\(\int \frac {b^4+a^4 x^4}{\sqrt {-b^2 x+a^2 x^3} (-b^4+a^4 x^4)} \, dx\) [2309]

Optimal result

Integrand size = 45, antiderivative size = 177 \[ \int \frac {b^4+a^4 x^4}{\sqrt {-b^2 x+a^2 x^3} \left (-b^4+a^4 x^4\right )} \, dx=\frac {\sqrt {-b^2 x+a^2 x^3}}{b^2-a^2 x^2}-\frac {\arctan \left (\frac {2 \sqrt {a} \sqrt {b} \sqrt {-b^2 x+a^2 x^3}}{-b^2-2 a b x+a^2 x^2}\right )}{4 \sqrt {a} \sqrt {b}}-\frac {\text {arctanh}\left (\frac {-\frac {b^{3/2}}{2 \sqrt {a}}+\sqrt {a} \sqrt {b} x+\frac {a^{3/2} x^2}{2 \sqrt {b}}}{\sqrt {-b^2 x+a^2 x^3}}\right )}{4 \sqrt {a} \sqrt {b}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.99 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.32, number of steps used = 22, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.289, Rules used = {2081, 6847, 6857, 230, 227, 1418, 425, 537, 418, 1225, 1713, 209, 212} \[ \int \frac {b^4+a^4 x^4}{\sqrt {-b^2 x+a^2 x^3} \left (-b^4+a^4 x^4\right )} \, dx=-\frac {\sqrt {x} \sqrt {a^2 x^2-b^2} \arctan \left (\frac {\sqrt {2} \sqrt [4]{-a^2} \sqrt {b} \sqrt {x}}{\sqrt {a^2 x^2-b^2}}\right )}{2 \sqrt {2} \sqrt [4]{-a^2} \sqrt {b} \sqrt {a^2 x^3-b^2 x}}-\frac {\sqrt {x} \sqrt {a^2 x^2-b^2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{-a^2} \sqrt {b} \sqrt {x}}{\sqrt {a^2 x^2-b^2}}\right )}{2 \sqrt {2} \sqrt [4]{-a^2} \sqrt {b} \sqrt {a^2 x^3-b^2 x}}-\frac {x}{\sqrt {a^2 x^3-b^2 x}} \]

[In]

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

[Out]

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

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

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 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 425

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(-b)*x*(a + b*x^n)^(p + 1)*
((c + d*x^n)^(q + 1)/(a*n*(p + 1)*(b*c - a*d))), x] + Dist[1/(a*n*(p + 1)*(b*c - a*d)), Int[(a + b*x^n)^(p + 1
)*(c + d*x^n)^q*Simp[b*c + n*(p + 1)*(b*c - a*d) + d*b*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c,
d, n, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] &&  !( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomi
alQ[a, b, c, d, n, p, q, x]

Rule 537

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*Sqrt[(c_) + (d_.)*(x_)^(n_)]), x_Symbol] :> Dist[f/b, I
nt[1/Sqrt[c + d*x^n], x], x] + Dist[(b*e - a*f)/b, Int[1/((a + b*x^n)*Sqrt[c + d*x^n]), x], x] /; FreeQ[{a, b,
c, d, e, f, n}, x]

Rule 1225

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

Rule 1418

Int[((d_) + (e_.)*(x_)^(n_))^(q_.)*((a_) + (c_.)*(x_)^(n2_))^(p_.), x_Symbol] :> Int[(d + e*x^n)^(p + q)*(a/d
+ (c/e)*x^n)^p, x] /; FreeQ[{a, c, d, e, n, q}, x] && EqQ[n2, 2*n] && EqQ[c*d^2 + a*e^2, 0] && IntegerQ[p]

