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

Optimal. Leaf size=113 \[ \frac {2 \tanh ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{x \left (2 a b+b^2\right )-a b^2+x^2 (-a-2 b)+x^3}}{b-x}\right )}{d^{3/4}}-\frac {2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{x \left (2 a b+b^2\right )-a b^2+x^2 (-a-2 b)+x^3}}{b-x}\right )}{d^{3/4}} \]

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Rubi [C]  time = 65.52, antiderivative size = 3261, normalized size of antiderivative = 28.86, number of steps used = 21, number of rules used = 8, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {6719, 6728, 107, 106, 490, 1217, 220, 1707}

result too large to display

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

Rule 106

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 107

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 220

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)]*EllipticF[2*ArcTan[q*x], 1/2])/(2*q*Sqrt[a + b*x^4]), x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rule 490

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), In
t[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 1217

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 1707

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)]*EllipticPi[Cancel[-((B*d - A*e)^2
/(4*d*e*A*B))], 2*ArcTan[q*x], 1/2])/(4*d*e*A*q*Sqrt[a + c*x^4]), 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 6719

Int[(u_.)*((a_.)*(v_)^(m_.)*(w_)^(n_.))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a*v^m*w^n)^FracPart[p])/(v^(m*F
racPart[p])*w^(n*FracPart[p])), Int[u*v^(m*p)*w^(n*p), x], x] /; FreeQ[{a, m, n, p}, x] &&  !IntegerQ[p] &&  !
FreeQ[v, x] &&  !FreeQ[w, x]

