\(\int \frac {\sqrt {a+\frac {b}{x^2}}}{(c+\frac {d}{x^2})^{3/2}} \, dx\) [53]

Optimal result

Integrand size = 23, antiderivative size = 234 \[ \int \frac {\sqrt {a+\frac {b}{x^2}}}{\left (c+\frac {d}{x^2}\right )^{3/2}} \, dx=-\frac {\sqrt {a+\frac {b}{x^2}} x}{c \sqrt {c+\frac {d}{x^2}}}+\frac {2 a \sqrt {c+\frac {d}{x^2}} x}{c^2 \sqrt {a+\frac {b}{x^2}}}+\frac {2 \sqrt {a} \sqrt {b} \sqrt {c+\frac {d}{x^2}} E\left (\cot ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )|1-\frac {a d}{b c}\right )}{c^2 \sqrt {a+\frac {b}{x^2}} \sqrt {\frac {a \left (c+\frac {d}{x^2}\right )}{c \left (a+\frac {b}{x^2}\right )}}}-\frac {\sqrt {a} \sqrt {b} \sqrt {c+\frac {d}{x^2}} \operatorname {EllipticF}\left (\cot ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right ),1-\frac {a d}{b c}\right )}{c^2 \sqrt {a+\frac {b}{x^2}} \sqrt {\frac {a \left (c+\frac {d}{x^2}\right )}{c \left (a+\frac {b}{x^2}\right )}}} \] Output:

-(a+b/x^2)^(1/2)*x/c/(c+d/x^2)^(1/2)+2*a*(c+d/x^2)^(1/2)*x/c^2/(a+b/x^2)^( 
1/2)+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))/c^2/(a+b/x^2)^(1/2)/(a*(c+d/x^2)/c/(a+b/x^2))^(1/2)-a^(1/2)* 
b^(1/2)*(c+d/x^2)^(1/2)*InverseJacobiAM(arccot(a^(1/2)*x/b^(1/2)),(1-a*d/b 
/c)^(1/2))/c^2/(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 = 3.46 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.82 \[ \int \frac {\sqrt {a+\frac {b}{x^2}}}{\left (c+\frac {d}{x^2}\right )^{3/2}} \, dx=-\frac {\sqrt {a+\frac {b}{x^2}} \left (\sqrt {\frac {a}{b}} c x \left (b+a x^2\right )+2 i a d \sqrt {1+\frac {a x^2}{b}} \sqrt {1+\frac {c x^2}{d}} E\left (i \text {arcsinh}\left (\sqrt {\frac {a}{b}} x\right )|\frac {b c}{a d}\right )+i (b c-2 a d) \sqrt {1+\frac {a x^2}{b}} \sqrt {1+\frac {c x^2}{d}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {a}{b}} x\right ),\frac {b c}{a d}\right )\right )}{\sqrt {\frac {a}{b}} c^2 \sqrt {c+\frac {d}{x^2}} \left (b+a x^2\right )} \] Input:

Integrate[Sqrt[a + b/x^2]/(c + d/x^2)^(3/2),x]
 

Output:

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

Rubi [A] (verified)

Time = 0.74 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.20, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {899, 371, 25, 445, 25, 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.

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

\(\Big \downarrow \) 899

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

\(\Big \downarrow \) 371

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 445

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 406

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

\(\Big \downarrow \) 320

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

\(\Big \downarrow \) 388

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

\(\Big \downarrow \) 313

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

Input:

Int[Sqrt[a + b/x^2]/(c + d/x^2)^(3/2),x]
 

Output:

-((Sqrt[a + b/x^2]*x)/(c*Sqrt[c + d/x^2])) - ((-2*Sqrt[a + b/x^2]*Sqrt[c + 
 d/x^2]*x)/c + (b*(2*d*(Sqrt[a + b/x^2]/(b*Sqrt[c + d/x^2]*x) - (Sqrt[c]*S 
qrt[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))]*Sqrt[c + d/x^2])))/c)/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 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 371

