3.28.19 \(\int \frac {-a b x^2+x^4}{(x^2 (-a+x) (-b+x))^{2/3} (a^2 b^2-2 a b (a+b) x+(a^2+4 a b+b^2-d) x^2-2 (a+b) x^3+x^4)} \, dx\)

Optimal. Leaf size=249 \[ -\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{d} x}{\sqrt [6]{d} x-2 \sqrt [3]{x^3 (-a-b)+a b x^2+x^4}}\right )}{2 d^{5/6}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{d} x}{2 \sqrt [3]{x^3 (-a-b)+a b x^2+x^4}+\sqrt [6]{d} x}\right )}{2 d^{5/6}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [6]{d} x}{\sqrt [3]{x^3 (-a-b)+a b x^2+x^4}}\right )}{d^{5/6}}-\frac {\tanh ^{-1}\left (\frac {\frac {\left (x^3 (-a-b)+a b x^2+x^4\right )^{2/3}}{\sqrt [6]{d}}+\sqrt [6]{d} x^2}{x \sqrt [3]{x^3 (-a-b)+a b x^2+x^4}}\right )}{2 d^{5/6}} \]

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Rubi [F]  time = 19.26, 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 {-a b x^2+x^4}{\left (x^2 (-a+x) (-b+x)\right )^{2/3} \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[(-(a*b*x^2) + x^4)/((x^2*(-a + x)*(-b + x))^(2/3)*(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^(4/3)*(-a + x)^(2/3)*(-b + x)^(2/3)*Defer[Subst][Defer[Int][x^4/((-a + x^3)^(2/3)*(-b + x^3)^(2/3)*(-
(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)^(2/3) + (3*x^(4/3)*(-a + x)^(2/3)*(-b + x)^(2/3)*Defer[Subst][Defer[Int][x^10/(
(-a + x^3)^(2/3)*(-b + x^3)^(2/3)*(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)^(2/3)

Rubi steps

\begin {align*} \int \frac {-a b x^2+x^4}{\left (x^2 (-a+x) (-b+x)\right )^{2/3} \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 &=\int \frac {x^2 \left (-a b+x^2\right )}{\left (x^2 (-a+x) (-b+x)\right )^{2/3} \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^{4/3} (-a+x)^{2/3} (-b+x)^{2/3}\right ) \int \frac {x^{2/3} \left (-a b+x^2\right )}{(-a+x)^{2/3} (-b+x)^{2/3} \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}{\left (x^2 (-a+x) (-b+x)\right )^{2/3}}\\ &=\frac {\left (3 x^{4/3} (-a+x)^{2/3} (-b+x)^{2/3}\right ) \operatorname {Subst}\left (\int \frac {x^4 \left (-a b+x^6\right )}{\left (-a+x^3\right )^{2/3} \left (-b+x^3\right )^{2/3} \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 )}{\left (x^2 (-a+x) (-b+x)\right )^{2/3}}\\ &=\frac {\left (3 x^{4/3} (-a+x)^{2/3} (-b+x)^{2/3}\right ) \operatorname {Subst}\left (\int \left (\frac {a b x^4}{\left (-a+x^3\right )^{2/3} \left (-b+x^3\right )^{2/3} \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^{10}}{\left (-a+x^3\right )^{2/3} \left (-b+x^3\right )^{2/3} \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 )}{\left (x^2 (-a+x) (-b+x)\right )^{2/3}}\\ &=\frac {\left (3 x^{4/3} (-a+x)^{2/3} (-b+x)^{2/3}\right ) \operatorname {Subst}\left (\int \frac {x^{10}}{\left (-a+x^3\right )^{2/3} \left (-b+x^3\right )^{2/3} \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 )}{\left (x^2 (-a+x) (-b+x)\right )^{2/3}}+\frac {\left (3 a b x^{4/3} (-a+x)^{2/3} (-b+x)^{2/3}\right ) \operatorname {Subst}\left (\int \frac {x^4}{\left (-a+x^3\right )^{2/3} \left (-b+x^3\right )^{2/3} \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 )}{\left (x^2 (-a+x) (-b+x)\right )^{2/3}}\\ \end {align*}

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Mathematica [F]  time = 3.25, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-a b x^2+x^4}{\left (x^2 (-a+x) (-b+x)\right )^{2/3} \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[(-(a*b*x^2) + x^4)/((x^2*(-a + x)*(-b + x))^(2/3)*(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[(-(a*b*x^2) + x^4)/((x^2*(-a + x)*(-b + x))^(2/3)*(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 = 1.42, size = 249, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{d} x}{\sqrt [6]{d} x-2 \sqrt [3]{a b x^2+(-a-b) x^3+x^4}}\right )}{2 d^{5/6}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{d} x}{\sqrt [6]{d} x+2 \sqrt [3]{a b x^2+(-a-b) x^3+x^4}}\right )}{2 d^{5/6}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [6]{d} x}{\sqrt [3]{a b x^2+(-a-b) x^3+x^4}}\right )}{d^{5/6}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [6]{d} x^2+\frac {\left (a b x^2+(-a-b) x^3+x^4\right )^{2/3}}{\sqrt [6]{d}}}{x \sqrt [3]{a b x^2+(-a-b) x^3+x^4}}\right )}{2 d^{5/6}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(-(a*b*x^2) + x^4)/((x^2*(-a + x)*(-b + x))^(2/3)*(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[(Sqrt[3]*d^(1/6)*x)/(d^(1/6)*x - 2*(a*b*x^2 + (-a - b)*x^3 + x^4)^(1/3))])/d^(5/6) + (Sqr
t[3]*ArcTan[(Sqrt[3]*d^(1/6)*x)/(d^(1/6)*x + 2*(a*b*x^2 + (-a - b)*x^3 + x^4)^(1/3))])/(2*d^(5/6)) - ArcTanh[(
d^(1/6)*x)/(a*b*x^2 + (-a - b)*x^3 + x^4)^(1/3)]/d^(5/6) - ArcTanh[(d^(1/6)*x^2 + (a*b*x^2 + (-a - b)*x^3 + x^
4)^(2/3)/d^(1/6))/(x*(a*b*x^2 + (-a - b)*x^3 + x^4)^(1/3))]/(2*d^(5/6))

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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*b*x^2+x^4)/(x^2*(-a+x)*(-b+x))^(2/3)/(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 {a b x^{2} - x^{4}}{{\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 {2}{3}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-a*b*x^2+x^4)/(x^2*(-a+x)*(-b+x))^(2/3)/(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((-a*b*x^2+x^4)/(x^2*(-a+x)*(-b+x))^(2/3)/(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 {a b x^{2} - x^{4}}{{\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 {2}{3}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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