3.8.69 \(\int \frac {(2+x^3) \sqrt [3]{x+x^3-x^4}}{1+x^2-2 x^3+x^4-x^5+x^6} \, dx\) [769]
Optimal. Leaf size=59 \[ -\text {RootSum}\left [1-\text {$\#$1}^3+\text {$\#$1}^6\& ,\frac {-\log (x) \text {$\#$1}+\log \left (\sqrt [3]{x+x^3-x^4}-x \text {$\#$1}\right ) \text {$\#$1}}{-1+2 \text {$\#$1}^3}\& \right ] \]
[Out]
Unintegrable
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Rubi [F]
time = 1.03, 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 [3]{x+x^3-x^4}}{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)*(x + x^3 - x^4)^(1/3))/(1 + x^2 - 2*x^3 + x^4 - x^5 + x^6),x]
[Out]
(6*(x + x^3 - x^4)^(1/3)*Defer[Subst][Defer[Int][(x^3*(1 + x^6 - x^9)^(1/3))/(1 + x^6 - 2*x^9 + x^12 - x^15 +
x^18), x], x, x^(1/3)])/(x^(1/3)*(1 + x^2 - x^3)^(1/3)) + (3*(x + x^3 - x^4)^(1/3)*Defer[Subst][Defer[Int][(x^
12*(1 + x^6 - x^9)^(1/3))/(1 + x^6 - 2*x^9 + x^12 - x^15 + x^18), x], x, x^(1/3)])/(x^(1/3)*(1 + x^2 - x^3)^(1
/3))
Rubi steps
\begin {align*} \int \frac {\left (2+x^3\right ) \sqrt [3]{x+x^3-x^4}}{1+x^2-2 x^3+x^4-x^5+x^6} \, dx &=\frac {\sqrt [3]{x+x^3-x^4} \int \frac {\sqrt [3]{x} \sqrt [3]{1+x^2-x^3} \left (2+x^3\right )}{1+x^2-2 x^3+x^4-x^5+x^6} \, dx}{\sqrt [3]{x} \sqrt [3]{1+x^2-x^3}}\\ &=\frac {\left (3 \sqrt [3]{x+x^3-x^4}\right ) \text {Subst}\left (\int \frac {x^3 \sqrt [3]{1+x^6-x^9} \left (2+x^9\right )}{1+x^6-2 x^9+x^{12}-x^{15}+x^{18}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+x^2-x^3}}\\ &=\frac {\left (3 \sqrt [3]{x+x^3-x^4}\right ) \text {Subst}\left (\int \left (\frac {2 x^3 \sqrt [3]{1+x^6-x^9}}{1+x^6-2 x^9+x^{12}-x^{15}+x^{18}}+\frac {x^{12} \sqrt [3]{1+x^6-x^9}}{1+x^6-2 x^9+x^{12}-x^{15}+x^{18}}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+x^2-x^3}}\\ &=\frac {\left (3 \sqrt [3]{x+x^3-x^4}\right ) \text {Subst}\left (\int \frac {x^{12} \sqrt [3]{1+x^6-x^9}}{1+x^6-2 x^9+x^{12}-x^{15}+x^{18}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+x^2-x^3}}+\frac {\left (6 \sqrt [3]{x+x^3-x^4}\right ) \text {Subst}\left (\int \frac {x^3 \sqrt [3]{1+x^6-x^9}}{1+x^6-2 x^9+x^{12}-x^{15}+x^{18}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+x^2-x^3}}\\ \end {align*}
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Mathematica [A]
time = 2.70, size = 97, normalized size = 1.64 \begin {gather*} \frac {\sqrt [3]{x+x^3-x^4} \text {RootSum}\left [1+\text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-2 \log \left (\sqrt [3]{x}\right ) \text {$\#$1}+\log \left (\sqrt [3]{-1-x^2+x^3}-x^{2/3} \text {$\#$1}\right ) \text {$\#$1}}{1+2 \text {$\#$1}^3}\&\right ]}{\sqrt [3]{x} \sqrt [3]{-1-x^2+x^3}} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[((2 + x^3)*(x + x^3 - x^4)^(1/3))/(1 + x^2 - 2*x^3 + x^4 - x^5 + x^6),x]
[Out]
((x + x^3 - x^4)^(1/3)*RootSum[1 + #1^3 + #1^6 & , (-2*Log[x^(1/3)]*#1 + Log[(-1 - x^2 + x^3)^(1/3) - x^(2/3)*
#1]*#1)/(1 + 2*#1^3) & ])/(x^(1/3)*(-1 - x^2 + x^3)^(1/3))
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
1.
