3.8.81 \(\int \frac {a (a b+a c-3 b c)+(-2 a^2+a b+a c+3 b c) x+(a-2 b-2 c) x^2+x^3}{\sqrt {(-a+x) (-b+x) (-c+x)} (-b c-a^3 d+(b+c+3 a^2 d) x-(1+3 a d) x^2+d x^3)} \, dx\)

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

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Rubi [F]  time = 53.09, 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 (-b c-a^3 d+\left (b+c+3 a^2 d\right ) x-(1+3 a d) x^2+d 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)]*(-(b*c) - a^3*d + (b + c + 3*a^2*d)*x - (1 + 3*a*d)*x^2 + d*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)
*d*Sqrt[((a - c)*(b - x))/((a - b)*(c - x))]*Sqrt[-((a - x)*(b - x)*(c - x))]) - (2*(a - b)*(a - c)*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) + 2*a*(1 - (b + c)/(2*a))*x^2 + x^4 - d*x^6)), x], x, Sqrt[-a + x]])/(d*Sqrt[-((a - x)*(b - x)*
(c - x))]) - (2*(2*a - b - c + a^2*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) + 2*a*(1 - (b + c)/(2*a))*x^2 + x^4 - d*x^6))
, x], x, Sqrt[-a + x]])/(d*Sqrt[-((a - x)*(b - x)*(c - x))]) - (2*(1 + 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) + 2*a
*(1 - (b + c)/(2*a))*x^2 + x^4 - d*x^6)), x], x, Sqrt[-a + x]])/(d*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)) - 2*a*(1 - (b + c)/(2*a))*x^2 - x^4 + d*x^6)), x], x, Sqrt[-a + x]])/S
qrt[-((a - x)*(b - x)*(c - x))] + (4*(a - b - c)*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)) - 2*a*(1 - (b + c)/(2*a))*x^2 -
x^4 + d*x^6)), x], x, Sqrt[-a + x]])/Sqrt[-((a - x)*(b - x)*(c - x))] + (2*(3*b*c - a*(b + c))*Sqrt[-a + x]*S
qrt[-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^
2)) + (b + c)*x^2 - x^4 + d*x^6 + a*(b + c - 2*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 (-b c-a^3 d+\left (b+c+3 a^2 d\right ) x-(1+3 a d) x^2+d 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 (-b c-a^3 d+\left (b+c+3 a^2 d\right ) x-(1+3 a d) x^2+d 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 (-b c-a^3 d+\left (b+c+3 a^2 d\right ) x-(1+3 a d) x^2+d 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 (b c+a^3 d-\left (b+c+3 a^2 d\right ) x+(1+3 a d) x^2-d 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 (b c+a^3 d-\left (b+c+3 a^2 d\right ) x+(1+3 a d) x^2-d x^3\right )}+\frac {2 (-a+b+c) x \sqrt {-a+x}}{\sqrt {-b+x} \sqrt {-c+x} \left (b c+a^3 d-\left (b+c+3 a^2 d\right ) x+(1+3 a d) x^2-d x^3\right )}+\frac {x^2 \sqrt {-a+x}}{\sqrt {-b+x} \sqrt {-c+x} \left (-b c-a^3 d+\left (b+c+3 a^2 d\right ) x-(1+3 a d) x^2+d 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 (-b c-a^3 d+\left (b+c+3 a^2 d\right ) x-(1+3 a d) x^2+d 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 (b c+a^3 d-\left (b+c+3 a^2 d\right ) x+(1+3 a d) x^2-d 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 (b c+a^3 d-\left (b+c+3 a^2 d\right ) x+(1+3 a d) x^2-d 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-b c+b x^2+c x^2-x^4+d x^6+a \left (b+c-2 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-b c+b x^2+c x^2-x^4+d x^6+a \left (b+c-2 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-b c+b x^2+c x^2-x^4+d x^6+a \left (b+c-2 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 {x^2 \left (a+x^2\right )^2}{\sqrt {a-b+x^2} \sqrt {a-c+x^2} \left (-a^2-b c+(b+c) x^2-x^4+d x^6+a \left (b+c-2 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-b c+(b+c) x^2-x^4+d x^6+a \left (b+c-2 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-b c+(b+c) x^2-x^4+d x^6+a \left (b+c-2 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 {x^2 \left (a+x^2\right )^2}{\sqrt {a-b+x^2} \sqrt {a-c+x^2} \left (-a^2 \left (1+\frac {b