3.379 \(\int x^2 (a+b x)^{2/3} \, dx\)

Optimal. Leaf size=53 \[ \frac{3 a^2 (a+b x)^{5/3}}{5 b^3}+\frac{3 (a+b x)^{11/3}}{11 b^3}-\frac{3 a (a+b x)^{8/3}}{4 b^3} \]

[Out]

(3*a^2*(a + b*x)^(5/3))/(5*b^3) - (3*a*(a + b*x)^(8/3))/(4*b^3) + (3*(a + b*x)^(
11/3))/(11*b^3)

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Rubi [A]  time = 0.0393378, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{3 a^2 (a+b x)^{5/3}}{5 b^3}+\frac{3 (a+b x)^{11/3}}{11 b^3}-\frac{3 a (a+b x)^{8/3}}{4 b^3} \]

Antiderivative was successfully verified.

[In]  Int[x^2*(a + b*x)^(2/3),x]

[Out]

(3*a^2*(a + b*x)^(5/3))/(5*b^3) - (3*a*(a + b*x)^(8/3))/(4*b^3) + (3*(a + b*x)^(
11/3))/(11*b^3)

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Rubi in Sympy [A]  time = 7.93304, size = 49, normalized size = 0.92 \[ \frac{3 a^{2} \left (a + b x\right )^{\frac{5}{3}}}{5 b^{3}} - \frac{3 a \left (a + b x\right )^{\frac{8}{3}}}{4 b^{3}} + \frac{3 \left (a + b x\right )^{\frac{11}{3}}}{11 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*a**2*(a + b*x)**(5/3)/(5*b**3) - 3*a*(a + b*x)**(8/3)/(4*b**3) + 3*(a + b*x)**
(11/3)/(11*b**3)

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Mathematica [A]  time = 0.0175047, size = 46, normalized size = 0.87 \[ \frac{3 (a+b x)^{2/3} \left (9 a^3-6 a^2 b x+5 a b^2 x^2+20 b^3 x^3\right )}{220 b^3} \]

Antiderivative was successfully verified.

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

[Out]

(3*(a + b*x)^(2/3)*(9*a^3 - 6*a^2*b*x + 5*a*b^2*x^2 + 20*b^3*x^3))/(220*b^3)

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Maple [A]  time = 0.007, size = 32, normalized size = 0.6 \[{\frac{60\,{b}^{2}{x}^{2}-45\,abx+27\,{a}^{2}}{220\,{b}^{3}} \left ( bx+a \right ) ^{{\frac{5}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(b*x+a)^(2/3),x)

[Out]

3/220*(b*x+a)^(5/3)*(20*b^2*x^2-15*a*b*x+9*a^2)/b^3

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Maxima [A]  time = 1.34847, size = 55, normalized size = 1.04 \[ \frac{3 \,{\left (b x + a\right )}^{\frac{11}{3}}}{11 \, b^{3}} - \frac{3 \,{\left (b x + a\right )}^{\frac{8}{3}} a}{4 \, b^{3}} + \frac{3 \,{\left (b x + a\right )}^{\frac{5}{3}} a^{2}}{5 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/11*(b*x + a)^(11/3)/b^3 - 3/4*(b*x + a)^(8/3)*a/b^3 + 3/5*(b*x + a)^(5/3)*a^2/
b^3

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Fricas [A]  time = 0.207486, size = 57, normalized size = 1.08 \[ \frac{3 \,{\left (20 \, b^{3} x^{3} + 5 \, a b^{2} x^{2} - 6 \, a^{2} b x + 9 \, a^{3}\right )}{\left (b x + a\right )}^{\frac{2}{3}}}{220 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/220*(20*b^3*x^3 + 5*a*b^2*x^2 - 6*a^2*b*x + 9*a^3)*(b*x + a)^(2/3)/b^3

