3.4.53 \(\int \frac {1}{x^2 \sqrt {a+\frac {b}{c+d x^2}}} \, dx\) [353]

3.4.53.1 Optimal result

Integrand size = 21, antiderivative size = 343 \[ \int \frac {1}{x^2 \sqrt {a+\frac {b}{c+d x^2}}} \, dx=-\frac {b+a c+a d x^2}{(b+a c) x \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}+\frac {d x \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}-\frac {\sqrt {c} \sqrt {d} \left (b+a c+a d x^2\right ) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{(b+a c) \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}} \sqrt {\frac {c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}}}+\frac {\sqrt {c} \sqrt {d} \left (b+a c+a d x^2\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),\frac {b}{b+a c}\right )}{(b+a c) \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}} \sqrt {\frac {c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}}} \]

output

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

3.4.53.2 Mathematica [A] (verified)

Time = 8.74 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.44 \[ \int \frac {1}{x^2 \sqrt {a+\frac {b}{c+d x^2}}} \, dx=-\frac {\left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{(b+a c) x}+\frac {d \sqrt {\frac {c+d x^2}{c}} \sqrt {\frac {b+a c+a d x^2}{c+d x^2}} E\left (\arcsin \left (\sqrt {-\frac {d}{c}} x\right )|\frac {a c}{b+a c}\right )}{(b+a c) \sqrt {-\frac {d}{c}} \sqrt {\frac {b+a c+a d x^2}{b+a c}}} \]

input

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

output

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

3.4.53.3 Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 329, normalized size of antiderivative = 0.96, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {2057, 2058, 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.

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

\(\Big \downarrow \) 2057

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

\(\Big \downarrow \) 2058

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

\(\Big \downarrow \) 377

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 324

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

\(\Big \downarrow \) 320

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

\(\Big \downarrow \) 388

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

\(\Big \downarrow \) 313

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

input

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

output

(Sqrt[b + a*c + a*d*x^2]*(-((Sqrt[c + d*x^2]*Sqrt[b + a*c + a*d*x^2])/((b 
+ a*c)*x)) + (d*(a*d*((x*Sqrt[b + a*c + a*d*x^2])/(a*d*Sqrt[c + d*x^2]) - 
(Sqrt[c]*Sqrt[b + a*c + a*d*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], b/ 
(b + a*c)])/(a*d^(3/2)*Sqrt[c + d*x^2]*Sqrt[(c*(b + a*c + a*d*x^2))/((b + 
a*c)*(c + d*x^2))])) + (Sqrt[c]*Sqrt[b + a*c + a*d*x^2]*EllipticF[ArcTan[( 
Sqrt[d]*x)/Sqrt[c]], b/(b + a*c)])/(Sqrt[d]*Sqrt[c + d*x^2]*Sqrt[(c*(b + a 
*c + a*d*x^2))/((b + a*c)*(c + d*x^2))])))/(b + a*c)))/(Sqrt[c + d*x^2]*Sq 
rt[(b + a*c + a*d*x^2)/(c + d*x^2)])
 

3.4.53.3.1 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 2057

Int[(u_.)*((a_) + (b_.)/((c_) + (d_.)*(x_)^(n_)))^(p_), x_Symbol] :> Int[u* 
((b + a*c + a*d*x^n)/(c + d*x^n))^p, x] /; FreeQ[{a, b, c, d, n, p}, x]
 

rule 2058

Int[(u_.)*((e_.)*((a_.) + (b_.)*(x_)^(n_.))^(q_.)*((c_) + (d_.)*(x_)^(n_))^ 
(r_.))^(p_), x_Symbol] :> Simp[Simp[(e*(a + b*x^n)^q*(c + d*x^n)^r)^p/((a + 
 b*x^n)^(p*q)*(c + d*x^n)^(p*r))]   Int[u*(a + b*x^n)^(p*q)*(c + d*x^n)^(p* 
r), x], x] /; FreeQ[{a, b, c, d, e, n, p, q, r}, x]
 

