3.6.44 \(\int \frac {x^{11}}{\sqrt [3]{a+b x^3}} \, dx\) [544]

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

[Out]

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

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Rubi [A]
time = 0.04, antiderivative size = 80, 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 = {272, 45} \begin {gather*} -\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 {\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} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^11/(a + b*x^3)^(1/3),x]

[Out]

-1/2*(a^3*(a + b*x^3)^(2/3))/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 45

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 272

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}}{\sqrt [3]{a+b x^3}} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {x^3}{\sqrt [3]{a+b x}} \, dx,x,x^3\right )\\ &=\frac {1}{3} \text {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*}

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Mathematica [A]
time = 0.03, size = 50, normalized size = 0.62 \begin {gather*} \frac {\left (a+b x^3\right )^{2/3} \left (-81 a^3+54 a^2 b x^3-45 a b^2 x^6+40 b^3 x^9\right )}{440 b^4} \end {gather*}

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 = 0.14, size = 47, normalized size = 0.59

method	result	size
gosper	\(-\frac {\left (b \,x^{3}+a \right )^{\frac {2}{3}} \left (-40 b^{3} x^{9}+45 a \,b^{2} x^{6}-54 a^{2} b \,x^{3}+81 a^{3}\right )}{440 b^{4}}\)	\(47\)
trager	\(-\frac {\left (b \,x^{3}+a \right )^{\frac {2}{3}} \left (-40 b^{3} x^{9}+45 a \,b^{2} x^{6}-54 a^{2} b \,x^{3}+81 a^{3}\right )}{440 b^{4}}\)	\(47\)
risch	\(-\frac {\left (b \,x^{3}+a \right )^{\frac {2}{3}} \left (-40 b^{3} x^{9}+45 a \,b^{2} x^{6}-54 a^{2} b \,x^{3}+81 a^{3}\right )}{440 b^{4}}\)	\(47\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 = 0.29, size = 64, normalized size = 0.80 \begin {gather*} \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 {gather*}

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 = 0.36, size = 46, normalized size = 0.58 \begin {gather*} \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 {gather*}

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 = 0.41, size = 92, normalized size = 1.15 \begin {gather*} \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 {gather*}

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 = 2.70, size = 61, normalized size = 0.76 \begin {gather*} -\frac {{\left (b x^{3} + a\right )}^{\frac {2}{3}} a^{3}}{2 \, b^{4}} + \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}}{440 \, b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 1.11, size = 48, normalized size = 0.60 \begin {gather*} -{\left (b\,x^3+a\right )}^{2/3}\,\left (\frac {81\,a^3}{440\,b^4}-\frac {x^9}{11\,b}+\frac {9\,a\,x^6}{88\,b^2}-\frac {27\,a^2\,x^3}{220\,b^3}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^11/(a + b*x^3)^(1/3),x)

[Out]

-(a + b*x^3)^(2/3)*((81*a^3)/(440*b^4) - x^9/(11*b) + (9*a*x^6)/(88*b^2) - (27*a^2*x^3)/(220*b^3))

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