3.387 \(\int x^{11} \left (a+b x^3\right )^{3/2} \, dx\)

Optimal. Leaf size=80 \[ -\frac{2 a^3 \left (a+b x^3\right )^{5/2}}{15 b^4}+\frac{2 a^2 \left (a+b x^3\right )^{7/2}}{7 b^4}+\frac{2 \left (a+b x^3\right )^{11/2}}{33 b^4}-\frac{2 a \left (a+b x^3\right )^{9/2}}{9 b^4} \]

[Out]

(-2*a^3*(a + b*x^3)^(5/2))/(15*b^4) + (2*a^2*(a + b*x^3)^(7/2))/(7*b^4) - (2*a*(
a + b*x^3)^(9/2))/(9*b^4) + (2*(a + b*x^3)^(11/2))/(33*b^4)

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Rubi [A]  time = 0.110949, 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 \[ -\frac{2 a^3 \left (a+b x^3\right )^{5/2}}{15 b^4}+\frac{2 a^2 \left (a+b x^3\right )^{7/2}}{7 b^4}+\frac{2 \left (a+b x^3\right )^{11/2}}{33 b^4}-\frac{2 a \left (a+b x^3\right )^{9/2}}{9 b^4} \]

Antiderivative was successfully verified.

[In]  Int[x^11*(a + b*x^3)^(3/2),x]

[Out]

(-2*a^3*(a + b*x^3)^(5/2))/(15*b^4) + (2*a^2*(a + b*x^3)^(7/2))/(7*b^4) - (2*a*(
a + b*x^3)^(9/2))/(9*b^4) + (2*(a + b*x^3)^(11/2))/(33*b^4)

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Rubi in Sympy [A]  time = 14.8682, size = 75, normalized size = 0.94 \[ - \frac{2 a^{3} \left (a + b x^{3}\right )^{\frac{5}{2}}}{15 b^{4}} + \frac{2 a^{2} \left (a + b x^{3}\right )^{\frac{7}{2}}}{7 b^{4}} - \frac{2 a \left (a + b x^{3}\right )^{\frac{9}{2}}}{9 b^{4}} + \frac{2 \left (a + b x^{3}\right )^{\frac{11}{2}}}{33 b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*a**3*(a + b*x**3)**(5/2)/(15*b**4) + 2*a**2*(a + b*x**3)**(7/2)/(7*b**4) - 2*
a*(a + b*x**3)**(9/2)/(9*b**4) + 2*(a + b*x**3)**(11/2)/(33*b**4)

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Mathematica [A]  time = 0.050446, size = 50, normalized size = 0.62 \[ \frac{2 \left (a+b x^3\right )^{5/2} \left (-16 a^3+40 a^2 b x^3-70 a b^2 x^6+105 b^3 x^9\right )}{3465 b^4} \]

Antiderivative was successfully verified.

[In]  Integrate[x^11*(a + b*x^3)^(3/2),x]

[Out]

(2*(a + b*x^3)^(5/2)*(-16*a^3 + 40*a^2*b*x^3 - 70*a*b^2*x^6 + 105*b^3*x^9))/(346
5*b^4)

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Maple [A]  time = 0.009, size = 47, normalized size = 0.6 \[ -{\frac{-210\,{b}^{3}{x}^{9}+140\,a{b}^{2}{x}^{6}-80\,{a}^{2}b{x}^{3}+32\,{a}^{3}}{3465\,{b}^{4}} \left ( b{x}^{3}+a \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^11*(b*x^3+a)^(3/2),x)

[Out]

