3.9.79 \(\int \frac {x}{\sqrt {1+4 x+3 x^2-2 x^3+x^4}} \, dx\)

Optimal. Leaf size=67 \[ \frac {1}{3} \tanh ^{-1}\left (\frac {(x+1) \sqrt {x^4-2 x^3+3 x^2+4 x+1}}{x^3}\right )+\tanh ^{-1}\left (\frac {\sqrt {x^4-2 x^3+3 x^2+4 x+1}-2 x-1}{x^2}\right ) \]

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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 {1+4 x+3 x^2-2 x^3+x^4}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[x/Sqrt[1 + 4*x + 3*x^2 - 2*x^3 + x^4],x]

[Out]

Defer[Int][x/Sqrt[1 + 4*x + 3*x^2 - 2*x^3 + x^4], x]

Rubi steps

\begin {align*} \int \frac {x}{\sqrt {1+4 x+3 x^2-2 x^3+x^4}} \, dx &=\int \frac {x}{\sqrt {1+4 x+3 x^2-2 x^3+x^4}} \, dx\\ \end {align*}

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Mathematica [C]  time = 2.25, size = 1105, normalized size = 16.49

result too large to display

Warning: Unable to verify antiderivative.

[In]

Integrate[x/Sqrt[1 + 4*x + 3*x^2 - 2*x^3 + x^4],x]

[Out]

((-1)^(1/4)*(1 + 2*(-3)^(1/4) + I*Sqrt[3] - 2*x)*(-I + (2*I)*(-3)^(1/4) + Sqrt[3] + (2*I)*x)*(-2*EllipticF[Arc
Sin[Sqrt[-(((-1)^(3/4)*(-1 + 2*(-3)^(1/4) - I*Sqrt[3] + 2*x)*(1 + 2*(-3)^(1/4) + I*Sqrt[3] - 2*Root[1 + 4*#1 +
3*#1^2 - 2*#1^3 + #1^4 & , 2, 0]))/(x - Root[1 + 4*#1 + 3*#1^2 - 2*#1^3 + #1^4 & , 2, 0]))]/(2*Sqrt[2]*3^(1/8
))], (-8*(-3)^(1/4)*(Root[1 + 4*#1 + 3*#1^2 - 2*#1^3 + #1^4 & , 2, 0] - Root[1 + 4*#1 + 3*#1^2 - 2*#1^3 + #1^4
& , 3, 0]))/((1 + 2*(-3)^(1/4) + I*Sqrt[3] - 2*Root[1 + 4*#1 + 3*#1^2 - 2*#1^3 + #1^4 & , 2, 0])*(-1 + 2*(-3)
^(1/4) - I*Sqrt[3] + 2*Root[1 + 4*#1 + 3*#1^2 - 2*#1^3 + #1^4 & , 3, 0]))]*Root[1 + 4*#1 + 3*#1^2 - 2*#1^3 + #
1^4 & , 2, 0] + EllipticPi[(4*(-3)^(1/4))/(1 + 2*(-3)^(1/4) + I*Sqrt[3] - 2*Root[1 + 4*#1 + 3*#1^2 - 2*#1^3 +
#1^4 & , 2, 0]), ArcSin[Sqrt[-(((-1)^(3/4)*(-1 + 2*(-3)^(1/4) - I*Sqrt[3] + 2*x)*(1 + 2*(-3)^(1/4) + I*Sqrt[3]
- 2*Root[1 + 4*#1 + 3*#1^2 - 2*#1^3 + #1^4 & , 2, 0]))/(x - Root[1 + 4*#1 + 3*#1^2 - 2*#1^3 + #1^4 & , 2, 0])
)]/(2*Sqrt[2]*3^(1/8))], (-8*(-3)^(1/4)*(Root[1 + 4*#1 + 3*#1^2 - 2*#1^3 + #1^4 & , 2, 0] - Root[1 + 4*#1 + 3*
#1^2 - 2*#1^3 + #1^4 & , 3, 0]))/((1 + 2*(-3)^(1/4) + I*Sqrt[3] - 2*Root[1 + 4*#1 + 3*#1^2 - 2*#1^3 + #1^4 & ,
2, 0])*(-1 + 2*(-3)^(1/4) - I*Sqrt[3] + 2*Root[1 + 4*#1 + 3*#1^2 - 2*#1^3 + #1^4 & , 3, 0]))]*(-1 + 2*(-3)^(1
/4) - I*Sqrt[3] + 2*Root[1 + 4*#1 + 3*#1^2 - 2*#1^3 + #1^4 & , 2, 0]))*Sqrt[((-1 + 2*(-3)^(1/4) - I*Sqrt[3] +
2*Root[1 + 4*#1 + 3*#1^2 - 2*#1^3 + #1^4 & , 2, 0])*(x - Root[1 + 4*#1 + 3*#1^2 - 2*#1^3 + #1^4 & , 3, 0]))/((
x - Root[1 + 4*#1 + 3*#1^2 - 2*#1^3 + #1^4 & , 2, 0])*(-1 + 2*(-3)^(1/4) - I*Sqrt[3] + 2*Root[1 + 4*#1 + 3*#1^
2 - 2*#1^3 + #1^4 & , 3, 0]))])/(Sqrt[1 + 4*x + 3*x^2 - 2*x^3 + x^4]*Sqrt[((1 + 2*(-3)^(1/4) + I*Sqrt[3] - 2*x
)*(-I + (2*I)*(-3)^(1/4) + Sqrt[3] + (2*I)*x)*(1 + 2*(-3)^(1/4) + I*Sqrt[3] - 2*Root[1 + 4*#1 + 3*#1^2 - 2*#1^
3 + #1^4 & , 2, 0])*(-1 + 2*(-3)^(1/4) - I*Sqrt[3] + 2*Root[1 + 4*#1 + 3*#1^2 - 2*#1^3 + #1^4 & , 2, 0]))/(x -
Root[1 + 4*#1 + 3*#1^2 - 2*#1^3 + #1^4 & , 2, 0])^2])

