\(\int \frac {x^2}{(a+b x^3)^3} \, dx\) [343]

Optimal result

Integrand size = 13, antiderivative size = 16 \[ \int \frac {x^2}{\left (a+b x^3\right )^3} \, dx=-\frac {1}{6 b \left (a+b x^3\right )^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {267} \[ \int \frac {x^2}{\left (a+b x^3\right )^3} \, dx=-\frac {1}{6 b \left (a+b x^3\right )^2} \]

[In]

Int[x^2/(a + b*x^3)^3,x]

[Out]

-1/6*1/(b*(a + b*x^3)^2)

Rule 267

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rubi steps \begin{align*} \text {integral}& = -\frac {1}{6 b \left (a+b x^3\right )^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {x^2}{\left (a+b x^3\right )^3} \, dx=-\frac {1}{6 b \left (a+b x^3\right )^2} \]

[In]

Integrate[x^2/(a + b*x^3)^3,x]

[Out]

-1/6*1/(b*(a + b*x^3)^2)

Maple [A] (verified)

Time = 3.63 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.94

method	result	size
gosper	\(-\frac {1}{6 b \left (b \,x^{3}+a \right )^{2}}\)	\(15\)
derivativedivides	\(-\frac {1}{6 b \left (b \,x^{3}+a \right )^{2}}\)	\(15\)
default	\(-\frac {1}{6 b \left (b \,x^{3}+a \right )^{2}}\)	\(15\)
norman	\(-\frac {1}{6 b \left (b \,x^{3}+a \right )^{2}}\)	\(15\)
risch	\(-\frac {1}{6 b \left (b \,x^{3}+a \right )^{2}}\)	\(15\)
parallelrisch	\(-\frac {1}{6 b \left (b \,x^{3}+a \right )^{2}}\)	\(15\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.62 \[ \int \frac {x^2}{\left (a+b x^3\right )^3} \, dx=-\frac {1}{6 \, {\left (b^{3} x^{6} + 2 \, a b^{2} x^{3} + a^{2} b\right )}} \]

[In]

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

[Out]

-1/6/(b^3*x^6 + 2*a*b^2*x^3 + a^2*b)

Sympy [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.69 \[ \int \frac {x^2}{\left (a+b x^3\right )^3} \, dx=- \frac {1}{6 a^{2} b + 12 a b^{2} x^{3} + 6 b^{3} x^{6}} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {x^2}{\left (a+b x^3\right )^3} \, dx=-\frac {1}{6 \, {\left (b x^{3} + a\right )}^{2} b} \]

[In]

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

[Out]

-1/6/((b*x^3 + a)^2*b)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {x^2}{\left (a+b x^3\right )^3} \, dx=-\frac {1}{6 \, {\left (b x^{3} + a\right )}^{2} b} \]

[In]

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

[Out]

-1/6/((b*x^3 + a)^2*b)

Mupad [B] (verification not implemented)

Time = 5.43 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.75 \[ \int \frac {x^2}{\left (a+b x^3\right )^3} \, dx=-\frac {1}{6\,a^2\,b+12\,a\,b^2\,x^3+6\,b^3\,x^6} \]

[In]

int(x^2/(a + b*x^3)^3,x)

[Out]

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