3.1818 \(\int (a+\frac{b}{x^2})^2 x^5 \, dx\)

Optimal. Leaf size=16 \[ \frac{\left (a x^2+b\right )^3}{6 a} \]

[Out]

(b + a*x^2)^3/(6*a)

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Rubi [A]  time = 0.004456, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {263, 261} \[ \frac{\left (a x^2+b\right )^3}{6 a} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x^2)^2*x^5,x]

[Out]

(b + a*x^2)^3/(6*a)

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 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rubi steps

\begin{align*} \int \left (a+\frac{b}{x^2}\right )^2 x^5 \, dx &=\int x \left (b+a x^2\right )^2 \, dx\\ &=\frac{\left (b+a x^2\right )^3}{6 a}\\ \end{align*}

Mathematica [A]  time = 0.0020138, size = 16, normalized size = 1. \[ \frac{\left (a x^2+b\right )^3}{6 a} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x^2)^2*x^5,x]

[Out]

(b + a*x^2)^3/(6*a)

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Maple [A]  time = 0., size = 25, normalized size = 1.6 \begin{align*}{\frac{{a}^{2}{x}^{6}}{6}}+{\frac{ab{x}^{4}}{2}}+{\frac{{b}^{2}{x}^{2}}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+1/x^2*b)^2*x^5,x)

[Out]

1/6*a^2*x^6+1/2*a*b*x^4+1/2*b^2*x^2

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Maxima [A]  time = 0.949848, size = 32, normalized size = 2. \begin{align*} \frac{1}{6} \, a^{2} x^{6} + \frac{1}{2} \, a b x^{4} + \frac{1}{2} \, b^{2} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^2)^2*x^5,x, algorithm="maxima")

[Out]

1/6*a^2*x^6 + 1/2*a*b*x^4 + 1/2*b^2*x^2

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Fricas [A]  time = 1.32867, size = 55, normalized size = 3.44 \begin{align*} \frac{1}{6} \, a^{2} x^{6} + \frac{1}{2} \, a b x^{4} + \frac{1}{2} \, b^{2} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^2)^2*x^5,x, algorithm="fricas")

[Out]

1/6*a^2*x^6 + 1/2*a*b*x^4 + 1/2*b^2*x^2

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Sympy [B]  time = 0.060206, size = 24, normalized size = 1.5 \begin{align*} \frac{a^{2} x^{6}}{6} + \frac{a b x^{4}}{2} + \frac{b^{2} x^{2}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x**2)**2*x**5,x)

[Out]

a**2*x**6/6 + a*b*x**4/2 + b**2*x**2/2

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Giac [A]  time = 1.18804, size = 32, normalized size = 2. \begin{align*} \frac{1}{6} \, a^{2} x^{6} + \frac{1}{2} \, a b x^{4} + \frac{1}{2} \, b^{2} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^2)^2*x^5,x, algorithm="giac")

[Out]

1/6*a^2*x^6 + 1/2*a*b*x^4 + 1/2*b^2*x^2