∖int x____________√ −71−96x+10x2+x4 ∖,dx

3.6 \(\int \frac{x}{\sqrt{-71-96 x+10 x^2+x^4}} \, dx\)

Optimal. Leaf size=78 \[ -\frac{1}{8} \log \left (x^8+20 x^6-128 x^5+54 x^4-1408 x^3+3124 x^2+\sqrt{x^4+10 x^2-96 x-71} \left (-x^6-15 x^4+80 x^3-27 x^2+528 x-781\right )+10001\right ) \]

[Out]

-Log[10001 + 3124*x^2 - 1408*x^3 + 54*x^4 - 128*x^5 + 20*x^6 + x^8 + Sqrt[-71 -

96*x + 10*x^2 + x^4]*(-781 + 528*x - 27*x^2 + 80*x^3 - 15*x^4 - x^6)]/8

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Rubi [F] time = 0.0757832, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0. \[ \text{Int}\left (\frac{x}{\sqrt{-71-96 x+10 x^2+x^4}},x\right ) \]

Veriﬁcation is Not applicable to the result.

[In] Int[x/Sqrt[-71 - 96*x + 10*x^2 + x^4],x]

[Out]

Defer[Int][x/Sqrt[-71 - 96*x + 10*x^2 + x^4], x]

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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{\sqrt{x^{4} + 10 x^{2} - 96 x - 71}}\, dx \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] rubi_integrate(x/(x**4+10*x**2-96*x-71)**(1/2),x)

[Out]

Integral(x/sqrt(x**4 + 10*x**2 - 96*x - 71), x)

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Mathematica [C] time = 2.86858, size = 1226, normalized size = 15.72 \[ \text{result too large to display} \]

Warning: Unable to verify antiderivative.

[In] Integrate[x/Sqrt[-71 - 96*x + 10*x^2 + x^4],x]

[Out]

(-2*(Sqrt[3] + 2*Sqrt[2*(-1 + Sqrt[3])] - x)*((Sqrt[3] + 2*Sqrt[2*(-1 + Sqrt[3])

])*EllipticF[ArcSin[Sqrt[((Sqrt[3] - 2*Sqrt[2*(-1 + Sqrt[3])] - x)*(Sqrt[3] + 2*

Sqrt[2*(-1 + Sqrt[3])] - Root[-71 - 96*#1 + 10*#1^2 + #1^4 & , 4, 0]))/((Sqrt[3]

+ 2*Sqrt[2*(-1 + Sqrt[3])] - x)*(Sqrt[3] - 2*Sqrt[2*(-1 + Sqrt[3])] - Root[-71

- 96*#1 + 10*#1^2 + #1^4 & , 4, 0]))]], ((Sqrt[3] + 2*Sqrt[2*(-1 + Sqrt[3])] - R

oot[-71 - 96*#1 + 10*#1^2 + #1^4 & , 3, 0])*(Sqrt[3] - 2*Sqrt[2*(-1 + Sqrt[3])]

