\(\int \frac {1}{\sqrt {a+\frac {b}{x}} (c+\frac {d}{x})^2} \, dx\) [30]

Optimal result

Integrand size = 21, antiderivative size = 172 \[ \int \frac {1}{\sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )^2} \, dx=\frac {d (b c-2 a d) \sqrt {a+\frac {b}{x}}}{a c^2 (b c-a d) \left (c+\frac {d}{x}\right )}+\frac {\sqrt {a+\frac {b}{x}} x}{a c \left (c+\frac {d}{x}\right )}-\frac {d^{3/2} (5 b c-4 a d) \arctan \left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b c-a d}}\right )}{c^3 (b c-a d)^{3/2}}-\frac {(b c+4 a d) \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{3/2} c^3} \] Output:

d*(-2*a*d+b*c)*(a+b/x)^(1/2)/a/c^2/(-a*d+b*c)/(c+d/x)+(a+b/x)^(1/2)*x/a/c/ 
(c+d/x)-d^(3/2)*(-4*a*d+5*b*c)*arctan(d^(1/2)*(a+b/x)^(1/2)/(-a*d+b*c)^(1/ 
2))/c^3/(-a*d+b*c)^(3/2)-(4*a*d+b*c)*arctanh((a+b/x)^(1/2)/a^(1/2))/a^(3/2 
)/c^3
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.82 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.87 \[ \int \frac {1}{\sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )^2} \, dx=\frac {\frac {c \sqrt {a+\frac {b}{x}} x (-b c (d+c x)+a d (2 d+c x))}{a (-b c+a d) (d+c x)}+\frac {d^{3/2} (-5 b c+4 a d) \arctan \left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b c-a d}}\right )}{(b c-a d)^{3/2}}-\frac {(b c+4 a d) \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{3/2}}}{c^3} \] Input:

Integrate[1/(Sqrt[a + b/x]*(c + d/x)^2),x]
 

Output:

((c*Sqrt[a + b/x]*x*(-(b*c*(d + c*x)) + a*d*(2*d + c*x)))/(a*(-(b*c) + a*d 
)*(d + c*x)) + (d^(3/2)*(-5*b*c + 4*a*d)*ArcTan[(Sqrt[d]*Sqrt[a + b/x])/Sq 
rt[b*c - a*d]])/(b*c - a*d)^(3/2) - ((b*c + 4*a*d)*ArcTanh[Sqrt[a + b/x]/S 
qrt[a]])/a^(3/2))/c^3
 

Rubi [A] (verified)

Time = 0.63 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.19, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {899, 114, 27, 168, 25, 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.

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

\(\Big \downarrow \) 899

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

\(\Big \downarrow \) 114

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 168

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 174

\(\displaystyle \frac {\frac {\frac {(b c-a d) (4 a d+b c) \int \frac {x}{\sqrt {a+\frac {b}{x}}}d\frac {1}{x}}{c}-\frac {a d^2 (5 b c-4 a d) \int \frac {1}{\sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )}d\frac {1}{x}}{c}}{c (b c-a d)}+\frac {2 d \sqrt {a+\frac {b}{x}} (b c-2 a d)}{c \left (c+\frac {d}{x}\right ) (b c-a d)}}{2 a c}+\frac {x \sqrt {a+\frac {b}{x}}}{a c \left (c+\frac {d}{x}\right )}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {\frac {\frac {2 (b c-a d) (4 a d+b c) \int \frac {1}{\frac {1}{b x^2}-\frac {a}{b}}d\sqrt {a+\frac {b}{x}}}{b c}-\frac {2 a d^2 (5 b c-4 a d) \int \frac {1}{c-\frac {a d}{b}+\frac {d}{b x^2}}d\sqrt {a+\frac {b}{x}}}{b c}}{c (b c-a d)}+\frac {2 d \sqrt {a+\frac {b}{x}} (b c-2 a d)}{c \left (c+\frac {d}{x}\right ) (b c-a d)}}{2 a c}+\frac {x \sqrt {a+\frac {b}{x}}}{a c \left (c+\frac {d}{x}\right )}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {\frac {\frac {2 (b c-a d) (4 a d+b c) \int \frac {1}{\frac {1}{b x^2}-\frac {a}{b}}d\sqrt {a+\frac {b}{x}}}{b c}-\frac {2 a d^{3/2} (5 b c-4 a d) \arctan \left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b c-a d}}\right )}{c \sqrt {b c-a d}}}{c (b c-a d)}+\frac {2 d \sqrt {a+\frac {b}{x}} (b c-2 a d)}{c \left (c+\frac {d}{x}\right ) (b c-a d)}}{2 a c}+\frac {x \sqrt {a+\frac {b}{x}}}{a c \left (c+\frac {d}{x}\right )}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {-\frac {2 a d^{3/2} (5 b c-4 a d) \arctan \left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b c-a d}}\right )}{c \sqrt {b c-a d}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right ) (b c-a d) (4 a d+b c)}{\sqrt {a} c}}{c (b c-a d)}+\frac {2 d \sqrt {a+\frac {b}{x}} (b c-2 a d)}{c \left (c+\frac {d}{x}\right ) (b c-a d)}}{2 a c}+\frac {x \sqrt {a+\frac {b}{x}}}{a c \left (c+\frac {d}{x}\right )}\)

