3.6.92 \(\int \frac {-1-2 x+3 x^2}{\sqrt {-3-2 x-x^2+4 x^3-x^4-2 x^5+x^6}} \, dx\)

Optimal. Leaf size=46 \[ -\log \left (-x^3+x^2+\sqrt {x^6-2 x^5-x^4+4 x^3-x^2-2 x-3}+x-1\right ) \]

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Rubi [F]  time = 0.31, 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 {-1-2 x+3 x^2}{\sqrt {-3-2 x-x^2+4 x^3-x^4-2 x^5+x^6}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-1 - 2*x + 3*x^2)/Sqrt[-3 - 2*x - x^2 + 4*x^3 - x^4 - 2*x^5 + x^6],x]

[Out]

-Defer[Int][1/Sqrt[-3 - 2*x - x^2 + 4*x^3 - x^4 - 2*x^5 + x^6], x] - 2*Defer[Int][x/Sqrt[-3 - 2*x - x^2 + 4*x^
3 - x^4 - 2*x^5 + x^6], x] + 3*Defer[Int][x^2/Sqrt[-3 - 2*x - x^2 + 4*x^3 - x^4 - 2*x^5 + x^6], x]

Rubi steps

\begin {align*} \int \frac {-1-2 x+3 x^2}{\sqrt {-3-2 x-x^2+4 x^3-x^4-2 x^5+x^6}} \, dx &=\int \left (-\frac {1}{\sqrt {-3-2 x-x^2+4 x^3-x^4-2 x^5+x^6}}-\frac {2 x}{\sqrt {-3-2 x-x^2+4 x^3-x^4-2 x^5+x^6}}+\frac {3 x^2}{\sqrt {-3-2 x-x^2+4 x^3-x^4-2 x^5+x^6}}\right ) \, dx\\ &=-\left (2 \int \frac {x}{\sqrt {-3-2 x-x^2+4 x^3-x^4-2 x^5+x^6}} \, dx\right )+3 \int \frac {x^2}{\sqrt {-3-2 x-x^2+4 x^3-x^4-2 x^5+x^6}} \, dx-\int \frac {1}{\sqrt {-3-2 x-x^2+4 x^3-x^4-2 x^5+x^6}} \, dx\\ \end {align*}

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Mathematica [F]  time = 0.16, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-1-2 x+3 x^2}{\sqrt {-3-2 x-x^2+4 x^3-x^4-2 x^5+x^6}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(-1 - 2*x + 3*x^2)/Sqrt[-3 - 2*x - x^2 + 4*x^3 - x^4 - 2*x^5 + x^6],x]

[Out]

Integrate[(-1 - 2*x + 3*x^2)/Sqrt[-3 - 2*x - x^2 + 4*x^3 - x^4 - 2*x^5 + x^6], x]

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IntegrateAlgebraic [A]  time = 3.74, size = 46, normalized size = 1.00 \begin {gather*} -\log \left (-1+x+x^2-x^3+\sqrt {-3-2 x-x^2+4 x^3-x^4-2 x^5+x^6}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(-1 - 2*x + 3*x^2)/Sqrt[-3 - 2*x - x^2 + 4*x^3 - x^4 - 2*x^5 + x^6],x]

[Out]

-Log[-1 + x + x^2 - x^3 + Sqrt[-3 - 2*x - x^2 + 4*x^3 - x^4 - 2*x^5 + x^6]]

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fricas [A]  time = 0.49, size = 44, normalized size = 0.96 \begin {gather*} \log \left (-x^{3} + x^{2} + x - \sqrt {x^{6} - 2 \, x^{5} - x^{4} + 4 \, x^{3} - x^{2} - 2 \, x - 3} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^2-2*x-1)/(x^6-2*x^5-x^4+4*x^3-x^2-2*x-3)^(1/2),x, algorithm="fricas")

[Out]

log(-x^3 + x^2 + x - sqrt(x^6 - 2*x^5 - x^4 + 4*x^3 - x^2 - 2*x - 3) - 1)

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giac [A]  time = 0.23, size = 46, normalized size = 1.00 \begin {gather*} -\log \left ({\left | -x^{3} + x^{2} + x + \sqrt {2 \, x^{3} + {\left (x^{3} - x^{2} - x\right )}^{2} - 2 \, x^{2} - 2 \, x - 3} - 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^2-2*x-1)/(x^6-2*x^5-x^4+4*x^3-x^2-2*x-3)^(1/2),x, algorithm="giac")

[Out]

-log(abs(-x^3 + x^2 + x + sqrt(2*x^3 + (x^3 - x^2 - x)^2 - 2*x^2 - 2*x - 3) - 1))

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maple [A]  time = 0.20, size = 45, normalized size = 0.98

method	result	size
trager	\(\ln \left (-x^{3}+x^{2}-\sqrt {x^{6}-2 x^{5}-x^{4}+4 x^{3}-x^{2}-2 x -3}+x -1\right )\)	\(45\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x^2-2*x-1)/(x^6-2*x^5-x^4+4*x^3-x^2-2*x-3)^(1/2),x,method=_RETURNVERBOSE)

[Out]

ln(-x^3+x^2-(x^6-2*x^5-x^4+4*x^3-x^2-2*x-3)^(1/2)+x-1)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {3 \, x^{2} - 2 \, x - 1}{\sqrt {x^{6} - 2 \, x^{5} - x^{4} + 4 \, x^{3} - x^{2} - 2 \, x - 3}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^2-2*x-1)/(x^6-2*x^5-x^4+4*x^3-x^2-2*x-3)^(1/2),x, algorithm="maxima")

[Out]

integrate((3*x^2 - 2*x - 1)/sqrt(x^6 - 2*x^5 - x^4 + 4*x^3 - x^2 - 2*x - 3), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int -\frac {-3\,x^2+2\,x+1}{\sqrt {x^6-2\,x^5-x^4+4\,x^3-x^2-2\,x-3}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(2*x - 3*x^2 + 1)/(4*x^3 - x^2 - 2*x - x^4 - 2*x^5 + x^6 - 3)^(1/2),x)

[Out]

int(-(2*x - 3*x^2 + 1)/(4*x^3 - x^2 - 2*x - x^4 - 2*x^5 + x^6 - 3)^(1/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x**2-2*x-1)/(x**6-2*x**5-x**4+4*x**3-x**2-2*x-3)**(1/2),x)

[Out]

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

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