3.13.91 \(\int \frac {(2+x^3) \sqrt {-1-x^2+x^3}}{1-x^2-2 x^3-x^4+x^5+x^6} \, dx\)
Optimal. Leaf size=93 \[ \frac {1}{5} \left (-5-\sqrt {5}\right ) \tan ^{-1}\left (\frac {\sqrt {\frac {3}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt {x^3-x^2-1}}\right )+\frac {1}{5} \left (5-\sqrt {5}\right ) \tan ^{-1}\left (\frac {\sqrt {\frac {2}{3+\sqrt {5}}} x}{\sqrt {x^3-x^2-1}}\right ) \]
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Rubi [F] time = 0.67, antiderivative size = 0, normalized size of antiderivative = 0.00,
number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used =
{} \begin {gather*} \int \frac {\left (2+x^3\right ) \sqrt {-1-x^2+x^3}}{1-x^2-2 x^3-x^4+x^5+x^6} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
Int[((2 + x^3)*Sqrt[-1 - x^2 + x^3])/(1 - x^2 - 2*x^3 - x^4 + x^5 + x^6),x]
[Out]
2*Defer[Int][Sqrt[-1 - x^2 + x^3]/(1 - x^2 - 2*x^3 - x^4 + x^5 + x^6), x] + Defer[Int][(x^3*Sqrt[-1 - x^2 + x^
3])/(1 - x^2 - 2*x^3 - x^4 + x^5 + x^6), x]
Rubi steps
\begin {align*} \int \frac {\left (2+x^3\right ) \sqrt {-1-x^2+x^3}}{1-x^2-2 x^3-x^4+x^5+x^6} \, dx &=\int \left (\frac {2 \sqrt {-1-x^2+x^3}}{1-x^2-2 x^3-x^4+x^5+x^6}+\frac {x^3 \sqrt {-1-x^2+x^3}}{1-x^2-2 x^3-x^4+x^5+x^6}\right ) \, dx\\ &=2 \int \frac {\sqrt {-1-x^2+x^3}}{1-x^2-2 x^3-x^4+x^5+x^6} \, dx+\int \frac {x^3 \sqrt {-1-x^2+x^3}}{1-x^2-2 x^3-x^4+x^5+x^6} \, dx\\ \end {align*}
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Mathematica [C] time = 5.18, size = 25746, normalized size = 276.84 \begin {gather*} \text {Result too large to show} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
Integrate[((2 + x^3)*Sqrt[-1 - x^2 + x^3])/(1 - x^2 - 2*x^3 - x^4 + x^5 + x^6),x]
[Out]
Result too large to show
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IntegrateAlgebraic [A] time = 0.50, size = 81, normalized size = 0.87 \begin {gather*} \frac {1}{5} \left (5-\sqrt {5}\right ) \tan ^{-1}\left (\frac {\left (-1+\sqrt {5}\right ) x}{2 \sqrt {-1-x^2+x^3}}\right )+\frac {1}{5} \left (-5-\sqrt {5}\right ) \tan ^{-1}\left (\frac {\left (1+\sqrt {5}\right ) x}{2 \sqrt {-1-x^2+x^3}}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[((2 + x^3)*Sqrt[-1 - x^2 + x^3])/(1 - x^2 - 2*x^3 - x^4 + x^5 + x^6),x]
[Out]
((5 - Sqrt[5])*ArcTan[((-1 + Sqrt[5])*x)/(2*Sqrt[-1 - x^2 + x^3])])/5 + ((-5 - Sqrt[5])*ArcTan[((1 + Sqrt[5])*
x)/(2*Sqrt[-1 - x^2 + x^3])])/5
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fricas [A] time = 0.52, size = 79, normalized size = 0.85 \begin {gather*} -\frac {1}{5} \, \sqrt {5} \sqrt {2 \, \sqrt {5} + 6} \arctan \left (\frac {x \sqrt {2 \, \sqrt {5} + 6}}{2 \, \sqrt {x^{3} - x^{2} - 1}}\right ) + \frac {1}{5} \, \sqrt {5} \sqrt {-2 \, \sqrt {5} + 6} \arctan \left (\frac {x \sqrt {-2 \, \sqrt {5} + 6}}{2 \, \sqrt {x^{3} - x^{2} - 1}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((x^3+2)*(x^3-x^2-1)^(1/2)/(x^6+x^5-x^4-2*x^3-x^2+1),x, algorithm="fricas")
[Out]
-1/5*sqrt(5)*sqrt(2*sqrt(5) + 6)*arctan(1/2*x*sqrt(2*sqrt(5) + 6)/sqrt(x^3 - x^2 - 1)) + 1/5*sqrt(5)*sqrt(-2*s