c}{a^2}\right )+(b+c) x^2-x^4+d x^6+a \left (b+c-2 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 \left (1+\frac {b c}{a^2}\right )-(b+c) x^2+x^4-d x^6-a \left (b+c-2 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 \left (1+\frac {b c}{a^2}\right )+(b+c) x^2-x^4+d x^6+a \left (b+c-2 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}{d \sqrt {a-b+x^2} \sqrt {a-c+x^2}}+\frac {(a-b) (a-c)+\left (2 a-b-c+a^2 d\right ) x^2+(1+2 a d) x^4}{d \sqrt {a-b+x^2} \sqrt {a-c+x^2} \left (-a^2 \left (1+\frac {b c}{a^2}\right )+(b+c) x^2-x^4+d x^6+a \left (b+c-2 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 )-2 a \left (1-\frac {b+c}{2 a}\right ) x^2-x^4+d 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 )-2 a \left (1-\frac {b+c}{2 a}\right ) x^2-x^4+d 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 \left (1+\frac {b c}{a^2}\right )+(b+c) x^2-x^4+d x^6+a \left (b+c-2 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 )-2 a \left (1-\frac {b+c}{2 a}\right ) x^2-x^4+d 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 )-2 a \left (1-\frac {b+c}{2 a}\right ) x^2-x^4+d 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 \left (1+\frac {b c}{a^2}\right )+(b+c) x^2-x^4+d x^6+a \left (b+c-2 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 )}{d \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)+\left (2 a-b-c+a^2 d\right ) x^2+(1+2 a d) x^4}{\sqrt {a-b+x^2} \sqrt {a-c+x^2} \left (-a^2 \left (1+\frac {b c}{a^2}\right )+(b+c) x^2-x^4+d x^6+a \left (b+c-2 x^2\right )\right )} \, dx,x,\sqrt {-a+x}\right )}{d \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) d \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 )-2 a \left (1-\frac {b+c}{2 a}\right ) x^2-x^4+d 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 )-2 a \left (1-\frac {b+c}{2 a}\right ) x^2-x^4+d 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 \left (1+\frac {b c}{a^2}\right )+(b+c) x^2-x^4+d x^6+a \left (b+c-2 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 {(a-b) (-a+c)}{\sqrt {a-b+x^2} \sqrt {a-c+x^2} \left (a^2 \left (1+\frac {b c-a (b+c)}{a^2}\right )+2 a \left (1-\frac {b+c}{2 a}\right ) x^2+x^4-d x^6\right )}+\frac {\left (-2 a+b+c-a^2 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 )+2 a \left (1-\frac {b+c}{2 a}\right ) x^2+x^4-d x^6\right )}+\frac {(-1-2 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 )+2 a \left (1-\frac {b+c}{2 a}\right ) x^2+x^4-d x^6\right )}\right ) \, dx,x,\sqrt {-a+x}\right )}{d \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) d \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 )-2 a \left (1-\frac {b+c}{2 a}\right ) x^2-x^4+d 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 )-2 a \left (1-\frac {b+c}{2 a}\right ) x^2-x^4+d 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 \left (1+\frac {b c}{a^2}\right )+(b+c) x^2-x^4+d x^6+a \left (b+c-2 x^2\right )\right )} \, dx,x,\sqrt {-a+x}\right )}{\sqrt {(-a+x) (-b+x) (-c+x)}}-\frac {\left (2 (a-b) (a-c) \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 )+2 a \left (1-\frac {b+c}{2 a}\right ) x^2+x^4-d x^6\right )} \, dx,x,\sqrt {-a+x}\right )}{d \sqrt {(-a+x) (-b+x) (-c+x)}}+\frac {\left (2 (-1-2 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 )+2 a \left (1-\frac {b+c}{2 a}\right ) x^2+x^4-d x^6\right )} \, dx,x,\sqrt {-a+x}\right )}{d \sqrt {(-a+x) (-b+x) (-c+x)}}+\frac {\left (2 \left (-2 a+b+c-a^2 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 )+2 a \left (1-\frac {b+c}{2 a}\right ) x^2+x^4-d x^6\right )} \, dx,x,\sqrt {-a+x}\right )}{d \sqrt {(-a+x) (-b+x) (-c+x)}}\\ \end {align*}

