3.393 \(\int \frac{x^2}{\sqrt [3]{a+b x}} \, dx\)

Optimal. Leaf size=53 \[ \frac{3 a^2 (a+b x)^{2/3}}{2 b^3}+\frac{3 (a+b x)^{8/3}}{8 b^3}-\frac{6 a (a+b x)^{5/3}}{5 b^3} \]

[Out]

(3*a^2*(a + b*x)^(2/3))/(2*b^3) - (6*a*(a + b*x)^(5/3))/(5*b^3) + (3*(a + b*x)^(
8/3))/(8*b^3)

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Rubi [A]  time = 0.0386107, 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 \[ \frac{3 a^2 (a+b x)^{2/3}}{2 b^3}+\frac{3 (a+b x)^{8/3}}{8 b^3}-\frac{6 a (a+b x)^{5/3}}{5 b^3} \]

Antiderivative was successfully verified.

[In]  Int[x^2/(a + b*x)^(1/3),x]

[Out]

(3*a^2*(a + b*x)^(2/3))/(2*b^3) - (6*a*(a + b*x)^(5/3))/(5*b^3) + (3*(a + b*x)^(
8/3))/(8*b^3)

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Rubi in Sympy [A]  time = 7.85877, size = 49, normalized size = 0.92 \[ \frac{3 a^{2} \left (a + b x\right )^{\frac{2}{3}}}{2 b^{3}} - \frac{6 a \left (a + b x\right )^{\frac{5}{3}}}{5 b^{3}} + \frac{3 \left (a + b x\right )^{\frac{8}{3}}}{8 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2/(b*x+a)**(1/3),x)

[Out]

3*a**2*(a + b*x)**(2/3)/(2*b**3) - 6*a*(a + b*x)**(5/3)/(5*b**3) + 3*(a + b*x)**
(8/3)/(8*b**3)

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Mathematica [A]  time = 0.018934, size = 35, normalized size = 0.66 \[ \frac{3 (a+b x)^{2/3} \left (9 a^2-6 a b x+5 b^2 x^2\right )}{40 b^3} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2/(a + b*x)^(1/3),x]

[Out]

(3*(a + b*x)^(2/3)*(9*a^2 - 6*a*b*x + 5*b^2*x^2))/(40*b^3)

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Maple [A]  time = 0.006, size = 32, normalized size = 0.6 \[{\frac{15\,{b}^{2}{x}^{2}-18\,abx+27\,{a}^{2}}{40\,{b}^{3}} \left ( bx+a \right ) ^{{\frac{2}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(b*x+a)^(1/3),x)

[Out]

3/40*(b*x+a)^(2/3)*(5*b^2*x^2-6*a*b*x+9*a^2)/b^3

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Maxima [A]  time = 1.35349, size = 55, normalized size = 1.04 \[ \frac{3 \,{\left (b x + a\right )}^{\frac{8}{3}}}{8 \, b^{3}} - \frac{6 \,{\left (b x + a\right )}^{\frac{5}{3}} a}{5 \, b^{3}} + \frac{3 \,{\left (b x + a\right )}^{\frac{2}{3}} a^{2}}{2 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(b*x + a)^(1/3),x, algorithm="maxima")

[Out]

3/8*(b*x + a)^(8/3)/b^3 - 6/5*(b*x + a)^(5/3)*a/b^3 + 3/2*(b*x + a)^(2/3)*a^2/b^
3

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Fricas [A]  time = 0.227355, size = 42, normalized size = 0.79 \[ \frac{3 \,{\left (5 \, b^{2} x^{2} - 6 \, a b x + 9 \, a^{2}\right )}{\left (b x + a\right )}^{\frac{2}{3}}}{40 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(b*x + a)^(1/3),x, algorithm="fricas")

[Out]

