3.5.22 \(\int \frac {x}{\sqrt {11-11 x-3 x^2+2 x^3+x^4}} \, dx\)

Optimal. Leaf size=34 \[ \frac {2}{3} \tanh ^{-1}\left (\frac {x^2+x-2}{\sqrt {x^4+2 x^3-3 x^2-11 x+11}}\right ) \]

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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 {11-11 x-3 x^2+2 x^3+x^4}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[x/Sqrt[11 - 11*x - 3*x^2 + 2*x^3 + x^4],x]

[Out]

Defer[Int][x/Sqrt[11 - 11*x - 3*x^2 + 2*x^3 + x^4], x]

Rubi steps

\begin {align*} \int \frac {x}{\sqrt {11-11 x-3 x^2+2 x^3+x^4}} \, dx &=\int \frac {x}{\sqrt {11-11 x-3 x^2+2 x^3+x^4}} \, dx\\ \end {align*}

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Mathematica [C]  time = 0.56, size = 759, normalized size = 22.32 \begin {gather*} \frac {2 (x-1) \left (x-\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2-11\&,1\right ]\right ) \sqrt {\frac {x-\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2-11\&,2\right ]}{\left (-1+\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2-11\&,2\right ]\right ) \left (x-\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2-11\&,1\right ]\right )}} \sqrt {\frac {x-\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2-11\&,3\right ]}{\left (-1+\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2-11\&,3\right ]\right ) \left (x-\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2-11\&,1\right ]\right )}} \left (\left (\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2-11\&,1\right ]-1\right ) \Pi \left (\frac {-1+\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2-11\&,3\right ]}{-\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2-11\&,1\right ]+\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2-11\&,3\right ]};\sin ^{-1}\left (\sqrt {-\frac {(x-1) \left (\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2-11\&,1\right ]-\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2-11\&,3\right ]\right )}{\left (x-\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2-11\&,1\right ]\right ) \left (-1+\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2-11\&,3\right ]\right )}}\right )|\frac {\left (\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2-11\&,1\right ]-\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2-11\&,2\right ]\right ) \left (-1+\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2-11\&,3\right ]\right )}{\left (-1+\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2-11\&,2\right ]\right ) \left (\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2-11\&,1\right ]-\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2-11\&,3\right ]\right )}\right )-\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2-11\&,1\right ] F\left (\sin ^{-1}\left (\sqrt {-\frac {(x-1) \left (\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2-11\&,1\right ]-\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2-11\&,3\right ]\right )}{\left (x-\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2-11\&,1\right ]\right ) \left (-1+\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2-11\&,3\right ]\right )}}\right )|\frac {\left (\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2-11\&,1\right ]-\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2-11\&,2\right ]\right ) \left (-1+\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2-11\&,3\right ]\right )}{\left (-1+\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2-11\&,2\right ]\right ) \left (\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2-11\&,1\right ]-\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2-11\&,3\right ]\right )}\right )\right )}{\sqrt {x^4+2 x^3-3 x^2-11 x+11} \sqrt {-\frac {(x-1) \left (\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2-11\&,1\right ]-\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2-11\&,3\right ]\right )}{\left (-1+\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2-11\&,3\right ]\right ) \left (x-\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2-11\&,1\right ]\right )}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x/Sqrt[11 - 11*x - 3*x^2 + 2*x^3 + x^4],x]

[Out]

