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

Optimal. Leaf size=311 \[ \frac {\log \left (\sqrt [3]{x^3 (-a-b)+a b x^2+x^4}-\sqrt [6]{d} x\right )}{2 d^{2/3}}+\frac {\log \left (\sqrt [3]{x^3 (-a-b)+a b x^2+x^4}+\sqrt [6]{d} x\right )}{2 d^{2/3}}-\frac {\log \left (-\sqrt [6]{d} x \sqrt [3]{x^3 (-a-b)+a b x^2+x^4}+\left (x^3 (-a-b)+a b x^2+x^4\right )^{2/3}+\sqrt [3]{d} x^2\right )}{4 d^{2/3}}-\frac {\log \left (\sqrt [6]{d} x \sqrt [3]{x^3 (-a-b)+a b x^2+x^4}+\left (x^3 (-a-b)+a b x^2+x^4\right )^{2/3}+\sqrt [3]{d} x^2\right )}{4 d^{2/3}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{d} x^2}{2 \left (x^3 (-a-b)+a b x^2+x^4\right )^{2/3}+\sqrt [3]{d} x^2}\right )}{2 d^{2/3}} \]

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Rubi [F]  time = 15.35, 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 (-a b+x^2\right )}{\sqrt [3]{x^2 (-a+x) (-b+x)} \left (a^2 b^2-2 a b (a+b) x+\left (a^2+4 a b+b^2-d\right ) x^2-2 (a+b) x^3+x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(3*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^2*b^2) + 2*a^2*b*(1 + b/a)*x^3 - a^2*(1 + (4*a*b + b^2 - d)/a^2)*x^6 + 2*a*(1 + b/a)*x^9 - x^12)), x], x, x
^(1/3)])/((a - x)*(b - x)*x^2)^(1/3) + (3*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^2*b^2 - 2*a^2*b*(1 + b/a)*x^3 + a^2*(1 + (4*a*b + b^2 - d)/a^2)*x^6 - 2*a*
(1 + b/a)*x^9 + x^12)), x], x, x^(1/3)])/((a - x)*(b - x)*x^2)^(1/3)

