3.646 \(\int \frac{1}{\left (8+24 x+8 x^2-15 x^3+8 x^4\right )^{5/2}} \, dx\)

Optimal. Leaf size=577 \[ -\frac{\left (124415-6308 \left (\frac{4}{x}+3\right )^2\right ) x^2}{97344 \sqrt{8 x^4-15 x^3+8 x^2+24 x+8}}+\frac{\left (18932921731-1086525994 \left (\frac{4}{x}+3\right )^2\right ) \left (\frac{4}{x}+3\right ) x^2}{78056941248 \sqrt{8 x^4-15 x^3+8 x^2+24 x+8}}+\frac{543262997 \left (\left (\frac{4}{x}+3\right )^4-38 \left (\frac{4}{x}+3\right )^2+517\right ) \left (\frac{4}{x}+3\right ) x^2}{39028470624 \left (\left (\frac{4}{x}+3\right )^2+\sqrt{517}\right ) \sqrt{8 x^4-15 x^3+8 x^2+24 x+8}}+\frac{\left (11921698-359497 \left (\frac{4}{x}+3\right )^2\right ) \left (\frac{4}{x}+3\right ) x^2}{483912 \left (\left (\frac{4}{x}+3\right )^4-38 \left (\frac{4}{x}+3\right )^2+517\right ) \sqrt{8 x^4-15 x^3+8 x^2+24 x+8}}-\frac{\left (64489-1399 \left (\frac{4}{x}+3\right )^2\right ) x^2}{624 \left (\left (\frac{4}{x}+3\right )^4-38 \left (\frac{4}{x}+3\right )^2+517\right ) \sqrt{8 x^4-15 x^3+8 x^2+24 x+8}}+\frac{\left (4346103976-175318963 \sqrt{517}\right ) \left (\left (\frac{4}{x}+3\right )^2+\sqrt{517}\right ) \sqrt{\frac{\left (\frac{4}{x}+3\right )^4-38 \left (\frac{4}{x}+3\right )^2+517}{\left (\left (\frac{4}{x}+3\right )^2+\sqrt{517}\right )^2}} x^2 F\left (2 \tan ^{-1}\left (\frac{3 x+4}{\sqrt [4]{517} x}\right )|\frac{517+19 \sqrt{517}}{1034}\right )}{1207844352\ 517^{3/4} \sqrt{8 x^4-15 x^3+8 x^2+24 x+8}}-\frac{543262997 \left (\left (\frac{4}{x}+3\right )^2+\sqrt{517}\right ) \sqrt{\frac{\left (\frac{4}{x}+3\right )^4-38 \left (\frac{4}{x}+3\right )^2+517}{\left (\left (\frac{4}{x}+3\right )^2+\sqrt{517}\right )^2}} x^2 E\left (2 \tan ^{-1}\left (\frac{3 x+4}{\sqrt [4]{517} x}\right )|\frac{517+19 \sqrt{517}}{1034}\right )}{75490272\ 517^{3/4} \sqrt{8 x^4-15 x^3+8 x^2+24 x+8}} \]

[Out]

-((124415 - 6308*(3 + 4/x)^2)*x^2)/(97344*Sqrt[8 + 24*x + 8*x^2 - 15*x^3 + 8*x^4
]) - ((64489 - 1399*(3 + 4/x)^2)*x^2)/(624*(517 - 38*(3 + 4/x)^2 + (3 + 4/x)^4)*
Sqrt[8 + 24*x + 8*x^2 - 15*x^3 + 8*x^4]) + ((18932921731 - 1086525994*(3 + 4/x)^
2)*(3 + 4/x)*x^2)/(78056941248*Sqrt[8 + 24*x + 8*x^2 - 15*x^3 + 8*x^4]) + ((1192
1698 - 359497*(3 + 4/x)^2)*(3 + 4/x)*x^2)/(483912*(517 - 38*(3 + 4/x)^2 + (3 + 4
/x)^4)*Sqrt[8 + 24*x + 8*x^2 - 15*x^3 + 8*x^4]) + (543262997*(517 - 38*(3 + 4/x)
^2 + (3 + 4/x)^4)*(3 + 4/x)*x^2)/(39028470624*(Sqrt[517] + (3 + 4/x)^2)*Sqrt[8 +
24*x + 8*x^2 - 15*x^3 + 8*x^4]) - (543262997*(Sqrt[517] + (3 + 4/x)^2)*Sqrt[(51
7 - 38*(3 + 4/x)^2 + (3 + 4/x)^4)/(Sqrt[517] + (3 + 4/x)^2)^2]*x^2*EllipticE[2*A
rcTan[(4 + 3*x)/(517^(1/4)*x)], (517 + 19*Sqrt[517])/1034])/(75490272*517^(3/4)*
Sqrt[8 + 24*x + 8*x^2 - 15*x^3 + 8*x^4]) + ((4346103976 - 175318963*Sqrt[517])*(
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*Sqrt[
517])/1034])/(1207844352*517^(3/4)*Sqrt[8 + 24*x + 8*x^2 - 15*x^3 + 8*x^4])

