3.1834 \(\int (a+\frac{b}{x^2})^3 x^2 \, dx\)

Optimal. Leaf size=37 \[ 3 a^2 b x+\frac{a^3 x^3}{3}-\frac{3 a b^2}{x}-\frac{b^3}{3 x^3} \]

[Out]

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

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Rubi [A]  time = 0.0153822, antiderivative size = 37, 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} \[ 3 a^2 b x+\frac{a^3 x^3}{3}-\frac{3 a b^2}{x}-\frac{b^3}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x^2)^3*x^2,x]

[Out]

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

Mathematica [A]  time = 0.0039753, size = 37, normalized size = 1. \[ 3 a^2 b x+\frac{a^3 x^3}{3}-\frac{3 a b^2}{x}-\frac{b^3}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x^2)^3*x^2,x]

[Out]

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

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Maple [A]  time = 0.004, size = 34, normalized size = 0.9 \begin{align*} -{\frac{{b}^{3}}{3\,{x}^{3}}}-3\,{\frac{{b}^{2}a}{x}}+3\,{a}^{2}bx+{\frac{{a}^{3}{x}^{3}}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.03813, size = 46, normalized size = 1.24 \begin{align*} \frac{1}{3} \, a^{3} x^{3} + 3 \, a^{2} b x - \frac{9 \, a b^{2} x^{2} + b^{3}}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.42852, size = 72, normalized size = 1.95 \begin{align*} \frac{a^{3} x^{6} + 9 \, a^{2} b x^{4} - 9 \, a b^{2} x^{2} - b^{3}}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.295007, size = 34, normalized size = 0.92 \begin{align*} \frac{a^{3} x^{3}}{3} + 3 a^{2} b x - \frac{9 a b^{2} x^{2} + b^{3}}{3 x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.14276, size = 46, normalized size = 1.24 \begin{align*} \frac{1}{3} \, a^{3} x^{3} + 3 \, a^{2} b x - \frac{9 \, a b^{2} x^{2} + b^{3}}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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