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

Optimal. Leaf size=68 \[ -\frac{4 b^2 \sqrt{a+b x^4}}{15 a^3 x^2}+\frac{2 b \sqrt{a+b x^4}}{15 a^2 x^6}-\frac{\sqrt{a+b x^4}}{10 a x^{10}} \]

[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 = 0.0658592, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ -\frac{4 b^2 \sqrt{a+b x^4}}{15 a^3 x^2}+\frac{2 b \sqrt{a+b x^4}}{15 a^2 x^6}-\frac{\sqrt{a+b x^4}}{10 a x^{10}} \]

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 = 6.8055, size = 61, normalized size = 0.9 \[ - \frac{\sqrt{a + b x^{4}}}{10 a x^{10}} + \frac{2 b \sqrt{a + b x^{4}}}{15 a^{2} x^{6}} - \frac{4 b^{2} \sqrt{a + b x^{4}}}{15 a^{3} x^{2}} \]

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 = 0.0340686, size = 42, normalized size = 0.62 \[ -\frac{\sqrt{a+b x^4} \left (3 a^2-4 a b x^4+8 b^2 x^8\right )}{30 a^3 x^{10}} \]

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 = 0.007, size = 39, 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 = 1.43589, size = 70, normalized size = 1.03 \[ -\frac{\frac{15 \, \sqrt{b x^{4} + a} b^{2}}{x^{2}} - \frac{10 \,{\left (b x^{4} + a\right )}^{\frac{3}{2}} b}{x^{6}} + \frac{3 \,{\left (b x^{4} + a\right )}^{\frac{5}{2}}}{x^{10}}}{30 \, a^{3}} \]

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 = 0.267401, size = 51, 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 = 8.00128, size = 298, normalized size = 4.38 \[ - \frac{3 a^{4} b^{\frac{9}{2}} \sqrt{\frac{a}{b x^{4}} + 1}}{30 a^{5} b^{4} x^{8} + 60 a^{4} b^{5} x^{12} + 30 a^{3} b^{6} x^{16}} - \frac{2 a^{3} b^{\frac{11}{2}} x^{4} \sqrt{\frac{a}{b x^{4}} + 1}}{30 a^{5} b^{4} x^{8} + 60 a^{4} b^{5} x^{12} + 30 a^{3} b^{6} x^{16}} - \frac{3 a^{2} b^{\frac{13}{2}} x^{8} \sqrt{\frac{a}{b x^{4}} + 1}}{30 a^{5} b^{4} x^{8} + 60 a^{4} b^{5} x^{12} + 30 a^{3} b^{6} x^{16}} - \frac{12 a b^{\frac{15}{2}} x^{12} \sqrt{\frac{a}{b x^{4}} + 1}}{30 a^{5} b^{4} x^{8} + 60 a^{4} b^{5} x^{12} + 30 a^{3} b^{6} x^{16}} - \frac{8 b^{\frac{17}{2}} x^{16} \sqrt{\frac{a}{b x^{4}} + 1}}{30 a^{5} b^{4} x^{8} + 60 a^{4} b^{5} x^{12} + 30 a^{3} b^{6} x^{16}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-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)

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GIAC/XCAS [A]  time = 0.217594, size = 58, normalized size = 0.85 \[ -\frac{3 \,{\left (b + \frac{a}{x^{4}}\right )}^{\frac{5}{2}} - 10 \,{\left (b + \frac{a}{x^{4}}\right )}^{\frac{3}{2}} b + 15 \, \sqrt{b + \frac{a}{x^{4}}} b^{2}}{30 \, a^{3}} \]

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