(2*(-1 + x)*(x - Root[-11 + 3*#1^2 + #1^3 & , 1, 0])*(EllipticPi[(-1 + Root[-11 + 3*#1^2 + #1^3 & , 3, 0])/(-R
oot[-11 + 3*#1^2 + #1^3 & , 1, 0] + Root[-11 + 3*#1^2 + #1^3 & , 3, 0]), ArcSin[Sqrt[-(((-1 + x)*(Root[-11 + 3
*#1^2 + #1^3 & , 1, 0] - Root[-11 + 3*#1^2 + #1^3 & , 3, 0]))/((x - Root[-11 + 3*#1^2 + #1^3 & , 1, 0])*(-1 +
Root[-11 + 3*#1^2 + #1^3 & , 3, 0])))]], ((Root[-11 + 3*#1^2 + #1^3 & , 1, 0] - Root[-11 + 3*#1^2 + #1^3 & , 2
, 0])*(-1 + Root[-11 + 3*#1^2 + #1^3 & , 3, 0]))/((-1 + Root[-11 + 3*#1^2 + #1^3 & , 2, 0])*(Root[-11 + 3*#1^2
+ #1^3 & , 1, 0] - Root[-11 + 3*#1^2 + #1^3 & , 3, 0]))]*(-1 + Root[-11 + 3*#1^2 + #1^3 & , 1, 0]) - Elliptic
F[ArcSin[Sqrt[-(((-1 + x)*(Root[-11 + 3*#1^2 + #1^3 & , 1, 0] - Root[-11 + 3*#1^2 + #1^3 & , 3, 0]))/((x - Roo
t[-11 + 3*#1^2 + #1^3 & , 1, 0])*(-1 + Root[-11 + 3*#1^2 + #1^3 & , 3, 0])))]], ((Root[-11 + 3*#1^2 + #1^3 & ,
1, 0] - Root[-11 + 3*#1^2 + #1^3 & , 2, 0])*(-1 + Root[-11 + 3*#1^2 + #1^3 & , 3, 0]))/((-1 + Root[-11 + 3*#1
^2 + #1^3 & , 2, 0])*(Root[-11 + 3*#1^2 + #1^3 & , 1, 0] - Root[-11 + 3*#1^2 + #1^3 & , 3, 0]))]*Root[-11 + 3*
#1^2 + #1^3 & , 1, 0])*Sqrt[(x - Root[-11 + 3*#1^2 + #1^3 & , 2, 0])/((x - Root[-11 + 3*#1^2 + #1^3 & , 1, 0])
*(-1 + Root[-11 + 3*#1^2 + #1^3 & , 2, 0]))]*Sqrt[(x - Root[-11 + 3*#1^2 + #1^3 & , 3, 0])/((x - Root[-11 + 3*
#1^2 + #1^3 & , 1, 0])*(-1 + Root[-11 + 3*#1^2 + #1^3 & , 3, 0]))])/(Sqrt[11 - 11*x - 3*x^2 + 2*x^3 + x^4]*Sqr
t[-(((-1 + x)*(Root[-11 + 3*#1^2 + #1^3 & , 1, 0] - Root[-11 + 3*#1^2 + #1^3 & , 3, 0]))/((x - Root[-11 + 3*#1
^2 + #1^3 & , 1, 0])*(-1 + Root[-11 + 3*#1^2 + #1^3 & , 3, 0])))])

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IntegrateAlgebraic [A]  time = 0.16, size = 34, normalized size = 1.00 \begin {gather*} \frac {2}{3} \tanh ^{-1}\left (\frac {x^2+x-2}{\sqrt {x^4+2 x^3-3 x^2-11 x+11}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x/Sqrt[11 - 11*x - 3*x^2 + 2*x^3 + x^4],x]

[Out]

(2*ArcTanh[(-2 + x + x^2)/Sqrt[11 - 11*x - 3*x^2 + 2*x^3 + x^4]])/3

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fricas [A]  time = 0.43, size = 40, normalized size = 1.18 \begin {gather*} \frac {1}{3} \, \log \left (2 \, x^{3} + 6 \, x^{2} + 2 \, \sqrt {x^{4} + 2 \, x^{3} - 3 \, x^{2} - 11 \, x + 11} {\left (x + 2\right )} - 15\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(x^4+2*x^3-3*x^2-11*x+11)^(1/2),x, algorithm="fricas")

[Out]

1/3*log(2*x^3 + 6*x^2 + 2*sqrt(x^4 + 2*x^3 - 3*x^2 - 11*x + 11)*(x + 2) - 15)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{\sqrt {x^{4} + 2 \, x^{3} - 3 \, x^{2} - 11 \, x + 11}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(x^4+2*x^3-3*x^2-11*x+11)^(1/2),x, algorithm="giac")

[Out]

integrate(x/sqrt(x^4 + 2*x^3 - 3*x^2 - 11*x + 11), x)

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maple [C]  time = 1.32, size = 1625, normalized size = 47.79

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(x^4+2*x^3-3*x^2-11*x+11)^(1/2),x)

[Out]

