3.659 \(\int x^{4/3} (a+b x)^2 \, dx\)

Optimal. Leaf size=36 \[ \frac{3}{7} a^2 x^{7/3}+\frac{3}{5} a b x^{10/3}+\frac{3}{13} b^2 x^{13/3} \]

[Out]

(3*a^2*x^(7/3))/7 + (3*a*b*x^(10/3))/5 + (3*b^2*x^(13/3))/13

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Rubi [A]  time = 0.0224497, 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 \[ \frac{3}{7} a^2 x^{7/3}+\frac{3}{5} a b x^{10/3}+\frac{3}{13} b^2 x^{13/3} \]

Antiderivative was successfully verified.

[In]  Int[x^(4/3)*(a + b*x)^2,x]

[Out]

(3*a^2*x^(7/3))/7 + (3*a*b*x^(10/3))/5 + (3*b^2*x^(13/3))/13

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Rubi in Sympy [A]  time = 4.23525, size = 34, normalized size = 0.94 \[ \frac{3 a^{2} x^{\frac{7}{3}}}{7} + \frac{3 a b x^{\frac{10}{3}}}{5} + \frac{3 b^{2} x^{\frac{13}{3}}}{13} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(4/3)*(b*x+a)**2,x)

[Out]

3*a**2*x**(7/3)/7 + 3*a*b*x**(10/3)/5 + 3*b**2*x**(13/3)/13

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Mathematica [A]  time = 0.00915503, size = 28, normalized size = 0.78 \[ \frac{3}{455} x^{7/3} \left (65 a^2+91 a b x+35 b^2 x^2\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^(4/3)*(a + b*x)^2,x]

[Out]

(3*x^(7/3)*(65*a^2 + 91*a*b*x + 35*b^2*x^2))/455

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Maple [A]  time = 0.007, size = 25, normalized size = 0.7 \[{\frac{105\,{b}^{2}{x}^{2}+273\,abx+195\,{a}^{2}}{455}{x}^{{\frac{7}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(4/3)*(b*x+a)^2,x)

[Out]

3/455*x^(7/3)*(35*b^2*x^2+91*a*b*x+65*a^2)

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Maxima [A]  time = 1.34612, size = 32, normalized size = 0.89 \[ \frac{3}{13} \, b^{2} x^{\frac{13}{3}} + \frac{3}{5} \, a b x^{\frac{10}{3}} + \frac{3}{7} \, a^{2} x^{\frac{7}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^2*x^(4/3),x, algorithm="maxima")

[Out]

3/13*b^2*x^(13/3) + 3/5*a*b*x^(10/3) + 3/7*a^2*x^(7/3)

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Fricas [A]  time = 0.205314, size = 39, normalized size = 1.08 \[ \frac{3}{455} \,{\left (35 \, b^{2} x^{4} + 91 \, a b x^{3} + 65 \, a^{2} x^{2}\right )} x^{\frac{1}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^2*x^(4/3),x, algorithm="fricas")

[Out]

3/455*(35*b^2*x^4 + 91*a*b*x^3 + 65*a^2*x^2)*x^(1/3)

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Sympy [A]  time = 8.83111, size = 2142, normalized size = 59.5 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(4/3)*(b*x+a)**2,x)

[Out]

