\(\int \frac {-2 b+a x}{(-b+a x+x^2) \sqrt [4]{-b x^2+a x^3}} \, dx\) [1698]

Optimal result

Integrand size = 36, antiderivative size = 114 \[ \int \frac {-2 b+a x}{\left (-b+a x+x^2\right ) \sqrt [4]{-b x^2+a x^3}} \, dx=-\sqrt {2} \arctan \left (\frac {-\frac {x^2}{\sqrt {2}}+\frac {\sqrt {-b x^2+a x^3}}{\sqrt {2}}}{x \sqrt [4]{-b x^2+a x^3}}\right )+\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} x \sqrt [4]{-b x^2+a x^3}}{x^2+\sqrt {-b x^2+a x^3}}\right ) \]

[Out]

-2^(1/2)*arctan((-1/2*2^(1/2)*x^2+1/2*(a*x^3-b*x^2)^(1/2)*2^(1/2))/x/(a*x^3-b*x^2)^(1/4))+2^(1/2)*arctanh(2^(1
/2)*x*(a*x^3-b*x^2)^(1/4)/(x^2+(a*x^3-b*x^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 = 6.74 (sec) , antiderivative size = 2624, normalized size of antiderivative = 23.02, number of steps used = 21, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2081, 6860, 108, 107, 504, 1231, 226, 1721} \[ \int \frac {-2 b+a x}{\left (-b+a x+x^2\right ) \sqrt [4]{-b x^2+a x^3}} \, dx=-\frac {\left (a+\sqrt {a^2+4 b}\right ) \sqrt [4]{a x-b} \sqrt {\frac {a x}{\left (\sqrt {b}+\sqrt {a x-b}\right )^2}} \left (\sqrt {b}+\sqrt {a x-b}\right ) \operatorname {EllipticPi}\left (\frac {\left (\sqrt {2} \sqrt {b}+\sqrt {-a^2-\sqrt {a^2+4 b} a-2 b}\right )^2}{4 \sqrt {2} \sqrt {b} \sqrt {-a^2-\sqrt {a^2+4 b} a-2 b}},2 \arctan \left (\frac {\sqrt [4]{a x-b}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right ) \left (\sqrt {2} \sqrt {b}-\sqrt {-a^2-\sqrt {a^2+4 b} a-2 b}\right )^2}{2 \sqrt {2} \sqrt [4]{b} \sqrt {-a^2-\sqrt {a^2+4 b} a-2 b} \left (a^2+\sqrt {a^2+4 b} a+4 b\right ) \sqrt [4]{a x^3-b x^2}}-\frac {\sqrt {b} \sqrt {-a-\sqrt {a^2+4 b}} \sqrt {\frac {a x}{b}} \sqrt [4]{a x-b} \arctan \left (\frac {\sqrt {a} \sqrt {-a-\sqrt {a^2+4 b}} \sqrt [4]{a x-b}}{\sqrt [4]{2} \sqrt {b} \sqrt [4]{-a^2-\sqrt {a^2+4 b} a-2 b} \sqrt {\frac {a x}{b}}}\right )}{\sqrt [4]{2} \sqrt {a} \sqrt [4]{-a^2-\sqrt {a^2+4 b} a-2 b} \sqrt [4]{a x^3-b x^2}}-\frac {\sqrt {b} \sqrt {a+\sqrt {a^2+4 b}} \sqrt {\frac {a x}{b}} \sqrt [4]{a x-b} \arctan \left (\frac {\sqrt {a} \sqrt {a+\sqrt {a^2+4 b}} \sqrt [4]{a x-b}}{\sqrt [4]{2} \sqrt {b} \sqrt [4]{-a^2-\sqrt {a^2+4 b} a-2 b} \sqrt {\frac {a x}{b}}}\right )}{\sqrt [4]{2} \sqrt {a} \sqrt [4]{-a^2-\sqrt {a^2+4 b} a-2 b} \sqrt [4]{a x^3-b x^2}}-\frac {\sqrt {b} \sqrt {a-\sqrt {a^2+4 b}} \sqrt {\frac {a x}{b}} \sqrt [4]{a x-b} \arctan \left (\frac {\sqrt {a} \sqrt {a-\sqrt {a^2+4 b}} \sqrt [4]{a x-b}}{\sqrt [4]{2} \sqrt {b} \sqrt [4]{-a^2+\sqrt {a^2+4 b} a-2 b} \sqrt {\frac {a x}{b}}}\right )}{\sqrt [4]{2} \sqrt {a} \sqrt [4]{-a^2+\sqrt {a^2+4 b} a-2 b} \sqrt [4]{a x^3-b x^2}}-\frac {\sqrt {b} \sqrt {\sqrt {a^2+4 b}-a} \sqrt {\frac {a x}{b}} \sqrt [4]{a x-b} \arctan \left (\frac {\sqrt {a} \sqrt {\sqrt {a^2+4 b}-a} \sqrt [4]{a x-b}}{\sqrt [4]{2} \sqrt {b} \sqrt [4]{-a^2+\sqrt {a^2+4 b} a-2 b} \sqrt {\frac {a x}{b}}}\right )}{\sqrt [4]{2} \sqrt {a} \sqrt [4]{-a^2+\sqrt {a^2+4 b} a-2 b} \sqrt [4]{a x^3-b x^2}}-\frac {\left (a+\sqrt {a^2+4 b}\right ) \left (2 \sqrt {b}-\sqrt {2} \sqrt {-a^2-\sqrt {a^2+4 b} a-2 b}\right ) \sqrt [4]{a x-b} \sqrt {\frac {a x}{\left (\sqrt {b}+\sqrt {a x-b}\right )^2}} \left (\sqrt {b}+\sqrt {a x-b}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a x-b}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{b} \left (a^2+\sqrt {a^2+4 b} a+4 b\right ) \sqrt [4]{a x^3-b x^2}}-\frac {\left (a+\sqrt {a^2+4 b}\right ) \left (2 \sqrt {b}+\sqrt {2} \sqrt {-a^2-\sqrt {a^2+4 b} a-2 b}\right ) \sqrt [4]{a x-b} \sqrt {\frac {a x}{\left (\sqrt {b}+\sqrt {a x-b}\right )^2}} \left (\sqrt {b}+\sqrt {a x-b}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a x-b}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{b} \left (a^2+\sqrt {a^2+4 b} a+4 b\right ) \sqrt [4]{a x^3-b x^2}}-\frac {\left (a-\sqrt {a^2+4 b}\right ) \left (2 \sqrt {b}-\sqrt {-2 a^2+2 \sqrt {a^2+4 b} a-4 b}\right ) \sqrt [4]{a x-b} \sqrt {\frac {a x}{\left (\sqrt {b}+\sqrt {a x-b}\right )^2}} \left (\sqrt {b}+\sqrt {a x-b}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a x-b}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{b} \left (a^2-\sqrt {a^2+4 b} a+4 b\right ) \sqrt [4]{a x^3-b x^2}}-\frac {\left (a-\sqrt {a^2+4 b}\right ) \left (2 \sqrt {b}+\sqrt {-2 a^2+2 \sqrt {a^2+4 b} a-4 b}\right ) \sqrt [4]{a