3.647 \(\int \frac{1}{\sqrt{9-6 x-44 x^2+15 x^3+3 x^4}} \, dx\)

Optimal. Leaf size=130 \[ -\frac{\sqrt{\frac{\left (\frac{6}{x}-1\right )^4-182 \left (1-\frac{6}{x}\right )^2+613}{\left (\frac{(6-x)^2}{x^2}+\sqrt{613}\right )^2}} \left (\frac{(6-x)^2}{x^2}+\sqrt{613}\right ) x^2 F\left (2 \tan ^{-1}\left (\frac{6-x}{\sqrt [4]{613} x}\right )|\frac{613+91 \sqrt{613}}{1226}\right )}{12 \sqrt [4]{613} \sqrt{3 x^4+15 x^3-44 x^2-6 x+9}} \]

[Out]

-(Sqrt[(613 - 182*(1 - 6/x)^2 + (-1 + 6/x)^4)/(Sqrt[613] + (6 - x)^2/x^2)^2]*(Sq
rt[613] + (6 - x)^2/x^2)*x^2*EllipticF[2*ArcTan[(6 - x)/(613^(1/4)*x)], (613 + 9
1*Sqrt[613])/1226])/(12*613^(1/4)*Sqrt[9 - 6*x - 44*x^2 + 15*x^3 + 3*x^4])

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Rubi [A]  time = 0.428497, antiderivative size = 130, 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 \[ -\frac{\sqrt{\frac{\left (\frac{6}{x}-1\right )^4-182 \left (1-\frac{6}{x}\right )^2+613}{\left (\frac{(6-x)^2}{x^2}+\sqrt{613}\right )^2}} \left (\frac{(6-x)^2}{x^2}+\sqrt{613}\right ) x^2 F\left (2 \tan ^{-1}\left (\frac{6-x}{\sqrt [4]{613} x}\right )|\frac{613+91 \sqrt{613}}{1226}\right )}{12 \sqrt [4]{613} \sqrt{3 x^4+15 x^3-44 x^2-6 x+9}} \]

Antiderivative was successfully verified.

[In]  Int[1/Sqrt[9 - 6*x - 44*x^2 + 15*x^3 + 3*x^4],x]

[Out]

-(Sqrt[(613 - 182*(1 - 6/x)^2 + (-1 + 6/x)^4)/(Sqrt[613] + (6 - x)^2/x^2)^2]*(Sq
rt[613] + (6 - x)^2/x^2)*x^2*EllipticF[2*ArcTan[(6 - x)/(613^(1/4)*x)], (613 + 9
1*Sqrt[613])/1226])/(12*613^(1/4)*Sqrt[9 - 6*x - 44*x^2 + 15*x^3 + 3*x^4])

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - 1296 \int ^{- \frac{1}{6} + \frac{1}{x}} \frac{1}{\sqrt{\frac{15116544 x^{4} - 76422528 x^{2} + 7150032}{\left (- 36 x - 6\right )^{4}}} \left (- 36 x - 6\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(3*x**4+15*x**3-44*x**2-6*x+9)**(1/2),x)

[Out]

-1296*Integral(1/(sqrt((15116544*x**4 - 76422528*x**2 + 7150032)/(-36*x - 6)**4)
*(-36*x - 6)**2), (x, -1/6 + 1/x))

