Integrand size = 20, antiderivative size = 115 \[ \int \frac {\left (a+\frac {b}{x}\right )^n}{x^3 (c+d x)} \, dx=\frac {(a c+b d) \left (a+\frac {b}{x}\right )^{1+n}}{b^2 c^2 (1+n)}-\frac {\left (a+\frac {b}{x}\right )^{2+n}}{b^2 c (2+n)}+\frac {d^2 \left (a+\frac {b}{x}\right )^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {c \left (a+\frac {b}{x}\right )}{a c-b d}\right )}{c^2 (a c-b d) (1+n)} \]
(a*c+b*d)*(a+b/x)^(1+n)/b^2/c^2/(1+n)-(a+b/x)^(2+n)/b^2/c/(2+n)+d^2*(a+b/x )^(1+n)*hypergeom([1, 1+n],[2+n],c*(a+b/x)/(a*c-b*d))/c^2/(a*c-b*d)/(1+n)
Time = 0.30 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.97 \[ \int \frac {\left (a+\frac {b}{x}\right )^n}{x^3 (c+d x)} \, dx=-\frac {\left (a+\frac {b}{x}\right )^n (b+a x) \left ((a c-b d) (-b c (1+n)+a c x+b d (2+n) x)+b^2 d^2 (2+n) x \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {c \left (a+\frac {b}{x}\right )}{a c-b d}\right )\right )}{b^2 c^2 (-a c+b d) (1+n) (2+n) x^2} \]
-(((a + b/x)^n*(b + a*x)*((a*c - b*d)*(-(b*c*(1 + n)) + a*c*x + b*d*(2 + n )*x) + b^2*d^2*(2 + n)*x*Hypergeometric2F1[1, 1 + n, 2 + n, (c*(a + b/x))/ (a*c - b*d)]))/(b^2*c^2*(-(a*c) + b*d)*(1 + n)*(2 + n)*x^2))
Time = 0.30 (sec) , antiderivative size = 115, 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, 99, 2009}
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.
\(\displaystyle \int \frac {\left (a+\frac {b}{x}\right )^n}{x^3 (c+d x)} \, dx\) |
\(\Big \downarrow \) 1016 |
\(\displaystyle \int \frac {\left (a+\frac {b}{x}\right )^n}{x^4 \left (\frac {c}{x}+d\right )}dx\) |
\(\Big \downarrow \) 948 |
\(\displaystyle -\int \frac {\left (a+\frac {b}{x}\right )^n}{\left (\frac {c}{x}+d\right ) x^2}d\frac {1}{x}\) |
\(\Big \downarrow \) 99 |
\(\displaystyle -\int \left (\frac {(-a c-b d) \left (a+\frac {b}{x}\right )^n}{b c^2}+\frac {d^2 \left (a+\frac {b}{x}\right )^n}{c^2 \left (\frac {c}{x}+d\right )}+\frac {\left (a+\frac {b}{x}\right )^{n+1}}{b c}\right )d\frac {1}{x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {(a c+b d) \left (a+\frac {b}{x}\right )^{n+1}}{b^2 c^2 (n+1)}-\frac {\left (a+\frac {b}{x}\right )^{n+2}}{b^2 c (n+2)}+\frac {d^2 \left (a+\frac {b}{x}\right )^{n+1} \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {c \left (a+\frac {b}{x}\right )}{a c-b d}\right )}{c^2 (n+1) (a c-b d)}\) |
((a*c + b*d)*(a + b/x)^(1 + n))/(b^2*c^2*(1 + n)) - (a + b/x)^(2 + n)/(b^2 *c*(2 + n)) + (d^2*(a + b/x)^(1 + n)*Hypergeometric2F1[1, 1 + n, 2 + n, (c *(a + b/x))/(a*c - b*d)])/(c^2*(a*c - b*d)*(1 + n))
3.3.90.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | | (GtQ[m, 0] && GeQ[n, -1]))
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])
\[\int \frac {\left (a +\frac {b}{x}\right )^{n}}{x^{3} \left (d x +c \right )}d x\]
\[ \int \frac {\left (a+\frac {b}{x}\right )^n}{x^3 (c+d x)} \, dx=\int { \frac {{\left (a + \frac {b}{x}\right )}^{n}}{{\left (d x + c\right )} x^{3}} \,d x } \]
\[ \int \frac {\left (a+\frac {b}{x}\right )^n}{x^3 (c+d x)} \, dx=\int \frac {\left (a + \frac {b}{x}\right )^{n}}{x^{3} \left (c + d x\right )}\, dx \]
\[ \int \frac {\left (a+\frac {b}{x}\right )^n}{x^3 (c+d x)} \, dx=\int { \frac {{\left (a + \frac {b}{x}\right )}^{n}}{{\left (d x + c\right )} x^{3}} \,d x } \]
\[ \int \frac {\left (a+\frac {b}{x}\right )^n}{x^3 (c+d x)} \, dx=\int { \frac {{\left (a + \frac {b}{x}\right )}^{n}}{{\left (d x + c\right )} x^{3}} \,d x } \]
Timed out. \[ \int \frac {\left (a+\frac {b}{x}\right )^n}{x^3 (c+d x)} \, dx=\int \frac {{\left (a+\frac {b}{x}\right )}^n}{x^3\,\left (c+d\,x\right )} \,d x \]