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

Optimal. Leaf size=36 \[ \frac{3}{5} a^2 x^{5/3}+\frac{3}{4} a b x^{8/3}+\frac{3}{11} b^2 x^{11/3} \]

[Out]

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

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Rubi [A]  time = 0.0220964, antiderivative size = 36, 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 \[ \frac{3}{5} a^2 x^{5/3}+\frac{3}{4} a b x^{8/3}+\frac{3}{11} b^2 x^{11/3} \]

Antiderivative was successfully verified.

[In]  Int[x^(2/3)*(a + b*x)^2,x]

[Out]

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

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Rubi in Sympy [A]  time = 3.93903, size = 34, normalized size = 0.94 \[ \frac{3 a^{2} x^{\frac{5}{3}}}{5} + \frac{3 a b x^{\frac{8}{3}}}{4} + \frac{3 b^{2} x^{\frac{11}{3}}}{11} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.00943086, size = 28, normalized size = 0.78 \[ \frac{3}{220} x^{5/3} \left (44 a^2+55 a b x+20 b^2 x^2\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.007, size = 25, normalized size = 0.7 \[{\frac{60\,{b}^{2}{x}^{2}+165\,abx+132\,{a}^{2}}{220}{x}^{{\frac{5}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(2/3)*(b*x+a)^2,x)

[Out]

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

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Maxima [A]  time = 1.34944, size = 32, normalized size = 0.89 \[ \frac{3}{11} \, b^{2} x^{\frac{11}{3}} + \frac{3}{4} \, a b x^{\frac{8}{3}} + \frac{3}{5} \, a^{2} x^{\frac{5}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.203381, size = 36, normalized size = 1. \[ \frac{3}{220} \,{\left (20 \, b^{2} x^{3} + 55 \, a b x^{2} + 44 \, a^{2} x\right )} x^{\frac{2}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/220*(20*b^2*x^3 + 55*a*b*x^2 + 44*a^2*x)*x^(2/3)

