3.8.78 \(\int \frac {a (a b+a c-3 b c)+(-2 a^2+a b+a c+3 b c) x+(a-2 b-2 c) x^2+x^3}{\sqrt {(-a+x) (-b+x) (-c+x)} (-a^3-b c d+(3 a^2+b d+c d) x-(3 a+d) x^2+x^3)} \, dx\)

Optimal. Leaf size=60 \[ -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {x^2 (-a-b-c)+x (a b+a c+b c)-a b c+x^3}}{(a-x)^2}\right )}{\sqrt {d}} \]

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Rubi [F]  time = 48.22, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {a (a b+a c-3 b c)+\left (-2 a^2+a b+a c+3 b c\right ) x+(a-2 b-2 c) x^2+x^3}{\sqrt {(-a+x) (-b+x) (-c+x)} \left (-a^3-b c d+\left (3 a^2+b d+c d\right ) x-(3 a+d) x^2+x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(-2*Sqrt[a - c]*(b - x)*Sqrt[-a + x]*EllipticF[ArcTan[Sqrt[-a + x]/Sqrt[a - c]], -((b - c)/(a - b))])/((a - b)
*Sqrt[((a - c)*(b - x))/((a - b)*(c - x))]*Sqrt[-((a - x)*(b - x)*(c - x))]) - (2*(a - b)*(a - c)*d*Sqrt[-a +
x]*Sqrt[-b + x]*Sqrt[-c + x]*Defer[Subst][Defer[Int][1/(Sqrt[a - b + x^2]*Sqrt[a - c + x^2]*(a^2*(1 + (b*c - a
*(b + c))/a^2)*d + 2*a*(1 - (b + c)/(2*a))*d*x^2 + d*x^4 - x^6)), x], x, Sqrt[-a + x]])/Sqrt[-((a - x)*(b - x)
*(c - x))] - (2*(a^2 + 2*a*d - (b + c)*d)*Sqrt[-a + x]*Sqrt[-b + x]*Sqrt[-c + x]*Defer[Subst][Defer[Int][x^2/(
Sqrt[a - b + x^2]*Sqrt[a - c + x^2]*(a^2*(1 + (b*c - a*(b + c))/a^2)*d + 2*a*(1 - (b + c)/(2*a))*d*x^2 + d*x^4
- x^6)), x], x, Sqrt[-a + x]])/Sqrt[-((a - x)*(b - x)*(c - x))] - (2*(2*a + d)*Sqrt[-a + x]*Sqrt[-b + x]*Sqrt
[-c + x]*Defer[Subst][Defer[Int][x^4/(Sqrt[a - b + x^2]*Sqrt[a - c + x^2]*(a^2*(1 + (b*c - a*(b + c))/a^2)*d +
2*a*(1 - (b + c)/(2*a))*d*x^2 + d*x^4 - x^6)), x], x, Sqrt[-a + x]])/Sqrt[-((a - x)*(b - x)*(c - x))] + (4*a*
(a - b - c)*Sqrt[-a + x]*Sqrt[-b + x]*Sqrt[-c + x]*Defer[Subst][Defer[Int][x^2/(Sqrt[a - b + x^2]*Sqrt[a - c +
x^2]*(-(a^2*(1 + (b*c - a*(b + c))/a^2)*d) - 2*a*(1 - (b + c)/(2*a))*d*x^2 - d*x^4 + x^6)), x], x, Sqrt[-a +
x]])/Sqrt[-((a - x)*(b - x)*(c - x))] + (4*(a - b - c)*Sqrt[-a + x]*Sqrt[-b + x]*Sqrt[-c + x]*Defer[Subst][Def
er[Int][x^4/(Sqrt[a - b + x^2]*Sqrt[a - c + x^2]*(-(a^2*(1 + (b*c - a*(b + c))/a^2)*d) - 2*a*(1 - (b + c)/(2*a
))*d*x^2 - d*x^4 + x^6)), x], x, Sqrt[-a + x]])/Sqrt[-((a - x)*(b - x)*(c - x))] + (2*(3*b*c - a*(b + c))*Sqrt
[-a + x]*Sqrt[-b + x]*Sqrt[-c + x]*Defer[Subst][Defer[Int][x^2/(Sqrt[a - b + x^2]*Sqrt[a - c + x^2]*(-(a^2*d)
+ c*d*x^2 - d*x^4 + x^6 + a*d*(b + c - 2*x^2) + b*d*(-c + x^2))), x], x, Sqrt[-a + x]])/Sqrt[-((a - x)*(b - x)
*(c - x))]

