\(\int \frac {-2 b+a x^2}{\sqrt [4]{-b+a x^2} (-b+a x^2+c x^4)} \, dx\) [1915]

Optimal result

Integrand size = 39, antiderivative size = 133 \[ \int \frac {-2 b+a x^2}{\sqrt [4]{-b+a x^2} \left (-b+a x^2+c x^4\right )} \, dx=-\frac {\arctan \left (\frac {-\frac {\sqrt [4]{c} x^2}{\sqrt {2}}+\frac {\sqrt {-b+a x^2}}{\sqrt {2} \sqrt [4]{c}}}{x \sqrt [4]{-b+a x^2}}\right )}{\sqrt {2} \sqrt [4]{c}}+\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{c} x \sqrt [4]{-b+a x^2}}{\sqrt {c} x^2+\sqrt {-b+a x^2}}\right )}{\sqrt {2} \sqrt [4]{c}} \]

[Out]

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

Rubi [C] (warning: unable to verify)

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

Time = 8.50 (sec) , antiderivative size = 2670, normalized size of antiderivative = 20.08, number of steps used = 18, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {1706, 408, 504, 1231, 226, 1721} \[ \int \frac {-2 b+a x^2}{\sqrt [4]{-b+a x^2} \left (-b+a x^2+c x^4\right )} \, dx=-\frac {\left (a+\sqrt {a^2+4 b c}\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 {EllipticPi}\left (\frac {\left (\sqrt {2} \sqrt {b} \sqrt {c}+\sqrt {-a^2-\sqrt {a^2+4 b c} a-2 b c}\right )^2}{4 \sqrt {2} \sqrt {b} \sqrt {c} \sqrt {-a^2-\sqrt {a^2+4 b c} a-2 b c}},2 \arctan \left (\frac {\sqrt [4]{a x^2-b}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right ) \left (\sqrt {2} \sqrt {b} \sqrt {c}-\sqrt {-a^2-\sqrt {a^2+4 b c} a-2 b c}\right )^2}{4 \sqrt {2} \sqrt [4]{b} \sqrt {c} \sqrt {-a^2-\sqrt {a^2+4 b c} a-2 b c} \left (a^2+\sqrt {a^2+4 b c} a+4 b c\right ) x}-\frac {\sqrt {b} \sqrt {-a-\sqrt {a^2+4 b c}} \sqrt {\frac {a x^2}{b}} \arctan \left (\frac {\sqrt {a} \sqrt {-a-\sqrt {a^2+4 b c}} \sqrt [4]{a x^2-b}}{\sqrt [4]{2} \sqrt {b} \sqrt [4]{c} \sqrt [4]{-a^2-\sqrt {a^2+4 b c} a-2 b c} \sqrt {\frac {a x^2}{b}}}\right )}{2 \sqrt [4]{2} \sqrt {a} \sqrt [4]{c} \sqrt [4]{-a^2-\sqrt {a^2+4 b c} a-2 b c} x}-\frac {\sqrt {b} \sqrt {a+\sqrt {a^2+4 b c}} \sqrt {\frac {a x^2}{b}} \arctan \left (\frac {\sqrt {a} \sqrt {a+\sqrt {a^2+4 b c}} \sqrt [4]{a x^2-b}}{\sqrt [4]{2} \sqrt {b} \sqrt [4]{c} \sqrt [4]{-a^2-\sqrt {a^2+4 b c} a-2 b c} \sqrt {\frac {a x^2}{b}}}\right )}{2 \sqrt [4]{2} \sqrt {a} \sqrt [4]{c} \sqrt [4]{-a^2-\sqrt {a^2+4 b c} a-2 b c} x}-\frac {\sqrt {b} \sqrt {a-\sqrt {a^2+4 b c}} \sqrt {\frac {a x^2}{b}} \arctan \left (\frac {\sqrt {a} \sqrt {a-\sqrt {a^2+4 b c}} \sqrt [4]{a x^2-b}}{\sqrt [4]{2} \sqrt {b} \sqrt [4]{c} \sqrt [4]{-a^2+\sqrt {a^2+4 b c} a-2 b c} \sqrt {\frac {a x^2}{b}}}\right )}{2 \sqrt [4]{2} \sqrt {a} \sqrt [4]{c} \sqrt [4]{-a^2+\sqrt {a^2+4 b c} a-2 b c} x}-\frac {\sqrt {b} \sqrt {\sqrt {a^2+4 b c}-a} \sqrt {\frac {a x^2}{b}} \arctan \left (\frac {\sqrt {a} \sqrt {\sqrt {a^2+4 b c}-a} \sqrt [4]{a