3.278 \(\int \frac{-8-8 x-x^2-3 x^3+7 x^4+4 x^5+2 x^6}{\left (-1+2 x^2\right )^2 \sqrt{1+2 x^2+4 x^3+x^4}} \, dx\)

Optimal. Leaf size=94 \[ \frac{(2 x+1) \sqrt{x^4+4 x^3+2 x^2+1}}{2 \left (2 x^2-1\right )}-\tanh ^{-1}\left (\frac{x (x+2) \left (33 x^3+27 x^2-x+7\right )}{\left (31 x^3+37 x^2+2\right ) \sqrt{x^4+4 x^3+2 x^2+1}}\right ) \]

[Out]

((1 + 2*x)*Sqrt[1 + 2*x^2 + 4*x^3 + x^4])/(2*(-1 + 2*x^2)) - ArcTanh[(x*(2 + x)*
(7 - x + 27*x^2 + 33*x^3))/((2 + 37*x^2 + 31*x^3)*Sqrt[1 + 2*x^2 + 4*x^3 + x^4])
]

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Rubi [F]  time = 2.88034, 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. \[ \text{Int}\left (\frac{-8-8 x-x^2-3 x^3+7 x^4+4 x^5+2 x^6}{\left (-1+2 x^2\right )^2 \sqrt{1+2 x^2+4 x^3+x^4}},x\right ) \]

Verification is Not applicable to the result.

[In]  Int[(-8 - 8*x - x^2 - 3*x^3 + 7*x^4 + 4*x^5 + 2*x^6)/((-1 + 2*x^2)^2*Sqrt[1 + 2*x^2 + 4*x^3 + x^4]),x]

[Out]

(9*Defer[Int][1/Sqrt[1 + 2*x^2 + 4*x^3 + x^4], x])/4 - (13*Defer[Int][1/((Sqrt[2
] - 2*x)^2*Sqrt[1 + 2*x^2 + 4*x^3 + x^4]), x])/4 + Defer[Int][x/Sqrt[1 + 2*x^2 +
4*x^3 + x^4], x] + Defer[Int][x^2/Sqrt[1 + 2*x^2 + 4*x^3 + x^4], x]/2 - (13*Def
er[Int][1/((Sqrt[2] + 2*x)^2*Sqrt[1 + 2*x^2 + 4*x^3 + x^4]), x])/4 - (13*Defer[I
nt][1/((1 - Sqrt[2]*x)*Sqrt[1 + 2*x^2 + 4*x^3 + x^4]), x])/8 - ((15 + Sqrt[2])*D
efer[Int][1/((1 - Sqrt[2]*x)*Sqrt[1 + 2*x^2 + 4*x^3 + x^4]), x])/8 - (13*Defer[I
nt][1/((1 + Sqrt[2]*x)*Sqrt[1 + 2*x^2 + 4*x^3 + x^4]), x])/8 - ((15 - Sqrt[2])*D
efer[Int][1/((1 + Sqrt[2]*x)*Sqrt[1 + 2*x^2 + 4*x^3 + x^4]), x])/8 - (17*Defer[I
nt][x/((-1 + 2*x^2)^2*Sqrt[1 + 2*x^2 + 4*x^3 + x^4]), x])/2

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2*x**6+4*x**5+7*x**4-3*x**3-x**2-8*x-8)/(2*x**2-1)**2/(x**4+4*x**3+2*x**2+1)**(1/2),x)

[Out]

Timed out

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Mathematica [C]  time = 6.13796, size = 5137, normalized size = 54.65 \[ \text{Result too large to show} \]

Antiderivative was successfully verified.

[In]  Integrate[(-8 - 8*x - x^2 - 3*x^3 + 7*x^4 + 4*x^5 + 2*x^6)/((-1 + 2*x^2)^2*Sqrt[1 + 2*x^2 + 4*x^3 + x^4]),x]

[Out]

Result too large to show

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Maple [B]  time = 0.703, size = 1197351, normalized size = 12737.8 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2*x^6+4*x^5+7*x^4-3*x^3-x^2-8*x-8)/(2*x^2-1)^2/(x^4+4*x^3+2*x^2+1)^(1/2),x)

