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

Optimal. Leaf size=159 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{2 a b+c} \sqrt {a x^3-b x}}{x \sqrt {2 a b+c}-a x^2+b}\right )}{\sqrt {2} \sqrt [4]{2 a b+c}}-\frac {\tanh ^{-1}\left (\frac {\frac {a x^2}{\sqrt {2} \sqrt [4]{2 a b+c}}+\frac {x \sqrt [4]{2 a b+c}}{\sqrt {2}}-\frac {b}{\sqrt {2} \sqrt [4]{2 a b+c}}}{\sqrt {a x^3-b x}}\right )}{\sqrt {2} \sqrt [4]{2 a b+c}} \]

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Rubi [C]  time = 3.22, antiderivative size = 880, normalized size of antiderivative = 5.53, number of steps used = 21, number of rules used = 10, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {2056, 1586, 6715, 6728, 406, 224, 221, 409, 1219, 1218} \begin {gather*} \frac {\sqrt [4]{b} \left (1-\frac {2 a b-c}{\sqrt {c^2-4 a^2 b^2}}\right ) \sqrt {x} \sqrt {1-\frac {a x^2}{b}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right )\right |-1\right )}{\sqrt [4]{a} \sqrt {a x^3-b x}}+\frac {\sqrt [4]{b} \left (\frac {2 a b-c}{\sqrt {c^2-4 a^2 b^2}}+1\right ) \sqrt {x} \sqrt {1-\frac {a x^2}{b}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right )\right |-1\right )}{\sqrt [4]{a} \sqrt {a x^3-b x}}+\frac {\sqrt [4]{b} \left (4 a^2 b^2-c \left (c+\sqrt {c^2-4 a^2 b^2}\right )\right ) \sqrt {x} \sqrt {1-\frac {a x^2}{b}} \Pi \left (-\frac {\sqrt {2} \sqrt {a} \sqrt {b}}{\sqrt {-c-\sqrt {c^2-4 a^2 b^2}}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right )\right |-1\right )}{\sqrt [4]{a} \sqrt {c^2-4 a^2 b^2} \left (c+\sqrt {c^2-4 a^2 b^2}\right ) \sqrt {a x^3-b x}}+\frac {\sqrt [4]{b} \left (4 a^2 b^2-c \left (c+\sqrt {c^2-4 a^2 b^2}\right )\right ) \sqrt {x} \sqrt {1-\frac {a x^2}{b}} \Pi \left (\frac {\sqrt {2} \sqrt {a} \sqrt {b}}{\sqrt {-c-\sqrt {c^2-4 a^2 b^2}}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right )\right |-1\right )}{\sqrt [4]{a} \sqrt {c^2-4 a^2 b^2} \left (c+\sqrt {c^2-4 a^2 b^2}\right ) \sqrt {a x^3-b x}}-\frac {\sqrt [4]{b} \left (4 a^2 b^2-c \left (c-\sqrt {c^2-4 a^2 b^2}\right )\right ) \sqrt {x} \sqrt {1-\frac {a x^2}{b}} \Pi \left (-\frac {\sqrt {2} \sqrt {a} \sqrt {b}}{\sqrt {\sqrt {c^2-4 a^2 b^2}-c}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right )\right |-1\right )}{\sqrt [4]{a} \sqrt {c^2-4 a^2 b^2} \left (c-\sqrt {c^2-4 a^2 b^2}\right ) \sqrt {a x^3-b x}}-\frac {\sqrt [4]{b} \left (4 a^2 b^2-c \left (c-\sqrt {c^2-4 a^2 b^2}\right )\right ) \sqrt {x} \sqrt {1-\frac {a x^2}{b}} \Pi \left (\frac {\sqrt {2} \sqrt {a} \sqrt {b}}{\sqrt {\sqrt {c^2-4 a^2 b^2}-c}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right )\right |-1\right )}{\sqrt [4]{a} \sqrt {c^2-4 a^2 b^2} \left (c-\sqrt {c^2-4 a^2 b^2}\right ) \sqrt {a x^3-b x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(b^(1/4)*(1 - (2*a*b - c)/Sqrt[-4*a^2*b^2 + c^2])*Sqrt[x]*Sqrt[1 - (a*x^2)/b]*EllipticF[ArcSin[(a^(1/4)*Sqrt[x
])/b^(1/4)], -1])/(a^(1/4)*Sqrt[-(b*x) + a*x^3]) + (b^(1/4)*(1 + (2*a*b - c)/Sqrt[-4*a^2*b^2 + c^2])*Sqrt[x]*S
qrt[1 - (a*x^2)/b]*EllipticF[ArcSin[(a^(1/4)*Sqrt[x])/b^(1/4)], -1])/(a^(1/4)*Sqrt[-(b*x) + a*x^3]) + (b^(1/4)
*(4*a^2*b^2 - c*(c + Sqrt[-4*a^2*b^2 + c^2]))*Sqrt[x]*Sqrt[1 - (a*x^2)/b]*EllipticPi[-((Sqrt[2]*Sqrt[a]*Sqrt[b
])/Sqrt[-c - Sqrt[-4*a^2*b^2 + c^2]]), ArcSin[(a^(1/4)*Sqrt[x])/b^(1/4)], -1])/(a^(1/4)*Sqrt[-4*a^2*b^2 + c^2]
*(c + Sqrt[-4*a^2*b^2 + c^2])*Sqrt[-(b*x) + a*x^3]) + (b^(1/4)*(4*a^2*b^2 - c*(c + Sqrt[-4*a^2*b^2 + c^2]))*Sq
rt[x]*Sqrt[1 - (a*x^2)/b]*EllipticPi[(Sqrt[2]*Sqrt[a]*Sqrt[b])/Sqrt[-c - Sqrt[-4*a^2*b^2 + c^2]], ArcSin[(a^(1
/4)*Sqrt[x])/b^(1/4)], -1])/(a^(1/4)*Sqrt[-4*a^2*b^2 + c^2]*(c + Sqrt[-4*a^2*b^2 + c^2])*Sqrt[-(b*x) + a*x^3])
- (b^(1/4)*(4*a^2*b^2 - c*(c - Sqrt[-4*a^2*b^2 + c^2]))*Sqrt[x]*Sqrt[1 - (a*x^2)/b]*EllipticPi[-((Sqrt[2]*Sqr
t[a]*Sqrt[b])/Sqrt[-c + Sqrt[-4*a^2*b^2 + c^2]]), ArcSin[(a^(1/4)*Sqrt[x])/b^(1/4)], -1])/(a^(1/4)*Sqrt[-4*a^2
*b^2 + c^2]*(c - Sqrt[-4*a^2*b^2 + c^2])*Sqrt[-(b*x) + a*x^3]) - (b^(1/4)*(4*a^2*b^2 - c*(c - Sqrt[-4*a^2*b^2
+ c^2]))*Sqrt[x]*Sqrt[1 - (a*x^2)/b]*EllipticPi[(Sqrt[2]*Sqrt[a]*Sqrt[b])/Sqrt[-c + Sqrt[-4*a^2*b^2 + c^2]], A
rcSin[(a^(1/4)*Sqrt[x])/b^(1/4)], -1])/(a^(1/4)*Sqrt[-4*a^2*b^2 + c^2]*(c - Sqrt[-4*a^2*b^2 + c^2])*Sqrt[-(b*x
) + a*x^3])

