3.4.28 \(\int \frac {\sqrt {a+\frac {b}{c+d x^2}}}{x^2} \, dx\) [328]

3.4.28.1 Optimal result

Integrand size = 21, antiderivative size = 265 \[ \int \frac {\sqrt {a+\frac {b}{c+d x^2}}}{x^2} \, dx=\frac {d x \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{c}-\frac {\left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{c x}-\frac {\sqrt {d} \sqrt {\frac {b+a c+a d x^2}{c+d x^2}} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{\sqrt {c} \sqrt {\frac {c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}}}+\frac {a \sqrt {c} \sqrt {d} \sqrt {\frac {b+a c+a d x^2}{c+d x^2}} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),\frac {b}{b+a c}\right )}{(b+a c) \sqrt {\frac {c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}}} \]

output

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

3.4.28.2 Mathematica [A] (verified)

Time = 9.54 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.51 \[ \int \frac {\sqrt {a+\frac {b}{c+d x^2}}}{x^2} \, dx=\sqrt {\frac {b+a c+a d x^2}{c+d x^2}} \left (-\frac {1}{x}-\frac {d x}{c}+\frac {a d \sqrt {\frac {b+a c+a d x^2}{b+a c}} \sqrt {1+\frac {d x^2}{c}} E\left (\arcsin \left (\sqrt {-\frac {a d}{b+a c}} x\right )|1+\frac {b}{a c}\right )}{\sqrt {-\frac {a d}{b+a c}} \left (b+a \left (c+d x^2\right )\right )}\right ) \]

input

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

output

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

3.4.28.3 Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 328, normalized size of antiderivative = 1.24, 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.

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

\(\Big \downarrow \) 2057

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

\(\Big \downarrow \) 2058

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

\(\Big \downarrow \) 377

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 324

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

\(\Big \downarrow \) 320

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

\(\Big \downarrow \) 388

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

\(\Big \downarrow \) 313

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

input

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

output

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

3.4.28.3.1 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]
 

3.4.28.4 Maple [A] (verified)

Time = 6.06 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.03

method	result	size
default	\(-\frac {\left (\sqrt {-\frac {a d}{a c +b}}\, a \,d^{2} x^{4}-a d c \sqrt {\frac {a d \,x^{2}+a c +b}{a c +b}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, x E\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {\frac {a c +b}{a c}}\right )+2 \sqrt {-\frac {a d}{a c +b}}\, a c d \,x^{2}+\sqrt {-\frac {a d}{a c +b}}\, b d \,x^{2}+\sqrt {-\frac {a d}{a c +b}}\, a \,c^{2}+\sqrt {-\frac {a d}{a c +b}}\, b c \right ) \left (d \,x^{2}+c \right ) \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}}{\sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \sqrt {-\frac {a d}{a c +b}}\, x c \sqrt {\left (a d \,x^{2}+a c +b \right ) \left (d \,x^{2}+c \right )}}\)	\(272\)
risch	\(-\frac {\left (d \,x^{2}+c \right ) \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}}{c x}+\frac {a d \left (\frac {c \sqrt {1+\frac {a d \,x^{2}}{a c +b}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {-1+\frac {2 a c d +b d}{d c a}}\right )}{\sqrt {-\frac {a d}{a c +b}}\, \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}}-\frac {2 d \left (a \,c^{2}+b c \right ) \sqrt {1+\frac {a d \,x^{2}}{a c +b}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (F\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {-1+\frac {2 a c d +b d}{d c a}}\right )-E\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {-1+\frac {2 a c d +b d}{d c a}}\right )\right )}{\sqrt {-\frac {a d}{a c +b}}\, \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \left (2 a c d +2 b d \right )}\right ) \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}\, \sqrt {\left (a d \,x^{2}+a c +b \right ) \left (d \,x^{2}+c \right )}}{c \left (a d \,x^{2}+a c +b \right )}\)	\(400\)

input

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

3.4.28.5 Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.74 \[ \int \frac {\sqrt {a+\frac {b}{c+d x^2}}}{x^2} \, dx=\frac {a \sqrt {-\frac {a d}{a c + b}} d^{2} x \sqrt {\frac {a c^{2} + b c}{d^{2}}} E(\arcsin \left (\sqrt {-\frac {a d}{a c + b}} x\right )\,|\,\frac {a c + b}{a c}) - {\left (a d^{2} + {\left (a c + b\right )} d\right )} \sqrt {-\frac {a d}{a c + b}} x \sqrt {\frac {a c^{2} + b c}{d^{2}}} F(\arcsin \left (\sqrt {-\frac {a d}{a c + b}} x\right )\,|\,\frac {a c + b}{a c}) - {\left ({\left (a c + b\right )} d x^{2} + a c^{2} + b c\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{{\left (a c^{2} + b c\right )} x} \]

input

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

output

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

3.4.28.6 Sympy [F]

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

input

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

output

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

3.4.28.7 Maxima [F]

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

input

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

output

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

3.4.28.8 Giac [F]

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

input

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

output

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

3.4.28.9 Mupad [F(-1)]

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

input

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

output

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