3.1819 \(\int (a+\frac{b}{x^2})^2 x^4 \, dx\)

Optimal. Leaf size=25 \[ \frac{a^2 x^5}{5}+\frac{2}{3} a b x^3+b^2 x \]

[Out]

b^2*x + (2*a*b*x^3)/3 + (a^2*x^5)/5

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Rubi [A]  time = 0.0089196, antiderivative size = 25, 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} \[ \frac{a^2 x^5}{5}+\frac{2}{3} a b x^3+b^2 x \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x^2)^2*x^4,x]

[Out]

b^2*x + (2*a*b*x^3)/3 + (a^2*x^5)/5

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

Mathematica [A]  time = 0.0010089, size = 25, normalized size = 1. \[ \frac{a^2 x^5}{5}+\frac{2}{3} a b x^3+b^2 x \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x^2)^2*x^4,x]

[Out]

b^2*x + (2*a*b*x^3)/3 + (a^2*x^5)/5

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Maple [A]  time = 0.001, size = 22, normalized size = 0.9 \begin{align*}{b}^{2}x+{\frac{2\,{x}^{3}ab}{3}}+{\frac{{x}^{5}{a}^{2}}{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.961557, size = 28, normalized size = 1.12 \begin{align*} \frac{1}{5} \, a^{2} x^{5} + \frac{2}{3} \, a b x^{3} + b^{2} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*a^2*x^5 + 2/3*a*b*x^3 + b^2*x

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Fricas [A]  time = 1.38841, size = 47, normalized size = 1.88 \begin{align*} \frac{1}{5} \, a^{2} x^{5} + \frac{2}{3} \, a b x^{3} + b^{2} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*a^2*x^5 + 2/3*a*b*x^3 + b^2*x

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Sympy [A]  time = 0.059351, size = 22, normalized size = 0.88 \begin{align*} \frac{a^{2} x^{5}}{5} + \frac{2 a b x^{3}}{3} + b^{2} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**2*x**5/5 + 2*a*b*x**3/3 + b**2*x

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Giac [A]  time = 1.17198, size = 28, normalized size = 1.12 \begin{align*} \frac{1}{5} \, a^{2} x^{5} + \frac{2}{3} \, a b x^{3} + b^{2} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*a^2*x^5 + 2/3*a*b*x^3 + b^2*x