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Rubi [A]  time = 1.27776, antiderivative size = 577, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 11, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.458 \[ -\frac{\left (124415-6308 \left (\frac{4}{x}+3\right )^2\right ) x^2}{97344 \sqrt{8 x^4-15 x^3+8 x^2+24 x+8}}+\frac{\left (18932921731-1086525994 \left (\frac{4}{x}+3\right )^2\right ) \left (\frac{4}{x}+3\right ) x^2}{78056941248 \sqrt{8 x^4-15 x^3+8 x^2+24 x+8}}+\frac{543262997 \left (\left (\frac{4}{x}+3\right )^4-38 \left (\frac{4}{x}+3\right )^2+517\right ) \left (\frac{4}{x}+3\right ) x^2}{39028470624 \left (\left (\frac{4}{x}+3\right )^2+\sqrt{517}\right ) \sqrt{8 x^4-15 x^3+8 x^2+24 x+8}}+\frac{\left (11921698-359497 \left (\frac{4}{x}+3\right )^2\right ) \left (\frac{4}{x}+3\right ) x^2}{483912 \left (\left (\frac{4}{x}+3\right )^4-38 \left (\frac{4}{x}+3\right )^2+517\right ) \sqrt{8 x^4-15 x^3+8 x^2+24 x+8}}-\frac{\left (64489-1399 \left (\frac{4}{x}+3\right )^2\right ) x^2}{624 \left (\left (\frac{4}{x}+3\right )^4-38 \left (\frac{4}{x}+3\right )^2+517\right ) \sqrt{8 x^4-15 x^3+8 x^2+24 x+8}}+\frac{\left (4346103976-175318963 \sqrt{517}\right ) \left (\left (\frac{4}{x}+3\right )^2+\sqrt{517}\right ) \sqrt{\frac{\left (\frac{4}{x}+3\right )^4-38 \left (\frac{4}{x}+3\right )^2+517}{\left (\left (\frac{4}{x}+3\right )^2+\sqrt{517}\right )^2}} x^2 F\left (2 \tan ^{-1}\left (\frac{3 x+4}{\sqrt [4]{517} x}\right )|\frac{517+19 \sqrt{517}}{1034}\right )}{1207844352\ 517^{3/4} \sqrt{8 x^4-15 x^3+8 x^2+24 x+8}}-\frac{543262997 \left (\left (\frac{4}{x}+3\right )^2+\sqrt{517}\right ) \sqrt{\frac{\left (\frac{4}{x}+3\right )^4-38 \left (\frac{4}{x}+3\right )^2+517}{\left (\left (\frac{4}{x}+3\right )^2+\sqrt{517}\right )^2}} x^2 E\left (2 \tan ^{-1}\left (\frac{3 x+4}{\sqrt [4]{517} x}\right )|\frac{517+19 \sqrt{517}}{1034}\right )}{75490272\ 517^{3/4} \sqrt{8 x^4-15 x^3+8 x^2+24 x+8}} \]

Antiderivative was successfully verified.

