3.233 \(\int \frac{(a+b x^3)^2}{x^{11}} \, dx\)

Optimal. Leaf size=30 \[ -\frac{a^2}{10 x^{10}}-\frac{2 a b}{7 x^7}-\frac{b^2}{4 x^4} \]

[Out]

-a^2/(10*x^10) - (2*a*b)/(7*x^7) - b^2/(4*x^4)

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Rubi [A]  time = 0.0103513, 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} \[ -\frac{a^2}{10 x^{10}}-\frac{2 a b}{7 x^7}-\frac{b^2}{4 x^4} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^2/x^11,x]

[Out]

-a^2/(10*x^10) - (2*a*b)/(7*x^7) - b^2/(4*x^4)

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^{11}} \, dx &=\int \left (\frac{a^2}{x^{11}}+\frac{2 a b}{x^8}+\frac{b^2}{x^5}\right ) \, dx\\ &=-\frac{a^2}{10 x^{10}}-\frac{2 a b}{7 x^7}-\frac{b^2}{4 x^4}\\ \end{align*}

Mathematica [A]  time = 0.000869, size = 30, normalized size = 1. \[ -\frac{a^2}{10 x^{10}}-\frac{2 a b}{7 x^7}-\frac{b^2}{4 x^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)^2/x^11,x]

[Out]

-a^2/(10*x^10) - (2*a*b)/(7*x^7) - b^2/(4*x^4)

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Maple [A]  time = 0.006, size = 25, normalized size = 0.8 \begin{align*} -{\frac{{a}^{2}}{10\,{x}^{10}}}-{\frac{2\,ab}{7\,{x}^{7}}}-{\frac{{b}^{2}}{4\,{x}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^2/x^11,x)

[Out]

-1/10*a^2/x^10-2/7*a*b/x^7-1/4*b^2/x^4

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Maxima [A]  time = 0.983578, size = 35, normalized size = 1.17 \begin{align*} -\frac{35 \, b^{2} x^{6} + 40 \, a b x^{3} + 14 \, a^{2}}{140 \, x^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^2/x^11,x, algorithm="maxima")

[Out]

-1/140*(35*b^2*x^6 + 40*a*b*x^3 + 14*a^2)/x^10

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Fricas [A]  time = 1.92519, size = 65, normalized size = 2.17 \begin{align*} -\frac{35 \, b^{2} x^{6} + 40 \, a b x^{3} + 14 \, a^{2}}{140 \, x^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^2/x^11,x, algorithm="fricas")

[Out]

-1/140*(35*b^2*x^6 + 40*a*b*x^3 + 14*a^2)/x^10

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Sympy [A]  time = 0.453512, size = 27, normalized size = 0.9 \begin{align*} - \frac{14 a^{2} + 40 a b x^{3} + 35 b^{2} x^{6}}{140 x^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**2/x**11,x)

[Out]

-(14*a**2 + 40*a*b*x**3 + 35*b**2*x**6)/(140*x**10)

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Giac [A]  time = 1.10009, size = 35, normalized size = 1.17 \begin{align*} -\frac{35 \, b^{2} x^{6} + 40 \, a b x^{3} + 14 \, a^{2}}{140 \, x^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^2/x^11,x, algorithm="giac")

[Out]

-1/140*(35*b^2*x^6 + 40*a*b*x^3 + 14*a^2)/x^10