3.2.46 \(\int (a+b x)^m (A+B x) (c+d x)^n (e+f x)^{-2-m-n} \, dx\) [146]

3.2.46.1 Optimal result

Integrand size = 34, antiderivative size = 277 \[ \int (a+b x)^m (A+B x) (c+d x)^n (e+f x)^{-2-m-n} \, dx=\frac {B (a+b x)^{1+m} (c+d x)^n \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} (e+f x)^{-m-n} \left (\frac {b (e+f x)}{b e-a f}\right )^{m+n} \operatorname {AppellF1}\left (1+m,-n,1+m+n,2+m,-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{f (b e-a f) (1+m)}-\frac {(B e-A f) (a+b x)^{1+m} (c+d x)^n \left (\frac {(b e-a f) (c+d x)}{(b c-a d) (e+f x)}\right )^{-n} (e+f x)^{-1-m-n} \operatorname {Hypergeometric2F1}\left (1+m,-n,2+m,-\frac {(d e-c f) (a+b x)}{(b c-a d) (e+f x)}\right )}{f (b e-a f) (1+m)} \]

output

B*(b*x+a)^(1+m)*(d*x+c)^n*(f*x+e)^(-m-n)*(b*(f*x+e)/(-a*f+b*e))^(m+n)*Appe 
llF1(1+m,-n,1+m+n,2+m,-d*(b*x+a)/(-a*d+b*c),-f*(b*x+a)/(-a*f+b*e))/f/(-a*f 
+b*e)/(1+m)/((b*(d*x+c)/(-a*d+b*c))^n)-(-A*f+B*e)*(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))/f/(-a*f+b*e)/(1+m)/(((-a*f+b*e)*(d*x+c)/(-a*d+b*c)/(f*x+e))^n)
 

3.2.46.2 Mathematica [A] (warning: unable to verify)

Time = 0.35 (sec) , antiderivative size = 215, normalized size of antiderivative = 0.78 \[ \int (a+b x)^m (A+B x) (c+d x)^n (e+f x)^{-2-m-n} \, dx=-\frac {(a+b x)^{1+m} (c+d x)^n \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} (e+f x)^{-1-m-n} \left (\frac {b (e+f x)}{b e-a f}\right )^n \left (B (e+f x) \left (\frac {b (e+f x)}{b e-a f}\right )^m \operatorname {AppellF1}\left (1+m,-n,1+m+n,2+m,\frac {d (a+b x)}{-b c+a d},\frac {f (a+b x)}{-b e+a f}\right )+(-B e+A f) \operatorname {Hypergeometric2F1}\left (1+m,-n,2+m,\frac {(-d e+c f) (a+b x)}{(b c-a d) (e+f x)}\right )\right )}{f (-b e+a f) (1+m)} \]

input

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

output

-(((a + b*x)^(1 + m)*(c + d*x)^n*(e + f*x)^(-1 - m - n)*((b*(e + f*x))/(b* 
e - a*f))^n*(B*(e + f*x)*((b*(e + f*x))/(b*e - a*f))^m*AppellF1[1 + m, -n, 
 1 + m + n, 2 + m, (d*(a + b*x))/(-(b*c) + a*d), (f*(a + b*x))/(-(b*e) + a 
*f)] + (-(B*e) + A*f)*Hypergeometric2F1[1 + m, -n, 2 + m, ((-(d*e) + c*f)* 
(a + b*x))/((b*c - a*d)*(e + f*x))]))/(f*(-(b*e) + a*f)*(1 + m)*((b*(c + d 
*x))/(b*c - a*d))^n))
 

3.2.46.3 Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.147, Rules used = {177, 142, 157, 156, 155}

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.

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

\(\Big \downarrow \) 177

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

\(\Big \downarrow \) 142

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

\(\Big \downarrow \) 157

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

\(\Big \downarrow \) 156

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

\(\Big \downarrow \) 155

\(\displaystyle \frac {B (a+b x)^{m+1} (c+d x)^n (e+f x)^{-m-n} \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} \left (\frac {b (e+f x)}{b e-a f}\right )^{m+n} \operatorname {AppellF1}\left (m+1,-n,m+n+1,m+2,-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{f (m+1) (b e-a f)}-\frac {(a+b x)^{m+1} (B e-A f) (c+d x)^n (e+f x)^{-m-n-1} \left (\frac {(c+d x) (b e-a f)}{(e+f x) (b c-a d)}\right )^{-n} \operatorname {Hypergeometric2F1}\left (m+1,-n,m+2,-\frac {(d e-c f) (a+b x)}{(b c-a d) (e+f x)}\right )}{f (m+1) (b e-a f)}\)

input

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

output

(B*(a + b*x)^(1 + m)*(c + d*x)^n*(e + f*x)^(-m - n)*((b*(e + f*x))/(b*e - 
a*f))^(m + n)*AppellF1[1 + m, -n, 1 + m + n, 2 + m, -((d*(a + b*x))/(b*c - 
 a*d)), -((f*(a + b*x))/(b*e - a*f))])/(f*(b*e - a*f)*(1 + m)*((b*(c + d*x 
))/(b*c - a*d))^n) - ((B*e - A*f)*(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)))])/(f*(b*e - a*f)*(1 + m)*(((b*e - a*f)*(c + d*x) 
)/((b*c - a*d)*(e + f*x)))^n)
 

3.2.46.3.1 Defintions of rubi rules used

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 155

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_)) 
^(p_), x_] :> Simp[((a + b*x)^(m + 1)/(b*(m + 1)*Simplify[b/(b*c - a*d)]^n* 
Simplify[b/(b*e - a*f)]^p))*AppellF1[m + 1, -n, -p, m + 2, (-d)*((a + b*x)/ 
(b*c - a*d)), (-f)*((a + b*x)/(b*e - a*f))], x] /; FreeQ[{a, b, c, d, e, f, 
 m, n, p}, x] &&  !IntegerQ[m] &&  !IntegerQ[n] &&  !IntegerQ[p] && GtQ[Sim 
plify[b/(b*c - a*d)], 0] && GtQ[Simplify[b/(b*e - a*f)], 0] &&  !(GtQ[Simpl 
ify[d/(d*a - c*b)], 0] && GtQ[Simplify[d/(d*e - c*f)], 0] && SimplerQ[c + d 
*x, a + b*x]) &&  !(GtQ[Simplify[f/(f*a - e*b)], 0] && GtQ[Simplify[f/(f*c 
- e*d)], 0] && SimplerQ[e + f*x, a + b*x])
 

rule 156

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

rule 157

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

rule 177

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

3.2.46.4 Maple [F]

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

input

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

output

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

3.2.46.5 Fricas [F]

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

input

integrate((b*x+a)^m*(B*x+A)*(d*x+c)^n*(f*x+e)^(-2-m-n),x, algorithm="frica 
s")
 

output

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

3.2.46.6 Sympy [F(-1)]

Timed out. \[ \int (a+b x)^m (A+B x) (c+d x)^n (e+f x)^{-2-m-n} \, dx=\text {Timed out} \]

input

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

output

Timed out
 

3.2.46.7 Maxima [F]

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

input

integrate((b*x+a)^m*(B*x+A)*(d*x+c)^n*(f*x+e)^(-2-m-n),x, algorithm="maxim 
a")
 

output

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

3.2.46.8 Giac [F]

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

input

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

output

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

3.2.46.9 Mupad [F(-1)]

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

input

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

output

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