3.251 \(\int \frac{1}{x^6 \sqrt{2+x^2}} \, dx\)

Optimal. Leaf size=49 \[ -\frac{\sqrt{x^2+2}}{15 x}-\frac{\sqrt{x^2+2}}{10 x^5}+\frac{\sqrt{x^2+2}}{15 x^3} \]

[Out]

-Sqrt[2 + x^2]/(10*x^5) + Sqrt[2 + x^2]/(15*x^3) - Sqrt[2 + x^2]/(15*x)

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Rubi [A]  time = 0.0355364, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ -\frac{\sqrt{x^2+2}}{15 x}-\frac{\sqrt{x^2+2}}{10 x^5}+\frac{\sqrt{x^2+2}}{15 x^3} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^6*Sqrt[2 + x^2]),x]

[Out]

-Sqrt[2 + x^2]/(10*x^5) + Sqrt[2 + x^2]/(15*x^3) - Sqrt[2 + x^2]/(15*x)

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Rubi in Sympy [A]  time = 2.65416, size = 37, normalized size = 0.76 \[ - \frac{\sqrt{x^{2} + 2}}{15 x} + \frac{\sqrt{x^{2} + 2}}{15 x^{3}} - \frac{\sqrt{x^{2} + 2}}{10 x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**6/(x**2+2)**(1/2),x)

[Out]

-sqrt(x**2 + 2)/(15*x) + sqrt(x**2 + 2)/(15*x**3) - sqrt(x**2 + 2)/(10*x**5)

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Mathematica [A]  time = 0.0134748, size = 28, normalized size = 0.57 \[ -\frac{\sqrt{x^2+2} \left (2 x^4-2 x^2+3\right )}{30 x^5} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^6*Sqrt[2 + x^2]),x]

[Out]

-(Sqrt[2 + x^2]*(3 - 2*x^2 + 2*x^4))/(30*x^5)

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Maple [A]  time = 0.006, size = 25, normalized size = 0.5 \[ -{\frac{2\,{x}^{4}-2\,{x}^{2}+3}{30\,{x}^{5}}\sqrt{{x}^{2}+2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^6/(x^2+2)^(1/2),x)

[Out]

-1/30*(x^2+2)^(1/2)*(2*x^4-2*x^2+3)/x^5

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Maxima [A]  time = 1.50153, size = 50, normalized size = 1.02 \[ -\frac{\sqrt{x^{2} + 2}}{15 \, x} + \frac{\sqrt{x^{2} + 2}}{15 \, x^{3}} - \frac{\sqrt{x^{2} + 2}}{10 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x^2 + 2)*x^6),x, algorithm="maxima")

[Out]

-1/15*sqrt(x^2 + 2)/x + 1/15*sqrt(x^2 + 2)/x^3 - 1/10*sqrt(x^2 + 2)/x^5

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Fricas [A]  time = 0.30163, size = 99, normalized size = 2.02 \[ \frac{20 \, x^{4} + 35 \, x^{2} - 5 \,{\left (4 \, x^{3} + 3 \, x\right )} \sqrt{x^{2} + 2} + 6}{30 \,{\left (4 \, x^{10} + 10 \, x^{8} + 5 \, x^{6} -{\left (4 \, x^{9} + 6 \, x^{7} + x^{5}\right )} \sqrt{x^{2} + 2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x^2 + 2)*x^6),x, algorithm="fricas")

[Out]

1/30*(20*x^4 + 35*x^2 - 5*(4*x^3 + 3*x)*sqrt(x^2 + 2) + 6)/(4*x^10 + 10*x^8 + 5*
x^6 - (4*x^9 + 6*x^7 + x^5)*sqrt(x^2 + 2))

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Sympy [A]  time = 8.40544, size = 41, normalized size = 0.84 \[ - \frac{\sqrt{1 + \frac{2}{x^{2}}}}{15} + \frac{\sqrt{1 + \frac{2}{x^{2}}}}{15 x^{2}} - \frac{\sqrt{1 + \frac{2}{x^{2}}}}{10 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**6/(x**2+2)**(1/2),x)

[Out]

-sqrt(1 + 2/x**2)/15 + sqrt(1 + 2/x**2)/(15*x**2) - sqrt(1 + 2/x**2)/(10*x**4)

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GIAC/XCAS [A]  time = 0.2088, size = 69, normalized size = 1.41 \[ \frac{32 \,{\left (5 \,{\left (x - \sqrt{x^{2} + 2}\right )}^{4} - 5 \,{\left (x - \sqrt{x^{2} + 2}\right )}^{2} + 2\right )}}{15 \,{\left ({\left (x - \sqrt{x^{2} + 2}\right )}^{2} - 2\right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x^2 + 2)*x^6),x, algorithm="giac")

[Out]

32/15*(5*(x - sqrt(x^2 + 2))^4 - 5*(x - sqrt(x^2 + 2))^2 + 2)/((x - sqrt(x^2 + 2
))^2 - 2)^5