Rubi steps

\begin {align*} \int \frac {a (a b+a c-3 b c)+\left (-2 a^2+a b+a c+3 b c\right ) x+(a-2 b-2 c) x^2+x^3}{\sqrt {(-a+x) (-b+x) (-c+x)} \left (-a^3-b c d+\left (3 a^2+b d+c d\right ) x-(3 a+d) x^2+x^3\right )} \, dx &=\frac {\left (\sqrt {-a+x} \sqrt {-b+x} \sqrt {-c+x}\right ) \int \frac {a (a b+a c-3 b c)+\left (-2 a^2+a b+a c+3 b c\right ) x+(a-2 b-2 c) x^2+x^3}{\sqrt {-a+x} \sqrt {-b+x} \sqrt {-c+x} \left (-a^3-b c d+\left (3 a^2+b d+c d\right ) x-(3 a+d) x^2+x^3\right )} \, dx}{\sqrt {(-a+x) (-b+x) (-c+x)}}\\ &=\frac {\left (\sqrt {-a+x} \sqrt {-b+x} \sqrt {-c+x}\right ) \int \frac {\sqrt {-a+x} \left (-a b-a c+3 b c+(2 a-2 b-2 c) x+x^2\right )}{\sqrt {-b+x} \sqrt {-c+x} \left (-a^3-b c d+\left (3 a^2+b d+c d\right ) x-(3 a+d) x^2+x^3\right )} \, dx}{\sqrt {(-a+x) (-b+x) (-c+x)}}\\ &=\frac {\left (\sqrt {-a+x} \sqrt {-b+x} \sqrt {-c+x}\right ) \int \frac {\sqrt {-a+x} \left (-3 b c+a (b+c)-2 (a-b-c) x-x^2\right )}{\sqrt {-b+x} \sqrt {-c+x} \left (a^3+b c d-\left (3 a^2+(b+c) d\right ) x+(3 a+d) x^2-x^3\right )} \, dx}{\sqrt {(-a+x) (-b+x) (-c+x)}}\\ &=\frac {\left (\sqrt {-a+x} \sqrt {-b+x} \sqrt {-c+x}\right ) \int \left (\frac {a (b+c) \left (1-\frac {3 b c}{a b+a c}\right ) \sqrt {-a+x}}{\sqrt {-b+x} \sqrt {-c+x} \left (a^3+b c d-\left (3 a^2+(b+c) d\right ) x+(3 a+d) x^2-x^3\right )}+\frac {2 (-a+b+c) x \sqrt {-a+x}}{\sqrt {-b+x} \sqrt {-c+x} \left (a^3+b c d-\left (3 a^2+(b+c) d\right ) x+(3 a+d) x^2-x^3\right )}+\frac {x^2 \sqrt {-a+x}}{\sqrt {-b+x} \sqrt {-c+x} \left (-a^3-b c d+\left (3 a^2+(b+c) d\right ) x-(3 a+d) x^2+x^3\right )}\right ) \, dx}{\sqrt {(-a+x) (-b+x) (-c+x)}}\\ &=\frac {\left (\sqrt {-a+x} \sqrt {-b+x} \sqrt {-c+x}\right ) \int \frac {x^2 \sqrt {-a+x}}{\sqrt {-b+x} \sqrt {-c+x} \left (-a^3-b c d+\left (3 a^2+(b+c) d\right ) x-(3 a+d) x^2+x^3\right )} \, dx}{\sqrt {(-a+x) (-b+x) (-c+x)}}-\frac {\left (2 (a-b-c) \sqrt {-a+x} \sqrt {-b+x} \sqrt {-c+x}\right ) \int \frac {x \sqrt {-a+x}}{\sqrt {-b+x} \sqrt {-c+x} \left (a^3+b c d-\left (3 a^2+(b+c) d\right ) x+(3 a+d) x^2-x^3\right )} \, dx}{\sqrt {(-a+x) (-b+x) (-c+x)}}+\frac {\left ((-3 b c+a (b+c)) \sqrt {-a+x} \sqrt {-b+x} \sqrt {-c+x}\right ) \int \frac {\sqrt {-a+x}}{\sqrt {-b+x} \sqrt {-c+x} \left (a^3+b c d-\left (3 a^2+(b+c) d\right ) x+(3 a+d) x^2-x^3\right )} \, dx}{\sqrt {(-a+x) (-b+x) (-c+x)}}\\ &=\frac {\left (2 \sqrt {-a+x} \sqrt {-b+x} \sqrt {-c+x}\right ) \operatorname {Subst}\left (\int \frac {x^2 \left (a+x^2\right )^2}{\sqrt {a-b+x^2} \sqrt {a-c+x^2} \left (-a^2 d+c d x^2-d x^4+x^6+a d \left (b+c-2 x^2\right )+b d \left (-c+x^2\right )\right )} \, dx,x,\sqrt {-a+x}\right )}{\sqrt {(-a+x) (-b+x) (-c+x)}}+\frac {\left (4 (a-b-c) \sqrt {-a+x} \sqrt {-b+x} \sqrt {-c+x}\right ) \operatorname {Subst}\left (\int \frac {x^2 \left (a+x^2\right )}{\sqrt {a-b+x^2} \sqrt {a-c+x^2} \left (-a^2 d+c d x^2-d x^4+x^6+a d \left (b+c-2 x^2\right )+b d \left (-c+x^2\right )\right )} \, dx,x,\sqrt {-a+x}\right )}{\sqrt {(-a+x) (-b+x) (-c+x)}}-\frac {\left (2 (-3 b c+a (b+c)) \sqrt {-a+x} \sqrt {-b+x} \sqrt {-c+x}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {a-b+x^2} \sqrt {a-c+x^2} \left (-a^2 d+c d x^2-d x^4+x^6+a d \left (b+c-2 x^2\right )+b d \left (-c+x^2\right )\right )} \, dx,x,\sqrt {-a+x}\right )}{\sqrt {(-a+x) (-b+x) (-c+x)}}\\ &=\frac {\left (2 \sqrt {-a+x} \sqrt {-b+x} \sqrt {-c+x}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{\sqrt {a-b+x^2} \sqrt {a-c+x^2}}+\frac {(a-b) (a-c) d+\left (a^2+2 a d-(b+c) d\right ) x^2+(2 a+d) x^4}{\sqrt {a-b+x^2} \sqrt {a-c+x^2} \left (-a^2 d+c d x^2-d x^4+x^6+a d \left (b+c-2 x^2\right )+b d \left (-c+x^2\right )\right )}\right ) \, dx,x,\sqrt {-a+x}\right )}{\sqrt {(-a+x) (-b+x) (-c+x)}}+\frac {\left (4 (a-b-c) \sqrt {-a+x} \sqrt {-b+x} \sqrt {-c+x}\right ) \operatorname {Subst}\left (\int \left (\frac {a x^2}{\sqrt {a-b+x^2} \sqrt {a-c+x^2} \left (-a^2 \left (1+\frac {b c-a (b+c)}{a^2}\right ) d-2 a \left (1-\frac {b+c}{2 a}\right ) d x^2-d x^4+x^6\right )}+\frac {x^4}{\sqrt {a-b+x^2} \sqrt {a-c+x^2} \left (-a^2 \left (1+\frac {b c-a (b+c)}{a^2}\right ) d-2 a \left (1-\frac {b+c}{2 a}\right ) d x^2-d x^4+x^6\right )}\right ) \, dx,x,\sqrt {-a+x}\right )}{\sqrt {(-a+x) (-b+x) (-c+x)}}-\frac {\left (2 (-3 b c+a (b+c)) \sqrt {-a+x} \sqrt {-b+x} \sqrt {-c+x}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {a-b+x^2} \sqrt {a-c+x^2} \left (-a^2 d+c d x^2-d x^4+x^6+a d \left (b+c-2 x^2\right )+b d \left (-c+x^2\right )\right )} \, dx,x,\sqrt {-a+x}\right )}{\sqrt {(-a+x) (-b+x) (-c+x)}}\\ &=\frac {\left (2 \sqrt {-a+x} \sqrt {-b+x} \sqrt {-c+x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a-b+x^2} \sqrt {a-c+x^2}} \, dx,x,\sqrt {-a+x}\right )}{\sqrt {(-a+x) (-b+x) (-c+x)}}+\frac {\left (2 \sqrt {-a+x} \sqrt {-b+x} \sqrt {-c+x}\right ) \operatorname {Subst}\left (\int \frac {(a-b) (a-c) d+\left (a^2+2 a d-(b+c) d\right ) x^2+(2 a+d) x^4}{\sqrt {a-b+x^2} \sqrt {a-c+x^2} \left (-a^2 d+c d x^2-d x^4+x^6+a d \left (b+c-2 x^2\right )+b d \left (-c+x^2\right )\right )} \, dx,x,\sqrt {-a+x}\right )}{\sqrt {(-a+x) (-b+x) (-c+x)}}+\frac {\left (4 (a-b-c) \sqrt {-a+x} \sqrt {-b+x} \sqrt {-c+x}\right ) \operatorname {Subst}\left (\int \frac {x^4}{\sqrt {a-b+x^2} \sqrt {a-c+x^2} \left (-a^2 \left (1+\frac {b c-a (b+c)}{a^2}\right ) d-2 a \left (1-\frac {b+c}{2 a}\right ) d x^2-d x^4+x^6\right )} \, dx,x,\sqrt {-a+x}\right )}{\sqrt {(-a+x) (-b+x) (-c+x)}}+\frac {\left (4 a (a-b-c) \sqrt {-a+x} \sqrt {-b+x} \sqrt {-c+x}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {a-b+x^2} \sqrt {a-c+x^2} \left (-a^2 \left (1+\frac {b c-a (b+c)}{a^2}\right ) d-2 a \left (1-\frac {b+c}{2 a}\right ) d x^2-d x^4+x^6\right )} \, dx,x,\sqrt {-a+x}\right )}{\sqrt {(-a+x) (-b+x) (-c+x)}}-\frac {\left (2 (-3 b c+a (b+c)) \sqrt {-a+x} \sqrt {-b+x} \sqrt {-c+x}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {a-b+x^2} \sqrt {a-c+x^2} \left (-a^2 d+c d x^2-d x^4+x^6+a d \left (b+c-2 x^2\right )+b d \left (-c+x^2\right )\right )} \, dx,x,\sqrt {-a+x}\right )}{\sqrt {(-a+x) (-b+x) (-c+x)}}\\ &=-\frac {2 \sqrt {a-c} (b-x) \sqrt {-a+x} F\left (\tan ^{-1}\left (\frac {\sqrt {-a+x}}{\sqrt {a-c}}\right )|-\frac {b-c}{a-b}\right )}{(a-b) \sqrt {\frac {(a-c) (b-x)}{(a-b) (c-x)}} \sqrt {-((a-x) (b-x) (c-x))}}+\frac {\left (2 \sqrt {-a+x} \sqrt {-b+x} \sqrt {-c+x}\right ) \operatorname {Subst}\left (\int \left (\frac {(a-b) (-a+c) d}{\sqrt {a-b+x^2} \sqrt {a-c+x^2} \left (a^2 \left (1+\frac {b c-a (b+c)}{a^2}\right ) d+2 a \left (1-\frac {b+c}{2 a}\right ) d x^2+d x^4-x^6\right )}+\frac {\left (-a^2-2 a d+(b+c) d\right ) x^2}{\sqrt {a-b+x^2} \sqrt {a-c+x^2} \left (a^2 \left (1+\frac {b c-a (b+c)}{a^2}\right ) d+2 a \left (1-\frac {b+c}{2 a}\right ) d x^2+d x^4-x^6\right )}+\frac {(-2 a-d) x^4}{\sqrt {a-b+x^2} \sqrt {a-c+x^2} \left (a^2 \left (1+\frac {b c-a (b+c)}{a^2}\right ) d+2 a \left (1-\frac {b+c}{2 a}\right ) d x^2+d x^4-x^6\right )}\right ) \, dx,x,\sqrt {-a+x}\right )}{\sqrt {(-a+x) (-b+x) (-c+x)}}+\frac {\left (4 (a-b-c) \sqrt {-a+x} \sqrt {-b+x} \sqrt {-c+x}\right ) \operatorname {Subst}\left (\int \frac {x^4}{\sqrt {a-b+x^2} \sqrt {a-c+x^2} \left (-a^2 \left (1+\frac {b c-a (b+c)}{a^2}\right ) d-2 a \left (1-\frac {b+c}{2 a}\right ) d x^2-d x^4+x^6\right )} \, dx,x,\sqrt {-a+x}\right )}{\sqrt {(-a+x) (-b+x) (-c+x)}}+\frac {\left (4 a (a-b-c) \sqrt {-a+x} \sqrt {-b+x} \sqrt {-c+x}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {a-b+x^2} \sqrt {a-c+x^2} \left (-a^2 \left (1+\frac {b c-a (b+c)}{a^2}\right ) d-2 a \left (1-\frac {b+c}{2 a}\right ) d x^2-d x^4+x^6\right )} \, dx,x,\sqrt {-a+x}\right )}{\sqrt {(-a+x) (-b+x) (-c+x)}}-\frac {\left (2 (-3 b c+a (b+c)) \sqrt {-a+x} \sqrt {-b+x} \sqrt {-c+x}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {a-b+x^2} \sqrt {a-c+x^2} \left (-a^2 d+c d x^2-d x^4+x^6+a d \left (b+c-2 x^2\right )+b d \left (-c+x^2\right )\right )} \, dx,x,\sqrt {-a+x}\right )}{\sqrt {(-a+x) (-b+x) (-c+x)}}\\ &=-\frac {2 \sqrt {a-c} (b-x) \sqrt {-a+x} F\left (\tan ^{-1}\left (\frac {\sqrt {-a+x}}{\sqrt {a-c}}\right )|-\frac {b-c}{a-b}\right )}{(a-b) \sqrt {\frac {(a-c) (b-x)}{(a-b) (c-x)}} \sqrt {-((a-x) (b-x) (c-x))}}+\frac {\left (4 (a-b-c) \sqrt {-a+x} \sqrt {-b+x} \sqrt {-c+x}\right ) \operatorname {Subst}\left (\int \frac {x^4}{\sqrt {a-b+x^2} \sqrt {a-c+x^2} \left (-a^2 \left (1+\frac {b c-a (b+c)}{a^2}\right ) d-2 a \left (1-\frac {b+c}{2 a}\right ) d x^2-d x^4+x^6\right )} \, dx,x,\sqrt {-a+x}\right )}{\sqrt {(-a+x) (-b+x) (-c+x)}}+\frac {\left (4 a (a-b-c) \sqrt {-a+x} \sqrt {-b+x} \sqrt {-c+x}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {a-b+x^2} \sqrt {a-c+x^2} \left (-a^2 \left (1+\frac {b c-a (b+c)}{a^2}\right ) d-2 a \left (1-\frac {b+c}{2 a}\right ) d x^2-d x^4+x^6\right )} \, dx,x,\sqrt {-a+x}\right )}{\sqrt {(-a+x) (-b+x) (-c+x)}}-\frac {\left (2 (-3 b c+a (b+c)) \sqrt {-a+x} \sqrt {-b+x} \sqrt {-c+x}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {a-b+x^2} \sqrt {a-c+x^2} \left (-a^2 d+c d x^2-d x^4+x^6+a d \left (b+c-2 x^2\right )+b d \left (-c+x^2\right )\right )} \, dx,x,\sqrt {-a+x}\right )}{\sqrt {(-a+x) (-b+x) (-c+x)}}+\frac {\left (2 (-2 a-d) \sqrt {-a+x} \sqrt {-b+x} \sqrt {-c+x}\right ) \operatorname {Subst}\left (\int \frac {x^4}{\sqrt {a-b+x^2} \sqrt {a-c+x^2} \left (a^2 \left (1+\frac {b c-a (b+c)}{a^2}\right ) d+2 a \left (1-\frac {b+c}{2 a}\right ) d x^2+d x^4-x^6\right )} \, dx,x,\sqrt {-a+x}\right )}{\sqrt {(-a+x) (-b+x) (-c+x)}}-\frac {\left (2 (a-b) (a-c) d \sqrt {-a+x} \sqrt {-b+x} \sqrt {-c+x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a-b+x^2} \sqrt {a-c+x^2} \left (a^2 \left (1+\frac {b c-a (b+c)}{a^2}\right ) d+2 a \left (1-\frac {b+c}{2 a}\right ) d x^2+d x^4-x^6\right )} \, dx,x,\sqrt {-a+x}\right )}{\sqrt {(-a+x) (-b+x) (-c+x)}}+\frac {\left (2 \left (-a^2-2 a d+(b+c) d\right ) \sqrt {-a+x} \sqrt {-b+x} \sqrt {-c+x}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {a-b+x^2} \sqrt {a-c+x^2} \left (a^2 \left (1+\frac {b c-a (b+c)}{a^2}\right ) d+2 a \left (1-\frac {b+c}{2 a}\right ) d x^2+d x^4-x^6\right )} \, dx,x,\sqrt {-a+x}\right )}{\sqrt {(-a+x) (-b+x) (-c+x)}}\\ \end {align*}

