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

Optimal. Leaf size=296 \[ \frac {\log \left (a^2 d^{2/3}+\left (\sqrt [3]{d} x-a \sqrt [3]{d}\right ) \sqrt [3]{x^2 (-a-b-c)+x (a b+a c+b c)-a b c+x^3}+\left (x^2 (-a-b-c)+x (a b+a c+b c)-a b c+x^3\right )^{2/3}-2 a d^{2/3} x+d^{2/3} x^2\right )}{2 d^{2/3}}-\frac {\log \left (\sqrt [3]{x^2 (-a-b-c)+x (a b+a c+b c)-a b c+x^3}+a \sqrt [3]{d}-\sqrt [3]{d} x\right )}{d^{2/3}}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} a \sqrt [3]{d}-\sqrt {3} \sqrt [3]{d} x}{-2 \sqrt [3]{x^2 (-a-b-c)+x (a b+a c+b c)-a b c+x^3}+a \sqrt [3]{d}-\sqrt [3]{d} x}\right )}{d^{2/3}} \]

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Rubi [F]  time = 8.97, 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 (-b c+a^2 d+(b+c-2 a 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)*(-(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)^(2/3)*(-b + x)^(2/3)
*(-c + x)^(2/3)*Defer[Int][(-a + x)^(1/3)/((-b + x)^(2/3)*(-c + x)^(2/3)*(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)))^(2/3)) - ((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)^(2/3)*(-b + x)^(2/3)*(-c + x
)^(2/3)*Defer[Int][(-a + x)^(1/3)/((-b + x)^(2/3)*(-c + x)^(2/3)*(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)))^(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 (-b c+a^2 d+(b+c-2 a 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 (-b c+a^2 d+(b+c-2 a 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 (-b c+a^2 d+(b+c-2 a 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-\sqrt {b^2-2 b c+c^2+4 a^2 d-4 a b d-4 a c d+4 b c d}\right ) \sqrt [3]{-a+x}}{(-b+x)^{2/3} (-c+x)^{2/3} \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 {\left (-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}\right ) \sqrt [3]{-a+x}}{(-b+x)^{2/3} (-c+x)^{2/3} \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}{((-a+x) (-b+x) (-c+x))^{2/3}}\\ &=\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 ) (-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 (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}{((-a+x) (-b+x) (-c+x))^{2/3}}+\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 ) (-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 (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}{((-a+x) (-b+x) (-c+x))^{2/3}}\\ \end {align*}

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Mathematica [F]  time = 3.29, 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 (-b c+a^2 d+(b+c-2 a 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
)*(-(b*c) + a^2*d + (b + c - 2*a*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
)*(-(b*c) + a^2*d + (b + c - 2*a*d)*x + (-1 + d)*x^2)), x]

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IntegrateAlgebraic [A]  time = 7.11, size = 296, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} a \sqrt [3]{d}-\sqrt {3} \sqrt [3]{d} x}{a \sqrt [3]{d}-\sqrt [3]{d} x-2 \sqrt [3]{-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3}}\right )}{d^{2/3}}-\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 )}{d^{2/3}}+\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 d^{2/3}} \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)*(-(b*c) + a^2*d + (b + c - 2*a*d)*x + (-1 + d)*x^2)),x]

[Out]

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

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

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

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

[Out]

Timed out

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