Integrand size = 21, antiderivative size = 147 \[ \int \frac {\sqrt {a+\frac {b}{x}}}{\left (c+\frac {d}{x}\right )^2} \, dx=\frac {2 d \sqrt {a+\frac {b}{x}}}{c^2 \left (c+\frac {d}{x}\right )}+\frac {\sqrt {a+\frac {b}{x}} x}{c \left (c+\frac {d}{x}\right )}+\frac {\sqrt {d} (3 b c-4 a d) \arctan \left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b c-a d}}\right )}{c^3 \sqrt {b c-a d}}+\frac {(b c-4 a d) \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{\sqrt {a} c^3} \]
(-4*a*d+b*c)*arctanh((a+b/x)^(1/2)/a^(1/2))/c^3/a^(1/2)+(-4*a*d+3*b*c)*arc tan(d^(1/2)*(a+b/x)^(1/2)/(-a*d+b*c)^(1/2))*d^(1/2)/c^3/(-a*d+b*c)^(1/2)+2 *d*(a+b/x)^(1/2)/c^2/(c+d/x)+x*(a+b/x)^(1/2)/c/(c+d/x)
Time = 0.53 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.83 \[ \int \frac {\sqrt {a+\frac {b}{x}}}{\left (c+\frac {d}{x}\right )^2} \, dx=\frac {\frac {c \sqrt {a+\frac {b}{x}} x (2 d+c x)}{d+c x}+\frac {\sqrt {d} (3 b c-4 a d) \arctan \left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b c-a d}}\right )}{\sqrt {b c-a d}}+\frac {(b c-4 a d) \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{\sqrt {a}}}{c^3} \]
((c*Sqrt[a + b/x]*x*(2*d + c*x))/(d + c*x) + (Sqrt[d]*(3*b*c - 4*a*d)*ArcT an[(Sqrt[d]*Sqrt[a + b/x])/Sqrt[b*c - a*d]])/Sqrt[b*c - a*d] + ((b*c - 4*a *d)*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/Sqrt[a])/c^3
Time = 0.32 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.10, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {899, 110, 27, 168, 25, 27, 174, 73, 218, 221}
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.
| \(\displaystyle \int \frac {\sqrt {a+\frac {b}{x}}}{\left (c+\frac {d}{x}\right )^2} \, dx\) |
| \(\Big \downarrow \) 899 |
| \(\displaystyle -\int \frac {\sqrt {a+\frac {b}{x}} x^2}{\left (c+\frac {d}{x}\right )^2}d\frac {1}{x}\) |
| \(\Big \downarrow \) 110 |
| \(\displaystyle \frac {x \sqrt {a+\frac {b}{x}}}{c \left (c+\frac {d}{x}\right )}-\frac {\int \frac {\left (b c-4 a d-\frac {3 b d}{x}\right ) x}{2 \sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )^2}d\frac {1}{x}}{c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x \sqrt {a+\frac {b}{x}}}{c \left (c+\frac {d}{x}\right )}-\frac {\int \frac {\left (b c-4 a d-\frac {3 b d}{x}\right ) x}{\sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )^2}d\frac {1}{x}}{2 c}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {x \sqrt {a+\frac {b}{x}}}{c \left (c+\frac {d}{x}\right )}-\frac {-\frac {\int -\frac {(b c-a d) \left (b c-4 a d-\frac {2 b d}{x}\right ) x}{\sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )}d\frac {1}{x}}{c (b c-a d)}-\frac {4 d \sqrt {a+\frac {b}{x}}}{c \left (c+\frac {d}{x}\right )}}{2 c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {x \sqrt {a+\frac {b}{x}}}{c \left (c+\frac {d}{x}\right )}-\frac {\frac {\int \frac {(b c-a d) \left (b c-4 a d-\frac {2 b d}{x}\right ) x}{\sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )}d\frac {1}{x}}{c (b c-a d)}-\frac {4 d \sqrt {a+\frac {b}{x}}}{c \left (c+\frac {d}{x}\right )}}{2 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x \sqrt {a+\frac {b}{x}}}{c \left (c+\frac {d}{x}\right )}-\frac {\frac {\int \frac {\left (b c-4 a d-\frac {2 b d}{x}\right ) x}{\sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )}d\frac {1}{x}}{c}-\frac {4 d \sqrt {a+\frac {b}{x}}}{c \left (c+\frac {d}{x}\right )}}{2 c}\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle \frac {x \sqrt {a+\frac {b}{x}}}{c \left (c+\frac {d}{x}\right )}-\frac {\frac {\frac {(b c-4 a d) \int \frac {x}{\sqrt {a+\frac {b}{x}}}d\frac {1}{x}}{c}-\frac {d (3 b c-4 a d) \int \frac {1}{\sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )}d\frac {1}{x}}{c}}{c}-\frac {4 d \sqrt {a+\frac {b}{x}}}{c \left (c+\frac {d}{x}\right )}}{2 c}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {x \sqrt {a+\frac {b}{x}}}{c \left (c+\frac {d}{x}\right )}-\frac {\frac {\frac {2 (b c-4 a d) \int \frac {1}{\frac {1}{b x^2}-\frac {a}{b}}d\sqrt {a+\frac {b}{x}}}{b c}-\frac {2 d (3 b c-4 a d) \int \frac {1}{c-\frac {a d}{b}+\frac {d}{b x^2}}d\sqrt {a+\frac {b}{x}}}{b c}}{c}-\frac {4 d \sqrt {a+\frac {b}{x}}}{c \left (c+\frac {d}{x}\right )}}{2 c}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {x \sqrt {a+\frac {b}{x}}}{c \left (c+\frac {d}{x}\right )}-\frac {\frac {\frac {2 (b c-4 a d) \int \frac {1}{\frac {1}{b x^2}-\frac {a}{b}}d\sqrt {a+\frac {b}{x}}}{b c}-\frac {2 \sqrt {d} (3 b c-4 a d) \arctan \left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b c-a d}}\right )}{c \sqrt {b c-a d}}}{c}-\frac {4 d \sqrt {a+\frac {b}{x}}}{c \left (c+\frac {d}{x}\right )}}{2 c}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {x \sqrt {a+\frac {b}{x}}}{c \left (c+\frac {d}{x}\right )}-\frac {\frac {-\frac {2 \sqrt {d} (3 b c-4 a d) \arctan \left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b c-a d}}\right )}{c \sqrt {b c-a d}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right ) (b c-4 a d)}{\sqrt {a} c}}{c}-\frac {4 d \sqrt {a+\frac {b}{x}}}{c \left (c+\frac {d}{x}\right )}}{2 c}\) |
(Sqrt[a + b/x]*x)/(c*(c + d/x)) - ((-4*d*Sqrt[a + b/x])/(c*(c + d/x)) + (( -2*Sqrt[d]*(3*b*c - 4*a*d)*ArcTan[(Sqrt[d]*Sqrt[a + b/x])/Sqrt[b*c - a*d]] )/(c*Sqrt[b*c - a*d]) - (2*(b*c - 4*a*d)*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/( Sqrt[a]*c))/c)/(2*c)
3.3.29.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
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)/((m + 1)*(b*e - a*f))), x] - Simp[1/((m + 1)*(b*e - a*f)) Int[(a + b*x)^(m + 1) *(c + d*x)^(n - 1)*(e + f*x)^p*Simp[d*e*n + c*f*(m + p + 2) + d*f*(m + n + p + 2)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && Gt Q[n, 0] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])
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_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
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]
Leaf count of result is larger than twice the leaf count of optimal. \(476\) vs. \(2(127)=254\).
