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

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

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Rubi [F]  time = 0.14, 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 [C]  time = 2.13, size = 1333, normalized size = 23.80

result too large to display

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(2*(5 + Sqrt[17] + 4*Sqrt[4 + Sqrt[17]] - 2*x)*(-((4 + Sqrt[17] + 4*Sqrt[4 + Sqrt[17]])*EllipticF[ArcSin[Sqrt[
((5 + Sqrt[17] - 4*Sqrt[4 + Sqrt[17]] - 2*x)*(5 + Sqrt[17] + 4*Sqrt[4 + Sqrt[17]] - 2*Root[-8 + 4*#1 - 3*#1^2
- 10*#1^3 + #1^4 & , 4, 0]))/((5 + Sqrt[17] + 4*Sqrt[4 + Sqrt[17]] - 2*x)*(5 + Sqrt[17] - 4*Sqrt[4 + Sqrt[17]]
- 2*Root[-8 + 4*#1 - 3*#1^2 - 10*#1^3 + #1^4 & , 4, 0]))]], ((5 + Sqrt[17] + 4*Sqrt[4 + Sqrt[17]] - 2*Root[-8
+ 4*#1 - 3*#1^2 - 10*#1^3 + #1^4 & , 3, 0])*(5 + Sqrt[17] - 4*Sqrt[4 + Sqrt[17]] - 2*Root[-8 + 4*#1 - 3*#1^2
- 10*#1^3 + #1^4 & , 4, 0]))/((5 + Sqrt[17] - 4*Sqrt[4 + Sqrt[17]] - 2*Root[-8 + 4*#1 - 3*#1^2 - 10*#1^3 + #1^
4 & , 3, 0])*(5 + Sqrt[17] + 4*Sqrt[4 + Sqrt[17]] - 2*Root[-8 + 4*#1 - 3*#1^2 - 10*#1^3 + #1^4 & , 4, 0]))]) +
8*Sqrt[4 + Sqrt[17]]*EllipticPi[(5 + Sqrt[17] - 4*Sqrt[4 + Sqrt[17]] - 2*Root[-8 + 4*#1 - 3*#1^2 - 10*#1^3 +
#1^4 & , 4, 0])/(5 + Sqrt[17] + 4*Sqrt[4 + Sqrt[17]] - 2*Root[-8 + 4*#1 - 3*#1^2 - 10*#1^3 + #1^4 & , 4, 0]),
ArcSin[Sqrt[((5 + Sqrt[17] - 4*Sqrt[4 + Sqrt[17]] - 2*x)*(5 + Sqrt[17] + 4*Sqrt[4 + Sqrt[17]] - 2*Root[-8 + 4*
#1 - 3*#1^2 - 10*#1^3 + #1^4 & , 4, 0]))/((5 + Sqrt[17] + 4*Sqrt[4 + Sqrt[17]] - 2*x)*(5 + Sqrt[17] - 4*Sqrt[4
+ Sqrt[17]] - 2*Root[-8 + 4*#1 - 3*#1^2 - 10*#1^3 + #1^4 & , 4, 0]))]], ((5 + Sqrt[17] + 4*Sqrt[4 + Sqrt[17]]
- 2*Root[-8 + 4*#1 - 3*#1^2 - 10*#1^3 + #1^4 & , 3, 0])*(5 + Sqrt[17] - 4*Sqrt[4 + Sqrt[17]] - 2*Root[-8 + 4*
#1 - 3*#1^2 - 10*#1^3 + #1^4 & , 4, 0]))/((5 + Sqrt[17] - 4*Sqrt[4 + Sqrt[17]] - 2*Root[-8 + 4*#1 - 3*#1^2 - 1
0*#1^3 + #1^4 & , 3, 0])*(5 + Sqrt[17] + 4*Sqrt[4 + Sqrt[17]] - 2*Root[-8 + 4*#1 - 3*#1^2 - 10*#1^3 + #1^4 & ,
4, 0]))])*Sqrt[(x - Root[-8 + 4*#1 - 3*#1^2 - 10*#1^3 + #1^4 & , 3, 0])/((5 + Sqrt[17] + 4*Sqrt[4 + Sqrt[17]]
- 2*x)*(5 + Sqrt[17] - 4*Sqrt[4 + Sqrt[17]] - 2*Root[-8 + 4*#1 - 3*#1^2 - 10*#1^3 + #1^4 & , 3, 0]))]*Sqrt[((
5 + Sqrt[17] - 4*Sqrt[4 + Sqrt[17]] - 2*x)*(5 + Sqrt[17] + 4*Sqrt[4 + Sqrt[17]] - 2*Root[-8 + 4*#1 - 3*#1^2 -
10*#1^3 + #1^4 & , 4, 0]))/((5 + Sqrt[17] + 4*Sqrt[4 + Sqrt[17]] - 2*x)*(5 + Sqrt[17] - 4*Sqrt[4 + Sqrt[17]] -
2*Root[-8 + 4*#1 - 3*#1^2 - 10*#1^3 + #1^4 & , 4, 0]))]*(x - Root[-8 + 4*#1 - 3*#1^2 - 10*#1^3 + #1^4 & , 4,
0]))/(Sqrt[-8 + 4*x - 3*x^2 - 10*x^3 + x^4]*(5 + Sqrt[17] + 4*Sqrt[4 + Sqrt[17]] - 2*Root[-8 + 4*#1 - 3*#1^2 -
10*#1^3 + #1^4 & , 4, 0])*Sqrt[(x - Root[-8 + 4*#1 - 3*#1^2 - 10*#1^3 + #1^4 & , 4, 0])/((5 + Sqrt[17] + 4*Sq
rt[4 + Sqrt[17]] - 2*x)*(5 + Sqrt[17] - 4*Sqrt[4 + Sqrt[17]] - 2*Root[-8 + 4*#1 - 3*#1^2 - 10*#1^3 + #1^4 & ,
4, 0]))])

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IntegrateAlgebraic [A]  time = 4.84, 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]

IntegrateAlgebraic[(-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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fricas [A]  time = 0.51, 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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giac [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}}\,{d x} \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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maple [A]  time = 1.20, size = 91, normalized size = 1.62

method	result	size
trager	\(\frac {\ln \left (x^{4}+\sqrt {x^{4}-10 x^{3}-3 x^{2}+4 x -8}\, x^{2}-18 x^{3}-13 \sqrt {x^{4}-10 x^{3}-3 x^{2}+4 x -8}\, x +89 x^{2}+38 \sqrt {x^{4}-10 x^{3}-3 x^{2}+4 x -8}-76 x -90\right )}{2}\)	\(91\)
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]

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

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maxima [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}}\,{d x} \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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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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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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