3.1.89 \(\int \frac {x}{\sqrt {-1+x^3} (-10+6 \sqrt {3}+x^3)} \, dx\) [89]

Optimal. Leaf size=214 \[ -\frac {\left (2+\sqrt {3}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{3} \left (1-\sqrt {3}\right ) (1-x)}{\sqrt {2} \sqrt {-1+x^3}}\right )}{2 \sqrt {2} 3^{3/4}}+\frac {\left (2+\sqrt {3}\right ) \tan ^{-1}\left (\frac {\left (1+\sqrt {3}\right ) \sqrt {-1+x^3}}{\sqrt {2} 3^{3/4}}\right )}{3 \sqrt {2} 3^{3/4}}+\frac {\left (2+\sqrt {3}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{3} \left (1+\sqrt {3}\right ) (1-x)}{\sqrt {2} \sqrt {-1+x^3}}\right )}{6 \sqrt {2} \sqrt [4]{3}}+\frac {\left (2+\sqrt {3}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{3} \left (1-\sqrt {3}+2 x\right )}{\sqrt {2} \sqrt {-1+x^3}}\right )}{3 \sqrt {2} \sqrt [4]{3}} \]

[Out]

-1/12*arctan(1/2*3^(1/4)*(1-x)*(1-3^(1/2))*2^(1/2)/(x^3-1)^(1/2))*(2+3^(1/2))*3^(1/4)*2^(1/2)+1/18*arctan(1/6*
(1+3^(1/2))*(x^3-1)^(1/2)*3^(1/4)*2^(1/2))*(2+3^(1/2))*3^(1/4)*2^(1/2)+1/18*arctanh(1/2*3^(1/4)*(1+2*x-3^(1/2)
)*2^(1/2)/(x^3-1)^(1/2))*(2+3^(1/2))*3^(3/4)*2^(1/2)+1/36*arctanh(1/2*3^(1/4)*(1-x)*(1+3^(1/2))*2^(1/2)/(x^3-1
)^(1/2))*(2+3^(1/2))*3^(3/4)*2^(1/2)

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Rubi [A]
time = 0.02, antiderivative size = 214, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {501} \begin {gather*} -\frac {\left (2+\sqrt {3}\right ) \text {ArcTan}\left (\frac {\sqrt [4]{3} \left (1-\sqrt {3}\right ) (1-x)}{\sqrt {2} \sqrt {x^3-1}}\right )}{2 \sqrt {2} 3^{3/4}}+\frac {\left (2+\sqrt {3}\right ) \text {ArcTan}\left (\frac {\left (1+\sqrt {3}\right ) \sqrt {x^3-1}}{\sqrt {2} 3^{3/4}}\right )}{3 \sqrt {2} 3^{3/4}}+\frac {\left (2+\sqrt {3}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{3} \left (1+\sqrt {3}\right ) (1-x)}{\sqrt {2} \sqrt {x^3-1}}\right )}{6 \sqrt {2} \sqrt [4]{3}}+\frac {\left (2+\sqrt {3}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{3} \left (2 x-\sqrt {3}+1\right )}{\sqrt {2} \sqrt {x^3-1}}\right )}{3 \sqrt {2} \sqrt [4]{3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x/(Sqrt[-1 + x^3]*(-10 + 6*Sqrt[3] + x^3)),x]

[Out]

-1/2*((2 + Sqrt[3])*ArcTan[(3^(1/4)*(1 - Sqrt[3])*(1 - x))/(Sqrt[2]*Sqrt[-1 + x^3])])/(Sqrt[2]*3^(3/4)) + ((2
+ Sqrt[3])*ArcTan[((1 + Sqrt[3])*Sqrt[-1 + x^3])/(Sqrt[2]*3^(3/4))])/(3*Sqrt[2]*3^(3/4)) + ((2 + Sqrt[3])*ArcT
anh[(3^(1/4)*(1 + Sqrt[3])*(1 - x))/(Sqrt[2]*Sqrt[-1 + x^3])])/(6*Sqrt[2]*3^(1/4)) + ((2 + Sqrt[3])*ArcTanh[(3
^(1/4)*(1 - Sqrt[3] + 2*x))/(Sqrt[2]*Sqrt[-1 + x^3])])/(3*Sqrt[2]*3^(1/4))

