3.26.15 \(\int \frac {x (4 a b-3 (a+b) x+2 x^2)}{\sqrt [3]{x^2 (-a+x) (-b+x)} (-a b d+(a+b) d x-d x^2+x^4)} \, dx\)

Optimal. Leaf size=210 \[ \frac {\log \left (x^2-\sqrt [3]{d} \sqrt [3]{x^3 (-a-b)+a b x^2+x^4}\right )}{d^{2/3}}-\frac {\log \left (d^{2/3} \left (x^3 (-a-b)+a b x^2+x^4\right )^{2/3}+\sqrt [3]{d} x^2 \sqrt [3]{x^3 (-a-b)+a b x^2+x^4}+x^4\right )}{2 d^{2/3}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{d} \sqrt [3]{x^3 (-a-b)+a b x^2+x^4}}{\sqrt [3]{d} \sqrt [3]{x^3 (-a-b)+a b x^2+x^4}+2 x^2}\right )}{d^{2/3}} \]

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Rubi [F]  time = 16.81, 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 {x \left (4 a b-3 (a+b) x+2 x^2\right )}{\sqrt [3]{x^2 (-a+x) (-b+x)} \left (-a b d+(a+b) d x-d x^2+x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(9*(a + b)*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*b*d - a*(1 + b/a)*d*x^3 + d*x^6 - x^12)), x], x, x^(1/3)])/((a - x)*(b - x)*x^2)^(1/3) + (12*a*b*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*b*d) + a*(1
+ b/a)*d*x^3 - d*x^6 + x^12)), x], x, x^(1/3)])/((a - x)*(b - x)*x^2)^(1/3) + (6*x^(2/3)*(-a + x)^(1/3)*(-b +
x)^(1/3)*Defer[Subst][Defer[Int][x^9/((-a + x^3)^(1/3)*(-b + x^3)^(1/3)*(-(a*b*d) + a*(1 + b/a)*d*x^3 - d*x^6
+ x^12)), x], x, x^(1/3)])/((a - x)*(b - x)*x^2)^(1/3)

Rubi steps

\begin {align*} \int \frac {x \left (4 a b-3 (a+b) x+2 x^2\right )}{\sqrt [3]{x^2 (-a+x) (-b+x)} \left (-a b d+(a+b) d x-d x^2+x^4\right )} \, dx &=\frac {\left (x^{2/3} \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \int \frac {\sqrt [3]{x} \left (4 a b-3 (a+b) x+2 x^2\right )}{\sqrt [3]{-a+x} \sqrt [3]{-b+x} \left (-a b d+(a+b) d x-d x^2+x^4\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 (4 a b-3 (a+b) x^3+2 x^6\right )}{\sqrt [3]{-a+x^3} \sqrt [3]{-b+x^3} \left (-a b d+(a+b) d x^3-d x^6+x^{12}\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 {3 (a+b) x^6}{\sqrt [3]{-a+x^3} \sqrt [3]{-b+x^3} \left (a b d-a \left (1+\frac {b}{a}\right ) d x^3+d x^6-x^{12}\right )}+\frac {4 a b x^3}{\sqrt [3]{-a+x^3} \sqrt [3]{-b+x^3} \left (-a b d+a \left (1+\frac {b}{a}\right ) d x^3-d x^6+x^{12}\right )}+\frac {2 x^9}{\sqrt [3]{-a+x^3} \sqrt [3]{-b+x^3} \left (-a b d+a \left (1+\frac {b}{a}\right ) d x^3-d x^6+x^{12}\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2 (-a+x) (-b+x)}}\\ &=\frac {\left (6 x^{2/3} \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^9}{\sqrt [3]{-a+x^3} \sqrt [3]{-b+x^3} \left (-a b d+a \left (1+\frac {b}{a}\right ) d x^3-d x^6+x^{12}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2 (-a+x) (-b+x)}}+\frac {\left (12 a b 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 b d+a \left (1+\frac {b}{a}\right ) d x^3-d x^6+x^{12}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2 (-a+x) (-b+x)}}+\frac {\left (9 (a+b) 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 b d-a \left (1+\frac {b}{a}\right ) d x^3+d x^6-x^{12}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2 (-a+x) (-b+x)}}\\ \end {align*}

