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

Optimal. Leaf size=16 \[ -\frac{1}{6 b \left (a+b x^3\right )^2} \]

[Out]

-1/(6*b*(a + b*x^3)^2)

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Rubi [A]  time = 0.0032999, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {261} \[ -\frac{1}{6 b \left (a+b x^3\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[x^2/(a + b*x^3)^3,x]

[Out]

-1/(6*b*(a + b*x^3)^2)

Rule 261

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*} \int \frac{x^2}{\left (a+b x^3\right )^3} \, dx &=-\frac{1}{6 b \left (a+b x^3\right )^2}\\ \end{align*}

Mathematica [A]  time = 0.0044564, size = 16, normalized size = 1. \[ -\frac{1}{6 b \left (a+b x^3\right )^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/(a + b*x^3)^3,x]

[Out]

-1/(6*b*(a + b*x^3)^2)

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Maple [A]  time = 0.002, size = 15, normalized size = 0.9 \begin{align*} -{\frac{1}{6\,b \left ( b{x}^{3}+a \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.02945, size = 19, normalized size = 1.19 \begin{align*} -\frac{1}{6 \,{\left (b x^{3} + a\right )}^{2} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6/((b*x^3 + a)^2*b)

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Fricas [A]  time = 1.34509, size = 51, normalized size = 3.19 \begin{align*} -\frac{1}{6 \,{\left (b^{3} x^{6} + 2 \, a b^{2} x^{3} + a^{2} b\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [A]  time = 0.70314, size = 27, normalized size = 1.69 \begin{align*} - \frac{1}{6 a^{2} b + 12 a b^{2} x^{3} + 6 b^{3} x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [A]  time = 1.10763, size = 19, normalized size = 1.19 \begin{align*} -\frac{1}{6 \,{\left (b x^{3} + a\right )}^{2} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6/((b*x^3 + a)^2*b)