Time = 0.24 (sec) , antiderivative size = 477, normalized size of antiderivative = 3.24
| method | result | size |
| risch | \(\frac {x \sqrt {\frac {a x +b}{x}}}{c^{2}}-\frac {\left (\frac {\left (4 a d -b c \right ) \ln \left (\frac {\frac {b}{2}+a x}{\sqrt {a}}+\sqrt {a \,x^{2}+b x}\right )}{c \sqrt {a}}+\frac {2 d^{2} \left (a d -b c \right ) \left (-\frac {c^{2} \sqrt {a \left (x +\frac {d}{c}\right )^{2}-\frac {\left (2 a d -b c \right ) \left (x +\frac {d}{c}\right )}{c}+\frac {\left (a d -b c \right ) d}{c^{2}}}}{\left (a d -b c \right ) d \left (x +\frac {d}{c}\right )}-\frac {\left (2 a d -b c \right ) c \ln \left (\frac {\frac {2 \left (a d -b c \right ) d}{c^{2}}-\frac {\left (2 a d -b c \right ) \left (x +\frac {d}{c}\right )}{c}+2 \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, \sqrt {a \left (x +\frac {d}{c}\right )^{2}-\frac {\left (2 a d -b c \right ) \left (x +\frac {d}{c}\right )}{c}+\frac {\left (a d -b c \right ) d}{c^{2}}}}{x +\frac {d}{c}}\right )}{2 \left (a d -b c \right ) d \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}}\right )}{c^{3}}+\frac {2 d \left (3 a d -2 b c \right ) \ln \left (\frac {\frac {2 \left (a d -b c \right ) d}{c^{2}}-\frac {\left (2 a d -b c \right ) \left (x +\frac {d}{c}\right )}{c}+2 \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, \sqrt {a \left (x +\frac {d}{c}\right )^{2}-\frac {\left (2 a d -b c \right ) \left (x +\frac {d}{c}\right )}{c}+\frac {\left (a d -b c \right ) d}{c^{2}}}}{x +\frac {d}{c}}\right )}{c^{2} \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}}\right ) \sqrt {\frac {a x +b}{x}}\, \sqrt {x \left (a x +b \right )}}{2 c^{2} \left (a x +b \right )}\) | \(477\) |
| default | \(-\frac {\sqrt {\frac {a x +b}{x}}\, x \left (4 a^{\frac {7}{2}} \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, c -2 a d x +b c x -b d}{c x +d}\right ) c \,d^{3} x +2 a^{\frac {5}{2}} \sqrt {x \left (a x +b \right )}\, \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, c^{4} x^{2}+4 a^{\frac {7}{2}} \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, c -2 a d x +b c x -b d}{c x +d}\right ) d^{4}-2 a^{\frac {5}{2}} \sqrt {x \left (a x +b \right )}\, \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, c^{3} d x -7 a^{\frac {5}{2}} \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, c -2 a d x +b c x -b d}{c x +d}\right ) b \,c^{2} d^{2} x -4 a^{\frac {5}{2}} \sqrt {x \left (a x +b \right )}\, \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, c^{2} d^{2}-7 a^{\frac {5}{2}} \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, c -2 a d x +b c x -b d}{c x +d}\right ) b c \,d^{3}-2 c^{4} \left (x \left (a x +b \right )\right )^{\frac {3}{2}} a^{\frac {3}{2}} \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}+4 a^{\frac {3}{2}} \sqrt {x \left (a x +b \right )}\, \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, b \,c^{4} x +3 a^{\frac {3}{2}} \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, c -2 a d x +b c x -b d}{c x +d}\right ) b^{2} c^{3} d x +4 a^{\frac {3}{2}} \sqrt {x \left (a x +b \right )}\, \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, b \,c^{3} d +3 a^{\frac {3}{2}} \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, c -2 a d x +b c x -b d}{c x +d}\right ) b^{2} c^{2} d^{2}+4 a^{3} \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, c^{2} d^{2} x -5 a^{2} \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, b \,c^{3} d x +a \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, b^{2} c^{4} x +4 a^{3} \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, c \,d^{3}-5 a^{2} \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, b \,c^{2} d^{2}+a \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, b^{2} c^{3} d \right )}{2 c^{4} \sqrt {x \left (a x +b \right )}\, \left (a d -b c \right ) \left (c x +d \right ) a^{\frac {3}{2}} \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}}\) | \(943\) |
1/c^2*x*((a*x+b)/x)^(1/2)-1/2/c^2*((4*a*d-b*c)/c*ln((1/2*b+a*x)/a^(1/2)+(a *x^2+b*x)^(1/2))/a^(1/2)+2*d^2*(a*d-b*c)/c^3*(-1/(a*d-b*c)/d*c^2/(x+d/c)*( a*(x+d/c)^2-(2*a*d-b*c)/c*(x+d/c)+(a*d-b*c)*d/c^2)^(1/2)-1/2*(2*a*d-b*c)*c /(a*d-b*c)/d/((a*d-b*c)*d/c^2)^(1/2)*ln((2*(a*d-b*c)*d/c^2-(2*a*d-b*c)/c*( x+d/c)+2*((a*d-b*c)*d/c^2)^(1/2)*(a*(x+d/c)^2-(2*a*d-b*c)/c*(x+d/c)+(a*d-b *c)*d/c^2)^(1/2))/(x+d/c)))+2/c^2*d*(3*a*d-2*b*c)/((a*d-b*c)*d/c^2)^(1/2)* ln((2*(a*d-b*c)*d/c^2-(2*a*d-b*c)/c*(x+d/c)+2*((a*d-b*c)*d/c^2)^(1/2)*(a*( x+d/c)^2-(2*a*d-b*c)/c*(x+d/c)+(a*d-b*c)*d/c^2)^(1/2))/(x+d/c)))*((a*x+b)/ x)^(1/2)*(x*(a*x+b))^(1/2)/(a*x+b)
Time = 0.30 (sec) , antiderivative size = 801, normalized size of antiderivative = 5.45 \[ \int \frac {\sqrt {a+\frac {b}{x}}}{\left (c+\frac {d}{x}\right )^2} \, dx=\left [-\frac {{\left (b c d - 4 \, a d^{2} + {\left (b c^{2} - 4 \, a c d\right )} x\right )} \sqrt {a} \log \left (2 \, a x - 2 \, \sqrt {a} x \sqrt {\frac {a x + b}{x}} + b\right ) + {\left (3 \, a b c d - 4 \, a^{2} d^{2} + {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x\right )} \sqrt {-\frac {d}{b c - a d}} \log \left (-\frac {2 \, {\left (b c - a d\right )} x \sqrt {-\frac {d}{b c - a d}} \sqrt {\frac {a x + b}{x}} - b d + {\left (b c - 2 \, a d\right )} x}{c x + d}\right ) - 2 \, {\left (a c^{2} x^{2} + 2 \, a c d x\right )} \sqrt {\frac {a x + b}{x}}}{2 \, {\left (a c^{4} x + a c^{3} d\right )}}, -\frac {2 \, {\left (b c d - 4 \, a d^{2} + {\left (b c^{2} - 4 \, a c d\right )} x\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a} \sqrt {\frac {a x + b}{x}}}{a}\right ) + {\left (3 \, a b c d - 4 \, a^{2} d^{2} + {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x\right )} \sqrt {-\frac {d}{b c - a d}} \log \left (-\frac {2 \, {\left (b c - a d\right )} x \sqrt {-\frac {d}{b c - a d}} \sqrt {\frac {a x + b}{x}} - b d + {\left (b c - 2 \, a d\right )} x}{c x + d}\right ) - 2 \, {\left (a c^{2} x^{2} + 2 \, a c d x\right )} \sqrt {\frac {a x + b}{x}}}{2 \, {\left (a c^{4} x + a c^{3} d\right )}}, \frac {2 \, {\left (3 \, a b c d - 4 \, a^{2} d^{2} + {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x\right )} \sqrt {\frac {d}{b c - a d}} \arctan \left (-\frac {{\left (b c - a d\right )} x \sqrt {\frac {d}{b c - a d}} \sqrt {\frac {a x + b}{x}}}{a d x + b d}\right ) - {\left (b c d - 4 \, a d^{2} + {\left (b c^{2} - 4 \, a c d\right )} x\right )} \sqrt {a} \log \left (2 \, a x - 2 \, \sqrt {a} x \sqrt {\frac {a x + b}{x}} + b\right ) + 2 \, {\left (a c^{2} x^{2} + 2 \, a c d x\right )} \sqrt {\frac {a x + b}{x}}}{2 \, {\left (a c^{4} x + a c^{3} d\right )}}, \frac {{\left (3 \, a b c d - 4 \, a^{2} d^{2} + {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x\right )} \sqrt {\frac {d}{b c - a d}} \arctan \left (-\frac {{\left (b c - a d\right )} x \sqrt {\frac {d}{b c - a d}} \sqrt {\frac {a x + b}{x}}}{a d x + b d}\right ) - {\left (b c d - 4 \, a d^{2} + {\left (b c^{2} - 4 \, a c d\right )} x\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a} \sqrt {\frac {a x + b}{x}}}{a}\right ) + {\left (a c^{2} x^{2} + 2 \, a c d x\right )} \sqrt {\frac {a x + b}{x}}}{a c^{4} x + a c^{3} d}\right ] \]
[-1/2*((b*c*d - 4*a*d^2 + (b*c^2 - 4*a*c*d)*x)*sqrt(a)*log(2*a*x - 2*sqrt( a)*x*sqrt((a*x + b)/x) + b) + (3*a*b*c*d - 4*a^2*d^2 + (3*a*b*c^2 - 4*a^2* c*d)*x)*sqrt(-d/(b*c - a*d))*log(-(2*(b*c - a*d)*x*sqrt(-d/(b*c - a*d))*sq rt((a*x + b)/x) - b*d + (b*c - 2*a*d)*x)/(c*x + d)) - 2*(a*c^2*x^2 + 2*a*c *d*x)*sqrt((a*x + b)/x))/(a*c^4*x + a*c^3*d), -1/2*(2*(b*c*d - 4*a*d^2 + ( b*c^2 - 4*a*c*d)*x)*sqrt(-a)*arctan(sqrt(-a)*sqrt((a*x + b)/x)/a) + (3*a*b *c*d - 4*a^2*d^2 + (3*a*b*c^2 - 4*a^2*c*d)*x)*sqrt(-d/(b*c - a*d))*log(-(2 *(b*c - a*d)*x*sqrt(-d/(b*c - a*d))*sqrt((a*x + b)/x) - b*d + (b*c - 2*a*d )*x)/(c*x + d)) - 2*(a*c^2*x^2 + 2*a*c*d*x)*sqrt((a*x + b)/x))/(a*c^4*x + a*c^3*d), 1/2*(2*(3*a*b*c*d - 4*a^2*d^2 + (3*a*b*c^2 - 4*a^2*c*d)*x)*sqrt( d/(b*c - a*d))*arctan(-(b*c - a*d)*x*sqrt(d/(b*c - a*d))*sqrt((a*x + b)/x) /(a*d*x + b*d)) - (b*c*d - 4*a*d^2 + (b*c^2 - 4*a*c*d)*x)*sqrt(a)*log(2*a* x - 2*sqrt(a)*x*sqrt((a*x + b)/x) + b) + 2*(a*c^2*x^2 + 2*a*c*d*x)*sqrt((a *x + b)/x))/(a*c^4*x + a*c^3*d), ((3*a*b*c*d - 4*a^2*d^2 + (3*a*b*c^2 - 4* a^2*c*d)*x)*sqrt(d/(b*c - a*d))*arctan(-(b*c - a*d)*x*sqrt(d/(b*c - a*d))* sqrt((a*x + b)/x)/(a*d*x + b*d)) - (b*c*d - 4*a*d^2 + (b*c^2 - 4*a*c*d)*x) *sqrt(-a)*arctan(sqrt(-a)*sqrt((a*x + b)/x)/a) + (a*c^2*x^2 + 2*a*c*d*x)*s qrt((a*x + b)/x))/(a*c^4*x + a*c^3*d)]
\[ \int \frac {\sqrt {a+\frac {b}{x}}}{\left (c+\frac {d}{x}\right )^2} \, dx=\int \frac {x^{2} \sqrt {a + \frac {b}{x}}}{\left (c x + d\right )^{2}}\, dx \]
\[ \int \frac {\sqrt {a+\frac {b}{x}}}{\left (c+\frac {d}{x}\right )^2} \, dx=\int { \frac {\sqrt {a + \frac {b}{x}}}{{\left (c + \frac {d}{x}\right )}^{2}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 401 vs. \(2 (127) = 254\).
