\(\int \frac {x^2 (-2 b+a x^2)}{(-b+a x^2)^{3/4} (4 b-4 a x^2+x^4)} \, dx\) [999]

Optimal result

Integrand size = 41, antiderivative size = 76 \[ \int \frac {x^2 \left (-2 b+a x^2\right )}{\left (-b+a x^2\right )^{3/4} \left (4 b-4 a x^2+x^4\right )} \, dx=-\frac {\arctan \left (\frac {x \left (-b+a x^2\right )^{3/4}}{\sqrt {2} \left (b-a x^2\right )}\right )}{\sqrt {2}}+\frac {\text {arctanh}\left (\frac {x \left (-b+a x^2\right )^{3/4}}{\sqrt {2} \left (b-a x^2\right )}\right )}{\sqrt {2}} \]

[Out]

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

Rubi [C] (warning: unable to verify)

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

Time = 29.39 (sec) , antiderivative size = 2421, normalized size of antiderivative = 31.86, number of steps used = 24, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.195, Rules used = {1706, 240, 226, 410, 109, 418, 1231, 1721} \[ \int \frac {x^2 \left (-2 b+a x^2\right )}{\left (-b+a x^2\right )^{3/4} \left (4 b-4 a x^2+x^4\right )} \, dx=\frac {\sqrt {\frac {a x^2}{\left (\sqrt {b}+\sqrt {a x^2-b}\right )^2}} \left (\sqrt {b}+\sqrt {a x^2-b}\right ) \operatorname {EllipticPi}\left (\frac {\left (\sqrt {2 a^2-2 \sqrt {a^2-b} a-b}+\sqrt {b}\right )^2}{4 \sqrt {2 a^2-2 \sqrt {a^2-b} a-b} \sqrt {b}},2 \arctan \left (\frac {\sqrt [4]{a x^2-b}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right ) \left (\sqrt {2 a^2-2 \sqrt {a^2-b} a-b}-\sqrt {b}\right )^2}{8 \left (a^2-\sqrt {a^2-b} a-b\right ) \sqrt [4]{b} x}-\frac {\sqrt {b} \left (4 a^4-5 b a^2-\left (4 a^2-3 b\right ) \sqrt {a^2-b} a+b^2\right ) \sqrt {\frac {a x^2}{b}} \arctan \left (\frac {\sqrt {2} \sqrt {a} \sqrt {a-\sqrt {a^2-b}} \sqrt [4]{a x^2-b}}{\sqrt [4]{2 a^2-2 \sqrt {a^2-b} a-b} \sqrt {b} \sqrt {\frac {a x^2}{b}}}\right )}{2 \sqrt {2} \sqrt {a} \sqrt {a-\sqrt {a^2-b}} \left (2 a^2-2 \sqrt {a^2-b} a-b\right )^{3/4} \left (a^2-\sqrt {a^2-b} a-b\right ) x}-\frac {\sqrt {b} \left (4 a^4-5 b a^2-\left (4 a^2-3 b\right ) \sqrt {a^2-b} a+b^2\right ) \sqrt {\frac {a x^2}{b}} \arctan \left (\frac {\sqrt {2} \sqrt {a} \sqrt {\sqrt {a^2-b}-a} \sqrt [4]{a x^2-b}}{\sqrt [4]{2 a^2-2 \sqrt {a^2-b} a-b} \sqrt {b} \sqrt {\frac {a x^2}{b}}}\right )}{2 \sqrt {2} \sqrt {a} \sqrt {\sqrt {a^2-b}-a} \left (2 a^2-2 \sqrt {a^2-b} a-b\right )^{3/4} \left (a^2-\sqrt {a^2-b} a-b\right ) x}-\frac {\sqrt [4]{2 a^2+2 \sqrt {a^2-b} a-b} \sqrt {b} \sqrt {\frac {a x^2}{b}} \arctan \left (\frac {\sqrt {2} \sqrt {a} \sqrt {-a-\sqrt {a^2-b}} \sqrt [4]{a x^2-b}}{\sqrt [4]{2 a^2+2 \sqrt {a^2-b} a-b} \sqrt {b} \sqrt {\frac {a x^2}{b}}}\right )}{2 \sqrt {2} \sqrt {a} \sqrt {-a-\sqrt {a^2-b}} x}-\frac {\sqrt [4]{2 a^2+2 \sqrt {a^2-b} a-b} \sqrt {b} \sqrt {\frac {a x^2}{b}} \arctan \left (\frac {\sqrt {2} \sqrt {a} \sqrt {a+\sqrt {a^2-b}} \sqrt [4]{a x^2-b}}{\sqrt [4]{2 a^2+2 \sqrt {a^2-b} a-b} \sqrt {b} \sqrt {\frac {a x^2}{b}}}\right )}{2 \sqrt {2} \sqrt {a} \sqrt {a+\sqrt {a^2-b}} x}-\frac {\left (1-\frac {\sqrt {b}}{\sqrt {2 a^2+2 \sqrt {a^2-b} a-b}}\right ) \left (2 a^2+2 \sqrt {a^2-b} a-b\right ) \sqrt {\frac {a x^2}{\left (\sqrt {b}+\sqrt {a x^2-b}\right )^2}} \left (\sqrt {b}+\sqrt {a x^2-b}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a x^2-b}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{4 \left (a^2+\sqrt {a^2-b} a-b\right ) \sqrt [4]{b} x}-\frac {\left (\frac {\sqrt {b}}{\sqrt {2 a^2+2 \sqrt {a^2-b} a-b}}+1\right ) \left (2 a^2+2 \sqrt {a^2-b} a-b\right ) \sqrt {\frac {a x^2}{\left (\sqrt {b}+\sqrt {a x^2-b}\right )^2}} \left (\sqrt {b}+\sqrt {a x^2-b}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a x^2-b}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{4 \left (a^2+\sqrt {a^2-b} a-b\right ) \sqrt [4]{b} x}-\frac {\left (1-\frac {\sqrt {b}}{\sqrt {2 a^2-2 \sqrt {a^2-b} a-b}}\right ) \left (2 a^2-2 \sqrt {a^2-b} a-b\right ) \sqrt {\frac {a x^2}{\left (\sqrt {b}+\sqrt {a x^2-b}\right )^2}} \left (\sqrt {b}+\sqrt {a x^2-b}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a x^2-b}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{4 \left (a^2-\sqrt {a^2-b} a-b\right ) \sqrt [4]{b} x}-\frac {\left (\frac {\sqrt {b}}{\sqrt {2 a^2-2 \sqrt {a^2-b} a-b}}+1\right ) \left (2 a^2-2 \sqrt {a^2-b} a-b\right ) \sqrt {\frac {a x^2}{\left (\sqrt {b}+\sqrt {a x^2-b}\right )^2}} \left (\sqrt {b}+\sqrt {a x^2-b}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a x^2-b}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{4 \left (a^2-\sqrt {a^2-b} a-b\right ) \sqrt [4]{b} x}+\frac {\sqrt {\frac {a x^2}{\left (\sqrt {b}+\sqrt {a x^2-b}\right )^2}} \left (\sqrt {b}+\sqrt {a x^2-b}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a x^2-b}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{\sqrt [4]{b} x}+\frac {\left (\sqrt {2 a^2-2 \sqrt {a^2-b} a-b}+\sqrt {b}\right )^2 \sqrt {\frac {a x^2}{\left (\sqrt {b}+\sqrt {a x^2-b}\right )^2}} \left (\sqrt {b}+\sqrt {a x^2-b}\right ) \operatorname {EllipticPi}\left (-\frac {\left (\sqrt {2 a^2-2 \sqrt {a^2-b} a-b}-\sqrt {b}\right )^2}{4 \sqrt {2 a^2-2 \sqrt {a^2-b} a-b} \sqrt {b}},2 \arctan \left (\frac {\sqrt [4]{a x^2-b}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{8 \left (a^2-\sqrt {a^2-b} a-b\right ) \sqrt [4]{b} x}+\frac {\left (\sqrt {2 a^2+2 \sqrt {a^2-b} a-b}+\sqrt {b}\right )^2 \sqrt {\frac {a x^2}{\left (\sqrt {b}+\sqrt {a x^2-b}\right )^2}} \left (\sqrt {b}+\sqrt {a x^2-b}\right ) \operatorname {EllipticPi}\left (-\frac {\left (\sqrt {2 a^2+2 \sqrt {a^2-b} a-b}-\sqrt {b}\right )^2}{4 \sqrt {2 a^2+2 \sqrt {a^2-b} a-b} \sqrt {b}},2 \arctan \left (\frac {\sqrt [4]{a x^2-b}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{8 \left (a^2+\sqrt {a^2-b} a-b\right ) \sqrt [4]{b} x}+\frac {\left (\sqrt {2 a^2+2 \sqrt {a^2-b} a-b}-\sqrt {b}\right )^2 \sqrt {\frac {a x^2}{\left (\sqrt {b}+\sqrt {a x^2-b}\right )^2}} \left (\sqrt {b}+\sqrt {a x^2-b}\right ) \operatorname {EllipticPi}\left (\frac {\left (\sqrt {2 a^2+2 \sqrt {a^2-b} a-b}+\sqrt {b}\right )^2}{4 \sqrt {2 a^2+2 \sqrt {a^2-b} a-b} \sqrt {b}},2 \arctan \left (\frac {\sqrt [4]{a x^2-b}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{8 \left (a^2+\sqrt {a^2-b} a-b\right ) \sqrt [4]{b} x} \]

