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3.8.25 \(\int \frac {-1+2 x}{\sqrt {-8+4 x-3 x^2-10 x^3+x^4}} \, dx\) [725]

Optimal. Leaf size=56 \[ -\frac {1}{2} \log \left (-90-76 x+89 x^2-18 x^3+x^4+\left (-38+13 x-x^2\right ) \sqrt {-8+4 x-3 x^2-10 x^3+x^4}\right ) \]

[Out]

-1/2*ln(-90-76*x+89*x^2-18*x^3+x^4+(-x^2+13*x-38)*(x^4-10*x^3-3*x^2+4*x-8)^(1/2))

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Rubi [F]
time = 0.09, 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}{\sqrt {-8+4 x-3 x^2-10 x^3+x^4}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-1 + 2*x)/Sqrt[-8 + 4*x - 3*x^2 - 10*x^3 + x^4],x]

[Out]

-Defer[Int][1/Sqrt[-8 + 4*x - 3*x^2 - 10*x^3 + x^4], x] + 2*Defer[Int][x/Sqrt[-8 + 4*x - 3*x^2 - 10*x^3 + x^4]
, x]

Rubi steps

\begin {align*} \int \frac {-1+2 x}{\sqrt {-8+4 x-3 x^2-10 x^3+x^4}} \, dx &=\int \left (-\frac {1}{\sqrt {-8+4 x-3 x^2-10 x^3+x^4}}+\frac {2 x}{\sqrt {-8+4 x-3 x^2-10 x^3+x^4}}\right ) \, dx\\ &=2 \int \frac {x}{\sqrt {-8+4 x-3 x^2-10 x^3+x^4}} \, dx-\int \frac {1}{\sqrt {-8+4 x-3 x^2-10 x^3+x^4}} \, dx\\ \end {align*}

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Mathematica [A]
time = 4.83, size = 56, normalized size = 1.00 \begin {gather*} -\frac {1}{2} \log \left (-90-76 x+89 x^2-18 x^3+x^4+\left (-38+13 x-x^2\right ) \sqrt {-8+4 x-3 x^2-10 x^3+x^4}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1 + 2*x)/Sqrt[-8 + 4*x - 3*x^2 - 10*x^3 + x^4],x]

[Out]

-1/2*Log[-90 - 76*x + 89*x^2 - 18*x^3 + x^4 + (-38 + 13*x - x^2)*Sqrt[-8 + 4*x - 3*x^2 - 10*x^3 + x^4]]

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 1.63, size = 2702, normalized size = 48.25

