3.1831 \(\int (a+\frac {b}{x^2})^3 x^5 \, dx\)

Optimal. Leaf size=39 \[ \frac {a^3 x^6}{6}+\frac {3}{4} a^2 b x^4+\frac {3}{2} a b^2 x^2+b^3 \log (x) \]

[Out]

3/2*a*b^2*x^2+3/4*a^2*b*x^4+1/6*a^3*x^6+b^3*ln(x)

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Rubi [A]  time = 0.02, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {263, 266, 43} \[ \frac {3}{4} a^2 b x^4+\frac {a^3 x^6}{6}+\frac {3}{2} a b^2 x^2+b^3 \log (x) \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x^2)^3*x^5,x]

[Out]

(3*a*b^2*x^2)/2 + (3*a^2*b*x^4)/4 + (a^3*x^6)/6 + b^3*Log[x]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

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 266

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

Rubi steps

\begin {align*} \int \left (a+\frac {b}{x^2}\right )^3 x^5 \, dx &=\int \frac {\left (b+a x^2\right )^3}{x} \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(b+a x)^3}{x} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (3 a b^2+\frac {b^3}{x}+3 a^2 b x+a^3 x^2\right ) \, dx,x,x^2\right )\\ &=\frac {3}{2} a b^2 x^2+\frac {3}{4} a^2 b x^4+\frac {a^3 x^6}{6}+b^3 \log (x)\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 39, normalized size = 1.00 \[ \frac {a^3 x^6}{6}+\frac {3}{4} a^2 b x^4+\frac {3}{2} a b^2 x^2+b^3 \log (x) \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x^2)^3*x^5,x]

[Out]

(3*a*b^2*x^2)/2 + (3*a^2*b*x^4)/4 + (a^3*x^6)/6 + b^3*Log[x]

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fricas [A]  time = 0.89, size = 33, normalized size = 0.85 \[ \frac {1}{6} \, a^{3} x^{6} + \frac {3}{4} \, a^{2} b x^{4} + \frac {3}{2} \, a b^{2} x^{2} + b^{3} \log \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^2)^3*x^5,x, algorithm="fricas")

[Out]

1/6*a^3*x^6 + 3/4*a^2*b*x^4 + 3/2*a*b^2*x^2 + b^3*log(x)

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giac [A]  time = 0.16, size = 36, normalized size = 0.92 \[ \frac {1}{6} \, a^{3} x^{6} + \frac {3}{4} \, a^{2} b x^{4} + \frac {3}{2} \, a b^{2} x^{2} + \frac {1}{2} \, b^{3} \log \left (x^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^2)^3*x^5,x, algorithm="giac")

[Out]

1/6*a^3*x^6 + 3/4*a^2*b*x^4 + 3/2*a*b^2*x^2 + 1/2*b^3*log(x^2)

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maple [A]  time = 0.00, size = 34, normalized size = 0.87 \[ \frac {a^{3} x^{6}}{6}+\frac {3 a^{2} b \,x^{4}}{4}+\frac {3 a \,b^{2} x^{2}}{2}+b^{3} \ln \relax (x ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/2*a*b^2*x^2+3/4*a^2*b*x^4+1/6*a^3*x^6+b^3*ln(x)

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maxima [A]  time = 0.87, size = 36, normalized size = 0.92 \[ \frac {1}{6} \, a^{3} x^{6} + \frac {3}{4} \, a^{2} b x^{4} + \frac {3}{2} \, a b^{2} x^{2} + \frac {1}{2} \, b^{3} \log \left (x^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^2)^3*x^5,x, algorithm="maxima")

[Out]

1/6*a^3*x^6 + 3/4*a^2*b*x^4 + 3/2*a*b^2*x^2 + 1/2*b^3*log(x^2)

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mupad [B]  time = 0.04, size = 33, normalized size = 0.85 \[ b^3\,\ln \relax (x)+\frac {a^3\,x^6}{6}+\frac {3\,a\,b^2\,x^2}{2}+\frac {3\,a^2\,b\,x^4}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(a + b/x^2)^3,x)

[Out]

b^3*log(x) + (a^3*x^6)/6 + (3*a*b^2*x^2)/2 + (3*a^2*b*x^4)/4

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sympy [A]  time = 0.12, size = 37, normalized size = 0.95 \[ \frac {a^{3} x^{6}}{6} + \frac {3 a^{2} b x^{4}}{4} + \frac {3 a b^{2} x^{2}}{2} + b^{3} \log {\relax (x )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x**2)**3*x**5,x)

[Out]

a**3*x**6/6 + 3*a**2*b*x**4/4 + 3*a*b**2*x**2/2 + b**3*log(x)

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