3.9.2 \(\int \frac {(-3+x^2) (1-x^2+x^3)^{2/3}}{1-2 x^2-x^3+x^4+x^5+x^6} \, dx\)
Optimal. Leaf size=61 \[ \text {RootSum}\left [\text {$\#$1}^6-3 \text {$\#$1}^3+3\& ,\frac {\text {$\#$1}^2 \log \left (\sqrt [3]{x^3-x^2+1}-\text {$\#$1} x\right )-\text {$\#$1}^2 \log (x)}{2 \text {$\#$1}^3-3}\& \right ] \]
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Rubi [F] time = 0.51, 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 (-3+x^2\right ) \left (1-x^2+x^3\right )^{2/3}}{1-2 x^2-x^3+x^4+x^5+x^6} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
Int[((-3 + x^2)*(1 - x^2 + x^3)^(2/3))/(1 - 2*x^2 - x^3 + x^4 + x^5 + x^6),x]
[Out]
-3*Defer[Int][(1 - x^2 + x^3)^(2/3)/(1 - 2*x^2 - x^3 + x^4 + x^5 + x^6), x] + Defer[Int][(x^2*(1 - x^2 + x^3)^
(2/3))/(1 - 2*x^2 - x^3 + x^4 + x^5 + x^6), x]
Rubi steps
\begin {align*} \int \frac {\left (-3+x^2\right ) \left (1-x^2+x^3\right )^{2/3}}{1-2 x^2-x^3+x^4+x^5+x^6} \, dx &=\int \left (-\frac {3 \left (1-x^2+x^3\right )^{2/3}}{1-2 x^2-x^3+x^4+x^5+x^6}+\frac {x^2 \left (1-x^2+x^3\right )^{2/3}}{1-2 x^2-x^3+x^4+x^5+x^6}\right ) \, dx\\ &=-\left (3 \int \frac {\left (1-x^2+x^3\right )^{2/3}}{1-2 x^2-x^3+x^4+x^5+x^6} \, dx\right )+\int \frac {x^2 \left (1-x^2+x^3\right )^{2/3}}{1-2 x^2-x^3+x^4+x^5+x^6} \, dx\\ \end {align*}
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Mathematica [F] time = 0.09, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-3+x^2\right ) \left (1-x^2+x^3\right )^{2/3}}{1-2 x^2-x^3+x^4+x^5+x^6} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
Integrate[((-3 + x^2)*(1 - x^2 + x^3)^(2/3))/(1 - 2*x^2 - x^3 + x^4 + x^5 + x^6),x]
[Out]
Integrate[((-3 + x^2)*(1 - x^2 + x^3)^(2/3))/(1 - 2*x^2 - x^3 + x^4 + x^5 + x^6), x]
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IntegrateAlgebraic [A] time = 0.00, size = 61, normalized size = 1.00 \begin {gather*} \text {RootSum}\left [3-3 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log (x) \text {$\#$1}^2+\log \left (\sqrt [3]{1-x^2+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^2}{-3+2 \text {$\#$1}^3}\&\right ] \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[((-3 + x^2)*(1 - x^2 + x^3)^(2/3))/(1 - 2*x^2 - x^3 + x^4 + x^5 + x^6),x]
[Out]
RootSum[3 - 3*#1^3 + #1^6 & , (-(Log[x]*#1^2) + Log[(1 - x^2 + x^3)^(1/3) - x*#1]*#1^2)/(-3 + 2*#1^3) & ]
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((x^2-3)*(x^3-x^2+1)^(2/3)/(x^6+x^5+x^4-x^3-2*x^2+1),x, algorithm="fricas")
[Out]
Exception raised: TypeError >> Error detected within library code: integrate: implementation incomplete (tr
ace 0)
