3.2303 \(\int \frac{\left (a+b \sqrt [3]{x}\right )^3}{x^3} \, dx\)

Optimal. Leaf size=45 \[ -\frac{a^3}{2 x^2}-\frac{9 a^2 b}{5 x^{5/3}}-\frac{9 a b^2}{4 x^{4/3}}-\frac{b^3}{x} \]

[Out]

-a^3/(2*x^2) - (9*a^2*b)/(5*x^(5/3)) - (9*a*b^2)/(4*x^(4/3)) - b^3/x

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Rubi [A]  time = 0.0554227, antiderivative size = 45, 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 \[ -\frac{a^3}{2 x^2}-\frac{9 a^2 b}{5 x^{5/3}}-\frac{9 a b^2}{4 x^{4/3}}-\frac{b^3}{x} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^(1/3))^3/x^3,x]

[Out]

-a^3/(2*x^2) - (9*a^2*b)/(5*x^(5/3)) - (9*a*b^2)/(4*x^(4/3)) - b^3/x

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Rubi in Sympy [A]  time = 8.97845, size = 41, normalized size = 0.91 \[ - \frac{a^{3}}{2 x^{2}} - \frac{9 a^{2} b}{5 x^{\frac{5}{3}}} - \frac{9 a b^{2}}{4 x^{\frac{4}{3}}} - \frac{b^{3}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a**3/(2*x**2) - 9*a**2*b/(5*x**(5/3)) - 9*a*b**2/(4*x**(4/3)) - b**3/x

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Mathematica [A]  time = 0.0159044, size = 41, normalized size = 0.91 \[ -\frac{10 a^3+36 a^2 b \sqrt [3]{x}+45 a b^2 x^{2/3}+20 b^3 x}{20 x^2} \]

Antiderivative was successfully verified.

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

[Out]

-(10*a^3 + 36*a^2*b*x^(1/3) + 45*a*b^2*x^(2/3) + 20*b^3*x)/(20*x^2)

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Maple [A]  time = 0.008, size = 36, normalized size = 0.8 \[ -{\frac{{a}^{3}}{2\,{x}^{2}}}-{\frac{9\,{a}^{2}b}{5}{x}^{-{\frac{5}{3}}}}-{\frac{9\,a{b}^{2}}{4}{x}^{-{\frac{4}{3}}}}-{\frac{{b}^{3}}{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*x^(1/3))^3/x^3,x)

[Out]

-1/2*a^3/x^2-9/5*a^2*b/x^(5/3)-9/4*a*b^2/x^(4/3)-b^3/x

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Maxima [A]  time = 1.44767, size = 47, normalized size = 1.04 \[ -\frac{20 \, b^{3} x + 45 \, a b^{2} x^{\frac{2}{3}} + 36 \, a^{2} b x^{\frac{1}{3}} + 10 \, a^{3}}{20 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/20*(20*b^3*x + 45*a*b^2*x^(2/3) + 36*a^2*b*x^(1/3) + 10*a^3)/x^2

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Fricas [A]  time = 0.213877, size = 47, normalized size = 1.04 \[ -\frac{20 \, b^{3} x + 45 \, a b^{2} x^{\frac{2}{3}} + 36 \, a^{2} b x^{\frac{1}{3}} + 10 \, a^{3}}{20 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/20*(20*b^3*x + 45*a*b^2*x^(2/3) + 36*a^2*b*x^(1/3) + 10*a^3)/x^2

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Sympy [A]  time = 4.10645, size = 41, normalized size = 0.91 \[ - \frac{a^{3}}{2 x^{2}} - \frac{9 a^{2} b}{5 x^{\frac{5}{3}}} - \frac{9 a b^{2}}{4 x^{\frac{4}{3}}} - \frac{b^{3}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a**3/(2*x**2) - 9*a**2*b/(5*x**(5/3)) - 9*a*b**2/(4*x**(4/3)) - b**3/x

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GIAC/XCAS [A]  time = 0.220307, size = 47, normalized size = 1.04 \[ -\frac{20 \, b^{3} x + 45 \, a b^{2} x^{\frac{2}{3}} + 36 \, a^{2} b x^{\frac{1}{3}} + 10 \, a^{3}}{20 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/20*(20*b^3*x + 45*a*b^2*x^(2/3) + 36*a^2*b*x^(1/3) + 10*a^3)/x^2