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IntegrateAlgebraic [A]  time = 13.56, size = 67, normalized size = 1.00 \begin {gather*} \frac {1}{3} \tanh ^{-1}\left (\frac {(1+x) \sqrt {1+4 x+3 x^2-2 x^3+x^4}}{x^3}\right )+\tanh ^{-1}\left (\frac {-1-2 x+\sqrt {1+4 x+3 x^2-2 x^3+x^4}}{x^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x/Sqrt[1 + 4*x + 3*x^2 - 2*x^3 + x^4],x]

[Out]

ArcTanh[((1 + x)*Sqrt[1 + 4*x + 3*x^2 - 2*x^3 + x^4])/x^3]/3 + ArcTanh[(-1 - 2*x + Sqrt[1 + 4*x + 3*x^2 - 2*x^
3 + x^4])/x^2]

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fricas [A]  time = 0.51, size = 70, normalized size = 1.04 \begin {gather*} \frac {1}{6} \, \log \left (2 \, x^{6} - 12 \, x^{5} + 36 \, x^{4} - 56 \, x^{3} + 42 \, x^{2} + 2 \, \sqrt {x^{4} - 2 \, x^{3} + 3 \, x^{2} + 4 \, x + 1} {\left (x^{4} - 5 \, x^{3} + 12 \, x^{2} - 14 \, x + 7\right )} - 13\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*log(2*x^6 - 12*x^5 + 36*x^4 - 56*x^3 + 42*x^2 + 2*sqrt(x^4 - 2*x^3 + 3*x^2 + 4*x + 1)*(x^4 - 5*x^3 + 12*x^
2 - 14*x + 7) - 13)

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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} + 4 \, x + 1}}\,{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+4*x+1)^(1/2),x, algorithm="giac")

[Out]

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

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maple [B]  time = 1.68, size = 151, normalized size = 2.25

method	result	size
trager	\(-\frac {\ln \left (-2 x^{6}+2 \sqrt {x^{4}-2 x^{3}+3 x^{2}+4 x +1}\, x^{4}+12 x^{5}-10 \sqrt {x^{4}-2 x^{3}+3 x^{2}+4 x +1}\, x^{3}-36 x^{4}+24 \sqrt {x^{4}-2 x^{3}+3 x^{2}+4 x +1}\, x^{2}+56 x^{3}-28 x \sqrt {x^{4}-2 x^{3}+3 x^{2}+4 x +1}-42 x^{2}+14 \sqrt {x^{4}-2 x^{3}+3 x^{2}+4 x +1}+13\right )}{6}\)	\(151\)
default	\(\text {Expression too large to display}\)	\(1609\)
elliptic	\(\text {Expression too large to display}\)	\(1609\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*ln(-2*x^6+2*(x^4-2*x^3+3*x^2+4*x+1)^(1/2)*x^4+12*x^5-10*(x^4-2*x^3+3*x^2+4*x+1)^(1/2)*x^3-36*x^4+24*(x^4-
2*x^3+3*x^2+4*x+1)^(1/2)*x^2+56*x^3-28*x*(x^4-2*x^3+3*x^2+4*x+1)^(1/2)-42*x^2+14*(x^4-2*x^3+3*x^2+4*x+1)^(1/2)
+13)

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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} + 4 \, x + 1}}\,{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+4*x+1)^(1/2),x, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x}{\sqrt {x^4-2\,x^3+3\,x^2+4\,x+1}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x/sqrt(x**4 - 2*x**3 + 3*x**2 + 4*x + 1), x)

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