x^2-b}}{\sqrt [4]{2} \sqrt {b} \sqrt [4]{c} \sqrt [4]{-a^2+\sqrt {a^2+4 b c} a-2 b c} \sqrt {\frac {a x^2}{b}}}\right )}{2 \sqrt [4]{2} \sqrt {a} \sqrt [4]{c} \sqrt [4]{-a^2+\sqrt {a^2+4 b c} a-2 b c} x}-\frac {\left (a+\sqrt {a^2+4 b c}\right ) \left (2 \sqrt {b}-\frac {\sqrt {2} \sqrt {-a^2-\sqrt {a^2+4 b c} a-2 b c}}{\sqrt {c}}\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 \sqrt [4]{b} \left (a^2+\sqrt {a^2+4 b c} a+4 b c\right ) x}-\frac {\left (a+\sqrt {a^2+4 b c}\right ) \left (2 \sqrt {b}+\frac {\sqrt {2} \sqrt {-a^2-\sqrt {a^2+4 b c} a-2 b c}}{\sqrt {c}}\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 \sqrt [4]{b} \left (a^2+\sqrt {a^2+4 b c} a+4 b c\right ) x}-\frac {\left (a-\sqrt {a^2+4 b c}\right ) \left (2 \sqrt {b} \sqrt {c}-\sqrt {-2 a^2+2 \sqrt {a^2+4 b c} a-4 b c}\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 \sqrt [4]{b} \sqrt {c} \left (a^2-\sqrt {a^2+4 b c} a+4 b c\right ) x}-\frac {\left (a-\sqrt {a^2+4 b c}\right ) \left (2 \sqrt {b} \sqrt {c}+\sqrt {-2 a^2+2 \sqrt {a^2+4 b c} a-4 b c}\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 \sqrt [4]{b} \sqrt {c} \left (a^2-\sqrt {a^2+4 b c} a+4 b c\right ) x}+\frac {\left (a+\sqrt {a^2+4 b c}\right ) \left (\sqrt {2} \sqrt {b} \sqrt {c}+\sqrt {-a^2-\sqrt {a^2+4 b c} a-2 b c}\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} \sqrt {b} \sqrt {c}-\sqrt {-a^2-\sqrt {a^2+4 b c} a-2 b c}\right )^2}{4 \sqrt {2} \sqrt {b} \sqrt {c} \sqrt {-a^2-\sqrt {a^2+4 b c} a-2 b c}},2 \arctan \left (\frac {\sqrt [4]{a x^2-b}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{4 \sqrt {2} \sqrt [4]{b} \sqrt {c} \sqrt {-a^2-\sqrt {a^2+4 b c} a-2 b c} \left (a^2+\sqrt {a^2+4 b c} a+4 b c\right ) x}+\frac {\left (a-\sqrt {a^2+4 b c}\right ) \left (\sqrt {2} \sqrt {b} \sqrt {c}+\sqrt {-a^2+\sqrt {a^2+4 b c} a-2 b c}\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} \sqrt {b} \sqrt {c}-\sqrt {-a^2+\sqrt {a^2+4 b c} a-2 b c}\right )^2}{4 \sqrt {2} \sqrt {b} \sqrt {c} \sqrt {-a^2+\sqrt {a^2+4 b c} a-2 b c}},2 \arctan \left (\frac {\sqrt [4]{a x^2-b}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{4 \sqrt {2} \sqrt [4]{b} \sqrt {c} \left (a^2-\sqrt {a^2+4 b c} a+4 b c\right ) \sqrt {-a^2+\sqrt {a^2+4 b c} a-2 b c} x}-\frac {\left (a-\sqrt {a^2+4 b c}\right ) \left (\sqrt {2} \sqrt {b} \sqrt {c}-\sqrt {-a^2+\sqrt {a^2+4 b c} a-2 b c}\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} \sqrt {b} \sqrt {c}+\sqrt {-a^2+\sqrt {a^2+4 b c} a-2 b c}\right )^2}{4 \sqrt {2} \sqrt {b} \sqrt {c} \sqrt {-a^2+\sqrt {a^2+4 b c} a-2 b c}},2 \arctan \left (\frac {\sqrt [4]{a x^2-b}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{4 \sqrt {2} \sqrt [4]{b} \sqrt {c} \left (a^2-\sqrt {a^2+4 b c} a+4 b c\right ) \sqrt {-a^2+\sqrt {a^2+4 b c} a-2 b c} x} \]