Input:

Int[1/(Sqrt[a + b/x]*(c + d/x)^2),x]
 

Output:

(Sqrt[a + b/x]*x)/(a*c*(c + d/x)) + ((2*d*(b*c - 2*a*d)*Sqrt[a + b/x])/(c* 
(b*c - a*d)*(c + d/x)) + ((-2*a*d^(3/2)*(5*b*c - 4*a*d)*ArcTan[(Sqrt[d]*Sq 
rt[a + b/x])/Sqrt[b*c - a*d]])/(c*Sqrt[b*c - a*d]) - (2*(b*c - a*d)*(b*c + 
 4*a*d)*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/(Sqrt[a]*c))/(c*(b*c - a*d)))/(2*a 
*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 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 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. \(468\) vs. \(2(152)=304\).

Time = 0.60 (sec) , antiderivative size = 469, normalized size of antiderivative = 2.73

method	result	size
risch	\(\frac {a x +b}{a \,c^{2} \sqrt {\frac {a x +b}{x}}}-\frac {\left (\frac {\left (4 a d +b c \right ) \ln \left (\frac {\frac {b}{2}+a x}{\sqrt {a}}+\sqrt {a \,x^{2}+b x}\right )}{c \sqrt {a}}+\frac {6 a \,d^{2} \ln \left (\frac {\frac {2 \left (a d -b c \right ) d}{c^{2}}-\frac {\left (2 a d -b c \right ) \left (x +\frac {d}{c}\right )}{c}+2 \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, \sqrt {a \left (x +\frac {d}{c}\right )^{2}-\frac {\left (2 a d -b c \right ) \left (x +\frac {d}{c}\right )}{c}+\frac {\left (a d -b c \right ) d}{c^{2}}}}{x +\frac {d}{c}}\right )}{c^{2} \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}}+\frac {2 a \,d^{3} \left (-\frac {c^{2} \sqrt {a \left (x +\frac {d}{c}\right )^{2}-\frac {\left (2 a d -b c \right ) \left (x +\frac {d}{c}\right )}{c}+\frac {\left (a d -b c \right ) d}{c^{2}}}}{\left (a d -b c \right ) d \left (x +\frac {d}{c}\right )}-\frac {\left (2 a d -b c \right ) c \ln \left (\frac {\frac {2 \left (a d -b c \right ) d}{c^{2}}-\frac {\left (2 a d -b c \right ) \left (x +\frac {d}{c}\right )}{c}+2 \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, \sqrt {a \left (x +\frac {d}{c}\right )^{2}-\frac {\left (2 a d -b c \right ) \left (x +\frac {d}{c}\right )}{c}+\frac {\left (a d -b c \right ) d}{c^{2}}}}{x +\frac {d}{c}}\right )}{2 \left (a d -b c \right ) d \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}}\right )}{c^{3}}\right ) \sqrt {x \left (a x +b \right )}}{2 c^{2} a x \sqrt {\frac {a x +b}{x}}}\)	\(469\)
default	\(\text {Expression too large to display}\)	\(1135\)

Input:

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

Output:

1/a/c^2*(a*x+b)/((a*x+b)/x)^(1/2)-1/2/c^2/a*((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)+6*a*d^2/c^2/((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*a*d 
^3/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 
))))/x/((a*x+b)/x)^(1/2)*(x*(a*x+b))^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 1131, normalized size of antiderivative = 6.58 \[ \int \frac {1}{\sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

