\(\int (a+b x)^m (A+B x) (c+d x)^n (e+f x)^{-3-m-n} \, dx\) [235]

Optimal result

Integrand size = 34, antiderivative size = 261 \[ \int (a+b x)^m (A+B x) (c+d x)^n (e+f x)^{-3-m-n} \, dx=\frac {(B e-A f) (a+b x)^{1+m} (c+d x)^{1+n} (e+f x)^{-2-m-n}}{(b e-a f) (d e-c f) (2+m+n)}-\frac {(b (B c e (1+m)+A c f (1+n)-A d e (2+m+n))+a (A d f (1+m)+B d e (1+n)-B c f (2+m+n))) (a+b x)^{1+m} (c+d x)^n \left (\frac {(b e-a f) (c+d x)}{(b c-a d) (e+f x)}\right )^{-n} (e+f x)^{-1-m-n} \operatorname {Hypergeometric2F1}\left (1+m,-n,2+m,-\frac {(d e-c f) (a+b x)}{(b c-a d) (e+f x)}\right )}{(b e-a f)^2 (d e-c f) (1+m) (2+m+n)} \] Output:

(-A*f+B*e)*(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)-(b*(B*c*e*(1+m)+A*c*f*(1+n)-A*d*e*(2+m+n))+a*(A*d*f*(1+m)+B*d* 
e*(1+n)-B*c*f*(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 = 0.24 (sec) , antiderivative size = 223, normalized size of antiderivative = 0.85 \[ \int (a+b x)^m (A+B x) (c+d x)^n (e+f x)^{-3-m-n} \, dx=-\frac {(a+b x)^{1+m} (c+d x)^n (e+f x)^{-2-m-n} \left ((-B e+A f) (c+d x)+\frac {(b (B c e (1+m)+A c f (1+n)-A d e (2+m+n))+a (A d f (1+m)+B d e (1+n)-B c f (2+m+n))) \left (\frac {(b e-a f) (c+d x)}{(b c-a d) (e+f x)}\right )^{-n} (e+f x) \operatorname {Hypergeometric2F1}\left (1+m,-n,2+m,\frac {(-d e+c f) (a+b x)}{(b c-a d) (e+f x)}\right )}{(b e-a f) (1+m)}\right )}{(b e-a f) (d e-c f) (2+m+n)} \] Input:

Integrate[(a + b*x)^m*(A + B*x)*(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)*((-(B*e) + A*f)*(c 
 + d*x) + ((b*(B*c*e*(1 + m) + A*c*f*(1 + n) - A*d*e*(2 + m + n)) + a*(A*d 
*f*(1 + m) + B*d*e*(1 + n) - B*c*f*(2 + m + n)))*(e + f*x)*Hypergeometric2 
F1[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 = 0.41 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.088, Rules used = {172, 27, 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.

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

\(\Big \downarrow \) 172

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 142

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

Input:

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

Output:

((B*e - A*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)) - ((b*(B*c*e*(1 + m) + A*c*f*(1 + n) 
- A*d*e*(2 + m + n)) + a*(A*d*f*(1 + m) + B*d*e*(1 + n) - B*c*f*(2 + m + n 
)))*(a + b*x)^(1 + m)*(c + d*x)^n*(e + f*x)^(-1 - m - n)*Hypergeometric2F1 
[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 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

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]
 

rule 172

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> With[{mnp = Simplify[m + n + p]}, Simp[ 
(b*g - a*h)*(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[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*Simp[(a*d*f*g - b*(d*e + c*f 
)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g 
 - a*h)*(mnp + 3)*x, x], x], x] /; ILtQ[mnp + 2, 0] && (SumSimplerQ[m, 1] | 
| ( !(NeQ[n, -1] && SumSimplerQ[n, 1]) &&  !(NeQ[p, -1] && SumSimplerQ[p, 1 
])))] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && NeQ[m, -1]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

int(((c + d*x)**n*(a + b*x)**m*x)/((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)*b + 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)*a