\(\int \frac {(1+x^3) \sqrt {x-x^4}}{2+4 x^3+3 x^6} \, dx\) [1895]

Optimal result

Integrand size = 31, antiderivative size = 131 \[ \int \frac {\left (1+x^3\right ) \sqrt {x-x^4}}{2+4 x^3+3 x^6} \, dx=-\frac {2}{9} \arctan \left (\frac {x^2}{\sqrt {x-x^4}}\right )-\frac {1}{18} \sqrt {8+7 i \sqrt {2}} \arctan \left (\frac {\sqrt {2-\frac {i}{\sqrt {2}}} x \sqrt {x-x^4}}{-1+x^3}\right )-\frac {1}{18} \sqrt {8-7 i \sqrt {2}} \arctan \left (\frac {\sqrt {2+\frac {i}{\sqrt {2}}} x \sqrt {x-x^4}}{-1+x^3}\right ) \]

[Out]

-2/9*arctan(x^2/(-x^4+x)^(1/2))-1/18*(8+7*I*2^(1/2))^(1/2)*arctan(1/2*(8-2*I*2^(1/2))^(1/2)*x*(-x^4+x)^(1/2)/(
x^3-1))-1/18*(8-7*I*2^(1/2))^(1/2)*arctan(1/2*(8+2*I*2^(1/2))^(1/2)*x*(-x^4+x)^(1/2)/(x^3-1))

Rubi [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(323\) vs. \(2(131)=262\).

Time = 0.60 (sec) , antiderivative size = 323, normalized size of antiderivative = 2.47, number of steps used = 12, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {2081, 6847, 1706, 399, 222, 385, 211} \[ \int \frac {\left (1+x^3\right ) \sqrt {x-x^4}}{2+4 x^3+3 x^6} \, dx=-\frac {\left (2+i \sqrt {2}\right ) \sqrt {x-x^4} \arcsin \left (x^{3/2}\right )}{18 \sqrt {x} \sqrt {1-x^3}}-\frac {\left (2-i \sqrt {2}\right ) \sqrt {x-x^4} \arcsin \left (x^{3/2}\right )}{18 \sqrt {x} \sqrt {1-x^3}}+\frac {\left (\sqrt {2}+i\right ) \sqrt {x-x^4} \arctan \left (\frac {x^{3/2}}{\sqrt {\frac {-\sqrt {2}+2 i}{-\sqrt {2}+5 i}} \sqrt {1-x^3}}\right )}{9 \sqrt {2} \sqrt {\frac {-\sqrt {2}+2 i}{-\sqrt {2}+5 i}} \sqrt {x} \sqrt {1-x^3}}+\frac {\left (7+4 i \sqrt {2}\right ) \sqrt {x-x^4} \arctan \left (\frac {x^{3/2}}{\sqrt {\frac {\sqrt {2}+2 i}{\sqrt {2}+5 i}} \sqrt {1-x^3}}\right )}{9 \sqrt {2} \sqrt {-8+7 i \sqrt {2}} \sqrt {x} \sqrt {1-x^3}} \]

[In]

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

[Out]

-1/18*((2 - I*Sqrt[2])*Sqrt[x - x^4]*ArcSin[x^(3/2)])/(Sqrt[x]*Sqrt[1 - x^3]) - ((2 + I*Sqrt[2])*Sqrt[x - x^4]
*ArcSin[x^(3/2)])/(18*Sqrt[x]*Sqrt[1 - x^3]) + ((I + Sqrt[2])*Sqrt[x - x^4]*ArcTan[x^(3/2)/(Sqrt[(2*I - Sqrt[2
])/(5*I - Sqrt[2])]*Sqrt[1 - x^3])])/(9*Sqrt[2]*Sqrt[(2*I - Sqrt[2])/(5*I - Sqrt[2])]*Sqrt[x]*Sqrt[1 - x^3]) +
((7 + (4*I)*Sqrt[2])*Sqrt[x - x^4]*ArcTan[x^(3/2)/(Sqrt[(2*I + Sqrt[2])/(5*I + Sqrt[2])]*Sqrt[1 - x^3])])/(9*
Sqrt[2]*Sqrt[-8 + (7*I)*Sqrt[2]]*Sqrt[x]*Sqrt[1 - x^3])

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 222

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt[a])]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rule 385

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]

