^3 + a^2*b^ 
2*d*f^4)*n^2 - (6*b^4*d*e^2*f^2 + (3*b^4*c - 10*a*b^3*d)*e*f^3 - (3*a*b^3* 
c - 4*a^2*b^2*d)*f^4)*n)*x^4 + (a*b^3*c*e^4 - 3*a^2*b^2*c*e^3*f + 3*a^3*b* 
c*e^2*f^2 - a^4*c*e*f^3)*n^3 - (20*b^4*d*e^3*f + 20*(3*b^4*c - 4*a*b^3*d)* 
e^2*f^2 - (b^4*d*e^3*f - 3*a*b^3*d*e^2*f^2 + 3*a^2*b^2*d*e*f^3 - a^3*b*d*f 
^4)*n^3 + (10*b^4*d*e^3*f + (3*b^4*c - 25*a*b^3*d)*e^2*f^2 - 2*(3*a*b^3*c 
- 10*a^2*b^2*d)*e*f^3 + (3*a^2*b^2*c - 5*a^3*b*d)*f^4)*n^2 - (29*b^4*d*e^3 
*f + 3*(9*b^4*c - 22*a*b^3*d)*e^2*f^2 - (30*a*b^3*c - 41*a^2*b^2*d)*e*f^3 
+ (3*a^2*b^2*c - 4*a^3*b*d)*f^4)*n)*x^3 + (6*a^4*c*e*f^3 - (9*a*b^3*c - a^ 
2*b^2*d)*e^4 + 2*(12*a^2*b^2*c - a^3*b*d)*e^3*f - (21*a^3*b*c - a^4*d)*e^2 
*f^2)*n^2 - (12*b^4*d*e^4 - 48*a^2*b^2*d*e^2*f^2 + 32*a^3*b*d*e*f^3 - 8*a^ 
4*d*f^4 + 12*(5*b^4*c - 4*a*b^3*d)*e^3*f - (b^4*d*e^4 - 3*a*b^3*c*e^2*f^2 
+ (b^4*c - 2*a*b^3*d)*e^3*f + (3*a^2*b^2*c + 2*a^3*b*d)*e*f^3 - (a^3*b*c + 
 a^4*d)*f^4)*n^3 + (8*b^4*d*e^4 + 2*(6*b^4*c - 7*a*b^3*d)*e^3*f - 3*(9*a*b 
^3*c + a^2*b^2*d)*e^2*f^2 + 2*(9*a^2*b^2*c + 8*a^3*b*d)*e*f^3 - (3*a^3*b*c 
 + 7*a^4*d)*f^4)*n^2 - (19*b^4*d*e^4 + (47*b^4*c - 36*a*b^3*d)*e^3*f - 15* 
(4*a*b^3*c + a^2*b^2*d)*e^2*f^2 + (15*a^2*b^2*c + 46*a^3*b*d)*e*f^3 - 2...
 

Sympy [F(-2)]

Exception generated. \[ \int (a+b x)^{-n} (c+d x) (e+f x)^{-5+n} \, dx=\text {Exception raised: HeuristicGCDFailed} \] Input:

integrate((d*x+c)*(f*x+e)**(-5+n)/((b*x+a)**n),x)
 

Output:

Exception raised: HeuristicGCDFailed >> no luck
 

Maxima [F]

\[ \int (a+b x)^{-n} (c+d x) (e+f x)^{-5+n} \, dx=\int { \frac {{\left (d x + c\right )} {\left (f x + e\right )}^{n - 5}}{{\left (b x + a\right )}^{n}} \,d x } \] Input:

integrate((d*x+c)*(f*x+e)^(-5+n)/((b*x+a)^n),x, algorithm="maxima")
 

Output:

integrate((d*x + c)*(f*x + e)^(n - 5)/(b*x + a)^n, x)
 

Giac [F]

\[ \int (a+b x)^{-n} (c+d x) (e+f x)^{-5+n} \, dx=\int { \frac {{\left (d x + c\right )} {\left (f x + e\right )}^{n - 5}}{{\left (b x + a\right )}^{n}} \,d x } \] Input:

integrate((d*x+c)*(f*x+e)^(-5+n)/((b*x+a)^n),x, algorithm="giac")
 

Output:

integrate((d*x + c)*(f*x + e)^(n - 5)/(b*x + a)^n, x)
 

Mupad [B] (verification not implemented)

Time = 2.39 (sec) , antiderivative size = 1659, normalized size of antiderivative = 5.59 \[ \int (a+b x)^{-n} (c+d x) (e+f x)^{-5+n} \, dx=\text {Too large to display} \] Input:

int(((e + f*x)^(n - 5)*(c + d*x))/(a + b*x)^n,x)
 

Output:

