3.838 \(\int \frac{1}{x^{11} \sqrt{a-b x^4}} \, dx\)

Optimal. Leaf size=71 \[ -\frac{4 b^2 \sqrt{a-b x^4}}{15 a^3 x^2}-\frac{2 b \sqrt{a-b x^4}}{15 a^2 x^6}-\frac{\sqrt{a-b x^4}}{10 a x^{10}} \]

[Out]

-Sqrt[a - b*x^4]/(10*a*x^10) - (2*b*Sqrt[a - b*x^4])/(15*a^2*x^6) - (4*b^2*Sqrt[
a - b*x^4])/(15*a^3*x^2)

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Rubi [A]  time = 0.0700337, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{4 b^2 \sqrt{a-b x^4}}{15 a^3 x^2}-\frac{2 b \sqrt{a-b x^4}}{15 a^2 x^6}-\frac{\sqrt{a-b x^4}}{10 a x^{10}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^11*Sqrt[a - b*x^4]),x]

[Out]

-Sqrt[a - b*x^4]/(10*a*x^10) - (2*b*Sqrt[a - b*x^4])/(15*a^2*x^6) - (4*b^2*Sqrt[
a - b*x^4])/(15*a^3*x^2)

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Rubi in Sympy [A]  time = 7.63583, size = 63, normalized size = 0.89 \[ - \frac{\sqrt{a - b x^{4}}}{10 a x^{10}} - \frac{2 b \sqrt{a - b x^{4}}}{15 a^{2} x^{6}} - \frac{4 b^{2} \sqrt{a - b x^{4}}}{15 a^{3} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**11/(-b*x**4+a)**(1/2),x)

[Out]

-sqrt(a - b*x**4)/(10*a*x**10) - 2*b*sqrt(a - b*x**4)/(15*a**2*x**6) - 4*b**2*sq
rt(a - b*x**4)/(15*a**3*x**2)

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Mathematica [A]  time = 0.0332488, size = 43, normalized size = 0.61 \[ -\frac{\sqrt{a-b x^4} \left (3 a^2+4 a b x^4+8 b^2 x^8\right )}{30 a^3 x^{10}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^11*Sqrt[a - b*x^4]),x]

[Out]

-(Sqrt[a - b*x^4]*(3*a^2 + 4*a*b*x^4 + 8*b^2*x^8))/(30*a^3*x^10)

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Maple [A]  time = 0.008, size = 40, normalized size = 0.6 \[ -{\frac{8\,{b}^{2}{x}^{8}+4\,ab{x}^{4}+3\,{a}^{2}}{30\,{x}^{10}{a}^{3}}\sqrt{-b{x}^{4}+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^11/(-b*x^4+a)^(1/2),x)

[Out]

-1/30*(-b*x^4+a)^(1/2)*(8*b^2*x^8+4*a*b*x^4+3*a^2)/x^10/a^3

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Maxima [A]  time = 1.5259, size = 74, normalized size = 1.04 \[ -\frac{\frac{15 \, \sqrt{-b x^{4} + a} b^{2}}{x^{2}} + \frac{10 \,{\left (-b x^{4} + a\right )}^{\frac{3}{2}} b}{x^{6}} + \frac{3 \,{\left (-b x^{4} + a\right )}^{\frac{5}{2}}}{x^{10}}}{30 \, a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-b*x^4 + a)*x^11),x, algorithm="maxima")

[Out]

-1/30*(15*sqrt(-b*x^4 + a)*b^2/x^2 + 10*(-b*x^4 + a)^(3/2)*b/x^6 + 3*(-b*x^4 + a
)^(5/2)/x^10)/a^3

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Fricas [A]  time = 0.273372, size = 53, normalized size = 0.75 \[ -\frac{{\left (8 \, b^{2} x^{8} + 4 \, a b x^{4} + 3 \, a^{2}\right )} \sqrt{-b x^{4} + a}}{30 \, a^{3} x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-b*x^4 + a)*x^11),x, algorithm="fricas")

[Out]

-1/30*(8*b^2*x^8 + 4*a*b*x^4 + 3*a^2)*sqrt(-b*x^4 + a)/(a^3*x^10)