Rule 399

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Dist[b/d, Int[(a + b*x^n)^(p - 1), x]
, x] - Dist[(b*c - a*d)/d, Int[(a + b*x^n)^(p - 1)/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b*c
- a*d, 0] && EqQ[n*(p - 1) + 1, 0] && IntegerQ[n]

Rule 1706

Int[(Px_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandInteg
rand[Px*(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e, q}, x] && PolyQ[Px, x^2] && NeQ[b
^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p]

Rule 2081

Int[(u_.)*(P_)^(p_.), x_Symbol] :> With[{m = MinimumMonomialExponent[P, x]}, Dist[P^FracPart[p]/(x^(m*FracPart
[p])*Distrib[1/x^m, P]^FracPart[p]), Int[u*x^(m*p)*Distrib[1/x^m, P]^p, x], x]] /; FreeQ[p, x] &&  !IntegerQ[p
] && SumQ[P] && EveryQ[BinomialQ[#1, x] & , P] &&  !PolyQ[P, x, 2]

Rule 6847

Int[(u_)*(x_)^(m_.), x_Symbol] :> Dist[1/(m + 1), Subst[Int[SubstFor[x^(m + 1), u, x], x], x, x^(m + 1)], x] /
; FreeQ[m, x] && NeQ[m, -1] && FunctionOfQ[x^(m + 1), u, x]

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {x-x^4} \int \frac {\sqrt {x} \sqrt {1-x^3} \left (1+x^3\right )}{2+4 x^3+3 x^6} \, dx}{\sqrt {x} \sqrt {1-x^3}} \\ & = \frac {\left (2 \sqrt {x-x^4}\right ) \text {Subst}\left (\int \frac {\sqrt {1-x^2} \left (1+x^2\right )}{2+4 x^2+3 x^4} \, dx,x,x^{3/2}\right )}{3 \sqrt {x} \sqrt {1-x^3}} \\ & = \frac {\left (2 \sqrt {x-x^4}\right ) \text {Subst}\left (\int \left (\frac {\left (1-\frac {i}{\sqrt {2}}\right ) \sqrt {1-x^2}}{4-2 i \sqrt {2}+6 x^2}+\frac {\left (1+\frac {i}{\sqrt {2}}\right ) \sqrt {1-x^2}}{4+2 i \sqrt {2}+6 x^2}\right ) \, dx,x,x^{3/2}\right )}{3 \sqrt {x} \sqrt {1-x^3}} \\ & = \frac {\left (\left (2-i \sqrt {2}\right ) \sqrt {x-x^4}\right ) \text {Subst}\left (\int \frac {\sqrt {1-x^2}}{4-2 i \sqrt {2}+6 x^2} \, dx,x,x^{3/2}\right )}{3 \sqrt {x} \sqrt {1-x^3}}+\frac {\left (\left (2+i \sqrt {2}\right ) \sqrt {x-x^4}\right ) \text {Subst}\left (\int \frac {\sqrt {1-x^2}}{4+2 i \sqrt {2}+6 x^2} \, dx,x,x^{3/2}\right )}{3 \sqrt {x} \sqrt {1-x^3}} \\ & = -\frac {\left (\left (2-i \sqrt {2}\right ) \sqrt {x-x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2}} \, dx,x,x^{3/2}\right )}{18 \sqrt {x} \sqrt {1-x^3}}-\frac {\left (\left (2-i \sqrt {2}\right ) \left (-5+i \sqrt {2}\right ) \sqrt {x-x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \left (4-2 i \sqrt {2}+6 x^2\right )} \, dx,x,x^{3/2}\right )}{9 \sqrt {x} \sqrt {1-x^3}}-\frac {\left (\left (2+i \sqrt {2}\right ) \sqrt {x-x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2}} \, dx,x,x^{3/2}\right )}{18 \sqrt {x} \sqrt {1-x^3}}+\frac {\left (\left (2+i \sqrt {2}\right ) \left (5+i \sqrt {2}\right ) \sqrt {x-x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \left (4+2 i \sqrt {2}+6 x^2\right )} \, dx,x,x^{3/2}\right )}{9 \sqrt {x} \sqrt {1-x^3}} \\ & = -\frac {\left (2-i \sqrt {2}\right ) \sqrt {x-x^4} \arcsin \left (x^{3/2}\right )}{18 \sqrt {x} \sqrt {1-x^3}}-\frac {\left (2+i \sqrt {2}\right ) \sqrt {x-x^4} \arcsin \left (x^{3/2}\right )}{18 \sqrt {x} \sqrt {1-x^3}}-\frac {\left (\left (2-i \sqrt {2}\right ) \left (-5+i \sqrt {2}\right ) \sqrt {x-x^4}\right ) \text {Subst}\left (\int \frac {1}{4-2 i \sqrt {2}-\left (-10+2 i \sqrt {2}\right ) x^2} \, dx,x,\frac {x^{3/2}}{\sqrt {1-x^3}}\right )}{9 \sqrt {x} \sqrt {1-x^3}}+\frac {\left (\left (2+i \sqrt {2}\right ) \left (5+i \sqrt {2}\right ) \sqrt {x-x^4}\right ) \text {Subst}\left (\int \frac {1}{4+2 i \sqrt {2}-\left (-10-2 i \sqrt {2}\right ) x^2} \, dx,x,\frac {x^{3/2}}{\sqrt {1-x^3}}\right )}{9 \sqrt {x} \sqrt {1-x^3}} \\ & = -\frac {\left (2-i \sqrt {2}\right ) \sqrt {x-x^4} \arcsin \left (x^{3/2}\right )}{18 \sqrt {x} \sqrt {1-x^3}}-\frac {\left (2+i \sqrt {2}\right ) \sqrt {x-x^4} \arcsin \left (x^{3/2}\right )}{18 \sqrt {x} \sqrt {1-x^3}}+\frac {\sqrt {\frac {2 i-\sqrt {2}}{5 i-\sqrt {2}}} \left (5+i \sqrt {2}\right ) \sqrt {x-x^4} \arctan \left (\frac {x^{3/2}}{\sqrt {\frac {2 i-\sqrt {2}}{5 i-\sqrt {2}}} \sqrt {1-x^3}}\right )}{18 \sqrt {x} \sqrt {1-x^3}}+\frac {\left (7+4 i \sqrt {2}\right ) \sqrt {x-x^4} \arctan \left (\frac {x^{3/2}}{\sqrt {\frac {2 i+\sqrt {2}}{5 i+\sqrt {2}}} \sqrt {1-x^3}}\right )}{9 \sqrt {2} \sqrt {-8+7 i \sqrt {2}} \sqrt {x} \sqrt {1-x^3}} \\ \end{align*}