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{3} - x^{2} + 1\right )}^{\frac {2}{3}} {\left (x^{2} - 3\right )}}{x^{6} + x^{5} + x^{4} - x^{3} - 2 \, x^{2} + 1}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((x^2-3)*(x^3-x^2+1)^(2/3)/(x^6+x^5+x^4-x^3-2*x^2+1),x, algorithm="giac")
[Out]
integrate((x^3 - x^2 + 1)^(2/3)*(x^2 - 3)/(x^6 + x^5 + x^4 - x^3 - 2*x^2 + 1), x)
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maple [B] time = 19.79, size = 2003, normalized size = 32.84
| | |
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result |
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| trager |
\(\text {Expression too large to display}\) |
\(2003\) |
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int((x^2-3)*(x^3-x^2+1)^(2/3)/(x^6+x^5+x^4-x^3-2*x^2+1),x,method=_RETURNVERBOSE)
[Out]
-3*RootOf(3*_Z^6+3*_Z^3+1)^4*ln(-(30*RootOf(3*_Z^6+3*_Z^3+1)^7*x^3-12*(x^3-x^2+1)^(1/3)*RootOf(3*_Z^6+3*_Z^3+1
)^5*x^2+38*RootOf(3*_Z^6+3*_Z^3+1)^4*x^3-5*RootOf(3*_Z^6+3*_Z^3+1)^4*x^2-9*(x^3-x^2+1)^(2/3)*RootOf(3*_Z^6+3*_
Z^3+1)^3*x-3*(x^3-x^2+1)^(1/3)*RootOf(3*_Z^6+3*_Z^3+1)^2*x^2+5*RootOf(3*_Z^6+3*_Z^3+1)^4+12*RootOf(3*_Z^6+3*_Z
^3+1)*x^3-3*RootOf(3*_Z^6+3*_Z^3+1)*x^2-4*(x^3-x^2+1)^(2/3)*x+3*RootOf(3*_Z^6+3*_Z^3+1))/(3*RootOf(3*_Z^6+3*_Z
^3+1)^3*x^3+2*x^3+x^2-1))+3*RootOf(3*_Z^6+3*_Z^3+1)^3*RootOf(RootOf(3*_Z^6+3*_Z^3+1)^3+_Z^3+1)*ln(-(30*RootOf(
RootOf(3*_Z^6+3*_Z^3+1)^3+_Z^3+1)*RootOf(3*_Z^6+3*_Z^3+1)^6*x^3+12*(x^3-x^2+1)^(1/3)*RootOf(3*_Z^6+3*_Z^3+1)^3
*RootOf(RootOf(3*_Z^6+3*_Z^3+1)^3+_Z^3+1)^2*x^2+22*RootOf(RootOf(3*_Z^6+3*_Z^3+1)^3+_Z^3+1)*RootOf(3*_Z^6+3*_Z
^3+1)^3*x^3+5*RootOf(RootOf(3*_Z^6+3*_Z^3+1)^3+_Z^3+1)*RootOf(3*_Z^6+3*_Z^3+1)^3*x^2+9*(x^3-x^2+1)^(2/3)*RootO
f(3*_Z^6+3*_Z^3+1)^3*x+9*(x^3-x^2+1)^(1/3)*RootOf(RootOf(3*_Z^6+3*_Z^3+1)^3+_Z^3+1)^2*x^2-5*RootOf(3*_Z^6+3*_Z
^3+1)^3*RootOf(RootOf(3*_Z^6+3*_Z^3+1)^3+_Z^3+1)+4*RootOf(RootOf(3*_Z^6+3*_Z^3+1)^3+_Z^3+1)*x^3+2*RootOf(RootO
f(3*_Z^6+3*_Z^3+1)^3+_Z^3+1)*x^2+5*(x^3-x^2+1)^(2/3)*x-2*RootOf(RootOf(3*_Z^6+3*_Z^3+1)^3+_Z^3+1))/(3*RootOf(3
*_Z^6+3*_Z^3+1)^3*x^3+x^3-x^2+1))-2*RootOf(3*_Z^6+3*_Z^3+1)*ln(-(30*RootOf(3*_Z^6+3*_Z^3+1)^7*x^3-12*(x^3-x^2+
1)^(1/3)*RootOf(3*_Z^6+3*_Z^3+1)^5*x^2+38*RootOf(3*_Z^6+3*_Z^3+1)^4*x^3-5*RootOf(3*_Z^6+3*_Z^3+1)^4*x^2-9*(x^3
-x^2+1)^(2/3)*RootOf(3*_Z^6+3*_Z^3+1)^3*x-3*(x^3-x^2+1)^(1/3)*RootOf(3*_Z^6+3*_Z^3+1)^2*x^2+5*RootOf(3*_Z^6+3*
_Z^3+1)^4+12*RootOf(3*_Z^6+3*_Z^3+1)*x^3-3*RootOf(3*_Z^6+3*_Z^3+1)*x^2-4*(x^3-x^2+1)^(2/3)*x+3*RootOf(3*_Z^6+3
*_Z^3+1))/(3*RootOf(3*_Z^6+3*_Z^3+1)^3*x^3+2*x^3+x^2-1))+RootOf(3*_Z^6+3*_Z^3+1)*ln((3*RootOf(3*_Z^6+3*_Z^3+1)
^7*x^3+3*(x^3-x^2+1)^(1/3)*RootOf(3*_Z^6+3*_Z^3+1)^5*x^2+2*RootOf(3*_Z^6+3*_Z^3+1)^4*x^3+RootOf(3*_Z^6+3*_Z^3+
