\(\int \frac {x^4}{\sqrt {a+b x^2-c x^4}} \, dx\) [1003]

Optimal result

Integrand size = 21, antiderivative size = 409 \[ \int \frac {x^4}{\sqrt {a+b x^2-c x^4}} \, dx=-\frac {x \sqrt {a+b x^2-c x^4}}{3 c}-\frac {b \left (b-\sqrt {b^2+4 a c}\right ) \sqrt {b+\sqrt {b^2+4 a c}} \sqrt {1-\frac {2 c x^2}{b-\sqrt {b^2+4 a c}}} \sqrt {1-\frac {2 c x^2}{b+\sqrt {b^2+4 a c}}} E\left (\arcsin \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2+4 a c}}}\right )|\frac {b+\sqrt {b^2+4 a c}}{b-\sqrt {b^2+4 a c}}\right )}{3 \sqrt {2} c^{5/2} \sqrt {a+b x^2-c x^4}}+\frac {\sqrt {b+\sqrt {b^2+4 a c}} \left (b^2+a c-b \sqrt {b^2+4 a c}\right ) \sqrt {1-\frac {2 c x^2}{b-\sqrt {b^2+4 a c}}} \sqrt {1-\frac {2 c x^2}{b+\sqrt {b^2+4 a c}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2+4 a c}}}\right ),\frac {b+\sqrt {b^2+4 a c}}{b-\sqrt {b^2+4 a c}}\right )}{3 \sqrt {2} c^{5/2} \sqrt {a+b x^2-c x^4}} \] Output:

-1/3*x*(-c*x^4+b*x^2+a)^(1/2)/c-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)/c^(5/2)/(-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)/c^(5/2)/(-c*x^4+b 
*x^2+a)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 10.53 (sec) , antiderivative size = 459, normalized size of antiderivative = 1.12 \[ \int \frac {x^4}{\sqrt {a+b x^2-c x^4}} \, dx=\frac {2 c \sqrt {-\frac {c}{b+\sqrt {b^2+4 a c}}} x \left (-a-b x^2+c x^4\right )-i \sqrt {2} b \left (-b+\sqrt {b^2+4 a c}\right ) \sqrt {\frac {b+\sqrt {b^2+4 a c}-2 c x^2}{b+\sqrt {b^2+4 a c}}} \sqrt {\frac {-b+\sqrt {b^2+4 a c}+2 c x^2}{-b+\sqrt {b^2+4 a c}}} E\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {-\frac {c}{b+\sqrt {b^2+4 a c}}} x\right )|\frac {b+\sqrt {b^2+4 a c}}{b-\sqrt {b^2+4 a c}}\right )+i \sqrt {2} \left (-b^2-a c+b \sqrt {b^2+4 a c}\right ) \sqrt {\frac {b+\sqrt {b^2+4 a c}-2 c x^2}{b+\sqrt {b^2+4 a c}}} \sqrt {\frac {-b+\sqrt {b^2+4 a c}+2 c x^2}{-b+\sqrt {b^2+4 a c}}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {-\frac {c}{b+\sqrt {b^2+4 a c}}} x\right ),\frac {b+\sqrt {b^2+4 a c}}{b-\sqrt {b^2+4 a c}}\right )}{6 c^2 \sqrt {-\frac {c}{b+\sqrt {b^2+4 a c}}} \sqrt {a+b x^2-c x^4}} \] Input:

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

Output:

(2*c*Sqrt[-(c/(b + Sqrt[b^2 + 4*a*c]))]*x*(-a - b*x^2 + c*x^4) - I*Sqrt[2] 
*b*(-b + Sqrt[b^2 + 4*a*c])*Sqrt[(b + Sqrt[b^2 + 4*a*c] - 2*c*x^2)/(b + Sq 
rt[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])*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*c^2*Sqrt[-(c/(b + Sqr 
t[b^2 + 4*a*c]))]*Sqrt[a + b*x^2 - c*x^4])
 

Rubi [A] (verified)

Time = 1.01 (sec) , antiderivative size = 337, normalized size of antiderivative = 0.82, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {1442, 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.

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

\(\Big \downarrow \) 1442

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

\(\Big \downarrow \) 1514

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

\(\Big \downarrow \) 399

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

\(\Big \downarrow \) 321

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

\(\Big \downarrow \) 327

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

Input:

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

Output:

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

Defintions of rubi rules used

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 1442

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

Maple [A] (verified)

