\(\int \frac {-b^4+a^4 x^4}{\sqrt {-b^2 x+a^2 x^3} (b^4+a^4 x^4)} \, dx\) [2228]

Optimal result

Integrand size = 45, antiderivative size = 166 \[ \int \frac {-b^4+a^4 x^4}{\sqrt {-b^2 x+a^2 x^3} \left (b^4+a^4 x^4\right )} \, dx=\frac {\arctan \left (\frac {2^{3/4} \sqrt {a} \sqrt {b} \sqrt {-b^2 x+a^2 x^3}}{b^2+\sqrt {2} a b x-a^2 x^2}\right )}{2^{3/4} \sqrt {a} \sqrt {b}}-\frac {\text {arctanh}\left (\frac {-\frac {b^{3/2}}{2^{3/4} \sqrt {a}}+\frac {\sqrt {a} \sqrt {b} x}{\sqrt [4]{2}}+\frac {a^{3/2} x^2}{2^{3/4} \sqrt {b}}}{\sqrt {-b^2 x+a^2 x^3}}\right )}{2^{3/4} \sqrt {a} \sqrt {b}} \]

[Out]

1/2*arctan(2^(3/4)*a^(1/2)*b^(1/2)*(a^2*x^3-b^2*x)^(1/2)/(b^2+2^(1/2)*a*b*x-a^2*x^2))*2^(1/4)/a^(1/2)/b^(1/2)-
1/2*arctanh((-1/2*b^(3/2)*2^(1/4)/a^(1/2)+1/2*a^(1/2)*b^(1/2)*x*2^(3/4)+1/2*a^(3/2)*x^2*2^(1/4)/b^(1/2))/(a^2*
x^3-b^2*x)^(1/2))*2^(1/4)/a^(1/2)/b^(1/2)

Rubi [C] (verified)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 1.05 (sec) , antiderivative size = 518, normalized size of antiderivative = 3.12, number of steps used = 21, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2081, 1600, 6847, 6857, 415, 230, 227, 418, 1233, 1232} \[ \int \frac {-b^4+a^4 x^4}{\sqrt {-b^2 x+a^2 x^3} \left (b^4+a^4 x^4\right )} \, dx=-\frac {\sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \operatorname {EllipticPi}\left (\frac {\sqrt [4]{-a^4}}{a},\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),-1\right )}{\sqrt {a} \sqrt {a^2 x^3-b^2 x}}-\frac {\sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \operatorname {EllipticPi}\left (-\frac {\sqrt {-\sqrt {-a^4}}}{a},\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),-1\right )}{\sqrt {a} \sqrt {a^2 x^3-b^2 x}}-\frac {\sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \operatorname {EllipticPi}\left (\frac {\sqrt {-\sqrt {-a^4}}}{a},\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),-1\right )}{\sqrt {a} \sqrt {a^2 x^3-b^2 x}}+\frac {\left (a^2-\sqrt {-a^4}\right ) \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),-1\right )}{a^{5/2} \sqrt {a^2 x^3-b^2 x}}+\frac {\left (\sqrt {-a^4}+a^2\right ) \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),-1\right )}{a^{5/2} \sqrt {a^2 x^3-b^2 x}}-\frac {\sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \operatorname {EllipticPi}\left (\frac {a^3}{\left (-a^4\right )^{3/4}},\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),-1\right )}{\sqrt {a} \sqrt {a^2 x^3-b^2 x}} \]

