3.9.34 \(\int \frac {\sqrt {2-x^3-x^4} (4+x^3+2 x^4)}{(-2-3 x^2+x^3+x^4) (-2-x^2+x^3+x^4)} \, dx\)

Optimal. Leaf size=63 \[ -\tan ^{-1}\left (\frac {x}{\sqrt {-x^4-x^3+2}}\right )-\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x \sqrt {-x^4-x^3+2}}{x^4+x^3-2}\right ) \]

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Rubi [F]  time = 1.69, 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 {\sqrt {2-x^3-x^4} \left (4+x^3+2 x^4\right )}{\left (-2-3 x^2+x^3+x^4\right ) \left (-2-x^2+x^3+x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(Sqrt[2 - x^3 - x^4]*(4 + x^3 + 2*x^4))/((-2 - 3*x^2 + x^3 + x^4)*(-2 - x^2 + x^3 + x^4)),x]

[Out]

-3*Defer[Int][Sqrt[2 - x^3 - x^4]/(-2 - 3*x^2 + x^3 + x^4), x] + (3*Defer[Int][(x*Sqrt[2 - x^3 - x^4])/(-2 - 3
*x^2 + x^3 + x^4), x])/2 + 2*Defer[Int][(x^2*Sqrt[2 - x^3 - x^4])/(-2 - 3*x^2 + x^3 + x^4), x] + Defer[Int][Sq
rt[2 - x^3 - x^4]/(-2 - x^2 + x^3 + x^4), x] - (3*Defer[Int][(x*Sqrt[2 - x^3 - x^4])/(-2 - x^2 + x^3 + x^4), x
])/2 - 2*Defer[Int][(x^2*Sqrt[2 - x^3 - x^4])/(-2 - x^2 + x^3 + x^4), x]

Rubi steps

\begin {align*} \int \frac {\sqrt {2-x^3-x^4} \left (4+x^3+2 x^4\right )}{\left (-2-3 x^2+x^3+x^4\right ) \left (-2-x^2+x^3+x^4\right )} \, dx &=\int \left (\frac {\left (-6+3 x+4 x^2\right ) \sqrt {2-x^3-x^4}}{2 \left (-2-3 x^2+x^3+x^4\right )}+\frac {\left (2-3 x-4 x^2\right ) \sqrt {2-x^3-x^4}}{2 \left (-2-x^2+x^3+x^4\right )}\right ) \, dx\\ &=\frac {1}{2} \int \frac {\left (-6+3 x+4 x^2\right ) \sqrt {2-x^3-x^4}}{-2-3 x^2+x^3+x^4} \, dx+\frac {1}{2} \int \frac {\left (2-3 x-4 x^2\right ) \sqrt {2-x^3-x^4}}{-2-x^2+x^3+x^4} \, dx\\ &=\frac {1}{2} \int \left (-\frac {6 \sqrt {2-x^3-x^4}}{-2-3 x^2+x^3+x^4}+\frac {3 x \sqrt {2-x^3-x^4}}{-2-3 x^2+x^3+x^4}+\frac {4 x^2 \sqrt {2-x^3-x^4}}{-2-3 x^2+x^3+x^4}\right ) \, dx+\frac {1}{2} \int \left (\frac {2 \sqrt {2-x^3-x^4}}{-2-x^2+x^3+x^4}-\frac {3 x \sqrt {2-x^3-x^4}}{-2-x^2+x^3+x^4}-\frac {4 x^2 \sqrt {2-x^3-x^4}}{-2-x^2+x^3+x^4}\right ) \, dx\\ &=\frac {3}{2} \int \frac {x \sqrt {2-x^3-x^4}}{-2-3 x^2+x^3+x^4} \, dx-\frac {3}{2} \int \frac {x \sqrt {2-x^3-x^4}}{-2-x^2+x^3+x^4} \, dx+2 \int \frac {x^2 \sqrt {2-x^3-x^4}}{-2-3 x^2+x^3+x^4} \, dx-2 \int \frac {x^2 \sqrt {2-x^3-x^4}}{-2-x^2+x^3+x^4} \, dx-3 \int \frac {\sqrt {2-x^3-x^4}}{-2-3 x^2+x^3+x^4} \, dx+\int \frac {\sqrt {2-x^3-x^4}}{-2-x^2+x^3+x^4} \, dx\\ \end {align*}

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Mathematica [C]  time = 6.63, size = 61074, normalized size = 969.43 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[(Sqrt[2 - x^3 - x^4]*(4 + x^3 + 2*x^4))/((-2 - 3*x^2 + x^3 + x^4)*(-2 - x^2 + x^3 + x^4)),x]

[Out]

Result too large to show

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IntegrateAlgebraic [A]  time = 1.35, size = 63, normalized size = 1.00 \begin {gather*} -\tan ^{-1}\left (\frac {x}{\sqrt {2-x^3-x^4}}\right )-\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x \sqrt {2-x^3-x^4}}{-2+x^3+x^4}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(Sqrt[2 - x^3 - x^4]*(4 + x^3 + 2*x^4))/((-2 - 3*x^2 + x^3 + x^4)*(-2 - x^2 + x^3 + x^4)),x
]

