\(\int (a+b x)^m (A+B x) (c+d x)^n (e+f x)^{-4-m-n} \, dx\) [148]

Optimal result

Integrand size = 34, antiderivative size = 558 \[ \int (a+b x)^m (A+B x) (c+d x)^n (e+f x)^{-4-m-n} \, dx=\frac {(B e-A f) (a+b x)^{1+m} (c+d x)^{1+n} (e+f x)^{-3-m-n}}{(b e-a f) (d e-c f) (3+m+n)}+\frac {(a f (A d f (2+m)+B (d e (1+n)-c f (3+m+n)))+b (B e (d e+c f (1+m))+A f (c f (2+n)-d e (4+m+n)))) (a+b x)^{1+m} (c+d x)^{1+n} (e+f x)^{-2-m-n}}{(b e-a f)^2 (d e-c f)^2 (2+m+n) (3+m+n)}+\frac {((2+m+n) (a b c d f (B e-A f)+b d e ((a B c f+A (b d e-b c f-a d f)) (3+m+n)-(B e-A f) (b c (1+m)+a d (1+n)))-(b c+a d) f ((a B c f+A (b d e-b c f-a d f)) (3+m+n)-(B e-A f) (b c (1+m)+a d (1+n))))-(b c (1+m)+a d (1+n)) (a f (A d f (2+m)+B (d e (1+n)-c f (3+m+n)))+b (B e (d e+c f (1+m))+A f (c f (2+n)-d e (4+m+n))))) (a+b x)^{1+m} (c+d x)^n \left (\frac {(b e-a f) (c+d x)}{(b c-a d) (e+f x)}\right )^{-n} (e+f x)^{-1-m-n} \operatorname {Hypergeometric2F1}\left (1+m,-n,2+m,-\frac {(d e-c f) (a+b x)}{(b c-a d) (e+f x)}\right )}{(b e-a f)^3 (d e-c f)^2 (1+m) (2+m+n) (3+m+n)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.64 (sec) , antiderivative size = 558, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.088, Rules used = {160, 12, 134} \[ \int (a+b x)^m (A+B x) (c+d x)^n (e+f x)^{-4-m-n} \, dx=\frac {(a+b x)^{m+1} (c+d x)^n (e+f x)^{-m-n-1} \left (\frac {(c+d x) (b e-a f)}{(e+f x) (b c-a d)}\right )^{-n} ((m+n+2) (-b d e (a (A d f (m+2)-B c f (m+n+3)+B d e (n+1))+b (A c f (n+2)-A d e (m+n+3)+B c e (m+1)))+f (a d+b c) (a (A d f (m+2)-B c f (m+n+3)+B d e (n+1))+b (A c f (n+2)-A d e (m+n+3)+B c e (m+1)))+a b c d f (B e-A f))-(a d (n+1)+b c (m+1)) (a f (A d f (m+2)-B c f (m+n+3)+B d e (n+1))+b (A f (c f (n+2)-d e (m+n+4))+B e (c f (m+1)+d e)))) \operatorname {Hypergeometric2F1}\left (m+1,-n,m+2,-\frac {(d e-c f) (a+b x)}{(b c-a d) (e+f x)}\right )}{(m+1) (m+n+2) (m+n+3) (b e-a f)^3 (d e-c f)^2}+\frac {(a+b x)^{m+1} (B e-A f) (c+d x)^{n+1} (e+f x)^{-m-n-3}}{(m+n+3) (b e-a f) (d e-c f)}+\frac {(a+b x)^{m+1} (c+d x)^{n+1} (e+f x)^{-m-n-2} (a f (A d f (m+2)-B c f (m+n+3)+B d e (n+1))+b (A f (c f (n+2)-d e (m+n+4))+B e (c f (m+1)+d e)))}{(m+n+2) (m+n+3) (b e-a f)^2 (d e-c f)^2} \]

[In]

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

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 134

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> 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 160