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Mathematica [F]  time = 3.61, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x \left (4 a b-3 (a+b) x+2 x^2\right )}{\sqrt [3]{x^2 (-a+x) (-b+x)} \left (-a b d+(a+b) d x-d x^2+x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.83, size = 210, 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 x^2+\sqrt [3]{d} \sqrt [3]{a b x^2+(-a-b) x^3+x^4}}\right )}{d^{2/3}}+\frac {\log \left (x^2-\sqrt [3]{d} \sqrt [3]{a b x^2+(-a-b) x^3+x^4}\right )}{d^{2/3}}-\frac {\log \left (x^4+\sqrt [3]{d} x^2 \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[(x*(4*a*b - 3*(a + b)*x + 2*x^2))/((x^2*(-a + x)*(-b + x))^(1/3)*(-(a*b*d) + (a + b)*d*x -
d*x^2 + x^4)),x]

[Out]

(Sqrt[3]*ArcTan[(Sqrt[3]*d^(1/3)*(a*b*x^2 + (-a - b)*x^3 + x^4)^(1/3))/(2*x^2 + d^(1/3)*(a*b*x^2 + (-a - b)*x^
3 + x^4)^(1/3))])/d^(2/3) + Log[x^2 - d^(1/3)*(a*b*x^2 + (-a - b)*x^3 + x^4)^(1/3)]/d^(2/3) - Log[x^4 + d^(1/3
)*x^2*(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(x*(4*a*b-3*(a+b)*x+2*x^2)/(x^2*(-a+x)*(-b+x))^(1/3)/(-a*b*d+(a+b)*d*x-d*x^2+x^4),x, algorithm="frica
s")

[Out]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (4 \, a b - 3 \, {\left (a + b\right )} x + 2 \, x^{2}\right )} x}{\left ({\left (a - x\right )} {\left (b - x\right )} x^{2}\right )^{\frac {1}{3}} {\left (x^{4} - a b d + {\left (a + b\right )} d x - d x^{2}\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(4*a*b-3*(a+b)*x+2*x^2)/(x^2*(-a+x)*(-b+x))^(1/3)/(-a*b*d+(a+b)*d*x-d*x^2+x^4),x, algorithm="giac"
)

[Out]

integrate((4*a*b - 3*(a + b)*x + 2*x^2)*x/(((a - x)*(b - x)*x^2)^(1/3)*(x^4 - a*b*d + (a + b)*d*x - d*x^2)), x
)

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maple [F]  time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {x \left (4 a b -3 \left (a +b \right ) x +2 x^{2}\right )}{\left (x^{2} \left (-a +x \right ) \left (-b +x \right )\right )^{\frac {1}{3}} \left (-a b d +\left (a +b \right ) d x -d \,x^{2}+x^{4}\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(4*a*b-3*(a+b)*x+2*x^2)/(x^2*(-a+x)*(-b+x))^(1/3)/(-a*b*d+(a+b)*d*x-d*x^2+x^4),x)

[Out]

int(x*(4*a*b-3*(a+b)*x+2*x^2)/(x^2*(-a+x)*(-b+x))^(1/3)/(-a*b*d+(a+b)*d*x-d*x^2+x^4),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (4 \, a b - 3 \, {\left (a + b\right )} x + 2 \, x^{2}\right )} x}{\left ({\left (a - x\right )} {\left (b - x\right )} x^{2}\right )^{\frac {1}{3}} {\left (x^{4} - a b d + {\left (a + b\right )} d x - d x^{2}\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(4*a*b-3*(a+b)*x+2*x^2)/(x^2*(-a+x)*(-b+x))^(1/3)/(-a*b*d+(a+b)*d*x-d*x^2+x^4),x, algorithm="maxim
a")

[Out]

integrate((4*a*b - 3*(a + b)*x + 2*x^2)*x/(((a - x)*(b - x)*x^2)^(1/3)*(x^4 - a*b*d + (a + b)*d*x - d*x^2)), x
)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} -\int \frac {x\,\left (4\,a\,b+2\,x^2-3\,x\,\left (a+b\right )\right )}{{\left (x^2\,\left (a-x\right )\,\left (b-x\right )\right )}^{1/3}\,\left (-x^4+d\,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*(4*a*b + 2*x^2 - 3*x*(a + b)))/((x^2*(a - x)*(b - x))^(1/3)*(d*x^2 - x^4 - d*x*(a + b) + a*b*d)),x)

[Out]

-int((x*(4*a*b + 2*x^2 - 3*x*(a + b)))/((x^2*(a - x)*(b - x))^(1/3)*(d*x^2 - x^4 - 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(x*(4*a*b-3*(a+b)*x+2*x**2)/(x**2*(-a+x)*(-b+x))**(1/3)/(-a*b*d+(a+b)*d*x-d*x**2+x**4),x)

[Out]

Timed out

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