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

Optimal. Leaf size=59 \[ \frac{3 a^2 \sqrt [3]{a+b x^2}}{2 b^3}+\frac{3 \left (a+b x^2\right )^{7/3}}{14 b^3}-\frac{3 a \left (a+b x^2\right )^{4/3}}{4 b^3} \]

[Out]

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

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Rubi [A]  time = 0.101157, 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 \[ \frac{3 a^2 \sqrt [3]{a+b x^2}}{2 b^3}+\frac{3 \left (a+b x^2\right )^{7/3}}{14 b^3}-\frac{3 a \left (a+b x^2\right )^{4/3}}{4 b^3} \]

Antiderivative was successfully verified.

[In]  Int[x^5/(a + b*x^2)^(2/3),x]

[Out]

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

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Rubi in Sympy [A]  time = 11.4539, size = 54, normalized size = 0.92 \[ \frac{3 a^{2} \sqrt [3]{a + b x^{2}}}{2 b^{3}} - \frac{3 a \left (a + b x^{2}\right )^{\frac{4}{3}}}{4 b^{3}} + \frac{3 \left (a + b x^{2}\right )^{\frac{7}{3}}}{14 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0241341, size = 39, normalized size = 0.66 \[ \frac{3 \sqrt [3]{a+b x^2} \left (9 a^2-3 a b x^2+2 b^2 x^4\right )}{28 b^3} \]

Antiderivative was successfully verified.

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

[Out]

(3*(a + b*x^2)^(1/3)*(9*a^2 - 3*a*b*x^2 + 2*b^2*x^4))/(28*b^3)

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Maple [A]  time = 0.008, size = 36, normalized size = 0.6 \[{\frac{6\,{b}^{2}{x}^{4}-9\,ab{x}^{2}+27\,{a}^{2}}{28\,{b}^{3}}\sqrt [3]{b{x}^{2}+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5/(b*x^2+a)^(2/3),x)

[Out]

3/28*(b*x^2+a)^(1/3)*(2*b^2*x^4-3*a*b*x^2+9*a^2)/b^3

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Maxima [A]  time = 1.33148, size = 63, normalized size = 1.07 \[ \frac{3 \,{\left (b x^{2} + a\right )}^{\frac{7}{3}}}{14 \, b^{3}} - \frac{3 \,{\left (b x^{2} + a\right )}^{\frac{4}{3}} a}{4 \, b^{3}} + \frac{3 \,{\left (b x^{2} + a\right )}^{\frac{1}{3}} a^{2}}{2 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.21553, size = 47, normalized size = 0.8 \[ \frac{3 \,{\left (2 \, b^{2} x^{4} - 3 \, a b x^{2} + 9 \, a^{2}\right )}{\left (b x^{2} + a\right )}^{\frac{1}{3}}}{28 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/28*(2*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 = 6.04621, size = 631, normalized size = 10.69 \[ \frac{27 a^{\frac{31}{3}} \sqrt [3]{1 + \frac{b x^{2}}{a}}}{28 a^{8} b^{3} + 84 a^{7} b^{4} x^{2} + 84 a^{6} b^{5} x^{4} + 28 a^{5} b^{6} x^{6}} - \frac{27 a^{\frac{31}{3}}}{28 a^{8} b^{3} + 84 a^{7} b^{4} x^{2} + 84 a^{6} b^{5} x^{4} + 28 a^{5} b^{6} x^{6}} + \frac{72 a^{\frac{28}{3}} b x^{2} \sqrt [3]{1 + \frac{b x^{2}}{a}}}{28 a^{8} b^{3} + 84 a^{7} b^{4} x^{2} + 84 a^{6} b^{5} x^{4} + 28 a^{5} b^{6} x^{6}} - \frac{81 a^{\frac{28}{3}} b x^{2}}{28 a^{8} b^{3} + 84 a^{7} b^{4} x^{2} + 84 a^{6} b^{5} x^{4} + 28 a^{5} b^{6} x^{6}} + \frac{60 a^{\frac{25}{3}} b^{2} x^{4} \sqrt [3]{1 + \frac{b x^{2}}{a}}}{28 a^{8} b^{3} + 84 a^{7} b^{4} x^{2} + 84 a^{6} b^{5} x^{4} + 28 a^{5} b^{6} x^{6}} - \frac{81 a^{\frac{25}{3}} b^{2} x^{4}}{28 a^{8} b^{3} + 84 a^{7} b^{4} x^{2} + 84 a^{6} b^{5} x^{4} + 28 a^{5} b^{6} x^{6}} + \frac{18 a^{\frac{22}{3}} b^{3} x^{6} \sqrt [3]{1 + \frac{b x^{2}}{a}}}{28 a^{8} b^{3} + 84 a^{7} b^{4} x^{2} + 84 a^{6} b^{5} x^{4} + 28 a^{5} b^{6} x^{6}} - \frac{27 a^{\frac{22}{3}} b^{3} x^{6}}{28 a^{8} b^{3} + 84 a^{7} b^{4} x^{2} + 84 a^{6} b^{5} x^{4} + 28 a^{5} b^{6} x^{6}} + \frac{9 a^{\frac{19}{3}} b^{4} x^{8} \sqrt [3]{1 + \frac{b x^{2}}{a}}}{28 a^{8} b^{3} + 84 a^{7} b^{4} x^{2} + 84 a^{6} b^{5} x^{4} + 28 a^{5} b^{6} x^{6}} + \frac{6 a^{\frac{16}{3}} b^{5} x^{10} \sqrt [3]{1 + \frac{b x^{2}}{a}}}{28 a^{8} b^{3} + 84 a^{7} b^{4} x^{2} + 84 a^{6} b^{5} x^{4} + 28 a^{5} b^{6} x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

