\(\int \frac {\sqrt {a+\frac {b}{x}}}{(c+\frac {d}{x})^2} \, dx\) [9]

Optimal result

Integrand size = 21, antiderivative size = 147 \[ \int \frac {\sqrt {a+\frac {b}{x}}}{\left (c+\frac {d}{x}\right )^2} \, dx=\frac {2 d \sqrt {a+\frac {b}{x}}}{c^2 \left (c+\frac {d}{x}\right )}+\frac {\sqrt {a+\frac {b}{x}} x}{c \left (c+\frac {d}{x}\right )}+\frac {\sqrt {d} (3 b c-4 a d) \arctan \left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b c-a d}}\right )}{c^3 \sqrt {b c-a d}}+\frac {(b c-4 a d) \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{\sqrt {a} c^3} \] Output:

2*d*(a+b/x)^(1/2)/c^2/(c+d/x)+(a+b/x)^(1/2)*x/c/(c+d/x)+d^(1/2)*(-4*a*d+3* 
b*c)*arctan(d^(1/2)*(a+b/x)^(1/2)/(-a*d+b*c)^(1/2))/c^3/(-a*d+b*c)^(1/2)+( 
-4*a*d+b*c)*arctanh((a+b/x)^(1/2)/a^(1/2))/a^(1/2)/c^3
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.60 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.83 \[ \int \frac {\sqrt {a+\frac {b}{x}}}{\left (c+\frac {d}{x}\right )^2} \, dx=\frac {\frac {c \sqrt {a+\frac {b}{x}} x (2 d+c x)}{d+c x}+\frac {\sqrt {d} (3 b c-4 a d) \arctan \left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b c-a d}}\right )}{\sqrt {b c-a d}}+\frac {(b c-4 a d) \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{\sqrt {a}}}{c^3} \] Input:

Integrate[Sqrt[a + b/x]/(c + d/x)^2,x]
 

Output:

((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
 

Rubi [A] (verified)

Time = 0.54 (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

\(\displaystyle \frac {x \sqrt {a+\frac {b}{x}}}{c \left (c+\frac {d}{x}\right )}-\frac {\frac {-\frac {2 \sqrt {d} (3 b c-4 a d) \arctan \left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b c-a d}}\right )}{c \sqrt {b c-a d}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right ) (b c-4 a d)}{\sqrt {a} c}}{c}-\frac {4 d \sqrt {a+\frac {b}{x}}}{c \left (c+\frac {d}{x}\right )}}{2 c}\)

Input:

Int[Sqrt[a + b/x]/(c + d/x)^2,x]
 

Output:

(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)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 73

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]
 

rule 110

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])
 

rule 168

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]
 

rule 174

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]
 

rule 218

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]
 

rule 221

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]
 

rule 899

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(476\) vs. \(2(127)=254\).

Time = 0.52 (sec) , antiderivative size = 477, normalized size of antiderivative = 3.24

