\(\int (a+b x) (c+d x)^{-2+n} (e+f x)^{-n} \, dx\) [254]

Optimal result

Integrand size = 24, antiderivative size = 110 \[ \int (a+b x) (c+d x)^{-2+n} (e+f x)^{-n} \, dx=\frac {(b c-a d) (c+d x)^{-1+n} (e+f x)^{1-n}}{d (d e-c f) (1-n)}+\frac {b (c+d x)^n (e+f x)^{1-n} \operatorname {Hypergeometric2F1}\left (1,1,1+n,-\frac {f (c+d x)}{d e-c f}\right )}{d (d e-c f) n} \] Output:

(-a*d+b*c)*(d*x+c)^(-1+n)*(f*x+e)^(1-n)/d/(-c*f+d*e)/(1-n)+b*(d*x+c)^n*(f* 
x+e)^(1-n)*hypergeom([1, 1],[1+n],-f*(d*x+c)/(-c*f+d*e))/d/(-c*f+d*e)/n
 

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.05 \[ \int (a+b x) (c+d x)^{-2+n} (e+f x)^{-n} \, dx=\frac {(c+d x)^{-1+n} (e+f x)^{-n} \left (d^2 (b e-a f) (e+f x)-b (d e-c f)^2 \left (\frac {d (e+f x)}{d e-c f}\right )^n \operatorname {Hypergeometric2F1}\left (-1+n,-1+n,n,\frac {f (c+d x)}{-d e+c f}\right )\right )}{d^2 f (-d e+c f) (-1+n)} \] Input:

Integrate[((a + b*x)*(c + d*x)^(-2 + n))/(e + f*x)^n,x]
 

Output:

((c + d*x)^(-1 + n)*(d^2*(b*e - a*f)*(e + f*x) - b*(d*e - c*f)^2*((d*(e + 
f*x))/(d*e - c*f))^n*Hypergeometric2F1[-1 + n, -1 + n, n, (f*(c + d*x))/(- 
(d*e) + c*f)]))/(d^2*f*(-(d*e) + c*f)*(-1 + n)*(e + f*x)^n)
 

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.06, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {88, 80, 79}

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.

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

\(\Big \downarrow \) 88

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

\(\Big \downarrow \) 80

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

\(\Big \downarrow \) 79

\(\displaystyle \frac {(b c-a d) (c+d x)^{n-1} (e+f x)^{1-n}}{d (1-n) (d e-c f)}+\frac {b (c+d x)^n (e+f x)^{-n} \left (\frac {d (e+f x)}{d e-c f}\right )^n \operatorname {Hypergeometric2F1}\left (n,n,n+1,-\frac {f (c+d x)}{d e-c f}\right )}{d^2 n}\)

Input:

Int[((a + b*x)*(c + d*x)^(-2 + n))/(e + f*x)^n,x]
 

Output:

((b*c - a*d)*(c + d*x)^(-1 + n)*(e + f*x)^(1 - n))/(d*(d*e - c*f)*(1 - n)) 
 + (b*(c + d*x)^n*((d*(e + f*x))/(d*e - c*f))^n*Hypergeometric2F1[n, n, 1 
+ n, -((f*(c + d*x))/(d*e - c*f))])/(d^2*n*(e + f*x)^n)
 

Defintions of rubi rules used

rule 79

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(( 
a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c - a*d))^n))*Hypergeometric2F1[-n, m + 1 
, m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m, n}, x] 
&&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] 
 ||  !(RationalQ[n] && GtQ[-d/(b*c - a*d), 0]))
 

rule 80

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(c 
 + d*x)^FracPart[n]/((b/(b*c - a*d))^IntPart[n]*(b*((c + d*x)/(b*c - a*d))) 
^FracPart[n])   Int[(a + b*x)^m*Simp[b*(c/(b*c - a*d)) + b*d*(x/(b*c - a*d) 
), x]^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] &&  !IntegerQ[m] &&  !Integ 
erQ[n] && (RationalQ[m] ||  !SimplerQ[n + 1, m + 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 [F]

\[\int \left (b x +a \right ) \left (x d +c \right )^{n -2} \left (f x +e \right )^{-n}d x\]

Input:

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

Output:

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

Fricas [F]

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

integrate((b*x+a)*(d*x+c)^(-2+n)/((f*x+e)^n),x, algorithm="fricas")
 

Output:

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

Sympy [F(-2)]

Exception generated. \[ \int (a+b x) (c+d x)^{-2+n} (e+f x)^{-n} \, dx=\text {Exception raised: HeuristicGCDFailed} \] Input:

integrate((b*x+a)*(d*x+c)**(-2+n)/((f*x+e)**n),x)
 

Output:

Exception raised: HeuristicGCDFailed >> no luck
 

Maxima [F]

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

integrate((b*x+a)*(d*x+c)^(-2+n)/((f*x+e)^n),x, algorithm="maxima")
 

Output:

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

Giac [F]

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

integrate((b*x+a)*(d*x+c)^(-2+n)/((f*x+e)^n),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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