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

Optimal. Leaf size=226 \[ -\frac {\log \left (a^2+d^{2/3} \left (x^3 (-a-b)+a b x^2+x^4\right )^{2/3}+\sqrt [3]{x^3 (-a-b)+a b x^2+x^4} \left (\sqrt [3]{d} x-a \sqrt [3]{d}\right )-2 a x+x^2\right )}{2 d^{2/3}}+\frac {\log \left (\sqrt [3]{d} \sqrt [3]{x^3 (-a-b)+a b x^2+x^4}+a-x\right )}{d^{2/3}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{d} \sqrt [3]{x^3 (-a-b)+a b x^2+x^4}}{\sqrt [3]{d} \sqrt [3]{x^3 (-a-b)+a b x^2+x^4}-2 a+2 x}\right )}{d^{2/3}} \]

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Rubi [F]  time = 13.26, 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-3 a x^2+x^3}{\sqrt [3]{x^2 (-a+x) (-b+x)} \left (-a^2+2 a x-(1+b d) x^2+d x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(3*x*(1 - x/a)^(1/3)*(1 - x/b)^(1/3)*AppellF1[1/3, 1/3, 1/3, 4/3, x/a, x/b])/(d*((a - x)*(b - x)*x^2)^(1/3)) +
(6*a*(1 - b*d)*x^(2/3)*(-a + x)^(1/3)*(-b + x)^(1/3)*Defer[Subst][Defer[Int][x^3/((-a + x^3)^(1/3)*(-b + x^3)
^(1/3)*(a^2 - 2*a*x^3 + (1 + b*d)*x^6 - d*x^9)), x], x, x^(1/3)])/(d*((a - x)*(b - x)*x^2)^(1/3)) - (3*(1 - 3*
a*d + b*d)*x^(2/3)*(-a + x)^(1/3)*(-b + x)^(1/3)*Defer[Subst][Defer[Int][x^6/((-a + x^3)^(1/3)*(-b + x^3)^(1/3
)*(a^2 - 2*a*x^3 + (1 + b*d)*x^6 - d*x^9)), x], x, x^(1/3)])/(d*((a - x)*(b - x)*x^2)^(1/3)) + (3*a^2*x^(2/3)*
(-a + x)^(1/3)*(-b + x)^(1/3)*Defer[Subst][Defer[Int][1/((-a + x^3)^(1/3)*(-b + x^3)^(1/3)*(-a^2 + 2*a*x^3 - (
1 + b*d)*x^6 + d*x^9)), x], x, x^(1/3)])/(d*((a - x)*(b - x)*x^2)^(1/3))

