\(\int (a+b x)^m (c+d x)^n (e+f x)^{-3-m-n} \, dx\) [1827]

Optimal result

Integrand size = 29, antiderivative size = 226 \[ \int (a+b x)^m (c+d x)^n (e+f x)^{-3-m-n} \, dx=-\frac {f (a+b x)^{1+m} (c+d x)^{1+n} (e+f x)^{-2-m-n}}{(b e-a f) (d e-c f) (2+m+n)}-\frac {(a d f (1+m)+b c f (1+n)-b d e (2+m+n)) (a+b x)^{1+m} (c+d x)^n \left (\frac {(b e-a f) (c+d x)}{(b c-a d) (e+f x)}\right )^{-n} (e+f x)^{-1-m-n} \operatorname {Hypergeometric2F1}\left (1+m,-n,2+m,-\frac {(d e-c f) (a+b x)}{(b c-a d) (e+f x)}\right )}{(b e-a f)^2 (d e-c f) (1+m) (2+m+n)} \] Output:

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

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.84 \[ \int (a+b x)^m (c+d x)^n (e+f x)^{-3-m-n} \, dx=-\frac {(a+b x)^{1+m} (c+d x)^n (e+f x)^{-2-m-n} \left (f (c+d x)-\frac {(-a d f (1+m)-b c f (1+n)+b d e (2+m+n)) \left (\frac {(b e-a f) (c+d x)}{(b c-a d) (e+f x)}\right )^{-n} (e+f x) \operatorname {Hypergeometric2F1}\left (1+m,-n,2+m,\frac {(-d e+c f) (a+b x)}{(b c-a d) (e+f x)}\right )}{(b e-a f) (1+m)}\right )}{(b e-a f) (d e-c f) (2+m+n)} \] Input:

Integrate[(a + b*x)^m*(c + d*x)^n*(e + f*x)^(-3 - m - n),x]
 

Output:

-(((a + b*x)^(1 + m)*(c + d*x)^n*(e + f*x)^(-2 - m - n)*(f*(c + d*x) - ((- 
(a*d*f*(1 + m)) - b*c*f*(1 + n) + b*d*e*(2 + m + n))*(e + f*x)*Hypergeomet 
ric2F1[1 + m, -n, 2 + m, ((-(d*e) + c*f)*(a + b*x))/((b*c - a*d)*(e + f*x) 
)])/((b*e - a*f)*(1 + m)*(((b*e - a*f)*(c + d*x))/((b*c - a*d)*(e + f*x))) 
^n)))/((b*e - a*f)*(d*e - c*f)*(2 + m + n)))
 

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {107, 142}

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.

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

\(\Big \downarrow \) 107

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

\(\Big \downarrow \) 142

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

Input:

Int[(a + b*x)^m*(c + d*x)^n*(e + f*x)^(-3 - m - n),x]
 

Output:

-((f*(a + b*x)^(1 + m)*(c + d*x)^(1 + n)*(e + f*x)^(-2 - m - n))/((b*e - a 
*f)*(d*e - c*f)*(2 + m + n))) - ((a*d*f*(1 + m) + b*c*f*(1 + n) - b*d*e*(2 
 + m + n))*(a + b*x)^(1 + m)*(c + d*x)^n*(e + f*x)^(-1 - m - n)*Hypergeome 
tric2F1[1 + m, -n, 2 + m, -(((d*e - c*f)*(a + b*x))/((b*c - a*d)*(e + f*x) 
))])/((b*e - a*f)^2*(d*e - c*f)*(1 + m)*(2 + m + n)*(((b*e - a*f)*(c + d*x 
))/((b*c - a*d)*(e + f*x)))^n)
 

Defintions of rubi rules used

rule 107

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 
)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[(a*d*f*(m + 1) + b*c*f*(n + 
 1) + b*d*e*(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f))   Int[(a + b*x)^(m + 
 1)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x 
] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || SumSimplerQ[m, 1])
 

rule 142

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((b*e 
 - a*f)*(m + 1)))*Hypergeometric2F1[m + 1, -n, m + 2, (-(d*e - c*f))*((a + 
b*x)/((b*c - a*d)*(e + f*x)))])/((b*e - a*f)*((c + d*x)/((b*c - a*d)*(e + f 
*x))))^n, x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[m + n + p + 2, 
 0] &&  !IntegerQ[n]
 

Maple [F]

\[\int \left (b x +a \right )^{m} \left (x d +c \right )^{n} \left (f x +e \right )^{-3-m -n}d x\]

Input:

int((b*x+a)^m*(d*x+c)^n*(f*x+e)^(-3-m-n),x)
 

Output:

int((b*x+a)^m*(d*x+c)^n*(f*x+e)^(-3-m-n),x)
 

Fricas [F]

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

integrate((b*x+a)^m*(d*x+c)^n*(f*x+e)^(-3-m-n),x, algorithm="fricas")
 

Output:

integral((b*x + a)^m*(d*x + c)^n*(f*x + e)^(-m - n - 3), x)
 

Sympy [F(-1)]

Timed out. \[ \int (a+b x)^m (c+d x)^n (e+f x)^{-3-m-n} \, dx=\text {Timed out} \] Input:

integrate((b*x+a)**m*(d*x+c)**n*(f*x+e)**(-3-m-n),x)
 

Output:

Timed out
 

Maxima [F]

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

integrate((b*x+a)^m*(d*x+c)^n*(f*x+e)^(-3-m-n),x, algorithm="maxima")
 

Output:

integrate((b*x + a)^m*(d*x + c)^n*(f*x + e)^(-m - n - 3), x)
 

Giac [F]

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

integrate((b*x+a)^m*(d*x+c)^n*(f*x+e)^(-3-m-n),x, algorithm="giac")
 

Output:

integrate((b*x + a)^m*(d*x + c)^n*(f*x + e)^(-m - n - 3), x)
 

Mupad [F(-1)]

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

int(((a + b*x)^m*(c + d*x)^n)/(e + f*x)^(m + n + 3),x)
 

Output:

int(((a + b*x)^m*(c + d*x)^n)/(e + f*x)^(m + n + 3), x)
 

Reduce [F]

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

int((b*x+a)^m*(d*x+c)^n*(f*x+e)^(-3-m-n),x)
 

Output:

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