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

Optimal. Leaf size=210 \[ -\frac {\left (2+\sqrt {3}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{3} \left (1-\sqrt {3}-2 x\right )}{\sqrt {2} \sqrt {1+x^3}}\right )}{3 \sqrt {2} \sqrt [4]{3}}-\frac {\left (2+\sqrt {3}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{3} \left (1+\sqrt {3}\right ) (1+x)}{\sqrt {2} \sqrt {1+x^3}}\right )}{6 \sqrt {2} \sqrt [4]{3}}+\frac {\left (2+\sqrt {3}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{3} \left (1-\sqrt {3}\right ) (1+x)}{\sqrt {2} \sqrt {1+x^3}}\right )}{2 \sqrt {2} 3^{3/4}}+\frac {\left (2+\sqrt {3}\right ) \tanh ^{-1}\left (\frac {\left (1+\sqrt {3}\right ) \sqrt {1+x^3}}{\sqrt {2} 3^{3/4}}\right )}{3 \sqrt {2} 3^{3/4}} \]

[Out]

-1/18*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)

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Rubi [A]
time = 0.02, antiderivative size = 210, 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 (-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 ) \text {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 ) \tanh ^{-1}\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 ) \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]

-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*} \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}-2 x\right )}{\sqrt {2} \sqrt {1+x^3}}\right )}{3 \sqrt {2} \sqrt [4]{3}}-\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 ) \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*}

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Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
time = 10.07, size = 50, normalized size = 0.24 \begin {gather*} -\frac {x^2 F_1\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 )} \end {gather*}

Antiderivative was successfully verified.

[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])

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 37.87, size = 350, normalized size = 1.67

method	result	size
default	\(\frac {\left (\sqrt {3}-1\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {3}\, \EllipticPi \left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, -\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}{3}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{9 \left (-2+\sqrt {3}\right ) \sqrt {x^{3}+1}}-\frac {\sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{2}+\left (\sqrt {3}-1\right ) \textit {\_Z} -2 \sqrt {3}+4\right )}{\sum }\frac {\left (-\sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha -\underline {\hspace {1.25 ex}}\alpha +2\right ) \left (3-i \sqrt {3}\right ) \sqrt {\frac {1+x}{3-i \sqrt {3}}}\, \sqrt {\frac {-i \sqrt {3}+2 x -1}{-3-i \sqrt {3}}}\, \sqrt {\frac {i \sqrt {3}+2 x -1}{i \sqrt {3}-3}}\, \left (2 \underline {\hspace {1.25 ex}}\alpha -1+\sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha \right ) \EllipticPi \left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {i \underline {\hspace {1.25 ex}}\alpha \sqrt {3}}{3}+\frac {i \underline {\hspace {1.25 ex}}\alpha }{2}-\frac {\sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha }{2}-\underline {\hspace {1.25 ex}}\alpha -\frac {i \sqrt {3}}{6}+\frac {1}{2}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\left (1-2 \underline {\hspace {1.25 ex}}\alpha -\sqrt {3}\right ) \sqrt {x^{3}+1}}\right )}{18}\)	\(350\)
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 (\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 ) \EllipticPi \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 =\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 ) \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}\)	\(483\)
trager	\(\text {Expression too large to display}\)	\(4171\)

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)/(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)*(2*_alpha-
1+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+(3^(1/2)-
1)*_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. 8237 vs. \(2 (146) = 292\).
time = 5.48, size = 8237, normalized size = 39.22 \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]

-1/108*sqrt(3)*sqrt(sqrt(3)*sqrt(56*sqrt(3) + 97)*(7*sqrt(3) - 12) + 6)*(672*sqrt(3) + 1164)^(1/4)*(56*sqrt(3)
+ 97)*(56*sqrt(3) - 97)*arctan(1/324*(216*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 + 9
9812*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)*(56*sqrt(3) + 97) - 36*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 - 76603680*
x^10 - 16678512*x^9 + 139922496*x^8 - 106227360*x^7 - 42453216*x^6 + 113269536*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 - 34
464704*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 + 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(56*sqrt(3) + 97))*sqrt(56*sqrt(3) + 97) + 3*sqrt(sqrt(3)*sqrt(56*sqrt(3) + 97)*(7*sqrt
(3) - 12) + 6)*((2*sqrt(3)*(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 + 7598336) + 21204736*x + 13160704)*sqrt(x^3 + 1)*(56*
sqrt(3) + 97) + (459*x^16 - 13425*x^15 - 33201*x^14 + 950652*x^13 - 997302*x^12 - 14760972*x^11 + 47069892*x^1
0 - 49762248*x^9 - 8212536*x^8 + 84377808*x^7 - 88427328*x^6 + 25613856*x^5 + 27458496*x^4 - 36433344*x^3 + 12
609792*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 + 14788128*x^5 + 15853184*x^4 - 21034816*x^3 +
7280256*x^2 - 2488832*x - 1889792) - 4310784*x - 3273216)*sqrt(x^3 + 1)*sqrt(56*sqrt(3) + 97))*(672*sqrt(3) +
1164)^(3/4) + 6*(sqrt(3)*(4945*x^15 - 88617*x^14 + 738528*x^13 - 1860046*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 - 452980*x^11 + 4427548*x^10 - 3793592*x^9 - 39
85944*x^8 + 8916720*x^7 - 2490016*x^6 - 5519008*x^5 + 5403904*x^4 - 90048*x^3 - 1741696*x^2 + 1543936*x + 5455
36) + 2674176*x + 944896)*sqrt(x^3 + 1)*(56*sqrt(3) + 97) + 2*(246*x^15 - 7653*x^14 + 41169*x^13 - 51342*x^12
- 72300*x^11 + 45930*x^10 + 221688*x^9 - 17892*x^8 - 490248*x^7 + 462360*x^6 + 389616*x^5 - 619728*x^4 + 16608
*x^3 + 187584*x^2 - 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 + 224784*x^5 - 357936*x^4 + 9632*x^3 + 108288*x^2 - 96000*x - 33920
) - 166272*x - 58752)*sqrt(x^3 + 1)*sqrt(56*sqrt(3) + 97))*(672*sqrt(3) + 1164)^(1/4)) + 108*(12*x^17 - 498*x^
16 + 462*x^15 + 24972*x^14 - 88530*x^13 + 9726*x^12 + 300000*x^11 - 396768*x^10 - 87216*x^9 + 723072*x^8 - 549
408*x^7 - 220128*x^6 + 584736*x^5 - 308256*x^4 - 155136*x^3 + 136704*x^2 - 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 - 1
25600*x^6 + 336608*x^5 - 177344*x^4 - 89152*x^3 + 78784*x^2 - 39040*x - 18176) - 67584*x - 31488)*sqrt(56*sqrt
(3) + 97) + (144*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 - 53232*x^5 + 139664*x^4 - 33280*x^3 - 31344*x^2 + 37024*x + 11584) + 1282
56*x + 40128)*(56*sqrt(3) + 97) + 12*sqrt(3)*(sqrt(3)*(2340*x^17 - 35850*x^16 - 106410*x^15 - 2064744*x^14 + 1
1945946*x^13 - 1710042*x^12 - 46293732*x^11 + 59161524*x^10 + 18480192*x^9 - 122366520*x^8 + 81203856*x^7 + 45
222000*x^6 - 100598112*x^5 + 42207168*x^4 + 296...

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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} - 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)

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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)*(x^3 - 6*3^(1/2) + 10)),x)

[Out]

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

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