\(\int \frac {1}{\sqrt {a+\frac {b}{x}} (c+\frac {d}{x})} \, dx\) [29]

Optimal result

Integrand size = 21, antiderivative size = 108 \[ \int \frac {1}{\sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )} \, dx=\frac {\sqrt {a+\frac {b}{x}} x}{a c}-\frac {2 d^{3/2} \arctan \left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b c-a d}}\right )}{c^2 \sqrt {b c-a d}}-\frac {(b c+2 a d) \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{3/2} c^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.39 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.96 \[ \int \frac {1}{\sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )} \, dx=\frac {\frac {c \sqrt {a+\frac {b}{x}} x}{a}-\frac {2 d^{3/2} \arctan \left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b c-a d}}\right )}{\sqrt {b c-a d}}-\frac {(b c+2 a d) \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{3/2}}}{c^2} \] Input:

Integrate[1/(Sqrt[a + b/x]*(c + d/x)),x]
 

Output:

((c*Sqrt[a + b/x]*x)/a - (2*d^(3/2)*ArcTan[(Sqrt[d]*Sqrt[a + b/x])/Sqrt[b* 
c - a*d]])/Sqrt[b*c - a*d] - ((b*c + 2*a*d)*ArcTanh[Sqrt[a + b/x]/Sqrt[a]] 
)/a^(3/2))/c^2
 

Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.11, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {899, 114, 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.

\(\displaystyle \int \frac {1}{\sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )} \, dx\)

\(\Big \downarrow \) 899

\(\displaystyle -\int \frac {x^2}{\sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )}d\frac {1}{x}\)

\(\Big \downarrow \) 114

\(\displaystyle \frac {\int \frac {\left (b c+2 a d+\frac {b d}{x}\right ) x}{2 \sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )}d\frac {1}{x}}{a c}+\frac {x \sqrt {a+\frac {b}{x}}}{a c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {\left (b c+2 a d+\frac {b d}{x}\right ) x}{\sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )}d\frac {1}{x}}{2 a c}+\frac {x \sqrt {a+\frac {b}{x}}}{a c}\)

\(\Big \downarrow \) 174

\(\displaystyle \frac {\frac {(2 a d+b c) \int \frac {x}{\sqrt {a+\frac {b}{x}}}d\frac {1}{x}}{c}-\frac {2 a d^2 \int \frac {1}{\sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )}d\frac {1}{x}}{c}}{2 a c}+\frac {x \sqrt {a+\frac {b}{x}}}{a c}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {\frac {2 (2 a d+b c) \int \frac {1}{\frac {1}{b x^2}-\frac {a}{b}}d\sqrt {a+\frac {b}{x}}}{b c}-\frac {4 a d^2 \int \frac {1}{c-\frac {a d}{b}+\frac {d}{b x^2}}d\sqrt {a+\frac {b}{x}}}{b c}}{2 a c}+\frac {x \sqrt {a+\frac {b}{x}}}{a c}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {\frac {2 (2 a d+b c) \int \frac {1}{\frac {1}{b x^2}-\frac {a}{b}}d\sqrt {a+\frac {b}{x}}}{b c}-\frac {4 a d^{3/2} \arctan \left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b c-a d}}\right )}{c \sqrt {b c-a d}}}{2 a c}+\frac {x \sqrt {a+\frac {b}{x}}}{a c}\)

\(\Big \downarrow \) 221

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

Input:

Int[1/(Sqrt[a + b/x]*(c + d/x)),x]
 

Output:

(Sqrt[a + b/x]*x)/(a*c) + ((-4*a*d^(3/2)*ArcTan[(Sqrt[d]*Sqrt[a + b/x])/Sq 
rt[b*c - a*d]])/(c*Sqrt[b*c - a*d]) - (2*(b*c + 2*a*d)*ArcTanh[Sqrt[a + b/ 
x]/Sqrt[a]])/(Sqrt[a]*c))/(2*a*c)
 

Defintions of rubi rules used

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 114

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])
 

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. \(228\) vs. \(2(90)=180\).