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Mathematica [C]  time = 6.21, size = 3908, normalized size = 65.13 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((2*I)*Sqrt[(a - x)/(b - x)]*(b - x)*Sqrt[(-c + x)/(a - c)]*(3*(a - b)*(a - c)*d*(EllipticF[I*ArcSinh[Sqrt[(-a
+ x)/(a - b)]], (a - b)/(a - c)] - EllipticPi[(a - b)/Root[a^2*d - a*b*d - a*c*d + b*c*d + (-2*a*d + b*d + c*
d)*#1 + d*#1^2 + #1^3 & , 1], I*ArcSinh[Sqrt[(-a + x)/(a - b)]], (a - b)/(a - c)]) + (2*a - b - c)*d^2*(Ellipt
icF[I*ArcSinh[Sqrt[(-a + x)/(a - b)]], (a - b)/(a - c)] - EllipticPi[(a - b)/Root[a^2*d - a*b*d - a*c*d + b*c*
d + (-2*a*d + b*d + c*d)*#1 + d*#1^2 + #1^3 & , 1], I*ArcSinh[Sqrt[(-a + x)/(a - b)]], (a - b)/(a - c)]) - 2*(
-2*a + b + c)^2*d*(EllipticPi[(a - b)/Root[a^2*d - a*b*d - a*c*d + b*c*d + (-2*a*d + b*d + c*d)*#1 + d*#1^2 +
#1^3 & , 1], I*ArcSinh[Sqrt[(-a + x)/(a - b)]], (a - b)/(a - c)] - EllipticPi[(a - b)/Root[a^2*d - a*b*d - a*c
*d + b*c*d + (-2*a*d + b*d + c*d)*#1 + d*#1^2 + #1^3 & , 2], I*ArcSinh[Sqrt[(-a + x)/(a - b)]], (a - b)/(a - c
)]) - 3*(a - b)*(a - c)*d*(EllipticPi[(a - b)/Root[a^2*d - a*b*d - a*c*d + b*c*d + (-2*a*d + b*d + c*d)*#1 + d
*#1^2 + #1^3 & , 2], I*ArcSinh[Sqrt[(-a + x)/(a - b)]], (a - b)/(a - c)] - EllipticPi[(a - b)/Root[a^2*d - a*b
*d - a*c*d + b*c*d + (-2*a*d + b*d + c*d)*#1 + d*#1^2 + #1^3 & , 3], I*ArcSinh[Sqrt[(-a + x)/(a - b)]], (a - b
)/(a - c)]) - 3*(a - b)*(a - c)*(EllipticPi[(a - b)/Root[a^2*d - a*b*d - a*c*d + b*c*d + (-2*a*d + b*d + c*d)*
#1 + d*#1^2 + #1^3 & , 1], I*ArcSinh[Sqrt[(-a + x)/(a - b)]], (a - b)/(a - c)] + EllipticPi[(a - b)/Root[a^2*d
- a*b*d - a*c*d + b*c*d + (-2*a*d + b*d + c*d)*#1 + d*#1^2 + #1^3 & , 2], I*ArcSinh[Sqrt[(-a + x)/(a - b)]],
(a - b)/(a - c)] - 2*EllipticPi[(a - b)/Root[a^2*d - a*b*d - a*c*d + b*c*d + (-2*a*d + b*d + c*d)*#1 + d*#1^2
+ #1^3 & , 3], I*ArcSinh[Sqrt[(-a + x)/(a - b)]], (a - b)/(a - c)])*Root[a^2*d - a*b*d - a*c*d + b*c*d + (-2*a
*d + b*d + c*d)*#1 + d*#1^2 + #1^3 & , 2] + (-2*EllipticF[I*ArcSinh[Sqrt[(-a + x)/(a - b)]], (a - b)/(a - c)]
+ EllipticPi[(a - b)/Root[a^2*d - a*b*d - a*c*d + b*c*d + (-2*a*d + b*d + c*d)*#1 + d*#1^2 + #1^3 & , 1], I*Ar
cSinh[Sqrt[(-a + x)/(a - b)]], (a - b)/(a - c)] + EllipticPi[(a - b)/Root[a^2*d - a*b*d - a*c*d + b*c*d + (-2*
a*d + b*d + c*d)*#1 + d*#1^2 + #1^3 & , 2], I*ArcSinh[Sqrt[(-a + x)/(a - b)]], (a - b)/(a - c)])*Root[a^2*d -
a*b*d - a*c*d + b*c*d + (-2*a*d + b*d + c*d)*#1 + d*#1^2 + #1^3 & , 1]*Root[a^2*d - a*b*d - a*c*d + b*c*d + (-
2*a*d + b*d + c*d)*#1 + d*#1^2 + #1^3 & , 2]^2 + 3*(a - b)*(a - c)*(EllipticPi[(a - b)/Root[a^2*d - a*b*d - a*
c*d + b*c*d + (-2*a*d + b*d + c*d)*#1 + d*#1^2 + #1^3 & , 1], I*ArcSinh[Sqrt[(-a + x)/(a - b)]], (a - b)/(a -
c)] - 2*EllipticPi[(a - b)/Root[a^2*d - a*b*d - a*c*d + b*c*d + (-2*a*d + b*d + c*d)*#1 + d*#1^2 + #1^3 & , 2]
, I*ArcSinh[Sqrt[(-a + x)/(a - b)]], (a - b)/(a - c)] + EllipticPi[(a - b)/Root[a^2*d - a*b*d - a*c*d + b*c*d
+ (-2*a*d + b*d + c*d)*#1 + d*#1^2 + #1^3 & , 3], I*ArcSinh[Sqrt[(-a + x)/(a - b)]], (a - b)/(a - c)])*Root[a^
2*d - a*b*d - a*c*d + b*c*d + (-2*a*d + b*d + c*d)*#1 + d*#1^2 + #1^3 & , 3] - 2*(2*a - b - c)*(2*EllipticPi[(
a - b)/Root[a^2*d - a*b*d - a*c*d + b*c*d + (-2*a*d + b*d + c*d)*#1 + d*#1^2 + #1^3 & , 1], I*ArcSinh[Sqrt[(-a
+ x)/(a - b)]], (a - b)/(a - c)] - EllipticPi[(a - b)/Root[a^2*d - a*b*d - a*c*d + b*c*d + (-2*a*d + b*d + c*
d)*#1 + d*#1^2 + #1^3 & , 2], I*ArcSinh[Sqrt[(-a + x)/(a - b)]], (a - b)/(a - c)] - EllipticPi[(a - b)/Root[a^
