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3.644 \(\int \frac{1}{\sqrt{8+24 x+8 x^2-15 x^3+8 x^4}} \, dx\)

Optimal. Leaf size=126 \[ -\frac{\left (\left (\frac{4}{x}+3\right )^2+\sqrt{517}\right ) \sqrt{\frac{\left (\frac{4}{x}+3\right )^4-38 \left (\frac{4}{x}+3\right )^2+517}{\left (\left (\frac{4}{x}+3\right )^2+\sqrt{517}\right )^2}} x^2 F\left (2 \tan ^{-1}\left (\frac{3 x+4}{\sqrt [4]{517} x}\right )|\frac{517+19 \sqrt{517}}{1034}\right )}{8 \sqrt [4]{517} \sqrt{8 x^4-15 x^3+8 x^2+24 x+8}} \]

[Out]

-((Sqrt[517] + (3 + 4/x)^2)*Sqrt[(517 - 38*(3 + 4/x)^2 + (3 + 4/x)^4)/(Sqrt[517]
 + (3 + 4/x)^2)^2]*x^2*EllipticF[2*ArcTan[(4 + 3*x)/(517^(1/4)*x)], (517 + 19*Sq
rt[517])/1034])/(8*517^(1/4)*Sqrt[8 + 24*x + 8*x^2 - 15*x^3 + 8*x^4])

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Rubi [A]  time = 0.559767, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{\left (\left (\frac{4}{x}+3\right )^2+\sqrt{517}\right ) \sqrt{\frac{\left (\frac{4}{x}+3\right )^4-38 \left (\frac{4}{x}+3\right )^2+517}{\left (\left (\frac{4}{x}+3\right )^2+\sqrt{517}\right )^2}} x^2 F\left (2 \tan ^{-1}\left (\frac{3 x+4}{\sqrt [4]{517} x}\right )|\frac{517+19 \sqrt{517}}{1034}\right )}{8 \sqrt [4]{517} \sqrt{8 x^4-15 x^3+8 x^2+24 x+8}} \]

Antiderivative was successfully verified.

[In]  Int[1/Sqrt[8 + 24*x + 8*x^2 - 15*x^3 + 8*x^4],x]

[Out]

-((Sqrt[517] + (3 + 4/x)^2)*Sqrt[(517 - 38*(3 + 4/x)^2 + (3 + 4/x)^4)/(Sqrt[517]
 + (3 + 4/x)^2)^2]*x^2*EllipticF[2*ArcTan[(4 + 3*x)/(517^(1/4)*x)], (517 + 19*Sq
rt[517])/1034])/(8*517^(1/4)*Sqrt[8 + 24*x + 8*x^2 - 15*x^3 + 8*x^4])

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - 1024 \int ^{\frac{3}{4} + \frac{1}{x}} \frac{1}{\sqrt{\frac{8388608 x^{4} - 19922944 x^{2} + 16941056}{\left (- 32 x + 24\right )^{4}}} \left (- 32 x + 24\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(8*x**4-15*x**3+8*x**2+24*x+8)**(1/2),x)

[Out]

-1024*Integral(1/(sqrt((8388608*x**4 - 19922944*x**2 + 16941056)/(-32*x + 24)**4
)*(-32*x + 24)**2), (x, 3/4 + 1/x))

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Mathematica [C]  time = 0.59733, size = 1148, normalized size = 9.11 \[ \text{result too large to display} \]

Antiderivative was successfully verified.

[In]  Integrate[1/Sqrt[8 + 24*x + 8*x^2 - 15*x^3 + 8*x^4],x]

[Out]