method result size
trager \(-\frac {\ln \left (-x^{4}+\sqrt {x^{4}-10 x^{3}-3 x^{2}+4 x -8}\, x^{2}+18 x^{3}-13 \sqrt {x^{4}-10 x^{3}-3 x^{2}+4 x -8}\, x -89 x^{2}+38 \sqrt {x^{4}-10 x^{3}-3 x^{2}+4 x -8}+76 x +90\right )}{2}\) \(93\)
default \(\text {Expression too large to display}\) \(2702\)
elliptic \(\text {Expression too large to display}\) \(2702\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-1+2*x)/(x^4-10*x^3-3*x^2+4*x-8)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-2*(RootOf(_Z^4-10*_Z^3-3*_Z^2+4*_Z-8,index=1)-RootOf(_Z^4-10*_Z^3-3*_Z^2+4*_Z-8,index=4))*((RootOf(_Z^4-10*_Z
^3-3*_Z^2+4*_Z-8,index=4)-RootOf(_Z^4-10*_Z^3-3*_Z^2+4*_Z-8,index=2))*(x-RootOf(_Z^4-10*_Z^3-3*_Z^2+4*_Z-8,ind
ex=1))/(RootOf(_Z^4-10*_Z^3-3*_Z^2+4*_Z-8,index=4)-RootOf(_Z^4-10*_Z^3-3*_Z^2+4*_Z-8,index=1))/(x-RootOf(_Z^4-
10*_Z^3-3*_Z^2+4*_Z-8,index=2)))^(1/2)*(x-RootOf(_Z^4-10*_Z^3-3*_Z^2+4*_Z-8,index=2))^2*(-(RootOf(_Z^4-10*_Z^3
-3*_Z^2+4*_Z-8,index=2)-RootOf(_Z^4-10*_Z^3-3*_Z^2+4*_Z-8,index=1))*(x-RootOf(_Z^4-10*_Z^3-3*_Z^2+4*_Z-8,index
=3))/(-RootOf(_Z^4-10*_Z^3-3*_Z^2+4*_Z-8,index=3)+RootOf(_Z^4-10*_Z^3-3*_Z^2+4*_Z-8,index=1))/(x-RootOf(_Z^4-1
0*_Z^3-3*_Z^2+4*_Z-8,index=2)))^(1/2)*((RootOf(_Z^4-10*_Z^3-3*_Z^2+4*_Z-8,index=2)-RootOf(_Z^4-10*_Z^3-3*_Z^2+
4*_Z-8,index=1))*(x-RootOf(_Z^4-10*_Z^3-3*_Z^2+4*_Z-8,index=4))/(RootOf(_Z^4-10*_Z^3-3*_Z^2+4*_Z-8,index=4)-Ro
otOf(_Z^4-10*_Z^3-3*_Z^2+4*_Z-8,index=1))/(x-RootOf(_Z^4-10*_Z^3-3*_Z^2+4*_Z-8,index=2)))^(1/2)/(RootOf(_Z^4-1
0*_Z^3-3*_Z^2+4*_Z-8,index=4)-RootOf(_Z^4-10*_Z^3-3*_Z^2+4*_Z-8,index=2))/(RootOf(_Z^4-10*_Z^3-3*_Z^2+4*_Z-8,i
ndex=2)-RootOf(_Z^4-10*_Z^3-3*_Z^2+4*_Z-8,index=1))/((x-RootOf(_Z^4-10*_Z^3-3*_Z^2+4*_Z-8,index=1))*(x-RootOf(
_Z^4-10*_Z^3-3*_Z^2+4*_Z-8,index=2))*(x-RootOf(_Z^4-10*_Z^3-3*_Z^2+4*_Z-8,index=3))*(x-RootOf(_Z^4-10*_Z^3-3*_
Z^2+4*_Z-8,index=4)))^(1/2)*EllipticF(((RootOf(_Z^4-10*_Z^3-3*_Z^2+4*_Z-8,index=4)-RootOf(_Z^4-10*_Z^3-3*_Z^2+
4*_Z-8,index=2))*(x-RootOf(_Z^4-10*_Z^3-3*_Z^2+4*_Z-8,index=1))/(RootOf(_Z^4-10*_Z^3-3*_Z^2+4*_Z-8,index=4)-Ro
otOf(_Z^4-10*_Z^3-3*_Z^2+4*_Z-8,index=1))/(x-RootOf(_Z^4-10*_Z^3-3*_Z^2+4*_Z-8,index=2)))^(1/2),((RootOf(_Z^4-
10*_Z^3-3*_Z^2+4*_Z-8,index=2)-RootOf(_Z^4-10*_Z^3-3*_Z^2+4*_Z-8,index=3))*(RootOf(_Z^4-10*_Z^3-3*_Z^2+4*_Z-8,
index=1)-RootOf(_Z^4-10*_Z^3-3*_Z^2+4*_Z-8,index=4))/(-RootOf(_Z^4-10*_Z^3-3*_Z^2+4*_Z-8,index=3)+RootOf(_Z^4-
10*_Z^3-3*_Z^2+4*_Z-8,index=1))/(RootOf(_Z^4-10*_Z^3-3*_Z^2+4*_Z-8,index=2)-RootOf(_Z^4-10*_Z^3-3*_Z^2+4*_Z-8,
index=4)))^(1/2))+4*(RootOf(_Z^4-10*_Z^3-3*_Z^2+4*_Z-8,index=1)-RootOf(_Z^4-10*_Z^3-3*_Z^2+4*_Z-8,index=4))*((
RootOf(_Z^4-10*_Z^3-3*_Z^2+4*_Z-8,index=4)-RootOf(_Z^4-10*_Z^3-3*_Z^2+4*_Z-8,index=2))*(x-RootOf(_Z^4-10*_Z^3-
3*_Z^2+4*_Z-8,index=1))/(RootOf(_Z^4-10*_Z^3-3*_Z^2+4*_Z-8,index=4)-RootOf(_Z^4-10*_Z^3-3*_Z^2+4*_Z-8,index=1)
