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

Optimal. Leaf size=827 \[ \frac {x \left (i a^{3/2} x^6-i \sqrt {a} c x^4-a \sqrt {b} x^4+i a \sqrt {a x^4-c x^2-b} x^4-i \sqrt {a} b x^2+\sqrt {b} c x^2-\frac {1}{2} i c \sqrt {a x^4-c x^2-b} x^2-\sqrt {a} \sqrt {b} \sqrt {a x^4-c x^2-b} x^2+b^{3/2}-i b \sqrt {a x^4-c x^2-b}\right )}{\left (a x^4-b\right ) \left (-2 i a x^4+i c x^2+2 \sqrt {a} \sqrt {b} x^2-2 i \sqrt {a} \sqrt {a x^4-c x^2-b} x^2+2 i b+2 \sqrt {b} \sqrt {a x^4-c x^2-b}\right )}+\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt [4]{a} \sqrt [4]{b} \tan ^{-1}\left (\frac {\sqrt {c-(2-2 i) \sqrt [4]{a} \sqrt [4]{b} \sqrt {c}-2 i \sqrt {a} \sqrt {b}} x}{\sqrt {a} x^2+i \sqrt {b}+\sqrt {a x^4-c x^2-b}}\right )}{\sqrt {c} \sqrt {c-(2-2 i) \sqrt [4]{a} \sqrt [4]{b} \sqrt {c}-2 i \sqrt {a} \sqrt {b}}}-\frac {\tan ^{-1}\left (\frac {\sqrt {c-(2-2 i) \sqrt [4]{a} \sqrt [4]{b} \sqrt {c}-2 i \sqrt {a} \sqrt {b}} x}{\sqrt {a} x^2+i \sqrt {b}+\sqrt {a x^4-c x^2-b}}\right )}{2 \sqrt {c-(2-2 i) \sqrt [4]{a} \sqrt [4]{b} \sqrt {c}-2 i \sqrt {a} \sqrt {b}}}-\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt [4]{a} \sqrt [4]{b} \tan ^{-1}\left (\frac {\sqrt {c+(2-2 i) \sqrt [4]{a} \sqrt [4]{b} \sqrt {c}-2 i \sqrt {a} \sqrt {b}} x}{\sqrt {a} x^2+i \sqrt {b}+\sqrt {a x^4-c x^2-b}}\right )}{\sqrt {c} \sqrt {c+(2-2 i) \sqrt [4]{a} \sqrt [4]{b} \sqrt {c}-2 i \sqrt {a} \sqrt {b}}}-\frac {\tan ^{-1}\left (\frac {\sqrt {c+(2-2 i) \sqrt [4]{a} \sqrt [4]{b} \sqrt {c}-2 i \sqrt {a} \sqrt {b}} x}{\sqrt {a} x^2+i \sqrt {b}+\sqrt {a x^4-c x^2-b}}\right )}{2 \sqrt {c+(2-2 i) \sqrt [4]{a} \sqrt [4]{b} \sqrt {c}-2 i \sqrt {a} \sqrt {b}}} \]

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Rubi [C]  time = 5.57, antiderivative size = 908, normalized size of antiderivative = 1.10, number of steps used = 50, number of rules used = 10, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {6742, 1226, 1202, 524, 424, 419, 1220, 537, 6725, 1208} \begin {gather*} \frac {\sqrt {a x^4-c x^2-b} x}{4 \sqrt {b} \left (\sqrt {b}-\sqrt {a} x^2\right )}+\frac {\sqrt {a x^4-c x^2-b} x}{4 \sqrt {b} \left (\sqrt {a} x^2+\sqrt {b}\right )}+\frac {\left (c+2 \sqrt {a} \sqrt {b}-\sqrt {c^2+4 a b}\right ) \sqrt {c+\sqrt {c^2+4 a b}} \sqrt {1-\frac {2 a x^2}{c-\sqrt {c^2+4 a b}}} \sqrt {1-\frac {2 a x^2}{c+\sqrt {c^2+4 a b}}} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {a} x}{\sqrt {c+\sqrt {c^2+4 a b}}}\right )|\frac {c+\sqrt {c^2+4 a b}}{c-\sqrt {c^2+4 a b}}\right )}{8 \sqrt {2} a \sqrt {b} \sqrt {a x^4-c x^2-b}}+\frac {\left (-c+2 \sqrt {a} \sqrt {b}+\sqrt {c^2+4 a b}\right ) \sqrt {c+\sqrt {c^2+4 a b}} \sqrt {1-\frac {2 a x^2}{c-\sqrt {c^2+4 a b}}} \sqrt {1-\frac {2 a x^2}{c+\sqrt {c^2+4 a b}}} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {a} x}{\sqrt {c+\sqrt {c^2+4 a b}}}\right )|\frac {c+\sqrt {c^2+4 a b}}{c-\sqrt {c^2+4 a b}}\right )}{8 \sqrt {2} a \sqrt {b} \sqrt {a x^4-c x^2-b}}-\frac {\sqrt {c+\sqrt {c^2+4 a b}} \sqrt {1-\frac {2 a x^2}{c-\sqrt {c^2+4 a b}}} \sqrt {1-\frac {2 a x^2}{c+\sqrt {c^2+4 a b}}} \Pi \left (-\frac {c+\sqrt {c^2+4 a b}}{2 \sqrt {a} \sqrt {b}};\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {a} x}{\sqrt {c+\sqrt {c^2+4 a b}}}\right )|\frac {c+\sqrt {c^2+4 a b}}{c-\sqrt {c^2+4 a b}}\right )}{2 \sqrt {2} \sqrt {a} \sqrt {a x^4-c x^2-b}}-\frac {\sqrt {c+\sqrt {c^2+4 a b}} \sqrt {1-\frac {2 a x^2}{c-\sqrt {c^2+4 a b}}} \sqrt {1-\frac {2 a x^2}{c+\sqrt {c^2+4 a b}}} \Pi \left (\frac {c+\sqrt {c^2+4 a b}}{2 \sqrt {a} \sqrt {b}};\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {a} x}{\sqrt {c+\sqrt {c^2+4 a b}}}\right )|\frac {c+\sqrt {c^2+4 a b}}{c-\sqrt {c^2+4 a b}}\right )}{2 \sqrt {2} \sqrt {a} \sqrt {a x^4-c x^2-b}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x*Sqrt[-b - c*x^2 + a*x^4])/(4*Sqrt[b]*(Sqrt[b] - Sqrt[a]*x^2)) + (x*Sqrt[-b - c*x^2 + a*x^4])/(4*Sqrt[b]*(Sq
rt[b] + Sqrt[a]*x^2)) + ((2*Sqrt[a]*Sqrt[b] + c - Sqrt[4*a*b + c^2])*Sqrt[c + Sqrt[4*a*b + c^2]]*Sqrt[1 - (2*a
*x^2)/(c - Sqrt[4*a*b + c^2])]*Sqrt[1 - (2*a*x^2)/(c + Sqrt[4*a*b + c^2])]*EllipticF[ArcSin[(Sqrt[2]*Sqrt[a]*x
)/Sqrt[c + Sqrt[4*a*b + c^2]]], (c + Sqrt[4*a*b + c^2])/(c - Sqrt[4*a*b + c^2])])/(8*Sqrt[2]*a*Sqrt[b]*Sqrt[-b
- c*x^2 + a*x^4]) + ((2*Sqrt[a]*Sqrt[b] - c + Sqrt[4*a*b + c^2])*Sqrt[c + Sqrt[4*a*b + c^2]]*Sqrt[1 - (2*a*x^
2)/(c - Sqrt[4*a*b + c^2])]*Sqrt[1 - (2*a*x^2)/(c + Sqrt[4*a*b + c^2])]*EllipticF[ArcSin[(Sqrt[2]*Sqrt[a]*x)/S
qrt[c + Sqrt[4*a*b + c^2]]], (c + Sqrt[4*a*b + c^2])/(c - Sqrt[4*a*b + c^2])])/(8*Sqrt[2]*a*Sqrt[b]*Sqrt[-b -
c*x^2 + a*x^4]) - (Sqrt[c + Sqrt[4*a*b + c^2]]*Sqrt[1 - (2*a*x^2)/(c - Sqrt[4*a*b + c^2])]*Sqrt[1 - (2*a*x^2)/
(c + Sqrt[4*a*b + c^2])]*EllipticPi[-1/2*(c + Sqrt[4*a*b + c^2])/(Sqrt[a]*Sqrt[b]), ArcSin[(Sqrt[2]*Sqrt[a]*x)
/Sqrt[c + Sqrt[4*a*b + c^2]]], (c + Sqrt[4*a*b + c^2])/(c - Sqrt[4*a*b + c^2])])/(2*Sqrt[2]*Sqrt[a]*Sqrt[-b -
c*x^2 + a*x^4]) - (Sqrt[c + Sqrt[4*a*b + c^2]]*Sqrt[1 - (2*a*x^2)/(c - Sqrt[4*a*b + c^2])]*Sqrt[1 - (2*a*x^2)/
(c + Sqrt[4*a*b + c^2])]*EllipticPi[(c + Sqrt[4*a*b + c^2])/(2*Sqrt[a]*Sqrt[b]), ArcSin[(Sqrt[2]*Sqrt[a]*x)/Sq
rt[c + Sqrt[4*a*b + c^2]]], (c + Sqrt[4*a*b + c^2])/(c - Sqrt[4*a*b + c^2])])/(2*Sqrt[2]*Sqrt[a]*Sqrt[-b - c*x
^2 + a*x^4])

