\(\int \frac {-b^2+a^4 x^4}{\sqrt {b x+a^2 x^3} (b^2+a^4 x^4)} \, dx\) [1791]

Optimal result

Integrand size = 42, antiderivative size = 121 \[ \int \frac {-b^2+a^4 x^4}{\sqrt {b x+a^2 x^3} \left (b^2+a^4 x^4\right )} \, dx=-\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt {a} \sqrt [4]{b} \sqrt {b x+a^2 x^3}}{b+a^2 x^2}\right )}{\sqrt [4]{2} \sqrt {a} \sqrt [4]{b}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt {a} \sqrt [4]{b} \sqrt {b x+a^2 x^3}}{b+a^2 x^2}\right )}{\sqrt [4]{2} \sqrt {a} \sqrt [4]{b}} \]

[Out]

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

Rubi [C] (warning: unable to verify)

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

Time = 3.54 (sec) , antiderivative size = 1906, normalized size of antiderivative = 15.75, number of steps used = 23, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {2081, 1600, 6847, 6857, 415, 226, 418, 1231, 1721} \[ \int \frac {-b^2+a^4 x^4}{\sqrt {b x+a^2 x^3} \left (b^2+a^4 x^4\right )} \, dx=-\frac {\left (\sqrt {-a^4}-a^2\right )^{3/2} \sqrt {x} \sqrt {a^2 x^2+b} \arctan \left (\frac {\sqrt [8]{-a^4} \sqrt {\sqrt {-a^4}-a^2} \sqrt [4]{b} \sqrt {x}}{a \sqrt {a^2 x^2+b}}\right ) a^3}{4 \left (-a^4\right )^{13/8} \sqrt [4]{b} \sqrt {a^2 x^3+b x}}-\frac {\left (\sqrt {-a^4}-a^2\right )^{3/2} \sqrt {x} \sqrt {a^2 x^2+b} \arctan \left (\frac {\sqrt {\sqrt {-a^4}-a^2} \sqrt [4]{-b} \sqrt {x}}{\sqrt [8]{-a^4} \sqrt {a^2 x^2+b}}\right ) a^2}{4 \left (-a^4\right )^{11/8} \sqrt [4]{-b} \sqrt {a^2 x^3+b x}}+\frac {\left (a^2-\sqrt {-a^4}\right )^{3/2} \sqrt {x} \sqrt {a^2 x^2+b} \arctan \left (\frac {\sqrt {a^2-\sqrt {-a^4}} \sqrt [4]{-b} \sqrt {x}}{\sqrt [8]{-a^4} \sqrt {a^2 x^2+b}}\right )}{4 \left (-a^4\right )^{3/8} \sqrt [4]{-b} \sqrt {a^2 x^3+b x} a^2}-\frac {\left (a^2+\sqrt {-a^4}\right )^{3/2} \sqrt {x} \sqrt {a^2 x^2+b} \arctan \left (\frac {\sqrt {a^2+\sqrt {-a^4}} \sqrt [4]{b} \sqrt {x}}{\sqrt [8]{-a^4} \sqrt {a^2 x^2+b}}\right )}{4 \left (-a^4\right )^{3/8} \sqrt [4]{b} \sqrt {a^2 x^3+b x} a^2}+\frac {\left (a^2-\sqrt {-a^4}\right ) \sqrt {x} \left (a x+\sqrt {b}\right ) \sqrt {\frac {a^2 x^2+b}{\left (a x+\sqrt {b}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{b} \sqrt {a^2 x^3+b x} a^{5/2}}+\frac {\left (a^2+\sqrt {-a^4}\right ) \sqrt {x} \left (a x+\sqrt {b}\right ) \sqrt {\frac {a^2 x^2+b}{\left (a x+\sqrt {b}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{b} \sqrt {a^2 x^3+b x} a^{5/2}}-\frac {\left (a-\sqrt [4]{-a^4}\right ) \left (a^2+\sqrt {-a^4}\right ) \sqrt {x} \left (a x+\sqrt {b}\right ) \sqrt {\frac {a^2 x^2+b}{\left (a x+\sqrt {b}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{b} \sqrt {a^2 x^3+b x} a^{7/2}}-\frac {\left (a+\sqrt [4]{-a^4}\right ) \left (a^2+\sqrt {-a^4}\right ) \sqrt {x} \left (a x+\sqrt {b}\right ) \sqrt {\frac {a^2 x^2+b}{\left (a x+\sqrt {b}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{b} \sqrt {a^2 x^3+b x} a^{7/2}}-\frac {\left (a^2-\sqrt {-a^4}\right ) \left (a+\frac {\sqrt [4]{-a^4} \sqrt {b}}{\sqrt {-b}}\right ) \sqrt {x} \left (a x+\sqrt {b}\right ) \sqrt {\frac {a^2 x^2+b}{\left (a x+\sqrt {b}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{b} \sqrt {a^2 x^3+b x} a^{7/2}}-\frac {\left (a^2-\sqrt {-a^4}\right ) \left (\sqrt {b} a+\sqrt [4]{-a^4} \sqrt {-b}\right ) \sqrt {x} \left (a x+\sqrt {b}\right ) \sqrt {\frac {a^2 x^2+b}{\left (a x+\sqrt {b}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{4 b^{3/4} \sqrt {a^2 x^3+b x} a^{7/2}}+\frac {\left (a+\sqrt [4]{-a^4}\right )^2 \left (a^2+\sqrt {-a^4}\right ) \sqrt {x} \left (a x+\sqrt {b}\right ) \sqrt {\frac {a^2 x^2+b}{\left (a x+\sqrt {b}\right )^2}} \operatorname {EllipticPi}\left (\frac {a^3 \left (a-\sqrt [4]{-a^4}\right )^2}{4 \left (-a^4\right )^{5/4}},2 \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{8 \sqrt [4]{b} \sqrt {a^2 x^3+b x} a^{9/2}}+\frac {\left (a-\sqrt [4]{-a^4}\right )^2 \left (a^2+\sqrt {-a^4}\right ) \sqrt {x} \left (a x+\sqrt {b}\right ) \sqrt {\frac {a^2 x^2+b}{\left (a x+\sqrt {b}\right )^2}} \operatorname {EllipticPi}\left (\frac {\left (a+\sqrt [4]{-a^4}\right )^2}{4 a \sqrt [4]{-a^4}},2 \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{8 \sqrt [4]{b} \sqrt {a^2 x^3+b x} a^{9/2}}+\frac {\left (a^2-\sqrt {-a^4}\right ) \left (b a^2-2 \sqrt [4]{-a^4} \sqrt {-b^2} a-\sqrt {-a^4} b\right ) \sqrt {x} \left (a x+\sqrt {b}\right ) \sqrt {\frac {a^2 x^2+b}{\left (a x+\sqrt {b}\right )^2}} \operatorname {EllipticPi}\left (\frac {a^3 \left (a \sqrt {-b}-\sqrt [4]{-a^4} \sqrt {b}\right )^2}{4 \left (-a^4\right )^{5/4} \sqrt {-b^2}},2 \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{8 b^{5/4} \sqrt {a^2 x^3+b x} a^{9/2}}+\frac {\left (a^2-\sqrt {-a^4}\right ) \left (b a^2+2 \sqrt [4]{-a^4} \sqrt {-b^2} a-\sqrt {-a^4} b\right ) \sqrt {x} \left (a x+\sqrt {b}\right ) \sqrt {\frac {a^2 x^2+b}{\left (a x+\sqrt {b}\right )^2}} \operatorname {EllipticPi}\left (\frac {\left (\sqrt {-b} a+\sqrt [4]{-a^4} \sqrt {b}\right )^2}{4 a \sqrt [4]{-a^4} \sqrt {-b^2}},2 \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{8 b^{5/4} \sqrt {a^2 x^3+b x} a^{9/2}} \]

