\(\int \frac {-a b-a c+2 b c+(2 a-b-c) x}{\sqrt [3]{(-a+x) (-b+x) (-c+x)} (-b c+a^2 d+(b+c-2 a d) x+(-1+d) x^2)} \, dx\) [2880]

Optimal result

Integrand size = 74, antiderivative size = 311 \[ \int \frac {-a b-a c+2 b c+(2 a-b-c) x}{\sqrt [3]{(-a+x) (-b+x) (-c+x)} \left (-b c+a^2 d+(b+c-2 a d) x+(-1+d) x^2\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3}}{-2 a \sqrt [3]{d}+2 \sqrt [3]{d} x+\sqrt [3]{-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3}}\right )}{\sqrt [3]{d}}+\frac {\log \left (a \sqrt [3]{d}-\sqrt [3]{d} x+\sqrt [3]{-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3}\right )}{\sqrt [3]{d}}-\frac {\log \left (a^2 d^{2/3}-2 a d^{2/3} x+d^{2/3} x^2+\left (-a \sqrt [3]{d}+\sqrt [3]{d} x\right ) \sqrt [3]{-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3}+\left (-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3\right )^{2/3}\right )}{2 \sqrt [3]{d}} \]

[Out]

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

Rubi [F]

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

[In]

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

[Out]

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

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

Mathematica [F]

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

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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