method	result	size
risch	\(\frac {x \sqrt {\frac {a x +b}{x}}}{c^{2}}-\frac {\left (\frac {\left (4 a d -b c \right ) \ln \left (\frac {\frac {b}{2}+a x}{\sqrt {a}}+\sqrt {a \,x^{2}+b x}\right )}{c \sqrt {a}}+\frac {2 d \left (3 a d -2 b c \right ) \ln \left (\frac {\frac {2 \left (a d -b c \right ) d}{c^{2}}-\frac {\left (2 a d -b c \right ) \left (x +\frac {d}{c}\right )}{c}+2 \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, \sqrt {a \left (x +\frac {d}{c}\right )^{2}-\frac {\left (2 a d -b c \right ) \left (x +\frac {d}{c}\right )}{c}+\frac {\left (a d -b c \right ) d}{c^{2}}}}{x +\frac {d}{c}}\right )}{c^{2} \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}}+\frac {2 d^{2} \left (a d -b c \right ) \left (-\frac {c^{2} \sqrt {a \left (x +\frac {d}{c}\right )^{2}-\frac {\left (2 a d -b c \right ) \left (x +\frac {d}{c}\right )}{c}+\frac {\left (a d -b c \right ) d}{c^{2}}}}{\left (a d -b c \right ) d \left (x +\frac {d}{c}\right )}-\frac {\left (2 a d -b c \right ) c \ln \left (\frac {\frac {2 \left (a d -b c \right ) d}{c^{2}}-\frac {\left (2 a d -b c \right ) \left (x +\frac {d}{c}\right )}{c}+2 \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, \sqrt {a \left (x +\frac {d}{c}\right )^{2}-\frac {\left (2 a d -b c \right ) \left (x +\frac {d}{c}\right )}{c}+\frac {\left (a d -b c \right ) d}{c^{2}}}}{x +\frac {d}{c}}\right )}{2 \left (a d -b c \right ) d \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}}\right )}{c^{3}}\right ) \sqrt {\frac {a x +b}{x}}\, \sqrt {x \left (a x +b \right )}}{2 c^{2} \left (a x +b \right )}\)	\(477\)
default	\(-\frac {\sqrt {\frac {a x +b}{x}}\, x \left (4 a^{\frac {7}{2}} \ln \left (\frac {2 \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, \sqrt {x \left (a x +b \right )}\, c -2 a d x +b c x -b d}{c x +d}\right ) c \,d^{3} x +2 a^{\frac {5}{2}} \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, \sqrt {x \left (a x +b \right )}\, c^{4} x^{2}+4 a^{\frac {7}{2}} \ln \left (\frac {2 \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, \sqrt {x \left (a x +b \right )}\, c -2 a d x +b c x -b d}{c x +d}\right ) d^{4}-2 a^{\frac {5}{2}} \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, \sqrt {x \left (a x +b \right )}\, c^{3} d x -7 a^{\frac {5}{2}} \ln \left (\frac {2 \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, \sqrt {x \left (a x +b \right )}\, 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 {\frac {\left (a d -b c \right ) d}{c^{2}}}\, \sqrt {x \left (a x +b \right )}\, c^{2} d^{2}-7 a^{\frac {5}{2}} \ln \left (\frac {2 \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, \sqrt {x \left (a x +b \right )}\, 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 {\frac {\left (a d -b c \right ) d}{c^{2}}}\, \sqrt {x \left (a x +b \right )}\, b \,c^{4} x +3 a^{\frac {3}{2}} \ln \left (\frac {2 \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, \sqrt {x \left (a x +b \right )}\, c -2 a d x +b c x -b d}{c x +d}\right ) b^{2} c^{3} d x +4 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{3} \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, c^{2} d^{2} x -5 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{2} \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, b \,c^{3} d x +\ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, b^{2} c^{4} x +4 a^{\frac {3}{2}} \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, \sqrt {x \left (a x +b \right )}\, b \,c^{3} d +3 a^{\frac {3}{2}} \ln \left (\frac {2 \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, \sqrt {x \left (a x +b \right )}\, c -2 a d x +b c x -b d}{c x +d}\right ) b^{2} c^{2} d^{2}+4 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{3} \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, c \,d^{3}-5 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{2} \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, b \,c^{2} d^{2}+\ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a \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}}}}\)	\(943\)

Input:

int((a+b/x)^(1/2)/(c+1/x*d)^2,x,method=_RETURNVERBOSE)
 

Output:

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/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+1/c*d)+2*((a*d-b*c)*d/c^2)^(1/2)*(a*(x 
+1/c*d)^2-(2*a*d-b*c)/c*(x+1/c*d)+(a*d-b*c)*d/c^2)^(1/2))/(x+1/c*d))+2*d^2 
*(a*d-b*c)/c^3*(-1/(a*d-b*c)/d*c^2/(x+1/c*d)*(a*(x+1/c*d)^2-(2*a*d-b*c)/c* 
(x+1/c*d)+(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+1/c*d)+2*((a*d-b*c)*d/ 
c^2)^(1/2)*(a*(x+1/c*d)^2-(2*a*d-b*c)/c*(x+1/c*d)+(a*d-b*c)*d/c^2)^(1/2))/ 
(x+1/c*d))))*((a*x+b)/x)^(1/2)*(x*(a*x+b))^(1/2)/(a*x+b)
 

