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

Optimal. Leaf size=16 \[ \frac{\left (a+b x^2\right )^4}{8 b} \]

[Out]

(a + b*x^2)^4/(8*b)

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Rubi [A]  time = 0.0023568, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {261} \[ \frac{\left (a+b x^2\right )^4}{8 b} \]

Antiderivative was successfully verified.

[In]

Int[x*(a + b*x^2)^3,x]

[Out]

(a + b*x^2)^4/(8*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 \left (a+b x^2\right )^3 \, dx &=\frac{\left (a+b x^2\right )^4}{8 b}\\ \end{align*}

Mathematica [A]  time = 0.0018186, size = 16, normalized size = 1. \[ \frac{\left (a+b x^2\right )^4}{8 b} \]

Antiderivative was successfully verified.

[In]

Integrate[x*(a + b*x^2)^3,x]

[Out]

(a + b*x^2)^4/(8*b)

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Maple [B]  time = 0.001, size = 36, normalized size = 2.3 \begin{align*}{\frac{{b}^{3}{x}^{8}}{8}}+{\frac{a{b}^{2}{x}^{6}}{2}}+{\frac{3\,{a}^{2}b{x}^{4}}{4}}+{\frac{{a}^{3}{x}^{2}}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*b^3*x^8+1/2*a*b^2*x^6+3/4*a^2*b*x^4+1/2*a^3*x^2

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Maxima [A]  time = 1.78371, size = 19, normalized size = 1.19 \begin{align*} \frac{{\left (b x^{2} + a\right )}^{4}}{8 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(b*x^2 + a)^4/b

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Fricas [B]  time = 1.81334, size = 80, normalized size = 5. \begin{align*} \frac{1}{8} x^{8} b^{3} + \frac{1}{2} x^{6} b^{2} a + \frac{3}{4} x^{4} b a^{2} + \frac{1}{2} x^{2} a^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*x^8*b^3 + 1/2*x^6*b^2*a + 3/4*x^4*b*a^2 + 1/2*x^2*a^3

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Sympy [B]  time = 0.06199, size = 37, normalized size = 2.31 \begin{align*} \frac{a^{3} x^{2}}{2} + \frac{3 a^{2} b x^{4}}{4} + \frac{a b^{2} x^{6}}{2} + \frac{b^{3} x^{8}}{8} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**3*x**2/2 + 3*a**2*b*x**4/4 + a*b**2*x**6/2 + b**3*x**8/8

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Giac [A]  time = 2.70175, size = 19, normalized size = 1.19 \begin{align*} \frac{{\left (b x^{2} + a\right )}^{4}}{8 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(b*x^2 + a)^4/b