3.30.7 \(\int \frac {x^3 (-b+x) (2 a b-3 a x+x^2)}{\sqrt [3]{x^2 (-a+x) (-b+x)} (-a^4+4 a^3 x-6 a^2 x^2+4 a x^3+(-1+b^2 d) x^4-2 b d x^5+d x^6)} \, dx\)

Optimal. Leaf size=324 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt [3]{x^3 (-a-b)+a b x^2+x^4} \left (a \sqrt [6]{d}-\sqrt [6]{d} x\right )}{a^2+\sqrt [3]{d} \left (x^3 (-a-b)+a b x^2+x^4\right )^{2/3}-2 a x+x^2}\right )}{2 d^{5/6}}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{d} \sqrt [3]{x^3 (-a-b)+a b x^2+x^4}}{\sqrt [6]{d} \sqrt [3]{x^3 (-a-b)+a b x^2+x^4}+2 a-2 x}\right )}{2 d^{5/6}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{d} \sqrt [3]{x^3 (-a-b)+a b x^2+x^4}}{\sqrt [6]{d} \sqrt [3]{x^3 (-a-b)+a b x^2+x^4}-2 a+2 x}\right )}{2 d^{5/6}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [6]{d} \sqrt [3]{x^3 (-a-b)+a b x^2+x^4}}{a-x}\right )}{d^{5/6}} \]

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Rubi [F]  time = 52.34, 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^3 (-b+x) \left (2 a b-3 a x+x^2\right )}{\sqrt [3]{x^2 (-a+x) (-b+x)} \left (-a^4+4 a^3 x-6 a^2 x^2+4 a x^3+\left (-1+b^2 d\right ) x^4-2 b d x^5+d x^6\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(x^3*(-b + x)*(2*a*b - 3*a*x + x^2))/((x^2*(-a + x)*(-b + x))^(1/3)*(-a^4 + 4*a^3*x - 6*a^2*x^2 + 4*a*x^3
+ (-1 + b^2*d)*x^4 - 2*b*d*x^5 + d*x^6)),x]

[Out]

(9*a*x^(2/3)*(-a + x)^(1/3)*(-b + x)^(1/3)*Defer[Subst][Defer[Int][(x^12*(-b + x^3)^(2/3))/((-a + x^3)^(1/3)*(
a^4 - 4*a^3*x^3 + 6*a^2*x^6 - 4*a*x^9 + (1 - b^2*d)*x^12 + 2*b*d*x^15 - d*x^18)), x], x, x^(1/3)])/((a - x)*(b
- x)*x^2)^(1/3) + (6*a*b*x^(2/3)*(-a + x)^(1/3)*(-b + x)^(1/3)*Defer[Subst][Defer[Int][(x^9*(-b + x^3)^(2/3))
/((-a + x^3)^(1/3)*(-a^4 + 4*a^3*x^3 - 6*a^2*x^6 + 4*a*x^9 - (1 - b^2*d)*x^12 - 2*b*d*x^15 + d*x^18)), 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^15
*(-b + x^3)^(2/3))/((-a + x^3)^(1/3)*(-a^4 + 4*a^3*x^3 - 6*a^2*x^6 + 4*a*x^9 - (1 - b^2*d)*x^12 - 2*b*d*x^15 +
d*x^18)), x], x, x^(1/3)])/((a - x)*(b - x)*x^2)^(1/3)

