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

Optimal. Leaf size=166 \[ -\frac {\log \left (d^{2/3} \left (x^2 (-a-b)+a b x+x^3\right )^{2/3}+\sqrt [3]{d} x \sqrt [3]{x^2 (-a-b)+a b x+x^3}+x^2\right )}{2 \sqrt [3]{d}}+\frac {\log \left (x-\sqrt [3]{d} \sqrt [3]{x^2 (-a-b)+a b x+x^3}\right )}{\sqrt [3]{d}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{d} \sqrt [3]{x^2 (-a-b)+a b x+x^3}+x}\right )}{\sqrt [3]{d}} \]

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Rubi [F]  time = 6.08, 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 d-(a+b) d 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*d - (a + b)*d*x + (-1 + d)*x^2)),x]

[Out]

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

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Mathematica [F]  time = 10.17, 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 d-(a+b) d 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*d - (a + b)*d*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*d - (a + b)*d*x + (-1 + d)*x^2)), x]

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IntegrateAlgebraic [A]  time = 0.79, size = 166, normalized size = 1.00 \begin {gather*} \frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{d} \sqrt [3]{a b x+(-a-b) x^2+x^3}}\right )}{\sqrt [3]{d}}+\frac {\log \left (x-\sqrt [3]{d} \sqrt [3]{a b x+(-a-b) x^2+x^3}\right )}{\sqrt [3]{d}}-\frac {\log \left (x^2+\sqrt [3]{d} x \sqrt [3]{a b x+(-a-b) x^2+x^3}+d^{2/3} \left (a b x+(-a-b) x^2+x^3\right )^{2/3}\right )}{2 \sqrt [3]{d}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*d^(1/3)*(a*b*x + (-a - b)*x^2 + x^3)^(1/3))])/d^(1/3) + Log[x - d^(1/3)*(a*
b*x + (-a - b)*x^2 + x^3)^(1/3)]/d^(1/3) - Log[x^2 + d^(1/3)*x*(a*b*x + (-a - b)*x^2 + x^3)^(1/3) + d^(2/3)*(a
*b*x + (-a - b)*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((-2*a*b*x+(a+b)*x^2)/(x*(-a+x)*(-b+x))^(2/3)/(a*b*d-(a+b)*d*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 {2 \, a b x - {\left (a + b\right )} x^{2}}{{\left (a b d - {\left (a + b\right )} d x + {\left (d - 1\right )} x^{2}\right )} \left ({\left (a - x\right )} {\left (b - x\right )} x\right )^{\frac {2}{3}}}\,{d x} \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*d-(a+b)*d*x+(-1+d)*x^2),x, algorithm="giac")

[Out]

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

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maple [F]  time = 0.14, size = 0, normalized size = 0.00 \[\int \frac {-2 a b x +\left (a +b \right ) x^{2}}{\left (x \left (-a +x \right ) \left (-b +x \right )\right )^{\frac {2}{3}} \left (a b d -\left (a +b \right ) d x +\left (-1+d \right ) x^{2}\right )}\, dx\]

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*d-(a+b)*d*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*d-(a+b)*d*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 {2 \, a b x - {\left (a + b\right )} x^{2}}{{\left (a b d - {\left (a + b\right )} d x + {\left (d - 1\right )} x^{2}\right )} \left ({\left (a - x\right )} {\left (b - x\right )} x\right )^{\frac {2}{3}}}\,{d x} \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*d-(a+b)*d*x+(-1+d)*x^2),x, algorithm="maxima")

[Out]

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

[Out]

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

[Out]

Timed out

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