3.678 \(\int x^5 \left (a+b x^2\right )^{2/3} \, dx\)

Optimal. Leaf size=59 \[ \frac{3 a^2 \left (a+b x^2\right )^{5/3}}{10 b^3}+\frac{3 \left (a+b x^2\right )^{11/3}}{22 b^3}-\frac{3 a \left (a+b x^2\right )^{8/3}}{8 b^3} \]

[Out]

(3*a^2*(a + b*x^2)^(5/3))/(10*b^3) - (3*a*(a + b*x^2)^(8/3))/(8*b^3) + (3*(a + b
*x^2)^(11/3))/(22*b^3)

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Rubi [A]  time = 0.0957364, antiderivative size = 59, 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 \[ \frac{3 a^2 \left (a+b x^2\right )^{5/3}}{10 b^3}+\frac{3 \left (a+b x^2\right )^{11/3}}{22 b^3}-\frac{3 a \left (a+b x^2\right )^{8/3}}{8 b^3} \]

Antiderivative was successfully verified.

[In]  Int[x^5*(a + b*x^2)^(2/3),x]

[Out]

(3*a^2*(a + b*x^2)^(5/3))/(10*b^3) - (3*a*(a + b*x^2)^(8/3))/(8*b^3) + (3*(a + b
*x^2)^(11/3))/(22*b^3)

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Rubi in Sympy [A]  time = 11.6083, size = 54, normalized size = 0.92 \[ \frac{3 a^{2} \left (a + b x^{2}\right )^{\frac{5}{3}}}{10 b^{3}} - \frac{3 a \left (a + b x^{2}\right )^{\frac{8}{3}}}{8 b^{3}} + \frac{3 \left (a + b x^{2}\right )^{\frac{11}{3}}}{22 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5*(b*x**2+a)**(2/3),x)

[Out]

3*a**2*(a + b*x**2)**(5/3)/(10*b**3) - 3*a*(a + b*x**2)**(8/3)/(8*b**3) + 3*(a +
b*x**2)**(11/3)/(22*b**3)

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Mathematica [A]  time = 0.0236759, size = 50, normalized size = 0.85 \[ \frac{3 \left (a+b x^2\right )^{2/3} \left (9 a^3-6 a^2 b x^2+5 a b^2 x^4+20 b^3 x^6\right )}{440 b^3} \]

Antiderivative was successfully verified.

[In]  Integrate[x^5*(a + b*x^2)^(2/3),x]

[Out]

(3*(a + b*x^2)^(2/3)*(9*a^3 - 6*a^2*b*x^2 + 5*a*b^2*x^4 + 20*b^3*x^6))/(440*b^3)

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Maple [A]  time = 0.007, size = 36, normalized size = 0.6 \[{\frac{60\,{b}^{2}{x}^{4}-45\,ab{x}^{2}+27\,{a}^{2}}{440\,{b}^{3}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5*(b*x^2+a)^(2/3),x)

[Out]

3/440*(b*x^2+a)^(5/3)*(20*b^2*x^4-15*a*b*x^2+9*a^2)/b^3

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Maxima [A]  time = 1.3386, size = 63, normalized size = 1.07 \[ \frac{3 \,{\left (b x^{2} + a\right )}^{\frac{11}{3}}}{22 \, b^{3}} - \frac{3 \,{\left (b x^{2} + a\right )}^{\frac{8}{3}} a}{8 \, b^{3}} + \frac{3 \,{\left (b x^{2} + a\right )}^{\frac{5}{3}} a^{2}}{10 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(2/3)*x^5,x, algorithm="maxima")

[Out]

3/22*(b*x^2 + a)^(11/3)/b^3 - 3/8*(b*x^2 + a)^(8/3)*a/b^3 + 3/10*(b*x^2 + a)^(5/
3)*a^2/b^3

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Fricas [A]  time = 0.210268, size = 62, normalized size = 1.05 \[ \frac{3 \,{\left (20 \, b^{3} x^{6} + 5 \, a b^{2} x^{4} - 6 \, a^{2} b x^{2} + 9 \, a^{3}\right )}{\left (b x^{2} + a\right )}^{\frac{2}{3}}}{440 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(2/3)*x^5,x, algorithm="fricas")

[Out]

