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

Optimal. Leaf size=59 \[ \frac{3 a^2 (b x-a)^{2/3}}{2 b^3}+\frac{3 (b x-a)^{8/3}}{8 b^3}+\frac{6 a (b x-a)^{5/3}}{5 b^3} \]

[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 = 0.0417991, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ \frac{3 a^2 (b x-a)^{2/3}}{2 b^3}+\frac{3 (b x-a)^{8/3}}{8 b^3}+\frac{6 a (b x-a)^{5/3}}{5 b^3} \]

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 = 8.67808, size = 49, normalized size = 0.83 \[ \frac{3 a^{2} \left (- a + b x\right )^{\frac{2}{3}}}{2 b^{3}} + \frac{6 a \left (- a + b x\right )^{\frac{5}{3}}}{5 b^{3}} + \frac{3 \left (- a + b x\right )^{\frac{8}{3}}}{8 b^{3}} \]

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 = 0.0200665, size = 37, normalized size = 0.63 \[ \frac{3 (b x-a)^{2/3} \left (9 a^2+6 a b x+5 b^2 x^2\right )}{40 b^3} \]

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 = 0.007, size = 34, 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*(5*b^2*x^2+6*a*b*x+9*a^2)/b^3*(b*x-a)^(2/3)

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Maxima [A]  time = 1.33124, size = 63, normalized size = 1.07 \[ \frac{3 \,{\left (b x - a\right )}^{\frac{8}{3}}}{8 \, b^{3}} + \frac{6 \,{\left (b x - a\right )}^{\frac{5}{3}} a}{5 \, b^{3}} + \frac{3 \,{\left (b x - a\right )}^{\frac{2}{3}} a^{2}}{2 \, b^{3}} \]

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 = 0.245096, size = 45, normalized size = 0.76 \[ \frac{3 \,{\left (5 \, b^{2} x^{2} + 6 \, a b x + 9 \, a^{2}\right )}{\left (b x - a\right )}^{\frac{2}{3}}}{40 \, b^{3}} \]

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 = 5.87803, size = 1326, normalized size = 22.47 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-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)*exp(8*I*pi/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 + 4
0*a**5*b**6*x**3) - 81*a**(29/3)*b*x*exp(8*I*pi/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) - 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 - 120*a**6*b**5*x**2 + 40*a**5*b**6*
x**3) + 81*a**(26/3)*b**2*x**2*exp(8*I*pi/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) + 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*exp(8*I*pi/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)/(
-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) + 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), Abs(b*x/a) > 1), (-27*a**(32/3)*(1 - b*x/a)**
(2/3)*exp(8*I*pi/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)*exp(8*I*pi/3)/(-40*a**8*b**3 + 120*a**7*b**4*x - 1
20*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)*exp
(8*I*pi/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*exp(8*I*pi/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) - 42*a**(26/3)*b**2*x**2*(1 - b*x/a)**(2/3)*ex
p(8*I*pi/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**(26/3)*b**2*x**2*exp(8*I*pi/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) + 18*a**(23/3)*b**3*x**3*(1 - b*x/a)**(
2/3)*exp(8*I*pi/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*exp(8*I*pi/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)*exp(8*I*pi/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) + 15*a**(17/3)*b**5*x**5*(1 - b*x/a)**(2/3)*exp(8*I*pi/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), True)
)

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GIAC/XCAS [A]  time = 0.206796, size = 70, normalized size = 1.19 \[ \frac{3 \,{\left (5 \,{\left (b x - a\right )}^{\frac{8}{3}} b^{14} + 16 \,{\left (b x - a\right )}^{\frac{5}{3}} a b^{14} + 20 \,{\left (b x - a\right )}^{\frac{2}{3}} a^{2} b^{14}\right )}}{40 \, b^{17}} \]

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