Rubi steps

\begin {align*} \int \frac {x^3 (-b+x) \left (2 a b-3 a x+x^2\right )}{\sqrt [3]{x^2 (-a+x) (-b+x)} \left (-a^4+4 a^3 x-6 a^2 x^2+4 a x^3+\left (-1+b^2 d\right ) x^4-2 b d x^5+d x^6\right )} \, dx &=\frac {\left (x^{2/3} \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \int \frac {x^{7/3} (-b+x)^{2/3} \left (2 a b-3 a x+x^2\right )}{\sqrt [3]{-a+x} \left (-a^4+4 a^3 x-6 a^2 x^2+4 a x^3+\left (-1+b^2 d\right ) x^4-2 b d x^5+d x^6\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 ) \operatorname {Subst}\left (\int \frac {x^9 \left (-b+x^3\right )^{2/3} \left (2 a b-3 a x^3+x^6\right )}{\sqrt [3]{-a+x^3} \left (-a^4+4 a^3 x^3-6 a^2 x^6+4 a x^9+\left (-1+b^2 d\right ) x^{12}-2 b d x^{15}+d x^{18}\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 ) \operatorname {Subst}\left (\int \left (\frac {3 a x^{12} \left (-b+x^3\right )^{2/3}}{\sqrt [3]{-a+x^3} \left (a^4-4 a^3 x^3+6 a^2 x^6-4 a x^9+\left (1-b^2 d\right ) x^{12}+2 b d x^{15}-d x^{18}\right )}+\frac {2 a b x^9 \left (-b+x^3\right )^{2/3}}{\sqrt [3]{-a+x^3} \left (-a^4+4 a^3 x^3-6 a^2 x^6+4 a x^9-\left (1-b^2 d\right ) x^{12}-2 b d x^{15}+d x^{18}\right )}+\frac {x^{15} \left (-b+x^3\right )^{2/3}}{\sqrt [3]{-a+x^3} \left (-a^4+4 a^3 x^3-6 a^2 x^6+4 a x^9-\left (1-b^2 d\right ) x^{12}-2 b d x^{15}+d x^{18}\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 ) \operatorname {Subst}\left (\int \frac {x^{15} \left (-b+x^3\right )^{2/3}}{\sqrt [3]{-a+x^3} \left (-a^4+4 a^3 x^3-6 a^2 x^6+4 a x^9-\left (1-b^2 d\right ) x^{12}-2 b d x^{15}+d x^{18}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2 (-a+x) (-b+x)}}+\frac {\left (9 a x^{2/3} \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^{12} \left (-b+x^3\right )^{2/3}}{\sqrt [3]{-a+x^3} \left (a^4-4 a^3 x^3+6 a^2 x^6-4 a x^9+\left (1-b^2 d\right ) x^{12}+2 b d x^{15}-d x^{18}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2 (-a+x) (-b+x)}}+\frac {\left (6 a b x^{2/3} \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^9 \left (-b+x^3\right )^{2/3}}{\sqrt [3]{-a+x^3} \left (-a^4+4 a^3 x^3-6 a^2 x^6+4 a x^9-\left (1-b^2 d\right ) x^{12}-2 b d x^{15}+d x^{18}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2 (-a+x) (-b+x)}}\\ \end {align*}

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Mathematica [F]  time = 3.34, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^3 (-b+x) \left (2 a b-3 a x+x^2\right )}{\sqrt [3]{x^2 (-a+x) (-b+x)} \left (-a^4+4 a^3 x-6 a^2 x^2+4 a x^3+\left (-1+b^2 d\right ) x^4-2 b d x^5+d x^6\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(x^3*(-b + x)*(2*a*b - 3*a*x + x^2))/((x^2*(-a + x)*(-b + x))^(1/3)*(-a^4 + 4*a^3*x - 6*a^2*x^2 + 4*
a*x^3 + (-1 + b^2*d)*x^4 - 2*b*d*x^5 + d*x^6)),x]

[Out]

Integrate[(x^3*(-b + x)*(2*a*b - 3*a*x + x^2))/((x^2*(-a + x)*(-b + x))^(1/3)*(-a^4 + 4*a^3*x - 6*a^2*x^2 + 4*
a*x^3 + (-1 + b^2*d)*x^4 - 2*b*d*x^5 + d*x^6)), x]

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IntegrateAlgebraic [A]  time = 4.35, size = 324, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{d} \sqrt [3]{a b x^2+(-a-b) x^3+x^4}}{2 a-2 x+\sqrt [6]{d} \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} \sqrt [3]{a b x^2+(-a-b) x^3+x^4}}{-2 a+2 x+\sqrt [6]{d} \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} \sqrt [3]{a b x^2+(-a-b) x^3+x^4}}{a-x}\right )}{d^{5/6}}+\frac {\tanh ^{-1}\left (\frac {\left (a \sqrt [6]{d}-\sqrt [6]{d} x\right ) \sqrt [3]{a b x^2+(-a-b) x^3+x^4}}{a^2-2 a x+x^2+\sqrt [3]{d} \left (a b x^2+(-a-b) x^3+x^4\right )^{2/3}}\right )}{2 d^{5/6}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(x^3*(-b + x)*(2*a*b - 3*a*x + x^2))/((x^2*(-a + x)*(-b + x))^(1/3)*(-a^4 + 4*a^3*x - 6*a^2
*x^2 + 4*a*x^3 + (-1 + b^2*d)*x^4 - 2*b*d*x^5 + d*x^6)),x]

[Out]

-1/2*(Sqrt[3]*ArcTan[(Sqrt[3]*d^(1/6)*(a*b*x^2 + (-a - b)*x^3 + x^4)^(1/3))/(2*a - 2*x + d^(1/6)*(a*b*x^2 + (-
a - b)*x^3 + x^4)^(1/3))])/d^(5/6) + (Sqrt[3]*ArcTan[(Sqrt[3]*d^(1/6)*(a*b*x^2 + (-a - b)*x^3 + x^4)^(1/3))/(-
2*a + 2*x + d^(1/6)*(a*b*x^2 + (-a - b)*x^3 + x^4)^(1/3))])/(2*d^(5/6)) + ArcTanh[(d^(1/6)*(a*b*x^2 + (-a - b)
*x^3 + x^4)^(1/3))/(a - x)]/d^(5/6) + ArcTanh[((a*d^(1/6) - d^(1/6)*x)*(a*b*x^2 + (-a - b)*x^3 + x^4)^(1/3))/(
a^2 - 2*a*x + x^2 + d^(1/3)*(a*b*x^2 + (-a - 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^3*(-b+x)*(2*a*b-3*a*x+x^2)/(x^2*(-a+x)*(-b+x))^(1/3)/(-a^4+4*a^3*x-6*a^2*x^2+4*a*x^3+(b^2*d-1)*x^4
-2*b*d*x^5+d*x^6),x, algorithm="fricas")