qrt(5) + 6)*arctan(1/2*x*sqrt(-2*sqrt(5) + 6)/sqrt(x^3 - x^2 - 1))
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x^{3} - x^{2} - 1} {\left (x^{3} + 2\right )}}{x^{6} + x^{5} - x^{4} - 2 \, x^{3} - x^{2} + 1}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((x^3+2)*(x^3-x^2-1)^(1/2)/(x^6+x^5-x^4-2*x^3-x^2+1),x, algorithm="giac")
[Out]
integrate(sqrt(x^3 - x^2 - 1)*(x^3 + 2)/(x^6 + x^5 - x^4 - 2*x^3 - x^2 + 1), x)
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maple [C] time = 5.05, size = 674, normalized size = 7.25
| | |
| method |
result |
size |
| | |
| trager |
\(-5 \ln \left (-\frac {175 \RootOf \left (25 \textit {\_Z}^{4}+15 \textit {\_Z}^{2}+1\right )^{5} x^{2}-35 \RootOf \left (25 \textit {\_Z}^{4}+15 \textit {\_Z}^{2}+1\right )^{3} x^{3}+230 \RootOf \left (25 \textit {\_Z}^{4}+15 \textit {\_Z}^{2}+1\right )^{3} x^{2}-30 \sqrt {x^{3}-x^{2}-1}\, \RootOf \left (25 \textit {\_Z}^{4}+15 \textit {\_Z}^{2}+1\right )^{2} x -18 \RootOf \left (25 \textit {\_Z}^{4}+15 \textit {\_Z}^{2}+1\right ) x^{3}+35 \RootOf \left (25 \textit {\_Z}^{4}+15 \textit {\_Z}^{2}+1\right )^{3}+72 \RootOf \left (25 \textit {\_Z}^{4}+15 \textit {\_Z}^{2}+1\right ) x^{2}-16 \sqrt {x^{3}-x^{2}-1}\, x +18 \RootOf \left (25 \textit {\_Z}^{4}+15 \textit {\_Z}^{2}+1\right )}{5 \RootOf \left (25 \textit {\_Z}^{4}+15 \textit {\_Z}^{2}+1\right )^{2} x^{2}+x^{3}+2 x^{2}-1}\right ) \RootOf \left (25 \textit {\_Z}^{4}+15 \textit {\_Z}^{2}+1\right )^{3}+\RootOf \left (25 \textit {\_Z}^{4}+15 \textit {\_Z}^{2}+1\right ) \ln \left (\frac {75 \RootOf \left (25 \textit {\_Z}^{4}+15 \textit {\_Z}^{2}+1\right )^{5} x^{2}+15 \RootOf \left (25 \textit {\_Z}^{4}+15 \textit {\_Z}^{2}+1\right )^{3} x^{3}-5 \RootOf \left (25 \textit {\_Z}^{4}+15 \textit {\_Z}^{2}+1\right )^{3} x^{2}-30 \sqrt {x^{3}-x^{2}-1}\, \RootOf \left (25 \textit {\_Z}^{4}+15 \textit {\_Z}^{2}+1\right )^{2} x +2 \RootOf \left (25 \textit {\_Z}^{4}+15 \textit {\_Z}^{2}+1\right ) x^{3}-15 \RootOf \left (25 \textit {\_Z}^{4}+15 \textit {\_Z}^{2}+1\right )^{3}-2 \RootOf \left (25 \textit {\_Z}^{4}+15 \textit {\_Z}^{2}+1\right ) x^{2}-2 \sqrt {x^{3}-x^{2}-1}\, x -2 \RootOf \left (25 \textit {\_Z}^{4}+15 \textit {\_Z}^{2}+1\right )}{5 \RootOf \left (25 \textit {\_Z}^{4}+15 \textit {\_Z}^{2}+1\right )^{2} x^{2}-x^{3}+x^{2}+1}\right )-3 \ln \left (-\frac {175 \RootOf \left (25 \textit {\_Z}^{4}+15 \textit {\_Z}^{2}+1\right )^{5} x^{2}-35 \RootOf \left (25 \textit {\_Z}^{4}+15 \textit {\_Z}^{2}+1\right )^{3} x^{3}+230 \RootOf \left (25 \textit {\_Z}^{4}+15 \textit {\_Z}^{2}+1\right )^{3} x^{2}-30 \sqrt {x^{3}-x^{2}-1}\, \RootOf \left (25 \textit {\_Z}^{4}+15 \textit {\_Z}^{2}+1\right )^{2} x -18 \RootOf \left (25 \textit {\_Z}^{4}+15 \textit {\_Z}^{2}+1\right ) x^{3}+35 \RootOf \left (25 \textit {\_Z}^{4}+15 \textit {\_Z}^{2}+1\right )^{3}+72 \RootOf \left (25 \textit {\_Z}^{4}+15 \textit {\_Z}^{2}+1\right ) x^{2}-16 \sqrt {x^{3}-x^{2}-1}\, x +18 \RootOf \left (25 \textit {\_Z}^{4}+15 \textit {\_Z}^{2}+1\right )}{5 \RootOf \left (25 \textit {\_Z}^{4}+15 \textit {\_Z}^{2}+1\right )^{2} x^{2}+x^{3}+2 x^{2}-1}\right ) \RootOf \left (25 \textit {\_Z}^{4}+15 \textit {\_Z}^{2}+1\right )\) |
