3.29.81 \(\int \frac {x (-a b+x^2)}{\sqrt [3]{x^2 (-a+x) (-b+x)} (a^2 b^2-2 a b (a+b) x+(a^2+4 a b+b^2-d) x^2-2 (a+b) x^3+x^4)} \, dx\) [2881]

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

[Out]

1/2*3^(1/2)*arctan(3^(1/2)*d^(1/3)*x^2/(d^(1/3)*x^2+2*(a*b*x^2+(-a-b)*x^3+x^4)^(2/3)))/d^(2/3)+1/2*ln(-d^(1/6)
*x+(a*b*x^2+(-a-b)*x^3+x^4)^(1/3))/d^(2/3)+1/2*ln(d^(1/6)*x+(a*b*x^2+(-a-b)*x^3+x^4)^(1/3))/d^(2/3)-1/4*ln(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))/d^(2/3)-1/4*ln(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))/d^(2/3)

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Rubi [F]
time = 11.45, 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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 = 33.37, 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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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*} \text {Failed to integrate} \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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Fricas [F(-1)] Timed out
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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Sympy [F(-1)] Timed out
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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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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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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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