x-b} \sqrt {\frac {a x}{\left (\sqrt {b}+\sqrt {a x-b}\right )^2}} \left (\sqrt {b}+\sqrt {a x-b}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a x-b}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{b} \left (a^2-\sqrt {a^2+4 b} a+4 b\right ) \sqrt [4]{a x^3-b x^2}}+\frac {\left (a+\sqrt {a^2+4 b}\right ) \left (\sqrt {2} \sqrt {b}+\sqrt {-a^2-\sqrt {a^2+4 b} a-2 b}\right )^2 \sqrt [4]{a x-b} \sqrt {\frac {a x}{\left (\sqrt {b}+\sqrt {a x-b}\right )^2}} \left (\sqrt {b}+\sqrt {a x-b}\right ) \operatorname {EllipticPi}\left (-\frac {\left (\sqrt {2} \sqrt {b}-\sqrt {-a^2-\sqrt {a^2+4 b} a-2 b}\right )^2}{4 \sqrt {2} \sqrt {b} \sqrt {-a^2-\sqrt {a^2+4 b} a-2 b}},2 \arctan \left (\frac {\sqrt [4]{a x-b}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{2 \sqrt {2} \sqrt [4]{b} \sqrt {-a^2-\sqrt {a^2+4 b} a-2 b} \left (a^2+\sqrt {a^2+4 b} a+4 b\right ) \sqrt [4]{a x^3-b x^2}}+\frac {\left (a-\sqrt {a^2+4 b}\right ) \left (\sqrt {2} \sqrt {b}+\sqrt {-a^2+\sqrt {a^2+4 b} a-2 b}\right )^2 \sqrt [4]{a x-b} \sqrt {\frac {a x}{\left (\sqrt {b}+\sqrt {a x-b}\right )^2}} \left (\sqrt {b}+\sqrt {a x-b}\right ) \operatorname {EllipticPi}\left (-\frac {\left (\sqrt {2} \sqrt {b}-\sqrt {-a^2+\sqrt {a^2+4 b} a-2 b}\right )^2}{4 \sqrt {2} \sqrt {b} \sqrt {-a^2+\sqrt {a^2+4 b} a-2 b}},2 \arctan \left (\frac {\sqrt [4]{a x-b}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{2 \sqrt {2} \sqrt [4]{b} \left (a^2-\sqrt {a^2+4 b} a+4 b\right ) \sqrt {-a^2+\sqrt {a^2+4 b} a-2 b} \sqrt [4]{a x^3-b x^2}}-\frac {\left (a-\sqrt {a^2+4 b}\right ) \left (\sqrt {2} \sqrt {b}-\sqrt {-a^2+\sqrt {a^2+4 b} a-2 b}\right )^2 \sqrt [4]{a x-b} \sqrt {\frac {a x}{\left (\sqrt {b}+\sqrt {a x-b}\right )^2}} \left (\sqrt {b}+\sqrt {a x-b}\right ) \operatorname {EllipticPi}\left (\frac {\left (\sqrt {2} \sqrt {b}+\sqrt {-a^2+\sqrt {a^2+4 b} a-2 b}\right )^2}{4 \sqrt {2} \sqrt {b} \sqrt {-a^2+\sqrt {a^2+4 b} a-2 b}},2 \arctan \left (\frac {\sqrt [4]{a x-b}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{2 \sqrt {2} \sqrt [4]{b} \left (a^2-\sqrt {a^2+4 b} a+4 b\right ) \sqrt {-a^2+\sqrt {a^2+4 b} a-2 b} \sqrt [4]{a x^3-b x^2}} \]