2*(2+1/4*(36+4*77^(1/2))^(1/3)+1/(36+4*77^(1/2))^(1/3)+1/2*I*3^(1/2)*(1/2*(36+4*77^(1/2))^(1/3)-2/(36+4*77^(1/
2))^(1/3)))*((-3/4*(36+4*77^(1/2))^(1/3)-3/(36+4*77^(1/2))^(1/3)-1/2*I*3^(1/2)*(1/2*(36+4*77^(1/2))^(1/3)-2/(3
6+4*77^(1/2))^(1/3)))*(-1+x)/(-1/4*(36+4*77^(1/2))^(1/3)-1/(36+4*77^(1/2))^(1/3)-2-1/2*I*3^(1/2)*(1/2*(36+4*77
^(1/2))^(1/3)-2/(36+4*77^(1/2))^(1/3)))/(x-1/2*(36+4*77^(1/2))^(1/3)-2/(36+4*77^(1/2))^(1/3)+1))^(1/2)*(x-1/2*
(36+4*77^(1/2))^(1/3)-2/(36+4*77^(1/2))^(1/3)+1)^2*((1/2*(36+4*77^(1/2))^(1/3)+2/(36+4*77^(1/2))^(1/3)-2)*(x+1
/4*(36+4*77^(1/2))^(1/3)+1/(36+4*77^(1/2))^(1/3)+1-1/2*I*3^(1/2)*(1/2*(36+4*77^(1/2))^(1/3)-2/(36+4*77^(1/2))^
(1/3)))/(-1/4*(36+4*77^(1/2))^(1/3)-1/(36+4*77^(1/2))^(1/3)-2+1/2*I*3^(1/2)*(1/2*(36+4*77^(1/2))^(1/3)-2/(36+4
*77^(1/2))^(1/3)))/(x-1/2*(36+4*77^(1/2))^(1/3)-2/(36+4*77^(1/2))^(1/3)+1))^(1/2)*((1/2*(36+4*77^(1/2))^(1/3)+
2/(36+4*77^(1/2))^(1/3)-2)*(x+1/4*(36+4*77^(1/2))^(1/3)+1/(36+4*77^(1/2))^(1/3)+1+1/2*I*3^(1/2)*(1/2*(36+4*77^
(1/2))^(1/3)-2/(36+4*77^(1/2))^(1/3)))/(-1/4*(36+4*77^(1/2))^(1/3)-1/(36+4*77^(1/2))^(1/3)-2-1/2*I*3^(1/2)*(1/
2*(36+4*77^(1/2))^(1/3)-2/(36+4*77^(1/2))^(1/3)))/(x-1/2*(36+4*77^(1/2))^(1/3)-2/(36+4*77^(1/2))^(1/3)+1))^(1/
2)/(-3/4*(36+4*77^(1/2))^(1/3)-3/(36+4*77^(1/2))^(1/3)-1/2*I*3^(1/2)*(1/2*(36+4*77^(1/2))^(1/3)-2/(36+4*77^(1/
2))^(1/3)))/(1/2*(36+4*77^(1/2))^(1/3)+2/(36+4*77^(1/2))^(1/3)-2)/((-1+x)*(x-1/2*(36+4*77^(1/2))^(1/3)-2/(36+4
*77^(1/2))^(1/3)+1)*(x+1/4*(36+4*77^(1/2))^(1/3)+1/(36+4*77^(1/2))^(1/3)+1-1/2*I*3^(1/2)*(1/2*(36+4*77^(1/2))^
(1/3)-2/(36+4*77^(1/2))^(1/3)))*(x+1/4*(36+4*77^(1/2))^(1/3)+1/(36+4*77^(1/2))^(1/3)+1+1/2*I*3^(1/2)*(1/2*(36+
4*77^(1/2))^(1/3)-2/(36+4*77^(1/2))^(1/3))))^(1/2)*((1/2*(36+4*77^(1/2))^(1/3)+2/(36+4*77^(1/2))^(1/3)-1)*Elli
pticF(((-3/4*(36+4*77^(1/2))^(1/3)-3/(36+4*77^(1/2))^(1/3)-1/2*I*3^(1/2)*(1/2*(36+4*77^(1/2))^(1/3)-2/(36+4*77
^(1/2))^(1/3)))*(-1+x)/(-1/4*(36+4*77^(1/2))^(1/3)-1/(36+4*77^(1/2))^(1/3)-2-1/2*I*3^(1/2)*(1/2*(36+4*77^(1/2)
)^(1/3)-2/(36+4*77^(1/2))^(1/3)))/(x-1/2*(36+4*77^(1/2))^(1/3)-2/(36+4*77^(1/2))^(1/3)+1))^(1/2),((3/4*(36+4*7
7^(1/2))^(1/3)+3/(36+4*77^(1/2))^(1/3)-1/2*I*3^(1/2)*(1/2*(36+4*77^(1/2))^(1/3)-2/(36+4*77^(1/2))^(1/3)))*(2+1