Rule 221

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[EllipticF[ArcSin[(Rt[-b, 4]*x)/Rt[a, 4]], -1]/(Rt[a, 4]*Rt[
-b, 4]), x] /; FreeQ[{a, b}, x] && NegQ[b/a] && GtQ[a, 0]

Rule 224

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Dist[Sqrt[1 + (b*x^4)/a]/Sqrt[a + b*x^4], Int[1/Sqrt[1 + (b*x^4)
/a], x], x] /; FreeQ[{a, b}, x] && NegQ[b/a] &&  !GtQ[a, 0]

Rule 406

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

Rule 409

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

Rule 1218

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[-(c/a), 4]}, Simp[(1*Ellipt
icPi[-(e/(d*q^2)), ArcSin[q*x], -1])/(d*Sqrt[a]*q), x]] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] && GtQ[a, 0]

Rule 1219

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

Rule 1586

Int[(u_.)*(Px_)^(p_.)*(Qx_)^(q_.), x_Symbol] :> Int[u*PolynomialQuotient[Px, Qx, x]^p*Qx^(p + q), x] /; FreeQ[
q, x] && PolyQ[Px, x] && PolyQ[Qx, x] && EqQ[PolynomialRemainder[Px, Qx, x], 0] && IntegerQ[p] && LtQ[p*q, 0]

