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

Optimal. Leaf size=131 \[ -\frac {1}{18} \sqrt {8+7 i \sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2-\frac {i}{\sqrt {2}}} x \sqrt {x-x^4}}{x^3-1}\right )-\frac {1}{18} \sqrt {8-7 i \sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2+\frac {i}{\sqrt {2}}} x \sqrt {x-x^4}}{x^3-1}\right )-\frac {2}{9} \tan ^{-1}\left (\frac {x^2}{\sqrt {x-x^4}}\right ) \]

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Rubi [B]  time = 0.86, antiderivative size = 323, normalized size of antiderivative = 2.47, number of steps used = 12, number of rules used = 7, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {2056, 6715, 1692, 402, 216, 377, 205} \begin {gather*} -\frac {\left (2+i \sqrt {2}\right ) \sqrt {x-x^4} \sin ^{-1}\left (x^{3/2}\right )}{18 \sqrt {x} \sqrt {1-x^3}}-\frac {\left (2-i \sqrt {2}\right ) \sqrt {x-x^4} \sin ^{-1}\left (x^{3/2}\right )}{18 \sqrt {x} \sqrt {1-x^3}}+\frac {\left (\sqrt {2}+i\right ) \sqrt {x-x^4} \tan ^{-1}\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} \tan ^{-1}\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}} \end {gather*}

Warning: Unable to verify antiderivative.

[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 205

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

Rule 216

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 377

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 402

Int[((a_) + (b_.)*(x_)^2)^(p_.)/((c_) + (d_.)*(x_)^2), x_Symbol] :> Dist[b/d, Int[(a + b*x^2)^(p - 1), x], x]
- Dist[(b*c - a*d)/d, Int[(a + b*x^2)^(p - 1)/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d,
0] && GtQ[p, 0] && (EqQ[p, 1/2] || EqQ[Denominator[p], 4])

Rule 1692

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 2056

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 6715

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*} \int \frac {\left (1+x^3\right ) \sqrt {x-x^4}}{2+4 x^3+3 x^6} \, dx &=\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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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} \sin ^{-1}\left (x^{3/2}\right )}{18 \sqrt {x} \sqrt {1-x^3}}-\frac {\left (2+i \sqrt {2}\right ) \sqrt {x-x^4} \sin ^{-1}\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 ) \operatorname {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 ) \operatorname {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} \sin ^{-1}\left (x^{3/2}\right )}{18 \sqrt {x} \sqrt {1-x^3}}-\frac {\left (2+i \sqrt {2}\right ) \sqrt {x-x^4} \sin ^{-1}\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} \tan ^{-1}\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} \tan ^{-1}\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*}