[In]

Int[(-2*b + a*x^2)/((-b + a*x^2)^(1/4)*(-b + a*x^2 + c*x^4)),x]

[Out]

-1/2*(Sqrt[b]*Sqrt[-a - Sqrt[a^2 + 4*b*c]]*Sqrt[(a*x^2)/b]*ArcTan[(Sqrt[a]*Sqrt[-a - Sqrt[a^2 + 4*b*c]]*(-b +
a*x^2)^(1/4))/(2^(1/4)*Sqrt[b]*c^(1/4)*(-a^2 - 2*b*c - a*Sqrt[a^2 + 4*b*c])^(1/4)*Sqrt[(a*x^2)/b])])/(2^(1/4)*
Sqrt[a]*c^(1/4)*(-a^2 - 2*b*c - a*Sqrt[a^2 + 4*b*c])^(1/4)*x) - (Sqrt[b]*Sqrt[a + Sqrt[a^2 + 4*b*c]]*Sqrt[(a*x
^2)/b]*ArcTan[(Sqrt[a]*Sqrt[a + Sqrt[a^2 + 4*b*c]]*(-b + a*x^2)^(1/4))/(2^(1/4)*Sqrt[b]*c^(1/4)*(-a^2 - 2*b*c
- a*Sqrt[a^2 + 4*b*c])^(1/4)*Sqrt[(a*x^2)/b])])/(2*2^(1/4)*Sqrt[a]*c^(1/4)*(-a^2 - 2*b*c - a*Sqrt[a^2 + 4*b*c]
)^(1/4)*x) - (Sqrt[b]*Sqrt[a - Sqrt[a^2 + 4*b*c]]*Sqrt[(a*x^2)/b]*ArcTan[(Sqrt[a]*Sqrt[a - Sqrt[a^2 + 4*b*c]]*
(-b + a*x^2)^(1/4))/(2^(1/4)*Sqrt[b]*c^(1/4)*(-a^2 - 2*b*c + a*Sqrt[a^2 + 4*b*c])^(1/4)*Sqrt[(a*x^2)/b])])/(2*
2^(1/4)*Sqrt[a]*c^(1/4)*(-a^2 - 2*b*c + a*Sqrt[a^2 + 4*b*c])^(1/4)*x) - (Sqrt[b]*Sqrt[-a + Sqrt[a^2 + 4*b*c]]*
Sqrt[(a*x^2)/b]*ArcTan[(Sqrt[a]*Sqrt[-a + Sqrt[a^2 + 4*b*c]]*(-b + a*x^2)^(1/4))/(2^(1/4)*Sqrt[b]*c^(1/4)*(-a^
2 - 2*b*c + a*Sqrt[a^2 + 4*b*c])^(1/4)*Sqrt[(a*x^2)/b])])/(2*2^(1/4)*Sqrt[a]*c^(1/4)*(-a^2 - 2*b*c + a*Sqrt[a^
2 + 4*b*c])^(1/4)*x) - ((a + Sqrt[a^2 + 4*b*c])*(2*Sqrt[b] - (Sqrt[2]*Sqrt[-a^2 - 2*b*c - a*Sqrt[a^2 + 4*b*c]]
)/Sqrt[c])*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*b^(1/4)*(a^2 + 4*b*c + a*Sqrt[a^2 + 4*b*c])*x) - ((a + Sqrt[a^2 + 4*b*c])*(2*S
qrt[b] + (Sqrt[2]*Sqrt[-a^2 - 2*b*c - a*Sqrt[a^2 + 4*b*c]])/Sqrt[c])*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*b^(1/4)*(a^2 + 4*b*c
+ a*Sqrt[a^2 + 4*b*c])*x) - ((a - Sqrt[a^2 + 4*b*c])*(2*Sqrt[b]*Sqrt[c] - Sqrt[-2*a^2 - 4*b*c + 2*a*Sqrt[a^2
+ 4*b*c]])*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*b^(1/4)*Sqrt[c]*(a^2 + 4*b*c - a*Sqrt[a^2 + 4*b*c])*x) - ((a - Sqrt[a^2 + 4*b*
c])*(2*Sqrt[b]*Sqrt[c] + Sqrt[-2*a^2 - 4*b*c + 2*a*Sqrt[a^2 + 4*b*c]])*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*b^(1/4)*Sqrt[c]*(a
