Integrand size = 20, antiderivative size = 105 \[ \int \frac {\left (a+\frac {b}{x}\right )^n}{x (c+d x)^2} \, dx=-\frac {d \left (a+\frac {b}{x}\right )^{1+n}}{c (a c-b d) \left (d+\frac {c}{x}\right )}+\frac {(a c-b d (1+n)) \left (a+\frac {b}{x}\right )^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {c \left (a+\frac {b}{x}\right )}{a c-b d}\right )}{c (a c-b d)^2 (1+n)} \]
-d*(a+b/x)^(1+n)/c/(a*c-b*d)/(d+c/x)+(a*c-b*d*(1+n))*(a+b/x)^(1+n)*hyperge om([1, 1+n],[2+n],c*(a+b/x)/(a*c-b*d))/c/(a*c-b*d)^2/(1+n)
Time = 0.22 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.84 \[ \int \frac {\left (a+\frac {b}{x}\right )^n}{x (c+d x)^2} \, dx=\frac {\left (a+\frac {b}{x}\right )^{1+n} \left (\frac {d (-a c+b d) x}{c+d x}+\frac {(a c-b d (1+n)) \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {c \left (a+\frac {b}{x}\right )}{a c-b d}\right )}{1+n}\right )}{c (a c-b d)^2} \]
((a + b/x)^(1 + n)*((d*(-(a*c) + b*d)*x)/(c + d*x) + ((a*c - b*d*(1 + n))* Hypergeometric2F1[1, 1 + n, 2 + n, (c*(a + b/x))/(a*c - b*d)])/(1 + n)))/( c*(a*c - b*d)^2)
Time = 0.24 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1016, 948, 87, 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.
\(\displaystyle \int \frac {\left (a+\frac {b}{x}\right )^n}{x (c+d x)^2} \, dx\) |
\(\Big \downarrow \) 1016 |
\(\displaystyle \int \frac {\left (a+\frac {b}{x}\right )^n}{x^3 \left (\frac {c}{x}+d\right )^2}dx\) |
\(\Big \downarrow \) 948 |
\(\displaystyle -\int \frac {\left (a+\frac {b}{x}\right )^n}{\left (\frac {c}{x}+d\right )^2 x}d\frac {1}{x}\) |
\(\Big \downarrow \) 87 |
\(\displaystyle -\frac {(a c-b d (n+1)) \int \frac {\left (a+\frac {b}{x}\right )^n}{\frac {c}{x}+d}d\frac {1}{x}}{c (a c-b d)}-\frac {d \left (a+\frac {b}{x}\right )^{n+1}}{c \left (\frac {c}{x}+d\right ) (a c-b d)}\) |
\(\Big \downarrow \) 78 |
\(\displaystyle \frac {\left (a+\frac {b}{x}\right )^{n+1} (a c-b d (n+1)) \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {c \left (a+\frac {b}{x}\right )}{a c-b d}\right )}{c (n+1) (a c-b d)^2}-\frac {d \left (a+\frac {b}{x}\right )^{n+1}}{c \left (\frac {c}{x}+d\right ) (a c-b d)}\) |
-((d*(a + b/x)^(1 + n))/(c*(a*c - b*d)*(d + c/x))) + ((a*c - b*d*(1 + n))* (a + b/x)^(1 + n)*Hypergeometric2F1[1, 1 + n, 2 + n, (c*(a + b/x))/(a*c - b*d)])/(c*(a*c - b*d)^2*(1 + n))
3.3.96.3.1 Defintions of rubi rules used
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[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
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[(x_)^(m_.)*((c_) + (d_.)*(x_)^(mn_.))^(q_.)*((a_) + (b_.)*(x_)^(n_.))^( p_.), x_Symbol] :> Int[x^(m - n*q)*(a + b*x^n)^p*(d + c*x^n)^q, x] /; FreeQ [{a, b, c, d, m, n, p}, x] && EqQ[mn, -n] && IntegerQ[q] && (PosQ[n] || !I ntegerQ[p])
\[\int \frac {\left (a +\frac {b}{x}\right )^{n}}{x \left (d x +c \right )^{2}}d x\]
\[ \int \frac {\left (a+\frac {b}{x}\right )^n}{x (c+d x)^2} \, dx=\int { \frac {{\left (a + \frac {b}{x}\right )}^{n}}{{\left (d x + c\right )}^{2} x} \,d x } \]
\[ \int \frac {\left (a+\frac {b}{x}\right )^n}{x (c+d x)^2} \, dx=\int \frac {\left (a + \frac {b}{x}\right )^{n}}{x \left (c + d x\right )^{2}}\, dx \]
\[ \int \frac {\left (a+\frac {b}{x}\right )^n}{x (c+d x)^2} \, dx=\int { \frac {{\left (a + \frac {b}{x}\right )}^{n}}{{\left (d x + c\right )}^{2} x} \,d x } \]
\[ \int \frac {\left (a+\frac {b}{x}\right )^n}{x (c+d x)^2} \, dx=\int { \frac {{\left (a + \frac {b}{x}\right )}^{n}}{{\left (d x + c\right )}^{2} x} \,d x } \]
Timed out. \[ \int \frac {\left (a+\frac {b}{x}\right )^n}{x (c+d x)^2} \, dx=\int \frac {{\left (a+\frac {b}{x}\right )}^n}{x\,{\left (c+d\,x\right )}^2} \,d x \]