3.9.80 \(\int \frac {x}{\sqrt {1-4 x+3 x^2+2 x^3+x^4}} \, dx\) [880]

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

[Out]

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

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Rubi [F]
time = 0.04, 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] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 11.16, size = 1189, normalized size = 17.75 \begin {gather*} \frac {\sqrt [4]{-1} \left (x-\text {Root}\left [\text {$\#$1}^4+2 \text {$\#$1}^3+3 \text {$\#$1}^2-4 \text {$\#$1}+1\&,2\right ]\right )^2 \sqrt {\frac {\left (-1+2 \sqrt [4]{-3}-i \sqrt {3}-2 x\right ) \left (1+2 \sqrt [4]{-3}+i \sqrt {3}+2 x\right ) \left (-i+2 i \sqrt [4]{-3}+\sqrt {3}-2 i \text {Root}\left [\text {$\#$1}^4+2 \text {$\#$1}^3+3 \text {$\#$1}^2-4 \text {$\#$1}+1\&,2\right ]\right ) \left (1+2 \sqrt [4]{-3}+i \sqrt {3}+2 \text {Root}\left [\text {$\#$1}^4+2 \text {$\#$1}^3+3 \text {$\#$1}^2-4 \text {$\#$1}+1\&,2\right ]\right )}{\left (x-\text {Root}\left [\text {$\#$1}^4+2 \text {$\#$1}^3+3 \text {$\#$1}^2-4 \text {$\#$1}+1\&,2\right ]\right )^2}} \left (-2 F\left (\text {ArcSin}\left (\frac {\sqrt {\frac {(-1)^{3/4} \left (\frac {1}{2}+\sqrt [4]{-3}+\frac {i \sqrt {3}}{2}+x\right ) \left (\frac {1}{2}-\sqrt [4]{-3}+\frac {i \sqrt {3}}{2}+\text {Root}\left [\text {$\#$1}^4+2 \text {$\#$1}^3+3 \text {$\#$1}^2-4 \text {$\#$1}+1\&,2\right ]\right )}{x-\text {Root}\left [\text {$\#$1}^4+2 \text {$\#$1}^3+3 \text {$\#$1}^2-4 \text {$\#$1}+1\&,2\right ]}}}{\sqrt {2} \sqrt [8]{3}}\right )|\frac {2 \sqrt [4]{-3} \left (\text {Root}\left [\text {$\#$1}^4+2 \text {$\#$1}^3+3 \text {$\#$1}^2-4 \text {$\#$1}+1\&,2\right ]-\text {Root}\left [\text {$\#$1}^4+2 \text {$\#$1}^3+3 \text {$\#$1}^2-4 \text {$\#$1}+1\&,3\right ]\right )}{\left (\frac {1}{2}-\sqrt [4]{-3}+\frac {i \sqrt {3}}{2}+\text {Root}\left [\text {$\#$1}^4+2 \text {$\#$1}^3+3 \text {$\#$1}^2-4 \text {$\#$1}+1\&,2\right ]\right ) \left (\frac {1}{2}+\sqrt [4]{-3}+\frac {i \sqrt {3}}{2}+\text {Root}\left [\text {$\#$1}^4+2 \text {$\#$1}^3+3 \text {$\#$1}^2-4 \text {$\#$1}+1\&,3\right ]\right )}\right ) \text {Root}\left [\text {$\#$1}^4+2 \text {$\#$1}^3+3 \text {$\#$1}^2-4 \text {$\#$1}+1\&,2\right ]+\Pi \left (\frac {4 \sqrt [4]{-3}}{-1+2 \sqrt [4]{-3}-i \sqrt {3}-2 \text {Root}\left [\text {$\#$1}^4+2 \text {$\#$1}^3+3 \text {$\#$1}^2-4 \text {$\#$1}+1\&,2\right ]};\text {ArcSin}\left (\frac {\sqrt {\frac {(-1)^{3/4} \left (\frac {1}{2}+\sqrt [4]{-3}+\frac {i \sqrt {3}}{2}+x\right ) \left (\frac {1}{2}-\sqrt [4]{-3}+\frac {i \sqrt {3}}{2}+\text {Root}\left [\text {$\#$1}^4+2 \text {$\#$1}^3+3 \text {$\#$1}^2-4 \text {$\#$1}+1\&,2\right ]\right )}{x-\text {Root}\left [\text {$\#$1}^4+2 \text {$\#$1}^3+3 \text {$\#$1}^2-4 \text {$\#$1}+1\&,2\right ]}}}{\sqrt {2} \sqrt [8]{3}}\right )|\frac {2 \sqrt [4]{-3} \left (\text {Root}\left [\text {$\#$1}^4+2 \text {$\#$1}^3+3 \text {$\#$1}^2-4 \text {$\#$1}+1\&,2\right ]-\text {Root}\left [\text {$\#$1}^4+2 \text {$\#$1}^3+3 \text {$\#$1}^2-4 \text {$\#$1}+1\&,3\right ]\right )}{\left (\frac {1}{2}-\sqrt [4]{-3}+\frac {i \sqrt {3}}{2}+\text {Root}\left [\text {$\#$1}^4+2 \text {$\#$1}^3+3 \text {$\#$1}^2-4 \text {$\#$1}+1\&,2\right ]\right ) \left (\frac {1}{2}+\sqrt [4]{-3}+\frac {i \sqrt {3}}{2}+\text {Root}\left [\text {$\#$1}^4+2 \text {$\#$1}^3+3 \text {$\#$1}^2-4 \text {$\#$1}+1\&,3\right ]\right )}\right ) \left (1+2 \sqrt [4]{-3}+i \sqrt {3}+2 \text {Root}\left [\text {$\#$1}^4+2 \text {$\#$1}^3+3 \text {$\#$1}^2-4 \text {$\#$1}+1\&,2\right ]\right )\right ) \sqrt {\frac {\left (1+2 \sqrt [4]{-3}+i \sqrt {3}+2 \text {Root}\left [\text {$\#$1}^4+2 \text {$\#$1}^3+3 \text {$\#$1}^2-4 \text {$\#$1}+1\&,2\right ]\right ) \left (x-\text {Root}\left [\text {$\#$1}^4+2 \text {$\#$1}^3+3 \text {$\#$1}^2-4 \text {$\#$1}+1\&,3\right ]\right )}{\left (x-\text {Root}\left [\text {$\#$1}^4+2 \text {$\#$1}^3+3 \text {$\#$1}^2-4 \text {$\#$1}+1\&,2\right ]\right ) \left (1+2 \sqrt [4]{-3}+i \sqrt {3}+2 \text {Root}\left [\text {$\#$1}^4+2 \text {$\#$1}^3+3 \text {$\#$1}^2-4 \text {$\#$1}+1\&,3\right ]\right )}}}{\sqrt {1-4 x+3 x^2+2 x^3+x^4} \left (-1+2 \sqrt [4]{-3}-i \sqrt {3}-2 \text {Root}\left [\text {$\#$1}^4+2 \text {$\#$1}^3+3 \text {$\#$1}^2-4 \text {$\#$1}+1\&,2\right ]\right ) \left (1+2 \sqrt [4]{-3}+i \sqrt {3}+2 \text {Root}\left [\text {$\#$1}^4+2 \text {$\#$1}^3+3 \text {$\#$1}^2-4 \text {$\#$1}+1\&,2\right ]\right )} \end {gather*}

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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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 3.04, size = 1609, normalized size = 24.01

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]

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,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,ind
ex=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,in
dex=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)*(Ro
otOf(_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,ind
ex=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=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=1)-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=2)-RootOf(_Z^4+2*_Z^3+3*_Z^2-4*_Z+1,in
dex=4)))^(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))*Ellipti
cPi(((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,in
dex=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=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,inde
x=1)-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=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*} \text {Failed to integrate} \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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Fricas [A]
time = 0.42, 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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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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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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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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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