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

Optimal. Leaf size=39 \[ 3 a^2 b x-\frac{a^3}{2 x^2}+\frac{3}{4} a b^2 x^4+\frac{b^3 x^7}{7} \]

[Out]

-a^3/(2*x^2) + 3*a^2*b*x + (3*a*b^2*x^4)/4 + (b^3*x^7)/7

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Rubi [A]  time = 0.0132737, 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} \[ 3 a^2 b x-\frac{a^3}{2 x^2}+\frac{3}{4} a b^2 x^4+\frac{b^3 x^7}{7} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^3/x^3,x]

[Out]

-a^3/(2*x^2) + 3*a^2*b*x + (3*a*b^2*x^4)/4 + (b^3*x^7)/7

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^3} \, dx &=\int \left (3 a^2 b+\frac{a^3}{x^3}+3 a b^2 x^3+b^3 x^6\right ) \, dx\\ &=-\frac{a^3}{2 x^2}+3 a^2 b x+\frac{3}{4} a b^2 x^4+\frac{b^3 x^7}{7}\\ \end{align*}

Mathematica [A]  time = 0.0061984, size = 39, normalized size = 1. \[ 3 a^2 b x-\frac{a^3}{2 x^2}+\frac{3}{4} a b^2 x^4+\frac{b^3 x^7}{7} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)^3/x^3,x]

[Out]

-a^3/(2*x^2) + 3*a^2*b*x + (3*a*b^2*x^4)/4 + (b^3*x^7)/7

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Maple [A]  time = 0.003, size = 34, normalized size = 0.9 \begin{align*} -{\frac{{a}^{3}}{2\,{x}^{2}}}+3\,{a}^{2}bx+{\frac{3\,a{b}^{2}{x}^{4}}{4}}+{\frac{{b}^{3}{x}^{7}}{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*a^3/x^2+3*a^2*b*x+3/4*a*b^2*x^4+1/7*b^3*x^7

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Maxima [A]  time = 0.974756, size = 45, normalized size = 1.15 \begin{align*} \frac{1}{7} \, b^{3} x^{7} + \frac{3}{4} \, a b^{2} x^{4} + 3 \, a^{2} b x - \frac{a^{3}}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*b^3*x^7 + 3/4*a*b^2*x^4 + 3*a^2*b*x - 1/2*a^3/x^2

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Fricas [A]  time = 1.65929, size = 82, normalized size = 2.1 \begin{align*} \frac{4 \, b^{3} x^{9} + 21 \, a b^{2} x^{6} + 84 \, a^{2} b x^{3} - 14 \, a^{3}}{28 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/28*(4*b^3*x^9 + 21*a*b^2*x^6 + 84*a^2*b*x^3 - 14*a^3)/x^2

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Sympy [A]  time = 0.351378, size = 36, normalized size = 0.92 \begin{align*} - \frac{a^{3}}{2 x^{2}} + 3 a^{2} b x + \frac{3 a b^{2} x^{4}}{4} + \frac{b^{3} x^{7}}{7} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a**3/(2*x**2) + 3*a**2*b*x + 3*a*b**2*x**4/4 + b**3*x**7/7

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Giac [A]  time = 1.12737, size = 45, normalized size = 1.15 \begin{align*} \frac{1}{7} \, b^{3} x^{7} + \frac{3}{4} \, a b^{2} x^{4} + 3 \, a^{2} b x - \frac{a^{3}}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*b^3*x^7 + 3/4*a*b^2*x^4 + 3*a^2*b*x - 1/2*a^3/x^2