3.1981 \(\int \frac{x^2}{(a+\frac{b}{x^3})^2} \, dx\)

Optimal. Leaf size=46 \[ -\frac{b^2}{3 a^3 \left (a x^3+b\right )}-\frac{2 b \log \left (a x^3+b\right )}{3 a^3}+\frac{x^3}{3 a^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0315299, antiderivative size = 46, 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} \[ -\frac{b^2}{3 a^3 \left (a x^3+b\right )}-\frac{2 b \log \left (a x^3+b\right )}{3 a^3}+\frac{x^3}{3 a^2} \]

Antiderivative was successfully verified.

[In]

Int[x^2/(a + b/x^3)^2,x]

[Out]

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

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{x^2}{\left (a+\frac{b}{x^3}\right )^2} \, dx &=\int \frac{x^8}{\left (b+a x^3\right )^2} \, dx\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{x^2}{(b+a x)^2} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (\frac{1}{a^2}+\frac{b^2}{a^2 (b+a x)^2}-\frac{2 b}{a^2 (b+a x)}\right ) \, dx,x,x^3\right )\\ &=\frac{x^3}{3 a^2}-\frac{b^2}{3 a^3 \left (b+a x^3\right )}-\frac{2 b \log \left (b+a x^3\right )}{3 a^3}\\ \end{align*}

Mathematica [A]  time = 0.0152276, size = 38, normalized size = 0.83 \[ \frac{-\frac{b^2}{a x^3+b}-2 b \log \left (a x^3+b\right )+a x^3}{3 a^3} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/(a + b/x^3)^2,x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.005, size = 41, normalized size = 0.9 \begin{align*}{\frac{{x}^{3}}{3\,{a}^{2}}}-{\frac{{b}^{2}}{3\,{a}^{3} \left ( a{x}^{3}+b \right ) }}-{\frac{2\,b\ln \left ( a{x}^{3}+b \right ) }{3\,{a}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 0.98249, size = 58, normalized size = 1.26 \begin{align*} -\frac{b^{2}}{3 \,{\left (a^{4} x^{3} + a^{3} b\right )}} + \frac{x^{3}}{3 \, a^{2}} - \frac{2 \, b \log \left (a x^{3} + b\right )}{3 \, a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.43891, size = 113, normalized size = 2.46 \begin{align*} \frac{a^{2} x^{6} + a b x^{3} - b^{2} - 2 \,{\left (a b x^{3} + b^{2}\right )} \log \left (a x^{3} + b\right )}{3 \,{\left (a^{4} x^{3} + a^{3} b\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 0.453041, size = 42, normalized size = 0.91 \begin{align*} - \frac{b^{2}}{3 a^{4} x^{3} + 3 a^{3} b} + \frac{x^{3}}{3 a^{2}} - \frac{2 b \log{\left (a x^{3} + b \right )}}{3 a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.14552, size = 55, normalized size = 1.2 \begin{align*} \frac{x^{3}}{3 \, a^{2}} - \frac{2 \, b \log \left ({\left | a x^{3} + b \right |}\right )}{3 \, a^{3}} - \frac{b^{2}}{3 \,{\left (a x^{3} + b\right )} a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*x^3/a^2 - 2/3*b*log(abs(a*x^3 + b))/a^3 - 1/3*b^2/((a*x^3 + b)*a^3)