Rule 2056

Int[(u_.)*(P_)^(p_.), x_Symbol] :> With[{m = MinimumMonomialExponent[P, x]}, Dist[P^FracPart[p]/(x^(m*FracPart
[p])*Distrib[1/x^m, P]^FracPart[p]), Int[u*x^(m*p)*Distrib[1/x^m, P]^p, x], x]] /; FreeQ[p, x] &&  !IntegerQ[p
] && SumQ[P] && EveryQ[BinomialQ[#1, x] & , P] &&  !PolyQ[P, x, 2]

Rule 6715

Int[(u_)*(x_)^(m_.), x_Symbol] :> Dist[1/(m + 1), Subst[Int[SubstFor[x^(m + 1), u, x], x], x, x^(m + 1)], x] /
; FreeQ[m, x] && NeQ[m, -1] && FunctionOfQ[x^(m + 1), u, x]

Rule 6728

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rubi steps

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

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Mathematica [C]  time = 1.49, size = 396, normalized size = 2.49 \begin {gather*} -\frac {i \sqrt {1-\frac {b}{a x^2}} \sqrt {a x^3-b x} \left (-\Pi \left (-\frac {\sqrt {2} \sqrt {a}}{\sqrt {b} \sqrt {\frac {\sqrt {c^2-4 a^2 b^2}-c}{b^2}}};\left .i \sinh ^{-1}\left (\frac {\sqrt {-\frac {\sqrt {b}}{\sqrt {a}}}}{\sqrt {x}}\right )\right |-1\right )-\Pi \left (\frac {\sqrt {2} \sqrt {a}}{\sqrt {b} \sqrt {\frac {\sqrt {c^2-4 a^2 b^2}-c}{b^2}}};\left .i \sinh ^{-1}\left (\frac {\sqrt {-\frac {\sqrt {b}}{\sqrt {a}}}}{\sqrt {x}}\right )\right |-1\right )-\Pi \left (-\frac {\sqrt {2} \sqrt {a}}{\sqrt {b} \sqrt {-\frac {c+\sqrt {c^2-4 a^2 b^2}}{b^2}}};\left .i \sinh ^{-1}\left (\frac {\sqrt {-\frac {\sqrt {b}}{\sqrt {a}}}}{\sqrt {x}}\right )\right |-1\right )-\Pi \left (\frac {\sqrt {2} \sqrt {a}}{\sqrt {b} \sqrt {-\frac {c+\sqrt {c^2-4 a^2 b^2}}{b^2}}};\left .i \sinh ^{-1}\left (\frac {\sqrt {-\frac {\sqrt {b}}{\sqrt {a}}}}{\sqrt {x}}\right )\right |-1\right )+2 F\left (\left .i \sinh ^{-1}\left (\frac {\sqrt {-\frac {\sqrt {b}}{\sqrt {a}}}}{\sqrt {x}}\right )\right |-1\right )\right )}{x^{3/2} \sqrt {-\frac {\sqrt {b}}{\sqrt {a}}} \left (a-\frac {b}{x^2}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-I)*Sqrt[1 - b/(a*x^2)]*Sqrt[-(b*x) + a*x^3]*(2*EllipticF[I*ArcSinh[Sqrt[-(Sqrt[b]/Sqrt[a])]/Sqrt[x]], -1] -
EllipticPi[-((Sqrt[2]*Sqrt[a])/(Sqrt[b]*Sqrt[(-c + Sqrt[-4*a^2*b^2 + c^2])/b^2])), I*ArcSinh[Sqrt[-(Sqrt[b]/S
qrt[a])]/Sqrt[x]], -1] - EllipticPi[(Sqrt[2]*Sqrt[a])/(Sqrt[b]*Sqrt[(-c + Sqrt[-4*a^2*b^2 + c^2])/b^2]), I*Arc
Sinh[Sqrt[-(Sqrt[b]/Sqrt[a])]/Sqrt[x]], -1] - EllipticPi[-((Sqrt[2]*Sqrt[a])/(Sqrt[b]*Sqrt[-((c + Sqrt[-4*a^2*
b^2 + c^2])/b^2)])), I*ArcSinh[Sqrt[-(Sqrt[b]/Sqrt[a])]/Sqrt[x]], -1] - EllipticPi[(Sqrt[2]*Sqrt[a])/(Sqrt[b]*
Sqrt[-((c + Sqrt[-4*a^2*b^2 + c^2])/b^2)]), I*ArcSinh[Sqrt[-(Sqrt[b]/Sqrt[a])]/Sqrt[x]], -1]))/(Sqrt[-(Sqrt[b]
/Sqrt[a])]*(a - b/x^2)*x^(3/2))

