3.8.9 \(\int \frac {x}{\sqrt {-17+18 x-11 x^2+6 x^3+x^4}} \, dx\) [709]
Optimal. Leaf size=55 \[ -\frac {1}{4} \log \left (-11-68 x^2-24 x^3-2 x^4+\left (34+18 x+2 x^2\right ) \sqrt {-17+18 x-11 x^2+6 x^3+x^4}\right ) \]
[Out]
-1/4*ln(-11-68*x^2-24*x^3-2*x^4+(2*x^2+18*x+34)*(x^4+6*x^3-11*x^2+18*x-17)^(1/2))
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Rubi [F]
time = 0.04, 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 [A]
time = 4.75, size = 55, normalized size = 1.00 \begin {gather*} -\frac {1}{4} \log \left (-11-68 x^2-24 x^3-2 x^4+\left (34+18 x+2 x^2\right ) \sqrt {-17+18 x-11 x^2+6 x^3+x^4}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[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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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 1.83, size = 1610, normalized size = 29.27
| | |
| method |
result |
size |
| | |
| trager |
\(-\frac {\ln \left (-2 x^{4}+2 \sqrt {x^{4}+6 x^{3}-11 x^{2}+18 x -17}\, x^{2}-24 x^{3}+18 x \sqrt {x^{4}+6 x^{3}-11 x^{2}+18 x -17}-68 x^{2}+34 \sqrt {x^{4}+6 x^{3}-11 x^{2}+18 x -17}-11\right )}{4}\) |
\(91\) |
| default |
\(\text {Expression too large to display}\) |
\(1610\) |
| elliptic |
\(\text {Expression too large to display}\) |
\(1610\) |
| | |
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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,method=_RETURNVERBOSE)
[Out]
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))*(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*(-(R
ootOf(_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-1
7,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,inde
x=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,
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=1))/((x-Root
Of(_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)-
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=3))*(-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,inde
x=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=2)+RootOf(_Z^4+6*_Z^3-11*
_Z^2+18*_Z-17,index=1))*EllipticPi(((RootOf(_Z^4+6*_Z^3-11*_Z^2+18*_Z-17,index=4)-RootOf(_Z^4+6*_Z^3-11*_Z^2+1
8*_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),(Root
Of(_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,ind
ex=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=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=3)+RootOf(_Z^4+6*_Z^3-11*_Z^2+1
8*_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*} \text {Failed to integrate} \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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Fricas [A]
time = 0.38, 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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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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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(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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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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