3.561 \(\int \frac {x^{11}}{(a+b x^3)^{2/3}} \, dx\)

Optimal. Leaf size=78 \[ -\frac {a^3 \sqrt [3]{a+b x^3}}{b^4}+\frac {3 a^2 \left (a+b x^3\right )^{4/3}}{4 b^4}+\frac {\left (a+b x^3\right )^{10/3}}{10 b^4}-\frac {3 a \left (a+b x^3\right )^{7/3}}{7 b^4} \]

[Out]

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

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Rubi [A]  time = 0.04, antiderivative size = 78, normalized size of antiderivative = 1.00, 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} \[ \frac {3 a^2 \left (a+b x^3\right )^{4/3}}{4 b^4}-\frac {a^3 \sqrt [3]{a+b x^3}}{b^4}+\frac {\left (a+b x^3\right )^{10/3}}{10 b^4}-\frac {3 a \left (a+b x^3\right )^{7/3}}{7 b^4} \]

Antiderivative was successfully verified.

[In]

Int[x^11/(a + b*x^3)^(2/3),x]

[Out]

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

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])

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]]

Rubi steps

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

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Mathematica [A]  time = 0.03, size = 50, normalized size = 0.64 \[ \frac {\sqrt [3]{a+b x^3} \left (-81 a^3+27 a^2 b x^3-18 a b^2 x^6+14 b^3 x^9\right )}{140 b^4} \]

Antiderivative was successfully verified.

[In]

Integrate[x^11/(a + b*x^3)^(2/3),x]

[Out]

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

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fricas [A]  time = 0.62, size = 46, normalized size = 0.59 \[ \frac {{\left (14 \, b^{3} x^{9} - 18 \, a b^{2} x^{6} + 27 \, a^{2} b x^{3} - 81 \, a^{3}\right )} {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{140 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11/(b*x^3+a)^(2/3),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.16, size = 61, normalized size = 0.78 \[ -\frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}} a^{3}}{b^{4}} + \frac {14 \, {\left (b x^{3} + a\right )}^{\frac {10}{3}} - 60 \, {\left (b x^{3} + a\right )}^{\frac {7}{3}} a + 105 \, {\left (b x^{3} + a\right )}^{\frac {4}{3}} a^{2}}{140 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11/(b*x^3+a)^(2/3),x, algorithm="giac")

[Out]

-(b*x^3 + a)^(1/3)*a^3/b^4 + 1/140*(14*(b*x^3 + a)^(10/3) - 60*(b*x^3 + a)^(7/3)*a + 105*(b*x^3 + a)^(4/3)*a^2
)/b^4

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maple [A]  time = 0.01, size = 47, normalized size = 0.60 \[ -\frac {\left (b \,x^{3}+a \right )^{\frac {1}{3}} \left (-14 b^{3} x^{9}+18 a \,b^{2} x^{6}-27 a^{2} b \,x^{3}+81 a^{3}\right )}{140 b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^11/(b*x^3+a)^(2/3),x)

[Out]

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

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maxima [A]  time = 1.35, size = 64, normalized size = 0.82 \[ \frac {{\left (b x^{3} + a\right )}^{\frac {10}{3}}}{10 \, b^{4}} - \frac {3 \, {\left (b x^{3} + a\right )}^{\frac {7}{3}} a}{7 \, b^{4}} + \frac {3 \, {\left (b x^{3} + a\right )}^{\frac {4}{3}} a^{2}}{4 \, b^{4}} - \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}} a^{3}}{b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11/(b*x^3+a)^(2/3),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 1.10, size = 48, normalized size = 0.62 \[ -{\left (b\,x^3+a\right )}^{1/3}\,\left (\frac {81\,a^3}{140\,b^4}-\frac {x^9}{10\,b}+\frac {9\,a\,x^6}{70\,b^2}-\frac {27\,a^2\,x^3}{140\,b^3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^11/(a + b*x^3)^(2/3),x)

[Out]

-(a + b*x^3)^(1/3)*((81*a^3)/(140*b^4) - x^9/(10*b) + (9*a*x^6)/(70*b^2) - (27*a^2*x^3)/(140*b^3))

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sympy [A]  time = 3.36, size = 92, normalized size = 1.18 \[ \begin {cases} - \frac {81 a^{3} \sqrt [3]{a + b x^{3}}}{140 b^{4}} + \frac {27 a^{2} x^{3} \sqrt [3]{a + b x^{3}}}{140 b^{3}} - \frac {9 a x^{6} \sqrt [3]{a + b x^{3}}}{70 b^{2}} + \frac {x^{9} \sqrt [3]{a + b x^{3}}}{10 b} & \text {for}\: b \neq 0 \\\frac {x^{12}}{12 a^{\frac {2}{3}}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**11/(b*x**3+a)**(2/3),x)

[Out]

Piecewise((-81*a**3*(a + b*x**3)**(1/3)/(140*b**4) + 27*a**2*x**3*(a + b*x**3)**(1/3)/(140*b**3) - 9*a*x**6*(a
+ b*x**3)**(1/3)/(70*b**2) + x**9*(a + b*x**3)**(1/3)/(10*b), Ne(b, 0)), (x**12/(12*a**(2/3)), True))

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