1)^4*x^2+3*(x^3-x^2+1)^(1/3)*RootOf(3*_Z^6+3*_Z^3+1)^2*x^2-RootOf(3*_Z^6+3*_Z^3+1)^4-RootOf(3*_Z^6+3*_Z^3+1)*x
^3+RootOf(3*_Z^6+3*_Z^3+1)*x^2+(x^3-x^2+1)^(2/3)*x-RootOf(3*_Z^6+3*_Z^3+1))/(3*RootOf(3*_Z^6+3*_Z^3+1)^3*x^3+2
*x^3+x^2-1))+RootOf(RootOf(3*_Z^6+3*_Z^3+1)^3+_Z^3+1)*ln((3*RootOf(RootOf(3*_Z^6+3*_Z^3+1)^3+_Z^3+1)*RootOf(3*
_Z^6+3*_Z^3+1)^6*x^3-3*(x^3-x^2+1)^(1/3)*RootOf(3*_Z^6+3*_Z^3+1)^3*RootOf(RootOf(3*_Z^6+3*_Z^3+1)^3+_Z^3+1)^2*
x^2+4*RootOf(RootOf(3*_Z^6+3*_Z^3+1)^3+_Z^3+1)*RootOf(3*_Z^6+3*_Z^3+1)^3*x^3-RootOf(RootOf(3*_Z^6+3*_Z^3+1)^3+
_Z^3+1)*RootOf(3*_Z^6+3*_Z^3+1)^3*x^2+RootOf(3*_Z^6+3*_Z^3+1)^3*RootOf(RootOf(3*_Z^6+3*_Z^3+1)^3+_Z^3+1)+(x^3-
x^2+1)^(2/3)*x)/(3*RootOf(3*_Z^6+3*_Z^3+1)^3*x^3+x^3-x^2+1))+RootOf(RootOf(3*_Z^6+3*_Z^3+1)^3+_Z^3+1)*ln(-(30*
RootOf(RootOf(3*_Z^6+3*_Z^3+1)^3+_Z^3+1)*RootOf(3*_Z^6+3*_Z^3+1)^6*x^3+12*(x^3-x^2+1)^(1/3)*RootOf(3*_Z^6+3*_Z
^3+1)^3*RootOf(RootOf(3*_Z^6+3*_Z^3+1)^3+_Z^3+1)^2*x^2+22*RootOf(RootOf(3*_Z^6+3*_Z^3+1)^3+_Z^3+1)*RootOf(3*_Z
^6+3*_Z^3+1)^3*x^3+5*RootOf(RootOf(3*_Z^6+3*_Z^3+1)^3+_Z^3+1)*RootOf(3*_Z^6+3*_Z^3+1)^3*x^2+9*(x^3-x^2+1)^(2/3
)*RootOf(3*_Z^6+3*_Z^3+1)^3*x+9*(x^3-x^2+1)^(1/3)*RootOf(RootOf(3*_Z^6+3*_Z^3+1)^3+_Z^3+1)^2*x^2-5*RootOf(3*_Z
^6+3*_Z^3+1)^3*RootOf(RootOf(3*_Z^6+3*_Z^3+1)^3+_Z^3+1)+4*RootOf(RootOf(3*_Z^6+3*_Z^3+1)^3+_Z^3+1)*x^3+2*RootO
f(RootOf(3*_Z^6+3*_Z^3+1)^3+_Z^3+1)*x^2+5*(x^3-x^2+1)^(2/3)*x-2*RootOf(RootOf(3*_Z^6+3*_Z^3+1)^3+_Z^3+1))/(3*R
ootOf(3*_Z^6+3*_Z^3+1)^3*x^3+x^3-x^2+1))
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{3} - x^{2} + 1\right )}^{\frac {2}{3}} {\left (x^{2} - 3\right )}}{x^{6} + x^{5} + x^{4} - x^{3} - 2 \, x^{2} + 1}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((x^2-3)*(x^3-x^2+1)^(2/3)/(x^6+x^5+x^4-x^3-2*x^2+1),x, algorithm="maxima")
[Out]
integrate((x^3 - x^2 + 1)^(2/3)*(x^2 - 3)/(x^6 + x^5 + x^4 - x^3 - 2*x^2 + 1), x)
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {\left (x^2-3\right )\,{\left (x^3-x^2+1\right )}^{2/3}}{x^6+x^5+x^4-x^3-2\,x^2+1} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(((x^2 - 3)*(x^3 - x^2 + 1)^(2/3))/(x^4 - x^3 - 2*x^2 + x^5 + x^6 + 1),x)
[Out]
int(((x^2 - 3)*(x^3 - x^2 + 1)^(2/3))/(x^4 - x^3 - 2*x^2 + x^5 + x^6 + 1), x)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (x^{2} - 3\right ) \left (x^{3} - x^{2} + 1\right )^{\frac {2}{3}}}{x^{6} + x^{5} + x^{4} - x^{3} - 2 x^{2} + 1}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((x**2-3)*(x**3-x**2+1)**(2/3)/(x**6+x**5+x**4-x**3-2*x**2+1),x)
[Out]
Integral((x**2 - 3)*(x**3 - x**2 + 1)**(2/3)/(x**6 + x**5 + x**4 - x**3 - 2*x**2 + 1), x)
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