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Mathematica [B]  time = 3.32, size = 1192, normalized size = 9.10 \begin {gather*} \frac {\sqrt {x-x^4} \left (-4 \sqrt {2 \left (-5+i \sqrt {2}\right )} \sqrt {-2+i \sqrt {2}} \log \left (x^{3/2}-\sqrt {-\frac {2}{3}-\frac {i \sqrt {2}}{3}}\right )+4 \sqrt {2 \left (-5+i \sqrt {2}\right )} \sqrt {-2+i \sqrt {2}} \log \left (x^{3/2}+\sqrt {-\frac {2}{3}-\frac {i \sqrt {2}}{3}}\right )-7 i \sqrt {-5+i \sqrt {2}} \sqrt {-2+i \sqrt {2}} \log \left (3 x^{3/2}-\sqrt {-6-3 i \sqrt {2}}\right )+7 i \sqrt {-5+i \sqrt {2}} \sqrt {-2+i \sqrt {2}} \log \left (3 x^{3/2}+\sqrt {-6-3 i \sqrt {2}}\right )-4 \sqrt {-2-i \sqrt {2}} \sqrt {-10-2 i \sqrt {2}} \log \left (3 x^{3/2}-\sqrt {-6+3 i \sqrt {2}}\right )+7 i \sqrt {-5-i \sqrt {2}} \sqrt {-2-i \sqrt {2}} \log \left (3 x^{3/2}-\sqrt {-6+3 i \sqrt {2}}\right )+4 \sqrt {-2-i \sqrt {2}} \sqrt {-10-2 i \sqrt {2}} \log \left (3 x^{3/2}+\sqrt {-6+3 i \sqrt {2}}\right )-7 i \sqrt {-5-i \sqrt {2}} \sqrt {-2-i \sqrt {2}} \log \left (3 x^{3/2}+\sqrt {-6+3 i \sqrt {2}}\right )+72 \log \left (x^{3/2}+\sqrt {x^3-1}\right )-4 \sqrt {2 \left (-5+i \sqrt {2}\right )} \sqrt {-2+i \sqrt {2}} \log \left (-\sqrt {-6-3 i \sqrt {2}} x^{3/2}+\sqrt {-15-3 i \sqrt {2}} \sqrt {x^3-1}-3\right )-7 i \sqrt {-5+i \sqrt {2}} \sqrt {-2+i \sqrt {2}} \log \left (-\sqrt {-6-3 i \sqrt {2}} x^{3/2}+\sqrt {-15-3 i \sqrt {2}} \sqrt {x^3-1}-3\right )+4 \sqrt {2 \left (-5+i \sqrt {2}\right )} \sqrt {-2+i \sqrt {2}} \log \left (\sqrt {-6-3 i \sqrt {2}} x^{3/2}+\sqrt {-15-3 i \sqrt {2}} \sqrt {x^3-1}-3\right )+7 i \sqrt {-5+i \sqrt {2}} \sqrt {-2+i \sqrt {2}} \log \left (\sqrt {-6-3 i \sqrt {2}} x^{3/2}+\sqrt {-15-3 i \sqrt {2}} \sqrt {x^3-1}-3\right )-4 \sqrt {-2-i \sqrt {2}} \sqrt {-10-2 i \sqrt {2}} \log \left (-\sqrt {-6+3 i \sqrt {2}} x^{3/2}+\sqrt {3 i \left (5 i+\sqrt {2}\right )} \sqrt {x^3-1}-3\right )+7 i \sqrt {-5-i \sqrt {2}} \sqrt {-2-i \sqrt {2}} \log \left (-\sqrt {-6+3 i \sqrt {2}} x^{3/2}+\sqrt {3 i \left (5 i+\sqrt {2}\right )} \sqrt {x^3-1}-3\right )+4 \sqrt {-2-i \sqrt {2}} \sqrt {-10-2 i \sqrt {2}} \log \left (\sqrt {-6+3 i \sqrt {2}} x^{3/2}+\sqrt {3 i \left (5 i+\sqrt {2}\right )} \sqrt {x^3-1}-3\right )-7 i \sqrt {-5-i \sqrt {2}} \sqrt {-2-i \sqrt {2}} \log \left (\sqrt {-6+3 i \sqrt {2}} x^{3/2}+\sqrt {3 i \left (5 i+\sqrt {2}\right )} \sqrt {x^3-1}-3\right )\right )}{324 \sqrt {x} \sqrt {x^3-1}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Sqrt[x - x^4]*(-4*Sqrt[2*(-5 + I*Sqrt[2])]*Sqrt[-2 + I*Sqrt[2]]*Log[-Sqrt[-2/3 - (I/3)*Sqrt[2]] + x^(3/2)] +
4*Sqrt[2*(-5 + I*Sqrt[2])]*Sqrt[-2 + I*Sqrt[2]]*Log[Sqrt[-2/3 - (I/3)*Sqrt[2]] + x^(3/2)] - (7*I)*Sqrt[-5 + I*
Sqrt[2]]*Sqrt[-2 + I*Sqrt[2]]*Log[-Sqrt[-6 - (3*I)*Sqrt[2]] + 3*x^(3/2)] + (7*I)*Sqrt[-5 + I*Sqrt[2]]*Sqrt[-2
+ I*Sqrt[2]]*Log[Sqrt[-6 - (3*I)*Sqrt[2]] + 3*x^(3/2)] + (7*I)*Sqrt[-5 - I*Sqrt[2]]*Sqrt[-2 - I*Sqrt[2]]*Log[-
Sqrt[-6 + (3*I)*Sqrt[2]] + 3*x^(3/2)] - 4*Sqrt[-2 - I*Sqrt[2]]*Sqrt[-10 - (2*I)*Sqrt[2]]*Log[-Sqrt[-6 + (3*I)*
Sqrt[2]] + 3*x^(3/2)] - (7*I)*Sqrt[-5 - I*Sqrt[2]]*Sqrt[-2 - I*Sqrt[2]]*Log[Sqrt[-6 + (3*I)*Sqrt[2]] + 3*x^(3/
2)] + 4*Sqrt[-2 - I*Sqrt[2]]*Sqrt[-10 - (2*I)*Sqrt[2]]*Log[Sqrt[-6 + (3*I)*Sqrt[2]] + 3*x^(3/2)] + 72*Log[x^(3
/2) + Sqrt[-1 + x^3]] - (7*I)*Sqrt[-5 + I*Sqrt[2]]*Sqrt[-2 + I*Sqrt[2]]*Log[-3 - Sqrt[-6 - (3*I)*Sqrt[2]]*x^(3
/2) + Sqrt[-15 - (3*I)*Sqrt[2]]*Sqrt[-1 + x^3]] - 4*Sqrt[2*(-5 + I*Sqrt[2])]*Sqrt[-2 + I*Sqrt[2]]*Log[-3 - Sqr
t[-6 - (3*I)*Sqrt[2]]*x^(3/2) + Sqrt[-15 - (3*I)*Sqrt[2]]*Sqrt[-1 + x^3]] + (7*I)*Sqrt[-5 + I*Sqrt[2]]*Sqrt[-2
+ I*Sqrt[2]]*Log[-3 + Sqrt[-6 - (3*I)*Sqrt[2]]*x^(3/2) + Sqrt[-15 - (3*I)*Sqrt[2]]*Sqrt[-1 + x^3]] + 4*Sqrt[2
*(-5 + I*Sqrt[2])]*Sqrt[-2 + I*Sqrt[2]]*Log[-3 + Sqrt[-6 - (3*I)*Sqrt[2]]*x^(3/2) + Sqrt[-15 - (3*I)*Sqrt[2]]*
Sqrt[-1 + x^3]] + (7*I)*Sqrt[-5 - I*Sqrt[2]]*Sqrt[-2 - I*Sqrt[2]]*Log[-3 - Sqrt[-6 + (3*I)*Sqrt[2]]*x^(3/2) +
Sqrt[(3*I)*(5*I + Sqrt[2])]*Sqrt[-1 + x^3]] - 4*Sqrt[-2 - I*Sqrt[2]]*Sqrt[-10 - (2*I)*Sqrt[2]]*Log[-3 - Sqrt[-
6 + (3*I)*Sqrt[2]]*x^(3/2) + Sqrt[(3*I)*(5*I + Sqrt[2])]*Sqrt[-1 + x^3]] - (7*I)*Sqrt[-5 - I*Sqrt[2]]*Sqrt[-2
- I*Sqrt[2]]*Log[-3 + Sqrt[-6 + (3*I)*Sqrt[2]]*x^(3/2) + Sqrt[(3*I)*(5*I + Sqrt[2])]*Sqrt[-1 + x^3]] + 4*Sqrt[
-2 - I*Sqrt[2]]*Sqrt[-10 - (2*I)*Sqrt[2]]*Log[-3 + Sqrt[-6 + (3*I)*Sqrt[2]]*x^(3/2) + Sqrt[(3*I)*(5*I + Sqrt[2
])]*Sqrt[-1 + x^3]]))/(324*Sqrt[x]*Sqrt[-1 + x^3])