Rule 1713

Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> Dist[A, Subst[Int[1/
(d + 2*a*e*x^2), x], x, x/Sqrt[a + c*x^4]], x] /; FreeQ[{a, c, d, e, A, B}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ
[c*d^2 - a*e^2, 0] && EqQ[B*d + A*e, 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 (2 \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {b^4+a^4 x^8}{\sqrt {-b^2+a^2 x^4} \left (-b^4+a^4 x^8\right )} \, 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 {1}{\sqrt {-b^2+a^2 x^4}}+\frac {2 b^4}{\sqrt {-b^2+a^2 x^4} \left (-b^4+a^4 x^8\right )}\right ) \, 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 \frac {1}{\sqrt {-b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {-b^2 x+a^2 x^3}}+\frac {\left (4 b^4 \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^4+a^4 x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {-b^2 x+a^2 x^3}} \\ & = \frac {\left (4 b^4 \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (-b^2+a^2 x^4\right )^{3/2} \left (b^2+a^2 x^4\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {-b^2 x+a^2 x^3}}+\frac {\left (2 \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 {-b^2 x+a^2 x^3}} \\ & = -\frac {x}{\sqrt {-b^2 x+a^2 x^3}}+\frac {2 \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 {-b^2 x+a^2 x^3}}+\frac {\left (\sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {-3 a^2 b^2-a^4 x^4}{\sqrt {-b^2+a^2 x^4} \left (b^2+a^2 x^4\right )} \, dx,x,\sqrt {x}\right )}{a^2 \sqrt {-b^2 x+a^2 x^3}} \\ & = -\frac {x}{\sqrt {-b^2 x+a^2 x^3}}+\frac {2 \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 {-b^2 x+a^2 x^3}}-\frac {\left (\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 {-b^2 x+a^2 x^3}}-\frac {\left (2 b^2 \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+a^2 x^4\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {-b^2 x+a^2 x^3}} \\ & = -\frac {x}{\sqrt {-b^2 x+a^2 x^3}}+\frac {2 \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 {-b^2 x+a^2 x^3}}-\frac {\left (\sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (1-\frac {\sqrt {-a^2} x^2}{b}\right ) \sqrt {-b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {-b^2 x+a^2 x^3}}-\frac {\left (\sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (1+\frac {\sqrt {-a^2} x^2}{b}\right ) \sqrt {-b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {-b^2 x+a^2 x^3}}-\frac {\left (\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 {-b^2 x+a^2 x^3}} \\ & = -\frac {x}{\sqrt {-b^2 x+a^2 x^3}}+\frac {\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 {-b^2 x+a^2 x^3}}-2 \frac {\left (\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 )}{2 \sqrt {-b^2 x+a^2 x^3}}-\frac {\left (\sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1-\frac {\sqrt {-a^2} x^2}{b}}{\left (1+\frac {\sqrt {-a^2} x^2}{b}\right ) \sqrt {-b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{2 \sqrt {-b^2 x+a^2 x^3}}-\frac {\left (\sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1+\frac {\sqrt {-a^2} x^2}{b}}{\left (1-\frac {\sqrt {-a^2} x^2}{b}\right ) \sqrt {-b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{2 \sqrt {-b^2 x+a^2 x^3}} \\ & = -\frac {x}{\sqrt {-b^2 x+a^2 x^3}}+\frac {\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 {-b^2 x+a^2 x^3}}-\frac {\left (\sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{1-2 \sqrt {-a^2} b x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {-b^2+a^2 x^2}}\right )}{2 \sqrt {-b^2 x+a^2 x^3}}-\frac {\left (\sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{1+2 \sqrt {-a^2} b x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {-b^2+a^2 x^2}}\right )}{2 \sqrt {-b^2 x+a^2 x^3}}-2 \frac {\left (\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 )}{2 \sqrt {-b^2 x+a^2 x^3}} \\ & = -\frac {x}{\sqrt {-b^2 x+a^2 x^3}}-\frac {\sqrt {x} \sqrt {-b^2+a^2 x^2} \arctan \left (\frac {\sqrt {2} \sqrt [4]{-a^2} \sqrt {b} \sqrt {x}}{\sqrt {-b^2+a^2 x^2}}\right )}{2 \sqrt {2} \sqrt [4]{-a^2} \sqrt {b} \sqrt {-b^2 x+a^2 x^3}}-\frac {\sqrt {x} \sqrt {-b^2+a^2 x^2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{-a^2} \sqrt {b} \sqrt {x}}{\sqrt {-b^2+a^2 x^2}}\right )}{2 \sqrt {2} \sqrt [4]{-a^2} \sqrt {b} \sqrt {-b^2 x+a^2 x^3}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.57 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.99 \[ \int \frac {b^4+a^4 x^4}{\sqrt {-b^2 x+a^2 x^3} \left (-b^4+a^4 x^4\right )} \, dx=\frac {-4 \sqrt {a} \sqrt {b} x-(1-i) \sqrt {x} \sqrt {-b^2+a^2 x^2} \arctan \left (\frac {(1+i) \sqrt {a} \sqrt {b} \sqrt {x}}{\sqrt {-b^2+a^2 x^2}}\right )+(1+i) \sqrt {x} \sqrt {-b^2+a^2 x^2} \arctan \left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {-b^2+a^2 x^2}}{\sqrt {a} \sqrt {b} \sqrt {x}}\right )}{4 \sqrt {a} \sqrt {b} \sqrt {-b^2 x+a^2 x^3}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 1.86 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.45