[In]

Int[(-2*b + a*x)/((-b + a*x + x^2)*(-(b*x^2) + a*x^3)^(1/4)),x]

[Out]

-((Sqrt[b]*Sqrt[-a - Sqrt[a^2 + 4*b]]*Sqrt[(a*x)/b]*(-b + a*x)^(1/4)*ArcTan[(Sqrt[a]*Sqrt[-a - Sqrt[a^2 + 4*b]
]*(-b + a*x)^(1/4))/(2^(1/4)*Sqrt[b]*(-a^2 - 2*b - a*Sqrt[a^2 + 4*b])^(1/4)*Sqrt[(a*x)/b])])/(2^(1/4)*Sqrt[a]*
(-a^2 - 2*b - a*Sqrt[a^2 + 4*b])^(1/4)*(-(b*x^2) + a*x^3)^(1/4))) - (Sqrt[b]*Sqrt[a + Sqrt[a^2 + 4*b]]*Sqrt[(a
*x)/b]*(-b + a*x)^(1/4)*ArcTan[(Sqrt[a]*Sqrt[a + Sqrt[a^2 + 4*b]]*(-b + a*x)^(1/4))/(2^(1/4)*Sqrt[b]*(-a^2 - 2
*b - a*Sqrt[a^2 + 4*b])^(1/4)*Sqrt[(a*x)/b])])/(2^(1/4)*Sqrt[a]*(-a^2 - 2*b - a*Sqrt[a^2 + 4*b])^(1/4)*(-(b*x^
2) + a*x^3)^(1/4)) - (Sqrt[b]*Sqrt[a - Sqrt[a^2 + 4*b]]*Sqrt[(a*x)/b]*(-b + a*x)^(1/4)*ArcTan[(Sqrt[a]*Sqrt[a
- Sqrt[a^2 + 4*b]]*(-b + a*x)^(1/4))/(2^(1/4)*Sqrt[b]*(-a^2 - 2*b + a*Sqrt[a^2 + 4*b])^(1/4)*Sqrt[(a*x)/b])])/
(2^(1/4)*Sqrt[a]*(-a^2 - 2*b + a*Sqrt[a^2 + 4*b])^(1/4)*(-(b*x^2) + a*x^3)^(1/4)) - (Sqrt[b]*Sqrt[-a + Sqrt[a^
2 + 4*b]]*Sqrt[(a*x)/b]*(-b + a*x)^(1/4)*ArcTan[(Sqrt[a]*Sqrt[-a + Sqrt[a^2 + 4*b]]*(-b + a*x)^(1/4))/(2^(1/4)
*Sqrt[b]*(-a^2 - 2*b + a*Sqrt[a^2 + 4*b])^(1/4)*Sqrt[(a*x)/b])])/(2^(1/4)*Sqrt[a]*(-a^2 - 2*b + a*Sqrt[a^2 + 4
*b])^(1/4)*(-(b*x^2) + a*x^3)^(1/4)) - ((a + Sqrt[a^2 + 4*b])*(2*Sqrt[b] - Sqrt[2]*Sqrt[-a^2 - 2*b - a*Sqrt[a^
2 + 4*b]])*(-b + a*x)^(1/4)*Sqrt[(a*x)/(Sqrt[b] + Sqrt[-b + a*x])^2]*(Sqrt[b] + Sqrt[-b + a*x])*EllipticF[2*Ar
cTan[(-b + a*x)^(1/4)/b^(1/4)], 1/2])/(2*b^(1/4)*(a^2 + 4*b + a*Sqrt[a^2 + 4*b])*(-(b*x^2) + a*x^3)^(1/4)) - (
(a + Sqrt[a^2 + 4*b])*(2*Sqrt[b] + Sqrt[2]*Sqrt[-a^2 - 2*b - a*Sqrt[a^2 + 4*b]])*(-b + a*x)^(1/4)*Sqrt[(a*x)/(
Sqrt[b] + Sqrt[-b + a*x])^2]*(Sqrt[b] + Sqrt[-b + a*x])*EllipticF[2*ArcTan[(-b + a*x)^(1/4)/b^(1/4)], 1/2])/(2
*b^(1/4)*(a^2 + 4*b + a*Sqrt[a^2 + 4*b])*(-(b*x^2) + a*x^3)^(1/4)) - ((a - Sqrt[a^2 + 4*b])*(2*Sqrt[b] - Sqrt[
-2*a^2 - 4*b + 2*a*Sqrt[a^2 + 4*b]])*(-b + a*x)^(1/4)*Sqrt[(a*x)/(Sqrt[b] + Sqrt[-b + a*x])^2]*(Sqrt[b] + Sqrt
[-b + a*x])*EllipticF[2*ArcTan[(-b + a*x)^(1/4)/b^(1/4)], 1/2])/(2*b^(1/4)*(a^2 + 4*b - a*Sqrt[a^2 + 4*b])*(-(