Time = 0.34 (sec) , antiderivative size = 401, normalized size of antiderivative = 2.73 \[ \int \frac {\sqrt {a+\frac {b}{x}}}{\left (c+\frac {d}{x}\right )^2} \, dx=\frac {\sqrt {a x^{2} + b x} \mathrm {sgn}\left (x\right )}{c^{2}} - \frac {{\left (3 \, b c d \mathrm {sgn}\left (x\right ) - 4 \, a d^{2} \mathrm {sgn}\left (x\right )\right )} \arctan \left (-\frac {{\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} c + \sqrt {a} d}{\sqrt {b c d - a d^{2}}}\right )}{\sqrt {b c d - a d^{2}} c^{3}} - \frac {{\left (b c \mathrm {sgn}\left (x\right ) - 4 \, a d \mathrm {sgn}\left (x\right )\right )} \log \left ({\left | 2 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} \sqrt {a} + b \right |}\right )}{2 \, \sqrt {a} c^{3}} - \frac {{\left (6 \, \sqrt {a} b c d \arctan \left (\frac {\sqrt {a} d}{\sqrt {b c d - a d^{2}}}\right ) - 8 \, a^{\frac {3}{2}} d^{2} \arctan \left (\frac {\sqrt {a} d}{\sqrt {b c d - a d^{2}}}\right ) - \sqrt {b c d - a d^{2}} b c \log \left ({\left | b \right |}\right ) + 4 \, \sqrt {b c d - a d^{2}} a d \log \left ({\left | b \right |}\right ) + 2 \, \sqrt {b c d - a d^{2}} a d\right )} \mathrm {sgn}\left (x\right )}{2 \, \sqrt {b c d - a d^{2}} \sqrt {a} c^{3}} - \frac {{\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} b c d \mathrm {sgn}\left (x\right ) - 2 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} a d^{2} \mathrm {sgn}\left (x\right ) - \sqrt {a} b d^{2} \mathrm {sgn}\left (x\right )}{{\left ({\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{2} c + 2 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} \sqrt {a} d + b d\right )} c^{3}} \]
sqrt(a*x^2 + b*x)*sgn(x)/c^2 - (3*b*c*d*sgn(x) - 4*a*d^2*sgn(x))*arctan(-( (sqrt(a)*x - sqrt(a*x^2 + b*x))*c + sqrt(a)*d)/sqrt(b*c*d - a*d^2))/(sqrt( b*c*d - a*d^2)*c^3) - 1/2*(b*c*sgn(x) - 4*a*d*sgn(x))*log(abs(2*(sqrt(a)*x - sqrt(a*x^2 + b*x))*sqrt(a) + b))/(sqrt(a)*c^3) - 1/2*(6*sqrt(a)*b*c*d*a rctan(sqrt(a)*d/sqrt(b*c*d - a*d^2)) - 8*a^(3/2)*d^2*arctan(sqrt(a)*d/sqrt (b*c*d - a*d^2)) - sqrt(b*c*d - a*d^2)*b*c*log(abs(b)) + 4*sqrt(b*c*d - a* d^2)*a*d*log(abs(b)) + 2*sqrt(b*c*d - a*d^2)*a*d)*sgn(x)/(sqrt(b*c*d - a*d ^2)*sqrt(a)*c^3) - ((sqrt(a)*x - sqrt(a*x^2 + b*x))*b*c*d*sgn(x) - 2*(sqrt (a)*x - sqrt(a*x^2 + b*x))*a*d^2*sgn(x) - sqrt(a)*b*d^2*sgn(x))/(((sqrt(a) *x - sqrt(a*x^2 + b*x))^2*c + 2*(sqrt(a)*x - sqrt(a*x^2 + b*x))*sqrt(a)*d + b*d)*c^3)
Time = 6.43 (sec) , antiderivative size = 1195, normalized size of antiderivative = 8.13 \[ \int \frac {\sqrt {a+\frac {b}{x}}}{\left (c+\frac {d}{x}\right )^2} \, dx=\text {Too large to display} \]
- ((2*b*d*(a + b/x)^(3/2))/c^2 - (b*(a + b/x)^(1/2)*(2*a*d - b*c))/c^2)/(( a + b/x)*(2*a*d - b*c) - d*(a + b/x)^2 - a^2*d + a*b*c) - (atanh((8*b^5*d^ 3*(a + b/x)^(1/2))/(a^(1/2)*(8*b^5*d^3 - (2*b^6*c*d^2)/a)) + (2*b^6*d^2*(a + b/x)^(1/2))/(a^(3/2)*((2*b^6*d^2)/a - (8*b^5*d^3)/c)))*(4*a*d - b*c))/( a^(1/2)*c^3) - (atan((((d*(a*d - b*c))^(1/2)*((4*(a + b/x)^(1/2)*(16*a^2*b ^2*d^5 + 5*b^4*c^2*d^3 - 16*a*b^3*c*d^4))/c^4 - (((2*(2*b^4*c^7*d^2 - 4*a* b^3*c^6*d^3))/c^6 - (2*(2*b^3*c^7*d^2 - 4*a*b^2*c^6*d^3)*(a + b/x)^(1/2)*( d*(a*d - b*c))^(1/2)*(4*a*d - 3*b*c))/(c^4*(b*c^4 - a*c^3*d)))*(d*(a*d - b *c))^(1/2)*(4*a*d - 3*b*c))/(2*(b*c^4 - a*c^3*d)))*(4*a*d - 3*b*c)*1i)/(2* (b*c^4 - a*c^3*d)) + ((d*(a*d - b*c))^(1/2)*((4*(a + b/x)^(1/2)*(16*a^2*b^ 2*d^5 + 5*b^4*c^2*d^3 - 16*a*b^3*c*d^4))/c^4 + (((2*(2*b^4*c^7*d^2 - 4*a*b ^3*c^6*d^3))/c^6 + (2*(2*b^3*c^7*d^2 - 4*a*b^2*c^6*d^3)*(a + b/x)^(1/2)*(d *(a*d - b*c))^(1/2)*(4*a*d - 3*b*c))/(c^4*(b*c^4 - a*c^3*d)))*(d*(a*d - b* c))^(1/2)*(4*a*d - 3*b*c))/(2*(b*c^4 - a*c^3*d)))*(4*a*d - 3*b*c)*1i)/(2*( b*c^4 - a*c^3*d)))/((4*(16*a^2*b^3*d^5 + 3*b^5*c^2*d^3 - 16*a*b^4*c*d^4))/ c^6 - ((d*(a*d - b*c))^(1/2)*((4*(a + b/x)^(1/2)*(16*a^2*b^2*d^5 + 5*b^4*c ^2*d^3 - 16*a*b^3*c*d^4))/c^4 - (((2*(2*b^4*c^7*d^2 - 4*a*b^3*c^6*d^3))/c^ 6 - (2*(2*b^3*c^7*d^2 - 4*a*b^2*c^6*d^3)*(a + b/x)^(1/2)*(d*(a*d - b*c))^( 1/2)*(4*a*d - 3*b*c))/(c^4*(b*c^4 - a*c^3*d)))*(d*(a*d - b*c))^(1/2)*(4*a* d - 3*b*c))/(2*(b*c^4 - a*c^3*d)))*(4*a*d - 3*b*c))/(2*(b*c^4 - a*c^3*d...