time = 9.52, size = 2069, normalized size = 35.07
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\(\text {Expression too large to display}\) |
\(2069\) |
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int((x^3+2)*(-x^4+x^3+x)^(1/3)/(x^6-x^5+x^4-2*x^3+x^2+1),x,method=_RETURNVERBOSE)
[Out]
18*RootOf(27*_Z^6-9*_Z^3+1)^5*ln((-5751*RootOf(27*_Z^6-9*_Z^3+1)^5*(-x^4+x^3+x)^(1/3)*x+819*RootOf(27*_Z^6-9*_
Z^3+1)^4*x^3-1638*RootOf(27*_Z^6-9*_Z^3+1)^4*x^2+2736*RootOf(27*_Z^6-9*_Z^3+1)^3*(-x^4+x^3+x)^(2/3)+1551*RootO
f(27*_Z^6-9*_Z^3+1)^2*(-x^4+x^3+x)^(1/3)*x-819*RootOf(27*_Z^6-9*_Z^3+1)^4+122*RootOf(27*_Z^6-9*_Z^3+1)*x^3-244
*RootOf(27*_Z^6-9*_Z^3+1)*x^2-395*(-x^4+x^3+x)^(2/3)-122*RootOf(27*_Z^6-9*_Z^3+1))/(9*RootOf(27*_Z^6-9*_Z^3+1)
^3*x^3-9*RootOf(27*_Z^6-9*_Z^3+1)^3-x^3-x^2+1))-9*RootOf(27*_Z^6-9*_Z^3+1)^5*ln((81*RootOf(27*_Z^6-9*_Z^3+1)^8
*x^3-81*RootOf(27*_Z^6-9*_Z^3+1)^8-45*RootOf(27*_Z^6-9*_Z^3+1)^5*x^3+9*RootOf(27*_Z^6-9*_Z^3+1)^5*x^2+9*(-x^4+
x^3+x)^(2/3)*RootOf(27*_Z^6-9*_Z^3+1)^4-9*(-x^4+x^3+x)^(1/3)*RootOf(27*_Z^6-9*_Z^3+1)^3*x+45*RootOf(27*_Z^6-9*
_Z^3+1)^5+6*RootOf(27*_Z^6-9*_Z^3+1)^2*x^3-3*RootOf(27*_Z^6-9*_Z^3+1)^2*x^2+2*(-x^4+x^3+x)^(1/3)*x-6*RootOf(27
*_Z^6-9*_Z^3+1)^2)/(9*RootOf(27*_Z^6-9*_Z^3+1)^3*x^3-9*RootOf(27*_Z^6-9*_Z^3+1)^3-x^3-x^2+1))-9*RootOf(27*_Z^6
-9*_Z^3+1)^4*ln(-(3555*RootOf(27*_Z^6-9*_Z^3+1)^5*x^3-7110*RootOf(27*_Z^6-9*_Z^3+1)^5*x^2-1098*(-x^4+x^3+x)^(1
/3)*RootOf(27*_Z^6-9*_Z^3+1)^4*x+2736*RootOf(27*_Z^6-9*_Z^3+1)^3*(-x^4+x^3+x)^(2/3)-3555*RootOf(27*_Z^6-9*_Z^3
+1)^5-912*RootOf(27*_Z^6-9*_Z^3+1)^2*x^3+1824*RootOf(27*_Z^6-9*_Z^3+1)^2*x^2-273*(-x^4+x^3+x)^(1/3)*RootOf(27*
_Z^6-9*_Z^3+1)*x-517*(-x^4+x^3+x)^(2/3)+912*RootOf(27*_Z^6-9*_Z^3+1)^2)/(9*RootOf(27*_Z^6-9*_Z^3+1)^3*x^3-9*Ro
otOf(27*_Z^6-9*_Z^3+1)^3-2*x^3+x^2+2))+9*RootOf(27*_Z^6-9*_Z^3+1)^4*ln((10530*RootOf(27*_Z^6-9*_Z^3+1)^8*x^3-1
0530*RootOf(27*_Z^6-9*_Z^3+1)^8-4491*RootOf(27*_Z^6-9*_Z^3+1)^5*x^3-1170*RootOf(27*_Z^6-9*_Z^3+1)^5*x^2-981*(-
x^4+x^3+x)^(1/3)*RootOf(27*_Z^6-9*_Z^3+1)^4*x+1107*RootOf(27*_Z^6-9*_Z^3+1)^3*(-x^4+x^3+x)^(2/3)+4491*RootOf(2
7*_Z^6-9*_Z^3+1)^5+369*RootOf(27*_Z^6-9*_Z^3+1)^2*x^3+369*RootOf(27*_Z^6-9*_Z^3+1)^2*x^2-21*(-x^4+x^3+x)^(1/3)
*RootOf(27*_Z^6-9*_Z^3+1)*x-239*(-x^4+x^3+x)^(2/3)-369*RootOf(27*_Z^6-9*_Z^3+1)^2)/(9*RootOf(27*_Z^6-9*_Z^3+1)
^3*x^3-9*RootOf(27*_Z^6-9*_Z^3+1)^3-2*x^3+x^2+2))-3*RootOf(27*_Z^6-9*_Z^3+1)^2*ln((-5751*RootOf(27*_Z^6-9*_Z^3
+1)^5*(-x^4+x^3+x)^(1/3)*x+819*RootOf(27*_Z^6-9*_Z^3+1)^4*x^3-1638*RootOf(27*_Z^6-9*_Z^3+1)^4*x^2+2736*RootOf(
27*_Z^6-9*_Z^3+1)^3*(-x^4+x^3+x)^(2/3)+1551*RootOf(27*_Z^6-9*_Z^3+1)^2*(-x^4+x^3+x)^(1/3)*x-819*RootOf(27*_Z^6
