3.837 \(\int \frac{1}{x^7 \sqrt{a-b x^4}} \, dx\)

Optimal. Leaf size=46 \[ -\frac{b \sqrt{a-b x^4}}{3 a^2 x^2}-\frac{\sqrt{a-b x^4}}{6 a x^6} \]

[Out]

-Sqrt[a - b*x^4]/(6*a*x^6) - (b*Sqrt[a - b*x^4])/(3*a^2*x^2)

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Rubi [A]  time = 0.0442738, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{b \sqrt{a-b x^4}}{3 a^2 x^2}-\frac{\sqrt{a-b x^4}}{6 a x^6} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^7*Sqrt[a - b*x^4]),x]

[Out]

-Sqrt[a - b*x^4]/(6*a*x^6) - (b*Sqrt[a - b*x^4])/(3*a^2*x^2)

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Rubi in Sympy [A]  time = 4.89733, size = 37, normalized size = 0.8 \[ - \frac{\sqrt{a - b x^{4}}}{6 a x^{6}} - \frac{b \sqrt{a - b x^{4}}}{3 a^{2} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(a - b*x**4)/(6*a*x**6) - b*sqrt(a - b*x**4)/(3*a**2*x**2)

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Mathematica [A]  time = 0.026636, size = 30, normalized size = 0.65 \[ -\frac{\sqrt{a-b x^4} \left (a+2 b x^4\right )}{6 a^2 x^6} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[a - b*x^4]*(a + 2*b*x^4))/(6*a^2*x^6)

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Maple [A]  time = 0.007, size = 27, normalized size = 0.6 \[ -{\frac{2\,b{x}^{4}+a}{6\,{a}^{2}{x}^{6}}\sqrt{-b{x}^{4}+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^7/(-b*x^4+a)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.4143, size = 49, normalized size = 1.07 \[ -\frac{\frac{3 \, \sqrt{-b x^{4} + a} b}{x^{2}} + \frac{{\left (-b x^{4} + a\right )}^{\frac{3}{2}}}{x^{6}}}{6 \, a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.251375, size = 35, normalized size = 0.76 \[ -\frac{{\left (2 \, b x^{4} + a\right )} \sqrt{-b x^{4} + a}}{6 \, a^{2} x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 3.81942, size = 189, normalized size = 4.11 \[ \begin{cases} - \frac{\sqrt{b} \sqrt{\frac{a}{b x^{4}} - 1}}{6 a x^{4}} - \frac{b^{\frac{3}{2}} \sqrt{\frac{a}{b x^{4}} - 1}}{3 a^{2}} & \text{for}\: \left |{\frac{a}{b x^{4}}}\right | > 1 \\\frac{i a^{2} b^{\frac{3}{2}} \sqrt{- \frac{a}{b x^{4}} + 1}}{- 6 a^{3} b x^{4} + 6 a^{2} b^{2} x^{8}} + \frac{i a b^{\frac{5}{2}} x^{4} \sqrt{- \frac{a}{b x^{4}} + 1}}{- 6 a^{3} b x^{4} + 6 a^{2} b^{2} x^{8}} - \frac{2 i b^{\frac{7}{2}} x^{8} \sqrt{- \frac{a}{b x^{4}} + 1}}{- 6 a^{3} b x^{4} + 6 a^{2} b^{2} x^{8}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-sqrt(b)*sqrt(a/(b*x**4) - 1)/(6*a*x**4) - b**(3/2)*sqrt(a/(b*x**4) -
1)/(3*a**2), Abs(a/(b*x**4)) > 1), (I*a**2*b**(3/2)*sqrt(-a/(b*x**4) + 1)/(-6*a
**3*b*x**4 + 6*a**2*b**2*x**8) + I*a*b**(5/2)*x**4*sqrt(-a/(b*x**4) + 1)/(-6*a**
3*b*x**4 + 6*a**2*b**2*x**8) - 2*I*b**(7/2)*x**8*sqrt(-a/(b*x**4) + 1)/(-6*a**3*
b*x**4 + 6*a**2*b**2*x**8), True))

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GIAC/XCAS [A]  time = 0.2161, size = 42, normalized size = 0.91 \[ -\frac{3 \, b \sqrt{-b + \frac{a}{x^{4}}} +{\left (-b + \frac{a}{x^{4}}\right )}^{\frac{3}{2}}}{6 \, a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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