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

Optimal. Leaf size=166 \[ \frac {\tan ^{-1}\left (\frac {2^{3/4} \sqrt {a} \sqrt {b} \sqrt {a^2 x^3-b^2 x}}{-a^2 x^2+\sqrt {2} a b x+b^2}\right )}{2^{3/4} \sqrt {a} \sqrt {b}}-\frac {\tanh ^{-1}\left (\frac {\frac {a^{3/2} x^2}{2^{3/4} \sqrt {b}}-\frac {b^{3/2}}{2^{3/4} \sqrt {a}}+\frac {\sqrt {a} \sqrt {b} x}{\sqrt [4]{2}}}{\sqrt {a^2 x^3-b^2 x}}\right )}{2^{3/4} \sqrt {a} \sqrt {b}} \]

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Rubi [C]  time = 1.71, antiderivative size = 518, normalized size of antiderivative = 3.12, number of steps used = 21, number of rules used = 10, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2056, 1586, 6715, 6725, 406, 224, 221, 409, 1219, 1218} \begin {gather*} -\frac {\sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \Pi \left (\frac {\sqrt [4]{-a^4}}{a};\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{\sqrt {a} \sqrt {a^2 x^3-b^2 x}}-\frac {\sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \Pi \left (-\frac {\sqrt {-\sqrt {-a^4}}}{a};\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{\sqrt {a} \sqrt {a^2 x^3-b^2 x}}-\frac {\sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \Pi \left (\frac {\sqrt {-\sqrt {-a^4}}}{a};\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{\sqrt {a} \sqrt {a^2 x^3-b^2 x}}+\frac {\left (a^2-\sqrt {-a^4}\right ) \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{a^{5/2} \sqrt {a^2 x^3-b^2 x}}+\frac {\left (\sqrt {-a^4}+a^2\right ) \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{a^{5/2} \sqrt {a^2 x^3-b^2 x}}-\frac {\sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \Pi \left (\frac {a^3}{\left (-a^4\right )^{3/4}};\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{\sqrt {a} \sqrt {a^2 x^3-b^2 x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((a^2 - Sqrt[-a^4])*Sqrt[b]*Sqrt[x]*Sqrt[1 - (a^2*x^2)/b^2]*EllipticF[ArcSin[(Sqrt[a]*Sqrt[x])/Sqrt[b]], -1])/
(a^(5/2)*Sqrt[-(b^2*x) + a^2*x^3]) + ((a^2 + Sqrt[-a^4])*Sqrt[b]*Sqrt[x]*Sqrt[1 - (a^2*x^2)/b^2]*EllipticF[Arc
Sin[(Sqrt[a]*Sqrt[x])/Sqrt[b]], -1])/(a^(5/2)*Sqrt[-(b^2*x) + a^2*x^3]) - (Sqrt[b]*Sqrt[x]*Sqrt[1 - (a^2*x^2)/
b^2]*EllipticPi[a^3/(-a^4)^(3/4), ArcSin[(Sqrt[a]*Sqrt[x])/Sqrt[b]], -1])/(Sqrt[a]*Sqrt[-(b^2*x) + a^2*x^3]) -
(Sqrt[b]*Sqrt[x]*Sqrt[1 - (a^2*x^2)/b^2]*EllipticPi[(-a^4)^(1/4)/a, ArcSin[(Sqrt[a]*Sqrt[x])/Sqrt[b]], -1])/(
Sqrt[a]*Sqrt[-(b^2*x) + a^2*x^3]) - (Sqrt[b]*Sqrt[x]*Sqrt[1 - (a^2*x^2)/b^2]*EllipticPi[-(Sqrt[-Sqrt[-a^4]]/a)
, ArcSin[(Sqrt[a]*Sqrt[x])/Sqrt[b]], -1])/(Sqrt[a]*Sqrt[-(b^2*x) + a^2*x^3]) - (Sqrt[b]*Sqrt[x]*Sqrt[1 - (a^2*
x^2)/b^2]*EllipticPi[Sqrt[-Sqrt[-a^4]]/a, ArcSin[(Sqrt[a]*Sqrt[x])/Sqrt[b]], -1])/(Sqrt[a]*Sqrt[-(b^2*x) + a^2