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Mathematica [C]  time = 4.95, size = 4752, normalized size = 79.20 \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)]*(-(b*c) - a^3*d + (b + c + 3*a^2*d)*x - (1 + 3*a*d)*x^2 + d*x^3)),x]

[Out]

((2*I)*(-a + x)*Sqrt[(-c + x)/(a - c)]*(-2*a*EllipticF[I*ArcSinh[Sqrt[(-a + x)/(a - b)]], (a - b)/(a - c)] + b
*EllipticF[I*ArcSinh[Sqrt[(-a + x)/(a - b)]], (a - b)/(a - c)] + c*EllipticF[I*ArcSinh[Sqrt[(-a + x)/(a - b)]]
, (a - b)/(a - c)] - 3*a^2*d*EllipticF[I*ArcSinh[Sqrt[(-a + x)/(a - b)]], (a - b)/(a - c)] + 3*a*b*d*EllipticF
[I*ArcSinh[Sqrt[(-a + x)/(a - b)]], (a - b)/(a - c)] + 3*a*c*d*EllipticF[I*ArcSinh[Sqrt[(-a + x)/(a - b)]], (a
- b)/(a - c)] - 3*b*c*d*EllipticF[I*ArcSinh[Sqrt[(-a + x)/(a - b)]], (a - b)/(a - c)] + 2*a*EllipticPi[(a - b
)/Root[a^2 - a*b - a*c + b*c + (-2*a + b + c)*#1 + #1^2 + d*#1^3 & , 1], I*ArcSinh[Sqrt[(-a + x)/(a - b)]], (a
- b)/(a - c)] - b*EllipticPi[(a - b)/Root[a^2 - a*b - a*c + b*c + (-2*a + b + c)*#1 + #1^2 + d*#1^3 & , 1], I
*ArcSinh[Sqrt[(-a + x)/(a - b)]], (a - b)/(a - c)] - c*EllipticPi[(a - b)/Root[a^2 - a*b - a*c + b*c + (-2*a +
b + c)*#1 + #1^2 + d*#1^3 & , 1], I*ArcSinh[Sqrt[(-a + x)/(a - b)]], (a - b)/(a - c)] + 11*a^2*d*EllipticPi[(
a - b)/Root[a^2 - a*b - a*c + b*c + (-2*a + b + c)*#1 + #1^2 + d*#1^3 & , 1], I*ArcSinh[Sqrt[(-a + x)/(a - b)]
], (a - b)/(a - c)] - 11*a*b*d*EllipticPi[(a - b)/Root[a^2 - a*b - a*c + b*c + (-2*a + b + c)*#1 + #1^2 + d*#1
^3 & , 1], I*ArcSinh[Sqrt[(-a + x)/(a - b)]], (a - b)/(a - c)] + 2*b^2*d*EllipticPi[(a - b)/Root[a^2 - a*b - a
*c + b*c + (-2*a + b + c)*#1 + #1^2 + d*#1^3 & , 1], I*ArcSinh[Sqrt[(-a + x)/(a - b)]], (a - b)/(a - c)] - 11*
a*c*d*EllipticPi[(a - b)/Root[a^2 - a*b - a*c + b*c + (-2*a + b + c)*#1 + #1^2 + d*#1^3 & , 1], I*ArcSinh[Sqrt
[(-a + x)/(a - b)]], (a - b)/(a - c)] + 7*b*c*d*EllipticPi[(a - b)/Root[a^2 - a*b - a*c + b*c + (-2*a + b + c)
*#1 + #1^2 + d*#1^3 & , 1], I*ArcSinh[Sqrt[(-a + x)/(a - b)]], (a - b)/(a - c)] + 2*c^2*d*EllipticPi[(a - b)/R
oot[a^2 - a*b - a*c + b*c + (-2*a + b + c)*#1 + #1^2 + d*#1^3 & , 1], I*ArcSinh[Sqrt[(-a + x)/(a - b)]], (a -
b)/(a - c)] - 5*a^2*d*EllipticPi[(a - b)/Root[a^2 - a*b - a*c + b*c + (-2*a + b + c)*#1 + #1^2 + d*#1^3 & , 2]
, I*ArcSinh[Sqrt[(-a + x)/(a - b)]], (a - b)/(a - c)] + 5*a*b*d*EllipticPi[(a - b)/Root[a^2 - a*b - a*c + b*c
+ (-2*a + b + c)*#1 + #1^2 + d*#1^3 & , 2], I*ArcSinh[Sqrt[(-a + x)/(a - b)]], (a - b)/(a - c)] - 2*b^2*d*Elli
pticPi[(a - b)/Root[a^2 - a*b - a*c + b*c + (-2*a + b + c)*#1 + #1^2 + d*#1^3 & , 2], I*ArcSinh[Sqrt[(-a + x)/
(a - b)]], (a - b)/(a - c)] + 5*a*c*d*EllipticPi[(a - b)/Root[a^2 - a*b - a*c + b*c + (-2*a + b + c)*#1 + #1^2
+ d*#1^3 & , 2], I*ArcSinh[Sqrt[(-a + x)/(a - b)]], (a - b)/(a - c)] - b*c*d*EllipticPi[(a - b)/Root[a^2 - a*
b - a*c + b*c + (-2*a + b + c)*#1 + #1^2 + d*#1^3 & , 2], I*ArcSinh[Sqrt[(-a + x)/(a - b)]], (a - b)/(a - c)]
