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]

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

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Rubi [A]  time = 0.00, antiderivative size = 16, normalized size of antiderivative = 1.00, 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 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]

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]

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*}

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Mathematica [A]  time = 0.00, size = 16, normalized size = 1.00 \[ \frac {\left (a x^2+b\right )^3}{6 a} \]

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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fricas [A]  time = 0.76, size = 24, normalized size = 1.50 \[ \frac {1}{6} \, a^{2} x^{6} + \frac {1}{2} \, a b x^{4} + \frac {1}{2} \, b^{2} x^{2} \]

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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giac [A]  time = 0.16, size = 24, normalized size = 1.50 \[ \frac {1}{6} \, a^{2} x^{6} + \frac {1}{2} \, a b x^{4} + \frac {1}{2} \, b^{2} x^{2} \]

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

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maple [A]  time = 0.00, size = 25, normalized size = 1.56 \[ \frac {1}{6} a^{2} x^{6}+\frac {1}{2} a b \,x^{4}+\frac {1}{2} b^{2} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b/x^2)^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 = 0.88, size = 24, normalized size = 1.50 \[ \frac {1}{6} \, a^{2} x^{6} + \frac {1}{2} \, a b x^{4} + \frac {1}{2} \, b^{2} x^{2} \]

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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mupad [B]  time = 0.03, size = 24, normalized size = 1.50 \[ \frac {a^2\,x^6}{6}+\frac {a\,b\,x^4}{2}+\frac {b^2\,x^2}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(a + b/x^2)^2,x)

[Out]

(a^2*x^6)/6 + (b^2*x^2)/2 + (a*b*x^4)/2

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sympy [B]  time = 0.07, size = 24, normalized size = 1.50 \[ \frac {a^{2} x^{6}}{6} + \frac {a b x^{4}}{2} + \frac {b^{2} x^{2}}{2} \]

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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