Mathematica [C] (verified)

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

Time = 0.33 (sec) , antiderivative size = 270, normalized size of antiderivative = 2.06 \[ \int \frac {\left (1+x^3\right ) \sqrt {x-x^4}}{2+4 x^3+3 x^6} \, dx=\frac {\sqrt {x-x^4} \left (8 \text {arctanh}\left (\frac {1+x^{3/2}}{\sqrt {-1+x^3}}\right )-\text {RootSum}\left [9+4 \text {$\#$1}^2+22 \text {$\#$1}^4+4 \text {$\#$1}^6+9 \text {$\#$1}^8\&,\frac {9 \log \left (-1+x^{3/2}\right )-9 \log \left (\sqrt {-1+x^3}-\text {$\#$1}+x^{3/2} \text {$\#$1}\right )-11 \log \left (-1+x^{3/2}\right ) \text {$\#$1}^2+11 \log \left (\sqrt {-1+x^3}-\text {$\#$1}+x^{3/2} \text {$\#$1}\right ) \text {$\#$1}^2+11 \log \left (-1+x^{3/2}\right ) \text {$\#$1}^4-11 \log \left (\sqrt {-1+x^3}-\text {$\#$1}+x^{3/2} \text {$\#$1}\right ) \text {$\#$1}^4-9 \log \left (-1+x^{3/2}\right ) \text {$\#$1}^6+9 \log \left (\sqrt {-1+x^3}-\text {$\#$1}+x^{3/2} \text {$\#$1}\right ) \text {$\#$1}^6}{\text {$\#$1}+11 \text {$\#$1}^3+3 \text {$\#$1}^5+9 \text {$\#$1}^7}\&\right ]\right )}{18 \sqrt {x} \sqrt {-1+x^3}} \]

[In]

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

[Out]

