\(\int \frac {\sqrt {a+b x^2+c x^4}}{x^6} \, dx\) [963]

Optimal result

Integrand size = 20, antiderivative size = 363 \[ \int \frac {\sqrt {a+b x^2+c x^4}}{x^6} \, dx=-\frac {\sqrt {a+b x^2+c x^4}}{5 x^5}-\frac {b \sqrt {a+b x^2+c x^4}}{15 a x^3}+\frac {2 \left (b^2-3 a c\right ) \sqrt {a+b x^2+c x^4}}{15 a^{3/2} x \left (\sqrt {a}+\sqrt {c} x^2\right )}+\frac {2 \sqrt [4]{c} \left (b^2-3 a c\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{15 a^{7/4} \sqrt {a+b x^2+c x^4}}-\frac {\sqrt [4]{c} \left (b \sqrt {c}+\frac {2 \left (b^2-3 a c\right )}{\sqrt {a}}\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{30 a^{5/4} \sqrt {a+b x^2+c x^4}} \] Output:

-1/5*(c*x^4+b*x^2+a)^(1/2)/x^5-1/15*b*(c*x^4+b*x^2+a)^(1/2)/a/x^3+2/15*(-3 
*a*c+b^2)*(c*x^4+b*x^2+a)^(1/2)/a^(3/2)/x/(a^(1/2)+c^(1/2)*x^2)+2/15*c^(1/ 
4)*(-3*a*c+b^2)*(a^(1/2)+c^(1/2)*x^2)*((c*x^4+b*x^2+a)/(a^(1/2)+c^(1/2)*x^ 
2)^2)^(1/2)*EllipticE(sin(2*arctan(c^(1/4)*x/a^(1/4))),1/2*(2-b/a^(1/2)/c^ 
(1/2))^(1/2))/a^(7/4)/(c*x^4+b*x^2+a)^(1/2)-1/30*c^(1/4)*(b*c^(1/2)+2*(-3* 
a*c+b^2)/a^(1/2))*(a^(1/2)+c^(1/2)*x^2)*((c*x^4+b*x^2+a)/(a^(1/2)+c^(1/2)* 
x^2)^2)^(1/2)*InverseJacobiAM(2*arctan(c^(1/4)*x/a^(1/4)),1/2*(2-b/a^(1/2) 
/c^(1/2))^(1/2))/a^(5/4)/(c*x^4+b*x^2+a)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 10.90 (sec) , antiderivative size = 530, normalized size of antiderivative = 1.46 \[ \int \frac {\sqrt {a+b x^2+c x^4}}{x^6} \, dx=\frac {-2 \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} \left (3 a^3-2 b^2 x^6 \left (b+c x^2\right )+a^2 \left (4 b x^2+9 c x^4\right )+a \left (-b^2 x^4+7 b c x^6+6 c^2 x^8\right )\right )-i \left (b^2-3 a c\right ) \left (-b+\sqrt {b^2-4 a c}\right ) x^5 \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} \sqrt {\frac {2 b-2 \sqrt {b^2-4 a c}+4 c x^2}{b-\sqrt {b^2-4 a c}}} E\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )+i \left (-b^3+4 a b c+b^2 \sqrt {b^2-4 a c}-3 a c \sqrt {b^2-4 a c}\right ) x^5 \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} \sqrt {\frac {2 b-2 \sqrt {b^2-4 a c}+4 c x^2}{b-\sqrt {b^2-4 a c}}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right ),\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )}{30 a^2 \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x^5 \sqrt {a+b x^2+c x^4}} \] Input:

Integrate[Sqrt[a + b*x^2 + c*x^4]/x^6,x]
 

Output:

(-2*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*(3*a^3 - 2*b^2*x^6*(b + c*x^2) + a^2*( 
4*b*x^2 + 9*c*x^4) + a*(-(b^2*x^4) + 7*b*c*x^6 + 6*c^2*x^8)) - I*(b^2 - 3* 
a*c)*(-b + Sqrt[b^2 - 4*a*c])*x^5*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/( 
b + Sqrt[b^2 - 4*a*c])]*Sqrt[(2*b - 2*Sqrt[b^2 - 4*a*c] + 4*c*x^2)/(b - Sq 
rt[b^2 - 4*a*c])]*EllipticE[I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c 
])]*x], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])] + I*(-b^3 + 4*a*b 
*c + b^2*Sqrt[b^2 - 4*a*c] - 3*a*c*Sqrt[b^2 - 4*a*c])*x^5*Sqrt[(b + Sqrt[b 
^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[(2*b - 2*Sqrt[b^2 - 4 
*a*c] + 4*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]*EllipticF[I*ArcSinh[Sqrt[2]*Sqrt 
[c/(b + Sqrt[b^2 - 4*a*c])]*x], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4* 
a*c])])/(30*a^2*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x^5*Sqrt[a + b*x^2 + c*x^4 
])
 

Rubi [A] (verified)

Time = 0.91 (sec) , antiderivative size = 402, normalized size of antiderivative = 1.11, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {1437, 1604, 1604, 25, 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.

