\(\int \frac {1}{x^2 \sqrt {(a+b x^2) (c+d x^2)}} \, dx\) [705]

Optimal result

Integrand size = 23, antiderivative size = 138 \[ \int \frac {1}{x^2 \sqrt {\left (a+b x^2\right ) \left (c+d x^2\right )}} \, dx=-\frac {\sqrt {a c+(b c+a d) x^2+b d x^4}}{c x \left (a+b x^2\right )}-\frac {\sqrt {b} \left (a+b x^2\right ) \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{a^{3/2} \sqrt {a c+(b c+a d) x^2+b d x^4}} \] Output:

-(a*c+(a*d+b*c)*x^2+b*d*x^4)^(1/2)/c/x/(b*x^2+a)-b^(1/2)*(b*x^2+a)*(a*(d*x 
^2+c)/c/(b*x^2+a))^(1/2)*EllipticE(b^(1/2)*x/a^(1/2)/(1+b*x^2/a)^(1/2),(1- 
a*d/b/c)^(1/2))/a^(3/2)/(a*c+(a*d+b*c)*x^2+b*d*x^4)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 2.14 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.04 \[ \int \frac {1}{x^2 \sqrt {\left (a+b x^2\right ) \left (c+d x^2\right )}} \, dx=\frac {-\frac {\left (a+b x^2\right ) \left (c+d x^2\right )}{c x}-i a \sqrt {\frac {b}{a}} \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \left (E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-\operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )\right )}{a \sqrt {\left (a+b x^2\right ) \left (c+d x^2\right )}} \] Input:

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

Output:

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

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(493\) vs. \(2(138)=276\).

Time = 0.99 (sec) , antiderivative size = 493, normalized size of antiderivative = 3.57, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {2048, 1443, 27, 1459, 27, 1416, 1509}

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.

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

\(\Big \downarrow \) 2048

\(\displaystyle \int \frac {1}{x^2 \sqrt {x^2 (a d+b c)+a c+b d x^4}}dx\)

\(\Big \downarrow \) 1443

\(\displaystyle \frac {\int \frac {b d x^2}{\sqrt {b d x^4+(b c+a d) x^2+a c}}dx}{a c}-\frac {\sqrt {x^2 (a d+b c)+a c+b d x^4}}{a c x}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {b d \int \frac {x^2}{\sqrt {b d x^4+(b c+a d) x^2+a c}}dx}{a c}-\frac {\sqrt {x^2 (a d+b c)+a c+b d x^4}}{a c x}\)

\(\Big \downarrow \) 1459

\(\displaystyle \frac {b d \left (\frac {\sqrt {a} \sqrt {c} \int \frac {1}{\sqrt {b d x^4+(b c+a d) x^2+a c}}dx}{\sqrt {b} \sqrt {d}}-\frac {\sqrt {a} \sqrt {c} \int \frac {\sqrt {a} \sqrt {c}-\sqrt {b} \sqrt {d} x^2}{\sqrt {a} \sqrt {c} \sqrt {b d x^4+(b c+a d) x^2+a c}}dx}{\sqrt {b} \sqrt {d}}\right )}{a c}-\frac {\sqrt {x^2 (a d+b c)+a c+b d x^4}}{a c x}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {b d \left (\frac {\sqrt {a} \sqrt {c} \int \frac {1}{\sqrt {b d x^4+(b c+a d) x^2+a c}}dx}{\sqrt {b} \sqrt {d}}-\frac {\int \frac {\sqrt {a} \sqrt {c}-\sqrt {b} \sqrt {d} x^2}{\sqrt {b d x^4+(b c+a d) x^2+a c}}dx}{\sqrt {b} \sqrt {d}}\right )}{a c}-\frac {\sqrt {x^2 (a d+b c)+a c+b d x^4}}{a c x}\)

\(\Big \downarrow \) 1416

\(\displaystyle \frac {b d \left (\frac {\sqrt [4]{a} \sqrt [4]{c} \left (\sqrt {a} \sqrt {c}+\sqrt {b} \sqrt {d} x^2\right ) \sqrt {\frac {x^2 (a d+b c)+a c+b d x^4}{\left (\sqrt {a} \sqrt {c}+\sqrt {b} \sqrt {d} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt [4]{d} x}{\sqrt [4]{a} \sqrt [4]{c}}\right ),\frac {1}{4} \left (2-\frac {b c+a d}{\sqrt {a} \sqrt {b} \sqrt {c} \sqrt {d}}\right )\right )}{2 b^{3/4} d^{3/4} \sqrt {x^2 (a d+b c)+a c+b d x^4}}-\frac {\int \frac {\sqrt {a} \sqrt {c}-\sqrt {b} \sqrt {d} x^2}{\sqrt {b d x^4+(b c+a d) x^2+a c}}dx}{\sqrt {b} \sqrt {d}}\right )}{a c}-\frac {\sqrt {x^2 (a d+b c)+a c+b d x^4}}{a c x}\)

