3.1.86 \(\int \frac {x}{\sqrt {1+x^3} (10+6 \sqrt {3}+x^3)} \, dx\) [86]

Optimal. Leaf size=218 \[ -\frac {\left (2-\sqrt {3}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{3} \left (1+\sqrt {3}\right ) (1+x)}{\sqrt {2} \sqrt {1+x^3}}\right )}{2 \sqrt {2} 3^{3/4}}-\frac {\left (2-\sqrt {3}\right ) \tan ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt {1+x^3}}{\sqrt {2} 3^{3/4}}\right )}{3 \sqrt {2} 3^{3/4}}-\frac {\left (2-\sqrt {3}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{3} \left (1+\sqrt {3}-2 x\right )}{\sqrt {2} \sqrt {1+x^3}}\right )}{3 \sqrt {2} \sqrt [4]{3}}-\frac {\left (2-\sqrt {3}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{3} \left (1-\sqrt {3}\right ) (1+x)}{\sqrt {2} \sqrt {1+x^3}}\right )}{6 \sqrt {2} \sqrt [4]{3}} \]

[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)-1/36*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)-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)

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Rubi [A]
time = 0.03, antiderivative size = 218, 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 = {500} \begin {gather*} -\frac {\left (2-\sqrt {3}\right ) \text {ArcTan}\left (\frac {\sqrt [4]{3} \left (1+\sqrt {3}\right ) (x+1)}{\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 (-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 ) (x+1)}{\sqrt {2} \sqrt {x^3+1}}\right )}{6 \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])*ArcTan
h[(3^(1/4)*(1 + Sqrt[3] - 2*x))/(Sqrt[2]*Sqrt[1 + x^3])])/(3*Sqrt[2]*3^(1/4)) - ((2 - Sqrt[3])*ArcTanh[(3^(1/4
)*(1 - Sqrt[3])*(1 + x))/(Sqrt[2]*Sqrt[1 + x^3])])/(6*Sqrt[2]*3^(1/4))

Rule 500

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)*(ArcTan[(1 - r)*(Sqrt[a + b*x^3]/(Sqrt[2]*Rt[a, 2]*r^(3/2)))]/(3*Sqr
t[2]*Rt[a, 2]*d*r^(3/2))), x] + (-Simp[q*(2 - r)*(ArcTan[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)*(ArcTanh[Rt[a, 2]*Sqrt[r]*((1 + r - 2*q*x)/(Sq
rt[2]*Sqrt[a + b*x^3]))]/(3*Sqrt[2]*Rt[a, 2]*d*Sqrt[r])), x] - Simp[q*(2 - r)*(ArcTanh[Rt[a, 2]*(1 - r)*Sqrt[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] && Ne
Q[b*c - a*d, 0] && EqQ[b^2*c^2 - 20*a*b*c*d - 8*a^2*d^2, 0] && PosQ[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}-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 )}{6 \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 = 47, normalized size = 0.22 \begin {gather*} \frac {x^2 F_1\left (\frac {2}{3};\frac {1}{2},1;\frac {5}{3};-x^3,-\frac {x^3}{10+6 \sqrt {3}}\right )}{20+12 \sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x^2*AppellF1[2/3, 1/2, 1, 5/3, -x^3, -(x^3/(10 + 6*Sqrt[3]))])/(20 + 12*Sqrt[3])

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 37.47, size = 353, normalized size = 1.62

method	result	size
default	\(-\frac {\sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{2}+\left (-1-\sqrt {3}\right ) \textit {\_Z} +2 \sqrt {3}+4\right )}{\sum }\frac {\left (-\sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha +\underline {\hspace {1.25 ex}}\alpha -2\right ) \left (3-i \sqrt {3}\right ) \sqrt {\frac {1+x}{3-i \sqrt {3}}}\, \sqrt {\frac {-i \sqrt {3}+2 x -1}{-3-i \sqrt {3}}}\, \sqrt {\frac {i \sqrt {3}+2 x -1}{i \sqrt {3}-3}}\, \left (-1+2 \underline {\hspace {1.25 ex}}\alpha -\sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha \right ) \EllipticPi \left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, -\frac {i \underline {\hspace {1.25 ex}}\alpha }{2}+\frac {i \underline {\hspace {1.25 ex}}\alpha \sqrt {3}}{3}+\frac {\sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha }{2}-\underline {\hspace {1.25 ex}}\alpha -\frac {i \sqrt {3}}{6}+\frac {1}{2}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\left (-1+2 \underline {\hspace {1.25 ex}}\alpha -\sqrt {3}\right ) \sqrt {x^{3}+1}}\right )}{18}+\frac {\left (-1-\sqrt {3}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {3}\, \EllipticPi \left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}{3}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{9 \left (2+\sqrt {3}\right ) \sqrt {x^{3}+1}}\)	\(353\)
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 {\left (-1-\sqrt {3}\right )^{2}}{3}+\frac {2 \left (-1-\sqrt {3}\right )^{2} \sqrt {3}}{9}-\frac {2}{3}-\frac {\sqrt {3}}{9}-\frac {2 \sqrt {3}\, \left (-1-\sqrt {3}\right )}{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 {\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}}{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}\)	\(507\)
trager	\(\text {Expression too large to display}\)	\(4185\)