[Out]

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

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fricas [A]  time = 0.79, size = 75, normalized size = 1.19 \begin {gather*} -\frac {1}{2} \, \sqrt {3} \arctan \left (\frac {2 \, \sqrt {3} \sqrt {-x^{4} - x^{3} + 2} x}{x^{4} + x^{3} + 3 \, x^{2} - 2}\right ) + \frac {1}{2} \, \arctan \left (\frac {2 \, \sqrt {-x^{4} - x^{3} + 2} x}{x^{4} + x^{3} + x^{2} - 2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^4-x^3+2)^(1/2)*(2*x^4+x^3+4)/(x^4+x^3-3*x^2-2)/(x^4+x^3-x^2-2),x, algorithm="fricas")

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (2 \, x^{4} + x^{3} + 4\right )} \sqrt {-x^{4} - x^{3} + 2}}{{\left (x^{4} + x^{3} - x^{2} - 2\right )} {\left (x^{4} + x^{3} - 3 \, x^{2} - 2\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^4-x^3+2)^(1/2)*(2*x^4+x^3+4)/(x^4+x^3-3*x^2-2)/(x^4+x^3-x^2-2),x, algorithm="giac")

[Out]

integrate((2*x^4 + x^3 + 4)*sqrt(-x^4 - x^3 + 2)/((x^4 + x^3 - x^2 - 2)*(x^4 + x^3 - 3*x^2 - 2)), x)

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maple [C]  time = 4.77, size = 167, normalized size = 2.65

method	result	size
trager	\(-\frac {\RootOf \left (\textit {\_Z}^{2}+3\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}+3\right ) x^{4}+\RootOf \left (\textit {\_Z}^{2}+3\right ) x^{3}+3 \RootOf \left (\textit {\_Z}^{2}+3\right ) x^{2}+6 \sqrt {-x^{4}-x^{3}+2}\, x -2 \RootOf \left (\textit {\_Z}^{2}+3\right )}{x^{4}+x^{3}-3 x^{2}-2}\right )}{2}+\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{4}+\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{3}+\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{2}+2 \sqrt {-x^{4}-x^{3}+2}\, x -2 \RootOf \left (\textit {\_Z}^{2}+1\right )}{x^{4}+x^{3}-x^{2}-2}\right )}{2}\)	\(167\)
default	\(\text {Expression too large to display}\)	\(5959\)
elliptic	\(\text {Expression too large to display}\)	\(433723\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-x^4-x^3+2)^(1/2)*(2*x^4+x^3+4)/(x^4+x^3-3*x^2-2)/(x^4+x^3-x^2-2),x,method=_RETURNVERBOSE)

[Out]

-1/2*RootOf(_Z^2+3)*ln(-(RootOf(_Z^2+3)*x^4+RootOf(_Z^2+3)*x^3+3*RootOf(_Z^2+3)*x^2+6*(-x^4-x^3+2)^(1/2)*x-2*R
ootOf(_Z^2+3))/(x^4+x^3-3*x^2-2))+1/2*RootOf(_Z^2+1)*ln(-(RootOf(_Z^2+1)*x^4+RootOf(_Z^2+1)*x^3+RootOf(_Z^2+1)
*x^2+2*(-x^4-x^3+2)^(1/2)*x-2*RootOf(_Z^2+1))/(x^4+x^3-x^2-2))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (2 \, x^{4} + x^{3} + 4\right )} \sqrt {-x^{4} - x^{3} + 2}}{{\left (x^{4} + x^{3} - x^{2} - 2\right )} {\left (x^{4} + x^{3} - 3 \, x^{2} - 2\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^4-x^3+2)^(1/2)*(2*x^4+x^3+4)/(x^4+x^3-3*x^2-2)/(x^4+x^3-x^2-2),x, algorithm="maxima")

[Out]

integrate((2*x^4 + x^3 + 4)*sqrt(-x^4 - x^3 + 2)/((x^4 + x^3 - x^2 - 2)*(x^4 + x^3 - 3*x^2 - 2)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {\sqrt {-x^4-x^3+2}\,\left (2\,x^4+x^3+4\right )}{\left (-x^4-x^3+3\,x^2+2\right )\,\left (-x^4-x^3+x^2+2\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2 - x^4 - x^3)^(1/2)*(x^3 + 2*x^4 + 4))/((3*x^2 - x^3 - x^4 + 2)*(x^2 - x^3 - x^4 + 2)),x)

[Out]

int(((2 - x^4 - x^3)^(1/2)*(x^3 + 2*x^4 + 4))/((3*x^2 - x^3 - x^4 + 2)*(x^2 - x^3 - x^4 + 2)), 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**4-x**3+2)**(1/2)*(2*x**4+x**3+4)/(x**4+x**3-3*x**2-2)/(x**4+x**3-x**2-2),x)

[Out]

Timed out

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