3.666 \(\int x^5 \sqrt [3]{a+b x^2} \, dx\)

Optimal. Leaf size=59 \[ \frac{3 a^2 \left (a+b x^2\right )^{4/3}}{8 b^3}+\frac{3 \left (a+b x^2\right )^{10/3}}{20 b^3}-\frac{3 a \left (a+b x^2\right )^{7/3}}{7 b^3} \]

[Out]

(3*a^2*(a + b*x^2)^(4/3))/(8*b^3) - (3*a*(a + b*x^2)^(7/3))/(7*b^3) + (3*(a + b*
x^2)^(10/3))/(20*b^3)

_______________________________________________________________________________________

Rubi [A]  time = 0.0959325, 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 \[ \frac{3 a^2 \left (a+b x^2\right )^{4/3}}{8 b^3}+\frac{3 \left (a+b x^2\right )^{10/3}}{20 b^3}-\frac{3 a \left (a+b x^2\right )^{7/3}}{7 b^3} \]

Antiderivative was successfully verified.

[In]  Int[x^5*(a + b*x^2)^(1/3),x]

[Out]

(3*a^2*(a + b*x^2)^(4/3))/(8*b^3) - (3*a*(a + b*x^2)^(7/3))/(7*b^3) + (3*(a + b*
x^2)^(10/3))/(20*b^3)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 11.5431, size = 54, normalized size = 0.92 \[ \frac{3 a^{2} \left (a + b x^{2}\right )^{\frac{4}{3}}}{8 b^{3}} - \frac{3 a \left (a + b x^{2}\right )^{\frac{7}{3}}}{7 b^{3}} + \frac{3 \left (a + b x^{2}\right )^{\frac{10}{3}}}{20 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5*(b*x**2+a)**(1/3),x)

[Out]

3*a**2*(a + b*x**2)**(4/3)/(8*b**3) - 3*a*(a + b*x**2)**(7/3)/(7*b**3) + 3*(a +
b*x**2)**(10/3)/(20*b**3)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0236736, size = 50, normalized size = 0.85 \[ \frac{3 \sqrt [3]{a+b x^2} \left (9 a^3-3 a^2 b x^2+2 a b^2 x^4+14 b^3 x^6\right )}{280 b^3} \]

Antiderivative was successfully verified.

[In]  Integrate[x^5*(a + b*x^2)^(1/3),x]

[Out]

(3*(a + b*x^2)^(1/3)*(9*a^3 - 3*a^2*b*x^2 + 2*a*b^2*x^4 + 14*b^3*x^6))/(280*b^3)

_______________________________________________________________________________________

Maple [A]  time = 0.008, size = 36, normalized size = 0.6 \[{\frac{42\,{b}^{2}{x}^{4}-36\,ab{x}^{2}+27\,{a}^{2}}{280\,{b}^{3}} \left ( b{x}^{2}+a \right ) ^{{\frac{4}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5*(b*x^2+a)^(1/3),x)

[Out]

3/280*(b*x^2+a)^(4/3)*(14*b^2*x^4-12*a*b*x^2+9*a^2)/b^3

_______________________________________________________________________________________

Maxima [A]  time = 1.35652, size = 63, normalized size = 1.07 \[ \frac{3 \,{\left (b x^{2} + a\right )}^{\frac{10}{3}}}{20 \, b^{3}} - \frac{3 \,{\left (b x^{2} + a\right )}^{\frac{7}{3}} a}{7 \, b^{3}} + \frac{3 \,{\left (b x^{2} + a\right )}^{\frac{4}{3}} a^{2}}{8 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(1/3)*x^5,x, algorithm="maxima")

[Out]

3/20*(b*x^2 + a)^(10/3)/b^3 - 3/7*(b*x^2 + a)^(7/3)*a/b^3 + 3/8*(b*x^2 + a)^(4/3
)*a^2/b^3

_______________________________________________________________________________________

Fricas [A]  time = 0.203324, size = 62, normalized size = 1.05 \[ \frac{3 \,{\left (14 \, b^{3} x^{6} + 2 \, a b^{2} x^{4} - 3 \, a^{2} b x^{2} + 9 \, a^{3}\right )}{\left (b x^{2} + a\right )}^{\frac{1}{3}}}{280 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(1/3)*x^5,x, algorithm="fricas")

[Out]