(Sqrt[x - x^4]*(8*ArcTanh[(1 + x^(3/2))/Sqrt[-1 + x^3]] - RootSum[9 + 4*#1^2 + 22*#1^4 + 4*#1^6 + 9*#1^8 & , (
9*Log[-1 + x^(3/2)] - 9*Log[Sqrt[-1 + x^3] - #1 + x^(3/2)*#1] - 11*Log[-1 + x^(3/2)]*#1^2 + 11*Log[Sqrt[-1 + x
^3] - #1 + x^(3/2)*#1]*#1^2 + 11*Log[-1 + x^(3/2)]*#1^4 - 11*Log[Sqrt[-1 + x^3] - #1 + x^(3/2)*#1]*#1^4 - 9*Lo
g[-1 + x^(3/2)]*#1^6 + 9*Log[Sqrt[-1 + x^3] - #1 + x^(3/2)*#1]*#1^6)/(#1 + 11*#1^3 + 3*#1^5 + 9*#1^7) & ]))/(1
8*Sqrt[x]*Sqrt[-1 + x^3])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(367\) vs. \(2(101)=202\).

Time = 4.40 (sec) , antiderivative size = 368, normalized size of antiderivative = 2.81

method	result	size
default	\(\frac {\left (-\sqrt {2}-3\right ) \ln \left (\frac {-\sqrt {-x^{4}+x}\, \sqrt {2}\, \sqrt {6 \sqrt {2}-8}\, x +3 \sqrt {2}\, x^{3}-2 x^{3}+2}{x^{3}}\right )+6 \arctan \left (\frac {\sqrt {2}\, \sqrt {6 \sqrt {2}-8}\, x^{2}-4 \sqrt {-x^{4}+x}}{2 x^{2} \sqrt {4+3 \sqrt {2}}}\right ) \sqrt {2}-6 \arctan \left (\frac {\sqrt {2}\, \sqrt {6 \sqrt {2}-8}\, x^{2}+4 \sqrt {-x^{4}+x}}{2 x^{2} \sqrt {4+3 \sqrt {2}}}\right ) \sqrt {2}+\sqrt {2}\, \ln \left (\frac {\sqrt {-x^{4}+x}\, \sqrt {2}\, \sqrt {6 \sqrt {2}-8}\, x +3 \sqrt {2}\, x^{3}-2 x^{3}+2}{x^{3}}\right )+8 \arctan \left (\frac {\sqrt {-x^{4}+x}}{x^{2}}\right ) \sqrt {4+3 \sqrt {2}}+10 \arctan \left (\frac {\sqrt {2}\, \sqrt {6 \sqrt {2}-8}\, x^{2}-4 \sqrt {-x^{4}+x}}{2 x^{2} \sqrt {4+3 \sqrt {2}}}\right )-10 \arctan \left (\frac {\sqrt {2}\, \sqrt {6 \sqrt {2}-8}\, x^{2}+4 \sqrt {-x^{4}+x}}{2 x^{2} \sqrt {4+3 \sqrt {2}}}\right )+3 \ln \left (\frac {\sqrt {-x^{4}+x}\, \sqrt {2}\, \sqrt {6 \sqrt {2}-8}\, x +3 \sqrt {2}\, x^{3}-2 x^{3}+2}{x^{3}}\right )}{36 \sqrt {4+3 \sqrt {2}}}\)	\(368\)
pseudoelliptic	\(\frac {\left (-\sqrt {2}-3\right ) \ln \left (\frac {-\sqrt {-x^{4}+x}\, \sqrt {2}\, \sqrt {6 \sqrt {2}-8}\, x +3 \sqrt {2}\, x^{3}-2 x^{3}+2}{x^{3}}\right )+6 \arctan \left (\frac {\sqrt {2}\, \sqrt {6 \sqrt {2}-8}\, x^{2}-4 \sqrt {-x^{4}+x}}{2 x^{2} \sqrt {4+3 \sqrt {2}}}\right ) \sqrt {2}-6 \arctan \left (\frac {\sqrt {2}\, \sqrt {6 \sqrt {2}-8}\, x^{2}+4 \sqrt {-x^{4}+x}}{2 x^{2} \sqrt {4+3 \sqrt {2}}}\right ) \sqrt {2}+\sqrt {2}\, \ln \left (\frac {\sqrt {-x^{4}+x}\, \sqrt {2}\, \sqrt {6 \sqrt {2}-8}\, x +3 \sqrt {2}\, x^{3}-2 x^{3}+2}{x^{3}}\right )+8 \arctan \left (\frac {\sqrt {-x^{4}+x}}{x^{2}}\right ) \sqrt {4+3 \sqrt {2}}+10 \arctan \left (\frac {\sqrt {2}\, \sqrt {6 \sqrt {2}-8}\, x^{2}-4 \sqrt {-x^{4}+x}}{2 x^{2} \sqrt {4+3 \sqrt {2}}}\right )-10 \arctan \left (\frac {\sqrt {2}\, \sqrt {6 \sqrt {2}-8}\, x^{2}+4 \sqrt {-x^{4}+x}}{2 x^{2} \sqrt {4+3 \sqrt {2}}}\right )+3 \ln \left (\frac {\sqrt {-x^{4}+x}\, \sqrt {2}\, \sqrt {6 \sqrt {2}-8}\, x +3 \sqrt {2}\, x^{3}-2 x^{3}+2}{x^{3}}\right )}{36 \sqrt {4+3 \sqrt {2}}}\)	\(368\)
elliptic	\(\text {Expression too large to display}\)	\(667\)

