3.9.25 \(\int \frac {x}{\sqrt {1-4 x+3 x^2+2 x^3+x^4}} \, dx\)
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 = 1.97, size = 1189, normalized size = 17.75
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)*(x - Root[1 - 4*#1 + 3*#1^2 + 2*#1^3 + #1^4 & , 2, 0])^2*Sqrt[((-1 + 2*(-3)^(1/4) - I*Sqrt[3] - 2*
x)*(1 + 2*(-3)^(1/4) + I*Sqrt[3] + 2*x)*(-I + (2*I)*(-3)^(1/4) + Sqrt[3] - (2*I)*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]*(-2*EllipticF[ArcSin[Sqrt[((-1)^(3/4)*(1/2 + (-3)^(1/4)
+ (I/2)*Sqrt[3] + x)*(1/2 - (-3)^(1/4) + (I/2)*Sqrt[3] + 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])]/(Sqrt[2]*3^(1/8))], (2*(-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/2)*Sq
rt[3] + Root[1 - 4*#1 + 3*#1^2 + 2*#1^3 + #1^4 & , 2, 0])*(1/2 + (-3)^(1/4) + (I/2)*Sqrt[3] + 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/2)*Sqrt[3] + x)*(1/2 - (-3)^(1/4) + (I/2)*Sqrt[3] + 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])]/(Sqrt[2]*3^(1/8))], (2*(-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/2)*Sqrt[3] + Root[1 - 4*#1 + 3*#1^2 + 2*#1^3 + #1^4 & , 2, 0])*(1/2 + (-3)^(1/4) + (I/2)*Sqrt[
3] + 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]*(-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]))
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IntegrateAlgebraic [A] time = 13.19, size = 67, normalized size = 1.00 \begin {gather*} \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 ) \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 = 0.46, 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 \, {\left (x^{4} + 5 \, x^{3} + 12 \, x^{2} + 14 \, x + 7\right )} \sqrt {x^{4} + 2 \, x^{3} + 3 \, x^{2} - 4 \, x + 1} - 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*(x^4 + 5*x^3 + 12*x^2 + 14*x + 7)*sqrt(x^4 + 2*x^3 + 3*x
^2 - 4*x + 1) - 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 [C] time = 1.05, size = 1609, normalized size = 24.01 \begin {gather*} \text {Expression too large to display} \end {gather*}
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)
[Out]
2*(RootOf(_Z^4+2*_Z^3+3*_Z^2-4*_Z+1,index=1)-RootOf(_Z^4+2*_Z^3+3*_Z^2-4*_Z+1,index=4))*((RootOf(_Z^4+2*_Z^3+3
*_Z^2-4*_Z+1,index=4)-RootOf(_Z^4+2*_Z^3+3*_Z^2-4*_Z+1,index=2))*(x-RootOf(_Z^4+2*_Z^3+3*_Z^2-4*_Z+1,index=1))
/(RootOf(_Z^4+2*_Z^3+3*_Z^2-4*_Z+1,index=4)-RootOf(_Z^4+2*_Z^3+3*_Z^2-4*_Z+1,index=1))/(x-RootOf(_Z^4+2*_Z^3+3
*_Z^2-4*_Z+1,index=2)))^(1/2)*(x-RootOf(_Z^4+2*_Z^3+3*_Z^2-4*_Z+1,index=2))^2*((RootOf(_Z^4+2*_Z^3+3*_Z^2-4*_Z
+1,index=2)-RootOf(_Z^4+2*_Z^3+3*_Z^2-4*_Z+1,index=1))*(x-RootOf(_Z^4+2*_Z^3+3*_Z^2-4*_Z+1,index=3))/(RootOf(_