-9*_Z^3+1)^4+122*RootOf(27*_Z^6-9*_Z^3+1)*x^3-244*RootOf(27*_Z^6-9*_Z^3+1)*x^2-395*(-x^4+x^3+x)^(2/3)-122*Root
Of(27*_Z^6-9*_Z^3+1))/(9*RootOf(27*_Z^6-9*_Z^3+1)^3*x^3-9*RootOf(27*_Z^6-9*_Z^3+1)^3-x^3-x^2+1))+RootOf(27*_Z^
6-9*_Z^3+1)*ln(-(3555*RootOf(27*_Z^6-9*_Z^3+1)^5*x^3-7110*RootOf(27*_Z^6-9*_Z^3+1)^5*x^2-1098*(-x^4+x^3+x)^(1/
3)*RootOf(27*_Z^6-9*_Z^3+1)^4*x+2736*RootOf(27*_Z^6-9*_Z^3+1)^3*(-x^4+x^3+x)^(2/3)-3555*RootOf(27*_Z^6-9*_Z^3+
1)^5-912*RootOf(27*_Z^6-9*_Z^3+1)^2*x^3+1824*RootOf(27*_Z^6-9*_Z^3+1)^2*x^2-273*(-x^4+x^3+x)^(1/3)*RootOf(27*_
Z^6-9*_Z^3+1)*x-517*(-x^4+x^3+x)^(2/3)+912*RootOf(27*_Z^6-9*_Z^3+1)^2)/(9*RootOf(27*_Z^6-9*_Z^3+1)^3*x^3-9*Roo
tOf(27*_Z^6-9*_Z^3+1)^3-2*x^3+x^2+2))-2*RootOf(27*_Z^6-9*_Z^3+1)*ln((10530*RootOf(27*_Z^6-9*_Z^3+1)^8*x^3-1053
0*RootOf(27*_Z^6-9*_Z^3+1)^8-4491*RootOf(27*_Z^6-9*_Z^3+1)^5*x^3-1170*RootOf(27*_Z^6-9*_Z^3+1)^5*x^2-981*(-x^4
+x^3+x)^(1/3)*RootOf(27*_Z^6-9*_Z^3+1)^4*x+1107*RootOf(27*_Z^6-9*_Z^3+1)^3*(-x^4+x^3+x)^(2/3)+4491*RootOf(27*_
Z^6-9*_Z^3+1)^5+369*RootOf(27*_Z^6-9*_Z^3+1)^2*x^3+369*RootOf(27*_Z^6-9*_Z^3+1)^2*x^2-21*(-x^4+x^3+x)^(1/3)*Ro
otOf(27*_Z^6-9*_Z^3+1)*x-239*(-x^4+x^3+x)^(2/3)-369*RootOf(27*_Z^6-9*_Z^3+1)^2)/(9*RootOf(27*_Z^6-9*_Z^3+1)^3*
x^3-9*RootOf(27*_Z^6-9*_Z^3+1)^3-2*x^3+x^2+2))
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((x^3+2)*(-x^4+x^3+x)^(1/3)/(x^6-x^5+x^4-2*x^3+x^2+1),x, algorithm="maxima")
[Out]
integrate((-x^4 + x^3 + x)^(1/3)*(x^3 + 2)/(x^6 - x^5 + x^4 - 2*x^3 + x^2 + 1), x)
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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^3+2)*(-x^4+x^3+x)^(1/3)/(x^6-x^5+x^4-2*x^3+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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [3]{- x \left (x^{3} - x^{2} - 1\right )} \left (x^{3} + 2\right )}{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**4+x**3+x)**(1/3)/(x**6-x**5+x**4-2*x**3+x**2+1),x)
[Out]
Integral((-x*(x**3 - x**2 - 1))**(1/3)*(x**3 + 2)/(x**6 - x**5 + x**4 - 2*x**3 + x**2 + 1), x)
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((x^3+2)*(-x^4+x^3+x)^(1/3)/(x^6-x^5+x^4-2*x^3+x^2+1),x, algorithm="giac")
[Out]
integrate((-x^4 + x^3 + x)^(1/3)*(x^3 + 2)/(x^6 - x^5 + x^4 - 2*x^3 + 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^3+2\right )\,{\left (-x^4+x^3+x\right )}^{1/3}}{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]
int(((x^3 + 2)*(x + x^3 - x^4)^(1/3))/(x^2 - 2*x^3 + x^4 - x^5 + x^6 + 1),x)
[Out]
int(((x^3 + 2)*(x + x^3 - x^4)^(1/3))/(x^2 - 2*x^3 + x^4 - x^5 + x^6 + 1), x)
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