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

Optimal. Leaf size=152 \[ -\frac {\log \left (a^2 d^{2/3} \left (x^3-a x^2\right )^{2/3}+a^2 \sqrt [3]{d} x^2 \sqrt [3]{x^3-a x^2}+a^2 x^4\right )}{2 d^{2/3}}+\frac {\log \left (a x^2-a \sqrt [3]{d} \sqrt [3]{x^3-a x^2}\right )}{d^{2/3}}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x^2}{2 \sqrt [3]{d} \sqrt [3]{x^3-a x^2}+x^2}\right )}{d^{2/3}} \]

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Rubi [F]  time = 1.90, 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 (-4 a+3 x)}{\sqrt [3]{x^2 (-a+x)} \left (a d-d x+x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [C]  time = 1.43, size = 646, normalized size = 4.25 \begin {gather*} \frac {a x \left (4 \text {RootSum}\left [\text {$\#$1}^4 \left (-a^3\right )+\text {$\#$1}^3 d-3 \text {$\#$1}^2 d+3 \text {$\#$1} d-d\&,\frac {-\sqrt [3]{\text {$\#$1}} \log \left (\text {$\#$1}^{2/3}+\sqrt [3]{\text {$\#$1}} \sqrt [3]{\frac {x}{x-a}}+\left (\frac {x}{x-a}\right )^{2/3}\right )+2 \sqrt [3]{\text {$\#$1}} \log \left (\sqrt [3]{\text {$\#$1}}-\sqrt [3]{\frac {x}{x-a}}\right )-2 \sqrt {3} \sqrt [3]{\text {$\#$1}} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{\frac {x}{x-a}}}{\sqrt [3]{\text {$\#$1}}}+1}{\sqrt {3}}\right )+6 \sqrt [3]{\frac {x}{x-a}}}{4 \text {$\#$1}^3 a^3-3 \text {$\#$1}^2 d+6 \text {$\#$1} d-3 d}\&\right ]+5 \text {RootSum}\left [\text {$\#$1}^4 \left (-a^3\right )+\text {$\#$1}^3 d-3 \text {$\#$1}^2 d+3 \text {$\#$1} d-d\&,\frac {-2 \text {$\#$1}^{4/3} \log \left (\sqrt [3]{\text {$\#$1}}-\sqrt [3]{\frac {x}{x-a}}\right )+\text {$\#$1}^{4/3} \log \left (\text {$\#$1}^{2/3}+\sqrt [3]{\text {$\#$1}} \sqrt [3]{\frac {x}{x-a}}+\left (\frac {x}{x-a}\right )^{2/3}\right )+2 \sqrt {3} \text {$\#$1}^{4/3} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{\frac {x}{x-a}}}{\sqrt [3]{\text {$\#$1}}}+1}{\sqrt {3}}\right )-6 \text {$\#$1} \sqrt [3]{\frac {x}{x-a}}}{4 \text {$\#$1}^3 a^3-3 \text {$\#$1}^2 d+6 \text {$\#$1} d-3 d}\&\right ]-\text {RootSum}\left [\text {$\#$1}^4 \left (-a^3\right )+\text {$\#$1}^3 d-3 \text {$\#$1}^2 d+3 \text {$\#$1} d-d\&,\frac {-2 \text {$\#$1}^{7/3} \log \left (\sqrt [3]{\text {$\#$1}}-\sqrt [3]{\frac {x}{x-a}}\right )+\text {$\#$1}^{7/3} \log \left (\text {$\#$1}^{2/3}+\sqrt [3]{\text {$\#$1}} \sqrt [3]{\frac {x}{x-a}}+\left (\frac {x}{x-a}\right )^{2/3}\right )+2 \sqrt {3} \text {$\#$1}^{7/3} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{\frac {x}{x-a}}}{\sqrt [3]{\text {$\#$1}}}+1}{\sqrt {3}}\right )-6 \text {$\#$1}^2 \sqrt [3]{\frac {x}{x-a}}}{4 \text {$\#$1}^3 a^3-3 \text {$\#$1}^2 d+6 \text {$\#$1} d-3 d}\&\right ]\right )}{2 \sqrt [3]{\frac {x}{x-a}} \sqrt [3]{x^2 (x-a)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a*x*(4*RootSum[-d + 3*d*#1 - 3*d*#1^2 + d*#1^3 - a^3*#1^4 & , (6*(x/(-a + x))^(1/3) - 2*Sqrt[3]*ArcTan[(1 + (
2*(x/(-a + x))^(1/3))/#1^(1/3))/Sqrt[3]]*#1^(1/3) + 2*Log[-(x/(-a + x))^(1/3) + #1^(1/3)]*#1^(1/3) - Log[(x/(-
a + x))^(2/3) + (x/(-a + x))^(1/3)*#1^(1/3) + #1^(2/3)]*#1^(1/3))/(-3*d + 6*d*#1 - 3*d*#1^2 + 4*a^3*#1^3) & ]
+ 5*RootSum[-d + 3*d*#1 - 3*d*#1^2 + d*#1^3 - a^3*#1^4 & , (-6*(x/(-a + x))^(1/3)*#1 + 2*Sqrt[3]*ArcTan[(1 + (
2*(x/(-a + x))^(1/3))/#1^(1/3))/Sqrt[3]]*#1^(4/3) - 2*Log[-(x/(-a + x))^(1/3) + #1^(1/3)]*#1^(4/3) + Log[(x/(-
a + x))^(2/3) + (x/(-a + x))^(1/3)*#1^(1/3) + #1^(2/3)]*#1^(4/3))/(-3*d + 6*d*#1 - 3*d*#1^2 + 4*a^3*#1^3) & ]
- RootSum[-d + 3*d*#1 - 3*d*#1^2 + d*#1^3 - a^3*#1^4 & , (-6*(x/(-a + x))^(1/3)*#1^2 + 2*Sqrt[3]*ArcTan[(1 + (
2*(x/(-a + x))^(1/3))/#1^(1/3))/Sqrt[3]]*#1^(7/3) - 2*Log[-(x/(-a + x))^(1/3) + #1^(1/3)]*#1^(7/3) + Log[(x/(-
a + x))^(2/3) + (x/(-a + x))^(1/3)*#1^(1/3) + #1^(2/3)]*#1^(7/3))/(-3*d + 6*d*#1 - 3*d*#1^2 + 4*a^3*#1^3) & ])
)/(2*(x/(-a + x))^(1/3)*(x^2*(-a + x))^(1/3))

