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

Optimal. Leaf size=34 \[ \frac{3 (a+b x)^{8/3}}{8 b^2}-\frac{3 a (a+b x)^{5/3}}{5 b^2} \]

[Out]

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

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Rubi [A]  time = 0.0255698, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{3 (a+b x)^{8/3}}{8 b^2}-\frac{3 a (a+b x)^{5/3}}{5 b^2} \]

Antiderivative was successfully verified.

[In]  Int[x*(a + b*x)^(2/3),x]

[Out]

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

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Rubi in Sympy [A]  time = 4.94114, size = 31, normalized size = 0.91 \[ - \frac{3 a \left (a + b x\right )^{\frac{5}{3}}}{5 b^{2}} + \frac{3 \left (a + b x\right )^{\frac{8}{3}}}{8 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0138223, size = 37, normalized size = 1.09 \[ (a+b x)^{2/3} \left (-\frac{9 a^2}{40 b^2}+\frac{3 a x}{20 b}+\frac{3 x^2}{8}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x*(a + b*x)^(2/3),x]

[Out]

(a + b*x)^(2/3)*((-9*a^2)/(40*b^2) + (3*a*x)/(20*b) + (3*x^2)/8)

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Maple [A]  time = 0.006, size = 21, normalized size = 0.6 \[ -{\frac{-15\,bx+9\,a}{40\,{b}^{2}} \left ( bx+a \right ) ^{{\frac{5}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(b*x+a)^(2/3),x)

[Out]

-3/40*(b*x+a)^(5/3)*(-5*b*x+3*a)/b^2

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Maxima [A]  time = 1.34762, size = 35, normalized size = 1.03 \[ \frac{3 \,{\left (b x + a\right )}^{\frac{8}{3}}}{8 \, b^{2}} - \frac{3 \,{\left (b x + a\right )}^{\frac{5}{3}} a}{5 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.206669, size = 42, normalized size = 1.24 \[ \frac{3 \,{\left (5 \, b^{2} x^{2} + 2 \, a b x - 3 \, a^{2}\right )}{\left (b x + a\right )}^{\frac{2}{3}}}{40 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/40*(5*b^2*x^2 + 2*a*b*x - 3*a^2)*(b*x + a)^(2/3)/b^2

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Sympy [A]  time = 3.88778, size = 202, normalized size = 5.94 \[ - \frac{9 a^{\frac{14}{3}} \left (1 + \frac{b x}{a}\right )^{\frac{2}{3}}}{40 a^{2} b^{2} + 40 a b^{3} x} + \frac{9 a^{\frac{14}{3}}}{40 a^{2} b^{2} + 40 a b^{3} x} - \frac{3 a^{\frac{11}{3}} b x \left (1 + \frac{b x}{a}\right )^{\frac{2}{3}}}{40 a^{2} b^{2} + 40 a b^{3} x} + \frac{9 a^{\frac{11}{3}} b x}{40 a^{2} b^{2} + 40 a b^{3} x} + \frac{21 a^{\frac{8}{3}} b^{2} x^{2} \left (1 + \frac{b x}{a}\right )^{\frac{2}{3}}}{40 a^{2} b^{2} + 40 a b^{3} x} + \frac{15 a^{\frac{5}{3}} b^{3} x^{3} \left (1 + \frac{b x}{a}\right )^{\frac{2}{3}}}{40 a^{2} b^{2} + 40 a b^{3} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*(b*x+a)**(2/3),x)

[Out]

-9*a**(14/3)*(1 + b*x/a)**(2/3)/(40*a**2*b**2 + 40*a*b**3*x) + 9*a**(14/3)/(40*a
**2*b**2 + 40*a*b**3*x) - 3*a**(11/3)*b*x*(1 + b*x/a)**(2/3)/(40*a**2*b**2 + 40*
a*b**3*x) + 9*a**(11/3)*b*x/(40*a**2*b**2 + 40*a*b**3*x) + 21*a**(8/3)*b**2*x**2
*(1 + b*x/a)**(2/3)/(40*a**2*b**2 + 40*a*b**3*x) + 15*a**(5/3)*b**3*x**3*(1 + b*
x/a)**(2/3)/(40*a**2*b**2 + 40*a*b**3*x)

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GIAC/XCAS [A]  time = 0.210603, size = 34, normalized size = 1. \[ \frac{3 \,{\left (5 \,{\left (b x + a\right )}^{\frac{8}{3}} - 8 \,{\left (b x + a\right )}^{\frac{5}{3}} a\right )}}{40 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/40*(5*(b*x + a)^(8/3) - 8*(b*x + a)^(5/3)*a)/b^2