Time = 0.59 (sec) , antiderivative size = 229, normalized size of antiderivative = 2.12

method	result	size
default	\(\frac {\left (2 \sqrt {x \left (a x +b \right )}\, c^{2} \sqrt {a}\, \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}-2 \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a c d -\sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) b \,c^{2}-2 a^{\frac {3}{2}} \ln \left (\frac {2 \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, \sqrt {x \left (a x +b \right )}\, c -2 a d x +b c x -b d}{c x +d}\right ) d^{2}\right ) x \sqrt {\frac {a x +b}{x}}}{2 a^{\frac {3}{2}} \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, c^{3} \sqrt {x \left (a x +b \right )}}\)	\(229\)
risch	\(\frac {a x +b}{a c \sqrt {\frac {a x +b}{x}}}-\frac {\left (\frac {\left (2 a d +b c \right ) \ln \left (\frac {\frac {b}{2}+a x}{\sqrt {a}}+\sqrt {a \,x^{2}+b x}\right )}{c \sqrt {a}}+\frac {2 a \,d^{2} \ln \left (\frac {\frac {2 \left (a d -b c \right ) d}{c^{2}}-\frac {\left (2 a d -b c \right ) \left (x +\frac {d}{c}\right )}{c}+2 \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, \sqrt {a \left (x +\frac {d}{c}\right )^{2}-\frac {\left (2 a d -b c \right ) \left (x +\frac {d}{c}\right )}{c}+\frac {\left (a d -b c \right ) d}{c^{2}}}}{x +\frac {d}{c}}\right )}{c^{2} \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}}\right ) \sqrt {x \left (a x +b \right )}}{2 a c x \sqrt {\frac {a x +b}{x}}}\)	\(232\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 512, normalized size of antiderivative = 4.74 \[ \int \frac {1}{\sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )} \, dx=\left [\frac {2 \, a^{2} d \sqrt {-\frac {d}{b c - a d}} \log \left (-\frac {2 \, {\left (b c - a d\right )} x \sqrt {-\frac {d}{b c - a d}} \sqrt {\frac {a x + b}{x}} - b d + {\left (b c - 2 \, a d\right )} x}{c x + d}\right ) + 2 \, a c x \sqrt {\frac {a x + b}{x}} + {\left (b c + 2 \, a d\right )} \sqrt {a} \log \left (2 \, a x - 2 \, \sqrt {a} x \sqrt {\frac {a x + b}{x}} + b\right )}{2 \, a^{2} c^{2}}, \frac {a^{2} d \sqrt {-\frac {d}{b c - a d}} \log \left (-\frac {2 \, {\left (b c - a d\right )} x \sqrt {-\frac {d}{b c - a d}} \sqrt {\frac {a x + b}{x}} - b d + {\left (b c - 2 \, a d\right )} x}{c x + d}\right ) + a c x \sqrt {\frac {a x + b}{x}} + {\left (b c + 2 \, a d\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a} x \sqrt {\frac {a x + b}{x}}}{a x + b}\right )}{a^{2} c^{2}}, -\frac {4 \, a^{2} d \sqrt {\frac {d}{b c - a d}} \arctan \left (\sqrt {\frac {d}{b c - a d}} \sqrt {\frac {a x + b}{x}}\right ) - 2 \, a c x \sqrt {\frac {a x + b}{x}} - {\left (b c + 2 \, a d\right )} \sqrt {a} \log \left (2 \, a x - 2 \, \sqrt {a} x \sqrt {\frac {a x + b}{x}} + b\right )}{2 \, a^{2} c^{2}}, -\frac {2 \, a^{2} d \sqrt {\frac {d}{b c - a d}} \arctan \left (\sqrt {\frac {d}{b c - a d}} \sqrt {\frac {a x + b}{x}}\right ) - a c x \sqrt {\frac {a x + b}{x}} - {\left (b c + 2 \, a d\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a} x \sqrt {\frac {a x + b}{x}}}{a x + b}\right )}{a^{2} c^{2}}\right ] \] Input:

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

Output:

[1/2*(2*a^2*d*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*x*sqrt(( 
a*x + b)/x) + (b*c + 2*a*d)*sqrt(a)*log(2*a*x - 2*sqrt(a)*x*sqrt((a*x + b) 
/x) + b))/(a^2*c^2), (a^2*d*sqrt(-d/(b*c - a*d))*log(-(2*(b*c - a*d)*x*sqr 
t(-d/(b*c - a*d))*sqrt((a*x + b)/x) - b*d + (b*c - 2*a*d)*x)/(c*x + d)) + 
a*c*x*sqrt((a*x + b)/x) + (b*c + 2*a*d)*sqrt(-a)*arctan(sqrt(-a)*x*sqrt((a 
*x + b)/x)/(a*x + b)))/(a^2*c^2), -1/2*(4*a^2*d*sqrt(d/(b*c - a*d))*arctan 
(sqrt(d/(b*c - a*d))*sqrt((a*x + b)/x)) - 2*a*c*x*sqrt((a*x + b)/x) - (b*c 
 + 2*a*d)*sqrt(a)*log(2*a*x - 2*sqrt(a)*x*sqrt((a*x + b)/x) + b))/(a^2*c^2 
), -(2*a^2*d*sqrt(d/(b*c - a*d))*arctan(sqrt(d/(b*c - a*d))*sqrt((a*x + b) 
/x)) - a*c*x*sqrt((a*x + b)/x) - (b*c + 2*a*d)*sqrt(-a)*arctan(sqrt(-a)*x* 
sqrt((a*x + b)/x)/(a*x + b)))/(a^2*c^2)]
 

Sympy [F]

\[ \int \frac {1}{\sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )} \, dx=\int \frac {x}{\sqrt {a + \frac {b}{x}} \left (c x + d\right )}\, dx \] Input:

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

Output:

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

Maxima [F]

\[ \int \frac {1}{\sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )} \, dx=\int { \frac {1}{\sqrt {a + \frac {b}{x}} {\left (c + \frac {d}{x}\right )}} \,d x } \] Input:

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

Output:

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

Giac [F(-2)]

Exception generated. \[ \int \frac {1}{\sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )} \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument Type
 

Mupad [B] (verification not implemented)

Time = 1.28 (sec) , antiderivative size = 1183, normalized size of antiderivative = 10.95 \[ \int \frac {1}{\sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )} \, dx =\text {Too large to display} \] Input:

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

Output:

(x*(a + b/x)^(1/2))/(a*c) - (atan(((((((2*(2*a*b^4*c^5*d^2 + 2*a^2*b^3*c^4 
*d^3))/(a^2*c^3) - (2*(4*a^2*b^3*c^5*d^2 - 8*a^3*b^2*c^4*d^3)*(a + b/x)^(1 
/2)*(a*d^4 - b*c*d^3)^(1/2))/(a^2*c^2*(b*c^3 - a*c^2*d)))*(a*d^4 - b*c*d^3 
)^(1/2))/(b*c^3 - a*c^2*d) - (2*(a + b/x)^(1/2)*(8*a^2*b^2*d^5 + b^4*c^2*d 
^3 + 4*a*b^3*c*d^4))/(a^2*c^2))*(a*d^4 - b*c*d^3)^(1/2)*1i)/(b*c^3 - a*c^2 
*d) - (((((2*(2*a*b^4*c^5*d^2 + 2*a^2*b^3*c^4*d^3))/(a^2*c^3) + (2*(4*a^2* 
b^3*c^5*d^2 - 8*a^3*b^2*c^4*d^3)*(a + b/x)^(1/2)*(a*d^4 - b*c*d^3)^(1/2))/ 
(a^2*c^2*(b*c^3 - a*c^2*d)))*(a*d^4 - b*c*d^3)^(1/2))/(b*c^3 - a*c^2*d) + 
(2*(a + b/x)^(1/2)*(8*a^2*b^2*d^5 + b^4*c^2*d^3 + 4*a*b^3*c*d^4))/(a^2*c^2 
))*(a*d^4 - b*c*d^3)^(1/2)*1i)/(b*c^3 - a*c^2*d))/((((((2*(2*a*b^4*c^5*d^2 
 + 2*a^2*b^3*c^4*d^3))/(a^2*c^3) - (2*(4*a^2*b^3*c^5*d^2 - 8*a^3*b^2*c^4*d 
^3)*(a + b/x)^(1/2)*(a*d^4 - b*c*d^3)^(1/2))/(a^2*c^2*(b*c^3 - a*c^2*d)))* 
(a*d^4 - b*c*d^3)^(1/2))/(b*c^3 - a*c^2*d) - (2*(a + b/x)^(1/2)*(8*a^2*b^2 
*d^5 + b^4*c^2*d^3 + 4*a*b^3*c*d^4))/(a^2*c^2))*(a*d^4 - b*c*d^3)^(1/2))/( 
b*c^3 - a*c^2*d) - (4*(2*a*b^3*d^5 + b^4*c*d^4))/(a^2*c^3) + (((((2*(2*a*b 
^4*c^5*d^2 + 2*a^2*b^3*c^4*d^3))/(a^2*c^3) + (2*(4*a^2*b^3*c^5*d^2 - 8*a^3 
*b^2*c^4*d^3)*(a + b/x)^(1/2)*(a*d^4 - b*c*d^3)^(1/2))/(a^2*c^2*(b*c^3 - a 
*c^2*d)))*(a*d^4 - b*c*d^3)^(1/2))/(b*c^3 - a*c^2*d) + (2*(a + b/x)^(1/2)* 
(8*a^2*b^2*d^5 + b^4*c^2*d^3 + 4*a*b^3*c*d^4))/(a^2*c^2))*(a*d^4 - b*c*d^3 
)^(1/2))/(b*c^3 - a*c^2*d)))*(a*d^4 - b*c*d^3)^(1/2)*2i)/(b*c^3 - a*c^2...
 