[In]

int((x^3+1)*(-x^4+x)^(1/2)/(3*x^6+4*x^3+2),x,method=_RETURNVERBOSE)

[Out]

1/36/(4+3*2^(1/2))^(1/2)*((-2^(1/2)-3)*ln((-(-x^4+x)^(1/2)*2^(1/2)*(6*2^(1/2)-8)^(1/2)*x+3*2^(1/2)*x^3-2*x^3+2
)/x^3)+6*arctan(1/2*(2^(1/2)*(6*2^(1/2)-8)^(1/2)*x^2-4*(-x^4+x)^(1/2))/x^2/(4+3*2^(1/2))^(1/2))*2^(1/2)-6*arct
an(1/2*(2^(1/2)*(6*2^(1/2)-8)^(1/2)*x^2+4*(-x^4+x)^(1/2))/x^2/(4+3*2^(1/2))^(1/2))*2^(1/2)+2^(1/2)*ln(((-x^4+x
)^(1/2)*2^(1/2)*(6*2^(1/2)-8)^(1/2)*x+3*2^(1/2)*x^3-2*x^3+2)/x^3)+8*arctan(1/x^2*(-x^4+x)^(1/2))*(4+3*2^(1/2))
^(1/2)+10*arctan(1/2*(2^(1/2)*(6*2^(1/2)-8)^(1/2)*x^2-4*(-x^4+x)^(1/2))/x^2/(4+3*2^(1/2))^(1/2))-10*arctan(1/2
*(2^(1/2)*(6*2^(1/2)-8)^(1/2)*x^2+4*(-x^4+x)^(1/2))/x^2/(4+3*2^(1/2))^(1/2))+3*ln(((-x^4+x)^(1/2)*2^(1/2)*(6*2
^(1/2)-8)^(1/2)*x+3*2^(1/2)*x^3-2*x^3+2)/x^3))

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 431 vs. \(2 (97) = 194\).

Time = 0.48 (sec) , antiderivative size = 431, normalized size of antiderivative = 3.29 \[ \int \frac {\left (1+x^3\right ) \sqrt {x-x^4}}{2+4 x^3+3 x^6} \, dx=-\frac {1}{72} \, \sqrt {7 i \, \sqrt {2} - 8} \log \left (-\frac {2 \, {\left (4160 \, x^{4} + \sqrt {2} {\left (6971 i \, x^{4} + 6034 i \, x\right )} - 1874 \, x\right )} \sqrt {-x^{4} + x} - {\left (683 \, x^{6} + 3684 \, x^{3} - 2 \, \sqrt {2} {\left (2231 i \, x^{6} + 651 i \, x^{3} - 524 i\right )} - 794\right )} \sqrt {7 i \, \sqrt {2} - 8}}{3 \, x^{6} + 4 \, x^{3} + 2}\right ) + \frac {1}{72} \, \sqrt {7 i \, \sqrt {2} - 8} \log \left (-\frac {2 \, {\left (4160 \, x^{4} + \sqrt {2} {\left (6971 i \, x^{4} + 6034 i \, x\right )} - 1874 \, x\right )} \sqrt {-x^{4} + x} + {\left (683 \, x^{6} + 3684 \, x^{3} + 2 \, \sqrt {2} {\left (-2231 i \, x^{6} - 651 i \, x^{3} + 524 i\right )} - 794\right )} \sqrt {7 i \, \sqrt {2} - 8}}{3 \, x^{6} + 4 \, x^{3} + 2}\right ) + \frac {1}{72} \, \sqrt {-7 i \, \sqrt {2} - 8} \log \left (-\frac {2 \, {\left (4160 \, x^{4} + \sqrt {2} {\left (-6971 i \, x^{4} - 6034 i \, x\right )} - 1874 \, x\right )} \sqrt {-x^{4} + x} + {\left (683 \, x^{6} + 3684 \, x^{3} + 2 \, \sqrt {2} {\left (2231 i \, x^{6} + 651 i \, x^{3} - 524 i\right )} - 794\right )} \sqrt {-7 i \, \sqrt {2} - 8}}{3 \, x^{6} + 4 \, x^{3} + 2}\right ) - \frac {1}{72} \, \sqrt {-7 i \, \sqrt {2} - 8} \log \left (-\frac {2 \, {\left (4160 \, x^{4} + \sqrt {2} {\left (-6971 i \, x^{4} - 6034 i \, x\right )} - 1874 \, x\right )} \sqrt {-x^{4} + x} - {\left (683 \, x^{6} + 3684 \, x^{3} - 2 \, \sqrt {2} {\left (-2231 i \, x^{6} - 651 i \, x^{3} + 524 i\right )} - 794\right )} \sqrt {-7 i \, \sqrt {2} - 8}}{3 \, x^{6} + 4 \, x^{3} + 2}\right ) + \frac {1}{9} \, \arctan \left (\frac {2 \, \sqrt {-x^{4} + x} x}{2 \, x^{3} - 1}\right ) \]

