3.3.3 \(\int \frac {x^2 (-2+x^3) \sqrt {1+x^3}}{4+12 x^3+13 x^6+4 x^9} \, dx\)
Optimal. Leaf size=21 \[ -\frac {1}{3} \tan ^{-1}\left (\frac {x^3}{2 \left (x^3+1\right )^{3/2}}\right ) \]
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Rubi [A] time = 0.44, antiderivative size = 19, normalized size of antiderivative = 0.90,
number of steps used = 4, number of rules used = 3, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.081, Rules used =
{6715, 2094, 203} \begin {gather*} \frac {1}{3} \tan ^{-1}\left (\frac {2 \left (x^3+1\right )^{3/2}}{x^3}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(x^2*(-2 + x^3)*Sqrt[1 + x^3])/(4 + 12*x^3 + 13*x^6 + 4*x^9),x]
[Out]
ArcTan[(2*(1 + x^3)^(3/2))/x^3]/3
Rule 203
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])
Rule 2094
Int[((x_)^(m_.)*((A_) + (B_.)*(x_)^(n_.)))/((a_) + (b_.)*(x_)^(k_.) + (c_.)*(x_)^(n_.) + (d_.)*(x_)^(n2_)), x_
Symbol] :> Dist[(A^2*(m - n + 1))/(m + 1), Subst[Int[1/(a + A^2*b*(m - n + 1)^2*x^2), x], x, x^(m + 1)/(A*(m -
n + 1) + B*(m + 1)*x^n)], x] /; FreeQ[{a, b, c, d, A, B, m, n}, x] && EqQ[n2, 2*n] && EqQ[k, 2*(m + 1)] && Eq
Q[a*B^2*(m + 1)^2 - A^2*d*(m - n + 1)^2, 0] && EqQ[B*c*(m + 1) - 2*A*d*(m - n + 1), 0]
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 {x^2 \left (-2+x^3\right ) \sqrt {1+x^3}}{4+12 x^3+13 x^6+4 x^9} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {(-2+x) \sqrt {1+x}}{4+12 x+13 x^2+4 x^3} \, dx,x,x^3\right )\\ &=\frac {2}{3} \operatorname {Subst}\left (\int \frac {x^2 \left (-3+x^2\right )}{1-2 x^2+x^4+4 x^6} \, dx,x,\sqrt {1+x^3}\right )\\ &=2 \operatorname {Subst}\left (\int \frac {1}{1+36 x^2} \, dx,x,\frac {\left (1+x^3\right )^{3/2}}{3 x^3}\right )\\ &=\frac {1}{3} \tan ^{-1}\left (\frac {2 \left (1+x^3\right )^{3/2}}{x^3}\right )\\ \end {align*}
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Mathematica [A] time = 0.05, size = 27, normalized size = 1.29 \begin {gather*} \frac {1}{3} \tan ^{-1}\left (\frac {6 \left (x^3+1\right )^{3/2}}{3 \left (x^3+1\right )-3}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(x^2*(-2 + x^3)*Sqrt[1 + x^3])/(4 + 12*x^3 + 13*x^6 + 4*x^9),x]
[Out]
ArcTan[(6*(1 + x^3)^(3/2))/(-3 + 3*(1 + x^3))]/3
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IntegrateAlgebraic [A] time = 0.05, size = 37, normalized size = 1.76 \begin {gather*} \frac {1}{3} \tan ^{-1}\left (\sqrt {1+x^3}\right )-\frac {1}{3} \tan ^{-1}\left (\frac {1+2 x^3}{\sqrt {1+x^3}}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[(x^2*(-2 + x^3)*Sqrt[1 + x^3])/(4 + 12*x^3 + 13*x^6 + 4*x^9),x]
[Out]