- 2*c^2*d*EllipticPi[(a - b)/Root[a^2 - a*b - a*c + b*c + (-2*a + b + c)*#1 + #1^2 + d*#1^3 & , 2], I*ArcSinh[
Sqrt[(-a + x)/(a - b)]], (a - b)/(a - c)] - 3*a^2*d*EllipticPi[(a - b)/Root[a^2 - a*b - a*c + b*c + (-2*a + b
+ c)*#1 + #1^2 + d*#1^3 & , 3], I*ArcSinh[Sqrt[(-a + x)/(a - b)]], (a - b)/(a - c)] + 3*a*b*d*EllipticPi[(a -
b)/Root[a^2 - a*b - a*c + b*c + (-2*a + b + c)*#1 + #1^2 + d*#1^3 & , 3], I*ArcSinh[Sqrt[(-a + x)/(a - b)]], (
a - b)/(a - c)] + 3*a*c*d*EllipticPi[(a - b)/Root[a^2 - a*b - a*c + b*c + (-2*a + b + c)*#1 + #1^2 + d*#1^3 &
, 3], I*ArcSinh[Sqrt[(-a + x)/(a - b)]], (a - b)/(a - c)] - 3*b*c*d*EllipticPi[(a - b)/Root[a^2 - a*b - a*c +
b*c + (-2*a + b + c)*#1 + #1^2 + d*#1^3 & , 3], I*ArcSinh[Sqrt[(-a + x)/(a - b)]], (a - b)/(a - c)] + 3*(a - b
)*(a - c)*d^2*(EllipticPi[(a - b)/Root[a^2 - a*b - a*c + b*c + (-2*a + b + c)*#1 + #1^2 + d*#1^3 & , 1], I*Arc
Sinh[Sqrt[(-a + x)/(a - b)]], (a - b)/(a - c)] + EllipticPi[(a - b)/Root[a^2 - a*b - a*c + b*c + (-2*a + b + c
)*#1 + #1^2 + d*#1^3 & , 2], I*ArcSinh[Sqrt[(-a + x)/(a - b)]], (a - b)/(a - c)] - 2*EllipticPi[(a - b)/Root[a
^2 - a*b - a*c + b*c + (-2*a + b + c)*#1 + #1^2 + d*#1^3 & , 3], I*ArcSinh[Sqrt[(-a + x)/(a - b)]], (a - b)/(a
- c)])*Root[a^2 - a*b - a*c + b*c + (-2*a + b + c)*#1 + #1^2 + d*#1^3 & , 2] + d^2*(2*EllipticF[I*ArcSinh[Sqr
t[(-a + x)/(a - b)]], (a - b)/(a - c)] - EllipticPi[(a - b)/Root[a^2 - a*b - a*c + b*c + (-2*a + b + c)*#1 + #
1^2 + d*#1^3 & , 1], I*ArcSinh[Sqrt[(-a + x)/(a - b)]], (a - b)/(a - c)] - EllipticPi[(a - b)/Root[a^2 - a*b -
a*c + b*c + (-2*a + b + c)*#1 + #1^2 + d*#1^3 & , 2], I*ArcSinh[Sqrt[(-a + x)/(a - b)]], (a - b)/(a - c)])*Ro
ot[a^2 - a*b - a*c + b*c + (-2*a + b + c)*#1 + #1^2 + d*#1^3 & , 1]*Root[a^2 - a*b - a*c + b*c + (-2*a + b + c
)*#1 + #1^2 + d*#1^3 & , 2]^2 - 3*(a - b)*(a - c)*d^2*(EllipticPi[(a - b)/Root[a^2 - a*b - a*c + b*c + (-2*a +
b + c)*#1 + #1^2 + d*#1^3 & , 1], I*ArcSinh[Sqrt[(-a + x)/(a - b)]], (a - b)/(a - c)] - 2*EllipticPi[(a - b)/
Root[a^2 - a*b - a*c + b*c + (-2*a + b + c)*#1 + #1^2 + d*#1^3 & , 2], I*ArcSinh[Sqrt[(-a + x)/(a - b)]], (a -
b)/(a - c)] + EllipticPi[(a - b)/Root[a^2 - a*b - a*c + b*c + (-2*a + b + c)*#1 + #1^2 + d*#1^3 & , 3], I*Arc
Sinh[Sqrt[(-a + x)/(a - b)]], (a - b)/(a - c)])*Root[a^2 - a*b - a*c + b*c + (-2*a + b + c)*#1 + #1^2 + d*#1^3
& , 3] + 2*(2*a - b - c)*d^2*(2*EllipticPi[(a - b)/Root[a^2 - a*b - a*c + b*c + (-2*a + b + c)*#1 + #1^2 + d*
#1^3 & , 1], I*ArcSinh[Sqrt[(-a + x)/(a - b)]], (a - b)/(a - c)] - EllipticPi[(a - b)/Root[a^2 - a*b - a*c + b
*c + (-2*a + b + c)*#1 + #1^2 + d*#1^3 & , 2], I*ArcSinh[Sqrt[(-a + x)/(a - b)]], (a - b)/(a - c)] - EllipticP
i[(a - b)/Root[a^2 - a*b - a*c + b*c + (-2*a + b + c)*#1 + #1^2 + d*#1^3 & , 3], I*ArcSinh[Sqrt[(-a + x)/(a -
b)]], (a - b)/(a - c)])*Root[a^2 - a*b - a*c + b*c + (-2*a + b + c)*#1 + #1^2 + d*#1^3 & , 1]*Root[a^2 - a*b -
