3.17.32 \(\int \frac {-2+x}{(2+x^2) \sqrt [3]{-1+x+2 x^2}} \, dx\)

Optimal. Leaf size=111 \[ \frac {1}{2} \log \left (\sqrt [3]{2 x^2+x-1}-x-1\right )-\frac {1}{4} \log \left (x^2+\left (2 x^2+x-1\right )^{2/3}+(x+1) \sqrt [3]{2 x^2+x-1}+2 x+1\right )+\frac {1}{2} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{2 x^2+x-1}}{\sqrt [3]{2 x^2+x-1}+2 x+2}\right ) \]

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Rubi [F]  time = 0.01, 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 {-2+x}{\left (2+x^2\right ) \sqrt [3]{-1+x+2 x^2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-2 + x)/((2 + x^2)*(-1 + x + 2*x^2)^(1/3)),x]

[Out]

Defer[Int][(-2 + x)/((2 + x^2)*(-1 + x + 2*x^2)^(1/3)), x]

Rubi steps

\begin {align*} \int \frac {-2+x}{\left (2+x^2\right ) \sqrt [3]{-1+x+2 x^2}} \, dx &=\int \frac {-2+x}{\left (2+x^2\right ) \sqrt [3]{-1+x+2 x^2}} \, dx\\ \end {align*}

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Mathematica [F]  time = 0.18, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-2+x}{\left (2+x^2\right ) \sqrt [3]{-1+x+2 x^2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.12, size = 111, normalized size = 1.00 \begin {gather*} \frac {1}{2} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{-1+x+2 x^2}}{2+2 x+\sqrt [3]{-1+x+2 x^2}}\right )+\frac {1}{2} \log \left (-1-x+\sqrt [3]{-1+x+2 x^2}\right )-\frac {1}{4} \log \left (1+2 x+x^2+(1+x) \sqrt [3]{-1+x+2 x^2}+\left (-1+x+2 x^2\right )^{2/3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[3]*ArcTan[(Sqrt[3]*(-1 + x + 2*x^2)^(1/3))/(2 + 2*x + (-1 + x + 2*x^2)^(1/3))])/2 + Log[-1 - x + (-1 + x
+ 2*x^2)^(1/3)]/2 - Log[1 + 2*x + x^2 + (1 + x)*(-1 + x + 2*x^2)^(1/3) + (-1 + x + 2*x^2)^(2/3)]/4

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fricas [A]  time = 1.09, size = 107, normalized size = 0.96 \begin {gather*} -\frac {1}{2} \, \sqrt {3} \arctan \left (-\frac {4 \, \sqrt {3} {\left (2 \, x^{2} + x - 1\right )}^{\frac {1}{3}} {\left (x + 1\right )} + \sqrt {3} {\left (2 \, x - 1\right )} - 2 \, \sqrt {3} {\left (2 \, x^{2} + x - 1\right )}^{\frac {2}{3}}}{8 \, x^{2} + 18 \, x + 7}\right ) + \frac {1}{4} \, \log \left (\frac {x^{2} - 3 \, {\left (2 \, x^{2} + x - 1\right )}^{\frac {1}{3}} {\left (x + 1\right )} + 3 \, {\left (2 \, x^{2} + x - 1\right )}^{\frac {2}{3}} + 2}{x^{2} + 2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*sqrt(3)*arctan(-(4*sqrt(3)*(2*x^2 + x - 1)^(1/3)*(x + 1) + sqrt(3)*(2*x - 1) - 2*sqrt(3)*(2*x^2 + x - 1)^
(2/3))/(8*x^2 + 18*x + 7)) + 1/4*log((x^2 - 3*(2*x^2 + x - 1)^(1/3)*(x + 1) + 3*(2*x^2 + x - 1)^(2/3) + 2)/(x^
2 + 2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 2.24, size = 552, normalized size = 4.97