[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 227

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 230

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 415

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 418

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 1232

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

Rule 1233

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 1600

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 2081

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 6847

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 6857

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*} \text {integral}& = \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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\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}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\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 ) \text {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 ) \text {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 ) \text {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 ) \text {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}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\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}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\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}} \operatorname {EllipticPi}\left (\frac {a^3}{\left (-a^4\right )^{3/4}},\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\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}} \operatorname {EllipticPi}\left (\frac {\sqrt [4]{-a^4}}{a},\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\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}} \operatorname {EllipticPi}\left (-\frac {\sqrt {-\sqrt {-a^4}}}{a},\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\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}} \operatorname {EllipticPi}\left (\frac {\sqrt {-\sqrt {-a^4}}}{a},\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),-1\right )}{\sqrt {a} \sqrt {-b^2 x+a^2 x^3}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.79 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.09 \[ \int \frac {-b^4+a^4 x^4}{\sqrt {-b^2 x+a^2 x^3} \left (b^4+a^4 x^4\right )} \, dx=-\frac {\sqrt {x} \sqrt {-b^2+a^2 x^2} \left (\arctan \left (\frac {b^2+\sqrt {2} a b x-a^2 x^2}{2^{3/4} \sqrt {a} \sqrt {b} \sqrt {x} \sqrt {-b^2+a^2 x^2}}\right )+\text {arctanh}\left (\frac {2^{3/4} \sqrt {a} \sqrt {b} \sqrt {x} \sqrt {-b^2+a^2 x^2}}{-b^2+\sqrt {2} a b x+a^2 x^2}\right )\right )}{2^{3/4} \sqrt {a} \sqrt {b} \sqrt {-b^2 x+a^2 x^3}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 1.91 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.39

