3.7.98 \(\int \frac {x}{\sqrt {-17+18 x-11 x^2+6 x^3+x^4}} \, dx\)
Optimal. Leaf size=55 \[ -\frac {1}{4} \log \left (-2 x^4-24 x^3-68 x^2+\left (2 x^2+18 x+34\right ) \sqrt {x^4+6 x^3-11 x^2+18 x-17}-11\right ) \]
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Rubi [F] time = 0.06, 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 {x}{\sqrt {-17+18 x-11 x^2+6 x^3+x^4}} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
Int[x/Sqrt[-17 + 18*x - 11*x^2 + 6*x^3 + x^4],x]
[Out]
Defer[Int][x/Sqrt[-17 + 18*x - 11*x^2 + 6*x^3 + x^4], x]
Rubi steps
\begin {align*} \int \frac {x}{\sqrt {-17+18 x-11 x^2+6 x^3+x^4}} \, dx &=\int \frac {x}{\sqrt {-17+18 x-11 x^2+6 x^3+x^4}} \, dx\\ \end {align*}
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Mathematica [C] time = 0.88, size = 1522, normalized size = 27.67
result too large to display
Antiderivative was successfully verified.
[In]
Integrate[x/Sqrt[-17 + 18*x - 11*x^2 + 6*x^3 + x^4],x]
[Out]
(-2*(x - Root[-17 + 18*#1 - 11*#1^2 + 6*#1^3 + #1^4 & , 1, 0])*(x - Root[-17 + 18*#1 - 11*#1^2 + 6*#1^3 + #1^4
& , 2, 0])*(EllipticPi[(Root[-17 + 18*#1 - 11*#1^2 + 6*#1^3 + #1^4 & , 1, 0] - Root[-17 + 18*#1 - 11*#1^2 + 6
*#1^3 + #1^4 & , 4, 0])/(Root[-17 + 18*#1 - 11*#1^2 + 6*#1^3 + #1^4 & , 2, 0] - Root[-17 + 18*#1 - 11*#1^2 + 6
*#1^3 + #1^4 & , 4, 0]), ArcSin[Sqrt[((x - Root[-17 + 18*#1 - 11*#1^2 + 6*#1^3 + #1^4 & , 1, 0])*(Root[-17 + 1
8*#1 - 11*#1^2 + 6*#1^3 + #1^4 & , 2, 0] - Root[-17 + 18*#1 - 11*#1^2 + 6*#1^3 + #1^4 & , 4, 0]))/((x - Root[-
17 + 18*#1 - 11*#1^2 + 6*#1^3 + #1^4 & , 2, 0])*(Root[-17 + 18*#1 - 11*#1^2 + 6*#1^3 + #1^4 & , 1, 0] - Root[-
17 + 18*#1 - 11*#1^2 + 6*#1^3 + #1^4 & , 4, 0]))]], -(((Root[-17 + 18*#1 - 11*#1^2 + 6*#1^3 + #1^4 & , 2, 0] -
Root[-17 + 18*#1 - 11*#1^2 + 6*#1^3 + #1^4 & , 3, 0])*(Root[-17 + 18*#1 - 11*#1^2 + 6*#1^3 + #1^4 & , 1, 0] -
Root[-17 + 18*#1 - 11*#1^2 + 6*#1^3 + #1^4 & , 4, 0]))/((-Root[-17 + 18*#1 - 11*#1^2 + 6*#1^3 + #1^4 & , 1, 0
] + Root[-17 + 18*#1 - 11*#1^2 + 6*#1^3 + #1^4 & , 3, 0])*(Root[-17 + 18*#1 - 11*#1^2 + 6*#1^3 + #1^4 & , 2, 0
] - Root[-17 + 18*#1 - 11*#1^2 + 6*#1^3 + #1^4 & , 4, 0])))]*(Root[-17 + 18*#1 - 11*#1^2 + 6*#1^3 + #1^4 & , 1
, 0] - Root[-17 + 18*#1 - 11*#1^2 + 6*#1^3 + #1^4 & , 2, 0]) + EllipticF[ArcSin[Sqrt[((x - Root[-17 + 18*#1 -
11*#1^2 + 6*#1^3 + #1^4 & , 1, 0])*(Root[-17 + 18*#1 - 11*#1^2 + 6*#1^3 + #1^4 & , 2, 0] - Root[-17 + 18*#1 -