/4*(36+4*77^(1/2))^(1/3)+1/(36+4*77^(1/2))^(1/3)+1/2*I*3^(1/2)*(1/2*(36+4*77^(1/2))^(1/3)-2/(36+4*77^(1/2))^(1
/3)))/(2+1/4*(36+4*77^(1/2))^(1/3)+1/(36+4*77^(1/2))^(1/3)-1/2*I*3^(1/2)*(1/2*(36+4*77^(1/2))^(1/3)-2/(36+4*77
^(1/2))^(1/3)))/(3/4*(36+4*77^(1/2))^(1/3)+3/(36+4*77^(1/2))^(1/3)+1/2*I*3^(1/2)*(1/2*(36+4*77^(1/2))^(1/3)-2/
(36+4*77^(1/2))^(1/3))))^(1/2))+(2-1/2*(36+4*77^(1/2))^(1/3)-2/(36+4*77^(1/2))^(1/3))*EllipticPi(((-3/4*(36+4*
77^(1/2))^(1/3)-3/(36+4*77^(1/2))^(1/3)-1/2*I*3^(1/2)*(1/2*(36+4*77^(1/2))^(1/3)-2/(36+4*77^(1/2))^(1/3)))*(-1
+x)/(-1/4*(36+4*77^(1/2))^(1/3)-1/(36+4*77^(1/2))^(1/3)-2-1/2*I*3^(1/2)*(1/2*(36+4*77^(1/2))^(1/3)-2/(36+4*77^
(1/2))^(1/3)))/(x-1/2*(36+4*77^(1/2))^(1/3)-2/(36+4*77^(1/2))^(1/3)+1))^(1/2),(-1/4*(36+4*77^(1/2))^(1/3)-1/(3
6+4*77^(1/2))^(1/3)-2-1/2*I*3^(1/2)*(1/2*(36+4*77^(1/2))^(1/3)-2/(36+4*77^(1/2))^(1/3)))/(-3/4*(36+4*77^(1/2))
^(1/3)-3/(36+4*77^(1/2))^(1/3)-1/2*I*3^(1/2)*(1/2*(36+4*77^(1/2))^(1/3)-2/(36+4*77^(1/2))^(1/3))),((3/4*(36+4*
77^(1/2))^(1/3)+3/(36+4*77^(1/2))^(1/3)-1/2*I*3^(1/2)*(1/2*(36+4*77^(1/2))^(1/3)-2/(36+4*77^(1/2))^(1/3)))*(2+
1/4*(36+4*77^(1/2))^(1/3)+1/(36+4*77^(1/2))^(1/3)+1/2*I*3^(1/2)*(1/2*(36+4*77^(1/2))^(1/3)-2/(36+4*77^(1/2))^(
1/3)))/(2+1/4*(36+4*77^(1/2))^(1/3)+1/(36+4*77^(1/2))^(1/3)-1/2*I*3^(1/2)*(1/2*(36+4*77^(1/2))^(1/3)-2/(36+4*7
7^(1/2))^(1/3)))/(3/4*(36+4*77^(1/2))^(1/3)+3/(36+4*77^(1/2))^(1/3)+1/2*I*3^(1/2)*(1/2*(36+4*77^(1/2))^(1/3)-2
/(36+4*77^(1/2))^(1/3))))^(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} + 2 \, x^{3} - 3 \, x^{2} - 11 \, x + 11}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(x^4+2*x^3-3*x^2-11*x+11)^(1/2),x, algorithm="maxima")

[Out]

integrate(x/sqrt(x^4 + 2*x^3 - 3*x^2 - 11*x + 11), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {x}{\sqrt {x^4+2\,x^3-3\,x^2-11\,x+11}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(2*x^3 - 3*x^2 - 11*x + x^4 + 11)^(1/2),x)

[Out]

int(x/(2*x^3 - 3*x^2 - 11*x + x^4 + 11)^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{\sqrt {\left (x - 1\right ) \left (x^{3} + 3 x^{2} - 11\right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(x**4+2*x**3-3*x**2-11*x+11)**(1/2),x)

[Out]

Integral(x/sqrt((x - 1)*(x**3 + 3*x**2 - 11)), x)

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