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

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

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Rubi [A]  time = 0.00, antiderivative size = 19, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\frac {\left (a+b x^2\right )^4}{8 a x^8} \end {gather*}

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*}

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Mathematica [B]  time = 0.01, size = 43, normalized size = 2.26 \begin {gather*} -\frac {a^3}{8 x^8}-\frac {a^2 b}{2 x^6}-\frac {3 a b^2}{4 x^4}-\frac {b^3}{2 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^3/x^9,x]

[Out]

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

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a+b x^2\right )^3}{x^9} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a + b*x^2)^3/x^9,x]

[Out]

IntegrateAlgebraic[(a + b*x^2)^3/x^9, x]

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fricas [B]  time = 1.74, size = 35, normalized size = 1.84 \begin {gather*} -\frac {4 \, b^{3} x^{6} + 6 \, a b^{2} x^{4} + 4 \, a^{2} b x^{2} + a^{3}}{8 \, x^{8}} \end {gather*}

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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giac [B]  time = 1.04, size = 35, normalized size = 1.84 \begin {gather*} -\frac {4 \, b^{3} x^{6} + 6 \, a b^{2} x^{4} + 4 \, a^{2} b x^{2} + a^{3}}{8 \, x^{8}} \end {gather*}

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

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maple [B]  time = 0.00, size = 36, normalized size = 1.89 \begin {gather*} -\frac {b^{3}}{2 x^{2}}-\frac {3 a \,b^{2}}{4 x^{4}}-\frac {a^{2} b}{2 x^{6}}-\frac {a^{3}}{8 x^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 1.32, size = 35, normalized size = 1.84 \begin {gather*} -\frac {4 \, b^{3} x^{6} + 6 \, a b^{2} x^{4} + 4 \, a^{2} b x^{2} + a^{3}}{8 \, x^{8}} \end {gather*}

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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mupad [B]  time = 0.03, size = 37, normalized size = 1.95 \begin {gather*} -\frac {\frac {a^3}{8}+\frac {a^2\,b\,x^2}{2}+\frac {3\,a\,b^2\,x^4}{4}+\frac {b^3\,x^6}{2}}{x^8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)^3/x^9,x)

[Out]

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

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sympy [B]  time = 0.27, size = 37, normalized size = 1.95 \begin {gather*} \frac {- a^{3} - 4 a^{2} b x^{2} - 6 a b^{2} x^{4} - 4 b^{3} x^{6}}{8 x^{8}} \end {gather*}

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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