[In]

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

[Out]

-1/2*(Sqrt[b]*(4*a^4 - a*(4*a^2 - 3*b)*Sqrt[a^2 - b] - 5*a^2*b + b^2)*Sqrt[(a*x^2)/b]*ArcTan[(Sqrt[2]*Sqrt[a]*
Sqrt[a - Sqrt[a^2 - b]]*(-b + a*x^2)^(1/4))/((2*a^2 - 2*a*Sqrt[a^2 - b] - b)^(1/4)*Sqrt[b]*Sqrt[(a*x^2)/b])])/
(Sqrt[2]*Sqrt[a]*Sqrt[a - Sqrt[a^2 - b]]*(2*a^2 - 2*a*Sqrt[a^2 - b] - b)^(3/4)*(a^2 - a*Sqrt[a^2 - b] - b)*x)
- (Sqrt[b]*(4*a^4 - a*(4*a^2 - 3*b)*Sqrt[a^2 - b] - 5*a^2*b + b^2)*Sqrt[(a*x^2)/b]*ArcTan[(Sqrt[2]*Sqrt[a]*Sqr
t[-a + Sqrt[a^2 - b]]*(-b + a*x^2)^(1/4))/((2*a^2 - 2*a*Sqrt[a^2 - b] - b)^(1/4)*Sqrt[b]*Sqrt[(a*x^2)/b])])/(2
*Sqrt[2]*Sqrt[a]*Sqrt[-a + Sqrt[a^2 - b]]*(2*a^2 - 2*a*Sqrt[a^2 - b] - b)^(3/4)*(a^2 - a*Sqrt[a^2 - b] - b)*x)
- ((2*a^2 + 2*a*Sqrt[a^2 - b] - b)^(1/4)*Sqrt[b]*Sqrt[(a*x^2)/b]*ArcTan[(Sqrt[2]*Sqrt[a]*Sqrt[-a - Sqrt[a^2 -
b]]*(-b + a*x^2)^(1/4))/((2*a^2 + 2*a*Sqrt[a^2 - b] - b)^(1/4)*Sqrt[b]*Sqrt[(a*x^2)/b])])/(2*Sqrt[2]*Sqrt[a]*
Sqrt[-a - Sqrt[a^2 - b]]*x) - ((2*a^2 + 2*a*Sqrt[a^2 - b] - b)^(1/4)*Sqrt[b]*Sqrt[(a*x^2)/b]*ArcTan[(Sqrt[2]*S
qrt[a]*Sqrt[a + Sqrt[a^2 - b]]*(-b + a*x^2)^(1/4))/((2*a^2 + 2*a*Sqrt[a^2 - b] - b)^(1/4)*Sqrt[b]*Sqrt[(a*x^2)
/b])])/(2*Sqrt[2]*Sqrt[a]*Sqrt[a + Sqrt[a^2 - b]]*x) + (Sqrt[(a*x^2)/(Sqrt[b] + Sqrt[-b + a*x^2])^2]*(Sqrt[b]
+ Sqrt[-b + a*x^2])*EllipticF[2*ArcTan[(-b + a*x^2)^(1/4)/b^(1/4)], 1/2])/(b^(1/4)*x) - ((1 - Sqrt[b]/Sqrt[2*a
^2 - 2*a*Sqrt[a^2 - b] - b])*(2*a^2 - 2*a*Sqrt[a^2 - b] - b)*Sqrt[(a*x^2)/(Sqrt[b] + Sqrt[-b + a*x^2])^2]*(Sqr
t[b] + Sqrt[-b + a*x^2])*EllipticF[2*ArcTan[(-b + a*x^2)^(1/4)/b^(1/4)], 1/2])/(4*(a^2 - a*Sqrt[a^2 - b] - b)*
b^(1/4)*x) - ((1 + Sqrt[b]/Sqrt[2*a^2 - 2*a*Sqrt[a^2 - b] - b])*(2*a^2 - 2*a*Sqrt[a^2 - b] - b)*Sqrt[(a*x^2)/(
Sqrt[b] + Sqrt[-b + a*x^2])^2]*(Sqrt[b] + Sqrt[-b + a*x^2])*EllipticF[2*ArcTan[(-b + a*x^2)^(1/4)/b^(1/4)], 1/