11*#1^2 + 6*#1^3 + #1^4 & , 4, 0]))/((x - Root[-17 + 18*#1 - 11*#1^2 + 6*#1^3 + #1^4 & , 2, 0])*(Root[-17 + 18
*#1 - 11*#1^2 + 6*#1^3 + #1^4 & , 1, 0] - Root[-17 + 18*#1 - 11*#1^2 + 6*#1^3 + #1^4 & , 4, 0]))]], -(((Root[-
17 + 18*#1 - 11*#1^2 + 6*#1^3 + #1^4 & , 2, 0] - Root[-17 + 18*#1 - 11*#1^2 + 6*#1^3 + #1^4 & , 3, 0])*(Root[-
17 + 18*#1 - 11*#1^2 + 6*#1^3 + #1^4 & , 1, 0] - Root[-17 + 18*#1 - 11*#1^2 + 6*#1^3 + #1^4 & , 4, 0]))/((-Roo
t[-17 + 18*#1 - 11*#1^2 + 6*#1^3 + #1^4 & , 1, 0] + Root[-17 + 18*#1 - 11*#1^2 + 6*#1^3 + #1^4 & , 3, 0])*(Roo
t[-17 + 18*#1 - 11*#1^2 + 6*#1^3 + #1^4 & , 2, 0] - Root[-17 + 18*#1 - 11*#1^2 + 6*#1^3 + #1^4 & , 4, 0])))]*R
oot[-17 + 18*#1 - 11*#1^2 + 6*#1^3 + #1^4 & , 2, 0])*Sqrt[(x - Root[-17 + 18*#1 - 11*#1^2 + 6*#1^3 + #1^4 & ,
3, 0])/((x - Root[-17 + 18*#1 - 11*#1^2 + 6*#1^3 + #1^4 & , 2, 0])*(-Root[-17 + 18*#1 - 11*#1^2 + 6*#1^3 + #1^
4 & , 1, 0] + Root[-17 + 18*#1 - 11*#1^2 + 6*#1^3 + #1^4 & , 3, 0]))]*Sqrt[(x - Root[-17 + 18*#1 - 11*#1^2 + 6
*#1^3 + #1^4 & , 4, 0])/((x - Root[-17 + 18*#1 - 11*#1^2 + 6*#1^3 + #1^4 & , 2, 0])*(-Root[-17 + 18*#1 - 11*#1
^2 + 6*#1^3 + #1^4 & , 1, 0] + Root[-17 + 18*#1 - 11*#1^2 + 6*#1^3 + #1^4 & , 4, 0]))])/(Sqrt[-17 + 18*x - 11*
x^2 + 6*x^3 + x^4]*Sqrt[((x - Root[-17 + 18*#1 - 11*#1^2 + 6*#1^3 + #1^4 & , 1, 0])*(Root[-17 + 18*#1 - 11*#1^
2 + 6*#1^3 + #1^4 & , 2, 0] - Root[-17 + 18*#1 - 11*#1^2 + 6*#1^3 + #1^4 & , 4, 0]))/((x - Root[-17 + 18*#1 -
11*#1^2 + 6*#1^3 + #1^4 & , 2, 0])*(Root[-17 + 18*#1 - 11*#1^2 + 6*#1^3 + #1^4 & , 1, 0] - Root[-17 + 18*#1 -
11*#1^2 + 6*#1^3 + #1^4 & , 4, 0]))])
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IntegrateAlgebraic [A] time = 4.71, size = 55, normalized size = 1.00 \begin {gather*} -\frac {1}{4} \log \left (-2 x^4-24 x^3-68 x^2+\left (2 x^2+18 x+34\right ) \sqrt {x^4+6 x^3-11 x^2+18 x-17}-11\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[x/Sqrt[-17 + 18*x - 11*x^2 + 6*x^3 + x^4],x]
[Out]
-1/4*Log[-11 - 68*x^2 - 24*x^3 - 2*x^4 + (34 + 18*x + 2*x^2)*Sqrt[-17 + 18*x - 11*x^2 + 6*x^3 + x^4]]
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fricas [A] time = 0.44, size = 50, normalized size = 0.91 \begin {gather*} \frac {1}{4} \, \log \left (2 \, x^{4} + 24 \, x^{3} + 68 \, x^{2} + 2 \, \sqrt {x^{4} + 6 \, x^{3} - 11 \, x^{2} + 18 \, x - 17} {\left (x^{2} + 9 \, x + 17\right )} + 11\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x/(x^4+6*x^3-11*x^2+18*x-17)^(1/2),x, algorithm="fricas")
