3.6.64 \(\int \frac {x^2}{(a+b x^3)^{2/3}} \, dx\) [564]

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

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 15, normalized size of antiderivative = 1.00, 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 = {267} \begin {gather*} \frac {\sqrt [3]{a+b x^3}}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2/(a + b*x^3)^(2/3),x]

[Out]

(a + b*x^3)^(1/3)/b

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

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Mathematica [A]
time = 0.02, size = 15, normalized size = 1.00 \begin {gather*} \frac {\sqrt [3]{a+b x^3}}{b} \end {gather*}

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 = 0.14, size = 14, normalized size = 0.93

method	result	size
gosper	\(\frac {\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{b}\)	\(14\)
derivativedivides	\(\frac {\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{b}\)	\(14\)
default	\(\frac {\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{b}\)	\(14\)
trager	\(\frac {\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{b}\)	\(14\)
risch	\(\frac {\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{b}\)	\(14\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.30, size = 13, normalized size = 0.87 \begin {gather*} \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}}}{b} \end {gather*}

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 = 0.36, size = 13, normalized size = 0.87 \begin {gather*} \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}}}{b} \end {gather*}

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 = 0.18, size = 20, normalized size = 1.33 \begin {gather*} \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 {gather*}

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 = 1.89, size = 13, normalized size = 0.87 \begin {gather*} \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}}}{b} \end {gather*}

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

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Mupad [B]
time = 1.05, size = 13, normalized size = 0.87 \begin {gather*} \frac {{\left (b\,x^3+a\right )}^{1/3}}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(a + b*x^3)^(2/3),x)

[Out]

(a + b*x^3)^(1/3)/b

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