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

Optimal. Leaf size=51 \[ \frac{3 a^2 \sqrt [3]{a+b x}}{b^3}+\frac{3 (a+b x)^{7/3}}{7 b^3}-\frac{3 a (a+b x)^{4/3}}{2 b^3} \]

[Out]

(3*a^2*(a + b*x)^(1/3))/b^3 - (3*a*(a + b*x)^(4/3))/(2*b^3) + (3*(a + b*x)^(7/3)
)/(7*b^3)

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Rubi [A]  time = 0.0382485, antiderivative size = 51, 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 \[ \frac{3 a^2 \sqrt [3]{a+b x}}{b^3}+\frac{3 (a+b x)^{7/3}}{7 b^3}-\frac{3 a (a+b x)^{4/3}}{2 b^3} \]

Antiderivative was successfully verified.

[In]  Int[x^2/(a + b*x)^(2/3),x]

[Out]

(3*a^2*(a + b*x)^(1/3))/b^3 - (3*a*(a + b*x)^(4/3))/(2*b^3) + (3*(a + b*x)^(7/3)
)/(7*b^3)

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Rubi in Sympy [A]  time = 7.84987, size = 48, normalized size = 0.94 \[ \frac{3 a^{2} \sqrt [3]{a + b x}}{b^{3}} - \frac{3 a \left (a + b x\right )^{\frac{4}{3}}}{2 b^{3}} + \frac{3 \left (a + b x\right )^{\frac{7}{3}}}{7 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*a**2*(a + b*x)**(1/3)/b**3 - 3*a*(a + b*x)**(4/3)/(2*b**3) + 3*(a + b*x)**(7/3
)/(7*b**3)

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Mathematica [A]  time = 0.0182218, size = 35, normalized size = 0.69 \[ \frac{3 \sqrt [3]{a+b x} \left (9 a^2-3 a b x+2 b^2 x^2\right )}{14 b^3} \]

Antiderivative was successfully verified.

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

[Out]

(3*(a + b*x)^(1/3)*(9*a^2 - 3*a*b*x + 2*b^2*x^2))/(14*b^3)

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Maple [A]  time = 0.007, size = 32, normalized size = 0.6 \[{\frac{6\,{b}^{2}{x}^{2}-9\,abx+27\,{a}^{2}}{14\,{b}^{3}}\sqrt [3]{bx+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(b*x+a)^(2/3),x)

[Out]

3/14*(b*x+a)^(1/3)*(2*b^2*x^2-3*a*b*x+9*a^2)/b^3

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Maxima [A]  time = 1.34147, size = 55, normalized size = 1.08 \[ \frac{3 \,{\left (b x + a\right )}^{\frac{7}{3}}}{7 \, b^{3}} - \frac{3 \,{\left (b x + a\right )}^{\frac{4}{3}} a}{2 \, b^{3}} + \frac{3 \,{\left (b x + a\right )}^{\frac{1}{3}} a^{2}}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/7*(b*x + a)^(7/3)/b^3 - 3/2*(b*x + a)^(4/3)*a/b^3 + 3*(b*x + a)^(1/3)*a^2/b^3

