3.529 \(\int x^5 (a+b x^3)^{2/3} \, dx\)

Optimal. Leaf size=38 \[ \frac{\left (a+b x^3\right )^{8/3}}{8 b^2}-\frac{a \left (a+b x^3\right )^{5/3}}{5 b^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0228097, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {266, 43} \[ \frac{\left (a+b x^3\right )^{8/3}}{8 b^2}-\frac{a \left (a+b x^3\right )^{5/3}}{5 b^2} \]

Antiderivative was successfully verified.

[In]

Int[x^5*(a + b*x^3)^(2/3),x]

[Out]

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

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int x^5 \left (a+b x^3\right )^{2/3} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int x (a+b x)^{2/3} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (-\frac{a (a+b x)^{2/3}}{b}+\frac{(a+b x)^{5/3}}{b}\right ) \, dx,x,x^3\right )\\ &=-\frac{a \left (a+b x^3\right )^{5/3}}{5 b^2}+\frac{\left (a+b x^3\right )^{8/3}}{8 b^2}\\ \end{align*}

Mathematica [A]  time = 0.0133506, size = 28, normalized size = 0.74 \[ \frac{\left (a+b x^3\right )^{5/3} \left (5 b x^3-3 a\right )}{40 b^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5*(a + b*x^3)^(2/3),x]

[Out]

((a + b*x^3)^(5/3)*(-3*a + 5*b*x^3))/(40*b^2)

________________________________________________________________________________________

Maple [A]  time = 0.003, size = 25, normalized size = 0.7 \begin{align*} -{\frac{-5\,b{x}^{3}+3\,a}{40\,{b}^{2}} \left ( b{x}^{3}+a \right ) ^{{\frac{5}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(b*x^3+a)^(2/3),x)

[Out]

-1/40*(b*x^3+a)^(5/3)*(-5*b*x^3+3*a)/b^2

________________________________________________________________________________________

Maxima [A]  time = 1.0115, size = 41, normalized size = 1.08 \begin{align*} \frac{{\left (b x^{3} + a\right )}^{\frac{8}{3}}}{8 \, b^{2}} - \frac{{\left (b x^{3} + a\right )}^{\frac{5}{3}} a}{5 \, b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(b*x^3+a)^(2/3),x, algorithm="maxima")

[Out]

1/8*(b*x^3 + a)^(8/3)/b^2 - 1/5*(b*x^3 + a)^(5/3)*a/b^2

________________________________________________________________________________________

Fricas [A]  time = 1.67089, size = 81, normalized size = 2.13 \begin{align*} \frac{{\left (5 \, b^{2} x^{6} + 2 \, a b x^{3} - 3 \, a^{2}\right )}{\left (b x^{3} + a\right )}^{\frac{2}{3}}}{40 \, b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(b*x^3+a)^(2/3),x, algorithm="fricas")

[Out]

1/40*(5*b^2*x^6 + 2*a*b*x^3 - 3*a^2)*(b*x^3 + a)^(2/3)/b^2

________________________________________________________________________________________

Sympy [A]  time = 1.7894, size = 63, normalized size = 1.66 \begin{align*} \begin{cases} - \frac{3 a^{2} \left (a + b x^{3}\right )^{\frac{2}{3}}}{40 b^{2}} + \frac{a x^{3} \left (a + b x^{3}\right )^{\frac{2}{3}}}{20 b} + \frac{x^{6} \left (a + b x^{3}\right )^{\frac{2}{3}}}{8} & \text{for}\: b \neq 0 \\\frac{a^{\frac{2}{3}} x^{6}}{6} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*(b*x**3+a)**(2/3),x)

[Out]

Piecewise((-3*a**2*(a + b*x**3)**(2/3)/(40*b**2) + a*x**3*(a + b*x**3)**(2/3)/(20*b) + x**6*(a + b*x**3)**(2/3
)/8, Ne(b, 0)), (a**(2/3)*x**6/6, True))

________________________________________________________________________________________

Giac [A]  time = 1.13585, size = 39, normalized size = 1.03 \begin{align*} \frac{5 \,{\left (b x^{3} + a\right )}^{\frac{8}{3}} - 8 \,{\left (b x^{3} + a\right )}^{\frac{5}{3}} a}{40 \, b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(b*x^3+a)^(2/3),x, algorithm="giac")

[Out]

1/40*(5*(b*x^3 + a)^(8/3) - 8*(b*x^3 + a)^(5/3)*a)/b^2