Rubi steps

\begin {align*} \int \frac {2 a b x-3 a x^2+x^3}{\sqrt [3]{x^2 (-a+x) (-b+x)} \left (-a^2+2 a x-(1+b d) x^2+d x^3\right )} \, dx &=\int \frac {x \left (2 a b-3 a x+x^2\right )}{\sqrt [3]{x^2 (-a+x) (-b+x)} \left (-a^2+2 a x-(1+b d) x^2+d x^3\right )} \, dx\\ &=\frac {\left (x^{2/3} \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \int \frac {\sqrt [3]{x} \left (2 a b-3 a x+x^2\right )}{\sqrt [3]{-a+x} \sqrt [3]{-b+x} \left (-a^2+2 a x-(1+b d) x^2+d x^3\right )} \, dx}{\sqrt [3]{x^2 (-a+x) (-b+x)}}\\ &=\frac {\left (3 x^{2/3} \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^3 \left (2 a b-3 a x^3+x^6\right )}{\sqrt [3]{-a+x^3} \sqrt [3]{-b+x^3} \left (-a^2+2 a x^3-(1+b d) x^6+d x^9\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2 (-a+x) (-b+x)}}\\ &=\frac {\left (3 x^{2/3} \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{d \sqrt [3]{-a+x^3} \sqrt [3]{-b+x^3}}+\frac {a^2-2 a (1-b d) x^3+(1-3 a d+b d) x^6}{d \sqrt [3]{-a+x^3} \sqrt [3]{-b+x^3} \left (-a^2+2 a x^3-(1+b d) x^6+d x^9\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2 (-a+x) (-b+x)}}\\ &=\frac {\left (3 x^{2/3} \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{-a+x^3} \sqrt [3]{-b+x^3}} \, dx,x,\sqrt [3]{x}\right )}{d \sqrt [3]{x^2 (-a+x) (-b+x)}}+\frac {\left (3 x^{2/3} \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \operatorname {Subst}\left (\int \frac {a^2-2 a (1-b d) x^3+(1-3 a d+b d) x^6}{\sqrt [3]{-a+x^3} \sqrt [3]{-b+x^3} \left (-a^2+2 a x^3-(1+b d) x^6+d x^9\right )} \, dx,x,\sqrt [3]{x}\right )}{d \sqrt [3]{x^2 (-a+x) (-b+x)}}\\ &=\frac {\left (3 x^{2/3} \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \operatorname {Subst}\left (\int \left (\frac {2 a (1-b d) x^3}{\sqrt [3]{-a+x^3} \sqrt [3]{-b+x^3} \left (a^2-2 a x^3+(1+b d) x^6-d x^9\right )}+\frac {(-1+3 a d-b d) x^6}{\sqrt [3]{-a+x^3} \sqrt [3]{-b+x^3} \left (a^2-2 a x^3+(1+b d) x^6-d x^9\right )}+\frac {a^2}{\sqrt [3]{-a+x^3} \sqrt [3]{-b+x^3} \left (-a^2+2 a x^3-(1+b d) x^6+d x^9\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{d \sqrt [3]{x^2 (-a+x) (-b+x)}}+\frac {\left (3 x^{2/3} \sqrt [3]{-b+x} \sqrt [3]{1-\frac {x}{a}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{-b+x^3} \sqrt [3]{1-\frac {x^3}{a}}} \, dx,x,\sqrt [3]{x}\right )}{d \sqrt [3]{x^2 (-a+x) (-b+x)}}\\ &=\frac {\left (3 a^2 x^{2/3} \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{-a+x^3} \sqrt [3]{-b+x^3} \left (-a^2+2 a x^3-(1+b d) x^6+d x^9\right )} \, dx,x,\sqrt [3]{x}\right )}{d \sqrt [3]{x^2 (-a+x) (-b+x)}}+\frac {\left (6 a (1-b d) x^{2/3} \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^3}{\sqrt [3]{-a+x^3} \sqrt [3]{-b+x^3} \left (a^2-2 a x^3+(1+b d) x^6-d x^9\right )} \, dx,x,\sqrt [3]{x}\right )}{d \sqrt [3]{x^2 (-a+x) (-b+x)}}+\frac {\left (3 (-1+3 a d-b d) x^{2/3} \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^6}{\sqrt [3]{-a+x^3} \sqrt [3]{-b+x^3} \left (a^2-2 a x^3+(1+b d) x^6-d x^9\right )} \, dx,x,\sqrt [3]{x}\right )}{d \sqrt [3]{x^2 (-a+x) (-b+x)}}+\frac {\left (3 x^{2/3} \sqrt [3]{1-\frac {x}{a}} \sqrt [3]{1-\frac {x}{b}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{1-\frac {x^3}{a}} \sqrt [3]{1-\frac {x^3}{b}}} \, dx,x,\sqrt [3]{x}\right )}{d \sqrt [3]{x^2 (-a+x) (-b+x)}}\\ &=\frac {3 x \sqrt [3]{1-\frac {x}{a}} \sqrt [3]{1-\frac {x}{b}} F_1\left (\frac {1}{3};\frac {1}{3},\frac {1}{3};\frac {4}{3};\frac {x}{a},\frac {x}{b}\right )}{d \sqrt [3]{(a-x) (b-x) x^2}}+\frac {\left (3 a^2 x^{2/3} \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{-a+x^3} \sqrt [3]{-b+x^3} \left (-a^2+2 a x^3-(1+b d) x^6+d x^9\right )} \, dx,x,\sqrt [3]{x}\right )}{d \sqrt [3]{x^2 (-a+x) (-b+x)}}+\frac {\left (6 a (1-b d) x^{2/3} \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^3}{\sqrt [3]{-a+x^3} \sqrt [3]{-b+x^3} \left (a^2-2 a x^3+(1+b d) x^6-d x^9\right )} \, dx,x,\sqrt [3]{x}\right )}{d \sqrt [3]{x^2 (-a+x) (-b+x)}}+\frac {\left (3 (-1+3 a d-b d) x^{2/3} \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^6}{\sqrt [3]{-a+x^3} \sqrt [3]{-b+x^3} \left (a^2-2 a x^3+(1+b d) x^6-d x^9\right )} \, dx,x,\sqrt [3]{x}\right )}{d \sqrt [3]{x^2 (-a+x) (-b+x)}}\\ \end {align*}

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Mathematica [F]  time = 4.24, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 a b x-3 a x^2+x^3}{\sqrt [3]{x^2 (-a+x) (-b+x)} \left (-a^2+2 a x-(1+b d) x^2+d x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 1.37, size = 226, normalized size = 1.00 \begin {gather*} \frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{d} \sqrt [3]{a b x^2+(-a-b) x^3+x^4}}{-2 a+2 x+\sqrt [3]{d} \sqrt [3]{a b x^2+(-a-b) x^3+x^4}}\right )}{d^{2/3}}+\frac {\log \left (a-x+\sqrt [3]{d} \sqrt [3]{a b x^2+(-a-b) x^3+x^4}\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 x^2+(-a-b) x^3+x^4}+d^{2/3} \left (a b x^2+(-a-b) x^3+x^4\right )^{2/3}\right )}{2 d^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[3]*ArcTan[(Sqrt[3]*d^(1/3)*(a*b*x^2 + (-a - b)*x^3 + x^4)^(1/3))/(-2*a + 2*x + d^(1/3)*(a*b*x^2 + (-a -
b)*x^3 + x^4)^(1/3))])/d^(2/3) + Log[a - x + d^(1/3)*(a*b*x^2 + (-a - b)*x^3 + x^4)^(1/3)]/d^(2/3) - Log[a^2 -
2*a*x + x^2 + (-(a*d^(1/3)) + d^(1/3)*x)*(a*b*x^2 + (-a - b)*x^3 + x^4)^(1/3) + d^(2/3)*(a*b*x^2 + (-a - b)*x
^3 + x^4)^(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((2*a*b*x-3*a*x^2+x^3)/(x^2*(-a+x)*(-b+x))^(1/3)/(-a^2+2*a*x-(b*d+1)*x^2+d*x^3),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 - 3 \, a x^{2} + x^{3}}{\left ({\left (a - x\right )} {\left (b - x\right )} x^{2}\right )^{\frac {1}{3}} {\left (d x^{3} - {\left (b d + 1\right )} x^{2} - a^{2} + 2 \, a x\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {2 a b x -3 a \,x^{2}+x^{3}}{\left (x^{2} \left (-a +x \right ) \left (-b +x \right )\right )^{\frac {1}{3}} \left (-a^{2}+2 a x -\left (b d +1\right ) x^{2}+d \,x^{3}\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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