[In]

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

[Out]

((a^2 - Sqrt[-a^4])^(3/2)*Sqrt[x]*Sqrt[b + a^2*x^2]*ArcTan[(Sqrt[a^2 - Sqrt[-a^4]]*(-b)^(1/4)*Sqrt[x])/((-a^4)
^(1/8)*Sqrt[b + a^2*x^2])])/(4*a^2*(-a^4)^(3/8)*(-b)^(1/4)*Sqrt[b*x + a^2*x^3]) - (a^2*(-a^2 + Sqrt[-a^4])^(3/
2)*Sqrt[x]*Sqrt[b + a^2*x^2]*ArcTan[(Sqrt[-a^2 + Sqrt[-a^4]]*(-b)^(1/4)*Sqrt[x])/((-a^4)^(1/8)*Sqrt[b + a^2*x^
2])])/(4*(-a^4)^(11/8)*(-b)^(1/4)*Sqrt[b*x + a^2*x^3]) - (a^3*(-a^2 + Sqrt[-a^4])^(3/2)*Sqrt[x]*Sqrt[b + a^2*x
^2]*ArcTan[((-a^4)^(1/8)*Sqrt[-a^2 + Sqrt[-a^4]]*b^(1/4)*Sqrt[x])/(a*Sqrt[b + a^2*x^2])])/(4*(-a^4)^(13/8)*b^(
1/4)*Sqrt[b*x + a^2*x^3]) - ((a^2 + Sqrt[-a^4])^(3/2)*Sqrt[x]*Sqrt[b + a^2*x^2]*ArcTan[(Sqrt[a^2 + Sqrt[-a^4]]
*b^(1/4)*Sqrt[x])/((-a^4)^(1/8)*Sqrt[b + a^2*x^2])])/(4*a^2*(-a^4)^(3/8)*b^(1/4)*Sqrt[b*x + a^2*x^3]) - ((a^2
- Sqrt[-a^4])*((-a^4)^(1/4)*Sqrt[-b] + a*Sqrt[b])*Sqrt[x]*(Sqrt[b] + a*x)*Sqrt[(b + a^2*x^2)/(Sqrt[b] + a*x)^2
]*EllipticF[2*ArcTan[(Sqrt[a]*Sqrt[x])/b^(1/4)], 1/2])/(4*a^(7/2)*b^(3/4)*Sqrt[b*x + a^2*x^3]) + ((a^2 - Sqrt[
-a^4])*Sqrt[x]*(Sqrt[b] + a*x)*Sqrt[(b + a^2*x^2)/(Sqrt[b] + a*x)^2]*EllipticF[2*ArcTan[(Sqrt[a]*Sqrt[x])/b^(1
/4)], 1/2])/(2*a^(5/2)*b^(1/4)*Sqrt[b*x + a^2*x^3]) + ((a^2 + Sqrt[-a^4])*Sqrt[x]*(Sqrt[b] + a*x)*Sqrt[(b + a^
2*x^2)/(Sqrt[b] + a*x)^2]*EllipticF[2*ArcTan[(Sqrt[a]*Sqrt[x])/b^(1/4)], 1/2])/(2*a^(5/2)*b^(1/4)*Sqrt[b*x + a
^2*x^3]) - ((a - (-a^4)^(1/4))*(a^2 + Sqrt[-a^4])*Sqrt[x]*(Sqrt[b] + a*x)*Sqrt[(b + a^2*x^2)/(Sqrt[b] + a*x)^2
