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

Optimal. Leaf size=445 \[ \frac{\sqrt{\sqrt{4 a c+b^2}+b} \left (-b \sqrt{4 a c+b^2}+a c+b^2\right ) \sqrt{1-\frac{2 c x^2}{b-\sqrt{4 a c+b^2}}} \sqrt{1-\frac{2 c x^2}{\sqrt{4 a c+b^2}+b}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{4 a c+b^2}+b}}\right ),\frac{\sqrt{4 a c+b^2}+b}{b-\sqrt{4 a c+b^2}}\right )}{3 \sqrt{2} a^2 \sqrt{c} \sqrt{a+b x^2-c x^4}}-\frac{b \left (b-\sqrt{4 a c+b^2}\right ) \sqrt{\sqrt{4 a c+b^2}+b} \sqrt{1-\frac{2 c x^2}{b-\sqrt{4 a c+b^2}}} \sqrt{1-\frac{2 c x^2}{\sqrt{4 a c+b^2}+b}} E\left (\sin ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2+4 a c}}}\right )|\frac{b+\sqrt{b^2+4 a c}}{b-\sqrt{b^2+4 a c}}\right )}{3 \sqrt{2} a^2 \sqrt{c} \sqrt{a+b x^2-c x^4}}+\frac{2 b \sqrt{a+b x^2-c x^4}}{3 a^2 x}-\frac{\sqrt{a+b x^2-c x^4}}{3 a x^3} \]

[Out]

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

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Rubi [A]  time = 0.469404, antiderivative size = 445, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {1123, 1281, 1202, 524, 424, 419} \[ \frac{\sqrt{\sqrt{4 a c+b^2}+b} \left (-b \sqrt{4 a c+b^2}+a c+b^2\right ) \sqrt{1-\frac{2 c x^2}{b-\sqrt{4 a c+b^2}}} \sqrt{1-\frac{2 c x^2}{\sqrt{4 a c+b^2}+b}} F\left (\sin ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2+4 a c}}}\right )|\frac{b+\sqrt{b^2+4 a c}}{b-\sqrt{b^2+4 a c}}\right )}{3 \sqrt{2} a^2 \sqrt{c} \sqrt{a+b x^2-c x^4}}-\frac{b \left (b-\sqrt{4 a c+b^2}\right ) \sqrt{\sqrt{4 a c+b^2}+b} \sqrt{1-\frac{2 c x^2}{b-\sqrt{4 a c+b^2}}} \sqrt{1-\frac{2 c x^2}{\sqrt{4 a c+b^2}+b}} E\left (\sin ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2+4 a c}}}\right )|\frac{b+\sqrt{b^2+4 a c}}{b-\sqrt{b^2+4 a c}}\right )}{3 \sqrt{2} a^2 \sqrt{c} \sqrt{a+b x^2-c x^4}}+\frac{2 b \sqrt{a+b x^2-c x^4}}{3 a^2 x}-\frac{\sqrt{a+b x^2-c x^4}}{3 a x^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 1123

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 1281

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])

Rule 1202

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 524

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 424

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

Rule 419

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

Rubi steps

\begin{align*} \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{\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]  time = 0.743167, size = 472, normalized size = 1.06 \[ \frac{i \sqrt{2} x^3 \left (b \sqrt{4 a c+b^2}-a c-b^2\right ) \sqrt{\frac{\sqrt{4 a c+b^2}+b-2 c x^2}{\sqrt{4 a c+b^2}+b}} \sqrt{\frac{\sqrt{4 a c+b^2}-b+2 c x^2}{\sqrt{4 a c+b^2}-b}} \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{2} x \sqrt{-\frac{c}{\sqrt{4 a c+b^2}+b}}\right ),\frac{\sqrt{4 a c+b^2}+b}{b-\sqrt{4 a c+b^2}}\right )-2 \sqrt{-\frac{c}{\sqrt{4 a c+b^2}+b}} \left (a-2 b x^2\right ) \left (a+b x^2-c x^4\right )-i \sqrt{2} b x^3 \left (\sqrt{4 a c+b^2}-b\right ) \sqrt{\frac{\sqrt{4 a c+b^2}+b-2 c x^2}{\sqrt{4 a c+b^2}+b}} \sqrt{\frac{\sqrt{4 a c+b^2}-b+2 c x^2}{\sqrt{4 a c+b^2}-b}} E\left (i \sinh ^{-1}\left (\sqrt{2} \sqrt{-\frac{c}{b+\sqrt{b^2+4 a c}}} x\right )|\frac{b+\sqrt{b^2+4 a c}}{b-\sqrt{b^2+4 a c}}\right )}{6 a^2 x^3 \sqrt{-\frac{c}{\sqrt{4 a c+b^2}+b}} \sqrt{a+b x^2-c x^4}} \]