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)])

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 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 537

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

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 1208

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

Rule 1220

Int[1/(((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[1/((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 1226

Int[Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]/((d_) + (e_.)*(x_)^2)^2, x_Symbol] :> Simp[(x*Sqrt[a + b*x^2 + c*
x^4])/(2*d*(d + e*x^2)), x] + (Dist[c/(2*d*e^2), Int[(d - e*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x] - Dist[(c*d^2
- a*e^2)/(2*d*e^2), Int[1/((d + e*x^2)*Sqrt[a + b*x^2 + c*x^4]), x], x]) /; FreeQ[{a, b, c, d, e}, x] && NeQ[
b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]

Rule 6725

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {align*} \int \frac {\left (b+a x^4\right ) \sqrt {-b-c x^2+a x^4}}{\left (-b+a x^4\right )^2} \, dx &=\int \left (\frac {2 b \sqrt {-b-c x^2+a x^4}}{\left (-b+a x^4\right )^2}+\frac {\sqrt {-b-c x^2+a x^4}}{-b+a x^4}\right ) \, dx\\ &=(2 b) \int \frac {\sqrt {-b-c x^2+a x^4}}{\left (-b+a x^4\right )^2} \, dx+\int \frac {\sqrt {-b-c x^2+a x^4}}{-b+a x^4} \, dx\\ &=(2 b) \int \left (\frac {a \sqrt {-b-c x^2+a x^4}}{4 b \left (\sqrt {a} \sqrt {b}-a x^2\right )^2}+\frac {a \sqrt {-b-c x^2+a x^4}}{4 b \left (\sqrt {a} \sqrt {b}+a x^2\right )^2}+\frac {a \sqrt {-b-c x^2+a x^4}}{2 b \left (a b-a^2 x^4\right )}\right ) \, dx+\int \left (-\frac {\sqrt {-b-c x^2+a x^4}}{2 \sqrt {b} \left (\sqrt {b}-\sqrt {a} x^2\right )}-\frac {\sqrt {-b-c x^2+a x^4}}{2 \sqrt {b} \left (\sqrt {b}+\sqrt {a} x^2\right )}\right ) \, dx\\ &=\frac {1}{2} a \int \frac {\sqrt {-b-c x^2+a x^4}}{\left (\sqrt {a} \sqrt {b}-a x^2\right )^2} \, dx+\frac {1}{2} a \int \frac {\sqrt {-b-c x^2+a x^4}}{\left (\sqrt {a} \sqrt {b}+a x^2\right )^2} \, dx+a \int \frac {\sqrt {-b-c x^2+a x^4}}{a b-a^2 x^4} \, dx-\frac {\int \frac {\sqrt {-b-c x^2+a x^4}}{\sqrt {b}-\sqrt {a} x^2} \, dx}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {-b-c x^2+a x^4}}{\sqrt {b}+\sqrt {a} x^2} \, dx}{2 \sqrt {b}}\\ &=\frac {x \sqrt {-b-c x^2+a x^4}}{4 \sqrt {b} \left (\sqrt {b}-\sqrt {a} x^2\right )}+\frac {x \sqrt {-b-c x^2+a x^4}}{4 \sqrt {b} \left (\sqrt {b}+\sqrt {a} x^2\right )}+a \int \left (\frac {\sqrt {-b-c x^2+a x^4}}{2 a \sqrt {b} \left (\sqrt {b}-\sqrt {a} x^2\right )}+\frac {\sqrt {-b-c x^2+a x^4}}{2 a \sqrt {b} \left (\sqrt {b}+\sqrt {a} x^2\right )}\right ) \, dx+\frac {\int \frac {a \sqrt {b}+\sqrt {a} c-a^{3/2} x^2}{\sqrt {-b-c x^2+a x^4}} \, dx}{2 a \sqrt {b}}+\frac {\int \frac {a \sqrt {b}-\sqrt {a} c+a^{3/2} x^2}{\sqrt {-b-c x^2+a x^4}} \, dx}{2 a \sqrt {b}}+\frac {\int \frac {\sqrt {a} \sqrt {b}-a x^2}{\sqrt {-b-c x^2+a x^4}} \, dx}{4 \sqrt {a} \sqrt {b}}+\frac {\int \frac {\sqrt {a} \sqrt {b}+a x^2}{\sqrt {-b-c x^2+a x^4}} \, dx}{4 \sqrt {a} \sqrt {b}}-\frac {1}{2} \left (\sqrt {a} \sqrt {b}\right ) \int \frac {1}{\left (\sqrt {a} \sqrt {b}-a x^2\right ) \sqrt {-b-c x^2+a x^4}} \, dx-\frac {1}{2} \left (\sqrt {a} \sqrt {b}\right ) \int \frac {1}{\left (\sqrt {a} \sqrt {b}+a x^2\right ) \sqrt {-b-c x^2+a x^4}} \, dx+\frac {c \int \frac {1}{\left (\sqrt {b}-\sqrt {a} x^2\right ) \sqrt {-b-c x^2+a x^4}} \, dx}{2 \sqrt {a}}-\frac {c \int \frac {1}{\left (\sqrt {b}+\sqrt {a} x^2\right ) \sqrt {-b-c x^2+a x^4}} \, dx}{2 \sqrt {a}}\\ &=\frac {x \sqrt {-b-c x^2+a x^4}}{4 \sqrt {b} \left (\sqrt {b}-\sqrt {a} x^2\right )}+\frac {x \sqrt {-b-c x^2+a x^4}}{4 \sqrt {b} \left (\sqrt {b}+\sqrt {a} x^2\right )}+\frac {\int \frac {\sqrt {-b-c x^2+a x^4}}{\sqrt {b}-\sqrt {a} x^2} \, dx}{2 \sqrt {b}}+\frac {\int \frac {\sqrt {-b-c x^2+a x^4}}{\sqrt {b}+\sqrt {a} x^2} \, dx}{2 \sqrt {b}}+\frac {\left (\sqrt {1+\frac {2 a x^2}{-c-\sqrt {4 a b+c^2}}} \sqrt {1+\frac {2 a x^2}{-c+\sqrt {4 a b+c^2}}}\right ) \int \frac {a \sqrt {b}+\sqrt {a} c-a^{3/2} x^2}{\sqrt {1+\frac {2 a x^2}{-c-\sqrt {4 a b+c^2}}} \sqrt {1+\frac {2 a x^2}{-c+\sqrt {4 a b+c^2}}}} \, dx}{2 a \sqrt {b} \sqrt {-b-c x^2+a x^4}}+\frac {\left (\sqrt {1+\frac {2 a x^2}{-c-\sqrt {4 a b+c^2}}} \sqrt {1+\frac {2 a x^2}{-c+\sqrt {4 a b+c^2}}}\right ) \int \frac {a \sqrt {b}-\sqrt {a} c+a^{3/2} x^2}{\sqrt {1+\frac {2 a x^2}{-c-\sqrt {4 a b+c^2}}} \sqrt {1+\frac {2 a x^2}{-c+\sqrt {4 a b+c^2}}}} \, dx}{2 a \sqrt {b} \sqrt {-b-c x^2+a x^4}}+\frac {\left (\sqrt {1+\frac {2 a x^2}{-c-\sqrt {4 a b+c^2}}} \sqrt {1+\frac {2 a x^2}{-c+\sqrt {4 a b+c^2}}}\right ) \int \frac {\sqrt {a} \sqrt {b}-a x^2}{\sqrt {1+\frac {2 a x^2}{-c-\sqrt {4 a b+c^2}}} \sqrt {1+\frac {2 a x^2}{-c+\sqrt {4 a b+c^2}}}} \, dx}{4 \sqrt {a} \sqrt {b} \sqrt {-b-c x^2+a x^4}}+\frac {\left (\sqrt {1+\frac {2 a x^2}{-c-\sqrt {4 a b+c^2}}} \sqrt {1+\frac {2 a x^2}{-c+\sqrt {4 a b+c^2}}}\right ) \int \frac {\sqrt {a} \sqrt {b}+a x^2}{\sqrt {1+\frac {2 a x^2}{-c-\sqrt {4 a b+c^2}}} \sqrt {1+\frac {2 a x^2}{-c+\sqrt {4 a b+c^2}}}} \, dx}{4 \sqrt {a} \sqrt {b} \sqrt {-b-c x^2+a x^4}}-\frac {\left (\sqrt {a} \sqrt {b} \sqrt {1+\frac {2 a x^2}{-c-\sqrt {4 a b+c^2}}} \sqrt {1+\frac {2 a x^2}{-c+\sqrt {4 a b+c^2}}}\right ) \int \frac {1}{\left (\sqrt {a} \sqrt {b}-a x^2\right ) \sqrt {1+\frac {2 a x^2}{-c-\sqrt {4 a b+c^2}}} \sqrt {1+\frac {2 a x^2}{-c+\sqrt {4 a b+c^2}}}} \, dx}{2 \sqrt {-b-c x^2+a x^4}}-\frac {\left (\sqrt {a} \sqrt {b} \sqrt {1+\frac {2 a x^2}{-c-\sqrt {4 a b+c^2}}} \sqrt {1+\frac {2 a x^2}{-c+\sqrt {4 a b+c^2}}}\right ) \int \frac {1}{\left (\sqrt {a} \sqrt {b}+a x^2\right ) \sqrt {1+\frac {2 a x^2}{-c-\sqrt {4 a b+c^2}}} \sqrt {1+\frac {2 a x^2}{-c+\sqrt {4 a b+c^2}}}} \, dx}{2 \sqrt {-b-c x^2+a x^4}}+\frac {\left (c \sqrt {1+\frac {2 a x^2}{-c-\sqrt {4 a b+c^2}}} \sqrt {1+\frac {2 a x^2}{-c+\sqrt {4 a b+c^2}}}\right ) \int \frac {1}{\left (\sqrt {b}-\sqrt {a} x^2\right ) \sqrt {1+\frac {2 a x^2}{-c-\sqrt {4 a b+c^2}}} \sqrt {1+\frac {2 a x^2}{-c+\sqrt {4 a b+c^2}}}} \, dx}{2 \sqrt {a} \sqrt {-b-c x^2+a x^4}}-\frac {\left (c \sqrt {1+\frac {2 a x^2}{-c-\sqrt {4 a b+c^2}}} \sqrt {1+\frac {2 a x^2}{-c+\sqrt {4 a b+c^2}}}\right ) \int \frac {1}{\left (\sqrt {b}+\sqrt {a} x^2\right ) \sqrt {1+\frac {2 a x^2}{-c-\sqrt {4 a b+c^2}}} \sqrt {1+\frac {2 a x^2}{-c+\sqrt {4 a b+c^2}}}} \, dx}{2 \sqrt {a} \sqrt {-b-c x^2+a x^4}}\\ &=\frac {x \sqrt {-b-c x^2+a x^4}}{4 \sqrt {b} \left (\sqrt {b}-\sqrt {a} x^2\right )}+\frac {x \sqrt {-b-c x^2+a x^4}}{4 \sqrt {b} \left (\sqrt {b}+\sqrt {a} x^2\right )}-\frac {\sqrt {c+\sqrt {4 a b+c^2}} \sqrt {1-\frac {2 a x^2}{c-\sqrt {4 a b+c^2}}} \sqrt {1-\frac {2 a x^2}{c+\sqrt {4 a b+c^2}}} \Pi \left (-\frac {c+\sqrt {4 a b+c^2}}{2 \sqrt {a} \sqrt {b}};\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {a} x}{\sqrt {c+\sqrt {4 a b+c^2}}}\right )|\frac {c+\sqrt {4 a b+c^2}}{c-\sqrt {4 a b+c^2}}\right )}{2 \sqrt {2} \sqrt {a} \sqrt {-b-c x^2+a x^4}}-\frac {c \sqrt {c+\sqrt {4 a b+c^2}} \sqrt {1-\frac {2 a x^2}{c-\sqrt {4 a b+c^2}}} \sqrt {1-\frac {2 a x^2}{c+\sqrt {4 a b+c^2}}} \Pi \left (-\frac {c+\sqrt {4 a b+c^2}}{2 \sqrt {a} \sqrt {b}};\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {a} x}{\sqrt {c+\sqrt {4 a b+c^2}}}\right )|\frac {c+\sqrt {4 a b+c^2}}{c-\sqrt {4 a b+c^2}}\right )}{2 \sqrt {2} a \sqrt {b} \sqrt {-b-c x^2+a x^4}}-\frac {\sqrt {c+\sqrt {4 a b+c^2}} \sqrt {1-\frac {2 a x^2}{c-\sqrt {4 a b+c^2}}} \sqrt {1-\frac {2 a x^2}{c+\sqrt {4 a b+c^2}}} \Pi \left (\frac {c+\sqrt {4 a b+c^2}}{2 \sqrt {a} \sqrt {b}};\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {a} x}{\sqrt {c+\sqrt {4 a b+c^2}}}\right )|\frac {c+\sqrt {4 a b+c^2}}{c-\sqrt {4 a b+c^2}}\right )}{2 \sqrt {2} \sqrt {a} \sqrt {-b-c x^2+a x^4}}+\frac {c \sqrt {c+\sqrt {4 a b+c^2}} \sqrt {1-\frac {2 a x^2}{c-\sqrt {4 a b+c^2}}} \sqrt {1-\frac {2 a x^2}{c+\sqrt {4 a b+c^2}}} \Pi \left (\frac {c+\sqrt {4 a b+c^2}}{2 \sqrt {a} \sqrt {b}};\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {a} x}{\sqrt {c+\sqrt {4 a b+c^2}}}\right )|\frac {c+\sqrt {4 a b+c^2}}{c-\sqrt {4 a b+c^2}}\right )}{2 \sqrt {2} a \sqrt {b} \sqrt {-b-c x^2+a x^4}}-\frac {\int \frac {a \sqrt {b}+\sqrt {a} c-a^{3/2} x^2}{\sqrt {-b-c x^2+a x^4}} \, dx}{2 a \sqrt {b}}-\frac {\int \frac {a \sqrt {b}-\sqrt {a} c+a^{3/2} x^2}{\sqrt {-b-c x^2+a x^4}} \, dx}{2 a \sqrt {b}}-\frac {c \int \frac {1}{\left (\sqrt {b}-\sqrt {a} x^2\right ) \sqrt {-b-c x^2+a x^4}} \, dx}{2 \sqrt {a}}+\frac {c \int \frac {1}{\left (\sqrt {b}+\sqrt {a} x^2\right ) \sqrt {-b-c x^2+a x^4}} \, dx}{2 \sqrt {a}}+\frac {\left (\left (2 \sqrt {a} \sqrt {b}-c-\sqrt {4 a b+c^2}\right ) \sqrt {1+\frac {2 a x^2}{-c-\sqrt {4 a b+c^2}}} \sqrt {1+\frac {2 a x^2}{-c+\sqrt {4 a b+c^2}}}\right ) \int \frac {1}{\sqrt {1+\frac {2 a x^2}{-c-\sqrt {4 a b+c^2}}} \sqrt {1+\frac {2 a x^2}{-c+\sqrt {4 a b+c^2}}}} \, dx}{4 \sqrt {a} \sqrt {b} \sqrt {-b-c x^2+a x^4}}+\frac {\left (\left (c-\sqrt {4 a b+c^2}\right ) \sqrt {1+\frac {2 a x^2}{-c-\sqrt {4 a b+c^2}}} \sqrt {1+\frac {2 a x^2}{-c+\sqrt {4 a b+c^2}}}\right ) \int \frac {\sqrt {1+\frac {2 a x^2}{-c+\sqrt {4 a b+c^2}}}}{\sqrt {1+\frac {2 a x^2}{-c-\sqrt {4 a b+c^2}}}} \, dx}{8 \sqrt {a} \sqrt {b} \sqrt {-b-c x^2+a x^4}}+\frac {\left (\left (2 \sqrt {a} \sqrt {b}+c-\sqrt {4 a b+c^2}\right ) \sqrt {1+\frac {2 a x^2}{-c-\sqrt {4 a b+c^2}}} \sqrt {1+\frac {2 a x^2}{-c+\sqrt {4 a b+c^2}}}\right ) \int \frac {1}{\sqrt {1+\frac {2 a x^2}{-c-\sqrt {4 a b+c^2}}} \sqrt {1+\frac {2 a x^2}{-c+\sqrt {4 a b+c^2}}}} \, dx}{8 \sqrt {a} \sqrt {b} \sqrt {-b-c x^2+a x^4}}+\frac {\left (\left (-c+\sqrt {4 a b+c^2}\right ) \sqrt {1+\frac {2 a x^2}{-c-\sqrt {4 a b+c^2}}} \sqrt {1+\frac {2 a x^2}{-c+\sqrt {4 a b+c^2}}}\right ) \int \frac {\sqrt {1+\frac {2 a x^2}{-c+\sqrt {4 a b+c^2}}}}{\sqrt {1+\frac {2 a x^2}{-c-\sqrt {4 a b+c^2}}}} \, dx}{8 \sqrt {a} \sqrt {b} \sqrt {-b-c x^2+a x^4}}+\frac {\left (\left (2 \sqrt {a} \sqrt {b}-c+\sqrt {4 a b+c^2}\right ) \sqrt {1+\frac {2 a x^2}{-c-\sqrt {4 a b+c^2}}} \sqrt {1+\frac {2 a x^2}{-c+\sqrt {4 a b+c^2}}}\right ) \int \frac {1}{\sqrt {1+\frac {2 a x^2}{-c-\sqrt {4 a b+c^2}}} \sqrt {1+\frac {2 a x^2}{-c+\sqrt {4 a b+c^2}}}} \, dx}{8 \sqrt {a} \sqrt {b} \sqrt {-b-c x^2+a x^4}}+\frac {\left (\left (2 \sqrt {a} \sqrt {b}+c+\sqrt {4 a b+c^2}\right ) \sqrt {1+\frac {2 a x^2}{-c-\sqrt {4 a b+c^2}}} \sqrt {1+\frac {2 a x^2}{-c+\sqrt {4 a b+c^2}}}\right ) \int \frac {1}{\sqrt {1+\frac {2 a x^2}{-c-\sqrt {4 a b+c^2}}} \sqrt {1+\frac {2 a x^2}{-c+\sqrt {4 a b+c^2}}}} \, dx}{4 \sqrt {a} \sqrt {b} \sqrt {-b-c x^2+a x^4}}\\ &=\frac {x \sqrt {-b-c x^2+a x^4}}{4 \sqrt {b} \left (\sqrt {b}-\sqrt {a} x^2\right )}+\frac {x \sqrt {-b-c x^2+a x^4}}{4 \sqrt {b} \left (\sqrt {b}+\sqrt {a} x^2\right )}+\frac {\left (2 \sqrt {a} \sqrt {b}-c-\sqrt {4 a b+c^2}\right ) \sqrt {c+\sqrt {4 a b+c^2}} \sqrt {1-\frac {2 a x^2}{c-\sqrt {4 a b+c^2}}} \sqrt {1-\frac {2 a x^2}{c+\sqrt {4 a b+c^2}}} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {a} x}{\sqrt {c+\sqrt {4 a b+c^2}}}\right )|\frac {c+\sqrt {4 a b+c^2}}{c-\sqrt {4 a b+c^2}}\right )}{4 \sqrt {2} a \sqrt {b} \sqrt {-b-c x^2+a x^4}}+\frac {\left (2 \sqrt {a} \sqrt {b}+c-\sqrt {4 a b+c^2}\right ) \sqrt {c+\sqrt {4 a b+c^2}} \sqrt {1-\frac {2 a x^2}{c-\sqrt {4 a b+c^2}}} \sqrt {1-\frac {2 a x^2}{c+\sqrt {4 a b+c^2}}} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {a} x}{\sqrt {c+\sqrt {4 a b+c^2}}}\right )|\frac {c+\sqrt {4 a b+c^2}}{c-\sqrt {4 a b+c^2}}\right )}{8 \sqrt {2} a \sqrt {b} \sqrt {-b-c x^2+a x^4}}+\frac {\left (2 \sqrt {a} \sqrt {b}-c+\sqrt {4 a b+c^2}\right ) \sqrt {c+\sqrt {4 a b+c^2}} \sqrt {1-\frac {2 a x^2}{c-\sqrt {4 a b+c^2}}} \sqrt {1-\frac {2 a x^2}{c+\sqrt {4 a b+c^2}}} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {a} x}{\sqrt {c+\sqrt {4 a b+c^2}}}\right )|\frac {c+\sqrt {4 a b+c^2}}{c-\sqrt {4 a b+c^2}}\right )}{8 \sqrt {2} a \sqrt {b} \sqrt {-b-c x^2+a x^4}}+\frac {\sqrt {c+\sqrt {4 a b+c^2}} \left (2 \sqrt {a} \sqrt {b}+c+\sqrt {4 a b+c^2}\right ) \sqrt {1-\frac {2 a x^2}{c-\sqrt {4 a b+c^2}}} \sqrt {1-\frac {2 a x^2}{c+\sqrt {4 a b+c^2}}} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {a} x}{\sqrt {c+\sqrt {4 a b+c^2}}}\right )|\frac {c+\sqrt {4 a b+c^2}}{c-\sqrt {4 a b+c^2}}\right )}{4 \sqrt {2} a \sqrt {b} \sqrt {-b-c x^2+a x^4}}-\frac {\sqrt {c+\sqrt {4 a b+c^2}} \sqrt {1-\frac {2 a x^2}{c-\sqrt {4 a b+c^2}}} \sqrt {1-\frac {2 a x^2}{c+\sqrt {4 a b+c^2}}} \Pi \left (-\frac {c+\sqrt {4 a b+c^2}}{2 \sqrt {a} \sqrt {b}};\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {a} x}{\sqrt {c+\sqrt {4 a b+c^2}}}\right )|\frac {c+\sqrt {4 a b+c^2}}{c-\sqrt {4 a b+c^2}}\right )}{2 \sqrt {2} \sqrt {a} \sqrt {-b-c x^2+a x^4}}-\frac {c \sqrt {c+\sqrt {4 a b+c^2}} \sqrt {1-\frac {2 a x^2}{c-\sqrt {4 a b+c^2}}} \sqrt {1-\frac {2 a x^2}{c+\sqrt {4 a b+c^2}}} \Pi \left (-\frac {c+\sqrt {4 a b+c^2}}{2 \sqrt {a} \sqrt {b}};\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {a} x}{\sqrt {c+\sqrt {4 a b+c^2}}}\right )|\frac {c+\sqrt {4 a b+c^2}}{c-\sqrt {4 a b+c^2}}\right )}{2 \sqrt {2} a \sqrt {b} \sqrt {-b-c x^2+a x^4}}-\frac {\sqrt {c+\sqrt {4 a b+c^2}} \sqrt {1-\frac {2 a x^2}{c-\sqrt {4 a b+c^2}}} \sqrt {1-\frac {2 a x^2}{c+\sqrt {4 a b+c^2}}} \Pi \left (\frac {c+\sqrt {4 a b+c^2}}{2 \sqrt {a} \sqrt {b}};\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {a} x}{\sqrt {c+\sqrt {4 a b+c^2}}}\right )|\frac {c+\sqrt {4 a b+c^2}}{c-\sqrt {4 a b+c^2}}\right )}{2 \sqrt {2} \sqrt {a} \sqrt {-b-c x^2+a x^4}}+\frac {c \sqrt {c+\sqrt {4 a b+c^2}} \sqrt {1-\frac {2 a x^2}{c-\sqrt {4 a b+c^2}}} \sqrt {1-\frac {2 a x^2}{c+\sqrt {4 a b+c^2}}} \Pi \left (\frac {c+\sqrt {4 a b+c^2}}{2 \sqrt {a} \sqrt {b}};\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {a} x}{\sqrt {c+\sqrt {4 a b+c^2}}}\right )|\frac {c+\sqrt {4 a b+c^2}}{c-\sqrt {4 a b+c^2}}\right )}{2 \sqrt {2} a \sqrt {b} \sqrt {-b-c x^2+a x^4}}-\frac {\left (\sqrt {1+\frac {2 a x^2}{-c-\sqrt {4 a b+c^2}}} \sqrt {1+\frac {2 a x^2}{-c+\sqrt {4 a b+c^2}}}\right ) \int \frac {a \sqrt {b}+\sqrt {a} c-a^{3/2} x^2}{\sqrt {1+\frac {2 a x^2}{-c-\sqrt {4 a b+c^2}}} \sqrt {1+\frac {2 a x^2}{-c+\sqrt {4 a b+c^2}}}} \, dx}{2 a \sqrt {b} \sqrt {-b-c x^2+a x^4}}-\frac {\left (\sqrt {1+\frac {2 a x^2}{-c-\sqrt {4 a b+c^2}}} \sqrt {1+\frac {2 a x^2}{-c+\sqrt {4 a b+c^2}}}\right ) \int \frac {a \sqrt {b}-\sqrt {a} c+a^{3/2} x^2}{\sqrt {1+\frac {2 a x^2}{-c-\sqrt {4 a b+c^2}}} \sqrt {1+\frac {2 a x^2}{-c+\sqrt {4 a b+c^2}}}} \, dx}{2 a \sqrt {b} \sqrt {-b-c x^2+a x^4}}-\frac {\left (c \sqrt {1+\frac {2 a x^2}{-c-\sqrt {4 a b+c^2}}} \sqrt {1+\frac {2 a x^2}{-c+\sqrt {4 a b+c^2}}}\right ) \int \frac {1}{\left (\sqrt {b}-\sqrt {a} x^2\right ) \sqrt {1+\frac {2 a x^2}{-c-\sqrt {4 a b+c^2}}} \sqrt {1+\frac {2 a x^2}{-c+\sqrt {4 a b+c^2}}}} \, dx}{2 \sqrt {a} \sqrt {-b-c x^2+a x^4}}+\frac {\left (c \sqrt {1+\frac {2 a x^2}{-c-\sqrt {4 a b+c^2}}} \sqrt {1+\frac {2 a x^2}{-c+\sqrt {4 a b+c^2}}}\right ) \int \frac {1}{\left (\sqrt {b}+\sqrt {a} x^2\right ) \sqrt {1+\frac {2 a x^2}{-c-\sqrt {4 a b+c^2}}} \sqrt {1+\frac {2 a x^2}{-c+\sqrt {4 a b+c^2}}}} \, dx}{2 \sqrt {a} \sqrt {-b-c x^2+a x^4}}\\ &=\frac {x \sqrt {-b-c x^2+a x^4}}{4 \sqrt {b} \left (\sqrt {b}-\sqrt {a} x^2\right )}+\frac {x \sqrt {-b-c x^2+a x^4}}{4 \sqrt {b} \left (\sqrt {b}+\sqrt {a} x^2\right )}+\frac {\left (2 \sqrt {a} \sqrt {b}-c-\sqrt {4 a b+c^2}\right ) \sqrt {c+\sqrt {4 a b+c^2}} \sqrt {1-\frac {2 a x^2}{c-\sqrt {4 a b+c^2}}} \sqrt {1-\frac {2 a x^2}{c+\sqrt {4 a b+c^2}}} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {a} x}{\sqrt {c+\sqrt {4 a b+c^2}}}\right )|\frac {c+\sqrt {4 a b+c^2}}{c-\sqrt {4 a b+c^2}}\right )}{4 \sqrt {2} a \sqrt {b} \sqrt {-b-c x^2+a x^4}}+\frac {\left (2 \sqrt {a} \sqrt {b}+c-\sqrt {4 a b+c^2}\right ) \sqrt {c+\sqrt {4 a b+c^2}} \sqrt {1-\frac {2 a x^2}{c-\sqrt {4 a b+c^2}}} \sqrt {1-\frac {2 a x^2}{c+\sqrt {4 a b+c^2}}} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {a} x}{\sqrt {c+\sqrt {4 a b+c^2}}}\right )|\frac {c+\sqrt {4 a b+c^2}}{c-\sqrt {4 a b+c^2}}\right )}{8 \sqrt {2} a \sqrt {b} \sqrt {-b-c x^2+a x^4}}+\frac {\left (2 \sqrt {a} \sqrt {b}-c+\sqrt {4 a b+c^2}\right ) \sqrt {c+\sqrt {4 a b+c^2}} \sqrt {1-\frac {2 a x^2}{c-\sqrt {4 a b+c^2}}} \sqrt {1-\frac {2 a x^2}{c+\sqrt {4 a b+c^2}}} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {a} x}{\sqrt {c+\sqrt {4 a b+c^2}}}\right )|\frac {c+\sqrt {4 a b+c^2}}{c-\sqrt {4 a b+c^2}}\right )}{8 \sqrt {2} a \sqrt {b} \sqrt {-b-c x^2+a x^4}}+\frac {\sqrt {c+\sqrt {4 a b+c^2}} \left (2 \sqrt {a} \sqrt {b}+c+\sqrt {4 a b+c^2}\right ) \sqrt {1-\frac {2 a x^2}{c-\sqrt {4 a b+c^2}}} \sqrt {1-\frac {2 a x^2}{c+\sqrt {4 a b+c^2}}} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {a} x}{\sqrt {c+\sqrt {4 a b+c^2}}}\right )|\frac {c+\sqrt {4 a b+c^2}}{c-\sqrt {4 a b+c^2}}\right )}{4 \sqrt {2} a \sqrt {b} \sqrt {-b-c x^2+a x^4}}-\frac {\sqrt {c+\sqrt {4 a b+c^2}} \sqrt {1-\frac {2 a x^2}{c-\sqrt {4 a b+c^2}}} \sqrt {1-\frac {2 a x^2}{c+\sqrt {4 a b+c^2}}} \Pi \left (-\frac {c+\sqrt {4 a b+c^2}}{2 \sqrt {a} \sqrt {b}};\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {a} x}{\sqrt {c+\sqrt {4 a b+c^2}}}\right )|\frac {c+\sqrt {4 a b+c^2}}{c-\sqrt {4 a b+c^2}}\right )}{2 \sqrt {2} \sqrt {a} \sqrt {-b-c x^2+a x^4}}-\frac {\sqrt {c+\sqrt {4 a b+c^2}} \sqrt {1-\frac {2 a x^2}{c-\sqrt {4 a b+c^2}}} \sqrt {1-\frac {2 a x^2}{c+\sqrt {4 a b+c^2}}} \Pi \left (\frac {c+\sqrt {4 a b+c^2}}{2 \sqrt {a} \sqrt {b}};\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {a} x}{\sqrt {c+\sqrt {4 a b+c^2}}}\right )|\frac {c+\sqrt {4 a b+c^2}}{c-\sqrt {4 a b+c^2}}\right )}{2 \sqrt {2} \sqrt {a} \sqrt {-b-c x^2+a x^4}}-\frac {\left (\left (2 \sqrt {a} \sqrt {b}-c-\sqrt {4 a b+c^2}\right ) \sqrt {1+\frac {2 a x^2}{-c-\sqrt {4 a b+c^2}}} \sqrt {1+\frac {2 a x^2}{-c+\sqrt {4 a b+c^2}}}\right ) \int \frac {1}{\sqrt {1+\frac {2 a x^2}{-c-\sqrt {4 a b+c^2}}} \sqrt {1+\frac {2 a x^2}{-c+\sqrt {4 a b+c^2}}}} \, dx}{4 \sqrt {a} \sqrt {b} \sqrt {-b-c x^2+a x^4}}-\frac {\left (\left (2 \sqrt {a} \sqrt {b}+c+\sqrt {4 a b+c^2}\right ) \sqrt {1+\frac {2 a x^2}{-c-\sqrt {4 a b+c^2}}} \sqrt {1+\frac {2 a x^2}{-c+\sqrt {4 a b+c^2}}}\right ) \int \frac {1}{\sqrt {1+\frac {2 a x^2}{-c-\sqrt {4 a b+c^2}}} \sqrt {1+\frac {2 a x^2}{-c+\sqrt {4 a b+c^2}}}} \, dx}{4 \sqrt {a} \sqrt {b} \sqrt {-b-c x^2+a x^4}}\\ &=\frac {x \sqrt {-b-c x^2+a x^4}}{4 \sqrt {b} \left (\sqrt {b}-\sqrt {a} x^2\right )}+\frac {x \sqrt {-b-c x^2+a x^4}}{4 \sqrt {b} \left (\sqrt {b}+\sqrt {a} x^2\right )}+\frac {\left (2 \sqrt {a} \sqrt {b}+c-\sqrt {4 a b+c^2}\right ) \sqrt {c+\sqrt {4 a b+c^2}} \sqrt {1-\frac {2 a x^2}{c-\sqrt {4 a b+c^2}}} \sqrt {1-\frac {2 a x^2}{c+\sqrt {4 a b+c^2}}} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {a} x}{\sqrt {c+\sqrt {4 a b+c^2}}}\right )|\frac {c+\sqrt {4 a b+c^2}}{c-\sqrt {4 a b+c^2}}\right )}{8 \sqrt {2} a \sqrt {b} \sqrt {-b-c x^2+a x^4}}+\frac {\left (2 \sqrt {a} \sqrt {b}-c+\sqrt {4 a b+c^2}\right ) \sqrt {c+\sqrt {4 a b+c^2}} \sqrt {1-\frac {2 a x^2}{c-\sqrt {4 a b+c^2}}} \sqrt {1-\frac {2 a x^2}{c+\sqrt {4 a b+c^2}}} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {a} x}{\sqrt {c+\sqrt {4 a b+c^2}}}\right )|\frac {c+\sqrt {4 a b+c^2}}{c-\sqrt {4 a b+c^2}}\right )}{8 \sqrt {2} a \sqrt {b} \sqrt {-b-c x^2+a x^4}}-\frac {\sqrt {c+\sqrt {4 a b+c^2}} \sqrt {1-\frac {2 a x^2}{c-\sqrt {4 a b+c^2}}} \sqrt {1-\frac {2 a x^2}{c+\sqrt {4 a b+c^2}}} \Pi \left (-\frac {c+\sqrt {4 a b+c^2}}{2 \sqrt {a} \sqrt {b}};\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {a} x}{\sqrt {c+\sqrt {4 a b+c^2}}}\right )|\frac {c+\sqrt {4 a b+c^2}}{c-\sqrt {4 a b+c^2}}\right )}{2 \sqrt {2} \sqrt {a} \sqrt {-b-c x^2+a x^4}}-\frac {\sqrt {c+\sqrt {4 a b+c^2}} \sqrt {1-\frac {2 a x^2}{c-\sqrt {4 a b+c^2}}} \sqrt {1-\frac {2 a x^2}{c+\sqrt {4 a b+c^2}}} \Pi \left (\frac {c+\sqrt {4 a b+c^2}}{2 \sqrt {a} \sqrt {b}};\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {a} x}{\sqrt {c+\sqrt {4 a b+c^2}}}\right )|\frac {c+\sqrt {4 a b+c^2}}{c-\sqrt {4 a b+c^2}}\right )}{2 \sqrt {2} \sqrt {a} \sqrt {-b-c x^2+a x^4}}\\ \end {align*}