2*d - a*b*d - a*c*d + b*c*d + (-2*a*d + b*d + c*d)*#1 + d*#1^2 + #1^3 & , 3], I*ArcSinh[Sqrt[(-a + x)/(a - b)]
], (a - b)/(a - c)])*Root[a^2*d - a*b*d - a*c*d + b*c*d + (-2*a*d + b*d + c*d)*#1 + d*#1^2 + #1^3 & , 1]*Root[
a^2*d - a*b*d - a*c*d + b*c*d + (-2*a*d + b*d + c*d)*#1 + d*#1^2 + #1^3 & , 3] - 2*(EllipticF[I*ArcSinh[Sqrt[(
-a + x)/(a - b)]], (a - b)/(a - c)] - EllipticPi[(a - b)/Root[a^2*d - a*b*d - a*c*d + b*c*d + (-2*a*d + b*d +
c*d)*#1 + d*#1^2 + #1^3 & , 1], I*ArcSinh[Sqrt[(-a + x)/(a - b)]], (a - b)/(a - c)])*Root[a^2*d - a*b*d - a*c*
d + b*c*d + (-2*a*d + b*d + c*d)*#1 + d*#1^2 + #1^3 & , 1]^2*Root[a^2*d - a*b*d - a*c*d + b*c*d + (-2*a*d + b*
d + c*d)*#1 + d*#1^2 + #1^3 & , 3] - 2*(2*a - b - c)*(EllipticPi[(a - b)/Root[a^2*d - a*b*d - a*c*d + b*c*d +
(-2*a*d + b*d + c*d)*#1 + d*#1^2 + #1^3 & , 1], I*ArcSinh[Sqrt[(-a + x)/(a - b)]], (a - b)/(a - c)] - 2*Ellipt
icPi[(a - b)/Root[a^2*d - a*b*d - a*c*d + b*c*d + (-2*a*d + b*d + c*d)*#1 + d*#1^2 + #1^3 & , 2], I*ArcSinh[Sq
rt[(-a + x)/(a - b)]], (a - b)/(a - c)] + EllipticPi[(a - b)/Root[a^2*d - a*b*d - a*c*d + b*c*d + (-2*a*d + b*
d + c*d)*#1 + d*#1^2 + #1^3 & , 3], I*ArcSinh[Sqrt[(-a + x)/(a - b)]], (a - b)/(a - c)])*Root[a^2*d - a*b*d -
a*c*d + b*c*d + (-2*a*d + b*d + c*d)*#1 + d*#1^2 + #1^3 & , 2]*Root[a^2*d - a*b*d - a*c*d + b*c*d + (-2*a*d +
b*d + c*d)*#1 + d*#1^2 + #1^3 & , 3] + (EllipticPi[(a - b)/Root[a^2*d - a*b*d - a*c*d + b*c*d + (-2*a*d + b*d
+ c*d)*#1 + d*#1^2 + #1^3 & , 1], I*ArcSinh[Sqrt[(-a + x)/(a - b)]], (a - b)/(a - c)] - EllipticPi[(a - b)/Roo
t[a^2*d - a*b*d - a*c*d + b*c*d + (-2*a*d + b*d + c*d)*#1 + d*#1^2 + #1^3 & , 2], I*ArcSinh[Sqrt[(-a + x)/(a -
b)]], (a - b)/(a - c)])*Root[a^2*d - a*b*d - a*c*d + b*c*d + (-2*a*d + b*d + c*d)*#1 + d*#1^2 + #1^3 & , 2]^2
*Root[a^2*d - a*b*d - a*c*d + b*c*d + (-2*a*d + b*d + c*d)*#1 + d*#1^2 + #1^3 & , 3] + (EllipticPi[(a - b)/Roo
t[a^2*d - a*b*d - a*c*d + b*c*d + (-2*a*d + b*d + c*d)*#1 + d*#1^2 + #1^3 & , 1], I*ArcSinh[Sqrt[(-a + x)/(a -
b)]], (a - b)/(a - c)] - EllipticPi[(a - b)/Root[a^2*d - a*b*d - a*c*d + b*c*d + (-2*a*d + b*d + c*d)*#1 + d*
#1^2 + #1^3 & , 3], I*ArcSinh[Sqrt[(-a + x)/(a - b)]], (a - b)/(a - c)])*Root[a^2*d - a*b*d - a*c*d + b*c*d +
(-2*a*d + b*d + c*d)*#1 + d*#1^2 + #1^3 & , 1]*Root[a^2*d - a*b*d - a*c*d + b*c*d + (-2*a*d + b*d + c*d)*#1 +
d*#1^2 + #1^3 & , 3]^2 + (-2*EllipticF[I*ArcSinh[Sqrt[(-a + x)/(a - b)]], (a - b)/(a - c)] + EllipticPi[(a - b
)/Root[a^2*d - a*b*d - a*c*d + b*c*d + (-2*a*d + b*d + c*d)*#1 + d*#1^2 + #1^3 & , 1], I*ArcSinh[Sqrt[(-a + x)
/(a - b)]], (a - b)/(a - c)] + EllipticPi[(a - b)/Root[a^2*d - a*b*d - a*c*d + b*c*d + (-2*a*d + b*d + c*d)*#1
+ d*#1^2 + #1^3 & , 3], I*ArcSinh[Sqrt[(-a + x)/(a - b)]], (a - b)/(a - c)])*Root[a^2*d - a*b*d - a*c*d + b*c
*d + (-2*a*d + b*d + c*d)*#1 + d*#1^2 + #1^3 & , 2]*Root[a^2*d - a*b*d - a*c*d + b*c*d + (-2*a*d + b*d + c*d)*
#1 + d*#1^2 + #1^3 & , 3]^2))/(Sqrt[(-a + x)*(-b + x)*(-c + x)]*(Root[a^2*d - a*b*d - a*c*d + b*c*d + (-2*a*d
+ b*d + c*d)*#1 + d*#1^2 + #1^3 & , 1] - Root[a^2*d - a*b*d - a*c*d + b*c*d + (-2*a*d + b*d + c*d)*#1 + d*#1^2
+ #1^3 & , 2])*(Root[a^2*d - a*b*d - a*c*d + b*c*d + (-2*a*d + b*d + c*d)*#1 + d*#1^2 + #1^3 & , 1] - Root[a^
2*d - a*b*d - a*c*d + b*c*d + (-2*a*d + b*d + c*d)*#1 + d*#1^2 + #1^3 & , 3])*(Root[a^2*d - a*b*d - a*c*d + b*
c*d + (-2*a*d + b*d + c*d)*#1 + d*#1^2 + #1^3 & , 2] - Root[a^2*d - a*b*d - a*c*d + b*c*d + (-2*a*d + b*d + c*
d)*#1 + d*#1^2 + #1^3 & , 3]))