3.4.53.4 Maple [A] (verified)

Time = 6.06 (sec) , antiderivative size = 345, normalized size of antiderivative = 1.01

method	result	size
default	\(-\frac {\left (\sqrt {-\frac {a d}{a c +b}}\, a \,d^{2} x^{4}-a d c \sqrt {\frac {a d \,x^{2}+a c +b}{a c +b}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, x E\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {\frac {a c +b}{a c}}\right )+2 \sqrt {-\frac {a d}{a c +b}}\, a c d \,x^{2}-\sqrt {\frac {a d \,x^{2}+a c +b}{a c +b}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, F\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {\frac {a c +b}{a c}}\right ) b d x +\sqrt {-\frac {a d}{a c +b}}\, b d \,x^{2}+\sqrt {-\frac {a d}{a c +b}}\, a \,c^{2}+\sqrt {-\frac {a d}{a c +b}}\, b c \right ) \left (d \,x^{2}+c \right ) \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}}{\sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \sqrt {-\frac {a d}{a c +b}}\, x \left (a c +b \right ) \sqrt {\left (a d \,x^{2}+a c +b \right ) \left (d \,x^{2}+c \right )}}\)	\(345\)
risch	\(-\frac {a d \,x^{2}+a c +b}{\left (a c +b \right ) x \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}}+\frac {d \left (\frac {b \sqrt {1+\frac {a d \,x^{2}}{a c +b}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {-1+\frac {2 a c d +b d}{d c a}}\right )}{\sqrt {-\frac {a d}{a c +b}}\, \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}}+\frac {a c \sqrt {1+\frac {a d \,x^{2}}{a c +b}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {-1+\frac {2 a c d +b d}{d c a}}\right )}{\sqrt {-\frac {a d}{a c +b}}\, \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}}-\frac {2 a d \left (a \,c^{2}+b c \right ) \sqrt {1+\frac {a d \,x^{2}}{a c +b}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (F\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {-1+\frac {2 a c d +b d}{d c a}}\right )-E\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {-1+\frac {2 a c d +b d}{d c a}}\right )\right )}{\sqrt {-\frac {a d}{a c +b}}\, \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \left (2 a c d +2 b d \right )}\right ) \sqrt {\left (a d \,x^{2}+a c +b \right ) \left (d \,x^{2}+c \right )}}{\left (a c +b \right ) \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}\, \left (d \,x^{2}+c \right )}\)	\(525\)

input

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

output

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

3.4.53.5 Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 239, normalized size of antiderivative = 0.70 \[ \int \frac {1}{x^2 \sqrt {a+\frac {b}{c+d x^2}}} \, dx=\frac {a^{2} c \sqrt {-\frac {a d}{a c + b}} d^{2} x \sqrt {\frac {a c^{2} + b c}{d^{2}}} E(\arcsin \left (\sqrt {-\frac {a d}{a c + b}} x\right )\,|\,\frac {a c + b}{a c}) - {\left (a^{2} c d^{2} + {\left (a^{2} c^{2} + 2 \, a b c + b^{2}\right )} d\right )} \sqrt {-\frac {a d}{a c + b}} x \sqrt {\frac {a c^{2} + b c}{d^{2}}} F(\arcsin \left (\sqrt {-\frac {a d}{a c + b}} x\right )\,|\,\frac {a c + b}{a c}) - {\left (a^{2} c^{3} + a b c^{2} + {\left (a^{2} c^{2} + a b c\right )} d x^{2}\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{{\left (a^{3} c^{3} + 2 \, a^{2} b c^{2} + a b^{2} c\right )} x} \]

input

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

output

(a^2*c*sqrt(-a*d/(a*c + b))*d^2*x*sqrt((a*c^2 + b*c)/d^2)*elliptic_e(arcsi 
n(sqrt(-a*d/(a*c + b))*x), (a*c + b)/(a*c)) - (a^2*c*d^2 + (a^2*c^2 + 2*a* 
b*c + b^2)*d)*sqrt(-a*d/(a*c + b))*x*sqrt((a*c^2 + b*c)/d^2)*elliptic_f(ar 
csin(sqrt(-a*d/(a*c + b))*x), (a*c + b)/(a*c)) - (a^2*c^3 + a*b*c^2 + (a^2 
*c^2 + a*b*c)*d*x^2)*sqrt((a*d*x^2 + a*c + b)/(d*x^2 + c)))/((a^3*c^3 + 2* 
a^2*b*c^2 + a*b^2*c)*x)
 

3.4.53.6 Sympy [F]

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

input

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

output

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

3.4.53.7 Maxima [F]

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

input

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

output

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

3.4.53.8 Giac [F]

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

input

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

output

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

3.4.53.9 Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x^2 \sqrt {a+\frac {b}{c+d x^2}}} \, dx=\int \frac {1}{x^2\,\sqrt {a+\frac {b}{d\,x^2+c}}} \,d x \]

input

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

output

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