\(\int \frac {1}{x^6 \sqrt {(a+b x^2) (c+d x^2)}} \, dx\) [707]

Optimal result

Integrand size = 23, antiderivative size = 375 \[ \int \frac {1}{x^6 \sqrt {\left (a+b x^2\right ) \left (c+d x^2\right )}} \, dx=-\frac {\sqrt {a c+(b c+a d) x^2+b d x^4}}{5 a c x^5}+\frac {4 (b c+a d) \sqrt {a c+(b c+a d) x^2+b d x^4}}{15 a^2 c^2 x^3}+\frac {\left (9 a b c d-8 (b c+a d)^2\right ) \sqrt {a c+(b c+a d) x^2+b d x^4}}{15 a^2 c^3 x \left (a+b x^2\right )}+\frac {\sqrt {b} \left (9 a b c d-8 (b c+a d)^2\right ) \left (a+b x^2\right ) \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{15 a^{7/2} c^2 \sqrt {a c+(b c+a d) x^2+b d x^4}}+\frac {4 \sqrt {b} d (b c+a d) \left (a+b x^2\right ) \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{15 a^{5/2} c^2 \sqrt {a c+(b c+a d) x^2+b d x^4}} \] Output:

-1/5*(a*c+(a*d+b*c)*x^2+b*d*x^4)^(1/2)/a/c/x^5+4/15*(a*d+b*c)*(a*c+(a*d+b* 
c)*x^2+b*d*x^4)^(1/2)/a^2/c^2/x^3+1/15*(9*a*b*c*d-8*(a*d+b*c)^2)*(a*c+(a*d 
+b*c)*x^2+b*d*x^4)^(1/2)/a^2/c^3/x/(b*x^2+a)+1/15*b^(1/2)*(9*a*b*c*d-8*(a* 
d+b*c)^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^(7/2)/c^2/(a*c+(a*d+b*c)*x^2+ 
b*d*x^4)^(1/2)+4/15*b^(1/2)*d*(a*d+b*c)*(b*x^2+a)*(a*(d*x^2+c)/c/(b*x^2+a) 
)^(1/2)*InverseJacobiAM(arctan(b^(1/2)*x/a^(1/2)),(1-a*d/b/c)^(1/2))/a^(5/ 
2)/c^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 = 3.31 (sec) , antiderivative size = 293, normalized size of antiderivative = 0.78 \[ \int \frac {1}{x^6 \sqrt {\left (a+b x^2\right ) \left (c+d x^2\right )}} \, dx=\frac {-\sqrt {\frac {b}{a}} \left (a+b x^2\right ) \left (c+d x^2\right ) \left (8 b^2 c^2 x^4+a b c x^2 \left (-4 c+7 d x^2\right )+a^2 \left (3 c^2-4 c d x^2+8 d^2 x^4\right )\right )-i b c \left (8 b^2 c^2+7 a b c d+8 a^2 d^2\right ) x^5 \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )+i b c \left (8 b^2 c^2+3 a b c d+4 a^2 d^2\right ) x^5 \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{15 a^3 \sqrt {\frac {b}{a}} c^3 x^5 \sqrt {\left (a+b x^2\right ) \left (c+d x^2\right )}} \] Input:

Integrate[1/(x^6*Sqrt[(a + b*x^2)*(c + d*x^2)]),x]
 

Output:

(-(Sqrt[b/a]*(a + b*x^2)*(c + d*x^2)*(8*b^2*c^2*x^4 + a*b*c*x^2*(-4*c + 7* 
d*x^2) + a^2*(3*c^2 - 4*c*d*x^2 + 8*d^2*x^4))) - I*b*c*(8*b^2*c^2 + 7*a*b* 
c*d + 8*a^2*d^2)*x^5*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*A 
rcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] + I*b*c*(8*b^2*c^2 + 3*a*b*c*d + 4*a^2*d 
^2)*x^5*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b 
/a]*x], (a*d)/(b*c)])/(15*a^3*Sqrt[b/a]*c^3*x^5*Sqrt[(a + b*x^2)*(c + d*x^ 
2)])
 

Rubi [A] (verified)

Time = 1.50 (sec) , antiderivative size = 681, normalized size of antiderivative = 1.82, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {2048, 1443, 25, 1604, 25, 1604, 27, 1511, 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.