method	result	size
default	\(\frac {\left (\arctan \left (\frac {\left (a^{2} b^{2}\right )^{\frac {1}{4}} x +\sqrt {a^{2} x^{3}-b^{2} x}}{\left (a^{2} b^{2}\right )^{\frac {1}{4}} x}\right )+\frac {\ln \left (\frac {a^{2} x^{2}+2 \sqrt {a^{2} b^{2}}\, x -b^{2}-2 \left (a^{2} b^{2}\right )^{\frac {1}{4}} \sqrt {a^{2} x^{3}-b^{2} x}}{a^{2} x^{2}+2 \left (a^{2} b^{2}\right )^{\frac {1}{4}} \sqrt {a^{2} x^{3}-b^{2} x}+2 \sqrt {a^{2} b^{2}}\, x -b^{2}}\right )}{2}-\arctan \left (\frac {\left (a^{2} b^{2}\right )^{\frac {1}{4}} x -\sqrt {a^{2} x^{3}-b^{2} x}}{\left (a^{2} b^{2}\right )^{\frac {1}{4}} x}\right )\right ) \sqrt {a^{2} x^{3}-b^{2} x}-4 \left (a^{2} b^{2}\right )^{\frac {1}{4}} x}{4 \sqrt {a^{2} x^{3}-b^{2} x}\, \left (a^{2} b^{2}\right )^{\frac {1}{4}}}\)	\(257\)
pseudoelliptic	\(\frac {\left (\arctan \left (\frac {\left (a^{2} b^{2}\right )^{\frac {1}{4}} x +\sqrt {a^{2} x^{3}-b^{2} x}}{\left (a^{2} b^{2}\right )^{\frac {1}{4}} x}\right )+\frac {\ln \left (\frac {a^{2} x^{2}+2 \sqrt {a^{2} b^{2}}\, x -b^{2}-2 \left (a^{2} b^{2}\right )^{\frac {1}{4}} \sqrt {a^{2} x^{3}-b^{2} x}}{a^{2} x^{2}+2 \left (a^{2} b^{2}\right )^{\frac {1}{4}} \sqrt {a^{2} x^{3}-b^{2} x}+2 \sqrt {a^{2} b^{2}}\, x -b^{2}}\right )}{2}-\arctan \left (\frac {\left (a^{2} b^{2}\right )^{\frac {1}{4}} x -\sqrt {a^{2} x^{3}-b^{2} x}}{\left (a^{2} b^{2}\right )^{\frac {1}{4}} x}\right )\right ) \sqrt {a^{2} x^{3}-b^{2} x}-4 \left (a^{2} b^{2}\right )^{\frac {1}{4}} x}{4 \sqrt {a^{2} x^{3}-b^{2} x}\, \left (a^{2} b^{2}\right )^{\frac {1}{4}}}\)	\(257\)
elliptic	\(-\frac {x}{\sqrt {\left (x^{2}-\frac {b^{2}}{a^{2}}\right ) a^{2} x}}+\frac {b \sqrt {1+\frac {a x}{b}}\, \sqrt {-\frac {2 a x}{b}+2}\, \sqrt {-\frac {a x}{b}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {b}{a}\right ) a}{b}}, \frac {\sqrt {2}}{2}\right )}{2 a \sqrt {a^{2} x^{3}-b^{2} x}}+\frac {i b^{2} \sqrt {1+\frac {a x}{b}}\, \sqrt {-\frac {2 a x}{b}+2}\, \sqrt {-\frac {a x}{b}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {\left (x +\frac {b}{a}\right ) a}{b}}, -\frac {b}{a \left (-\frac {i b}{a}-\frac {b}{a}\right )}, \frac {\sqrt {2}}{2}\right )}{2 a^{2} \sqrt {a^{2} x^{3}-b^{2} x}\, \left (-\frac {i b}{a}-\frac {b}{a}\right )}-\frac {i b^{2} \sqrt {1+\frac {a x}{b}}\, \sqrt {-\frac {2 a x}{b}+2}\, \sqrt {-\frac {a x}{b}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {\left (x +\frac {b}{a}\right ) a}{b}}, -\frac {b}{a \left (-\frac {b}{a}+\frac {i b}{a}\right )}, \frac {\sqrt {2}}{2}\right )}{2 a^{2} \sqrt {a^{2} x^{3}-b^{2} x}\, \left (-\frac {b}{a}+\frac {i b}{a}\right )}\)	\(322\)

