3.1837 \(\int \frac{(a+\frac{b}{x^2})^3}{x} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0208135, antiderivative size = 39, normalized size of antiderivative = 1., 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} \[ -\frac{3 a^2 b}{2 x^2}+a^3 \log (x)-\frac{3 a b^2}{4 x^4}-\frac{b^3}{6 x^6} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x^2)^3/x,x]

[Out]

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

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]]

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])

Rubi steps

\begin{align*} \int \frac{\left (a+\frac{b}{x^2}\right )^3}{x} \, dx &=\int \frac{\left (b+a x^2\right )^3}{x^7} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(b+a x)^3}{x^4} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{b^3}{x^4}+\frac{3 a b^2}{x^3}+\frac{3 a^2 b}{x^2}+\frac{a^3}{x}\right ) \, dx,x,x^2\right )\\ &=-\frac{b^3}{6 x^6}-\frac{3 a b^2}{4 x^4}-\frac{3 a^2 b}{2 x^2}+a^3 \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0041217, size = 39, normalized size = 1. \[ -\frac{3 a^2 b}{2 x^2}+a^3 \log (x)-\frac{3 a b^2}{4 x^4}-\frac{b^3}{6 x^6} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x^2)^3/x,x]

[Out]

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

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Maple [A]  time = 0.006, size = 34, normalized size = 0.9 \begin{align*} -{\frac{{b}^{3}}{6\,{x}^{6}}}-{\frac{3\,{b}^{2}a}{4\,{x}^{4}}}-{\frac{3\,{a}^{2}b}{2\,{x}^{2}}}+{a}^{3}\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.980571, size = 53, normalized size = 1.36 \begin{align*} \frac{1}{2} \, a^{3} \log \left (x^{2}\right ) - \frac{18 \, a^{2} b x^{4} + 9 \, a b^{2} x^{2} + 2 \, b^{3}}{12 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*a^3*log(x^2) - 1/12*(18*a^2*b*x^4 + 9*a*b^2*x^2 + 2*b^3)/x^6

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Fricas [A]  time = 1.47874, size = 90, normalized size = 2.31 \begin{align*} \frac{12 \, a^{3} x^{6} \log \left (x\right ) - 18 \, a^{2} b x^{4} - 9 \, a b^{2} x^{2} - 2 \, b^{3}}{12 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(12*a^3*x^6*log(x) - 18*a^2*b*x^4 - 9*a*b^2*x^2 - 2*b^3)/x^6

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Sympy [A]  time = 0.380377, size = 36, normalized size = 0.92 \begin{align*} a^{3} \log{\left (x \right )} - \frac{18 a^{2} b x^{4} + 9 a b^{2} x^{2} + 2 b^{3}}{12 x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**3*log(x) - (18*a**2*b*x**4 + 9*a*b**2*x**2 + 2*b**3)/(12*x**6)

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Giac [A]  time = 1.15831, size = 63, normalized size = 1.62 \begin{align*} \frac{1}{2} \, a^{3} \log \left (x^{2}\right ) - \frac{11 \, a^{3} x^{6} + 18 \, a^{2} b x^{4} + 9 \, a b^{2} x^{2} + 2 \, b^{3}}{12 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*a^3*log(x^2) - 1/12*(11*a^3*x^6 + 18*a^2*b*x^4 + 9*a*b^2*x^2 + 2*b^3)/x^6