Piecewise((-27*a**(37/3)*(-1 + b*(a/b + x)/a)**(1/3)/(-455*a**8*b**(7/3) + 1365*
a**7*b**(10/3)*(a/b + x) - 1365*a**6*b**(13/3)*(a/b + x)**2 + 455*a**5*b**(16/3)
*(a/b + x)**3) + 27*a**(37/3)*exp(13*I*pi/3)/(-455*a**8*b**(7/3) + 1365*a**7*b**
(10/3)*(a/b + x) - 1365*a**6*b**(13/3)*(a/b + x)**2 + 455*a**5*b**(16/3)*(a/b +
x)**3) + 72*a**(34/3)*b*(-1 + b*(a/b + x)/a)**(1/3)*(a/b + x)/(-455*a**8*b**(7/3
) + 1365*a**7*b**(10/3)*(a/b + x) - 1365*a**6*b**(13/3)*(a/b + x)**2 + 455*a**5*
b**(16/3)*(a/b + x)**3) - 81*a**(34/3)*b*(a/b + x)*exp(13*I*pi/3)/(-455*a**8*b**
(7/3) + 1365*a**7*b**(10/3)*(a/b + x) - 1365*a**6*b**(13/3)*(a/b + x)**2 + 455*a
**5*b**(16/3)*(a/b + x)**3) - 60*a**(31/3)*b**2*(-1 + b*(a/b + x)/a)**(1/3)*(a/b
+ x)**2/(-455*a**8*b**(7/3) + 1365*a**7*b**(10/3)*(a/b + x) - 1365*a**6*b**(13/
3)*(a/b + x)**2 + 455*a**5*b**(16/3)*(a/b + x)**3) + 81*a**(31/3)*b**2*(a/b + x)
**2*exp(13*I*pi/3)/(-455*a**8*b**(7/3) + 1365*a**7*b**(10/3)*(a/b + x) - 1365*a*
*6*b**(13/3)*(a/b + x)**2 + 455*a**5*b**(16/3)*(a/b + x)**3) + 165*a**(28/3)*b**
3*(-1 + b*(a/b + x)/a)**(1/3)*(a/b + x)**3/(-455*a**8*b**(7/3) + 1365*a**7*b**(1
0/3)*(a/b + x) - 1365*a**6*b**(13/3)*(a/b + x)**2 + 455*a**5*b**(16/3)*(a/b + x)
**3) - 27*a**(28/3)*b**3*(a/b + x)**3*exp(13*I*pi/3)/(-455*a**8*b**(7/3) + 1365*
a**7*b**(10/3)*(a/b + x) - 1365*a**6*b**(13/3)*(a/b + x)**2 + 455*a**5*b**(16/3)
*(a/b + x)**3) - 555*a**(25/3)*b**4*(-1 + b*(a/b + x)/a)**(1/3)*(a/b + x)**4/(-4
55*a**8*b**(7/3) + 1365*a**7*b**(10/3)*(a/b + x) - 1365*a**6*b**(13/3)*(a/b + x)
**2 + 455*a**5*b**(16/3)*(a/b + x)**3) + 762*a**(22/3)*b**5*(-1 + b*(a/b + x)/a)
**(1/3)*(a/b + x)**5/(-455*a**8*b**(7/3) + 1365*a**7*b**(10/3)*(a/b + x) - 1365*
a**6*b**(13/3)*(a/b + x)**2 + 455*a**5*b**(16/3)*(a/b + x)**3) - 462*a**(19/3)*b
**6*(-1 + b*(a/b + x)/a)**(1/3)*(a/b + x)**6/(-455*a**8*b**(7/3) + 1365*a**7*b**
(10/3)*(a/b + x) - 1365*a**6*b**(13/3)*(a/b + x)**2 + 455*a**5*b**(16/3)*(a/b +
x)**3) + 105*a**(16/3)*b**7*(-1 + b*(a/b + x)/a)**(1/3)*(a/b + x)**7/(-455*a**8*
b**(7/3) + 1365*a**7*b**(10/3)*(a/b + x) - 1365*a**6*b**(13/3)*(a/b + x)**2 + 45
5*a**5*b**(16/3)*(a/b + x)**3), Abs(b*(a/b + x)/a) > 1), (-27*a**(37/3)*(1 - b*(
a/b + x)/a)**(1/3)*exp(13*I*pi/3)/(-455*a**8*b**(7/3) + 1365*a**7*b**(10/3)*(a/b
+ x) - 1365*a**6*b**(13/3)*(a/b + x)**2 + 455*a**5*b**(16/3)*(a/b + x)**3) + 27
*a**(37/3)*exp(13*I*pi/3)/(-455*a**8*b**(7/3) + 1365*a**7*b**(10/3)*(a/b + x) -
