3.256 \(\int \frac{(a+b x^3)^3}{x^6} \, dx\)

Optimal. Leaf size=39 \[ -\frac{3 a^2 b}{2 x^2}-\frac{a^3}{5 x^5}+3 a b^2 x+\frac{b^3 x^4}{4} \]

[Out]

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

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Rubi [A]  time = 0.012512, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {270} \[ -\frac{3 a^2 b}{2 x^2}-\frac{a^3}{5 x^5}+3 a b^2 x+\frac{b^3 x^4}{4} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^3/x^6,x]

[Out]

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

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int \frac{\left (a+b x^3\right )^3}{x^6} \, dx &=\int \left (3 a b^2+\frac{a^3}{x^6}+\frac{3 a^2 b}{x^3}+b^3 x^3\right ) \, dx\\ &=-\frac{a^3}{5 x^5}-\frac{3 a^2 b}{2 x^2}+3 a b^2 x+\frac{b^3 x^4}{4}\\ \end{align*}

Mathematica [A]  time = 0.0062322, size = 39, normalized size = 1. \[ -\frac{3 a^2 b}{2 x^2}-\frac{a^3}{5 x^5}+3 a b^2 x+\frac{b^3 x^4}{4} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)^3/x^6,x]

[Out]

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

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Maple [A]  time = 0.005, size = 34, normalized size = 0.9 \begin{align*} -{\frac{{a}^{3}}{5\,{x}^{5}}}-{\frac{3\,{a}^{2}b}{2\,{x}^{2}}}+3\,xa{b}^{2}+{\frac{{b}^{3}{x}^{4}}{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^3/x^6,x)

[Out]

-1/5*a^3/x^5-3/2*a^2*b/x^2+3*x*a*b^2+1/4*b^3*x^4

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Maxima [A]  time = 0.987241, size = 49, normalized size = 1.26 \begin{align*} \frac{1}{4} \, b^{3} x^{4} + 3 \, a b^{2} x - \frac{15 \, a^{2} b x^{3} + 2 \, a^{3}}{10 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^3/x^6,x, algorithm="maxima")

[Out]

1/4*b^3*x^4 + 3*a*b^2*x - 1/10*(15*a^2*b*x^3 + 2*a^3)/x^5

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Fricas [A]  time = 1.5816, size = 81, normalized size = 2.08 \begin{align*} \frac{5 \, b^{3} x^{9} + 60 \, a b^{2} x^{6} - 30 \, a^{2} b x^{3} - 4 \, a^{3}}{20 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^3/x^6,x, algorithm="fricas")

[Out]

1/20*(5*b^3*x^9 + 60*a*b^2*x^6 - 30*a^2*b*x^3 - 4*a^3)/x^5

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Sympy [A]  time = 0.438506, size = 36, normalized size = 0.92 \begin{align*} 3 a b^{2} x + \frac{b^{3} x^{4}}{4} - \frac{2 a^{3} + 15 a^{2} b x^{3}}{10 x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**3/x**6,x)

[Out]

3*a*b**2*x + b**3*x**4/4 - (2*a**3 + 15*a**2*b*x**3)/(10*x**5)

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Giac [A]  time = 1.11974, size = 49, normalized size = 1.26 \begin{align*} \frac{1}{4} \, b^{3} x^{4} + 3 \, a b^{2} x - \frac{15 \, a^{2} b x^{3} + 2 \, a^{3}}{10 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^3/x^6,x, algorithm="giac")

[Out]

1/4*b^3*x^4 + 3*a*b^2*x - 1/10*(15*a^2*b*x^3 + 2*a^3)/x^5