^2 + 4*b*c - a*Sqrt[a^2 + 4*b*c])*x) + ((a + Sqrt[a^2 + 4*b*c])*(Sqrt[2]*Sqrt[b]*Sqrt[c] + Sqrt[-a^2 - 2*b*c -
a*Sqrt[a^2 + 4*b*c]])^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]*Sqrt[b]*Sqrt[c] - Sqrt[-a^2 - 2*b*c - a*Sqrt[a^2 + 4*b*c]])^2/(Sqrt[2]*Sqrt[b]*Sqrt[c]*Sqrt[-a^2
- 2*b*c - a*Sqrt[a^2 + 4*b*c]]), 2*ArcTan[(-b + a*x^2)^(1/4)/b^(1/4)], 1/2])/(4*Sqrt[2]*b^(1/4)*Sqrt[c]*Sqrt[
-a^2 - 2*b*c - a*Sqrt[a^2 + 4*b*c]]*(a^2 + 4*b*c + a*Sqrt[a^2 + 4*b*c])*x) - ((a + Sqrt[a^2 + 4*b*c])*(Sqrt[2]
*Sqrt[b]*Sqrt[c] - Sqrt[-a^2 - 2*b*c - a*Sqrt[a^2 + 4*b*c]])^2*Sqrt[(a*x^2)/(Sqrt[b] + Sqrt[-b + a*x^2])^2]*(S
qrt[b] + Sqrt[-b + a*x^2])*EllipticPi[(Sqrt[2]*Sqrt[b]*Sqrt[c] + Sqrt[-a^2 - 2*b*c - a*Sqrt[a^2 + 4*b*c]])^2/(
4*Sqrt[2]*Sqrt[b]*Sqrt[c]*Sqrt[-a^2 - 2*b*c - a*Sqrt[a^2 + 4*b*c]]), 2*ArcTan[(-b + a*x^2)^(1/4)/b^(1/4)], 1/2
])/(4*Sqrt[2]*b^(1/4)*Sqrt[c]*Sqrt[-a^2 - 2*b*c - a*Sqrt[a^2 + 4*b*c]]*(a^2 + 4*b*c + a*Sqrt[a^2 + 4*b*c])*x)
+ ((a - Sqrt[a^2 + 4*b*c])*(Sqrt[2]*Sqrt[b]*Sqrt[c] + Sqrt[-a^2 - 2*b*c + a*Sqrt[a^2 + 4*b*c]])^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]*Sqrt[b]*Sqrt[c] - Sqrt[
-a^2 - 2*b*c + a*Sqrt[a^2 + 4*b*c]])^2/(Sqrt[2]*Sqrt[b]*Sqrt[c]*Sqrt[-a^2 - 2*b*c + a*Sqrt[a^2 + 4*b*c]]), 2*A
rcTan[(-b + a*x^2)^(1/4)/b^(1/4)], 1/2])/(4*Sqrt[2]*b^(1/4)*Sqrt[c]*(a^2 + 4*b*c - a*Sqrt[a^2 + 4*b*c])*Sqrt[-
a^2 - 2*b*c + a*Sqrt[a^2 + 4*b*c]]*x) - ((a - Sqrt[a^2 + 4*b*c])*(Sqrt[2]*Sqrt[b]*Sqrt[c] - Sqrt[-a^2 - 2*b*c
+ a*Sqrt[a^2 + 4*b*c]])^2*Sqrt[(a*x^2)/(Sqrt[b] + Sqrt[-b + a*x^2])^2]*(Sqrt[b] + Sqrt[-b + a*x^2])*EllipticPi
[(Sqrt[2]*Sqrt[b]*Sqrt[c] + Sqrt[-a^2 - 2*b*c + a*Sqrt[a^2 + 4*b*c]])^2/(4*Sqrt[2]*Sqrt[b]*Sqrt[c]*Sqrt[-a^2 -
2*b*c + a*Sqrt[a^2 + 4*b*c]]), 2*ArcTan[(-b + a*x^2)^(1/4)/b^(1/4)], 1/2])/(4*Sqrt[2]*b^(1/4)*Sqrt[c]*(a^2 +
4*b*c - a*Sqrt[a^2 + 4*b*c])*Sqrt[-a^2 - 2*b*c + a*Sqrt[a^2 + 4*b*c]]*x)

