3.14.90 \(\int \frac {6+2 x+x^2}{\sqrt [3]{2+x^2} (1+3 x-2 x^2+x^3)} \, dx\)

Optimal. Leaf size=100 \[ \log \left (\sqrt [3]{x^2+2}+x-1\right )-\frac {1}{2} \log \left (x^2+\left (x^2+2\right )^{2/3}+(1-x) \sqrt [3]{x^2+2}-2 x+1\right )-\sqrt {3} \tan ^{-1}\left (\frac {\frac {\sqrt [3]{x^2+2}}{\sqrt {3}}-\frac {2 x}{\sqrt {3}}+\frac {2}{\sqrt {3}}}{\sqrt [3]{x^2+2}}\right ) \]

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Rubi [F]  time = 0.59, 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 {6+2 x+x^2}{\sqrt [3]{2+x^2} \left (1+3 x-2 x^2+x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(6 + 2*x + x^2)/((2 + x^2)^(1/3)*(1 + 3*x - 2*x^2 + x^3)),x]

[Out]

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

Rubi steps

\begin {align*} \int \frac {6+2 x+x^2}{\sqrt [3]{2+x^2} \left (1+3 x-2 x^2+x^3\right )} \, dx &=\int \left (\frac {6}{\sqrt [3]{2+x^2} \left (1+3 x-2 x^2+x^3\right )}+\frac {2 x}{\sqrt [3]{2+x^2} \left (1+3 x-2 x^2+x^3\right )}+\frac {x^2}{\sqrt [3]{2+x^2} \left (1+3 x-2 x^2+x^3\right )}\right ) \, dx\\ &=2 \int \frac {x}{\sqrt [3]{2+x^2} \left (1+3 x-2 x^2+x^3\right )} \, dx+6 \int \frac {1}{\sqrt [3]{2+x^2} \left (1+3 x-2 x^2+x^3\right )} \, dx+\int \frac {x^2}{\sqrt [3]{2+x^2} \left (1+3 x-2 x^2+x^3\right )} \, dx\\ \end {align*}

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Mathematica [F]  time = 0.16, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {6+2 x+x^2}{\sqrt [3]{2+x^2} \left (1+3 x-2 x^2+x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(6 + 2*x + x^2)/((2 + x^2)^(1/3)*(1 + 3*x - 2*x^2 + x^3)),x]

[Out]

Integrate[(6 + 2*x + x^2)/((2 + x^2)^(1/3)*(1 + 3*x - 2*x^2 + x^3)), x]

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IntegrateAlgebraic [A]  time = 0.14, size = 100, normalized size = 1.00 \begin {gather*} -\sqrt {3} \tan ^{-1}\left (\frac {\frac {2}{\sqrt {3}}-\frac {2 x}{\sqrt {3}}+\frac {\sqrt [3]{2+x^2}}{\sqrt {3}}}{\sqrt [3]{2+x^2}}\right )+\log \left (-1+x+\sqrt [3]{2+x^2}\right )-\frac {1}{2} \log \left (1-2 x+x^2+(1-x) \sqrt [3]{2+x^2}+\left (2+x^2\right )^{2/3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(6 + 2*x + x^2)/((2 + x^2)^(1/3)*(1 + 3*x - 2*x^2 + x^3)),x]

[Out]

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

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fricas [A]  time = 1.24, size = 138, normalized size = 1.38 \begin {gather*} -\sqrt {3} \arctan \left (\frac {2 \, \sqrt {3} {\left (x^{2} + 2\right )}^{\frac {2}{3}} {\left (x - 1\right )} + 2 \, \sqrt {3} {\left (x^{2} - 2 \, x + 1\right )} {\left (x^{2} + 2\right )}^{\frac {1}{3}} + \sqrt {3} {\left (x^{3} - 2 \, x^{2} + 3 \, x + 1\right )}}{3 \, {\left (x^{3} - 4 \, x^{2} + 3 \, x - 3\right )}}\right ) + \frac {1}{2} \, \log \left (\frac {x^{3} - 2 \, x^{2} + 3 \, {\left (x^{2} + 2\right )}^{\frac {2}{3}} {\left (x - 1\right )} + 3 \, {\left (x^{2} - 2 \, x + 1\right )} {\left (x^{2} + 2\right )}^{\frac {1}{3}} + 3 \, x + 1}{x^{3} - 2 \, x^{2} + 3 \, x + 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+2*x+6)/(x^2+2)^(1/3)/(x^3-2*x^2+3*x+1),x, algorithm="fricas")