Rule 501

Int[(x_)/(Sqrt[(a_) + (b_.)*(x_)^3]*((c_) + (d_.)*(x_)^3)), x_Symbol] :> With[{q = Rt[b/a, 3], r = Simplify[(b
*c - 10*a*d)/(6*a*d)]}, Simp[q*(2 - r)*(ArcTanh[(1 - r)*(Sqrt[a + b*x^3]/(Sqrt[2]*Rt[-a, 2]*r^(3/2)))]/(3*Sqrt
[2]*Rt[-a, 2]*d*r^(3/2))), x] + (-Simp[q*(2 - r)*(ArcTanh[Rt[-a, 2]*Sqrt[r]*(1 + r)*((1 + q*x)/(Sqrt[2]*Sqrt[a
+ b*x^3]))]/(2*Sqrt[2]*Rt[-a, 2]*d*r^(3/2))), x] - Simp[q*(2 - r)*(ArcTan[Rt[-a, 2]*Sqrt[r]*((1 + r - 2*q*x)/
(Sqrt[2]*Sqrt[a + b*x^3]))]/(3*Sqrt[2]*Rt[-a, 2]*d*Sqrt[r])), x] - Simp[q*(2 - r)*(ArcTan[Rt[-a, 2]*(1 - r)*Sq
rt[r]*((1 + q*x)/(Sqrt[2]*Sqrt[a + b*x^3]))]/(6*Sqrt[2]*Rt[-a, 2]*d*Sqrt[r])), x])] /; FreeQ[{a, b, c, d}, x]
&& NeQ[b*c - a*d, 0] && EqQ[b^2*c^2 - 20*a*b*c*d - 8*a^2*d^2, 0] && NegQ[a]

Rubi steps

\begin {align*} \int \frac {x}{\sqrt {-1+x^3} \left (-10+6 \sqrt {3}+x^3\right )} \, dx &=-\frac {\left (2+\sqrt {3}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{3} \left (1-\sqrt {3}\right ) (1-x)}{\sqrt {2} \sqrt {-1+x^3}}\right )}{2 \sqrt {2} 3^{3/4}}+\frac {\left (2+\sqrt {3}\right ) \tan ^{-1}\left (\frac {\left (1+\sqrt {3}\right ) \sqrt {-1+x^3}}{\sqrt {2} 3^{3/4}}\right )}{3 \sqrt {2} 3^{3/4}}+\frac {\left (2+\sqrt {3}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{3} \left (1+\sqrt {3}\right ) (1-x)}{\sqrt {2} \sqrt {-1+x^3}}\right )}{6 \sqrt {2} \sqrt [4]{3}}+\frac {\left (2+\sqrt {3}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{3} \left (1-\sqrt {3}+2 x\right )}{\sqrt {2} \sqrt {-1+x^3}}\right )}{3 \sqrt {2} \sqrt [4]{3}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
time = 10.05, size = 68, normalized size = 0.32 \begin {gather*} \frac {x^2 \sqrt {1-x^3} F_1\left (\frac {2}{3};\frac {1}{2},1;\frac {5}{3};x^3,-\frac {x^3}{-10+6 \sqrt {3}}\right )}{4 \left (-5+3 \sqrt {3}\right ) \sqrt {-1+x^3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x/(Sqrt[-1 + x^3]*(-10 + 6*Sqrt[3] + x^3)),x]

[Out]

(x^2*Sqrt[1 - x^3]*AppellF1[2/3, 1/2, 1, 5/3, x^3, -(x^3/(-10 + 6*Sqrt[3]))])/(4*(-5 + 3*Sqrt[3])*Sqrt[-1 + x^
3])