)/(x-RootOf(_Z^4-10*_Z^3-3*_Z^2+4*_Z-8,index=2)))^(1/2)*(x-RootOf(_Z^4-10*_Z^3-3*_Z^2+4*_Z-8,index=2))^2*(-(Ro
otOf(_Z^4-10*_Z^3-3*_Z^2+4*_Z-8,index=2)-RootOf(_Z^4-10*_Z^3-3*_Z^2+4*_Z-8,index=1))*(x-RootOf(_Z^4-10*_Z^3-3*
_Z^2+4*_Z-8,index=3))/(-RootOf(_Z^4-10*_Z^3-3*_Z^2+4*_Z-8,index=3)+RootOf(_Z^4-10*_Z^3-3*_Z^2+4*_Z-8,index=1))
/(x-RootOf(_Z^4-10*_Z^3-3*_Z^2+4*_Z-8,index=2)))^(1/2)*((RootOf(_Z^4-10*_Z^3-3*_Z^2+4*_Z-8,index=2)-RootOf(_Z^
4-10*_Z^3-3*_Z^2+4*_Z-8,index=1))*(x-RootOf(_Z^4-10*_Z^3-3*_Z^2+4*_Z-8,index=4))/(RootOf(_Z^4-10*_Z^3-3*_Z^2+4
*_Z-8,index=4)-RootOf(_Z^4-10*_Z^3-3*_Z^2+4*_Z-8,index=1))/(x-RootOf(_Z^4-10*_Z^3-3*_Z^2+4*_Z-8,index=2)))^(1/
2)/(RootOf(_Z^4-10*_Z^3-3*_Z^2+4*_Z-8,index=4)-RootOf(_Z^4-10*_Z^3-3*_Z^2+4*_Z-8,index=2))/(RootOf(_Z^4-10*_Z^
3-3*_Z^2+4*_Z-8,index=2)-RootOf(_Z^4-10*_Z^3-3*_Z^2+4*_Z-8,index=1))/((x-RootOf(_Z^4-10*_Z^3-3*_Z^2+4*_Z-8,ind
ex=1))*(x-RootOf(_Z^4-10*_Z^3-3*_Z^2+4*_Z-8,index=2))*(x-RootOf(_Z^4-10*_Z^3-3*_Z^2+4*_Z-8,index=3))*(x-RootOf
(_Z^4-10*_Z^3-3*_Z^2+4*_Z-8,index=4)))^(1/2)*(RootOf(_Z^4-10*_Z^3-3*_Z^2+4*_Z-8,index=2)*EllipticF(((RootOf(_Z
^4-10*_Z^3-3*_Z^2+4*_Z-8,index=4)-RootOf(_Z^4-10*_Z^3-3*_Z^2+4*_Z-8,index=2))*(x-RootOf(_Z^4-10*_Z^3-3*_Z^2+4*
_Z-8,index=1))/(RootOf(_Z^4-10*_Z^3-3*_Z^2+4*_Z-8,index=4)-RootOf(_Z^4-10*_Z^3-3*_Z^2+4*_Z-8,index=1))/(x-Root
Of(_Z^4-10*_Z^3-3*_Z^2+4*_Z-8,index=2)))^(1/2),((RootOf(_Z^4-10*_Z^3-3*_Z^2+4*_Z-8,index=2)-RootOf(_Z^4-10*_Z^
3-3*_Z^2+4*_Z-8,index=3))*(RootOf(_Z^4-10*_Z^3-3*_Z^2+4*_Z-8,index=1)-RootOf(_Z^4-10*_Z^3-3*_Z^2+4*_Z-8,index=
4))/(-RootOf(_Z^4-10*_Z^3-3*_Z^2+4*_Z-8,index=3)+RootOf(_Z^4-10*_Z^3-3*_Z^2+4*_Z-8,index=1))/(RootOf(_Z^4-10*_
Z^3-3*_Z^2+4*_Z-8,index=2)-RootOf(_Z^4-10*_Z^3-3*_Z^2+4*_Z-8,index=4)))^(1/2))+(RootOf(_Z^4-10*_Z^3-3*_Z^2+4*_
Z-8,index=1)-RootOf(_Z^4-10*_Z^3-3*_Z^2+4*_Z-8,index=2))*EllipticPi(((RootOf(_Z^4-10*_Z^3-3*_Z^2+4*_Z-8,index=
4)-RootOf(_Z^4-10*_Z^3-3*_Z^2+4*_Z-8,index=2))*(x-RootOf(_Z^4-10*_Z^3-3*_Z^2+4*_Z-8,index=1))/(RootOf(_Z^4-10*
_Z^3-3*_Z^2+4*_Z-8,index=4)-RootOf(_Z^4-10*_Z^3-3*_Z^2+4*_Z-8,index=1))/(x-RootOf(_Z^4-10*_Z^3-3*_Z^2+4*_Z-8,i
ndex=2)))^(1/2),(RootOf(_Z^4-10*_Z^3-3*_Z^2+4*_Z-8,index=4)-RootOf(_Z^4-10*_Z^3-3*_Z^2+4*_Z-8,index=1))/(RootO
f(_Z^4-10*_Z^3-3*_Z^2+4*_Z-8,index=4)-RootOf(_Z^4-10*_Z^3-3*_Z^2+4*_Z-8,index=2)),((RootOf(_Z^4-10*_Z^3-3*_Z^2
+4*_Z-8,index=2)-RootOf(_Z^4-10*_Z^3-3*_Z^2+4*_Z-8,index=3))*(RootOf(_Z^4-10*_Z^3-3*_Z^2+4*_Z-8,index=1)-RootO
f(_Z^4-10*_Z^3-3*_Z^2+4*_Z-8,index=4))/(-RootOf(_Z^4-10*_Z^3-3*_Z^2+4*_Z-8,index=3)+RootOf(_Z^4-10*_Z^3-3*_Z^2
+4*_Z-8,index=1))/(RootOf(_Z^4-10*_Z^3-3*_Z^2+4*_Z-8,index=2)-RootOf(_Z^4-10*_Z^3-3*_Z^2+4*_Z-8,index=4)))^(1/
2)))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1+2*x)/(x^4-10*x^3-3*x^2+4*x-8)^(1/2),x, algorithm="maxima")

