3.1817 \(\int (a+\frac{b}{x^2})^2 x^6 \, dx\)

Optimal. Leaf size=30 \[ \frac{a^2 x^7}{7}+\frac{2}{5} a b x^5+\frac{b^2 x^3}{3} \]

[Out]

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

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Rubi [A]  time = 0.0117358, antiderivative size = 30, 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} \[ \frac{a^2 x^7}{7}+\frac{2}{5} a b x^5+\frac{b^2 x^3}{3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x^2)^2*x^6,x]

[Out]

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

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

Mathematica [A]  time = 0.0010932, size = 30, normalized size = 1. \[ \frac{a^2 x^7}{7}+\frac{2}{5} a b x^5+\frac{b^2 x^3}{3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x^2)^2*x^6,x]

[Out]

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

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Maple [A]  time = 0., size = 25, normalized size = 0.8 \begin{align*}{\frac{{b}^{2}{x}^{3}}{3}}+{\frac{2\,{x}^{5}ab}{5}}+{\frac{{a}^{2}{x}^{7}}{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+1/x^2*b)^2*x^6,x)

[Out]

1/3*b^2*x^3+2/5*x^5*a*b+1/7*a^2*x^7

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Maxima [A]  time = 0.951734, size = 32, normalized size = 1.07 \begin{align*} \frac{1}{7} \, a^{2} x^{7} + \frac{2}{5} \, a b x^{5} + \frac{1}{3} \, b^{2} x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^2)^2*x^6,x, algorithm="maxima")

[Out]

1/7*a^2*x^7 + 2/5*a*b*x^5 + 1/3*b^2*x^3

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Fricas [A]  time = 1.36482, size = 55, normalized size = 1.83 \begin{align*} \frac{1}{7} \, a^{2} x^{7} + \frac{2}{5} \, a b x^{5} + \frac{1}{3} \, b^{2} x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^2)^2*x^6,x, algorithm="fricas")

[Out]

1/7*a^2*x^7 + 2/5*a*b*x^5 + 1/3*b^2*x^3

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Sympy [A]  time = 0.059942, size = 26, normalized size = 0.87 \begin{align*} \frac{a^{2} x^{7}}{7} + \frac{2 a b x^{5}}{5} + \frac{b^{2} x^{3}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x**2)**2*x**6,x)

[Out]

a**2*x**7/7 + 2*a*b*x**5/5 + b**2*x**3/3

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Giac [A]  time = 1.13259, size = 32, normalized size = 1.07 \begin{align*} \frac{1}{7} \, a^{2} x^{7} + \frac{2}{5} \, a b x^{5} + \frac{1}{3} \, b^{2} x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^2)^2*x^6,x, algorithm="giac")

[Out]

1/7*a^2*x^7 + 2/5*a*b*x^5 + 1/3*b^2*x^3