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 37.46, size = 350, normalized size = 1.64

method	result	size
default	\(\frac {\left (\sqrt {3}-1\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {3}\, \EllipticPi \left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}{3}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{9 \left (-2+\sqrt {3}\right ) \sqrt {x^{3}-1}}-\frac {\sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{2}+\left (1-\sqrt {3}\right ) \textit {\_Z} -2 \sqrt {3}+4\right )}{\sum }\frac {\left (-\sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha -\underline {\hspace {1.25 ex}}\alpha -2\right ) \left (-3-i \sqrt {3}\right ) \sqrt {\frac {-1+x}{-3-i \sqrt {3}}}\, \sqrt {\frac {-i \sqrt {3}+2 x +1}{3-i \sqrt {3}}}\, \sqrt {\frac {i \sqrt {3}+2 x +1}{i \sqrt {3}+3}}\, \left (1+2 \underline {\hspace {1.25 ex}}\alpha +\sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha \right ) \EllipticPi \left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {i \underline {\hspace {1.25 ex}}\alpha \sqrt {3}}{3}+\frac {i \underline {\hspace {1.25 ex}}\alpha }{2}+\frac {\sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha }{2}+\underline {\hspace {1.25 ex}}\alpha +\frac {i \sqrt {3}}{6}+\frac {1}{2}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\left (-\sqrt {3}+2 \underline {\hspace {1.25 ex}}\alpha +1\right ) \sqrt {x^{3}-1}}\right )}{18}\)	\(350\)
elliptic	\(\frac {2 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \left (\frac {2 \left (1-\sqrt {3}\right )^{2} \sqrt {3}}{9}+\frac {\left (1-\sqrt {3}\right )^{2}}{3}+\frac {2 \sqrt {3}\, \left (1-\sqrt {3}\right )}{9}+\frac {2}{3}-\frac {\sqrt {3}}{9}\right ) \EllipticPi \left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {i \left (1-\sqrt {3}\right )^{2} \sqrt {3}}{6}+\frac {i \left (1-\sqrt {3}\right )^{2}}{3}+\frac {\left (1-\sqrt {3}\right )^{2} \sqrt {3}}{3}+\frac {\left (1-\sqrt {3}\right )^{2}}{2}+\frac {i \left (1-\sqrt {3}\right ) \sqrt {3}}{6}+\frac {i \left (1-\sqrt {3}\right )}{3}+\frac {\sqrt {3}\, \left (1-\sqrt {3}\right )}{3}+1+\frac {i}{3}-\frac {\sqrt {3}}{6}+\frac {i \sqrt {3}}{6}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{3 \left (1-\sqrt {3}\right ) \sqrt {x^{3}-1}}+\frac {\sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{2}+\left (1-\sqrt {3}\right ) \textit {\_Z} -2 \sqrt {3}+4\right )}{\sum }\frac {\left (-3-i \sqrt {3}\right ) \sqrt {\frac {-1+x}{-3-i \sqrt {3}}}\, \sqrt {\frac {-i \sqrt {3}+2 x +1}{3-i \sqrt {3}}}\, \sqrt {\frac {i \sqrt {3}+2 x +1}{i \sqrt {3}+3}}\, \left (3 \underline {\hspace {1.25 ex}}\alpha ^{2}+3 \underline {\hspace {1.25 ex}}\alpha +3+2 \sqrt {3}\, \left (\underline {\hspace {1.25 ex}}\alpha ^{2}+\underline {\hspace {1.25 ex}}\alpha +1\right )\right ) \EllipticPi \left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {i \underline {\hspace {1.25 ex}}\alpha ^{2} \sqrt {3}}{6}+\frac {i \underline {\hspace {1.25 ex}}\alpha ^{2}}{3}+\frac {\underline {\hspace {1.25 ex}}\alpha ^{2} \sqrt {3}}{3}+\frac {\underline {\hspace {1.25 ex}}\alpha ^{2}}{2}+\frac {i \underline {\hspace {1.25 ex}}\alpha \sqrt {3}}{6}+\frac {i \underline {\hspace {1.25 ex}}\alpha }{3}+\frac {\sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha }{3}+\frac {\underline {\hspace {1.25 ex}}\alpha }{2}+\frac {i \sqrt {3}}{6}+\frac {1}{2}+\frac {i}{3}+\frac {\sqrt {3}}{3}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\underline {\hspace {1.25 ex}}\alpha \sqrt {x^{3}-1}}\right )}{27}\)	\(505\)
trager	\(\text {Expression too large to display}\)	\(4178\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(-10+x^3+6*3^(1/2))/(x^3-1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/9*(3^(1/2)-1)/(-2+3^(1/2))*(-3/2-1/2*I*3^(1/2))*((-1+x)/(-3/2-1/2*I*3^(1/2)))^(1/2)*((x+1/2-1/2*I*3^(1/2))/(
3/2-1/2*I*3^(1/2)))^(1/2)*((x+1/2+1/2*I*3^(1/2))/(3/2+1/2*I*3^(1/2)))^(1/2)/(x^3-1)^(1/2)*3^(1/2)*EllipticPi((
(-1+x)/(-3/2-1/2*I*3^(1/2)))^(1/2),1/3*(3/2+1/2*I*3^(1/2))*3^(1/2),((3/2+1/2*I*3^(1/2))/(3/2-1/2*I*3^(1/2)))^(
1/2))-1/18*2^(1/2)*sum((-3^(1/2)*_alpha-_alpha-2)/(-3^(1/2)+2*_alpha+1)*(-3-I*3^(1/2))*((-1+x)/(-3-I*3^(1/2)))
^(1/2)*((-I*3^(1/2)+2*x+1)/(3-I*3^(1/2)))^(1/2)*((I*3^(1/2)+2*x+1)/(I*3^(1/2)+3))^(1/2)/(x^3-1)^(1/2)*(1+2*_al
pha+3^(1/2)*_alpha)*EllipticPi(((-1+x)/(-3/2-1/2*I*3^(1/2)))^(1/2),1/3*I*_alpha*3^(1/2)+1/2*I*_alpha+1/2*3^(1/
2)*_alpha+_alpha+1/6*I*3^(1/2)+1/2,((3/2+1/2*I*3^(1/2))/(3/2-1/2*I*3^(1/2)))^(1/2)),_alpha=RootOf(_Z^2+(1-3^(1
/2))*_Z-2*3^(1/2)+4))

