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

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0199808, 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} \[ \frac{3}{4} a^2 b x^4+\frac{a^3 x^6}{6}+\frac{3}{2} a b^2 x^2+b^3 \log (x) \]

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 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^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*}

Mathematica [A]  time = 0.0040285, size = 39, normalized size = 1. \[ \frac{3}{4} a^2 b x^4+\frac{a^3 x^6}{6}+\frac{3}{2} a b^2 x^2+b^3 \log (x) \]

Antiderivative was successfully verified.

[In]

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]

________________________________________________________________________________________

Maple [A]  time = 0.002, size = 34, normalized size = 0.9 \begin{align*}{\frac{3\,a{b}^{2}{x}^{2}}{2}}+{\frac{3\,{a}^{2}b{x}^{4}}{4}}+{\frac{{a}^{3}{x}^{6}}{6}}+{b}^{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^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)

________________________________________________________________________________________

Maxima [A]  time = 0.966368, size = 49, normalized size = 1.26 \begin{align*} \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 ) \end{align*}

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)

________________________________________________________________________________________

Fricas [A]  time = 1.4256, size = 78, normalized size = 2. \begin{align*} \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 \left (x\right ) \end{align*}

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)

________________________________________________________________________________________

Sympy [A]  time = 0.25561, size = 37, normalized size = 0.95 \begin{align*} \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{\left (x \right )} \end{align*}

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)

________________________________________________________________________________________

Giac [A]  time = 1.17138, size = 49, normalized size = 1.26 \begin{align*} \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 ) \end{align*}

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)