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*2*(p + 1))), x] + Simp[1/(a*2*(p + 1))   Int[(e*x)^m*(a + b*x^2)^(p + 1) 
*(c + d*x^2)^(q - 1)*Simp[c*(m + 2*(p + 1) + 1) + d*(m + 2*(p + q + 1) + 1) 
*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && Lt 
Q[p, -1] && LtQ[0, q, 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 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 445

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

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 = 5.30 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.79

method	result	size
default	\(-\frac {\sqrt {\frac {a \,x^{2}+b}{x^{2}}}\, \left (\sqrt {-\frac {c}{d}}\, a \,x^{3}+b \sqrt {\frac {c \,x^{2}+d}{d}}\, \sqrt {\frac {a \,x^{2}+b}{b}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {c}{d}}, \sqrt {\frac {a d}{b c}}\right )-2 b \sqrt {\frac {c \,x^{2}+d}{d}}\, \sqrt {\frac {a \,x^{2}+b}{b}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {c}{d}}, \sqrt {\frac {a d}{b c}}\right )+\sqrt {-\frac {c}{d}}\, b x \right ) \left (c \,x^{2}+d \right )}{x^{2} \left (a \,x^{2}+b \right ) \sqrt {-\frac {c}{d}}\, c \left (\frac {c \,x^{2}+d}{x^{2}}\right )^{\frac {3}{2}}}\)	\(185\)

Input:

int((a+b/x^2)^(1/2)/(c+1/x^2*d)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.70 \[ \int \frac {\sqrt {a+\frac {b}{x^2}}}{\left (c+\frac {d}{x^2}\right )^{3/2}} \, dx=-\frac {2 \, {\left (b c x^{2} + b d\right )} \sqrt {a c} \sqrt {-\frac {b}{a}} E(\arcsin \left (\frac {\sqrt {-\frac {b}{a}}}{x}\right )\,|\,\frac {a d}{b c}) - {\left ({\left (a + 2 \, b\right )} c x^{2} + {\left (a + 2 \, b\right )} d\right )} \sqrt {a c} \sqrt {-\frac {b}{a}} F(\arcsin \left (\frac {\sqrt {-\frac {b}{a}}}{x}\right )\,|\,\frac {a d}{b c}) - {\left (a c x^{3} + 2 \, a d x\right )} \sqrt {\frac {a x^{2} + b}{x^{2}}} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{a c^{3} x^{2} + a c^{2} d} \] Input:

integrate((a+b/x^2)^(1/2)/(c+d/x^2)^(3/2),x, algorithm="fricas")
 

Output:

-(2*(b*c*x^2 + b*d)*sqrt(a*c)*sqrt(-b/a)*elliptic_e(arcsin(sqrt(-b/a)/x), 
a*d/(b*c)) - ((a + 2*b)*c*x^2 + (a + 2*b)*d)*sqrt(a*c)*sqrt(-b/a)*elliptic 
_f(arcsin(sqrt(-b/a)/x), a*d/(b*c)) - (a*c*x^3 + 2*a*d*x)*sqrt((a*x^2 + b) 
/x^2)*sqrt((c*x^2 + d)/x^2))/(a*c^3*x^2 + a*c^2*d)
 

Sympy [F]

\[ \int \frac {\sqrt {a+\frac {b}{x^2}}}{\left (c+\frac {d}{x^2}\right )^{3/2}} \, dx=\int \frac {\sqrt {a + \frac {b}{x^{2}}}}{\left (c + \frac {d}{x^{2}}\right )^{\frac {3}{2}}}\, dx \] Input:

integrate((a+b/x**2)**(1/2)/(c+d/x**2)**(3/2),x)
 

Output:

Integral(sqrt(a + b/x**2)/(c + d/x**2)**(3/2), x)
 

Maxima [F]

\[ \int \frac {\sqrt {a+\frac {b}{x^2}}}{\left (c+\frac {d}{x^2}\right )^{3/2}} \, dx=\int { \frac {\sqrt {a + \frac {b}{x^{2}}}}{{\left (c + \frac {d}{x^{2}}\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate((a+b/x^2)^(1/2)/(c+d/x^2)^(3/2),x, algorithm="maxima")
 