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2} + 2 \, x + 6}{{\left (x^{3} - 2 \, x^{2} + 3 \, x + 1\right )} {\left (x^{2} + 2\right )}^{\frac {1}{3}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+2*x+6)/(x^2+2)^(1/3)/(x^3-2*x^2+3*x+1),x, algorithm="giac")

[Out]

integrate((x^2 + 2*x + 6)/((x^3 - 2*x^2 + 3*x + 1)*(x^2 + 2)^(1/3)), x)

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maple [C]  time = 3.23, size = 772, normalized size = 7.72

method	result	size
trager	\(\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (-\frac {188 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}+573 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{2}+2\right )^{\frac {2}{3}} x -573 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{2}+2\right )^{\frac {1}{3}} x^{2}-470 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{2}+676 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}-573 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{2}+2\right )^{\frac {2}{3}}+891 \left (x^{2}+2\right )^{\frac {2}{3}} x +1146 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{2}+2\right )^{\frac {1}{3}} x -891 \left (x^{2}+2\right )^{\frac {1}{3}} x^{2}+564 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x -2019 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}+591 x^{3}-891 \left (x^{2}+2\right )^{\frac {2}{3}}-573 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{2}+2\right )^{\frac {1}{3}}+1782 \left (x^{2}+2\right )^{\frac {1}{3}} x +2028 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x -2167 x^{2}-891 \left (x^{2}+2\right )^{\frac {1}{3}}-658 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+1773 x -1379}{x^{3}-2 x^{2}+3 x +1}\right )-\ln \left (\frac {-188 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}+573 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{2}+2\right )^{\frac {2}{3}} x -573 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{2}+2\right )^{\frac {1}{3}} x^{2}+470 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{2}+300 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}-573 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{2}+2\right )^{\frac {2}{3}}-318 \left (x^{2}+2\right )^{\frac {2}{3}} x +1146 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{2}+2\right )^{\frac {1}{3}} x +318 \left (x^{2}+2\right )^{\frac {1}{3}} x^{2}-564 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x -1079 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}-103 x^{3}+318 \left (x^{2}+2\right )^{\frac {2}{3}}-573 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{2}+2\right )^{\frac {1}{3}}-636 \left (x^{2}+2\right )^{\frac {1}{3}} x +900 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x +618 x^{2}+318 \left (x^{2}+2\right )^{\frac {1}{3}}-658 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-309 x +721}{x^{3}-2 x^{2}+3 x +1}\right ) \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-\ln \left (\frac {-188 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}+573 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{2}+2\right )^{\frac {2}{3}} x -573 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{2}+2\right )^{\frac {1}{3}} x^{2}+470 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{2}+300 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}-573 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{2}+2\right )^{\frac {2}{3}}-318 \left (x^{2}+2\right )^{\frac {2}{3}} x +1146 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{2}+2\right )^{\frac {1}{3}} x +318 \left (x^{2}+2\right )^{\frac {1}{3}} x^{2}-564 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x -1079 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}-103 x^{3}+318 \left (x^{2}+2\right )^{\frac {2}{3}}-573 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{2}+2\right )^{\frac {1}{3}}-636 \left (x^{2}+2\right )^{\frac {1}{3}} x +900 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x +618 x^{2}+318 \left (x^{2}+2\right )^{\frac {1}{3}}-658 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-309 x +721}{x^{3}-2 x^{2}+3 x +1}\right )\)	\(772\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2+2*x+6)/(x^2+2)^(1/3)/(x^3-2*x^2+3*x+1),x,method=_RETURNVERBOSE)

[Out]

