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

Optimal. Leaf size=289 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt [3]{x (-a-b)+a b+x^2} \left (a \sqrt [6]{d}-\sqrt [6]{d} x\right )}{a^2+\sqrt [3]{d} \left (x (-a-b)+a b+x^2\right )^{2/3}-2 a x+x^2}\right )}{2 d^{5/6}}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{d} \sqrt [3]{x (-a-b)+a b+x^2}}{\sqrt [6]{d} \sqrt [3]{x (-a-b)+a b+x^2}+2 a-2 x}\right )}{2 d^{5/6}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{d} \sqrt [3]{x (-a-b)+a b+x^2}}{\sqrt [6]{d} \sqrt [3]{x (-a-b)+a b+x^2}-2 a+2 x}\right )}{2 d^{5/6}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [6]{d} \sqrt [3]{x (-a-b)+a b+x^2}}{a-x}\right )}{d^{5/6}} \]

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Rubi [F]  time = 6.62, 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-2 b+x) (-b+x)}{\sqrt [3]{(-a+x) (-b+x)} \left (a^4-b^2 d-2 \left (2 a^3-b d\right ) x+\left (6 a^2-d\right ) x^2-4 a x^3+x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [F]  time = 2.58, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a-2 b+x) (-b+x)}{\sqrt [3]{(-a+x) (-b+x)} \left (a^4-b^2 d-2 \left (2 a^3-b d\right ) x+\left (6 a^2-d\right ) x^2-4 a x^3+x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 1.08, size = 289, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{d} \sqrt [3]{a b+(-a-b) x+x^2}}{2 a-2 x+\sqrt [6]{d} \sqrt [3]{a b+(-a-b) x+x^2}}\right )}{2 d^{5/6}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{d} \sqrt [3]{a b+(-a-b) x+x^2}}{-2 a+2 x+\sqrt [6]{d} \sqrt [3]{a b+(-a-b) x+x^2}}\right )}{2 d^{5/6}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [6]{d} \sqrt [3]{a b+(-a-b) x+x^2}}{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+(-a-b) x+x^2}}{a^2-2 a x+x^2+\sqrt [3]{d} \left (a b+(-a-b) x+x^2\right )^{2/3}}\right )}{2 d^{5/6}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.12, size = 0, normalized size = 0.00 \[\int \frac {\left (a -2 b +x \right ) \left (-b +x \right )}{\left (\left (-a +x \right ) \left (-b +x \right )\right )^{\frac {1}{3}} \left (a^{4}-b^{2} d -2 \left (2 a^{3}-b d \right ) x +\left (6 a^{2}-d \right ) x^{2}-4 a \,x^{3}+x^{4}\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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