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Mathematica [C]  time = 0.202398, size = 826, normalized size = 6.35 \[ -\frac{2 F\left (\sin ^{-1}\left (\sqrt{\frac{\left (x-\text{Root}\left [3 \text{$\#$1}^4+15 \text{$\#$1}^3-44 \text{$\#$1}^2-6 \text{$\#$1}+9\&,1\right ]\right ) \left (\text{Root}\left [3 \text{$\#$1}^4+15 \text{$\#$1}^3-44 \text{$\#$1}^2-6 \text{$\#$1}+9\&,2\right ]-\text{Root}\left [3 \text{$\#$1}^4+15 \text{$\#$1}^3-44 \text{$\#$1}^2-6 \text{$\#$1}+9\&,4\right ]\right )}{\left (x-\text{Root}\left [3 \text{$\#$1}^4+15 \text{$\#$1}^3-44 \text{$\#$1}^2-6 \text{$\#$1}+9\&,2\right ]\right ) \left (\text{Root}\left [3 \text{$\#$1}^4+15 \text{$\#$1}^3-44 \text{$\#$1}^2-6 \text{$\#$1}+9\&,1\right ]-\text{Root}\left [3 \text{$\#$1}^4+15 \text{$\#$1}^3-44 \text{$\#$1}^2-6 \text{$\#$1}+9\&,4\right ]\right )}}\right )|\frac{\left (\text{Root}\left [3 \text{$\#$1}^4+15 \text{$\#$1}^3-44 \text{$\#$1}^2-6 \text{$\#$1}+9\&,2\right ]-\text{Root}\left [3 \text{$\#$1}^4+15 \text{$\#$1}^3-44 \text{$\#$1}^2-6 \text{$\#$1}+9\&,3\right ]\right ) \left (\text{Root}\left [3 \text{$\#$1}^4+15 \text{$\#$1}^3-44 \text{$\#$1}^2-6 \text{$\#$1}+9\&,1\right ]-\text{Root}\left [3 \text{$\#$1}^4+15 \text{$\#$1}^3-44 \text{$\#$1}^2-6 \text{$\#$1}+9\&,4\right ]\right )}{\left (\text{Root}\left [3 \text{$\#$1}^4+15 \text{$\#$1}^3-44 \text{$\#$1}^2-6 \text{$\#$1}+9\&,1\right ]-\text{Root}\left [3 \text{$\#$1}^4+15 \text{$\#$1}^3-44 \text{$\#$1}^2-6 \text{$\#$1}+9\&,3\right ]\right ) \left (\text{Root}\left [3 \text{$\#$1}^4+15 \text{$\#$1}^3-44 \text{$\#$1}^2-6 \text{$\#$1}+9\&,2\right ]-\text{Root}\left [3 \text{$\#$1}^4+15 \text{$\#$1}^3-44 \text{$\#$1}^2-6 \text{$\#$1}+9\&,4\right ]\right )}\right ) \sqrt{\frac{x-\text{Root}\left [3 \text{$\#$1}^4+15 \text{$\#$1}^3-44 \text{$\#$1}^2-6 \text{$\#$1}+9\&,1\right ]}{x-\text{Root}\left [3 \text{$\#$1}^4+15 \text{$\#$1}^3-44 \text{$\#$1}^2-6 \text{$\#$1}+9\&,2\right ]}} \left (x-\text{Root}\left [3 \text{$\#$1}^4+15 \text{$\#$1}^3-44 \text{$\#$1}^2-6 \text{$\#$1}+9\&,2\right ]\right )^2 \sqrt{\frac{x-\text{Root}\left [3 \text{$\#$1}^4+15 \text{$\#$1}^3-44 \text{$\#$1}^2-6 \text{$\#$1}+9\&,3\right ]}{x-\text{Root}\left [3 \text{$\#$1}^4+15 \text{$\#$1}^3-44 \text{$\#$1}^2-6 \text{$\#$1}+9\&,2\right ]}} \sqrt{\frac{x-\text{Root}\left [3 \text{$\#$1}^4+15 \text{$\#$1}^3-44 \text{$\#$1}^2-6 \text{$\#$1}+9\&,4\right ]}{x-\text{Root}\left [3 \text{$\#$1}^4+15 \text{$\#$1}^3-44 \text{$\#$1}^2-6 \text{$\#$1}+9\&,2\right ]}}}{\sqrt{\left (3 x^4+15 x^3-44 x^2-6 x+9\right ) \left (\text{Root}\left [3 \text{$\#$1}^4+15 \text{$\#$1}^3-44 \text{$\#$1}^2-6 \text{$\#$1}+9\&,1\right ]-\text{Root}\left [3 \text{$\#$1}^4+15 \text{$\#$1}^3-44 \text{$\#$1}^2-6 \text{$\#$1}+9\&,3\right ]\right ) \left (\text{Root}\left [3 \text{$\#$1}^4+15 \text{$\#$1}^3-44 \text{$\#$1}^2-6 \text{$\#$1}+9\&,2\right ]-\text{Root}\left [3 \text{$\#$1}^4+15 \text{$\#$1}^3-44 \text{$\#$1}^2-6 \text{$\#$1}+9\&,4\right ]\right )}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/Sqrt[9 - 6*x - 44*x^2 + 15*x^3 + 3*x^4],x]

