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

Optimal. Leaf size=80 \[ -\frac {2 a^3 \left (a+b x^3\right )^{3/2}}{9 b^4}+\frac {2 a^2 \left (a+b x^3\right )^{5/2}}{5 b^4}+\frac {2 \left (a+b x^3\right )^{9/2}}{27 b^4}-\frac {2 a \left (a+b x^3\right )^{7/2}}{7 b^4} \]

[Out]

-2/9*a^3*(b*x^3+a)^(3/2)/b^4+2/5*a^2*(b*x^3+a)^(5/2)/b^4-2/7*a*(b*x^3+a)^(7/2)/b^4+2/27*(b*x^3+a)^(9/2)/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 = {266, 43} \[ \frac {2 a^2 \left (a+b x^3\right )^{5/2}}{5 b^4}-\frac {2 a^3 \left (a+b x^3\right )^{3/2}}{9 b^4}+\frac {2 \left (a+b x^3\right )^{9/2}}{27 b^4}-\frac {2 a \left (a+b x^3\right )^{7/2}}{7 b^4} \]

Antiderivative was successfully verified.

[In]

Int[x^11*Sqrt[a + b*x^3],x]

[Out]

(-2*a^3*(a + b*x^3)^(3/2))/(9*b^4) + (2*a^2*(a + b*x^3)^(5/2))/(5*b^4) - (2*a*(a + b*x^3)^(7/2))/(7*b^4) + (2*
(a + b*x^3)^(9/2))/(27*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 x^{11} \sqrt {a+b x^3} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int x^3 \sqrt {a+b x} \, dx,x,x^3\right )\\ &=\frac {1}{3} \operatorname {Subst}\left (\int \left (-\frac {a^3 \sqrt {a+b x}}{b^3}+\frac {3 a^2 (a+b x)^{3/2}}{b^3}-\frac {3 a (a+b x)^{5/2}}{b^3}+\frac {(a+b x)^{7/2}}{b^3}\right ) \, dx,x,x^3\right )\\ &=-\frac {2 a^3 \left (a+b x^3\right )^{3/2}}{9 b^4}+\frac {2 a^2 \left (a+b x^3\right )^{5/2}}{5 b^4}-\frac {2 a \left (a+b x^3\right )^{7/2}}{7 b^4}+\frac {2 \left (a+b x^3\right )^{9/2}}{27 b^4}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 50, normalized size = 0.62 \[ \frac {2 \left (a+b x^3\right )^{3/2} \left (-16 a^3+24 a^2 b x^3-30 a b^2 x^6+35 b^3 x^9\right )}{945 b^4} \]

Antiderivative was successfully verified.

[In]

Integrate[x^11*Sqrt[a + b*x^3],x]

[Out]

(2*(a + b*x^3)^(3/2)*(-16*a^3 + 24*a^2*b*x^3 - 30*a*b^2*x^6 + 35*b^3*x^9))/(945*b^4)

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fricas [A]  time = 0.76, size = 57, normalized size = 0.71 \[ \frac {2 \, {\left (35 \, b^{4} x^{12} + 5 \, a b^{3} x^{9} - 6 \, a^{2} b^{2} x^{6} + 8 \, a^{3} b x^{3} - 16 \, a^{4}\right )} \sqrt {b x^{3} + a}}{945 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11*(b*x^3+a)^(1/2),x, algorithm="fricas")

[Out]

2/945*(35*b^4*x^12 + 5*a*b^3*x^9 - 6*a^2*b^2*x^6 + 8*a^3*b*x^3 - 16*a^4)*sqrt(b*x^3 + a)/b^4

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giac [A]  time = 0.16, size = 57, normalized size = 0.71 \[ \frac {2 \, {\left (35 \, {\left (b x^{3} + a\right )}^{\frac {9}{2}} - 135 \, {\left (b x^{3} + a\right )}^{\frac {7}{2}} a + 189 \, {\left (b x^{3} + a\right )}^{\frac {5}{2}} a^{2} - 105 \, {\left (b x^{3} + a\right )}^{\frac {3}{2}} a^{3}\right )}}{945 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11*(b*x^3+a)^(1/2),x, algorithm="giac")

