3.343 \(\int \frac{1}{\left (\frac{b}{x^5}+a x^3\right )^3} \, dx\)

Optimal. Leaf size=19 \[ \frac{x^{16}}{16 b \left (a x^8+b\right )^2} \]

[Out]

x^16/(16*b*(b + a*x^8)^2)

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Rubi [A]  time = 0.017464, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{x^{16}}{16 b \left (a x^8+b\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[(b/x^5 + a*x^3)^(-3),x]

[Out]

x^16/(16*b*(b + a*x^8)^2)

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Rubi in Sympy [A]  time = 3.29014, size = 14, normalized size = 0.74 \[ \frac{x^{16}}{16 b \left (a x^{8} + b\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b/x**5+a*x**3)**3,x)

[Out]

x**16/(16*b*(a*x**8 + b)**2)

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Mathematica [A]  time = 0.0170535, size = 24, normalized size = 1.26 \[ -\frac{2 a x^8+b}{16 a^2 \left (a x^8+b\right )^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(b/x^5 + a*x^3)^(-3),x]

[Out]

-(b + 2*a*x^8)/(16*a^2*(b + a*x^8)^2)

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Maple [A]  time = 0.013, size = 31, normalized size = 1.6 \[ -{\frac{1}{ \left ( 8\,a{x}^{8}+8\,b \right ){a}^{2}}}+{\frac{b}{16\,{a}^{2} \left ( a{x}^{8}+b \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b/x^5+a*x^3)^3,x)

[Out]

-1/8/(a*x^8+b)/a^2+1/16*b/a^2/(a*x^8+b)^2

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Maxima [A]  time = 1.37092, size = 49, normalized size = 2.58 \[ -\frac{2 \, a x^{8} + b}{16 \,{\left (a^{4} x^{16} + 2 \, a^{3} b x^{8} + a^{2} b^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a*x^3 + b/x^5)^(-3),x, algorithm="maxima")

[Out]

-1/16*(2*a*x^8 + b)/(a^4*x^16 + 2*a^3*b*x^8 + a^2*b^2)

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Fricas [A]  time = 0.205732, size = 49, normalized size = 2.58 \[ -\frac{2 \, a x^{8} + b}{16 \,{\left (a^{4} x^{16} + 2 \, a^{3} b x^{8} + a^{2} b^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a*x^3 + b/x^5)^(-3),x, algorithm="fricas")

[Out]

-1/16*(2*a*x^8 + b)/(a^4*x^16 + 2*a^3*b*x^8 + a^2*b^2)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b/x**5+a*x**3)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.215441, size = 30, normalized size = 1.58 \[ -\frac{2 \, a x^{8} + b}{16 \,{\left (a x^{8} + b\right )}^{2} a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a*x^3 + b/x^5)^(-3),x, algorithm="giac")

[Out]

-1/16*(2*a*x^8 + b)/((a*x^8 + b)^2*a^2)