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

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

[Out]

(a + b*x^4)^4/(16*b)

________________________________________________________________________________________

Rubi [A]  time = 0.0029122, 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} \[ \frac{\left (a+b x^4\right )^4}{16 b} \]

Antiderivative was successfully verified.

[In]

Int[x^3*(a + b*x^4)^3,x]

[Out]

(a + b*x^4)^4/(16*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 )^3 \, dx &=\frac{\left (a+b x^4\right )^4}{16 b}\\ \end{align*}

Mathematica [B]  time = 0.001719, size = 43, normalized size = 2.69 \[ \frac{3}{8} a^2 b x^8+\frac{a^3 x^4}{4}+\frac{1}{4} a b^2 x^{12}+\frac{b^3 x^{16}}{16} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*(a + b*x^4)^3,x]

[Out]

(a^3*x^4)/4 + (3*a^2*b*x^8)/8 + (a*b^2*x^12)/4 + (b^3*x^16)/16

________________________________________________________________________________________

Maple [B]  time = 0., size = 36, normalized size = 2.3 \begin{align*}{\frac{{b}^{3}{x}^{16}}{16}}+{\frac{a{b}^{2}{x}^{12}}{4}}+{\frac{3\,{a}^{2}b{x}^{8}}{8}}+{\frac{{a}^{3}{x}^{4}}{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*b^3*x^16+1/4*a*b^2*x^12+3/8*a^2*b*x^8+1/4*a^3*x^4

________________________________________________________________________________________

Maxima [A]  time = 0.970116, size = 19, normalized size = 1.19 \begin{align*} \frac{{\left (b x^{4} + a\right )}^{4}}{16 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*(b*x^4 + a)^4/b

________________________________________________________________________________________

Fricas [B]  time = 1.22993, size = 84, normalized size = 5.25 \begin{align*} \frac{1}{16} x^{16} b^{3} + \frac{1}{4} x^{12} b^{2} a + \frac{3}{8} x^{8} b a^{2} + \frac{1}{4} x^{4} a^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*x^16*b^3 + 1/4*x^12*b^2*a + 3/8*x^8*b*a^2 + 1/4*x^4*a^3

________________________________________________________________________________________

Sympy [B]  time = 0.068845, size = 37, normalized size = 2.31 \begin{align*} \frac{a^{3} x^{4}}{4} + \frac{3 a^{2} b x^{8}}{8} + \frac{a b^{2} x^{12}}{4} + \frac{b^{3} x^{16}}{16} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**3*x**4/4 + 3*a**2*b*x**8/8 + a*b**2*x**12/4 + b**3*x**16/16

________________________________________________________________________________________

Giac [A]  time = 1.11926, size = 19, normalized size = 1.19 \begin{align*} \frac{{\left (b x^{4} + a\right )}^{4}}{16 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*(b*x^4 + a)^4/b