3.253 \(\int \frac{(a+b x^3)^3}{x^2} \, dx\)

Optimal. Leaf size=41 \[ \frac{3}{2} a^2 b x^2-\frac{a^3}{x}+\frac{3}{5} a b^2 x^5+\frac{b^3 x^8}{8} \]

[Out]

-(a^3/x) + (3*a^2*b*x^2)/2 + (3*a*b^2*x^5)/5 + (b^3*x^8)/8

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Rubi [A]  time = 0.0137518, antiderivative size = 41, 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} \[ \frac{3}{2} a^2 b x^2-\frac{a^3}{x}+\frac{3}{5} a b^2 x^5+\frac{b^3 x^8}{8} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^3/x^2,x]

[Out]

-(a^3/x) + (3*a^2*b*x^2)/2 + (3*a*b^2*x^5)/5 + (b^3*x^8)/8

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^2} \, dx &=\int \left (\frac{a^3}{x^2}+3 a^2 b x+3 a b^2 x^4+b^3 x^7\right ) \, dx\\ &=-\frac{a^3}{x}+\frac{3}{2} a^2 b x^2+\frac{3}{5} a b^2 x^5+\frac{b^3 x^8}{8}\\ \end{align*}

Mathematica [A]  time = 0.0035876, size = 41, normalized size = 1. \[ \frac{3}{2} a^2 b x^2-\frac{a^3}{x}+\frac{3}{5} a b^2 x^5+\frac{b^3 x^8}{8} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)^3/x^2,x]

[Out]

-(a^3/x) + (3*a^2*b*x^2)/2 + (3*a*b^2*x^5)/5 + (b^3*x^8)/8

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Maple [A]  time = 0.003, size = 36, normalized size = 0.9 \begin{align*} -{\frac{{a}^{3}}{x}}+{\frac{3\,{a}^{2}b{x}^{2}}{2}}+{\frac{3\,{x}^{5}a{b}^{2}}{5}}+{\frac{{b}^{3}{x}^{8}}{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^3/x^2,x)

[Out]

-a^3/x+3/2*a^2*b*x^2+3/5*x^5*a*b^2+1/8*b^3*x^8

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Maxima [A]  time = 0.977069, size = 47, normalized size = 1.15 \begin{align*} \frac{1}{8} \, b^{3} x^{8} + \frac{3}{5} \, a b^{2} x^{5} + \frac{3}{2} \, a^{2} b x^{2} - \frac{a^{3}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*b^3*x^8 + 3/5*a*b^2*x^5 + 3/2*a^2*b*x^2 - a^3/x

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Fricas [A]  time = 1.60814, size = 80, normalized size = 1.95 \begin{align*} \frac{5 \, b^{3} x^{9} + 24 \, a b^{2} x^{6} + 60 \, a^{2} b x^{3} - 40 \, a^{3}}{40 \, x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/40*(5*b^3*x^9 + 24*a*b^2*x^6 + 60*a^2*b*x^3 - 40*a^3)/x

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Sympy [A]  time = 0.348536, size = 36, normalized size = 0.88 \begin{align*} - \frac{a^{3}}{x} + \frac{3 a^{2} b x^{2}}{2} + \frac{3 a b^{2} x^{5}}{5} + \frac{b^{3} x^{8}}{8} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**3/x**2,x)

[Out]

-a**3/x + 3*a**2*b*x**2/2 + 3*a*b**2*x**5/5 + b**3*x**8/8

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Giac [A]  time = 1.09239, size = 47, normalized size = 1.15 \begin{align*} \frac{1}{8} \, b^{3} x^{8} + \frac{3}{5} \, a b^{2} x^{5} + \frac{3}{2} \, a^{2} b x^{2} - \frac{a^{3}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*b^3*x^8 + 3/5*a*b^2*x^5 + 3/2*a^2*b*x^2 - a^3/x