3.1968 \(\int \frac{x^2}{a+\frac{b}{x^3}} \, dx\)

Optimal. Leaf size=27 \[ \frac{x^3}{3 a}-\frac{b \log \left (a x^3+b\right )}{3 a^2} \]

[Out]

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

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Rubi [A]  time = 0.0205104, antiderivative size = 27, 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} \[ \frac{x^3}{3 a}-\frac{b \log \left (a x^3+b\right )}{3 a^2} \]

Antiderivative was successfully verified.

[In]

Int[x^2/(a + b/x^3),x]

[Out]

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

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

Mathematica [A]  time = 0.0043059, size = 27, normalized size = 1. \[ \frac{x^3}{3 a}-\frac{b \log \left (a x^3+b\right )}{3 a^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/(a + b/x^3),x]

[Out]

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

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Maple [A]  time = 0.003, size = 24, normalized size = 0.9 \begin{align*}{\frac{{x}^{3}}{3\,a}}-{\frac{b\ln \left ( a{x}^{3}+b \right ) }{3\,{a}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.974833, size = 31, normalized size = 1.15 \begin{align*} \frac{x^{3}}{3 \, a} - \frac{b \log \left (a x^{3} + b\right )}{3 \, a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*x^3/a - 1/3*b*log(a*x^3 + b)/a^2

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Fricas [A]  time = 1.42037, size = 49, normalized size = 1.81 \begin{align*} \frac{a x^{3} - b \log \left (a x^{3} + b\right )}{3 \, a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(a*x^3 - b*log(a*x^3 + b))/a^2

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Sympy [A]  time = 0.408818, size = 20, normalized size = 0.74 \begin{align*} \frac{x^{3}}{3 a} - \frac{b \log{\left (a x^{3} + b \right )}}{3 a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.16879, size = 32, normalized size = 1.19 \begin{align*} \frac{x^{3}}{3 \, a} - \frac{b \log \left ({\left | a x^{3} + b \right |}\right )}{3 \, a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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