2])/(4*(a^2 - a*Sqrt[a^2 - b] - b)*b^(1/4)*x) - ((1 - Sqrt[b]/Sqrt[2*a^2 + 2*a*Sqrt[a^2 - b] - b])*(2*a^2 + 2*
a*Sqrt[a^2 - b] - b)*Sqrt[(a*x^2)/(Sqrt[b] + Sqrt[-b + a*x^2])^2]*(Sqrt[b] + Sqrt[-b + a*x^2])*EllipticF[2*Arc
Tan[(-b + a*x^2)^(1/4)/b^(1/4)], 1/2])/(4*(a^2 + a*Sqrt[a^2 - b] - b)*b^(1/4)*x) - ((1 + Sqrt[b]/Sqrt[2*a^2 +
2*a*Sqrt[a^2 - b] - b])*(2*a^2 + 2*a*Sqrt[a^2 - b] - b)*Sqrt[(a*x^2)/(Sqrt[b] + Sqrt[-b + a*x^2])^2]*(Sqrt[b]
+ Sqrt[-b + a*x^2])*EllipticF[2*ArcTan[(-b + a*x^2)^(1/4)/b^(1/4)], 1/2])/(4*(a^2 + a*Sqrt[a^2 - b] - b)*b^(1/
4)*x) + ((Sqrt[2*a^2 - 2*a*Sqrt[a^2 - b] - b] + Sqrt[b])^2*Sqrt[(a*x^2)/(Sqrt[b] + Sqrt[-b + a*x^2])^2]*(Sqrt[
b] + Sqrt[-b + a*x^2])*EllipticPi[-1/4*(Sqrt[2*a^2 - 2*a*Sqrt[a^2 - b] - b] - Sqrt[b])^2/(Sqrt[2*a^2 - 2*a*Sqr
t[a^2 - b] - b]*Sqrt[b]), 2*ArcTan[(-b + a*x^2)^(1/4)/b^(1/4)], 1/2])/(8*(a^2 - a*Sqrt[a^2 - b] - b)*b^(1/4)*x
) + ((Sqrt[2*a^2 - 2*a*Sqrt[a^2 - b] - b] - Sqrt[b])^2*Sqrt[(a*x^2)/(Sqrt[b] + Sqrt[-b + a*x^2])^2]*(Sqrt[b] +
Sqrt[-b + a*x^2])*EllipticPi[(Sqrt[2*a^2 - 2*a*Sqrt[a^2 - b] - b] + Sqrt[b])^2/(4*Sqrt[2*a^2 - 2*a*Sqrt[a^2 -
b] - b]*Sqrt[b]), 2*ArcTan[(-b + a*x^2)^(1/4)/b^(1/4)], 1/2])/(8*(a^2 - a*Sqrt[a^2 - b] - b)*b^(1/4)*x) + ((S
qrt[2*a^2 + 2*a*Sqrt[a^2 - b] - b] + Sqrt[b])^2*Sqrt[(a*x^2)/(Sqrt[b] + Sqrt[-b + a*x^2])^2]*(Sqrt[b] + Sqrt[-
b + a*x^2])*EllipticPi[-1/4*(Sqrt[2*a^2 + 2*a*Sqrt[a^2 - b] - b] - Sqrt[b])^2/(Sqrt[2*a^2 + 2*a*Sqrt[a^2 - b]
- b]*Sqrt[b]), 2*ArcTan[(-b + a*x^2)^(1/4)/b^(1/4)], 1/2])/(8*(a^2 + a*Sqrt[a^2 - b] - b)*b^(1/4)*x) + ((Sqrt[
2*a^2 + 2*a*Sqrt[a^2 - b] - b] - Sqrt[b])^2*Sqrt[(a*x^2)/(Sqrt[b] + Sqrt[-b + a*x^2])^2]*(Sqrt[b] + Sqrt[-b +
a*x^2])*EllipticPi[(Sqrt[2*a^2 + 2*a*Sqrt[a^2 - b] - b] + Sqrt[b])^2/(4*Sqrt[2*a^2 + 2*a*Sqrt[a^2 - b] - b]*Sq
rt[b]), 2*ArcTan[(-b + a*x^2)^(1/4)/b^(1/4)], 1/2])/(8*(a^2 + a*Sqrt[a^2 - b] - b)*b^(1/4)*x)