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Fricas [A]  time = 0.207488, size = 42, normalized size = 0.82 \[ \frac{3 \,{\left (2 \, b^{2} x^{2} - 3 \, a b x + 9 \, a^{2}\right )}{\left (b x + a\right )}^{\frac{1}{3}}}{14 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/14*(2*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 = 5.42302, size = 600, normalized size = 11.76 \[ \frac{27 a^{\frac{31}{3}} \sqrt [3]{1 + \frac{b x}{a}}}{14 a^{8} b^{3} + 42 a^{7} b^{4} x + 42 a^{6} b^{5} x^{2} + 14 a^{5} b^{6} x^{3}} - \frac{27 a^{\frac{31}{3}}}{14 a^{8} b^{3} + 42 a^{7} b^{4} x + 42 a^{6} b^{5} x^{2} + 14 a^{5} b^{6} x^{3}} + \frac{72 a^{\frac{28}{3}} b x \sqrt [3]{1 + \frac{b x}{a}}}{14 a^{8} b^{3} + 42 a^{7} b^{4} x + 42 a^{6} b^{5} x^{2} + 14 a^{5} b^{6} x^{3}} - \frac{81 a^{\frac{28}{3}} b x}{14 a^{8} b^{3} + 42 a^{7} b^{4} x + 42 a^{6} b^{5} x^{2} + 14 a^{5} b^{6} x^{3}} + \frac{60 a^{\frac{25}{3}} b^{2} x^{2} \sqrt [3]{1 + \frac{b x}{a}}}{14 a^{8} b^{3} + 42 a^{7} b^{4} x + 42 a^{6} b^{5} x^{2} + 14 a^{5} b^{6} x^{3}} - \frac{81 a^{\frac{25}{3}} b^{2} x^{2}}{14 a^{8} b^{3} + 42 a^{7} b^{4} x + 42 a^{6} b^{5} x^{2} + 14 a^{5} b^{6} x^{3}} + \frac{18 a^{\frac{22}{3}} b^{3} x^{3} \sqrt [3]{1 + \frac{b x}{a}}}{14 a^{8} b^{3} + 42 a^{7} b^{4} x + 42 a^{6} b^{5} x^{2} + 14 a^{5} b^{6} x^{3}} - \frac{27 a^{\frac{22}{3}} b^{3} x^{3}}{14 a^{8} b^{3} + 42 a^{7} b^{4} x + 42 a^{6} b^{5} x^{2} + 14 a^{5} b^{6} x^{3}} + \frac{9 a^{\frac{19}{3}} b^{4} x^{4} \sqrt [3]{1 + \frac{b x}{a}}}{14 a^{8} b^{3} + 42 a^{7} b^{4} x + 42 a^{6} b^{5} x^{2} + 14 a^{5} b^{6} x^{3}} + \frac{6 a^{\frac{16}{3}} b^{5} x^{5} \sqrt [3]{1 + \frac{b x}{a}}}{14 a^{8} b^{3} + 42 a^{7} b^{4} x + 42 a^{6} b^{5} x^{2} + 14 a^{5} b^{6} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

27*a**(31/3)*(1 + b*x/a)**(1/3)/(14*a**8*b**3 + 42*a**7*b**4*x + 42*a**6*b**5*x*
*2 + 14*a**5*b**6*x**3) - 27*a**(31/3)/(14*a**8*b**3 + 42*a**7*b**4*x + 42*a**6*
b**5*x**2 + 14*a**5*b**6*x**3) + 72*a**(28/3)*b*x*(1 + b*x/a)**(1/3)/(14*a**8*b*
*3 + 42*a**7*b**4*x + 42*a**6*b**5*x**2 + 14*a**5*b**6*x**3) - 81*a**(28/3)*b*x/
(14*a**8*b**3 + 42*a**7*b**4*x + 42*a**6*b**5*x**2 + 14*a**5*b**6*x**3) + 60*a**
(25/3)*b**2*x**2*(1 + b*x/a)**(1/3)/(14*a**8*b**3 + 42*a**7*b**4*x + 42*a**6*b**
5*x**2 + 14*a**5*b**6*x**3) - 81*a**(25/3)*b**2*x**2/(14*a**8*b**3 + 42*a**7*b**
4*x + 42*a**6*b**5*x**2 + 14*a**5*b**6*x**3) + 18*a**(22/3)*b**3*x**3*(1 + b*x/a
)**(1/3)/(14*a**8*b**3 + 42*a**7*b**4*x + 42*a**6*b**5*x**2 + 14*a**5*b**6*x**3)
- 27*a**(22/3)*b**3*x**3/(14*a**8*b**3 + 42*a**7*b**4*x + 42*a**6*b**5*x**2 + 1
4*a**5*b**6*x**3) + 9*a**(19/3)*b**4*x**4*(1 + b*x/a)**(1/3)/(14*a**8*b**3 + 42*
a**7*b**4*x + 42*a**6*b**5*x**2 + 14*a**5*b**6*x**3) + 6*a**(16/3)*b**5*x**5*(1
+ b*x/a)**(1/3)/(14*a**8*b**3 + 42*a**7*b**4*x + 42*a**6*b**5*x**2 + 14*a**5*b**
6*x**3)

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GIAC/XCAS [A]  time = 0.203474, size = 62, normalized size = 1.22 \[ \frac{3 \,{\left (2 \,{\left (b x + a\right )}^{\frac{7}{3}} b^{12} - 7 \,{\left (b x + a\right )}^{\frac{4}{3}} a b^{12} + 14 \,{\left (b x + a\right )}^{\frac{1}{3}} a^{2} b^{12}\right )}}{14 \, b^{15}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/14*(2*(b*x + a)^(7/3)*b^12 - 7*(b*x + a)^(4/3)*a*b^12 + 14*(b*x + a)^(1/3)*a^2
*b^12)/b^15