\(\int \frac {1}{x^4 \sqrt {a+b x^2-c x^4}} \, dx\) [1007]

Optimal result

Integrand size = 21, antiderivative size = 445 \[ \int \frac {1}{x^4 \sqrt {a+b x^2-c x^4}} \, dx=-\frac {\sqrt {a+b x^2-c x^4}}{3 a x^3}+\frac {2 b \sqrt {a+b x^2-c x^4}}{3 a^2 x}-\frac {b \left (b-\sqrt {b^2+4 a c}\right ) \sqrt {b+\sqrt {b^2+4 a c}} \sqrt {1-\frac {2 c x^2}{b-\sqrt {b^2+4 a c}}} \sqrt {1-\frac {2 c x^2}{b+\sqrt {b^2+4 a c}}} E\left (\arcsin \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2+4 a c}}}\right )|\frac {b+\sqrt {b^2+4 a c}}{b-\sqrt {b^2+4 a c}}\right )}{3 \sqrt {2} a^2 \sqrt {c} \sqrt {a+b x^2-c x^4}}+\frac {\sqrt {b+\sqrt {b^2+4 a c}} \left (b^2+a c-b \sqrt {b^2+4 a c}\right ) \sqrt {1-\frac {2 c x^2}{b-\sqrt {b^2+4 a c}}} \sqrt {1-\frac {2 c x^2}{b+\sqrt {b^2+4 a c}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2+4 a c}}}\right ),\frac {b+\sqrt {b^2+4 a c}}{b-\sqrt {b^2+4 a c}}\right )}{3 \sqrt {2} a^2 \sqrt {c} \sqrt {a+b x^2-c x^4}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 10.50 (sec) , antiderivative size = 472, normalized size of antiderivative = 1.06 \[ \int \frac {1}{x^4 \sqrt {a+b x^2-c x^4}} \, dx=\frac {-2 \sqrt {-\frac {c}{b+\sqrt {b^2+4 a c}}} \left (a-2 b x^2\right ) \left (a+b x^2-c x^4\right )-i \sqrt {2} b \left (-b+\sqrt {b^2+4 a c}\right ) x^3 \sqrt {\frac {b+\sqrt {b^2+4 a c}-2 c x^2}{b+\sqrt {b^2+4 a c}}} \sqrt {\frac {-b+\sqrt {b^2+4 a c}+2 c x^2}{-b+\sqrt {b^2+4 a c}}} E\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {-\frac {c}{b+\sqrt {b^2+4 a c}}} x\right )|\frac {b+\sqrt {b^2+4 a c}}{b-\sqrt {b^2+4 a c}}\right )+i \sqrt {2} \left (-b^2-a c+b \sqrt {b^2+4 a c}\right ) x^3 \sqrt {\frac {b+\sqrt {b^2+4 a c}-2 c x^2}{b+\sqrt {b^2+4 a c}}} \sqrt {\frac {-b+\sqrt {b^2+4 a c}+2 c x^2}{-b+\sqrt {b^2+4 a c}}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {-\frac {c}{b+\sqrt {b^2+4 a c}}} x\right ),\frac {b+\sqrt {b^2+4 a c}}{b-\sqrt {b^2+4 a c}}\right )}{6 a^2 \sqrt {-\frac {c}{b+\sqrt {b^2+4 a c}}} x^3 \sqrt {a+b x^2-c x^4}} \] Input:

Integrate[1/(x^4*Sqrt[a + b*x^2 - c*x^4]),x]
 

Output:

(-2*Sqrt[-(c/(b + Sqrt[b^2 + 4*a*c]))]*(a - 2*b*x^2)*(a + b*x^2 - c*x^4) - 
 I*Sqrt[2]*b*(-b + Sqrt[b^2 + 4*a*c])*x^3*Sqrt[(b + Sqrt[b^2 + 4*a*c] - 2* 
c*x^2)/(b + Sqrt[b^2 + 4*a*c])]*Sqrt[(-b + Sqrt[b^2 + 4*a*c] + 2*c*x^2)/(- 
b + Sqrt[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*Sqrt 
[2]*(-b^2 - a*c + b*Sqrt[b^2 + 4*a*c])*x^3*Sqrt[(b + Sqrt[b^2 + 4*a*c] - 2 
*c*x^2)/(b + Sqrt[b^2 + 4*a*c])]*Sqrt[(-b + Sqrt[b^2 + 4*a*c] + 2*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])])/(6*a^2 
*Sqrt[-(c/(b + Sqrt[b^2 + 4*a*c]))]*x^3*Sqrt[a + b*x^2 - c*x^4])
 

Rubi [A] (verified)

Time = 0.94 (sec) , antiderivative size = 372, normalized size of antiderivative = 0.84, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {1443, 25, 1604, 27, 1514, 399, 321, 327}

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.

