\(\int \frac {\sqrt {a+\frac {b}{x^2}}}{\sqrt {c+\frac {d}{x^2}}} \, dx\) [52]

Optimal result

Integrand size = 23, antiderivative size = 198 \[ \int \frac {\sqrt {a+\frac {b}{x^2}}}{\sqrt {c+\frac {d}{x^2}}} \, dx=\frac {\sqrt {a+\frac {b}{x^2}} x}{\sqrt {c+\frac {d}{x^2}}}+\frac {\sqrt {d} \sqrt {a+\frac {b}{x^2}} E\left (\cot ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {d}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {\frac {c \left (a+\frac {b}{x^2}\right )}{a \left (c+\frac {d}{x^2}\right )}} \sqrt {c+\frac {d}{x^2}}}-\frac {b \sqrt {c} \sqrt {a+\frac {b}{x^2}} \operatorname {EllipticF}\left (\cot ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {d}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {\frac {c \left (a+\frac {b}{x^2}\right )}{a \left (c+\frac {d}{x^2}\right )}} \sqrt {c+\frac {d}{x^2}}} \] Output:

(a+b/x^2)^(1/2)*x/(c+d/x^2)^(1/2)+d^(1/2)*(a+b/x^2)^(1/2)*EllipticE(1/(1+c 
*x^2/d)^(1/2),(1-b*c/a/d)^(1/2))/c^(1/2)/(c*(a+b/x^2)/a/(c+d/x^2))^(1/2)/( 
c+d/x^2)^(1/2)-b*c^(1/2)*(a+b/x^2)^(1/2)*InverseJacobiAM(arccot(c^(1/2)*x/ 
d^(1/2)),(1-b*c/a/d)^(1/2))/a/d^(1/2)/(c*(a+b/x^2)/a/(c+d/x^2))^(1/2)/(c+d 
/x^2)^(1/2)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 2.14 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.43 \[ \int \frac {\sqrt {a+\frac {b}{x^2}}}{\sqrt {c+\frac {d}{x^2}}} \, dx=\frac {\sqrt {a+\frac {b}{x^2}} \sqrt {\frac {d+c x^2}{d}} E\left (\arcsin \left (\sqrt {-\frac {c}{d}} x\right )|\frac {a d}{b c}\right )}{\sqrt {-\frac {c}{d}} \sqrt {c+\frac {d}{x^2}} \sqrt {\frac {b+a x^2}{b}}} \] Input:

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

Output:

(Sqrt[a + b/x^2]*Sqrt[(d + c*x^2)/d]*EllipticE[ArcSin[Sqrt[-(c/d)]*x], (a* 
d)/(b*c)])/(Sqrt[-(c/d)]*Sqrt[c + d/x^2]*Sqrt[(b + a*x^2)/b])
 

Rubi [A] (verified)

Time = 0.61 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.24, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {899, 377, 27, 324, 320, 388, 313}

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.

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

\(\Big \downarrow \) 899

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

\(\Big \downarrow \) 377

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 324

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

\(\Big \downarrow \) 320

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

\(\Big \downarrow \) 388

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

\(\Big \downarrow \) 313

\(\displaystyle \frac {x \sqrt {a+\frac {b}{x^2}} \sqrt {c+\frac {d}{x^2}}}{c}-\frac {b \left (\frac {c^{3/2} \sqrt {a+\frac {b}{x^2}} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d}}{\sqrt {c} x}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+\frac {d}{x^2}} \sqrt {\frac {c \left (a+\frac {b}{x^2}\right )}{a \left (c+\frac {d}{x^2}\right )}}}+d \left (\frac {\sqrt {a+\frac {b}{x^2}}}{b x \sqrt {c+\frac {d}{x^2}}}-\frac {\sqrt {c} \sqrt {a+\frac {b}{x^2}} E\left (\arctan \left (\frac {\sqrt {d}}{\sqrt {c} x}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+\frac {d}{x^2}} \sqrt {\frac {c \left (a+\frac {b}{x^2}\right )}{a \left (c+\frac {d}{x^2}\right )}}}\right )\right )}{c}\)

Input:

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

Output:

(Sqrt[a + b/x^2]*Sqrt[c + d/x^2]*x)/c - (b*(d*(Sqrt[a + b/x^2]/(b*Sqrt[c + 
 d/x^2]*x) - (Sqrt[c]*Sqrt[a + b/x^2]*EllipticE[ArcTan[Sqrt[d]/(Sqrt[c]*x) 
], 1 - (b*c)/(a*d)])/(b*Sqrt[d]*Sqrt[(c*(a + b/x^2))/(a*(c + d/x^2))]*Sqrt 
[c + d/x^2])) + (c^(3/2)*Sqrt[a + b/x^2]*EllipticF[ArcTan[Sqrt[d]/(Sqrt[c] 
*x)], 1 - (b*c)/(a*d)])/(a*Sqrt[d]*Sqrt[(c*(a + b/x^2))/(a*(c + d/x^2))]*S 
qrt[c + d/x^2])))/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 313

Int[Sqrt[(a_) + (b_.)*(x_)^2]/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Sim 
p[(Sqrt[a + b*x^2]/(c*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*(c 
+ d*x^2)))]))*EllipticE[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; FreeQ 
[{a, b, c, d}, x] && PosQ[b/a] && PosQ[d/c]
 

rule 320

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[(Sqrt[a + b*x^2]/(a*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*( 
c + d*x^2)))]))*EllipticF[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; Fre 
eQ[{a, b, c, d}, x] && PosQ[d/c] && PosQ[b/a] &&  !SimplerSqrtQ[b/a, d/c]
 

rule 324

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
a   Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] + Simp[b   Int[x^2/(Sqr 
t[a + b*x^2]*Sqrt[c + d*x^2]), x], x] /; FreeQ[{a, b, c, d}, x] && PosQ[d/c 
] && PosQ[b/a]
 

rule 377

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_) 
, x_Symbol] :> Simp[(e*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(a*e*( 
m + 1))), x] - Simp[1/(a*e^2*(m + 1))   Int[(e*x)^(m + 2)*(a + b*x^2)^p*(c 
+ d*x^2)^(q - 1)*Simp[b*c*(m + 1) + 2*(b*c*(p + 1) + a*d*q) + d*(b*(m + 1) 
+ 2*b*(p + q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b 
*c - a*d, 0] && LtQ[0, q, 1] && LtQ[m, -1] && IntBinomialQ[a, b, c, d, e, m 
, 2, p, q, x]
 

rule 388

Int[(x_)^2/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] 
 :> Simp[x*(Sqrt[a + b*x^2]/(b*Sqrt[c + d*x^2])), x] - Simp[c/b   Int[Sqrt[ 
a + b*x^2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - 
 a*d, 0] && PosQ[b/a] && PosQ[d/c] &&  !SimplerSqrtQ[b/a, d/c]
 

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 [A] (verified)

Time = 1.22 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.47

method	result	size
default	\(\frac {\sqrt {\frac {a \,x^{2}+b}{x^{2}}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {c}{d}}, \sqrt {\frac {a d}{b c}}\right ) \sqrt {\frac {a \,x^{2}+b}{b}}\, \sqrt {\frac {c \,x^{2}+d}{d}}\, b}{\left (a \,x^{2}+b \right ) \sqrt {-\frac {c}{d}}\, \sqrt {\frac {c \,x^{2}+d}{x^{2}}}}\)	\(94\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.59 \[ \int \frac {\sqrt {a+\frac {b}{x^2}}}{\sqrt {c+\frac {d}{x^2}}} \, dx=\frac {a x \sqrt {\frac {a x^{2} + b}{x^{2}}} \sqrt {\frac {c x^{2} + d}{x^{2}}} - \sqrt {a c} b \sqrt {-\frac {b}{a}} E(\arcsin \left (\frac {\sqrt {-\frac {b}{a}}}{x}\right )\,|\,\frac {a d}{b c}) + \sqrt {a c} {\left (a + b\right )} \sqrt {-\frac {b}{a}} F(\arcsin \left (\frac {\sqrt {-\frac {b}{a}}}{x}\right )\,|\,\frac {a d}{b c})}{a c} \] Input:

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

Output:

(a*x*sqrt((a*x^2 + b)/x^2)*sqrt((c*x^2 + d)/x^2) - sqrt(a*c)*b*sqrt(-b/a)* 
elliptic_e(arcsin(sqrt(-b/a)/x), a*d/(b*c)) + sqrt(a*c)*(a + b)*sqrt(-b/a) 
*elliptic_f(arcsin(sqrt(-b/a)/x), a*d/(b*c)))/(a*c)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+\frac {b}{x^2}}}{\sqrt {c+\frac {d}{x^2}}} \, dx=\int \frac {\sqrt {a+\frac {b}{x^2}}}{\sqrt {c+\frac {d}{x^2}}} \,d x \] Input:

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

Output:

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

Reduce [F]

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

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

Output:

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