3.1835 \(\int (a+\frac{b}{x^2})^3 x \, dx\)

Optimal. Leaf size=40 \[ 3 a^2 b \log (x)+\frac{a^3 x^2}{2}-\frac{3 a b^2}{2 x^2}-\frac{b^3}{4 x^4} \]

[Out]

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

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Rubi [A]  time = 0.0196545, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {263, 266, 43} \[ 3 a^2 b \log (x)+\frac{a^3 x^2}{2}-\frac{3 a b^2}{2 x^2}-\frac{b^3}{4 x^4} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x^2)^3*x,x]

[Out]

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

Mathematica [A]  time = 0.0048519, size = 40, normalized size = 1. \[ 3 a^2 b \log (x)+\frac{a^3 x^2}{2}-\frac{3 a b^2}{2 x^2}-\frac{b^3}{4 x^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x^2)^3*x,x]

[Out]

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

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Maple [A]  time = 0.004, size = 35, normalized size = 0.9 \begin{align*} -{\frac{{b}^{3}}{4\,{x}^{4}}}-{\frac{3\,{b}^{2}a}{2\,{x}^{2}}}+{\frac{{x}^{2}{a}^{3}}{2}}+3\,{a}^{2}b\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]

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

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Maxima [A]  time = 1.03739, size = 50, normalized size = 1.25 \begin{align*} \frac{1}{2} \, a^{3} x^{2} + \frac{3}{2} \, a^{2} b \log \left (x^{2}\right ) - \frac{6 \, a b^{2} x^{2} + b^{3}}{4 \, x^{4}} \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*x^2 + 3/2*a^2*b*log(x^2) - 1/4*(6*a*b^2*x^2 + b^3)/x^4

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Fricas [A]  time = 1.46929, size = 85, normalized size = 2.12 \begin{align*} \frac{2 \, a^{3} x^{6} + 12 \, a^{2} b x^{4} \log \left (x\right ) - 6 \, a b^{2} x^{2} - b^{3}}{4 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.332937, size = 36, normalized size = 0.9 \begin{align*} \frac{a^{3} x^{2}}{2} + 3 a^{2} b \log{\left (x \right )} - \frac{6 a b^{2} x^{2} + b^{3}}{4 x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.1878, size = 49, normalized size = 1.22 \begin{align*} \frac{1}{2} \, a^{3} x^{2} + 3 \, a^{2} b \log \left ({\left | x \right |}\right ) - \frac{6 \, a b^{2} x^{2} + b^{3}}{4 \, x^{4}} \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*x^2 + 3*a^2*b*log(abs(x)) - 1/4*(6*a*b^2*x^2 + b^3)/x^4