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

\(\Big \downarrow \) 1443

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 1604

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1514

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

\(\Big \downarrow \) 399

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

\(\Big \downarrow \) 321

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

\(\Big \downarrow \) 327

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

Input:

Int[1/(x^4*Sqrt[a + b*x^2 - c*x^4]),x]
 

Output:

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

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 321

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c 
/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 
0] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
 

rule 327

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) 
)], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
 

rule 399

Int[((e_) + (f_.)*(x_)^2)/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_) 
^2]), x_Symbol] :> Simp[f/b   Int[Sqrt[a + b*x^2]/Sqrt[c + d*x^2], x], x] + 
 Simp[(b*e - a*f)/b   Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] /; Fr 
eeQ[{a, b, c, d, e, f}, x] &&  !((PosQ[b/a] && PosQ[d/c]) || (NegQ[b/a] && 
(PosQ[d/c] || (GtQ[a, 0] && ( !GtQ[c, 0] || SimplerSqrtQ[-b/a, -d/c])))))
 

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 1514

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo 
l] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[Sqrt[1 + 2*c*(x^2/(b - q))]*(Sqrt 
[1 + 2*c*(x^2/(b + q))]/Sqrt[a + b*x^2 + c*x^4])   Int[(d + e*x^2)/(Sqrt[1 
+ 2*c*(x^2/(b - q))]*Sqrt[1 + 2*c*(x^2/(b + q))]), x], x]] /; FreeQ[{a, b, 
c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NegQ[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 = 2.88 (sec) , antiderivative size = 402, normalized size of antiderivative = 0.90

method	result	size
risch	\(-\frac {\sqrt {-c \,x^{4}+b \,x^{2}+a}\, \left (-2 b \,x^{2}+a \right )}{3 a^{2} x^{3}}+\frac {c \left (\frac {a \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {4 a c +b^{2}}\right ) x^{2}}{a}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4-\frac {2 b \left (b +\sqrt {4 a c +b^{2}}\right )}{a c}}}{2}\right )}{4 \sqrt {\frac {-b +\sqrt {4 a c +b^{2}}}{a}}\, \sqrt {-c \,x^{4}+b \,x^{2}+a}}-\frac {b a \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {4 a c +b^{2}}\right ) x^{2}}{a}}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4-\frac {2 b \left (b +\sqrt {4 a c +b^{2}}\right )}{a c}}}{2}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4-\frac {2 b \left (b +\sqrt {4 a c +b^{2}}\right )}{a c}}}{2}\right )\right )}{\sqrt {\frac {-b +\sqrt {4 a c +b^{2}}}{a}}\, \sqrt {-c \,x^{4}+b \,x^{2}+a}\, \left (b +\sqrt {4 a c +b^{2}}\right )}\right )}{3 a^{2}}\)	\(402\)
default	\(-\frac {\sqrt {-c \,x^{4}+b \,x^{2}+a}}{3 a \,x^{3}}+\frac {2 b \sqrt {-c \,x^{4}+b \,x^{2}+a}}{3 a^{2} x}+\frac {c \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {4 a c +b^{2}}\right ) x^{2}}{a}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4-\frac {2 b \left (b +\sqrt {4 a c +b^{2}}\right )}{a c}}}{2}\right )}{12 a \sqrt {\frac {-b +\sqrt {4 a c +b^{2}}}{a}}\, \sqrt {-c \,x^{4}+b \,x^{2}+a}}-\frac {b c \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {4 a c +b^{2}}\right ) x^{2}}{a}}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4-\frac {2 b \left (b +\sqrt {4 a c +b^{2}}\right )}{a c}}}{2}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4-\frac {2 b \left (b +\sqrt {4 a c +b^{2}}\right )}{a c}}}{2}\right )\right )}{3 a \sqrt {\frac {-b +\sqrt {4 a c +b^{2}}}{a}}\, \sqrt {-c \,x^{4}+b \,x^{2}+a}\, \left (b +\sqrt {4 a c +b^{2}}\right )}\)	\(417\)
elliptic	\(-\frac {\sqrt {-c \,x^{4}+b \,x^{2}+a}}{3 a \,x^{3}}+\frac {2 b \sqrt {-c \,x^{4}+b \,x^{2}+a}}{3 a^{2} x}+\frac {c \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {4 a c +b^{2}}\right ) x^{2}}{a}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4-\frac {2 b \left (b +\sqrt {4 a c +b^{2}}\right )}{a c}}}{2}\right )}{12 a \sqrt {\frac {-b +\sqrt {4 a c +b^{2}}}{a}}\, \sqrt {-c \,x^{4}+b \,x^{2}+a}}-\frac {b c \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {4 a c +b^{2}}\right ) x^{2}}{a}}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4-\frac {2 b \left (b +\sqrt {4 a c +b^{2}}\right )}{a c}}}{2}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4-\frac {2 b \left (b +\sqrt {4 a c +b^{2}}\right )}{a c}}}{2}\right )\right )}{3 a \sqrt {\frac {-b +\sqrt {4 a c +b^{2}}}{a}}\, \sqrt {-c \,x^{4}+b \,x^{2}+a}\, \left (b +\sqrt {4 a c +b^{2}}\right )}\)	\(417\)

