3.28.73 \(\int \frac {x (-b+x) (-a^2 b+2 a^2 x+(-2 a+b) x^2)}{(x (-a+x) (-b+x))^{2/3} (-a^4+4 a^3 x+(-6 a^2+b^2 d) x^2+2 (2 a-b d) x^3+(-1+d) x^4)} \, dx\)

Optimal. Leaf size=265 \[ \frac {\log \left (a^2-\sqrt [3]{d} \left (x^2 (-a-b)+a b x+x^3\right )^{2/3}-2 a x+x^2\right )}{2 d^{2/3}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{d} \left (x^2 (-a-b)+a b x+x^3\right )^{2/3}}{2 a^2+\sqrt [3]{d} \left (x^2 (-a-b)+a b x+x^3\right )^{2/3}-4 a x+2 x^2}\right )}{2 d^{2/3}}-\frac {\log \left (a^4-4 a^3 x+\left (x^2 (-a-b)+a b x+x^3\right )^{2/3} \left (a^2 \sqrt [3]{d}-2 a \sqrt [3]{d} x+\sqrt [3]{d} x^2\right )+6 a^2 x^2+d^{2/3} \left (x^2 (-a-b)+a b x+x^3\right )^{4/3}-4 a x^3+x^4\right )}{4 d^{2/3}} \]

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Rubi [F]  time = 14.06, 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 (-b+x) \left (-a^2 b+2 a^2 x+(-2 a+b) x^2\right )}{(x (-a+x) (-b+x))^{2/3} \left (-a^4+4 a^3 x+\left (-6 a^2+b^2 d\right ) x^2+2 (2 a-b d) x^3+(-1+d) x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [F]  time = 1.93, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x (-b+x) \left (-a^2 b+2 a^2 x+(-2 a+b) x^2\right )}{(x (-a+x) (-b+x))^{2/3} \left (-a^4+4 a^3 x+\left (-6 a^2+b^2 d\right ) x^2+2 (2 a-b d) x^3+(-1+d) x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 3.38, size = 265, normalized size = 1.00 \begin {gather*} \frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{d} \left (a b x+(-a-b) x^2+x^3\right )^{2/3}}{2 a^2-4 a x+2 x^2+\sqrt [3]{d} \left (a b x+(-a-b) x^2+x^3\right )^{2/3}}\right )}{2 d^{2/3}}+\frac {\log \left (a^2-2 a x+x^2-\sqrt [3]{d} \left (a b x+(-a-b) x^2+x^3\right )^{2/3}\right )}{2 d^{2/3}}-\frac {\log \left (a^4-4 a^3 x+6 a^2 x^2-4 a x^3+x^4+\left (a^2 \sqrt [3]{d}-2 a \sqrt [3]{d} x+\sqrt [3]{d} x^2\right ) \left (a b x+(-a-b) x^2+x^3\right )^{2/3}+d^{2/3} \left (a b x+(-a-b) x^2+x^3\right )^{4/3}\right )}{4 d^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(-b+x)*(-a^2*b+2*a^2*x+(-2*a+b)*x^2)/(x*(-a+x)*(-b+x))^(2/3)/(-a^4+4*a^3*x+(b^2*d-6*a^2)*x^2+2*(-b
*d+2*a)*x^3+(-1+d)*x^4),x, algorithm="giac")

[Out]

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

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maple [F]  time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {x \left (-b +x \right ) \left (-a^{2} b +2 a^{2} x +\left (-2 a +b \right ) x^{2}\right )}{\left (x \left (-a +x \right ) \left (-b +x \right )\right )^{\frac {2}{3}} \left (-a^{4}+4 a^{3} x +\left (b^{2} d -6 a^{2}\right ) x^{2}+2 \left (-b d +2 a \right ) x^{3}+\left (-1+d \right ) x^{4}\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (a^{2} b - 2 \, a^{2} x + {\left (2 \, a - b\right )} x^{2}\right )} {\left (b - x\right )} x}{{\left ({\left (d - 1\right )} x^{4} - a^{4} + 4 \, a^{3} x - 2 \, {\left (b d - 2 \, a\right )} x^{3} + {\left (b^{2} d - 6 \, a^{2}\right )} x^{2}\right )} \left ({\left (a - x\right )} {\left (b - x\right )} x\right )^{\frac {2}{3}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(-b+x)*(-a^2*b+2*a^2*x+(-2*a+b)*x^2)/(x*(-a+x)*(-b+x))^(2/3)/(-a^4+4*a^3*x+(b^2*d-6*a^2)*x^2+2*(-b
*d+2*a)*x^3+(-1+d)*x^4),x, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} -\int -\frac {x\,\left (b-x\right )\,\left (x^2\,\left (2\,a-b\right )+a^2\,b-2\,a^2\,x\right )}{{\left (x\,\left (a-x\right )\,\left (b-x\right )\right )}^{2/3}\,\left (x^2\,\left (b^2\,d-6\,a^2\right )+2\,x^3\,\left (2\,a-b\,d\right )+4\,a^3\,x-a^4+x^4\,\left (d-1\right )\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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