Fricas [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 773, normalized size of antiderivative = 5.26 \[ \int \frac {\sqrt {a+\frac {b}{x}}}{\left (c+\frac {d}{x}\right )^2} \, dx =\text {Too large to display} \] Input:

integrate((a+b/x)^(1/2)/(c+d/x)^2,x, algorithm="fricas")
 

Output:

[-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)*x*sqrt((a*x + b)/x)/(a*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))*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(sqrt(d/(b*c - a*d))*sqrt((a*x + b)/x)) - ( 
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*sqr 
t((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), -((b*c*d - 4*a*d^2 + (b*c^2 - 4*a*c*d)*x)*sqrt(-a)*arctan(sq 
rt(-a)*x*sqrt((a*x + b)/x)/(a*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))*arctan(sqrt(d/(b*c - a*d))*sqrt((a*x 
 + b)/x)) - (a*c^2*x^2 + 2*a*c*d*x)*sqrt((a*x + b)/x))/(a*c^4*x + a*c^3*d) 
]
 

Sympy [F]

\[ \int \frac {\sqrt {a+\frac {b}{x}}}{\left (c+\frac {d}{x}\right )^2} \, dx=\int \frac {x^{2} \sqrt {a + \frac {b}{x}}}{\left (c x + d\right )^{2}}\, dx \] Input:

integrate((a+b/x)**(1/2)/(c+d/x)**2,x)
 

Output:

Integral(x**2*sqrt(a + b/x)/(c*x + d)**2, x)
 

Maxima [F]

\[ \int \frac {\sqrt {a+\frac {b}{x}}}{\left (c+\frac {d}{x}\right )^2} \, dx=\int { \frac {\sqrt {a + \frac {b}{x}}}{{\left (c + \frac {d}{x}\right )}^{2}} \,d x } \] Input:

integrate((a+b/x)^(1/2)/(c+d/x)^2,x, algorithm="maxima")
 

Output:

integrate(sqrt(a + b/x)/(c + d/x)^2, x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 401 vs. \(2 (127) = 254\).

Time = 0.16 (sec) , antiderivative size = 401, normalized size of antiderivative = 2.73 \[ \int \frac {\sqrt {a+\frac {b}{x}}}{\left (c+\frac {d}{x}\right )^2} \, dx=\frac {\sqrt {a x^{2} + b x} \mathrm {sgn}\left (x\right )}{c^{2}} - \frac {{\left (3 \, b c d \mathrm {sgn}\left (x\right ) - 4 \, a d^{2} \mathrm {sgn}\left (x\right )\right )} \arctan \left (-\frac {{\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} c + \sqrt {a} d}{\sqrt {b c d - a d^{2}}}\right )}{\sqrt {b c d - a d^{2}} c^{3}} - \frac {{\left (b c \mathrm {sgn}\left (x\right ) - 4 \, a d \mathrm {sgn}\left (x\right )\right )} \log \left ({\left | 2 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} \sqrt {a} + b \right |}\right )}{2 \, \sqrt {a} c^{3}} - \frac {{\left (6 \, \sqrt {a} b c d \arctan \left (\frac {\sqrt {a} d}{\sqrt {b c d - a d^{2}}}\right ) - 8 \, a^{\frac {3}{2}} d^{2} \arctan \left (\frac {\sqrt {a} d}{\sqrt {b c d - a d^{2}}}\right ) - \sqrt {b c d - a d^{2}} b c \log \left ({\left | b \right |}\right ) + 4 \, \sqrt {b c d - a d^{2}} a d \log \left ({\left | b \right |}\right ) + 2 \, \sqrt {b c d - a d^{2}} a d\right )} \mathrm {sgn}\left (x\right )}{2 \, \sqrt {b c d - a d^{2}} \sqrt {a} c^{3}} - \frac {{\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} b c d \mathrm {sgn}\left (x\right ) - 2 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} a d^{2} \mathrm {sgn}\left (x\right ) - \sqrt {a} b d^{2} \mathrm {sgn}\left (x\right )}{{\left ({\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{2} c + 2 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} \sqrt {a} d + b d\right )} c^{3}} \] Input:

integrate((a+b/x)^(1/2)/(c+d/x)^2,x, algorithm="giac")
 

Output:

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)
 

Mupad [B] (verification not implemented)

Time = 1.75 (sec) , antiderivative size = 1195, normalized size of antiderivative = 8.13 \[ \int \frac {\sqrt {a+\frac {b}{x}}}{\left (c+\frac {d}{x}\right )^2} \, dx=\text {Too large to display} \] Input:

int((a + b/x)^(1/2)/(c + d/x)^2,x)
 

Output:

- ((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...
 