Z^4+2*_Z^3+3*_Z^2-4*_Z+1,index=3)-RootOf(_Z^4+2*_Z^3+3*_Z^2-4*_Z+1,index=1))/(x-RootOf(_Z^4+2*_Z^3+3*_Z^2-4*_Z
+1,index=2)))^(1/2)*((RootOf(_Z^4+2*_Z^3+3*_Z^2-4*_Z+1,index=2)-RootOf(_Z^4+2*_Z^3+3*_Z^2-4*_Z+1,index=1))*(x-
RootOf(_Z^4+2*_Z^3+3*_Z^2-4*_Z+1,index=4))/(RootOf(_Z^4+2*_Z^3+3*_Z^2-4*_Z+1,index=4)-RootOf(_Z^4+2*_Z^3+3*_Z^
2-4*_Z+1,index=1))/(x-RootOf(_Z^4+2*_Z^3+3*_Z^2-4*_Z+1,index=2)))^(1/2)/(RootOf(_Z^4+2*_Z^3+3*_Z^2-4*_Z+1,inde
x=4)-RootOf(_Z^4+2*_Z^3+3*_Z^2-4*_Z+1,index=2))/(RootOf(_Z^4+2*_Z^3+3*_Z^2-4*_Z+1,index=2)-RootOf(_Z^4+2*_Z^3+
3*_Z^2-4*_Z+1,index=1))/((x-RootOf(_Z^4+2*_Z^3+3*_Z^2-4*_Z+1,index=1))*(x-RootOf(_Z^4+2*_Z^3+3*_Z^2-4*_Z+1,ind
ex=2))*(x-RootOf(_Z^4+2*_Z^3+3*_Z^2-4*_Z+1,index=3))*(x-RootOf(_Z^4+2*_Z^3+3*_Z^2-4*_Z+1,index=4)))^(1/2)*(Roo
tOf(_Z^4+2*_Z^3+3*_Z^2-4*_Z+1,index=2)*EllipticF(((RootOf(_Z^4+2*_Z^3+3*_Z^2-4*_Z+1,index=4)-RootOf(_Z^4+2*_Z^
3+3*_Z^2-4*_Z+1,index=2))*(x-RootOf(_Z^4+2*_Z^3+3*_Z^2-4*_Z+1,index=1))/(RootOf(_Z^4+2*_Z^3+3*_Z^2-4*_Z+1,inde
x=4)-RootOf(_Z^4+2*_Z^3+3*_Z^2-4*_Z+1,index=1))/(x-RootOf(_Z^4+2*_Z^3+3*_Z^2-4*_Z+1,index=2)))^(1/2),((RootOf(
_Z^4+2*_Z^3+3*_Z^2-4*_Z+1,index=2)-RootOf(_Z^4+2*_Z^3+3*_Z^2-4*_Z+1,index=3))*(RootOf(_Z^4+2*_Z^3+3*_Z^2-4*_Z+
1,index=1)-RootOf(_Z^4+2*_Z^3+3*_Z^2-4*_Z+1,index=4))/(-RootOf(_Z^4+2*_Z^3+3*_Z^2-4*_Z+1,index=3)+RootOf(_Z^4+
2*_Z^3+3*_Z^2-4*_Z+1,index=1))/(RootOf(_Z^4+2*_Z^3+3*_Z^2-4*_Z+1,index=2)-RootOf(_Z^4+2*_Z^3+3*_Z^2-4*_Z+1,ind
ex=4)))^(1/2))+(RootOf(_Z^4+2*_Z^3+3*_Z^2-4*_Z+1,index=1)-RootOf(_Z^4+2*_Z^3+3*_Z^2-4*_Z+1,index=2))*EllipticP
i(((RootOf(_Z^4+2*_Z^3+3*_Z^2-4*_Z+1,index=4)-RootOf(_Z^4+2*_Z^3+3*_Z^2-4*_Z+1,index=2))*(x-RootOf(_Z^4+2*_Z^3
+3*_Z^2-4*_Z+1,index=1))/(RootOf(_Z^4+2*_Z^3+3*_Z^2-4*_Z+1,index=4)-RootOf(_Z^4+2*_Z^3+3*_Z^2-4*_Z+1,index=1))
/(x-RootOf(_Z^4+2*_Z^3+3*_Z^2-4*_Z+1,index=2)))^(1/2),(RootOf(_Z^4+2*_Z^3+3*_Z^2-4*_Z+1,index=4)-RootOf(_Z^4+2
*_Z^3+3*_Z^2-4*_Z+1,index=1))/(RootOf(_Z^4+2*_Z^3+3*_Z^2-4*_Z+1,index=4)-RootOf(_Z^4+2*_Z^3+3*_Z^2-4*_Z+1,inde
x=2)),((RootOf(_Z^4+2*_Z^3+3*_Z^2-4*_Z+1,index=2)-RootOf(_Z^4+2*_Z^3+3*_Z^2-4*_Z+1,index=3))*(RootOf(_Z^4+2*_Z
^3+3*_Z^2-4*_Z+1,index=1)-RootOf(_Z^4+2*_Z^3+3*_Z^2-4*_Z+1,index=4))/(-RootOf(_Z^4+2*_Z^3+3*_Z^2-4*_Z+1,index=
3)+RootOf(_Z^4+2*_Z^3+3*_Z^2-4*_Z+1,index=1))/(RootOf(_Z^4+2*_Z^3+3*_Z^2-4*_Z+1,index=2)-RootOf(_Z^4+2*_Z^3+3*
_Z^2-4*_Z+1,index=4)))^(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} - 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/(3*x^2 - 4*x + 2*x^3 + x^4 + 1)^(1/2),x)
[Out]
int(x/(3*x^2 - 4*x + 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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