3.1830 \(\int (a+\frac{b}{x^2})^3 x^6 \, dx\)

Optimal. Leaf size=35 \[ \frac{3}{5} a^2 b x^5+\frac{a^3 x^7}{7}+a b^2 x^3+b^3 x \]

[Out]

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

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Rubi [A]  time = 0.0138719, antiderivative size = 35, 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, 194} \[ \frac{3}{5} a^2 b x^5+\frac{a^3 x^7}{7}+a b^2 x^3+b^3 x \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x^2)^3*x^6,x]

[Out]

b^3*x + a*b^2*x^3 + (3*a^2*b*x^5)/5 + (a^3*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 194

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n)^p, x], x] /; FreeQ[{a, b}, x]
&& IGtQ[n, 0] && IGtQ[p, 0]

Rubi steps

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

Mathematica [A]  time = 0.0010596, size = 35, normalized size = 1. \[ \frac{3}{5} a^2 b x^5+\frac{a^3 x^7}{7}+a b^2 x^3+b^3 x \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x^2)^3*x^6,x]

[Out]

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

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Maple [A]  time = 0.001, size = 32, normalized size = 0.9 \begin{align*}{b}^{3}x+{x}^{3}a{b}^{2}+{\frac{3\,{a}^{2}b{x}^{5}}{5}}+{\frac{{a}^{3}{x}^{7}}{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+1/x^2*b)^3*x^6,x)

[Out]

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

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Maxima [A]  time = 1.09346, size = 42, normalized size = 1.2 \begin{align*} \frac{1}{7} \, a^{3} x^{7} + \frac{3}{5} \, a^{2} b x^{5} + a b^{2} x^{3} + b^{3} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^2)^3*x^6,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.44121, size = 66, normalized size = 1.89 \begin{align*} \frac{1}{7} \, a^{3} x^{7} + \frac{3}{5} \, a^{2} b x^{5} + a b^{2} x^{3} + b^{3} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^2)^3*x^6,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.062721, size = 32, normalized size = 0.91 \begin{align*} \frac{a^{3} x^{7}}{7} + \frac{3 a^{2} b x^{5}}{5} + a b^{2} x^{3} + b^{3} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x**2)**3*x**6,x)

[Out]

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

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Giac [A]  time = 1.1397, size = 42, normalized size = 1.2 \begin{align*} \frac{1}{7} \, a^{3} x^{7} + \frac{3}{5} \, a^{2} b x^{5} + a b^{2} x^{3} + b^{3} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^2)^3*x^6,x, algorithm="giac")

[Out]

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