27*a**(31/3)*(1 + b*x**2/a)**(1/3)/(28*a**8*b**3 + 84*a**7*b**4*x**2 + 84*a**6*b
**5*x**4 + 28*a**5*b**6*x**6) - 27*a**(31/3)/(28*a**8*b**3 + 84*a**7*b**4*x**2 +
84*a**6*b**5*x**4 + 28*a**5*b**6*x**6) + 72*a**(28/3)*b*x**2*(1 + b*x**2/a)**(1
/3)/(28*a**8*b**3 + 84*a**7*b**4*x**2 + 84*a**6*b**5*x**4 + 28*a**5*b**6*x**6) -
81*a**(28/3)*b*x**2/(28*a**8*b**3 + 84*a**7*b**4*x**2 + 84*a**6*b**5*x**4 + 28*
a**5*b**6*x**6) + 60*a**(25/3)*b**2*x**4*(1 + b*x**2/a)**(1/3)/(28*a**8*b**3 + 8
4*a**7*b**4*x**2 + 84*a**6*b**5*x**4 + 28*a**5*b**6*x**6) - 81*a**(25/3)*b**2*x*
*4/(28*a**8*b**3 + 84*a**7*b**4*x**2 + 84*a**6*b**5*x**4 + 28*a**5*b**6*x**6) +
18*a**(22/3)*b**3*x**6*(1 + b*x**2/a)**(1/3)/(28*a**8*b**3 + 84*a**7*b**4*x**2 +
84*a**6*b**5*x**4 + 28*a**5*b**6*x**6) - 27*a**(22/3)*b**3*x**6/(28*a**8*b**3 +
84*a**7*b**4*x**2 + 84*a**6*b**5*x**4 + 28*a**5*b**6*x**6) + 9*a**(19/3)*b**4*x
**8*(1 + b*x**2/a)**(1/3)/(28*a**8*b**3 + 84*a**7*b**4*x**2 + 84*a**6*b**5*x**4
+ 28*a**5*b**6*x**6) + 6*a**(16/3)*b**5*x**10*(1 + b*x**2/a)**(1/3)/(28*a**8*b**
3 + 84*a**7*b**4*x**2 + 84*a**6*b**5*x**4 + 28*a**5*b**6*x**6)

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GIAC/XCAS [A]  time = 0.212791, size = 58, normalized size = 0.98 \[ \frac{3 \,{\left (2 \,{\left (b x^{2} + a\right )}^{\frac{7}{3}} - 7 \,{\left (b x^{2} + a\right )}^{\frac{4}{3}} a + 14 \,{\left (b x^{2} + a\right )}^{\frac{1}{3}} a^{2}\right )}}{28 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/28*(2*(b*x^2 + a)^(7/3) - 7*(b*x^2 + a)^(4/3)*a + 14*(b*x^2 + a)^(1/3)*a^2)/b^
3