3.944 \(\int \frac{1}{x^7 \left (1+x^4\right )^{3/2}} \, dx\)

Optimal. Leaf size=49 \[ -\frac{1}{6 \sqrt{x^4+1} x^6}+\frac{4 x^2}{3 \sqrt{x^4+1}}+\frac{2}{3 \sqrt{x^4+1} x^2} \]

[Out]

-1/(6*x^6*Sqrt[1 + x^4]) + 2/(3*x^2*Sqrt[1 + x^4]) + (4*x^2)/(3*Sqrt[1 + x^4])

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Rubi [A]  time = 0.03545, 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 \[ -\frac{1}{6 \sqrt{x^4+1} x^6}+\frac{4 x^2}{3 \sqrt{x^4+1}}+\frac{2}{3 \sqrt{x^4+1} x^2} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^7*(1 + x^4)^(3/2)),x]

[Out]

-1/(6*x^6*Sqrt[1 + x^4]) + 2/(3*x^2*Sqrt[1 + x^4]) + (4*x^2)/(3*Sqrt[1 + x^4])

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Rubi in Sympy [A]  time = 3.7896, size = 44, normalized size = 0.9 \[ \frac{4 x^{2}}{3 \sqrt{x^{4} + 1}} + \frac{2}{3 x^{2} \sqrt{x^{4} + 1}} - \frac{1}{6 x^{6} \sqrt{x^{4} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4*x**2/(3*sqrt(x**4 + 1)) + 2/(3*x**2*sqrt(x**4 + 1)) - 1/(6*x**6*sqrt(x**4 + 1)
)

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Mathematica [A]  time = 0.0155781, size = 28, normalized size = 0.57 \[ \frac{8 x^8+4 x^4-1}{6 x^6 \sqrt{x^4+1}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^7*(1 + x^4)^(3/2)),x]

[Out]

(-1 + 4*x^4 + 8*x^8)/(6*x^6*Sqrt[1 + x^4])

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Maple [A]  time = 0.006, size = 25, normalized size = 0.5 \[{\frac{8\,{x}^{8}+4\,{x}^{4}-1}{6\,{x}^{6}}{\frac{1}{\sqrt{{x}^{4}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^7/(x^4+1)^(3/2),x)

[Out]

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

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Maxima [A]  time = 1.48871, size = 49, normalized size = 1. \[ \frac{x^{2}}{2 \, \sqrt{x^{4} + 1}} + \frac{\sqrt{x^{4} + 1}}{x^{2}} - \frac{{\left (x^{4} + 1\right )}^{\frac{3}{2}}}{6 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.264475, size = 84, normalized size = 1.71 \[ \frac{4 \, x^{4} - 4 \, \sqrt{x^{4} + 1} x^{2} + 1}{6 \,{\left (8 \, x^{16} + 12 \, x^{12} + 4 \, x^{8} -{\left (8 \, x^{14} + 8 \, x^{10} + x^{6}\right )} \sqrt{x^{4} + 1}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*(4*x^4 - 4*sqrt(x^4 + 1)*x^2 + 1)/(8*x^16 + 12*x^12 + 4*x^8 - (8*x^14 + 8*x^
10 + x^6)*sqrt(x^4 + 1))

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Sympy [A]  time = 5.06567, size = 70, normalized size = 1.43 \[ \frac{8 x^{8} \sqrt{1 + \frac{1}{x^{4}}}}{6 x^{8} + 6 x^{4}} + \frac{4 x^{4} \sqrt{1 + \frac{1}{x^{4}}}}{6 x^{8} + 6 x^{4}} - \frac{\sqrt{1 + \frac{1}{x^{4}}}}{6 x^{8} + 6 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

8*x**8*sqrt(1 + x**(-4))/(6*x**8 + 6*x**4) + 4*x**4*sqrt(1 + x**(-4))/(6*x**8 +
6*x**4) - sqrt(1 + x**(-4))/(6*x**8 + 6*x**4)

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GIAC/XCAS [A]  time = 0.233008, size = 39, normalized size = 0.8 \[ \frac{x^{2}}{2 \, \sqrt{x^{4} + 1}} - \frac{1}{6} \,{\left (\frac{1}{x^{4}} + 1\right )}^{\frac{3}{2}} + \sqrt{\frac{1}{x^{4}} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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