3.31.15 \(\int \frac {x (-a+x) (a b-2 b x+x^2)}{\sqrt [3]{x (-a+x) (-b+x)^2} (-b^2+2 b x+(-1+a^2 d) x^2-2 a d x^3+d x^4)} \, dx\)

Optimal. Leaf size=415 \[ -\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{d} \sqrt [3]{x^2 \left (2 a b+b^2\right )-a b^2 x+x^3 (-a-2 b)+x^4}}{\sqrt [6]{d} \sqrt [3]{x^2 \left (2 a b+b^2\right )-a b^2 x+x^3 (-a-2 b)+x^4}+2 b-2 x}\right )}{2 d^{5/6}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{d} \sqrt [3]{x^2 \left (2 a b+b^2\right )-a b^2 x+x^3 (-a-2 b)+x^4}}{\sqrt [6]{d} \sqrt [3]{x^2 \left (2 a b+b^2\right )-a b^2 x+x^3 (-a-2 b)+x^4}-2 b+2 x}\right )}{2 d^{5/6}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [6]{d} \sqrt [3]{x^2 \left (2 a b+b^2\right )-a b^2 x+x^3 (-a-2 b)+x^4}}{b-x}\right )}{d^{5/6}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [3]{x^2 \left (2 a b+b^2\right )-a b^2 x+x^3 (-a-2 b)+x^4} \left (b \sqrt [6]{d}-\sqrt [6]{d} x\right )}{\sqrt [3]{d} \left (x^2 \left (2 a b+b^2\right )-a b^2 x+x^3 (-a-2 b)+x^4\right )^{2/3}+b^2-2 b x+x^2}\right )}{2 d^{5/6}} \]

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Rubi [F]  time = 17.54, 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) \left (a b-2 b x+x^2\right )}{\sqrt [3]{x (-a+x) (-b+x)^2} \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)*(a*b - 2*b*x + x^2))/((x*(-a + x)*(-b + x)^2)^(1/3)*(-b^2 + 2*b*x + (-1 + a^2*d)*x^2 - 2*a*d*x
^3 + d*x^4)),x]

[Out]

(6*b*x^(1/3)*(-a + x)^(1/3)*(-b + x)^(2/3)*Defer[Subst][Defer[Int][(x^7*(-a + x^3)^(2/3))/((-b + x^3)^(2/3)*(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))^(1/3) + (3*a*
b*x^(1/3)*(-a + x)^(1/3)*(-b + x)^(2/3)*Defer[Subst][Defer[Int][(x^4*(-a + x^3)^(2/3))/((-b + x^3)^(2/3)*(-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))^(1/3) + (3*x^(1
/3)*(-a + x)^(1/3)*(-b + x)^(2/3)*Defer[Subst][Defer[Int][(x^10*(-a + x^3)^(2/3))/((-b + x^3)^(2/3)*(-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))^(1/3)

