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

Optimal. Leaf size=294 \[ -\frac {\log \left (\sqrt [3]{x^3-a x^2}-\sqrt [12]{d} x\right )}{4 a d^{5/6}}-\frac {\log \left (\sqrt [3]{x^3-a x^2}+\sqrt [12]{d} x\right )}{4 a d^{5/6}}+\frac {\log \left (\left (x^3-a x^2\right )^{2/3}+\sqrt [6]{d} x^2\right )}{4 a d^{5/6}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{d} x^2}{\sqrt [6]{d} x^2-2 \left (x^3-a x^2\right )^{2/3}}\right )}{4 a d^{5/6}}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{d} x^2}{2 \left (x^3-a x^2\right )^{2/3}+\sqrt [6]{d} x^2}\right )}{4 a d^{5/6}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [6]{d} \left (x^3-a x^2\right )^{2/3}}{(x-a) \sqrt [3]{x^3-a x^2}+\sqrt [3]{d} x^2}\right )}{4 a d^{5/6}} \]

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Rubi [F]  time = 1.79, 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}{\sqrt [3]{x^2 (-a+x)} \left (-a^4+4 a^3 x-6 a^2 x^2+4 a x^3+(-1+d) x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.70, size = 219, normalized size = 0.74 \begin {gather*} \frac {x \left (-\log \left (\sqrt [3]{d} \left (\frac {x}{x-a}\right )^{4/3}-\sqrt [6]{d} \left (\frac {x}{x-a}\right )^{2/3}+1\right )+\log \left (\sqrt [3]{d} \left (\frac {x}{x-a}\right )^{4/3}+\sqrt [6]{d} \left (\frac {x}{x-a}\right )^{2/3}+1\right )+2 \sqrt {3} \tan ^{-1}\left (\frac {1-2 \sqrt [6]{d} \left (\frac {x}{x-a}\right )^{2/3}}{\sqrt {3}}\right )-2 \sqrt {3} \tan ^{-1}\left (\frac {2 \sqrt [6]{d} \left (\frac {x}{x-a}\right )^{2/3}+1}{\sqrt {3}}\right )+4 \tanh ^{-1}\left (\sqrt [6]{d} \left (\frac {x}{x-a}\right )^{2/3}\right )\right )}{8 a d^{5/6} \sqrt [3]{\frac {x}{x-a}} \sqrt [3]{x^2 (x-a)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(2*Sqrt[3]*ArcTan[(1 - 2*d^(1/6)*(x/(-a + x))^(2/3))/Sqrt[3]] - 2*Sqrt[3]*ArcTan[(1 + 2*d^(1/6)*(x/(-a + x)
)^(2/3))/Sqrt[3]] + 4*ArcTanh[d^(1/6)*(x/(-a + x))^(2/3)] - Log[1 - d^(1/6)*(x/(-a + x))^(2/3) + d^(1/3)*(x/(-
a + x))^(4/3)] + Log[1 + d^(1/6)*(x/(-a + x))^(2/3) + d^(1/3)*(x/(-a + x))^(4/3)]))/(8*a*d^(5/6)*(x/(-a + x))^
(1/3)*(x^2*(-a + x))^(1/3))

