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

Optimal. Leaf size=201 \[ -\frac {\log \left (a^2+d^{2/3} \left (x (-a-b)+a b+x^2\right )^{2/3}+\sqrt [3]{x (-a-b)+a b+x^2} \left (\sqrt [3]{d} x-a \sqrt [3]{d}\right )-2 a x+x^2\right )}{2 d^{2/3}}+\frac {\log \left (\sqrt [3]{d} \sqrt [3]{x (-a-b)+a b+x^2}+a-x\right )}{d^{2/3}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{d} \sqrt [3]{x (-a-b)+a b+x^2}}{\sqrt [3]{d} \sqrt [3]{x (-a-b)+a b+x^2}-2 a+2 x}\right )}{d^{2/3}} \]

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Rubi [F]  time = 0.16, 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-2 b+x}{\sqrt [3]{(-a+x) (-b+x)} \left (a^2+b d-(2 a+d) x+x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [F]  time = 11.11, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a-2 b+x}{\sqrt [3]{(-a+x) (-b+x)} \left (a^2+b d-(2 a+d) x+x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.43, size = 201, normalized size = 1.00 \begin {gather*} \frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{d} \sqrt [3]{a b+(-a-b) x+x^2}}{-2 a+2 x+\sqrt [3]{d} \sqrt [3]{a b+(-a-b) x+x^2}}\right )}{d^{2/3}}+\frac {\log \left (a-x+\sqrt [3]{d} \sqrt [3]{a b+(-a-b) x+x^2}\right )}{d^{2/3}}-\frac {\log \left (a^2-2 a x+x^2+\left (-a \sqrt [3]{d}+\sqrt [3]{d} x\right ) \sqrt [3]{a b+(-a-b) x+x^2}+d^{2/3} \left (a b+(-a-b) x+x^2\right )^{2/3}\right )}{2 d^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.24, size = 0, normalized size = 0.00 \[\int \frac {a -2 b +x}{\left (\left (-a +x \right ) \left (-b +x \right )\right )^{\frac {1}{3}} \left (a^{2}+b d -\left (2 a +d \right ) x +x^{2}\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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