\(\int (a+b x)^m (c+d x)^{-m-n} (e+f x)^{-3+n} \, dx\) [3150]

Optimal result

Integrand size = 30, antiderivative size = 237 \[ \int (a+b x)^m (c+d x)^{-m-n} (e+f x)^{-3+n} \, dx=-\frac {f (a+b x)^{1+m} (c+d x)^{1-m-n} (e+f x)^{-2+n}}{(b e-a f) (d e-c f) (2-n)}-\frac {(a d f (1+m)-b (d e (2-n)-c f (1-m-n))) (a+b x)^{1+m} (c+d x)^{-m-n} \left (\frac {(b e-a f) (c+d x)}{(b c-a d) (e+f x)}\right )^{m+n} (e+f x)^{-1+n} \operatorname {Hypergeometric2F1}\left (1+m,m+n,2+m,-\frac {(d e-c f) (a+b x)}{(b c-a d) (e+f x)}\right )}{(b e-a f)^2 (d e-c f) (1+m) (2-n)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 235, normalized size of antiderivative = 0.99, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {98, 134} \[ \int (a+b x)^m (c+d x)^{-m-n} (e+f x)^{-3+n} \, dx=-\frac {(a+b x)^{m+1} (e+f x)^{n-1} (c+d x)^{-m-n} (a d f (m+1)+b c f (-m-n+1)-b d e (2-n)) \left (\frac {(c+d x) (b e-a f)}{(e+f x) (b c-a d)}\right )^{m+n} \operatorname {Hypergeometric2F1}\left (m+1,m+n,m+2,-\frac {(d e-c f) (a+b x)}{(b c-a d) (e+f x)}\right )}{(m+1) (2-n) (b e-a f)^2 (d e-c f)}-\frac {f (a+b x)^{m+1} (e+f x)^{n-2} (c+d x)^{-m-n+1}}{(2-n) (b e-a f) (d e-c f)} \]

[In]

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

[Out]

-((f*(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))) - ((a*d*f*
(1 + m) - b*d*e*(2 - n) + b*c*f*(1 - m - 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))

Rule 98

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> 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] + Dist[(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] || Sum
SimplerQ[m, 1])

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]

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

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.78 \[ \int (a+b x)^m (c+d x)^{-m-n} (e+f x)^{-3+n} \, dx=\frac {(a+b x)^{1+m} (c+d x)^{-m-n} (e+f x)^{-2+n} \left (f (c+d x)+\frac {(a d f (1+m)+b d e (-2+n)-b c f (-1+m+n)) \left (\frac {(b e-a f) (c+d x)}{(b c-a d) (e+f x)}\right )^{m+n} (e+f x) \operatorname {Hypergeometric2F1}\left (1+m,m+n,2+m,\frac {(-d e+c f) (a+b x)}{(b c-a d) (e+f x)}\right )}{(b e-a f) (1+m)}\right )}{(b e-a f) (d e-c f) (-2+n)} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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