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

Optimal. Leaf size=80 \[ \frac{3 a^2 \left (a+b x^3\right )^{7/3}}{7 b^4}-\frac{a^3 \left (a+b x^3\right )^{4/3}}{4 b^4}+\frac{\left (a+b x^3\right )^{13/3}}{13 b^4}-\frac{3 a \left (a+b x^3\right )^{10/3}}{10 b^4} \]

[Out]

-(a^3*(a + b*x^3)^(4/3))/(4*b^4) + (3*a^2*(a + b*x^3)^(7/3))/(7*b^4) - (3*a*(a + b*x^3)^(10/3))/(10*b^4) + (a
+ b*x^3)^(13/3)/(13*b^4)

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Rubi [A]  time = 0.04404, 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} \[ \frac{3 a^2 \left (a+b x^3\right )^{7/3}}{7 b^4}-\frac{a^3 \left (a+b x^3\right )^{4/3}}{4 b^4}+\frac{\left (a+b x^3\right )^{13/3}}{13 b^4}-\frac{3 a \left (a+b x^3\right )^{10/3}}{10 b^4} \]

Antiderivative was successfully verified.

[In]

Int[x^11*(a + b*x^3)^(1/3),x]

[Out]

-(a^3*(a + b*x^3)^(4/3))/(4*b^4) + (3*a^2*(a + b*x^3)^(7/3))/(7*b^4) - (3*a*(a + b*x^3)^(10/3))/(10*b^4) + (a
+ b*x^3)^(13/3)/(13*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 x^{11} \sqrt [3]{a+b x^3} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int x^3 \sqrt [3]{a+b x} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (-\frac{a^3 \sqrt [3]{a+b x}}{b^3}+\frac{3 a^2 (a+b x)^{4/3}}{b^3}-\frac{3 a (a+b x)^{7/3}}{b^3}+\frac{(a+b x)^{10/3}}{b^3}\right ) \, dx,x,x^3\right )\\ &=-\frac{a^3 \left (a+b x^3\right )^{4/3}}{4 b^4}+\frac{3 a^2 \left (a+b x^3\right )^{7/3}}{7 b^4}-\frac{3 a \left (a+b x^3\right )^{10/3}}{10 b^4}+\frac{\left (a+b x^3\right )^{13/3}}{13 b^4}\\ \end{align*}

Mathematica [A]  time = 0.0257377, size = 50, normalized size = 0.62 \[ \frac{\left (a+b x^3\right )^{4/3} \left (108 a^2 b x^3-81 a^3-126 a b^2 x^6+140 b^3 x^9\right )}{1820 b^4} \]

Antiderivative was successfully verified.

[In]

Integrate[x^11*(a + b*x^3)^(1/3),x]

[Out]

((a + b*x^3)^(4/3)*(-81*a^3 + 108*a^2*b*x^3 - 126*a*b^2*x^6 + 140*b^3*x^9))/(1820*b^4)

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Maple [A]  time = 0.005, size = 47, normalized size = 0.6 \begin{align*} -{\frac{-140\,{b}^{3}{x}^{9}+126\,a{b}^{2}{x}^{6}-108\,{a}^{2}b{x}^{3}+81\,{a}^{3}}{1820\,{b}^{4}} \left ( b{x}^{3}+a \right ) ^{{\frac{4}{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]

-1/1820*(b*x^3+a)^(4/3)*(-140*b^3*x^9+126*a*b^2*x^6-108*a^2*b*x^3+81*a^3)/b^4

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Maxima [A]  time = 0.976842, size = 86, normalized size = 1.08 \begin{align*} \frac{{\left (b x^{3} + a\right )}^{\frac{13}{3}}}{13 \, b^{4}} - \frac{3 \,{\left (b x^{3} + a\right )}^{\frac{10}{3}} a}{10 \, b^{4}} + \frac{3 \,{\left (b x^{3} + a\right )}^{\frac{7}{3}} a^{2}}{7 \, b^{4}} - \frac{{\left (b x^{3} + a\right )}^{\frac{4}{3}} a^{3}}{4 \, 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]

1/13*(b*x^3 + a)^(13/3)/b^4 - 3/10*(b*x^3 + a)^(10/3)*a/b^4 + 3/7*(b*x^3 + a)^(7/3)*a^2/b^4 - 1/4*(b*x^3 + a)^
(4/3)*a^3/b^4

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Fricas [A]  time = 1.70515, size = 136, normalized size = 1.7 \begin{align*} \frac{{\left (140 \, b^{4} x^{12} + 14 \, a b^{3} x^{9} - 18 \, a^{2} b^{2} x^{6} + 27 \, a^{3} b x^{3} - 81 \, a^{4}\right )}{\left (b x^{3} + a\right )}^{\frac{1}{3}}}{1820 \, 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]

1/1820*(140*b^4*x^12 + 14*a*b^3*x^9 - 18*a^2*b^2*x^6 + 27*a^3*b*x^3 - 81*a^4)*(b*x^3 + a)^(1/3)/b^4

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Sympy [A]  time = 4.97147, size = 110, normalized size = 1.38 \begin{align*} \begin{cases} - \frac{81 a^{4} \sqrt [3]{a + b x^{3}}}{1820 b^{4}} + \frac{27 a^{3} x^{3} \sqrt [3]{a + b x^{3}}}{1820 b^{3}} - \frac{9 a^{2} x^{6} \sqrt [3]{a + b x^{3}}}{910 b^{2}} + \frac{a x^{9} \sqrt [3]{a + b x^{3}}}{130 b} + \frac{x^{12} \sqrt [3]{a + b x^{3}}}{13} & \text{for}\: b \neq 0 \\\frac{\sqrt [3]{a} x^{12}}{12} & \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**4*(a + b*x**3)**(1/3)/(1820*b**4) + 27*a**3*x**3*(a + b*x**3)**(1/3)/(1820*b**3) - 9*a**2*x*
*6*(a + b*x**3)**(1/3)/(910*b**2) + a*x**9*(a + b*x**3)**(1/3)/(130*b) + x**12*(a + b*x**3)**(1/3)/13, Ne(b, 0
)), (a**(1/3)*x**12/12, True))

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Giac [A]  time = 1.10155, size = 77, normalized size = 0.96 \begin{align*} \frac{140 \,{\left (b x^{3} + a\right )}^{\frac{13}{3}} - 546 \,{\left (b x^{3} + a\right )}^{\frac{10}{3}} a + 780 \,{\left (b x^{3} + a\right )}^{\frac{7}{3}} a^{2} - 455 \,{\left (b x^{3} + a\right )}^{\frac{4}{3}} a^{3}}{1820 \, 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]

1/1820*(140*(b*x^3 + a)^(13/3) - 546*(b*x^3 + a)^(10/3)*a + 780*(b*x^3 + a)^(7/3)*a^2 - 455*(b*x^3 + a)^(4/3)*
a^3)/b^4