- Root[-71 - 96*#1 + 10*#1^2 + #1^4 & , 4, 0]))/((Sqrt[3] - 2*Sqrt[2*(-1 + Sqrt[

3])] - Root[-71 - 96*#1 + 10*#1^2 + #1^4 & , 3, 0])*(Sqrt[3] + 2*Sqrt[2*(-1 + Sq

rt[3])] - Root[-71 - 96*#1 + 10*#1^2 + #1^4 & , 4, 0]))] - 4*Sqrt[2*(-1 + Sqrt[3

])]*EllipticPi[(Sqrt[3] - 2*Sqrt[2*(-1 + Sqrt[3])] - Root[-71 - 96*#1 + 10*#1^2

+ #1^4 & , 4, 0])/(Sqrt[3] + 2*Sqrt[2*(-1 + Sqrt[3])] - Root[-71 - 96*#1 + 10*#1

^2 + #1^4 & , 4, 0]), ArcSin[Sqrt[((Sqrt[3] - 2*Sqrt[2*(-1 + Sqrt[3])] - x)*(Sqr

t[3] + 2*Sqrt[2*(-1 + Sqrt[3])] - Root[-71 - 96*#1 + 10*#1^2 + #1^4 & , 4, 0]))/

((Sqrt[3] + 2*Sqrt[2*(-1 + Sqrt[3])] - x)*(Sqrt[3] - 2*Sqrt[2*(-1 + Sqrt[3])] -

Root[-71 - 96*#1 + 10*#1^2 + #1^4 & , 4, 0]))]], ((Sqrt[3] + 2*Sqrt[2*(-1 + Sqrt

[3])] - Root[-71 - 96*#1 + 10*#1^2 + #1^4 & , 3, 0])*(Sqrt[3] - 2*Sqrt[2*(-1 + S

qrt[3])] - Root[-71 - 96*#1 + 10*#1^2 + #1^4 & , 4, 0]))/((Sqrt[3] - 2*Sqrt[2*(-

1 + Sqrt[3])] - Root[-71 - 96*#1 + 10*#1^2 + #1^4 & , 3, 0])*(Sqrt[3] + 2*Sqrt[2

*(-1 + Sqrt[3])] - Root[-71 - 96*#1 + 10*#1^2 + #1^4 & , 4, 0]))])*Sqrt[(x - Roo

t[-71 - 96*#1 + 10*#1^2 + #1^4 & , 3, 0])/((Sqrt[3] + 2*Sqrt[2*(-1 + Sqrt[3])] -

x)*(Sqrt[3] - 2*Sqrt[2*(-1 + Sqrt[3])] - Root[-71 - 96*#1 + 10*#1^2 + #1^4 & ,

3, 0]))]*Sqrt[((Sqrt[3] - 2*Sqrt[2*(-1 + Sqrt[3])] - x)*(Sqrt[3] + 2*Sqrt[2*(-1

+ Sqrt[3])] - Root[-71 - 96*#1 + 10*#1^2 + #1^4 & , 4, 0]))/((Sqrt[3] + 2*Sqrt[2

*(-1 + Sqrt[3])] - x)*(Sqrt[3] - 2*Sqrt[2*(-1 + Sqrt[3])] - Root[-71 - 96*#1 + 1

0*#1^2 + #1^4 & , 4, 0]))]*(x - Root[-71 - 96*#1 + 10*#1^2 + #1^4 & , 4, 0]))/(S

qrt[-71 - 96*x + 10*x^2 + x^4]*(Sqrt[3] + 2*Sqrt[2*(-1 + Sqrt[3])] - Root[-71 -

96*#1 + 10*#1^2 + #1^4 & , 4, 0])*Sqrt[(x - Root[-71 - 96*#1 + 10*#1^2 + #1^4 &

, 4, 0])/((Sqrt[3] + 2*Sqrt[2*(-1 + Sqrt[3])] - x)*(Sqrt[3] - 2*Sqrt[2*(-1 + Sqr

t[3])] - Root[-71 - 96*#1 + 10*#1^2 + #1^4 & , 4, 0]))])

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Maple [C] time = 1.283, size = 1290, normalized size = 16.5 \[ \text{result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] int(x/(x^4+10*x^2-96*x-71)^(1/2),x)

[Out]

2*(-RootOf(_Z^4+10*_Z^2-96*_Z-71,index=4)+RootOf(_Z^4+10*_Z^2-96*_Z-71,index=1))

*((RootOf(_Z^4+10*_Z^2-96*_Z-71,index=4)-RootOf(_Z^4+10*_Z^2-96*_Z-71,index=2))*

(x-RootOf(_Z^4+10*_Z^2-96*_Z-71,index=1))/(RootOf(_Z^4+10*_Z^2-96*_Z-71,index=4)

-RootOf(_Z^4+10*_Z^2-96*_Z-71,index=1))/(x-RootOf(_Z^4+10*_Z^2-96*_Z-71,index=2)

