\(\int \frac {x}{\sqrt {1+x^3} (10-6 \sqrt {3}+x^3)} \, dx\) [87]

Optimal result

Integrand size = 25, antiderivative size = 210 \[ \int \frac {x}{\sqrt {1+x^3} \left (10-6 \sqrt {3}+x^3\right )} \, dx=-\frac {\left (2+\sqrt {3}\right ) \arctan \left (\frac {\sqrt [4]{3} \left (1-\sqrt {3}-2 x\right )}{\sqrt {2} \sqrt {1+x^3}}\right )}{3 \sqrt {2} \sqrt [4]{3}}-\frac {\left (2+\sqrt {3}\right ) \arctan \left (\frac {\sqrt [4]{3} \left (1+\sqrt {3}\right ) (1+x)}{\sqrt {2} \sqrt {1+x^3}}\right )}{6 \sqrt {2} \sqrt [4]{3}}+\frac {\left (2+\sqrt {3}\right ) \text {arctanh}\left (\frac {\sqrt [4]{3} \left (1-\sqrt {3}\right ) (1+x)}{\sqrt {2} \sqrt {1+x^3}}\right )}{2 \sqrt {2} 3^{3/4}}+\frac {\left (2+\sqrt {3}\right ) \text {arctanh}\left (\frac {\left (1+\sqrt {3}\right ) \sqrt {1+x^3}}{\sqrt {2} 3^{3/4}}\right )}{3 \sqrt {2} 3^{3/4}} \]

[Out]

-1/18*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)-1/36*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)+1/12*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] (verified)

Time = 0.03 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {500} \[ \int \frac {x}{\sqrt {1+x^3} \left (10-6 \sqrt {3}+x^3\right )} \, dx=-\frac {\left (2+\sqrt {3}\right ) \arctan \left (\frac {\sqrt [4]{3} \left (-2 x-\sqrt {3}+1\right )}{\sqrt {2} \sqrt {x^3+1}}\right )}{3 \sqrt {2} \sqrt [4]{3}}-\frac {\left (2+\sqrt {3}\right ) \arctan \left (\frac {\sqrt [4]{3} \left (1+\sqrt {3}\right ) (x+1)}{\sqrt {2} \sqrt {x^3+1}}\right )}{6 \sqrt {2} \sqrt [4]{3}}+\frac {\left (2+\sqrt {3}\right ) \text {arctanh}\left (\frac {\sqrt [4]{3} \left (1-\sqrt {3}\right ) (x+1)}{\sqrt {2} \sqrt {x^3+1}}\right )}{2 \sqrt {2} 3^{3/4}}+\frac {\left (2+\sqrt {3}\right ) \text {arctanh}\left (\frac {\left (1+\sqrt {3}\right ) \sqrt {x^3+1}}{\sqrt {2} 3^{3/4}}\right )}{3 \sqrt {2} 3^{3/4}} \]

[In]

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

[Out]

-1/3*((2 + Sqrt[3])*ArcTan[(3^(1/4)*(1 - Sqrt[3] - 2*x))/(Sqrt[2]*Sqrt[1 + x^3])])/(Sqrt[2]*3^(1/4)) - ((2 + S
qrt[3])*ArcTan[(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])*(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 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*} \text {integral}& = -\frac {\left (2+\sqrt {3}\right ) \arctan \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 ) \arctan \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 ) \text {arctanh}\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 ) \text {arctanh}\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] (verified)

Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.

Time = 10.05 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.24 \[ \int \frac {x}{\sqrt {1+x^3} \left (10-6 \sqrt {3}+x^3\right )} \, dx=-\frac {x^2 \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{2},1,\frac {5}{3},-x^3,\frac {1}{4} \left (5+3 \sqrt {3}\right ) x^3\right )}{4 \left (-5+3 \sqrt {3}\right )} \]

[In]

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

[Out]

