3.1.88 \(\int \frac {x}{\sqrt {-1+x^3} (-10-6 \sqrt {3}+x^3)} \, dx\) [88]

Optimal. Leaf size=222 \[ \frac {\left (2-\sqrt {3}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{3} \left (1-\sqrt {3}\right ) (1-x)}{\sqrt {2} \sqrt {-1+x^3}}\right )}{6 \sqrt {2} \sqrt [4]{3}}+\frac {\left (2-\sqrt {3}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{3} \left (1+\sqrt {3}+2 x\right )}{\sqrt {2} \sqrt {-1+x^3}}\right )}{3 \sqrt {2} \sqrt [4]{3}}+\frac {\left (2-\sqrt {3}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{3} \left (1+\sqrt {3}\right ) (1-x)}{\sqrt {2} \sqrt {-1+x^3}}\right )}{2 \sqrt {2} 3^{3/4}}-\frac {\left (2-\sqrt {3}\right ) \tanh ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt {-1+x^3}}{\sqrt {2} 3^{3/4}}\right )}{3 \sqrt {2} 3^{3/4}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.02, antiderivative size = 222, 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 )}{6 \sqrt {2} \sqrt [4]{3}}+\frac {\left (2-\sqrt {3}\right ) \text {ArcTan}\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}}+\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 )}{2 \sqrt {2} 3^{3/4}}-\frac {\left (2-\sqrt {3}\right ) \tanh ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt {x^3-1}}{\sqrt {2} 3^{3/4}}\right )}{3 \sqrt {2} 3^{3/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((2 - Sqrt[3])*ArcTan[(3^(1/4)*(1 - Sqrt[3])*(1 - x))/(Sqrt[2]*Sqrt[-1 + x^3])])/(6*Sqrt[2]*3^(1/4)) + ((2 - S
qrt[3])*ArcTan[(3^(1/4)*(1 + Sqrt[3] + 2*x))/(Sqrt[2]*Sqrt[-1 + x^3])])/(3*Sqrt[2]*3^(1/4)) + ((2 - Sqrt[3])*A
rcTanh[(3^(1/4)*(1 + Sqrt[3])*(1 - x))/(Sqrt[2]*Sqrt[-1 + x^3])])/(2*Sqrt[2]*3^(3/4)) - ((2 - Sqrt[3])*ArcTanh
[((1 - Sqrt[3])*Sqrt[-1 + x^3])/(Sqrt[2]*3^(3/4))])/(3*Sqrt[2]*3^(3/4))

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 )}{6 \sqrt {2} \sqrt [4]{3}}+\frac {\left (2-\sqrt {3}\right ) \tan ^{-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}}+\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 )}{2 \sqrt {2} 3^{3/4}}-\frac {\left (2-\sqrt {3}\right ) \tanh ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt {-1+x^3}}{\sqrt {2} 3^{3/4}}\right )}{3 \sqrt {2} 3^{3/4}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
time = 10.05, size = 65, normalized size = 0.29 \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 )}{\left (20+12 \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])])/((20 + 12*Sqrt[3])*Sqrt[-1 + x^3])
)

________________________________________________________________________________________

Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 37.97, size = 349, normalized size = 1.57