Rubi steps

\begin {align*} \int \frac {x \left (-a b+x^2\right )}{\sqrt [3]{x^2 (-a+x) (-b+x)} \left (a^2 b^2-2 a b (a+b) x+\left (a^2+4 a b+b^2-d\right ) x^2-2 (a+b) x^3+x^4\right )} \, dx &=\frac {\left (x^{2/3} \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \int \frac {\sqrt [3]{x} \left (-a b+x^2\right )}{\sqrt [3]{-a+x} \sqrt [3]{-b+x} \left (a^2 b^2-2 a b (a+b) x+\left (a^2+4 a b+b^2-d\right ) x^2-2 (a+b) x^3+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 (-a b+x^6\right )}{\sqrt [3]{-a+x^3} \sqrt [3]{-b+x^3} \left (a^2 b^2-2 a b (a+b) x^3+\left (a^2+4 a b+b^2-d\right ) x^6-2 (a+b) x^9+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 {a b x^3}{\sqrt [3]{-a+x^3} \sqrt [3]{-b+x^3} \left (-a^2 b^2+2 a^2 b \left (1+\frac {b}{a}\right ) x^3-a^2 \left (1+\frac {4 a b+b^2-d}{a^2}\right ) x^6+2 a \left (1+\frac {b}{a}\right ) x^9-x^{12}\right )}+\frac {x^9}{\sqrt [3]{-a+x^3} \sqrt [3]{-b+x^3} \left (a^2 b^2-2 a^2 b \left (1+\frac {b}{a}\right ) x^3+a^2 \left (1+\frac {4 a b+b^2-d}{a^2}\right ) x^6-2 a \left (1+\frac {b}{a}\right ) x^9+x^{12}\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 {x^9}{\sqrt [3]{-a+x^3} \sqrt [3]{-b+x^3} \left (a^2 b^2-2 a^2 b \left (1+\frac {b}{a}\right ) x^3+a^2 \left (1+\frac {4 a b+b^2-d}{a^2}\right ) x^6-2 a \left (1+\frac {b}{a}\right ) x^9+x^{12}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2 (-a+x) (-b+x)}}+\frac {\left (3 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^2 b^2+2 a^2 b \left (1+\frac {b}{a}\right ) x^3-a^2 \left (1+\frac {4 a b+b^2-d}{a^2}\right ) x^6+2 a \left (1+\frac {b}{a}\right ) x^9-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 = 3.16, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x \left (-a b+x^2\right )}{\sqrt [3]{x^2 (-a+x) (-b+x)} \left (a^2 b^2-2 a b (a+b) x+\left (a^2+4 a b+b^2-d\right ) x^2-2 (a+b) x^3+x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.78, size = 309, normalized size = 0.99 \begin {gather*} -\frac {\sqrt {3} \tan ^{-1}\left (\frac {\frac {x^2}{\sqrt {3}}+\frac {2 \left (a b x^2+(-a-b) x^3+x^4\right )^{2/3}}{\sqrt {3} \sqrt [3]{d}}}{x^2}\right )}{2 d^{2/3}}+\frac {\log \left (-\sqrt [6]{d} x+\sqrt [3]{a b x^2+(-a-b) x^3+x^4}\right )}{2 d^{2/3}}+\frac {\log \left (\sqrt [6]{d} x+\sqrt [3]{a b x^2+(-a-b) x^3+x^4}\right )}{2 d^{2/3}}-\frac {\log \left (\sqrt [3]{d} x^2-\sqrt [6]{d} x \sqrt [3]{a b x^2+(-a-b) x^3+x^4}+\left (a b x^2+(-a-b) x^3+x^4\right )^{2/3}\right )}{4 d^{2/3}}-\frac {\log \left (\sqrt [3]{d} x^2+\sqrt [6]{d} x \sqrt [3]{a b x^2+(-a-b) x^3+x^4}+\left (a b x^2+(-a-b) x^3+x^4\right )^{2/3}\right )}{4 d^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*(Sqrt[3]*ArcTan[(x^2/Sqrt[3] + (2*(a*b*x^2 + (-a - b)*x^3 + x^4)^(2/3))/(Sqrt[3]*d^(1/3)))/x^2])/d^(2/3)
+ Log[-(d^(1/6)*x) + (a*b*x^2 + (-a - b)*x^3 + x^4)^(1/3)]/(2*d^(2/3)) + Log[d^(1/6)*x + (a*b*x^2 + (-a - b)*x
^3 + x^4)^(1/3)]/(2*d^(2/3)) - Log[d^(1/3)*x^2 - d^(1/6)*x*(a*b*x^2 + (-a - b)*x^3 + x^4)^(1/3) + (a*b*x^2 + (
-a - b)*x^3 + x^4)^(2/3)]/(4*d^(2/3)) - Log[d^(1/3)*x^2 + d^(1/6)*x*(a*b*x^2 + (-a - b)*x^3 + x^4)^(1/3) + (a*
b*x^2 + (-a - b)*x^3 + x^4)^(2/3)]/(4*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*(-a*b+x^2)/(x^2*(-a+x)*(-b+x))^(1/3)/(a^2*b^2-2*a*b*(a+b)*x+(a^2+4*a*b+b^2-d)*x^2-2*(a+b)*x^3+x^4)
,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 {{\left (a b - x^{2}\right )} x}{{\left (a^{2} b^{2} - 2 \, {\left (a + b\right )} a b x - 2 \, {\left (a + b\right )} x^{3} + x^{4} + {\left (a^{2} + 4 \, a b + b^{2} - d\right )} x^{2}\right )} \left ({\left (a - x\right )} {\left (b - x\right )} x^{2}\right )^{\frac {1}{3}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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