3.6.10 \(\int x^{11} \sqrt [3]{a+b x^3} \, dx\) [510]

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

[Out]

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

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Rubi [A]
time = 0.03, 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 )^{4/3}}{4 b^4}+\frac {3 a^2 \left (a+b x^3\right )^{7/3}}{7 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} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^11*(a + b*x^3)^(1/3),x]

[Out]

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

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Mathematica [A]
time = 0.03, size = 50, normalized size = 0.62 \begin {gather*} \frac {\left (a+b x^3\right )^{4/3} \left (-81 a^3+108 a^2 b x^3-126 a b^2 x^6+140 b^3 x^9\right )}{1820 b^4} \end {gather*}

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 = 0.14, size = 47, normalized size = 0.59

method	result	size
gosper	\(-\frac {\left (b \,x^{3}+a \right )^{\frac {4}{3}} \left (-140 b^{3} x^{9}+126 a \,b^{2} x^{6}-108 a^{2} b \,x^{3}+81 a^{3}\right )}{1820 b^{4}}\)	\(47\)
trager	\(-\frac {\left (-140 b^{4} x^{12}-14 a \,b^{3} x^{9}+18 a^{2} b^{2} x^{6}-27 a^{3} b \,x^{3}+81 a^{4}\right ) \left (b \,x^{3}+a \right )^{\frac {1}{3}}}{1820 b^{4}}\)	\(58\)
risch	\(-\frac {\left (-140 b^{4} x^{12}-14 a \,b^{3} x^{9}+18 a^{2} b^{2} x^{6}-27 a^{3} b \,x^{3}+81 a^{4}\right ) \left (b \,x^{3}+a \right )^{\frac {1}{3}}}{1820 b^{4}}\)	\(58\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1820*(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 = 0.29, size = 64, normalized size = 0.80 \begin {gather*} \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 {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]

1/13*(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 = 0.35, size = 57, normalized size = 0.71 \begin {gather*} \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 {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]

1/1820*(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 = 0.38, size = 110, normalized size = 1.38 \begin {gather*} \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 {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**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 = 1.42, size = 57, normalized size = 0.71 \begin {gather*} \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 {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]

1/1820*(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

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Mupad [B]
time = 1.11, size = 55, normalized size = 0.69 \begin {gather*} {\left (b\,x^3+a\right )}^{1/3}\,\left (\frac {x^{12}}{13}-\frac {81\,a^4}{1820\,b^4}+\frac {a\,x^9}{130\,b}+\frac {27\,a^3\,x^3}{1820\,b^3}-\frac {9\,a^2\,x^6}{910\,b^2}\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)^(1/3)*(x^12/13 - (81*a^4)/(1820*b^4) + (a*x^9)/(130*b) + (27*a^3*x^3)/(1820*b^3) - (9*a^2*x^6)/(91
0*b^2))

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