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Mathematica [C]  time = 2.96, size = 416, normalized size = 0.50 \begin {gather*} \frac {1}{2} \sqrt {a x^4-b-c x^2} \left (\frac {x}{b-a x^4}-\frac {i \sqrt {\frac {4 a x^2}{\sqrt {4 a b+c^2}-c}+2} \sqrt {1-\frac {2 a x^2}{\sqrt {4 a b+c^2}+c}} \left (F\left (i \sinh ^{-1}\left (\sqrt {2} \sqrt {\frac {a}{\sqrt {c^2+4 a b}-c}} x\right )|\frac {c-\sqrt {c^2+4 a b}}{c+\sqrt {c^2+4 a b}}\right )-\Pi \left (\frac {c-\sqrt {c^2+4 a b}}{2 \sqrt {a} \sqrt {b}};i \sinh ^{-1}\left (\sqrt {2} \sqrt {\frac {a}{\sqrt {c^2+4 a b}-c}} x\right )|\frac {c-\sqrt {c^2+4 a b}}{c+\sqrt {c^2+4 a b}}\right )-\Pi \left (\frac {\sqrt {c^2+4 a b}-c}{2 \sqrt {a} \sqrt {b}};i \sinh ^{-1}\left (\sqrt {2} \sqrt {\frac {a}{\sqrt {c^2+4 a b}-c}} x\right )|\frac {c-\sqrt {c^2+4 a b}}{c+\sqrt {c^2+4 a b}}\right )\right )}{2 \sqrt {\frac {a}{\sqrt {4 a b+c^2}-c}} \left (a x^4-b-c x^2\right )}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[-b - c*x^2 + a*x^4]*(x/(b - a*x^4) - ((I/2)*Sqrt[2 + (4*a*x^2)/(-c + Sqrt[4*a*b + c^2])]*Sqrt[1 - (2*a*x
^2)/(c + Sqrt[4*a*b + c^2])]*(EllipticF[I*ArcSinh[Sqrt[2]*Sqrt[a/(-c + Sqrt[4*a*b + c^2])]*x], (c - Sqrt[4*a*b
+ c^2])/(c + Sqrt[4*a*b + c^2])] - EllipticPi[(c - Sqrt[4*a*b + c^2])/(2*Sqrt[a]*Sqrt[b]), I*ArcSinh[Sqrt[2]*
Sqrt[a/(-c + Sqrt[4*a*b + c^2])]*x], (c - Sqrt[4*a*b + c^2])/(c + Sqrt[4*a*b + c^2])] - EllipticPi[(-c + Sqrt[
4*a*b + c^2])/(2*Sqrt[a]*Sqrt[b]), I*ArcSinh[Sqrt[2]*Sqrt[a/(-c + Sqrt[4*a*b + c^2])]*x], (c - Sqrt[4*a*b + c^
2])/(c + Sqrt[4*a*b + c^2])]))/(Sqrt[a/(-c + Sqrt[4*a*b + c^2])]*(-b - c*x^2 + a*x^4))))/2