[1/2*((b^2*c^2*d + 3*a*b*c*d^2 - 4*a^2*d^3 + (b^2*c^3 + 3*a*b*c^2*d - 4*a^ 
2*c*d^2)*x)*sqrt(a)*log(2*a*x - 2*sqrt(a)*x*sqrt((a*x + b)/x) + b) + (5*a^ 
2*b*c*d^2 - 4*a^3*d^3 + (5*a^2*b*c^2*d - 4*a^3*c*d^2)*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*b*c^3 - a^2*c^2*d)*x^2 + (a*b*c^2*d - 2 
*a^2*c*d^2)*x)*sqrt((a*x + b)/x))/(a^2*b*c^4*d - a^3*c^3*d^2 + (a^2*b*c^5 
- a^3*c^4*d)*x), -1/2*(2*(5*a^2*b*c*d^2 - 4*a^3*d^3 + (5*a^2*b*c^2*d - 4*a 
^3*c*d^2)*x)*sqrt(d/(b*c - a*d))*arctan(sqrt(d/(b*c - a*d))*sqrt((a*x + b) 
/x)) - (b^2*c^2*d + 3*a*b*c*d^2 - 4*a^2*d^3 + (b^2*c^3 + 3*a*b*c^2*d - 4*a 
^2*c*d^2)*x)*sqrt(a)*log(2*a*x - 2*sqrt(a)*x*sqrt((a*x + b)/x) + b) - 2*(( 
a*b*c^3 - a^2*c^2*d)*x^2 + (a*b*c^2*d - 2*a^2*c*d^2)*x)*sqrt((a*x + b)/x)) 
/(a^2*b*c^4*d - a^3*c^3*d^2 + (a^2*b*c^5 - a^3*c^4*d)*x), 1/2*(2*(b^2*c^2* 
d + 3*a*b*c*d^2 - 4*a^2*d^3 + (b^2*c^3 + 3*a*b*c^2*d - 4*a^2*c*d^2)*x)*sqr 
t(-a)*arctan(sqrt(-a)*x*sqrt((a*x + b)/x)/(a*x + b)) + (5*a^2*b*c*d^2 - 4* 
a^3*d^3 + (5*a^2*b*c^2*d - 4*a^3*c*d^2)*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*b*c^3 - a^2*c^2*d)*x^2 + (a*b*c^2*d - 2*a^2*c*d^2)*x) 
*sqrt((a*x + b)/x))/(a^2*b*c^4*d - a^3*c^3*d^2 + (a^2*b*c^5 - a^3*c^4*d)*x 
), ((b^2*c^2*d + 3*a*b*c*d^2 - 4*a^2*d^3 + (b^2*c^3 + 3*a*b*c^2*d - 4*a^2* 
c*d^2)*x)*sqrt(-a)*arctan(sqrt(-a)*x*sqrt((a*x + b)/x)/(a*x + b)) - (5*...
 

Sympy [F]

\[ \int \frac {1}{\sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )^2} \, dx=\int \frac {x^{2}}{\sqrt {a + \frac {b}{x}} \left (c x + d\right )^{2}}\, dx \] Input:

integrate(1/(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]

\[ \int \frac {1}{\sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )^2} \, dx=\int { \frac {1}{\sqrt {a + \frac {b}{x}} {\left (c + \frac {d}{x}\right )}^{2}} \,d x } \] Input:

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

Output:

integrate(1/(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. 493 vs. \(2 (152) = 304\).