[Out]

(-2*EllipticF[ArcSin[Sqrt[((x - Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 1
, 0])*(Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 2, 0] - Root[9 - 6*#1 - 44
*#1^2 + 15*#1^3 + 3*#1^4 & , 4, 0]))/((x - Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3
*#1^4 & , 2, 0])*(Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 1, 0] - Root[9
- 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 4, 0]))]], ((Root[9 - 6*#1 - 44*#1^2 + 1
5*#1^3 + 3*#1^4 & , 2, 0] - Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 3, 0]
)*(Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 1, 0] - Root[9 - 6*#1 - 44*#1^
2 + 15*#1^3 + 3*#1^4 & , 4, 0]))/((Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 &
, 1, 0] - Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 3, 0])*(Root[9 - 6*#1 -
44*#1^2 + 15*#1^3 + 3*#1^4 & , 2, 0] - Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1
^4 & , 4, 0]))]*Sqrt[(x - Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 1, 0])/
(x - Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 2, 0])]*(x - Root[9 - 6*#1 -
44*#1^2 + 15*#1^3 + 3*#1^4 & , 2, 0])^2*Sqrt[(x - Root[9 - 6*#1 - 44*#1^2 + 15*
#1^3 + 3*#1^4 & , 3, 0])/(x - Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 2,
0])]*Sqrt[(x - Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 4, 0])/(x - Root[9
- 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 2, 0])])/Sqrt[(9 - 6*x - 44*x^2 + 15*x^
3 + 3*x^4)*(Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 1, 0] - Root[9 - 6*#1
- 44*#1^2 + 15*#1^3 + 3*#1^4 & , 3, 0])*(Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*
#1^4 & , 2, 0] - Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 4, 0])]

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Maple [C]  time = 0.911, size = 1182, normalized size = 9.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(3*x^4+15*x^3-44*x^2-6*x+9)^(1/2),x)

[Out]