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IntegrateAlgebraic [A]  time = 1.18, size = 426, normalized size = 1.45 \begin {gather*} -\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{d} x^2 \left (-a x^2+x^3\right )^{2/3}}{-\sqrt [3]{d} x^4+\left (-a x^2+x^3\right )^{4/3}}\right )}{4 a d^{5/6}}-\frac {\log \left (-\sqrt [12]{d} x+\sqrt [3]{-a x^2+x^3}\right )}{4 a d^{5/6}}-\frac {\log \left (\sqrt [12]{d} x+\sqrt [3]{-a x^2+x^3}\right )}{4 a d^{5/6}}+\frac {\log \left (\sqrt [6]{d} x^2+\left (-a x^2+x^3\right )^{2/3}\right )}{4 a d^{5/6}}+\frac {\log \left (\sqrt [6]{d} x^2-\sqrt [12]{d} x \sqrt [3]{-a x^2+x^3}+\left (-a x^2+x^3\right )^{2/3}\right )}{8 a d^{5/6}}+\frac {\log \left (\sqrt [6]{d} x^2+\sqrt [12]{d} x \sqrt [3]{-a x^2+x^3}+\left (-a x^2+x^3\right )^{2/3}\right )}{8 a d^{5/6}}-\frac {\log \left (\sqrt [6]{d} x^2-\sqrt {3} \sqrt [12]{d} x \sqrt [3]{-a x^2+x^3}+\left (-a x^2+x^3\right )^{2/3}\right )}{8 a d^{5/6}}-\frac {\log \left (\sqrt [6]{d} x^2+\sqrt {3} \sqrt [12]{d} x \sqrt [3]{-a x^2+x^3}+\left (-a x^2+x^3\right )^{2/3}\right )}{8 a d^{5/6}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/4*(Sqrt[3]*ArcTan[(Sqrt[3]*d^(1/6)*x^2*(-(a*x^2) + x^3)^(2/3))/(-(d^(1/3)*x^4) + (-(a*x^2) + x^3)^(4/3))])/
(a*d^(5/6)) - Log[-(d^(1/12)*x) + (-(a*x^2) + x^3)^(1/3)]/(4*a*d^(5/6)) - Log[d^(1/12)*x + (-(a*x^2) + x^3)^(1
/3)]/(4*a*d^(5/6)) + Log[d^(1/6)*x^2 + (-(a*x^2) + x^3)^(2/3)]/(4*a*d^(5/6)) + Log[d^(1/6)*x^2 - d^(1/12)*x*(-
(a*x^2) + x^3)^(1/3) + (-(a*x^2) + x^3)^(2/3)]/(8*a*d^(5/6)) + Log[d^(1/6)*x^2 + d^(1/12)*x*(-(a*x^2) + x^3)^(
1/3) + (-(a*x^2) + x^3)^(2/3)]/(8*a*d^(5/6)) - Log[d^(1/6)*x^2 - Sqrt[3]*d^(1/12)*x*(-(a*x^2) + x^3)^(1/3) + (
-(a*x^2) + x^3)^(2/3)]/(8*a*d^(5/6)) - Log[d^(1/6)*x^2 + Sqrt[3]*d^(1/12)*x*(-(a*x^2) + x^3)^(1/3) + (-(a*x^2)
+ x^3)^(2/3)]/(8*a*d^(5/6))

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fricas [B]  time = 0.51, size = 557, normalized size = 1.89 \begin {gather*} -\frac {1}{2} \, \sqrt {3} \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{6}} \arctan \left (\frac {2 \, \sqrt {3} a^{5} d^{4} x^{2} \sqrt {\frac {a^{2} d^{2} x^{2} \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{3}} + {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}} a d \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{6}} - {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} {\left (a - x\right )}}{x^{2}}} \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {5}{6}} - 2 \, \sqrt {3} {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}} a^{5} d^{4} \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {5}{6}} - \sqrt {3} x^{2}}{3 \, x^{2}}\right ) - \frac {1}{2} \, \sqrt {3} \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{6}} \arctan \left (\frac {2 \, \sqrt {3} a^{5} d^{4} x^{2} \sqrt {\frac {a^{2} d^{2} x^{2} \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{3}} - {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}} a d \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{6}} - {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} {\left (a - x\right )}}{x^{2}}} \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {5}{6}} - 2 \, \sqrt {3} {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}} a^{5} d^{4} \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {5}{6}} + \sqrt {3} x^{2}}{3 \, x^{2}}\right ) + \frac {1}{8} \, \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{6}} \log \left (\frac {a^{2} d^{2} x^{2} \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{3}} + {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}} a d \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{6}} - {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} {\left (a - x\right )}}{x^{2}}\right ) - \frac {1}{8} \, \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{6}} \log \left (\frac {a^{2} d^{2} x^{2} \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{3}} - {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}} a d \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{6}} - {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} {\left (a - x\right )}}{x^{2}}\right ) + \frac {1}{4} \, \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{6}} \log \left (\frac {a d x^{2} \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{6}} + {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}}}{x^{2}}\right ) - \frac {1}{4} \, \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{6}} \log \left (-\frac {a d x^{2} \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{6}} - {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}}}{x^{2}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*sqrt(3)*(1/(a^6*d^5))^(1/6)*arctan(1/3*(2*sqrt(3)*a^5*d^4*x^2*sqrt((a^2*d^2*x^2*(1/(a^6*d^5))^(1/3) + (-a
*x^2 + x^3)^(2/3)*a*d*(1/(a^6*d^5))^(1/6) - (-a*x^2 + x^3)^(1/3)*(a - x))/x^2)*(1/(a^6*d^5))^(5/6) - 2*sqrt(3)
*(-a*x^2 + x^3)^(2/3)*a^5*d^4*(1/(a^6*d^5))^(5/6) - sqrt(3)*x^2)/x^2) - 1/2*sqrt(3)*(1/(a^6*d^5))^(1/6)*arctan
(1/3*(2*sqrt(3)*a^5*d^4*x^2*sqrt((a^2*d^2*x^2*(1/(a^6*d^5))^(1/3) - (-a*x^2 + x^3)^(2/3)*a*d*(1/(a^6*d^5))^(1/
6) - (-a*x^2 + x^3)^(1/3)*(a - x))/x^2)*(1/(a^6*d^5))^(5/6) - 2*sqrt(3)*(-a*x^2 + x^3)^(2/3)*a^5*d^4*(1/(a^6*d
^5))^(5/6) + sqrt(3)*x^2)/x^2) + 1/8*(1/(a^6*d^5))^(1/6)*log((a^2*d^2*x^2*(1/(a^6*d^5))^(1/3) + (-a*x^2 + x^3)
^(2/3)*a*d*(1/(a^6*d^5))^(1/6) - (-a*x^2 + x^3)^(1/3)*(a - x))/x^2) - 1/8*(1/(a^6*d^5))^(1/6)*log((a^2*d^2*x^2
*(1/(a^6*d^5))^(1/3) - (-a*x^2 + x^3)^(2/3)*a*d*(1/(a^6*d^5))^(1/6) - (-a*x^2 + x^3)^(1/3)*(a - x))/x^2) + 1/4
*(1/(a^6*d^5))^(1/6)*log((a*d*x^2*(1/(a^6*d^5))^(1/6) + (-a*x^2 + x^3)^(2/3))/x^2) - 1/4*(1/(a^6*d^5))^(1/6)*l
og(-(a*d*x^2*(1/(a^6*d^5))^(1/6) - (-a*x^2 + x^3)^(2/3))/x^2)

