3.4.52 \(\int \frac {1}{\sqrt {a+\frac {b}{c+d x^2}}} \, dx\) [352]

3.4.52.1 Optimal result

Integrand size = 17, antiderivative size = 286 \[ \int \frac {1}{\sqrt {a+\frac {b}{c+d x^2}}} \, dx=\frac {x \left (b+a c+a d x^2\right )}{a \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}-\frac {\sqrt {c} \left (b+a c+a d x^2\right ) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{a \sqrt {d} \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}} \sqrt {\frac {c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}}}+\frac {c^{3/2} \left (b+a c+a d x^2\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),\frac {b}{b+a c}\right )}{(b+a c) \sqrt {d} \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}} \sqrt {\frac {c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}}} \]

output

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

3.4.52.2 Mathematica [A] (verified)

Time = 7.33 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.37 \[ \int \frac {1}{\sqrt {a+\frac {b}{c+d x^2}}} \, dx=\frac {\sqrt {\frac {b+a c+a d x^2}{b+a c}} E\left (\arcsin \left (\sqrt {-\frac {a d}{b+a c}} x\right )|1+\frac {b}{a c}\right )}{\sqrt {-\frac {a d}{b+a c}} \sqrt {\frac {b+a c+a d x^2}{c+d x^2}} \sqrt {1+\frac {d x^2}{c}}} \]

input

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

output

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

3.4.52.3 Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {2057, 2058, 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.

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

\(\Big \downarrow \) 2057

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

\(\Big \downarrow \) 2058

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

\(\Big \downarrow \) 324

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

\(\Big \downarrow \) 320

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

\(\Big \downarrow \) 388

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

\(\Big \downarrow \) 313

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

input

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

output

(Sqrt[b + a*c + a*d*x^2]*(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))])) + (c^(3/2)*Sqrt[b + a*c + a*d*x^2]*EllipticF[A 
rcTan[(Sqrt[d]*x)/Sqrt[c]], b/(b + a*c)])/((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))])))/(Sqrt[c + d*x^ 
2]*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)])
 

3.4.52.3.1 Defintions of rubi rules used

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 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]
 

3.4.52.4 Maple [A] (verified)

Time = 1.15 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.57

method	result	size
default	\(\frac {E\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {\frac {a c +b}{a c}}\right ) \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {a d \,x^{2}+a c +b}{a c +b}}\, c \left (d \,x^{2}+c \right ) \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}}{\sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \sqrt {-\frac {a d}{a c +b}}\, \sqrt {\left (a d \,x^{2}+a c +b \right ) \left (d \,x^{2}+c \right )}}\)	\(164\)

input

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

output

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

3.4.52.5 Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.44 \[ \int \frac {1}{\sqrt {a+\frac {b}{c+d x^2}}} \, dx=-\frac {\sqrt {a} c x \sqrt {-\frac {c}{d}} E(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a c + b}{a c}) - \sqrt {a} {\left (c + d\right )} x \sqrt {-\frac {c}{d}} F(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a c + b}{a c}) - {\left (d x^{2} + c\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{a d x} \]

input

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

output

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

3.4.52.6 Sympy [F]

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

input

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

output

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

3.4.52.7 Maxima [F]

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

input

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

output

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

3.4.52.8 Giac [F]

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

input

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

output

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

3.4.52.9 Mupad [F(-1)]

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

input

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

output

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