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Sympy [A]  time = 5.92255, size = 666, normalized size = 12.57 \[ \frac{27 a^{\frac{35}{3}} \left (1 + \frac{b x}{a}\right )^{\frac{2}{3}}}{220 a^{8} b^{3} + 660 a^{7} b^{4} x + 660 a^{6} b^{5} x^{2} + 220 a^{5} b^{6} x^{3}} - \frac{27 a^{\frac{35}{3}}}{220 a^{8} b^{3} + 660 a^{7} b^{4} x + 660 a^{6} b^{5} x^{2} + 220 a^{5} b^{6} x^{3}} + \frac{63 a^{\frac{32}{3}} b x \left (1 + \frac{b x}{a}\right )^{\frac{2}{3}}}{220 a^{8} b^{3} + 660 a^{7} b^{4} x + 660 a^{6} b^{5} x^{2} + 220 a^{5} b^{6} x^{3}} - \frac{81 a^{\frac{32}{3}} b x}{220 a^{8} b^{3} + 660 a^{7} b^{4} x + 660 a^{6} b^{5} x^{2} + 220 a^{5} b^{6} x^{3}} + \frac{42 a^{\frac{29}{3}} b^{2} x^{2} \left (1 + \frac{b x}{a}\right )^{\frac{2}{3}}}{220 a^{8} b^{3} + 660 a^{7} b^{4} x + 660 a^{6} b^{5} x^{2} + 220 a^{5} b^{6} x^{3}} - \frac{81 a^{\frac{29}{3}} b^{2} x^{2}}{220 a^{8} b^{3} + 660 a^{7} b^{4} x + 660 a^{6} b^{5} x^{2} + 220 a^{5} b^{6} x^{3}} + \frac{78 a^{\frac{26}{3}} b^{3} x^{3} \left (1 + \frac{b x}{a}\right )^{\frac{2}{3}}}{220 a^{8} b^{3} + 660 a^{7} b^{4} x + 660 a^{6} b^{5} x^{2} + 220 a^{5} b^{6} x^{3}} - \frac{27 a^{\frac{26}{3}} b^{3} x^{3}}{220 a^{8} b^{3} + 660 a^{7} b^{4} x + 660 a^{6} b^{5} x^{2} + 220 a^{5} b^{6} x^{3}} + \frac{207 a^{\frac{23}{3}} b^{4} x^{4} \left (1 + \frac{b x}{a}\right )^{\frac{2}{3}}}{220 a^{8} b^{3} + 660 a^{7} b^{4} x + 660 a^{6} b^{5} x^{2} + 220 a^{5} b^{6} x^{3}} + \frac{195 a^{\frac{20}{3}} b^{5} x^{5} \left (1 + \frac{b x}{a}\right )^{\frac{2}{3}}}{220 a^{8} b^{3} + 660 a^{7} b^{4} x + 660 a^{6} b^{5} x^{2} + 220 a^{5} b^{6} x^{3}} + \frac{60 a^{\frac{17}{3}} b^{6} x^{6} \left (1 + \frac{b x}{a}\right )^{\frac{2}{3}}}{220 a^{8} b^{3} + 660 a^{7} b^{4} x + 660 a^{6} b^{5} x^{2} + 220 a^{5} b^{6} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

27*a**(35/3)*(1 + b*x/a)**(2/3)/(220*a**8*b**3 + 660*a**7*b**4*x + 660*a**6*b**5
*x**2 + 220*a**5*b**6*x**3) - 27*a**(35/3)/(220*a**8*b**3 + 660*a**7*b**4*x + 66
0*a**6*b**5*x**2 + 220*a**5*b**6*x**3) + 63*a**(32/3)*b*x*(1 + b*x/a)**(2/3)/(22
0*a**8*b**3 + 660*a**7*b**4*x + 660*a**6*b**5*x**2 + 220*a**5*b**6*x**3) - 81*a*
*(32/3)*b*x/(220*a**8*b**3 + 660*a**7*b**4*x + 660*a**6*b**5*x**2 + 220*a**5*b**
6*x**3) + 42*a**(29/3)*b**2*x**2*(1 + b*x/a)**(2/3)/(220*a**8*b**3 + 660*a**7*b*
*4*x + 660*a**6*b**5*x**2 + 220*a**5*b**6*x**3) - 81*a**(29/3)*b**2*x**2/(220*a*
*8*b**3 + 660*a**7*b**4*x + 660*a**6*b**5*x**2 + 220*a**5*b**6*x**3) + 78*a**(26
/3)*b**3*x**3*(1 + b*x/a)**(2/3)/(220*a**8*b**3 + 660*a**7*b**4*x + 660*a**6*b**
5*x**2 + 220*a**5*b**6*x**3) - 27*a**(26/3)*b**3*x**3/(220*a**8*b**3 + 660*a**7*
b**4*x + 660*a**6*b**5*x**2 + 220*a**5*b**6*x**3) + 207*a**(23/3)*b**4*x**4*(1 +
b*x/a)**(2/3)/(220*a**8*b**3 + 660*a**7*b**4*x + 660*a**6*b**5*x**2 + 220*a**5*
b**6*x**3) + 195*a**(20/3)*b**5*x**5*(1 + b*x/a)**(2/3)/(220*a**8*b**3 + 660*a**
7*b**4*x + 660*a**6*b**5*x**2 + 220*a**5*b**6*x**3) + 60*a**(17/3)*b**6*x**6*(1
+ b*x/a)**(2/3)/(220*a**8*b**3 + 660*a**7*b**4*x + 660*a**6*b**5*x**2 + 220*a**5
*b**6*x**3)

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GIAC/XCAS [A]  time = 0.223572, size = 62, normalized size = 1.17 \[ \frac{3 \,{\left (20 \,{\left (b x + a\right )}^{\frac{11}{3}} b^{20} - 55 \,{\left (b x + a\right )}^{\frac{8}{3}} a b^{20} + 44 \,{\left (b x + a\right )}^{\frac{5}{3}} a^{2} b^{20}\right )}}{220 \, b^{23}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/220*(20*(b*x + a)^(11/3)*b^20 - 55*(b*x + a)^(8/3)*a*b^20 + 44*(b*x + a)^(5/3)
*a^2*b^20)/b^23