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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/(-10+x^3+6*3^(1/2))/(x^3-1)^(1/2),x, algorithm="maxima")

[Out]

integrate(x/((x^3 + 6*sqrt(3) - 10)*sqrt(x^3 - 1)), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 8105 vs. \(2 (148) = 296\).
time = 4.90, size = 8105, normalized size = 37.87 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(-10+x^3+6*3^(1/2))/(x^3-1)^(1/2),x, algorithm="fricas")

[Out]

1/216*sqrt(3)*sqrt(-4*sqrt(3)*sqrt(56*sqrt(3) + 97)*(7*sqrt(3) - 12) + 24)*(672*sqrt(3) + 1164)^(1/4)*(56*sqrt
(3) + 97)*(56*sqrt(3) - 97)*arctan(-1/648*(432*sqrt(3)*(97*x^17 + 523*x^16 - 2171*x^15 - 27737*x^14 - 136013*x
^13 - 345761*x^12 - 483752*x^11 - 26558*x^10 + 1051756*x^9 + 1656560*x^8 + 801584*x^7 - 1113424*x^6 - 1680688*
x^5 - 911344*x^4 + 536192*x^3 + 535520*x^2 - 2*sqrt(3)*(28*x^17 + 151*x^16 - 626*x^15 - 8006*x^14 - 39266*x^13
- 99812*x^12 - 139652*x^11 - 7661*x^10 + 303610*x^9 + 478214*x^8 + 231392*x^7 - 321412*x^6 - 485176*x^5 - 263
080*x^4 + 154784*x^3 + 154592*x^2 + 78464*x - 36544) + 271808*x - 126592)*(56*sqrt(3) + 97) + 72*sqrt(3)*(sqrt
(3)*(2340*x^17 + 96354*x^16 + 84798*x^15 - 4817124*x^14 - 17052930*x^13 - 1941678*x^12 + 57963744*x^11 + 76603
680*x^10 - 16678512*x^9 - 139922496*x^8 - 106227360*x^7 + 42453216*x^6 + 113269536*x^5 + 59694624*x^4 - 300257
28*x^3 - 26496000*x^2 - sqrt(3)*(1351*x^17 + 55630*x^16 + 48958*x^15 - 2781167*x^14 - 9845510*x^13 - 1121030*x
^12 + 33465376*x^11 + 44227144*x^10 - 9629336*x^9 - 80784280*x^8 - 61330384*x^7 + 24510368*x^6 + 65396192*x^5
+ 34464704*x^4 - 17335360*x^3 - 15297472*x^2 - 7571584*x + 3526400) - 13114368*x + 6107904)*(56*sqrt(3) + 97)
- 6*(97*x^17 + 523*x^16 - 2171*x^15 - 27737*x^14 - 136013*x^13 - 345761*x^12 - 483752*x^11 - 26558*x^10 + 1051
756*x^9 + 1656560*x^8 + 801584*x^7 - 1113424*x^6 - 1680688*x^5 - 911344*x^4 + 536192*x^3 + 535520*x^2 - 2*sqrt
(3)*(28*x^17 + 151*x^16 - 626*x^15 - 8006*x^14 - 39266*x^13 - 99812*x^12 - 139652*x^11 - 7661*x^10 + 303610*x^
9 + 478214*x^8 + 231392*x^7 - 321412*x^6 - 485176*x^5 - 263080*x^4 + 154784*x^3 + 154592*x^2 + 78464*x - 36544
) + 271808*x - 126592)*sqrt(56*sqrt(3) + 97))*sqrt(56*sqrt(3) + 97) - sqrt(1/2)*(288*sqrt(3)*(627*x^16 + 14286
*x^15 + 39762*x^14 - 50142*x^13 - 216816*x^12 - 112284*x^11 + 325707*x^10 + 586326*x^9 - 3294*x^8 - 631752*x^7
- 539220*x^6 + 184392*x^5 + 483816*x^4 + 115296*x^3 - 108576*x^2 - 2*sqrt(3)*(181*x^16 + 4124*x^15 + 11478*x^
14 - 14474*x^13 - 62584*x^12 - 32412*x^11 + 94021*x^10 + 169244*x^9 - 954*x^8 - 182368*x^7 - 155648*x^6 + 5323
2*x^5 + 139664*x^4 + 33280*x^3 - 31344*x^2 - 37024*x + 11584) - 128256*x + 40128)*(56*sqrt(3) + 97) + 24*sqrt(
3)*(sqrt(3)*(2340*x^17 + 35850*x^16 - 106410*x^15 + 2064744*x^14 + 11945946*x^13 + 1710042*x^12 - 46293732*x^1
1 - 59161524*x^10 + 18480192*x^9 + 122366520*x^8 + 81203856*x^7 - 45222000*x^6 - 100598112*x^5 - 42207168*x^4
+ 29609472*x^3 + 22458240*x^2 - sqrt(3)*(1351*x^17 + 20698*x^16 - 61436*x^15 + 1192081*x^14 + 6896998*x^13 + 9
87292*x^12 - 26727704*x^11 - 34156928*x^10 + 10669552*x^9 + 70648352*x^8 + 46883072*x^7 - 26108944*x^6 - 58080
