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

Optimal. Leaf size=15 \[ \frac{\sqrt [3]{a+b x^3}}{b} \]

[Out]

(a + b*x^3)^(1/3)/b

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Rubi [A]  time = 0.004248, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {261} \[ \frac{\sqrt [3]{a+b x^3}}{b} \]

Antiderivative was successfully verified.

[In]

Int[x^2/(a + b*x^3)^(2/3),x]

[Out]

(a + b*x^3)^(1/3)/b

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 )^{2/3}} \, dx &=\frac{\sqrt [3]{a+b x^3}}{b}\\ \end{align*}

Mathematica [A]  time = 0.0030014, size = 15, normalized size = 1. \[ \frac{\sqrt [3]{a+b x^3}}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/(a + b*x^3)^(2/3),x]

[Out]

(a + b*x^3)^(1/3)/b

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Maple [A]  time = 0.003, size = 14, normalized size = 0.9 \begin{align*}{\frac{1}{b}\sqrt [3]{b{x}^{3}+a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*x^3+a)^(1/3)/b

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Maxima [A]  time = 1.05325, size = 18, normalized size = 1.2 \begin{align*} \frac{{\left (b x^{3} + a\right )}^{\frac{1}{3}}}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*x^3 + a)^(1/3)/b

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Fricas [A]  time = 1.43783, size = 28, normalized size = 1.87 \begin{align*} \frac{{\left (b x^{3} + a\right )}^{\frac{1}{3}}}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*x^3 + a)^(1/3)/b

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Sympy [A]  time = 0.538253, size = 20, normalized size = 1.33 \begin{align*} \begin{cases} \frac{\sqrt [3]{a + b x^{3}}}{b} & \text{for}\: b \neq 0 \\\frac{x^{3}}{3 a^{\frac{2}{3}}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise(((a + b*x**3)**(1/3)/b, Ne(b, 0)), (x**3/(3*a**(2/3)), True))

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Giac [A]  time = 1.10198, size = 18, normalized size = 1.2 \begin{align*} \frac{{\left (b x^{3} + a\right )}^{\frac{1}{3}}}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*x^3 + a)^(1/3)/b