3.2404 \(\int \frac{\left (a+\frac{b}{\sqrt [3]{x}}\right )^2}{x^4} \, dx\)

Optimal. Leaf size=34 \[ -\frac{a^2}{3 x^3}-\frac{3 a b}{5 x^{10/3}}-\frac{3 b^2}{11 x^{11/3}} \]

[Out]

(-3*b^2)/(11*x^(11/3)) - (3*a*b)/(5*x^(10/3)) - a^2/(3*x^3)

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Rubi [A]  time = 0.0501065, 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 \[ -\frac{a^2}{3 x^3}-\frac{3 a b}{5 x^{10/3}}-\frac{3 b^2}{11 x^{11/3}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^(1/3))^2/x^4,x]

[Out]

(-3*b^2)/(11*x^(11/3)) - (3*a*b)/(5*x^(10/3)) - a^2/(3*x^3)

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Rubi in Sympy [A]  time = 8.28133, size = 32, normalized size = 0.94 \[ - \frac{a^{2}}{3 x^{3}} - \frac{3 a b}{5 x^{\frac{10}{3}}} - \frac{3 b^{2}}{11 x^{\frac{11}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x**(1/3))**2/x**4,x)

[Out]

-a**2/(3*x**3) - 3*a*b/(5*x**(10/3)) - 3*b**2/(11*x**(11/3))

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Mathematica [A]  time = 0.0180864, size = 34, normalized size = 1. \[ -\frac{a^2}{3 x^3}-\frac{3 a b}{5 x^{10/3}}-\frac{3 b^2}{11 x^{11/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x^(1/3))^2/x^4,x]

[Out]

(-3*b^2)/(11*x^(11/3)) - (3*a*b)/(5*x^(10/3)) - a^2/(3*x^3)

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Maple [A]  time = 0.008, size = 25, normalized size = 0.7 \[ -{\frac{3\,{b}^{2}}{11}{x}^{-{\frac{11}{3}}}}-{\frac{3\,ab}{5}{x}^{-{\frac{10}{3}}}}-{\frac{{a}^{2}}{3\,{x}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^(1/3))^2/x^4,x)

[Out]

-3/11*b^2/x^(11/3)-3/5*a*b/x^(10/3)-1/3*a^2/x^3

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Maxima [A]  time = 1.44088, size = 201, normalized size = 5.91 \[ -\frac{3 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{11}}{11 \, b^{9}} + \frac{12 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{10} a}{5 \, b^{9}} - \frac{28 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{9} a^{2}}{3 \, b^{9}} + \frac{21 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{8} a^{3}}{b^{9}} - \frac{30 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{7} a^{4}}{b^{9}} + \frac{28 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{6} a^{5}}{b^{9}} - \frac{84 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{5} a^{6}}{5 \, b^{9}} + \frac{6 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{4} a^{7}}{b^{9}} - \frac{{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{3} a^{8}}{b^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^(1/3))^2/x^4,x, algorithm="maxima")

[Out]

-3/11*(a + b/x^(1/3))^11/b^9 + 12/5*(a + b/x^(1/3))^10*a/b^9 - 28/3*(a + b/x^(1/
3))^9*a^2/b^9 + 21*(a + b/x^(1/3))^8*a^3/b^9 - 30*(a + b/x^(1/3))^7*a^4/b^9 + 28
*(a + b/x^(1/3))^6*a^5/b^9 - 84/5*(a + b/x^(1/3))^5*a^6/b^9 + 6*(a + b/x^(1/3))^
4*a^7/b^9 - (a + b/x^(1/3))^3*a^8/b^9

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Fricas [A]  time = 0.22193, size = 35, normalized size = 1.03 \[ -\frac{55 \, a^{2} x^{\frac{2}{3}} + 99 \, a b x^{\frac{1}{3}} + 45 \, b^{2}}{165 \, x^{\frac{11}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^(1/3))^2/x^4,x, algorithm="fricas")

[Out]

-1/165*(55*a^2*x^(2/3) + 99*a*b*x^(1/3) + 45*b^2)/x^(11/3)

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Sympy [A]  time = 11.4025, size = 32, normalized size = 0.94 \[ - \frac{a^{2}}{3 x^{3}} - \frac{3 a b}{5 x^{\frac{10}{3}}} - \frac{3 b^{2}}{11 x^{\frac{11}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x**(1/3))**2/x**4,x)

[Out]

-a**2/(3*x**3) - 3*a*b/(5*x**(10/3)) - 3*b**2/(11*x**(11/3))

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GIAC/XCAS [A]  time = 0.210804, size = 35, normalized size = 1.03 \[ -\frac{55 \, a^{2} x^{\frac{2}{3}} + 99 \, a b x^{\frac{1}{3}} + 45 \, b^{2}}{165 \, x^{\frac{11}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^(1/3))^2/x^4,x, algorithm="giac")

[Out]

-1/165*(55*a^2*x^(2/3) + 99*a*b*x^(1/3) + 45*b^2)/x^(11/3)