Time = 2.86 (sec) , antiderivative size = 391, normalized size of antiderivative = 0.96

method	result	size
default	\(-\frac {x \sqrt {-c \,x^{4}+b \,x^{2}+a}}{3 c}+\frac {a \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {4 a c +b^{2}}\right ) x^{2}}{a}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4-\frac {2 b \left (b +\sqrt {4 a c +b^{2}}\right )}{a c}}}{2}\right )}{12 c \sqrt {\frac {-b +\sqrt {4 a c +b^{2}}}{a}}\, \sqrt {-c \,x^{4}+b \,x^{2}+a}}-\frac {b a \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {4 a c +b^{2}}\right ) x^{2}}{a}}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4-\frac {2 b \left (b +\sqrt {4 a c +b^{2}}\right )}{a c}}}{2}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4-\frac {2 b \left (b +\sqrt {4 a c +b^{2}}\right )}{a c}}}{2}\right )\right )}{3 c \sqrt {\frac {-b +\sqrt {4 a c +b^{2}}}{a}}\, \sqrt {-c \,x^{4}+b \,x^{2}+a}\, \left (b +\sqrt {4 a c +b^{2}}\right )}\)	\(391\)
risch	\(-\frac {x \sqrt {-c \,x^{4}+b \,x^{2}+a}}{3 c}+\frac {\frac {a \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {4 a c +b^{2}}\right ) x^{2}}{a}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4-\frac {2 b \left (b +\sqrt {4 a c +b^{2}}\right )}{a c}}}{2}\right )}{4 \sqrt {\frac {-b +\sqrt {4 a c +b^{2}}}{a}}\, \sqrt {-c \,x^{4}+b \,x^{2}+a}}-\frac {b a \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {4 a c +b^{2}}\right ) x^{2}}{a}}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4-\frac {2 b \left (b +\sqrt {4 a c +b^{2}}\right )}{a c}}}{2}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4-\frac {2 b \left (b +\sqrt {4 a c +b^{2}}\right )}{a c}}}{2}\right )\right )}{\sqrt {\frac {-b +\sqrt {4 a c +b^{2}}}{a}}\, \sqrt {-c \,x^{4}+b \,x^{2}+a}\, \left (b +\sqrt {4 a c +b^{2}}\right )}}{3 c}\)	\(391\)
elliptic	\(-\frac {x \sqrt {-c \,x^{4}+b \,x^{2}+a}}{3 c}+\frac {a \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {4 a c +b^{2}}\right ) x^{2}}{a}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4-\frac {2 b \left (b +\sqrt {4 a c +b^{2}}\right )}{a c}}}{2}\right )}{12 c \sqrt {\frac {-b +\sqrt {4 a c +b^{2}}}{a}}\, \sqrt {-c \,x^{4}+b \,x^{2}+a}}-\frac {b a \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {4 a c +b^{2}}\right ) x^{2}}{a}}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4-\frac {2 b \left (b +\sqrt {4 a c +b^{2}}\right )}{a c}}}{2}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4-\frac {2 b \left (b +\sqrt {4 a c +b^{2}}\right )}{a c}}}{2}\right )\right )}{3 c \sqrt {\frac {-b +\sqrt {4 a c +b^{2}}}{a}}\, \sqrt {-c \,x^{4}+b \,x^{2}+a}\, \left (b +\sqrt {4 a c +b^{2}}\right )}\)	\(391\)

Input:

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

Output:

-1/3*x*(-c*x^4+b*x^2+a)^(1/2)/c+1/12/c*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))-1/3/c*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))-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)))
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 313, normalized size of antiderivative = 0.77 \[ \int \frac {x^4}{\sqrt {a+b x^2-c x^4}} \, dx=-\frac {2 \, \sqrt {\frac {1}{2}} {\left (b \sqrt {-c} c x \sqrt {\frac {b^{2} + 4 \, a c}{c^{2}}} + b^{2} \sqrt {-c} x\right )} \sqrt {\frac {c \sqrt {\frac {b^{2} + 4 \, a c}{c^{2}}} + b}{c}} E(\arcsin \left (\frac {\sqrt {\frac {1}{2}} \sqrt {\frac {c \sqrt {\frac {b^{2} + 4 \, a c}{c^{2}}} + b}{c}}}{x}\right )\,|\,\frac {b c \sqrt {\frac {b^{2} + 4 \, a c}{c^{2}}} - b^{2} - 2 \, a c}{2 \, a c}) - \sqrt {\frac {1}{2}} {\left ({\left (2 \, b c + c^{2}\right )} \sqrt {-c} x \sqrt {\frac {b^{2} + 4 \, a c}{c^{2}}} + {\left (2 \, b^{2} - b c\right )} \sqrt {-c} x\right )} \sqrt {\frac {c \sqrt {\frac {b^{2} + 4 \, a c}{c^{2}}} + b}{c}} F(\arcsin \left (\frac {\sqrt {\frac {1}{2}} \sqrt {\frac {c \sqrt {\frac {b^{2} + 4 \, a c}{c^{2}}} + b}{c}}}{x}\right )\,|\,\frac {b c \sqrt {\frac {b^{2} + 4 \, a c}{c^{2}}} - b^{2} - 2 \, a c}{2 \, a c}) + 2 \, \sqrt {-c x^{4} + b x^{2} + a} {\left (c^{2} x^{2} + 2 \, b c\right )}}{6 \, c^{3} x} \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

( - sqrt(a + b*x**2 - c*x**4)*x + int(sqrt(a + b*x**2 - c*x**4)/(a + b*x** 
2 - c*x**4),x)*a + 2*int((sqrt(a + b*x**2 - c*x**4)*x**2)/(a + b*x**2 - c* 
x**4),x)*b)/(3*c)