\(\int (a+b x)^m (c+d x)^{-m-n} (e+f x)^{-4+n} \, dx\) [1795]

Optimal result

Integrand size = 30, antiderivative size = 426 \[ \int (a+b x)^m (c+d x)^{-m-n} (e+f x)^{-4+n} \, dx=-\frac {f (a+b x)^{1+m} (c+d x)^{1-m-n} (e+f x)^{-3+n}}{(b e-a f) (d e-c f) (3-n)}+\frac {f (a d f (2+m)-b d e (4-n)+b c f (2-m-n)) (a+b x)^{1+m} (c+d x)^{1-m-n} (e+f x)^{-2+n}}{(b e-a f)^2 (d e-c f)^2 (2-n) (3-n)}+\frac {\left (a^2 d^2 f^2 \left (2+3 m+m^2\right )-2 a b d f (1+m) (d e (3-n)-c f (1-m-n))-b^2 \left (2 c d e f (3-n) (1-m-n)-d^2 e^2 \left (6-5 n+n^2\right )-c^2 f^2 \left (2+m^2-m (3-2 n)-3 n+n^2\right )\right )\right ) (a+b x)^{1+m} (c+d x)^{-m-n} \left (\frac {(b e-a f) (c+d x)}{(b c-a d) (e+f x)}\right )^{m+n} (e+f x)^{-1+n} \operatorname {Hypergeometric2F1}\left (1+m,m+n,2+m,-\frac {(d e-c f) (a+b x)}{(b c-a d) (e+f x)}\right )}{(b e-a f)^3 (d e-c f)^2 (1+m) (2-n) (3-n)} \] Output:

-f*(b*x+a)^(1+m)*(d*x+c)^(1-m-n)*(f*x+e)^(-3+n)/(-a*f+b*e)/(-c*f+d*e)/(3-n 
)+f*(a*d*f*(2+m)-b*d*e*(4-n)+b*c*f*(2-m-n))*(b*x+a)^(1+m)*(d*x+c)^(1-m-n)* 
(f*x+e)^(-2+n)/(-a*f+b*e)^2/(-c*f+d*e)^2/(2-n)/(3-n)+(a^2*d^2*f^2*(m^2+3*m 
+2)-2*a*b*d*f*(1+m)*(d*e*(3-n)-c*f*(1-m-n))-b^2*(2*c*d*e*f*(3-n)*(1-m-n)-d 
^2*e^2*(n^2-5*n+6)-c^2*f^2*(2+m^2-m*(3-2*n)-3*n+n^2)))*(b*x+a)^(1+m)*(d*x+ 
c)^(-m-n)*((-a*f+b*e)*(d*x+c)/(-a*d+b*c)/(f*x+e))^(m+n)*(f*x+e)^(-1+n)*hyp 
ergeom([m+n, 1+m],[2+m],-(-c*f+d*e)*(b*x+a)/(-a*d+b*c)/(f*x+e))/(-a*f+b*e) 
^3/(-c*f+d*e)^2/(1+m)/(2-n)/(3-n)
 

Mathematica [A] (verified)

Time = 0.59 (sec) , antiderivative size = 341, normalized size of antiderivative = 0.80 \[ \int (a+b x)^m (c+d x)^{-m-n} (e+f x)^{-4+n} \, dx=\frac {(a+b x)^{1+m} (c+d x)^{-m-n} (e+f x)^{-3+n} \left (f (c+d x)+\frac {f (a d f (2+m)+b d e (-4+n)-b c f (-2+m+n)) (c+d x) (e+f x)}{(b e-a f) (d e-c f) (-2+n)}+\frac {\left (a^2 d^2 f^2 \left (2+3 m+m^2\right )+2 a b d f (1+m) (d e (-3+n)-c f (-1+m+n))+b^2 \left (-2 c d e f (-3+n) (-1+m+n)+d^2 e^2 \left (6-5 n+n^2\right )+c^2 f^2 \left (2+m^2-3 n+n^2+m (-3+2 n)\right )\right )\right ) \left (\frac {(b e-a f) (c+d x)}{(b c-a d) (e+f x)}\right )^{m+n} (e+f x)^2 \operatorname {Hypergeometric2F1}\left (1+m,m+n,2+m,\frac {(-d e+c f) (a+b x)}{(b c-a d) (e+f x)}\right )}{(b e-a f)^2 (d e-c f) (1+m) (-2+n)}\right )}{(b e-a f) (d e-c f) (-3+n)} \] Input:

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

Output:

((a + b*x)^(1 + m)*(c + d*x)^(-m - n)*(e + f*x)^(-3 + n)*(f*(c + d*x) + (f 
*(a*d*f*(2 + m) + b*d*e*(-4 + n) - b*c*f*(-2 + m + n))*(c + d*x)*(e + f*x) 
)/((b*e - a*f)*(d*e - c*f)*(-2 + n)) + ((a^2*d^2*f^2*(2 + 3*m + m^2) + 2*a 
*b*d*f*(1 + m)*(d*e*(-3 + n) - c*f*(-1 + m + n)) + b^2*(-2*c*d*e*f*(-3 + n 
)*(-1 + m + n) + d^2*e^2*(6 - 5*n + n^2) + c^2*f^2*(2 + m^2 - 3*n + n^2 + 
m*(-3 + 2*n))))*(((b*e - a*f)*(c + d*x))/((b*c - a*d)*(e + f*x)))^(m + n)* 
(e + f*x)^2*Hypergeometric2F1[1 + m, 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 + n)) 
))/((b*e - a*f)*(d*e - c*f)*(-3 + n))
 

Rubi [A] (verified)

Time = 0.65 (sec) , antiderivative size = 466, normalized size of antiderivative = 1.09, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {144, 172, 25, 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.

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

\(\Big \downarrow \) 144

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

\(\Big \downarrow \) 172

\(\displaystyle -\frac {-\frac {\int -\left (\left (f (b c (m+1)+a d (-m-n+1)) (a d f (m+2)-b d e (4-n)+b c f (-m-n+2))+\left (a b c d f^2-(b c+a d) (a d f (m+2)-b d e (3-n)+b c f (-m-n+2)) f+b d e (a d f (m+2)-b d e (3-n)+b c f (-m-n+2))\right ) (2-n)\right ) (a+b x)^m (c+d x)^{-m-n} (e+f x)^{n-2}\right )dx}{(2-n) (b e-a f) (d e-c f)}-\frac {f (a+b x)^{m+1} (e+f x)^{n-2} (c+d x)^{-m-n+1} (a d f (m+2)+b c f (-m-n+2)-b d e (4-n))}{(2-n) (b e-a f) (d e-c f)}}{(3-n) (b e-a f) (d e-c f)}-\frac {f (a+b x)^{m+1} (e+f x)^{n-3} (c+d x)^{-m-n+1}}{(3-n) (b e-a f) (d e-c f)}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\frac {\int \left (f (b c (m+1)+a d (-m-n+1)) (a d f (m+2)-b d e (4-n)+b c f (-m-n+2))+\left (a b c d f^2-(b c+a d) (a d f (m+2)-b d e (3-n)+b c f (-m-n+2)) f+b d e (a d f (m+2)-b d e (3-n)+b c f (-m-n+2))\right ) (2-n)\right ) (a+b x)^m (c+d x)^{-m-n} (e+f x)^{n-2}dx}{(2-n) (b e-a f) (d e-c f)}-\frac {f (a+b x)^{m+1} (e+f x)^{n-2} (c+d x)^{-m-n+1} (a d f (m+2)+b c f (-m-n+2)-b d e (4-n))}{(2-n) (b e-a f) (d e-c f)}}{(3-n) (b e-a f) (d e-c f)}-\frac {f (a+b x)^{m+1} (e+f x)^{n-3} (c+d x)^{-m-n+1}}{(3-n) (b e-a f) (d e-c f)}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\frac {\left ((2-n) \left (-f (a d+b c) (a d f (m+2)+b c f (-m-n+2)-b d e (3-n))+b d e (a d f (m+2)+b c f (-m-n+2)-b d e (3-n))+a b c d f^2\right )+f (a d (-m-n+1)+b c (m+1)) (a d f (m+2)+b c f (-m-n+2)-b d e (4-n))\right ) \int (a+b x)^m (c+d x)^{-m-n} (e+f x)^{n-2}dx}{(2-n) (b e-a f) (d e-c f)}-\frac {f (a+b x)^{m+1} (e+f x)^{n-2} (c+d x)^{-m-n+1} (a d f (m+2)+b c f (-m-n+2)-b d e (4-n))}{(2-n) (b e-a f) (d e-c f)}}{(3-n) (b e-a f) (d e-c f)}-\frac {f (a+b x)^{m+1} (e+f x)^{n-3} (c+d x)^{-m-n+1}}{(3-n) (b e-a f) (d e-c f)}\)