a*c + b*c + (-2*a + b + c)*#1 + #1^2 + d*#1^3 & , 3] + 2*d^2*(EllipticF[I*ArcSinh[Sqrt[(-a + x)/(a - b)]], (a
- b)/(a - c)] - EllipticPi[(a - b)/Root[a^2 - a*b - a*c + b*c + (-2*a + b + c)*#1 + #1^2 + d*#1^3 & , 1], I*A
rcSinh[Sqrt[(-a + x)/(a - b)]], (a - b)/(a - c)])*Root[a^2 - a*b - a*c + b*c + (-2*a + b + c)*#1 + #1^2 + d*#1
^3 & , 1]^2*Root[a^2 - a*b - a*c + b*c + (-2*a + b + c)*#1 + #1^2 + d*#1^3 & , 3] + 2*(2*a - b - c)*d^2*(Ellip
ticPi[(a - b)/Root[a^2 - a*b - a*c + b*c + (-2*a + b + c)*#1 + #1^2 + d*#1^3 & , 1], I*ArcSinh[Sqrt[(-a + x)/(
a - b)]], (a - b)/(a - c)] - 2*EllipticPi[(a - b)/Root[a^2 - a*b - a*c + b*c + (-2*a + b + c)*#1 + #1^2 + d*#1
^3 & , 2], I*ArcSinh[Sqrt[(-a + x)/(a - b)]], (a - b)/(a - c)] + EllipticPi[(a - b)/Root[a^2 - a*b - a*c + b*c
+ (-2*a + b + c)*#1 + #1^2 + d*#1^3 & , 3], I*ArcSinh[Sqrt[(-a + x)/(a - b)]], (a - b)/(a - c)])*Root[a^2 - a
*b - a*c + b*c + (-2*a + b + c)*#1 + #1^2 + d*#1^3 & , 2]*Root[a^2 - a*b - a*c + b*c + (-2*a + b + c)*#1 + #1^
2 + d*#1^3 & , 3] - d^2*(EllipticPi[(a - b)/Root[a^2 - a*b - a*c + b*c + (-2*a + b + c)*#1 + #1^2 + d*#1^3 & ,
1], I*ArcSinh[Sqrt[(-a + x)/(a - b)]], (a - b)/(a - c)] - EllipticPi[(a - b)/Root[a^2 - a*b - a*c + b*c + (-2
*a + b + c)*#1 + #1^2 + d*#1^3 & , 2], I*ArcSinh[Sqrt[(-a + x)/(a - b)]], (a - b)/(a - c)])*Root[a^2 - a*b - a
*c + b*c + (-2*a + b + c)*#1 + #1^2 + d*#1^3 & , 2]^2*Root[a^2 - a*b - a*c + b*c + (-2*a + b + c)*#1 + #1^2 +
d*#1^3 & , 3] - d^2*(EllipticPi[(a - b)/Root[a^2 - a*b - a*c + b*c + (-2*a + b + c)*#1 + #1^2 + d*#1^3 & , 1],
I*ArcSinh[Sqrt[(-a + x)/(a - b)]], (a - b)/(a - c)] - EllipticPi[(a - b)/Root[a^2 - a*b - a*c + b*c + (-2*a +
b + c)*#1 + #1^2 + d*#1^3 & , 3], I*ArcSinh[Sqrt[(-a + x)/(a - b)]], (a - b)/(a - c)])*Root[a^2 - a*b - a*c +
b*c + (-2*a + b + c)*#1 + #1^2 + d*#1^3 & , 1]*Root[a^2 - a*b - a*c + b*c + (-2*a + b + c)*#1 + #1^2 + d*#1^3
& , 3]^2 + d^2*(2*EllipticF[I*ArcSinh[Sqrt[(-a + x)/(a - b)]], (a - b)/(a - c)] - EllipticPi[(a - b)/Root[a^2
- a*b - a*c + b*c + (-2*a + b + c)*#1 + #1^2 + d*#1^3 & , 1], I*ArcSinh[Sqrt[(-a + x)/(a - b)]], (a - b)/(a -
c)] - EllipticPi[(a - b)/Root[a^2 - a*b - a*c + b*c + (-2*a + b + c)*#1 + #1^2 + d*#1^3 & , 3], I*ArcSinh[Sqr
t[(-a + x)/(a - b)]], (a - b)/(a - c)])*Root[a^2 - a*b - a*c + b*c + (-2*a + b + c)*#1 + #1^2 + d*#1^3 & , 2]*
Root[a^2 - a*b - a*c + b*c + (-2*a + b + c)*#1 + #1^2 + d*#1^3 & , 3]^2))/(d^3*Sqrt[(a - x)/(b - x)]*Sqrt[(-a
+ x)*(-b + x)*(-c + x)]*(Root[a^2 - a*b - a*c + b*c + (-2*a + b + c)*#1 + #1^2 + d*#1^3 & , 1] - Root[a^2 - a*
b - a*c + b*c + (-2*a + b + c)*#1 + #1^2 + d*#1^3 & , 2])*(Root[a^2 - a*b - a*c + b*c + (-2*a + b + c)*#1 + #1
^2 + d*#1^3 & , 1] - Root[a^2 - a*b - a*c + b*c + (-2*a + b + c)*#1 + #1^2 + d*#1^3 & , 3])*(Root[a^2 - a*b -
a*c + b*c + (-2*a + b + c)*#1 + #1^2 + d*#1^3 & , 2] - Root[a^2 - a*b - a*c + b*c + (-2*a + b + c)*#1 + #1^2 +
d*#1^3 & , 3]))

