\(\int \frac {\sqrt {a+\frac {b}{x}}}{\sqrt {c+\frac {d}{x}}} \, dx\) [47]

Optimal result

Integrand size = 23, antiderivative size = 81 \[ \int \frac {\sqrt {a+\frac {b}{x}}}{\sqrt {c+\frac {d}{x}}} \, dx=\frac {\sqrt {a+\frac {b}{x}} \sqrt {c+\frac {d}{x}} x}{c}+\frac {(b c-a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+\frac {b}{x}}}{\sqrt {a} \sqrt {c+\frac {d}{x}}}\right )}{\sqrt {a} c^{3/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.39 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.40 \[ \int \frac {\sqrt {a+\frac {b}{x}}}{\sqrt {c+\frac {d}{x}}} \, dx=\frac {\sqrt {a+\frac {b}{x}} \sqrt {d+c x} \left (\frac {\sqrt {b+a x} \sqrt {d+c x}}{c}+\frac {(b c-a d) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {d+c x}}{\sqrt {c} \sqrt {b+a x}}\right )}{\sqrt {a} c^{3/2}}\right )}{\sqrt {c+\frac {d}{x}} \sqrt {b+a x}} \] Input:

Integrate[Sqrt[a + b/x]/Sqrt[c + d/x],x]
 

Output:

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

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {899, 105, 104, 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.

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

\(\Big \downarrow \) 899

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

\(\Big \downarrow \) 105

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

\(\Big \downarrow \) 104

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

\(\Big \downarrow \) 221

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

Input:

Int[Sqrt[a + b/x]/Sqrt[c + d/x],x]
 

Output:

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

Defintions of rubi rules used

rule 104

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x 
_)), x_] :> With[{q = Denominator[m]}, Simp[q   Subst[Int[x^(q*(m + 1) - 1) 
/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] 
] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L 
tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
 

rule 105

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[n*((d*e - c*f)/((m + 1)*(b*e - a*f)))   Int[(a 
+ b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, 
e, f, m, p}, x] && EqQ[m + n + p + 2, 0] && GtQ[n, 0] && (SumSimplerQ[m, 1] 
 ||  !SumSimplerQ[p, 1]) && NeQ[m, -1]
 

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. \(154\) vs. \(2(65)=130\).

Time = 0.29 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.91

method	result	size
default	\(-\frac {\sqrt {\frac {a x +b}{x}}\, x \sqrt {\frac {c x +d}{x}}\, \left (\ln \left (\frac {2 a c x +2 \sqrt {\left (a x +b \right ) \left (c x +d \right )}\, \sqrt {a c}+a d +b c}{2 \sqrt {a c}}\right ) a d -\ln \left (\frac {2 a c x +2 \sqrt {\left (a x +b \right ) \left (c x +d \right )}\, \sqrt {a c}+a d +b c}{2 \sqrt {a c}}\right ) b c -2 \sqrt {\left (a x +b \right ) \left (c x +d \right )}\, \sqrt {a c}\right )}{2 \sqrt {\left (a x +b \right ) \left (c x +d \right )}\, c \sqrt {a c}}\)	\(155\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 247, normalized size of antiderivative = 3.05 \[ \int \frac {\sqrt {a+\frac {b}{x}}}{\sqrt {c+\frac {d}{x}}} \, dx=\left [\frac {4 \, a c x \sqrt {\frac {a x + b}{x}} \sqrt {\frac {c x + d}{x}} - \sqrt {a c} {\left (b c - a d\right )} \log \left (-8 \, a^{2} c^{2} x^{2} - b^{2} c^{2} - 6 \, a b c d - a^{2} d^{2} + 4 \, {\left (2 \, a c x^{2} + {\left (b c + a d\right )} x\right )} \sqrt {a c} \sqrt {\frac {a x + b}{x}} \sqrt {\frac {c x + d}{x}} - 8 \, {\left (a b c^{2} + a^{2} c d\right )} x\right )}{4 \, a c^{2}}, \frac {2 \, a c x \sqrt {\frac {a x + b}{x}} \sqrt {\frac {c x + d}{x}} - \sqrt {-a c} {\left (b c - a d\right )} \arctan \left (\frac {2 \, \sqrt {-a c} x \sqrt {\frac {a x + b}{x}} \sqrt {\frac {c x + d}{x}}}{2 \, a c x + b c + a d}\right )}{2 \, a c^{2}}\right ] \] Input:

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

Output:

[1/4*(4*a*c*x*sqrt((a*x + b)/x)*sqrt((c*x + d)/x) - sqrt(a*c)*(b*c - a*d)* 
log(-8*a^2*c^2*x^2 - b^2*c^2 - 6*a*b*c*d - a^2*d^2 + 4*(2*a*c*x^2 + (b*c + 
 a*d)*x)*sqrt(a*c)*sqrt((a*x + b)/x)*sqrt((c*x + d)/x) - 8*(a*b*c^2 + a^2* 
c*d)*x))/(a*c^2), 1/2*(2*a*c*x*sqrt((a*x + b)/x)*sqrt((c*x + d)/x) - sqrt( 
-a*c)*(b*c - a*d)*arctan(2*sqrt(-a*c)*x*sqrt((a*x + b)/x)*sqrt((c*x + d)/x 
)/(2*a*c*x + b*c + a*d)))/(a*c^2)]
 