352*x^5 - 24368320*x^4 + 17095040*x^3 + 12966272*x^2 + 4724480*x - 2581504) + 8183040*x - 4471296)*(56*sqrt(3)
+ 97) - 6*(97*x^17 - 104*x^16 - 20510*x^15 - 43181*x^14 + 217294*x^13 + 691762*x^12 + 584800*x^11 - 521510*x^
10 - 1780028*x^9 - 1416580*x^8 + 80528*x^7 + 1518056*x^6 + 1321712*x^5 + 393392*x^4 - 501952*x^3 - 446848*x^2
- 4*sqrt(3)*(14*x^17 - 15*x^16 - 2960*x^15 - 6232*x^14 + 31362*x^13 + 99844*x^12 + 84404*x^11 - 75267*x^10 - 2
56916*x^9 - 204458*x^8 + 11616*x^7 + 219104*x^6 + 190768*x^5 + 56784*x^4 - 72448*x^3 - 64496*x^2 - 24480*x + 1
3376) - 169600*x + 92672)*sqrt(56*sqrt(3) + 97))*sqrt(56*sqrt(3) + 97) - sqrt(-4*sqrt(3)*sqrt(56*sqrt(3) + 97)
*(7*sqrt(3) - 12) + 24)*((2*sqrt(3)*(3691*x^16 - 17731*x^15 - 951114*x^14 - 450359*x^13 + 4370159*x^12 - 30318
522*x^11 - 78096668*x^10 - 9429316*x^9 + 146877876*x^8 + 197107784*x^7 - 30834152*x^6 - 185125776*x^5 - 132260
896*x^4 + 45545344*x^3 + 69517536*x^2 - sqrt(3)*(2131*x^16 - 10237*x^15 - 549126*x^14 - 260015*x^13 + 2523113*
x^12 - 17504406*x^11 - 45089132*x^10 - 5444020*x^9 + 84799980*x^8 + 113800232*x^7 - 17802104*x^6 - 106882416*x
^5 - 76360864*x^4 + 26295616*x^3 + 40135968*x^2 + 7907648*x - 5562368) + 13696448*x - 9634304)*sqrt(x^3 - 1)*(
56*sqrt(3) + 97) - (459*x^16 + 1557*x^15 - 26415*x^14 + 1449954*x^13 + 4677912*x^12 - 12651948*x^11 - 55684800
*x^10 - 62834256*x^9 + 8526168*x^8 + 105313392*x^7 + 99605088*x^6 + 18897984*x^5 - 42499296*x^4 - 37357632*x^3
- 8256960*x^2 - sqrt(3)*(265*x^16 + 899*x^15 - 15249*x^14 + 837130*x^13 + 2700776*x^12 - 7304604*x^11 - 32149
640*x^10 - 36277360*x^9 + 4922568*x^8 + 60802736*x^7 + 57507040*x^6 + 10910784*x^5 - 24536992*x^4 - 21568448*x
^3 - 4767168*x^2 - 1207168*x + 1383424) - 2090880*x + 2396160)*sqrt(x^3 - 1)*sqrt(56*sqrt(3) + 97))*(672*sqrt(
3) + 1164)^(3/4) + 6*(sqrt(3)*(4945*x^15 + 37473*x^14 - 490698*x^13 - 2249468*x^12 + 474132*x^11 + 8423784*x^1
0 + 5853520*x^9 - 8451720*x^8 - 15320016*x^7 - 768064*x^6 + 10405056*x^5 + 6627744*x^4 - 700480*x^3 - 2799552*
x^2 - sqrt(3)*(2855*x^15 + 21635*x^14 - 283306*x^13 - 1298732*x^12 + 273748*x^11 + 4863472*x^10 + 3379536*x^9
- 4879608*x^8 - 8845008*x^7 - 443456*x^6 + 6007360*x^5 + 3826528*x^4 - 404416*x^3 - 1616320*x^2 - 1003648*x +
399360) - 1738368*x + 691712)*sqrt(x^3 - 1)*(56*sqrt(3) + 97) - 2*(246*x^15 + 3678*x^14 - 13485*x^13 - 102933*
x^12 - 70062*x^11 + 81156*x^10 + 45204*x^9 + 12...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{\sqrt {\left (x - 1\right ) \left (x^{2} + x + 1\right )} \left (x^{3} - 10 + 6 \sqrt {3}\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(-10+x**3+6*3**(1/2))/(x**3-1)**(1/2),x)

[Out]

Integral(x/(sqrt((x - 1)*(x**2 + x + 1))*(x**3 - 10 + 6*sqrt(3))), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(-10+x^3+6*3^(1/2))/(x^3-1)^(1/2),x, algorithm="giac")

[Out]

integrate(x/((x^3 + 6*sqrt(3) - 10)*sqrt(x^3 - 1)), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x}{\sqrt {x^3-1}\,\left (x^3+6\,\sqrt {3}-10\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/((x^3 - 1)^(1/2)*(6*3^(1/2) + x^3 - 10)),x)

[Out]

int(x/((x^3 - 1)^(1/2)*(6*3^(1/2) + x^3 - 10)), x)

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