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

\(\Big \downarrow \) 1437

\(\displaystyle \frac {1}{5} \int \frac {2 c x^2+b}{x^4 \sqrt {c x^4+b x^2+a}}dx-\frac {\sqrt {a+b x^2+c x^4}}{5 x^5}\)

\(\Big \downarrow \) 1604

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

\(\Big \downarrow \) 1604

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1511

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1416

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

\(\Big \downarrow \) 1509

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

Input:

Int[Sqrt[a + b*x^2 + c*x^4]/x^6,x]
 

Output:

-1/5*Sqrt[a + b*x^2 + c*x^4]/x^5 + (-1/3*(b*Sqrt[a + b*x^2 + c*x^4])/(a*x^ 
3) - ((-2*(b^2 - 3*a*c)*Sqrt[a + b*x^2 + c*x^4])/(a*x) + (c*((-2*(b^2 - 3* 
a*c)*(-((x*Sqrt[a + b*x^2 + c*x^4])/(Sqrt[a] + Sqrt[c]*x^2)) + (a^(1/4)*(S 
qrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]* 
EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(c^ 
(1/4)*Sqrt[a + b*x^2 + c*x^4])))/Sqrt[c] + (a^(1/4)*(Sqrt[a]*b + (2*(b^2 - 
 3*a*c))/Sqrt[c])*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a 
] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[ 
a]*Sqrt[c]))/4])/(2*c^(1/4)*Sqrt[a + b*x^2 + c*x^4])))/a)/(3*a))/5
 

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 1437

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/(d*(m + 1))), x] - Simp[2*(p/( 
d^2*(m + 1)))   Int[(d*x)^(m + 2)*(b + 2*c*x^2)*(a + b*x^2 + c*x^4)^(p - 1) 
, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && L 
tQ[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])
 

Maple [A] (verified)

Time = 3.75 (sec) , antiderivative size = 428, normalized size of antiderivative = 1.18

method	result	size
risch	\(-\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (6 a c \,x^{4}-2 b^{2} x^{4}+a b \,x^{2}+3 a^{2}\right )}{15 x^{5} a^{2}}-\frac {c \left (\frac {\left (6 a c -2 b^{2}\right ) a \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )\right )}{2 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (b +\sqrt {-4 a c +b^{2}}\right )}+\frac {a b \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{4 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}\right )}{15 a^{2}}\)	\(428\)
default	\(-\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}}{5 x^{5}}-\frac {b \sqrt {c \,x^{4}+b \,x^{2}+a}}{15 a \,x^{3}}-\frac {2 \left (3 a c -b^{2}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}{15 a^{2} x}-\frac {b c \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{60 a \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {c \left (3 a c -b^{2}\right ) \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )\right )}{15 a \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (b +\sqrt {-4 a c +b^{2}}\right )}\)	\(452\)
elliptic	\(-\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}}{5 x^{5}}-\frac {b \sqrt {c \,x^{4}+b \,x^{2}+a}}{15 a \,x^{3}}-\frac {2 \left (3 a c -b^{2}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}{15 a^{2} x}-\frac {b c \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{60 a \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {c \left (3 a c -b^{2}\right ) \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )\right )}{15 a \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (b +\sqrt {-4 a c +b^{2}}\right )}\)	\(452\)

Input:

int((c*x^4+b*x^2+a)^(1/2)/x^6,x,method=_RETURNVERBOSE)
 

Output:

