\(\int \frac {1}{x^4 \sqrt {a+b x^2-c x^4}} \, dx\) [980]

Optimal result

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

[Out]

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

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 445, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {1137, 1295, 1216, 538, 435, 430} \[ \int \frac {1}{x^4 \sqrt {a+b x^2-c x^4}} \, dx=\frac {\sqrt {\sqrt {4 a c+b^2}+b} \left (-b \sqrt {4 a c+b^2}+a c+b^2\right ) \sqrt {1-\frac {2 c x^2}{b-\sqrt {4 a c+b^2}}} \sqrt {1-\frac {2 c x^2}{\sqrt {4 a c+b^2}+b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2+4 a c}}}\right ),\frac {b+\sqrt {b^2+4 a c}}{b-\sqrt {b^2+4 a c}}\right )}{3 \sqrt {2} a^2 \sqrt {c} \sqrt {a+b x^2-c x^4}}-\frac {b \left (b-\sqrt {4 a c+b^2}\right ) \sqrt {\sqrt {4 a c+b^2}+b} \sqrt {1-\frac {2 c x^2}{b-\sqrt {4 a c+b^2}}} \sqrt {1-\frac {2 c x^2}{\sqrt {4 a c+b^2}+b}} E\left (\arcsin \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2+4 a c}}}\right )|\frac {b+\sqrt {b^2+4 a c}}{b-\sqrt {b^2+4 a c}}\right )}{3 \sqrt {2} a^2 \sqrt {c} \sqrt {a+b x^2-c x^4}}+\frac {2 b \sqrt {a+b x^2-c x^4}}{3 a^2 x}-\frac {\sqrt {a+b x^2-c x^4}}{3 a x^3} \]

[In]

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

[Out]

-1/3*Sqrt[a + b*x^2 - c*x^4]/(a*x^3) + (2*b*Sqrt[a + b*x^2 - c*x^4])/(3*a^2*x) - (b*(b - Sqrt[b^2 + 4*a*c])*Sq
rt[b + Sqrt[b^2 + 4*a*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
])]*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])])/(3*Sqrt[2]*a^2*Sqrt[c]*Sqrt[a + b*x^2 - c*x^4]) + (Sqrt[b + Sqrt[b^2 + 4*a*c]]*(b^2 + a*c - b*Sqrt[
b^2 + 4*a*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])]*Elliptic
F[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])])/(
3*Sqrt[2]*a^2*Sqrt[c]*Sqrt[a + b*x^2 - c*x^4])

Rule 430

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(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] && Gt
Q[a, 0] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])

Rule 435

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*Ell
ipticE[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 538

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

Rule 1137

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] - Dist[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] && In
tegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])

Rule 1216

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}
, Dist[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)/(Sqr
t[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 1295

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] + Dist[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])

Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {a+b x^2-c x^4}}{3 a x^3}+\frac {\int \frac {-2 b+c x^2}{x^2 \sqrt {a+b x^2-c x^4}} \, dx}{3 a} \\ & = -\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 {\int \frac {-a c-2 b c x^2}{\sqrt {a+b x^2-c x^4}} \, dx}{3 a^2} \\ & = -\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 {\left (\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}}}\right ) \int \frac {-a c-2 b c x^2}{\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 a^2 \sqrt {a+b x^2-c x^4}} \\ & = -\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 {\left (b \left (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}}}\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}{3 a^2 \sqrt {a+b x^2-c x^4}}+\frac {\left (\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}}}\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}{3 a^2 \sqrt {a+b x^2-c x^4}} \\ & = -\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 (\sin ^{-1}\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}}} F\left (\sin ^{-1}\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}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

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

[In]

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

[Out]

(-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])]*Ellipti
cF[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])

Maple [A] (verified)

Time = 1.10 (sec) , antiderivative size = 402, normalized size of antiderivative = 0.90

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

[In]

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

[Out]

-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))-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)

none

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

[In]

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

[Out]

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*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 - 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]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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