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 408

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

Rule 504

Int[(x_)^2/(((a_) + (b_.)*(x_)^4)*Sqrt[(c_) + (d_.)*(x_)^4]), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s
= Denominator[Rt[-a/b, 2]]}, Dist[s/(2*b), Int[1/((r + s*x^2)*Sqrt[c + d*x^4]), x], x] - Dist[s/(2*b), Int[1/
((r - s*x^2)*Sqrt[c + d*x^4]), 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-\sqrt {a^2+4 b c}}{\sqrt [4]{-b+a x^2} \left (a-\sqrt {a^2+4 b c}+2 c x^2\right )}+\frac {a+\sqrt {a^2+4 b c}}{\sqrt [4]{-b+a x^2} \left (a+\sqrt {a^2+4 b c}+2 c x^2\right )}\right ) \, dx \\ & = \left (a-\sqrt {a^2+4 b c}\right ) \int \frac {1}{\sqrt [4]{-b+a x^2} \left (a-\sqrt {a^2+4 b c}+2 c x^2\right )} \, dx+\left (a+\sqrt {a^2+4 b c}\right ) \int \frac {1}{\sqrt [4]{-b+a x^2} \left (a+\sqrt {a^2+4 b c}+2 c x^2\right )} \, dx \\ & = \frac {\left (2 \left (a-\sqrt {a^2+4 b c}\right ) \sqrt {\frac {a x^2}{b}}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {1+\frac {x^4}{b}} \left (2 b c+a \left (a-\sqrt {a^2+4 b c}\right )+2 c x^4\right )} \, dx,x,\sqrt [4]{-b+a x^2}\right )}{x}+\frac {\left (2 \left (a+\sqrt {a^2+4 b c}\right ) \sqrt {\frac {a x^2}{b}}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {1+\frac {x^4}{b}} \left (2 b c+a \left (a+\sqrt {a^2+4 b c}\right )+2 c x^4\right )} \, dx,x,\sqrt [4]{-b+a x^2}\right )}{x} \\ & = -\frac {\left (\left (a-\sqrt {a^2+4 b c}\right ) \sqrt {\frac {a x^2}{b}}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {-a^2-2 b c+a \sqrt {a^2+4 b c}}-\sqrt {2} \sqrt {c} x^2\right ) \sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x^2}\right )}{\sqrt {2} \sqrt {c} x}+\frac {\left (\left (a-\sqrt {a^2+4 b c}\right ) \sqrt {\frac {a x^2}{b}}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {-a^2-2 b c+a \sqrt {a^2+4 b c}}+\sqrt {2} \sqrt {c} x^2\right ) \sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x^2}\right )}{\sqrt {2} \sqrt {c} x}-\frac {\left (\left (a+\sqrt {a^2+4 b c}\right ) \sqrt {\frac {a x^2}{b}}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {-a^2-2 b c-a \sqrt {a^2+4 b c}}-\sqrt {2} \sqrt {c} x^2\right ) \sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x^2}\right )}{\sqrt {2} \sqrt {c} x}+\frac {\left (\left (a+\sqrt {a^2+4 b c}\right ) \sqrt {\frac {a x^2}{b}}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {-a^2-2 b c-a \sqrt {a^2+4 b c}}+\sqrt {2} \sqrt {c} x^2\right ) \sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x^2}\right )}{\sqrt {2} \sqrt {c} x} \\ & = -\frac {\left (\left (a+\sqrt {a^2+4 b c}\right ) \left (\sqrt {2} \sqrt {b} \sqrt {c}-\sqrt {-a^2-2 b c-a \sqrt {a^2+4 b c}}\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 {2} \sqrt {c} \left (a^2+4 b c+a \sqrt {a^2+4 b c}\right ) x}-\frac {\left (\left (a+\sqrt {a^2+4 b c}\right ) \left (\sqrt {2} \sqrt {b} \sqrt {c}+\sqrt {-a^2-2 b c-a \sqrt {a^2+4 b c}}\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 {2} \sqrt {c} \left (a^2+4 b c+a \sqrt {a^2+4 b c}\right ) x}-\frac {\left (\sqrt {b} \left (a+\sqrt {a^2+4 b c}\right ) \left (2 \sqrt {b} \sqrt {c}-\sqrt {2} \sqrt {-a^2-2 b c-a \sqrt {a^2+4 b c}}\right ) \sqrt {\frac {a x^2}{b}}\right ) \text {Subst}\left (\int \frac {1+\frac {x^2}{\sqrt {b}}}{\left (\sqrt {-a^2-2 b c-a \sqrt {a^2+4 b c}}-\sqrt {2} \sqrt {c} x^2\right ) \sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x^2}\right )}{\sqrt {2} \left (a^2+4 b c+a \sqrt {a^2+4 b c}\right ) x}+\frac {\left (\sqrt {b} \left (a+\sqrt {a^2+4 b c}\right ) \left (2 \sqrt {b} \sqrt {c}+\sqrt {2} \sqrt {-a^2-2 b c-a \sqrt {a^2+4 b c}}\right ) \sqrt {\frac {a x^2}{b}}\right ) \text {Subst}\left (\int \frac {1+\frac {x^2}{\sqrt {b}}}{\left (\sqrt {-a^2-2 b c-a \sqrt {a^2+4 b c}}+\sqrt {2} \sqrt {c} x^2\right ) \sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x^2}\right )}{\sqrt {2} \left (a^2+4 b c+a \sqrt {a^2+4 b c}\right ) x}-\frac {\left (\left (a-\sqrt {a^2+4 b c}\right ) \left (\sqrt {2} \sqrt {b} \sqrt {c}-\sqrt {-a^2-2 b c+a \sqrt {a^2+4 b c}}\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 {2} \sqrt {c} \left (a^2+4 b c-a \sqrt {a^2+4 b c}\right ) x}-\frac {\left (\left (a-\sqrt {a^2+4 b c}\right ) \left (\sqrt {2} \sqrt {b} \sqrt {c}+\sqrt {-a^2-2 b c+a \sqrt {a^2+4 b c}}\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 {2} \sqrt {c} \left (a^2+4 b c-a \sqrt {a^2+4 b c}\right ) x}-\frac {\left (\sqrt {b} \left (a-\sqrt {a^2+4 b c}\right ) \left (2 \sqrt {b} \sqrt {c}-\sqrt {2} \sqrt {-a^2-2 b c+a \sqrt {a^2+4 b c}}\right ) \sqrt {\frac {a x^2}{b}}\right ) \text {Subst}\left (\int \frac {1+\frac {x^2}{\sqrt {b}}}{\left (\sqrt {-a^2-2 b c+a \sqrt {a^2+4 b c}}-\sqrt {2} \sqrt {c} x^2\right ) \sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x^2}\right )}{\sqrt {2} \left (a^2+4 b c-a \sqrt {a^2+4 b c}\right ) x}+\frac {\left (\sqrt {b} \left (a-\sqrt {a^2+4 b c}\right ) \left (2 \sqrt {b} \sqrt {c}+\sqrt {2} \sqrt {-a^2-2 b c+a \sqrt {a^2+4 b c}}\right ) \sqrt {\frac {a x^2}{b}}\right ) \text {Subst}\left (\int \frac {1+\frac {x^2}{\sqrt {b}}}{\left (\sqrt {-a^2-2 b c+a \sqrt {a^2+4 b c}}+\sqrt {2} \sqrt {c} x^2\right ) \sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x^2}\right )}{\sqrt {2} \left (a^2+4 b c-a \sqrt {a^2+4 b c}\right ) x} \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.29 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.88 \[ \int \frac {-2 b+a x^2}{\sqrt [4]{-b+a x^2} \left (-b+a x^2+c x^4\right )} \, dx=\frac {-\arctan \left (\frac {-\sqrt {c} x^2+\sqrt {-b+a x^2}}{\sqrt {2} \sqrt [4]{c} x \sqrt [4]{-b+a x^2}}\right )+\text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{c} x \sqrt [4]{-b+a x^2}}{\sqrt {c} x^2+\sqrt {-b+a x^2}}\right )}{\sqrt {2} \sqrt [4]{c}} \]