[Out]
1/4*log(2*x^4 + 24*x^3 + 68*x^2 + 2*sqrt(x^4 + 6*x^3 - 11*x^2 + 18*x - 17)*(x^2 + 9*x + 17) + 11)
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{\sqrt {x^{4} + 6 \, x^{3} - 11 \, x^{2} + 18 \, x - 17}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x/(x^4+6*x^3-11*x^2+18*x-17)^(1/2),x, algorithm="giac")
[Out]
integrate(x/sqrt(x^4 + 6*x^3 - 11*x^2 + 18*x - 17), x)
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maple [C] time = 1.05, size = 1610, normalized size = 29.27 \begin {gather*} \text {Expression too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x/(x^4+6*x^3-11*x^2+18*x-17)^(1/2),x)
[Out]
2*(RootOf(_Z^4+6*_Z^3-11*_Z^2+18*_Z-17,index=1)-RootOf(_Z^4+6*_Z^3-11*_Z^2+18*_Z-17,index=4))*((RootOf(_Z^4+6*
_Z^3-11*_Z^2+18*_Z-17,index=4)-RootOf(_Z^4+6*_Z^3-11*_Z^2+18*_Z-17,index=2))*(x-RootOf(_Z^4+6*_Z^3-11*_Z^2+18*
_Z-17,index=1))/(RootOf(_Z^4+6*_Z^3-11*_Z^2+18*_Z-17,index=4)-RootOf(_Z^4+6*_Z^3-11*_Z^2+18*_Z-17,index=1))/(x
-RootOf(_Z^4+6*_Z^3-11*_Z^2+18*_Z-17,index=2)))^(1/2)*(x-RootOf(_Z^4+6*_Z^3-11*_Z^2+18*_Z-17,index=2))^2*(-(Ro
otOf(_Z^4+6*_Z^3-11*_Z^2+18*_Z-17,index=2)-RootOf(_Z^4+6*_Z^3-11*_Z^2+18*_Z-17,index=1))*(x-RootOf(_Z^4+6*_Z^3
-11*_Z^2+18*_Z-17,index=3))/(-RootOf(_Z^4+6*_Z^3-11*_Z^2+18*_Z-17,index=3)+RootOf(_Z^4+6*_Z^3-11*_Z^2+18*_Z-17
,index=1))/(x-RootOf(_Z^4+6*_Z^3-11*_Z^2+18*_Z-17,index=2)))^(1/2)*((RootOf(_Z^4+6*_Z^3-11*_Z^2+18*_Z-17,index
=2)-RootOf(_Z^4+6*_Z^3-11*_Z^2+18*_Z-17,index=1))*(x-RootOf(_Z^4+6*_Z^3-11*_Z^2+18*_Z-17,index=4))/(RootOf(_Z^
4+6*_Z^3-11*_Z^2+18*_Z-17,index=4)-RootOf(_Z^4+6*_Z^3-11*_Z^2+18*_Z-17,index=1))/(x-RootOf(_Z^4+6*_Z^3-11*_Z^2
+18*_Z-17,index=2)))^(1/2)/(RootOf(_Z^4+6*_Z^3-11*_Z^2+18*_Z-17,index=4)-RootOf(_Z^4+6*_Z^3-11*_Z^2+18*_Z-17,i
ndex=2))/(RootOf(_Z^4+6*_Z^3-11*_Z^2+18*_Z-17,index=2)-RootOf(_Z^4+6*_Z^3-11*_Z^2+18*_Z-17,index=1))/((x-RootO
f(_Z^4+6*_Z^3-11*_Z^2+18*_Z-17,index=1))*(x-RootOf(_Z^4+6*_Z^3-11*_Z^2+18*_Z-17,index=2))*(x-RootOf(_Z^4+6*_Z^
3-11*_Z^2+18*_Z-17,index=3))*(x-RootOf(_Z^4+6*_Z^3-11*_Z^2+18*_Z-17,index=4)))^(1/2)*(RootOf(_Z^4+6*_Z^3-11*_Z
^2+18*_Z-17,index=2)*EllipticF(((RootOf(_Z^4+6*_Z^3-11*_Z^2+18*_Z-17,index=4)-RootOf(_Z^4+6*_Z^3-11*_Z^2+18*_Z