3/40*(5*b^2*x^2 - 6*a*b*x + 9*a^2)*(b*x + a)^(2/3)/b^3

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Sympy [A]  time = 5.33341, size = 600, normalized size = 11.32 \[ \frac{27 a^{\frac{32}{3}} \left (1 + \frac{b x}{a}\right )^{\frac{2}{3}}}{40 a^{8} b^{3} + 120 a^{7} b^{4} x + 120 a^{6} b^{5} x^{2} + 40 a^{5} b^{6} x^{3}} - \frac{27 a^{\frac{32}{3}}}{40 a^{8} b^{3} + 120 a^{7} b^{4} x + 120 a^{6} b^{5} x^{2} + 40 a^{5} b^{6} x^{3}} + \frac{63 a^{\frac{29}{3}} b x \left (1 + \frac{b x}{a}\right )^{\frac{2}{3}}}{40 a^{8} b^{3} + 120 a^{7} b^{4} x + 120 a^{6} b^{5} x^{2} + 40 a^{5} b^{6} x^{3}} - \frac{81 a^{\frac{29}{3}} b x}{40 a^{8} b^{3} + 120 a^{7} b^{4} x + 120 a^{6} b^{5} x^{2} + 40 a^{5} b^{6} x^{3}} + \frac{42 a^{\frac{26}{3}} b^{2} x^{2} \left (1 + \frac{b x}{a}\right )^{\frac{2}{3}}}{40 a^{8} b^{3} + 120 a^{7} b^{4} x + 120 a^{6} b^{5} x^{2} + 40 a^{5} b^{6} x^{3}} - \frac{81 a^{\frac{26}{3}} b^{2} x^{2}}{40 a^{8} b^{3} + 120 a^{7} b^{4} x + 120 a^{6} b^{5} x^{2} + 40 a^{5} b^{6} x^{3}} + \frac{18 a^{\frac{23}{3}} b^{3} x^{3} \left (1 + \frac{b x}{a}\right )^{\frac{2}{3}}}{40 a^{8} b^{3} + 120 a^{7} b^{4} x + 120 a^{6} b^{5} x^{2} + 40 a^{5} b^{6} x^{3}} - \frac{27 a^{\frac{23}{3}} b^{3} x^{3}}{40 a^{8} b^{3} + 120 a^{7} b^{4} x + 120 a^{6} b^{5} x^{2} + 40 a^{5} b^{6} x^{3}} + \frac{27 a^{\frac{20}{3}} b^{4} x^{4} \left (1 + \frac{b x}{a}\right )^{\frac{2}{3}}}{40 a^{8} b^{3} + 120 a^{7} b^{4} x + 120 a^{6} b^{5} x^{2} + 40 a^{5} b^{6} x^{3}} + \frac{15 a^{\frac{17}{3}} b^{5} x^{5} \left (1 + \frac{b x}{a}\right )^{\frac{2}{3}}}{40 a^{8} b^{3} + 120 a^{7} b^{4} x + 120 a^{6} b^{5} x^{2} + 40 a^{5} b^{6} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2/(b*x+a)**(1/3),x)

[Out]

27*a**(32/3)*(1 + b*x/a)**(2/3)/(40*a**8*b**3 + 120*a**7*b**4*x + 120*a**6*b**5*
x**2 + 40*a**5*b**6*x**3) - 27*a**(32/3)/(40*a**8*b**3 + 120*a**7*b**4*x + 120*a
**6*b**5*x**2 + 40*a**5*b**6*x**3) + 63*a**(29/3)*b*x*(1 + b*x/a)**(2/3)/(40*a**
8*b**3 + 120*a**7*b**4*x + 120*a**6*b**5*x**2 + 40*a**5*b**6*x**3) - 81*a**(29/3
)*b*x/(40*a**8*b**3 + 120*a**7*b**4*x + 120*a**6*b**5*x**2 + 40*a**5*b**6*x**3)
+ 42*a**(26/3)*b**2*x**2*(1 + b*x/a)**(2/3)/(40*a**8*b**3 + 120*a**7*b**4*x + 12
0*a**6*b**5*x**2 + 40*a**5*b**6*x**3) - 81*a**(26/3)*b**2*x**2/(40*a**8*b**3 + 1
20*a**7*b**4*x + 120*a**6*b**5*x**2 + 40*a**5*b**6*x**3) + 18*a**(23/3)*b**3*x**
3*(1 + b*x/a)**(2/3)/(40*a**8*b**3 + 120*a**7*b**4*x + 120*a**6*b**5*x**2 + 40*a
**5*b**6*x**3) - 27*a**(23/3)*b**3*x**3/(40*a**8*b**3 + 120*a**7*b**4*x + 120*a*
*6*b**5*x**2 + 40*a**5*b**6*x**3) + 27*a**(20/3)*b**4*x**4*(1 + b*x/a)**(2/3)/(4
0*a**8*b**3 + 120*a**7*b**4*x + 120*a**6*b**5*x**2 + 40*a**5*b**6*x**3) + 15*a**
(17/3)*b**5*x**5*(1 + b*x/a)**(2/3)/(40*a**8*b**3 + 120*a**7*b**4*x + 120*a**6*b
**5*x**2 + 40*a**5*b**6*x**3)

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GIAC/XCAS [A]  time = 0.207548, size = 62, normalized size = 1.17 \[ \frac{3 \,{\left (5 \,{\left (b x + a\right )}^{\frac{8}{3}} b^{14} - 16 \,{\left (b x + a\right )}^{\frac{5}{3}} a b^{14} + 20 \,{\left (b x + a\right )}^{\frac{2}{3}} a^{2} b^{14}\right )}}{40 \, b^{17}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(b*x + a)^(1/3),x, algorithm="giac")

[Out]

3/40*(5*(b*x + a)^(8/3)*b^14 - 16*(b*x + a)^(5/3)*a*b^14 + 20*(b*x + a)^(2/3)*a^
2*b^14)/b^17