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IntegrateAlgebraic [C]  time = 0.61, size = 78, normalized size = 0.51 \begin {gather*} -\frac {a \text {RootSum}\left [a^3-d \text {$\#$1}^3+3 d \text {$\#$1}^6-3 d \text {$\#$1}^9+d \text {$\#$1}^{12}\&,\frac {-\log (x)+\log \left (\sqrt [3]{-a x^2+x^3}-x \text {$\#$1}\right )}{-\text {$\#$1}+\text {$\#$1}^4}\&\right ]}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((a*RootSum[a^3 - d*#1^3 + 3*d*#1^6 - 3*d*#1^9 + d*#1^12 & , (-Log[x] + Log[(-(a*x^2) + x^3)^(1/3) - x*#1])/(
-#1 + #1^4) & ])/d)

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fricas [A]  time = 0.51, size = 156, normalized size = 1.03 \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}} x^{2} + 2 \, {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} d\right )} {\left (d^{2}\right )}^{\frac {1}{6}}}{3 \, d x^{2}}\right ) + 2 \, {\left (d^{2}\right )}^{\frac {2}{3}} \log \left (\frac {{\left (d^{2}\right )}^{\frac {1}{3}} x^{2} - {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} d}{x^{2}}\right ) - {\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 {1}{3}} {\left (d^{2}\right )}^{\frac {1}{3}} d x^{2} + {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}} d^{2}}{x^{4}}\right )}{2 \, d^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.26, size = 1, normalized size = 0.01 \begin {gather*} 0 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

0

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int -\frac {x\,\left (4\,a-3\,x\right )}{{\left (-x^2\,\left (a-x\right )\right )}^{1/3}\,\left (x^4-d\,x+a\,d\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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