\(674\) |
| default |
\(\text {Expression too large to display}\) |
\(2877\) |
| elliptic |
\(\text {Expression too large to display}\) |
\(2877\) |
| | |
|
|
|
|
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((x^3+2)*(x^3-x^2-1)^(1/2)/(x^6+x^5-x^4-2*x^3-x^2+1),x,method=_RETURNVERBOSE)
[Out]
-5*ln(-(175*RootOf(25*_Z^4+15*_Z^2+1)^5*x^2-35*RootOf(25*_Z^4+15*_Z^2+1)^3*x^3+230*RootOf(25*_Z^4+15*_Z^2+1)^3
*x^2-30*(x^3-x^2-1)^(1/2)*RootOf(25*_Z^4+15*_Z^2+1)^2*x-18*RootOf(25*_Z^4+15*_Z^2+1)*x^3+35*RootOf(25*_Z^4+15*
_Z^2+1)^3+72*RootOf(25*_Z^4+15*_Z^2+1)*x^2-16*(x^3-x^2-1)^(1/2)*x+18*RootOf(25*_Z^4+15*_Z^2+1))/(5*RootOf(25*_
Z^4+15*_Z^2+1)^2*x^2+x^3+2*x^2-1))*RootOf(25*_Z^4+15*_Z^2+1)^3+RootOf(25*_Z^4+15*_Z^2+1)*ln((75*RootOf(25*_Z^4
+15*_Z^2+1)^5*x^2+15*RootOf(25*_Z^4+15*_Z^2+1)^3*x^3-5*RootOf(25*_Z^4+15*_Z^2+1)^3*x^2-30*(x^3-x^2-1)^(1/2)*Ro
otOf(25*_Z^4+15*_Z^2+1)^2*x+2*RootOf(25*_Z^4+15*_Z^2+1)*x^3-15*RootOf(25*_Z^4+15*_Z^2+1)^3-2*RootOf(25*_Z^4+15
*_Z^2+1)*x^2-2*(x^3-x^2-1)^(1/2)*x-2*RootOf(25*_Z^4+15*_Z^2+1))/(5*RootOf(25*_Z^4+15*_Z^2+1)^2*x^2-x^3+x^2+1))
-3*ln(-(175*RootOf(25*_Z^4+15*_Z^2+1)^5*x^2-35*RootOf(25*_Z^4+15*_Z^2+1)^3*x^3+230*RootOf(25*_Z^4+15*_Z^2+1)^3
*x^2-30*(x^3-x^2-1)^(1/2)*RootOf(25*_Z^4+15*_Z^2+1)^2*x-18*RootOf(25*_Z^4+15*_Z^2+1)*x^3+35*RootOf(25*_Z^4+15*
_Z^2+1)^3+72*RootOf(25*_Z^4+15*_Z^2+1)*x^2-16*(x^3-x^2-1)^(1/2)*x+18*RootOf(25*_Z^4+15*_Z^2+1))/(5*RootOf(25*_
Z^4+15*_Z^2+1)^2*x^2+x^3+2*x^2-1))*RootOf(25*_Z^4+15*_Z^2+1)
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x^{3} - x^{2} - 1} {\left (x^{3} + 2\right )}}{x^{6} + x^{5} - x^{4} - 2 \, x^{3} - x^{2} + 1}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((x^3+2)*(x^3-x^2-1)^(1/2)/(x^6+x^5-x^4-2*x^3-x^2+1),x, algorithm="maxima")
[Out]
integrate(sqrt(x^3 - x^2 - 1)*(x^3 + 2)/(x^6 + x^5 - x^4 - 2*x^3 - x^2 + 1), x)
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mupad [B] time = 3.20, size = 2803, normalized size = 30.14
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(-((x^3 + 2)*(x^3 - x^2 - 1)^(1/2))/(x^2 + 2*x^3 + x^4 - x^5 - x^6 - 1),x)
[Out]
symsum(-(2*((x - (3^(1/2)*(1/(9*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - ((31^(1/2)*108^(1/2))/108 + 29/54)
^(1/3))*1i)/2 + 1/(18*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) + ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)/2 -
1/3)/(1/(6*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - (3^(1/2)*(1/(9*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3
)) - ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3))*1i)/2 + (3*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3))/2))^(1/2)*
((1/(9*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - x + ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3) + 1/3)/(1/(6*(
(31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - (3^(1/2)*(1/(9*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - ((31^(1/