(-2*EllipticF[ArcSin[Sqrt[((x - Root[8 + 24*#1 + 8*#1^2 - 15*#1^3 + 8*#1^4 & , 1
, 0])*(Root[8 + 24*#1 + 8*#1^2 - 15*#1^3 + 8*#1^4 & , 2, 0] - Root[8 + 24*#1 + 8
*#1^2 - 15*#1^3 + 8*#1^4 & , 4, 0]))/((x - Root[8 + 24*#1 + 8*#1^2 - 15*#1^3 + 8
*#1^4 & , 2, 0])*(Root[8 + 24*#1 + 8*#1^2 - 15*#1^3 + 8*#1^4 & , 1, 0] - Root[8
+ 24*#1 + 8*#1^2 - 15*#1^3 + 8*#1^4 & , 4, 0]))]], ((Root[8 + 24*#1 + 8*#1^2 - 1
5*#1^3 + 8*#1^4 & , 2, 0] - Root[8 + 24*#1 + 8*#1^2 - 15*#1^3 + 8*#1^4 & , 3, 0]
)*(Root[8 + 24*#1 + 8*#1^2 - 15*#1^3 + 8*#1^4 & , 1, 0] - Root[8 + 24*#1 + 8*#1^
2 - 15*#1^3 + 8*#1^4 & , 4, 0]))/((Root[8 + 24*#1 + 8*#1^2 - 15*#1^3 + 8*#1^4 &
, 1, 0] - Root[8 + 24*#1 + 8*#1^2 - 15*#1^3 + 8*#1^4 & , 3, 0])*(Root[8 + 24*#1
+ 8*#1^2 - 15*#1^3 + 8*#1^4 & , 2, 0] - Root[8 + 24*#1 + 8*#1^2 - 15*#1^3 + 8*#1
^4 & , 4, 0]))]*(x - Root[8 + 24*#1 + 8*#1^2 - 15*#1^3 + 8*#1^4 & , 2, 0])^2*Sqr
t[((Root[8 + 24*#1 + 8*#1^2 - 15*#1^3 + 8*#1^4 & , 1, 0] - Root[8 + 24*#1 + 8*#1
^2 - 15*#1^3 + 8*#1^4 & , 2, 0])*(x - Root[8 + 24*#1 + 8*#1^2 - 15*#1^3 + 8*#1^4
 & , 3, 0]))/((x - Root[8 + 24*#1 + 8*#1^2 - 15*#1^3 + 8*#1^4 & , 2, 0])*(Root[8
 + 24*#1 + 8*#1^2 - 15*#1^3 + 8*#1^4 & , 1, 0] - Root[8 + 24*#1 + 8*#1^2 - 15*#1
^3 + 8*#1^4 & , 3, 0]))]*(Root[8 + 24*#1 + 8*#1^2 - 15*#1^3 + 8*#1^4 & , 1, 0] -
 Root[8 + 24*#1 + 8*#1^2 - 15*#1^3 + 8*#1^4 & , 4, 0])*Sqrt[((x - Root[8 + 24*#1
 + 8*#1^2 - 15*#1^3 + 8*#1^4 & , 1, 0])*(Root[8 + 24*#1 + 8*#1^2 - 15*#1^3 + 8*#
1^4 & , 1, 0] - Root[8 + 24*#1 + 8*#1^2 - 15*#1^3 + 8*#1^4 & , 2, 0])*(x - Root[
8 + 24*#1 + 8*#1^2 - 15*#1^3 + 8*#1^4 & , 4, 0])*(Root[8 + 24*#1 + 8*#1^2 - 15*#
1^3 + 8*#1^4 & , 2, 0] - Root[8 + 24*#1 + 8*#1^2 - 15*#1^3 + 8*#1^4 & , 4, 0]))/
((x - Root[8 + 24*#1 + 8*#1^2 - 15*#1^3 + 8*#1^4 & , 2, 0])^2*(Root[8 + 24*#1 +
8*#1^2 - 15*#1^3 + 8*#1^4 & , 1, 0] - Root[8 + 24*#1 + 8*#1^2 - 15*#1^3 + 8*#1^4
 & , 4, 0])^2)])/(Sqrt[8 + 24*x + 8*x^2 - 15*x^3 + 8*x^4]*(-Root[8 + 24*#1 + 8*#
1^2 - 15*#1^3 + 8*#1^4 & , 1, 0] + Root[8 + 24*#1 + 8*#1^2 - 15*#1^3 + 8*#1^4 &
, 2, 0])*(Root[8 + 24*#1 + 8*#1^2 - 15*#1^3 + 8*#1^4 & , 2, 0] - Root[8 + 24*#1
+ 8*#1^2 - 15*#1^3 + 8*#1^4 & , 4, 0]))

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Maple [C]  time = 1.678, size = 1180, normalized size = 9.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(8*x^4-15*x^3+8*x^2+24*x+8)^(1/2),x)

[Out]

