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*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)

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Rubi [A]  time = 0.109074, 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 \[ -\frac{2 a^3 \left (a+b x^3\right )^{3/2}}{9 b^4}+\frac{2 a^2 \left (a+b x^3\right )^{5/2}}{5 b^4}+\frac{2 \left (a+b x^3\right )^{9/2}}{27 b^4}-\frac{2 a \left (a+b x^3\right )^{7/2}}{7 b^4} \]

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)

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Rubi in Sympy [A]  time = 14.7656, size = 75, normalized size = 0.94 \[ - \frac{2 a^{3} \left (a + b x^{3}\right )^{\frac{3}{2}}}{9 b^{4}} + \frac{2 a^{2} \left (a + b x^{3}\right )^{\frac{5}{2}}}{5 b^{4}} - \frac{2 a \left (a + b x^{3}\right )^{\frac{7}{2}}}{7 b^{4}} + \frac{2 \left (a + b x^{3}\right )^{\frac{9}{2}}}{27 b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**11*(b*x**3+a)**(1/2),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)

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Mathematica [A]  time = 0.0299107, size = 61, normalized size = 0.76 \[ \frac{2 \sqrt{a+b x^3} \left (-16 a^4+8 a^3 b x^3-6 a^2 b^2 x^6+5 a b^3 x^9+35 b^4 x^{12}\right )}{945 b^4} \]

Antiderivative was successfully verified.

[In]  Integrate[x^11*Sqrt[a + b*x^3],x]

[Out]

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

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Maple [A]  time = 0.01, size = 47, normalized size = 0.6 \[ -{\frac{-70\,{b}^{3}{x}^{9}+60\,a{b}^{2}{x}^{6}-48\,{a}^{2}b{x}^{3}+32\,{a}^{3}}{945\,{b}^{4}} \left ( b{x}^{3}+a \right ) ^{{\frac{3}{2}}}} \]

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 = 1.43903, size = 86, normalized size = 1.08 \[ \frac{2 \,{\left (b x^{3} + a\right )}^{\frac{9}{2}}}{27 \, b^{4}} - \frac{2 \,{\left (b x^{3} + a\right )}^{\frac{7}{2}} a}{7 \, b^{4}} + \frac{2 \,{\left (b x^{3} + a\right )}^{\frac{5}{2}} a^{2}}{5 \, b^{4}} - \frac{2 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}} a^{3}}{9 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x^3 + a)*x^11,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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Fricas [A]  time = 0.214362, size = 77, normalized size = 0.96 \[ \frac{2 \,{\left (35 \, b^{4} x^{12} + 5 \, a b^{3} x^{9} - 6 \, a^{2} b^{2} x^{6} + 8 \, a^{3} b x^{3} - 16 \, a^{4}\right )} \sqrt{b x^{3} + a}}{945 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x^3 + a)*x^11,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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Sympy [A]  time = 10.5404, 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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GIAC/XCAS [A]  time = 0.229034, size = 77, normalized size = 0.96 \[ \frac{2 \,{\left (35 \,{\left (b x^{3} + a\right )}^{\frac{9}{2}} - 135 \,{\left (b x^{3} + a\right )}^{\frac{7}{2}} a + 189 \,{\left (b x^{3} + a\right )}^{\frac{5}{2}} a^{2} - 105 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}} a^{3}\right )}}{945 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x^3 + a)*x^11,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