*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 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]

Rubi steps

\begin {align*} \int \frac {-b^4+a^4 x^4}{\sqrt {-b^2 x+a^2 x^3} \left (b^4+a^4 x^4\right )} \, dx &=\frac {\left (\sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \int \frac {-b^4+a^4 x^4}{\sqrt {x} \sqrt {-b^2+a^2 x^2} \left (b^4+a^4 x^4\right )} \, dx}{\sqrt {-b^2 x+a^2 x^3}}\\ &=\frac {\left (\sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \int \frac {\sqrt {-b^2+a^2 x^2} \left (b^2+a^2 x^2\right )}{\sqrt {x} \left (b^4+a^4 x^4\right )} \, dx}{\sqrt {-b^2 x+a^2 x^3}}\\ &=\frac {\left (2 \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-b^2+a^2 x^4} \left (b^2+a^2 x^4\right )}{b^4+a^4 x^8} \, dx,x,\sqrt {x}\right )}{\sqrt {-b^2 x+a^2 x^3}}\\ &=\frac {\left (2 \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \left (-\frac {\sqrt {-a^4} \left (a^2 b^2+\sqrt {-a^4} b^2\right ) \sqrt {-b^2+a^2 x^4}}{2 a^4 b^2 \left (b^2-\sqrt {-a^4} x^4\right )}+\frac {\sqrt {-a^4} \left (a^2 b^2-\sqrt {-a^4} b^2\right ) \sqrt {-b^2+a^2 x^4}}{2 a^4 b^2 \left (b^2+\sqrt {-a^4} x^4\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {-b^2 x+a^2 x^3}}\\ &=\frac {\left (\left (a^2+\sqrt {-a^4}\right ) \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-b^2+a^2 x^4}}{b^2+\sqrt {-a^4} x^4} \, dx,x,\sqrt {x}\right )}{a^2 \sqrt {-b^2 x+a^2 x^3}}-\frac {\left (\sqrt {-a^4} \left (a^2 b^2+\sqrt {-a^4} b^2\right ) \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-b^2+a^2 x^4}}{b^2-\sqrt {-a^4} x^4} \, dx,x,\sqrt {x}\right )}{a^4 b^2 \sqrt {-b^2 x+a^2 x^3}}\\ &=\frac {\left (\left (a^2+\sqrt {-a^4}\right ) \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {-a^4} \sqrt {-b^2 x+a^2 x^3}}-\frac {\left (\left (a^2+\sqrt {-a^4}\right ) \left (a^2 b^2+\sqrt {-a^4} b^2\right ) \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-b^2+a^2 x^4} \left (b^2+\sqrt {-a^4} x^4\right )} \, dx,x,\sqrt {x}\right )}{a^2 \sqrt {-a^4} \sqrt {-b^2 x+a^2 x^3}}+\frac {\left (\sqrt {-a^4} \left (a^2+\sqrt {-a^4}\right ) \left (a^2 b^2+\sqrt {-a^4} b^2\right ) \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-b^2+a^2 x^4} \left (b^2-\sqrt {-a^4} x^4\right )} \, dx,x,\sqrt {x}\right )}{a^6 \sqrt {-b^2 x+a^2 x^3}}+\frac {\left (\left (a^2 b^2+\sqrt {-a^4} b^2\right ) \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{a^2 b^2 \sqrt {-b^2 x+a^2 x^3}}\\ &=-\frac {\left (\left (a^2+\sqrt {-a^4}\right ) \left (a^2 b^2+\sqrt {-a^4} b^2\right ) \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-\frac {\sqrt {-\sqrt {-a^4}} x^2}{b}\right ) \sqrt {-b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{2 a^2 \sqrt {-a^4} b^2 \sqrt {-b^2 x+a^2 x^3}}-\frac {\left (\left (a^2+\sqrt {-a^4}\right ) \left (a^2 b^2+\sqrt {-a^4} b^2\right ) \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+\frac {\sqrt {-\sqrt {-a^4}} x^2}{b}\right ) \sqrt {-b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{2 a^2 \sqrt {-a^4} b^2 \sqrt {-b^2 x+a^2 x^3}}+\frac {\left (\sqrt {-a^4} \left (a^2+\sqrt {-a^4}\right ) \left (a^2 b^2+\sqrt {-a^4} b^2\right ) \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-\frac {\sqrt [4]{-a^4} x^2}{b}\right ) \sqrt {-b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{2 a^6 b^2 \sqrt {-b^2 x+a^2 x^3}}+\frac {\left (\sqrt {-a^4} \left (a^2+\sqrt {-a^4}\right ) \left (a^2 b^2+\sqrt {-a^4} b^2\right ) \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+\frac {\sqrt [4]{-a^4} x^2}{b}\right ) \sqrt {-b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{2 a^6 b^2 \sqrt {-b^2 x+a^2 x^3}}+\frac {\left (\left (a^2+\sqrt {-a^4}\right ) \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {a^2 x^4}{b^2}}} \, dx,x,\sqrt {x}\right )}{\sqrt {-a^4} \sqrt {-b^2 x+a^2 x^3}}+\frac {\left (\left (a^2 b^2+\sqrt {-a^4} b^2\right ) \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {a^2 x^4}{b^2}}} \, dx,x,\sqrt {x}\right )}{a^2 b^2 \sqrt {-b^2 x+a^2 x^3}}\\ &=\frac {\left (a^2+\sqrt {-a^4}\right ) \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{a^{5/2} \sqrt {-b^2 x+a^2 x^3}}+\frac {\left (a^2+\sqrt {-a^4}\right ) \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{\sqrt {a} \sqrt {-a^4} \sqrt {-b^2 x+a^2 x^3}}-\frac {\left (\left (a^2+\sqrt {-a^4}\right ) \left (a^2 b^2+\sqrt {-a^4} b^2\right ) \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-\frac {\sqrt {-\sqrt {-a^4}} x^2}{b}\right ) \sqrt {1-\frac {a^2 x^4}{b^2}}} \, dx,x,\sqrt {x}\right )}{2 a^2 \sqrt {-a^4} b^2 \sqrt {-b^2 x+a^2 x^3}}-\frac {\left (\left (a^2+\sqrt {-a^4}\right ) \left (a^2 b^2+\sqrt {-a^4} b^2\right ) \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+\frac {\sqrt {-\sqrt {-a^4}} x^2}{b}\right ) \sqrt {1-\frac {a^2 x^4}{b^2}}} \, dx,x,\sqrt {x}\right )}{2 a^2 \sqrt {-a^4} b^2 \sqrt {-b^2 x+a^2 x^3}}+\frac {\left (\sqrt {-a^4} \left (a^2+\sqrt {-a^4}\right ) \left (a^2 b^2+\sqrt {-a^4} b^2\right ) \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-\frac {\sqrt [4]{-a^4} x^2}{b}\right ) \sqrt {1-\frac {a^2 x^4}{b^2}}} \, dx,x,\sqrt {x}\right )}{2 a^6 b^2 \sqrt {-b^2 x+a^2 x^3}}+\frac {\left (\sqrt {-a^4} \left (a^2+\sqrt {-a^4}\right ) \left (a^2 b^2+\sqrt {-a^4} b^2\right ) \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+\frac {\sqrt [4]{-a^4} x^2}{b}\right ) \sqrt {1-\frac {a^2 x^4}{b^2}}} \, dx,x,\sqrt {x}\right )}{2 a^6 b^2 \sqrt {-b^2 x+a^2 x^3}}\\ &=\frac {\left (a^2+\sqrt {-a^4}\right ) \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{a^{5/2} \sqrt {-b^2 x+a^2 x^3}}+\frac {\left (a^2+\sqrt {-a^4}\right ) \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{\sqrt {a} \sqrt {-a^4} \sqrt {-b^2 x+a^2 x^3}}-\frac {\sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \Pi \left (\frac {a^3}{\left (-a^4\right )^{3/4}};\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{\sqrt {a} \sqrt {-b^2 x+a^2 x^3}}-\frac {\sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \Pi \left (\frac {\sqrt [4]{-a^4}}{a};\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{\sqrt {a} \sqrt {-b^2 x+a^2 x^3}}-\frac {\sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \Pi \left (-\frac {\sqrt {-\sqrt {-a^4}}}{a};\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{\sqrt {a} \sqrt {-b^2 x+a^2 x^3}}-\frac {\sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \Pi \left (\frac {\sqrt {-\sqrt {-a^4}}}{a};\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{\sqrt {a} \sqrt {-b^2 x+a^2 x^3}}\\ \end {align*}

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Mathematica [C]  time = 1.48, size = 204, normalized size = 1.23 \begin {gather*} -\frac {i x^{3/2} \sqrt {1-\frac {b^2}{a^2 x^2}} \left (2 F\left (\left .i \sinh ^{-1}\left (\frac {\sqrt {-\frac {b}{a}}}{\sqrt {x}}\right )\right |-1\right )-\Pi \left (-\sqrt [4]{-1};\left .i \sinh ^{-1}\left (\frac {\sqrt {-\frac {b}{a}}}{\sqrt {x}}\right )\right |-1\right )-\Pi \left (\sqrt [4]{-1};\left .i \sinh ^{-1}\left (\frac {\sqrt {-\frac {b}{a}}}{\sqrt {x}}\right )\right |-1\right )-\Pi \left (-(-1)^{3/4};\left .i \sinh ^{-1}\left (\frac {\sqrt {-\frac {b}{a}}}{\sqrt {x}}\right )\right |-1\right )-\Pi \left ((-1)^{3/4};\left .i \sinh ^{-1}\left (\frac {\sqrt {-\frac {b}{a}}}{\sqrt {x}}\right )\right |-1\right )\right )}{\sqrt {-\frac {b}{a}} \sqrt {a^2 x^3-b^2 x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-I)*Sqrt[1 - b^2/(a^2*x^2)]*x^(3/2)*(2*EllipticF[I*ArcSinh[Sqrt[-(b/a)]/Sqrt[x]], -1] - EllipticPi[-(-1)^(1/
4), I*ArcSinh[Sqrt[-(b/a)]/Sqrt[x]], -1] - EllipticPi[(-1)^(1/4), I*ArcSinh[Sqrt[-(b/a)]/Sqrt[x]], -1] - Ellip
ticPi[-(-1)^(3/4), I*ArcSinh[Sqrt[-(b/a)]/Sqrt[x]], -1] - EllipticPi[(-1)^(3/4), I*ArcSinh[Sqrt[-(b/a)]/Sqrt[x
]], -1]))/(Sqrt[-(b/a)]*Sqrt[-(b^2*x) + a^2*x^3])

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IntegrateAlgebraic [A]  time = 0.59, size = 166, normalized size = 1.00 \begin {gather*} \frac {\tan ^{-1}\left (\frac {2^{3/4} \sqrt {a} \sqrt {b} \sqrt {-b^2 x+a^2 x^3}}{b^2+\sqrt {2} a b x-a^2 x^2}\right )}{2^{3/4} \sqrt {a} \sqrt {b}}-\frac {\tanh ^{-1}\left (\frac {-\frac {b^{3/2}}{2^{3/4} \sqrt {a}}+\frac {\sqrt {a} \sqrt {b} x}{\sqrt [4]{2}}+\frac {a^{3/2} x^2}{2^{3/4} \sqrt {b}}}{\sqrt {-b^2 x+a^2 x^3}}\right )}{2^{3/4} \sqrt {a} \sqrt {b}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 1.04, size = 979, normalized size = 5.90