))^(1/2)*(x-RootOf(_Z^4+10*_Z^2-96*_Z-71,index=2))^2*(-(RootOf(_Z^4+10*_Z^2-96*_

Z-71,index=2)-RootOf(_Z^4+10*_Z^2-96*_Z-71,index=1))*(x-RootOf(_Z^4+10*_Z^2-96*_

Z-71,index=3))/(-RootOf(_Z^4+10*_Z^2-96*_Z-71,index=3)+RootOf(_Z^4+10*_Z^2-96*_Z

-71,index=1))/(x-RootOf(_Z^4+10*_Z^2-96*_Z-71,index=2)))^(1/2)*((RootOf(_Z^4+10*

_Z^2-96*_Z-71,index=2)-RootOf(_Z^4+10*_Z^2-96*_Z-71,index=1))*(x-RootOf(_Z^4+10*

_Z^2-96*_Z-71,index=4))/(RootOf(_Z^4+10*_Z^2-96*_Z-71,index=4)-RootOf(_Z^4+10*_Z

^2-96*_Z-71,index=1))/(x-RootOf(_Z^4+10*_Z^2-96*_Z-71,index=2)))^(1/2)/(RootOf(_

Z^4+10*_Z^2-96*_Z-71,index=4)-RootOf(_Z^4+10*_Z^2-96*_Z-71,index=2))/(RootOf(_Z^

4+10*_Z^2-96*_Z-71,index=2)-RootOf(_Z^4+10*_Z^2-96*_Z-71,index=1))/((x-RootOf(_Z

^4+10*_Z^2-96*_Z-71,index=1))*(x-RootOf(_Z^4+10*_Z^2-96*_Z-71,index=2))*(x-RootO

f(_Z^4+10*_Z^2-96*_Z-71,index=3))*(x-RootOf(_Z^4+10*_Z^2-96*_Z-71,index=4)))^(1/

2)*(RootOf(_Z^4+10*_Z^2-96*_Z-71,index=2)*EllipticF(((RootOf(_Z^4+10*_Z^2-96*_Z-

71,index=4)-RootOf(_Z^4+10*_Z^2-96*_Z-71,index=2))*(x-RootOf(_Z^4+10*_Z^2-96*_Z-

71,index=1))/(RootOf(_Z^4+10*_Z^2-96*_Z-71,index=4)-RootOf(_Z^4+10*_Z^2-96*_Z-71

,index=1))/(x-RootOf(_Z^4+10*_Z^2-96*_Z-71,index=2)))^(1/2),((RootOf(_Z^4+10*_Z^

2-96*_Z-71,index=2)-RootOf(_Z^4+10*_Z^2-96*_Z-71,index=3))*(-RootOf(_Z^4+10*_Z^2

-96*_Z-71,index=4)+RootOf(_Z^4+10*_Z^2-96*_Z-71,index=1))/(-RootOf(_Z^4+10*_Z^2-

96*_Z-71,index=3)+RootOf(_Z^4+10*_Z^2-96*_Z-71,index=1))/(RootOf(_Z^4+10*_Z^2-96

*_Z-71,index=2)-RootOf(_Z^4+10*_Z^2-96*_Z-71,index=4)))^(1/2))+(-RootOf(_Z^4+10*

_Z^2-96*_Z-71,index=2)+RootOf(_Z^4+10*_Z^2-96*_Z-71,index=1))*EllipticPi(((RootO

f(_Z^4+10*_Z^2-96*_Z-71,index=4)-RootOf(_Z^4+10*_Z^2-96*_Z-71,index=2))*(x-RootO

f(_Z^4+10*_Z^2-96*_Z-71,index=1))/(RootOf(_Z^4+10*_Z^2-96*_Z-71,index=4)-RootOf(

_Z^4+10*_Z^2-96*_Z-71,index=1))/(x-RootOf(_Z^4+10*_Z^2-96*_Z-71,index=2)))^(1/2)

,(RootOf(_Z^4+10*_Z^2-96*_Z-71,index=4)-RootOf(_Z^4+10*_Z^2-96*_Z-71,index=1))/(

RootOf(_Z^4+10*_Z^2-96*_Z-71,index=4)-RootOf(_Z^4+10*_Z^2-96*_Z-71,index=2)),((R

ootOf(_Z^4+10*_Z^2-96*_Z-71,index=2)-RootOf(_Z^4+10*_Z^2-96*_Z-71,index=3))*(-Ro

otOf(_Z^4+10*_Z^2-96*_Z-71,index=4)+RootOf(_Z^4+10*_Z^2-96*_Z-71,index=1))/(-Roo

tOf(_Z^4+10*_Z^2-96*_Z-71,index=3)+RootOf(_Z^4+10*_Z^2-96*_Z-71,index=1))/(RootO

f(_Z^4+10*_Z^2-96*_Z-71,index=2)-RootOf(_Z^4+10*_Z^2-96*_Z-71,index=4)))^(1/2)))

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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{\sqrt{x^{4} + 10 \, x^{2} - 96 \, x - 71}}\,{d x} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] integrate(x/sqrt(x^4 + 10*x^2 - 96*x - 71),x, algorithm="maxima")

[Out]

integrate(x/sqrt(x^4 + 10*x^2 - 96*x - 71), x)

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Fricas [A] time = 0.28659, size = 97, normalized size = 1.24 \[ \frac{1}{8} \, \log \left (x^{8} + 20 \, x^{6} - 128 \, x^{5} + 54 \, x^{4} - 1408 \, x^{3} + 3124 \, x^{2} +{\left (x^{6} + 15 \, x^{4} - 80 \, x^{3} + 27 \, x^{2} - 528 \, x + 781\right )} \sqrt{x^{4} + 10 \, x^{2} - 96 \, x - 71} + 10001\right ) \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] integrate(x/sqrt(x^4 + 10*x^2 - 96*x - 71),x, algorithm="fricas")

[Out]

1/8*log(x^8 + 20*x^6 - 128*x^5 + 54*x^4 - 1408*x^3 + 3124*x^2 + (x^6 + 15*x^4 -

80*x^3 + 27*x^2 - 528*x + 781)*sqrt(x^4 + 10*x^2 - 96*x - 71) + 10001)

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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{\sqrt{x^{4} + 10 x^{2} - 96 x - 71}}\, dx \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] integrate(x/(x**4+10*x**2-96*x-71)**(1/2),x)

[Out]

Integral(x/sqrt(x**4 + 10*x**2 - 96*x - 71), x)

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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{\sqrt{x^{4} + 10 \, x^{2} - 96 \, x - 71}}\,{d x} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] integrate(x/sqrt(x^4 + 10*x^2 - 96*x - 71),x, algorithm="giac")

[Out]

integrate(x/sqrt(x^4 + 10*x^2 - 96*x - 71), x)

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