\(\displaystyle \int \frac {1}{x^6 \sqrt {\left (a+b x^2\right ) \left (c+d x^2\right )}} \, dx\)

\(\Big \downarrow \) 2048

\(\displaystyle \int \frac {1}{x^6 \sqrt {x^2 (a d+b c)+a c+b d x^4}}dx\)

\(\Big \downarrow \) 1443

\(\displaystyle \frac {\int -\frac {3 b d x^2+4 (b c+a d)}{x^4 \sqrt {b d x^4+(b c+a d) x^2+a c}}dx}{5 a c}-\frac {\sqrt {x^2 (a d+b c)+a c+b d x^4}}{5 a c x^5}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\int \frac {3 b d x^2+4 (b c+a d)}{x^4 \sqrt {b d x^4+(b c+a d) x^2+a c}}dx}{5 a c}-\frac {\sqrt {x^2 (a d+b c)+a c+b d x^4}}{5 a c x^5}\)

\(\Big \downarrow \) 1604

\(\displaystyle -\frac {-\frac {\int -\frac {-8 (b c+a d)^2-4 b d x^2 (b c+a d)+9 a b c d}{x^2 \sqrt {b d x^4+(b c+a d) x^2+a c}}dx}{3 a c}-\frac {4 (a d+b c) \sqrt {x^2 (a d+b c)+a c+b d x^4}}{3 a c x^3}}{5 a c}-\frac {\sqrt {x^2 (a d+b c)+a c+b d x^4}}{5 a c x^5}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\frac {\int \frac {-8 (b c+a d)^2-4 b d x^2 (b c+a d)+9 a b c d}{x^2 \sqrt {b d x^4+(b c+a d) x^2+a c}}dx}{3 a c}-\frac {4 (a d+b c) \sqrt {x^2 (a d+b c)+a c+b d x^4}}{3 a c x^3}}{5 a c}-\frac {\sqrt {x^2 (a d+b c)+a c+b d x^4}}{5 a c x^5}\)

\(\Big \downarrow \) 1604

\(\displaystyle -\frac {\frac {-\frac {\int \frac {b d \left (4 a c (b c+a d)-\left (9 a b c d-8 (b c+a d)^2\right ) x^2\right )}{\sqrt {b d x^4+(b c+a d) x^2+a c}}dx}{a c}-\frac {\left (9 a b c d-8 (a d+b c)^2\right ) \sqrt {x^2 (a d+b c)+a c+b d x^4}}{a c x}}{3 a c}-\frac {4 (a d+b c) \sqrt {x^2 (a d+b c)+a c+b d x^4}}{3 a c x^3}}{5 a c}-\frac {\sqrt {x^2 (a d+b c)+a c+b d x^4}}{5 a c x^5}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\frac {-\frac {b d \int \frac {4 a c (b c+a d)-\left (9 a b c d-8 (b c+a d)^2\right ) x^2}{\sqrt {b d x^4+(b c+a d) x^2+a c}}dx}{a c}-\frac {\left (9 a b c d-8 (a d+b c)^2\right ) \sqrt {x^2 (a d+b c)+a c+b d x^4}}{a c x}}{3 a c}-\frac {4 (a d+b c) \sqrt {x^2 (a d+b c)+a c+b d x^4}}{3 a c x^3}}{5 a c}-\frac {\sqrt {x^2 (a d+b c)+a c+b d x^4}}{5 a c x^5}\)

\(\Big \downarrow \) 1511

\(\displaystyle -\frac {\frac {-\frac {b d \left (\frac {\sqrt {a} \sqrt {c} \left (9 a b c d-8 (a d+b c)^2\right ) \int \frac {\sqrt {a} \sqrt {c}-\sqrt {b} \sqrt {d} x^2}{\sqrt {a} \sqrt {c} \sqrt {b d x^4+(b c+a d) x^2+a c}}dx}{\sqrt {b} \sqrt {d}}-\frac {\sqrt {a} \sqrt {c} \left (-8 (a d+b c)^2-4 \sqrt {a} \sqrt {b} \sqrt {c} \sqrt {d} (a d+b c)+9 a b c d\right ) \int \frac {1}{\sqrt {b d x^4+(b c+a d) x^2+a c}}dx}{\sqrt {b} \sqrt {d}}\right )}{a c}-\frac {\left (9 a b c d-8 (a d+b c)^2\right ) \sqrt {x^2 (a d+b c)+a c+b d x^4}}{a c x}}{3 a c}-\frac {4 (a d+b c) \sqrt {x^2 (a d+b c)+a c+b d x^4}}{3 a c x^3}}{5 a c}-\frac {\sqrt {x^2 (a d+b c)+a c+b d x^4}}{5 a c x^5}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\frac {-\frac {b d \left (\frac {\left (9 a b c d-8 (a d+b c)^2\right ) \int \frac {\sqrt {a} \sqrt {c}-\sqrt {b} \sqrt {d} x^2}{\sqrt {b d x^4+(b c+a d) x^2+a c}}dx}{\sqrt {b} \sqrt {d}}-\frac {\sqrt {a} \sqrt {c} \left (-8 (a d+b c)^2-4 \sqrt {a} \sqrt {b} \sqrt {c} \sqrt {d} (a d+b c)+9 a b c d\right ) \int \frac {1}{\sqrt {b d x^4+(b c+a d) x^2+a c}}dx}{\sqrt {b} \sqrt {d}}\right )}{a c}-\frac {\left (9 a b c d-8 (a d+b c)^2\right ) \sqrt {x^2 (a d+b c)+a c+b d x^4}}{a c x}}{3 a c}-\frac {4 (a d+b c) \sqrt {x^2 (a d+b c)+a c+b d x^4}}{3 a c x^3}}{5 a c}-\frac {\sqrt {x^2 (a d+b c)+a c+b d x^4}}{5 a c x^5}\)