-2/3465*(b*x^3+a)^(5/2)*(-105*b^3*x^9+70*a*b^2*x^6-40*a^2*b*x^3+16*a^3)/b^4

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Maxima [A]  time = 1.45739, size = 86, normalized size = 1.08 \[ \frac{2 \,{\left (b x^{3} + a\right )}^{\frac{11}{2}}}{33 \, b^{4}} - \frac{2 \,{\left (b x^{3} + a\right )}^{\frac{9}{2}} a}{9 \, b^{4}} + \frac{2 \,{\left (b x^{3} + a\right )}^{\frac{7}{2}} a^{2}}{7 \, b^{4}} - \frac{2 \,{\left (b x^{3} + a\right )}^{\frac{5}{2}} a^{3}}{15 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/33*(b*x^3 + a)^(11/2)/b^4 - 2/9*(b*x^3 + a)^(9/2)*a/b^4 + 2/7*(b*x^3 + a)^(7/2
)*a^2/b^4 - 2/15*(b*x^3 + a)^(5/2)*a^3/b^4

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Fricas [A]  time = 0.242898, size = 92, normalized size = 1.15 \[ \frac{2 \,{\left (105 \, b^{5} x^{15} + 140 \, a b^{4} x^{12} + 5 \, a^{2} b^{3} x^{9} - 6 \, a^{3} b^{2} x^{6} + 8 \, a^{4} b x^{3} - 16 \, a^{5}\right )} \sqrt{b x^{3} + a}}{3465 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3465*(105*b^5*x^15 + 140*a*b^4*x^12 + 5*a^2*b^3*x^9 - 6*a^3*b^2*x^6 + 8*a^4*b*
x^3 - 16*a^5)*sqrt(b*x^3 + a)/b^4

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Sympy [A]  time = 24.8052, size = 136, normalized size = 1.7 \[ \begin{cases} - \frac{32 a^{5} \sqrt{a + b x^{3}}}{3465 b^{4}} + \frac{16 a^{4} x^{3} \sqrt{a + b x^{3}}}{3465 b^{3}} - \frac{4 a^{3} x^{6} \sqrt{a + b x^{3}}}{1155 b^{2}} + \frac{2 a^{2} x^{9} \sqrt{a + b x^{3}}}{693 b} + \frac{8 a x^{12} \sqrt{a + b x^{3}}}{99} + \frac{2 b x^{15} \sqrt{a + b x^{3}}}{33} & \text{for}\: b \neq 0 \\\frac{a^{\frac{3}{2}} x^{12}}{12} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-32*a**5*sqrt(a + b*x**3)/(3465*b**4) + 16*a**4*x**3*sqrt(a + b*x**3)
/(3465*b**3) - 4*a**3*x**6*sqrt(a + b*x**3)/(1155*b**2) + 2*a**2*x**9*sqrt(a + b
*x**3)/(693*b) + 8*a*x**12*sqrt(a + b*x**3)/99 + 2*b*x**15*sqrt(a + b*x**3)/33,
Ne(b, 0)), (a**(3/2)*x**12/12, True))

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GIAC/XCAS [A]  time = 0.245447, size = 181, normalized size = 2.26 \[ \frac{2 \,{\left (\frac{11 \,{\left (35 \,{\left (b x^{3} + a\right )}^{\frac{9}{2}} - 135 \,{\left (b x^{3} + a\right )}^{\frac{7}{2}} a + 189 \,{\left (b x^{3} + a\right )}^{\frac{5}{2}} a^{2} - 105 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}} a^{3}\right )} a}{b^{3}} + \frac{315 \,{\left (b x^{3} + a\right )}^{\frac{11}{2}} - 1540 \,{\left (b x^{3} + a\right )}^{\frac{9}{2}} a + 2970 \,{\left (b x^{3} + a\right )}^{\frac{7}{2}} a^{2} - 2772 \,{\left (b x^{3} + a\right )}^{\frac{5}{2}} a^{3} + 1155 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}} a^{4}}{b^{3}}\right )}}{10395 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/10395*(11*(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)*a/b^3 + (315*(b*x^3 + a)^(11/2) - 1540*(b*x
^3 + a)^(9/2)*a + 2970*(b*x^3 + a)^(7/2)*a^2 - 2772*(b*x^3 + a)^(5/2)*a^3 + 1155
*(b*x^3 + a)^(3/2)*a^4)/b^3)/b