3.645 \(\int \frac{1}{\left (8+24 x+8 x^2-15 x^3+8 x^4\right )^{3/2}} \, dx\)

Optimal. Leaf size=434 \[ -\frac{\left (172-7 \left (\frac{4}{x}+3\right )^2\right ) x^2}{208 \sqrt{8 x^4-15 x^3+8 x^2+24 x+8}}+\frac{\left (50896-2455 \left (\frac{4}{x}+3\right )^2\right ) \left (\frac{4}{x}+3\right ) x^2}{322608 \sqrt{8 x^4-15 x^3+8 x^2+24 x+8}}+\frac{2455 \left (\left (\frac{4}{x}+3\right )^4-38 \left (\frac{4}{x}+3\right )^2+517\right ) \left (\frac{4}{x}+3\right ) x^2}{322608 \left (\left (\frac{4}{x}+3\right )^2+\sqrt{517}\right ) \sqrt{8 x^4-15 x^3+8 x^2+24 x+8}}+\frac{\left (4910-203 \sqrt{517}\right ) \left (\left (\frac{4}{x}+3\right )^2+\sqrt{517}\right ) \sqrt{\frac{\left (\frac{4}{x}+3\right )^4-38 \left (\frac{4}{x}+3\right )^2+517}{\left (\left (\frac{4}{x}+3\right )^2+\sqrt{517}\right )^2}} x^2 F\left (2 \tan ^{-1}\left (\frac{3 x+4}{\sqrt [4]{517} x}\right )|\frac{517+19 \sqrt{517}}{1034}\right )}{2496\ 517^{3/4} \sqrt{8 x^4-15 x^3+8 x^2+24 x+8}}-\frac{2455 \left (\left (\frac{4}{x}+3\right )^2+\sqrt{517}\right ) \sqrt{\frac{\left (\frac{4}{x}+3\right )^4-38 \left (\frac{4}{x}+3\right )^2+517}{\left (\left (\frac{4}{x}+3\right )^2+\sqrt{517}\right )^2}} x^2 E\left (2 \tan ^{-1}\left (\frac{3 x+4}{\sqrt [4]{517} x}\right )|\frac{517+19 \sqrt{517}}{1034}\right )}{624\ 517^{3/4} \sqrt{8 x^4-15 x^3+8 x^2+24 x+8}} \]

[Out]

-((172 - 7*(3 + 4/x)^2)*x^2)/(208*Sqrt[8 + 24*x + 8*x^2 - 15*x^3 + 8*x^4]) + ((5
0896 - 2455*(3 + 4/x)^2)*(3 + 4/x)*x^2)/(322608*Sqrt[8 + 24*x + 8*x^2 - 15*x^3 +
8*x^4]) + (2455*(517 - 38*(3 + 4/x)^2 + (3 + 4/x)^4)*(3 + 4/x)*x^2)/(322608*(Sq
rt[517] + (3 + 4/x)^2)*Sqrt[8 + 24*x + 8*x^2 - 15*x^3 + 8*x^4]) - (2455*(Sqrt[51
7] + (3 + 4/x)^2)*Sqrt[(517 - 38*(3 + 4/x)^2 + (3 + 4/x)^4)/(Sqrt[517] + (3 + 4/
x)^2)^2]*x^2*EllipticE[2*ArcTan[(4 + 3*x)/(517^(1/4)*x)], (517 + 19*Sqrt[517])/1
034])/(624*517^(3/4)*Sqrt[8 + 24*x + 8*x^2 - 15*x^3 + 8*x^4]) + ((4910 - 203*Sqr
t[517])*(Sqrt[517] + (3 + 4/x)^2)*Sqrt[(517 - 38*(3 + 4/x)^2 + (3 + 4/x)^4)/(Sqr
t[517] + (3 + 4/x)^2)^2]*x^2*EllipticF[2*ArcTan[(4 + 3*x)/(517^(1/4)*x)], (517 +
19*Sqrt[517])/1034])/(2496*517^(3/4)*Sqrt[8 + 24*x + 8*x^2 - 15*x^3 + 8*x^4])