Reduce [B] (verification not implemented)

Time = 0.52 (sec) , antiderivative size = 996, normalized size of antiderivative = 6.78 \[ \int \frac {\sqrt {a+\frac {b}{x}}}{\left (c+\frac {d}{x}\right )^2} \, dx =\text {Too large to display} \] Input:

int((a+b/x)^(1/2)/(c+d/x)^2,x)
 

Output:

(4*sqrt(d)*sqrt(a*d - b*c)*log(sqrt(c)*sqrt(a*x + b) - sqrt(2*sqrt(d)*sqrt 
(a)*sqrt(a*d - b*c) - 2*a*d + b*c) + sqrt(x)*sqrt(c)*sqrt(a))*a**2*c*d*x + 
 4*sqrt(d)*sqrt(a*d - b*c)*log(sqrt(c)*sqrt(a*x + b) - sqrt(2*sqrt(d)*sqrt 
(a)*sqrt(a*d - b*c) - 2*a*d + b*c) + sqrt(x)*sqrt(c)*sqrt(a))*a**2*d**2 - 
3*sqrt(d)*sqrt(a*d - b*c)*log(sqrt(c)*sqrt(a*x + b) - sqrt(2*sqrt(d)*sqrt( 
a)*sqrt(a*d - b*c) - 2*a*d + b*c) + sqrt(x)*sqrt(c)*sqrt(a))*a*b*c**2*x - 
3*sqrt(d)*sqrt(a*d - b*c)*log(sqrt(c)*sqrt(a*x + b) - sqrt(2*sqrt(d)*sqrt( 
a)*sqrt(a*d - b*c) - 2*a*d + b*c) + sqrt(x)*sqrt(c)*sqrt(a))*a*b*c*d + 4*s 
qrt(d)*sqrt(a*d - b*c)*log(sqrt(c)*sqrt(a*x + b) + sqrt(2*sqrt(d)*sqrt(a)* 
sqrt(a*d - b*c) - 2*a*d + b*c) + sqrt(x)*sqrt(c)*sqrt(a))*a**2*c*d*x + 4*s 
qrt(d)*sqrt(a*d - b*c)*log(sqrt(c)*sqrt(a*x + b) + sqrt(2*sqrt(d)*sqrt(a)* 
sqrt(a*d - b*c) - 2*a*d + b*c) + sqrt(x)*sqrt(c)*sqrt(a))*a**2*d**2 - 3*sq 
rt(d)*sqrt(a*d - b*c)*log(sqrt(c)*sqrt(a*x + b) + sqrt(2*sqrt(d)*sqrt(a)*s 
qrt(a*d - b*c) - 2*a*d + b*c) + sqrt(x)*sqrt(c)*sqrt(a))*a*b*c**2*x - 3*sq 
rt(d)*sqrt(a*d - b*c)*log(sqrt(c)*sqrt(a*x + b) + sqrt(2*sqrt(d)*sqrt(a)*s 
qrt(a*d - b*c) - 2*a*d + b*c) + sqrt(x)*sqrt(c)*sqrt(a))*a*b*c*d - 4*sqrt( 
d)*sqrt(a*d - b*c)*log(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) + 2*sqrt(x)*sqrt( 
a)*sqrt(a*x + b)*c + 2*a*c*x + 2*a*d)*a**2*c*d*x - 4*sqrt(d)*sqrt(a*d - b* 
c)*log(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) + 2*sqrt(x)*sqrt(a)*sqrt(a*x + b) 
*c + 2*a*c*x + 2*a*d)*a**2*d**2 + 3*sqrt(d)*sqrt(a*d - b*c)*log(2*sqrt(...