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IntegrateAlgebraic [A]  time = 0.99, size = 159, normalized size = 1.00 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{2 a b+c} \sqrt {-b x+a x^3}}{b+\sqrt {2 a b+c} x-a x^2}\right )}{\sqrt {2} \sqrt [4]{2 a b+c}}-\frac {\tanh ^{-1}\left (\frac {-\frac {b}{\sqrt {2} \sqrt [4]{2 a b+c}}+\frac {\sqrt [4]{2 a b+c} x}{\sqrt {2}}+\frac {a x^2}{\sqrt {2} \sqrt [4]{2 a b+c}}}{\sqrt {-b x+a x^3}}\right )}{\sqrt {2} \sqrt [4]{2 a b+c}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

ArcTan[(Sqrt[2]*(2*a*b + c)^(1/4)*Sqrt[-(b*x) + a*x^3])/(b + Sqrt[2*a*b + c]*x - a*x^2)]/(Sqrt[2]*(2*a*b + c)^
(1/4)) - ArcTanh[(-(b/(Sqrt[2]*(2*a*b + c)^(1/4))) + ((2*a*b + c)^(1/4)*x)/Sqrt[2] + (a*x^2)/(Sqrt[2]*(2*a*b +
c)^(1/4)))/Sqrt[-(b*x) + a*x^3]]/(Sqrt[2]*(2*a*b + c)^(1/4))

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fricas [B]  time = 0.72, size = 401, normalized size = 2.52 \begin {gather*} \left (-\frac {1}{2 \, a b + c}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {a x^{3} - b x} {\left (2 \, a b + c\right )} \left (-\frac {1}{2 \, a b + c}\right )^{\frac {3}{4}}}{a x^{2} - b}\right ) + \frac {1}{4} \, \left (-\frac {1}{2 \, a b + c}\right )^{\frac {1}{4}} \log \left (\frac {a^{2} x^{4} - {\left (4 \, a b + c\right )} x^{2} + b^{2} + 2 \, \sqrt {a x^{3} - b x} {\left ({\left (2 \, a b + c\right )} x \left (-\frac {1}{2 \, a b + c}\right )^{\frac {1}{4}} - {\left (2 \, a b^{2} - {\left (2 \, a^{2} b + a c\right )} x^{2} + b c\right )} \left (-\frac {1}{2 \, a b + c}\right )^{\frac {3}{4}}\right )} - 2 \, {\left ({\left (2 \, a^{2} b + a c\right )} x^{3} - {\left (2 \, a b^{2} + b c\right )} x\right )} \sqrt {-\frac {1}{2 \, a b + c}}}{a^{2} x^{4} + c x^{2} + b^{2}}\right ) - \frac {1}{4} \, \left (-\frac {1}{2 \, a b + c}\right )^{\frac {1}{4}} \log \left (\frac {a^{2} x^{4} - {\left (4 \, a b + c\right )} x^{2} + b^{2} - 2 \, \sqrt {a x^{3} - b x} {\left ({\left (2 \, a b + c\right )} x \left (-\frac {1}{2 \, a b + c}\right )^{\frac {1}{4}} - {\left (2 \, a b^{2} - {\left (2 \, a^{2} b + a c\right )} x^{2} + b c\right )} \left (-\frac {1}{2 \, a b + c}\right )^{\frac {3}{4}}\right )} - 2 \, {\left ({\left (2 \, a^{2} b + a c\right )} x^{3} - {\left (2 \, a b^{2} + b c\right )} x\right )} \sqrt {-\frac {1}{2 \, a b + c}}}{a^{2} x^{4} + c x^{2} + b^{2}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a^2*x^4-b^2)/(a*x^3-b*x)^(1/2)/(a^2*x^4+c*x^2+b^2),x, algorithm="fricas")

[Out]

