3.38 \(\int \frac{(a+b x^2)^3}{x^9} \, dx\)

Optimal. Leaf size=19 \[ -\frac{\left (a+b x^2\right )^4}{8 a x^8} \]

[Out]

-(a + b*x^2)^4/(8*a*x^8)

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Rubi [A]  time = 0.0030151, 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} \[ -\frac{\left (a+b x^2\right )^4}{8 a x^8} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^3/x^9,x]

[Out]

-(a + b*x^2)^4/(8*a*x^8)

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^2\right )^3}{x^9} \, dx &=-\frac{\left (a+b x^2\right )^4}{8 a x^8}\\ \end{align*}

Mathematica [B]  time = 0.0062467, size = 43, normalized size = 2.26 \[ -\frac{a^2 b}{2 x^6}-\frac{a^3}{8 x^8}-\frac{3 a b^2}{4 x^4}-\frac{b^3}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^3/x^9,x]

[Out]

-a^3/(8*x^8) - (a^2*b)/(2*x^6) - (3*a*b^2)/(4*x^4) - b^3/(2*x^2)

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Maple [B]  time = 0.004, size = 36, normalized size = 1.9 \begin{align*} -{\frac{{a}^{3}}{8\,{x}^{8}}}-{\frac{3\,a{b}^{2}}{4\,{x}^{4}}}-{\frac{{b}^{3}}{2\,{x}^{2}}}-{\frac{{a}^{2}b}{2\,{x}^{6}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^3/x^9,x)

[Out]

-1/8*a^3/x^8-3/4*a*b^2/x^4-1/2*b^3/x^2-1/2*a^2*b/x^6

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Maxima [B]  time = 1.08615, size = 47, normalized size = 2.47 \begin{align*} -\frac{4 \, b^{3} x^{6} + 6 \, a b^{2} x^{4} + 4 \, a^{2} b x^{2} + a^{3}}{8 \, x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^3/x^9,x, algorithm="maxima")

[Out]

-1/8*(4*b^3*x^6 + 6*a*b^2*x^4 + 4*a^2*b*x^2 + a^3)/x^8

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Fricas [B]  time = 1.20025, size = 76, normalized size = 4. \begin{align*} -\frac{4 \, b^{3} x^{6} + 6 \, a b^{2} x^{4} + 4 \, a^{2} b x^{2} + a^{3}}{8 \, x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^3/x^9,x, algorithm="fricas")

[Out]

-1/8*(4*b^3*x^6 + 6*a*b^2*x^4 + 4*a^2*b*x^2 + a^3)/x^8

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Sympy [B]  time = 0.403944, size = 37, normalized size = 1.95 \begin{align*} - \frac{a^{3} + 4 a^{2} b x^{2} + 6 a b^{2} x^{4} + 4 b^{3} x^{6}}{8 x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**3/x**9,x)

[Out]

-(a**3 + 4*a**2*b*x**2 + 6*a*b**2*x**4 + 4*b**3*x**6)/(8*x**8)

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Giac [B]  time = 2.64129, size = 47, normalized size = 2.47 \begin{align*} -\frac{4 \, b^{3} x^{6} + 6 \, a b^{2} x^{4} + 4 \, a^{2} b x^{2} + a^{3}}{8 \, x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^3/x^9,x, algorithm="giac")

[Out]

-1/8*(4*b^3*x^6 + 6*a*b^2*x^4 + 4*a^2*b*x^2 + a^3)/x^8