3.234 \(\int \frac{(a+b x^3)^2}{x^{12}} \, dx\)

Optimal. Leaf size=30 \[ -\frac{a^2}{11 x^{11}}-\frac{a b}{4 x^8}-\frac{b^2}{5 x^5} \]

[Out]

-a^2/(11*x^11) - (a*b)/(4*x^8) - b^2/(5*x^5)

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Rubi [A]  time = 0.0098668, 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} \[ -\frac{a^2}{11 x^{11}}-\frac{a b}{4 x^8}-\frac{b^2}{5 x^5} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^2/x^12,x]

[Out]

-a^2/(11*x^11) - (a*b)/(4*x^8) - b^2/(5*x^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 )^2}{x^{12}} \, dx &=\int \left (\frac{a^2}{x^{12}}+\frac{2 a b}{x^9}+\frac{b^2}{x^6}\right ) \, dx\\ &=-\frac{a^2}{11 x^{11}}-\frac{a b}{4 x^8}-\frac{b^2}{5 x^5}\\ \end{align*}

Mathematica [A]  time = 0.0007645, size = 30, normalized size = 1. \[ -\frac{a^2}{11 x^{11}}-\frac{a b}{4 x^8}-\frac{b^2}{5 x^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)^2/x^12,x]

[Out]

-a^2/(11*x^11) - (a*b)/(4*x^8) - b^2/(5*x^5)

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Maple [A]  time = 0.006, size = 25, normalized size = 0.8 \begin{align*} -{\frac{{a}^{2}}{11\,{x}^{11}}}-{\frac{ab}{4\,{x}^{8}}}-{\frac{{b}^{2}}{5\,{x}^{5}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^2/x^12,x)

[Out]

-1/11*a^2/x^11-1/4*a*b/x^8-1/5*b^2/x^5

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Maxima [A]  time = 0.971097, size = 35, normalized size = 1.17 \begin{align*} -\frac{44 \, b^{2} x^{6} + 55 \, a b x^{3} + 20 \, a^{2}}{220 \, x^{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^2/x^12,x, algorithm="maxima")

[Out]

-1/220*(44*b^2*x^6 + 55*a*b*x^3 + 20*a^2)/x^11

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Fricas [A]  time = 1.88636, size = 65, normalized size = 2.17 \begin{align*} -\frac{44 \, b^{2} x^{6} + 55 \, a b x^{3} + 20 \, a^{2}}{220 \, x^{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^2/x^12,x, algorithm="fricas")

[Out]

-1/220*(44*b^2*x^6 + 55*a*b*x^3 + 20*a^2)/x^11

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Sympy [A]  time = 0.368011, size = 27, normalized size = 0.9 \begin{align*} - \frac{20 a^{2} + 55 a b x^{3} + 44 b^{2} x^{6}}{220 x^{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**2/x**12,x)

[Out]

-(20*a**2 + 55*a*b*x**3 + 44*b**2*x**6)/(220*x**11)

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Giac [A]  time = 1.10967, size = 35, normalized size = 1.17 \begin{align*} -\frac{44 \, b^{2} x^{6} + 55 \, a b x^{3} + 20 \, a^{2}}{220 \, x^{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^2/x^12,x, algorithm="giac")

[Out]

-1/220*(44*b^2*x^6 + 55*a*b*x^3 + 20*a^2)/x^11