(-1/(2*a*b + c))^(1/4)*arctan(sqrt(a*x^3 - b*x)*(2*a*b + c)*(-1/(2*a*b + c))^(3/4)/(a*x^2 - b)) + 1/4*(-1/(2*a
*b + c))^(1/4)*log((a^2*x^4 - (4*a*b + c)*x^2 + b^2 + 2*sqrt(a*x^3 - b*x)*((2*a*b + c)*x*(-1/(2*a*b + c))^(1/4
) - (2*a*b^2 - (2*a^2*b + a*c)*x^2 + b*c)*(-1/(2*a*b + c))^(3/4)) - 2*((2*a^2*b + a*c)*x^3 - (2*a*b^2 + b*c)*x
)*sqrt(-1/(2*a*b + c)))/(a^2*x^4 + c*x^2 + b^2)) - 1/4*(-1/(2*a*b + c))^(1/4)*log((a^2*x^4 - (4*a*b + c)*x^2 +
b^2 - 2*sqrt(a*x^3 - b*x)*((2*a*b + c)*x*(-1/(2*a*b + c))^(1/4) - (2*a*b^2 - (2*a^2*b + a*c)*x^2 + b*c)*(-1/(
2*a*b + c))^(3/4)) - 2*((2*a^2*b + a*c)*x^3 - (2*a*b^2 + b*c)*x)*sqrt(-1/(2*a*b + c)))/(a^2*x^4 + c*x^2 + b^2)
)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a^{2} x^{4} - b^{2}}{{\left (a^{2} x^{4} + c x^{2} + b^{2}\right )} \sqrt {a x^{3} - b x}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a^2*x^4-b^2)/(a*x^3-b*x)^(1/2)/(a^2*x^4+c*x^2+b^2),x, algorithm="giac")

[Out]

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

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maple [C]  time = 0.16, size = 374, normalized size = 2.35

method	result	size
elliptic	\(\frac {\sqrt {a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{a}\right ) a}{\sqrt {a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{a}\right ) a}{\sqrt {a b}}}\, \sqrt {-\frac {x a}{\sqrt {a b}}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{a}\right ) a}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right )}{a \sqrt {a \,x^{3}-b x}}-\frac {\sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (a^{2} \textit {\_Z}^{4}+c \,\textit {\_Z}^{2}+b^{2}\right )}{\sum }\frac {\left (\underline {\hspace {1.25 ex}}\alpha ^{2} c +2 b^{2}\right ) \sqrt {a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{a}\right ) a}{\sqrt {a b}}}\, \sqrt {-\frac {\left (x -\frac {\sqrt {a b}}{a}\right ) a}{\sqrt {a b}}}\, \sqrt {-\frac {x a}{\sqrt {a b}}}\, \left (a \left (a^{2} \underline {\hspace {1.25 ex}}\alpha ^{3}+a b \underline {\hspace {1.25 ex}}\alpha +c \underline {\hspace {1.25 ex}}\alpha \right )-a^{2} \sqrt {a b}\, \underline {\hspace {1.25 ex}}\alpha ^{2}-\sqrt {a b}\, a b -\sqrt {a b}\, c \right ) \EllipticPi \left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{a}\right ) a}{\sqrt {a b}}}, \frac {-\sqrt {a b}\, \underline {\hspace {1.25 ex}}\alpha ^{3} a^{2}+\underline {\hspace {1.25 ex}}\alpha ^{2} a^{2} b -\sqrt {a b}\, \underline {\hspace {1.25 ex}}\alpha a b -\sqrt {a b}\, \underline {\hspace {1.25 ex}}\alpha c +a \,b^{2}+b c}{b \left (2 a b +c \right )}, \frac {\sqrt {2}}{2}\right )}{\underline {\hspace {1.25 ex}}\alpha \left (2 \underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}+c \right ) \left (2 a b +c \right ) \sqrt {x \left (a \,x^{2}-b \right )}}\right )}{2 a b}\)	\(374\)
default	\(\frac {\sqrt {a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{a}\right ) a}{\sqrt {a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{a}\right ) a}{\sqrt {a b}}}\, \sqrt {-\frac {x a}{\sqrt {a b}}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{a}\right ) a}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right )}{a \sqrt {a \,x^{3}-b x}}+\frac {\sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (a^{2} \textit {\_Z}^{4}+c \,\textit {\_Z}^{2}+b^{2}\right )}{\sum }\frac {\left (-\underline {\hspace {1.25 ex}}\alpha ^{2} c -2 b^{2}\right ) \sqrt {a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{a}\right ) a}{\sqrt {a b}}}\, \sqrt {-\frac {\left (x -\frac {\sqrt {a b}}{a}\right ) a}{\sqrt {a b}}}\, \sqrt {-\frac {x a}{\sqrt {a b}}}\, \left (a \left (a^{2} \underline {\hspace {1.25 ex}}\alpha ^{3}+a b \underline {\hspace {1.25 ex}}\alpha +c \underline {\hspace {1.25 ex}}\alpha \right )-a^{2} \sqrt {a b}\, \underline {\hspace {1.25 ex}}\alpha ^{2}-\sqrt {a b}\, a b -\sqrt {a b}\, c \right ) \EllipticPi \left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{a}\right ) a}{\sqrt {a b}}}, \frac {-\sqrt {a b}\, \underline {\hspace {1.25 ex}}\alpha ^{3} a^{2}+\underline {\hspace {1.25 ex}}\alpha ^{2} a^{2} b -\sqrt {a b}\, \underline {\hspace {1.25 ex}}\alpha a b -\sqrt {a b}\, \underline {\hspace {1.25 ex}}\alpha c +a \,b^{2}+b c}{b \left (2 a b +c \right )}, \frac {\sqrt {2}}{2}\right )}{\underline {\hspace {1.25 ex}}\alpha \left (2 \underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}+c \right ) \left (2 a b +c \right ) \sqrt {x \left (a \,x^{2}-b \right )}}\right )}{2 a b}\)	\(375\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a^2*x^4-b^2)/(a*x^3-b*x)^(1/2)/(a^2*x^4+c*x^2+b^2),x,method=_RETURNVERBOSE)