2)*108^(1/2))/108 + 29/54)^(1/3))*1i)/2 + (3*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3))/2))^(1/2)*ellipticPi((1
/(6*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - (3^(1/2)*(1/(9*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - ((3
1^(1/2)*108^(1/2))/108 + 29/54)^(1/3))*1i)/2 + (3*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3))/2)/(root(z^6 + z^5
- z^4 - 2*z^3 - z^2 + 1, z, k) - (3^(1/2)*(1/(9*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - ((31^(1/2)*108^(1
/2))/108 + 29/54)^(1/3))*1i)/2 + 1/(18*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) + ((31^(1/2)*108^(1/2))/108 +
29/54)^(1/3)/2 - 1/3), asin(((x - (3^(1/2)*(1/(9*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - ((31^(1/2)*108^(
1/2))/108 + 29/54)^(1/3))*1i)/2 + 1/(18*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) + ((31^(1/2)*108^(1/2))/108
+ 29/54)^(1/3)/2 - 1/3)/(1/(6*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - (3^(1/2)*(1/(9*((31^(1/2)*108^(1/2))
/108 + 29/54)^(1/3)) - ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3))*1i)/2 + (3*((31^(1/2)*108^(1/2))/108 + 29/54)
^(1/3))/2))^(1/2)), (3^(1/2)*(1/(6*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - (3^(1/2)*(1/(9*((31^(1/2)*108^(
1/2))/108 + 29/54)^(1/3)) - ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3))*1i)/2 + (3*((31^(1/2)*108^(1/2))/108 + 2
9/54)^(1/3))/2)*1i)/(3*(1/(9*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - ((31^(1/2)*108^(1/2))/108 + 29/54)^(1
/3))))*(1/(6*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - (3^(1/2)*(1/(9*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/
3)) - ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3))*1i)/2 + (3*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3))/2)*(-(3^(
1/2)*(x + (3^(1/2)*(1/(9*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3))
*1i)/2 + 1/(18*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) + ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)/2 - 1/3)*1
i)/(3*(1/(9*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3))))^(1/2)*(roo
t(z^6 + z^5 - z^4 - 2*z^3 - z^2 + 1, z, k)^2 - 3*root(z^6 + z^5 - z^4 - 2*z^3 - z^2 + 1, z, k)^3 - root(z^6 +
z^5 - z^4 - 2*z^3 - z^2 + 1, z, k)^4 + 2*root(z^6 + z^5 - z^4 - 2*z^3 - z^2 + 1, z, k)^5 + 3))/((x^3 - x^2 - x
*(((3^(1/2)*(1/(9*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3))*1i)/2
- 1/(18*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)/2 + 1/3)*((3^(1/2
)*(1/(9*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3))*1i)/2 + 1/(18*((
31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) + ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)/2 - 1/3) - (1/(9*((31^(1/2)
*108^(1/2))/108 + 29/54)^(1/3)) + ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3) + 1/3)*((3^(1/2)*(1/(9*((31^(1/2)*1
08^(1/2))/108 + 29/54)^(1/3)) - ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3))*1i)/2 - 1/(18*((31^(1/2)*108^(1/2))/