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IntegrateAlgebraic [A]  time = 1.16, size = 121, normalized size = 0.92 \begin {gather*} -\frac {2}{9} \tan ^{-1}\left (\frac {x^2}{\sqrt {x-x^4}}\right )+\frac {1}{18} \sqrt {8+7 i \sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2-\frac {i}{\sqrt {2}}} x^2}{\sqrt {x-x^4}}\right )+\frac {1}{18} \sqrt {8-7 i \sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2+\frac {i}{\sqrt {2}}} x^2}{\sqrt {x-x^4}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 11.85, size = 2741, normalized size = 20.92

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1008*2^(3/4)*sqrt(4*sqrt(2) + 9)*(4*sqrt(2) - 9)*log(-6360849/7*(168*x^6 - 168*x^3 + 2*2^(3/4)*(28*x^4 - 3*
sqrt(2)*(7*x^4 - 2*x) - 10*x)*sqrt(-x^4 + x)*sqrt(4*sqrt(2) + 9) - 7*sqrt(2)*(19*x^6 - 12*x^3 + 2))/(3*x^6 + 4
*x^3 + 2)) + 1/1008*2^(3/4)*sqrt(4*sqrt(2) + 9)*(4*sqrt(2) - 9)*log(-6360849/7*(168*x^6 - 168*x^3 - 2*2^(3/4)*
(28*x^4 - 3*sqrt(2)*(7*x^4 - 2*x) - 10*x)*sqrt(-x^4 + x)*sqrt(4*sqrt(2) + 9) - 7*sqrt(2)*(19*x^6 - 12*x^3 + 2)
)/(3*x^6 + 4*x^3 + 2)) - 1/36*2^(3/4)*sqrt(4*sqrt(2) + 9)*arctan(-1/14*(8632050444092280152834837119864926*x^3
6 + 9161521558932209861778630592599792*x^33 - 17260935589581589566186845470001928*x^30 - 224175398714101147927
89642783506784*x^27 + 8632251370051221229521254573075784*x^24 + 17941208497396218206799599939395968*x^21 + 111
688867273889746553275629025344*x^18 - 5493502622574650172977792469639936*x^15 + 309053370024366207648382661924
16*x^12 + 790651863971902368236671518751488*x^9 - 138031505983935794529450862913664*x^6 + 12890476859438719685
2145177088*x^3 + 12721698*sqrt(2)*sqrt(-x^4 + x)*(2^(3/4)*(155068222869201603274217652*x^34 + 2528379567845730
83685349366*x^31 - 299319523809873331574781920*x^28 - 586004622273400652833815356*x^25 + 121997183777565423757
231456*x^22 + 419025032959227012317601072*x^19 + 28076058669471027380499328*x^16 - 90013541853774632322919520*
x^13 - 4918374089438875919755968*x^10 + 5625954326986205601627616*x^7 + 48339376368627709608960*x^4 - 3*sqrt(2
)*(16825352941418359482113253*x^34 + 59987578296866209972620350*x^31 - 6580181442381755563592634*x^28 - 133348
333567725981519681428*x^25 - 43202477060524375944383896*x^22 + 90505619247446912851428400*x^19 + 3824689815010
5540704718192*x^16 - 19238762657512258793540000*x^13 - 5649039556365590504631024*x^10 + 1892796862144660136321
888*x^7 - 11363374832743038268448*x^4 + 145841485650323828672*x) - 619706421311517572800*x) - 16*2^(1/4)*(3406
1651742756490436056644*x^34 + 95703917494121772333264282*x^31 + 22588579341510421818747546*x^28 - 144065524109
011381824443844*x^25 - 111261131573116451568361236*x^22 + 45474265224424729086049080*x^19 + 648826379473233914
31307656*x^16 + 5306970009315400116931728*x^13 - 10505618506466998937300016*x^10 - 2392945547527543470258048*x
^7 + 207197976670170578006208*x^4 + sqrt(2)*(12030880734706239561839937*x^34 + 44933301219178895918527241*x^31
+ 46944706396173846244715276*x^28 - 35613080256963383211529986*x^25 - 97068552946327814904168272*x^22 - 32620
008949099351619636380*x^19 + 38469908845526136232962336*x^16 + 24972611089770871113434152*x^13 - 5042714489256
87639314320*x^10 - 1692005295003728616548448*x^7 + 146510610963976919718464*x^4)))*sqrt(4*sqrt(2) + 9) + 99*sq
rt(649/7)*(sqrt(2)*(2^(3/4)*(3121596530260239669336820157913*x^36 + 5834255341413061687138179876792*x^33 - 688
8167414217170094657282052608*x^30 - 13995366187142943155635818597216*x^27 + 4523218796861744773348981882116*x^
24 + 10950989276996393893976849951808*x^21 - 1093549213499303367270942631488*x^18 - 30771128174530625415154357
58592*x^15 + 381460432275327319018147934640*x^12 + 300850897180239679913838976896*x^9 - 5855591447183549358755
4868992*x^6 - 5108884480838567481905664*x^3 - 2*sqrt(2)*(1086133862743827732006103951791*x^36 + 20785366287488
68754952479046197*x^33 - 2374524441070525146385161658610*x^30 - 4992736270966904035107639005666*x^27 + 1517995
330234711341166019538136*x^24 + 3915463113897739790410043247464*x^21 - 338072542128739686425119214608*x^18 - 1
104683428514778558567052777424*x^15 + 124434881558629873274651070128*x^12 + 108391044654310593595674374672*x^9
- 20827411787368918663045438624*x^6 + 1406128425854828903266144*x^3 - 13507123892806437370880) + 464047631432
