3.28.86 \(\int \frac {x^5 (-4 a+3 x)}{\sqrt [3]{x^2 (-a+x)} (-a^2+2 a x-x^2+d x^8)} \, dx\) [2786]

Optimal. Leaf size=268 \[ \frac {\sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} \sqrt [3]{d} x^4}{\sqrt [3]{d} x^4+2 \left (-a x^2+x^3\right )^{2/3}}\right )}{2 d^{2/3}}+\frac {\log \left (-a \sqrt [6]{d} x^2+a \sqrt [3]{-a x^2+x^3}\right )}{2 d^{2/3}}+\frac {\log \left (a \sqrt [6]{d} x^2+a \sqrt [3]{-a x^2+x^3}\right )}{2 d^{2/3}}-\frac {\log \left (a^2 \sqrt [3]{d} x^4-a^2 \sqrt [6]{d} x^2 \sqrt [3]{-a x^2+x^3}+a^2 \left (-a x^2+x^3\right )^{2/3}\right )}{4 d^{2/3}}-\frac {\log \left (a^2 \sqrt [3]{d} x^4+a^2 \sqrt [6]{d} x^2 \sqrt [3]{-a x^2+x^3}+a^2 \left (-a x^2+x^3\right )^{2/3}\right )}{4 d^{2/3}} \]

[Out]

1/2*3^(1/2)*arctan(3^(1/2)*d^(1/3)*x^4/(d^(1/3)*x^4+2*(-a*x^2+x^3)^(2/3)))/d^(2/3)+1/2*ln(-a*d^(1/6)*x^2+a*(-a
*x^2+x^3)^(1/3))/d^(2/3)+1/2*ln(a*d^(1/6)*x^2+a*(-a*x^2+x^3)^(1/3))/d^(2/3)-1/4*ln(a^2*d^(1/3)*x^4-a^2*d^(1/6)
*x^2*(-a*x^2+x^3)^(1/3)+a^2*(-a*x^2+x^3)^(2/3))/d^(2/3)-1/4*ln(a^2*d^(1/3)*x^4+a^2*d^(1/6)*x^2*(-a*x^2+x^3)^(1
/3)+a^2*(-a*x^2+x^3)^(2/3))/d^(2/3)

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Rubi [F]
time = 2.25, 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^5 (-4 a+3 x)}{\sqrt [3]{x^2 (-a+x)} \left (-a^2+2 a x-x^2+d x^8\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(x^5*(-4*a + 3*x))/((x^2*(-a + x))^(1/3)*(-a^2 + 2*a*x - x^2 + d*x^8)),x]

[Out]

(12*a*x^(2/3)*(-a + x)^(1/3)*Defer[Subst][Defer[Int][x^15/((-a + x^3)^(1/3)*(a^2 - 2*a*x^3 + x^6 - d*x^24)), x
], x, x^(1/3)])/(-((a - x)*x^2))^(1/3) + (9*x^(2/3)*(-a + x)^(1/3)*Defer[Subst][Defer[Int][x^18/((-a + x^3)^(1
/3)*(-a^2 + 2*a*x^3 - x^6 + d*x^24)), x], x, x^(1/3)])/(-((a - x)*x^2))^(1/3)