108 + 29/54)^(1/3)) - ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)/2 + 1/3) + (1/(9*((31^(1/2)*108^(1/2))/108 + 29
/54)^(1/3)) + ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3) + 1/3)*((3^(1/2)*(1/(9*((31^(1/2)*108^(1/2))/108 + 29/5
4)^(1/3)) - ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3))*1i)/2 + 1/(18*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3))
+ ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)/2 - 1/3)) + (1/(9*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) + ((31^
(1/2)*108^(1/2))/108 + 29/54)^(1/3) + 1/3)*((3^(1/2)*(1/(9*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - ((31^(1
/2)*108^(1/2))/108 + 29/54)^(1/3))*1i)/2 - 1/(18*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - ((31^(1/2)*108^(1
/2))/108 + 29/54)^(1/3)/2 + 1/3)*((3^(1/2)*(1/(9*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - ((31^(1/2)*108^(1
/2))/108 + 29/54)^(1/3))*1i)/2 + 1/(18*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) + ((31^(1/2)*108^(1/2))/108 +
29/54)^(1/3)/2 - 1/3))^(1/2)*(root(z^6 + z^5 - z^4 - 2*z^3 - z^2 + 1, z, k) - (3^(1/2)*(1/(9*((31^(1/2)*108^(
1/2))/108 + 29/54)^(1/3)) - ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3))*1i)/2 + 1/(18*((31^(1/2)*108^(1/2))/108
+ 29/54)^(1/3)) + ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)/2 - 1/3)*(6*root(z^6 + z^5 - z^4 - 2*z^3 - z^2 + 1,
z, k)^2 + 4*root(z^6 + z^5 - z^4 - 2*z^3 - z^2 + 1, z, k)^3 - 5*root(z^6 + z^5 - z^4 - 2*z^3 - z^2 + 1, z, k)
^4 - 6*root(z^6 + z^5 - z^4 - 2*z^3 - z^2 + 1, z, k)^5 + 2*root(z^6 + z^5 - z^4 - 2*z^3 - z^2 + 1, z, k))), k,
1, 6) + (2*((x - (3^(1/2)*(1/(9*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - ((31^(1/2)*108^(1/2))/108 + 29/54
)^(1/3))*1i)/2 + 1/(18*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) + ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)/2
- 1/3)/(1/(6*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - (3^(1/2)*(1/(9*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/
3)) - ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3))*1i)/2 + (3*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3))/2))^(1/2)
*((1/(9*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - x + ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3) + 1/3)/(1/(6*
((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - (3^(1/2)*(1/(9*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - ((31^(1
/2)*108^(1/2))/108 + 29/54)^(1/3))*1i)/2 + (3*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3))/2))^(1/2)*ellipticF(as
in(((x - (3^(1/2)*(1/(9*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3))*
1i)/2 + 1/(18*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) + ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)/2 - 1/3)/(1
/(6*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - (3^(1/2)*(1/(9*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - ((3
1^(1/2)*108^(1/2))/108 + 29/54)^(1/3))*1i)/2 + (3*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3))/2))^(1/2)), (3^(1/