[Out]

Timed out

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giac [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^3*(-b+x)*(2*a*b-3*a*x+x^2)/(x^2*(-a+x)*(-b+x))^(1/3)/(-a^4+4*a^3*x-6*a^2*x^2+4*a*x^3+(b^2*d-1)*x^4
-2*b*d*x^5+d*x^6),x, algorithm="giac")

[Out]

Timed out

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maple [F]  time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {x^{3} \left (-b +x \right ) \left (2 a b -3 a x +x^{2}\right )}{\left (x^{2} \left (-a +x \right ) \left (-b +x \right )\right )^{\frac {1}{3}} \left (-a^{4}+4 a^{3} x -6 a^{2} x^{2}+4 a \,x^{3}+\left (b^{2} d -1\right ) x^{4}-2 b d \,x^{5}+d \,x^{6}\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(-b+x)*(2*a*b-3*a*x+x^2)/(x^2*(-a+x)*(-b+x))^(1/3)/(-a^4+4*a^3*x-6*a^2*x^2+4*a*x^3+(b^2*d-1)*x^4-2*b*d
*x^5+d*x^6),x)

[Out]

int(x^3*(-b+x)*(2*a*b-3*a*x+x^2)/(x^2*(-a+x)*(-b+x))^(1/3)/(-a^4+4*a^3*x-6*a^2*x^2+4*a*x^3+(b^2*d-1)*x^4-2*b*d
*x^5+d*x^6),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (2 \, a b - 3 \, a x + x^{2}\right )} {\left (b - x\right )} x^{3}}{{\left (2 \, b d x^{5} - d x^{6} - {\left (b^{2} d - 1\right )} x^{4} + a^{4} - 4 \, a^{3} x + 6 \, a^{2} x^{2} - 4 \, a x^{3}\right )} \left ({\left (a - x\right )} {\left (b - x\right )} x^{2}\right )^{\frac {1}{3}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(-b+x)*(2*a*b-3*a*x+x^2)/(x^2*(-a+x)*(-b+x))^(1/3)/(-a^4+4*a^3*x-6*a^2*x^2+4*a*x^3+(b^2*d-1)*x^4
-2*b*d*x^5+d*x^6),x, algorithm="maxima")

[Out]

integrate((2*a*b - 3*a*x + x^2)*(b - x)*x^3/((2*b*d*x^5 - d*x^6 - (b^2*d - 1)*x^4 + a^4 - 4*a^3*x + 6*a^2*x^2
- 4*a*x^3)*((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^3\,\left (b-x\right )\,\left (x^2-3\,a\,x+2\,a\,b\right )}{{\left (x^2\,\left (a-x\right )\,\left (b-x\right )\right )}^{1/3}\,\left (-a^4+4\,a^3\,x-6\,a^2\,x^2+4\,a\,x^3+d\,x^6-2\,b\,d\,x^5+\left (b^2\,d-1\right )\,x^4\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x^3*(b - x)*(2*a*b - 3*a*x + x^2))/((x^2*(a - x)*(b - x))^(1/3)*(x^4*(b^2*d - 1) + 4*a*x^3 + 4*a^3*x + d
*x^6 - a^4 - 6*a^2*x^2 - 2*b*d*x^5)),x)

[Out]

int(-(x^3*(b - x)*(2*a*b - 3*a*x + x^2))/((x^2*(a - x)*(b - x))^(1/3)*(x^4*(b^2*d - 1) + 4*a*x^3 + 4*a^3*x + d
*x^6 - a^4 - 6*a^2*x^2 - 2*b*d*x^5)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{3} \left (- b + x\right ) \left (2 a b - 3 a x + x^{2}\right )}{\sqrt [3]{x^{2} \left (- a + x\right ) \left (- b + x\right )} \left (- a^{4} + 4 a^{3} x - 6 a^{2} x^{2} + 4 a x^{3} + b^{2} d x^{4} - 2 b d x^{5} + d x^{6} - x^{4}\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(-b+x)*(2*a*b-3*a*x+x**2)/(x**2*(-a+x)*(-b+x))**(1/3)/(-a**4+4*a**3*x-6*a**2*x**2+4*a*x**3+(b**
2*d-1)*x**4-2*b*d*x**5+d*x**6),x)

[Out]

Integral(x**3*(-b + x)*(2*a*b - 3*a*x + x**2)/((x**2*(-a + x)*(-b + x))**(1/3)*(-a**4 + 4*a**3*x - 6*a**2*x**2
+ 4*a*x**3 + b**2*d*x**4 - 2*b*d*x**5 + d*x**6 - x**4)), x)

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