RootOf(_Z^2+_Z+1)*ln(-(188*RootOf(_Z^2+_Z+1)^2*x^3+573*RootOf(_Z^2+_Z+1)*(x^2+2)^(2/3)*x-573*RootOf(_Z^2+_Z+1)
*(x^2+2)^(1/3)*x^2-470*RootOf(_Z^2+_Z+1)^2*x^2+676*RootOf(_Z^2+_Z+1)*x^3-573*RootOf(_Z^2+_Z+1)*(x^2+2)^(2/3)+8
91*(x^2+2)^(2/3)*x+1146*RootOf(_Z^2+_Z+1)*(x^2+2)^(1/3)*x-891*(x^2+2)^(1/3)*x^2+564*RootOf(_Z^2+_Z+1)^2*x-2019
*RootOf(_Z^2+_Z+1)*x^2+591*x^3-891*(x^2+2)^(2/3)-573*RootOf(_Z^2+_Z+1)*(x^2+2)^(1/3)+1782*(x^2+2)^(1/3)*x+2028
*RootOf(_Z^2+_Z+1)*x-2167*x^2-891*(x^2+2)^(1/3)-658*RootOf(_Z^2+_Z+1)+1773*x-1379)/(x^3-2*x^2+3*x+1))-ln((-188
*RootOf(_Z^2+_Z+1)^2*x^3+573*RootOf(_Z^2+_Z+1)*(x^2+2)^(2/3)*x-573*RootOf(_Z^2+_Z+1)*(x^2+2)^(1/3)*x^2+470*Roo
tOf(_Z^2+_Z+1)^2*x^2+300*RootOf(_Z^2+_Z+1)*x^3-573*RootOf(_Z^2+_Z+1)*(x^2+2)^(2/3)-318*(x^2+2)^(2/3)*x+1146*Ro
otOf(_Z^2+_Z+1)*(x^2+2)^(1/3)*x+318*(x^2+2)^(1/3)*x^2-564*RootOf(_Z^2+_Z+1)^2*x-1079*RootOf(_Z^2+_Z+1)*x^2-103
*x^3+318*(x^2+2)^(2/3)-573*RootOf(_Z^2+_Z+1)*(x^2+2)^(1/3)-636*(x^2+2)^(1/3)*x+900*RootOf(_Z^2+_Z+1)*x+618*x^2
+318*(x^2+2)^(1/3)-658*RootOf(_Z^2+_Z+1)-309*x+721)/(x^3-2*x^2+3*x+1))*RootOf(_Z^2+_Z+1)-ln((-188*RootOf(_Z^2+
_Z+1)^2*x^3+573*RootOf(_Z^2+_Z+1)*(x^2+2)^(2/3)*x-573*RootOf(_Z^2+_Z+1)*(x^2+2)^(1/3)*x^2+470*RootOf(_Z^2+_Z+1
)^2*x^2+300*RootOf(_Z^2+_Z+1)*x^3-573*RootOf(_Z^2+_Z+1)*(x^2+2)^(2/3)-318*(x^2+2)^(2/3)*x+1146*RootOf(_Z^2+_Z+
1)*(x^2+2)^(1/3)*x+318*(x^2+2)^(1/3)*x^2-564*RootOf(_Z^2+_Z+1)^2*x-1079*RootOf(_Z^2+_Z+1)*x^2-103*x^3+318*(x^2
+2)^(2/3)-573*RootOf(_Z^2+_Z+1)*(x^2+2)^(1/3)-636*(x^2+2)^(1/3)*x+900*RootOf(_Z^2+_Z+1)*x+618*x^2+318*(x^2+2)^
(1/3)-658*RootOf(_Z^2+_Z+1)-309*x+721)/(x^3-2*x^2+3*x+1))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2} + 2 \, x + 6}{{\left (x^{3} - 2 \, x^{2} + 3 \, x + 1\right )} {\left (x^{2} + 2\right )}^{\frac {1}{3}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+2*x+6)/(x^2+2)^(1/3)/(x^3-2*x^2+3*x+1),x, algorithm="maxima")

[Out]

integrate((x^2 + 2*x + 6)/((x^3 - 2*x^2 + 3*x + 1)*(x^2 + 2)^(1/3)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^2+2\,x+6}{{\left (x^2+2\right )}^{1/3}\,\left (x^3-2\,x^2+3\,x+1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x + x^2 + 6)/((x^2 + 2)^(1/3)*(3*x - 2*x^2 + x^3 + 1)),x)

[Out]

int((2*x + x^2 + 6)/((x^2 + 2)^(1/3)*(3*x - 2*x^2 + x^3 + 1)), x)

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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+2*x+6)/(x**2+2)**(1/3)/(x**3-2*x**2+3*x+1),x)

[Out]

Timed out

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