ArcTan[Sqrt[1 + x^3]]/3 - ArcTan[(1 + 2*x^3)/Sqrt[1 + x^3]]/3
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fricas [A] time = 0.46, size = 15, normalized size = 0.71 \begin {gather*} \frac {1}{3} \, \arctan \left (\frac {2 \, {\left (x^{3} + 1\right )}^{\frac {3}{2}}}{x^{3}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^2*(x^3-2)*(x^3+1)^(1/2)/(4*x^9+13*x^6+12*x^3+4),x, algorithm="fricas")
[Out]
1/3*arctan(2*(x^3 + 1)^(3/2)/x^3)
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giac [B] time = 0.32, size = 67, normalized size = 3.19 \begin {gather*} -\frac {1}{3} \, \arctan \left (2 \, \sqrt {2} \left (\frac {1}{4}\right )^{\frac {3}{4}} {\left (\sqrt {14} \left (\frac {1}{4}\right )^{\frac {1}{4}} + 4 \, \sqrt {x^{3} + 1}\right )}\right ) - \frac {1}{3} \, \arctan \left (-2 \, \sqrt {2} \left (\frac {1}{4}\right )^{\frac {3}{4}} {\left (\sqrt {14} \left (\frac {1}{4}\right )^{\frac {1}{4}} - 4 \, \sqrt {x^{3} + 1}\right )}\right ) + \frac {1}{3} \, \arctan \left (\sqrt {x^{3} + 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^2*(x^3-2)*(x^3+1)^(1/2)/(4*x^9+13*x^6+12*x^3+4),x, algorithm="giac")
[Out]
-1/3*arctan(2*sqrt(2)*(1/4)^(3/4)*(sqrt(14)*(1/4)^(1/4) + 4*sqrt(x^3 + 1))) - 1/3*arctan(-2*sqrt(2)*(1/4)^(3/4
)*(sqrt(14)*(1/4)^(1/4) - 4*sqrt(x^3 + 1))) + 1/3*arctan(sqrt(x^3 + 1))
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maple [C] time = 0.66, size = 99, normalized size = 4.71
| | |
| method |
result |
size |
| | |
| trager |
\(\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {-4 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{9}+4 \sqrt {x^{3}+1}\, x^{6}-11 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{6}+4 \sqrt {x^{3}+1}\, x^{3}-12 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{3}-4 \RootOf \left (\textit {\_Z}^{2}+1\right )}{\left (x^{3}+2\right ) \left (4 x^{6}+5 x^{3}+2\right )}\right )}{6}\) |
\(99\) |
| default |
\(\frac {\sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (4 \textit {\_Z}^{6}+5 \textit {\_Z}^{3}+2\right )}{\sum }\frac {\left (2 \underline {\hspace {1.25 ex}}\alpha ^{3}+1\right ) \left (4 \underline {\hspace {1.25 ex}}\alpha ^{5}-4 \underline {\hspace {1.25 ex}}\alpha ^{4}+4 \underline {\hspace {1.25 ex}}\alpha ^{3}+\underline {\hspace {1.25 ex}}\alpha ^{2}-\underline {\hspace {1.25 ex}}\alpha +1\right ) \left (3-i \sqrt {3}\right ) \sqrt {\frac {1+x}{3-i \sqrt {3}}}\, \sqrt {\frac {-1+2 x -i \sqrt {3}}{-3-i \sqrt {3}}}\, \sqrt {\frac {-1+2 x +i \sqrt {3}}{-3+i \sqrt {3}}}\, \EllipticPi \left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, -6 \underline {\hspace {1.25 ex}}\alpha ^{5}+6 \underline {\hspace {1.25 ex}}\alpha ^{4}-6 \underline {\hspace {1.25 ex}}\alpha ^{3}-\frac {3 \underline {\hspace {1.25 ex}}\alpha ^{2}}{2}+\frac {3 \underline {\hspace {1.25 ex}}\alpha }{2}-\frac {3}{2}+2 i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{5}-2 i \underline {\hspace {1.25 ex}}\alpha ^{4} \sqrt {3}+2 i \underline {\hspace {1.25 ex}}\alpha ^{3} \sqrt {3}+\frac {i \underline {\hspace {1.25 ex}}\alpha ^{2} \sqrt {3}}{2}-\frac {i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha }{2}+\frac {i \sqrt {3}}{2}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}+1}}\right )}{6}+\frac {\sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{3}+2\right )}{\sum }\frac {\left (\underline {\hspace {1.25 ex}}\alpha ^{2}-\underline {\hspace {1.25 ex}}\alpha +1\right ) \left (3-i \sqrt {3}\right ) \sqrt {\frac {1+x}{3-i \sqrt {3}}}\, \sqrt {\frac {-1+2 x -i \sqrt {3}}{-3-i \sqrt {3}}}\, \sqrt {\frac {-1+2 x +i \sqrt {3}}{-3+i \sqrt {3}}}\, \EllipticPi \left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, -\frac {3 \underline {\hspace {1.25 ex}}\alpha ^{2}}{2}+\frac {3 \underline {\hspace {1.25 ex}}\alpha }{2}-\frac {3}{2}+\frac {i \underline {\hspace {1.25 ex}}\alpha ^{2} \sqrt {3}}{2}-\frac {i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha }{2}+\frac {i \sqrt {3}}{2}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}+1}}\right )}{6}\) |
\(417\) |
| elliptic |
\(\text {Expression too large to display}\) |
\(2469\) |
| | |
|
|
|
|
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^2*(x^3-2)*(x^3+1)^(1/2)/(4*x^9+13*x^6+12*x^3+4),x,method=_RETURNVERBOSE)
[Out]
1/6*RootOf(_Z^2+1)*ln(-(-4*RootOf(_Z^2+1)*x^9+4*(x^3+1)^(1/2)*x^6-11*RootOf(_Z^2+1)*x^6+4*(x^3+1)^(1/2)*x^3-12
*RootOf(_Z^2+1)*x^3-4*RootOf(_Z^2+1))/(x^3+2)/(4*x^6+5*x^3+2))
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x^{3} + 1} {\left (x^{3} - 2\right )} x^{2}}{4 \, x^{9} + 13 \, x^{6} + 12 \, x^{3} + 4}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^2*(x^3-2)*(x^3+1)^(1/2)/(4*x^9+13*x^6+12*x^3+4),x, algorithm="maxima")
[Out]
integrate(sqrt(x^3 + 1)*(x^3 - 2)*x^2/(4*x^9 + 13*x^6 + 12*x^3 + 4), x)
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mupad [B] time = 0.10, size = 2737, normalized size = 130.33
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((x^2*(x^3 + 1)^(1/2)*(x^3 - 2))/(12*x^3 + 13*x^6 + 4*x^9 + 4),x)
[Out]
(2*((3^(1/2)*1i)/2 + 3/2)*((x + (3^(1/2)*1i)/2 - 1/2)/((3^(1/2)*1i)/2 - 3/2))^(1/2)*((x + 1)/((3^(1/2)*1i)/2 +