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IntegrateAlgebraic [A]  time = 3.13, size = 60, normalized size = 1.00 \begin {gather*} -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3}}{\sqrt {d} (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)]*(-(b*c) - a^3*d + (b + c + 3*a^2*d)*x - (1 + 3*a*d)*x^2 + d*x^3)),x]

[Out]

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

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fricas [B]  time = 89.10, size = 651, normalized size = 10.85 \begin {gather*} \left [\frac {\log \left (\frac {a^{6} d^{2} + d^{2} x^{6} - 6 \, a^{3} b c d - 6 \, {\left (a d^{2} - d\right )} x^{5} + {\left (15 \, a^{2} d^{2} - 6 \, {\left (3 \, a + b + c\right )} d + 1\right )} x^{4} + b^{2} c^{2} - 2 \, {\left (10 \, a^{3} d^{2} - 3 \, {\left (3 \, a^{2} + 3 \, a b + {\left (3 \, a + b\right )} c\right )} d + b + c\right )} x^{3} + {\left (15 \, a^{4} d^{2} + b^{2} + 4 \, b c + c^{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} d + d x^{4} - {\left (4 \, a d - 1\right )} x^{3} - a b c + {\left (6 \, a^{2} d - a - b - c\right )} x^{2} - {\left (4 \, a^{3} d - a b - {\left (a + b\right )} c\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} d^{2} + b^{2} c + b c^{2} - 3 \, {\left (a^{3} b + {\left (a^{3} + 3 \, a^{2} b\right )} c\right )} d\right )} x}{a^{6} d^{2} + d^{2} x^{6} + 2 \, a^{3} b c d - 2 \, {\left (3 \, a d^{2} + d\right )} x^{5} + {\left (15 \, a^{2} d^{2} + 2 \, {\left (3 \, a + b + c\right )} d + 1\right )} x^{4} + b^{2} c^{2} - 2 \, {\left (10 \, a^{3} d^{2} + {\left (3 \, a^{2} + 3 \, a b + {\left (3 \, a + b\right )} c\right )} d + b + c\right )} x^{3} + {\left (15 \, a^{4} d^{2} + b^{2} + 4 \, b c + c^{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} d^{2} + b^{2} c + b c^{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} d - d x^{3} + {\left (3 \, a d - 1\right )} x^{2} - b c - {\left (3 \, a^{2} d - b - c\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)/(-b*c-
a^3*d+(3*a^2*d+b+c)*x-(3*a*d+1)*x^2+d*x^3),x, algorithm="fricas")

[Out]

[1/2*log((a^6*d^2 + d^2*x^6 - 6*a^3*b*c*d - 6*(a*d^2 - d)*x^5 + (15*a^2*d^2 - 6*(3*a + b + c)*d + 1)*x^4 + b^2
*c^2 - 2*(10*a^3*d^2 - 3*(3*a^2 + 3*a*b + (3*a + b)*c)*d + b + c)*x^3 + (15*a^4*d^2 + b^2 + 4*b*c + c^2 - 6*(a
^3 + 3*a^2*b + 3*(a^2 + a*b)*c)*d)*x^2 - 4*(a^4*d + d*x^4 - (4*a*d - 1)*x^3 - a*b*c + (6*a^2*d - a - b - c)*x^
2 - (4*a^3*d - a*b - (a + b)*c)*x)*sqrt(-a*b*c - (a + b + c)*x^2 + x^3 + (a*b + (a + b)*c)*x)*sqrt(d) - 2*(3*a
^5*d^2 + b^2*c + b*c^2 - 3*(a^3*b + (a^3 + 3*a^2*b)*c)*d)*x)/(a^6*d^2 + d^2*x^6 + 2*a^3*b*c*d - 2*(3*a*d^2 + d
)*x^5 + (15*a^2*d^2 + 2*(3*a + b + c)*d + 1)*x^4 + b^2*c^2 - 2*(10*a^3*d^2 + (3*a^2 + 3*a*b + (3*a + b)*c)*d +
b + c)*x^3 + (15*a^4*d^2 + b^2 + 4*b*c + c^2 + 2*(a^3 + 3*a^2*b + 3*(a^2 + a*b)*c)*d)*x^2 - 2*(3*a^5*d^2 + b^
2*c + b*c^2 + (a^3*b + (a^3 + 3*a^2*b)*c)*d)*x))/sqrt(d), sqrt(-d)*arctan(-1/2*(a^3*d - d*x^3 + (3*a*d - 1)*x^
2 - b*c - (3*a^2*d - b - c)*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(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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)/(-b*c-
a^3*d+(3*a^2*d+b+c)*x-(3*a*d+1)*x^2+d*x^3),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Eval
uation time: 27.35Not invertible Error: Bad Argument Value