\(\Big \downarrow \) 142

\(\displaystyle -\frac {\frac {(a+b x)^{m+1} (e+f x)^{n-1} (c+d x)^{-m-n} \left ((2-n) \left (-f (a d+b c) (a d f (m+2)+b c f (-m-n+2)-b d e (3-n))+b d e (a d f (m+2)+b c f (-m-n+2)-b d e (3-n))+a b c d f^2\right )+f (a d (-m-n+1)+b c (m+1)) (a d f (m+2)+b c f (-m-n+2)-b d e (4-n))\right ) \left (\frac {(c+d x) (b e-a f)}{(e+f x) (b c-a d)}\right )^{m+n} \operatorname {Hypergeometric2F1}\left (m+1,m+n,m+2,-\frac {(d e-c f) (a+b x)}{(b c-a d) (e+f x)}\right )}{(m+1) (2-n) (b e-a f)^2 (d e-c f)}-\frac {f (a+b x)^{m+1} (e+f x)^{n-2} (c+d x)^{-m-n+1} (a d f (m+2)+b c f (-m-n+2)-b d e (4-n))}{(2-n) (b e-a f) (d e-c f)}}{(3-n) (b e-a f) (d e-c f)}-\frac {f (a+b x)^{m+1} (e+f x)^{n-3} (c+d x)^{-m-n+1}}{(3-n) (b e-a f) (d e-c f)}\)

Input:

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

Output:

-((f*(a + b*x)^(1 + m)*(c + d*x)^(1 - m - n)*(e + f*x)^(-3 + n))/((b*e - a 
*f)*(d*e - c*f)*(3 - n))) - (-((f*(a*d*f*(2 + m) - b*d*e*(4 - n) + b*c*f*( 
2 - m - n))*(a + b*x)^(1 + m)*(c + d*x)^(1 - m - n)*(e + f*x)^(-2 + n))/(( 
b*e - a*f)*(d*e - c*f)*(2 - n))) + ((f*(b*c*(1 + m) + a*d*(1 - m - n))*(a* 
d*f*(2 + m) - b*d*e*(4 - n) + b*c*f*(2 - m - n)) + (a*b*c*d*f^2 + b*d*e*(a 
*d*f*(2 + m) - b*d*e*(3 - n) + b*c*f*(2 - m - n)) - (b*c + a*d)*f*(a*d*f*( 
2 + m) - b*d*e*(3 - n) + b*c*f*(2 - m - n)))*(2 - n))*(a + b*x)^(1 + m)*(c 
 + d*x)^(-m - n)*(((b*e - a*f)*(c + d*x))/((b*c - a*d)*(e + f*x)))^(m + n) 
*(e + f*x)^(-1 + n)*Hypergeometric2F1[1 + m, 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 - n)))/((b*e - a*f)*(d*e - c*f)*(3 - n))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 144

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

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]

\[\int \left (b x +a \right )^{m} \left (x d +c \right )^{-m -n} \left (f x +e \right )^{-4+n}d x\]

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral((b*x + a)^m*(d*x + c)^(-m - n)*(f*x + e)^(n - 4), x)
 

Sympy [F(-1)]

Timed out. \[ \int (a+b x)^m (c+d x)^{-m-n} (e+f x)^{-4+n} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int (a+b x)^m (c+d x)^{-m-n} (e+f x)^{-4+n} \, dx=\int \frac {{\left (e+f\,x\right )}^{n-4}\,{\left (a+b\,x\right )}^m}{{\left (c+d\,x\right )}^{m+n}} \,d x \] Input:

int(((e + f*x)^(n - 4)*(a + b*x)^m)/(c + d*x)^(m + n),x)
 

Output:

int(((e + f*x)^(n - 4)*(a + b*x)^m)/(c + d*x)^(m + n), x)
 

Reduce [F]

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

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

Output:

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