3.23.83 \(\int \frac {-2 a b x+(a+b) x^2}{(x (-a+x) (-b+x))^{2/3} (-a b+(a+b) x+(-1+d) x^2)} \, dx\) [2283]

Optimal. Leaf size=174 \[ -\frac {\sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} \sqrt [3]{d} x}{\sqrt [3]{d} x+2 \sqrt [3]{a b x+(-a-b) x^2+x^3}}\right )}{d^{2/3}}-\frac {\log \left (-\sqrt [3]{d} x+\sqrt [3]{a b x+(-a-b) x^2+x^3}\right )}{d^{2/3}}+\frac {\log \left (d^{2/3} x^2+\sqrt [3]{d} x \sqrt [3]{a b x+(-a-b) x^2+x^3}+\left (a b x+(-a-b) x^2+x^3\right )^{2/3}\right )}{2 d^{2/3}} \]

[Out]

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

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Rubi [F]
time = 3.00, 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 {-2 a b x+(a+b) x^2}{(x (-a+x) (-b+x))^{2/3} \left (-a b+(a+b) x+(-1+d) x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [F]
time = 39.15, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-2 a b x+(a+b) x^2}{(x (-a+x) (-b+x))^{2/3} \left (-a b+(a+b) x+(-1+d) x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [F]
time = 0.07, size = 0, normalized size = 0.00 \[\int \frac {-2 a b x +\left (a +b \right ) x^{2}}{\left (x \left (-a +x \right ) \left (-b +x \right )\right )^{\frac {2}{3}} \left (-a b +\left (a +b \right ) x +\left (-1+d \right ) x^{2}\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F(-1)] Timed out
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((-2*a*b*x+(a+b)*x^2)/(x*(-a+x)*(-b+x))^(2/3)/(-a*b+(a+b)*x+(-1+d)*x^2),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F(-1)] Timed out
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((-2*a*b*x+(a+b)*x**2)/(x*(-a+x)*(-b+x))**(2/3)/(-a*b+(a+b)*x+(-1+d)*x**2),x)

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^2\,\left (a+b\right )-2\,a\,b\,x}{{\left (x\,\left (a-x\right )\,\left (b-x\right )\right )}^{2/3}\,\left (\left (d-1\right )\,x^2+\left (a+b\right )\,x-a\,b\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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