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

Optimal. Leaf size=38 \[ \frac{3 (b x-a)^{5/3}}{5 b^2}+\frac{3 a (b x-a)^{2/3}}{2 b^2} \]

[Out]

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

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Rubi [A]  time = 0.0273403, antiderivative size = 38, 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 \[ \frac{3 (b x-a)^{5/3}}{5 b^2}+\frac{3 a (b x-a)^{2/3}}{2 b^2} \]

Antiderivative was successfully verified.

[In]  Int[x/(-a + b*x)^(1/3),x]

[Out]

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

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Rubi in Sympy [A]  time = 5.19663, size = 31, normalized size = 0.82 \[ \frac{3 a \left (- a + b x\right )^{\frac{2}{3}}}{2 b^{2}} + \frac{3 \left (- a + b x\right )^{\frac{5}{3}}}{5 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0133036, size = 26, normalized size = 0.68 \[ \frac{3 (b x-a)^{2/3} (3 a+2 b x)}{10 b^2} \]

Antiderivative was successfully verified.

[In]  Integrate[x/(-a + b*x)^(1/3),x]

[Out]

(3*(-a + b*x)^(2/3)*(3*a + 2*b*x))/(10*b^2)

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Maple [A]  time = 0.004, size = 23, normalized size = 0.6 \[{\frac{6\,bx+9\,a}{10\,{b}^{2}} \left ( bx-a \right ) ^{{\frac{2}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(b*x-a)^(1/3),x)

[Out]

3/10*(2*b*x+3*a)/b^2*(b*x-a)^(2/3)

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Maxima [A]  time = 1.33791, size = 41, normalized size = 1.08 \[ \frac{3 \,{\left (b x - a\right )}^{\frac{5}{3}}}{5 \, b^{2}} + \frac{3 \,{\left (b x - a\right )}^{\frac{2}{3}} a}{2 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.20897, size = 30, normalized size = 0.79 \[ \frac{3 \,{\left (2 \, b x + 3 \, a\right )}{\left (b x - a\right )}^{\frac{2}{3}}}{10 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/10*(2*b*x + 3*a)*(b*x - a)^(2/3)/b^2

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Sympy [A]  time = 3.92547, size = 389, normalized size = 10.24 \[ \begin{cases} - \frac{9 a^{\frac{11}{3}} \left (-1 + \frac{b x}{a}\right )^{\frac{2}{3}}}{- 10 a^{2} b^{2} + 10 a b^{3} x} - \frac{9 a^{\frac{11}{3}} e^{\frac{5 i \pi }{3}}}{- 10 a^{2} b^{2} + 10 a b^{3} x} + \frac{3 a^{\frac{8}{3}} b x \left (-1 + \frac{b x}{a}\right )^{\frac{2}{3}}}{- 10 a^{2} b^{2} + 10 a b^{3} x} + \frac{9 a^{\frac{8}{3}} b x e^{\frac{5 i \pi }{3}}}{- 10 a^{2} b^{2} + 10 a b^{3} x} + \frac{6 a^{\frac{5}{3}} b^{2} x^{2} \left (-1 + \frac{b x}{a}\right )^{\frac{2}{3}}}{- 10 a^{2} b^{2} + 10 a b^{3} x} & \text{for}\: \left |{\frac{b x}{a}}\right | > 1 \\\frac{9 a^{\frac{11}{3}} \left (1 - \frac{b x}{a}\right )^{\frac{2}{3}} e^{\frac{5 i \pi }{3}}}{- 10 a^{2} b^{2} + 10 a b^{3} x} - \frac{9 a^{\frac{11}{3}} e^{\frac{5 i \pi }{3}}}{- 10 a^{2} b^{2} + 10 a b^{3} x} - \frac{3 a^{\frac{8}{3}} b x \left (1 - \frac{b x}{a}\right )^{\frac{2}{3}} e^{\frac{5 i \pi }{3}}}{- 10 a^{2} b^{2} + 10 a b^{3} x} + \frac{9 a^{\frac{8}{3}} b x e^{\frac{5 i \pi }{3}}}{- 10 a^{2} b^{2} + 10 a b^{3} x} - \frac{6 a^{\frac{5}{3}} b^{2} x^{2} \left (1 - \frac{b x}{a}\right )^{\frac{2}{3}} e^{\frac{5 i \pi }{3}}}{- 10 a^{2} b^{2} + 10 a b^{3} x} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(b*x-a)**(1/3),x)

[Out]

Piecewise((-9*a**(11/3)*(-1 + b*x/a)**(2/3)/(-10*a**2*b**2 + 10*a*b**3*x) - 9*a*
*(11/3)*exp(5*I*pi/3)/(-10*a**2*b**2 + 10*a*b**3*x) + 3*a**(8/3)*b*x*(-1 + b*x/a
)**(2/3)/(-10*a**2*b**2 + 10*a*b**3*x) + 9*a**(8/3)*b*x*exp(5*I*pi/3)/(-10*a**2*
b**2 + 10*a*b**3*x) + 6*a**(5/3)*b**2*x**2*(-1 + b*x/a)**(2/3)/(-10*a**2*b**2 +
10*a*b**3*x), Abs(b*x/a) > 1), (9*a**(11/3)*(1 - b*x/a)**(2/3)*exp(5*I*pi/3)/(-1
0*a**2*b**2 + 10*a*b**3*x) - 9*a**(11/3)*exp(5*I*pi/3)/(-10*a**2*b**2 + 10*a*b**
3*x) - 3*a**(8/3)*b*x*(1 - b*x/a)**(2/3)*exp(5*I*pi/3)/(-10*a**2*b**2 + 10*a*b**
3*x) + 9*a**(8/3)*b*x*exp(5*I*pi/3)/(-10*a**2*b**2 + 10*a*b**3*x) - 6*a**(5/3)*b
**2*x**2*(1 - b*x/a)**(2/3)*exp(5*I*pi/3)/(-10*a**2*b**2 + 10*a*b**3*x), True))

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GIAC/XCAS [A]  time = 0.204289, size = 39, normalized size = 1.03 \[ \frac{3 \,{\left (2 \,{\left (b x - a\right )}^{\frac{5}{3}} + 5 \,{\left (b x - a\right )}^{\frac{2}{3}} a\right )}}{10 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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