\(\int \sqrt {a+\frac {b}{x^2}} \sqrt {c+\frac {d}{x^2}} \, dx\) [51]

Optimal result

Integrand size = 23, antiderivative size = 233 \[ \int \sqrt {a+\frac {b}{x^2}} \sqrt {c+\frac {d}{x^2}} \, dx=-\frac {2 b \sqrt {c+\frac {d}{x^2}}}{\sqrt {a+\frac {b}{x^2}} x}+\sqrt {a+\frac {b}{x^2}} \sqrt {c+\frac {d}{x^2}} x+\frac {2 \sqrt {a} \sqrt {b} \sqrt {c+\frac {d}{x^2}} E\left (\cot ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )|1-\frac {a d}{b c}\right )}{\sqrt {a+\frac {b}{x^2}} \sqrt {\frac {a \left (c+\frac {d}{x^2}\right )}{c \left (a+\frac {b}{x^2}\right )}}}-\frac {\sqrt {a} (b c+a d) \sqrt {c+\frac {d}{x^2}} \operatorname {EllipticF}\left (\cot ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right ),1-\frac {a d}{b c}\right )}{\sqrt {b} c \sqrt {a+\frac {b}{x^2}} \sqrt {\frac {a \left (c+\frac {d}{x^2}\right )}{c \left (a+\frac {b}{x^2}\right )}}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 2.50 (sec) , antiderivative size = 205, normalized size of antiderivative = 0.88 \[ \int \sqrt {a+\frac {b}{x^2}} \sqrt {c+\frac {d}{x^2}} \, dx=-\frac {\sqrt {a+\frac {b}{x^2}} \sqrt {c+\frac {d}{x^2}} x \left (\sqrt {\frac {a}{b}} \left (b+a x^2\right ) \left (d+c x^2\right )+2 i a d x \sqrt {1+\frac {a x^2}{b}} \sqrt {1+\frac {c x^2}{d}} E\left (i \text {arcsinh}\left (\sqrt {\frac {a}{b}} x\right )|\frac {b c}{a d}\right )+i (b c-a d) x \sqrt {1+\frac {a x^2}{b}} \sqrt {1+\frac {c x^2}{d}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {a}{b}} x\right ),\frac {b c}{a d}\right )\right )}{\sqrt {\frac {a}{b}} \left (b+a x^2\right ) \left (d+c x^2\right )} \] Input:

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

Output:

-((Sqrt[a + b/x^2]*Sqrt[c + d/x^2]*x*(Sqrt[a/b]*(b + a*x^2)*(d + c*x^2) + 
(2*I)*a*d*x*Sqrt[1 + (a*x^2)/b]*Sqrt[1 + (c*x^2)/d]*EllipticE[I*ArcSinh[Sq 
rt[a/b]*x], (b*c)/(a*d)] + I*(b*c - a*d)*x*Sqrt[1 + (a*x^2)/b]*Sqrt[1 + (c 
*x^2)/d]*EllipticF[I*ArcSinh[Sqrt[a/b]*x], (b*c)/(a*d)]))/(Sqrt[a/b]*(b + 
a*x^2)*(d + c*x^2)))
 

Rubi [A] (verified)

Time = 0.65 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.06, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {899, 375, 27, 406, 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.

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

\(\Big \downarrow \) 899

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

\(\Big \downarrow \) 375

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 406

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

\(\Big \downarrow \) 320

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

\(\Big \downarrow \) 388

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

\(\Big \downarrow \) 313

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

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

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 375

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*((c + d*x^2)^q/(e*(m + 1))) 
, x] - Simp[2/(e^2*(m + 1))   Int[(e*x)^(m + 2)*(a + b*x^2)^(p - 1)*(c + d* 
x^2)^(q - 1)*Simp[b*c*p + a*d*q + b*d*(p + q)*x^2, x], x], x] /; FreeQ[{a, 
b, c, d, e}, x] && NeQ[b*c - a*d, 0] && GtQ[q, 0] && LtQ[m, -1] && GtQ[p, 0 
] && 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 406

Int[((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*( 
x_)^2), x_Symbol] :> Simp[e   Int[(a + b*x^2)^p*(c + d*x^2)^q, x], x] + Sim 
p[f   Int[x^2*(a + b*x^2)^p*(c + d*x^2)^q, x], x] /; FreeQ[{a, b, c, d, e, 
f, p, q}, x]
 