]*EllipticF[2*ArcTan[(Sqrt[a]*Sqrt[x])/b^(1/4)], 1/2])/(4*a^(7/2)*b^(1/4)*Sqrt[b*x + a^2*x^3]) - ((a + (-a^4)^
(1/4))*(a^2 + Sqrt[-a^4])*Sqrt[x]*(Sqrt[b] + a*x)*Sqrt[(b + a^2*x^2)/(Sqrt[b] + a*x)^2]*EllipticF[2*ArcTan[(Sq
rt[a]*Sqrt[x])/b^(1/4)], 1/2])/(4*a^(7/2)*b^(1/4)*Sqrt[b*x + a^2*x^3]) - ((a^2 - Sqrt[-a^4])*(a + ((-a^4)^(1/4
)*Sqrt[b])/Sqrt[-b])*Sqrt[x]*(Sqrt[b] + a*x)*Sqrt[(b + a^2*x^2)/(Sqrt[b] + a*x)^2]*EllipticF[2*ArcTan[(Sqrt[a]
*Sqrt[x])/b^(1/4)], 1/2])/(4*a^(7/2)*b^(1/4)*Sqrt[b*x + a^2*x^3]) + ((a + (-a^4)^(1/4))^2*(a^2 + Sqrt[-a^4])*S
qrt[x]*(Sqrt[b] + a*x)*Sqrt[(b + a^2*x^2)/(Sqrt[b] + a*x)^2]*EllipticPi[(a^3*(a - (-a^4)^(1/4))^2)/(4*(-a^4)^(
5/4)), 2*ArcTan[(Sqrt[a]*Sqrt[x])/b^(1/4)], 1/2])/(8*a^(9/2)*b^(1/4)*Sqrt[b*x + a^2*x^3]) + ((a - (-a^4)^(1/4)
)^2*(a^2 + Sqrt[-a^4])*Sqrt[x]*(Sqrt[b] + a*x)*Sqrt[(b + a^2*x^2)/(Sqrt[b] + a*x)^2]*EllipticPi[(a + (-a^4)^(1
/4))^2/(4*a*(-a^4)^(1/4)), 2*ArcTan[(Sqrt[a]*Sqrt[x])/b^(1/4)], 1/2])/(8*a^(9/2)*b^(1/4)*Sqrt[b*x + a^2*x^3])
+ ((a^2 - Sqrt[-a^4])*(a^2*b - Sqrt[-a^4]*b - 2*a*(-a^4)^(1/4)*Sqrt[-b^2])*Sqrt[x]*(Sqrt[b] + a*x)*Sqrt[(b + a
^2*x^2)/(Sqrt[b] + a*x)^2]*EllipticPi[(a^3*(a*Sqrt[-b] - (-a^4)^(1/4)*Sqrt[b])^2)/(4*(-a^4)^(5/4)*Sqrt[-b^2]),
2*ArcTan[(Sqrt[a]*Sqrt[x])/b^(1/4)], 1/2])/(8*a^(9/2)*b^(5/4)*Sqrt[b*x + a^2*x^3]) + ((a^2 - Sqrt[-a^4])*(a^2
*b - Sqrt[-a^4]*b + 2*a*(-a^4)^(1/4)*Sqrt[-b^2])*Sqrt[x]*(Sqrt[b] + a*x)*Sqrt[(b + a^2*x^2)/(Sqrt[b] + a*x)^2]
*EllipticPi[(a*Sqrt[-b] + (-a^4)^(1/4)*Sqrt[b])^2/(4*a*(-a^4)^(1/4)*Sqrt[-b^2]), 2*ArcTan[(Sqrt[a]*Sqrt[x])/b^
(1/4)], 1/2])/(8*a^(9/2)*b^(5/4)*Sqrt[b*x + a^2*x^3])