3/440*(20*b^3*x^6 + 5*a*b^2*x^4 - 6*a^2*b*x^2 + 9*a^3)*(b*x^2 + a)^(2/3)/b^3

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Sympy [A]  time = 7.0286, size = 700, normalized size = 11.86 \[ \frac{27 a^{\frac{35}{3}} \left (1 + \frac{b x^{2}}{a}\right )^{\frac{2}{3}}}{440 a^{8} b^{3} + 1320 a^{7} b^{4} x^{2} + 1320 a^{6} b^{5} x^{4} + 440 a^{5} b^{6} x^{6}} - \frac{27 a^{\frac{35}{3}}}{440 a^{8} b^{3} + 1320 a^{7} b^{4} x^{2} + 1320 a^{6} b^{5} x^{4} + 440 a^{5} b^{6} x^{6}} + \frac{63 a^{\frac{32}{3}} b x^{2} \left (1 + \frac{b x^{2}}{a}\right )^{\frac{2}{3}}}{440 a^{8} b^{3} + 1320 a^{7} b^{4} x^{2} + 1320 a^{6} b^{5} x^{4} + 440 a^{5} b^{6} x^{6}} - \frac{81 a^{\frac{32}{3}} b x^{2}}{440 a^{8} b^{3} + 1320 a^{7} b^{4} x^{2} + 1320 a^{6} b^{5} x^{4} + 440 a^{5} b^{6} x^{6}} + \frac{42 a^{\frac{29}{3}} b^{2} x^{4} \left (1 + \frac{b x^{2}}{a}\right )^{\frac{2}{3}}}{440 a^{8} b^{3} + 1320 a^{7} b^{4} x^{2} + 1320 a^{6} b^{5} x^{4} + 440 a^{5} b^{6} x^{6}} - \frac{81 a^{\frac{29}{3}} b^{2} x^{4}}{440 a^{8} b^{3} + 1320 a^{7} b^{4} x^{2} + 1320 a^{6} b^{5} x^{4} + 440 a^{5} b^{6} x^{6}} + \frac{78 a^{\frac{26}{3}} b^{3} x^{6} \left (1 + \frac{b x^{2}}{a}\right )^{\frac{2}{3}}}{440 a^{8} b^{3} + 1320 a^{7} b^{4} x^{2} + 1320 a^{6} b^{5} x^{4} + 440 a^{5} b^{6} x^{6}} - \frac{27 a^{\frac{26}{3}} b^{3} x^{6}}{440 a^{8} b^{3} + 1320 a^{7} b^{4} x^{2} + 1320 a^{6} b^{5} x^{4} + 440 a^{5} b^{6} x^{6}} + \frac{207 a^{\frac{23}{3}} b^{4} x^{8} \left (1 + \frac{b x^{2}}{a}\right )^{\frac{2}{3}}}{440 a^{8} b^{3} + 1320 a^{7} b^{4} x^{2} + 1320 a^{6} b^{5} x^{4} + 440 a^{5} b^{6} x^{6}} + \frac{195 a^{\frac{20}{3}} b^{5} x^{10} \left (1 + \frac{b x^{2}}{a}\right )^{\frac{2}{3}}}{440 a^{8} b^{3} + 1320 a^{7} b^{4} x^{2} + 1320 a^{6} b^{5} x^{4} + 440 a^{5} b^{6} x^{6}} + \frac{60 a^{\frac{17}{3}} b^{6} x^{12} \left (1 + \frac{b x^{2}}{a}\right )^{\frac{2}{3}}}{440 a^{8} b^{3} + 1320 a^{7} b^{4} x^{2} + 1320 a^{6} b^{5} x^{4} + 440 a^{5} b^{6} x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5*(b*x**2+a)**(2/3),x)

[Out]

27*a**(35/3)*(1 + b*x**2/a)**(2/3)/(440*a**8*b**3 + 1320*a**7*b**4*x**2 + 1320*a
**6*b**5*x**4 + 440*a**5*b**6*x**6) - 27*a**(35/3)/(440*a**8*b**3 + 1320*a**7*b*
*4*x**2 + 1320*a**6*b**5*x**4 + 440*a**5*b**6*x**6) + 63*a**(32/3)*b*x**2*(1 + b
*x**2/a)**(2/3)/(440*a**8*b**3 + 1320*a**7*b**4*x**2 + 1320*a**6*b**5*x**4 + 440
*a**5*b**6*x**6) - 81*a**(32/3)*b*x**2/(440*a**8*b**3 + 1320*a**7*b**4*x**2 + 13
20*a**6*b**5*x**4 + 440*a**5*b**6*x**6) + 42*a**(29/3)*b**2*x**4*(1 + b*x**2/a)*
*(2/3)/(440*a**8*b**3 + 1320*a**7*b**4*x**2 + 1320*a**6*b**5*x**4 + 440*a**5*b**
6*x**6) - 81*a**(29/3)*b**2*x**4/(440*a**8*b**3 + 1320*a**7*b**4*x**2 + 1320*a**
6*b**5*x**4 + 440*a**5*b**6*x**6) + 78*a**(26/3)*b**3*x**6*(1 + b*x**2/a)**(2/3)
/(440*a**8*b**3 + 1320*a**7*b**4*x**2 + 1320*a**6*b**5*x**4 + 440*a**5*b**6*x**6
) - 27*a**(26/3)*b**3*x**6/(440*a**8*b**3 + 1320*a**7*b**4*x**2 + 1320*a**6*b**5
*x**4 + 440*a**5*b**6*x**6) + 207*a**(23/3)*b**4*x**8*(1 + b*x**2/a)**(2/3)/(440
*a**8*b**3 + 1320*a**7*b**4*x**2 + 1320*a**6*b**5*x**4 + 440*a**5*b**6*x**6) + 1
95*a**(20/3)*b**5*x**10*(1 + b*x**2/a)**(2/3)/(440*a**8*b**3 + 1320*a**7*b**4*x*
*2 + 1320*a**6*b**5*x**4 + 440*a**5*b**6*x**6) + 60*a**(17/3)*b**6*x**12*(1 + b*
x**2/a)**(2/3)/(440*a**8*b**3 + 1320*a**7*b**4*x**2 + 1320*a**6*b**5*x**4 + 440*
a**5*b**6*x**6)

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GIAC/XCAS [A]  time = 0.214432, size = 58, normalized size = 0.98 \[ \frac{3 \,{\left (20 \,{\left (b x^{2} + a\right )}^{\frac{11}{3}} - 55 \,{\left (b x^{2} + a\right )}^{\frac{8}{3}} a + 44 \,{\left (b x^{2} + a\right )}^{\frac{5}{3}} a^{2}\right )}}{440 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(2/3)*x^5,x, algorithm="giac")

[Out]

3/440*(20*(b*x^2 + a)^(11/3) - 55*(b*x^2 + a)^(8/3)*a + 44*(b*x^2 + a)^(5/3)*a^2
)/b^3