66731467456) - 16*2^(1/4)*(248250202572925434417171709641*x^36 + 392158279593892776753649524707*x^33 - 7643655
10952128583802234715988*x^30 - 1158625043830399121658197199928*x^27 + 744974044625885970803079278080*x^24 + 11
86605786206243645868681816096*x^21 - 210612742993896435791727033808*x^18 - 457525832401022893299363583856*x^15
- 20029782308524211147631862800*x^12 + 38604316354119658596668068688*x^9 + 580441335212311981979699072*x^6 -
14158202308552722075699904*x^3 - 18*sqrt(2)*(8564251745904023585195931603*x^36 + 12750457151213196303514367536
*x^33 - 28534154646070291083552637573*x^30 - 38161333576607079640202027588*x^27 + 3188378469461408518469391979
4*x^24 + 40679760423015746022246526032*x^21 - 12754053545003855227347594068*x^18 - 170144727458124891976558566
88*x^15 + 815653447800013477852821480*x^12 + 1792322302234552578825256576*x^9 - 22771207005398409855974432*x^6
+ 555955717496406285267328*x^3)))*sqrt(4*sqrt(2) + 9) - 28*(524236500266665825730685749760*x^34 + 13324048450
15756903230965089536*x^31 - 130480878679520904884096757504*x^28 - 2107192014832089761023279214592*x^25 - 61249
2456088768126642309490688*x^22 + 988722784766968088259898358784*x^19 + 228997944867525635371563924480*x^16 - 2
35754311387964830721719050240*x^13 - 8539169464378485377627922432*x^10 + 23301104080816172474328858624*x^7 - 3
204348545010516418409545728*x^4 - sqrt(2)*(328841013685710897201295863120*x^34 + 71242362392872749735201317471
2*x^31 - 507875961967949035452802264160*x^28 - 1634077001729216627778034549952*x^25 - 301195528463815158794839
12064*x^22 + 1159152878738410570103304835584*x^19 + 252390387321045426549485332480*x^16 - 25458996850281287344
9866789248*x^13 - 43069085653914687948369003264*x^10 + 19342332894031279621629314176*x^7 - 4392886874881611701
6977920*x^4 - 6360849*sqrt(2)*(33227621146716099962831*x^34 + 75316178531714961613054*x^31 - 46850423411517051
236214*x^28 - 168426333235143900655124*x^25 - 10554911710188156761256*x^22 + 114645820898039007484144*x^19 + 2
7634437117784163769424*x^16 - 23483842011297496036128*x^13 - 3713017786705821859664*x^10 + 1934426021997052439
648*x^7 + 4921845856891760672*x^4 - 66678885297238592*x) + 597251310504684115083776*x) - 16*sqrt(2)*(102474016
36214771696674632303*x^34 + 3275263563747727892387456203*x^31 - 80962254423821149367897353376*x^28 - 875147205
07317003029708428470*x^25 + 80176236326587548057900990584*x^22 + 132642871210885978930084859404*x^19 + 8584531
730908213324425520080*x^16 - 47307947259275353642309640776*x^13 - 18253055338878700118700816272*x^10 - 1029941
776235616195421930656*x^7 + 141614837183582452564710976*x^4))*sqrt(-x^4 + x))*sqrt(-(168*x^6 - 168*x^3 + 2*2^(
3/4)*(28*x^4 - 3*sqrt(2)*(7*x^4 - 2*x) - 10*x)*sqrt(-x^4 + x)*sqrt(4*sqrt(2) + 9) - 7*sqrt(2)*(19*x^6 - 12*x^3
+ 2))/(3*x^6 + 4*x^3 + 2)) + 56*sqrt(2)*(218241848288644389389848879258650*x^36 + 251359916402604912653099443
236315*x^33 - 653777886516007933475373096546014*x^30 - 750700155453942593740040211672026*x^27 + 68686297097696
6659426192517349092*x^24 + 795085121396131507309277160898104*x^21 - 289632726736470239458313325939952*x^18 - 3
41211879567795800265863199007312*x^15 + 40768303155595910162237651899968*x^12 + 455193470777527498374226186356
32*x^9 - 2637210339419949781840773265504*x^6 + 1616891482648877834396339936*x^3 - 14541470867755473408047552)
- 712415088*sqrt(2)*(18379641660262029137485017*x^36 + 22231658947052511977755683*x^33 - 546621716360812935332
53176*x^30 - 68425267185349358200890180*x^27 + 52213949743345144992129576*x^24 + 71314761668688399743404920*x^
21 - 16482475575272462172115776*x^18 - 28047105217963056593939424*x^15 + 727273483738706512589904*x^12 + 29435
81370971382216583536*x^9 - 198227704559878256174208*x^6 + 4380445167874176424128*x^3 + sqrt(2)*(48817688806532
19801142320*x^36 + 3033693318603343422286517*x^33 - 12850910120393402178568823*x^30 - 105094765460557738388121
34*x^27 + 10106491657868602044833128*x^24 + 10804600615433251603874088*x^21 - 2531419719312256190585608*x^18 -
3921202481378418716194160*x^15 + 552408965265963731159328*x^12 + 571130043714430822119440*x^9 - 1401821623186
62558387184*x^6 + 3097547919702057133088*x^3)) - 1156571977971226039247832192)/(376242708176116008877087774703