3/2))^(1/2)*(((3^(1/2)*1i)/2 - x + 1/2)/((3^(1/2)*1i)/2 + 3/2))^(1/2)*ellipticPi(((3^(1/2)*1i)/2 + 3/2)/((- (
7^(1/2)*1i)/8 - 5/8)^(1/3) + 1), asin(((x + 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)), -((3^(1/2)*1i)/2 + 3/2)/((3^(1/
2)*1i)/2 - 3/2))*(2*(- (7^(1/2)*1i)/8 - 5/8)^(2/3) + (- (7^(1/2)*1i)/8 - 5/8)^(5/3) - (- (7^(1/2)*1i)/8 - 5/8)
^(8/3)))/(((- (7^(1/2)*1i)/8 - 5/8)^(1/3) + 1)*(x^3 - x*(((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2) + 1) -
((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2))^(1/2)*(36*(- (7^(1/2)*1i)/8 - 5/8)^(2/3) + 78*(- (7^(1/2)*1i)/8
- 5/8)^(5/3) + 36*(- (7^(1/2)*1i)/8 - 5/8)^(8/3))) - (((3^(1/2)*1i)/2 + 3/2)*((x + (3^(1/2)*1i)/2 - 1/2)/((3^
(1/2)*1i)/2 - 3/2))^(1/2)*((x + 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)*(((3^(1/2)*1i)/2 - x + 1/2)/((3^(1/2)*1i)/2 +
3/2))^(1/2)*ellipticPi(((3^(1/2)*1i)/2 + 3/2)/((-1)^(1/3)*2^(1/3) + 1), asin(((x + 1)/((3^(1/2)*1i)/2 + 3/2))
^(1/2)), -((3^(1/2)*1i)/2 + 3/2)/((3^(1/2)*1i)/2 - 3/2)))/(3*((-1)^(1/3)*2^(1/3) + 1)*(x^3 - x*(((3^(1/2)*1i)/
2 - 1/2)*((3^(1/2)*1i)/2 + 1/2) + 1) - ((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2))^(1/2)) + (2*((3^(1/2)*1i
)/2 + 3/2)*((x + (3^(1/2)*1i)/2 - 1/2)/((3^(1/2)*1i)/2 - 3/2))^(1/2)*((x + 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)*((
(3^(1/2)*1i)/2 - x + 1/2)/((3^(1/2)*1i)/2 + 3/2))^(1/2)*ellipticPi(((3^(1/2)*1i)/2 + 3/2)/(((7^(1/2)*1i)/8 - 5
/8)^(1/3) + 1), asin(((x + 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)), -((3^(1/2)*1i)/2 + 3/2)/((3^(1/2)*1i)/2 - 3/2))*
(2*((7^(1/2)*1i)/8 - 5/8)^(2/3) + ((7^(1/2)*1i)/8 - 5/8)^(5/3) - ((7^(1/2)*1i)/8 - 5/8)^(8/3)))/((((7^(1/2)*1i
)/8 - 5/8)^(1/3) + 1)*(x^3 - x*(((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2) + 1) - ((3^(1/2)*1i)/2 - 1/2)*((
3^(1/2)*1i)/2 + 1/2))^(1/2)*(36*((7^(1/2)*1i)/8 - 5/8)^(2/3) + 78*((7^(1/2)*1i)/8 - 5/8)^(5/3) + 36*((7^(1/2)*
1i)/8 - 5/8)^(8/3))) + (2*((3^(1/2)*1i)/2 + 3/2)*((x + (3^(1/2)*1i)/2 - 1/2)/((3^(1/2)*1i)/2 - 3/2))^(1/2)*((x
+ 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)*(((3^(1/2)*1i)/2 - x + 1/2)/((3^(1/2)*1i)/2 + 3/2))^(1/2)*ellipticPi(((3^(
1/2)*1i)/2 + 3/2)/(((3^(1/2)*1i)/2 - 1/2)*(- (7^(1/2)*1i)/8 - 5/8)^(1/3) + 1), asin(((x + 1)/((3^(1/2)*1i)/2 +
3/2))^(1/2)), -((3^(1/2)*1i)/2 + 3/2)/((3^(1/2)*1i)/2 - 3/2))*(2*((3^(1/2)*1i)/2 - 1/2)^2*(- (7^(1/2)*1i)/8 -
5/8)^(2/3) + ((3^(1/2)*1i)/2 - 1/2)^5*(- (7^(1/2)*1i)/8 - 5/8)^(5/3) - ((3^(1/2)*1i)/2 - 1/2)^8*(- (7^(1/2)*1
i)/8 - 5/8)^(8/3)))/((((3^(1/2)*1i)/2 - 1/2)*(- (7^(1/2)*1i)/8 - 5/8)^(1/3) + 1)*(x^3 - x*(((3^(1/2)*1i)/2 - 1
