3.544 \(\int \frac{x^{11}}{\sqrt [3]{a+b x^3}} \, dx\)

Optimal. Leaf size=80 \[ \frac{3 a^2 \left (a+b x^3\right )^{5/3}}{5 b^4}-\frac{a^3 \left (a+b x^3\right )^{2/3}}{2 b^4}+\frac{\left (a+b x^3\right )^{11/3}}{11 b^4}-\frac{3 a \left (a+b x^3\right )^{8/3}}{8 b^4} \]

[Out]

-(a^3*(a + b*x^3)^(2/3))/(2*b^4) + (3*a^2*(a + b*x^3)^(5/3))/(5*b^4) - (3*a*(a + b*x^3)^(8/3))/(8*b^4) + (a +
b*x^3)^(11/3)/(11*b^4)

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Rubi [A]  time = 0.0450665, antiderivative size = 80, 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} \[ \frac{3 a^2 \left (a+b x^3\right )^{5/3}}{5 b^4}-\frac{a^3 \left (a+b x^3\right )^{2/3}}{2 b^4}+\frac{\left (a+b x^3\right )^{11/3}}{11 b^4}-\frac{3 a \left (a+b x^3\right )^{8/3}}{8 b^4} \]

Antiderivative was successfully verified.

[In]

Int[x^11/(a + b*x^3)^(1/3),x]

[Out]

-(a^3*(a + b*x^3)^(2/3))/(2*b^4) + (3*a^2*(a + b*x^3)^(5/3))/(5*b^4) - (3*a*(a + b*x^3)^(8/3))/(8*b^4) + (a +
b*x^3)^(11/3)/(11*b^4)

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 \frac{x^{11}}{\sqrt [3]{a+b x^3}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{x^3}{\sqrt [3]{a+b x}} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (-\frac{a^3}{b^3 \sqrt [3]{a+b x}}+\frac{3 a^2 (a+b x)^{2/3}}{b^3}-\frac{3 a (a+b x)^{5/3}}{b^3}+\frac{(a+b x)^{8/3}}{b^3}\right ) \, dx,x,x^3\right )\\ &=-\frac{a^3 \left (a+b x^3\right )^{2/3}}{2 b^4}+\frac{3 a^2 \left (a+b x^3\right )^{5/3}}{5 b^4}-\frac{3 a \left (a+b x^3\right )^{8/3}}{8 b^4}+\frac{\left (a+b x^3\right )^{11/3}}{11 b^4}\\ \end{align*}

Mathematica [A]  time = 0.0234508, size = 50, normalized size = 0.62 \[ \frac{\left (a+b x^3\right )^{2/3} \left (54 a^2 b x^3-81 a^3-45 a b^2 x^6+40 b^3 x^9\right )}{440 b^4} \]

Antiderivative was successfully verified.

[In]

Integrate[x^11/(a + b*x^3)^(1/3),x]

[Out]

((a + b*x^3)^(2/3)*(-81*a^3 + 54*a^2*b*x^3 - 45*a*b^2*x^6 + 40*b^3*x^9))/(440*b^4)

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Maple [A]  time = 0.005, size = 47, normalized size = 0.6 \begin{align*} -{\frac{-40\,{b}^{3}{x}^{9}+45\,a{b}^{2}{x}^{6}-54\,{a}^{2}b{x}^{3}+81\,{a}^{3}}{440\,{b}^{4}} \left ( b{x}^{3}+a \right ) ^{{\frac{2}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^11/(b*x^3+a)^(1/3),x)

[Out]

-1/440*(b*x^3+a)^(2/3)*(-40*b^3*x^9+45*a*b^2*x^6-54*a^2*b*x^3+81*a^3)/b^4

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Maxima [A]  time = 1.02971, size = 86, normalized size = 1.08 \begin{align*} \frac{{\left (b x^{3} + a\right )}^{\frac{11}{3}}}{11 \, b^{4}} - \frac{3 \,{\left (b x^{3} + a\right )}^{\frac{8}{3}} a}{8 \, b^{4}} + \frac{3 \,{\left (b x^{3} + a\right )}^{\frac{5}{3}} a^{2}}{5 \, b^{4}} - \frac{{\left (b x^{3} + a\right )}^{\frac{2}{3}} a^{3}}{2 \, b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11/(b*x^3+a)^(1/3),x, algorithm="maxima")

[Out]

1/11*(b*x^3 + a)^(11/3)/b^4 - 3/8*(b*x^3 + a)^(8/3)*a/b^4 + 3/5*(b*x^3 + a)^(5/3)*a^2/b^4 - 1/2*(b*x^3 + a)^(2
/3)*a^3/b^4

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Fricas [A]  time = 1.64882, size = 109, normalized size = 1.36 \begin{align*} \frac{{\left (40 \, b^{3} x^{9} - 45 \, a b^{2} x^{6} + 54 \, a^{2} b x^{3} - 81 \, a^{3}\right )}{\left (b x^{3} + a\right )}^{\frac{2}{3}}}{440 \, b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11/(b*x^3+a)^(1/3),x, algorithm="fricas")

[Out]

1/440*(40*b^3*x^9 - 45*a*b^2*x^6 + 54*a^2*b*x^3 - 81*a^3)*(b*x^3 + a)^(2/3)/b^4

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Sympy [A]  time = 3.91322, size = 92, normalized size = 1.15 \begin{align*} \begin{cases} - \frac{81 a^{3} \left (a + b x^{3}\right )^{\frac{2}{3}}}{440 b^{4}} + \frac{27 a^{2} x^{3} \left (a + b x^{3}\right )^{\frac{2}{3}}}{220 b^{3}} - \frac{9 a x^{6} \left (a + b x^{3}\right )^{\frac{2}{3}}}{88 b^{2}} + \frac{x^{9} \left (a + b x^{3}\right )^{\frac{2}{3}}}{11 b} & \text{for}\: b \neq 0 \\\frac{x^{12}}{12 \sqrt [3]{a}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**11/(b*x**3+a)**(1/3),x)

[Out]

Piecewise((-81*a**3*(a + b*x**3)**(2/3)/(440*b**4) + 27*a**2*x**3*(a + b*x**3)**(2/3)/(220*b**3) - 9*a*x**6*(a
+ b*x**3)**(2/3)/(88*b**2) + x**9*(a + b*x**3)**(2/3)/(11*b), Ne(b, 0)), (x**12/(12*a**(1/3)), True))

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Giac [A]  time = 1.14018, size = 77, normalized size = 0.96 \begin{align*} \frac{40 \,{\left (b x^{3} + a\right )}^{\frac{11}{3}} - 165 \,{\left (b x^{3} + a\right )}^{\frac{8}{3}} a + 264 \,{\left (b x^{3} + a\right )}^{\frac{5}{3}} a^{2} - 220 \,{\left (b x^{3} + a\right )}^{\frac{2}{3}} a^{3}}{440 \, b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11/(b*x^3+a)^(1/3),x, algorithm="giac")

[Out]

1/440*(40*(b*x^3 + a)^(11/3) - 165*(b*x^3 + a)^(8/3)*a + 264*(b*x^3 + a)^(5/3)*a^2 - 220*(b*x^3 + a)^(2/3)*a^3
)/b^4