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IntegrateAlgebraic [A]  time = 3.15, size = 60, normalized size = 1.00 \begin {gather*} -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3}}{(a-x)^2}\right )}{\sqrt {d}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(a*(a*b + a*c - 3*b*c) + (-2*a^2 + a*b + a*c + 3*b*c)*x + (a - 2*b - 2*c)*x^2 + x^3)/(Sqrt[
(-a + x)*(-b + x)*(-c + x)]*(-a^3 - b*c*d + (3*a^2 + b*d + c*d)*x - (3*a + d)*x^2 + x^3)),x]

[Out]

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

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fricas [B]  time = 48.15, size = 638, normalized size = 10.63 \begin {gather*} \left [\frac {\log \left (\frac {a^{6} - 6 \, a^{3} b c d + b^{2} c^{2} d^{2} - 6 \, {\left (a - d\right )} x^{5} + x^{6} + {\left (15 \, a^{2} - 6 \, {\left (3 \, a + b + c\right )} d + d^{2}\right )} x^{4} - 2 \, {\left (10 \, a^{3} + {\left (b + c\right )} d^{2} - 3 \, {\left (3 \, a^{2} + 3 \, a b + {\left (3 \, a + b\right )} c\right )} d\right )} x^{3} + {\left (15 \, a^{4} + {\left (b^{2} + 4 \, b c + c^{2}\right )} d^{2} - 6 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, {\left (a^{2} + a b\right )} c\right )} d\right )} x^{2} - 4 \, {\left (a^{4} - a b c d - {\left (4 \, a - d\right )} x^{3} + x^{4} + {\left (6 \, a^{2} - {\left (a + b + c\right )} d\right )} x^{2} - {\left (4 \, a^{3} - {\left (a b + {\left (a + b\right )} c\right )} d\right )} x\right )} \sqrt {-a b c - {\left (a + b + c\right )} x^{2} + x^{3} + {\left (a b + {\left (a + b\right )} c\right )} x} \sqrt {d} - 2 \, {\left (3 \, a^{5} + {\left (b^{2} c + b c^{2}\right )} d^{2} - 3 \, {\left (a^{3} b + {\left (a^{3} + 3 \, a^{2} b\right )} c\right )} d\right )} x}{a^{6} + 2 \, a^{3} b c d + b^{2} c^{2} d^{2} - 2 \, {\left (3 \, a + d\right )} x^{5} + x^{6} + {\left (15 \, a^{2} + 2 \, {\left (3 \, a + b + c\right )} d + d^{2}\right )} x^{4} - 2 \, {\left (10 \, a^{3} + {\left (b + c\right )} d^{2} + {\left (3 \, a^{2} + 3 \, a b + {\left (3 \, a + b\right )} c\right )} d\right )} x^{3} + {\left (15 \, a^{4} + {\left (b^{2} + 4 \, b c + c^{2}\right )} d^{2} + 2 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, {\left (a^{2} + a b\right )} c\right )} d\right )} x^{2} - 2 \, {\left (3 \, a^{5} + {\left (b^{2} c + b c^{2}\right )} d^{2} + {\left (a^{3} b + {\left (a^{3} + 3 \, a^{2} b\right )} c\right )} d\right )} x}\right )}{2 \, \sqrt {d}}, \frac {\sqrt {-d} \arctan \left (-\frac {{\left (a^{3} - b c d + {\left (3 \, a - d\right )} x^{2} - x^{3} - {\left (3 \, a^{2} - {\left (b + c\right )} d\right )} x\right )} \sqrt {-a b c - {\left (a + b + c\right )} x^{2} + x^{3} + {\left (a b + {\left (a + b\right )} c\right )} x} \sqrt {-d}}{2 \, {\left (a^{2} b c d - {\left (2 \, a + b + c\right )} d x^{3} + d x^{4} + {\left (a^{2} + 2 \, a b + {\left (2 \, a + b\right )} c\right )} d x^{2} - {\left (a^{2} b + {\left (a^{2} + 2 \, a b\right )} c\right )} d x\right )}}\right )}{d}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*log((a^6 - 6*a^3*b*c*d + b^2*c^2*d^2 - 6*(a - d)*x^5 + x^6 + (15*a^2 - 6*(3*a + b + c)*d + d^2)*x^4 - 2*(
10*a^3 + (b + c)*d^2 - 3*(3*a^2 + 3*a*b + (3*a + b)*c)*d)*x^3 + (15*a^4 + (b^2 + 4*b*c + c^2)*d^2 - 6*(a^3 + 3
*a^2*b + 3*(a^2 + a*b)*c)*d)*x^2 - 4*(a^4 - a*b*c*d - (4*a - d)*x^3 + x^4 + (6*a^2 - (a + b + c)*d)*x^2 - (4*a
^3 - (a*b + (a + b)*c)*d)*x)*sqrt(-a*b*c - (a + b + c)*x^2 + x^3 + (a*b + (a + b)*c)*x)*sqrt(d) - 2*(3*a^5 + (
b^2*c + b*c^2)*d^2 - 3*(a^3*b + (a^3 + 3*a^2*b)*c)*d)*x)/(a^6 + 2*a^3*b*c*d + b^2*c^2*d^2 - 2*(3*a + d)*x^5 +
x^6 + (15*a^2 + 2*(3*a + b + c)*d + d^2)*x^4 - 2*(10*a^3 + (b + c)*d^2 + (3*a^2 + 3*a*b + (3*a + b)*c)*d)*x^3
+ (15*a^4 + (b^2 + 4*b*c + c^2)*d^2 + 2*(a^3 + 3*a^2*b + 3*(a^2 + a*b)*c)*d)*x^2 - 2*(3*a^5 + (b^2*c + b*c^2)*
d^2 + (a^3*b + (a^3 + 3*a^2*b)*c)*d)*x))/sqrt(d), sqrt(-d)*arctan(-1/2*(a^3 - b*c*d + (3*a - d)*x^2 - x^3 - (3
*a^2 - (b + c)*d)*x)*sqrt(-a*b*c - (a + b + c)*x^2 + x^3 + (a*b + (a + b)*c)*x)*sqrt(-d)/(a^2*b*c*d - (2*a + b
+ c)*d*x^3 + d*x^4 + (a^2 + 2*a*b + (2*a + b)*c)*d*x^2 - (a^2*b + (a^2 + 2*a*b)*c)*d*x))/d]

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giac [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((a*(a*b+a*c-3*b*c)+(-2*a^2+a*b+a*c+3*b*c)*x+(a-2*b-2*c)*x^2+x^3)/((-a+x)*(-b+x)*(-c+x))^(1/2)/(-a^3-
b*c*d+(3*a^2+b*d+c*d)*x-(3*a+d)*x^2+x^3),x, algorithm="giac")

[Out]

Timed out

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maple [C]  time = 0.13, size = 510, normalized size = 8.50

