3.1840 \(\int \frac{(a+\frac{b}{x^2})^3}{x^4} \, dx\)

Optimal. Leaf size=43 \[ -\frac{3 a^2 b}{5 x^5}-\frac{a^3}{3 x^3}-\frac{3 a b^2}{7 x^7}-\frac{b^3}{9 x^9} \]

[Out]

-b^3/(9*x^9) - (3*a*b^2)/(7*x^7) - (3*a^2*b)/(5*x^5) - a^3/(3*x^3)

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Rubi [A]  time = 0.0153341, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {263, 270} \[ -\frac{3 a^2 b}{5 x^5}-\frac{a^3}{3 x^3}-\frac{3 a b^2}{7 x^7}-\frac{b^3}{9 x^9} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x^2)^3/x^4,x]

[Out]

-b^3/(9*x^9) - (3*a*b^2)/(7*x^7) - (3*a^2*b)/(5*x^5) - a^3/(3*x^3)

Rule 263

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(m + n*p)*(b + a/x^n)^p, x] /; FreeQ[{a, b, m
, n}, x] && IntegerQ[p] && NegQ[n]

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+\frac{b}{x^2}\right )^3}{x^4} \, dx &=\int \frac{\left (b+a x^2\right )^3}{x^{10}} \, dx\\ &=\int \left (\frac{b^3}{x^{10}}+\frac{3 a b^2}{x^8}+\frac{3 a^2 b}{x^6}+\frac{a^3}{x^4}\right ) \, dx\\ &=-\frac{b^3}{9 x^9}-\frac{3 a b^2}{7 x^7}-\frac{3 a^2 b}{5 x^5}-\frac{a^3}{3 x^3}\\ \end{align*}

Mathematica [A]  time = 0.004138, size = 43, normalized size = 1. \[ -\frac{3 a^2 b}{5 x^5}-\frac{a^3}{3 x^3}-\frac{3 a b^2}{7 x^7}-\frac{b^3}{9 x^9} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x^2)^3/x^4,x]

[Out]

-b^3/(9*x^9) - (3*a*b^2)/(7*x^7) - (3*a^2*b)/(5*x^5) - a^3/(3*x^3)

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Maple [A]  time = 0.002, size = 36, normalized size = 0.8 \begin{align*} -{\frac{{b}^{3}}{9\,{x}^{9}}}-{\frac{3\,{b}^{2}a}{7\,{x}^{7}}}-{\frac{3\,{a}^{2}b}{5\,{x}^{5}}}-{\frac{{a}^{3}}{3\,{x}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+1/x^2*b)^3/x^4,x)

[Out]

-1/9*b^3/x^9-3/7*a*b^2/x^7-3/5*a^2*b/x^5-1/3*a^3/x^3

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Maxima [A]  time = 0.975167, size = 50, normalized size = 1.16 \begin{align*} -\frac{105 \, a^{3} x^{6} + 189 \, a^{2} b x^{4} + 135 \, a b^{2} x^{2} + 35 \, b^{3}}{315 \, x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^2)^3/x^4,x, algorithm="maxima")

[Out]

-1/315*(105*a^3*x^6 + 189*a^2*b*x^4 + 135*a*b^2*x^2 + 35*b^3)/x^9

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Fricas [A]  time = 1.38159, size = 90, normalized size = 2.09 \begin{align*} -\frac{105 \, a^{3} x^{6} + 189 \, a^{2} b x^{4} + 135 \, a b^{2} x^{2} + 35 \, b^{3}}{315 \, x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^2)^3/x^4,x, algorithm="fricas")

[Out]

-1/315*(105*a^3*x^6 + 189*a^2*b*x^4 + 135*a*b^2*x^2 + 35*b^3)/x^9

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Sympy [A]  time = 0.410442, size = 39, normalized size = 0.91 \begin{align*} - \frac{105 a^{3} x^{6} + 189 a^{2} b x^{4} + 135 a b^{2} x^{2} + 35 b^{3}}{315 x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x**2)**3/x**4,x)

[Out]

-(105*a**3*x**6 + 189*a**2*b*x**4 + 135*a*b**2*x**2 + 35*b**3)/(315*x**9)

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Giac [A]  time = 1.67962, size = 50, normalized size = 1.16 \begin{align*} -\frac{105 \, a^{3} x^{6} + 189 \, a^{2} b x^{4} + 135 \, a b^{2} x^{2} + 35 \, b^{3}}{315 \, x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^2)^3/x^4,x, algorithm="giac")

[Out]

-1/315*(105*a^3*x^6 + 189*a^2*b*x^4 + 135*a*b^2*x^2 + 35*b^3)/x^9