3.255 \(\int \frac{(a+b x^3)^3}{x^5} \, dx\)

Optimal. Leaf size=41 \[ -\frac{3 a^2 b}{x}-\frac{a^3}{4 x^4}+\frac{3}{2} a b^2 x^2+\frac{b^3 x^5}{5} \]

[Out]

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

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Rubi [A]  time = 0.0148853, 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} \[ -\frac{3 a^2 b}{x}-\frac{a^3}{4 x^4}+\frac{3}{2} a b^2 x^2+\frac{b^3 x^5}{5} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^3/x^5,x]

[Out]

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

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

Mathematica [A]  time = 0.0036775, size = 41, normalized size = 1. \[ -\frac{3 a^2 b}{x}-\frac{a^3}{4 x^4}+\frac{3}{2} a b^2 x^2+\frac{b^3 x^5}{5} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)^3/x^5,x]

[Out]

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

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Maple [A]  time = 0.005, size = 36, normalized size = 0.9 \begin{align*} -{\frac{{a}^{3}}{4\,{x}^{4}}}-3\,{\frac{{a}^{2}b}{x}}+{\frac{3\,a{b}^{2}{x}^{2}}{2}}+{\frac{{b}^{3}{x}^{5}}{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^3/x^5,x)

[Out]

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

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Maxima [A]  time = 0.967687, size = 49, normalized size = 1.2 \begin{align*} \frac{1}{5} \, b^{3} x^{5} + \frac{3}{2} \, a b^{2} x^{2} - \frac{12 \, a^{2} b x^{3} + a^{3}}{4 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^3/x^5,x, algorithm="maxima")

[Out]

1/5*b^3*x^5 + 3/2*a*b^2*x^2 - 1/4*(12*a^2*b*x^3 + a^3)/x^4

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Fricas [A]  time = 1.57176, size = 81, normalized size = 1.98 \begin{align*} \frac{4 \, b^{3} x^{9} + 30 \, a b^{2} x^{6} - 60 \, a^{2} b x^{3} - 5 \, a^{3}}{20 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^3/x^5,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.40157, size = 37, normalized size = 0.9 \begin{align*} \frac{3 a b^{2} x^{2}}{2} + \frac{b^{3} x^{5}}{5} - \frac{a^{3} + 12 a^{2} b x^{3}}{4 x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**3/x**5,x)

[Out]

3*a*b**2*x**2/2 + b**3*x**5/5 - (a**3 + 12*a**2*b*x**3)/(4*x**4)

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Giac [A]  time = 1.11872, size = 49, normalized size = 1.2 \begin{align*} \frac{1}{5} \, b^{3} x^{5} + \frac{3}{2} \, a b^{2} x^{2} - \frac{12 \, a^{2} b x^{3} + a^{3}}{4 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^3/x^5,x, algorithm="giac")

[Out]

1/5*b^3*x^5 + 3/2*a*b^2*x^2 - 1/4*(12*a^2*b*x^3 + a^3)/x^4