method	result	size
trager	\(\frac {\ln \left (-\frac {4 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x^{2}+16 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x -6 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (2 x^{2}+x -1\right )^{\frac {1}{3}} x +6 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{2}-6 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (2 x^{2}+x -1\right )^{\frac {1}{3}}+20 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x +3 \left (2 x^{2}+x -1\right )^{\frac {2}{3}}-3 x \left (2 x^{2}+x -1\right )^{\frac {1}{3}}+2 x^{2}+2 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )-3 \left (2 x^{2}+x -1\right )^{\frac {1}{3}}+4 x +2}{x^{2}+2}\right )}{2}-\frac {\ln \left (-\frac {4 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x^{2}+16 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x -6 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (2 x^{2}+x -1\right )^{\frac {1}{3}} x +4 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{2}-6 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (2 x^{2}+x -1\right )^{\frac {1}{3}}+20 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x +3 \left (2 x^{2}+x -1\right )^{\frac {2}{3}}-3 x \left (2 x^{2}+x -1\right )^{\frac {1}{3}}-2 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )-3 \left (2 x^{2}+x -1\right )^{\frac {1}{3}}+4 x -2}{x^{2}+2}\right )}{2}-\ln \left (-\frac {4 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x^{2}+16 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x -6 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (2 x^{2}+x -1\right )^{\frac {1}{3}} x +4 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{2}-6 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (2 x^{2}+x -1\right )^{\frac {1}{3}}+20 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x +3 \left (2 x^{2}+x -1\right )^{\frac {2}{3}}-3 x \left (2 x^{2}+x -1\right )^{\frac {1}{3}}-2 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )-3 \left (2 x^{2}+x -1\right )^{\frac {1}{3}}+4 x -2}{x^{2}+2}\right ) \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )\)	\(552\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*ln(-(4*RootOf(4*_Z^2+2*_Z+1)^2*x^2+16*RootOf(4*_Z^2+2*_Z+1)^2*x-6*RootOf(4*_Z^2+2*_Z+1)*(2*x^2+x-1)^(1/3)*
x+6*RootOf(4*_Z^2+2*_Z+1)*x^2-6*RootOf(4*_Z^2+2*_Z+1)*(2*x^2+x-1)^(1/3)+20*RootOf(4*_Z^2+2*_Z+1)*x+3*(2*x^2+x-
1)^(2/3)-3*x*(2*x^2+x-1)^(1/3)+2*x^2+2*RootOf(4*_Z^2+2*_Z+1)-3*(2*x^2+x-1)^(1/3)+4*x+2)/(x^2+2))-1/2*ln(-(4*Ro
otOf(4*_Z^2+2*_Z+1)^2*x^2+16*RootOf(4*_Z^2+2*_Z+1)^2*x-6*RootOf(4*_Z^2+2*_Z+1)*(2*x^2+x-1)^(1/3)*x+4*RootOf(4*
_Z^2+2*_Z+1)*x^2-6*RootOf(4*_Z^2+2*_Z+1)*(2*x^2+x-1)^(1/3)+20*RootOf(4*_Z^2+2*_Z+1)*x+3*(2*x^2+x-1)^(2/3)-3*x*
(2*x^2+x-1)^(1/3)-2*RootOf(4*_Z^2+2*_Z+1)-3*(2*x^2+x-1)^(1/3)+4*x-2)/(x^2+2))-ln(-(4*RootOf(4*_Z^2+2*_Z+1)^2*x
^2+16*RootOf(4*_Z^2+2*_Z+1)^2*x-6*RootOf(4*_Z^2+2*_Z+1)*(2*x^2+x-1)^(1/3)*x+4*RootOf(4*_Z^2+2*_Z+1)*x^2-6*Root
Of(4*_Z^2+2*_Z+1)*(2*x^2+x-1)^(1/3)+20*RootOf(4*_Z^2+2*_Z+1)*x+3*(2*x^2+x-1)^(2/3)-3*x*(2*x^2+x-1)^(1/3)-2*Roo
tOf(4*_Z^2+2*_Z+1)-3*(2*x^2+x-1)^(1/3)+4*x-2)/(x^2+2))*RootOf(4*_Z^2+2*_Z+1)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x - 2)/((x^2 + 2)*(x + 2*x^2 - 1)^(1/3)),x)

[Out]

int((x - 2)/((x^2 + 2)*(x + 2*x^2 - 1)^(1/3)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((x - 2)/(((x + 1)*(2*x - 1))**(1/3)*(x**2 + 2)), x)

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