\(\Big \downarrow \) 1509

\(\displaystyle \frac {b d \left (\frac {\sqrt [4]{a} \sqrt [4]{c} \left (\sqrt {a} \sqrt {c}+\sqrt {b} \sqrt {d} x^2\right ) \sqrt {\frac {x^2 (a d+b c)+a c+b d x^4}{\left (\sqrt {a} \sqrt {c}+\sqrt {b} \sqrt {d} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt [4]{d} x}{\sqrt [4]{a} \sqrt [4]{c}}\right ),\frac {1}{4} \left (2-\frac {b c+a d}{\sqrt {a} \sqrt {b} \sqrt {c} \sqrt {d}}\right )\right )}{2 b^{3/4} d^{3/4} \sqrt {x^2 (a d+b c)+a c+b d x^4}}-\frac {\frac {\sqrt [4]{a} \sqrt [4]{c} \left (\sqrt {a} \sqrt {c}+\sqrt {b} \sqrt {d} x^2\right ) \sqrt {\frac {x^2 (a d+b c)+a c+b d x^4}{\left (\sqrt {a} \sqrt {c}+\sqrt {b} \sqrt {d} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt [4]{d} x}{\sqrt [4]{a} \sqrt [4]{c}}\right )|\frac {1}{4} \left (2-\frac {b c+a d}{\sqrt {a} \sqrt {b} \sqrt {c} \sqrt {d}}\right )\right )}{\sqrt [4]{b} \sqrt [4]{d} \sqrt {x^2 (a d+b c)+a c+b d x^4}}-\frac {x \sqrt {x^2 (a d+b c)+a c+b d x^4}}{\sqrt {a} \sqrt {c}+\sqrt {b} \sqrt {d} x^2}}{\sqrt {b} \sqrt {d}}\right )}{a c}-\frac {\sqrt {x^2 (a d+b c)+a c+b d x^4}}{a c x}\)

Input:

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

Output:

-(Sqrt[a*c + (b*c + a*d)*x^2 + b*d*x^4]/(a*c*x)) + (b*d*(-((-((x*Sqrt[a*c 
+ (b*c + a*d)*x^2 + b*d*x^4])/(Sqrt[a]*Sqrt[c] + Sqrt[b]*Sqrt[d]*x^2)) + ( 
a^(1/4)*c^(1/4)*(Sqrt[a]*Sqrt[c] + Sqrt[b]*Sqrt[d]*x^2)*Sqrt[(a*c + (b*c + 
 a*d)*x^2 + b*d*x^4)/(Sqrt[a]*Sqrt[c] + Sqrt[b]*Sqrt[d]*x^2)^2]*EllipticE[ 
2*ArcTan[(b^(1/4)*d^(1/4)*x)/(a^(1/4)*c^(1/4))], (2 - (b*c + a*d)/(Sqrt[a] 
*Sqrt[b]*Sqrt[c]*Sqrt[d]))/4])/(b^(1/4)*d^(1/4)*Sqrt[a*c + (b*c + a*d)*x^2 
 + b*d*x^4]))/(Sqrt[b]*Sqrt[d])) + (a^(1/4)*c^(1/4)*(Sqrt[a]*Sqrt[c] + Sqr 
t[b]*Sqrt[d]*x^2)*Sqrt[(a*c + (b*c + a*d)*x^2 + b*d*x^4)/(Sqrt[a]*Sqrt[c] 
+ Sqrt[b]*Sqrt[d]*x^2)^2]*EllipticF[2*ArcTan[(b^(1/4)*d^(1/4)*x)/(a^(1/4)* 
c^(1/4))], (2 - (b*c + a*d)/(Sqrt[a]*Sqrt[b]*Sqrt[c]*Sqrt[d]))/4])/(2*b^(3 
/4)*d^(3/4)*Sqrt[a*c + (b*c + a*d)*x^2 + b*d*x^4])))/(a*c)
 

