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

Optimal. Leaf size=262 \[ -\frac {\log \left (a^2+d^{2/3} \left (x^2 (-a-b-c)+x (a b+a c+b c)-a b c+x^3\right )^{2/3}+\left (\sqrt [3]{d} x-a \sqrt [3]{d}\right ) \sqrt [3]{x^2 (-a-b-c)+x (a b+a c+b c)-a b c+x^3}-2 a x+x^2\right )}{2 \sqrt [3]{d}}+\frac {\log \left (\sqrt [3]{d} \sqrt [3]{x^2 (-a-b-c)+x (a b+a c+b c)-a b c+x^3}+a-x\right )}{\sqrt [3]{d}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} a-\sqrt {3} x}{-2 \sqrt [3]{d} \sqrt [3]{x^2 (-a-b-c)+x (a b+a c+b c)-a b c+x^3}+a-x}\right )}{\sqrt [3]{d}} \]

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Rubi [F]  time = 12.90, 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-2 b c)-2 \left (a^2-b c\right ) x+(2 a-b-c) x^2}{((-a+x) (-b+x) (-c+x))^{2/3} \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*(a*b + a*c - 2*b*c) - 2*(a^2 - b*c)*x + (2*a - b - c)*x^2)/(((-a + x)*(-b + x)*(-c + x))^(2/3)*(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)^(2/3)*(-b + x)^(2/
3)*(-c + x)^(2/3)*Defer[Int][(-a + x)^(1/3)/((-b + x)^(2/3)*(-c + x)^(2/3)*(-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)))^(2/3)
+ ((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)^(2/3)*(-b + x)^
(2/3)*(-c + x)^(2/3)*Defer[Int][(-a + x)^(1/3)/((-b + x)^(2/3)*(-c + x)^(2/3)*(-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)))^(2
/3)

Rubi steps

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

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Mathematica [F]  time = 3.50, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a (a b+a c-2 b c)-2 \left (a^2-b c\right ) x+(2 a-b-c) x^2}{((-a+x) (-b+x) (-c+x))^{2/3} \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*(a*b + a*c - 2*b*c) - 2*(a^2 - b*c)*x + (2*a - b - c)*x^2)/(((-a + x)*(-b + x)*(-c + x))^(2/3)*(a
^2 - b*c*d + (-2*a + b*d + c*d)*x + (1 - d)*x^2)),x]

[Out]

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

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IntegrateAlgebraic [A]  time = 7.17, size = 262, normalized size = 1.00 \begin {gather*} \frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} a-\sqrt {3} x}{a-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 )}{\sqrt [3]{d}}+\frac {\log \left (a-x+\sqrt [3]{d} \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^2-2 a x+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}+d^{2/3} \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}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

[Out]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {{\left (2 \, a - b - c\right )} x^{2} + {\left (a b + a c - 2 \, b c\right )} a - 2 \, {\left (a^{2} - b c\right )} x}{\left (-{\left (a - x\right )} {\left (b - x\right )} {\left (c - x\right )}\right )^{\frac {2}{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*(a*b+a*c-2*b*c)-2*(a^2-b*c)*x+(2*a-b-c)*x^2)/((-a+x)*(-b+x)*(-c+x))^(2/3)/(a^2-b*c*d+(b*d+c*d-2*a
)*x+(1-d)*x^2),x, algorithm="giac")

[Out]

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

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maple [F]  time = 0.27, size = 0, normalized size = 0.00 \[\int \frac {a \left (a b +a c -2 b c \right )-2 \left (a^{2}-b c \right ) x +\left (2 a -b -c \right ) x^{2}}{\left (\left (-a +x \right ) \left (-b +x \right ) \left (-c +x \right )\right )^{\frac {2}{3}} \left (a^{2}-b c d +\left (b d +c d -2 a \right ) x +\left (1-d \right ) x^{2}\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\int \frac {{\left (2 \, a - b - c\right )} x^{2} + {\left (a b + a c - 2 \, b c\right )} a - 2 \, {\left (a^{2} - b c\right )} x}{\left (-{\left (a - x\right )} {\left (b - x\right )} {\left (c - x\right )}\right )^{\frac {2}{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*(a*b+a*c-2*b*c)-2*(a^2-b*c)*x+(2*a-b-c)*x^2)/((-a+x)*(-b+x)*(-c+x))^(2/3)/(a^2-b*c*d+(b*d+c*d-2*a
)*x+(1-d)*x^2),x, algorithm="maxima")

[Out]

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

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

[Out]

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

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

[Out]

Timed out

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