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

Optimal. Leaf size=158 \[ -\frac {\log \left (a^2 d^{2/3} x^4+a^2 \sqrt [3]{d} x^2 \sqrt [3]{x^3-a x^2}+a^2 \left (x^3-a x^2\right )^{2/3}\right )}{2 \sqrt [3]{d}}+\frac {\log \left (a \sqrt [3]{x^3-a x^2}-a \sqrt [3]{d} x^2\right )}{\sqrt [3]{d}}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{d} x^2}{2 \sqrt [3]{x^3-a x^2}+\sqrt [3]{d} x^2}\right )}{\sqrt [3]{d}} \]

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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 (-4 a+3 x)}{\sqrt [3]{x^2 (-a+x)} \left (a-x+d 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 - x + d*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 - x^3 + d*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 - x^3
+ d*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-x+d 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-x+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^3 \left (-4 a+3 x^3\right )}{\sqrt [3]{-a+x^3} \left (a-x^3+d 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-x^3+d x^{12}\right )}+\frac {3 x^6}{\sqrt [3]{-a+x^3} \left (a-x^3+d 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-x^3+d 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-x^3+d x^{12}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2 (-a+x)}}\\ \end {align*}

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Mathematica [C]  time = 1.20, size = 625, normalized size = 3.96 \begin {gather*} \frac {a x \left (4 \text {RootSum}\left [\text {$\#$1}^4 a^3 d-\text {$\#$1}^3+3 \text {$\#$1}^2-3 \text {$\#$1}+1\&,\frac {-\sqrt [3]{\text {$\#$1}} \log \left (\text {$\#$1}^{2/3}+\sqrt [3]{\text {$\#$1}} \sqrt [3]{\frac {x}{x-a}}+\left (\frac {x}{x-a}\right )^{2/3}\right )+2 \sqrt [3]{\text {$\#$1}} \log \left (\sqrt [3]{\text {$\#$1}}-\sqrt [3]{\frac {x}{x-a}}\right )-2 \sqrt {3} \sqrt [3]{\text {$\#$1}} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{\frac {x}{x-a}}}{\sqrt [3]{\text {$\#$1}}}+1}{\sqrt {3}}\right )+6 \sqrt [3]{\frac {x}{x-a}}}{4 \text {$\#$1}^3 a^3 d-3 \text {$\#$1}^2+6 \text {$\#$1}-3}\&\right ]+5 \text {RootSum}\left [\text {$\#$1}^4 a^3 d-\text {$\#$1}^3+3 \text {$\#$1}^2-3 \text {$\#$1}+1\&,\frac {-2 \text {$\#$1}^{4/3} \log \left (\sqrt [3]{\text {$\#$1}}-\sqrt [3]{\frac {x}{x-a}}\right )+\text {$\#$1}^{4/3} \log \left (\text {$\#$1}^{2/3}+\sqrt [3]{\text {$\#$1}} \sqrt [3]{\frac {x}{x-a}}+\left (\frac {x}{x-a}\right )^{2/3}\right )+2 \sqrt {3} \text {$\#$1}^{4/3} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{\frac {x}{x-a}}}{\sqrt [3]{\text {$\#$1}}}+1}{\sqrt {3}}\right )-6 \text {$\#$1} \sqrt [3]{\frac {x}{x-a}}}{4 \text {$\#$1}^3 a^3 d-3 \text {$\#$1}^2+6 \text {$\#$1}-3}\&\right ]-\text {RootSum}\left [\text {$\#$1}^4 a^3 d-\text {$\#$1}^3+3 \text {$\#$1}^2-3 \text {$\#$1}+1\&,\frac {-2 \text {$\#$1}^{7/3} \log \left (\sqrt [3]{\text {$\#$1}}-\sqrt [3]{\frac {x}{x-a}}\right )+\text {$\#$1}^{7/3} \log \left (\text {$\#$1}^{2/3}+\sqrt [3]{\text {$\#$1}} \sqrt [3]{\frac {x}{x-a}}+\left (\frac {x}{x-a}\right )^{2/3}\right )+2 \sqrt {3} \text {$\#$1}^{7/3} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{\frac {x}{x-a}}}{\sqrt [3]{\text {$\#$1}}}+1}{\sqrt {3}}\right )-6 \text {$\#$1}^2 \sqrt [3]{\frac {x}{x-a}}}{4 \text {$\#$1}^3 a^3 d-3 \text {$\#$1}^2+6 \text {$\#$1}-3}\&\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 - x + d*x^4)),x]

[Out]

(a*x*(4*RootSum[1 - 3*#1 + 3*#1^2 - #1^3 + a^3*d*#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 + 6*#1 - 3*#1^2 + 4*a^3*d*#1^3) & ] + 5*RootS
um[1 - 3*#1 + 3*#1^2 - #1^3 + a^3*d*#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 + 6*#1 - 3*#1^2 + 4*a^3*d*#1^3) & ] - RootSum[1 - 3*#1
+ 3*#1^2 - #1^3 + a^3*d*#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 + 6*#1 - 3*#1^2 + 4*a^3*d*#1^3) & ]))/(2*(x/(-a + x))^(1/3)*(x^
2*(-a + x))^(1/3))