-17,index=2))*(x-RootOf(_Z^4+6*_Z^3-11*_Z^2+18*_Z-17,index=1))/(RootOf(_Z^4+6*_Z^3-11*_Z^2+18*_Z-17,index=4)-R
ootOf(_Z^4+6*_Z^3-11*_Z^2+18*_Z-17,index=1))/(x-RootOf(_Z^4+6*_Z^3-11*_Z^2+18*_Z-17,index=2)))^(1/2),((RootOf(
_Z^4+6*_Z^3-11*_Z^2+18*_Z-17,index=2)-RootOf(_Z^4+6*_Z^3-11*_Z^2+18*_Z-17,index=3))*(RootOf(_Z^4+6*_Z^3-11*_Z^
2+18*_Z-17,index=1)-RootOf(_Z^4+6*_Z^3-11*_Z^2+18*_Z-17,index=4))/(-RootOf(_Z^4+6*_Z^3-11*_Z^2+18*_Z-17,index=
3)+RootOf(_Z^4+6*_Z^3-11*_Z^2+18*_Z-17,index=1))/(RootOf(_Z^4+6*_Z^3-11*_Z^2+18*_Z-17,index=2)-RootOf(_Z^4+6*_
Z^3-11*_Z^2+18*_Z-17,index=4)))^(1/2))+(RootOf(_Z^4+6*_Z^3-11*_Z^2+18*_Z-17,index=1)-RootOf(_Z^4+6*_Z^3-11*_Z^
2+18*_Z-17,index=2))*EllipticPi(((RootOf(_Z^4+6*_Z^3-11*_Z^2+18*_Z-17,index=4)-RootOf(_Z^4+6*_Z^3-11*_Z^2+18*_
Z-17,index=2))*(x-RootOf(_Z^4+6*_Z^3-11*_Z^2+18*_Z-17,index=1))/(RootOf(_Z^4+6*_Z^3-11*_Z^2+18*_Z-17,index=4)-
RootOf(_Z^4+6*_Z^3-11*_Z^2+18*_Z-17,index=1))/(x-RootOf(_Z^4+6*_Z^3-11*_Z^2+18*_Z-17,index=2)))^(1/2),(RootOf(
_Z^4+6*_Z^3-11*_Z^2+18*_Z-17,index=4)-RootOf(_Z^4+6*_Z^3-11*_Z^2+18*_Z-17,index=1))/(RootOf(_Z^4+6*_Z^3-11*_Z^
2+18*_Z-17,index=4)-RootOf(_Z^4+6*_Z^3-11*_Z^2+18*_Z-17,index=2)),((RootOf(_Z^4+6*_Z^3-11*_Z^2+18*_Z-17,index=
2)-RootOf(_Z^4+6*_Z^3-11*_Z^2+18*_Z-17,index=3))*(RootOf(_Z^4+6*_Z^3-11*_Z^2+18*_Z-17,index=1)-RootOf(_Z^4+6*_
Z^3-11*_Z^2+18*_Z-17,index=4))/(-RootOf(_Z^4+6*_Z^3-11*_Z^2+18*_Z-17,index=3)+RootOf(_Z^4+6*_Z^3-11*_Z^2+18*_Z
-17,index=1))/(RootOf(_Z^4+6*_Z^3-11*_Z^2+18*_Z-17,index=2)-RootOf(_Z^4+6*_Z^3-11*_Z^2+18*_Z-17,index=4)))^(1/
2)))
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{\sqrt {x^{4} + 6 \, x^{3} - 11 \, x^{2} + 18 \, x - 17}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x/(x^4+6*x^3-11*x^2+18*x-17)^(1/2),x, algorithm="maxima")
[Out]
integrate(x/sqrt(x^4 + 6*x^3 - 11*x^2 + 18*x - 17), x)
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {x}{\sqrt {x^4+6\,x^3-11\,x^2+18\,x-17}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x/(18*x - 11*x^2 + 6*x^3 + x^4 - 17)^(1/2),x)
[Out]
int(x/(18*x - 11*x^2 + 6*x^3 + x^4 - 17)^(1/2), x)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{\sqrt {x^{4} + 6 x^{3} - 11 x^{2} + 18 x - 17}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x/(x**4+6*x**3-11*x**2+18*x-17)**(1/2),x)
[Out]
Integral(x/sqrt(x**4 + 6*x**3 - 11*x**2 + 18*x - 17), x)
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