Time = 0.17 (sec) , antiderivative size = 493, normalized size of antiderivative = 2.87 \[ \int \frac {1}{\sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )^2} \, dx=\frac {{\left (10 \, a^{\frac {3}{2}} b c d^{2} \arctan \left (\frac {\sqrt {a} d}{\sqrt {b c d - a d^{2}}}\right ) - 8 \, a^{\frac {5}{2}} d^{3} \arctan \left (\frac {\sqrt {a} d}{\sqrt {b c d - a d^{2}}}\right ) - \sqrt {b c d - a d^{2}} b^{2} c^{2} \log \left ({\left | b \right |}\right ) - 3 \, \sqrt {b c d - a d^{2}} a b c d \log \left ({\left | b \right |}\right ) + 4 \, \sqrt {b c d - a d^{2}} a^{2} d^{2} \log \left ({\left | b \right |}\right ) + 2 \, \sqrt {b c d - a d^{2}} a^{2} d^{2}\right )} \mathrm {sgn}\left (x\right )}{2 \, {\left (\sqrt {b c d - a d^{2}} a^{\frac {3}{2}} b c^{4} - \sqrt {b c d - a d^{2}} a^{\frac {5}{2}} c^{3} d\right )}} + \frac {{\left (5 \, b c d^{2} - 4 \, a d^{3}\right )} \arctan \left (-\frac {{\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} c + \sqrt {a} d}{\sqrt {b c d - a d^{2}}}\right )}{{\left (b c^{4} \mathrm {sgn}\left (x\right ) - a c^{3} d \mathrm {sgn}\left (x\right )\right )} \sqrt {b c d - a d^{2}}} + \frac {{\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} b c d^{2} - 2 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} a d^{3} - \sqrt {a} b d^{3}}{{\left (b c^{4} \mathrm {sgn}\left (x\right ) - a c^{3} d \mathrm {sgn}\left (x\right )\right )} {\left ({\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{2} c + 2 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} \sqrt {a} d + b d\right )}} + \frac {\sqrt {a x^{2} + b x}}{a c^{2} \mathrm {sgn}\left (x\right )} + \frac {{\left (b c + 4 \, a d\right )} \log \left ({\left | 2 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} \sqrt {a} + b \right |}\right )}{2 \, a^{\frac {3}{2}} c^{3} \mathrm {sgn}\left (x\right )} \] Input:

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

Output:

1/2*(10*a^(3/2)*b*c*d^2*arctan(sqrt(a)*d/sqrt(b*c*d - a*d^2)) - 8*a^(5/2)* 
d^3*arctan(sqrt(a)*d/sqrt(b*c*d - a*d^2)) - sqrt(b*c*d - a*d^2)*b^2*c^2*lo 
g(abs(b)) - 3*sqrt(b*c*d - a*d^2)*a*b*c*d*log(abs(b)) + 4*sqrt(b*c*d - a*d 
^2)*a^2*d^2*log(abs(b)) + 2*sqrt(b*c*d - a*d^2)*a^2*d^2)*sgn(x)/(sqrt(b*c* 
d - a*d^2)*a^(3/2)*b*c^4 - sqrt(b*c*d - a*d^2)*a^(5/2)*c^3*d) + (5*b*c*d^2 
 - 4*a*d^3)*arctan(-((sqrt(a)*x - sqrt(a*x^2 + b*x))*c + sqrt(a)*d)/sqrt(b 
*c*d - a*d^2))/((b*c^4*sgn(x) - a*c^3*d*sgn(x))*sqrt(b*c*d - a*d^2)) + ((s 
qrt(a)*x - sqrt(a*x^2 + b*x))*b*c*d^2 - 2*(sqrt(a)*x - sqrt(a*x^2 + b*x))* 
a*d^3 - sqrt(a)*b*d^3)/((b*c^4*sgn(x) - a*c^3*d*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)) + 
 sqrt(a*x^2 + b*x)/(a*c^2*sgn(x)) + 1/2*(b*c + 4*a*d)*log(abs(2*(sqrt(a)*x 
 - sqrt(a*x^2 + b*x))*sqrt(a) + b))/(a^(3/2)*c^3*sgn(x))
 

Mupad [B] (verification not implemented)

Time = 3.00 (sec) , antiderivative size = 3813, normalized size of antiderivative = 22.17 \[ \int \frac {1}{\sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

