\(\int (a+b x)^{-4-n} (c+d x)^n \, dx\) [784]

Optimal result

Integrand size = 19, antiderivative size = 131 \[ \int (a+b x)^{-4-n} (c+d x)^n \, dx=-\frac {(a+b x)^{-3-n} (c+d x)^{1+n}}{(b c-a d) (3+n)}+\frac {2 d (a+b x)^{-2-n} (c+d x)^{1+n}}{(b c-a d)^2 (2+n) (3+n)}-\frac {2 d^2 (a+b x)^{-1-n} (c+d x)^{1+n}}{(b c-a d)^3 (1+n) (2+n) (3+n)} \] Output:

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

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.86 \[ \int (a+b x)^{-4-n} (c+d x)^n \, dx=-\frac {(a+b x)^{-3-n} (c+d x)^{1+n} \left (a^2 d^2 \left (6+5 n+n^2\right )-2 a b d (3+n) (c+c n-d x)+b^2 \left (c^2 \left (2+3 n+n^2\right )-2 c d (1+n) x+2 d^2 x^2\right )\right )}{(b c-a d)^3 (1+n) (2+n) (3+n)} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.04, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {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.

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

\(\Big \downarrow \) 55

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

\(\Big \downarrow \) 55

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

\(\Big \downarrow \) 48

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

Input:

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

Output:

-(((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))
 

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])
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(317\) vs. \(2(131)=262\).

Time = 0.39 (sec) , antiderivative size = 318, normalized size of antiderivative = 2.43

method	result	size
gosper	\(\frac {\left (b x +a \right )^{-3-n} \left (x d +c \right )^{1+n} \left (a^{2} d^{2} n^{2}-2 a b c d \,n^{2}+2 a b \,d^{2} n x +b^{2} c^{2} n^{2}-2 b^{2} c d n x +2 d^{2} x^{2} b^{2}+5 a^{2} d^{2} n -8 a b c d n +6 x a b \,d^{2}+3 b^{2} c^{2} n -2 x \,b^{2} c d +6 a^{2} d^{2}-6 a b c d +2 b^{2} c^{2}\right )}{a^{3} d^{3} n^{3}-3 a^{2} b c \,d^{2} n^{3}+3 a \,b^{2} c^{2} d \,n^{3}-b^{3} c^{3} n^{3}+6 a^{3} d^{3} n^{2}-18 a^{2} b c \,d^{2} n^{2}+18 a \,b^{2} c^{2} d \,n^{2}-6 b^{3} c^{3} n^{2}+11 a^{3} d^{3} n -33 a^{2} b c \,d^{2} n +33 a \,b^{2} c^{2} d n -11 b^{3} c^{3} n +6 a^{3} d^{3}-18 a^{2} b c \,d^{2}+18 a \,b^{2} c^{2} d -6 b^{3} c^{3}}\)	\(318\)
orering	\(\frac {\left (b x +a \right ) \left (x d +c \right ) \left (a^{2} d^{2} n^{2}-2 a b c d \,n^{2}+2 a b \,d^{2} n x +b^{2} c^{2} n^{2}-2 b^{2} c d n x +2 d^{2} x^{2} b^{2}+5 a^{2} d^{2} n -8 a b c d n +6 x a b \,d^{2}+3 b^{2} c^{2} n -2 x \,b^{2} c d +6 a^{2} d^{2}-6 a b c d +2 b^{2} c^{2}\right ) \left (b x +a \right )^{-4-n} \left (x d +c \right )^{n}}{a^{3} d^{3} n^{3}-3 a^{2} b c \,d^{2} n^{3}+3 a \,b^{2} c^{2} d \,n^{3}-b^{3} c^{3} n^{3}+6 a^{3} d^{3} n^{2}-18 a^{2} b c \,d^{2} n^{2}+18 a \,b^{2} c^{2} d \,n^{2}-6 b^{3} c^{3} n^{2}+11 a^{3} d^{3} n -33 a^{2} b c \,d^{2} n +33 a \,b^{2} c^{2} d n -11 b^{3} c^{3} n +6 a^{3} d^{3}-18 a^{2} b c \,d^{2}+18 a \,b^{2} c^{2} d -6 b^{3} c^{3}}\)	\(326\)

Input:

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

Output:

(b*x+a)^(-3-n)*(d*x+c)^(1+n)*(a^2*d^2*n^2-2*a*b*c*d*n^2+2*a*b*d^2*n*x+b^2* 
c^2*n^2-2*b^2*c*d*n*x+2*b^2*d^2*x^2+5*a^2*d^2*n-8*a*b*c*d*n+6*a*b*d^2*x+3* 
b^2*c^2*n-2*b^2*c*d*x+6*a^2*d^2-6*a*b*c*d+2*b^2*c^2)/(a^3*d^3*n^3-3*a^2*b* 
c*d^2*n^3+3*a*b^2*c^2*d*n^3-b^3*c^3*n^3+6*a^3*d^3*n^2-18*a^2*b*c*d^2*n^2+1 
8*a*b^2*c^2*d*n^2-6*b^3*c^3*n^2+11*a^3*d^3*n-33*a^2*b*c*d^2*n+33*a*b^2*c^2 
*d*n-11*b^3*c^3*n+6*a^3*d^3-18*a^2*b*c*d^2+18*a*b^2*c^2*d-6*b^3*c^3)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 509 vs. \(2 (131) = 262\).

