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

Optimal. Leaf size=19 \[ \frac{x^4}{4 b \left (a x^2+b\right )^2} \]

[Out]

x^4/(4*b*(b + a*x^2)^2)

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Rubi [A]  time = 0.0171456, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{x^4}{4 b \left (a x^2+b\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[(b/x + a*x)^(-3),x]

[Out]

x^4/(4*b*(b + a*x^2)^2)

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Rubi in Sympy [A]  time = 3.52927, size = 14, normalized size = 0.74 \[ \frac{x^{4}}{4 b \left (a x^{2} + b\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**4/(4*b*(a*x**2 + b)**2)

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Mathematica [A]  time = 0.0146395, size = 24, normalized size = 1.26 \[ -\frac{2 a x^2+b}{4 a^2 \left (a x^2+b\right )^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(b/x + a*x)^(-3),x]

[Out]

-(b + 2*a*x^2)/(4*a^2*(b + a*x^2)^2)

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Maple [A]  time = 0.007, size = 31, normalized size = 1.6 \[ -{\frac{1}{ \left ( 2\,a{x}^{2}+2\,b \right ){a}^{2}}}+{\frac{b}{4\,{a}^{2} \left ( a{x}^{2}+b \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b/x+a*x)^3,x)

[Out]

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

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Maxima [A]  time = 1.37528, size = 49, normalized size = 2.58 \[ -\frac{2 \, a x^{2} + b}{4 \,{\left (a^{4} x^{4} + 2 \, a^{3} b x^{2} + a^{2} b^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.205972, size = 49, normalized size = 2.58 \[ -\frac{2 \, a x^{2} + b}{4 \,{\left (a^{4} x^{4} + 2 \, a^{3} b x^{2} + a^{2} b^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.64192, size = 36, normalized size = 1.89 \[ - \frac{2 a x^{2} + b}{4 a^{4} x^{4} + 8 a^{3} b x^{2} + 4 a^{2} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b/x+a*x)**3,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.21962, size = 30, normalized size = 1.58 \[ -\frac{2 \, a x^{2} + b}{4 \,{\left (a x^{2} + b\right )}^{2} a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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