Sympy [F]

\[ \int \frac {\sqrt {a+\frac {b}{x}}}{\sqrt {c+\frac {d}{x}}} \, dx=\int \frac {\sqrt {a + \frac {b}{x}}}{\sqrt {c + \frac {d}{x}}}\, dx \] Input:

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

Output:

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

Maxima [F]

\[ \int \frac {\sqrt {a+\frac {b}{x}}}{\sqrt {c+\frac {d}{x}}} \, dx=\int { \frac {\sqrt {a + \frac {b}{x}}}{\sqrt {c + \frac {d}{x}}} \,d x } \] Input:

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

Output:

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

Giac [F]

\[ \int \frac {\sqrt {a+\frac {b}{x}}}{\sqrt {c+\frac {d}{x}}} \, dx=\int { \frac {\sqrt {a + \frac {b}{x}}}{\sqrt {c + \frac {d}{x}}} \,d x } \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 6.33 (sec) , antiderivative size = 478, normalized size of antiderivative = 5.90 \[ \int \frac {\sqrt {a+\frac {b}{x}}}{\sqrt {c+\frac {d}{x}}} \, dx=\frac {d\,\left (\sqrt {a+\frac {b}{x}}-\sqrt {a}\right )}{4\,c\,\left (\sqrt {c+\frac {d}{x}}-\sqrt {c}\right )}-\frac {\frac {\left (\sqrt {a+\frac {b}{x}}-\sqrt {a}\right )\,\left (\frac {c\,b^2}{4}+\frac {a\,d\,b}{4}\right )}{\sqrt {a}\,c^{3/2}\,d\,\left (\sqrt {c+\frac {d}{x}}-\sqrt {c}\right )}-\frac {b^2}{4\,c\,d}+\frac {{\left (\sqrt {a+\frac {b}{x}}-\sqrt {a}\right )}^2\,\left (\frac {a^2\,d^2}{4}-\frac {3\,a\,b\,c\,d}{4}+\frac {b^2\,c^2}{4}\right )}{a\,c^2\,d\,{\left (\sqrt {c+\frac {d}{x}}-\sqrt {c}\right )}^2}}{\frac {{\left (\sqrt {a+\frac {b}{x}}-\sqrt {a}\right )}^3}{{\left (\sqrt {c+\frac {d}{x}}-\sqrt {c}\right )}^3}+\frac {b\,\left (\sqrt {a+\frac {b}{x}}-\sqrt {a}\right )}{d\,\left (\sqrt {c+\frac {d}{x}}-\sqrt {c}\right )}-\frac {{\left (\sqrt {a+\frac {b}{x}}-\sqrt {a}\right )}^2\,\left (a\,d+b\,c\right )}{\sqrt {a}\,\sqrt {c}\,d\,{\left (\sqrt {c+\frac {d}{x}}-\sqrt {c}\right )}^2}}+\frac {\ln \left (\frac {\sqrt {a+\frac {b}{x}}-\sqrt {a}}{\sqrt {c+\frac {d}{x}}-\sqrt {c}}\right )\,\left (\sqrt {a}\,b\,c^{3/2}-a^{3/2}\,\sqrt {c}\,d\right )}{2\,a\,c^2}-\frac {\ln \left (\frac {\left (\sqrt {c}\,\sqrt {a+\frac {b}{x}}-\sqrt {a}\,\sqrt {c+\frac {d}{x}}\right )\,\left (b\,\sqrt {c}-\frac {\sqrt {a}\,d\,\left (\sqrt {a+\frac {b}{x}}-\sqrt {a}\right )}{\sqrt {c+\frac {d}{x}}-\sqrt {c}}\right )}{\sqrt {c+\frac {d}{x}}-\sqrt {c}}\right )\,\left (\sqrt {a}\,b\,c^{3/2}-a^{3/2}\,\sqrt {c}\,d\right )}{2\,a\,c^2} \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.26 \[ \int \frac {\sqrt {a+\frac {b}{x}}}{\sqrt {c+\frac {d}{x}}} \, dx=\frac {\sqrt {c x +d}\, \sqrt {a x +b}\, a c -\sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {c}\, \sqrt {a x +b}+\sqrt {a}\, \sqrt {c x +d}}{\sqrt {a d -b c}}\right ) a d +\sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {c}\, \sqrt {a x +b}+\sqrt {a}\, \sqrt {c x +d}}{\sqrt {a d -b c}}\right ) b c}{a \,c^{2}} \] Input:

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

Output:

(sqrt(c*x + d)*sqrt(a*x + b)*a*c - sqrt(c)*sqrt(a)*log((sqrt(c)*sqrt(a*x + 
 b) + sqrt(a)*sqrt(c*x + d))/sqrt(a*d - b*c))*a*d + sqrt(c)*sqrt(a)*log((s 
qrt(c)*sqrt(a*x + b) + sqrt(a)*sqrt(c*x + d))/sqrt(a*d - b*c))*b*c)/(a*c** 
2)