[Out]

integrate((2*x - 1)/sqrt(x^4 - 10*x^3 - 3*x^2 + 4*x - 8), x)

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Fricas [A]
time = 0.42, size = 50, normalized size = 0.89 \begin {gather*} \frac {1}{2} \, \log \left (x^{4} - 18 \, x^{3} + 89 \, x^{2} + \sqrt {x^{4} - 10 \, x^{3} - 3 \, x^{2} + 4 \, x - 8} {\left (x^{2} - 13 \, x + 38\right )} - 76 \, x - 90\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1+2*x)/(x^4-10*x^3-3*x^2+4*x-8)^(1/2),x, algorithm="fricas")

[Out]

1/2*log(x^4 - 18*x^3 + 89*x^2 + sqrt(x^4 - 10*x^3 - 3*x^2 + 4*x - 8)*(x^2 - 13*x + 38) - 76*x - 90)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 x - 1}{\sqrt {x^{4} - 10 x^{3} - 3 x^{2} + 4 x - 8}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1+2*x)/(x**4-10*x**3-3*x**2+4*x-8)**(1/2),x)

[Out]

Integral((2*x - 1)/sqrt(x**4 - 10*x**3 - 3*x**2 + 4*x - 8), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1+2*x)/(x^4-10*x^3-3*x^2+4*x-8)^(1/2),x, algorithm="giac")

[Out]

integrate((2*x - 1)/sqrt(x^4 - 10*x^3 - 3*x^2 + 4*x - 8), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {2\,x-1}{\sqrt {x^4-10\,x^3-3\,x^2+4\,x-8}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x - 1)/(4*x - 3*x^2 - 10*x^3 + x^4 - 8)^(1/2),x)

[Out]

int((2*x - 1)/(4*x - 3*x^2 - 10*x^3 + x^4 - 8)^(1/2), x)

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