/2)*((3^(1/2)*1i)/2 + 1/2) + 1) - ((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2))^(1/2)*(36*((3^(1/2)*1i)/2 - 1
/2)^2*(- (7^(1/2)*1i)/8 - 5/8)^(2/3) + 78*((3^(1/2)*1i)/2 - 1/2)^5*(- (7^(1/2)*1i)/8 - 5/8)^(5/3) + 36*((3^(1/
2)*1i)/2 - 1/2)^8*(- (7^(1/2)*1i)/8 - 5/8)^(8/3))) + (2*((3^(1/2)*1i)/2 + 3/2)*((x + (3^(1/2)*1i)/2 - 1/2)/((3
^(1/2)*1i)/2 - 3/2))^(1/2)*((x + 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)*(((3^(1/2)*1i)/2 - x + 1/2)/((3^(1/2)*1i)/2
+ 3/2))^(1/2)*ellipticPi(((3^(1/2)*1i)/2 + 3/2)/(((3^(1/2)*1i)/2 - 1/2)*((7^(1/2)*1i)/8 - 5/8)^(1/3) + 1), asi
n(((x + 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)), -((3^(1/2)*1i)/2 + 3/2)/((3^(1/2)*1i)/2 - 3/2))*(2*((3^(1/2)*1i)/2
- 1/2)^2*((7^(1/2)*1i)/8 - 5/8)^(2/3) + ((3^(1/2)*1i)/2 - 1/2)^5*((7^(1/2)*1i)/8 - 5/8)^(5/3) - ((3^(1/2)*1i)/
2 - 1/2)^8*((7^(1/2)*1i)/8 - 5/8)^(8/3)))/((((3^(1/2)*1i)/2 - 1/2)*((7^(1/2)*1i)/8 - 5/8)^(1/3) + 1)*(x^3 - x*
(((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2) + 1) - ((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2))^(1/2)*(36
*((3^(1/2)*1i)/2 - 1/2)^2*((7^(1/2)*1i)/8 - 5/8)^(2/3) + 78*((3^(1/2)*1i)/2 - 1/2)^5*((7^(1/2)*1i)/8 - 5/8)^(5
/3) + 36*((3^(1/2)*1i)/2 - 1/2)^8*((7^(1/2)*1i)/8 - 5/8)^(8/3))) + (2*((3^(1/2)*1i)/2 + 3/2)*((x + (3^(1/2)*1i
)/2 - 1/2)/((3^(1/2)*1i)/2 - 3/2))^(1/2)*((x + 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)*(((3^(1/2)*1i)/2 - x + 1/2)/((
3^(1/2)*1i)/2 + 3/2))^(1/2)*ellipticPi(-((3^(1/2)*1i)/2 + 3/2)/(((3^(1/2)*1i)/2 + 1/2)*(- (7^(1/2)*1i)/8 - 5/8
)^(1/3) - 1), asin(((x + 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)), -((3^(1/2)*1i)/2 + 3/2)/((3^(1/2)*1i)/2 - 3/2))*((
(3^(1/2)*1i)/2 + 1/2)^5*(- (7^(1/2)*1i)/8 - 5/8)^(5/3) - 2*((3^(1/2)*1i)/2 + 1/2)^2*(- (7^(1/2)*1i)/8 - 5/8)^(
2/3) + ((3^(1/2)*1i)/2 + 1/2)^8*(- (7^(1/2)*1i)/8 - 5/8)^(8/3)))/((((3^(1/2)*1i)/2 + 1/2)*(- (7^(1/2)*1i)/8 -
5/8)^(1/3) - 1)*(x^3 - x*(((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2) + 1) - ((3^(1/2)*1i)/2 - 1/2)*((3^(1/2
)*1i)/2 + 1/2))^(1/2)*(36*((3^(1/2)*1i)/2 + 1/2)^2*(- (7^(1/2)*1i)/8 - 5/8)^(2/3) - 78*((3^(1/2)*1i)/2 + 1/2)^
5*(- (7^(1/2)*1i)/8 - 5/8)^(5/3) + 36*((3^(1/2)*1i)/2 + 1/2)^8*(- (7^(1/2)*1i)/8 - 5/8)^(8/3))) + (2*((3^(1/2)
*1i)/2 + 3/2)*((x + (3^(1/2)*1i)/2 - 1/2)/((3^(1/2)*1i)/2 - 3/2))^(1/2)*((x + 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)
*(((3^(1/2)*1i)/2 - x + 1/2)/((3^(1/2)*1i)/2 + 3/2))^(1/2)*ellipticPi(-((3^(1/2)*1i)/2 + 3/2)/(((3^(1/2)*1i)/2
+ 1/2)*((7^(1/2)*1i)/8 - 5/8)^(1/3) - 1), asin(((x + 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)), -((3^(1/2)*1i)/2 + 3/
