\(\int \frac {-2 a+b+x}{\sqrt [4]{(-a+x) (-b+x)^2} (b^2+a d-(2 b+d) x+x^2)} \, dx\) [1680]

Optimal result

Integrand size = 44, antiderivative size = 113 \[ \int \frac {-2 a+b+x}{\sqrt [4]{(-a+x) (-b+x)^2} \left (b^2+a d-(2 b+d) x+x^2\right )} \, dx=-\frac {2 \arctan \left (\frac {\sqrt [4]{d} \sqrt [4]{-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3}}{b-x}\right )}{d^{3/4}}+\frac {2 \text {arctanh}\left (\frac {\sqrt [4]{d} \sqrt [4]{-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3}}{b-x}\right )}{d^{3/4}} \]

[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)

Rubi [C] (warning: unable to verify)

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

Time = 37.53 (sec) , antiderivative size = 3261, normalized size of antiderivative = 28.86, number of steps used = 21, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {6851, 6860, 108, 107, 504, 1231, 226, 1721} \[ \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 {\sqrt [4]{a-b} \left (1-\frac {\sqrt {-4 a+4 b+d}}{\sqrt {d}}\right ) \sqrt [4]{x-a} \sqrt {-\frac {b-x}{(a-b) \left (\frac {\sqrt {x-a}}{\sqrt {a-b}}+1\right )^2}} \left (\frac {\sqrt {x-a}}{\sqrt {a-b}}+1\right ) \operatorname {EllipticPi}\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 \arctan \left (\frac {\sqrt [4]{x-a}}{\sqrt [4]{a-b}}\right ),\frac {1}{2}\right ) \left (\sqrt {2} \sqrt {a-b}-\sqrt {-2 a+2 b+d-\sqrt {d} \sqrt {-4 a+4 b+d}}\right )^2}{2 \sqrt {2} \sqrt {-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 {a-b} \sqrt {\sqrt {d}-\sqrt {-4 a+4 b+d}} \sqrt {-\frac {b-x}{a-b}} \sqrt [4]{x-a} \arctan \left (\frac {\sqrt [4]{d} \sqrt {\sqrt {d}-\sqrt {-4 a+4 b+d}} \sqrt [4]{x-a}}{\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 )}{\sqrt [4]{2} d^{3/4} \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 {-4 a+4 b+d}-\sqrt {d}} \sqrt {-\frac {b-x}{a-b}} \sqrt [4]{x-a} \arctan \left (\frac {\sqrt [4]{d} \sqrt {\sqrt {-4 a+4 b+d}-\sqrt {d}} \sqrt [4]{x-a}}{\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 )}{\sqrt [4]{2} d^{3/4} \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}} \sqrt {-\frac {b-x}{a-b}} \sqrt [4]{x-a} \arctan \left (\frac {\sqrt [4]{d} \sqrt {\sqrt {d}+\sqrt {-4 a+4 b+d}} \sqrt [4]{x-a}}{\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 )}{\sqrt [4]{2} d^{3/4} \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 {b-a} \sqrt {\sqrt {d}+\sqrt {-4 a+4 b+d}} \sqrt {-\frac {b-x}{a-b}} \sqrt [4]{x-a} \arctan \left (\frac {\sqrt [4]{d} \sqrt {\sqrt {d}+\sqrt {-4 a+4 b+d}} \sqrt [4]{x-a}}{\sqrt [4]{2} \sqrt {b-a} \sqrt [4]{-2 a+2 b+d+\sqrt {d} \sqrt {-4 a+4 b+d}} \sqrt {-\frac {b-x}{a-b}}}\right )}{\sqrt [4]{2} d^{3/4} \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 (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]{x-a} \sqrt {-\frac {b-x}{(a-b) \left (\frac {\sqrt {x-a}}{\sqrt {a-b}}+1\right )^2}} \left (\frac {\sqrt {x-a}}{\sqrt {a-b}}+1\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{x-a}}{\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-2 b+\sqrt {2} \sqrt {a-b} \sqrt {-2 a+2 b+d-\sqrt {d} \sqrt {-4 a+4 b+d}}\right ) \sqrt [4]{x-a} \sqrt {-\frac {b-x}{(a-b) \left (\frac {\sqrt {x-a}}{\sqrt {a-b}}+1\right )^2}} \left (\frac {\sqrt {x-a}}{\sqrt {a-b}}+1\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{x-a}}{\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 (\frac {\sqrt {-4 a+4 b+d}}{\sqrt {d}}+1\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]{x-a} \sqrt {-\frac {b-x}{(a-b) \left (\frac {\sqrt {x-a}}{\sqrt {a-b}}+1\right )^2}} \left (\frac {\sqrt {x-a}}{\sqrt {a-b}}+1\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{x-a}}{\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 (\frac {\sqrt {-4 a+4 b+d}}{\sqrt {d}}+1\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]{x-a} \sqrt {-\frac {b-x}{(a-b) \left (\frac {\sqrt {x-a}}{\sqrt {a-b}}+1\right )^2}} \left (\frac {\sqrt {x-a}}{\sqrt {a-b}}+1\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{x-a}}{\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 {\sqrt [4]{a-b} \left (1-\frac {\sqrt {-4 a+4 b+d}}{\sqrt {d}}\right ) \left (\sqrt {2} \sqrt {a-b}+\sqrt {-2 a+2 b+d-\sqrt {d} \sqrt {-4 a+4 b+d}}\right )^2 \sqrt [4]{x-a} \sqrt {-\frac {b-x}{(a-b) \left (\frac {\sqrt {x-a}}{\sqrt {a-b}}+1\right )^2}} \left (\frac {\sqrt {x-a}}{\sqrt {a-b}}+1\right ) \operatorname {EllipticPi}\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 \arctan \left (\frac {\sqrt [4]{x-a}}{\sqrt [4]{a-b}}\right ),\frac {1}{2}\right )}{2 \sqrt {2} \sqrt {-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 [4]{a-b} \left (\frac {\sqrt {-4 a+4 b+d}}{\sqrt {d}}+1\right ) \left (\sqrt {2} \sqrt {a-b}+\sqrt {-2 a+2 b+d+\sqrt {d} \sqrt {-4 a+4 b+d}}\right )^2 \sqrt [4]{x-a} \sqrt {-\frac {b-x}{(a-b) \left (\frac {\sqrt {x-a}}{\sqrt {a-b}}+1\right )^2}} \left (\frac {\sqrt {x-a}}{\sqrt {a-b}}+1\right ) \operatorname {EllipticPi}\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 \arctan \left (\frac {\sqrt [4]{x-a}}{\sqrt [4]{a-b}}\right ),\frac {1}{2}\right )}{2 \sqrt {2} \left (4 a-4 b-d-\sqrt {d} \sqrt {-4 a+4 b+d}\right ) \sqrt {-2 a+2 b+d+\sqrt {d} \sqrt {-4 a+4 b+d}} \sqrt [4]{-\left ((a-x) (b-x)^2\right )}}-\frac {\sqrt [4]{a-b} \left (\sqrt {d}+\sqrt {-4 a+4 b+d}\right ) \left (\sqrt {2} \sqrt {a-b}-\sqrt {-2 a+2 b+d+\sqrt {d} \sqrt {-4 a+4 b+d}}\right )^2 \sqrt [4]{x-a} \sqrt {-\frac {b-x}{(a-b) \left (\frac {\sqrt {x-a}}{\sqrt {a-b}}+1\right )^2}} \left (\frac {\sqrt {x-a}}{\sqrt {a-b}}+1\right ) \operatorname {EllipticPi}\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 \arctan \left (\frac {\sqrt [4]{x-a}}{\sqrt [4]{a-b}}\right ),\frac {1}{2}\right )}{2 \sqrt {2} \sqrt {d} \left (4 a-4 b-d-\sqrt {d} \sqrt {-4 a+4 b+d}\right ) \sqrt {-2 a+2 b+d+\sqrt {d} \sqrt {-4 a+4 b+d}} \sqrt [4]{-\left ((a-x) (b-x)^2\right )}} \]

[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 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 6851

Int[(u_.)*((a_.)*(v_)^(m_.)*(w_)^(n_.))^(p_), x_Symbol] :> Dist[a^IntPart[p]*((a*v^m*w^n)^FracPart[p]/(v^(m*Fr
acPart[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 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 [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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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}} \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [A] (verified)

Time = 3.29 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.84 \[ \int \frac {-2 a+b+x}{\sqrt [4]{(-a+x) (-b+x)^2} \left (b^2+a d-(2 b+d) x+x^2\right )} \, dx=\frac {2 \sqrt {b-x} \sqrt [4]{-a+x} \left (-\arctan \left (\frac {\sqrt [4]{d} \sqrt [4]{-a+x}}{\sqrt {b-x}}\right )+\text {arctanh}\left (\frac {\sqrt [4]{d} \sqrt [4]{-a+x}}{\sqrt {b-x}}\right )\right )}{d^{3/4} \sqrt [4]{(b-x)^2 (-a+x)}} \]

[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]

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

Maple [F]

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

[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)

Fricas [F(-1)]

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

[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

Sympy [F(-1)]

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

[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

Maxima [F]

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

[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)

Giac [F]

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

[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)

Mupad [F(-1)]

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

[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)