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

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 48.71 (sec) , antiderivative size = 352, normalized size of antiderivative = 1.68

method	result	size
default	\(\frac {2 \left (\sqrt {3}-1\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {3}\, \Pi \left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, -\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}{3}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{3 \left (6 \sqrt {3}-12\right ) \sqrt {x^{3}+1}}-\frac {\sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{2}+\left (\sqrt {3}-1\right ) \textit {\_Z} -2 \sqrt {3}+4\right )}{\sum }\frac {\left (-\underline {\hspace {1.25 ex}}\alpha \sqrt {3}-\underline {\hspace {1.25 ex}}\alpha +2\right ) \left (3-i \sqrt {3}\right ) \sqrt {\frac {1+x}{3-i \sqrt {3}}}\, \sqrt {\frac {-i \sqrt {3}+2 x -1}{-3-i \sqrt {3}}}\, \sqrt {\frac {i \sqrt {3}+2 x -1}{i \sqrt {3}-3}}\, \left (-1+2 \underline {\hspace {1.25 ex}}\alpha +\underline {\hspace {1.25 ex}}\alpha \sqrt {3}\right ) \Pi \left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {i \underline {\hspace {1.25 ex}}\alpha \sqrt {3}}{3}+\frac {i \underline {\hspace {1.25 ex}}\alpha }{2}-\frac {\underline {\hspace {1.25 ex}}\alpha \sqrt {3}}{2}-\underline {\hspace {1.25 ex}}\alpha -\frac {i \sqrt {3}}{6}+\frac {1}{2}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\left (1-2 \underline {\hspace {1.25 ex}}\alpha -\sqrt {3}\right ) \sqrt {x^{3}+1}}\right )}{18}\)	\(352\)
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 (\sqrt {3}-1\right )^{2} \sqrt {3}}{9}-\frac {\left (\sqrt {3}-1\right )^{2}}{3}+\frac {2 \left (\sqrt {3}-1\right ) \sqrt {3}}{9}+\frac {\sqrt {3}}{9}-\frac {2}{3}\right ) \Pi \left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, -\frac {i \left (\sqrt {3}-1\right )^{2} \sqrt {3}}{6}-\frac {i \left (\sqrt {3}-1\right )^{2}}{3}+\frac {\left (\sqrt {3}-1\right )^{2} \sqrt {3}}{3}+\frac {\left (\sqrt {3}-1\right )^{2}}{2}+\frac {i \left (\sqrt {3}-1\right ) \sqrt {3}}{6}+\frac {i \left (\sqrt {3}-1\right )}{3}-\frac {\left (\sqrt {3}-1\right ) \sqrt {3}}{3}-\frac {\sqrt {3}}{6}+1-\frac {i}{3}-\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 (\sqrt {3}-1\right ) \sqrt {x^{3}+1}}+\frac {\sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{2}+\left (\sqrt {3}-1\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 ) \Pi \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 {\underline {\hspace {1.25 ex}}\alpha \sqrt {3}}{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}\)	\(483\)
trager	\(\text {Expression too large to display}\)	\(4171\)

[In]

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

[Out]

2/3*(3^(1/2)-1)/(6*3^(1/2)-12)*(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((-_alpha*3^(1/2)-_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+_alpha*3^(1/2))*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*_alpha
*3^(1/2)-_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+(3^(1/
2)-1)*_Z-2*3^(1/2)+4))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1702 vs. \(2 (146) = 292\).

