3.623 \(\int x^3 (a+b x^4)^2 \, dx\)

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

[Out]

(a + b*x^4)^3/(12*b)

________________________________________________________________________________________

Rubi [A]  time = 0.0029175, 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} \[ \frac{\left (a+b x^4\right )^3}{12 b} \]

Antiderivative was successfully verified.

[In]

Int[x^3*(a + b*x^4)^2,x]

[Out]

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

Mathematica [A]  time = 0.0006971, size = 30, normalized size = 1.88 \[ \frac{a^2 x^4}{4}+\frac{1}{4} a b x^8+\frac{b^2 x^{12}}{12} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*(a + b*x^4)^2,x]

[Out]

(a^2*x^4)/4 + (a*b*x^8)/4 + (b^2*x^12)/12

________________________________________________________________________________________

Maple [A]  time = 0.001, size = 25, normalized size = 1.6 \begin{align*}{\frac{{b}^{2}{x}^{12}}{12}}+{\frac{ab{x}^{8}}{4}}+{\frac{{a}^{2}{x}^{4}}{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(b*x^4+a)^2,x)

[Out]

1/12*b^2*x^12+1/4*a*b*x^8+1/4*a^2*x^4

________________________________________________________________________________________

Maxima [A]  time = 0.962192, size = 19, normalized size = 1.19 \begin{align*} \frac{{\left (b x^{4} + a\right )}^{3}}{12 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(b*x^4+a)^2,x, algorithm="maxima")

[Out]

1/12*(b*x^4 + a)^3/b

________________________________________________________________________________________

Fricas [A]  time = 1.27111, size = 58, normalized size = 3.62 \begin{align*} \frac{1}{12} x^{12} b^{2} + \frac{1}{4} x^{8} b a + \frac{1}{4} x^{4} a^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(b*x^4+a)^2,x, algorithm="fricas")

[Out]

1/12*x^12*b^2 + 1/4*x^8*b*a + 1/4*x^4*a^2

________________________________________________________________________________________

Sympy [B]  time = 0.084215, size = 24, normalized size = 1.5 \begin{align*} \frac{a^{2} x^{4}}{4} + \frac{a b x^{8}}{4} + \frac{b^{2} x^{12}}{12} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(b*x**4+a)**2,x)

[Out]

a**2*x**4/4 + a*b*x**8/4 + b**2*x**12/12

________________________________________________________________________________________

Giac [A]  time = 1.0959, size = 19, normalized size = 1.19 \begin{align*} \frac{{\left (b x^{4} + a\right )}^{3}}{12 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(b*x^4+a)^2,x, algorithm="giac")

[Out]

1/12*(b*x^4 + a)^3/b