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

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.36 (sec) , antiderivative size = 763, normalized size of antiderivative = 4.31 \[ \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 (\frac {1}{4}\right )^{\frac {1}{4}} {\left (a^{2} x^{2} - b^{2}\right )} \left (-\frac {1}{a^{2} b^{2}}\right )^{\frac {1}{4}} \log \left (\frac {a^{4} x^{4} - 6 \, a^{2} b^{2} x^{2} + b^{4} + 8 \, {\left (\left (\frac {1}{4}\right )^{\frac {1}{4}} a^{2} b^{2} x \left (-\frac {1}{a^{2} b^{2}}\right )^{\frac {1}{4}} + \left (\frac {1}{4}\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} - 4 \, {\left (a^{4} b^{2} x^{3} - a^{2} b^{4} x\right )} \sqrt {-\frac {1}{a^{2} b^{2}}}}{a^{4} x^{4} + 2 \, a^{2} b^{2} x^{2} + b^{4}}\right ) - \left (\frac {1}{4}\right )^{\frac {1}{4}} {\left (a^{2} x^{2} - b^{2}\right )} \left (-\frac {1}{a^{2} b^{2}}\right )^{\frac {1}{4}} \log \left (\frac {a^{4} x^{4} - 6 \, a^{2} b^{2} x^{2} + b^{4} - 8 \, {\left (\left (\frac {1}{4}\right )^{\frac {1}{4}} a^{2} b^{2} x \left (-\frac {1}{a^{2} b^{2}}\right )^{\frac {1}{4}} + \left (\frac {1}{4}\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} - 4 \, {\left (a^{4} b^{2} x^{3} - a^{2} b^{4} x\right )} \sqrt {-\frac {1}{a^{2} b^{2}}}}{a^{4} x^{4} + 2 \, a^{2} b^{2} x^{2} + b^{4}}\right ) + \left (\frac {1}{4}\right )^{\frac {1}{4}} {\left (-i \, a^{2} x^{2} + i \, b^{2}\right )} \left (-\frac {1}{a^{2} b^{2}}\right )^{\frac {1}{4}} \log \left (\frac {a^{4} x^{4} - 6 \, a^{2} b^{2} x^{2} + b^{4} - 8 \, {\left (i \, \left (\frac {1}{4}\right )^{\frac {1}{4}} a^{2} b^{2} x \left (-\frac {1}{a^{2} b^{2}}\right )^{\frac {1}{4}} + \left (\frac {1}{4}\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} + 4 \, {\left (a^{4} b^{2} x^{3} - a^{2} b^{4} x\right )} \sqrt {-\frac {1}{a^{2} b^{2}}}}{a^{4} x^{4} + 2 \, a^{2} b^{2} x^{2} + b^{4}}\right ) + \left (\frac {1}{4}\right )^{\frac {1}{4}} {\left (i \, a^{2} x^{2} - i \, b^{2}\right )} \left (-\frac {1}{a^{2} b^{2}}\right )^{\frac {1}{4}} \log \left (\frac {a^{4} x^{4} - 6 \, a^{2} b^{2} x^{2} + b^{4} - 8 \, {\left (-i \, \left (\frac {1}{4}\right )^{\frac {1}{4}} a^{2} b^{2} x \left (-\frac {1}{a^{2} b^{2}}\right )^{\frac {1}{4}} + \left (\frac {1}{4}\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} + 4 \, {\left (a^{4} b^{2} x^{3} - a^{2} b^{4} x\right )} \sqrt {-\frac {1}{a^{2} b^{2}}}}{a^{4} x^{4} + 2 \, a^{2} b^{2} x^{2} + b^{4}}\right ) - 8 \, \sqrt {a^{2} x^{3} - b^{2} x}}{8 \, {\left (a^{2} x^{2} - b^{2}\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]

