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

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

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Rubi [F]  time = 0.07, 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 {-3+3 x-3 x^2+2 x^3+x^4}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [C]  time = 0.93, size = 768, normalized size = 22.59 \begin {gather*} \frac {2 (x-1) \left (x-\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+3\&,3\right ]\right ) \sqrt {\frac {\left (\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+3\&,1\right ]-1\right ) \left (x-\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+3\&,2\right ]\right )}{\left (-1+\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+3\&,2\right ]\right ) \left (x-\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+3\&,1\right ]\right )}} \left (\left (\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+3\&,1\right ]-1\right ) \Pi \left (\frac {-1+\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+3\&,3\right ]}{-\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+3\&,1\right ]+\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+3\&,3\right ]};\sin ^{-1}\left (\sqrt {-\frac {(x-1) \left (\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+3\&,1\right ]-\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+3\&,3\right ]\right )}{\left (x-\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+3\&,1\right ]\right ) \left (-1+\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+3\&,3\right ]\right )}}\right )|\frac {\left (\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+3\&,1\right ]-\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+3\&,2\right ]\right ) \left (-1+\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+3\&,3\right ]\right )}{\left (-1+\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+3\&,2\right ]\right ) \left (\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+3\&,1\right ]-\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+3\&,3\right ]\right )}\right )-\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+3\&,1\right ] F\left (\sin ^{-1}\left (\sqrt {-\frac {(x-1) \left (\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+3\&,1\right ]-\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+3\&,3\right ]\right )}{\left (x-\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+3\&,1\right ]\right ) \left (-1+\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+3\&,3\right ]\right )}}\right )|\frac {\left (\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+3\&,1\right ]-\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+3\&,2\right ]\right ) \left (-1+\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+3\&,3\right ]\right )}{\left (-1+\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+3\&,2\right ]\right ) \left (\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+3\&,1\right ]-\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+3\&,3\right ]\right )}\right )\right )}{\sqrt {x^4+2 x^3-3 x^2+3 x-3} \left (-1+\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+3\&,3\right ]\right ) \sqrt {-\frac {(x-1) \left (\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+3\&,1\right ]-1\right ) \left (\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+3\&,1\right ]-\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+3\&,3\right ]\right ) \left (x-\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+3\&,3\right ]\right )}{\left (-1+\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+3\&,3\right ]\right )^2 \left (x-\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+3\&,1\right ]\right )^2}}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(2*(-1 + x)*(EllipticPi[(-1 + Root[3 + 3*#1^2 + #1^3 & , 3, 0])/(-Root[3 + 3*#1^2 + #1^3 & , 1, 0] + Root[3 +
3*#1^2 + #1^3 & , 3, 0]), ArcSin[Sqrt[-(((-1 + x)*(Root[3 + 3*#1^2 + #1^3 & , 1, 0] - Root[3 + 3*#1^2 + #1^3 &
, 3, 0]))/((x - Root[3 + 3*#1^2 + #1^3 & , 1, 0])*(-1 + Root[3 + 3*#1^2 + #1^3 & , 3, 0])))]], ((Root[3 + 3*#
1^2 + #1^3 & , 1, 0] - Root[3 + 3*#1^2 + #1^3 & , 2, 0])*(-1 + Root[3 + 3*#1^2 + #1^3 & , 3, 0]))/((-1 + Root[
3 + 3*#1^2 + #1^3 & , 2, 0])*(Root[3 + 3*#1^2 + #1^3 & , 1, 0] - Root[3 + 3*#1^2 + #1^3 & , 3, 0]))]*(-1 + Roo
t[3 + 3*#1^2 + #1^3 & , 1, 0]) - EllipticF[ArcSin[Sqrt[-(((-1 + x)*(Root[3 + 3*#1^2 + #1^3 & , 1, 0] - Root[3
+ 3*#1^2 + #1^3 & , 3, 0]))/((x - Root[3 + 3*#1^2 + #1^3 & , 1, 0])*(-1 + Root[3 + 3*#1^2 + #1^3 & , 3, 0])))]
], ((Root[3 + 3*#1^2 + #1^3 & , 1, 0] - Root[3 + 3*#1^2 + #1^3 & , 2, 0])*(-1 + Root[3 + 3*#1^2 + #1^3 & , 3,
0]))/((-1 + Root[3 + 3*#1^2 + #1^3 & , 2, 0])*(Root[3 + 3*#1^2 + #1^3 & , 1, 0] - Root[3 + 3*#1^2 + #1^3 & , 3
, 0]))]*Root[3 + 3*#1^2 + #1^3 & , 1, 0])*Sqrt[((-1 + Root[3 + 3*#1^2 + #1^3 & , 1, 0])*(x - Root[3 + 3*#1^2 +
#1^3 & , 2, 0]))/((x - Root[3 + 3*#1^2 + #1^3 & , 1, 0])*(-1 + Root[3 + 3*#1^2 + #1^3 & , 2, 0]))]*(x - Root[
3 + 3*#1^2 + #1^3 & , 3, 0]))/(Sqrt[-3 + 3*x - 3*x^2 + 2*x^3 + x^4]*Sqrt[-(((-1 + x)*(-1 + Root[3 + 3*#1^2 + #
1^3 & , 1, 0])*(x - Root[3 + 3*#1^2 + #1^3 & , 3, 0])*(Root[3 + 3*#1^2 + #1^3 & , 1, 0] - Root[3 + 3*#1^2 + #1
^3 & , 3, 0]))/((x - Root[3 + 3*#1^2 + #1^3 & , 1, 0])^2*(-1 + Root[3 + 3*#1^2 + #1^3 & , 3, 0])^2))]*(-1 + Ro
ot[3 + 3*#1^2 + #1^3 & , 3, 0]))

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IntegrateAlgebraic [A]  time = 0.13, size = 34, normalized size = 1.00 \begin {gather*} \frac {2}{3} \tanh ^{-1}\left (\frac {-2+x+x^2}{\sqrt {-3+3 x-3 x^2+2 x^3+x^4}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.51, 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} + 3 \, x - 3} {\left (x + 2\right )} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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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} + 3 \, x - 3}}\,{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+3*x-3)^(1/2),x, algorithm="giac")

[Out]

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

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maple [A]  time = 1.39, size = 61, normalized size = 1.79

method	result	size
trager	\(\frac {\ln \left (2 x^{3}+2 x \sqrt {x^{4}+2 x^{3}-3 x^{2}+3 x -3}+6 x^{2}+4 \sqrt {x^{4}+2 x^{3}-3 x^{2}+3 x -3}-1\right )}{3}\)	\(61\)
default	\(\text {Expression too large to display}\)	\(1633\)
elliptic	\(\text {Expression too large to display}\)	\(1633\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(x^4+2*x^3-3*x^2+3*x-3)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/3*ln(2*x^3+2*x*(x^4+2*x^3-3*x^2+3*x-3)^(1/2)+6*x^2+4*(x^4+2*x^3-3*x^2+3*x-3)^(1/2)-1)

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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} + 3 \, x - 3}}\,{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+3*x-3)^(1/2),x, algorithm="maxima")

[Out]

integrate(x/sqrt(x^4 + 2*x^3 - 3*x^2 + 3*x - 3), 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+3\,x-3}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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