Integrand size = 20, antiderivative size = 202 \[ \int \frac {\left (a+\frac {b}{x}\right )^n x^2}{(c+d x)^2} \, dx=\frac {c (2 a c-b d) \left (a+\frac {b}{x}\right )^{1+n}}{a d^2 (a c-b d) \left (d+\frac {c}{x}\right )}+\frac {\left (a+\frac {b}{x}\right )^{1+n} x}{a d \left (d+\frac {c}{x}\right )}+\frac {c^2 (2 a c-b d (2-n)) \left (a+\frac {b}{x}\right )^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {c \left (a+\frac {b}{x}\right )}{a c-b d}\right )}{d^3 (a c-b d)^2 (1+n)}-\frac {(2 a c-b d n) \left (a+\frac {b}{x}\right )^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,1+\frac {b}{a x}\right )}{a^2 d^3 (1+n)} \]
c*(2*a*c-b*d)*(a+b/x)^(1+n)/a/d^2/(a*c-b*d)/(d+c/x)+(a+b/x)^(1+n)*x/a/d/(d +c/x)+c^2*(2*a*c-b*d*(2-n))*(a+b/x)^(1+n)*hypergeom([1, 1+n],[2+n],c*(a+b/ x)/(a*c-b*d))/d^3/(a*c-b*d)^2/(1+n)-(-b*d*n+2*a*c)*(a+b/x)^(1+n)*hypergeom ([1, 1+n],[2+n],1+b/a/x)/a^2/d^3/(1+n)
Time = 0.34 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.89 \[ \int \frac {\left (a+\frac {b}{x}\right )^n x^2}{(c+d x)^2} \, dx=\frac {\left (a+\frac {b}{x}\right )^{1+n} \left (a c d (a c-b d) (2 a c-b d) (1+n) x+a d^2 (a c-b d)^2 (1+n) x^2+(c+d x) \left (a^2 c^2 (2 a c+b d (-2+n)) \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {c \left (a+\frac {b}{x}\right )}{a c-b d}\right )-(a c-b d)^2 (2 a c-b d n) \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,1+\frac {b}{a x}\right )\right )\right )}{a^2 d^3 (a c-b d)^2 (1+n) (c+d x)} \]
((a + b/x)^(1 + n)*(a*c*d*(a*c - b*d)*(2*a*c - b*d)*(1 + n)*x + a*d^2*(a*c - b*d)^2*(1 + n)*x^2 + (c + d*x)*(a^2*c^2*(2*a*c + b*d*(-2 + n))*Hypergeo metric2F1[1, 1 + n, 2 + n, (c*(a + b/x))/(a*c - b*d)] - (a*c - b*d)^2*(2*a *c - b*d*n)*Hypergeometric2F1[1, 1 + n, 2 + n, 1 + b/(a*x)])))/(a^2*d^3*(a *c - b*d)^2*(1 + n)*(c + d*x))
Time = 0.38 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.14, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {1016, 899, 114, 168, 174, 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.
\(\displaystyle \int \frac {x^2 \left (a+\frac {b}{x}\right )^n}{(c+d x)^2} \, dx\) |
\(\Big \downarrow \) 1016 |
\(\displaystyle \int \frac {\left (a+\frac {b}{x}\right )^n}{\left (\frac {c}{x}+d\right )^2}dx\) |
\(\Big \downarrow \) 899 |
\(\displaystyle -\int \frac {\left (a+\frac {b}{x}\right )^n x^2}{\left (\frac {c}{x}+d\right )^2}d\frac {1}{x}\) |
\(\Big \downarrow \) 114 |
\(\displaystyle \frac {\int \frac {\left (a+\frac {b}{x}\right )^n \left (2 a c+\frac {b (1-n) c}{x}-b d n\right ) x}{\left (\frac {c}{x}+d\right )^2}d\frac {1}{x}}{a d}+\frac {x \left (a+\frac {b}{x}\right )^{n+1}}{a d \left (\frac {c}{x}+d\right )}\) |
\(\Big \downarrow \) 168 |
\(\displaystyle \frac {\frac {\int \frac {\left (a+\frac {b}{x}\right )^n \left ((a c-b d) (2 a c-b d n)-\frac {b c (2 a c-b d) n}{x}\right ) x}{\frac {c}{x}+d}d\frac {1}{x}}{d (a c-b d)}+\frac {c (2 a c-b d) \left (a+\frac {b}{x}\right )^{n+1}}{d \left (\frac {c}{x}+d\right ) (a c-b d)}}{a d}+\frac {x \left (a+\frac {b}{x}\right )^{n+1}}{a d \left (\frac {c}{x}+d\right )}\) |
\(\Big \downarrow \) 174 |
\(\displaystyle \frac {\frac {\frac {(a c-b d) (2 a c-b d n) \int \left (a+\frac {b}{x}\right )^n xd\frac {1}{x}}{d}-\frac {a c^2 (2 a c-b d (2-n)) \int \frac {\left (a+\frac {b}{x}\right )^n}{\frac {c}{x}+d}d\frac {1}{x}}{d}}{d (a c-b d)}+\frac {c (2 a c-b d) \left (a+\frac {b}{x}\right )^{n+1}}{d \left (\frac {c}{x}+d\right ) (a c-b d)}}{a d}+\frac {x \left (a+\frac {b}{x}\right )^{n+1}}{a d \left (\frac {c}{x}+d\right )}\) |