Rule 226

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^4)/(a*(
1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^4]))*EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

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 1231

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

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 1721

Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[B/A, 2]
}, Simp[(-(B*d - A*e))*(ArcTan[Rt[c*(d/e) + a*(e/d), 2]*(x/Sqrt[a + c*x^4])]/(2*d*e*Rt[c*(d/e) + a*(e/d), 2]))
, x] + Simp[(B*d + A*e)*(A + B*x^2)*(Sqrt[A^2*((a + c*x^4)/(a*(A + B*x^2)^2))]/(4*d*e*A*q*Sqrt[a + c*x^4]))*El
lipticPi[Cancel[-(B*d - A*e)^2/(4*d*e*A*B)], 2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, c, d, e, A, B}, x] && NeQ[c
*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a] && EqQ[c*A^2 - a*B^2, 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+a^2 x^2}\right ) \int \frac {-b^2+a^4 x^4}{\sqrt {x} \sqrt {b+a^2 x^2} \left (b^2+a^4 x^4\right )} \, dx}{\sqrt {b x+a^2 x^3}} \\ & = \frac {\left (\sqrt {x} \sqrt {b+a^2 x^2}\right ) \int \frac {\left (-b+a^2 x^2\right ) \sqrt {b+a^2 x^2}}{\sqrt {x} \left (b^2+a^4 x^4\right )} \, dx}{\sqrt {b x+a^2 x^3}} \\ & = \frac {\left (2 \sqrt {x} \sqrt {b+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\left (-b+a^2 x^4\right ) \sqrt {b+a^2 x^4}}{b^2+a^4 x^8} \, dx,x,\sqrt {x}\right )}{\sqrt {b x+a^2 x^3}} \\ & = \frac {\left (2 \sqrt {x} \sqrt {b+a^2 x^2}\right ) \text {Subst}\left (\int \left (-\frac {\sqrt {-a^4} \left (a^2 b-\sqrt {-a^4} b\right ) \sqrt {b+a^2 x^4}}{2 a^4 b \left (b-\sqrt {-a^4} x^4\right )}+\frac {\sqrt {-a^4} \left (a^2 b+\sqrt {-a^4} b\right ) \sqrt {b+a^2 x^4}}{2 a^4 b \left (b+\sqrt {-a^4} x^4\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {b x+a^2 x^3}} \\ & = -\frac {\left (\left (a^2+\sqrt {-a^4}\right ) \sqrt {x} \sqrt {b+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\sqrt {b+a^2 x^4}}{b-\sqrt {-a^4} x^4} \, dx,x,\sqrt {x}\right )}{a^2 \sqrt {b x+a^2 x^3}}+\frac {\left (\sqrt {-a^4} \left (a^2 b+\sqrt {-a^4} b\right ) \sqrt {x} \sqrt {b+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\sqrt {b+a^2 x^4}}{b+\sqrt {-a^4} x^4} \, dx,x,\sqrt {x}\right )}{a^4 b \sqrt {b x+a^2 x^3}} \\ & = \frac {\left (\left (a^2+\sqrt {-a^4}\right ) \sqrt {x} \sqrt {b+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {-a^4} \sqrt {b x+a^2 x^3}}-\frac {\left (\left (a^2+\sqrt {-a^4}\right ) \left (a^2 b+\sqrt {-a^4} b\right ) \sqrt {x} \sqrt {b+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b+a^2 x^4} \left (b-\sqrt {-a^4} x^4\right )} \, dx,x,\sqrt {x}\right )}{a^2 \sqrt {-a^4} \sqrt {b x+a^2 x^3}}+\frac {\left (\sqrt {-a^4} \left (a^2+\sqrt {-a^4}\right ) \left (a^2 b+\sqrt {-a^4} b\right ) \sqrt {x} \sqrt {b+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b+a^2 x^4} \left (b+\sqrt {-a^4} x^4\right )} \, dx,x,\sqrt {x}\right )}{a^6 \sqrt {b x+a^2 x^3}}+\frac {\left (\left (a^2 b+\sqrt {-a^4} b\right ) \sqrt {x} \sqrt {b+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{a^2 b \sqrt {b x+a^2 x^3}} \\ & = \frac {\left (a^2+\sqrt {-a^4}\right ) \sqrt {x} \left (\sqrt {b}+a x\right ) \sqrt {\frac {b+a^2 x^2}{\left (\sqrt {b}+a x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{2 a^{5/2} \sqrt [4]{b} \sqrt {b x+a^2 x^3}}+\frac {\left (a^2+\sqrt {-a^4}\right ) \sqrt {x} \left (\sqrt {b}+a x\right ) \sqrt {\frac {b+a^2 x^2}{\left (\sqrt {b}+a x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{2 \sqrt {a} \sqrt {-a^4} \sqrt [4]{b} \sqrt {b x+a^2 x^3}}-\frac {\left (\left (a^2+\sqrt {-a^4}\right ) \left (a^2 b+\sqrt {-a^4} b\right ) \sqrt {x} \sqrt {b+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (1-\frac {\sqrt [4]{-a^4} x^2}{\sqrt {b}}\right ) \sqrt {b+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{2 a^2 \sqrt {-a^4} b \sqrt {b x+a^2 x^3}}-\frac {\left (\left (a^2+\sqrt {-a^4}\right ) \left (a^2 b+\sqrt {-a^4} b\right ) \sqrt {x} \sqrt {b+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (1+\frac {\sqrt [4]{-a^4} x^2}{\sqrt {b}}\right ) \sqrt {b+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{2 a^2 \sqrt {-a^4} b \sqrt {b x+a^2 x^3}}+\frac {\left (\sqrt {-a^4} \left (a^2+\sqrt {-a^4}\right ) \left (a^2 b+\sqrt {-a^4} b\right ) \sqrt {x} \sqrt {b+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (1-\frac {\sqrt [4]{-a^4} x^2}{\sqrt {-b}}\right ) \sqrt {b+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{2 a^6 b \sqrt {b x+a^2 x^3}}+\frac {\left (\sqrt {-a^4} \left (a^2+\sqrt {-a^4}\right ) \left (a^2 b+\sqrt {-a^4} b\right ) \sqrt {x} \sqrt {b+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (1+\frac {\sqrt [4]{-a^4} x^2}{\sqrt {-b}}\right ) \sqrt {b+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{2 a^6 b \sqrt {b x+a^2 x^3}} \\ & = \frac {\left (a^2+\sqrt {-a^4}\right ) \sqrt {x} \left (\sqrt {b}+a x\right ) \sqrt {\frac {b+a^2 x^2}{\left (\sqrt {b}+a x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{2 a^{5/2} \sqrt [4]{b} \sqrt {b x+a^2 x^3}}+\frac {\left (a^2+\sqrt {-a^4}\right ) \sqrt {x} \left (\sqrt {b}+a x\right ) \sqrt {\frac {b+a^2 x^2}{\left (\sqrt {b}+a x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{2 \sqrt {a} \sqrt {-a^4} \sqrt [4]{b} \sqrt {b x+a^2 x^3}}-\frac {\left (\left (a^2+\sqrt {-a^4}\right ) \left (a^2 b+\sqrt {-a^4} b\right ) \sqrt {x} \sqrt {b+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{2 a \sqrt {-a^4} \left (a-\sqrt [4]{-a^4}\right ) b \sqrt {b x+a^2 x^3}}+\frac {\left (\left (a^2+\sqrt {-a^4}\right ) \left (a^2 b+\sqrt {-a^4} b\right ) \sqrt {x} \sqrt {b+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1+\frac {a x^2}{\sqrt {b}}}{\left (1+\frac {\sqrt [4]{-a^4} x^2}{\sqrt {b}}\right ) \sqrt {b+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{2 a^2 \sqrt [4]{-a^4} \left (a-\sqrt [4]{-a^4}\right ) b \sqrt {b x+a^2 x^3}}-\frac {\left (\left (a^2+\sqrt {-a^4}\right ) \left (a^2 b+\sqrt {-a^4} b\right ) \sqrt {x} \sqrt {b+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{2 a \sqrt {-a^4} \left (a+\sqrt [4]{-a^4}\right ) b \sqrt {b x+a^2 x^3}}-\frac {\left (\left (a^2+\sqrt {-a^4}\right ) \left (a^2 b+\sqrt {-a^4} b\right ) \sqrt {x} \sqrt {b+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1+\frac {a x^2}{\sqrt {b}}}{\left (1-\frac {\sqrt [4]{-a^4} x^2}{\sqrt {b}}\right ) \sqrt {b+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{2 a^2 \sqrt [4]{-a^4} \left (a+\sqrt [4]{-a^4}\right ) b \sqrt {b x+a^2 x^3}}+\frac {\left (\left (-a^4\right )^{3/4} \left (\sqrt [4]{-a^4}+\frac {a \sqrt {-b}}{\sqrt {b}}\right ) \left (a^2 b+\sqrt {-a^4} b\right ) \sqrt {x} \sqrt {b+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1+\frac {a x^2}{\sqrt {b}}}{\left (1+\frac {\sqrt [4]{-a^4} x^2}{\sqrt {-b}}\right ) \sqrt {b+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{2 a^6 b \sqrt {b x+a^2 x^3}}+\frac {\left (\sqrt {-a^4} \left (a+\frac {\sqrt [4]{-a^4} \sqrt {-b}}{\sqrt {b}}\right ) \left (a^2 b+\sqrt {-a^4} b\right ) \sqrt {x} \sqrt {b+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{2 a^5 b \sqrt {b x+a^2 x^3}}+\frac {\left (\left (-a^4\right )^{3/4} \left (\sqrt [4]{-a^4}+\frac {a \sqrt {b}}{\sqrt {-b}}\right ) \left (a^2 b+\sqrt {-a^4} b\right ) \sqrt {x} \sqrt {b+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1+\frac {a x^2}{\sqrt {b}}}{\left (1-\frac {\sqrt [4]{-a^4} x^2}{\sqrt {-b}}\right ) \sqrt {b+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{2 a^6 b \sqrt {b x+a^2 x^3}}+\frac {\left (\sqrt {-a^4} \left (a+\frac {\sqrt [4]{-a^4} \sqrt {b}}{\sqrt {-b}}\right ) \left (a^2 b+\sqrt {-a^4} b\right ) \sqrt {x} \sqrt {b+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{2 a^5 b \sqrt {b x+a^2 x^3}} \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.59 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.00 \[ \int \frac {-b^2+a^4 x^4}{\sqrt {b x+a^2 x^3} \left (b^2+a^4 x^4\right )} \, dx=-\frac {\sqrt {x} \sqrt {b+a^2 x^2} \left (\arctan \left (\frac {\sqrt [4]{2} \sqrt {a} \sqrt [4]{b} \sqrt {x}}{\sqrt {b+a^2 x^2}}\right )+\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt {a} \sqrt [4]{b} \sqrt {x}}{\sqrt {b+a^2 x^2}}\right )\right )}{\sqrt [4]{2} \sqrt {a} \sqrt [4]{b} \sqrt {x \left (b+a^2 x^2\right )}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.83 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.88