1/2*(RootOf(8*_Z^4-15*_Z^3+8*_Z^2+24*_Z+8,index=1)-RootOf(8*_Z^4-15*_Z^3+8*_Z^2+
24*_Z+8,index=4))*((RootOf(8*_Z^4-15*_Z^3+8*_Z^2+24*_Z+8,index=4)-RootOf(8*_Z^4-
15*_Z^3+8*_Z^2+24*_Z+8,index=2))*(x-RootOf(8*_Z^4-15*_Z^3+8*_Z^2+24*_Z+8,index=1
))/(RootOf(8*_Z^4-15*_Z^3+8*_Z^2+24*_Z+8,index=4)-RootOf(8*_Z^4-15*_Z^3+8*_Z^2+2
4*_Z+8,index=1))/(x-RootOf(8*_Z^4-15*_Z^3+8*_Z^2+24*_Z+8,index=2)))^(1/2)*(x-Roo
tOf(8*_Z^4-15*_Z^3+8*_Z^2+24*_Z+8,index=2))^2*((RootOf(8*_Z^4-15*_Z^3+8*_Z^2+24*
_Z+8,index=2)-RootOf(8*_Z^4-15*_Z^3+8*_Z^2+24*_Z+8,index=1))*(x-RootOf(8*_Z^4-15
*_Z^3+8*_Z^2+24*_Z+8,index=3))/(RootOf(8*_Z^4-15*_Z^3+8*_Z^2+24*_Z+8,index=3)-Ro
otOf(8*_Z^4-15*_Z^3+8*_Z^2+24*_Z+8,index=1))/(x-RootOf(8*_Z^4-15*_Z^3+8*_Z^2+24*
_Z+8,index=2)))^(1/2)*((RootOf(8*_Z^4-15*_Z^3+8*_Z^2+24*_Z+8,index=2)-RootOf(8*_
Z^4-15*_Z^3+8*_Z^2+24*_Z+8,index=1))*(x-RootOf(8*_Z^4-15*_Z^3+8*_Z^2+24*_Z+8,ind
ex=4))/(RootOf(8*_Z^4-15*_Z^3+8*_Z^2+24*_Z+8,index=4)-RootOf(8*_Z^4-15*_Z^3+8*_Z
^2+24*_Z+8,index=1))/(x-RootOf(8*_Z^4-15*_Z^3+8*_Z^2+24*_Z+8,index=2)))^(1/2)/(R
ootOf(8*_Z^4-15*_Z^3+8*_Z^2+24*_Z+8,index=4)-RootOf(8*_Z^4-15*_Z^3+8*_Z^2+24*_Z+
8,index=2))/(RootOf(8*_Z^4-15*_Z^3+8*_Z^2+24*_Z+8,index=2)-RootOf(8*_Z^4-15*_Z^3
+8*_Z^2+24*_Z+8,index=1))*2^(1/2)/((x-RootOf(8*_Z^4-15*_Z^3+8*_Z^2+24*_Z+8,index
=1))*(x-RootOf(8*_Z^4-15*_Z^3+8*_Z^2+24*_Z+8,index=2))*(x-RootOf(8*_Z^4-15*_Z^3+
8*_Z^2+24*_Z+8,index=3))*(x-RootOf(8*_Z^4-15*_Z^3+8*_Z^2+24*_Z+8,index=4)))^(1/2
)*EllipticF(((RootOf(8*_Z^4-15*_Z^3+8*_Z^2+24*_Z+8,index=4)-RootOf(8*_Z^4-15*_Z^
3+8*_Z^2+24*_Z+8,index=2))*(x-RootOf(8*_Z^4-15*_Z^3+8*_Z^2+24*_Z+8,index=1))/(Ro
otOf(8*_Z^4-15*_Z^3+8*_Z^2+24*_Z+8,index=4)-RootOf(8*_Z^4-15*_Z^3+8*_Z^2+24*_Z+8
,index=1))/(x-RootOf(8*_Z^4-15*_Z^3+8*_Z^2+24*_Z+8,index=2)))^(1/2),((RootOf(8*_
Z^4-15*_Z^3+8*_Z^2+24*_Z+8,index=2)-RootOf(8*_Z^4-15*_Z^3+8*_Z^2+24*_Z+8,index=3
))*(RootOf(8*_Z^4-15*_Z^3+8*_Z^2+24*_Z+8,index=1)-RootOf(8*_Z^4-15*_Z^3+8*_Z^2+2
4*_Z+8,index=4))/(-RootOf(8*_Z^4-15*_Z^3+8*_Z^2+24*_Z+8,index=3)+RootOf(8*_Z^4-1
5*_Z^3+8*_Z^2+24*_Z+8,index=1))/(RootOf(8*_Z^4-15*_Z^3+8*_Z^2+24*_Z+8,index=2)-R
ootOf(8*_Z^4-15*_Z^3+8*_Z^2+24*_Z+8,index=4)))^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{8 \, x^{4} - 15 \, x^{3} + 8 \, x^{2} + 24 \, x + 8}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(8*x^4 - 15*x^3 + 8*x^2 + 24*x + 8),x, algorithm="maxima")

[Out]

integrate(1/sqrt(8*x^4 - 15*x^3 + 8*x^2 + 24*x + 8), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{\sqrt{8 \, x^{4} - 15 \, x^{3} + 8 \, x^{2} + 24 \, x + 8}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(8*x^4 - 15*x^3 + 8*x^2 + 24*x + 8),x, algorithm="fricas")

[Out]

integral(1/sqrt(8*x^4 - 15*x^3 + 8*x^2 + 24*x + 8), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{8 x^{4} - 15 x^{3} + 8 x^{2} + 24 x + 8}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(8*x**4-15*x**3+8*x**2+24*x+8)**(1/2),x)

[Out]

Integral(1/sqrt(8*x**4 - 15*x**3 + 8*x**2 + 24*x + 8), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{8 \, x^{4} - 15 \, x^{3} + 8 \, x^{2} + 24 \, x + 8}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(8*x^4 - 15*x^3 + 8*x^2 + 24*x + 8),x, algorithm="giac")

[Out]

integrate(1/sqrt(8*x^4 - 15*x^3 + 8*x^2 + 24*x + 8), x)

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