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maple [C]  time = 0.14, size = 534, normalized size = 8.90

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 )}{d \sqrt {-a b c +a b x +a c x -a \,x^{2}+b c x -b \,x^{2}-c \,x^{2}+x^{3}}}+\frac {2 \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (d \,\textit {\_Z}^{3}+\left (-3 a d -1\right ) \textit {\_Z}^{2}+\left (3 a^{2} d +b +c \right ) \textit {\_Z} -a^{3} d -b c \right )}{\sum }\frac {\left (4 \underline {\hspace {1.25 ex}}\alpha ^{2} a d -2 \underline {\hspace {1.25 ex}}\alpha ^{2} b d -2 \underline {\hspace {1.25 ex}}\alpha ^{2} c d -5 \underline {\hspace {1.25 ex}}\alpha \,a^{2} d +\underline {\hspace {1.25 ex}}\alpha a b d +\underline {\hspace {1.25 ex}}\alpha a c d +3 \underline {\hspace {1.25 ex}}\alpha b c d +a^{3} d +a^{2} b d +a^{2} c d -3 a b c d +\underline {\hspace {1.25 ex}}\alpha ^{2}-\underline {\hspace {1.25 ex}}\alpha b -\underline {\hspace {1.25 ex}}\alpha c +b c \right ) \left (b -c \right ) \left (\underline {\hspace {1.25 ex}}\alpha ^{2} d -3 \underline {\hspace {1.25 ex}}\alpha a d +\underline {\hspace {1.25 ex}}\alpha c d +3 a^{2} d -3 a c d +c^{2} d -\underline {\hspace {1.25 ex}}\alpha +b \right ) \sqrt {\frac {-c +x}{b -c}}\, \sqrt {\frac {-a +x}{c -a}}\, \sqrt {\frac {-b +x}{-b +c}}\, \EllipticPi \left (\sqrt {\frac {-c +x}{b -c}}, \frac {\left (\underline {\hspace {1.25 ex}}\alpha ^{2} d -3 \underline {\hspace {1.25 ex}}\alpha a d +\underline {\hspace {1.25 ex}}\alpha c d +3 a^{2} d -3 a c d +c^{2} d -\underline {\hspace {1.25 ex}}\alpha +b \right ) \left (b -c \right )}{d \left (a^{3}-3 a^{2} c +3 a \,c^{2}-c^{3}\right )}, \sqrt {\frac {-b +c}{c -a}}\right )}{\left (-3 \underline {\hspace {1.25 ex}}\alpha ^{2} d +6 \underline {\hspace {1.25 ex}}\alpha a d -3 a^{2} d +2 \underline {\hspace {1.25 ex}}\alpha -b -c \right ) \left (a^{3}-3 a^{2} c +3 a \,c^{2}-c^{3}\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}}}\right )}{d^{2}}\)	\(534\)
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 )}{d \sqrt {-a b c +a b x +a c x -a \,x^{2}+b c x -b \,x^{2}-c \,x^{2}+x^{3}}}+\frac {2 \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (d \,\textit {\_Z}^{3}+\left (-3 a d -1\right ) \textit {\_Z}^{2}+\left (3 a^{2} d +b +c \right ) \textit {\_Z} -a^{3} d -b c \right )}{\sum }\frac {\left (-4 \underline {\hspace {1.25 ex}}\alpha ^{2} a d +2 \underline {\hspace {1.25 ex}}\alpha ^{2} b d +2 \underline {\hspace {1.25 ex}}\alpha ^{2} c d +5 \underline {\hspace {1.25 ex}}\alpha \,a^{2} d -\underline {\hspace {1.25 ex}}\alpha a b d -\underline {\hspace {1.25 ex}}\alpha a c d -3 \underline {\hspace {1.25 ex}}\alpha b c d -a^{3} d -a^{2} b d -a^{2} c d +3 a b c d -\underline {\hspace {1.25 ex}}\alpha ^{2}+\underline {\hspace {1.25 ex}}\alpha b +\underline {\hspace {1.25 ex}}\alpha c -b c \right ) \left (b -c \right ) \left (\underline {\hspace {1.25 ex}}\alpha ^{2} d -3 \underline {\hspace {1.25 ex}}\alpha a d +\underline {\hspace {1.25 ex}}\alpha c d +3 a^{2} d -3 a c d +c^{2} d -\underline {\hspace {1.25 ex}}\alpha +b \right ) \sqrt {\frac {-c +x}{b -c}}\, \sqrt {\frac {-a +x}{c -a}}\, \sqrt {\frac {-b +x}{-b +c}}\, \EllipticPi \left (\sqrt {\frac {-c +x}{b -c}}, \frac {\left (\underline {\hspace {1.25 ex}}\alpha ^{2} d -3 \underline {\hspace {1.25 ex}}\alpha a d +\underline {\hspace {1.25 ex}}\alpha c d +3 a^{2} d -3 a c d +c^{2} d -\underline {\hspace {1.25 ex}}\alpha +b \right ) \left (b -c \right )}{d \left (a^{3}-3 a^{2} c +3 a \,c^{2}-c^{3}\right )}, \sqrt {\frac {-b +c}{c -a}}\right )}{\left (3 \underline {\hspace {1.25 ex}}\alpha ^{2} d -6 \underline {\hspace {1.25 ex}}\alpha a d +3 a^{2} d -2 \underline {\hspace {1.25 ex}}\alpha +b +c \right ) \left (a^{3}-3 a^{2} c +3 a \,c^{2}-c^{3}\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}}}\right )}{d^{2}}\)	\(536\)