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 = 3.44 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.19

method	result	size
default	\(\frac {\sqrt {\frac {a \,x^{2}+b}{x^{2}}}\, x \sqrt {\frac {c \,x^{2}+d}{x^{2}}}\, \left (-\sqrt {-\frac {c}{d}}\, a c \,x^{4}+\sqrt {\frac {c \,x^{2}+d}{d}}\, \sqrt {\frac {a \,x^{2}+b}{b}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {c}{d}}, \sqrt {\frac {a d}{b c}}\right ) a d x -c b \sqrt {\frac {c \,x^{2}+d}{d}}\, \sqrt {\frac {a \,x^{2}+b}{b}}\, x \operatorname {EllipticF}\left (x \sqrt {-\frac {c}{d}}, \sqrt {\frac {a d}{b c}}\right )+2 c b \sqrt {\frac {c \,x^{2}+d}{d}}\, \sqrt {\frac {a \,x^{2}+b}{b}}\, x \operatorname {EllipticE}\left (x \sqrt {-\frac {c}{d}}, \sqrt {\frac {a d}{b c}}\right )-\sqrt {-\frac {c}{d}}\, a d \,x^{2}-\sqrt {-\frac {c}{d}}\, b c \,x^{2}-\sqrt {-\frac {c}{d}}\, b d \right )}{\left (a \,x^{4} c +a d \,x^{2}+x^{2} b c +b d \right ) \sqrt {-\frac {c}{d}}}\)	\(277\)
risch	\(-x \sqrt {\frac {a \,x^{2}+b}{x^{2}}}\, \sqrt {\frac {c \,x^{2}+d}{x^{2}}}+\frac {\left (\frac {a d \sqrt {1+\frac {c \,x^{2}}{d}}\, \sqrt {1+\frac {a \,x^{2}}{b}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {c}{d}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {c}{d}}\, \sqrt {a \,x^{4} c +a d \,x^{2}+x^{2} b c +b d}}+\frac {b c \sqrt {1+\frac {c \,x^{2}}{d}}\, \sqrt {1+\frac {a \,x^{2}}{b}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {c}{d}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {c}{d}}\, \sqrt {a \,x^{4} c +a d \,x^{2}+x^{2} b c +b d}}-\frac {2 c b \sqrt {1+\frac {c \,x^{2}}{d}}\, \sqrt {1+\frac {a \,x^{2}}{b}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {c}{d}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {c}{d}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {c}{d}}\, \sqrt {a \,x^{4} c +a d \,x^{2}+x^{2} b c +b d}}\right ) \sqrt {\frac {a \,x^{2}+b}{x^{2}}}\, x^{2} \sqrt {\frac {c \,x^{2}+d}{x^{2}}}\, \sqrt {\left (a \,x^{2}+b \right ) \left (c \,x^{2}+d \right )}}{\left (a \,x^{2}+b \right ) \left (c \,x^{2}+d \right )}\)	\(394\)

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)*x*((c*x^2+d)/x^2)^(1/2)*(-(-c/d)^(1/2)*a*c*x^4+((c*x 
^2+d)/d)^(1/2)*((a*x^2+b)/b)^(1/2)*EllipticF(x*(-c/d)^(1/2),(a*d/b/c)^(1/2 
))*a*d*x-c*b*((c*x^2+d)/d)^(1/2)*((a*x^2+b)/b)^(1/2)*x*EllipticF(x*(-c/d)^ 
(1/2),(a*d/b/c)^(1/2))+2*c*b*((c*x^2+d)/d)^(1/2)*((a*x^2+b)/b)^(1/2)*x*Ell 
ipticE(x*(-c/d)^(1/2),(a*d/b/c)^(1/2))-(-c/d)^(1/2)*a*d*x^2-(-c/d)^(1/2)*b 
*c*x^2-(-c/d)^(1/2)*b*d)/(a*c*x^4+a*d*x^2+b*c*x^2+b*d)/(-c/d)^(1/2)
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

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

\[ \int \sqrt {a+\frac {b}{x^2}} \sqrt {c+\frac {d}{x^2}} \, dx=\frac {\sqrt {c \,x^{2}+d}\, \sqrt {a \,x^{2}+b}+2 \left (\int \frac {\sqrt {c \,x^{2}+d}\, \sqrt {a \,x^{2}+b}}{a c \,x^{6}+a d \,x^{4}+b c \,x^{4}+b d \,x^{2}}d x \right ) b d x +\left (\int \frac {\sqrt {c \,x^{2}+d}\, \sqrt {a \,x^{2}+b}}{a c \,x^{4}+a d \,x^{2}+b c \,x^{2}+b d}d x \right ) a d x +\left (\int \frac {\sqrt {c \,x^{2}+d}\, \sqrt {a \,x^{2}+b}}{a c \,x^{4}+a d \,x^{2}+b c \,x^{2}+b d}d x \right ) b c x}{x} \] Input:

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

Output:

(sqrt(c*x**2 + d)*sqrt(a*x**2 + b) + 2*int((sqrt(c*x**2 + d)*sqrt(a*x**2 + 
 b))/(a*c*x**6 + a*d*x**4 + b*c*x**4 + b*d*x**2),x)*b*d*x + int((sqrt(c*x* 
*2 + d)*sqrt(a*x**2 + b))/(a*c*x**4 + a*d*x**2 + b*c*x**2 + b*d),x)*a*d*x 
+ int((sqrt(c*x**2 + d)*sqrt(a*x**2 + b))/(a*c*x**4 + a*d*x**2 + b*c*x**2 
+ b*d),x)*b*c*x)/x