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

Optimal. Leaf size=43 \[ \frac{3}{5} a^2 b x^5+\frac{a^3 x^2}{2}+\frac{3}{8} a b^2 x^8+\frac{b^3 x^{11}}{11} \]

[Out]

(a^3*x^2)/2 + (3*a^2*b*x^5)/5 + (3*a*b^2*x^8)/8 + (b^3*x^11)/11

________________________________________________________________________________________

Rubi [A]  time = 0.0136601, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {270} \[ \frac{3}{5} a^2 b x^5+\frac{a^3 x^2}{2}+\frac{3}{8} a b^2 x^8+\frac{b^3 x^{11}}{11} \]

Antiderivative was successfully verified.

[In]

Int[x*(a + b*x^3)^3,x]

[Out]

(a^3*x^2)/2 + (3*a^2*b*x^5)/5 + (3*a*b^2*x^8)/8 + (b^3*x^11)/11

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int x \left (a+b x^3\right )^3 \, dx &=\int \left (a^3 x+3 a^2 b x^4+3 a b^2 x^7+b^3 x^{10}\right ) \, dx\\ &=\frac{a^3 x^2}{2}+\frac{3}{5} a^2 b x^5+\frac{3}{8} a b^2 x^8+\frac{b^3 x^{11}}{11}\\ \end{align*}

Mathematica [A]  time = 0.0015159, size = 43, normalized size = 1. \[ \frac{3}{5} a^2 b x^5+\frac{a^3 x^2}{2}+\frac{3}{8} a b^2 x^8+\frac{b^3 x^{11}}{11} \]

Antiderivative was successfully verified.

[In]

Integrate[x*(a + b*x^3)^3,x]

[Out]

(a^3*x^2)/2 + (3*a^2*b*x^5)/5 + (3*a*b^2*x^8)/8 + (b^3*x^11)/11

________________________________________________________________________________________

Maple [A]  time = 0., size = 36, normalized size = 0.8 \begin{align*}{\frac{{x}^{2}{a}^{3}}{2}}+{\frac{3\,{a}^{2}b{x}^{5}}{5}}+{\frac{3\,a{b}^{2}{x}^{8}}{8}}+{\frac{{b}^{3}{x}^{11}}{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^2*a^3+3/5*a^2*b*x^5+3/8*a*b^2*x^8+1/11*b^3*x^11

________________________________________________________________________________________

Maxima [A]  time = 0.994227, size = 47, normalized size = 1.09 \begin{align*} \frac{1}{11} \, b^{3} x^{11} + \frac{3}{8} \, a b^{2} x^{8} + \frac{3}{5} \, a^{2} b x^{5} + \frac{1}{2} \, a^{3} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11*b^3*x^11 + 3/8*a*b^2*x^8 + 3/5*a^2*b*x^5 + 1/2*a^3*x^2

________________________________________________________________________________________

Fricas [A]  time = 1.523, size = 82, normalized size = 1.91 \begin{align*} \frac{1}{11} x^{11} b^{3} + \frac{3}{8} x^{8} b^{2} a + \frac{3}{5} x^{5} 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^3+a)^3,x, algorithm="fricas")

[Out]

1/11*x^11*b^3 + 3/8*x^8*b^2*a + 3/5*x^5*b*a^2 + 1/2*x^2*a^3

________________________________________________________________________________________

Sympy [A]  time = 0.075378, size = 39, normalized size = 0.91 \begin{align*} \frac{a^{3} x^{2}}{2} + \frac{3 a^{2} b x^{5}}{5} + \frac{3 a b^{2} x^{8}}{8} + \frac{b^{3} x^{11}}{11} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**3*x**2/2 + 3*a**2*b*x**5/5 + 3*a*b**2*x**8/8 + b**3*x**11/11

________________________________________________________________________________________

Giac [A]  time = 1.14247, size = 47, normalized size = 1.09 \begin{align*} \frac{1}{11} \, b^{3} x^{11} + \frac{3}{8} \, a b^{2} x^{8} + \frac{3}{5} \, a^{2} b x^{5} + \frac{1}{2} \, a^{3} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11*b^3*x^11 + 3/8*a*b^2*x^8 + 3/5*a^2*b*x^5 + 1/2*a^3*x^2