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*sqrt(2)*(1/2)^(1/4)*(1/(a^2*b^2))^(1/4)*arctan(1/2*((2*sqrt(2)*(1/2)^(3/4)*a^2*b^2*x*(1/(a^2*b^2))^(3/4)
- sqrt(2)*(1/2)^(1/4)*(a^2*x^2 - b^2)*(1/(a^2*b^2))^(1/4))*sqrt(a^2*x^3 - b^2*x) - (2*a^2*x^3 - 2*b^2*x - (2*s
qrt(2)*(1/2)^(3/4)*a^2*b^2*x*(1/(a^2*b^2))^(3/4) + sqrt(2)*(1/2)^(1/4)*(a^2*x^2 - b^2)*(1/(a^2*b^2))^(1/4))*sq
rt(a^2*x^3 - b^2*x))*sqrt((a^4*x^4 + b^4 + 8*sqrt(1/2)*(a^4*b^2*x^3 - a^2*b^4*x)*sqrt(1/(a^2*b^2)) + 4*(sqrt(2
)*(1/2)^(1/4)*a^2*b^2*x*(1/(a^2*b^2))^(1/4) + sqrt(2)*(1/2)^(3/4)*(a^4*b^2*x^2 - a^2*b^4)*(1/(a^2*b^2))^(3/4))
*sqrt(a^2*x^3 - b^2*x))/(a^4*x^4 + b^4)))/(a^2*x^3 - b^2*x)) - 1/2*sqrt(2)*(1/2)^(1/4)*(1/(a^2*b^2))^(1/4)*arc
tan(1/2*((2*sqrt(2)*(1/2)^(3/4)*a^2*b^2*x*(1/(a^2*b^2))^(3/4) - sqrt(2)*(1/2)^(1/4)*(a^2*x^2 - b^2)*(1/(a^2*b^
2))^(1/4))*sqrt(a^2*x^3 - b^2*x) + (2*a^2*x^3 - 2*b^2*x + (2*sqrt(2)*(1/2)^(3/4)*a^2*b^2*x*(1/(a^2*b^2))^(3/4)
+ sqrt(2)*(1/2)^(1/4)*(a^2*x^2 - b^2)*(1/(a^2*b^2))^(1/4))*sqrt(a^2*x^3 - b^2*x))*sqrt((a^4*x^4 + b^4 + 8*sqr
t(1/2)*(a^4*b^2*x^3 - a^2*b^4*x)*sqrt(1/(a^2*b^2)) - 4*(sqrt(2)*(1/2)^(1/4)*a^2*b^2*x*(1/(a^2*b^2))^(1/4) + sq
rt(2)*(1/2)^(3/4)*(a^4*b^2*x^2 - a^2*b^4)*(1/(a^2*b^2))^(3/4))*sqrt(a^2*x^3 - b^2*x))/(a^4*x^4 + b^4)))/(a^2*x
^3 - b^2*x)) - 1/8*sqrt(2)*(1/2)^(1/4)*(1/(a^2*b^2))^(1/4)*log((a^4*x^4 + b^4 + 8*sqrt(1/2)*(a^4*b^2*x^3 - a^2
*b^4*x)*sqrt(1/(a^2*b^2)) + 4*(sqrt(2)*(1/2)^(1/4)*a^2*b^2*x*(1/(a^2*b^2))^(1/4) + sqrt(2)*(1/2)^(3/4)*(a^4*b^
2*x^2 - a^2*b^4)*(1/(a^2*b^2))^(3/4))*sqrt(a^2*x^3 - b^2*x))/(a^4*x^4 + b^4)) + 1/8*sqrt(2)*(1/2)^(1/4)*(1/(a^
2*b^2))^(1/4)*log((a^4*x^4 + b^4 + 8*sqrt(1/2)*(a^4*b^2*x^3 - a^2*b^4*x)*sqrt(1/(a^2*b^2)) - 4*(sqrt(2)*(1/2)^
(1/4)*a^2*b^2*x*(1/(a^2*b^2))^(1/4) + sqrt(2)*(1/2)^(3/4)*(a^4*b^2*x^2 - a^2*b^4)*(1/(a^2*b^2))^(3/4))*sqrt(a^
2*x^3 - b^2*x))/(a^4*x^4 + b^4))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.17, size = 247, normalized size = 1.49

method	result	size
default	\(\frac {b \sqrt {\frac {\left (x +\frac {b}{a}\right ) a}{b}}\, \sqrt {-\frac {2 \left (x -\frac {b}{a}\right ) a}{b}}\, \sqrt {-\frac {a x}{b}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {b}{a}\right ) a}{b}}, \frac {\sqrt {2}}{2}\right )}{a \sqrt {a^{2} x^{3}-b^{2} x}}-\frac {b \sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (a^{4} \textit {\_Z}^{4}+b^{4}\right )}{\sum }\frac {\left (a^{3} \underline {\hspace {1.25 ex}}\alpha ^{3}-\underline {\hspace {1.25 ex}}\alpha ^{2} a^{2} b +\underline {\hspace {1.25 ex}}\alpha a \,b^{2}-b^{3}\right ) \sqrt {\frac {\left (x +\frac {b}{a}\right ) a}{b}}\, \sqrt {-\frac {\left (x -\frac {b}{a}\right ) a}{b}}\, \sqrt {-\frac {a x}{b}}\, \EllipticPi \left (\sqrt {\frac {\left (x +\frac {b}{a}\right ) a}{b}}, -\frac {a^{3} \underline {\hspace {1.25 ex}}\alpha ^{3}-\underline {\hspace {1.25 ex}}\alpha ^{2} a^{2} b +\underline {\hspace {1.25 ex}}\alpha a \,b^{2}-b^{3}}{2 b^{3}}, \frac {\sqrt {2}}{2}\right )}{\underline {\hspace {1.25 ex}}\alpha ^{3} \sqrt {x \left (a^{2} x^{2}-b^{2}\right )}}\right )}{4 a^{4}}\)	\(247\)