9271*x^36 + 11176455591989619085097268601346040*x^33 - 405206787589120595364629876997268*x^30 - 26441727421386
436860245914517793136*x^27 - 15623066860355145469583071278966908*x^24 + 18378215491781049218505676436316864*x^
21 + 17501761125917080810108705358759584*x^18 - 2116382992907014932708830590815104*x^15 - 53463699449399794675
16460009090672*x^12 - 995484862411051605673619483583104*x^9 + 109315127011405789944450185284288*x^6 + 10832968
43887043076993611008*x^3 - 7632135417534638943928384)) - 1/36*2^(3/4)*sqrt(4*sqrt(2) + 9)*arctan(1/14*(8632050
444092280152834837119864926*x^36 + 9161521558932209861778630592599792*x^33 - 172609355895815895661868454700019
28*x^30 - 22417539871410114792789642783506784*x^27 + 8632251370051221229521254573075784*x^24 + 179412084973962
18206799599939395968*x^21 + 111688867273889746553275629025344*x^18 - 5493502622574650172977792469639936*x^15 +
30905337002436620764838266192416*x^12 + 790651863971902368236671518751488*x^9 - 13803150598393579452945086291
3664*x^6 + 128904768594387196852145177088*x^3 - 12721698*sqrt(2)*sqrt(-x^4 + x)*(2^(3/4)*(15506822286920160327
4217652*x^34 + 252837956784573083685349366*x^31 - 299319523809873331574781920*x^28 - 5860046222734006528338153
56*x^25 + 121997183777565423757231456*x^22 + 419025032959227012317601072*x^19 + 28076058669471027380499328*x^1
6 - 90013541853774632322919520*x^13 - 4918374089438875919755968*x^10 + 5625954326986205601627616*x^7 + 4833937
6368627709608960*x^4 - 3*sqrt(2)*(16825352941418359482113253*x^34 + 59987578296866209972620350*x^31 - 65801814
42381755563592634*x^28 - 133348333567725981519681428*x^25 - 43202477060524375944383896*x^22 + 9050561924744691
2851428400*x^19 + 38246898150105540704718192*x^16 - 19238762657512258793540000*x^13 - 564903955636559050463102
4*x^10 + 1892796862144660136321888*x^7 - 11363374832743038268448*x^4 + 145841485650323828672*x) - 619706421311
517572800*x) - 16*2^(1/4)*(34061651742756490436056644*x^34 + 95703917494121772333264282*x^31 + 225885793415104
21818747546*x^28 - 144065524109011381824443844*x^25 - 111261131573116451568361236*x^22 + 454742652244247290860
49080*x^19 + 64882637947323391431307656*x^16 + 5306970009315400116931728*x^13 - 10505618506466998937300016*x^1
0 - 2392945547527543470258048*x^7 + 207197976670170578006208*x^4 + sqrt(2)*(12030880734706239561839937*x^34 +
44933301219178895918527241*x^31 + 46944706396173846244715276*x^28 - 35613080256963383211529986*x^25 - 97068552
946327814904168272*x^22 - 32620008949099351619636380*x^19 + 38469908845526136232962336*x^16 + 2497261108977087
1113434152*x^13 - 504271448925687639314320*x^10 - 1692005295003728616548448*x^7 + 146510610963976919718464*x^4
)))*sqrt(4*sqrt(2) + 9) - 99*sqrt(649/7)*(sqrt(2)*(2^(3/4)*(3121596530260239669336820157913*x^36 + 58342553414
13061687138179876792*x^33 - 6888167414217170094657282052608*x^30 - 13995366187142943155635818597216*x^27 + 452
3218796861744773348981882116*x^24 + 10950989276996393893976849951808*x^21 - 1093549213499303367270942631488*x^
18 - 3077112817453062541515435758592*x^15 + 381460432275327319018147934640*x^12 + 3008508971802396799138389768
96*x^9 - 58555914471835493587554868992*x^6 - 5108884480838567481905664*x^3 - 2*sqrt(2)*(1086133862743827732006
103951791*x^36 + 2078536628748868754952479046197*x^33 - 2374524441070525146385161658610*x^30 - 499273627096690
4035107639005666*x^27 + 1517995330234711341166019538136*x^24 + 3915463113897739790410043247464*x^21 - 33807254
2128739686425119214608*x^18 - 1104683428514778558567052777424*x^15 + 124434881558629873274651070128*x^12 + 108
391044654310593595674374672*x^9 - 20827411787368918663045438624*x^6 + 1406128425854828903266144*x^3 - 13507123
892806437370880) + 46404763143266731467456) - 16*2^(1/4)*(248250202572925434417171709641*x^36 + 39215827959389
2776753649524707*x^33 - 764365510952128583802234715988*x^30 - 1158625043830399121658197199928*x^27 + 744974044
625885970803079278080*x^24 + 1186605786206243645868681816096*x^21 - 210612742993896435791727033808*x^18 - 4575
25832401022893299363583856*x^15 - 20029782308524211147631862800*x^12 + 38604316354119658596668068688*x^9 + 580
441335212311981979699072*x^6 - 14158202308552722075699904*x^3 - 18*sqrt(2)*(8564251745904023585195931603*x^36