2)*(1/(6*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - (3^(1/2)*(1/(9*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3))
- ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3))*1i)/2 + (3*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3))/2)*1i)/(3*(1/
(9*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3))))*(1/(6*((31^(1/2)*10
8^(1/2))/108 + 29/54)^(1/3)) - (3^(1/2)*(1/(9*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - ((31^(1/2)*108^(1/2)
)/108 + 29/54)^(1/3))*1i)/2 + (3*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3))/2)*(-(3^(1/2)*(x + (3^(1/2)*(1/(9*(
(31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3))*1i)/2 + 1/(18*((31^(1/2)*
108^(1/2))/108 + 29/54)^(1/3)) + ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)/2 - 1/3)*1i)/(3*(1/(9*((31^(1/2)*108
^(1/2))/108 + 29/54)^(1/3)) - ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3))))^(1/2))/(x^3 - x^2 - x*(((3^(1/2)*(1/
(9*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3))*1i)/2 - 1/(18*((31^(1
/2)*108^(1/2))/108 + 29/54)^(1/3)) - ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)/2 + 1/3)*((3^(1/2)*(1/(9*((31^(1
/2)*108^(1/2))/108 + 29/54)^(1/3)) - ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3))*1i)/2 + 1/(18*((31^(1/2)*108^(1
/2))/108 + 29/54)^(1/3)) + ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)/2 - 1/3) - (1/(9*((31^(1/2)*108^(1/2))/108
+ 29/54)^(1/3)) + ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3) + 1/3)*((3^(1/2)*(1/(9*((31^(1/2)*108^(1/2))/108 +
29/54)^(1/3)) - ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3))*1i)/2 - 1/(18*((31^(1/2)*108^(1/2))/108 + 29/54)^(1
/3)) - ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)/2 + 1/3) + (1/(9*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) + (
(31^(1/2)*108^(1/2))/108 + 29/54)^(1/3) + 1/3)*((3^(1/2)*(1/(9*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - ((3
1^(1/2)*108^(1/2))/108 + 29/54)^(1/3))*1i)/2 + 1/(18*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) + ((31^(1/2)*10
8^(1/2))/108 + 29/54)^(1/3)/2 - 1/3)) + (1/(9*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) + ((31^(1/2)*108^(1/2)
)/108 + 29/54)^(1/3) + 1/3)*((3^(1/2)*(1/(9*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - ((31^(1/2)*108^(1/2))/
108 + 29/54)^(1/3))*1i)/2 - 1/(18*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - ((31^(1/2)*108^(1/2))/108 + 29/5
4)^(1/3)/2 + 1/3)*((3^(1/2)*(1/(9*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - ((31^(1/2)*108^(1/2))/108 + 29/5
4)^(1/3))*1i)/2 + 1/(18*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) + ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)/2
- 1/3))^(1/2)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (x^{3} + 2\right ) \sqrt {x^{3} - x^{2} - 1}}{x^{6} + x^{5} - x^{4} - 2 x^{3} - x^{2} + 1}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((x**3+2)*(x**3-x**2-1)**(1/2)/(x**6+x**5-x**4-2*x**3-x**2+1),x)
[Out]
Integral((x**3 + 2)*sqrt(x**3 - x**2 - 1)/(x**6 + x**5 - x**4 - 2*x**3 - x**2 + 1), x)
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