elliptic	\(\frac {b \sqrt {\frac {\left (x +\frac {b}{a}\right ) a}{b}}\, \sqrt {-\frac {2 \left (x -\frac {b}{a}\right ) a}{b}}\, \sqrt {-\frac {a x}{b}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {b}{a}\right ) a}{b}}, \frac {\sqrt {2}}{2}\right )}{a \sqrt {a^{2} x^{3}-b^{2} x}}-\frac {b \sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (a^{4} \textit {\_Z}^{4}+b^{4}\right )}{\sum }\frac {\left (a^{3} \underline {\hspace {1.25 ex}}\alpha ^{3}-\underline {\hspace {1.25 ex}}\alpha ^{2} a^{2} b +\underline {\hspace {1.25 ex}}\alpha a \,b^{2}-b^{3}\right ) \sqrt {\frac {\left (x +\frac {b}{a}\right ) a}{b}}\, \sqrt {-\frac {\left (x -\frac {b}{a}\right ) a}{b}}\, \sqrt {-\frac {a x}{b}}\, \EllipticPi \left (\sqrt {\frac {\left (x +\frac {b}{a}\right ) a}{b}}, -\frac {a^{3} \underline {\hspace {1.25 ex}}\alpha ^{3}-\underline {\hspace {1.25 ex}}\alpha ^{2} a^{2} b +\underline {\hspace {1.25 ex}}\alpha a \,b^{2}-b^{3}}{2 b^{3}}, \frac {\sqrt {2}}{2}\right )}{\underline {\hspace {1.25 ex}}\alpha ^{3} \sqrt {x \left (a^{2} x^{2}-b^{2}\right )}}\right )}{4 a^{4}}\)	\(247\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 7.93, size = 202, normalized size = 1.22 \begin {gather*} \frac {2^{1/4}\,\sqrt {-\frac {1}{8}{}\mathrm {i}}\,\ln \left (\frac {{\left (-1\right )}^{1/4}\,2^{3/4}\,b^2-{\left (-1\right )}^{1/4}\,2^{3/4}\,a^2\,x^2-2\,{\left (-1\right )}^{3/4}\,2^{1/4}\,a\,b\,x+\sqrt {a}\,\sqrt {b}\,\sqrt {a^2\,x^3-b^2\,x}\,4{}\mathrm {i}}{-a^2\,x^2+1{}\mathrm {i}\,\sqrt {2}\,a\,b\,x+b^2}\right )}{\sqrt {a}\,\sqrt {b}}+\frac {2^{1/4}\,\sqrt {\frac {1}{8}{}\mathrm {i}}\,\ln \left (\frac {{\left (-1\right )}^{3/4}\,2^{3/4}\,b^2-{\left (-1\right )}^{3/4}\,2^{3/4}\,a^2\,x^2-2\,{\left (-1\right )}^{1/4}\,2^{1/4}\,a\,b\,x+\sqrt {a}\,\sqrt {b}\,\sqrt {a^2\,x^3-b^2\,x}\,4{}\mathrm {i}}{a^2\,x^2+1{}\mathrm {i}\,\sqrt {2}\,a\,b\,x-b^2}\right )}{\sqrt {a}\,\sqrt {b}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2^(1/4)*(-1i/8)^(1/2)*log((a^(1/2)*b^(1/2)*(a^2*x^3 - b^2*x)^(1/2)*4i + (-1)^(1/4)*2^(3/4)*b^2 - (-1)^(1/4)*2
^(3/4)*a^2*x^2 - 2*(-1)^(3/4)*2^(1/4)*a*b*x)/(b^2 - a^2*x^2 + 2^(1/2)*a*b*x*1i)))/(a^(1/2)*b^(1/2)) + (2^(1/4)
*(1i/8)^(1/2)*log((a^(1/2)*b^(1/2)*(a^2*x^3 - b^2*x)^(1/2)*4i + (-1)^(3/4)*2^(3/4)*b^2 - (-1)^(3/4)*2^(3/4)*a^
2*x^2 - 2*(-1)^(1/4)*2^(1/4)*a*b*x)/(a^2*x^2 - b^2 + 2^(1/2)*a*b*x*1i)))/(a^(1/2)*b^(1/2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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