Rule 6728

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*} \int \frac {-2 a+b+x}{\sqrt [4]{(-a+x) (-b+x)^2} \left (b^2+a d-(2 b+d) x+x^2\right )} \, dx &=\frac {\left (\sqrt [4]{-a+x} \sqrt {-b+x}\right ) \int \frac {-2 a+b+x}{\sqrt [4]{-a+x} \sqrt {-b+x} \left (b^2+a d-(2 b+d) x+x^2\right )} \, dx}{\sqrt [4]{(-a+x) (-b+x)^2}}\\ &=\frac {\left (\sqrt [4]{-a+x} \sqrt {-b+x}\right ) \int \left (\frac {1+\frac {\sqrt {-4 a+4 b+d}}{\sqrt {d}}}{\sqrt [4]{-a+x} \sqrt {-b+x} \left (-2 b-d-\sqrt {d} \sqrt {-4 a+4 b+d}+2 x\right )}+\frac {1-\frac {\sqrt {-4 a+4 b+d}}{\sqrt {d}}}{\sqrt [4]{-a+x} \sqrt {-b+x} \left (-2 b-d+\sqrt {d} \sqrt {-4 a+4 b+d}+2 x\right )}\right ) \, dx}{\sqrt [4]{(-a+x) (-b+x)^2}}\\ &=\frac {\left (\left (1-\frac {\sqrt {-4 a+4 b+d}}{\sqrt {d}}\right ) \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \int \frac {1}{\sqrt [4]{-a+x} \sqrt {-b+x} \left (-2 b-d+\sqrt {d} \sqrt {-4 a+4 b+d}+2 x\right )} \, dx}{\sqrt [4]{(-a+x) (-b+x)^2}}+\frac {\left (\left (1+\frac {\sqrt {-4 a+4 b+d}}{\sqrt {d}}\right ) \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \int \frac {1}{\sqrt [4]{-a+x} \sqrt {-b+x} \left (-2 b-d-\sqrt {d} \sqrt {-4 a+4 b+d}+2 x\right )} \, dx}{\sqrt [4]{(-a+x) (-b+x)^2}}\\ &=\frac {\left (\left (1-\frac {\sqrt {-4 a+4 b+d}}{\sqrt {d}}\right ) \sqrt [4]{-a+x} \sqrt {-\frac {-b+x}{-a+b}}\right ) \int \frac {1}{\sqrt [4]{-a+x} \left (-2 b-d+\sqrt {d} \sqrt {-4 a+4 b+d}+2 x\right ) \sqrt {\frac {b}{-a+b}-\frac {x}{-a+b}}} \, dx}{\sqrt [4]{(-a+x) (-b+x)^2}}+\frac {\left (\left (1+\frac {\sqrt {-4 a+4 b+d}}{\sqrt {d}}\right ) \sqrt [4]{-a+x} \sqrt {-\frac {-b+x}{-a+b}}\right ) \int \frac {1}{\sqrt [4]{-a+x} \left (-2 b-d-\sqrt {d} \sqrt {-4 a+4 b+d}+2 x\right ) \sqrt {\frac {b}{-a+b}-\frac {x}{-a+b}}} \, dx}{\sqrt [4]{(-a+x) (-b+x)^2}}\\ &=-\frac {\left (4 \left (1-\frac {\sqrt {-4 a+4 b+d}}{\sqrt {d}}\right ) \sqrt [4]{-a+x} \sqrt {-\frac {-b+x}{-a+b}}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (-2 a+2 b+d-\sqrt {d} \sqrt {-4 a+4 b+d}-2 x^4\right ) \sqrt {-\frac {a}{-a+b}+\frac {b}{-a+b}-\frac {x^4}{-a+b}}} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x)^2}}-\frac {\left (4 \left (1+\frac {\sqrt {-4 a+4 b+d}}{\sqrt {d}}\right ) \sqrt [4]{-a+x} \sqrt {-\frac {-b+x}{-a+b}}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (-2 a+2 b+d+\sqrt {d} \sqrt {-4 a+4 b+d}-2 x^4\right ) \sqrt {-\frac {a}{-a+b}+\frac {b}{-a+b}-\frac {x^4}{-a+b}}} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x)^2}}\\ &=-\frac {\left (\sqrt {2} \left (1-\frac {\sqrt {-4 a+4 b+d}}{\sqrt {d}}\right ) \sqrt [4]{-a+x} \sqrt {-\frac {-b+x}{-a+b}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\sqrt {-2 a+2 b+d-\sqrt {d} \sqrt {-4 a+4 b+d}}-\sqrt {2} x^2\right ) \sqrt {-\frac {a}{-a+b}+\frac {b}{-a+b}-\frac {x^4}{-a+b}}} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x)^2}}+\frac {\left (\sqrt {2} \left (1-\frac {\sqrt {-4 a+4 b+d}}{\sqrt {d}}\right ) \sqrt [4]{-a+x} \sqrt {-\frac {-b+x}{-a+b}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\sqrt {-2 a+2 b+d-\sqrt {d} \sqrt {-4 a+4 b+d}}+\sqrt {2} x^2\right ) \sqrt {-\frac {a}{-a+b}+\frac {b}{-a+b}-\frac {x^4}{-a+b}}} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x)^2}}-\frac {\left (\sqrt {2} \left (1+\frac {\sqrt {-4 a+4 b+d}}{\sqrt {d}}\right ) \sqrt [4]{-a+x} \sqrt {-\frac {-b+x}{-a+b}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\sqrt {-2 a+2 b+d+\sqrt {d} \sqrt {-4 a+4 b+d}}-\sqrt {2} x^2\right ) \sqrt {-\frac {a}{-a+b}+\frac {b}{-a+b}-\frac {x^4}{-a+b}}} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x)^2}}+\frac {\left (\sqrt {2} \left (1+\frac {\sqrt {-4 a+4 b+d}}{\sqrt {d}}\right ) \sqrt [4]{-a+x} \sqrt {-\frac {-b+x}{-a+b}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\sqrt {-2 a+2 b+d+\sqrt {d} \sqrt {-4 a+4 b+d}}+\sqrt {2} x^2\right ) \sqrt {-\frac {a}{-a+b}+\frac {b}{-a+b}-\frac {x^4}{-a+b}}} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x)^2}}\\ &=-\frac {\left (\sqrt {2} \left (1-\frac {\sqrt {-4 a+4 b+d}}{\sqrt {d}}\right ) \left (\sqrt {2} a-\sqrt {2} b-\sqrt {a-b} \sqrt {-2 a+2 b+d-\sqrt {d} \sqrt {-4 a+4 b+d}}\right ) \sqrt [4]{-a+x} \sqrt {-\frac {-b+x}{-a+b}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-\frac {a}{-a+b}+\frac {b}{-a+b}-\frac {x^4}{-a+b}}} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt {a-b} \left (4 a-4 b-d+\sqrt {d} \sqrt {-4 a+4 b+d}\right ) \sqrt [4]{(-a+x) (-b+x)^2}}-\frac {\left (\sqrt {2} \left (1-\frac {\sqrt {-4 a+4 b+d}}{\sqrt {d}}\right ) \left (2 a-2 b-\sqrt {2} \sqrt {a-b} \sqrt {-2 a+2 b+d-\sqrt {d} \sqrt {-4 a+4 b+d}}\right ) \sqrt [4]{-a+x} \sqrt {-\frac {-b+x}{-a+b}}\right ) \operatorname {Subst}\left (\int \frac {1+\frac {x^2}{\sqrt {a-b}}}{\left (\sqrt {-2 a+2 b+d-\sqrt {d} \sqrt {-4 a+4 b+d}}-\sqrt {2} x^2\right ) \sqrt {-\frac {a}{-a+b}+\frac {b}{-a+b}-\frac {x^4}{-a+b}}} \, dx,x,\sqrt [4]{-a+x}\right )}{\left (4 a-4 b-d+\sqrt {d} \sqrt {-4 a+4 b+d}\right ) \sqrt [4]{(-a+x) (-b+x)^2}}+\frac {\left (\sqrt {2} \left (1-\frac {\sqrt {-4 a+4 b+d}}{\sqrt {d}}\right ) \left (2 a-2 b+\sqrt {2} \sqrt {a-b} \sqrt {-2 a+2 b+d-\sqrt {d} \sqrt {-4 a+4 b+d}}\right ) \sqrt [4]{-a+x} \sqrt {-\frac {-b+x}{-a+b}}\right ) \operatorname {Subst}\left (\int \frac {1+\frac {x^2}{\sqrt {a-b}}}{\left (\sqrt {-2 a+2 b+d-\sqrt {d} \sqrt {-4 a+4 b+d}}+\sqrt {2} x^2\right ) \sqrt {-\frac {a}{-a+b}+\frac {b}{-a+b}-\frac {x^4}{-a+b}}} \, dx,x,\sqrt [4]{-a+x}\right )}{\left (4 a-4 b-d+\sqrt {d} \sqrt {-4 a+4 b+d}\right ) \sqrt [4]{(-a+x) (-b+x)^2}}+\frac {\left (\sqrt {2} \left (1-\frac {\sqrt {-4 a+4 b+d}}{\sqrt {d}}\right ) \left (\frac {\sqrt {2} \left (-\frac {a}{-a+b}+\frac {b}{-a+b}\right )}{\sqrt {a-b}}-\frac {\sqrt {-2 a+2 b+d-\sqrt {d} \sqrt {-4 a+4 b+d}}}{-a+b}\right ) \sqrt [4]{-a+x} \sqrt {-\frac {-b+x}{-a+b}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-\frac {a}{-a+b}+\frac {b}{-a+b}-\frac {x^4}{-a+b}}} \, dx,x,\sqrt [4]{-a+x}\right )}{\left (-2 \left (-\frac {a}{-a+b}+\frac {b}{-a+b}\right )-\frac {-2 a+2 b+d-\sqrt {d} \sqrt {-4 a+4 b+d}}{-a+b}\right ) \sqrt [4]{(-a+x) (-b+x)^2}}-\frac {\left (\sqrt {2} \left (1+\frac {\sqrt {-4 a+4 b+d}}{\sqrt {d}}\right ) \left (\sqrt {2} a-\sqrt {2} b-\sqrt {a-b} \sqrt {-2 a+2 b+d+\sqrt {d} \sqrt {-4 a+4 b+d}}\right ) \sqrt [4]{-a+x} \sqrt {-\frac {-b+x}{-a+b}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-\frac {a}{-a+b}+\frac {b}{-a+b}-\frac {x^4}{-a+b}}} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt {a-b} \left (4 a-4 b-d-\sqrt {d} \sqrt {-4 a+4 b+d}\right ) \sqrt [4]{(-a+x) (-b+x)^2}}-\frac {\left (\sqrt {2} \left (1+\frac {\sqrt {-4 a+4 b+d}}{\sqrt {d}}\right ) \left (\sqrt {2} a-\sqrt {2} b+\sqrt {a-b} \sqrt {-2 a+2 b+d+\sqrt {d} \sqrt {-4 a+4 b+d}}\right ) \sqrt [4]{-a+x} \sqrt {-\frac {-b+x}{-a+b}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-\frac {a}{-a+b}+\frac {b}{-a+b}-\frac {x^4}{-a+b}}} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt {a-b} \left (4 a-4 b-d-\sqrt {d} \sqrt {-4 a+4 b+d}\right ) \sqrt [4]{(-a+x) (-b+x)^2}}-\frac {\left (\sqrt {2} \left (1+\frac {\sqrt {-4 a+4 b+d}}{\sqrt {d}}\right ) \left (2 a-2 b-\sqrt {2} \sqrt {a-b} \sqrt {-2 a+2 b+d+\sqrt {d} \sqrt {-4 a+4 b+d}}\right ) \sqrt [4]{-a+x} \sqrt {-\frac {-b+x}{-a+b}}\right ) \operatorname {Subst}\left (\int \frac {1+\frac {x^2}{\sqrt {a-b}}}{\left (\sqrt {-2 a+2 b+d+\sqrt {d} \sqrt {-4 a+4 b+d}}-\sqrt {2} x^2\right ) \sqrt {-\frac {a}{-a+b}+\frac {b}{-a+b}-\frac {x^4}{-a+b}}} \, dx,x,\sqrt [4]{-a+x}\right )}{\left (4 a-4 b-d-\sqrt {d} \sqrt {-4 a+4 b+d}\right ) \sqrt [4]{(-a+x) (-b+x)^2}}+\frac {\left (\sqrt {2} \left (1+\frac {\sqrt {-4 a+4 b+d}}{\sqrt {d}}\right ) \left (2 a-2 b+\sqrt {2} \sqrt {a-b} \sqrt {-2 a+2 b+d+\sqrt {d} \sqrt {-4 a+4 b+d}}\right ) \sqrt [4]{-a+x} \sqrt {-\frac {-b+x}{-a+b}}\right ) \operatorname {Subst}\left (\int \frac {1+\frac {x^2}{\sqrt {a-b}}}{\left (\sqrt {-2 a+2 b+d+\sqrt {d} \sqrt {-4 a+4 b+d}}+\sqrt {2} x^2\right ) \sqrt {-\frac {a}{-a+b}+\frac {b}{-a+b}-\frac {x^4}{-a+b}}} \, dx,x,\sqrt [4]{-a+x}\right )}{\left (4 a-4 b-d-\sqrt {d} \sqrt {-4 a+4 b+d}\right ) \sqrt [4]{(-a+x) (-b+x)^2}}\\ &=\frac {\sqrt {\sqrt {d}-\sqrt {-4 a+4 b+d}} \left (\sqrt {2} \sqrt {a-b}-\sqrt {-2 a+2 b+d-\sqrt {d} \sqrt {-4 a+4 b+d}}\right ) \left (2 a-2 b+\sqrt {2} \sqrt {a-b} \sqrt {-2 a+2 b+d-\sqrt {d} \sqrt {-4 a+4 b+d}}\right ) \sqrt {-\frac {b-x}{a-b}} \sqrt [4]{-a+x} \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {\sqrt {d}-\sqrt {-4 a+4 b+d}} \sqrt [4]{-a+x}}{\sqrt [4]{2} \sqrt {a-b} \sqrt [4]{-2 a+2 b+d-\sqrt {d} \sqrt {-4 a+4 b+d}} \sqrt {-\frac {b-x}{a-b}}}\right )}{2^{3/4} d^{3/4} \sqrt [4]{-2 a+2 b+d-\sqrt {d} \sqrt {-4 a+4 b+d}} \left (4 a-4 b-d+\sqrt {d} \sqrt {-4 a+4 b+d}\right ) \sqrt [4]{-\left ((a-x) (b-x)^2\right )}}+\frac {\sqrt {-\sqrt {d}+\sqrt {-4 a+4 b+d}} \left (\sqrt {2} \sqrt {a-b}+\sqrt {-2 a+2 b+d-\sqrt {d} \sqrt {-4 a+4 b+d}}\right ) \left (2 a-2 b-\sqrt {2} \sqrt {a-b} \sqrt {-2 a+2 b+d-\sqrt {d} \sqrt {-4 a+4 b+d}}\right ) \sqrt {-\frac {b-x}{a-b}} \sqrt [4]{-a+x} \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {-\sqrt {d}+\sqrt {-4 a+4 b+d}} \sqrt [4]{-a+x}}{\sqrt [4]{2} \sqrt {a-b} \sqrt [4]{-2 a+2 b+d-\sqrt {d} \sqrt {-4 a+4 b+d}} \sqrt {-\frac {b-x}{a-b}}}\right )}{2^{3/4} d^{3/4} \sqrt [4]{-2 a+2 b+d-\sqrt {d} \sqrt {-4 a+4 b+d}} \left (4 a-4 b-d+\sqrt {d} \sqrt {-4 a+4 b+d}\right ) \sqrt [4]{-\left ((a-x) (b-x)^2\right )}}+\frac {\sqrt {\sqrt {d}+\sqrt {-4 a+4 b+d}} \left (\sqrt {2} \sqrt {a-b}-\sqrt {-2 a+2 b+d+\sqrt {d} \sqrt {-4 a+4 b+d}}\right ) \left (2 a-2 b+\sqrt {2} \sqrt {a-b} \sqrt {-2 a+2 b+d+\sqrt {d} \sqrt {-4 a+4 b+d}}\right ) \sqrt {-\frac {b-x}{a-b}} \sqrt [4]{-a+x} \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {\sqrt {d}+\sqrt {-4 a+4 b+d}} \sqrt [4]{-a+x}}{\sqrt [4]{2} \sqrt {a-b} \sqrt [4]{-2 a+2 b+d+\sqrt {d} \sqrt {-4 a+4 b+d}} \sqrt {-\frac {b-x}{a-b}}}\right )}{2^{3/4} d^{3/4} \left (4 a-4 b-d-\sqrt {d} \sqrt {-4 a+4 b+d}\right ) \sqrt [4]{-2 a+2 b+d+\sqrt {d} \sqrt {-4 a+4 b+d}} \sqrt [4]{-\left ((a-x) (b-x)^2\right )}}-\frac {\sqrt {-a+b} \sqrt {\sqrt {d}+\sqrt {-4 a+4 b+d}} \left (\sqrt {2} \sqrt {a-b}+\sqrt {-2 a+2 b+d+\sqrt {d} \sqrt {-4 a+4 b+d}}\right ) \left (2 a-2 b-\sqrt {2} \sqrt {a-b} \sqrt {-2 a+2 b+d+\sqrt {d} \sqrt {-4 a+4 b+d}}\right ) \sqrt {-\frac {b-x}{a-b}} \sqrt [4]{-a+x} \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {\sqrt {d}+\sqrt {-4 a+4 b+d}} \sqrt [4]{-a+x}}{\sqrt [4]{2} \sqrt {-a+b} \sqrt [4]{-2 a+2 b+d+\sqrt {d} \sqrt {-4 a+4 b+d}} \sqrt {-\frac {b-x}{a-b}}}\right )}{2^{3/4} \sqrt {a-b} d^{3/4} \left (4 a-4 b-d-\sqrt {d} \sqrt {-4 a+4 b+d}\right ) \sqrt [4]{-2 a+2 b+d+\sqrt {d} \sqrt {-4 a+4 b+d}} \sqrt [4]{-\left ((a-x) (b-x)^2\right )}}-\frac {\left (\sqrt {d}-\sqrt {-4 a+4 b+d}\right ) \left (2 a-\sqrt {2} \left (\sqrt {2} b-\sqrt {a-b} \sqrt {-2 a+2 b+d-\sqrt {d} \sqrt {-4 a+4 b+d}}\right )\right ) \sqrt [4]{-a+x} \sqrt {-\frac {b-x}{(a-b) \left (1+\frac {\sqrt {-a+x}}{\sqrt {a-b}}\right )^2}} \left (1+\frac {\sqrt {-a+x}}{\sqrt {a-b}}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{-a+x}}{\sqrt [4]{a-b}}\right )|\frac {1}{2}\right )}{2 \sqrt [4]{a-b} \sqrt {d} \left (4 a-4 b-d+\sqrt {d} \sqrt {-4 a+4 b+d}\right ) \sqrt [4]{-\left ((a-x) (b-x)^2\right )}}-\frac {\left (1-\frac {\sqrt {-4 a+4 b+d}}{\sqrt {d}}\right ) \left (2 a-\sqrt {2} \left (\sqrt {2} b+\sqrt {a-b} \sqrt {-2 a+2 b+d-\sqrt {d} \sqrt {-4 a+4 b+d}}\right )\right ) \sqrt [4]{-a+x} \sqrt {-\frac {b-x}{(a-b) \left (1+\frac {\sqrt {-a+x}}{\sqrt {a-b}}\right )^2}} \left (1+\frac {\sqrt {-a+x}}{\sqrt {a-b}}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{-a+x}}{\sqrt [4]{a-b}}\right )|\frac {1}{2}\right )}{2 \sqrt [4]{a-b} \left (4 a-4 b-d+\sqrt {d} \sqrt {-4 a+4 b+d}\right ) \sqrt [4]{-\left ((a-x) (b-x)^2\right )}}-\frac {\left (1+\frac {\sqrt {-4 a+4 b+d}}{\sqrt {d}}\right ) \left (2 a-\sqrt {2} \left (\sqrt {2} b-\sqrt {a-b} \sqrt {-2 a+2 b+d+\sqrt {d} \sqrt {-4 a+4 b+d}}\right )\right ) \sqrt [4]{-a+x} \sqrt {-\frac {b-x}{(a-b) \left (1+\frac {\sqrt {-a+x}}{\sqrt {a-b}}\right )^2}} \left (1+\frac {\sqrt {-a+x}}{\sqrt {a-b}}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{-a+x}}{\sqrt [4]{a-b}}\right )|\frac {1}{2}\right )}{2 \sqrt [4]{a-b} \left (4 a-4 b-d-\sqrt {d} \sqrt {-4 a+4 b+d}\right ) \sqrt [4]{-\left ((a-x) (b-x)^2\right )}}-\frac {\left (1+\frac {\sqrt {-4 a+4 b+d}}{\sqrt {d}}\right ) \left (2 a-\sqrt {2} \left (\sqrt {2} b+\sqrt {a-b} \sqrt {-2 a+2 b+d+\sqrt {d} \sqrt {-4 a+4 b+d}}\right )\right ) \sqrt [4]{-a+x} \sqrt {-\frac {b-x}{(a-b) \left (1+\frac {\sqrt {-a+x}}{\sqrt {a-b}}\right )^2}} \left (1+\frac {\sqrt {-a+x}}{\sqrt {a-b}}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{-a+x}}{\sqrt [4]{a-b}}\right )|\frac {1}{2}\right )}{2 \sqrt [4]{a-b} \left (4 a-4 b-d-\sqrt {d} \sqrt {-4 a+4 b+d}\right ) \sqrt [4]{-\left ((a-x) (b-x)^2\right )}}+\frac {\left (1-\frac {\sqrt {-4 a+4 b+d}}{\sqrt {d}}\right ) \left (1+\frac {\sqrt {2} \sqrt {a-b}}{\sqrt {-2 a+2 b+d-\sqrt {d} \sqrt {-4 a+4 b+d}}}\right ) \left (2 a-2 b+\sqrt {2} \sqrt {a-b} \sqrt {-2 a+2 b+d-\sqrt {d} \sqrt {-4 a+4 b+d}}\right ) \sqrt [4]{-a+x} \sqrt {-\frac {b-x}{(a-b) \left (1+\frac {\sqrt {-a+x}}{\sqrt {a-b}}\right )^2}} \left (1+\frac {\sqrt {-a+x}}{\sqrt {a-b}}\right ) \Pi \left (-\frac {\left (\sqrt {2} \sqrt {a-b}-\sqrt {-2 a+2 b+d-\sqrt {d} \sqrt {-4 a+4 b+d}}\right )^2}{4 \sqrt {2} \sqrt {a-b} \sqrt {-2 a+2 b+d-\sqrt {d} \sqrt {-4 a+4 b+d}}};2 \tan ^{-1}\left (\frac {\sqrt [4]{-a+x}}{\sqrt [4]{a-b}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{a-b} \left (4 a-4 b-d+\sqrt {d} \sqrt {-4 a+4 b+d}\right ) \sqrt [4]{-\left ((a-x) (b-x)^2\right )}}+\frac {\left (1-\frac {\sqrt {-4 a+4 b+d}}{\sqrt {d}}\right ) \left (1-\frac {\sqrt {2} \sqrt {a-b}}{\sqrt {-2 a+2 b+d-\sqrt {d} \sqrt {-4 a+4 b+d}}}\right ) \left (2 a-2 b-\sqrt {2} \sqrt {a-b} \sqrt {-2 a+2 b+d-\sqrt {d} \sqrt {-4 a+4 b+d}}\right ) \sqrt [4]{-a+x} \sqrt {-\frac {b-x}{(a-b) \left (1+\frac {\sqrt {-a+x}}{\sqrt {a-b}}\right )^2}} \left (1+\frac {\sqrt {-a+x}}{\sqrt {a-b}}\right ) \Pi \left (\frac {\left (\sqrt {2} \sqrt {a-b}+\sqrt {-2 a+2 b+d-\sqrt {d} \sqrt {-4 a+4 b+d}}\right )^2}{4 \sqrt {2} \sqrt {a-b} \sqrt {-2 a+2 b+d-\sqrt {d} \sqrt {-4 a+4 b+d}}};2 \tan ^{-1}\left (\frac {\sqrt [4]{-a+x}}{\sqrt [4]{a-b}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{a-b} \left (4 a-4 b-d+\sqrt {d} \sqrt {-4 a+4 b+d}\right ) \sqrt [4]{-\left ((a-x) (b-x)^2\right )}}+\frac {\left (1+\frac {\sqrt {-4 a+4 b+d}}{\sqrt {d}}\right ) \left (1+\frac {\sqrt {2} \sqrt {a-b}}{\sqrt {-2 a+2 b+d+\sqrt {d} \sqrt {-4 a+4 b+d}}}\right ) \left (2 a-2 b+\sqrt {2} \sqrt {a-b} \sqrt {-2 a+2 b+d+\sqrt {d} \sqrt {-4 a+4 b+d}}\right ) \sqrt [4]{-a+x} \sqrt {-\frac {b-x}{(a-b) \left (1+\frac {\sqrt {-a+x}}{\sqrt {a-b}}\right )^2}} \left (1+\frac {\sqrt {-a+x}}{\sqrt {a-b}}\right ) \Pi \left (-\frac {\left (\sqrt {2} \sqrt {a-b}-\sqrt {-2 a+2 b+d+\sqrt {d} \sqrt {-4 a+4 b+d}}\right )^2}{4 \sqrt {2} \sqrt {a-b} \sqrt {-2 a+2 b+d+\sqrt {d} \sqrt {-4 a+4 b+d}}};2 \tan ^{-1}\left (\frac {\sqrt [4]{-a+x}}{\sqrt [4]{a-b}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{a-b} \left (4 a-4 b-d-\sqrt {d} \sqrt {-4 a+4 b+d}\right ) \sqrt [4]{-\left ((a-x) (b-x)^2\right )}}+\frac {\left (1+\frac {\sqrt {-4 a+4 b+d}}{\sqrt {d}}\right ) \left (1-\frac {\sqrt {2} \sqrt {a-b}}{\sqrt {-2 a+2 b+d+\sqrt {d} \sqrt {-4 a+4 b+d}}}\right ) \left (2 a-2 b-\sqrt {2} \sqrt {a-b} \sqrt {-2 a+2 b+d+\sqrt {d} \sqrt {-4 a+4 b+d}}\right ) \sqrt [4]{-a+x} \sqrt {-\frac {b-x}{(a-b) \left (1+\frac {\sqrt {-a+x}}{\sqrt {a-b}}\right )^2}} \left (1+\frac {\sqrt {-a+x}}{\sqrt {a-b}}\right ) \Pi \left (\frac {\left (\sqrt {2} \sqrt {a-b}+\sqrt {-2 a+2 b+d+\sqrt {d} \sqrt {-4 a+4 b+d}}\right )^2}{4 \sqrt {2} \sqrt {a-b} \sqrt {-2 a+2 b+d+\sqrt {d} \sqrt {-4 a+4 b+d}}};2 \tan ^{-1}\left (\frac {\sqrt [4]{-a+x}}{\sqrt [4]{a-b}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{a-b} \left (4 a-4 b-d-\sqrt {d} \sqrt {-4 a+4 b+d}\right ) \sqrt [4]{-\left ((a-x) (b-x)^2\right )}}\\ \end {align*}