[Out]

2/945*(35*(b*x^3 + a)^(9/2) - 135*(b*x^3 + a)^(7/2)*a + 189*(b*x^3 + a)^(5/2)*a^2 - 105*(b*x^3 + a)^(3/2)*a^3)
/b^4

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maple [A]  time = 0.01, size = 47, normalized size = 0.59 \[ -\frac {2 \left (b \,x^{3}+a \right )^{\frac {3}{2}} \left (-35 b^{3} x^{9}+30 a \,b^{2} x^{6}-24 a^{2} b \,x^{3}+16 a^{3}\right )}{945 b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^11*(b*x^3+a)^(1/2),x)

[Out]

-2/945*(b*x^3+a)^(3/2)*(-35*b^3*x^9+30*a*b^2*x^6-24*a^2*b*x^3+16*a^3)/b^4

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maxima [A]  time = 1.28, size = 64, normalized size = 0.80 \[ \frac {2 \, {\left (b x^{3} + a\right )}^{\frac {9}{2}}}{27 \, b^{4}} - \frac {2 \, {\left (b x^{3} + a\right )}^{\frac {7}{2}} a}{7 \, b^{4}} + \frac {2 \, {\left (b x^{3} + a\right )}^{\frac {5}{2}} a^{2}}{5 \, b^{4}} - \frac {2 \, {\left (b x^{3} + a\right )}^{\frac {3}{2}} a^{3}}{9 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11*(b*x^3+a)^(1/2),x, algorithm="maxima")

[Out]

2/27*(b*x^3 + a)^(9/2)/b^4 - 2/7*(b*x^3 + a)^(7/2)*a/b^4 + 2/5*(b*x^3 + a)^(5/2)*a^2/b^4 - 2/9*(b*x^3 + a)^(3/
2)*a^3/b^4

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mupad [B]  time = 1.12, size = 90, normalized size = 1.12 \[ \frac {2\,x^{12}\,\sqrt {b\,x^3+a}}{27}-\frac {32\,a^4\,\sqrt {b\,x^3+a}}{945\,b^4}+\frac {2\,a\,x^9\,\sqrt {b\,x^3+a}}{189\,b}+\frac {16\,a^3\,x^3\,\sqrt {b\,x^3+a}}{945\,b^3}-\frac {4\,a^2\,x^6\,\sqrt {b\,x^3+a}}{315\,b^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^11*(a + b*x^3)^(1/2),x)

[Out]

(2*x^12*(a + b*x^3)^(1/2))/27 - (32*a^4*(a + b*x^3)^(1/2))/(945*b^4) + (2*a*x^9*(a + b*x^3)^(1/2))/(189*b) + (
16*a^3*x^3*(a + b*x^3)^(1/2))/(945*b^3) - (4*a^2*x^6*(a + b*x^3)^(1/2))/(315*b^2)

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sympy [A]  time = 4.98, size = 114, normalized size = 1.42 \[ \begin {cases} - \frac {32 a^{4} \sqrt {a + b x^{3}}}{945 b^{4}} + \frac {16 a^{3} x^{3} \sqrt {a + b x^{3}}}{945 b^{3}} - \frac {4 a^{2} x^{6} \sqrt {a + b x^{3}}}{315 b^{2}} + \frac {2 a x^{9} \sqrt {a + b x^{3}}}{189 b} + \frac {2 x^{12} \sqrt {a + b x^{3}}}{27} & \text {for}\: b \neq 0 \\\frac {\sqrt {a} x^{12}}{12} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**11*(b*x**3+a)**(1/2),x)

[Out]

Piecewise((-32*a**4*sqrt(a + b*x**3)/(945*b**4) + 16*a**3*x**3*sqrt(a + b*x**3)/(945*b**3) - 4*a**2*x**6*sqrt(
a + b*x**3)/(315*b**2) + 2*a*x**9*sqrt(a + b*x**3)/(189*b) + 2*x**12*sqrt(a + b*x**3)/27, Ne(b, 0)), (sqrt(a)*
x**12/12, True))

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