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

Rubi steps \begin{align*} \text {integral}& = \frac {(B e-A f) (a+b x)^{1+m} (c+d x)^{1+n} (e+f x)^{-3-m-n}}{(b e-a f) (d e-c f) (3+m+n)}-\frac {\int (a+b x)^m (c+d x)^n (e+f x)^{-3-m-n} (b (B c e (1+m)+A c f (2+n)-A d e (3+m+n))+a (A d f (2+m)+B d e (1+n)-B c f (3+m+n))-b d (B e-A f) x) \, dx}{(b e-a f) (d e-c f) (3+m+n)} \\ & = \frac {(B e-A f) (a+b x)^{1+m} (c+d x)^{1+n} (e+f x)^{-3-m-n}}{(b e-a f) (d e-c f) (3+m+n)}+\frac {(a f (A d f (2+m)+B d e (1+n)-B c f (3+m+n))+b (B e (d e+c f (1+m))+A f (c f (2+n)-d e (4+m+n)))) (a+b x)^{1+m} (c+d x)^{1+n} (e+f x)^{-2-m-n}}{(b e-a f)^2 (d e-c f)^2 (2+m+n) (3+m+n)}+\frac {\int ((2+m+n) (a b c d f (B e-A f)-b d e (b (B c e (1+m)+A c f (2+n)-A d e (3+m+n))+a (A d f (2+m)+B d e (1+n)-B c f (3+m+n)))+(b c+a d) f (b (B c e (1+m)+A c f (2+n)-A d e (3+m+n))+a (A d f (2+m)+B d e (1+n)-B c f (3+m+n))))-(b c (1+m)+a d (1+n)) (a f (A d f (2+m)+B d e (1+n)-B c f (3+m+n))+b (B e (d e+c f (1+m))+A f (c f (2+n)-d e (4+m+n))))) (a+b x)^m (c+d x)^n (e+f x)^{-2-m-n} \, dx}{(b e-a f)^2 (d e-c f)^2 (2+m+n) (3+m+n)} \\ & = \frac {(B e-A f) (a+b x)^{1+m} (c+d x)^{1+n} (e+f x)^{-3-m-n}}{(b e-a f) (d e-c f) (3+m+n)}+\frac {(a f (A d f (2+m)+B d e (1+n)-B c f (3+m+n))+b (B e (d e+c f (1+m))+A f (c f (2+n)-d e (4+m+n)))) (a+b x)^{1+m} (c+d x)^{1+n} (e+f x)^{-2-m-n}}{(b e-a f)^2 (d e-c f)^2 (2+m+n) (3+m+n)}+\frac {((2+m+n) (a b c d f (B e-A f)-b d e (b (B c e (1+m)+A c f (2+n)-A d e (3+m+n))+a (A d f (2+m)+B d e (1+n)-B c f (3+m+n)))+(b c+a d) f (b (B c e (1+m)+A c f (2+n)-A d e (3+m+n))+a (A d f (2+m)+B d e (1+n)-B c f (3+m+n))))-(b c (1+m)+a d (1+n)) (a f (A d f (2+m)+B d e (1+n)-B c f (3+m+n))+b (B e (d e+c f (1+m))+A f (c f (2+n)-d e (4+m+n))))) \int (a+b x)^m (c+d x)^n (e+f x)^{-2-m-n} \, dx}{(b e-a f)^2 (d e-c f)^2 (2+m+n) (3+m+n)} \\ & = \frac {(B e-A f) (a+b x)^{1+m} (c+d x)^{1+n} (e+f x)^{-3-m-n}}{(b e-a f) (d e-c f) (3+m+n)}+\frac {(a f (A d f (2+m)+B d e (1+n)-B c f (3+m+n))+b (B e (d e+c f (1+m))+A f (c f (2+n)-d e (4+m+n)))) (a+b x)^{1+m} (c+d x)^{1+n} (e+f x)^{-2-m-n}}{(b e-a f)^2 (d e-c f)^2 (2+m+n) (3+m+n)}+\frac {((2+m+n) (a b c d f (B e-A f)-b d e (b (B c e (1+m)+A c f (2+n)-A d e (3+m+n))+a (A d f (2+m)+B d e (1+n)-B c f (3+m+n)))+(b c+a d) f (b (B c e (1+m)+A c f (2+n)-A d e (3+m+n))+a (A d f (2+m)+B d e (1+n)-B c f (3+m+n))))-(b c (1+m)+a d (1+n)) (a f (A d f (2+m)+B d e (1+n)-B c f (3+m+n))+b (B e (d e+c f (1+m))+A f (c f (2+n)-d e (4+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} \, _2F_1\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)^3 (d e-c f)^2 (1+m) (2+m+n) (3+m+n)} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.09 (sec) , antiderivative size = 508, normalized size of antiderivative = 0.91 \[ \int (a+b x)^m (A+B x) (c+d x)^n (e+f x)^{-4-m-n} \, dx=-\frac {(a+b x)^{1+m} (c+d x)^n (e+f x)^{-3-m-n} \left (-((B e-A f) (c+d x))-\frac {(a f (A d f (2+m)+B d e (1+n)-B c f (3+m+n))+b (B e (d e+c f (1+m))+A f (c f (2+n)-d e (4+m+n)))) (c+d x) (e+f x)}{(b e-a f) (d e-c f) (2+m+n)}-\frac {((2+m+n) (a b c d f (B e-A f)-b d e (b (B c e (1+m)+A c f (2+n)-A d e (3+m+n))+a (A d f (2+m)+B d e (1+n)-B c f (3+m+n)))+(b c+a d) f (b (B c e (1+m)+A c f (2+n)-A d e (3+m+n))+a (A d f (2+m)+B d e (1+n)-B c f (3+m+n))))-(b c (1+m)+a d (1+n)) (a f (A d f (2+m)+B d e (1+n)-B c f (3+m+n))+b (B e (d e+c f (1+m))+A f (c f (2+n)-d e (4+m+n))))) \left (\frac {(b e-a f) (c+d x)}{(b c-a d) (e+f x)}\right )^{-n} (e+f x)^2 \operatorname {Hypergeometric2F1}\left (1+m,-n,2+m,\frac {(-d e+c f) (a+b x)}{(b c-a d) (e+f x)}\right )}{(b e-a f)^2 (d e-c f) (1+m) (2+m+n)}\right )}{(b e-a f) (d e-c f) (3+m+n)} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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