3.965 \(\int \frac{1}{x^7 \sqrt{16-x^4}} \, dx\)

Optimal. Leaf size=37 \[ -\frac{\sqrt{16-x^4}}{96 x^6}-\frac{\sqrt{16-x^4}}{768 x^2} \]

[Out]

-Sqrt[16 - x^4]/(96*x^6) - Sqrt[16 - x^4]/(768*x^2)

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Rubi [A]  time = 0.0289965, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ -\frac{\sqrt{16-x^4}}{96 x^6}-\frac{\sqrt{16-x^4}}{768 x^2} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^7*Sqrt[16 - x^4]),x]

[Out]

-Sqrt[16 - x^4]/(96*x^6) - Sqrt[16 - x^4]/(768*x^2)

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Rubi in Sympy [A]  time = 3.54588, size = 27, normalized size = 0.73 \[ - \frac{\sqrt{- x^{4} + 16}}{768 x^{2}} - \frac{\sqrt{- x^{4} + 16}}{96 x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**7/(-x**4+16)**(1/2),x)

[Out]

-sqrt(-x**4 + 16)/(768*x**2) - sqrt(-x**4 + 16)/(96*x**6)

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Mathematica [A]  time = 0.0130572, size = 23, normalized size = 0.62 \[ -\frac{\sqrt{16-x^4} \left (x^4+8\right )}{768 x^6} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^7*Sqrt[16 - x^4]),x]

[Out]

-(Sqrt[16 - x^4]*(8 + x^4))/(768*x^6)

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Maple [A]  time = 0.007, size = 31, normalized size = 0.8 \[{\frac{ \left ( -2+x \right ) \left ( 2+x \right ) \left ({x}^{2}+4 \right ) \left ({x}^{4}+8 \right ) }{768\,{x}^{6}}{\frac{1}{\sqrt{-{x}^{4}+16}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^7/(-x^4+16)^(1/2),x)

[Out]

1/768*(-2+x)*(2+x)*(x^2+4)*(x^4+8)/x^6/(-x^4+16)^(1/2)

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Maxima [A]  time = 1.43925, size = 39, normalized size = 1.05 \[ -\frac{\sqrt{-x^{4} + 16}}{512 \, x^{2}} - \frac{{\left (-x^{4} + 16\right )}^{\frac{3}{2}}}{1536 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-x^4 + 16)*x^7),x, algorithm="maxima")

[Out]

-1/512*sqrt(-x^4 + 16)/x^2 - 1/1536*(-x^4 + 16)^(3/2)/x^6

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Fricas [A]  time = 0.26191, size = 99, normalized size = 2.68 \[ -\frac{x^{12} - 72 \, x^{8} + 384 \, x^{4} + 4 \,{\left (3 \, x^{8} - 40 \, x^{4} - 512\right )} \sqrt{-x^{4} + 16} + 8192}{768 \,{\left (12 \, x^{10} - 256 \, x^{6} -{\left (x^{10} - 64 \, x^{6}\right )} \sqrt{-x^{4} + 16}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-x^4 + 16)*x^7),x, algorithm="fricas")

[Out]

-1/768*(x^12 - 72*x^8 + 384*x^4 + 4*(3*x^8 - 40*x^4 - 512)*sqrt(-x^4 + 16) + 819
2)/(12*x^10 - 256*x^6 - (x^10 - 64*x^6)*sqrt(-x^4 + 16))

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Sympy [A]  time = 3.10299, size = 66, normalized size = 1.78 \[ \begin{cases} - \frac{\sqrt{-1 + \frac{16}{x^{4}}}}{768} - \frac{\sqrt{-1 + \frac{16}{x^{4}}}}{96 x^{4}} & \text{for}\: 16 \left |{\frac{1}{x^{4}}}\right | > 1 \\- \frac{i \sqrt{1 - \frac{16}{x^{4}}}}{768} - \frac{i \sqrt{1 - \frac{16}{x^{4}}}}{96 x^{4}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**7/(-x**4+16)**(1/2),x)

[Out]

Piecewise((-sqrt(-1 + 16/x**4)/768 - sqrt(-1 + 16/x**4)/(96*x**4), 16*Abs(x**(-4
)) > 1), (-I*sqrt(1 - 16/x**4)/768 - I*sqrt(1 - 16/x**4)/(96*x**4), True))

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GIAC/XCAS [A]  time = 0.214914, size = 31, normalized size = 0.84 \[ -\frac{1}{1536} \,{\left (\frac{16}{x^{4}} - 1\right )}^{\frac{3}{2}} - \frac{1}{512} \, \sqrt{\frac{16}{x^{4}} - 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-x^4 + 16)*x^7),x, algorithm="giac")

[Out]

-1/1536*(16/x^4 - 1)^(3/2) - 1/512*sqrt(16/x^4 - 1)