b*x^2) + a*x^3)^(1/4)) - ((a - Sqrt[a^2 + 4*b])*(2*Sqrt[b] + Sqrt[-2*a^2 - 4*b + 2*a*Sqrt[a^2 + 4*b]])*(-b + a
*x)^(1/4)*Sqrt[(a*x)/(Sqrt[b] + Sqrt[-b + a*x])^2]*(Sqrt[b] + Sqrt[-b + a*x])*EllipticF[2*ArcTan[(-b + a*x)^(1
/4)/b^(1/4)], 1/2])/(2*b^(1/4)*(a^2 + 4*b - a*Sqrt[a^2 + 4*b])*(-(b*x^2) + a*x^3)^(1/4)) + ((a + Sqrt[a^2 + 4*
b])*(Sqrt[2]*Sqrt[b] + Sqrt[-a^2 - 2*b - a*Sqrt[a^2 + 4*b]])^2*(-b + a*x)^(1/4)*Sqrt[(a*x)/(Sqrt[b] + Sqrt[-b
+ a*x])^2]*(Sqrt[b] + Sqrt[-b + a*x])*EllipticPi[-1/4*(Sqrt[2]*Sqrt[b] - Sqrt[-a^2 - 2*b - a*Sqrt[a^2 + 4*b]])
^2/(Sqrt[2]*Sqrt[b]*Sqrt[-a^2 - 2*b - a*Sqrt[a^2 + 4*b]]), 2*ArcTan[(-b + a*x)^(1/4)/b^(1/4)], 1/2])/(2*Sqrt[2
]*b^(1/4)*Sqrt[-a^2 - 2*b - a*Sqrt[a^2 + 4*b]]*(a^2 + 4*b + a*Sqrt[a^2 + 4*b])*(-(b*x^2) + a*x^3)^(1/4)) - ((a
+ Sqrt[a^2 + 4*b])*(Sqrt[2]*Sqrt[b] - Sqrt[-a^2 - 2*b - a*Sqrt[a^2 + 4*b]])^2*(-b + a*x)^(1/4)*Sqrt[(a*x)/(Sq
rt[b] + Sqrt[-b + a*x])^2]*(Sqrt[b] + Sqrt[-b + a*x])*EllipticPi[(Sqrt[2]*Sqrt[b] + Sqrt[-a^2 - 2*b - a*Sqrt[a
^2 + 4*b]])^2/(4*Sqrt[2]*Sqrt[b]*Sqrt[-a^2 - 2*b - a*Sqrt[a^2 + 4*b]]), 2*ArcTan[(-b + a*x)^(1/4)/b^(1/4)], 1/
2])/(2*Sqrt[2]*b^(1/4)*Sqrt[-a^2 - 2*b - a*Sqrt[a^2 + 4*b]]*(a^2 + 4*b + a*Sqrt[a^2 + 4*b])*(-(b*x^2) + a*x^3)
^(1/4)) + ((a - Sqrt[a^2 + 4*b])*(Sqrt[2]*Sqrt[b] + Sqrt[-a^2 - 2*b + a*Sqrt[a^2 + 4*b]])^2*(-b + a*x)^(1/4)*S
qrt[(a*x)/(Sqrt[b] + Sqrt[-b + a*x])^2]*(Sqrt[b] + Sqrt[-b + a*x])*EllipticPi[-1/4*(Sqrt[2]*Sqrt[b] - Sqrt[-a^
2 - 2*b + a*Sqrt[a^2 + 4*b]])^2/(Sqrt[2]*Sqrt[b]*Sqrt[-a^2 - 2*b + a*Sqrt[a^2 + 4*b]]), 2*ArcTan[(-b + a*x)^(1
/4)/b^(1/4)], 1/2])/(2*Sqrt[2]*b^(1/4)*(a^2 + 4*b - a*Sqrt[a^2 + 4*b])*Sqrt[-a^2 - 2*b + a*Sqrt[a^2 + 4*b]]*(-
(b*x^2) + a*x^3)^(1/4)) - ((a - Sqrt[a^2 + 4*b])*(Sqrt[2]*Sqrt[b] - Sqrt[-a^2 - 2*b + a*Sqrt[a^2 + 4*b]])^2*(-
b + a*x)^(1/4)*Sqrt[(a*x)/(Sqrt[b] + Sqrt[-b + a*x])^2]*(Sqrt[b] + Sqrt[-b + a*x])*EllipticPi[(Sqrt[2]*Sqrt[b]
+ Sqrt[-a^2 - 2*b + a*Sqrt[a^2 + 4*b]])^2/(4*Sqrt[2]*Sqrt[b]*Sqrt[-a^2 - 2*b + a*Sqrt[a^2 + 4*b]]), 2*ArcTan[
(-b + a*x)^(1/4)/b^(1/4)], 1/2])/(2*Sqrt[2]*b^(1/4)*(a^2 + 4*b - a*Sqrt[a^2 + 4*b])*Sqrt[-a^2 - 2*b + a*Sqrt[a
^2 + 4*b]]*(-(b*x^2) + a*x^3)^(1/4))