\(\Big \downarrow \) 1416

\(\displaystyle -\frac {\frac {-\frac {b d \left (\frac {\left (9 a b c d-8 (a d+b c)^2\right ) \int \frac {\sqrt {a} \sqrt {c}-\sqrt {b} \sqrt {d} x^2}{\sqrt {b d x^4+(b c+a d) x^2+a c}}dx}{\sqrt {b} \sqrt {d}}-\frac {\sqrt [4]{a} \sqrt [4]{c} \left (-8 (a d+b c)^2-4 \sqrt {a} \sqrt {b} \sqrt {c} \sqrt {d} (a d+b c)+9 a b c d\right ) \left (\sqrt {a} \sqrt {c}+\sqrt {b} \sqrt {d} x^2\right ) \sqrt {\frac {x^2 (a d+b c)+a c+b d x^4}{\left (\sqrt {a} \sqrt {c}+\sqrt {b} \sqrt {d} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt [4]{d} x}{\sqrt [4]{a} \sqrt [4]{c}}\right ),\frac {1}{4} \left (2-\frac {b c+a d}{\sqrt {a} \sqrt {b} \sqrt {c} \sqrt {d}}\right )\right )}{2 b^{3/4} d^{3/4} \sqrt {x^2 (a d+b c)+a c+b d x^4}}\right )}{a c}-\frac {\left (9 a b c d-8 (a d+b c)^2\right ) \sqrt {x^2 (a d+b c)+a c+b d x^4}}{a c x}}{3 a c}-\frac {4 (a d+b c) \sqrt {x^2 (a d+b c)+a c+b d x^4}}{3 a c x^3}}{5 a c}-\frac {\sqrt {x^2 (a d+b c)+a c+b d x^4}}{5 a c x^5}\)

\(\Big \downarrow \) 1509

\(\displaystyle -\frac {\frac {-\frac {b d \left (\frac {\left (9 a b c d-8 (a d+b c)^2\right ) \left (\frac {\sqrt [4]{a} \sqrt [4]{c} \left (\sqrt {a} \sqrt {c}+\sqrt {b} \sqrt {d} x^2\right ) \sqrt {\frac {x^2 (a d+b c)+a c+b d x^4}{\left (\sqrt {a} \sqrt {c}+\sqrt {b} \sqrt {d} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt [4]{d} x}{\sqrt [4]{a} \sqrt [4]{c}}\right )|\frac {1}{4} \left (2-\frac {b c+a d}{\sqrt {a} \sqrt {b} \sqrt {c} \sqrt {d}}\right )\right )}{\sqrt [4]{b} \sqrt [4]{d} \sqrt {x^2 (a d+b c)+a c+b d x^4}}-\frac {x \sqrt {x^2 (a d+b c)+a c+b d x^4}}{\sqrt {a} \sqrt {c}+\sqrt {b} \sqrt {d} x^2}\right )}{\sqrt {b} \sqrt {d}}-\frac {\sqrt [4]{a} \sqrt [4]{c} \left (-8 (a d+b c)^2-4 \sqrt {a} \sqrt {b} \sqrt {c} \sqrt {d} (a d+b c)+9 a b c d\right ) \left (\sqrt {a} \sqrt {c}+\sqrt {b} \sqrt {d} x^2\right ) \sqrt {\frac {x^2 (a d+b c)+a c+b d x^4}{\left (\sqrt {a} \sqrt {c}+\sqrt {b} \sqrt {d} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt [4]{d} x}{\sqrt [4]{a} \sqrt [4]{c}}\right ),\frac {1}{4} \left (2-\frac {b c+a d}{\sqrt {a} \sqrt {b} \sqrt {c} \sqrt {d}}\right )\right )}{2 b^{3/4} d^{3/4} \sqrt {x^2 (a d+b c)+a c+b d x^4}}\right )}{a c}-\frac {\left (9 a b c d-8 (a d+b c)^2\right ) \sqrt {x^2 (a d+b c)+a c+b d x^4}}{a c x}}{3 a c}-\frac {4 (a d+b c) \sqrt {x^2 (a d+b c)+a c+b d x^4}}{3 a c x^3}}{5 a c}-\frac {\sqrt {x^2 (a d+b c)+a c+b d x^4}}{5 a c x^5}\)

