\(\int \frac {x^2}{a+\frac {b}{x^3}} \, dx\) [1968]

Optimal result

Integrand size = 13, antiderivative size = 27 \[ \int \frac {x^2}{a+\frac {b}{x^3}} \, dx=\frac {x^3}{3 a}-\frac {b \log \left (b+a x^3\right )}{3 a^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {269, 272, 45} \[ \int \frac {x^2}{a+\frac {b}{x^3}} \, dx=\frac {x^3}{3 a}-\frac {b \log \left (a x^3+b\right )}{3 a^2} \]

[In]

Int[x^2/(a + b/x^3),x]

[Out]

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

Rule 45

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])

Rule 269

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 272

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]]

Rubi steps \begin{align*} \text {integral}& = \int \frac {x^5}{b+a x^3} \, dx \\ & = \frac {1}{3} \text {Subst}\left (\int \frac {x}{b+a x} \, dx,x,x^3\right ) \\ & = \frac {1}{3} \text {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] (verified)

Time = 0.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {x^2}{a+\frac {b}{x^3}} \, dx=\frac {x^3}{3 a}-\frac {b \log \left (b+a x^3\right )}{3 a^2} \]

[In]

Integrate[x^2/(a + b/x^3),x]

[Out]

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

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.85

method	result	size
parallelrisch	\(-\frac {-a \,x^{3}+b \ln \left (a \,x^{3}+b \right )}{3 a^{2}}\)	\(23\)
default	\(\frac {x^{3}}{3 a}-\frac {b \ln \left (a \,x^{3}+b \right )}{3 a^{2}}\)	\(24\)
norman	\(\frac {x^{3}}{3 a}-\frac {b \ln \left (a \,x^{3}+b \right )}{3 a^{2}}\)	\(24\)
risch	\(\frac {x^{3}}{3 a}-\frac {b \ln \left (a \,x^{3}+b \right )}{3 a^{2}}\)	\(24\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.81 \[ \int \frac {x^2}{a+\frac {b}{x^3}} \, dx=\frac {a x^{3} - b \log \left (a x^{3} + b\right )}{3 \, a^{2}} \]

[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

Sympy [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.74 \[ \int \frac {x^2}{a+\frac {b}{x^3}} \, dx=\frac {x^{3}}{3 a} - \frac {b \log {\left (a x^{3} + b \right )}}{3 a^{2}} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.85 \[ \int \frac {x^2}{a+\frac {b}{x^3}} \, dx=\frac {x^{3}}{3 \, a} - \frac {b \log \left (a x^{3} + b\right )}{3 \, a^{2}} \]

[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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89 \[ \int \frac {x^2}{a+\frac {b}{x^3}} \, dx=\frac {x^{3}}{3 \, a} - \frac {b \log \left ({\left | a x^{3} + b \right |}\right )}{3 \, a^{2}} \]

[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

Mupad [B] (verification not implemented)

Time = 5.76 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.81 \[ \int \frac {x^2}{a+\frac {b}{x^3}} \, dx=-\frac {b\,\ln \left (a\,x^3+b\right )-a\,x^3}{3\,a^2} \]

[In]

int(x^2/(a + b/x^3),x)

[Out]

-(b*log(b + a*x^3) - a*x^3)/(3*a^2)