2/3*(-RootOf(3*_Z^4+15*_Z^3-44*_Z^2-6*_Z+9,index=4)+RootOf(3*_Z^4+15*_Z^3-44*_Z^
2-6*_Z+9,index=1))*((x-RootOf(3*_Z^4+15*_Z^3-44*_Z^2-6*_Z+9,index=1))/(-RootOf(3
*_Z^4+15*_Z^3-44*_Z^2-6*_Z+9,index=4)+RootOf(3*_Z^4+15*_Z^3-44*_Z^2-6*_Z+9,index
=1))/(x-RootOf(3*_Z^4+15*_Z^3-44*_Z^2-6*_Z+9,index=2))*(-RootOf(3*_Z^4+15*_Z^3-4
4*_Z^2-6*_Z+9,index=4)+RootOf(3*_Z^4+15*_Z^3-44*_Z^2-6*_Z+9,index=2)))^(1/2)*(x-
RootOf(3*_Z^4+15*_Z^3-44*_Z^2-6*_Z+9,index=2))^2*(-(x-RootOf(3*_Z^4+15*_Z^3-44*_
Z^2-6*_Z+9,index=3))/(-RootOf(3*_Z^4+15*_Z^3-44*_Z^2-6*_Z+9,index=3)+RootOf(3*_Z
^4+15*_Z^3-44*_Z^2-6*_Z+9,index=1))/(x-RootOf(3*_Z^4+15*_Z^3-44*_Z^2-6*_Z+9,inde
x=2))*(RootOf(3*_Z^4+15*_Z^3-44*_Z^2-6*_Z+9,index=2)-RootOf(3*_Z^4+15*_Z^3-44*_Z
^2-6*_Z+9,index=1)))^(1/2)*(-(x-RootOf(3*_Z^4+15*_Z^3-44*_Z^2-6*_Z+9,index=4))/(
-RootOf(3*_Z^4+15*_Z^3-44*_Z^2-6*_Z+9,index=4)+RootOf(3*_Z^4+15*_Z^3-44*_Z^2-6*_
Z+9,index=1))/(x-RootOf(3*_Z^4+15*_Z^3-44*_Z^2-6*_Z+9,index=2))*(RootOf(3*_Z^4+1
5*_Z^3-44*_Z^2-6*_Z+9,index=2)-RootOf(3*_Z^4+15*_Z^3-44*_Z^2-6*_Z+9,index=1)))^(
1/2)/(RootOf(3*_Z^4+15*_Z^3-44*_Z^2-6*_Z+9,index=4)-RootOf(3*_Z^4+15*_Z^3-44*_Z^
2-6*_Z+9,index=2))/(RootOf(3*_Z^4+15*_Z^3-44*_Z^2-6*_Z+9,index=2)-RootOf(3*_Z^4+
15*_Z^3-44*_Z^2-6*_Z+9,index=1))*3^(1/2)/((x-RootOf(3*_Z^4+15*_Z^3-44*_Z^2-6*_Z+
9,index=1))*(x-RootOf(3*_Z^4+15*_Z^3-44*_Z^2-6*_Z+9,index=2))*(x-RootOf(3*_Z^4+1
5*_Z^3-44*_Z^2-6*_Z+9,index=3))*(x-RootOf(3*_Z^4+15*_Z^3-44*_Z^2-6*_Z+9,index=4)
))^(1/2)*EllipticF(((x-RootOf(3*_Z^4+15*_Z^3-44*_Z^2-6*_Z+9,index=1))/(-RootOf(3
*_Z^4+15*_Z^3-44*_Z^2-6*_Z+9,index=4)+RootOf(3*_Z^4+15*_Z^3-44*_Z^2-6*_Z+9,index
=1))/(x-RootOf(3*_Z^4+15*_Z^3-44*_Z^2-6*_Z+9,index=2))*(-RootOf(3*_Z^4+15*_Z^3-4
4*_Z^2-6*_Z+9,index=4)+RootOf(3*_Z^4+15*_Z^3-44*_Z^2-6*_Z+9,index=2)))^(1/2),((R
ootOf(3*_Z^4+15*_Z^3-44*_Z^2-6*_Z+9,index=2)-RootOf(3*_Z^4+15*_Z^3-44*_Z^2-6*_Z+
9,index=3))*(-RootOf(3*_Z^4+15*_Z^3-44*_Z^2-6*_Z+9,index=4)+RootOf(3*_Z^4+15*_Z^
3-44*_Z^2-6*_Z+9,index=1))/(-RootOf(3*_Z^4+15*_Z^3-44*_Z^2-6*_Z+9,index=3)+RootO
f(3*_Z^4+15*_Z^3-44*_Z^2-6*_Z+9,index=1))/(-RootOf(3*_Z^4+15*_Z^3-44*_Z^2-6*_Z+9
,index=4)+RootOf(3*_Z^4+15*_Z^3-44*_Z^2-6*_Z+9,index=2)))^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{3 \, x^{4} + 15 \, x^{3} - 44 \, x^{2} - 6 \, x + 9}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(3*x^4 + 15*x^3 - 44*x^2 - 6*x + 9),x, algorithm="maxima")

[Out]

integrate(1/sqrt(3*x^4 + 15*x^3 - 44*x^2 - 6*x + 9), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{\sqrt{3 \, x^{4} + 15 \, x^{3} - 44 \, x^{2} - 6 \, x + 9}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(3*x^4 + 15*x^3 - 44*x^2 - 6*x + 9),x, algorithm="fricas")

[Out]

integral(1/sqrt(3*x^4 + 15*x^3 - 44*x^2 - 6*x + 9), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{3 x^{4} + 15 x^{3} - 44 x^{2} - 6 x + 9}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(3*x**4+15*x**3-44*x**2-6*x+9)**(1/2),x)

[Out]

Integral(1/sqrt(3*x**4 + 15*x**3 - 44*x**2 - 6*x + 9), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{3 \, x^{4} + 15 \, x^{3} - 44 \, x^{2} - 6 \, x + 9}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(3*x^4 + 15*x^3 - 44*x^2 - 6*x + 9),x, algorithm="giac")

[Out]

integrate(1/sqrt(3*x^4 + 15*x^3 - 44*x^2 - 6*x + 9), x)