Input:

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

Output:

-1/3*(-c*x^4+b*x^2+a)^(1/2)*(-2*b*x^2+a)/a^2/x^3+1/3*c/a^2*(1/4*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)*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))-b*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))-Ell 
ipticE(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 = 0.09 (sec) , antiderivative size = 312, normalized size of antiderivative = 0.70 \[ \int \frac {1}{x^4 \sqrt {a+b x^2-c x^4}} \, dx=\frac {2 \, \sqrt {\frac {1}{2}} {\left (a^{\frac {3}{2}} b x^{3} \sqrt {\frac {b^{2} + 4 \, a c}{a^{2}}} - \sqrt {a} b^{2} x^{3}\right )} \sqrt {\frac {a \sqrt {\frac {b^{2} + 4 \, a c}{a^{2}}} - b}{a}} E(\arcsin \left (\sqrt {\frac {1}{2}} x \sqrt {\frac {a \sqrt {\frac {b^{2} + 4 \, a c}{a^{2}}} - b}{a}}\right )\,|\,-\frac {a b \sqrt {\frac {b^{2} + 4 \, a c}{a^{2}}} + b^{2} + 2 \, a c}{2 \, a c}) + \sqrt {\frac {1}{2}} {\left ({\left (a^{2} - 2 \, a b\right )} \sqrt {a} x^{3} \sqrt {\frac {b^{2} + 4 \, a c}{a^{2}}} + {\left (a b + 2 \, b^{2}\right )} \sqrt {a} x^{3}\right )} \sqrt {\frac {a \sqrt {\frac {b^{2} + 4 \, a c}{a^{2}}} - b}{a}} F(\arcsin \left (\sqrt {\frac {1}{2}} x \sqrt {\frac {a \sqrt {\frac {b^{2} + 4 \, a c}{a^{2}}} - b}{a}}\right )\,|\,-\frac {a b \sqrt {\frac {b^{2} + 4 \, a c}{a^{2}}} + b^{2} + 2 \, a c}{2 \, a c}) + 2 \, \sqrt {-c x^{4} + b x^{2} + a} {\left (2 \, a b x^{2} - a^{2}\right )}}{6 \, a^{3} x^{3}} \] Input:

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

Output:

1/6*(2*sqrt(1/2)*(a^(3/2)*b*x^3*sqrt((b^2 + 4*a*c)/a^2) - sqrt(a)*b^2*x^3) 
*sqrt((a*sqrt((b^2 + 4*a*c)/a^2) - b)/a)*elliptic_e(arcsin(sqrt(1/2)*x*sqr 
t((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 - 2*a*b)*sqrt(a)*x^3*sqrt((b^2 + 4* 
a*c)/a^2) + (a*b + 2*b^2)*sqrt(a)*x^3)*sqrt((a*sqrt((b^2 + 4*a*c)/a^2) - b 
)/a)*elliptic_f(arcsin(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)) + 2*sqrt(-c*x^4 
 + b*x^2 + a)*(2*a*b*x^2 - a^2))/(a^3*x^3)
 

Sympy [F]

\[ \int \frac {1}{x^4 \sqrt {a+b x^2-c x^4}} \, dx=\int \frac {1}{x^{4} \sqrt {a + b x^{2} - c x^{4}}}\, dx \] Input:

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

Output:

Integral(1/(x**4*sqrt(a + b*x**2 - c*x**4)), x)
 

Maxima [F]

\[ \int \frac {1}{x^4 \sqrt {a+b x^2-c x^4}} \, dx=\int { \frac {1}{\sqrt {-c x^{4} + b x^{2} + a} x^{4}} \,d x } \] Input:

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

Output:

integrate(1/(sqrt(-c*x^4 + b*x^2 + a)*x^4), x)
 

Giac [F]

\[ \int \frac {1}{x^4 \sqrt {a+b x^2-c x^4}} \, dx=\int { \frac {1}{\sqrt {-c x^{4} + b x^{2} + a} x^{4}} \,d x } \] Input:

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

Output:

integrate(1/(sqrt(-c*x^4 + b*x^2 + a)*x^4), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x^4 \sqrt {a+b x^2-c x^4}} \, dx=\int \frac {1}{x^4\,\sqrt {-c\,x^4+b\,x^2+a}} \,d x \] Input:

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

Output:

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

Reduce [F]

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

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

Output:

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