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giac [A]  time = 0.29, size = 209, normalized size = 0.71 \begin {gather*} -\frac {\sqrt {3} \log \left (\sqrt {3} \left (-d\right )^{\frac {1}{6}} {\left (-\frac {a}{x} + 1\right )}^{\frac {2}{3}} + {\left (-\frac {a}{x} + 1\right )}^{\frac {4}{3}} + \left (-d\right )^{\frac {1}{3}}\right )}{8 \, a \left (-d\right )^{\frac {5}{6}}} - \frac {\sqrt {3} \left (-d\right )^{\frac {1}{6}} \log \left (-\sqrt {3} \left (-d\right )^{\frac {1}{6}} {\left (-\frac {a}{x} + 1\right )}^{\frac {2}{3}} + {\left (-\frac {a}{x} + 1\right )}^{\frac {4}{3}} + \left (-d\right )^{\frac {1}{3}}\right )}{8 \, a d} - \frac {\arctan \left (\frac {\sqrt {3} \left (-d\right )^{\frac {1}{6}} + 2 \, {\left (-\frac {a}{x} + 1\right )}^{\frac {2}{3}}}{\left (-d\right )^{\frac {1}{6}}}\right )}{4 \, a \left (-d\right )^{\frac {5}{6}}} - \frac {\arctan \left (-\frac {\sqrt {3} \left (-d\right )^{\frac {1}{6}} - 2 \, {\left (-\frac {a}{x} + 1\right )}^{\frac {2}{3}}}{\left (-d\right )^{\frac {1}{6}}}\right )}{4 \, a \left (-d\right )^{\frac {5}{6}}} - \frac {\arctan \left (\frac {{\left (-\frac {a}{x} + 1\right )}^{\frac {2}{3}}}{\left (-d\right )^{\frac {1}{6}}}\right )}{2 \, a \left (-d\right )^{\frac {5}{6}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*sqrt(3)*log(sqrt(3)*(-d)^(1/6)*(-a/x + 1)^(2/3) + (-a/x + 1)^(4/3) + (-d)^(1/3))/(a*(-d)^(5/6)) - 1/8*sqr
t(3)*(-d)^(1/6)*log(-sqrt(3)*(-d)^(1/6)*(-a/x + 1)^(2/3) + (-a/x + 1)^(4/3) + (-d)^(1/3))/(a*d) - 1/4*arctan((
sqrt(3)*(-d)^(1/6) + 2*(-a/x + 1)^(2/3))/(-d)^(1/6))/(a*(-d)^(5/6)) - 1/4*arctan(-(sqrt(3)*(-d)^(1/6) - 2*(-a/
x + 1)^(2/3))/(-d)^(1/6))/(a*(-d)^(5/6)) - 1/2*arctan((-a/x + 1)^(2/3)/(-d)^(1/6))/(a*(-d)^(5/6))

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maple [F]  time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {x^{3}}{\left (x^{2} \left (-a +x \right )\right )^{\frac {1}{3}} \left (-a^{4}+4 a^{3} x -6 a^{2} x^{2}+4 a \,x^{3}+\left (-1+d \right ) x^{4}\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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