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{2 \, x^{6} + 4 \, x^{5} + 7 \, x^{4} - 3 \, x^{3} - x^{2} - 8 \, x - 8}{\sqrt{x^{4} + 4 \, x^{3} + 2 \, x^{2} + 1}{\left (2 \, x^{2} - 1\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*x^6 + 4*x^5 + 7*x^4 - 3*x^3 - x^2 - 8*x - 8)/(sqrt(x^4 + 4*x^3 + 2*x^2 + 1)*(2*x^2 - 1)^2),x, algorithm="maxima")

[Out]

integrate((2*x^6 + 4*x^5 + 7*x^4 - 3*x^3 - x^2 - 8*x - 8)/(sqrt(x^4 + 4*x^3 + 2*
x^2 + 1)*(2*x^2 - 1)^2), x)

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Fricas [A]  time = 0.294648, size = 242, normalized size = 2.57 \[ \frac{{\left (2 \, x^{2} - 1\right )} \log \left (\frac{1025 \, x^{10} + 6138 \, x^{9} + 12307 \, x^{8} + 10188 \, x^{7} + 4503 \, x^{6} + 3134 \, x^{5} + 1589 \, x^{4} + 140 \, x^{3} + 176 \, x^{2} -{\left (1023 \, x^{8} + 4104 \, x^{7} + 5084 \, x^{6} + 2182 \, x^{5} + 805 \, x^{4} + 624 \, x^{3} + 10 \, x^{2} + 28 \, x\right )} \sqrt{x^{4} + 4 \, x^{3} + 2 \, x^{2} + 1} + 2}{32 \, x^{10} - 80 \, x^{8} + 80 \, x^{6} - 40 \, x^{4} + 10 \, x^{2} - 1}\right ) + \sqrt{x^{4} + 4 \, x^{3} + 2 \, x^{2} + 1}{\left (2 \, x + 1\right )}}{2 \,{\left (2 \, x^{2} - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*x^6 + 4*x^5 + 7*x^4 - 3*x^3 - x^2 - 8*x - 8)/(sqrt(x^4 + 4*x^3 + 2*x^2 + 1)*(2*x^2 - 1)^2),x, algorithm="fricas")

[Out]

1/2*((2*x^2 - 1)*log((1025*x^10 + 6138*x^9 + 12307*x^8 + 10188*x^7 + 4503*x^6 +
3134*x^5 + 1589*x^4 + 140*x^3 + 176*x^2 - (1023*x^8 + 4104*x^7 + 5084*x^6 + 2182
*x^5 + 805*x^4 + 624*x^3 + 10*x^2 + 28*x)*sqrt(x^4 + 4*x^3 + 2*x^2 + 1) + 2)/(32
*x^10 - 80*x^8 + 80*x^6 - 40*x^4 + 10*x^2 - 1)) + sqrt(x^4 + 4*x^3 + 2*x^2 + 1)*
(2*x + 1))/(2*x^2 - 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{2 x^{6} + 4 x^{5} + 7 x^{4} - 3 x^{3} - x^{2} - 8 x - 8}{\sqrt{\left (x + 1\right ) \left (x^{3} + 3 x^{2} - x + 1\right )} \left (2 x^{2} - 1\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*x**6+4*x**5+7*x**4-3*x**3-x**2-8*x-8)/(2*x**2-1)**2/(x**4+4*x**3+2*x**2+1)**(1/2),x)

[Out]

Integral((2*x**6 + 4*x**5 + 7*x**4 - 3*x**3 - x**2 - 8*x - 8)/(sqrt((x + 1)*(x**
3 + 3*x**2 - x + 1))*(2*x**2 - 1)**2), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{2 \, x^{6} + 4 \, x^{5} + 7 \, x^{4} - 3 \, x^{3} - x^{2} - 8 \, x - 8}{\sqrt{x^{4} + 4 \, x^{3} + 2 \, x^{2} + 1}{\left (2 \, x^{2} - 1\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*x^6 + 4*x^5 + 7*x^4 - 3*x^3 - x^2 - 8*x - 8)/(sqrt(x^4 + 4*x^3 + 2*x^2 + 1)*(2*x^2 - 1)^2),x, algorithm="giac")

[Out]

integrate((2*x^6 + 4*x^5 + 7*x^4 - 3*x^3 - x^2 - 8*x - 8)/(sqrt(x^4 + 4*x^3 + 2*
x^2 + 1)*(2*x^2 - 1)^2), x)