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

Optimal. Leaf size=369 \[ -\frac {i \left (\sqrt {3}-i\right ) \log \left (\sqrt [3]{-1} \left (a^2-2 a x+x^2\right )-d^{2/3} \left (x^2 (-a-b-c)+x (a b+a c+b c)-a b c+x^3\right )^{2/3}+(-1)^{2/3} \sqrt [3]{d} (a-x) \sqrt [3]{x^2 (-a-b-c)+x (a b+a c+b c)-a b c+x^3}\right )}{4 d^{2/3}}+\frac {\left (1+i \sqrt {3}\right ) \log \left (\sqrt [3]{d} \sqrt [3]{x^2 (-a-b-c)+x (a b+a c+b c)-a b c+x^3}+(-1)^{2/3} (a-x)\right )}{2 d^{2/3}}+\frac {\sqrt {\frac {1}{2} \left (-3+3 i \sqrt {3}\right )} \tan ^{-1}\left (\frac {3 \sqrt [3]{d} \sqrt [3]{x^2 (-a-b-c)+x (a b+a c+b c)-a b c+x^3}}{\sqrt {3} \sqrt [3]{d} \sqrt [3]{x^2 (-a-b-c)+x (a b+a c+b c)-a b c+x^3}+\sqrt {3} a-3 i a-\sqrt {3} x+3 i x}\right )}{d^{2/3}} \]

________________________________________________________________________________________

Rubi [F]  time = 10.15, 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 b+a c-2 b c+(-2 a+b+c) x}{\sqrt [3]{(-a+x) (-b+x) (-c+x)} \left (a^2-b c d+(-2 a+b d+c d) x+(1-d) x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [F]  time = 9.93, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a b+a c-2 b c+(-2 a+b+c) x}{\sqrt [3]{(-a+x) (-b+x) (-c+x)} \left (a^2-b c d+(-2 a+b d+c d) x+(1-d) x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 6.34, size = 455, normalized size = 1.23 \begin {gather*} \frac {\sqrt {\frac {1}{2} \left (-3+3 i \sqrt {3}\right )} \tan ^{-1}\left (\frac {3 \sqrt [3]{d} \sqrt [3]{-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3}}{-3 i a+\sqrt {3} a+3 i x-\sqrt {3} x+\sqrt {3} \sqrt [3]{d} \sqrt [3]{-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3}}\right )}{d^{2/3}}+\frac {\left (1+i \sqrt {3}\right ) \log \left (-a+\sqrt {3} (i a-i x)+x+2 \sqrt [3]{d} \sqrt [3]{-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3}\right )}{2 d^{2/3}}-\frac {i \left (-i+\sqrt {3}\right ) \log \left (-a^2+2 a x-x^2+\sqrt [3]{d} (a-x) \sqrt [3]{-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3}+2 d^{2/3} \left (-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3\right )^{2/3}+\sqrt {3} \left (-i a^2+2 i a x-i x^2+\sqrt [3]{d} (-i a+i x) \sqrt [3]{-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3}\right )\right )}{4 d^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 d + {\left (d - 1\right )} x^{2} - a^{2} - {\left (b d + c d - 2 \, a\right )} x\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [F]  time = 0.22, size = 0, normalized size = 0.00 \[\int \frac {a b +a c -2 b c +\left (-2 a +b +c \right ) x}{\left (\left (-a +x \right ) \left (-b +x \right ) \left (-c +x \right )\right )^{\frac {1}{3}} \left (a^{2}-b c d +\left (b d +c d -2 a \right ) x +\left (1-d \right ) x^{2}\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\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 d + {\left (d - 1\right )} x^{2} - a^{2} - {\left (b d + c d - 2 \, a\right )} x\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \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\,d-2\,a+c\,d\right )+a^2-x^2\,\left (d-1\right )-b\,c\,d\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________