\(\Big \downarrow \) 75 |
\(\displaystyle \frac {\frac {-\frac {a c^2 (2 a c-b d (2-n)) \int \frac {\left (a+\frac {b}{x}\right )^n}{\frac {c}{x}+d}d\frac {1}{x}}{d}-\frac {(a c-b d) \left (a+\frac {b}{x}\right )^{n+1} (2 a c-b d n) \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {b}{a x}+1\right )}{a d (n+1)}}{d (a c-b d)}+\frac {c (2 a c-b d) \left (a+\frac {b}{x}\right )^{n+1}}{d \left (\frac {c}{x}+d\right ) (a c-b d)}}{a d}+\frac {x \left (a+\frac {b}{x}\right )^{n+1}}{a d \left (\frac {c}{x}+d\right )}\) |
\(\Big \downarrow \) 78 |
\(\displaystyle \frac {\frac {\frac {a c^2 \left (a+\frac {b}{x}\right )^{n+1} (2 a c-b d (2-n)) \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {c \left (a+\frac {b}{x}\right )}{a c-b d}\right )}{d (n+1) (a c-b d)}-\frac {(a c-b d) \left (a+\frac {b}{x}\right )^{n+1} (2 a c-b d n) \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {b}{a x}+1\right )}{a d (n+1)}}{d (a c-b d)}+\frac {c (2 a c-b d) \left (a+\frac {b}{x}\right )^{n+1}}{d \left (\frac {c}{x}+d\right ) (a c-b d)}}{a d}+\frac {x \left (a+\frac {b}{x}\right )^{n+1}}{a d \left (\frac {c}{x}+d\right )}\) |
((a + b/x)^(1 + n)*x)/(a*d*(d + c/x)) + ((c*(2*a*c - b*d)*(a + b/x)^(1 + n ))/(d*(a*c - b*d)*(d + c/x)) + ((a*c^2*(2*a*c - b*d*(2 - n))*(a + b/x)^(1 + n)*Hypergeometric2F1[1, 1 + n, 2 + n, (c*(a + b/x))/(a*c - b*d)])/(d*(a* c - b*d)*(1 + n)) - ((a*c - b*d)*(2*a*c - b*d*n)*(a + b/x)^(1 + n)*Hyperge ometric2F1[1, 1 + n, 2 + n, 1 + b/(a*x)])/(a*d*(1 + n)))/(d*(a*c - b*d)))/ (a*d)
3.3.93.3.1 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[((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[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*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] && (IntegerQ[n] || IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(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] + S imp[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*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m, -1]
Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))* ((c_.) + (d_.)*(x_))), x_] :> Simp[(b*g - a*h)/(b*c - a*d) Int[(e + f*x)^ p/(a + b*x), x], x] - Simp[(d*g - c*h)/(b*c - a*d) Int[(e + f*x)^p/(c + d *x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]
Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol ] :> -Subst[Int[(a + b/x^n)^p*((c + d/x^n)^q/x^2), x], x, 1/x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && ILtQ[n, 0]
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])
\[\int \frac {\left (a +\frac {b}{x}\right )^{n} x^{2}}{\left (d x +c \right )^{2}}d x\]
\[ \int \frac {\left (a+\frac {b}{x}\right )^n x^2}{(c+d x)^2} \, dx=\int { \frac {{\left (a + \frac {b}{x}\right )}^{n} x^{2}}{{\left (d x + c\right )}^{2}} \,d x } \]
\[ \int \frac {\left (a+\frac {b}{x}\right )^n x^2}{(c+d x)^2} \, dx=\int \frac {x^{2} \left (a + \frac {b}{x}\right )^{n}}{\left (c + d x\right )^{2}}\, dx \]
\[ \int \frac {\left (a+\frac {b}{x}\right )^n x^2}{(c+d x)^2} \, dx=\int { \frac {{\left (a + \frac {b}{x}\right )}^{n} x^{2}}{{\left (d x + c\right )}^{2}} \,d x } \]
\[ \int \frac {\left (a+\frac {b}{x}\right )^n x^2}{(c+d x)^2} \, dx=\int { \frac {{\left (a + \frac {b}{x}\right )}^{n} x^{2}}{{\left (d x + c\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {\left (a+\frac {b}{x}\right )^n x^2}{(c+d x)^2} \, dx=\int \frac {x^2\,{\left (a+\frac {b}{x}\right )}^n}{{\left (c+d\,x\right )}^2} \,d x \]