[In]

integrate((x^3+1)*(-x^4+x)^(1/2)/(3*x^6+4*x^3+2),x, algorithm="fricas")

[Out]

-1/72*sqrt(7*I*sqrt(2) - 8)*log(-(2*(4160*x^4 + sqrt(2)*(6971*I*x^4 + 6034*I*x) - 1874*x)*sqrt(-x^4 + x) - (68
3*x^6 + 3684*x^3 - 2*sqrt(2)*(2231*I*x^6 + 651*I*x^3 - 524*I) - 794)*sqrt(7*I*sqrt(2) - 8))/(3*x^6 + 4*x^3 + 2
)) + 1/72*sqrt(7*I*sqrt(2) - 8)*log(-(2*(4160*x^4 + sqrt(2)*(6971*I*x^4 + 6034*I*x) - 1874*x)*sqrt(-x^4 + x) +
(683*x^6 + 3684*x^3 + 2*sqrt(2)*(-2231*I*x^6 - 651*I*x^3 + 524*I) - 794)*sqrt(7*I*sqrt(2) - 8))/(3*x^6 + 4*x^
3 + 2)) + 1/72*sqrt(-7*I*sqrt(2) - 8)*log(-(2*(4160*x^4 + sqrt(2)*(-6971*I*x^4 - 6034*I*x) - 1874*x)*sqrt(-x^4
+ x) + (683*x^6 + 3684*x^3 + 2*sqrt(2)*(2231*I*x^6 + 651*I*x^3 - 524*I) - 794)*sqrt(-7*I*sqrt(2) - 8))/(3*x^6
+ 4*x^3 + 2)) - 1/72*sqrt(-7*I*sqrt(2) - 8)*log(-(2*(4160*x^4 + sqrt(2)*(-6971*I*x^4 - 6034*I*x) - 1874*x)*sq
rt(-x^4 + x) - (683*x^6 + 3684*x^3 - 2*sqrt(2)*(-2231*I*x^6 - 651*I*x^3 + 524*I) - 794)*sqrt(-7*I*sqrt(2) - 8)
)/(3*x^6 + 4*x^3 + 2)) + 1/9*arctan(2*sqrt(-x^4 + x)*x/(2*x^3 - 1))

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (1+x^3\right ) \sqrt {x-x^4}}{2+4 x^3+3 x^6} \, dx=\text {Timed out} \]

[In]

integrate((x**3+1)*(-x**4+x)**(1/2)/(3*x**6+4*x**3+2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {\left (1+x^3\right ) \sqrt {x-x^4}}{2+4 x^3+3 x^6} \, dx=\int { \frac {\sqrt {-x^{4} + x} {\left (x^{3} + 1\right )}}{3 \, x^{6} + 4 \, x^{3} + 2} \,d x } \]

[In]

integrate((x^3+1)*(-x^4+x)^(1/2)/(3*x^6+4*x^3+2),x, algorithm="maxima")

[Out]

integrate(sqrt(-x^4 + x)*(x^3 + 1)/(3*x^6 + 4*x^3 + 2), x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 207 vs. \(2 (97) = 194\).