Output:

integrate(sqrt(a + b/x^2)/(c + d/x^2)^(3/2), x)
 

Giac [F]

\[ \int \frac {\sqrt {a+\frac {b}{x^2}}}{\left (c+\frac {d}{x^2}\right )^{3/2}} \, dx=\int { \frac {\sqrt {a + \frac {b}{x^{2}}}}{{\left (c + \frac {d}{x^{2}}\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate((a+b/x^2)^(1/2)/(c+d/x^2)^(3/2),x, algorithm="giac")
 

Output:

integrate(sqrt(a + b/x^2)/(c + d/x^2)^(3/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+\frac {b}{x^2}}}{\left (c+\frac {d}{x^2}\right )^{3/2}} \, dx=\int \frac {\sqrt {a+\frac {b}{x^2}}}{{\left (c+\frac {d}{x^2}\right )}^{3/2}} \,d x \] Input:

int((a + b/x^2)^(1/2)/(c + d/x^2)^(3/2),x)
 

Output:

int((a + b/x^2)^(1/2)/(c + d/x^2)^(3/2), x)
 

Reduce [F]

\[ \int \frac {\sqrt {a+\frac {b}{x^2}}}{\left (c+\frac {d}{x^2}\right )^{3/2}} \, dx=\frac {\sqrt {c \,x^{2}+d}\, \sqrt {a \,x^{2}+b}\, b x +2 \left (\int \frac {\sqrt {c \,x^{2}+d}\, \sqrt {a \,x^{2}+b}\, x^{4}}{a \,c^{2} x^{6}+2 a c d \,x^{4}+b \,c^{2} x^{4}+a \,d^{2} x^{2}+2 b c d \,x^{2}+b \,d^{2}}d x \right ) a^{2} c d \,x^{2}+2 \left (\int \frac {\sqrt {c \,x^{2}+d}\, \sqrt {a \,x^{2}+b}\, x^{4}}{a \,c^{2} x^{6}+2 a c d \,x^{4}+b \,c^{2} x^{4}+a \,d^{2} x^{2}+2 b c d \,x^{2}+b \,d^{2}}d x \right ) a^{2} d^{2}-\left (\int \frac {\sqrt {c \,x^{2}+d}\, \sqrt {a \,x^{2}+b}\, x^{4}}{a \,c^{2} x^{6}+2 a c d \,x^{4}+b \,c^{2} x^{4}+a \,d^{2} x^{2}+2 b c d \,x^{2}+b \,d^{2}}d x \right ) a b \,c^{2} x^{2}-\left (\int \frac {\sqrt {c \,x^{2}+d}\, \sqrt {a \,x^{2}+b}\, x^{4}}{a \,c^{2} x^{6}+2 a c d \,x^{4}+b \,c^{2} x^{4}+a \,d^{2} x^{2}+2 b c d \,x^{2}+b \,d^{2}}d x \right ) a b c d -\left (\int \frac {\sqrt {c \,x^{2}+d}\, \sqrt {a \,x^{2}+b}}{a \,c^{2} x^{6}+2 a c d \,x^{4}+b \,c^{2} x^{4}+a \,d^{2} x^{2}+2 b c d \,x^{2}+b \,d^{2}}d x \right ) b^{2} c d \,x^{2}-\left (\int \frac {\sqrt {c \,x^{2}+d}\, \sqrt {a \,x^{2}+b}}{a \,c^{2} x^{6}+2 a c d \,x^{4}+b \,c^{2} x^{4}+a \,d^{2} x^{2}+2 b c d \,x^{2}+b \,d^{2}}d x \right ) b^{2} d^{2}}{2 a d \left (c \,x^{2}+d \right )} \] Input:

int((a+b/x^2)^(1/2)/(c+d/x^2)^(3/2),x)
 

Output:

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