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Sympy [A]  time = 8.24177, size = 609, normalized size = 8.58 \[ \begin{cases} - \frac{3 a^{4} b^{\frac{9}{2}} \sqrt{\frac{a}{b x^{4}} - 1}}{30 a^{5} b^{4} x^{8} - 60 a^{4} b^{5} x^{12} + 30 a^{3} b^{6} x^{16}} + \frac{2 a^{3} b^{\frac{11}{2}} x^{4} \sqrt{\frac{a}{b x^{4}} - 1}}{30 a^{5} b^{4} x^{8} - 60 a^{4} b^{5} x^{12} + 30 a^{3} b^{6} x^{16}} - \frac{3 a^{2} b^{\frac{13}{2}} x^{8} \sqrt{\frac{a}{b x^{4}} - 1}}{30 a^{5} b^{4} x^{8} - 60 a^{4} b^{5} x^{12} + 30 a^{3} b^{6} x^{16}} + \frac{12 a b^{\frac{15}{2}} x^{12} \sqrt{\frac{a}{b x^{4}} - 1}}{30 a^{5} b^{4} x^{8} - 60 a^{4} b^{5} x^{12} + 30 a^{3} b^{6} x^{16}} - \frac{8 b^{\frac{17}{2}} x^{16} \sqrt{\frac{a}{b x^{4}} - 1}}{30 a^{5} b^{4} x^{8} - 60 a^{4} b^{5} x^{12} + 30 a^{3} b^{6} x^{16}} & \text{for}\: \left |{\frac{a}{b x^{4}}}\right | > 1 \\- \frac{3 i a^{4} b^{\frac{9}{2}} \sqrt{- \frac{a}{b x^{4}} + 1}}{30 a^{5} b^{4} x^{8} - 60 a^{4} b^{5} x^{12} + 30 a^{3} b^{6} x^{16}} + \frac{2 i a^{3} b^{\frac{11}{2}} x^{4} \sqrt{- \frac{a}{b x^{4}} + 1}}{30 a^{5} b^{4} x^{8} - 60 a^{4} b^{5} x^{12} + 30 a^{3} b^{6} x^{16}} - \frac{3 i a^{2} b^{\frac{13}{2}} x^{8} \sqrt{- \frac{a}{b x^{4}} + 1}}{30 a^{5} b^{4} x^{8} - 60 a^{4} b^{5} x^{12} + 30 a^{3} b^{6} x^{16}} + \frac{12 i a b^{\frac{15}{2}} x^{12} \sqrt{- \frac{a}{b x^{4}} + 1}}{30 a^{5} b^{4} x^{8} - 60 a^{4} b^{5} x^{12} + 30 a^{3} b^{6} x^{16}} - \frac{8 i b^{\frac{17}{2}} x^{16} \sqrt{- \frac{a}{b x^{4}} + 1}}{30 a^{5} b^{4} x^{8} - 60 a^{4} b^{5} x^{12} + 30 a^{3} b^{6} x^{16}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**11/(-b*x**4+a)**(1/2),x)

[Out]

Piecewise((-3*a**4*b**(9/2)*sqrt(a/(b*x**4) - 1)/(30*a**5*b**4*x**8 - 60*a**4*b*
*5*x**12 + 30*a**3*b**6*x**16) + 2*a**3*b**(11/2)*x**4*sqrt(a/(b*x**4) - 1)/(30*
a**5*b**4*x**8 - 60*a**4*b**5*x**12 + 30*a**3*b**6*x**16) - 3*a**2*b**(13/2)*x**
8*sqrt(a/(b*x**4) - 1)/(30*a**5*b**4*x**8 - 60*a**4*b**5*x**12 + 30*a**3*b**6*x*
*16) + 12*a*b**(15/2)*x**12*sqrt(a/(b*x**4) - 1)/(30*a**5*b**4*x**8 - 60*a**4*b*
*5*x**12 + 30*a**3*b**6*x**16) - 8*b**(17/2)*x**16*sqrt(a/(b*x**4) - 1)/(30*a**5
*b**4*x**8 - 60*a**4*b**5*x**12 + 30*a**3*b**6*x**16), Abs(a/(b*x**4)) > 1), (-3
*I*a**4*b**(9/2)*sqrt(-a/(b*x**4) + 1)/(30*a**5*b**4*x**8 - 60*a**4*b**5*x**12 +
30*a**3*b**6*x**16) + 2*I*a**3*b**(11/2)*x**4*sqrt(-a/(b*x**4) + 1)/(30*a**5*b*
*4*x**8 - 60*a**4*b**5*x**12 + 30*a**3*b**6*x**16) - 3*I*a**2*b**(13/2)*x**8*sqr
t(-a/(b*x**4) + 1)/(30*a**5*b**4*x**8 - 60*a**4*b**5*x**12 + 30*a**3*b**6*x**16)
+ 12*I*a*b**(15/2)*x**12*sqrt(-a/(b*x**4) + 1)/(30*a**5*b**4*x**8 - 60*a**4*b**
5*x**12 + 30*a**3*b**6*x**16) - 8*I*b**(17/2)*x**16*sqrt(-a/(b*x**4) + 1)/(30*a*
*5*b**4*x**8 - 60*a**4*b**5*x**12 + 30*a**3*b**6*x**16), True))

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GIAC/XCAS [A]  time = 0.219657, size = 80, normalized size = 1.13 \[ -\frac{3 \,{\left (b - \frac{a}{x^{4}}\right )}^{2} \sqrt{-b + \frac{a}{x^{4}}} + 15 \, b^{2} \sqrt{-b + \frac{a}{x^{4}}} + 10 \, b{\left (-b + \frac{a}{x^{4}}\right )}^{\frac{3}{2}}}{30 \, a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-b*x^4 + a)*x^11),x, algorithm="giac")

[Out]

-1/30*(3*(b - a/x^4)^2*sqrt(-b + a/x^4) + 15*b^2*sqrt(-b + a/x^4) + 10*b*(-b + a
/x^4)^(3/2))/a^3