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Rubi [A]  time = 1.01155, antiderivative size = 434, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 10, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417 \[ -\frac{\left (172-7 \left (\frac{4}{x}+3\right )^2\right ) x^2}{208 \sqrt{8 x^4-15 x^3+8 x^2+24 x+8}}+\frac{\left (50896-2455 \left (\frac{4}{x}+3\right )^2\right ) \left (\frac{4}{x}+3\right ) x^2}{322608 \sqrt{8 x^4-15 x^3+8 x^2+24 x+8}}+\frac{2455 \left (\left (\frac{4}{x}+3\right )^4-38 \left (\frac{4}{x}+3\right )^2+517\right ) \left (\frac{4}{x}+3\right ) x^2}{322608 \left (\left (\frac{4}{x}+3\right )^2+\sqrt{517}\right ) \sqrt{8 x^4-15 x^3+8 x^2+24 x+8}}+\frac{\left (4910-203 \sqrt{517}\right ) \left (\left (\frac{4}{x}+3\right )^2+\sqrt{517}\right ) \sqrt{\frac{\left (\frac{4}{x}+3\right )^4-38 \left (\frac{4}{x}+3\right )^2+517}{\left (\left (\frac{4}{x}+3\right )^2+\sqrt{517}\right )^2}} x^2 F\left (2 \tan ^{-1}\left (\frac{3 x+4}{\sqrt [4]{517} x}\right )|\frac{517+19 \sqrt{517}}{1034}\right )}{2496\ 517^{3/4} \sqrt{8 x^4-15 x^3+8 x^2+24 x+8}}-\frac{2455 \left (\left (\frac{4}{x}+3\right )^2+\sqrt{517}\right ) \sqrt{\frac{\left (\frac{4}{x}+3\right )^4-38 \left (\frac{4}{x}+3\right )^2+517}{\left (\left (\frac{4}{x}+3\right )^2+\sqrt{517}\right )^2}} x^2 E\left (2 \tan ^{-1}\left (\frac{3 x+4}{\sqrt [4]{517} x}\right )|\frac{517+19 \sqrt{517}}{1034}\right )}{624\ 517^{3/4} \sqrt{8 x^4-15 x^3+8 x^2+24 x+8}} \]

Warning: Unable to verify antiderivative.

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

[Out]

-((172 - 7*(3 + 4/x)^2)*x^2)/(208*Sqrt[8 + 24*x + 8*x^2 - 15*x^3 + 8*x^4]) + ((5
0896 - 2455*(3 + 4/x)^2)*(3 + 4/x)*x^2)/(322608*Sqrt[8 + 24*x + 8*x^2 - 15*x^3 +
8*x^4]) + (2455*(517 - 38*(3 + 4/x)^2 + (3 + 4/x)^4)*(3 + 4/x)*x^2)/(322608*(Sq
rt[517] + (3 + 4/x)^2)*Sqrt[8 + 24*x + 8*x^2 - 15*x^3 + 8*x^4]) - (2455*(Sqrt[51
7] + (3 + 4/x)^2)*Sqrt[(517 - 38*(3 + 4/x)^2 + (3 + 4/x)^4)/(Sqrt[517] + (3 + 4/
x)^2)^2]*x^2*EllipticE[2*ArcTan[(4 + 3*x)/(517^(1/4)*x)], (517 + 19*Sqrt[517])/1
034])/(624*517^(3/4)*Sqrt[8 + 24*x + 8*x^2 - 15*x^3 + 8*x^4]) + ((4910 - 203*Sqr
t[517])*(Sqrt[517] + (3 + 4/x)^2)*Sqrt[(517 - 38*(3 + 4/x)^2 + (3 + 4/x)^4)/(Sqr
t[517] + (3 + 4/x)^2)^2]*x^2*EllipticF[2*ArcTan[(4 + 3*x)/(517^(1/4)*x)], (517 +
19*Sqrt[517])/1034])/(2496*517^(3/4)*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. \[ - 1024 \int ^{\frac{3}{4} + \frac{1}{x}} \frac{1}{\left (\frac{8388608 x^{4} - 19922944 x^{2} + 16941056}{\left (- 32 x + 24\right )^{4}}\right )^{\frac{3}{2}} \left (- 32 x + 24\right )^{2}}\, dx \]

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)**(3/2),x)

[Out]

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

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Mathematica [C]  time = 6.07067, size = 6019, normalized size = 13.87 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

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

[Out]

Result too large to show

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Maple [C]  time = 0.03, size = 5421, normalized size = 12.5 \[ \text{output too large to display} \]

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)^(3/2),x)

[Out]

result too large to display

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

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)**(3/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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