method	result	size
default	\(\frac {\left (-1-\sqrt {3}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {3}\, \EllipticPi \left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, -\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}{3}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{9 \left (2+\sqrt {3}\right ) \sqrt {x^{3}-1}}-\frac {\sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{2}+\left (1+\sqrt {3}\right ) \textit {\_Z} +2 \sqrt {3}+4\right )}{\sum }\frac {\left (-\sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha +\underline {\hspace {1.25 ex}}\alpha +2\right ) \left (-3-i \sqrt {3}\right ) \sqrt {\frac {-1+x}{-3-i \sqrt {3}}}\, \sqrt {\frac {-i \sqrt {3}+2 x +1}{3-i \sqrt {3}}}\, \sqrt {\frac {i \sqrt {3}+2 x +1}{i \sqrt {3}+3}}\, \left (1+2 \underline {\hspace {1.25 ex}}\alpha -\sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha \right ) \EllipticPi \left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {i \underline {\hspace {1.25 ex}}\alpha \sqrt {3}}{3}-\frac {i \underline {\hspace {1.25 ex}}\alpha }{2}-\frac {\sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha }{2}+\underline {\hspace {1.25 ex}}\alpha +\frac {i \sqrt {3}}{6}+\frac {1}{2}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\left (-1-2 \underline {\hspace {1.25 ex}}\alpha -\sqrt {3}\right ) \sqrt {x^{3}-1}}\right )}{18}\)	\(349\)
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 \left (1+\sqrt {3}\right ) \sqrt {3}}{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}}{3}+\frac {i \left (1+\sqrt {3}\right )^{2} \sqrt {3}}{6}-\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 )}{3}+\frac {i \left (1+\sqrt {3}\right ) \sqrt {3}}{6}-\frac {\left (1+\sqrt {3}\right ) \sqrt {3}}{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}}{3}+\frac {i \underline {\hspace {1.25 ex}}\alpha ^{2} \sqrt {3}}{6}-\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 }{3}+\frac {i \underline {\hspace {1.25 ex}}\alpha \sqrt {3}}{6}-\frac {\sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha }{3}+\frac {\underline {\hspace {1.25 ex}}\alpha }{2}+\frac {1}{2}-\frac {i}{3}+\frac {i \sqrt {3}}{6}-\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}\)	\(485\)
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*(-1-3^(1/2))/(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)/(-1-2*_alpha-3^(1/2))*(-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*_a
lpha-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))

________________________________________________________________________________________

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)

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 7910 vs. \(2 (146) = 292\).
time = 4.94, size = 7910, normalized size = 35.63 \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]

