3.2403 \(\int \frac{\left (a+\frac{b}{\sqrt [3]{x}}\right )^2}{x^3} \, dx\)

Optimal. Leaf size=34 \[ -\frac{a^2}{2 x^2}-\frac{6 a b}{7 x^{7/3}}-\frac{3 b^2}{8 x^{8/3}} \]

[Out]

(-3*b^2)/(8*x^(8/3)) - (6*a*b)/(7*x^(7/3)) - a^2/(2*x^2)

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Rubi [A]  time = 0.0511726, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{a^2}{2 x^2}-\frac{6 a b}{7 x^{7/3}}-\frac{3 b^2}{8 x^{8/3}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^(1/3))^2/x^3,x]

[Out]

(-3*b^2)/(8*x^(8/3)) - (6*a*b)/(7*x^(7/3)) - a^2/(2*x^2)

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Rubi in Sympy [A]  time = 8.04342, size = 32, normalized size = 0.94 \[ - \frac{a^{2}}{2 x^{2}} - \frac{6 a b}{7 x^{\frac{7}{3}}} - \frac{3 b^{2}}{8 x^{\frac{8}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x**(1/3))**2/x**3,x)

[Out]

-a**2/(2*x**2) - 6*a*b/(7*x**(7/3)) - 3*b**2/(8*x**(8/3))

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Mathematica [A]  time = 0.0165527, size = 34, normalized size = 1. \[ -\frac{a^2}{2 x^2}-\frac{6 a b}{7 x^{7/3}}-\frac{3 b^2}{8 x^{8/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x^(1/3))^2/x^3,x]

[Out]

(-3*b^2)/(8*x^(8/3)) - (6*a*b)/(7*x^(7/3)) - a^2/(2*x^2)

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Maple [A]  time = 0.008, size = 25, normalized size = 0.7 \[ -{\frac{3\,{b}^{2}}{8}{x}^{-{\frac{8}{3}}}}-{\frac{6\,ab}{7}{x}^{-{\frac{7}{3}}}}-{\frac{{a}^{2}}{2\,{x}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^(1/3))^2/x^3,x)

[Out]

-3/8*b^2/x^(8/3)-6/7*a*b/x^(7/3)-1/2*a^2/x^2

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Maxima [A]  time = 1.44569, size = 131, normalized size = 3.85 \[ -\frac{3 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{8}}{8 \, b^{6}} + \frac{15 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{7} a}{7 \, b^{6}} - \frac{5 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{6} a^{2}}{b^{6}} + \frac{6 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{5} a^{3}}{b^{6}} - \frac{15 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{4} a^{4}}{4 \, b^{6}} + \frac{{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{3} a^{5}}{b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^(1/3))^2/x^3,x, algorithm="maxima")

[Out]

-3/8*(a + b/x^(1/3))^8/b^6 + 15/7*(a + b/x^(1/3))^7*a/b^6 - 5*(a + b/x^(1/3))^6*
a^2/b^6 + 6*(a + b/x^(1/3))^5*a^3/b^6 - 15/4*(a + b/x^(1/3))^4*a^4/b^6 + (a + b/
x^(1/3))^3*a^5/b^6

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Fricas [A]  time = 0.219145, size = 35, normalized size = 1.03 \[ -\frac{28 \, a^{2} x^{\frac{2}{3}} + 48 \, a b x^{\frac{1}{3}} + 21 \, b^{2}}{56 \, x^{\frac{8}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^(1/3))^2/x^3,x, algorithm="fricas")

[Out]

-1/56*(28*a^2*x^(2/3) + 48*a*b*x^(1/3) + 21*b^2)/x^(8/3)

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Sympy [A]  time = 6.05623, size = 32, normalized size = 0.94 \[ - \frac{a^{2}}{2 x^{2}} - \frac{6 a b}{7 x^{\frac{7}{3}}} - \frac{3 b^{2}}{8 x^{\frac{8}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x**(1/3))**2/x**3,x)

[Out]

-a**2/(2*x**2) - 6*a*b/(7*x**(7/3)) - 3*b**2/(8*x**(8/3))

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GIAC/XCAS [A]  time = 0.212049, size = 35, normalized size = 1.03 \[ -\frac{28 \, a^{2} x^{\frac{2}{3}} + 48 \, a b x^{\frac{1}{3}} + 21 \, b^{2}}{56 \, x^{\frac{8}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^(1/3))^2/x^3,x, algorithm="giac")

[Out]

-1/56*(28*a^2*x^(2/3) + 48*a*b*x^(1/3) + 21*b^2)/x^(8/3)