3.8.9 \(\int \frac {x}{\sqrt {-17+18 x-11 x^2+6 x^3+x^4}} \, dx\)

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 = 4.72, 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]

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 = 0.49, 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 [A]  time = 1.17, size = 91, normalized size = 1.65

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	\(\text {Expression too large to display}\)	\(1610\)

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]

-1/4*ln(-2*x^4+2*(x^4+6*x^3-11*x^2+18*x-17)^(1/2)*x^2-24*x^3+18*x*(x^4+6*x^3-11*x^2+18*x-17)^(1/2)-68*x^2+34*(
x^4+6*x^3-11*x^2+18*x-17)^(1/2)-11)

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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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