1/432*sqrt(2*(7*sqrt(3) + 12)*sqrt(-672*sqrt(3) + 1164) + 24)*(56*sqrt(3) + 97)*sqrt(-56*sqrt(3) + 97)*(-672*s
qrt(3) + 1164)^(3/4)*arctan(1/1296*(6*sqrt(x^3 - 1)*((459*x^16 + 13425*x^15 - 33201*x^14 - 950652*x^13 - 99730
2*x^12 + 14760972*x^11 + 47069892*x^10 + 49762248*x^9 - 8212536*x^8 - 84377808*x^7 - 88427328*x^6 - 25613856*x
^5 + 27458496*x^4 + 36433344*x^3 + 12609792*x^2 + sqrt(3)*(265*x^16 + 7751*x^15 - 19167*x^14 - 548864*x^13 - 5
75818*x^12 + 8522268*x^11 + 27175852*x^10 + 28730312*x^9 - 4741560*x^8 - 48715600*x^7 - 51053600*x^6 - 1478812
8*x^5 + 15853184*x^4 + 21034816*x^3 + 7280256*x^2 + 2488832*x - 1889792) - (3691*x^16 + 6128*x^15 - 537864*x^1
4 - 1586477*x^13 + 16210952*x^12 + 77181756*x^11 + 84218362*x^10 - 71018320*x^9 - 254455812*x^8 - 196076008*x^
7 + 120105208*x^6 + 256326864*x^5 + 134645168*x^4 - 78464672*x^3 - 78514944*x^2 + sqrt(3)*(2131*x^16 + 3538*x^
15 - 310536*x^14 - 915953*x^13 + 9359398*x^12 + 44560908*x^11 + 48623494*x^10 - 41002448*x^9 - 146910132*x^8 -
113204536*x^7 + 69342776*x^6 + 147990384*x^5 + 77737424*x^4 - 45301600*x^3 - 45330624*x^2 - 12242560*x + 7598
336) - 21204736*x + 13160704)*sqrt(-672*sqrt(3) + 1164) + 4310784*x - 3273216)*(-672*sqrt(3) + 1164)^(3/4) + 3
*(984*x^15 + 30612*x^14 + 164676*x^13 + 205368*x^12 - 289200*x^11 - 183720*x^10 + 886752*x^9 + 71568*x^8 - 196
0992*x^7 - 1849440*x^6 + 1558464*x^5 + 2478912*x^4 + 66432*x^3 - 750336*x^2 + 4*sqrt(3)*(142*x^15 + 4419*x^14
+ 23781*x^13 + 29608*x^12 - 41940*x^11 - 26454*x^10 + 128152*x^9 + 10692*x^8 - 283320*x^7 - 267064*x^6 + 22478
4*x^5 + 357936*x^4 + 9632*x^3 - 108288*x^2 - 96000*x + 33920) - (4945*x^15 + 88617*x^14 + 738528*x^13 + 186004
6*x^12 - 784596*x^11 - 7668708*x^10 - 6570680*x^9 + 6903864*x^8 + 15444144*x^7 + 4312832*x^6 - 9559200*x^5 - 9
359808*x^4 - 155968*x^3 + 3016704*x^2 + sqrt(3)*(2855*x^15 + 51163*x^14 + 426388*x^13 + 1073898*x^12 - 452980*
x^11 - 4427548*x^10 - 3793592*x^9 + 3985944*x^8 + 8916720*x^7 + 2490016*x^6 - 5519008*x^5 - 5403904*x^4 - 9004
8*x^3 + 1741696*x^2 + 1543936*x - 545536) + 2674176*x - 944896)*sqrt(-672*sqrt(3) + 1164) - 665088*x + 235008)
*(-672*sqrt(3) + 1164)^(1/4))*sqrt(2*(7*sqrt(3) + 12)*sqrt(-672*sqrt(3) + 1164) + 24)*sqrt(-56*sqrt(3) + 97) +
36*(144*x^17 + 5976*x^16 + 5544*x^15 - 299664*x^14 - 1062360*x^13 - 116712*x^12 + 3600000*x^11 + 4761216*x^10
- 1046592*x^9 - 8676864*x^8 - 6592896*x^7 + 2641536*x^6 + 7016832*x^5 + 3699072*x^4 - 1861632*x^3 - 1640448*x
^2 + 12*sqrt(3)*(7*x^17 + 286*x^16 + 238*x^15 - 14255*x^14 - 50390*x^13 - 5942*x^12 + 171808*x^11 + 226888*x^1
0 - 48920*x^9 - 415384*x^8 - 315088*x^7 + 125600*x^6 + 336608*x^5 + 177344*x^4 - 89152*x^3 - 78784*x^2 - 39040
*x + 18176) + (1164*x^17 + 6276*x^16 - 26052*x^15 - 332844*x^14 - 1632156*x^13 - 4149132*x^12 - 5805024*x^11 -
318696*x^10 + 12621072*x^9 + 19878720*x^8 + 9619008*x^7 - 13361088*x^6 - 20168256*x^5 - 10936128*x^4 + 643430
4*x^3 + 6426240*x^2 + 24*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) - (2340*x^17 + 96354*x^16 + 84798*x^15 - 4817124*x^14 - 17052930*x^13 - 1941678
*x^12 + 57963744*x^11 + 76603680*x^10 - 16678512*x^9 - 139922496*x^8 - 106227360*x^7 + 42453216*x^6 + 11326953
6*x^5 + 59694624*x^4 - 30025728*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)*sqrt(-672*sqrt(3) + 1164) + 3261696*x - 1519104)*sqrt(-672*sqrt(3) + 1164) - 12*(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 - 263080*x^4 + 154784*x^3 + 154592*x^2 + 78464*x - 36544) + 271808*x - 126592)*
sqrt(-672*sqrt(3) + 1164) - 811008*x + 377856)*sqrt(-56*sqrt(3) + 97) - (sqrt(x^3 - 1)*((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 + 1053
13392*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 - 32149640*x^10 - 36277360*x^9 + 4922568*x^8 + 6
0802736*x^7 + 57507040*x^6 + 10910784*x^5 - 24536992*x^4 - 21568448*x^3 - 4767168*x^2 - 1207168*x + 1383424) -
(3691*x^16 - 17731*x^15 - 951114*x^14 - 450359*x^13 + 4370159*x^12 - 30318522*x^11 - 78096668*x^10 - 9429316*
x^9 + 146877876*x^8 + 197107784*x^7 - 30834152*x^6 - 185125776*x^5 - 132260896*x^4 + 45545344*x^3 + 69517536*x
^2 + sqrt(3)*(2131*x^16 - 10237*x^15 - 549126*x...

________________________________________________________________________________________

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} - 6 \sqrt {3} - 10\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 - 6*sqrt(3) - 10)), x)

________________________________________________________________________________________

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)

________________________________________________________________________________________

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)

________________________________________________________________________________________