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]

-1/18*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*_alpha-3^
(1/2)*_alpha)*EllipticPi(((1+x)/(3/2-1/2*I*3^(1/2)))^(1/2),-1/2*I*_alpha+1/3*I*_alpha*3^(1/2)+1/2*3^(1/2)*_alp
ha-_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))+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))

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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. 7739 vs. \(2 (148) = 296\).
time = 6.26, size = 7739, normalized size = 35.50 \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
*sqrt(3) + 1164)^(3/4)*arctan(-1/1296*(6*sqrt(x^3 + 1)*((459*x^16 - 13425*x^15 - 33201*x^14 + 950652*x^13 - 99
7302*x^12 - 14760972*x^11 + 47069892*x^10 - 49762248*x^9 - 8212536*x^8 + 84377808*x^7 - 88427328*x^6 + 2561385
6*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
- 575818*x^12 - 8522268*x^11 + 27175852*x^10 - 28730312*x^9 - 4741560*x^8 + 48715600*x^7 - 51053600*x^6 + 1478
8128*x^5 + 15853184*x^4 - 21034816*x^3 + 7280256*x^2 - 2488832*x - 1889792) + (3691*x^16 - 6128*x^15 - 537864*
x^14 + 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 + 7
598336) + 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 -
1960992*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 + 22
4784*x^5 - 357936*x^4 + 9632*x^3 + 108288*x^2 - 96000*x - 33920) + (4945*x^15 - 88617*x^14 + 738528*x^13 - 186
0046*x^12 - 784596*x^11 + 7668708*x^10 - 6570680*x^9 - 6903864*x^8 + 15444144*x^7 - 4312832*x^6 - 9559200*x^5
+ 9359808*x^4 - 155968*x^3 - 3016704*x^2 + sqrt(3)*(2855*x^15 - 51163*x^14 + 426388*x^13 - 1073898*x^12 - 4529
80*x^11 + 4427548*x^10 - 3793592*x^9 - 3985944*x^8 + 8916720*x^7 - 2490016*x^6 - 5519008*x^5 + 5403904*x^4 - 9
0048*x^3 - 1741696*x^2 + 1543936*x + 545536) + 2674176*x + 944896)*sqrt(-672*sqrt(3) + 1164) - 665088*x - 2350
08)*(-672*sqrt(3) + 1164)^(1/4))*sqrt(-2*(7*sqrt(3) + 12)*sqrt(-672*sqrt(3) + 1164) + 24)*sqrt(-56*sqrt(3) + 9
7) + 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 + 16404
48*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^10 - 48920*x^9 + 415384*x^8 - 315088*x^7 - 125600*x^6 + 336608*x^5 - 177344*x^4 - 89152*x^3 + 78784*x^2 - 3
9040*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 + 64
34304*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 - 13
9652*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 + 194
1678*x^12 + 57963744*x^11 - 76603680*x^10 - 16678512*x^9 + 139922496*x^8 - 106227360*x^7 - 42453216*x^6 + 1132
69536*x^5 - 59694624*x^4 - 30025728*x^3 + 26496000*x^2 + sqrt(3)*(1351*x^17 - 55630*x^16 + 48958*x^15 + 278116
7*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) - 1311436
8*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 + 23
1392*x^7 + 321412*x^6 - 485176*x^5 + 263080*x^4 + 154784*x^3 - 154592*x^2 + 78464*x + 36544) + 271808*x + 1265
92)*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 -
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 - 32149640*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 + 138342
4) + (3691*x^16 + 17731*x^15 - 951114*x^14 + 450359*x^13 + 4370159*x^12 + 30318522*x^11 - 78096668*x^10 + 9429
316*x^9 + 146877876*x^8 - 197107784*x^7 - 30834152*x^6 + 185125776*x^5 - 132260896*x^4 - 45545344*x^3 + 695175
36*x^2 + sqrt(3)*(2131*x^16 + 10237*x^15 - 5491...

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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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