1/8*((1/4)^(1/4)*(a^2*x^2 - b^2)*(-1/(a^2*b^2))^(1/4)*log((a^4*x^4 - 6*a^2*b^2*x^2 + b^4 + 8*((1/4)^(1/4)*a^2*
b^2*x*(-1/(a^2*b^2))^(1/4) + (1/4)^(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) -
4*(a^4*b^2*x^3 - a^2*b^4*x)*sqrt(-1/(a^2*b^2)))/(a^4*x^4 + 2*a^2*b^2*x^2 + b^4)) - (1/4)^(1/4)*(a^2*x^2 - b^2
)*(-1/(a^2*b^2))^(1/4)*log((a^4*x^4 - 6*a^2*b^2*x^2 + b^4 - 8*((1/4)^(1/4)*a^2*b^2*x*(-1/(a^2*b^2))^(1/4) + (1
/4)^(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) - 4*(a^4*b^2*x^3 - a^2*b^4*x)*sq
rt(-1/(a^2*b^2)))/(a^4*x^4 + 2*a^2*b^2*x^2 + b^4)) + (1/4)^(1/4)*(-I*a^2*x^2 + I*b^2)*(-1/(a^2*b^2))^(1/4)*log
((a^4*x^4 - 6*a^2*b^2*x^2 + b^4 - 8*(I*(1/4)^(1/4)*a^2*b^2*x*(-1/(a^2*b^2))^(1/4) + (1/4)^(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) + 4*(a^4*b^2*x^3 - a^2*b^4*x)*sqrt(-1/(a^2*b^2)))/(
a^4*x^4 + 2*a^2*b^2*x^2 + b^4)) + (1/4)^(1/4)*(I*a^2*x^2 - I*b^2)*(-1/(a^2*b^2))^(1/4)*log((a^4*x^4 - 6*a^2*b^
2*x^2 + b^4 - 8*(-I*(1/4)^(1/4)*a^2*b^2*x*(-1/(a^2*b^2))^(1/4) + (1/4)^(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) + 4*(a^4*b^2*x^3 - a^2*b^4*x)*sqrt(-1/(a^2*b^2)))/(a^4*x^4 + 2*a^2*b^2*
x^2 + b^4)) - 8*sqrt(a^2*x^3 - b^2*x))/(a^2*x^2 - b^2)

Sympy [F]

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

[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**4*x**4 + b**4)/(sqrt(x*(a*x - b)*(a*x + b))*(a*x - b)*(a*x + b)*(a**2*x**2 + b**2)), 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 [F(-1)]

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

[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]

\text{Hanged}