\(\int \frac {-a b-a c+3 b c+2 (a-b-c) x+x^2}{\sqrt [4]{(-a+x) (-b+x) (-c+x)} (-a^3-b c d+(3 a^2+b d+c d) x-(3 a+d) x^2+x^3)} \, dx\) [1787]

Optimal result

Integrand size = 88, antiderivative size = 121 \[ \int \frac {-a b-a c+3 b c+2 (a-b-c) x+x^2}{\sqrt [4]{(-a+x) (-b+x) (-c+x)} \left (-a^3-b c d+\left (3 a^2+b d+c d\right ) x-(3 a+d) x^2+x^3\right )} \, dx=-\frac {2 \arctan \left (\frac {\sqrt [4]{d} \sqrt [4]{-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3}}{a-x}\right )}{d^{3/4}}+\frac {2 \text {arctanh}\left (\frac {\sqrt [4]{d} \sqrt [4]{-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3}}{a-x}\right )}{d^{3/4}} \]

[Out]

-2*arctan(d^(1/4)*(-a*b*c+(a*b+a*c+b*c)*x+(-a-b-c)*x^2+x^3)^(1/4)/(a-x))/d^(3/4)+2*arctanh(d^(1/4)*(-a*b*c+(a*
b+a*c+b*c)*x+(-a-b-c)*x^2+x^3)^(1/4)/(a-x))/d^(3/4)

Rubi [F]

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

[In]

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

[Out]

(-8*a*(a - b - c)*(-a + x)^(1/4)*(-b + x)^(1/4)*(-c + x)^(1/4)*Defer[Subst][Defer[Int][x^2/((a - b + x^4)^(1/4
)*(a - c + x^4)^(1/4)*(a^2*(1 + (b*c - a*(b + c))/a^2)*d + 2*a*(1 - (b + c)/(2*a))*d*x^4 + d*x^8 - x^12)), x],
x, (-a + x)^(1/4)])/(-((a - x)*(b - x)*(c - x)))^(1/4) - (8*(a - b - c)*(-a + x)^(1/4)*(-b + x)^(1/4)*(-c + x
)^(1/4)*Defer[Subst][Defer[Int][x^6/((a - b + x^4)^(1/4)*(a - c + x^4)^(1/4)*(a^2*(1 + (b*c - a*(b + c))/a^2)*
d + 2*a*(1 - (b + c)/(2*a))*d*x^4 + d*x^8 - x^12)), x], x, (-a + x)^(1/4)])/(-((a - x)*(b - x)*(c - x)))^(1/4)
+ (4*a^2*(-a + x)^(1/4)*(-b + x)^(1/4)*(-c + x)^(1/4)*Defer[Subst][Defer[Int][x^2/((a - b + x^4)^(1/4)*(a - c
+ x^4)^(1/4)*(-(a^2*(1 + (b*c - a*(b + c))/a^2)*d) - 2*a*(1 - (b + c)/(2*a))*d*x^4 - d*x^8 + x^12)), x], x, (
-a + x)^(1/4)])/(-((a - x)*(b - x)*(c - x)))^(1/4) + (8*a*(-a + x)^(1/4)*(-b + x)^(1/4)*(-c + x)^(1/4)*Defer[S
ubst][Defer[Int][x^6/((a - b + x^4)^(1/4)*(a - c + x^4)^(1/4)*(-(a^2*(1 + (b*c - a*(b + c))/a^2)*d) - 2*a*(1 -
(b + c)/(2*a))*d*x^4 - d*x^8 + x^12)), x], x, (-a + x)^(1/4)])/(-((a - x)*(b - x)*(c - x)))^(1/4) + (4*(-a +
x)^(1/4)*(-b + x)^(1/4)*(-c + x)^(1/4)*Defer[Subst][Defer[Int][x^10/((a - b + x^4)^(1/4)*(a - c + x^4)^(1/4)*(
-(a^2*(1 + (b*c - a*(b + c))/a^2)*d) - 2*a*(1 - (b + c)/(2*a))*d*x^4 - d*x^8 + x^12)), x], x, (-a + x)^(1/4)])
/(-((a - x)*(b - x)*(c - x)))^(1/4) - (4*(3*b*c - a*(b + c))*(-a + x)^(1/4)*(-b + x)^(1/4)*(-c + x)^(1/4)*Defe
r[Subst][Defer[Int][x^2/((a - b + x^4)^(1/4)*(a - c + x^4)^(1/4)*(a^3*(1 + (b*c*d)/a^3) - (3*a^2 + (b + c)*d)*
(a + x^4) + (3*a + d)*(a + x^4)^2 - (a + x^4)^3)), x], x, (-a + x)^(1/4)])/(-((a - x)*(b - x)*(c - x)))^(1/4)

Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt [4]{-a+x} \sqrt [4]{-b+x} \sqrt [4]{-c+x}\right ) \int \frac {-a b-a c+3 b c+2 (a-b-c) x+x^2}{\sqrt [4]{-a+x} \sqrt [4]{-b+x} \sqrt [4]{-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 [4]{(-a+x) (-b+x) (-c+x)}} \\ & = \frac {\left (\sqrt [4]{-a+x} \sqrt [4]{-b+x} \sqrt [4]{-c+x}\right ) \int \frac {-3 b c+a (b+c)-2 (a-b-c) x-x^2}{\sqrt [4]{-a+x} \sqrt [4]{-b+x} \sqrt [4]{-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 [4]{(-a+x) (-b+x) (-c+x)}} \\ & = \frac {\left (\sqrt [4]{-a+x} \sqrt [4]{-b+x} \sqrt [4]{-c+x}\right ) \int \left (\frac {a (b+c) \left (1-\frac {3 b c}{a b+a c}\right )}{\sqrt [4]{-a+x} \sqrt [4]{-b+x} \sqrt [4]{-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 [4]{-a+x} \sqrt [4]{-b+x} \sqrt [4]{-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 [4]{-a+x} \sqrt [4]{-b+x} \sqrt [4]{-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 [4]{(-a+x) (-b+x) (-c+x)}} \\ & = \frac {\left (\sqrt [4]{-a+x} \sqrt [4]{-b+x} \sqrt [4]{-c+x}\right ) \int \frac {x^2}{\sqrt [4]{-a+x} \sqrt [4]{-b+x} \sqrt [4]{-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 [4]{(-a+x) (-b+x) (-c+x)}}-\frac {\left (2 (a-b-c) \sqrt [4]{-a+x} \sqrt [4]{-b+x} \sqrt [4]{-c+x}\right ) \int \frac {x}{\sqrt [4]{-a+x} \sqrt [4]{-b+x} \sqrt [4]{-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 [4]{(-a+x) (-b+x) (-c+x)}}+\frac {\left ((-3 b c+a (b+c)) \sqrt [4]{-a+x} \sqrt [4]{-b+x} \sqrt [4]{-c+x}\right ) \int \frac {1}{\sqrt [4]{-a+x} \sqrt [4]{-b+x} \sqrt [4]{-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 [4]{(-a+x) (-b+x) (-c+x)}} \\ & = \frac {\left (4 \sqrt [4]{-a+x} \sqrt [4]{-b+x} \sqrt [4]{-c+x}\right ) \text {Subst}\left (\int \frac {x^2 \left (a+x^4\right )^2}{\sqrt [4]{a-b+x^4} \sqrt [4]{a-c+x^4} \left (-a^2 d+a d \left (b+c-2 x^4\right )+b d \left (-c+x^4\right )+x^4 \left (c d-d x^4+x^8\right )\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x) (-c+x)}}-\frac {\left (8 (a-b-c) \sqrt [4]{-a+x} \sqrt [4]{-b+x} \sqrt [4]{-c+x}\right ) \text {Subst}\left (\int \frac {x^2 \left (a+x^4\right )}{\sqrt [4]{a-b+x^4} \sqrt [4]{a-c+x^4} \left (a^3+b c d-\left (3 a^2+(b+c) d\right ) \left (a+x^4\right )+(3 a+d) \left (a+x^4\right )^2-\left (a+x^4\right )^3\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x) (-c+x)}}+\frac {\left (4 (-3 b c+a (b+c)) \sqrt [4]{-a+x} \sqrt [4]{-b+x} \sqrt [4]{-c+x}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt [4]{a-b+x^4} \sqrt [4]{a-c+x^4} \left (a^3+b c d-\left (3 a^2+(b+c) d\right ) \left (a+x^4\right )+(3 a+d) \left (a+x^4\right )^2-\left (a+x^4\right )^3\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x) (-c+x)}} \\ & = \frac {\left (4 \sqrt [4]{-a+x} \sqrt [4]{-b+x} \sqrt [4]{-c+x}\right ) \text {Subst}\left (\int \left (\frac {a^2 x^2}{\sqrt [4]{a-b+x^4} \sqrt [4]{a-c+x^4} \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^4-d x^8+x^{12}\right )}+\frac {2 a x^6}{\sqrt [4]{a-b+x^4} \sqrt [4]{a-c+x^4} \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^4-d x^8+x^{12}\right )}+\frac {x^{10}}{\sqrt [4]{a-b+x^4} \sqrt [4]{a-c+x^4} \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^4-d x^8+x^{12}\right )}\right ) \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x) (-c+x)}}-\frac {\left (8 (a-b-c) \sqrt [4]{-a+x} \sqrt [4]{-b+x} \sqrt [4]{-c+x}\right ) \text {Subst}\left (\int \frac {x^2 \left (a+x^4\right )}{\sqrt [4]{a-b+x^4} \sqrt [4]{a-c+x^4} \left (a^3 \left (1+\frac {b c d}{a^3}\right )-\left (3 a^2+(b+c) d\right ) \left (a+x^4\right )+(3 a+d) \left (a+x^4\right )^2-\left (a+x^4\right )^3\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x) (-c+x)}}+\frac {\left (4 (-3 b c+a (b+c)) \sqrt [4]{-a+x} \sqrt [4]{-b+x} \sqrt [4]{-c+x}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt [4]{a-b+x^4} \sqrt [4]{a-c+x^4} \left (a^3 \left (1+\frac {b c d}{a^3}\right )-\left (3 a^2+(b+c) d\right ) \left (a+x^4\right )+(3 a+d) \left (a+x^4\right )^2-\left (a+x^4\right )^3\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x) (-c+x)}} \\ & = \frac {\left (4 \sqrt [4]{-a+x} \sqrt [4]{-b+x} \sqrt [4]{-c+x}\right ) \text {Subst}\left (\int \frac {x^{10}}{\sqrt [4]{a-b+x^4} \sqrt [4]{a-c+x^4} \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^4-d x^8+x^{12}\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x) (-c+x)}}+\frac {\left (8 a \sqrt [4]{-a+x} \sqrt [4]{-b+x} \sqrt [4]{-c+x}\right ) \text {Subst}\left (\int \frac {x^6}{\sqrt [4]{a-b+x^4} \sqrt [4]{a-c+x^4} \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^4-d x^8+x^{12}\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x) (-c+x)}}+\frac {\left (4 a^2 \sqrt [4]{-a+x} \sqrt [4]{-b+x} \sqrt [4]{-c+x}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt [4]{a-b+x^4} \sqrt [4]{a-c+x^4} \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^4-d x^8+x^{12}\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x) (-c+x)}}-\frac {\left (8 (a-b-c) \sqrt [4]{-a+x} \sqrt [4]{-b+x} \sqrt [4]{-c+x}\right ) \text {Subst}\left (\int \left (\frac {a x^2}{\sqrt [4]{a-b+x^4} \sqrt [4]{a-c+x^4} \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^4+d x^8-x^{12}\right )}+\frac {x^6}{\sqrt [4]{a-b+x^4} \sqrt [4]{a-c+x^4} \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^4+d x^8-x^{12}\right )}\right ) \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x) (-c+x)}}+\frac {\left (4 (-3 b c+a (b+c)) \sqrt [4]{-a+x} \sqrt [4]{-b+x} \sqrt [4]{-c+x}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt [4]{a-b+x^4} \sqrt [4]{a-c+x^4} \left (a^3 \left (1+\frac {b c d}{a^3}\right )-\left (3 a^2+(b+c) d\right ) \left (a+x^4\right )+(3 a+d) \left (a+x^4\right )^2-\left (a+x^4\right )^3\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x) (-c+x)}} \\ & = \frac {\left (4 \sqrt [4]{-a+x} \sqrt [4]{-b+x} \sqrt [4]{-c+x}\right ) \text {Subst}\left (\int \frac {x^{10}}{\sqrt [4]{a-b+x^4} \sqrt [4]{a-c+x^4} \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^4-d x^8+x^{12}\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x) (-c+x)}}+\frac {\left (8 a \sqrt [4]{-a+x} \sqrt [4]{-b+x} \sqrt [4]{-c+x}\right ) \text {Subst}\left (\int \frac {x^6}{\sqrt [4]{a-b+x^4} \sqrt [4]{a-c+x^4} \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^4-d x^8+x^{12}\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x) (-c+x)}}+\frac {\left (4 a^2 \sqrt [4]{-a+x} \sqrt [4]{-b+x} \sqrt [4]{-c+x}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt [4]{a-b+x^4} \sqrt [4]{a-c+x^4} \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^4-d x^8+x^{12}\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x) (-c+x)}}-\frac {\left (8 (a-b-c) \sqrt [4]{-a+x} \sqrt [4]{-b+x} \sqrt [4]{-c+x}\right ) \text {Subst}\left (\int \frac {x^6}{\sqrt [4]{a-b+x^4} \sqrt [4]{a-c+x^4} \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^4+d x^8-x^{12}\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x) (-c+x)}}-\frac {\left (8 a (a-b-c) \sqrt [4]{-a+x} \sqrt [4]{-b+x} \sqrt [4]{-c+x}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt [4]{a-b+x^4} \sqrt [4]{a-c+x^4} \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^4+d x^8-x^{12}\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x) (-c+x)}}+\frac {\left (4 (-3 b c+a (b+c)) \sqrt [4]{-a+x} \sqrt [4]{-b+x} \sqrt [4]{-c+x}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt [4]{a-b+x^4} \sqrt [4]{a-c+x^4} \left (a^3 \left (1+\frac {b c d}{a^3}\right )-\left (3 a^2+(b+c) d\right ) \left (a+x^4\right )+(3 a+d) \left (a+x^4\right )^2-\left (a+x^4\right )^3\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x) (-c+x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 10.38 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.63 \[ \int \frac {-a b-a c+3 b c+2 (a-b-c) x+x^2}{\sqrt [4]{(-a+x) (-b+x) (-c+x)} \left (-a^3-b c d+\left (3 a^2+b d+c d\right ) x-(3 a+d) x^2+x^3\right )} \, dx=\frac {2 \left (\arctan \left (\frac {\sqrt [4]{d} \sqrt [4]{(-a+x) (-b+x) (-c+x)}}{-a+x}\right )+\text {arctanh}\left (\frac {\sqrt [4]{d} \sqrt [4]{(-a+x) (-b+x) (-c+x)}}{a-x}\right )\right )}{d^{3/4}} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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