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IntegrateAlgebraic [A]  time = 0.78, size = 87, normalized size = 0.11 \begin {gather*} -\frac {x \sqrt {-b-c x^2+a x^4}}{2 \left (-b+a x^4\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {c} x \sqrt {-b-c x^2+a x^4}}{b+c x^2-a x^4}\right )}{2 \sqrt {c}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.75, size = 246, normalized size = 0.30 \begin {gather*} \left [-\frac {4 \, \sqrt {a x^{4} - c x^{2} - b} c x + {\left (a x^{4} - b\right )} \sqrt {-c} \log \left (-\frac {a^{2} x^{8} - 8 \, a c x^{6} - 2 \, {\left (a b - 4 \, c^{2}\right )} x^{4} + 8 \, b c x^{2} + b^{2} - 4 \, {\left (a x^{5} - 2 \, c x^{3} - b x\right )} \sqrt {a x^{4} - c x^{2} - b} \sqrt {-c}}{a^{2} x^{8} - 2 \, a b x^{4} + b^{2}}\right )}{8 \, {\left (a c x^{4} - b c\right )}}, -\frac {2 \, \sqrt {a x^{4} - c x^{2} - b} c x + {\left (a x^{4} - b\right )} \sqrt {c} \arctan \left (\frac {2 \, \sqrt {a x^{4} - c x^{2} - b} \sqrt {c} x}{a x^{4} - 2 \, c x^{2} - b}\right )}{4 \, {\left (a c x^{4} - b c\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/8*(4*sqrt(a*x^4 - c*x^2 - b)*c*x + (a*x^4 - b)*sqrt(-c)*log(-(a^2*x^8 - 8*a*c*x^6 - 2*(a*b - 4*c^2)*x^4 +
8*b*c*x^2 + b^2 - 4*(a*x^5 - 2*c*x^3 - b*x)*sqrt(a*x^4 - c*x^2 - b)*sqrt(-c))/(a^2*x^8 - 2*a*b*x^4 + b^2)))/(a
*c*x^4 - b*c), -1/4*(2*sqrt(a*x^4 - c*x^2 - b)*c*x + (a*x^4 - b)*sqrt(c)*arctan(2*sqrt(a*x^4 - c*x^2 - b)*sqrt
(c)*x/(a*x^4 - 2*c*x^2 - b)))/(a*c*x^4 - b*c)]