(2*b^3*f^3*x^5*(e + f*x)^(n - 5)*(3*b*c*f - 4*a*d*f + b*d*e + a*d*f*n - b* 
d*e*n))/((a*f - b*e)^4*(a + b*x)^n*(35*n^2 - 50*n - 10*n^3 + n^4 + 24)) - 
(x*(e + f*x)^(n - 5)*(6*a^4*c*f^4 - 24*b^4*c*e^4 + 6*a^4*c*f^4*n^2 - 9*b^4 
*c*e^4*n^2 - a^4*c*f^4*n^3 + b^4*c*e^4*n^3 + 10*a^4*d*e*f^3 - 11*a^4*c*f^4 
*n + 26*b^4*c*e^4*n - 24*a*b^3*c*e^3*f - 24*a^3*b*c*e*f^3 + 12*a*b^3*d*e^4 
*n - 17*a^4*d*e*f^3*n + 60*a^2*b^2*d*e^3*f - 40*a^3*b*d*e^2*f^2 - 7*a*b^3* 
d*e^4*n^2 + a*b^3*d*e^4*n^3 + 8*a^4*d*e*f^3*n^2 - a^4*d*e*f^3*n^3 + 36*a^2 
*b^2*c*e^2*f^2 - 45*a^2*b^2*c*e^2*f^2*n + 22*a^2*b^2*d*e^3*f*n^2 - 23*a^3* 
b*d*e^2*f^2*n^2 - 3*a^2*b^2*d*e^3*f*n^3 + 3*a^3*b*d*e^2*f^2*n^3 - 10*a*b^3 
*c*e^3*f*n + 40*a^3*b*c*e*f^3*n + 9*a^2*b^2*c*e^2*f^2*n^2 + 12*a*b^3*c*e^3 
*f*n^2 - 18*a^3*b*c*e*f^3*n^2 - 2*a*b^3*c*e^3*f*n^3 + 2*a^3*b*c*e*f^3*n^3 
- 55*a^2*b^2*d*e^3*f*n + 60*a^3*b*d*e^2*f^2*n))/((a*f - b*e)^4*(a + b*x)^n 
*(35*n^2 - 50*n - 10*n^3 + n^4 + 24)) - (x^2*(e + f*x)^(n - 5)*(8*a^4*d*f^ 
4 - 12*b^4*d*e^4 + 7*a^4*d*f^4*n^2 - 8*b^4*d*e^4*n^2 - a^4*d*f^4*n^3 + b^4 
*d*e^4*n^3 - 60*b^4*c*e^3*f - 14*a^4*d*f^4*n + 19*b^4*d*e^4*n + 48*a*b^3*d 
*e^3*f - 32*a^3*b*d*e*f^3 - 2*a^3*b*c*f^4*n + 47*b^4*c*e^3*f*n + 3*a^3*b*c 
*f^4*n^2 - a^3*b*c*f^4*n^3 - 12*b^4*c*e^3*f*n^2 + b^4*c*e^3*f*n^3 + 48*a^2 
*b^2*d*e^2*f^2 + 27*a*b^3*c*e^2*f^2*n^2 - 18*a^2*b^2*c*e*f^3*n^2 - 3*a*b^3 
*c*e^2*f^2*n^3 + 3*a^2*b^2*c*e*f^3*n^3 - 15*a^2*b^2*d*e^2*f^2*n - 36*a*b^3 
*d*e^3*f*n + 46*a^3*b*d*e*f^3*n + 3*a^2*b^2*d*e^2*f^2*n^2 - 60*a*b^3*c*...
 

Reduce [F]

\[ \int (a+b x)^{-n} (c+d x) (e+f x)^{-5+n} \, dx=\left (\int \frac {\left (f x +e \right )^{n}}{\left (b x +a \right )^{n} e^{5}+5 \left (b x +a \right )^{n} e^{4} f x +10 \left (b x +a \right )^{n} e^{3} f^{2} x^{2}+10 \left (b x +a \right )^{n} e^{2} f^{3} x^{3}+5 \left (b x +a \right )^{n} e \,f^{4} x^{4}+\left (b x +a \right )^{n} f^{5} x^{5}}d x \right ) c +\left (\int \frac {\left (f x +e \right )^{n} x}{\left (b x +a \right )^{n} e^{5}+5 \left (b x +a \right )^{n} e^{4} f x +10 \left (b x +a \right )^{n} e^{3} f^{2} x^{2}+10 \left (b x +a \right )^{n} e^{2} f^{3} x^{3}+5 \left (b x +a \right )^{n} e \,f^{4} x^{4}+\left (b x +a \right )^{n} f^{5} x^{5}}d x \right ) d \] Input:

int((d*x+c)*(f*x+e)^(-5+n)/((b*x+a)^n),x)
 

Output:

int((e + f*x)**n/((a + b*x)**n*e**5 + 5*(a + b*x)**n*e**4*f*x + 10*(a + b* 
x)**n*e**3*f**2*x**2 + 10*(a + b*x)**n*e**2*f**3*x**3 + 5*(a + b*x)**n*e*f 
**4*x**4 + (a + b*x)**n*f**5*x**5),x)*c + int(((e + f*x)**n*x)/((a + b*x)* 
*n*e**5 + 5*(a + b*x)**n*e**4*f*x + 10*(a + b*x)**n*e**3*f**2*x**2 + 10*(a 
 + b*x)**n*e**2*f**3*x**3 + 5*(a + b*x)**n*e*f**4*x**4 + (a + b*x)**n*f**5 
*x**5),x)*d