1365*a**6*b**(13/3)*(a/b + x)**2 + 455*a**5*b**(16/3)*(a/b + x)**3) + 72*a**(34/
3)*b*(1 - b*(a/b + x)/a)**(1/3)*(a/b + x)*exp(13*I*pi/3)/(-455*a**8*b**(7/3) + 1
365*a**7*b**(10/3)*(a/b + x) - 1365*a**6*b**(13/3)*(a/b + x)**2 + 455*a**5*b**(1
6/3)*(a/b + x)**3) - 81*a**(34/3)*b*(a/b + x)*exp(13*I*pi/3)/(-455*a**8*b**(7/3)
+ 1365*a**7*b**(10/3)*(a/b + x) - 1365*a**6*b**(13/3)*(a/b + x)**2 + 455*a**5*b
**(16/3)*(a/b + x)**3) - 60*a**(31/3)*b**2*(1 - b*(a/b + x)/a)**(1/3)*(a/b + x)*
*2*exp(13*I*pi/3)/(-455*a**8*b**(7/3) + 1365*a**7*b**(10/3)*(a/b + x) - 1365*a**
6*b**(13/3)*(a/b + x)**2 + 455*a**5*b**(16/3)*(a/b + x)**3) + 81*a**(31/3)*b**2*
(a/b + x)**2*exp(13*I*pi/3)/(-455*a**8*b**(7/3) + 1365*a**7*b**(10/3)*(a/b + x)
- 1365*a**6*b**(13/3)*(a/b + x)**2 + 455*a**5*b**(16/3)*(a/b + x)**3) + 165*a**(
28/3)*b**3*(1 - b*(a/b + x)/a)**(1/3)*(a/b + x)**3*exp(13*I*pi/3)/(-455*a**8*b**
(7/3) + 1365*a**7*b**(10/3)*(a/b + x) - 1365*a**6*b**(13/3)*(a/b + x)**2 + 455*a
**5*b**(16/3)*(a/b + x)**3) - 27*a**(28/3)*b**3*(a/b + x)**3*exp(13*I*pi/3)/(-45
5*a**8*b**(7/3) + 1365*a**7*b**(10/3)*(a/b + x) - 1365*a**6*b**(13/3)*(a/b + x)*
*2 + 455*a**5*b**(16/3)*(a/b + x)**3) - 555*a**(25/3)*b**4*(1 - b*(a/b + x)/a)**
(1/3)*(a/b + x)**4*exp(13*I*pi/3)/(-455*a**8*b**(7/3) + 1365*a**7*b**(10/3)*(a/b
+ x) - 1365*a**6*b**(13/3)*(a/b + x)**2 + 455*a**5*b**(16/3)*(a/b + x)**3) + 76
2*a**(22/3)*b**5*(1 - b*(a/b + x)/a)**(1/3)*(a/b + x)**5*exp(13*I*pi/3)/(-455*a*
*8*b**(7/3) + 1365*a**7*b**(10/3)*(a/b + x) - 1365*a**6*b**(13/3)*(a/b + x)**2 +
455*a**5*b**(16/3)*(a/b + x)**3) - 462*a**(19/3)*b**6*(1 - b*(a/b + x)/a)**(1/3
)*(a/b + x)**6*exp(13*I*pi/3)/(-455*a**8*b**(7/3) + 1365*a**7*b**(10/3)*(a/b + x
) - 1365*a**6*b**(13/3)*(a/b + x)**2 + 455*a**5*b**(16/3)*(a/b + x)**3) + 105*a*
*(16/3)*b**7*(1 - b*(a/b + x)/a)**(1/3)*(a/b + x)**7*exp(13*I*pi/3)/(-455*a**8*b
**(7/3) + 1365*a**7*b**(10/3)*(a/b + x) - 1365*a**6*b**(13/3)*(a/b + x)**2 + 455
*a**5*b**(16/3)*(a/b + x)**3), True))

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GIAC/XCAS [A]  time = 0.205826, size = 32, normalized size = 0.89 \[ \frac{3}{13} \, b^{2} x^{\frac{13}{3}} + \frac{3}{5} \, a b x^{\frac{10}{3}} + \frac{3}{7} \, a^{2} x^{\frac{7}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^2*x^(4/3),x, algorithm="giac")

[Out]

3/13*b^2*x^(13/3) + 3/5*a*b*x^(10/3) + 3/7*a^2*x^(7/3)