method	result	size
default	\(-\frac {2^{\frac {3}{4}} \left (-2 \arctan \left (\frac {\sqrt {x \left (a^{2} x^{2}+b \right )}\, 2^{\frac {3}{4}}}{2 x \left (a^{2} b \right )^{\frac {1}{4}}}\right )+\ln \left (\frac {-x 2^{\frac {1}{4}} \left (a^{2} b \right )^{\frac {1}{4}}-\sqrt {x \left (a^{2} x^{2}+b \right )}}{x 2^{\frac {1}{4}} \left (a^{2} b \right )^{\frac {1}{4}}-\sqrt {x \left (a^{2} x^{2}+b \right )}}\right )\right )}{4 \left (a^{2} b \right )^{\frac {1}{4}}}\)	\(106\)
pseudoelliptic	\(-\frac {2^{\frac {3}{4}} \left (-2 \arctan \left (\frac {\sqrt {x \left (a^{2} x^{2}+b \right )}\, 2^{\frac {3}{4}}}{2 x \left (a^{2} b \right )^{\frac {1}{4}}}\right )+\ln \left (\frac {-x 2^{\frac {1}{4}} \left (a^{2} b \right )^{\frac {1}{4}}-\sqrt {x \left (a^{2} x^{2}+b \right )}}{x 2^{\frac {1}{4}} \left (a^{2} b \right )^{\frac {1}{4}}-\sqrt {x \left (a^{2} x^{2}+b \right )}}\right )\right )}{4 \left (a^{2} b \right )^{\frac {1}{4}}}\)	\(106\)
elliptic	\(\frac {\sqrt {-b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-b}}{a}\right ) a}{\sqrt {-b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-b}}{a}\right ) a}{\sqrt {-b}}}\, \sqrt {-\frac {x a}{\sqrt {-b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-b}}{a}\right ) a}{\sqrt {-b}}}, \frac {\sqrt {2}}{2}\right )}{a \sqrt {a^{2} x^{3}+b x}}-\frac {\sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (a^{4} \textit {\_Z}^{4}+b^{2}\right )}{\sum }\frac {\sqrt {-b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-b}}{a}\right ) a}{\sqrt {-b}}}\, \sqrt {-\frac {\left (x -\frac {\sqrt {-b}}{a}\right ) a}{\sqrt {-b}}}\, \sqrt {-\frac {x a}{\sqrt {-b}}}\, \left (a \left (a^{2} \underline {\hspace {1.25 ex}}\alpha ^{3}-b \underline {\hspace {1.25 ex}}\alpha \right )-a^{2} \sqrt {-b}\, \underline {\hspace {1.25 ex}}\alpha ^{2}+b \sqrt {-b}\right ) \operatorname {EllipticPi}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-b}}{a}\right ) a}{\sqrt {-b}}}, -\frac {\underline {\hspace {1.25 ex}}\alpha ^{3} \sqrt {-b}\, a^{3}+b \,\underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}-\underline {\hspace {1.25 ex}}\alpha \sqrt {-b}\, a b -b^{2}}{2 b^{2}}, \frac {\sqrt {2}}{2}\right )}{\underline {\hspace {1.25 ex}}\alpha ^{3} \sqrt {x \left (a^{2} x^{2}+b \right )}}\right )}{4 a^{4}}\)	\(303\)