2)/((3^(1/2)*1i)/2 - 3/2))*(((3^(1/2)*1i)/2 + 1/2)^5*((7^(1/2)*1i)/8 - 5/8)^(5/3) - 2*((3^(1/2)*1i)/2 + 1/2)^2
*((7^(1/2)*1i)/8 - 5/8)^(2/3) + ((3^(1/2)*1i)/2 + 1/2)^8*((7^(1/2)*1i)/8 - 5/8)^(8/3)))/((((3^(1/2)*1i)/2 + 1/
2)*((7^(1/2)*1i)/8 - 5/8)^(1/3) - 1)*(x^3 - x*(((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2) + 1) - ((3^(1/2)*
1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2))^(1/2)*(36*((3^(1/2)*1i)/2 + 1/2)^2*((7^(1/2)*1i)/8 - 5/8)^(2/3) - 78*((3^
(1/2)*1i)/2 + 1/2)^5*((7^(1/2)*1i)/8 - 5/8)^(5/3) + 36*((3^(1/2)*1i)/2 + 1/2)^8*((7^(1/2)*1i)/8 - 5/8)^(8/3)))
- (2*((3^(1/2)*1i)/2 + 3/2)*((x + (3^(1/2)*1i)/2 - 1/2)/((3^(1/2)*1i)/2 - 3/2))^(1/2)*((x + 1)/((3^(1/2)*1i)/
2 + 3/2))^(1/2)*(((3^(1/2)*1i)/2 - x + 1/2)/((3^(1/2)*1i)/2 + 3/2))^(1/2)*(2*(-1)^(2/3)*2^(2/3)*((3^(1/2)*1i)/
2 - 1/2)^5 - 2*(-1)^(2/3)*2^(2/3)*((3^(1/2)*1i)/2 - 1/2)^2 + 4*(-1)^(2/3)*2^(2/3)*((3^(1/2)*1i)/2 - 1/2)^8)*el
lipticPi(((3^(1/2)*1i)/2 + 3/2)/((-1)^(1/3)*2^(1/3)*((3^(1/2)*1i)/2 - 1/2) + 1), asin(((x + 1)/((3^(1/2)*1i)/2
+ 3/2))^(1/2)), -((3^(1/2)*1i)/2 + 3/2)/((3^(1/2)*1i)/2 - 3/2)))/(((-1)^(1/3)*2^(1/3)*((3^(1/2)*1i)/2 - 1/2)
+ 1)*(x^3 - x*(((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2) + 1) - ((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1
/2))^(1/2)*(36*(-1)^(2/3)*2^(2/3)*((3^(1/2)*1i)/2 - 1/2)^2 - 156*(-1)^(2/3)*2^(2/3)*((3^(1/2)*1i)/2 - 1/2)^5 +
144*(-1)^(2/3)*2^(2/3)*((3^(1/2)*1i)/2 - 1/2)^8)) - (2*((3^(1/2)*1i)/2 + 3/2)*((x + (3^(1/2)*1i)/2 - 1/2)/((3
^(1/2)*1i)/2 - 3/2))^(1/2)*((x + 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)*(((3^(1/2)*1i)/2 - x + 1/2)/((3^(1/2)*1i)/2
+ 3/2))^(1/2)*(2*(-1)^(2/3)*2^(2/3)*((3^(1/2)*1i)/2 + 1/2)^2 + 2*(-1)^(2/3)*2^(2/3)*((3^(1/2)*1i)/2 + 1/2)^5 -
4*(-1)^(2/3)*2^(2/3)*((3^(1/2)*1i)/2 + 1/2)^8)*ellipticPi(-((3^(1/2)*1i)/2 + 3/2)/((-1)^(1/3)*2^(1/3)*((3^(1/
2)*1i)/2 + 1/2) - 1), asin(((x + 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)), -((3^(1/2)*1i)/2 + 3/2)/((3^(1/2)*1i)/2 -
3/2)))/(((-1)^(1/3)*2^(1/3)*((3^(1/2)*1i)/2 + 1/2) - 1)*(x^3 - x*(((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2
) + 1) - ((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2))^(1/2)*(36*(-1)^(2/3)*2^(2/3)*((3^(1/2)*1i)/2 + 1/2)^2
+ 156*(-1)^(2/3)*2^(2/3)*((3^(1/2)*1i)/2 + 1/2)^5 + 144*(-1)^(2/3)*2^(2/3)*((3^(1/2)*1i)/2 + 1/2)^8))
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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**2*(x**3-2)*(x**3+1)**(1/2)/(4*x**9+13*x**6+12*x**3+4),x)
[Out]
Timed out
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