method	result	size
default	\(\frac {2^{\frac {1}{4}} \left (\ln \left (\frac {a^{2} x^{2}+\sqrt {2}\, \sqrt {a^{2} b^{2}}\, x -2^{\frac {3}{4}} \left (a^{2} b^{2}\right )^{\frac {1}{4}} \sqrt {a^{2} x^{3}-b^{2} x}-b^{2}}{a^{2} x^{2}+2^{\frac {3}{4}} \left (a^{2} b^{2}\right )^{\frac {1}{4}} \sqrt {a^{2} x^{3}-b^{2} x}+\sqrt {2}\, \sqrt {a^{2} b^{2}}\, x -b^{2}}\right )+2 \arctan \left (\frac {2^{\frac {1}{4}} \sqrt {a^{2} x^{3}-b^{2} x}+\left (a^{2} b^{2}\right )^{\frac {1}{4}} x}{\left (a^{2} b^{2}\right )^{\frac {1}{4}} x}\right )+2 \arctan \left (\frac {2^{\frac {1}{4}} \sqrt {a^{2} x^{3}-b^{2} x}-\left (a^{2} b^{2}\right )^{\frac {1}{4}} x}{\left (a^{2} b^{2}\right )^{\frac {1}{4}} x}\right )\right )}{4 \left (a^{2} b^{2}\right )^{\frac {1}{4}}}\)	\(230\)
pseudoelliptic	\(\frac {2^{\frac {1}{4}} \left (\ln \left (\frac {a^{2} x^{2}+\sqrt {2}\, \sqrt {a^{2} b^{2}}\, x -2^{\frac {3}{4}} \left (a^{2} b^{2}\right )^{\frac {1}{4}} \sqrt {a^{2} x^{3}-b^{2} x}-b^{2}}{a^{2} x^{2}+2^{\frac {3}{4}} \left (a^{2} b^{2}\right )^{\frac {1}{4}} \sqrt {a^{2} x^{3}-b^{2} x}+\sqrt {2}\, \sqrt {a^{2} b^{2}}\, x -b^{2}}\right )+2 \arctan \left (\frac {2^{\frac {1}{4}} \sqrt {a^{2} x^{3}-b^{2} x}+\left (a^{2} b^{2}\right )^{\frac {1}{4}} x}{\left (a^{2} b^{2}\right )^{\frac {1}{4}} x}\right )+2 \arctan \left (\frac {2^{\frac {1}{4}} \sqrt {a^{2} x^{3}-b^{2} x}-\left (a^{2} b^{2}\right )^{\frac {1}{4}} x}{\left (a^{2} b^{2}\right )^{\frac {1}{4}} x}\right )\right )}{4 \left (a^{2} b^{2}\right )^{\frac {1}{4}}}\)	\(230\)
elliptic	\(\frac {b \sqrt {\frac {\left (x +\frac {b}{a}\right ) a}{b}}\, \sqrt {-\frac {2 \left (x -\frac {b}{a}\right ) a}{b}}\, \sqrt {-\frac {a x}{b}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {b}{a}\right ) a}{b}}, \frac {\sqrt {2}}{2}\right )}{a \sqrt {a^{2} x^{3}-b^{2} x}}-\frac {b \sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (a^{4} \textit {\_Z}^{4}+b^{4}\right )}{\sum }\frac {\left (a^{3} \underline {\hspace {1.25 ex}}\alpha ^{3}-b \,\underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}+\underline {\hspace {1.25 ex}}\alpha a \,b^{2}-b^{3}\right ) \sqrt {\frac {\left (x +\frac {b}{a}\right ) a}{b}}\, \sqrt {-\frac {\left (x -\frac {b}{a}\right ) a}{b}}\, \sqrt {-\frac {a x}{b}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {\left (x +\frac {b}{a}\right ) a}{b}}, -\frac {a^{3} \underline {\hspace {1.25 ex}}\alpha ^{3}-b \,\underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}+\underline {\hspace {1.25 ex}}\alpha a \,b^{2}-b^{3}}{2 b^{3}}, \frac {\sqrt {2}}{2}\right )}{\underline {\hspace {1.25 ex}}\alpha ^{3} \sqrt {x \left (a^{2} x^{2}-b^{2}\right )}}\right )}{4 a^{4}}\)	\(247\)