[In]

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

[Out]

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

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.35 (sec) , antiderivative size = 579, normalized size of antiderivative = 4.79 \[ \int \frac {-b^2+a^4 x^4}{\sqrt {b x+a^2 x^3} \left (b^2+a^4 x^4\right )} \, dx=-\frac {1}{4} \, \left (\frac {1}{2}\right )^{\frac {1}{4}} \left (\frac {1}{a^{2} b}\right )^{\frac {1}{4}} \log \left (\frac {a^{4} x^{4} + 4 \, a^{2} b x^{2} + b^{2} + 4 \, \sqrt {\frac {1}{2}} {\left (a^{4} b x^{3} + a^{2} b^{2} x\right )} \sqrt {\frac {1}{a^{2} b}} + 4 \, \sqrt {a^{2} x^{3} + b x} {\left (\left (\frac {1}{2}\right )^{\frac {1}{4}} a^{2} b x \left (\frac {1}{a^{2} b}\right )^{\frac {1}{4}} + \left (\frac {1}{2}\right )^{\frac {3}{4}} {\left (a^{4} b x^{2} + a^{2} b^{2}\right )} \left (\frac {1}{a^{2} b}\right )^{\frac {3}{4}}\right )}}{a^{4} x^{4} + b^{2}}\right ) + \frac {1}{4} \, \left (\frac {1}{2}\right )^{\frac {1}{4}} \left (\frac {1}{a^{2} b}\right )^{\frac {1}{4}} \log \left (\frac {a^{4} x^{4} + 4 \, a^{2} b x^{2} + b^{2} + 4 \, \sqrt {\frac {1}{2}} {\left (a^{4} b x^{3} + a^{2} b^{2} x\right )} \sqrt {\frac {1}{a^{2} b}} - 4 \, \sqrt {a^{2} x^{3} + b x} {\left (\left (\frac {1}{2}\right )^{\frac {1}{4}} a^{2} b x \left (\frac {1}{a^{2} b}\right )^{\frac {1}{4}} + \left (\frac {1}{2}\right )^{\frac {3}{4}} {\left (a^{4} b x^{2} + a^{2} b^{2}\right )} \left (\frac {1}{a^{2} b}\right )^{\frac {3}{4}}\right )}}{a^{4} x^{4} + b^{2}}\right ) + \frac {1}{4} i \, \left (\frac {1}{2}\right )^{\frac {1}{4}} \left (\frac {1}{a^{2} b}\right )^{\frac {1}{4}} \log \left (\frac {a^{4} x^{4} + 4 \, a^{2} b x^{2} + b^{2} - 4 \, \sqrt {\frac {1}{2}} {\left (a^{4} b x^{3} + a^{2} b^{2} x\right )} \sqrt {\frac {1}{a^{2} b}} - 4 \, \sqrt {a^{2} x^{3} + b x} {\left (i \, \left (\frac {1}{2}\right )^{\frac {1}{4}} a^{2} b x \left (\frac {1}{a^{2} b}\right )^{\frac {1}{4}} + \left (\frac {1}{2}\right )^{\frac {3}{4}} {\left (-i \, a^{4} b x^{2} - i \, a^{2} b^{2}\right )} \left (\frac {1}{a^{2} b}\right )^{\frac {3}{4}}\right )}}{a^{4} x^{4} + b^{2}}\right ) - \frac {1}{4} i \, \left (\frac {1}{2}\right )^{\frac {1}{4}} \left (\frac {1}{a^{2} b}\right )^{\frac {1}{4}} \log \left (\frac {a^{4} x^{4} + 4 \, a^{2} b x^{2} + b^{2} - 4 \, \sqrt {\frac {1}{2}} {\left (a^{4} b x^{3} + a^{2} b^{2} x\right )} \sqrt {\frac {1}{a^{2} b}} - 4 \, \sqrt {a^{2} x^{3} + b x} {\left (-i \, \left (\frac {1}{2}\right )^{\frac {1}{4}} a^{2} b x \left (\frac {1}{a^{2} b}\right )^{\frac {1}{4}} + \left (\frac {1}{2}\right )^{\frac {3}{4}} {\left (i \, a^{4} b x^{2} + i \, a^{2} b^{2}\right )} \left (\frac {1}{a^{2} b}\right )^{\frac {3}{4}}\right )}}{a^{4} x^{4} + b^{2}}\right ) \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 11.24 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.36 \[ \int \frac {-b^2+a^4 x^4}{\sqrt {b x+a^2 x^3} \left (b^2+a^4 x^4\right )} \, dx=\frac {2^{3/4}\,\ln \left (\frac {2^{3/4}\,b+2^{3/4}\,a^2\,x^2-4\,\sqrt {a}\,b^{1/4}\,\sqrt {a^2\,x^3+b\,x}+2\,2^{1/4}\,a\,\sqrt {b}\,x}{b+a^2\,x^2-\sqrt {2}\,a\,\sqrt {b}\,x}\right )}{4\,\sqrt {a}\,b^{1/4}}+\frac {2^{3/4}\,\ln \left (\frac {2^{3/4}\,b+2^{3/4}\,a^2\,x^2-2\,2^{1/4}\,a\,\sqrt {b}\,x+\sqrt {a}\,b^{1/4}\,\sqrt {a^2\,x^3+b\,x}\,4{}\mathrm {i}}{b+a^2\,x^2+\sqrt {2}\,a\,\sqrt {b}\,x}\right )\,1{}\mathrm {i}}{4\,\sqrt {a}\,b^{1/4}} \]

[In]

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

[Out]

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