3.240 \(\int x^2 (a+b x^3)^3 \, dx\)

Optimal. Leaf size=16 \[ \frac{\left (a+b x^3\right )^4}{12 b} \]

[Out]

(a + b*x^3)^4/(12*b)

________________________________________________________________________________________

Rubi [A]  time = 0.0029147, 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} \[ \frac{\left (a+b x^3\right )^4}{12 b} \]

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*x^3)^3,x]

[Out]

(a + b*x^3)^4/(12*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 x^2 \left (a+b x^3\right )^3 \, dx &=\frac{\left (a+b x^3\right )^4}{12 b}\\ \end{align*}

Mathematica [B]  time = 0.0016595, size = 43, normalized size = 2.69 \[ \frac{1}{2} a^2 b x^6+\frac{a^3 x^3}{3}+\frac{1}{3} a b^2 x^9+\frac{b^3 x^{12}}{12} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*(a + b*x^3)^3,x]

[Out]

(a^3*x^3)/3 + (a^2*b*x^6)/2 + (a*b^2*x^9)/3 + (b^3*x^12)/12

________________________________________________________________________________________

Maple [B]  time = 0.002, size = 36, normalized size = 2.3 \begin{align*}{\frac{{b}^{3}{x}^{12}}{12}}+{\frac{a{b}^{2}{x}^{9}}{3}}+{\frac{{a}^{2}b{x}^{6}}{2}}+{\frac{{a}^{3}{x}^{3}}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*b^3*x^12+1/3*a*b^2*x^9+1/2*a^2*b*x^6+1/3*a^3*x^3

________________________________________________________________________________________

Maxima [A]  time = 0.957055, size = 19, normalized size = 1.19 \begin{align*} \frac{{\left (b x^{3} + a\right )}^{4}}{12 \, 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]

1/12*(b*x^3 + a)^4/b

________________________________________________________________________________________

Fricas [B]  time = 1.43057, size = 82, normalized size = 5.12 \begin{align*} \frac{1}{12} x^{12} b^{3} + \frac{1}{3} x^{9} b^{2} a + \frac{1}{2} x^{6} b a^{2} + \frac{1}{3} x^{3} a^{3} \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]

1/12*x^12*b^3 + 1/3*x^9*b^2*a + 1/2*x^6*b*a^2 + 1/3*x^3*a^3

________________________________________________________________________________________

Sympy [B]  time = 0.094796, size = 36, normalized size = 2.25 \begin{align*} \frac{a^{3} x^{3}}{3} + \frac{a^{2} b x^{6}}{2} + \frac{a b^{2} x^{9}}{3} + \frac{b^{3} x^{12}}{12} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**3*x**3/3 + a**2*b*x**6/2 + a*b**2*x**9/3 + b**3*x**12/12

________________________________________________________________________________________

Giac [A]  time = 1.12702, size = 19, normalized size = 1.19 \begin{align*} \frac{{\left (b x^{3} + a\right )}^{4}}{12 \, 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]

1/12*(b*x^3 + a)^4/b