Rubi steps

\begin {align*} \int \frac {x^5 (-4 a+3 x)}{\sqrt [3]{x^2 (-a+x)} \left (-a^2+2 a x-x^2+d x^8\right )} \, dx &=\frac {\left (x^{2/3} \sqrt [3]{-a+x}\right ) \int \frac {x^{13/3} (-4 a+3 x)}{\sqrt [3]{-a+x} \left (-a^2+2 a x-x^2+d x^8\right )} \, dx}{\sqrt [3]{x^2 (-a+x)}}\\ &=\frac {\left (3 x^{2/3} \sqrt [3]{-a+x}\right ) \text {Subst}\left (\int \frac {x^{15} \left (-4 a+3 x^3\right )}{\sqrt [3]{-a+x^3} \left (-a^2+2 a x^3-x^6+d x^{24}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2 (-a+x)}}\\ &=\frac {\left (3 x^{2/3} \sqrt [3]{-a+x}\right ) \text {Subst}\left (\int \left (\frac {4 a x^{15}}{\sqrt [3]{-a+x^3} \left (a^2-2 a x^3+x^6-d x^{24}\right )}+\frac {3 x^{18}}{\sqrt [3]{-a+x^3} \left (-a^2+2 a x^3-x^6+d x^{24}\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2 (-a+x)}}\\ &=\frac {\left (9 x^{2/3} \sqrt [3]{-a+x}\right ) \text {Subst}\left (\int \frac {x^{18}}{\sqrt [3]{-a+x^3} \left (-a^2+2 a x^3-x^6+d x^{24}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2 (-a+x)}}+\frac {\left (12 a x^{2/3} \sqrt [3]{-a+x}\right ) \text {Subst}\left (\int \frac {x^{15}}{\sqrt [3]{-a+x^3} \left (a^2-2 a x^3+x^6-d x^{24}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2 (-a+x)}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 31.10, size = 738, normalized size = 2.75 \begin {gather*} -\frac {a^4 x \left (4 \text {RootSum}\left [-1+6 \text {$\#$1}-15 \text {$\#$1}^2+20 \text {$\#$1}^3-15 \text {$\#$1}^4+6 \text {$\#$1}^5-\text {$\#$1}^6+a^6 d \text {$\#$1}^8\&,\frac {-6 \sqrt [3]{\frac {x}{-a+x}} \text {$\#$1}^4+2 \sqrt {3} \text {ArcTan}\left (\frac {1+\frac {2 \sqrt [3]{\frac {x}{-a+x}}}{\sqrt [3]{\text {$\#$1}}}}{\sqrt {3}}\right ) \text {$\#$1}^{13/3}-2 \log \left (-\sqrt [3]{\frac {x}{-a+x}}+\sqrt [3]{\text {$\#$1}}\right ) \text {$\#$1}^{13/3}+\log \left (\left (\frac {x}{-a+x}\right )^{2/3}+\sqrt [3]{\frac {x}{-a+x}} \sqrt [3]{\text {$\#$1}}+\text {$\#$1}^{2/3}\right ) \text {$\#$1}^{13/3}}{3-15 \text {$\#$1}+30 \text {$\#$1}^2-30 \text {$\#$1}^3+15 \text {$\#$1}^4-3 \text {$\#$1}^5+4 a^6 d \text {$\#$1}^7}\&\right ]-5 \text {RootSum}\left [-1+6 \text {$\#$1}-15 \text {$\#$1}^2+20 \text {$\#$1}^3-15 \text {$\#$1}^4+6 \text {$\#$1}^5-\text {$\#$1}^6+a^6 d \text {$\#$1}^8\&,\frac {-6 \sqrt [3]{\frac {x}{-a+x}} \text {$\#$1}^5+2 \sqrt {3} \text {ArcTan}\left (\frac {1+\frac {2 \sqrt [3]{\frac {x}{-a+x}}}{\sqrt [3]{\text {$\#$1}}}}{\sqrt {3}}\right ) \text {$\#$1}^{16/3}-2 \log \left (-\sqrt [3]{\frac {x}{-a+x}}+\sqrt [3]{\text {$\#$1}}\right ) \text {$\#$1}^{16/3}+\log \left (\left (\frac {x}{-a+x}\right )^{2/3}+\sqrt [3]{\frac {x}{-a+x}} \sqrt [3]{\text {$\#$1}}+\text {$\#$1}^{2/3}\right ) \text {$\#$1}^{16/3}}{3-15 \text {$\#$1}+30 \text {$\#$1}^2-30 \text {$\#$1}^3+15 \text {$\#$1}^4-3 \text {$\#$1}^5+4 a^6 d \text {$\#$1}^7}\&\right ]+\text {RootSum}\left [-1+6 \text {$\#$1}-15 \text {$\#$1}^2+20 \text {$\#$1}^3-15 \text {$\#$1}^4+6 \text {$\#$1}^5-\text {$\#$1}^6+a^6 d \text {$\#$1}^8\&,\frac {-6 \sqrt [3]{\frac {x}{-a+x}} \text {$\#$1}^6+2 \sqrt {3} \text {ArcTan}\left (\frac {1+\frac {2 \sqrt [3]{\frac {x}{-a+x}}}{\sqrt [3]{\text {$\#$1}}}}{\sqrt {3}}\right ) \text {$\#$1}^{19/3}-2 \log \left (-\sqrt [3]{\frac {x}{-a+x}}+\sqrt [3]{\text {$\#$1}}\right ) \text {$\#$1}^{19/3}+\log \left (\left (\frac {x}{-a+x}\right )^{2/3}+\sqrt [3]{\frac {x}{-a+x}} \sqrt [3]{\text {$\#$1}}+\text {$\#$1}^{2/3}\right ) \text {$\#$1}^{19/3}}{3-15 \text {$\#$1}+30 \text {$\#$1}^2-30 \text {$\#$1}^3+15 \text {$\#$1}^4-3 \text {$\#$1}^5+4 a^6 d \text {$\#$1}^7}\&\right ]\right )}{4 \sqrt [3]{\frac {x}{-a+x}} \sqrt [3]{x^2 (-a+x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^5*(-4*a + 3*x))/((x^2*(-a + x))^(1/3)*(-a^2 + 2*a*x - x^2 + d*x^8)),x]