method	result	size
default	\(\frac {2 \left (b -c \right ) \sqrt {\frac {-c +x}{b -c}}\, \sqrt {\frac {-a +x}{c -a}}\, \sqrt {\frac {-b +x}{-b +c}}\, \EllipticF \left (\sqrt {\frac {-c +x}{b -c}}, \sqrt {\frac {-b +c}{c -a}}\right )}{\sqrt {-a b c +a b x +a c x -a \,x^{2}+b c x -b \,x^{2}-c \,x^{2}+x^{3}}}+2 \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{3}+\left (-3 a -d \right ) \textit {\_Z}^{2}+\left (3 a^{2}+b d +c d \right ) \textit {\_Z} -a^{3}-b c d \right )}{\sum }\frac {\left (4 \underline {\hspace {1.25 ex}}\alpha ^{2} a -2 \underline {\hspace {1.25 ex}}\alpha ^{2} b -2 \underline {\hspace {1.25 ex}}\alpha ^{2} c +\underline {\hspace {1.25 ex}}\alpha ^{2} d -5 \underline {\hspace {1.25 ex}}\alpha \,a^{2}+\underline {\hspace {1.25 ex}}\alpha a b +\underline {\hspace {1.25 ex}}\alpha a c +3 \underline {\hspace {1.25 ex}}\alpha b c -\underline {\hspace {1.25 ex}}\alpha b d -\underline {\hspace {1.25 ex}}\alpha c d +a^{3}+a^{2} b +a^{2} c -3 a b c +b c d \right ) \left (b -c \right ) \sqrt {\frac {-c +x}{b -c}}\, \sqrt {\frac {-a +x}{c -a}}\, \sqrt {\frac {-b +x}{-b +c}}\, \left (\underline {\hspace {1.25 ex}}\alpha ^{2}-3 \underline {\hspace {1.25 ex}}\alpha a +\underline {\hspace {1.25 ex}}\alpha c -\underline {\hspace {1.25 ex}}\alpha d +3 a^{2}-3 a c +b d +c^{2}\right ) \EllipticPi \left (\sqrt {\frac {-c +x}{b -c}}, \frac {\left (\underline {\hspace {1.25 ex}}\alpha ^{2}-3 \underline {\hspace {1.25 ex}}\alpha a +\underline {\hspace {1.25 ex}}\alpha c -\underline {\hspace {1.25 ex}}\alpha d +3 a^{2}-3 a c +b d +c^{2}\right ) \left (b -c \right )}{a^{3}-3 a^{2} c +3 a \,c^{2}-c^{3}}, \sqrt {\frac {-b +c}{c -a}}\right )}{\left (-3 \underline {\hspace {1.25 ex}}\alpha ^{2}+6 \underline {\hspace {1.25 ex}}\alpha a +2 \underline {\hspace {1.25 ex}}\alpha d -3 a^{2}-b d -c d \right ) \sqrt {-a b c +a b x +a c x -a \,x^{2}+b c x -b \,x^{2}-c \,x^{2}+x^{3}}\, \left (a^{3}-3 a^{2} c +3 a \,c^{2}-c^{3}\right )}\right )\)	\(510\)
elliptic	\(\frac {2 \left (b -c \right ) \sqrt {\frac {-c +x}{b -c}}\, \sqrt {\frac {-a +x}{c -a}}\, \sqrt {\frac {-b +x}{-b +c}}\, \EllipticF \left (\sqrt {\frac {-c +x}{b -c}}, \sqrt {\frac {-b +c}{c -a}}\right )}{\sqrt {-a b c +a b x +a c x -a \,x^{2}+b c x -b \,x^{2}-c \,x^{2}+x^{3}}}-2 \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{3}+\left (-3 a -d \right ) \textit {\_Z}^{2}+\left (3 a^{2}+b d +c d \right ) \textit {\_Z} -a^{3}-b c d \right )}{\sum }\frac {\left (-4 \underline {\hspace {1.25 ex}}\alpha ^{2} a +2 \underline {\hspace {1.25 ex}}\alpha ^{2} b +2 \underline {\hspace {1.25 ex}}\alpha ^{2} c -\underline {\hspace {1.25 ex}}\alpha ^{2} d +5 \underline {\hspace {1.25 ex}}\alpha \,a^{2}-\underline {\hspace {1.25 ex}}\alpha a b -\underline {\hspace {1.25 ex}}\alpha a c -3 \underline {\hspace {1.25 ex}}\alpha b c +\underline {\hspace {1.25 ex}}\alpha b d +\underline {\hspace {1.25 ex}}\alpha c d -a^{3}-a^{2} b -a^{2} c +3 a b c -b c d \right ) \left (b -c \right ) \sqrt {\frac {-c +x}{b -c}}\, \sqrt {\frac {-a +x}{c -a}}\, \sqrt {\frac {-b +x}{-b +c}}\, \left (\underline {\hspace {1.25 ex}}\alpha ^{2}-3 \underline {\hspace {1.25 ex}}\alpha a +\underline {\hspace {1.25 ex}}\alpha c -\underline {\hspace {1.25 ex}}\alpha d +3 a^{2}-3 a c +b d +c^{2}\right ) \EllipticPi \left (\sqrt {\frac {-c +x}{b -c}}, \frac {\left (\underline {\hspace {1.25 ex}}\alpha ^{2}-3 \underline {\hspace {1.25 ex}}\alpha a +\underline {\hspace {1.25 ex}}\alpha c -\underline {\hspace {1.25 ex}}\alpha d +3 a^{2}-3 a c +b d +c^{2}\right ) \left (b -c \right )}{a^{3}-3 a^{2} c +3 a \,c^{2}-c^{3}}, \sqrt {\frac {-b +c}{c -a}}\right )}{\left (-3 \underline {\hspace {1.25 ex}}\alpha ^{2}+6 \underline {\hspace {1.25 ex}}\alpha a +2 \underline {\hspace {1.25 ex}}\alpha d -3 a^{2}-b d -c d \right ) \sqrt {-a b c +a b x +a c x -a \,x^{2}+b c x -b \,x^{2}-c \,x^{2}+x^{3}}\, \left (a^{3}-3 a^{2} c +3 a \,c^{2}-c^{3}\right )}\right )\)	\(516\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*(a*b+a*c-3*b*c)+(-2*a^2+a*b+a*c+3*b*c)*x+(a-2*b-2*c)*x^2+x^3)/((-a+x)*(-b+x)*(-c+x))^(1/2)/(-a^3-b*c*d+
(3*a^2+b*d+c*d)*x-(3*a+d)*x^2+x^3),x,method=_RETURNVERBOSE)

[Out]

2*(b-c)*((-c+x)/(b-c))^(1/2)*((-a+x)/(c-a))^(1/2)*((-b+x)/(-b+c))^(1/2)/(-a*b*c+a*b*x+a*c*x-a*x^2+b*c*x-b*x^2-
c*x^2+x^3)^(1/2)*EllipticF(((-c+x)/(b-c))^(1/2),((-b+c)/(c-a))^(1/2))+2*sum((4*_alpha^2*a-2*_alpha^2*b-2*_alph
a^2*c+_alpha^2*d-5*_alpha*a^2+_alpha*a*b+_alpha*a*c+3*_alpha*b*c-_alpha*b*d-_alpha*c*d+a^3+a^2*b+a^2*c-3*a*b*c
+b*c*d)/(-3*_alpha^2+6*_alpha*a+2*_alpha*d-3*a^2-b*d-c*d)*(b-c)*((-c+x)/(b-c))^(1/2)*((-a+x)/(c-a))^(1/2)*((-b
+x)/(-b+c))^(1/2)/(-a*b*c+a*b*x+a*c*x-a*x^2+b*c*x-b*x^2-c*x^2+x^3)^(1/2)*(_alpha^2-3*_alpha*a+_alpha*c-_alpha*
d+3*a^2-3*a*c+b*d+c^2)/(a^3-3*a^2*c+3*a*c^2-c^3)*EllipticPi(((-c+x)/(b-c))^(1/2),(_alpha^2-3*_alpha*a+_alpha*c
-_alpha*d+3*a^2-3*a*c+b*d+c^2)*(b-c)/(a^3-3*a^2*c+3*a*c^2-c^3),((-b+c)/(c-a))^(1/2)),_alpha=RootOf(_Z^3+(-3*a-
d)*_Z^2+(3*a^2+b*d+c*d)*_Z-a^3-b*c*d))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\int \frac {{\left (a - 2 \, b - 2 \, c\right )} x^{2} + x^{3} + {\left (a b + a c - 3 \, b c\right )} a - {\left (2 \, a^{2} - a b - a c - 3 \, b c\right )} x}{{\left (a^{3} + b c d + {\left (3 \, a + d\right )} x^{2} - x^{3} - {\left (3 \, a^{2} + b d + c d\right )} x\right )} \sqrt {-{\left (a - x\right )} {\left (b - x\right )} {\left (c - x\right )}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-integrate(((a - 2*b - 2*c)*x^2 + x^3 + (a*b + a*c - 3*b*c)*a - (2*a^2 - a*b - a*c - 3*b*c)*x)/((a^3 + b*c*d +
(3*a + d)*x^2 - x^3 - (3*a^2 + b*d + c*d)*x)*sqrt(-(a - x)*(b - x)*(c - x))), x)

