3.727 \(\int \frac{x^5}{\left (a+b x^2\right )^{4/3}} \, dx\)

Optimal. Leaf size=59 \[ -\frac{3 a^2}{2 b^3 \sqrt [3]{a+b x^2}}-\frac{3 a \left (a+b x^2\right )^{2/3}}{2 b^3}+\frac{3 \left (a+b x^2\right )^{5/3}}{10 b^3} \]

[Out]

(-3*a^2)/(2*b^3*(a + b*x^2)^(1/3)) - (3*a*(a + b*x^2)^(2/3))/(2*b^3) + (3*(a + b
*x^2)^(5/3))/(10*b^3)

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Rubi [A]  time = 0.0996635, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ -\frac{3 a^2}{2 b^3 \sqrt [3]{a+b x^2}}-\frac{3 a \left (a+b x^2\right )^{2/3}}{2 b^3}+\frac{3 \left (a+b x^2\right )^{5/3}}{10 b^3} \]

Antiderivative was successfully verified.

[In]  Int[x^5/(a + b*x^2)^(4/3),x]

[Out]

(-3*a^2)/(2*b^3*(a + b*x^2)^(1/3)) - (3*a*(a + b*x^2)^(2/3))/(2*b^3) + (3*(a + b
*x^2)^(5/3))/(10*b^3)

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Rubi in Sympy [A]  time = 11.4622, size = 54, normalized size = 0.92 \[ - \frac{3 a^{2}}{2 b^{3} \sqrt [3]{a + b x^{2}}} - \frac{3 a \left (a + b x^{2}\right )^{\frac{2}{3}}}{2 b^{3}} + \frac{3 \left (a + b x^{2}\right )^{\frac{5}{3}}}{10 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5/(b*x**2+a)**(4/3),x)

[Out]

-3*a**2/(2*b**3*(a + b*x**2)**(1/3)) - 3*a*(a + b*x**2)**(2/3)/(2*b**3) + 3*(a +
b*x**2)**(5/3)/(10*b**3)

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Mathematica [A]  time = 0.0287818, size = 38, normalized size = 0.64 \[ \frac{3 \left (-9 a^2-3 a b x^2+b^2 x^4\right )}{10 b^3 \sqrt [3]{a+b x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^5/(a + b*x^2)^(4/3),x]

[Out]

(3*(-9*a^2 - 3*a*b*x^2 + b^2*x^4))/(10*b^3*(a + b*x^2)^(1/3))

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Maple [A]  time = 0.008, size = 36, normalized size = 0.6 \[ -{\frac{-3\,{b}^{2}{x}^{4}+9\,ab{x}^{2}+27\,{a}^{2}}{10\,{b}^{3}}{\frac{1}{\sqrt [3]{b{x}^{2}+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5/(b*x^2+a)^(4/3),x)

[Out]

-3/10/(b*x^2+a)^(1/3)*(-b^2*x^4+3*a*b*x^2+9*a^2)/b^3

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Maxima [A]  time = 1.33747, size = 63, normalized size = 1.07 \[ \frac{3 \,{\left (b x^{2} + a\right )}^{\frac{5}{3}}}{10 \, b^{3}} - \frac{3 \,{\left (b x^{2} + a\right )}^{\frac{2}{3}} a}{2 \, b^{3}} - \frac{3 \, a^{2}}{2 \,{\left (b x^{2} + a\right )}^{\frac{1}{3}} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(b*x^2 + a)^(4/3),x, algorithm="maxima")

[Out]

3/10*(b*x^2 + a)^(5/3)/b^3 - 3/2*(b*x^2 + a)^(2/3)*a/b^3 - 3/2*a^2/((b*x^2 + a)^
(1/3)*b^3)

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Fricas [A]  time = 0.210167, size = 46, normalized size = 0.78 \[ \frac{3 \,{\left (b^{2} x^{4} - 3 \, a b x^{2} - 9 \, a^{2}\right )}}{10 \,{\left (b x^{2} + a\right )}^{\frac{1}{3}} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(b*x^2 + a)^(4/3),x, algorithm="fricas")

[Out]

3/10*(b^2*x^4 - 3*a*b*x^2 - 9*a^2)/((b*x^2 + a)^(1/3)*b^3)

