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

Optimal. Leaf size=41 \[ -\frac{3 a^2 b}{4 x^4}-\frac{a^3}{7 x^7}-\frac{3 a b^2}{x}+\frac{b^3 x^2}{2} \]

[Out]

-a^3/(7*x^7) - (3*a^2*b)/(4*x^4) - (3*a*b^2)/x + (b^3*x^2)/2

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Rubi [A]  time = 0.0129814, antiderivative size = 41, 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} \[ -\frac{3 a^2 b}{4 x^4}-\frac{a^3}{7 x^7}-\frac{3 a b^2}{x}+\frac{b^3 x^2}{2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^3/x^8,x]

[Out]

-a^3/(7*x^7) - (3*a^2*b)/(4*x^4) - (3*a*b^2)/x + (b^3*x^2)/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 )^3}{x^8} \, dx &=\int \left (\frac{a^3}{x^8}+\frac{3 a^2 b}{x^5}+\frac{3 a b^2}{x^2}+b^3 x\right ) \, dx\\ &=-\frac{a^3}{7 x^7}-\frac{3 a^2 b}{4 x^4}-\frac{3 a b^2}{x}+\frac{b^3 x^2}{2}\\ \end{align*}

Mathematica [A]  time = 0.003807, size = 41, normalized size = 1. \[ -\frac{3 a^2 b}{4 x^4}-\frac{a^3}{7 x^7}-\frac{3 a b^2}{x}+\frac{b^3 x^2}{2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)^3/x^8,x]

[Out]

-a^3/(7*x^7) - (3*a^2*b)/(4*x^4) - (3*a*b^2)/x + (b^3*x^2)/2

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Maple [A]  time = 0.006, size = 36, normalized size = 0.9 \begin{align*} -{\frac{{a}^{3}}{7\,{x}^{7}}}-{\frac{3\,{a}^{2}b}{4\,{x}^{4}}}-3\,{\frac{a{b}^{2}}{x}}+{\frac{{b}^{3}{x}^{2}}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/7*a^3/x^7-3/4*a^2*b/x^4-3*a*b^2/x+1/2*b^3*x^2

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Maxima [A]  time = 0.982407, size = 51, normalized size = 1.24 \begin{align*} \frac{1}{2} \, b^{3} x^{2} - \frac{84 \, a b^{2} x^{6} + 21 \, a^{2} b x^{3} + 4 \, a^{3}}{28 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*b^3*x^2 - 1/28*(84*a*b^2*x^6 + 21*a^2*b*x^3 + 4*a^3)/x^7

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Fricas [A]  time = 1.70178, size = 82, normalized size = 2. \begin{align*} \frac{14 \, b^{3} x^{9} - 84 \, a b^{2} x^{6} - 21 \, a^{2} b x^{3} - 4 \, a^{3}}{28 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/28*(14*b^3*x^9 - 84*a*b^2*x^6 - 21*a^2*b*x^3 - 4*a^3)/x^7

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Sympy [A]  time = 0.526814, size = 37, normalized size = 0.9 \begin{align*} \frac{b^{3} x^{2}}{2} - \frac{4 a^{3} + 21 a^{2} b x^{3} + 84 a b^{2} x^{6}}{28 x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b**3*x**2/2 - (4*a**3 + 21*a**2*b*x**3 + 84*a*b**2*x**6)/(28*x**7)

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Giac [A]  time = 1.1086, size = 51, normalized size = 1.24 \begin{align*} \frac{1}{2} \, b^{3} x^{2} - \frac{84 \, a b^{2} x^{6} + 21 \, a^{2} b x^{3} + 4 \, a^{3}}{28 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*b^3*x^2 - 1/28*(84*a*b^2*x^6 + 21*a^2*b*x^3 + 4*a^3)/x^7