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

[Out]

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

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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} d - d x^{3} + {\left (3 \, a d + 1\right )} x^{2} + b c - {\left (3 \, a^{2} d + b + c\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)/(-b*c-
a^3*d+(3*a^2*d+b+c)*x-(3*a*d+1)*x^2+d*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*d - d*x^3
+ (3*a*d + 1)*x^2 + b*c - (3*a^2*d + b + c)*x)*sqrt(-(a - x)*(b - x)*(c - x))), x)

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mupad [B]  time = 19.26, size = 569, normalized size = 9.48 \begin {gather*} \frac {\ln \left (\frac {\left (a-b-c+x+a^2\,d+d\,x^2-2\,\sqrt {d}\,\sqrt {-\left (a-x\right )\,\left (b-x\right )\,\left (c-x\right )}-2\,a\,d\,x\right )\,\left (b\,c^2+b^2\,c+a^4\,d-a\,x^2+2\,b\,x^2-b^2\,x+2\,c\,x^2-c^2\,x+2\,d\,x^4-x^3+a^5\,d^2-d^2\,x^5+3\,a^2\,d\,x^2+5\,a\,d^2\,x^4-5\,a^4\,d^2\,x-2\,a^2\,\sqrt {d}\,\sqrt {-\left (a-x\right )\,\left (b-x\right )\,\left (c-x\right )}-a\,b\,c+a\,b\,x+a\,c\,x-3\,b\,c\,x-10\,a^2\,d^2\,x^3+10\,a^3\,d^2\,x^2-a^3\,b\,d-a^3\,c\,d-4\,a\,d\,x^3-2\,a^3\,d\,x-2\,b\,d\,x^3-2\,c\,d\,x^3+3\,a^2\,b\,c\,d+3\,a\,b\,d\,x^2+3\,a\,c\,d\,x^2+3\,b\,c\,d\,x^2+2\,a\,b\,\sqrt {d}\,\sqrt {-\left (a-x\right )\,\left (b-x\right )\,\left (c-x\right )}+2\,a\,c\,\sqrt {d}\,\sqrt {-\left (a-x\right )\,\left (b-x\right )\,\left (c-x\right )}-2\,b\,c\,\sqrt {d}\,\sqrt {-\left (a-x\right )\,\left (b-x\right )\,\left (c-x\right )}-6\,a\,b\,c\,d\,x\right )}{\left (b\,c-b\,x-c\,x+a^3\,d-d\,x^3+x^2+3\,a\,d\,x^2-3\,a^2\,d\,x\right )\,\left (a^4\,d^2-4\,a^3\,d^2\,x+2\,a^3\,d-2\,a^2\,b\,d-2\,a^2\,c\,d+6\,a^2\,d^2\,x^2-2\,a^2\,d\,x+a^2+4\,a\,b\,c\,d-2\,a\,b-2\,a\,c-4\,a\,d^2\,x^3+2\,a\,d\,x^2+2\,a\,x+b^2-4\,b\,c\,d\,x+2\,b\,c+2\,b\,d\,x^2-2\,b\,x+c^2+2\,c\,d\,x^2-2\,c\,x+d^2\,x^4-2\,d\,x^3+x^2\right )}\right )}{\sqrt {d}} \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)*(b*c - x*(b + c + 3*a^2*d) + a^3*d - d*x^3 + x^2*(3*a*d + 1))),x)

[Out]

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

[Out]

Timed out

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