Time = 0.42 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.58 \[ \int \frac {\left (1+x^3\right ) \sqrt {x-x^4}}{2+4 x^3+3 x^6} \, dx=-\frac {1}{36} \, \sqrt {18 \, \sqrt {2} + 16} \arctan \left (\frac {2 \, \left (\frac {9}{2}\right )^{\frac {3}{4}} {\left (\left (\frac {9}{2}\right )^{\frac {1}{4}} {\left (\sqrt {6} - 2 \, \sqrt {3}\right )} + 6 \, \sqrt {\frac {1}{x^{3}} - 1}\right )}}{9 \, {\left (\sqrt {6} + 2 \, \sqrt {3}\right )}}\right ) + \frac {1}{36} \, \sqrt {18 \, \sqrt {2} + 16} \arctan \left (\frac {2 \, \left (\frac {9}{2}\right )^{\frac {3}{4}} {\left (\left (\frac {9}{2}\right )^{\frac {1}{4}} {\left (\sqrt {6} - 2 \, \sqrt {3}\right )} - 6 \, \sqrt {\frac {1}{x^{3}} - 1}\right )}}{9 \, {\left (\sqrt {6} + 2 \, \sqrt {3}\right )}}\right ) - \frac {1}{72} \, \sqrt {18 \, \sqrt {2} - 16} \log \left (\frac {1}{3} \, {\left (\sqrt {6} \left (\frac {9}{2}\right )^{\frac {1}{4}} - 2 \, \left (\frac {9}{2}\right )^{\frac {1}{4}} \sqrt {3}\right )} \sqrt {\frac {1}{x^{3}} - 1} + 3 \, \sqrt {\frac {1}{2}} + \frac {1}{x^{3}} - 1\right ) + \frac {1}{72} \, \sqrt {18 \, \sqrt {2} - 16} \log \left (-\frac {1}{3} \, {\left (\sqrt {6} \left (\frac {9}{2}\right )^{\frac {1}{4}} - 2 \, \left (\frac {9}{2}\right )^{\frac {1}{4}} \sqrt {3}\right )} \sqrt {\frac {1}{x^{3}} - 1} + 3 \, \sqrt {\frac {1}{2}} + \frac {1}{x^{3}} - 1\right ) + \frac {2}{9} \, \arctan \left (\sqrt {\frac {1}{x^{3}} - 1}\right ) \]

[In]

integrate((x^3+1)*(-x^4+x)^(1/2)/(3*x^6+4*x^3+2),x, algorithm="giac")

[Out]

-1/36*sqrt(18*sqrt(2) + 16)*arctan(2/9*(9/2)^(3/4)*((9/2)^(1/4)*(sqrt(6) - 2*sqrt(3)) + 6*sqrt(1/x^3 - 1))/(sq
rt(6) + 2*sqrt(3))) + 1/36*sqrt(18*sqrt(2) + 16)*arctan(2/9*(9/2)^(3/4)*((9/2)^(1/4)*(sqrt(6) - 2*sqrt(3)) - 6
*sqrt(1/x^3 - 1))/(sqrt(6) + 2*sqrt(3))) - 1/72*sqrt(18*sqrt(2) - 16)*log(1/3*(sqrt(6)*(9/2)^(1/4) - 2*(9/2)^(
1/4)*sqrt(3))*sqrt(1/x^3 - 1) + 3*sqrt(1/2) + 1/x^3 - 1) + 1/72*sqrt(18*sqrt(2) - 16)*log(-1/3*(sqrt(6)*(9/2)^
(1/4) - 2*(9/2)^(1/4)*sqrt(3))*sqrt(1/x^3 - 1) + 3*sqrt(1/2) + 1/x^3 - 1) + 2/9*arctan(sqrt(1/x^3 - 1))

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (1+x^3\right ) \sqrt {x-x^4}}{2+4 x^3+3 x^6} \, dx=\int \frac {\sqrt {x-x^4}\,\left (x^3+1\right )}{3\,x^6+4\,x^3+2} \,d x \]

[In]

int(((x - x^4)^(1/2)*(x^3 + 1))/(4*x^3 + 3*x^6 + 2),x)

[Out]

int(((x - x^4)^(1/2)*(x^3 + 1))/(4*x^3 + 3*x^6 + 2), x)