[Out]

1/a*(a*b)^(1/2)*((x+1/a*(a*b)^(1/2))*a/(a*b)^(1/2))^(1/2)*(-2*(x-1/a*(a*b)^(1/2))*a/(a*b)^(1/2))^(1/2)*(-x*a/(
a*b)^(1/2))^(1/2)/(a*x^3-b*x)^(1/2)*EllipticF(((x+1/a*(a*b)^(1/2))*a/(a*b)^(1/2))^(1/2),1/2*2^(1/2))-1/2/a/b*2
^(1/2)*sum((_alpha^2*c+2*b^2)/_alpha/(2*_alpha^2*a^2+c)/(2*a*b+c)*(a*b)^(1/2)*((x+1/a*(a*b)^(1/2))*a/(a*b)^(1/
2))^(1/2)*(-(x-1/a*(a*b)^(1/2))*a/(a*b)^(1/2))^(1/2)*(-x*a/(a*b)^(1/2))^(1/2)/(x*(a*x^2-b))^(1/2)*(a*(_alpha^3
*a^2+_alpha*a*b+_alpha*c)-a^2*(a*b)^(1/2)*_alpha^2-(a*b)^(1/2)*a*b-(a*b)^(1/2)*c)*EllipticPi(((x+1/a*(a*b)^(1/
2))*a/(a*b)^(1/2))^(1/2),(-(a*b)^(1/2)*_alpha^3*a^2+_alpha^2*a^2*b-(a*b)^(1/2)*_alpha*a*b-(a*b)^(1/2)*_alpha*c
+a*b^2+b*c)/b/(2*a*b+c),1/2*2^(1/2)),_alpha=RootOf(_Z^4*a^2+_Z^2*c+b^2))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a^{2} x^{4} - b^{2}}{{\left (a^{2} x^{4} + c x^{2} + b^{2}\right )} \sqrt {a x^{3} - b x}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a^2*x^4-b^2)/(a*x^3-b*x)^(1/2)/(a^2*x^4+c*x^2+b^2),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 4.81, size = 165, normalized size = 1.04 \begin {gather*} \frac {\ln \left (\frac {b-x\,\sqrt {-c-2\,a\,b}+2\,\sqrt {a\,x^3-b\,x}\,{\left (-c-2\,a\,b\right )}^{1/4}-a\,x^2}{b+x\,\sqrt {-c-2\,a\,b}-a\,x^2}\right )}{2\,{\left (-c-2\,a\,b\right )}^{1/4}}+\frac {\ln \left (\frac {b+x\,\sqrt {-c-2\,a\,b}-a\,x^2-\sqrt {a\,x^3-b\,x}\,{\left (-c-2\,a\,b\right )}^{1/4}\,2{}\mathrm {i}}{x\,\sqrt {-c-2\,a\,b}-b+a\,x^2}\right )\,1{}\mathrm {i}}{2\,{\left (-c-2\,a\,b\right )}^{1/4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a**2*x**4-b**2)/(a*x**3-b*x)**(1/2)/(a**2*x**4+c*x**2+b**2),x)

[Out]

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

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