+ 12750457151213196303514367536*x^33 - 28534154646070291083552637573*x^30 - 38161333576607079640202027588*x^27
+ 31883784694614085184693919794*x^24 + 40679760423015746022246526032*x^21 - 12754053545003855227347594068*x^1
8 - 17014472745812489197655856688*x^15 + 815653447800013477852821480*x^12 + 1792322302234552578825256576*x^9 -
22771207005398409855974432*x^6 + 555955717496406285267328*x^3)))*sqrt(4*sqrt(2) + 9) + 28*(524236500266665825
730685749760*x^34 + 1332404845015756903230965089536*x^31 - 130480878679520904884096757504*x^28 - 2107192014832
089761023279214592*x^25 - 612492456088768126642309490688*x^22 + 988722784766968088259898358784*x^19 + 22899794
4867525635371563924480*x^16 - 235754311387964830721719050240*x^13 - 8539169464378485377627922432*x^10 + 233011
04080816172474328858624*x^7 - 3204348545010516418409545728*x^4 - sqrt(2)*(328841013685710897201295863120*x^34
+ 712423623928727497352013174712*x^31 - 507875961967949035452802264160*x^28 - 1634077001729216627778034549952*
x^25 - 30119552846381515879483912064*x^22 + 1159152878738410570103304835584*x^19 + 252390387321045426549485332
480*x^16 - 254589968502812873449866789248*x^13 - 43069085653914687948369003264*x^10 + 193423328940312796216293
14176*x^7 - 43928868748816117016977920*x^4 - 6360849*sqrt(2)*(33227621146716099962831*x^34 + 75316178531714961
613054*x^31 - 46850423411517051236214*x^28 - 168426333235143900655124*x^25 - 10554911710188156761256*x^22 + 11
4645820898039007484144*x^19 + 27634437117784163769424*x^16 - 23483842011297496036128*x^13 - 371301778670582185
9664*x^10 + 1934426021997052439648*x^7 + 4921845856891760672*x^4 - 66678885297238592*x) + 59725131050468411508
3776*x) - 16*sqrt(2)*(10247401636214771696674632303*x^34 + 3275263563747727892387456203*x^31 - 809622544238211
49367897353376*x^28 - 87514720507317003029708428470*x^25 + 80176236326587548057900990584*x^22 + 13264287121088
5978930084859404*x^19 + 8584531730908213324425520080*x^16 - 47307947259275353642309640776*x^13 - 1825305533887
8700118700816272*x^10 - 1029941776235616195421930656*x^7 + 141614837183582452564710976*x^4))*sqrt(-x^4 + x))*s
qrt(-(168*x^6 - 168*x^3 - 2*2^(3/4)*(28*x^4 - 3*sqrt(2)*(7*x^4 - 2*x) - 10*x)*sqrt(-x^4 + x)*sqrt(4*sqrt(2) +
9) - 7*sqrt(2)*(19*x^6 - 12*x^3 + 2))/(3*x^6 + 4*x^3 + 2)) + 56*sqrt(2)*(218241848288644389389848879258650*x^3
6 + 251359916402604912653099443236315*x^33 - 653777886516007933475373096546014*x^30 - 750700155453942593740040
211672026*x^27 + 686862970976966659426192517349092*x^24 + 795085121396131507309277160898104*x^21 - 28963272673
6470239458313325939952*x^18 - 341211879567795800265863199007312*x^15 + 40768303155595910162237651899968*x^12 +
45519347077752749837422618635632*x^9 - 2637210339419949781840773265504*x^6 + 1616891482648877834396339936*x^3
- 14541470867755473408047552) - 712415088*sqrt(2)*(18379641660262029137485017*x^36 + 222316589470525119777556
83*x^33 - 54662171636081293533253176*x^30 - 68425267185349358200890180*x^27 + 52213949743345144992129576*x^24
+ 71314761668688399743404920*x^21 - 16482475575272462172115776*x^18 - 28047105217963056593939424*x^15 + 727273
483738706512589904*x^12 + 2943581370971382216583536*x^9 - 198227704559878256174208*x^6 + 438044516787417642412
8*x^3 + sqrt(2)*(4881768880653219801142320*x^36 + 3033693318603343422286517*x^33 - 12850910120393402178568823*
x^30 - 10509476546055773838812134*x^27 + 10106491657868602044833128*x^24 + 10804600615433251603874088*x^21 - 2
531419719312256190585608*x^18 - 3921202481378418716194160*x^15 + 552408965265963731159328*x^12 + 5711300437144
30822119440*x^9 - 140182162318662558387184*x^6 + 3097547919702057133088*x^3)) - 1156571977971226039247832192)/
(3762427081761160088770877747039271*x^36 + 11176455591989619085097268601346040*x^33 - 405206787589120595364629
876997268*x^30 - 26441727421386436860245914517793136*x^27 - 15623066860355145469583071278966908*x^24 + 1837821
5491781049218505676436316864*x^21 + 17501761125917080810108705358759584*x^18 - 2116382992907014932708830590815
104*x^15 - 5346369944939979467516460009090672*x^12 - 995484862411051605673619483583104*x^9 + 10931512701140578
9944450185284288*x^6 + 1083296843887043076993611008*x^3 - 7632135417534638943928384)) + 1/9*arctan(2*sqrt(-x^4
+ x)*x/(2*x^3 - 1))

