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

Optimal. Leaf size=340 \[ \frac {\log \left (-\sqrt [3]{d} \left (x^2 \left (2 a b+b^2\right )-a b^2 x+x^3 (-a-2 b)+x^4\right )^{2/3}+b^2-2 b x+x^2\right )}{2 d^{2/3}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{d} \left (x^2 \left (2 a b+b^2\right )-a b^2 x+x^3 (-a-2 b)+x^4\right )^{2/3}}{\sqrt [3]{d} \left (x^2 \left (2 a b+b^2\right )-a b^2 x+x^3 (-a-2 b)+x^4\right )^{2/3}+2 b^2-4 b x+2 x^2}\right )}{2 d^{2/3}}-\frac {\log \left (d^{2/3} \left (x^2 \left (2 a b+b^2\right )-a b^2 x+x^3 (-a-2 b)+x^4\right )^{4/3}+\left (x^2 \left (2 a b+b^2\right )-a b^2 x+x^3 (-a-2 b)+x^4\right )^{2/3} \left (b^2 \sqrt [3]{d}-2 b \sqrt [3]{d} x+\sqrt [3]{d} x^2\right )+b^4-4 b^3 x+6 b^2 x^2-4 b x^3+x^4\right )}{4 d^{2/3}} \]

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Rubi [F]  time = 23.64, 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 (-a+x) (-b+x) \left (a b-2 b x+x^2\right )}{\left (x (-a+x) (-b+x)^2\right )^{2/3} \left (-b^2+2 b x-\left (1-a^2 d\right ) x^2-2 a d x^3+d x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [F]  time = 3.37, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x (-a+x) (-b+x) \left (a b-2 b x+x^2\right )}{\left (x (-a+x) (-b+x)^2\right )^{2/3} \left (-b^2+2 b x-\left (1-a^2 d\right ) x^2-2 a d x^3+d x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.82, size = 340, normalized size = 1.00 \begin {gather*} \frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{d} \left (-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4\right )^{2/3}}{2 b^2-4 b x+2 x^2+\sqrt [3]{d} \left (-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4\right )^{2/3}}\right )}{2 d^{2/3}}+\frac {\log \left (b^2-2 b x+x^2-\sqrt [3]{d} \left (-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4\right )^{2/3}\right )}{2 d^{2/3}}-\frac {\log \left (b^4-4 b^3 x+6 b^2 x^2-4 b x^3+x^4+\left (b^2 \sqrt [3]{d}-2 b \sqrt [3]{d} x+\sqrt [3]{d} x^2\right ) \left (-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4\right )^{2/3}+d^{2/3} \left (-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4\right )^{4/3}\right )}{4 d^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {x \left (-a +x \right ) \left (-b +x \right ) \left (a b -2 b x +x^{2}\right )}{\left (x \left (-a +x \right ) \left (-b +x \right )^{2}\right )^{\frac {2}{3}} \left (-b^{2}+2 b x -\left (-a^{2} d +1\right ) x^{2}-2 a d \,x^{3}+d \,x^{4}\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x*(-a+x)*(-b+x)*(a*b-2*b*x+x^2)/(x*(-a+x)*(-b+x)^2)^(2/3)/(-b^2+2*b*x-(-a^2*d+1)*x^2-2*a*d*x^3+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 b - 2 \, b x + x^{2}\right )} {\left (a - x\right )} {\left (b - x\right )} x}{{\left (2 \, a d x^{3} - d x^{4} - {\left (a^{2} d - 1\right )} x^{2} + b^{2} - 2 \, b x\right )} \left (-{\left (a - x\right )} {\left (b - x\right )}^{2} x\right )^{\frac {2}{3}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-integrate((a*b - 2*b*x + x^2)*(a - x)*(b - x)*x/((2*a*d*x^3 - d*x^4 - (a^2*d - 1)*x^2 + b^2 - 2*b*x)*(-(a - x
)*(b - x)^2*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 (a-x\right )\,\left (b-x\right )\,\left (x^2-2\,b\,x+a\,b\right )}{{\left (-x\,\left (a-x\right )\,{\left (b-x\right )}^2\right )}^{2/3}\,\left (-b^2+2\,b\,x+d\,x^4-2\,a\,d\,x^3+\left (a^2\,d-1\right )\,x^2\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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