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IntegrateAlgebraic [C]  time = 0.47, size = 72, normalized size = 0.46 \begin {gather*} -a \text {RootSum}\left [a^3 d-\text {$\#$1}^3+3 \text {$\#$1}^6-3 \text {$\#$1}^9+\text {$\#$1}^{12}\&,\frac {-\log (x)+\log \left (\sqrt [3]{-a x^2+x^3}-x \text {$\#$1}\right )}{-\text {$\#$1}+\text {$\#$1}^4}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.48, size = 330, normalized size = 2.09 \begin {gather*} \left [\frac {\sqrt {3} d \sqrt {-\frac {1}{d^{\frac {2}{3}}}} \log \left (-\frac {d x^{4} - 3 \, {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} d^{\frac {2}{3}} x^{2} - \sqrt {3} {\left (d^{\frac {4}{3}} x^{4} + {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} d x^{2} - 2 \, {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}} d^{\frac {2}{3}}\right )} \sqrt {-\frac {1}{d^{\frac {2}{3}}}} - 2 \, a + 2 \, x}{d x^{4} + a - x}\right ) + 2 \, d^{\frac {2}{3}} \log \left (\frac {d^{\frac {1}{3}} x^{2} - {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}}}{x^{2}}\right ) - d^{\frac {2}{3}} \log \left (\frac {d^{\frac {2}{3}} x^{4} + {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} d^{\frac {1}{3}} x^{2} + {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}}}{x^{4}}\right )}{2 \, d}, \frac {2 \, \sqrt {3} d^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (d^{\frac {1}{3}} x^{2} + 2 \, {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}}\right )}}{3 \, d^{\frac {1}{3}} x^{2}}\right ) + 2 \, d^{\frac {2}{3}} \log \left (\frac {d^{\frac {1}{3}} x^{2} - {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}}}{x^{2}}\right ) - d^{\frac {2}{3}} \log \left (\frac {d^{\frac {2}{3}} x^{4} + {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} d^{\frac {1}{3}} x^{2} + {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}}}{x^{4}}\right )}{2 \, d}\right ] \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)/(d*x^4+a-x),x, algorithm="fricas")

[Out]

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

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \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)/(d*x^4+a-x),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: 1/2*(-1/d)^(1/3)*ln((sqrt(3)*(abs(d)^(1/
3))^2*sqrt(3)/2*(-a/x+1)^(1/3)*(-a/x+1)-sqrt(3)*(abs(d)^(1/3))^2*sqrt(3)/2*(-a/x+1)^(1/3)+(abs(d)^(1/3))^2/2*(
-a/x+1)^(1/3)*(-a/x+1)-(abs(d)^(1/3))^2/2*(-a/x+1)^(1/3)+2*d*a)*(sqrt(3)*(abs(d)^(1/3))^2*sqrt(3)/2*(-a/x+1)^(
1/3)*(-a/x+1)-sqrt(3)*(abs(d)^(1/3))^2*sqrt(3)/2*(-a/x+1)^(1/3)+(abs(d)^(1/3))^2/2*(-a/x+1)^(1/3)*(-a/x+1)-(ab
s(d)^(1/3))^2/2*(-a/x+1)^(1/3)+2*d*a)+(-sqrt(3)*(abs(d)^(1/3))^2/2*(-a/x+1)^(1/3)*(-a/x+1)+sqrt(3)*(abs(d)^(1/
3))^2/2*(-a/x+1)^(1/3)+(abs(d)^(1/3))^2*sqrt(3)/2*(-a/x+1)^(1/3)*(-a/x+1)-(abs(d)^(1/3))^2*sqrt(3)/2*(-a/x+1)^
(1/3))*(-sqrt(3)*(abs(d)^(1/3))^2/2*(-a/x+1)^(1/3)*(-a/x+1)+sqrt(3)*(abs(d)^(1/3))^2/2*(-a/x+1)^(1/3)+(abs(d)^
(1/3))^2*sqrt(3)/2*(-a/x+1)^(1/3)*(-a/x+1)-(abs(d)^(1/3))^2*sqrt(3)/2*(-a/x+1)^(1/3)))-sqrt(3)*(-1/d)^(1/3)*at
an((sqrt(3)*(abs(d)^(1/3))^2*sqrt(3)/2*(-a/x+1)^(1/3)*(-a/x+1)-sqrt(3)*(abs(d)^(1/3))^2*sqrt(3)/2*(-a/x+1)^(1/
3)+(abs(d)^(1/3))^2/2*(-a/x+1)^(1/3)*(-a/x+1)-(abs(d)^(1/3))^2/2*(-a/x+1)^(1/3)+2*d*a)/(-sqrt(3)*(abs(d)^(1/3)
)^2/2*(-a/x+1)^(1/3)*(-a/x+1)+sqrt(3)*(abs(d)^(1/3))^2/2*(-a/x+1)^(1/3)+(abs(d)^(1/3))^2*sqrt(3)/2*(-a/x+1)^(1
/3)*(-a/x+1)-(abs(d)^(1/3))^2*sqrt(3)/2*(-a/x+1)^(1/3)))-(-1/d)^(1/3)/2*ln((-(d^(1/3))^2/2*(-a/x+1)^(1/3)*(-a/
x+1)+(d^(1/3))^2/2*(-a/x+1)^(1/3)+d*a)^2+(-(d^(1/3))^2*sqrt(3)/2*(-a/x+1)^(1/3)*(-a/x+1)+(d^(1/3))^2*sqrt(3)/2
*(-a/x+1)^(1/3))^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x*(-4*a+3*x)/(x^2*(-a+x))^(1/3)/(d*x^4+a-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 (d x^{4} + a - 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)/(d*x^4+a-x),x, algorithm="maxima")

[Out]

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

[Out]

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

[Out]

Timed out

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