Time = 0.39 (sec) , antiderivative size = 1702, normalized size of antiderivative = 8.10 \[ \int \frac {x}{\sqrt {1+x^3} \left (10-6 \sqrt {3}+x^3\right )} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/72*sqrt(-7*sqrt(3) + 3*sqrt(-56*sqrt(3) - 97) - 12)*log((x^8 - x^7 - 11*x^6 - 16*x^5 - 20*x^4 + 32*x^3 - 44*
x^2 + (5*x^6 - 6*x^5 - 33*x^4 - 44*x^3 - 42*x^2 - sqrt(3)*(3*x^6 - 4*x^5 - 17*x^4 - 28*x^3 - 22*x^2 - 8*x - 4)
+ (336*x^5 + 33*x^4 - 132*x^3 - 474*x^2 - sqrt(3)*(194*x^5 + 19*x^4 - 76*x^3 - 274*x^2 - 152*x - 76) - 264*x
- 132)*sqrt(-56*sqrt(3) - 97) - 24*x - 4)*sqrt(x^3 + 1)*sqrt(-7*sqrt(3) + 3*sqrt(-56*sqrt(3) - 97) - 12) - 2*s
qrt(3)*(x^7 - 8*x^6 - 7*x^4 - 16*x^3 - 8*x - 8) + 3*(26*x^7 + 12*x^6 - 48*x^5 - 98*x^4 - 96*x^3 - 48*x^2 - sqr
t(3)*(15*x^7 + 7*x^6 - 28*x^5 - 56*x^4 - 56*x^3 - 28*x^2 - 8*x) - 16*x)*sqrt(-56*sqrt(3) - 97) + 8*x + 16)/(x^
8 - 4*x^7 + 16*x^6 - 16*x^5 + 28*x^4 + 32*x^3 + 64*x^2 + 32*x + 16)) - 1/72*sqrt(-7*sqrt(3) + 3*sqrt(-56*sqrt(
3) - 97) - 12)*log((x^8 - x^7 - 11*x^6 - 16*x^5 - 20*x^4 + 32*x^3 - 44*x^2 - (5*x^6 - 6*x^5 - 33*x^4 - 44*x^3
- 42*x^2 - sqrt(3)*(3*x^6 - 4*x^5 - 17*x^4 - 28*x^3 - 22*x^2 - 8*x - 4) + (336*x^5 + 33*x^4 - 132*x^3 - 474*x^
2 - sqrt(3)*(194*x^5 + 19*x^4 - 76*x^3 - 274*x^2 - 152*x - 76) - 264*x - 132)*sqrt(-56*sqrt(3) - 97) - 24*x -
4)*sqrt(x^3 + 1)*sqrt(-7*sqrt(3) + 3*sqrt(-56*sqrt(3) - 97) - 12) - 2*sqrt(3)*(x^7 - 8*x^6 - 7*x^4 - 16*x^3 -
8*x - 8) + 3*(26*x^7 + 12*x^6 - 48*x^5 - 98*x^4 - 96*x^3 - 48*x^2 - sqrt(3)*(15*x^7 + 7*x^6 - 28*x^5 - 56*x^4
- 56*x^3 - 28*x^2 - 8*x) - 16*x)*sqrt(-56*sqrt(3) - 97) + 8*x + 16)/(x^8 - 4*x^7 + 16*x^6 - 16*x^5 + 28*x^4 +
32*x^3 + 64*x^2 + 32*x + 16)) + 1/72*sqrt(-7*sqrt(3) - 3*sqrt(-56*sqrt(3) - 97) - 12)*log((x^8 - x^7 - 11*x^6
- 16*x^5 - 20*x^4 + 32*x^3 - 44*x^2 + (5*x^6 - 6*x^5 - 33*x^4 - 44*x^3 - 42*x^2 - sqrt(3)*(3*x^6 - 4*x^5 - 17*
x^4 - 28*x^3 - 22*x^2 - 8*x - 4) - (336*x^5 + 33*x^4 - 132*x^3 - 474*x^2 - sqrt(3)*(194*x^5 + 19*x^4 - 76*x^3
- 274*x^2 - 152*x - 76) - 264*x - 132)*sqrt(-56*sqrt(3) - 97) - 24*x - 4)*sqrt(x^3 + 1)*sqrt(-7*sqrt(3) - 3*sq
rt(-56*sqrt(3) - 97) - 12) - 2*sqrt(3)*(x^7 - 8*x^6 - 7*x^4 - 16*x^3 - 8*x - 8) - 3*(26*x^7 + 12*x^6 - 48*x^5
- 98*x^4 - 96*x^3 - 48*x^2 - sqrt(3)*(15*x^7 + 7*x^6 - 28*x^5 - 56*x^4 - 56*x^3 - 28*x^2 - 8*x) - 16*x)*sqrt(-
56*sqrt(3) - 97) + 8*x + 16)/(x^8 - 4*x^7 + 16*x^6 - 16*x^5 + 28*x^4 + 32*x^3 + 64*x^2 + 32*x + 16)) - 1/72*sq
rt(-7*sqrt(3) - 3*sqrt(-56*sqrt(3) - 97) - 12)*log((x^8 - x^7 - 11*x^6 - 16*x^5 - 20*x^4 + 32*x^3 - 44*x^2 - (
5*x^6 - 6*x^5 - 33*x^4 - 44*x^3 - 42*x^2 - sqrt(3)*(3*x^6 - 4*x^5 - 17*x^4 - 28*x^3 - 22*x^2 - 8*x - 4) - (336
*x^5 + 33*x^4 - 132*x^3 - 474*x^2 - sqrt(3)*(194*x^5 + 19*x^4 - 76*x^3 - 274*x^2 - 152*x - 76) - 264*x - 132)*
sqrt(-56*sqrt(3) - 97) - 24*x - 4)*sqrt(x^3 + 1)*sqrt(-7*sqrt(3) - 3*sqrt(-56*sqrt(3) - 97) - 12) - 2*sqrt(3)*
(x^7 - 8*x^6 - 7*x^4 - 16*x^3 - 8*x - 8) - 3*(26*x^7 + 12*x^6 - 48*x^5 - 98*x^4 - 96*x^3 - 48*x^2 - sqrt(3)*(1
5*x^7 + 7*x^6 - 28*x^5 - 56*x^4 - 56*x^3 - 28*x^2 - 8*x) - 16*x)*sqrt(-56*sqrt(3) - 97) + 8*x + 16)/(x^8 - 4*x
^7 + 16*x^6 - 16*x^5 + 28*x^4 + 32*x^3 + 64*x^2 + 32*x + 16)) + 1/72*sqrt(14*sqrt(3) + 24)*log((x^8 - 16*x^7 +
112*x^6 - 16*x^5 + 112*x^4 + 224*x^3 + 64*x^2 + 2*(5*x^6 - 54*x^5 + 96*x^4 - 56*x^3 - 36*x^2 - 3*sqrt(3)*(x^6
- 10*x^5 + 20*x^4 - 8*x^3 - 4*x^2 + 8*x) + 24*x - 16)*sqrt(x^3 + 1)*sqrt(14*sqrt(3) + 24) + 16*sqrt(3)*(x^7 -
2*x^6 + 6*x^5 + 5*x^4 + 2*x^3 + 6*x^2 + 4*x + 4) + 128*x + 112)/(x^8 + 8*x^7 + 16*x^6 - 16*x^5 - 56*x^4 + 32*
x^3 + 64*x^2 - 64*x + 16))

