3.232 \(\int \frac{(a+b x^3)^2}{x^{10}} \, dx\)

Optimal. Leaf size=19 \[ -\frac{\left (a+b x^3\right )^3}{9 a x^9} \]

[Out]

-(a + b*x^3)^3/(9*a*x^9)

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Rubi [A]  time = 0.0033423, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {264} \[ -\frac{\left (a+b x^3\right )^3}{9 a x^9} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^2/x^10,x]

[Out]

-(a + b*x^3)^3/(9*a*x^9)

Rule 264

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \frac{\left (a+b x^3\right )^2}{x^{10}} \, dx &=-\frac{\left (a+b x^3\right )^3}{9 a x^9}\\ \end{align*}

Mathematica [A]  time = 0.0007634, size = 30, normalized size = 1.58 \[ -\frac{a^2}{9 x^9}-\frac{a b}{3 x^6}-\frac{b^2}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)^2/x^10,x]

[Out]

-a^2/(9*x^9) - (a*b)/(3*x^6) - b^2/(3*x^3)

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Maple [A]  time = 0.004, size = 25, normalized size = 1.3 \begin{align*} -{\frac{{b}^{2}}{3\,{x}^{3}}}-{\frac{ab}{3\,{x}^{6}}}-{\frac{{a}^{2}}{9\,{x}^{9}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^2/x^10,x)

[Out]

-1/3*b^2/x^3-1/3*a*b/x^6-1/9*a^2/x^9

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Maxima [A]  time = 0.952874, size = 32, normalized size = 1.68 \begin{align*} -\frac{3 \, b^{2} x^{6} + 3 \, a b x^{3} + a^{2}}{9 \, x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^2/x^10,x, algorithm="maxima")

[Out]

-1/9*(3*b^2*x^6 + 3*a*b*x^3 + a^2)/x^9

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Fricas [A]  time = 1.82685, size = 54, normalized size = 2.84 \begin{align*} -\frac{3 \, b^{2} x^{6} + 3 \, a b x^{3} + a^{2}}{9 \, x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^2/x^10,x, algorithm="fricas")

[Out]

-1/9*(3*b^2*x^6 + 3*a*b*x^3 + a^2)/x^9

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Sympy [A]  time = 0.461835, size = 26, normalized size = 1.37 \begin{align*} - \frac{a^{2} + 3 a b x^{3} + 3 b^{2} x^{6}}{9 x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**2/x**10,x)

[Out]

-(a**2 + 3*a*b*x**3 + 3*b**2*x**6)/(9*x**9)

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Giac [A]  time = 1.11509, size = 32, normalized size = 1.68 \begin{align*} -\frac{3 \, b^{2} x^{6} + 3 \, a b x^{3} + a^{2}}{9 \, x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^2/x^10,x, algorithm="giac")

[Out]

-1/9*(3*b^2*x^6 + 3*a*b*x^3 + a^2)/x^9