Rubi steps

\begin {align*} \int \frac {x (-a+x) \left (a b-2 b x+x^2\right )}{\sqrt [3]{x (-a+x) (-b+x)^2} \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 (\sqrt [3]{x} \sqrt [3]{-a+x} (-b+x)^{2/3}\right ) \int \frac {x^{2/3} (-a+x)^{2/3} \left (a b-2 b x+x^2\right )}{(-b+x)^{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}{\sqrt [3]{x (-a+x) (-b+x)^2}}\\ &=\frac {\left (3 \sqrt [3]{x} \sqrt [3]{-a+x} (-b+x)^{2/3}\right ) \operatorname {Subst}\left (\int \frac {x^4 \left (-a+x^3\right )^{2/3} \left (a b-2 b x^3+x^6\right )}{\left (-b+x^3\right )^{2/3} \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 )}{\sqrt [3]{x (-a+x) (-b+x)^2}}\\ &=\frac {\left (3 \sqrt [3]{x} \sqrt [3]{-a+x} (-b+x)^{2/3}\right ) \operatorname {Subst}\left (\int \left (\frac {2 b x^7 \left (-a+x^3\right )^{2/3}}{\left (-b+x^3\right )^{2/3} \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^4 \left (-a+x^3\right )^{2/3}}{\left (-b+x^3\right )^{2/3} \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^{10} \left (-a+x^3\right )^{2/3}}{\left (-b+x^3\right )^{2/3} \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 )}{\sqrt [3]{x (-a+x) (-b+x)^2}}\\ &=\frac {\left (3 \sqrt [3]{x} \sqrt [3]{-a+x} (-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 (-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 )}{\sqrt [3]{x (-a+x) (-b+x)^2}}+\frac {\left (6 b \sqrt [3]{x} \sqrt [3]{-a+x} (-b+x)^{2/3}\right ) \operatorname {Subst}\left (\int \frac {x^7 \left (-a+x^3\right )^{2/3}}{\left (-b+x^3\right )^{2/3} \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 )}{\sqrt [3]{x (-a+x) (-b+x)^2}}+\frac {\left (3 a b \sqrt [3]{x} \sqrt [3]{-a+x} (-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 (-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 )}{\sqrt [3]{x (-a+x) (-b+x)^2}}\\ \end {align*}

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Mathematica [F]  time = 1.62, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x (-a+x) \left (a b-2 b x+x^2\right )}{\sqrt [3]{x (-a+x) (-b+x)^2} \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)*(a*b - 2*b*x + x^2))/((x*(-a + x)*(-b + x)^2)^(1/3)*(-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)*(a*b - 2*b*x + x^2))/((x*(-a + x)*(-b + x)^2)^(1/3)*(-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 = 1.69, size = 415, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{d} \sqrt [3]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}}{2 b-2 x+\sqrt [6]{d} \sqrt [3]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}}\right )}{2 d^{5/6}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{d} \sqrt [3]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}}{-2 b+2 x+\sqrt [6]{d} \sqrt [3]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}}\right )}{2 d^{5/6}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [6]{d} \sqrt [3]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}}{b-x}\right )}{d^{5/6}}+\frac {\tanh ^{-1}\left (\frac {\left (b \sqrt [6]{d}-\sqrt [6]{d} x\right ) \sqrt [3]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}}{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^{5/6}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*(Sqrt[3]*ArcTan[(Sqrt[3]*d^(1/6)*(-(a*b^2*x) + (2*a*b + b^2)*x^2 + (-a - 2*b)*x^3 + x^4)^(1/3))/(2*b - 2*
x + d^(1/6)*(-(a*b^2*x) + (2*a*b + b^2)*x^2 + (-a - 2*b)*x^3 + x^4)^(1/3))])/d^(5/6) + (Sqrt[3]*ArcTan[(Sqrt[3
]*d^(1/6)*(-(a*b^2*x) + (2*a*b + b^2)*x^2 + (-a - 2*b)*x^3 + x^4)^(1/3))/(-2*b + 2*x + d^(1/6)*(-(a*b^2*x) + (
2*a*b + b^2)*x^2 + (-a - 2*b)*x^3 + x^4)^(1/3))])/(2*d^(5/6)) + ArcTanh[(d^(1/6)*(-(a*b^2*x) + (2*a*b + b^2)*x
^2 + (-a - 2*b)*x^3 + x^4)^(1/3))/(b - x)]/d^(5/6) + ArcTanh[((b*d^(1/6) - d^(1/6)*x)*(-(a*b^2*x) + (2*a*b + b
^2)*x^2 + (-a - 2*b)*x^3 + x^4)^(1/3))/(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^(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(x*(-a+x)*(a*b-2*b*x+x^2)/(x*(-a+x)*(-b+x)^2)^(1/3)/(-b^2+2*b*x+(a^2*d-1)*x^2-2*a*d*x^3+d*x^4),x, alg
orithm="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 )} 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 {1}{3}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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