3.2294 \(\int \frac{\left (a+b \sqrt [3]{x}\right )^2}{x^3} \, dx\)

Optimal. Leaf size=34 \[ -\frac{a^2}{2 x^2}-\frac{6 a b}{5 x^{5/3}}-\frac{3 b^2}{4 x^{4/3}} \]

[Out]

-a^2/(2*x^2) - (6*a*b)/(5*x^(5/3)) - (3*b^2)/(4*x^(4/3))

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Rubi [A]  time = 0.0433039, antiderivative size = 34, 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 \[ -\frac{a^2}{2 x^2}-\frac{6 a b}{5 x^{5/3}}-\frac{3 b^2}{4 x^{4/3}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^(1/3))^2/x^3,x]

[Out]

-a^2/(2*x^2) - (6*a*b)/(5*x^(5/3)) - (3*b^2)/(4*x^(4/3))

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Rubi in Sympy [A]  time = 6.76713, size = 32, normalized size = 0.94 \[ - \frac{a^{2}}{2 x^{2}} - \frac{6 a b}{5 x^{\frac{5}{3}}} - \frac{3 b^{2}}{4 x^{\frac{4}{3}}} \]

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/(5*x**(5/3)) - 3*b**2/(4*x**(4/3))

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Mathematica [A]  time = 0.0128438, size = 34, normalized size = 1. \[ -\frac{a^2}{2 x^2}-\frac{6 a b}{5 x^{5/3}}-\frac{3 b^2}{4 x^{4/3}} \]

Antiderivative was successfully verified.

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

[Out]

-a^2/(2*x^2) - (6*a*b)/(5*x^(5/3)) - (3*b^2)/(4*x^(4/3))

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Maple [A]  time = 0.008, size = 25, normalized size = 0.7 \[ -{\frac{{a}^{2}}{2\,{x}^{2}}}-{\frac{6\,ab}{5}{x}^{-{\frac{5}{3}}}}-{\frac{3\,{b}^{2}}{4}{x}^{-{\frac{4}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*x^(1/3))^2/x^3,x)

[Out]

-1/2*a^2/x^2-6/5*a*b/x^(5/3)-3/4*b^2/x^(4/3)

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Maxima [A]  time = 1.42443, size = 35, normalized size = 1.03 \[ -\frac{15 \, b^{2} x^{\frac{2}{3}} + 24 \, a b x^{\frac{1}{3}} + 10 \, a^{2}}{20 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/20*(15*b^2*x^(2/3) + 24*a*b*x^(1/3) + 10*a^2)/x^2

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Fricas [A]  time = 0.212666, size = 35, normalized size = 1.03 \[ -\frac{15 \, b^{2} x^{\frac{2}{3}} + 24 \, a b x^{\frac{1}{3}} + 10 \, a^{2}}{20 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/20*(15*b^2*x^(2/3) + 24*a*b*x^(1/3) + 10*a^2)/x^2

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Sympy [A]  time = 3.99564, size = 32, normalized size = 0.94 \[ - \frac{a^{2}}{2 x^{2}} - \frac{6 a b}{5 x^{\frac{5}{3}}} - \frac{3 b^{2}}{4 x^{\frac{4}{3}}} \]

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/(5*x**(5/3)) - 3*b**2/(4*x**(4/3))

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GIAC/XCAS [A]  time = 0.25413, size = 35, normalized size = 1.03 \[ -\frac{15 \, b^{2} x^{\frac{2}{3}} + 24 \, a b x^{\frac{1}{3}} + 10 \, a^{2}}{20 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/20*(15*b^2*x^(2/3) + 24*a*b*x^(1/3) + 10*a^2)/x^2