3.8.9 \(\int \frac {-1+2 x}{\sqrt {-8+4 x-3 x^2-10 x^3+x^4}} \, dx\)
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 = 5.03, size = 56, normalized size = 1.00 \begin {gather*} -\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 ) \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 = 0.44, 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 [C] time = 1.14, size = 2702, normalized size = 48.25 \begin {gather*} \text {Expression too large to display} \end {gather*}
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)
[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*} \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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