Integrand size = 17, antiderivative size = 101 \[ \int \frac {\left (a+\frac {b}{x}\right )^m}{c+d x} \, dx=-\frac {c \left (a+\frac {b}{x}\right )^{1+m} \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {c \left (a+\frac {b}{x}\right )}{a c-b d}\right )}{d (a c-b d) (1+m)}+\frac {\left (a+\frac {b}{x}\right )^{1+m} \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,1+\frac {b}{a x}\right )}{a d (1+m)} \]
-c*(a+b/x)^(1+m)*hypergeom([1, 1+m],[2+m],c*(a+b/x)/(a*c-b*d))/d/(a*c-b*d) /(1+m)+(a+b/x)^(1+m)*hypergeom([1, 1+m],[2+m],1+b/a/x)/a/d/(1+m)
Time = 0.14 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.96 \[ \int \frac {\left (a+\frac {b}{x}\right )^m}{c+d x} \, dx=\frac {\left (a+\frac {b}{x}\right )^m (b+a x) \left (a c \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {c \left (a+\frac {b}{x}\right )}{a c-b d}\right )+(-a c+b d) \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,1+\frac {b}{a x}\right )\right )}{a d (-a c+b d) (1+m) x} \]
((a + b/x)^m*(b + a*x)*(a*c*Hypergeometric2F1[1, 1 + m, 2 + m, (c*(a + b/x ))/(a*c - b*d)] + (-(a*c) + b*d)*Hypergeometric2F1[1, 1 + m, 2 + m, 1 + b/ (a*x)]))/(a*d*(-(a*c) + b*d)*(1 + m)*x)
Time = 0.22 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {941, 948, 97, 75, 78}
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.
\(\displaystyle \int \frac {\left (a+\frac {b}{x}\right )^m}{c+d x} \, dx\) |
\(\Big \downarrow \) 941 |
\(\displaystyle \int \frac {\left (a+\frac {b}{x}\right )^m}{x \left (\frac {c}{x}+d\right )}dx\) |
\(\Big \downarrow \) 948 |
\(\displaystyle -\int \frac {\left (a+\frac {b}{x}\right )^m x}{\frac {c}{x}+d}d\frac {1}{x}\) |
\(\Big \downarrow \) 97 |
\(\displaystyle \frac {c \int \frac {\left (a+\frac {b}{x}\right )^m}{\frac {c}{x}+d}d\frac {1}{x}}{d}-\frac {\int \left (a+\frac {b}{x}\right )^m xd\frac {1}{x}}{d}\) |
\(\Big \downarrow \) 75 |
\(\displaystyle \frac {c \int \frac {\left (a+\frac {b}{x}\right )^m}{\frac {c}{x}+d}d\frac {1}{x}}{d}+\frac {\left (a+\frac {b}{x}\right )^{m+1} \operatorname {Hypergeometric2F1}\left (1,m+1,m+2,\frac {b}{a x}+1\right )}{a d (m+1)}\) |
\(\Big \downarrow \) 78 |
\(\displaystyle \frac {\left (a+\frac {b}{x}\right )^{m+1} \operatorname {Hypergeometric2F1}\left (1,m+1,m+2,\frac {b}{a x}+1\right )}{a d (m+1)}-\frac {c \left (a+\frac {b}{x}\right )^{m+1} \operatorname {Hypergeometric2F1}\left (1,m+1,m+2,\frac {c \left (a+\frac {b}{x}\right )}{a c-b d}\right )}{d (m+1) (a c-b d)}\) |
-((c*(a + b/x)^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, (c*(a + b/x))/(a *c - b*d)])/(d*(a*c - b*d)*(1 + m))) + ((a + b/x)^(1 + m)*Hypergeometric2F 1[1, 1 + m, 2 + m, 1 + b/(a*x)])/(a*d*(1 + m))
3.6.91.3.1 Defintions of rubi rules used
Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((c + d*x )^(n + 1)/(d*(n + 1)*(-d/(b*c))^m))*Hypergeometric2F1[-m, n + 1, n + 2, 1 + d*(x/c)], x] /; FreeQ[{b, c, d, m, n}, x] && !IntegerQ[n] && (IntegerQ[m] || GtQ[-d/(b*c), 0])
Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(b *c - a*d)^n*((a + b*x)^(m + 1)/(b^(n + 1)*(m + 1)))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m}, x] && !IntegerQ[m] && IntegerQ[n]
Int[((e_.) + (f_.)*(x_))^(p_)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_] :> Simp[b/(b*c - a*d) Int[(e + f*x)^p/(a + b*x), x], x] - Simp[d/(b*c - a*d) Int[(e + f*x)^p/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && !IntegerQ[p]
Int[((c_) + (d_.)*(x_)^(mn_.))^(q_.)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Sym bol] :> Int[(a + b*x^n)^p*((d + c*x^n)^q/x^(n*q)), x] /; FreeQ[{a, b, c, d, n, p}, x] && EqQ[mn, -n] && IntegerQ[q] && (PosQ[n] || !IntegerQ[p])
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_. ), x_Symbol] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^ p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ [b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]
\[\int \frac {\left (a +\frac {b}{x}\right )^{m}}{d x +c}d x\]
\[ \int \frac {\left (a+\frac {b}{x}\right )^m}{c+d x} \, dx=\int { \frac {{\left (a + \frac {b}{x}\right )}^{m}}{d x + c} \,d x } \]
\[ \int \frac {\left (a+\frac {b}{x}\right )^m}{c+d x} \, dx=\int \frac {\left (a + \frac {b}{x}\right )^{m}}{c + d x}\, dx \]
\[ \int \frac {\left (a+\frac {b}{x}\right )^m}{c+d x} \, dx=\int { \frac {{\left (a + \frac {b}{x}\right )}^{m}}{d x + c} \,d x } \]
\[ \int \frac {\left (a+\frac {b}{x}\right )^m}{c+d x} \, dx=\int { \frac {{\left (a + \frac {b}{x}\right )}^{m}}{d x + c} \,d x } \]
Timed out. \[ \int \frac {\left (a+\frac {b}{x}\right )^m}{c+d x} \, dx=\int \frac {{\left (a+\frac {b}{x}\right )}^m}{c+d\,x} \,d x \]