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Mathematica [C]  time = 25.69, size = 675, normalized size = 5.97 \begin {gather*} -\frac {i \sqrt {2} (x-a)^{3/4} \sqrt {\frac {b-x}{a-x}} \left (\left (\sqrt {d (-4 a+4 b+d)}+4 a-4 b-d\right ) \sqrt {\frac {\sqrt {d (-4 a+4 b+d)}-2 a+2 b+d}{(a-b)^2}} \Pi \left (-\frac {\sqrt {2}}{\sqrt {b-a} \sqrt {\frac {-2 a+2 b+d-\sqrt {d (-4 a+4 b+d)}}{(a-b)^2}}};\left .i \sinh ^{-1}\left (\frac {\sqrt {-\sqrt {b-a}}}{\sqrt [4]{x-a}}\right )\right |-1\right )-\left (\sqrt {d (-4 a+4 b+d)}+4 a-4 b-d\right ) \sqrt {\frac {\sqrt {d (-4 a+4 b+d)}-2 a+2 b+d}{(a-b)^2}} \Pi \left (\frac {\sqrt {2}}{\sqrt {b-a} \sqrt {\frac {-2 a+2 b+d-\sqrt {d (-4 a+4 b+d)}}{(a-b)^2}}};\left .i \sinh ^{-1}\left (\frac {\sqrt {-\sqrt {b-a}}}{\sqrt [4]{x-a}}\right )\right |-1\right )-\left (-\sqrt {d (-4 a+4 b+d)}+4 a-4 b-d\right ) \sqrt {\frac {-\sqrt {d (-4 a+4 b+d)}-2 a+2 b+d}{(a-b)^2}} \left (\Pi \left (-\frac {\sqrt {2}}{\sqrt {b-a} \sqrt {\frac {-2 a+2 b+d+\sqrt {d (-4 a+4 b+d)}}{(a-b)^2}}};\left .i \sinh ^{-1}\left (\frac {\sqrt {-\sqrt {b-a}}}{\sqrt [4]{x-a}}\right )\right |-1\right )-\Pi \left (\frac {\sqrt {2}}{\sqrt {b-a} \sqrt {\frac {-2 a+2 b+d+\sqrt {d (-4 a+4 b+d)}}{(a-b)^2}}};\left .i \sinh ^{-1}\left (\frac {\sqrt {-\sqrt {b-a}}}{\sqrt [4]{x-a}}\right )\right |-1\right )\right )\right )}{(a-b) \sqrt {-\sqrt {b-a}} \sqrt {d (-4 a+4 b+d)} \sqrt {\frac {-\sqrt {d (-4 a+4 b+d)}-2 a+2 b+d}{(a-b)^2}} \sqrt {\frac {\sqrt {d (-4 a+4 b+d)}-2 a+2 b+d}{(a-b)^2}} \sqrt [4]{(x-a) (b-x)^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-I)*Sqrt[2]*Sqrt[(b - x)/(a - x)]*(-a + x)^(3/4)*((4*a - 4*b - d + Sqrt[d*(-4*a + 4*b + d)])*Sqrt[(-2*a + 2*
b + d + Sqrt[d*(-4*a + 4*b + d)])/(a - b)^2]*EllipticPi[-(Sqrt[2]/(Sqrt[-a + b]*Sqrt[(-2*a + 2*b + d - Sqrt[d*
(-4*a + 4*b + d)])/(a - b)^2])), I*ArcSinh[Sqrt[-Sqrt[-a + b]]/(-a + x)^(1/4)], -1] - (4*a - 4*b - d + Sqrt[d*
(-4*a + 4*b + d)])*Sqrt[(-2*a + 2*b + d + Sqrt[d*(-4*a + 4*b + d)])/(a - b)^2]*EllipticPi[Sqrt[2]/(Sqrt[-a + b
]*Sqrt[(-2*a + 2*b + d - Sqrt[d*(-4*a + 4*b + d)])/(a - b)^2]), I*ArcSinh[Sqrt[-Sqrt[-a + b]]/(-a + x)^(1/4)],
-1] - (4*a - 4*b - d - Sqrt[d*(-4*a + 4*b + d)])*Sqrt[(-2*a + 2*b + d - Sqrt[d*(-4*a + 4*b + d)])/(a - b)^2]*
(EllipticPi[-(Sqrt[2]/(Sqrt[-a + b]*Sqrt[(-2*a + 2*b + d + Sqrt[d*(-4*a + 4*b + d)])/(a - b)^2])), I*ArcSinh[S
qrt[-Sqrt[-a + b]]/(-a + x)^(1/4)], -1] - EllipticPi[Sqrt[2]/(Sqrt[-a + b]*Sqrt[(-2*a + 2*b + d + Sqrt[d*(-4*a
+ 4*b + d)])/(a - b)^2]), I*ArcSinh[Sqrt[-Sqrt[-a + b]]/(-a + x)^(1/4)], -1])))/((a - b)*Sqrt[-Sqrt[-a + b]]*
Sqrt[d*(-4*a + 4*b + d)]*Sqrt[(-2*a + 2*b + d - Sqrt[d*(-4*a + 4*b + d)])/(a - b)^2]*Sqrt[(-2*a + 2*b + d + Sq
rt[d*(-4*a + 4*b + d)])/(a - b)^2]*((b - x)^2*(-a + x))^(1/4))

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IntegrateAlgebraic [A]  time = 0.29, size = 113, normalized size = 1.00 \begin {gather*} -\frac {2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3}}{b-x}\right )}{d^{3/4}}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3}}{b-x}\right )}{d^{3/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {2 \, a - b - x}{\left (-{\left (a - x\right )} {\left (b - x\right )}^{2}\right )^{\frac {1}{4}} {\left (b^{2} + a d - {\left (2 \, b + d\right )} x + x^{2}\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.09, size = 0, normalized size = 0.00 \[\int \frac {-2 a +b +x}{\left (\left (-a +x \right ) \left (-b +x \right )^{2}\right )^{\frac {1}{4}} \left (b^{2}+a d -\left (2 b +d \right ) x +x^{2}\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\int \frac {2 \, a - b - x}{\left (-{\left (a - x\right )} {\left (b - x\right )}^{2}\right )^{\frac {1}{4}} {\left (b^{2} + a d - {\left (2 \, b + d\right )} x + x^{2}\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {b-2\,a+x}{{\left (-\left (a-x\right )\,{\left (b-x\right )}^2\right )}^{1/4}\,\left (a\,d-x\,\left (2\,b+d\right )+b^2+x^2\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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