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

Optimal. Leaf size=49 \[ -\frac{3 a^2}{b^3 \sqrt [3]{a+b x}}-\frac{3 a (a+b x)^{2/3}}{b^3}+\frac{3 (a+b x)^{5/3}}{5 b^3} \]

[Out]

(-3*a^2)/(b^3*(a + b*x)^(1/3)) - (3*a*(a + b*x)^(2/3))/b^3 + (3*(a + b*x)^(5/3))
/(5*b^3)

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Rubi [A]  time = 0.0388706, antiderivative size = 49, 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 \[ -\frac{3 a^2}{b^3 \sqrt [3]{a+b x}}-\frac{3 a (a+b x)^{2/3}}{b^3}+\frac{3 (a+b x)^{5/3}}{5 b^3} \]

Antiderivative was successfully verified.

[In]  Int[x^2/(a + b*x)^(4/3),x]

[Out]

(-3*a^2)/(b^3*(a + b*x)^(1/3)) - (3*a*(a + b*x)^(2/3))/b^3 + (3*(a + b*x)^(5/3))
/(5*b^3)

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Rubi in Sympy [A]  time = 7.86509, size = 46, normalized size = 0.94 \[ - \frac{3 a^{2}}{b^{3} \sqrt [3]{a + b x}} - \frac{3 a \left (a + b x\right )^{\frac{2}{3}}}{b^{3}} + \frac{3 \left (a + b x\right )^{\frac{5}{3}}}{5 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*a**2/(b**3*(a + b*x)**(1/3)) - 3*a*(a + b*x)**(2/3)/b**3 + 3*(a + b*x)**(5/3)
/(5*b**3)

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Mathematica [A]  time = 0.0215493, size = 34, normalized size = 0.69 \[ \frac{3 \left (-9 a^2-3 a b x+b^2 x^2\right )}{5 b^3 \sqrt [3]{a+b x}} \]

Antiderivative was successfully verified.

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

[Out]

(3*(-9*a^2 - 3*a*b*x + b^2*x^2))/(5*b^3*(a + b*x)^(1/3))

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Maple [A]  time = 0.007, size = 32, normalized size = 0.7 \[ -{\frac{-3\,{b}^{2}{x}^{2}+9\,abx+27\,{a}^{2}}{5\,{b}^{3}}{\frac{1}{\sqrt [3]{bx+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(b*x+a)^(4/3),x)

[Out]

-3/5/(b*x+a)^(1/3)*(-b^2*x^2+3*a*b*x+9*a^2)/b^3

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Maxima [A]  time = 1.35337, size = 55, normalized size = 1.12 \[ \frac{3 \,{\left (b x + a\right )}^{\frac{5}{3}}}{5 \, b^{3}} - \frac{3 \,{\left (b x + a\right )}^{\frac{2}{3}} a}{b^{3}} - \frac{3 \, a^{2}}{{\left (b x + a\right )}^{\frac{1}{3}} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/5*(b*x + a)^(5/3)/b^3 - 3*(b*x + a)^(2/3)*a/b^3 - 3*a^2/((b*x + a)^(1/3)*b^3)

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Fricas [A]  time = 0.203969, size = 41, normalized size = 0.84 \[ \frac{3 \,{\left (b^{2} x^{2} - 3 \, a b x - 9 \, a^{2}\right )}}{5 \,{\left (b x + a\right )}^{\frac{1}{3}} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/5*(b^2*x^2 - 3*a*b*x - 9*a^2)/((b*x + a)^(1/3)*b^3)

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Sympy [A]  time = 5.76842, size = 534, normalized size = 10.9 \[ - \frac{27 a^{\frac{29}{3}} \left (1 + \frac{b x}{a}\right )^{\frac{2}{3}}}{5 a^{8} b^{3} + 15 a^{7} b^{4} x + 15 a^{6} b^{5} x^{2} + 5 a^{5} b^{6} x^{3}} + \frac{27 a^{\frac{29}{3}}}{5 a^{8} b^{3} + 15 a^{7} b^{4} x + 15 a^{6} b^{5} x^{2} + 5 a^{5} b^{6} x^{3}} - \frac{63 a^{\frac{26}{3}} b x \left (1 + \frac{b x}{a}\right )^{\frac{2}{3}}}{5 a^{8} b^{3} + 15 a^{7} b^{4} x + 15 a^{6} b^{5} x^{2} + 5 a^{5} b^{6} x^{3}} + \frac{81 a^{\frac{26}{3}} b x}{5 a^{8} b^{3} + 15 a^{7} b^{4} x + 15 a^{6} b^{5} x^{2} + 5 a^{5} b^{6} x^{3}} - \frac{42 a^{\frac{23}{3}} b^{2} x^{2} \left (1 + \frac{b x}{a}\right )^{\frac{2}{3}}}{5 a^{8} b^{3} + 15 a^{7} b^{4} x + 15 a^{6} b^{5} x^{2} + 5 a^{5} b^{6} x^{3}} + \frac{81 a^{\frac{23}{3}} b^{2} x^{2}}{5 a^{8} b^{3} + 15 a^{7} b^{4} x + 15 a^{6} b^{5} x^{2} + 5 a^{5} b^{6} x^{3}} - \frac{3 a^{\frac{20}{3}} b^{3} x^{3} \left (1 + \frac{b x}{a}\right )^{\frac{2}{3}}}{5 a^{8} b^{3} + 15 a^{7} b^{4} x + 15 a^{6} b^{5} x^{2} + 5 a^{5} b^{6} x^{3}} + \frac{27 a^{\frac{20}{3}} b^{3} x^{3}}{5 a^{8} b^{3} + 15 a^{7} b^{4} x + 15 a^{6} b^{5} x^{2} + 5 a^{5} b^{6} x^{3}} + \frac{3 a^{\frac{17}{3}} b^{4} x^{4} \left (1 + \frac{b x}{a}\right )^{\frac{2}{3}}}{5 a^{8} b^{3} + 15 a^{7} b^{4} x + 15 a^{6} b^{5} x^{2} + 5 a^{5} b^{6} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.215462, size = 62, normalized size = 1.27 \[ -\frac{3 \, a^{2}}{{\left (b x + a\right )}^{\frac{1}{3}} b^{3}} + \frac{3 \,{\left ({\left (b x + a\right )}^{\frac{5}{3}} b^{12} - 5 \,{\left (b x + a\right )}^{\frac{2}{3}} a b^{12}\right )}}{5 \, b^{15}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*a^2/((b*x + a)^(1/3)*b^3) + 3/5*((b*x + a)^(5/3)*b^12 - 5*(b*x + a)^(2/3)*a*b
^12)/b^15