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

Optimal. Leaf size=38 \[ \log \left (x^3-x^2+\sqrt {x^6-2 x^5+x^4+4 x^3-4 x^2+5}+2\right ) \]

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Rubi [F]  time = 0.40, 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+3 x^2}{\sqrt {5-4 x^2+4 x^3+x^4-2 x^5+x^6}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.06, size = 11, normalized size = 0.29 \begin {gather*} \sinh ^{-1}\left (x^3-x^2+2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

ArcSinh[2 - x^2 + x^3]

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IntegrateAlgebraic [A]  time = 0.15, size = 38, normalized size = 1.00 \begin {gather*} \log \left (2-x^2+x^3+\sqrt {5-4 x^2+4 x^3+x^4-2 x^5+x^6}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.50, size = 36, normalized size = 0.95 \begin {gather*} \log \left (x^{3} - x^{2} + \sqrt {x^{6} - 2 \, x^{5} + x^{4} + 4 \, x^{3} - 4 \, x^{2} + 5} + 2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.69, size = 38, normalized size = 1.00 \begin {gather*} -\log \left (-x^{3} + x^{2} + \sqrt {4 \, x^{3} + {\left (x^{3} - x^{2}\right )}^{2} - 4 \, x^{2} + 5} - 2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.18, size = 37, normalized size = 0.97

method	result	size
trager	\(\ln \left (2-x^{2}+x^{3}+\sqrt {x^{6}-2 x^{5}+x^{4}+4 x^{3}-4 x^{2}+5}\right )\)	\(37\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int -\frac {2\,x-3\,x^2}{\sqrt {x^6-2\,x^5+x^4+4\,x^3-4\,x^2+5}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x \left (3 x - 2\right )}{\sqrt {x^{6} - 2 x^{5} + x^{4} + 4 x^{3} - 4 x^{2} + 5}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x*(3*x - 2)/sqrt(x**6 - 2*x**5 + x**4 + 4*x**3 - 4*x**2 + 5), x)

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