[In]

Integrate[(-2*b + a*x^2)/((-b + a*x^2)^(1/4)*(-b + a*x^2 + c*x^4)),x]

[Out]

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

Maple [F]

\[\int \frac {a \,x^{2}-2 b}{\left (a \,x^{2}-b \right )^{\frac {1}{4}} \left (c \,x^{4}+a \,x^{2}-b \right )}d x\]

[In]

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

[Out]

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

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F]

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

[In]

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

[Out]

Integral((a*x**2 - 2*b)/((a*x**2 - b)**(1/4)*(a*x**2 - b + c*x**4)), x)

Maxima [F]

\[ \int \frac {-2 b+a x^2}{\sqrt [4]{-b+a x^2} \left (-b+a x^2+c x^4\right )} \, dx=\int { \frac {a x^{2} - 2 \, b}{{\left (c x^{4} + a x^{2} - b\right )} {\left (a x^{2} - b\right )}^{\frac {1}{4}}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {-2 b+a x^2}{\sqrt [4]{-b+a x^2} \left (-b+a x^2+c x^4\right )} \, dx=\int { \frac {a x^{2} - 2 \, b}{{\left (c x^{4} + a x^{2} - b\right )} {\left (a x^{2} - b\right )}^{\frac {1}{4}}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {-2 b+a x^2}{\sqrt [4]{-b+a x^2} \left (-b+a x^2+c x^4\right )} \, dx=\int -\frac {2\,b-a\,x^2}{{\left (a\,x^2-b\right )}^{1/4}\,\left (c\,x^4+a\,x^2-b\right )} \,d x \]

[In]

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

[Out]

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