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a x^{4} - c x^{2} - b} {\left (a x^{4} + b\right )}}{{\left (a x^{4} - b\right )}^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.14, size = 91, normalized size = 0.11

method	result	size
elliptic	\(\frac {\left (-\frac {\sqrt {a \,x^{4}-c \,x^{2}-b}\, \sqrt {2}}{4 x \left (\frac {a \,x^{4}-c \,x^{2}-b}{2 x^{2}}+\frac {c}{2}\right )}+\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {a \,x^{4}-c \,x^{2}-b}}{x \sqrt {c}}\right )}{2 \sqrt {c}}\right ) \sqrt {2}}{2}\)	\(91\)
default	\(2 b \left (-\frac {x \sqrt {a \,x^{4}-c \,x^{2}-b}}{4 b \left (a \,x^{4}-b \right )}-\frac {\sqrt {4+\frac {2 \left (c +\sqrt {4 a b +c^{2}}\right ) x^{2}}{b}}\, \sqrt {4-\frac {2 \left (-c +\sqrt {4 a b +c^{2}}\right ) x^{2}}{b}}\, \EllipticF \left (\frac {x \sqrt {-\frac {2 \left (c +\sqrt {4 a b +c^{2}}\right )}{b}}}{2}, \frac {\sqrt {-4+\frac {2 c \left (-c +\sqrt {4 a b +c^{2}}\right )}{b a}}}{2}\right )}{8 b \sqrt {-\frac {2 \left (c +\sqrt {4 a b +c^{2}}\right )}{b}}\, \sqrt {a \,x^{4}-c \,x^{2}-b}}-\frac {\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (a \,\textit {\_Z}^{4}-b \right )}{\sum }\frac {\left (-c \,\underline {\hspace {1.25 ex}}\alpha ^{2}-b \right ) \left (-\frac {\arctanh \left (\frac {2 a \,x^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}-c \,\underline {\hspace {1.25 ex}}\alpha ^{2}-c \,x^{2}-2 b}{2 \sqrt {-c \,\underline {\hspace {1.25 ex}}\alpha ^{2}}\, \sqrt {a \,x^{4}-c \,x^{2}-b}}\right )}{\sqrt {-c \,\underline {\hspace {1.25 ex}}\alpha ^{2}}}-\frac {a \sqrt {2}\, \underline {\hspace {1.25 ex}}\alpha ^{3} \sqrt {2+\frac {c \,x^{2}}{b}+\frac {x^{2} \sqrt {4 a b +c^{2}}}{b}}\, \sqrt {2+\frac {c \,x^{2}}{b}-\frac {x^{2} \sqrt {4 a b +c^{2}}}{b}}\, \EllipticPi \left (\sqrt {-\frac {c +\sqrt {4 a b +c^{2}}}{2 b}}\, x , \frac {\underline {\hspace {1.25 ex}}\alpha ^{2} \left (c -\sqrt {4 a b +c^{2}}\right )}{2 b}, \frac {\sqrt {2}\, \sqrt {\frac {-c +\sqrt {4 a b +c^{2}}}{b}}}{2 \sqrt {-\frac {c +\sqrt {4 a b +c^{2}}}{2 b}}}\right )}{b \sqrt {-\frac {c +\sqrt {4 a b +c^{2}}}{b}}\, \sqrt {a \,x^{4}-c \,x^{2}-b}}\right )}{\underline {\hspace {1.25 ex}}\alpha ^{3}}}{16 a b}\right )+\frac {\sqrt {4+\frac {2 \left (c +\sqrt {4 a b +c^{2}}\right ) x^{2}}{b}}\, \sqrt {4-\frac {2 \left (-c +\sqrt {4 a b +c^{2}}\right ) x^{2}}{b}}\, \EllipticF \left (\frac {x \sqrt {-\frac {2 \left (c +\sqrt {4 a b +c^{2}}\right )}{b}}}{2}, \frac {\sqrt {-4+\frac {2 c \left (-c +\sqrt {4 a b +c^{2}}\right )}{b a}}}{2}\right )}{2 \sqrt {-\frac {2 \left (c +\sqrt {4 a b +c^{2}}\right )}{b}}\, \sqrt {a \,x^{4}-c \,x^{2}-b}}-\frac {c \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (a \,\textit {\_Z}^{4}-b \right )}{\sum }\frac {-\frac {\arctanh \left (\frac {2 a \,x^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}-c \,\underline {\hspace {1.25 ex}}\alpha ^{2}-c \,x^{2}-2 b}{2 \sqrt {-c \,\underline {\hspace {1.25 ex}}\alpha ^{2}}\, \sqrt {a \,x^{4}-c \,x^{2}-b}}\right )}{\sqrt {-c \,\underline {\hspace {1.25 ex}}\alpha ^{2}}}-\frac {a \sqrt {2}\, \underline {\hspace {1.25 ex}}\alpha ^{3} \sqrt {2+\frac {c \,x^{2}}{b}+\frac {x^{2} \sqrt {4 a b +c^{2}}}{b}}\, \sqrt {2+\frac {c \,x^{2}}{b}-\frac {x^{2} \sqrt {4 a b +c^{2}}}{b}}\, \EllipticPi \left (\sqrt {-\frac {c +\sqrt {4 a b +c^{2}}}{2 b}}\, x , \frac {\underline {\hspace {1.25 ex}}\alpha ^{2} \left (c -\sqrt {4 a b +c^{2}}\right )}{2 b}, \frac {\sqrt {2}\, \sqrt {\frac {-c +\sqrt {4 a b +c^{2}}}{b}}}{2 \sqrt {-\frac {c +\sqrt {4 a b +c^{2}}}{2 b}}}\right )}{b \sqrt {-\frac {c +\sqrt {4 a b +c^{2}}}{b}}\, \sqrt {a \,x^{4}-c \,x^{2}-b}}}{\underline {\hspace {1.25 ex}}\alpha }\right )}{8 a}\)	\(899\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a x^{4} - c x^{2} - b} {\left (a x^{4} + b\right )}}{{\left (a x^{4} - b\right )}^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\left (a\,x^4+b\right )\,\sqrt {a\,x^4-c\,x^2-b}}{{\left (b-a\,x^4\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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