3.230 \(\int \frac{(a+b x^3)^2}{x^8} \, dx\)

Optimal. Leaf size=28 \[ -\frac{a^2}{7 x^7}-\frac{a b}{2 x^4}-\frac{b^2}{x} \]

[Out]

-a^2/(7*x^7) - (a*b)/(2*x^4) - b^2/x

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Rubi [A]  time = 0.0098012, antiderivative size = 28, 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} \[ -\frac{a^2}{7 x^7}-\frac{a b}{2 x^4}-\frac{b^2}{x} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^2/x^8,x]

[Out]

-a^2/(7*x^7) - (a*b)/(2*x^4) - b^2/x

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

Mathematica [A]  time = 0.0007933, size = 28, normalized size = 1. \[ -\frac{a^2}{7 x^7}-\frac{a b}{2 x^4}-\frac{b^2}{x} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)^2/x^8,x]

[Out]

-a^2/(7*x^7) - (a*b)/(2*x^4) - b^2/x

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Maple [A]  time = 0.004, size = 25, normalized size = 0.9 \begin{align*} -{\frac{{a}^{2}}{7\,{x}^{7}}}-{\frac{ab}{2\,{x}^{4}}}-{\frac{{b}^{2}}{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^2/x^8,x)

[Out]

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

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Maxima [A]  time = 0.952046, size = 35, normalized size = 1.25 \begin{align*} -\frac{14 \, b^{2} x^{6} + 7 \, a b x^{3} + 2 \, a^{2}}{14 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^2/x^8,x, algorithm="maxima")

[Out]

-1/14*(14*b^2*x^6 + 7*a*b*x^3 + 2*a^2)/x^7

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Fricas [A]  time = 1.59698, size = 59, normalized size = 2.11 \begin{align*} -\frac{14 \, b^{2} x^{6} + 7 \, a b x^{3} + 2 \, a^{2}}{14 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^2/x^8,x, algorithm="fricas")

[Out]

-1/14*(14*b^2*x^6 + 7*a*b*x^3 + 2*a^2)/x^7

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Sympy [A]  time = 0.519241, size = 27, normalized size = 0.96 \begin{align*} - \frac{2 a^{2} + 7 a b x^{3} + 14 b^{2} x^{6}}{14 x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**2/x**8,x)

[Out]

-(2*a**2 + 7*a*b*x**3 + 14*b**2*x**6)/(14*x**7)

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Giac [A]  time = 1.13846, size = 35, normalized size = 1.25 \begin{align*} -\frac{14 \, b^{2} x^{6} + 7 \, a b x^{3} + 2 \, a^{2}}{14 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^2/x^8,x, algorithm="giac")

[Out]

-1/14*(14*b^2*x^6 + 7*a*b*x^3 + 2*a^2)/x^7