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mupad [B]  time = 1.73, size = 946, normalized size = 15.77 \begin {gather*} \left (\sum _{k=1}^3\left (-\frac {2\,\left (a-c\right )\,\sqrt {\frac {a-x}{a-c}}\,\sqrt {-\frac {c-x}{a-c}}\,\sqrt {\frac {b-x}{b-c}}\,\Pi \left (\frac {a-c}{\mathrm {root}\left (z^3-z^2\,\left (3\,a+d\right )+z\,\left (b\,d+c\,d+3\,a^2\right )-b\,c\,d-a^3,z,k\right )-c};\mathrm {asin}\left (\sqrt {-\frac {c-x}{a-c}}\right )\middle |\frac {a-c}{b-c}\right )\,\left (a^2\,b+a^2\,c+4\,a\,{\mathrm {root}\left (z^3-z^2\,\left (3\,a+d\right )+z\,\left (b\,d+c\,d+3\,a^2\right )-b\,c\,d-a^3,z,k\right )}^2-5\,a^2\,\mathrm {root}\left (z^3-z^2\,\left (3\,a+d\right )+z\,\left (b\,d+c\,d+3\,a^2\right )-b\,c\,d-a^3,z,k\right )-2\,b\,{\mathrm {root}\left (z^3-z^2\,\left (3\,a+d\right )+z\,\left (b\,d+c\,d+3\,a^2\right )-b\,c\,d-a^3,z,k\right )}^2-2\,c\,{\mathrm {root}\left (z^3-z^2\,\left (3\,a+d\right )+z\,\left (b\,d+c\,d+3\,a^2\right )-b\,c\,d-a^3,z,k\right )}^2+d\,{\mathrm {root}\left (z^3-z^2\,\left (3\,a+d\right )+z\,\left (b\,d+c\,d+3\,a^2\right )-b\,c\,d-a^3,z,k\right )}^2+a^3-3\,a\,b\,c+b\,c\,d+a\,b\,\mathrm {root}\left (z^3-z^2\,\left (3\,a+d\right )+z\,\left (b\,d+c\,d+3\,a^2\right )-b\,c\,d-a^3,z,k\right )+a\,c\,\mathrm {root}\left (z^3-z^2\,\left (3\,a+d\right )+z\,\left (b\,d+c\,d+3\,a^2\right )-b\,c\,d-a^3,z,k\right )+3\,b\,c\,\mathrm {root}\left (z^3-z^2\,\left (3\,a+d\right )+z\,\left (b\,d+c\,d+3\,a^2\right )-b\,c\,d-a^3,z,k\right )-b\,d\,\mathrm {root}\left (z^3-z^2\,\left (3\,a+d\right )+z\,\left (b\,d+c\,d+3\,a^2\right )-b\,c\,d-a^3,z,k\right )-c\,d\,\mathrm {root}\left (z^3-z^2\,\left (3\,a+d\right )+z\,\left (b\,d+c\,d+3\,a^2\right )-b\,c\,d-a^3,z,k\right )\right )}{\left (\mathrm {root}\left (z^3-z^2\,\left (3\,a+d\right )+z\,\left (b\,d+c\,d+3\,a^2\right )-b\,c\,d-a^3,z,k\right )-c\right )\,\sqrt {-\left (a-x\right )\,\left (b-x\right )\,\left (c-x\right )}\,\left (3\,a^2-6\,a\,\mathrm {root}\left (z^3-z^2\,\left (3\,a+d\right )+z\,\left (b\,d+c\,d+3\,a^2\right )-b\,c\,d-a^3,z,k\right )+3\,{\mathrm {root}\left (z^3-z^2\,\left (3\,a+d\right )+z\,\left (b\,d+c\,d+3\,a^2\right )-b\,c\,d-a^3,z,k\right )}^2-2\,d\,\mathrm {root}\left (z^3-z^2\,\left (3\,a+d\right )+z\,\left (b\,d+c\,d+3\,a^2\right )-b\,c\,d-a^3,z,k\right )+b\,d+c\,d\right )}\right )\right )+\frac {2\,\left (a-c\right )\,\mathrm {F}\left (\mathrm {asin}\left (\sqrt {-\frac {c-x}{a-c}}\right )\middle |\frac {a-c}{b-c}\right )\,\sqrt {\frac {a-x}{a-c}}\,\sqrt {-\frac {c-x}{a-c}}\,\sqrt {\frac {b-x}{b-c}}}{\sqrt {x^3+\left (-a-b-c\right )\,x^2+\left (a\,b+a\,c+b\,c\right )\,x-a\,b\,c}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

symsum(-(2*(a - c)*((a - x)/(a - c))^(1/2)*(-(c - x)/(a - c))^(1/2)*((b - x)/(b - c))^(1/2)*ellipticPi((a - c)
/(root(z^3 - z^2*(3*a + d) + z*(b*d + c*d + 3*a^2) - b*c*d - a^3, z, k) - c), asin((-(c - x)/(a - c))^(1/2)),
(a - c)/(b - c))*(a^2*b + a^2*c + 4*a*root(z^3 - z^2*(3*a + d) + z*(b*d + c*d + 3*a^2) - b*c*d - a^3, z, k)^2
- 5*a^2*root(z^3 - z^2*(3*a + d) + z*(b*d + c*d + 3*a^2) - b*c*d - a^3, z, k) - 2*b*root(z^3 - z^2*(3*a + d) +
z*(b*d + c*d + 3*a^2) - b*c*d - a^3, z, k)^2 - 2*c*root(z^3 - z^2*(3*a + d) + z*(b*d + c*d + 3*a^2) - b*c*d -
a^3, z, k)^2 + d*root(z^3 - z^2*(3*a + d) + z*(b*d + c*d + 3*a^2) - b*c*d - a^3, z, k)^2 + a^3 - 3*a*b*c + b*
c*d + a*b*root(z^3 - z^2*(3*a + d) + z*(b*d + c*d + 3*a^2) - b*c*d - a^3, z, k) + a*c*root(z^3 - z^2*(3*a + d)
+ z*(b*d + c*d + 3*a^2) - b*c*d - a^3, z, k) + 3*b*c*root(z^3 - z^2*(3*a + d) + z*(b*d + c*d + 3*a^2) - b*c*d
- a^3, z, k) - b*d*root(z^3 - z^2*(3*a + d) + z*(b*d + c*d + 3*a^2) - b*c*d - a^3, z, k) - c*d*root(z^3 - z^2
*(3*a + d) + z*(b*d + c*d + 3*a^2) - b*c*d - a^3, z, k)))/((root(z^3 - z^2*(3*a + d) + z*(b*d + c*d + 3*a^2) -
b*c*d - a^3, z, k) - c)*(-(a - x)*(b - x)*(c - x))^(1/2)*(b*d + c*d - 6*a*root(z^3 - z^2*(3*a + d) + z*(b*d +
c*d + 3*a^2) - b*c*d - a^3, z, k) - 2*d*root(z^3 - z^2*(3*a + d) + z*(b*d + c*d + 3*a^2) - b*c*d - a^3, z, k)
+ 3*a^2 + 3*root(z^3 - z^2*(3*a + d) + z*(b*d + c*d + 3*a^2) - b*c*d - a^3, z, k)^2)), k, 1, 3) + (2*(a - c)*
ellipticF(asin((-(c - x)/(a - c))^(1/2)), (a - c)/(b - c))*((a - x)/(a - c))^(1/2)*(-(c - x)/(a - c))^(1/2)*((
b - x)/(b - c))^(1/2))/(x*(a*b + a*c + b*c) - x^2*(a + b + c) + x^3 - a*b*c)^(1/2)

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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((a*(a*b+a*c-3*b*c)+(-2*a**2+a*b+a*c+3*b*c)*x+(a-2*b-2*c)*x**2+x**3)/((-a+x)*(-b+x)*(-c+x))**(1/2)/(-
a**3-b*c*d+(3*a**2+b*d+c*d)*x-(3*a+d)*x**2+x**3),x)

[Out]

Timed out

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