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

Optimal. Leaf size=233 \[ \frac {\log \left (a^2 d^{2/3}+\sqrt [3]{x^2 (-a-b)+a b x+x^3} \left (\sqrt [3]{d} x-a \sqrt [3]{d}\right )+\left (x^2 (-a-b)+a b x+x^3\right )^{2/3}-2 a d^{2/3} x+d^{2/3} x^2\right )}{2 \sqrt [3]{d}}-\frac {\log \left (\sqrt [3]{x^2 (-a-b)+a b x+x^3}+a \sqrt [3]{d}-\sqrt [3]{d} x\right )}{\sqrt [3]{d}}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{x^2 (-a-b)+a b x+x^3}}{\sqrt [3]{x^2 (-a-b)+a b x+x^3}-2 a \sqrt [3]{d}+2 \sqrt [3]{d} x}\right )}{\sqrt [3]{d}} \]

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Rubi [F]  time = 3.69, 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+(-2 a+b) x}{\sqrt [3]{x (-a+x) (-b+x)} \left (a^2 d+(b-2 a d) x+(-1+d) x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [F]  time = 4.91, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a b+(-2 a+b) x}{\sqrt [3]{x (-a+x) (-b+x)} \left (a^2 d+(b-2 a d) x+(-1+d) x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 2.95, size = 233, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{a b x+(-a-b) x^2+x^3}}{-2 a \sqrt [3]{d}+2 \sqrt [3]{d} x+\sqrt [3]{a b x+(-a-b) x^2+x^3}}\right )}{\sqrt [3]{d}}-\frac {\log \left (a \sqrt [3]{d}-\sqrt [3]{d} x+\sqrt [3]{a b x+(-a-b) x^2+x^3}\right )}{\sqrt [3]{d}}+\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 x+(-a-b) x^2+x^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[(a*b + (-2*a + b)*x)/((x*(-a + x)*(-b + x))^(1/3)*(a^2*d + (b - 2*a*d)*x + (-1 + d)*x^2)),x
]

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.16, size = 0, normalized size = 0.00 \[\int \frac {a b +\left (-2 a +b \right ) x}{\left (x \left (-a +x \right ) \left (-b +x \right )\right )^{\frac {1}{3}} \left (a^{2} d +\left (-2 a d +b \right ) x +\left (-1+d \right ) x^{2}\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((a*b+(-2*a+b)*x)/(x*(-a+x)*(-b+x))^(1/3)/(a^2*d+(-2*a*d+b)*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 {a b - {\left (2 \, a - b\right )} x}{{\left (a^{2} d + {\left (d - 1\right )} x^{2} - {\left (2 \, a d - b\right )} x\right )} \left ({\left (a - x\right )} {\left (b - x\right )} x\right )^{\frac {1}{3}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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