3.6.13 \(\int x^2 \sqrt [3]{a+b x^3} \, dx\) [513]

Optimal. Leaf size=18 \[ \frac {\left (a+b x^3\right )^{4/3}}{4 b} \]

[Out]

1/4*(b*x^3+a)^(4/3)/b

________________________________________________________________________________________

Rubi [A]
time = 0.00, antiderivative size = 18, 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 {\left (a+b x^3\right )^{4/3}}{4 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*x^3)^(1/3),x]

[Out]

(a + b*x^3)^(4/3)/(4*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 x^2 \sqrt [3]{a+b x^3} \, dx &=\frac {\left (a+b x^3\right )^{4/3}}{4 b}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.01, size = 18, normalized size = 1.00 \begin {gather*} \frac {\left (a+b x^3\right )^{4/3}}{4 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^2*(a + b*x^3)^(1/3),x]

[Out]

(a + b*x^3)^(4/3)/(4*b)

________________________________________________________________________________________

Maple [A]
time = 0.13, size = 15, normalized size = 0.83

method	result	size
gosper	\(\frac {\left (b \,x^{3}+a \right )^{\frac {4}{3}}}{4 b}\)	\(15\)
derivativedivides	\(\frac {\left (b \,x^{3}+a \right )^{\frac {4}{3}}}{4 b}\)	\(15\)
default	\(\frac {\left (b \,x^{3}+a \right )^{\frac {4}{3}}}{4 b}\)	\(15\)
trager	\(\frac {\left (b \,x^{3}+a \right )^{\frac {4}{3}}}{4 b}\)	\(15\)
risch	\(\frac {\left (b \,x^{3}+a \right )^{\frac {4}{3}}}{4 b}\)	\(15\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(b*x^3+a)^(1/3),x,method=_RETURNVERBOSE)

[Out]

1/4*(b*x^3+a)^(4/3)/b

________________________________________________________________________________________

Maxima [A]
time = 0.29, size = 14, normalized size = 0.78 \begin {gather*} \frac {{\left (b x^{3} + a\right )}^{\frac {4}{3}}}{4 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(b*x^3+a)^(1/3),x, algorithm="maxima")

[Out]

1/4*(b*x^3 + a)^(4/3)/b

________________________________________________________________________________________

Fricas [A]
time = 0.36, size = 14, normalized size = 0.78 \begin {gather*} \frac {{\left (b x^{3} + a\right )}^{\frac {4}{3}}}{4 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(b*x^3+a)^(1/3),x, algorithm="fricas")

[Out]

1/4*(b*x^3 + a)^(4/3)/b

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 39 vs. \(2 (12) = 24\).
time = 0.08, size = 39, normalized size = 2.17 \begin {gather*} \begin {cases} \frac {a \sqrt [3]{a + b x^{3}}}{4 b} + \frac {x^{3} \sqrt [3]{a + b x^{3}}}{4} & \text {for}\: b \neq 0 \\\frac {\sqrt [3]{a} x^{3}}{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)**(1/3),x)

[Out]

Piecewise((a*(a + b*x**3)**(1/3)/(4*b) + x**3*(a + b*x**3)**(1/3)/4, Ne(b, 0)), (a**(1/3)*x**3/3, True))

________________________________________________________________________________________

Giac [A]
time = 2.62, size = 14, normalized size = 0.78 \begin {gather*} \frac {{\left (b x^{3} + a\right )}^{\frac {4}{3}}}{4 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(b*x^3+a)^(1/3),x, algorithm="giac")

[Out]

1/4*(b*x^3 + a)^(4/3)/b

________________________________________________________________________________________

Mupad [B]
time = 1.03, size = 14, normalized size = 0.78 \begin {gather*} \frac {{\left (b\,x^3+a\right )}^{4/3}}{4\,b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a + b*x^3)^(1/3),x)

[Out]

(a + b*x^3)^(4/3)/(4*b)

________________________________________________________________________________________