-1/15*(c*x^4+b*x^2+a)^(1/2)*(6*a*c*x^4-2*b^2*x^4+a*b*x^2+3*a^2)/x^5/a^2-1/ 
15*c/a^2*(1/2*(6*a*c-2*b^2)*a*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*(4 
-2*(-b+(-4*a*c+b^2)^(1/2))/a*x^2)^(1/2)*(4+2*(b+(-4*a*c+b^2)^(1/2))/a*x^2) 
^(1/2)/(c*x^4+b*x^2+a)^(1/2)/(b+(-4*a*c+b^2)^(1/2))*(EllipticF(1/2*x*2^(1/ 
2)*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2),1/2*(-4+2*b*(b+(-4*a*c+b^2)^(1/2))/a/ 
c)^(1/2))-EllipticE(1/2*x*2^(1/2)*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2),1/2*(- 
4+2*b*(b+(-4*a*c+b^2)^(1/2))/a/c)^(1/2)))+1/4*a*b*2^(1/2)/((-b+(-4*a*c+b^2 
)^(1/2))/a)^(1/2)*(4-2*(-b+(-4*a*c+b^2)^(1/2))/a*x^2)^(1/2)*(4+2*(b+(-4*a* 
c+b^2)^(1/2))/a*x^2)^(1/2)/(c*x^4+b*x^2+a)^(1/2)*EllipticF(1/2*x*2^(1/2)*( 
(-b+(-4*a*c+b^2)^(1/2))/a)^(1/2),1/2*(-4+2*b*(b+(-4*a*c+b^2)^(1/2))/a/c)^( 
1/2)))
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 357, normalized size of antiderivative = 0.98 \[ \int \frac {\sqrt {a+b x^2+c x^4}}{x^6} \, dx=\frac {2 \, \sqrt {\frac {1}{2}} {\left ({\left (a b^{2} - 3 \, a^{2} c\right )} x^{5} \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} - {\left (b^{3} - 3 \, a b c\right )} x^{5}\right )} \sqrt {a} \sqrt {\frac {a \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} - b}{a}} E(\arcsin \left (\sqrt {\frac {1}{2}} x \sqrt {\frac {a \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} - b}{a}}\right )\,|\,\frac {a b \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} + b^{2} - 2 \, a c}{2 \, a c}) + \sqrt {\frac {1}{2}} {\left ({\left (a^{2} b - 2 \, a b^{2} + 6 \, a^{2} c\right )} x^{5} \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} + {\left (a b^{2} + 2 \, b^{3} - 6 \, a b c\right )} x^{5}\right )} \sqrt {a} \sqrt {\frac {a \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} - b}{a}} F(\arcsin \left (\sqrt {\frac {1}{2}} x \sqrt {\frac {a \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} - b}{a}}\right )\,|\,\frac {a b \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} + b^{2} - 2 \, a c}{2 \, a c}) - 2 \, {\left (a^{2} b x^{2} - 2 \, {\left (a b^{2} - 3 \, a^{2} c\right )} x^{4} + 3 \, a^{3}\right )} \sqrt {c x^{4} + b x^{2} + a}}{30 \, a^{3} x^{5}} \] Input:

integrate((c*x^4+b*x^2+a)^(1/2)/x^6,x, algorithm="fricas")
 

Output:

1/30*(2*sqrt(1/2)*((a*b^2 - 3*a^2*c)*x^5*sqrt((b^2 - 4*a*c)/a^2) - (b^3 - 
3*a*b*c)*x^5)*sqrt(a)*sqrt((a*sqrt((b^2 - 4*a*c)/a^2) - b)/a)*elliptic_e(a 
rcsin(sqrt(1/2)*x*sqrt((a*sqrt((b^2 - 4*a*c)/a^2) - b)/a)), 1/2*(a*b*sqrt( 
(b^2 - 4*a*c)/a^2) + b^2 - 2*a*c)/(a*c)) + sqrt(1/2)*((a^2*b - 2*a*b^2 + 6 
*a^2*c)*x^5*sqrt((b^2 - 4*a*c)/a^2) + (a*b^2 + 2*b^3 - 6*a*b*c)*x^5)*sqrt( 
a)*sqrt((a*sqrt((b^2 - 4*a*c)/a^2) - b)/a)*elliptic_f(arcsin(sqrt(1/2)*x*s 
qrt((a*sqrt((b^2 - 4*a*c)/a^2) - b)/a)), 1/2*(a*b*sqrt((b^2 - 4*a*c)/a^2) 
+ b^2 - 2*a*c)/(a*c)) - 2*(a^2*b*x^2 - 2*(a*b^2 - 3*a^2*c)*x^4 + 3*a^3)*sq 
rt(c*x^4 + b*x^2 + a))/(a^3*x^5)
 

Sympy [F]

\[ \int \frac {\sqrt {a+b x^2+c x^4}}{x^6} \, dx=\int \frac {\sqrt {a + b x^{2} + c x^{4}}}{x^{6}}\, dx \] Input:

integrate((c*x**4+b*x**2+a)**(1/2)/x**6,x)
 

Output:

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

Maxima [F]

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

integrate((c*x^4+b*x^2+a)^(1/2)/x^6,x, algorithm="maxima")
 

Output:

integrate(sqrt(c*x^4 + b*x^2 + a)/x^6, x)
 

Giac [F]

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

integrate((c*x^4+b*x^2+a)^(1/2)/x^6,x, algorithm="giac")
 

Output:

integrate(sqrt(c*x^4 + b*x^2 + a)/x^6, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+b x^2+c x^4}}{x^6} \, dx=\int \frac {\sqrt {c\,x^4+b\,x^2+a}}{x^6} \,d x \] Input:

int((a + b*x^2 + c*x^4)^(1/2)/x^6,x)
 

Output:

int((a + b*x^2 + c*x^4)^(1/2)/x^6, x)
 

Reduce [F]

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

int((c*x^4+b*x^2+a)^(1/2)/x^6,x)
 

Output:

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