3.245 \(\int \frac{(a+b x^3)^3}{x^{13}} \, dx\)

Optimal. Leaf size=19 \[ -\frac{\left (a+b x^3\right )^4}{12 a x^{12}} \]

[Out]

-(a + b*x^3)^4/(12*a*x^12)

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Rubi [A]  time = 0.0030226, 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} \[ -\frac{\left (a+b x^3\right )^4}{12 a x^{12}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^3/x^13,x]

[Out]

-(a + b*x^3)^4/(12*a*x^12)

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 )^3}{x^{13}} \, dx &=-\frac{\left (a+b x^3\right )^4}{12 a x^{12}}\\ \end{align*}

Mathematica [B]  time = 0.0036476, size = 43, normalized size = 2.26 \[ -\frac{a^2 b}{3 x^9}-\frac{a^3}{12 x^{12}}-\frac{a b^2}{2 x^6}-\frac{b^3}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)^3/x^13,x]

[Out]

-a^3/(12*x^12) - (a^2*b)/(3*x^9) - (a*b^2)/(2*x^6) - b^3/(3*x^3)

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Maple [B]  time = 0.005, size = 36, normalized size = 1.9 \begin{align*} -{\frac{{b}^{3}}{3\,{x}^{3}}}-{\frac{{a}^{3}}{12\,{x}^{12}}}-{\frac{a{b}^{2}}{2\,{x}^{6}}}-{\frac{{a}^{2}b}{3\,{x}^{9}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^3/x^13,x)

[Out]

-1/3*b^3/x^3-1/12*a^3/x^12-1/2*a*b^2/x^6-1/3*a^2*b/x^9

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Maxima [B]  time = 0.99131, size = 47, normalized size = 2.47 \begin{align*} -\frac{4 \, b^{3} x^{9} + 6 \, a b^{2} x^{6} + 4 \, a^{2} b x^{3} + a^{3}}{12 \, x^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^3/x^13,x, algorithm="maxima")

[Out]

-1/12*(4*b^3*x^9 + 6*a*b^2*x^6 + 4*a^2*b*x^3 + a^3)/x^12

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Fricas [B]  time = 1.63048, size = 78, normalized size = 4.11 \begin{align*} -\frac{4 \, b^{3} x^{9} + 6 \, a b^{2} x^{6} + 4 \, a^{2} b x^{3} + a^{3}}{12 \, x^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^3/x^13,x, algorithm="fricas")

[Out]

-1/12*(4*b^3*x^9 + 6*a*b^2*x^6 + 4*a^2*b*x^3 + a^3)/x^12

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Sympy [B]  time = 0.611185, size = 37, normalized size = 1.95 \begin{align*} - \frac{a^{3} + 4 a^{2} b x^{3} + 6 a b^{2} x^{6} + 4 b^{3} x^{9}}{12 x^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**3/x**13,x)

[Out]

-(a**3 + 4*a**2*b*x**3 + 6*a*b**2*x**6 + 4*b**3*x**9)/(12*x**12)

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Giac [B]  time = 1.12031, size = 47, normalized size = 2.47 \begin{align*} -\frac{4 \, b^{3} x^{9} + 6 \, a b^{2} x^{6} + 4 \, a^{2} b x^{3} + a^{3}}{12 \, x^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^3/x^13,x, algorithm="giac")

[Out]

-1/12*(4*b^3*x^9 + 6*a*b^2*x^6 + 4*a^2*b*x^3 + a^3)/x^12