Rule 109

Int[1/(((a_.) + (b_.)*(x_))*Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(3/4)), x_Symbol] :> Dist[-4, Subst[
Int[1/((b*e - a*f - b*x^4)*Sqrt[c - d*(e/f) + d*(x^4/f)]), x], x, (e + f*x)^(1/4)], x] /; FreeQ[{a, b, c, d, e
, f}, x] && GtQ[-f/(d*e - c*f), 0]

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 240

Int[((a_) + (b_.)*(x_)^2)^(-3/4), x_Symbol] :> Dist[2*(Sqrt[(-b)*(x^2/a)]/(b*x)), Subst[Int[1/Sqrt[1 - x^4/a],
x], x, (a + b*x^2)^(1/4)], x] /; FreeQ[{a, b}, x] && NegQ[a]

Rule 410

Int[1/(((a_) + (b_.)*(x_)^2)^(3/4)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Dist[Sqrt[(-b)*(x^2/a)]/(2*x), Subst[I
nt[1/(Sqrt[(-b)*(x/a)]*(a + b*x)^(3/4)*(c + d*x)), x], x, x^2], 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 1706

Int[(Px_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandInteg
rand[Px*(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e, q}, x] && PolyQ[Px, x^2] && NeQ[b
^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p]

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]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a}{\left (-b+a x^2\right )^{3/4}}-\frac {2 \left (2 a b-\left (2 a^2-b\right ) x^2\right )}{\left (-b+a x^2\right )^{3/4} \left (4 b-4 a x^2+x^4\right )}\right ) \, dx \\ & = -\left (2 \int \frac {2 a b-\left (2 a^2-b\right ) x^2}{\left (-b+a x^2\right )^{3/4} \left (4 b-4 a x^2+x^4\right )} \, dx\right )+a \int \frac {1}{\left (-b+a x^2\right )^{3/4}} \, dx \\ & = -\left (2 \int \left (\frac {-2 a^2-2 a \sqrt {a^2-b}+b}{\left (-4 a-4 \sqrt {a^2-b}+2 x^2\right ) \left (-b+a x^2\right )^{3/4}}+\frac {-2 a^2+2 a \sqrt {a^2-b}+b}{\left (-4 a+4 \sqrt {a^2-b}+2 x^2\right ) \left (-b+a x^2\right )^{3/4}}\right ) \, dx\right )+\frac {\left (2 \sqrt {\frac {a x^2}{b}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x^2}\right )}{x} \\ & = \frac {\sqrt {\frac {a x^2}{\left (\sqrt {b}+\sqrt {-b+a x^2}\right )^2}} \left (\sqrt {b}+\sqrt {-b+a x^2}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{-b+a x^2}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{\sqrt [4]{b} x}+\left (2 \left (2 a^2-2 a \sqrt {a^2-b}-b\right )\right ) \int \frac {1}{\left (-4 a+4 \sqrt {a^2-b}+2 x^2\right ) \left (-b+a x^2\right )^{3/4}} \, dx+\left (2 \left (2 a^2+2 a \sqrt {a^2-b}-b\right )\right ) \int \frac {1}{\left (-4 a-4 \sqrt {a^2-b}+2 x^2\right ) \left (-b+a x^2\right )^{3/4}} \, dx \\ & = \frac {\sqrt {\frac {a x^2}{\left (\sqrt {b}+\sqrt {-b+a x^2}\right )^2}} \left (\sqrt {b}+\sqrt {-b+a x^2}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{-b+a x^2}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{\sqrt [4]{b} x}+\frac {\left (\left (2 a^2-2 a \sqrt {a^2-b}-b\right ) \sqrt {\frac {a x^2}{b}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {\frac {a x}{b}} \left (-4 a+4 \sqrt {a^2-b}+2 x\right ) (-b+a x)^{3/4}} \, dx,x,x^2\right )}{x}+\frac {\left (\left (2 a^2+2 a \sqrt {a^2-b}-b\right ) \sqrt {\frac {a x^2}{b}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {\frac {a x}{b}} \left (-4 a-4 \sqrt {a^2-b}+2 x\right ) (-b+a x)^{3/4}} \, dx,x,x^2\right )}{x} \\ & = \frac {\sqrt {\frac {a x^2}{\left (\sqrt {b}+\sqrt {-b+a x^2}\right )^2}} \left (\sqrt {b}+\sqrt {-b+a x^2}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{-b+a x^2}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{\sqrt [4]{b} x}-\frac {\left (4 \left (2 a^2-2 a \sqrt {a^2-b}-b\right ) \sqrt {\frac {a x^2}{b}}\right ) \text {Subst}\left (\int \frac {1}{\left (-a \left (-4 a+4 \sqrt {a^2-b}\right )-2 b-2 x^4\right ) \sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x^2}\right )}{x}-\frac {\left (4 \left (2 a^2+2 a \sqrt {a^2-b}-b\right ) \sqrt {\frac {a x^2}{b}}\right ) \text {Subst}\left (\int \frac {1}{\left (-a \left (-4 a-4 \sqrt {a^2-b}\right )-2 b-2 x^4\right ) \sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x^2}\right )}{x} \\ & = \frac {\sqrt {\frac {a x^2}{\left (\sqrt {b}+\sqrt {-b+a x^2}\right )^2}} \left (\sqrt {b}+\sqrt {-b+a x^2}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{-b+a x^2}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{\sqrt [4]{b} x}-\frac {\sqrt {\frac {a x^2}{b}} \text {Subst}\left (\int \frac {1}{\left (1-\frac {x^2}{\sqrt {2 a^2-2 a \sqrt {a^2-b}-b}}\right ) \sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x^2}\right )}{x}-\frac {\sqrt {\frac {a x^2}{b}} \text {Subst}\left (\int \frac {1}{\left (1+\frac {x^2}{\sqrt {2 a^2-2 a \sqrt {a^2-b}-b}}\right ) \sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x^2}\right )}{x}-\frac {\sqrt {\frac {a x^2}{b}} \text {Subst}\left (\int \frac {1}{\left (1-\frac {x^2}{\sqrt {2 a^2+2 a \sqrt {a^2-b}-b}}\right ) \sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x^2}\right )}{x}-\frac {\sqrt {\frac {a x^2}{b}} \text {Subst}\left (\int \frac {1}{\left (1+\frac {x^2}{\sqrt {2 a^2+2 a \sqrt {a^2-b}-b}}\right ) \sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x^2}\right )}{x} \\ & = \frac {\sqrt {\frac {a x^2}{\left (\sqrt {b}+\sqrt {-b+a x^2}\right )^2}} \left (\sqrt {b}+\sqrt {-b+a x^2}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{-b+a x^2}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{\sqrt [4]{b} x}-\frac {\left (\left (\frac {1}{\sqrt {2 a^2+2 a \sqrt {a^2-b}-b}}-\frac {1}{\sqrt {b}}\right ) \sqrt {\frac {a x^2}{b}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x^2}\right )}{\left (\frac {1}{2 a^2+2 a \sqrt {a^2-b}-b}-\frac {1}{b}\right ) \sqrt {b} x}+\frac {\left (\left (\frac {1}{\sqrt {2 a^2+2 a \sqrt {a^2-b}-b}}+\frac {1}{\sqrt {b}}\right ) \sqrt {\frac {a x^2}{b}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x^2}\right )}{\left (\frac {1}{2 a^2+2 a \sqrt {a^2-b}-b}-\frac {1}{b}\right ) \sqrt {b} x}-\frac {\left (\left (\sqrt {2 a^2-2 a \sqrt {a^2-b}-b}-\sqrt {b}\right ) \sqrt {b} \sqrt {\frac {a x^2}{b}}\right ) \text {Subst}\left (\int \frac {1+\frac {x^2}{\sqrt {b}}}{\left (1-\frac {x^2}{\sqrt {2 a^2-2 a \sqrt {a^2-b}-b}}\right ) \sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x^2}\right )}{2 \left (a^2-a \sqrt {a^2-b}-b\right ) x}+\frac {\left (\left (\sqrt {2 a^2-2 a \sqrt {a^2-b}-b}+\sqrt {b}\right ) \sqrt {b} \sqrt {\frac {a x^2}{b}}\right ) \text {Subst}\left (\int \frac {1+\frac {x^2}{\sqrt {b}}}{\left (1+\frac {x^2}{\sqrt {2 a^2-2 a \sqrt {a^2-b}-b}}\right ) \sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x^2}\right )}{2 \left (a^2-a \sqrt {a^2-b}-b\right ) x}-\frac {\left (\left (\sqrt {2 a^2+2 a \sqrt {a^2-b}-b}-\sqrt {b}\right ) \sqrt {b} \sqrt {\frac {a x^2}{b}}\right ) \text {Subst}\left (\int \frac {1+\frac {x^2}{\sqrt {b}}}{\left (1-\frac {x^2}{\sqrt {2 a^2+2 a \sqrt {a^2-b}-b}}\right ) \sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x^2}\right )}{2 \left (a^2+a \sqrt {a^2-b}-b\right ) x}+\frac {\left (\left (\sqrt {2 a^2+2 a \sqrt {a^2-b}-b}+\sqrt {b}\right ) \sqrt {b} \sqrt {\frac {a x^2}{b}}\right ) \text {Subst}\left (\int \frac {1+\frac {x^2}{\sqrt {b}}}{\left (1+\frac {x^2}{\sqrt {2 a^2+2 a \sqrt {a^2-b}-b}}\right ) \sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x^2}\right )}{2 \left (a^2+a \sqrt {a^2-b}-b\right ) x}+\frac {\left (\left (\frac {1}{\sqrt {2 a^2-2 a \sqrt {a^2-b}-b}}-\frac {1}{\sqrt {b}}\right ) \sqrt {\frac {a x^2}{b}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x^2}\right )}{\sqrt {b} \left (\frac {1}{b}+\frac {1}{-2 a^2+2 a \sqrt {a^2-b}+b}\right ) x}-\frac {\left (\left (\frac {1}{\sqrt {2 a^2-2 a \sqrt {a^2-b}-b}}+\frac {1}{\sqrt {b}}\right ) \sqrt {\frac {a x^2}{b}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x^2}\right )}{\sqrt {b} \left (\frac {1}{b}+\frac {1}{-2 a^2+2 a \sqrt {a^2-b}+b}\right ) x} \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.09 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.67 \[ \int \frac {x^2 \left (-2 b+a x^2\right )}{\left (-b+a x^2\right )^{3/4} \left (4 b-4 a x^2+x^4\right )} \, dx=\frac {\arctan \left (\frac {x}{\sqrt {2} \sqrt [4]{-b+a x^2}}\right )-\text {arctanh}\left (\frac {x}{\sqrt {2} \sqrt [4]{-b+a x^2}}\right )}{\sqrt {2}} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 113 vs. \(2 (47) = 94\).

Time = 0.27 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.49 \[ \int \frac {x^2 \left (-2 b+a x^2\right )}{\left (-b+a x^2\right )^{3/4} \left (4 b-4 a x^2+x^4\right )} \, dx=-\frac {1}{2} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (a x^{2} - b\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{4} \, \sqrt {2} \log \left (-\frac {x^{4} - 2 \, \sqrt {2} {\left (a x^{2} - b\right )}^{\frac {1}{4}} x^{3} + 4 \, a x^{2} + 4 \, \sqrt {a x^{2} - b} x^{2} - 4 \, \sqrt {2} {\left (a x^{2} - b\right )}^{\frac {3}{4}} x - 4 \, b}{x^{4} - 4 \, a x^{2} + 4 \, b}\right ) \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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