Sympy [F]

\[ \int \frac {x}{\sqrt {1+x^3} \left (10-6 \sqrt {3}+x^3\right )} \, dx=\int \frac {x}{\sqrt {\left (x + 1\right ) \left (x^{2} - x + 1\right )} \left (x^{3} - 6 \sqrt {3} + 10\right )}\, dx \]

[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)

Maxima [F]

\[ \int \frac {x}{\sqrt {1+x^3} \left (10-6 \sqrt {3}+x^3\right )} \, dx=\int { \frac {x}{{\left (x^{3} - 6 \, \sqrt {3} + 10\right )} \sqrt {x^{3} + 1}} \,d x } \]

[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)

Giac [F]

\[ \int \frac {x}{\sqrt {1+x^3} \left (10-6 \sqrt {3}+x^3\right )} \, dx=\int { \frac {x}{{\left (x^{3} - 6 \, \sqrt {3} + 10\right )} \sqrt {x^{3} + 1}} \,d x } \]

[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(-1)]

Timed out. \[ \int \frac {x}{\sqrt {1+x^3} \left (10-6 \sqrt {3}+x^3\right )} \, dx=\int \frac {x}{\sqrt {x^3+1}\,\left (x^3-6\,\sqrt {3}+10\right )} \,d x \]

[In]

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

[Out]

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