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 1416

Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c 
/a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/ 
(2*q*Sqrt[a + b*x^2 + c*x^4]))*EllipticF[2*ArcTan[q*x], 1/2 - b*(q^2/(4*c)) 
], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
 

rule 1443

Int[((d_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] 
:> Simp[(d*x)^(m + 1)*((a + b*x^2 + c*x^4)^(p + 1)/(a*d*(m + 1))), x] - Sim 
p[1/(a*d^2*(m + 1))   Int[(d*x)^(m + 2)*(b*(m + 2*p + 3) + c*(m + 4*p + 5)* 
x^2)*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b^2 
- 4*a*c, 0] && LtQ[m, -1] && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])
 

rule 1459

Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = 
 Rt[c/a, 2]}, Simp[1/q   Int[1/Sqrt[a + b*x^2 + c*x^4], x], x] - Simp[1/q 
 Int[(1 - q*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x]] /; FreeQ[{a, b, c}, x] && 
 NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
 

rule 1509

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo 
l] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + b*x^2 + c*x^4]/(a*(1 + q 
^2*x^2))), x] + Simp[d*(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2* 
x^2)^2)]/(q*Sqrt[a + b*x^2 + c*x^4]))*EllipticE[2*ArcTan[q*x], 1/2 - b*(q^2 
/(4*c))], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 
- 4*a*c, 0] && PosQ[c/a]
 

rule 2048

Int[(u_.)*((e_.)*((a_.) + (b_.)*(x_)^(n_.))*((c_) + (d_.)*(x_)^(n_.)))^(p_) 
, x_Symbol] :> Int[u*(a*c*e + (b*c + a*d)*e*x^n + b*d*e*x^(2*n))^p, x] /; F 
reeQ[{a, b, c, d, e, n, p}, x]
 

Maple [A] (verified)

Time = 1.82 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.16

method	result	size
default	\(-\frac {\sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}}{a c x}-\frac {b \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{a \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}}\)	\(160\)
elliptic	\(-\frac {\sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}}{a c x}-\frac {b \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{a \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}}\)	\(160\)
risch	\(-\frac {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}{a c x \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}}-\frac {b \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{a \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}}\)	\(167\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.81 \[ \int \frac {1}{x^2 \sqrt {\left (a+b x^2\right ) \left (c+d x^2\right )}} \, dx=\frac {\sqrt {a c} b x \sqrt {-\frac {b}{a}} E(\arcsin \left (x \sqrt {-\frac {b}{a}}\right )\,|\,\frac {a d}{b c}) - \sqrt {a c} b x \sqrt {-\frac {b}{a}} F(\arcsin \left (x \sqrt {-\frac {b}{a}}\right )\,|\,\frac {a d}{b c}) - \sqrt {b d x^{4} + {\left (b c + a d\right )} x^{2} + a c} a}{a^{2} c x} \] Input:

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

Output:

(sqrt(a*c)*b*x*sqrt(-b/a)*elliptic_e(arcsin(x*sqrt(-b/a)), a*d/(b*c)) - sq 
rt(a*c)*b*x*sqrt(-b/a)*elliptic_f(arcsin(x*sqrt(-b/a)), a*d/(b*c)) - sqrt( 
b*d*x^4 + (b*c + a*d)*x^2 + a*c)*a)/(a^2*c*x)
 

Sympy [F]

\[ \int \frac {1}{x^2 \sqrt {\left (a+b x^2\right ) \left (c+d x^2\right )}} \, dx=\int \frac {1}{x^{2} \sqrt {\left (a + b x^{2}\right ) \left (c + d x^{2}\right )}}\, dx \] Input:

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

Output:

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

Maxima [F]

\[ \int \frac {1}{x^2 \sqrt {\left (a+b x^2\right ) \left (c+d x^2\right )}} \, dx=\int { \frac {1}{\sqrt {{\left (b x^{2} + a\right )} {\left (d x^{2} + c\right )}} x^{2}} \,d x } \] Input:

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

Output:

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

Giac [F]

\[ \int \frac {1}{x^2 \sqrt {\left (a+b x^2\right ) \left (c+d x^2\right )}} \, dx=\int { \frac {1}{\sqrt {{\left (b x^{2} + a\right )} {\left (d x^{2} + c\right )}} x^{2}} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x^2 \sqrt {\left (a+b x^2\right ) \left (c+d x^2\right )}} \, dx=\int \frac {1}{x^2\,\sqrt {\left (b\,x^2+a\right )\,\left (d\,x^2+c\right )}} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {1}{x^2 \sqrt {\left (a+b x^2\right ) \left (c+d x^2\right )}} \, dx=\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b d \,x^{6}+a d \,x^{4}+b c \,x^{4}+a c \,x^{2}}d x \] Input:

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

Output:

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