(((a + b/x)^(1/2)*(b^3*c^2 + 2*a^2*b*d^2 - 2*a*b^2*c*d))/(c^2*(a^2*d - a*b 
*c)) + (d*(a + b/x)^(3/2)*(b^2*c - 2*a*b*d))/(c^2*(a^2*d - a*b*c)))/((a + 
b/x)*(2*a*d - b*c) - d*(a + b/x)^2 - a^2*d + a*b*c) - (atan(((((2*(a + b/x 
)^(1/2)*(32*a^4*b^2*d^7 + b^6*c^4*d^3 + 6*a*b^5*c^3*d^4 - 64*a^3*b^3*c*d^6 
 + 26*a^2*b^4*c^2*d^5))/(a^2*b^2*c^6 + a^4*c^4*d^2 - 2*a^3*b*c^5*d) + (((4 
*a*b^6*c^9*d^2 + 4*a^2*b^5*c^8*d^3 - 16*a^3*b^4*c^7*d^4 + 8*a^4*b^3*c^6*d^ 
5)/(a^2*b^2*c^8 + a^4*c^6*d^2 - 2*a^3*b*c^7*d) + ((a + b/x)^(1/2)*(4*a*d + 
 b*c)*(4*a^2*b^5*c^9*d^2 - 16*a^3*b^4*c^8*d^3 + 20*a^4*b^3*c^7*d^4 - 8*a^5 
*b^2*c^6*d^5))/(c^3*(a^3)^(1/2)*(a^2*b^2*c^6 + a^4*c^4*d^2 - 2*a^3*b*c^5*d 
)))*(4*a*d + b*c))/(2*c^3*(a^3)^(1/2)))*(4*a*d + b*c)*1i)/(2*c^3*(a^3)^(1/ 
2)) + (((2*(a + b/x)^(1/2)*(32*a^4*b^2*d^7 + b^6*c^4*d^3 + 6*a*b^5*c^3*d^4 
 - 64*a^3*b^3*c*d^6 + 26*a^2*b^4*c^2*d^5))/(a^2*b^2*c^6 + a^4*c^4*d^2 - 2* 
a^3*b*c^5*d) - (((4*a*b^6*c^9*d^2 + 4*a^2*b^5*c^8*d^3 - 16*a^3*b^4*c^7*d^4 
 + 8*a^4*b^3*c^6*d^5)/(a^2*b^2*c^8 + a^4*c^6*d^2 - 2*a^3*b*c^7*d) - ((a + 
b/x)^(1/2)*(4*a*d + b*c)*(4*a^2*b^5*c^9*d^2 - 16*a^3*b^4*c^8*d^3 + 20*a^4* 
b^3*c^7*d^4 - 8*a^5*b^2*c^6*d^5))/(c^3*(a^3)^(1/2)*(a^2*b^2*c^6 + a^4*c^4* 
d^2 - 2*a^3*b*c^5*d)))*(4*a*d + b*c))/(2*c^3*(a^3)^(1/2)))*(4*a*d + b*c)*1 
i)/(2*c^3*(a^3)^(1/2)))/((2*(32*a^3*b^3*d^7 + 5*b^6*c^3*d^4 + 6*a*b^5*c^2* 
d^5 - 48*a^2*b^4*c*d^6))/(a^2*b^2*c^8 + a^4*c^6*d^2 - 2*a^3*b*c^7*d) - ((( 
2*(a + b/x)^(1/2)*(32*a^4*b^2*d^7 + b^6*c^4*d^3 + 6*a*b^5*c^3*d^4 - 64*...
 

Reduce [B] (verification not implemented)

Time = 0.56 (sec) , antiderivative size = 1164, normalized size of antiderivative = 6.77 \[ \int \frac {1}{\sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )^2} \, dx =\text {Too large to display} \] Input:

int(1/(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**3*c*d**2* 
x + 4*sqrt(d)*sqrt(a*d - b*c)*log(sqrt(c)*sqrt(a*x + b) - sqrt(2*sqrt(d)*s 
qrt(a)*sqrt(a*d - b*c) - 2*a*d + b*c) + sqrt(x)*sqrt(c)*sqrt(a))*a**3*d**3 
 - 5*sqrt(d)*sqrt(a*d - b*c)*log(sqrt(c)*sqrt(a*x + b) - sqrt(2*sqrt(d)*sq 
rt(a)*sqrt(a*d - b*c) - 2*a*d + b*c) + sqrt(x)*sqrt(c)*sqrt(a))*a**2*b*c** 
2*d*x - 5*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* 
b*c*d**2 + 4*sqrt(d)*sqrt(a*d - b*c)*log(sqrt(c)*sqrt(a*x + b) + sqrt(2*sq 
rt(d)*sqrt(a)*sqrt(a*d - b*c) - 2*a*d + b*c) + sqrt(x)*sqrt(c)*sqrt(a))*a* 
*3*c*d**2*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**3*d**3 - 5*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*b*c**2*d*x - 5*sqrt(d)*sqrt(a*d - b*c)*log(sqrt(c)*sqrt(a*x + b) + sq 
rt(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) - 2*a*d + b*c) + sqrt(x)*sqrt(c)*sqrt 
(a))*a**2*b*c*d**2 - 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**3*c*d 
**2*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**3*d**3 + 5*sqrt...