Reduce [B] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 301, normalized size of antiderivative = 2.79 \[ \int \frac {1}{\sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )} \, dx=\frac {\sqrt {d}\, \sqrt {a d -b c}\, \mathrm {log}\left (\sqrt {c}\, \sqrt {a x +b}-\sqrt {2 \sqrt {d}\, \sqrt {a}\, \sqrt {a d -b c}-2 a d +b c}+\sqrt {x}\, \sqrt {c}\, \sqrt {a}\right ) a^{2} d +\sqrt {d}\, \sqrt {a d -b c}\, \mathrm {log}\left (\sqrt {c}\, \sqrt {a x +b}+\sqrt {2 \sqrt {d}\, \sqrt {a}\, \sqrt {a d -b c}-2 a d +b c}+\sqrt {x}\, \sqrt {c}\, \sqrt {a}\right ) a^{2} d -\sqrt {d}\, \sqrt {a d -b c}\, \mathrm {log}\left (2 \sqrt {d}\, \sqrt {a}\, \sqrt {a d -b c}+2 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +b}\, c +2 a c x +2 a d \right ) a^{2} d +\sqrt {x}\, \sqrt {a x +b}\, a^{2} c d -\sqrt {x}\, \sqrt {a x +b}\, a b \,c^{2}-2 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {a x +b}+\sqrt {x}\, \sqrt {a}}{\sqrt {b}}\right ) a^{2} d^{2}+\sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {a x +b}+\sqrt {x}\, \sqrt {a}}{\sqrt {b}}\right ) a b c d +\sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {a x +b}+\sqrt {x}\, \sqrt {a}}{\sqrt {b}}\right ) b^{2} c^{2}}{a^{2} c^{2} \left (a d -b c \right )} \] Input:

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

Output:

(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 + 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 - sqrt(d)*sqr 
t(a*d - b*c)*log(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) + 2*sqrt(x)*sqrt(a)*sqr 
t(a*x + b)*c + 2*a*c*x + 2*a*d)*a**2*d + sqrt(x)*sqrt(a*x + b)*a**2*c*d - 
sqrt(x)*sqrt(a*x + b)*a*b*c**2 - 2*sqrt(a)*log((sqrt(a*x + b) + sqrt(x)*sq 
rt(a))/sqrt(b))*a**2*d**2 + sqrt(a)*log((sqrt(a*x + b) + sqrt(x)*sqrt(a))/ 
sqrt(b))*a*b*c*d + sqrt(a)*log((sqrt(a*x + b) + sqrt(x)*sqrt(a))/sqrt(b))* 
b**2*c**2)/(a**2*c**2*(a*d - b*c))