Integrand size = 30, antiderivative size = 237 \[ \int (a+b x)^m (c+d x)^{-m-n} (e+f x)^{-3+n} \, dx=-\frac {f (a+b x)^{1+m} (c+d x)^{1-m-n} (e+f x)^{-2+n}}{(b e-a f) (d e-c f) (2-n)}-\frac {(a d f (1+m)-b (d e (2-n)-c f (1-m-n))) (a+b x)^{1+m} (c+d x)^{-m-n} \left (\frac {(b e-a f) (c+d x)}{(b c-a d) (e+f x)}\right )^{m+n} (e+f x)^{-1+n} \operatorname {Hypergeometric2F1}\left (1+m,m+n,2+m,-\frac {(d e-c f) (a+b x)}{(b c-a d) (e+f x)}\right )}{(b e-a f)^2 (d e-c f) (1+m) (2-n)} \]
-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([m+n, 1+m],[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)
Time = 0.19 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.78 \[ \int (a+b x)^m (c+d x)^{-m-n} (e+f x)^{-3+n} \, dx=\frac {(a+b x)^{1+m} (c+d x)^{-m-n} (e+f x)^{-2+n} \left (f (c+d x)+\frac {(a d f (1+m)+b d e (-2+n)-b c f (-1+m+n)) \left (\frac {(b e-a f) (c+d x)}{(b c-a d) (e+f x)}\right )^{m+n} (e+f x) \operatorname {Hypergeometric2F1}\left (1+m,m+n,2+m,\frac {(-d e+c f) (a+b x)}{(b c-a d) (e+f x)}\right )}{(b e-a f) (1+m)}\right )}{(b e-a f) (d e-c f) (-2+n)} \]
((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))
Time = 0.30 (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 = {107, 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.
\(\displaystyle \int (a+b x)^m (e+f x)^{n-3} (c+d x)^{-m-n} \, dx\) |
\(\Big \downarrow \) 107 |
\(\displaystyle -\frac {(a d f (m+1)+b c f (-m-n+1)-b d e (2-n)) \int (a+b x)^m (c+d x)^{-m-n} (e+f x)^{n-2}dx}{(2-n) (b e-a f) (d e-c f)}-\frac {f (a+b x)^{m+1} (e+f x)^{n-2} (c+d x)^{-m-n+1}}{(2-n) (b e-a f) (d e-c f)}\) |
\(\Big \downarrow \) 142 |
\(\displaystyle -\frac {(a+b x)^{m+1} (e+f x)^{n-1} (c+d x)^{-m-n} (a d f (m+1)+b c f (-m-n+1)-b d e (2-n)) \left (\frac {(c+d x) (b e-a f)}{(e+f x) (b c-a d)}\right )^{m+n} \operatorname {Hypergeometric2F1}\left (m+1,m+n,m+2,-\frac {(d e-c f) (a+b x)}{(b c-a d) (e+f x)}\right )}{(m+1) (2-n) (b e-a f)^2 (d e-c f)}-\frac {f (a+b x)^{m+1} (e+f x)^{n-2} (c+d x)^{-m-n+1}}{(2-n) (b e-a f) (d e-c f)}\) |
-((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))
3.32.50.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> 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[(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] || SumSimplerQ[m, 1])
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]
\[\int \left (b x +a \right )^{m} \left (d x +c \right )^{-n -m} \left (f x +e \right )^{-3+n}d x\]
\[ \int (a+b x)^m (c+d x)^{-m-n} (e+f x)^{-3+n} \, dx=\int { {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m - n} {\left (f x + e\right )}^{n - 3} \,d x } \]
Timed out. \[ \int (a+b x)^m (c+d x)^{-m-n} (e+f x)^{-3+n} \, dx=\text {Timed out} \]
\[ \int (a+b x)^m (c+d x)^{-m-n} (e+f x)^{-3+n} \, dx=\int { {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m - n} {\left (f x + e\right )}^{n - 3} \,d x } \]
\[ \int (a+b x)^m (c+d x)^{-m-n} (e+f x)^{-3+n} \, dx=\int { {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m - n} {\left (f x + e\right )}^{n - 3} \,d x } \]
Timed out. \[ \int (a+b x)^m (c+d x)^{-m-n} (e+f x)^{-3+n} \, dx=\int \frac {{\left (e+f\,x\right )}^{n-3}\,{\left (a+b\,x\right )}^m}{{\left (c+d\,x\right )}^{m+n}} \,d x \]