Time = 0.10 (sec) , antiderivative size = 509, normalized size of antiderivative = 3.89 \[ \int (a+b x)^{-4-n} (c+d x)^n \, dx=-\frac {{\left (2 \, b^{3} d^{3} x^{4} + 2 \, a b^{2} c^{3} - 6 \, a^{2} b c^{2} d + 6 \, a^{3} c d^{2} + 2 \, {\left (4 \, a b^{2} d^{3} - {\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} n\right )} x^{3} + {\left (a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2}\right )} n^{2} + {\left (12 \, a^{2} b d^{3} + {\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} n^{2} + {\left (b^{3} c^{2} d - 8 \, a b^{2} c d^{2} + 7 \, a^{2} b d^{3}\right )} n\right )} x^{2} + {\left (3 \, a b^{2} c^{3} - 8 \, a^{2} b c^{2} d + 5 \, a^{3} c d^{2}\right )} n + {\left (2 \, b^{3} c^{3} - 6 \, a b^{2} c^{2} d + 6 \, a^{2} b c d^{2} + 6 \, a^{3} d^{3} + {\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )} n^{2} + {\left (3 \, b^{3} c^{3} - 7 \, a b^{2} c^{2} d - a^{2} b c d^{2} + 5 \, a^{3} d^{3}\right )} n\right )} x\right )} {\left (b x + a\right )}^{-n - 4} {\left (d x + c\right )}^{n}}{6 \, b^{3} c^{3} - 18 \, a b^{2} c^{2} d + 18 \, a^{2} b c d^{2} - 6 \, a^{3} d^{3} + {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} n^{3} + 6 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} n^{2} + 11 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} n} \] Input:

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

Output:

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

Sympy [F(-2)]

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

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

Output:

Exception raised: HeuristicGCDFailed >> no luck
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [B] (verification not implemented)

Time = 0.58 (sec) , antiderivative size = 525, normalized size of antiderivative = 4.01 \[ \int (a+b x)^{-4-n} (c+d x)^n \, dx=\frac {x\,{\left (c+d\,x\right )}^n\,\left (a^3\,d^3\,n^2+5\,a^3\,d^3\,n+6\,a^3\,d^3-a^2\,b\,c\,d^2\,n^2-a^2\,b\,c\,d^2\,n+6\,a^2\,b\,c\,d^2-a\,b^2\,c^2\,d\,n^2-7\,a\,b^2\,c^2\,d\,n-6\,a\,b^2\,c^2\,d+b^3\,c^3\,n^2+3\,b^3\,c^3\,n+2\,b^3\,c^3\right )}{{\left (a\,d-b\,c\right )}^3\,{\left (a+b\,x\right )}^{n+4}\,\left (n^3+6\,n^2+11\,n+6\right )}+\frac {a\,c\,{\left (c+d\,x\right )}^n\,\left (a^2\,d^2\,n^2+5\,a^2\,d^2\,n+6\,a^2\,d^2-2\,a\,b\,c\,d\,n^2-8\,a\,b\,c\,d\,n-6\,a\,b\,c\,d+b^2\,c^2\,n^2+3\,b^2\,c^2\,n+2\,b^2\,c^2\right )}{{\left (a\,d-b\,c\right )}^3\,{\left (a+b\,x\right )}^{n+4}\,\left (n^3+6\,n^2+11\,n+6\right )}+\frac {2\,b^3\,d^3\,x^4\,{\left (c+d\,x\right )}^n}{{\left (a\,d-b\,c\right )}^3\,{\left (a+b\,x\right )}^{n+4}\,\left (n^3+6\,n^2+11\,n+6\right )}+\frac {b\,d\,x^2\,{\left (c+d\,x\right )}^n\,\left (a^2\,d^2\,n^2+7\,a^2\,d^2\,n+12\,a^2\,d^2-2\,a\,b\,c\,d\,n^2-8\,a\,b\,c\,d\,n+b^2\,c^2\,n^2+b^2\,c^2\,n\right )}{{\left (a\,d-b\,c\right )}^3\,{\left (a+b\,x\right )}^{n+4}\,\left (n^3+6\,n^2+11\,n+6\right )}+\frac {2\,b^2\,d^2\,x^3\,{\left (c+d\,x\right )}^n\,\left (4\,a\,d+a\,d\,n-b\,c\,n\right )}{{\left (a\,d-b\,c\right )}^3\,{\left (a+b\,x\right )}^{n+4}\,\left (n^3+6\,n^2+11\,n+6\right )} \] Input:

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

Output:

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

Reduce [F]

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

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

Output:

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