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Sympy [A]  time = 6.19344, size = 561, normalized size = 9.51 \[ - \frac{27 a^{\frac{29}{3}} \left (1 + \frac{b x^{2}}{a}\right )^{\frac{2}{3}}}{10 a^{8} b^{3} + 30 a^{7} b^{4} x^{2} + 30 a^{6} b^{5} x^{4} + 10 a^{5} b^{6} x^{6}} + \frac{27 a^{\frac{29}{3}}}{10 a^{8} b^{3} + 30 a^{7} b^{4} x^{2} + 30 a^{6} b^{5} x^{4} + 10 a^{5} b^{6} x^{6}} - \frac{63 a^{\frac{26}{3}} b x^{2} \left (1 + \frac{b x^{2}}{a}\right )^{\frac{2}{3}}}{10 a^{8} b^{3} + 30 a^{7} b^{4} x^{2} + 30 a^{6} b^{5} x^{4} + 10 a^{5} b^{6} x^{6}} + \frac{81 a^{\frac{26}{3}} b x^{2}}{10 a^{8} b^{3} + 30 a^{7} b^{4} x^{2} + 30 a^{6} b^{5} x^{4} + 10 a^{5} b^{6} x^{6}} - \frac{42 a^{\frac{23}{3}} b^{2} x^{4} \left (1 + \frac{b x^{2}}{a}\right )^{\frac{2}{3}}}{10 a^{8} b^{3} + 30 a^{7} b^{4} x^{2} + 30 a^{6} b^{5} x^{4} + 10 a^{5} b^{6} x^{6}} + \frac{81 a^{\frac{23}{3}} b^{2} x^{4}}{10 a^{8} b^{3} + 30 a^{7} b^{4} x^{2} + 30 a^{6} b^{5} x^{4} + 10 a^{5} b^{6} x^{6}} - \frac{3 a^{\frac{20}{3}} b^{3} x^{6} \left (1 + \frac{b x^{2}}{a}\right )^{\frac{2}{3}}}{10 a^{8} b^{3} + 30 a^{7} b^{4} x^{2} + 30 a^{6} b^{5} x^{4} + 10 a^{5} b^{6} x^{6}} + \frac{27 a^{\frac{20}{3}} b^{3} x^{6}}{10 a^{8} b^{3} + 30 a^{7} b^{4} x^{2} + 30 a^{6} b^{5} x^{4} + 10 a^{5} b^{6} x^{6}} + \frac{3 a^{\frac{17}{3}} b^{4} x^{8} \left (1 + \frac{b x^{2}}{a}\right )^{\frac{2}{3}}}{10 a^{8} b^{3} + 30 a^{7} b^{4} x^{2} + 30 a^{6} b^{5} x^{4} + 10 a^{5} b^{6} x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5/(b*x**2+a)**(4/3),x)

[Out]

-27*a**(29/3)*(1 + b*x**2/a)**(2/3)/(10*a**8*b**3 + 30*a**7*b**4*x**2 + 30*a**6*
b**5*x**4 + 10*a**5*b**6*x**6) + 27*a**(29/3)/(10*a**8*b**3 + 30*a**7*b**4*x**2
+ 30*a**6*b**5*x**4 + 10*a**5*b**6*x**6) - 63*a**(26/3)*b*x**2*(1 + b*x**2/a)**(
2/3)/(10*a**8*b**3 + 30*a**7*b**4*x**2 + 30*a**6*b**5*x**4 + 10*a**5*b**6*x**6)
+ 81*a**(26/3)*b*x**2/(10*a**8*b**3 + 30*a**7*b**4*x**2 + 30*a**6*b**5*x**4 + 10
*a**5*b**6*x**6) - 42*a**(23/3)*b**2*x**4*(1 + b*x**2/a)**(2/3)/(10*a**8*b**3 +
30*a**7*b**4*x**2 + 30*a**6*b**5*x**4 + 10*a**5*b**6*x**6) + 81*a**(23/3)*b**2*x
**4/(10*a**8*b**3 + 30*a**7*b**4*x**2 + 30*a**6*b**5*x**4 + 10*a**5*b**6*x**6) -
3*a**(20/3)*b**3*x**6*(1 + b*x**2/a)**(2/3)/(10*a**8*b**3 + 30*a**7*b**4*x**2 +
30*a**6*b**5*x**4 + 10*a**5*b**6*x**6) + 27*a**(20/3)*b**3*x**6/(10*a**8*b**3 +
30*a**7*b**4*x**2 + 30*a**6*b**5*x**4 + 10*a**5*b**6*x**6) + 3*a**(17/3)*b**4*x
**8*(1 + b*x**2/a)**(2/3)/(10*a**8*b**3 + 30*a**7*b**4*x**2 + 30*a**6*b**5*x**4
+ 10*a**5*b**6*x**6)

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GIAC/XCAS [A]  time = 0.215172, size = 55, normalized size = 0.93 \[ \frac{3 \,{\left ({\left (b x^{2} + a\right )}^{\frac{5}{3}} - 5 \,{\left (b x^{2} + a\right )}^{\frac{2}{3}} a - \frac{5 \, a^{2}}{{\left (b x^{2} + a\right )}^{\frac{1}{3}}}\right )}}{10 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(b*x^2 + a)^(4/3),x, algorithm="giac")

[Out]

3/10*((b*x^2 + a)^(5/3) - 5*(b*x^2 + a)^(2/3)*a - 5*a^2/(b*x^2 + a)^(1/3))/b^3