[Out]

-1/4*(a^4*x*(4*RootSum[-1 + 6*#1 - 15*#1^2 + 20*#1^3 - 15*#1^4 + 6*#1^5 - #1^6 + a^6*d*#1^8 & , (-6*(x/(-a + x
))^(1/3)*#1^4 + 2*Sqrt[3]*ArcTan[(1 + (2*(x/(-a + x))^(1/3))/#1^(1/3))/Sqrt[3]]*#1^(13/3) - 2*Log[-(x/(-a + x)
)^(1/3) + #1^(1/3)]*#1^(13/3) + Log[(x/(-a + x))^(2/3) + (x/(-a + x))^(1/3)*#1^(1/3) + #1^(2/3)]*#1^(13/3))/(3
- 15*#1 + 30*#1^2 - 30*#1^3 + 15*#1^4 - 3*#1^5 + 4*a^6*d*#1^7) & ] - 5*RootSum[-1 + 6*#1 - 15*#1^2 + 20*#1^3
- 15*#1^4 + 6*#1^5 - #1^6 + a^6*d*#1^8 & , (-6*(x/(-a + x))^(1/3)*#1^5 + 2*Sqrt[3]*ArcTan[(1 + (2*(x/(-a + x))
^(1/3))/#1^(1/3))/Sqrt[3]]*#1^(16/3) - 2*Log[-(x/(-a + x))^(1/3) + #1^(1/3)]*#1^(16/3) + Log[(x/(-a + x))^(2/3
) + (x/(-a + x))^(1/3)*#1^(1/3) + #1^(2/3)]*#1^(16/3))/(3 - 15*#1 + 30*#1^2 - 30*#1^3 + 15*#1^4 - 3*#1^5 + 4*a
^6*d*#1^7) & ] + RootSum[-1 + 6*#1 - 15*#1^2 + 20*#1^3 - 15*#1^4 + 6*#1^5 - #1^6 + a^6*d*#1^8 & , (-6*(x/(-a +
x))^(1/3)*#1^6 + 2*Sqrt[3]*ArcTan[(1 + (2*(x/(-a + x))^(1/3))/#1^(1/3))/Sqrt[3]]*#1^(19/3) - 2*Log[-(x/(-a +
x))^(1/3) + #1^(1/3)]*#1^(19/3) + Log[(x/(-a + x))^(2/3) + (x/(-a + x))^(1/3)*#1^(1/3) + #1^(2/3)]*#1^(19/3))/
(3 - 15*#1 + 30*#1^2 - 30*#1^3 + 15*#1^4 - 3*#1^5 + 4*a^6*d*#1^7) & ]))/((x/(-a + x))^(1/3)*(x^2*(-a + x))^(1/
3))