Antiderivative was successfully verified.

[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])

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Maple [A]  time = 0.233, size = 417, normalized size = 0.9 \begin{align*} -{\frac{1}{3\,a{x}^{3}}\sqrt{-c{x}^{4}+b{x}^{2}+a}}+{\frac{2\,b}{3\,{a}^{2}x}\sqrt{-c{x}^{4}+b{x}^{2}+a}}+{\frac{c\sqrt{2}}{12\,a}\sqrt{4-2\,{\frac{ \left ( -b+\sqrt{4\,ac+{b}^{2}} \right ){x}^{2}}{a}}}\sqrt{4+2\,{\frac{ \left ( b+\sqrt{4\,ac+{b}^{2}} \right ){x}^{2}}{a}}}{\it EllipticF} \left ({\frac{x\sqrt{2}}{2}\sqrt{{\frac{1}{a} \left ( -b+\sqrt{4\,ac+{b}^{2}} \right ) }}},{\frac{1}{2}\sqrt{-4-2\,{\frac{b \left ( b+\sqrt{4\,ac+{b}^{2}} \right ) }{ac}}}} \right ){\frac{1}{\sqrt{{\frac{1}{a} \left ( -b+\sqrt{4\,ac+{b}^{2}} \right ) }}}}{\frac{1}{\sqrt{-c{x}^{4}+b{x}^{2}+a}}}}-{\frac{bc\sqrt{2}}{3\,a}\sqrt{4-2\,{\frac{ \left ( -b+\sqrt{4\,ac+{b}^{2}} \right ){x}^{2}}{a}}}\sqrt{4+2\,{\frac{ \left ( b+\sqrt{4\,ac+{b}^{2}} \right ){x}^{2}}{a}}} \left ({\it EllipticF} \left ({\frac{x\sqrt{2}}{2}\sqrt{{\frac{1}{a} \left ( -b+\sqrt{4\,ac+{b}^{2}} \right ) }}},{\frac{1}{2}\sqrt{-4-2\,{\frac{b \left ( b+\sqrt{4\,ac+{b}^{2}} \right ) }{ac}}}} \right ) -{\it EllipticE} \left ({\frac{x\sqrt{2}}{2}\sqrt{{\frac{1}{a} \left ( -b+\sqrt{4\,ac+{b}^{2}} \right ) }}},{\frac{1}{2}\sqrt{-4-2\,{\frac{b \left ( b+\sqrt{4\,ac+{b}^{2}} \right ) }{ac}}}} \right ) \right ){\frac{1}{\sqrt{{\frac{1}{a} \left ( -b+\sqrt{4\,ac+{b}^{2}} \right ) }}}}{\frac{1}{\sqrt{-c{x}^{4}+b{x}^{2}+a}}} \left ( b+\sqrt{4\,ac+{b}^{2}} \right ) ^{-1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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+1/12*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*
b*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)/(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)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-c x^{4} + b x^{2} + a} x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-c x^{4} + b x^{2} + a}}{c x^{8} - b x^{6} - a x^{4}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{4} \sqrt{a + b x^{2} - c x^{4}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-c x^{4} + b x^{2} + a} x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)