[In]  Int[(8 + 24*x + 8*x^2 - 15*x^3 + 8*x^4)^(-5/2),x]

[Out]

-((124415 - 6308*(3 + 4/x)^2)*x^2)/(97344*Sqrt[8 + 24*x + 8*x^2 - 15*x^3 + 8*x^4
]) - ((64489 - 1399*(3 + 4/x)^2)*x^2)/(624*(517 - 38*(3 + 4/x)^2 + (3 + 4/x)^4)*
Sqrt[8 + 24*x + 8*x^2 - 15*x^3 + 8*x^4]) + ((18932921731 - 1086525994*(3 + 4/x)^
2)*(3 + 4/x)*x^2)/(78056941248*Sqrt[8 + 24*x + 8*x^2 - 15*x^3 + 8*x^4]) + ((1192
1698 - 359497*(3 + 4/x)^2)*(3 + 4/x)*x^2)/(483912*(517 - 38*(3 + 4/x)^2 + (3 + 4
/x)^4)*Sqrt[8 + 24*x + 8*x^2 - 15*x^3 + 8*x^4]) + (543262997*(517 - 38*(3 + 4/x)
^2 + (3 + 4/x)^4)*(3 + 4/x)*x^2)/(39028470624*(Sqrt[517] + (3 + 4/x)^2)*Sqrt[8 +
24*x + 8*x^2 - 15*x^3 + 8*x^4]) - (543262997*(Sqrt[517] + (3 + 4/x)^2)*Sqrt[(51
7 - 38*(3 + 4/x)^2 + (3 + 4/x)^4)/(Sqrt[517] + (3 + 4/x)^2)^2]*x^2*EllipticE[2*A
rcTan[(4 + 3*x)/(517^(1/4)*x)], (517 + 19*Sqrt[517])/1034])/(75490272*517^(3/4)*
Sqrt[8 + 24*x + 8*x^2 - 15*x^3 + 8*x^4]) + ((4346103976 - 175318963*Sqrt[517])*(
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*Sqrt[
517])/1034])/(1207844352*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. \[ - 1024 \int ^{\frac{3}{4} + \frac{1}{x}} \frac{1}{\left (\frac{8388608 x^{4} - 19922944 x^{2} + 16941056}{\left (- 32 x + 24\right )^{4}}\right )^{\frac{5}{2}} \left (- 32 x + 24\right )^{2}}\, dx \]

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)**(5/2),x)

[Out]

-1024*Integral(1/(((8388608*x**4 - 19922944*x**2 + 16941056)/(-32*x + 24)**4)**(
5/2)*(-32*x + 24)**2), (x, 3/4 + 1/x))

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Mathematica [C]  time = 6.0784, size = 6084, normalized size = 10.54 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(8 + 24*x + 8*x^2 - 15*x^3 + 8*x^4)^(-5/2),x]

[Out]

Result too large to show

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Maple [C]  time = 0.033, size = 5477, normalized size = 9.5 \[ \text{output too large to display} \]

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)^(5/2),x)

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (8 \, x^{4} - 15 \, x^{3} + 8 \, x^{2} + 24 \, x + 8\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((8*x^4 - 15*x^3 + 8*x^2 + 24*x + 8)^(-5/2),x, algorithm="maxima")

[Out]

integrate((8*x^4 - 15*x^3 + 8*x^2 + 24*x + 8)^(-5/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (64 \, x^{8} - 240 \, x^{7} + 353 \, x^{6} + 144 \, x^{5} - 528 \, x^{4} + 144 \, x^{3} + 704 \, x^{2} + 384 \, x + 64\right )} \sqrt{8 \, x^{4} - 15 \, x^{3} + 8 \, x^{2} + 24 \, x + 8}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((8*x^4 - 15*x^3 + 8*x^2 + 24*x + 8)^(-5/2),x, algorithm="fricas")

[Out]

integral(1/((64*x^8 - 240*x^7 + 353*x^6 + 144*x^5 - 528*x^4 + 144*x^3 + 704*x^2
+ 384*x + 64)*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. \[ \int \frac{1}{\left (8 x^{4} - 15 x^{3} + 8 x^{2} + 24 x + 8\right )^{\frac{5}{2}}}\, dx \]

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)**(5/2),x)

[Out]

Integral((8*x**4 - 15*x**3 + 8*x**2 + 24*x + 8)**(-5/2), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (8 \, x^{4} - 15 \, x^{3} + 8 \, x^{2} + 24 \, x + 8\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((8*x^4 - 15*x^3 + 8*x^2 + 24*x + 8)^(-5/2),x, algorithm="giac")

[Out]

integrate((8*x^4 - 15*x^3 + 8*x^2 + 24*x + 8)^(-5/2), x)