[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]

1/4*2^(1/4)/(a^2*b^2)^(1/4)*(ln((a^2*x^2+2^(1/2)*(a^2*b^2)^(1/2)*x-2^(3/4)*(a^2*b^2)^(1/4)*(a^2*x^3-b^2*x)^(1/
2)-b^2)/(a^2*x^2+2^(3/4)*(a^2*b^2)^(1/4)*(a^2*x^3-b^2*x)^(1/2)+2^(1/2)*(a^2*b^2)^(1/2)*x-b^2))+2*arctan((2^(1/
4)*(a^2*x^3-b^2*x)^(1/2)+(a^2*b^2)^(1/4)*x)/(a^2*b^2)^(1/4)/x)+2*arctan((2^(1/4)*(a^2*x^3-b^2*x)^(1/2)-(a^2*b^
2)^(1/4)*x)/(a^2*b^2)^(1/4)/x))

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.32 (sec) , antiderivative size = 645, normalized size of antiderivative = 3.89 \[ \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 {1}{4} i \, \left (\frac {1}{2}\right )^{\frac {1}{4}} \left (-\frac {1}{a^{2} b^{2}}\right )^{\frac {1}{4}} \log \left (\frac {a^{4} x^{4} - 4 \, a^{2} b^{2} x^{2} + b^{4} + 4 \, \sqrt {\frac {1}{2}} {\left (a^{4} b^{2} x^{3} - a^{2} b^{4} x\right )} \sqrt {-\frac {1}{a^{2} b^{2}}} - 4 \, {\left (i \, \left (\frac {1}{2}\right )^{\frac {1}{4}} a^{2} b^{2} x \left (-\frac {1}{a^{2} b^{2}}\right )^{\frac {1}{4}} + \left (\frac {1}{2}\right )^{\frac {3}{4}} {\left (-i \, a^{4} b^{2} x^{2} + i \, a^{2} b^{4}\right )} \left (-\frac {1}{a^{2} b^{2}}\right )^{\frac {3}{4}}\right )} \sqrt {a^{2} x^{3} - b^{2} x}}{a^{4} x^{4} + b^{4}}\right ) + \frac {1}{4} i \, \left (\frac {1}{2}\right )^{\frac {1}{4}} \left (-\frac {1}{a^{2} b^{2}}\right )^{\frac {1}{4}} \log \left (\frac {a^{4} x^{4} - 4 \, a^{2} b^{2} x^{2} + b^{4} + 4 \, \sqrt {\frac {1}{2}} {\left (a^{4} b^{2} x^{3} - a^{2} b^{4} x\right )} \sqrt {-\frac {1}{a^{2} b^{2}}} - 4 \, {\left (-i \, \left (\frac {1}{2}\right )^{\frac {1}{4}} a^{2} b^{2} x \left (-\frac {1}{a^{2} b^{2}}\right )^{\frac {1}{4}} + \left (\frac {1}{2}\right )^{\frac {3}{4}} {\left (i \, a^{4} b^{2} x^{2} - i \, a^{2} b^{4}\right )} \left (-\frac {1}{a^{2} b^{2}}\right )^{\frac {3}{4}}\right )} \sqrt {a^{2} x^{3} - b^{2} x}}{a^{4} x^{4} + b^{4}}\right ) + \frac {1}{4} \, \left (\frac {1}{2}\right )^{\frac {1}{4}} \left (-\frac {1}{a^{2} b^{2}}\right )^{\frac {1}{4}} \log \left (\frac {a^{4} x^{4} - 4 \, a^{2} b^{2} x^{2} + b^{4} - 4 \, \sqrt {\frac {1}{2}} {\left (a^{4} b^{2} x^{3} - a^{2} b^{4} x\right )} \sqrt {-\frac {1}{a^{2} b^{2}}} + 4 \, {\left (\left (\frac {1}{2}\right )^{\frac {1}{4}} a^{2} b^{2} x \left (-\frac {1}{a^{2} b^{2}}\right )^{\frac {1}{4}} + \left (\frac {1}{2}\right )^{\frac {3}{4}} {\left (a^{4} b^{2} x^{2} - a^{2} b^{4}\right )} \left (-\frac {1}{a^{2} b^{2}}\right )^{\frac {3}{4}}\right )} \sqrt {a^{2} x^{3} - b^{2} x}}{a^{4} x^{4} + b^{4}}\right ) - \frac {1}{4} \, \left (\frac {1}{2}\right )^{\frac {1}{4}} \left (-\frac {1}{a^{2} b^{2}}\right )^{\frac {1}{4}} \log \left (\frac {a^{4} x^{4} - 4 \, a^{2} b^{2} x^{2} + b^{4} - 4 \, \sqrt {\frac {1}{2}} {\left (a^{4} b^{2} x^{3} - a^{2} b^{4} x\right )} \sqrt {-\frac {1}{a^{2} b^{2}}} - 4 \, {\left (\left (\frac {1}{2}\right )^{\frac {1}{4}} a^{2} b^{2} x \left (-\frac {1}{a^{2} b^{2}}\right )^{\frac {1}{4}} + \left (\frac {1}{2}\right )^{\frac {3}{4}} {\left (a^{4} b^{2} x^{2} - a^{2} b^{4}\right )} \left (-\frac {1}{a^{2} b^{2}}\right )^{\frac {3}{4}}\right )} \sqrt {a^{2} x^{3} - b^{2} x}}{a^{4} x^{4} + b^{4}}\right ) \]

[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]

