3.231 \(\int \frac{(a+b x^3)^2}{x^9} \, dx\)

Optimal. Leaf size=30 \[ -\frac{a^2}{8 x^8}-\frac{2 a b}{5 x^5}-\frac{b^2}{2 x^2} \]

[Out]

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

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Rubi [A]  time = 0.0095963, antiderivative size = 30, 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} \[ -\frac{a^2}{8 x^8}-\frac{2 a b}{5 x^5}-\frac{b^2}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^2/x^9,x]

[Out]

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

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 )^2}{x^9} \, dx &=\int \left (\frac{a^2}{x^9}+\frac{2 a b}{x^6}+\frac{b^2}{x^3}\right ) \, dx\\ &=-\frac{a^2}{8 x^8}-\frac{2 a b}{5 x^5}-\frac{b^2}{2 x^2}\\ \end{align*}

Mathematica [A]  time = 0.0007312, size = 30, normalized size = 1. \[ -\frac{a^2}{8 x^8}-\frac{2 a b}{5 x^5}-\frac{b^2}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)^2/x^9,x]

[Out]

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

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Maple [A]  time = 0.004, size = 25, normalized size = 0.8 \begin{align*} -{\frac{{a}^{2}}{8\,{x}^{8}}}-{\frac{2\,ab}{5\,{x}^{5}}}-{\frac{{b}^{2}}{2\,{x}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^2/x^9,x)

[Out]

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

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Maxima [A]  time = 0.988919, size = 35, normalized size = 1.17 \begin{align*} -\frac{20 \, b^{2} x^{6} + 16 \, a b x^{3} + 5 \, a^{2}}{40 \, x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^2/x^9,x, algorithm="maxima")

[Out]

-1/40*(20*b^2*x^6 + 16*a*b*x^3 + 5*a^2)/x^8

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Fricas [A]  time = 1.64134, size = 61, normalized size = 2.03 \begin{align*} -\frac{20 \, b^{2} x^{6} + 16 \, a b x^{3} + 5 \, a^{2}}{40 \, x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^2/x^9,x, algorithm="fricas")

[Out]

-1/40*(20*b^2*x^6 + 16*a*b*x^3 + 5*a^2)/x^8

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Sympy [A]  time = 0.43763, size = 27, normalized size = 0.9 \begin{align*} - \frac{5 a^{2} + 16 a b x^{3} + 20 b^{2} x^{6}}{40 x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**2/x**9,x)

[Out]

-(5*a**2 + 16*a*b*x**3 + 20*b**2*x**6)/(40*x**8)

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Giac [A]  time = 1.11538, size = 35, normalized size = 1.17 \begin{align*} -\frac{20 \, b^{2} x^{6} + 16 \, a b x^{3} + 5 \, a^{2}}{40 \, x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^2/x^9,x, algorithm="giac")

[Out]

-1/40*(20*b^2*x^6 + 16*a*b*x^3 + 5*a^2)/x^8