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giac [B]  time = 0.35, size = 207, normalized size = 1.58 \begin {gather*} -\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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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maple [C]  time = 1.71, size = 667, normalized size = 5.09

method	result	size
default	\(-\frac {2 \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}\, \left (-1+x \right )^{2} \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}\, \left (\EllipticF \left (\sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}, \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )-\EllipticPi \left (\sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}, \frac {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )\right )}{3 \left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {-x \left (-1+x \right ) \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}+\frac {\sqrt {4}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (3 \textit {\_Z}^{6}+4 \textit {\_Z}^{3}+2\right )}{\sum }\frac {\underline {\hspace {1.25 ex}}\alpha ^{2} \left (\underline {\hspace {1.25 ex}}\alpha ^{3}-1\right ) \left (-1+x \right )^{2} \left (3 \underline {\hspace {1.25 ex}}\alpha ^{5}+3 \underline {\hspace {1.25 ex}}\alpha ^{4}+3 \underline {\hspace {1.25 ex}}\alpha ^{3}+7 \underline {\hspace {1.25 ex}}\alpha ^{2}+7 \underline {\hspace {1.25 ex}}\alpha +7\right ) \left (1-i \sqrt {3}\right ) \sqrt {\frac {x \left (-3+i \sqrt {3}\right )}{\left (-1+x \right ) \left (-1+i \sqrt {3}\right )}}\, \sqrt {\frac {i \sqrt {3}+2 x +1}{\left (-1+x \right ) \left (-1-i \sqrt {3}\right )}}\, \sqrt {\frac {1+2 x -i \sqrt {3}}{\left (-1+x \right ) \left (-1+i \sqrt {3}\right )}}\, \left (2 \EllipticF \left (\sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}, \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )-\left (-3 \underline {\hspace {1.25 ex}}\alpha ^{5}-4 \underline {\hspace {1.25 ex}}\alpha ^{2}\right ) \EllipticPi \left (\sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}, -\frac {i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{5}}{4}+\frac {3 \underline {\hspace {1.25 ex}}\alpha ^{5}}{4}-\frac {i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{2}}{3}+\underline {\hspace {1.25 ex}}\alpha ^{2}-\frac {i \sqrt {3}}{6}+\frac {1}{2}, \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )\right )}{\left (-3+i \sqrt {3}\right ) \sqrt {-x \left (-1+x \right ) \left (i \sqrt {3}+2 x +1\right ) \left (1+2 x -i \sqrt {3}\right )}}\right )}{108}\)	\(667\)
elliptic	\(-\frac {2 \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}\, \left (-1+x \right )^{2} \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}\, \left (\EllipticF \left (\sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}, \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )-\EllipticPi \left (\sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}, \frac {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )\right )}{3 \left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {-x \left (-1+x \right ) \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}+\frac {\sqrt {4}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (3 \textit {\_Z}^{6}+4 \textit {\_Z}^{3}+2\right )}{\sum }\frac {\underline {\hspace {1.25 ex}}\alpha ^{2} \left (\underline {\hspace {1.25 ex}}\alpha ^{3}-1\right ) \left (-1+x \right )^{2} \left (3 \underline {\hspace {1.25 ex}}\alpha ^{5}+3 \underline {\hspace {1.25 ex}}\alpha ^{4}+3 \underline {\hspace {1.25 ex}}\alpha ^{3}+7 \underline {\hspace {1.25 ex}}\alpha ^{2}+7 \underline {\hspace {1.25 ex}}\alpha +7\right ) \left (1-i \sqrt {3}\right ) \sqrt {\frac {x \left (-3+i \sqrt {3}\right )}{\left (-1+x \right ) \left (-1+i \sqrt {3}\right )}}\, \sqrt {\frac {i \sqrt {3}+2 x +1}{\left (-1+x \right ) \left (-1-i \sqrt {3}\right )}}\, \sqrt {\frac {1+2 x -i \sqrt {3}}{\left (-1+x \right ) \left (-1+i \sqrt {3}\right )}}\, \left (2 \EllipticF \left (\sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}, \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )-\left (-3 \underline {\hspace {1.25 ex}}\alpha ^{5}-4 \underline {\hspace {1.25 ex}}\alpha ^{2}\right ) \EllipticPi \left (\sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}, -\frac {i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{5}}{4}+\frac {3 \underline {\hspace {1.25 ex}}\alpha ^{5}}{4}-\frac {i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{2}}{3}+\underline {\hspace {1.25 ex}}\alpha ^{2}-\frac {i \sqrt {3}}{6}+\frac {1}{2}, \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )\right )}{\left (-3+i \sqrt {3}\right ) \sqrt {-x \left (-1+x \right ) \left (i \sqrt {3}+2 x +1\right ) \left (1+2 x -i \sqrt {3}\right )}}\right )}{108}\)	\(667\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*(1/2-1/2*I*3^(1/2))*((-3/2+1/2*I*3^(1/2))*x/(-1/2+1/2*I*3^(1/2))/(-1+x))^(1/2)*(-1+x)^2*((x+1/2+1/2*I*3^(
1/2))/(-1/2-1/2*I*3^(1/2))/(-1+x))^(1/2)*((x+1/2-1/2*I*3^(1/2))/(-1/2+1/2*I*3^(1/2))/(-1+x))^(1/2)/(-3/2+1/2*I
*3^(1/2))/(-x*(-1+x)*(x+1/2+1/2*I*3^(1/2))*(x+1/2-1/2*I*3^(1/2)))^(1/2)*(EllipticF(((-3/2+1/2*I*3^(1/2))*x/(-1
/2+1/2*I*3^(1/2))/(-1+x))^(1/2),((3/2+1/2*I*3^(1/2))*(1/2-1/2*I*3^(1/2))/(1/2+1/2*I*3^(1/2))/(3/2-1/2*I*3^(1/2
)))^(1/2))-EllipticPi(((-3/2+1/2*I*3^(1/2))*x/(-1/2+1/2*I*3^(1/2))/(-1+x))^(1/2),(-1/2+1/2*I*3^(1/2))/(-3/2+1/
2*I*3^(1/2)),((3/2+1/2*I*3^(1/2))*(1/2-1/2*I*3^(1/2))/(1/2+1/2*I*3^(1/2))/(3/2-1/2*I*3^(1/2)))^(1/2)))+1/108*4
^(1/2)*sum(_alpha^2*(_alpha^3-1)*(-1+x)^2*(3*_alpha^5+3*_alpha^4+3*_alpha^3+7*_alpha^2+7*_alpha+7)*(1-I*3^(1/2
))*(x/(-1+x)*(-3+I*3^(1/2))/(-1+I*3^(1/2)))^(1/2)*(1/(-1+x)*(I*3^(1/2)+2*x+1)/(-1-I*3^(1/2)))^(1/2)*(1/(-1+x)*
(1+2*x-I*3^(1/2))/(-1+I*3^(1/2)))^(1/2)/(-3+I*3^(1/2))/(-x*(-1+x)*(I*3^(1/2)+2*x+1)*(1+2*x-I*3^(1/2)))^(1/2)*(
2*EllipticF(((-3/2+1/2*I*3^(1/2))*x/(-1/2+1/2*I*3^(1/2))/(-1+x))^(1/2),((3/2+1/2*I*3^(1/2))*(1/2-1/2*I*3^(1/2)
)/(1/2+1/2*I*3^(1/2))/(3/2-1/2*I*3^(1/2)))^(1/2))-(-3*_alpha^5-4*_alpha^2)*EllipticPi(((-3/2+1/2*I*3^(1/2))*x/
(-1/2+1/2*I*3^(1/2))/(-1+x))^(1/2),-1/4*I*3^(1/2)*_alpha^5+3/4*_alpha^5-1/3*I*3^(1/2)*_alpha^2+_alpha^2-1/6*I*
3^(1/2)+1/2,((3/2+1/2*I*3^(1/2))*(1/2-1/2*I*3^(1/2))/(1/2+1/2*I*3^(1/2))/(3/2-1/2*I*3^(1/2)))^(1/2))),_alpha=R
ootOf(3*_Z^6+4*_Z^3+2))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {-x^{4} + x} {\left (x^{3} + 1\right )}}{3 \, x^{6} + 4 \, x^{3} + 2}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sqrt {x-x^4}\,\left (x^3+1\right )}{3\,x^6+4\,x^3+2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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