Input:

Int[1/(x^6*Sqrt[(a + b*x^2)*(c + d*x^2)]),x]
 

Output:

-1/5*Sqrt[a*c + (b*c + a*d)*x^2 + b*d*x^4]/(a*c*x^5) - ((-4*(b*c + a*d)*Sq 
rt[a*c + (b*c + a*d)*x^2 + b*d*x^4])/(3*a*c*x^3) + (-(((9*a*b*c*d - 8*(b*c 
 + a*d)^2)*Sqrt[a*c + (b*c + a*d)*x^2 + b*d*x^4])/(a*c*x)) - (b*d*(((9*a*b 
*c*d - 8*(b*c + a*d)^2)*(-((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)*(9*a*b*c*d - 4*Sqrt[a]*Sqrt[b]*Sqrt[c]*Sqrt[d]*(b*c + a*d 
) - 8*(b*c + a*d)^2)*(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]*Ellip 
ticF[2*ArcTan[(b^(1/4)*d^(1/4)*x)/(a^(1/4)*c^(1/4))], (2 - (b*c + a*d)/(Sq 
rt[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))/(3*a*c))/(5*a*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 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 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 1511

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

rule 1604

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

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 = 5.29 (sec) , antiderivative size = 388, normalized size of antiderivative = 1.03

method	result	size
default	\(-\frac {\sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}}{5 a c \,x^{5}}+\frac {4 \left (a d +b c \right ) \sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}}{15 a^{2} c^{2} x^{3}}-\frac {\left (8 a^{2} d^{2}+7 a b c d +8 b^{2} c^{2}\right ) \sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}}{15 a^{3} c^{3} x}+\frac {4 \left (a d +b c \right ) b d \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{15 a^{2} c^{2} \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}}-\frac {b \left (8 a^{2} d^{2}+7 a b c d +8 b^{2} c^{2}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{15 a^{3} c^{2} \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}}\)	\(388\)
elliptic	\(-\frac {\sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}}{5 a c \,x^{5}}+\frac {4 \left (a d +b c \right ) \sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}}{15 a^{2} c^{2} x^{3}}-\frac {\left (8 a^{2} d^{2}+7 a b c d +8 b^{2} c^{2}\right ) \sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}}{15 a^{3} c^{3} x}+\frac {4 \left (a d +b c \right ) b d \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{15 a^{2} c^{2} \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}}-\frac {b \left (8 a^{2} d^{2}+7 a b c d +8 b^{2} c^{2}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{15 a^{3} c^{2} \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}}\)	\(388\)
risch	\(-\frac {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right ) \left (8 a^{2} d^{2} x^{4}+7 a b c d \,x^{4}+8 b^{2} c^{2} x^{4}-4 a^{2} c d \,x^{2}-4 a b \,c^{2} x^{2}+3 a^{2} c^{2}\right )}{15 a^{3} c^{3} x^{5} \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}}+\frac {b d \left (-\frac {\left (8 a^{2} d^{2}+7 a b c d +8 b^{2} c^{2}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}\, d}+\frac {4 a b \,c^{2} \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}}+\frac {4 a^{2} c d \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}}\right )}{15 a^{3} c^{3}}\)	\(445\)

Input:

int(1/x^6/((b*x^2+a)*(d*x^2+c))^(1/2),x,method=_RETURNVERBOSE)
 

Output:

-1/5/a/c*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)/x^5+4/15*(a*d+b*c)/a^2/c^2*(b 
*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)/x^3-1/15*(8*a^2*d^2+7*a*b*c*d+8*b^2*c^2) 
/a^3/c^3*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)/x+4/15*(a*d+b*c)*b*d/a^2/c^2/ 
(-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))-1/15*b*(8*a^ 
2*d^2+7*a*b*c*d+8*b^2*c^2)/a^3/c^2/(-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 = 0.08 (sec) , antiderivative size = 239, normalized size of antiderivative = 0.64 \[ \int \frac {1}{x^6 \sqrt {\left (a+b x^2\right ) \left (c+d x^2\right )}} \, dx=\frac {{\left (8 \, b^{3} c^{2} + 7 \, a b^{2} c d + 8 \, a^{2} b d^{2}\right )} \sqrt {a c} x^{5} \sqrt {-\frac {b}{a}} E(\arcsin \left (x \sqrt {-\frac {b}{a}}\right )\,|\,\frac {a d}{b c}) - {\left (8 \, b^{3} c^{2} + {\left (4 \, a^{2} b + 7 \, a b^{2}\right )} c d + 4 \, {\left (a^{3} + 2 \, a^{2} b\right )} d^{2}\right )} \sqrt {a c} x^{5} \sqrt {-\frac {b}{a}} F(\arcsin \left (x \sqrt {-\frac {b}{a}}\right )\,|\,\frac {a d}{b c}) - \sqrt {b d x^{4} + {\left (b c + a d\right )} x^{2} + a c} {\left (3 \, a^{3} c^{2} + {\left (8 \, a b^{2} c^{2} + 7 \, a^{2} b c d + 8 \, a^{3} d^{2}\right )} x^{4} - 4 \, {\left (a^{2} b c^{2} + a^{3} c d\right )} x^{2}\right )}}{15 \, a^{4} c^{3} x^{5}} \] Input:

integrate(1/x^6/((b*x^2+a)*(d*x^2+c))^(1/2),x, algorithm="fricas")
 

Output:

1/15*((8*b^3*c^2 + 7*a*b^2*c*d + 8*a^2*b*d^2)*sqrt(a*c)*x^5*sqrt(-b/a)*ell 
iptic_e(arcsin(x*sqrt(-b/a)), a*d/(b*c)) - (8*b^3*c^2 + (4*a^2*b + 7*a*b^2 
)*c*d + 4*(a^3 + 2*a^2*b)*d^2)*sqrt(a*c)*x^5*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)*(3*a^3*c 
^2 + (8*a*b^2*c^2 + 7*a^2*b*c*d + 8*a^3*d^2)*x^4 - 4*(a^2*b*c^2 + a^3*c*d) 
*x^2))/(a^4*c^3*x^5)
 

Sympy [F]

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

integrate(1/x**6/((b*x**2+a)*(d*x**2+c))**(1/2),x)
 

Output:

Integral(1/(x**6*sqrt((a + b*x**2)*(c + d*x**2))), x)
 

Maxima [F]

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

integrate(1/x^6/((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^6), x)
 

Giac [F]

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

integrate(1/x^6/((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^6), x)
 

Mupad [F(-1)]

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

int(1/(x^6*((a + b*x^2)*(c + d*x^2))^(1/2)),x)
                                                                                    
                                                                                    
 

Output:

int(1/(x^6*((a + b*x^2)*(c + d*x^2))^(1/2)), x)
 

Reduce [F]

\[ \int \frac {1}{x^6 \sqrt {\left (a+b x^2\right ) \left (c+d x^2\right )}} \, dx=\frac {-\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, a c +3 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, b d \,x^{4}-3 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) b^{2} d^{2} x^{5}-4 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b d \,x^{8}+a d \,x^{6}+b c \,x^{6}+a c \,x^{4}}d x \right ) a^{2} c d \,x^{5}-4 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b d \,x^{8}+a d \,x^{6}+b c \,x^{6}+a c \,x^{4}}d x \right ) a b \,c^{2} x^{5}}{5 a^{2} c^{2} x^{5}} \] Input:

int(1/x^6/((b*x^2+a)*(d*x^2+c))^(1/2),x)
 

Output:

( - sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*c + 3*sqrt(c + d*x**2)*sqrt(a + b* 
x**2)*b*d*x**4 - 3*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**2)/(a*c + a*d 
*x**2 + b*c*x**2 + b*d*x**4),x)*b**2*d**2*x**5 - 4*int((sqrt(c + d*x**2)*s 
qrt(a + b*x**2))/(a*c*x**4 + a*d*x**6 + b*c*x**6 + b*d*x**8),x)*a**2*c*d*x 
**5 - 4*int((sqrt(c + d*x**2)*sqrt(a + b*x**2))/(a*c*x**4 + a*d*x**6 + b*c 
*x**6 + b*d*x**8),x)*a*b*c**2*x**5)/(5*a**2*c**2*x**5)