Rule 107

Int[1/(((a_.) + (b_.)*(x_))*Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(1/4)), x_Symbol] :> Dist[-4, Subst[
Int[x^2/((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 108

Int[1/(((a_.) + (b_.)*(x_))*Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(1/4)), x_Symbol] :> Dist[Sqrt[(-f)*
((c + d*x)/(d*e - c*f))]/Sqrt[c + d*x], Int[1/((a + b*x)*Sqrt[(-c)*(f/(d*e - c*f)) - d*f*(x/(d*e - c*f))]*(e +
f*x)^(1/4)), x], 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 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 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 6860

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

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

Mathematica [A] (verified)

Time = 0.28 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.02 \[ \int \frac {-2 b+a x}{\left (-b+a x+x^2\right ) \sqrt [4]{-b x^2+a x^3}} \, dx=\frac {\sqrt {2} \sqrt {x} \sqrt [4]{-b+a x} \left (-\arctan \left (\frac {-x+\sqrt {-b+a x}}{\sqrt {2} \sqrt {x} \sqrt [4]{-b+a x}}\right )+\text {arctanh}\left (\frac {\sqrt {2} \sqrt {x} \sqrt [4]{-b+a x}}{x+\sqrt {-b+a x}}\right )\right )}{\sqrt [4]{x^2 (-b+a x)}} \]

[In]

Integrate[(-2*b + a*x)/((-b + a*x + x^2)*(-(b*x^2) + a*x^3)^(1/4)),x]

[Out]

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

Maple [F]

\[\int \frac {a x -2 b}{\left (a x +x^{2}-b \right ) \left (a \,x^{3}-b \,x^{2}\right )^{\frac {1}{4}}}d x\]

[In]

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

[Out]

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

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {-2 b+a x}{\left (-b+a x+x^2\right ) \sqrt [4]{-b x^2+a x^3}} \, dx=\int { \frac {a x - 2 \, b}{{\left (a x^{3} - b x^{2}\right )}^{\frac {1}{4}} {\left (a x + x^{2} - b\right )}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {-2 b+a x}{\left (-b+a x+x^2\right ) \sqrt [4]{-b x^2+a x^3}} \, dx=\int { \frac {a x - 2 \, b}{{\left (a x^{3} - b x^{2}\right )}^{\frac {1}{4}} {\left (a x + x^{2} - b\right )}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {-2 b+a x}{\left (-b+a x+x^2\right ) \sqrt [4]{-b x^2+a x^3}} \, dx=\int -\frac {2\,b-a\,x}{{\left (a\,x^3-b\,x^2\right )}^{1/4}\,\left (x^2+a\,x-b\right )} \,d x \]

[In]

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

[Out]

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