Integrand size = 30, antiderivative size = 124 \[ \int (a+b x)^m (c+d x)^{-m-n} (e+f x)^{-2+n} \, dx=\frac {(a+b x)^{1+m} (c+d x)^{-m-n} \left (\frac {(b e-a f) (c+d x)}{(b c-a d) (e+f x)}\right )^{m+n} (e+f x)^{-1+n} \operatorname {Hypergeometric2F1}\left (1+m,m+n,2+m,-\frac {(d e-c f) (a+b x)}{(b c-a d) (e+f x)}\right )}{(b e-a f) (1+m)} \] Output:
(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([m+n, 1+m],[2+m],-(-c*f+d*e)*(b*x+a)/(-a*d+b*c)/ (f*x+e))/(-a*f+b*e)/(1+m)
Time = 0.09 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.99 \[ \int (a+b x)^m (c+d x)^{-m-n} (e+f x)^{-2+n} \, dx=\frac {(a+b x)^{1+m} (c+d x)^{-m-n} \left (\frac {(b e-a f) (c+d x)}{(b c-a d) (e+f x)}\right )^{m+n} (e+f x)^{-1+n} \operatorname {Hypergeometric2F1}\left (1+m,m+n,2+m,\frac {(-d e+c f) (a+b x)}{(b c-a d) (e+f x)}\right )}{(b e-a f) (1+m)} \] Input:
Integrate[(a + b*x)^m*(c + d*x)^(-m - n)*(e + f*x)^(-2 + n),x]
Output:
((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)*(1 + m))
Time = 0.22 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.033, Rules used = {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.
\(\displaystyle \int (a+b x)^m (e+f x)^{n-2} (c+d x)^{-m-n} \, dx\) |
\(\Big \downarrow \) 142 |
\(\displaystyle \frac {(a+b x)^{m+1} (e+f x)^{n-1} (c+d x)^{-m-n} \left (\frac {(c+d x) (b e-a f)}{(e+f x) (b c-a d)}\right )^{m+n} \operatorname {Hypergeometric2F1}\left (m+1,m+n,m+2,-\frac {(d e-c f) (a+b x)}{(b c-a d) (e+f x)}\right )}{(m+1) (b e-a f)}\) |
Input:
Int[(a + b*x)^m*(c + d*x)^(-m - n)*(e + f*x)^(-2 + n),x]
Output:
((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)*(1 + m))
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]
\[\int \left (b x +a \right )^{m} \left (x d +c \right )^{-m -n} \left (f x +e \right )^{n -2}d x\]
Input:
int((b*x+a)^m*(d*x+c)^(-m-n)*(f*x+e)^(n-2),x)
Output:
int((b*x+a)^m*(d*x+c)^(-m-n)*(f*x+e)^(n-2),x)
\[ \int (a+b x)^m (c+d x)^{-m-n} (e+f x)^{-2+n} \, dx=\int { {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m - n} {\left (f x + e\right )}^{n - 2} \,d x } \] Input:
integrate((b*x+a)^m*(d*x+c)^(-m-n)*(f*x+e)^(-2+n),x, algorithm="fricas")
Output:
integral((b*x + a)^m*(d*x + c)^(-m - n)*(f*x + e)^(n - 2), x)
Timed out. \[ \int (a+b x)^m (c+d x)^{-m-n} (e+f x)^{-2+n} \, dx=\text {Timed out} \] Input:
integrate((b*x+a)**m*(d*x+c)**(-m-n)*(f*x+e)**(-2+n),x)
Output:
Timed out
\[ \int (a+b x)^m (c+d x)^{-m-n} (e+f x)^{-2+n} \, dx=\int { {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m - n} {\left (f x + e\right )}^{n - 2} \,d x } \] Input:
integrate((b*x+a)^m*(d*x+c)^(-m-n)*(f*x+e)^(-2+n),x, algorithm="maxima")
Output:
integrate((b*x + a)^m*(d*x + c)^(-m - n)*(f*x + e)^(n - 2), x)
\[ \int (a+b x)^m (c+d x)^{-m-n} (e+f x)^{-2+n} \, dx=\int { {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m - n} {\left (f x + e\right )}^{n - 2} \,d x } \] Input:
integrate((b*x+a)^m*(d*x+c)^(-m-n)*(f*x+e)^(-2+n),x, algorithm="giac")
Output:
integrate((b*x + a)^m*(d*x + c)^(-m - n)*(f*x + e)^(n - 2), x)
Timed out. \[ \int (a+b x)^m (c+d x)^{-m-n} (e+f x)^{-2+n} \, dx=\int \frac {{\left (e+f\,x\right )}^{n-2}\,{\left (a+b\,x\right )}^m}{{\left (c+d\,x\right )}^{m+n}} \,d x \] Input:
int(((e + f*x)^(n - 2)*(a + b*x)^m)/(c + d*x)^(m + n),x)
Output:
int(((e + f*x)^(n - 2)*(a + b*x)^m)/(c + d*x)^(m + n), x)
\[ \int (a+b x)^m (c+d x)^{-m-n} (e+f x)^{-2+n} \, dx=\int \frac {\left (f x +e \right )^{n} \left (b x +a \right )^{m}}{\left (d x +c \right )^{m +n} e^{2}+2 \left (d x +c \right )^{m +n} e f x +\left (d x +c \right )^{m +n} f^{2} x^{2}}d x \] Input:
int((b*x+a)^m*(d*x+c)^(-m-n)*(f*x+e)^(-2+n),x)
Output:
int(((e + f*x)**n*(a + b*x)**m)/((c + d*x)**(m + n)*e**2 + 2*(c + d*x)**(m + n)*e*f*x + (c + d*x)**(m + n)*f**2*x**2),x)