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Sympy [A]  time = 7.1552, size = 1953, normalized size = 54.25 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((27*a**(35/3)*(-1 + b*(a/b + x)/a)**(2/3)/(-220*a**8*b**(5/3) + 660*a*
*7*b**(8/3)*(a/b + x) - 660*a**6*b**(11/3)*(a/b + x)**2 + 220*a**5*b**(14/3)*(a/
b + x)**3) + 27*a**(35/3)*exp(17*I*pi/3)/(-220*a**8*b**(5/3) + 660*a**7*b**(8/3)
*(a/b + x) - 660*a**6*b**(11/3)*(a/b + x)**2 + 220*a**5*b**(14/3)*(a/b + x)**3)
- 63*a**(32/3)*b*(-1 + b*(a/b + x)/a)**(2/3)*(a/b + x)/(-220*a**8*b**(5/3) + 660
*a**7*b**(8/3)*(a/b + x) - 660*a**6*b**(11/3)*(a/b + x)**2 + 220*a**5*b**(14/3)*
(a/b + x)**3) - 81*a**(32/3)*b*(a/b + x)*exp(17*I*pi/3)/(-220*a**8*b**(5/3) + 66
0*a**7*b**(8/3)*(a/b + x) - 660*a**6*b**(11/3)*(a/b + x)**2 + 220*a**5*b**(14/3)
*(a/b + x)**3) + 42*a**(29/3)*b**2*(-1 + b*(a/b + x)/a)**(2/3)*(a/b + x)**2/(-22
0*a**8*b**(5/3) + 660*a**7*b**(8/3)*(a/b + x) - 660*a**6*b**(11/3)*(a/b + x)**2
+ 220*a**5*b**(14/3)*(a/b + x)**3) + 81*a**(29/3)*b**2*(a/b + x)**2*exp(17*I*pi/
3)/(-220*a**8*b**(5/3) + 660*a**7*b**(8/3)*(a/b + x) - 660*a**6*b**(11/3)*(a/b +
x)**2 + 220*a**5*b**(14/3)*(a/b + x)**3) - 78*a**(26/3)*b**3*(-1 + b*(a/b + x)/
a)**(2/3)*(a/b + x)**3/(-220*a**8*b**(5/3) + 660*a**7*b**(8/3)*(a/b + x) - 660*a
**6*b**(11/3)*(a/b + x)**2 + 220*a**5*b**(14/3)*(a/b + x)**3) - 27*a**(26/3)*b**
3*(a/b + x)**3*exp(17*I*pi/3)/(-220*a**8*b**(5/3) + 660*a**7*b**(8/3)*(a/b + x)
- 660*a**6*b**(11/3)*(a/b + x)**2 + 220*a**5*b**(14/3)*(a/b + x)**3) + 207*a**(2
3/3)*b**4*(-1 + b*(a/b + x)/a)**(2/3)*(a/b + x)**4/(-220*a**8*b**(5/3) + 660*a**
7*b**(8/3)*(a/b + x) - 660*a**6*b**(11/3)*(a/b + x)**2 + 220*a**5*b**(14/3)*(a/b
+ x)**3) - 195*a**(20/3)*b**5*(-1 + b*(a/b + x)/a)**(2/3)*(a/b + x)**5/(-220*a*
*8*b**(5/3) + 660*a**7*b**(8/3)*(a/b + x) - 660*a**6*b**(11/3)*(a/b + x)**2 + 22
0*a**5*b**(14/3)*(a/b + x)**3) + 60*a**(17/3)*b**6*(-1 + b*(a/b + x)/a)**(2/3)*(
a/b + x)**6/(-220*a**8*b**(5/3) + 660*a**7*b**(8/3)*(a/b + x) - 660*a**6*b**(11/
3)*(a/b + x)**2 + 220*a**5*b**(14/3)*(a/b + x)**3), Abs(b*(a/b + x)/a) > 1), (-2
7*a**(35/3)*(1 - b*(a/b + x)/a)**(2/3)*exp(17*I*pi/3)/(-220*a**8*b**(5/3) + 660*
a**7*b**(8/3)*(a/b + x) - 660*a**6*b**(11/3)*(a/b + x)**2 + 220*a**5*b**(14/3)*(
a/b + x)**3) + 27*a**(35/3)*exp(17*I*pi/3)/(-220*a**8*b**(5/3) + 660*a**7*b**(8/
3)*(a/b + x) - 660*a**6*b**(11/3)*(a/b + x)**2 + 220*a**5*b**(14/3)*(a/b + x)**3
) + 63*a**(32/3)*b*(1 - b*(a/b + x)/a)**(2/3)*(a/b + x)*exp(17*I*pi/3)/(-220*a**
8*b**(5/3) + 660*a**7*b**(8/3)*(a/b + x) - 660*a**6*b**(11/3)*(a/b + x)**2 + 220
*a**5*b**(14/3)*(a/b + x)**3) - 81*a**(32/3)*b*(a/b + x)*exp(17*I*pi/3)/(-220*a*
*8*b**(5/3) + 660*a**7*b**(8/3)*(a/b + x) - 660*a**6*b**(11/3)*(a/b + x)**2 + 22
0*a**5*b**(14/3)*(a/b + x)**3) - 42*a**(29/3)*b**2*(1 - b*(a/b + x)/a)**(2/3)*(a
/b + x)**2*exp(17*I*pi/3)/(-220*a**8*b**(5/3) + 660*a**7*b**(8/3)*(a/b + x) - 66
0*a**6*b**(11/3)*(a/b + x)**2 + 220*a**5*b**(14/3)*(a/b + x)**3) + 81*a**(29/3)*
b**2*(a/b + x)**2*exp(17*I*pi/3)/(-220*a**8*b**(5/3) + 660*a**7*b**(8/3)*(a/b +
x) - 660*a**6*b**(11/3)*(a/b + x)**2 + 220*a**5*b**(14/3)*(a/b + x)**3) + 78*a**
(26/3)*b**3*(1 - b*(a/b + x)/a)**(2/3)*(a/b + x)**3*exp(17*I*pi/3)/(-220*a**8*b*
*(5/3) + 660*a**7*b**(8/3)*(a/b + x) - 660*a**6*b**(11/3)*(a/b + x)**2 + 220*a**
5*b**(14/3)*(a/b + x)**3) - 27*a**(26/3)*b**3*(a/b + x)**3*exp(17*I*pi/3)/(-220*
a**8*b**(5/3) + 660*a**7*b**(8/3)*(a/b + x) - 660*a**6*b**(11/3)*(a/b + x)**2 +
220*a**5*b**(14/3)*(a/b + x)**3) - 207*a**(23/3)*b**4*(1 - b*(a/b + x)/a)**(2/3)
*(a/b + x)**4*exp(17*I*pi/3)/(-220*a**8*b**(5/3) + 660*a**7*b**(8/3)*(a/b + x) -
660*a**6*b**(11/3)*(a/b + x)**2 + 220*a**5*b**(14/3)*(a/b + x)**3) + 195*a**(20
/3)*b**5*(1 - b*(a/b + x)/a)**(2/3)*(a/b + x)**5*exp(17*I*pi/3)/(-220*a**8*b**(5
/3) + 660*a**7*b**(8/3)*(a/b + x) - 660*a**6*b**(11/3)*(a/b + x)**2 + 220*a**5*b
**(14/3)*(a/b + x)**3) - 60*a**(17/3)*b**6*(1 - b*(a/b + x)/a)**(2/3)*(a/b + x)*
*6*exp(17*I*pi/3)/(-220*a**8*b**(5/3) + 660*a**7*b**(8/3)*(a/b + x) - 660*a**6*b
**(11/3)*(a/b + x)**2 + 220*a**5*b**(14/3)*(a/b + x)**3), True))

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GIAC/XCAS [A]  time = 0.202674, size = 32, normalized size = 0.89 \[ \frac{3}{11} \, b^{2} x^{\frac{11}{3}} + \frac{3}{4} \, a b x^{\frac{8}{3}} + \frac{3}{5} \, a^{2} x^{\frac{5}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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