-1/4*I*(1/2)^(1/4)*(-1/(a^2*b^2))^(1/4)*log((a^4*x^4 - 4*a^2*b^2*x^2 + b^4 + 4*sqrt(1/2)*(a^4*b^2*x^3 - a^2*b^
4*x)*sqrt(-1/(a^2*b^2)) - 4*(I*(1/2)^(1/4)*a^2*b^2*x*(-1/(a^2*b^2))^(1/4) + (1/2)^(3/4)*(-I*a^4*b^2*x^2 + I*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)) + 1/4*I*(1/2)^(1/4)*(-1/(a^2*b^2))^(1/4)*
log((a^4*x^4 - 4*a^2*b^2*x^2 + b^4 + 4*sqrt(1/2)*(a^4*b^2*x^3 - a^2*b^4*x)*sqrt(-1/(a^2*b^2)) - 4*(-I*(1/2)^(1
/4)*a^2*b^2*x*(-1/(a^2*b^2))^(1/4) + (1/2)^(3/4)*(I*a^4*b^2*x^2 - I*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)) + 1/4*(1/2)^(1/4)*(-1/(a^2*b^2))^(1/4)*log((a^4*x^4 - 4*a^2*b^2*x^2 + b^4 - 4*sqr
t(1/2)*(a^4*b^2*x^3 - a^2*b^4*x)*sqrt(-1/(a^2*b^2)) + 4*((1/2)^(1/4)*a^2*b^2*x*(-1/(a^2*b^2))^(1/4) + (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)) - 1/4*(1/2)^(1/4)*(-
1/(a^2*b^2))^(1/4)*log((a^4*x^4 - 4*a^2*b^2*x^2 + b^4 - 4*sqrt(1/2)*(a^4*b^2*x^3 - a^2*b^4*x)*sqrt(-1/(a^2*b^2
)) - 4*((1/2)^(1/4)*a^2*b^2*x*(-1/(a^2*b^2))^(1/4) + (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))

Sympy [F]

\[ \int \frac {-b^4+a^4 x^4}{\sqrt {-b^2 x+a^2 x^3} \left (b^4+a^4 x^4\right )} \, dx=\int \frac {\left (a x - b\right ) \left (a x + b\right ) \left (a^{2} x^{2} + b^{2}\right )}{\sqrt {x \left (a x - b\right ) \left (a x + b\right )} \left (a^{4} x^{4} + b^{4}\right )}\, dx \]

[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)

Maxima [F]

\[ \int \frac {-b^4+a^4 x^4}{\sqrt {-b^2 x+a^2 x^3} \left (b^4+a^4 x^4\right )} \, dx=\int { \frac {a^{4} x^{4} - b^{4}}{{\left (a^{4} x^{4} + b^{4}\right )} \sqrt {a^{2} x^{3} - b^{2} x}} \,d x } \]

[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)

Giac [F]

\[ \int \frac {-b^4+a^4 x^4}{\sqrt {-b^2 x+a^2 x^3} \left (b^4+a^4 x^4\right )} \, dx=\int { \frac {a^{4} x^{4} - b^{4}}{{\left (a^{4} x^{4} + b^{4}\right )} \sqrt {a^{2} x^{3} - b^{2} x}} \,d x } \]

[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)

Mupad [B] (verification not implemented)

Time = 13.71 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.22 \[ \int \frac {-b^4+a^4 x^4}{\sqrt {-b^2 x+a^2 x^3} \left (b^4+a^4 x^4\right )} \, dx=\frac {2^{1/4}\,\sqrt {-\frac {1}{8}{}\mathrm {i}}\,\ln \left (\frac {{\left (-1\right )}^{1/4}\,2^{3/4}\,b^2-{\left (-1\right )}^{1/4}\,2^{3/4}\,a^2\,x^2-2\,{\left (-1\right )}^{3/4}\,2^{1/4}\,a\,b\,x+\sqrt {a}\,\sqrt {b}\,\sqrt {a^2\,x^3-b^2\,x}\,4{}\mathrm {i}}{-a^2\,x^2+1{}\mathrm {i}\,\sqrt {2}\,a\,b\,x+b^2}\right )}{\sqrt {a}\,\sqrt {b}}+\frac {2^{1/4}\,\sqrt {\frac {1}{8}{}\mathrm {i}}\,\ln \left (\frac {{\left (-1\right )}^{3/4}\,2^{3/4}\,b^2-{\left (-1\right )}^{3/4}\,2^{3/4}\,a^2\,x^2-2\,{\left (-1\right )}^{1/4}\,2^{1/4}\,a\,b\,x+\sqrt {a}\,\sqrt {b}\,\sqrt {a^2\,x^3-b^2\,x}\,4{}\mathrm {i}}{a^2\,x^2+1{}\mathrm {i}\,\sqrt {2}\,a\,b\,x-b^2}\right )}{\sqrt {a}\,\sqrt {b}} \]

[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))