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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {x^{5} \left (-4 a +3 x \right )}{\left (x^{2} \left (-a +x \right )\right )^{\frac {1}{3}} \left (d \,x^{8}-a^{2}+2 a x -x^{2}\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(-4*a+3*x)/(x^2*(-a+x))^(1/3)/(d*x^8-a^2+2*a*x-x^2),x)

[Out]

int(x^5*(-4*a+3*x)/(x^2*(-a+x))^(1/3)/(d*x^8-a^2+2*a*x-x^2),x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(-4*a+3*x)/(x^2*(-a+x))^(1/3)/(d*x^8-a^2+2*a*x-x^2),x, algorithm="maxima")

[Out]

-integrate((4*a - 3*x)*x^5/((d*x^8 - a^2 + 2*a*x - x^2)*(-(a - x)*x^2)^(1/3)), x)

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Fricas [A]
time = 0.40, size = 167, normalized size = 0.62 \begin {gather*} -\frac {2 \, \sqrt {3} {\left (d^{2}\right )}^{\frac {1}{6}} d \arctan \left (\frac {\sqrt {3} {\left ({\left (d^{2}\right )}^{\frac {1}{3}} d x^{4} + 2 \, {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}} {\left (d^{2}\right )}^{\frac {2}{3}}\right )} {\left (d^{2}\right )}^{\frac {1}{6}}}{3 \, d^{2} x^{4}}\right ) - 2 \, {\left (d^{2}\right )}^{\frac {2}{3}} \log \left (-\frac {{\left (d^{2}\right )}^{\frac {2}{3}} x^{4} - {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}} d}{x^{4}}\right ) + {\left (d^{2}\right )}^{\frac {2}{3}} \log \left (\frac {{\left (d^{2}\right )}^{\frac {1}{3}} d x^{6} + {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}} {\left (d^{2}\right )}^{\frac {2}{3}} x^{2} - {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} {\left (a d - d x\right )}}{x^{6}}\right )}{4 \, d^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(-4*a+3*x)/(x^2*(-a+x))^(1/3)/(d*x^8-a^2+2*a*x-x^2),x, algorithm="fricas")

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{5} \left (- 4 a + 3 x\right )}{\sqrt [3]{x^{2} \left (- a + x\right )} \left (- a^{2} + 2 a x + d x^{8} - x^{2}\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*(-4*a+3*x)/(x**2*(-a+x))**(1/3)/(d*x**8-a**2+2*a*x-x**2),x)

[Out]

Integral(x**5*(-4*a + 3*x)/((x**2*(-a + x))**(1/3)*(-a**2 + 2*a*x + d*x**8 - x**2)), x)

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Giac [A]
time = 0.56, size = 1, normalized size = 0.00 \begin {gather*} 0 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(-4*a+3*x)/(x^2*(-a+x))^(1/3)/(d*x^8-a^2+2*a*x-x^2),x, algorithm="giac")

[Out]

0

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int -\frac {x^5\,\left (4\,a-3\,x\right )}{{\left (-x^2\,\left (a-x\right )\right )}^{1/3}\,\left (-a^2+2\,a\,x+d\,x^8-x^2\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x^5*(4*a - 3*x))/((-x^2*(a - x))^(1/3)*(2*a*x + d*x^8 - a^2 - x^2)),x)

[Out]

int(-(x^5*(4*a - 3*x))/((-x^2*(a - x))^(1/3)*(2*a*x + d*x^8 - a^2 - x^2)), x)

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