3.6.21 \(\int \frac {-1+x}{\sqrt {-7+4 x+14 x^2-12 x^3+x^4}} \, dx\)

Optimal. Leaf size=40 \[ \log (x-1)-\log \left (-x^2+\sqrt {x^4-12 x^3+14 x^2+4 x-7}+6 x-5\right ) \]

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Rubi [F]  time = 0.18, 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+x}{\sqrt {-7+4 x+14 x^2-12 x^3+x^4}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-1 + x)/Sqrt[-7 + 4*x + 14*x^2 - 12*x^3 + x^4],x]

[Out]

-Defer[Int][1/Sqrt[-7 + 4*x + 14*x^2 - 12*x^3 + x^4], x] + Defer[Int][x/Sqrt[-7 + 4*x + 14*x^2 - 12*x^3 + x^4]
, x]

Rubi steps

\begin {align*} \int \frac {-1+x}{\sqrt {-7+4 x+14 x^2-12 x^3+x^4}} \, dx &=\int \left (-\frac {1}{\sqrt {-7+4 x+14 x^2-12 x^3+x^4}}+\frac {x}{\sqrt {-7+4 x+14 x^2-12 x^3+x^4}}\right ) \, dx\\ &=-\int \frac {1}{\sqrt {-7+4 x+14 x^2-12 x^3+x^4}} \, dx+\int \frac {x}{\sqrt {-7+4 x+14 x^2-12 x^3+x^4}} \, dx\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 54, normalized size = 1.35 \begin {gather*} \frac {(x-1) \sqrt {x^2-10 x-7} \log \left (-\sqrt {x^2-10 x-7}-x+5\right )}{\sqrt {(x-1)^2 \left (x^2-10 x-7\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1 + x)/Sqrt[-7 + 4*x + 14*x^2 - 12*x^3 + x^4],x]

[Out]

((-1 + x)*Sqrt[-7 - 10*x + x^2]*Log[5 - x - Sqrt[-7 - 10*x + x^2]])/Sqrt[(-1 + x)^2*(-7 - 10*x + x^2)]

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IntegrateAlgebraic [A]  time = 0.13, size = 40, normalized size = 1.00 \begin {gather*} \log (-1+x)-\log \left (-5+6 x-x^2+\sqrt {-7+4 x+14 x^2-12 x^3+x^4}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(-1 + x)/Sqrt[-7 + 4*x + 14*x^2 - 12*x^3 + x^4],x]

[Out]

Log[-1 + x] - Log[-5 + 6*x - x^2 + Sqrt[-7 + 4*x + 14*x^2 - 12*x^3 + x^4]]

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fricas [A]  time = 0.45, size = 40, normalized size = 1.00 \begin {gather*} -\log \left (-\frac {x^{2} - 6 \, x - \sqrt {x^{4} - 12 \, x^{3} + 14 \, x^{2} + 4 \, x - 7} + 5}{x - 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1+x)/(x^4-12*x^3+14*x^2+4*x-7)^(1/2),x, algorithm="fricas")

[Out]

-log(-(x^2 - 6*x - sqrt(x^4 - 12*x^3 + 14*x^2 + 4*x - 7) + 5)/(x - 1))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \mathit {undef} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1+x)/(x^4-12*x^3+14*x^2+4*x-7)^(1/2),x, algorithm="giac")

[Out]

undef

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maple [A]  time = 0.09, size = 41, normalized size = 1.02

method	result	size
trager	\(-\ln \left (-\frac {-5+6 x -x^{2}+\sqrt {x^{4}-12 x^{3}+14 x^{2}+4 x -7}}{-1+x}\right )\)	\(41\)
default	\(\frac {\left (-1+x \right ) \sqrt {x^{2}-10 x -7}\, \ln \left (-5+x +\sqrt {x^{2}-10 x -7}\right )}{\sqrt {x^{4}-12 x^{3}+14 x^{2}+4 x -7}}\)	\(49\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-1+x)/(x^4-12*x^3+14*x^2+4*x-7)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-ln(-(-5+6*x-x^2+(x^4-12*x^3+14*x^2+4*x-7)^(1/2))/(-1+x))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x - 1}{\sqrt {x^{4} - 12 \, x^{3} + 14 \, x^{2} + 4 \, x - 7}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1+x)/(x^4-12*x^3+14*x^2+4*x-7)^(1/2),x, algorithm="maxima")

[Out]

integrate((x - 1)/sqrt(x^4 - 12*x^3 + 14*x^2 + 4*x - 7), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {x-1}{\sqrt {x^4-12\,x^3+14\,x^2+4\,x-7}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x - 1)/(4*x + 14*x^2 - 12*x^3 + x^4 - 7)^(1/2),x)

[Out]

int((x - 1)/(4*x + 14*x^2 - 12*x^3 + x^4 - 7)^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x - 1}{\sqrt {\left (x - 1\right )^{2} \left (x^{2} - 10 x - 7\right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1+x)/(x**4-12*x**3+14*x**2+4*x-7)**(1/2),x)

[Out]

Integral((x - 1)/sqrt((x - 1)**2*(x**2 - 10*x - 7)), x)

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