3/280*(14*b^3*x^6 + 2*a*b^2*x^4 - 3*a^2*b*x^2 + 9*a^3)*(b*x^2 + a)^(1/3)/b^3

_______________________________________________________________________________________

Sympy [A]  time = 6.42102, size = 700, normalized size = 11.86 \[ \frac{27 a^{\frac{34}{3}} \sqrt [3]{1 + \frac{b x^{2}}{a}}}{280 a^{8} b^{3} + 840 a^{7} b^{4} x^{2} + 840 a^{6} b^{5} x^{4} + 280 a^{5} b^{6} x^{6}} - \frac{27 a^{\frac{34}{3}}}{280 a^{8} b^{3} + 840 a^{7} b^{4} x^{2} + 840 a^{6} b^{5} x^{4} + 280 a^{5} b^{6} x^{6}} + \frac{72 a^{\frac{31}{3}} b x^{2} \sqrt [3]{1 + \frac{b x^{2}}{a}}}{280 a^{8} b^{3} + 840 a^{7} b^{4} x^{2} + 840 a^{6} b^{5} x^{4} + 280 a^{5} b^{6} x^{6}} - \frac{81 a^{\frac{31}{3}} b x^{2}}{280 a^{8} b^{3} + 840 a^{7} b^{4} x^{2} + 840 a^{6} b^{5} x^{4} + 280 a^{5} b^{6} x^{6}} + \frac{60 a^{\frac{28}{3}} b^{2} x^{4} \sqrt [3]{1 + \frac{b x^{2}}{a}}}{280 a^{8} b^{3} + 840 a^{7} b^{4} x^{2} + 840 a^{6} b^{5} x^{4} + 280 a^{5} b^{6} x^{6}} - \frac{81 a^{\frac{28}{3}} b^{2} x^{4}}{280 a^{8} b^{3} + 840 a^{7} b^{4} x^{2} + 840 a^{6} b^{5} x^{4} + 280 a^{5} b^{6} x^{6}} + \frac{60 a^{\frac{25}{3}} b^{3} x^{6} \sqrt [3]{1 + \frac{b x^{2}}{a}}}{280 a^{8} b^{3} + 840 a^{7} b^{4} x^{2} + 840 a^{6} b^{5} x^{4} + 280 a^{5} b^{6} x^{6}} - \frac{27 a^{\frac{25}{3}} b^{3} x^{6}}{280 a^{8} b^{3} + 840 a^{7} b^{4} x^{2} + 840 a^{6} b^{5} x^{4} + 280 a^{5} b^{6} x^{6}} + \frac{135 a^{\frac{22}{3}} b^{4} x^{8} \sqrt [3]{1 + \frac{b x^{2}}{a}}}{280 a^{8} b^{3} + 840 a^{7} b^{4} x^{2} + 840 a^{6} b^{5} x^{4} + 280 a^{5} b^{6} x^{6}} + \frac{132 a^{\frac{19}{3}} b^{5} x^{10} \sqrt [3]{1 + \frac{b x^{2}}{a}}}{280 a^{8} b^{3} + 840 a^{7} b^{4} x^{2} + 840 a^{6} b^{5} x^{4} + 280 a^{5} b^{6} x^{6}} + \frac{42 a^{\frac{16}{3}} b^{6} x^{12} \sqrt [3]{1 + \frac{b x^{2}}{a}}}{280 a^{8} b^{3} + 840 a^{7} b^{4} x^{2} + 840 a^{6} b^{5} x^{4} + 280 a^{5} b^{6} x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5*(b*x**2+a)**(1/3),x)

[Out]

27*a**(34/3)*(1 + b*x**2/a)**(1/3)/(280*a**8*b**3 + 840*a**7*b**4*x**2 + 840*a**
6*b**5*x**4 + 280*a**5*b**6*x**6) - 27*a**(34/3)/(280*a**8*b**3 + 840*a**7*b**4*
x**2 + 840*a**6*b**5*x**4 + 280*a**5*b**6*x**6) + 72*a**(31/3)*b*x**2*(1 + b*x**
2/a)**(1/3)/(280*a**8*b**3 + 840*a**7*b**4*x**2 + 840*a**6*b**5*x**4 + 280*a**5*
b**6*x**6) - 81*a**(31/3)*b*x**2/(280*a**8*b**3 + 840*a**7*b**4*x**2 + 840*a**6*
b**5*x**4 + 280*a**5*b**6*x**6) + 60*a**(28/3)*b**2*x**4*(1 + b*x**2/a)**(1/3)/(
280*a**8*b**3 + 840*a**7*b**4*x**2 + 840*a**6*b**5*x**4 + 280*a**5*b**6*x**6) -
81*a**(28/3)*b**2*x**4/(280*a**8*b**3 + 840*a**7*b**4*x**2 + 840*a**6*b**5*x**4
+ 280*a**5*b**6*x**6) + 60*a**(25/3)*b**3*x**6*(1 + b*x**2/a)**(1/3)/(280*a**8*b
**3 + 840*a**7*b**4*x**2 + 840*a**6*b**5*x**4 + 280*a**5*b**6*x**6) - 27*a**(25/
3)*b**3*x**6/(280*a**8*b**3 + 840*a**7*b**4*x**2 + 840*a**6*b**5*x**4 + 280*a**5
*b**6*x**6) + 135*a**(22/3)*b**4*x**8*(1 + b*x**2/a)**(1/3)/(280*a**8*b**3 + 840
*a**7*b**4*x**2 + 840*a**6*b**5*x**4 + 280*a**5*b**6*x**6) + 132*a**(19/3)*b**5*
x**10*(1 + b*x**2/a)**(1/3)/(280*a**8*b**3 + 840*a**7*b**4*x**2 + 840*a**6*b**5*
x**4 + 280*a**5*b**6*x**6) + 42*a**(16/3)*b**6*x**12*(1 + b*x**2/a)**(1/3)/(280*
a**8*b**3 + 840*a**7*b**4*x**2 + 840*a**6*b**5*x**4 + 280*a**5*b**6*x**6)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.21708, size = 58, normalized size = 0.98 \[ \frac{3 \,{\left (14 \,{\left (b x^{2} + a\right )}^{\frac{10}{3}} - 40 \,{\left (b x^{2} + a\right )}^{\frac{7}{3}} a + 35 \,{\left (b x^{2} + a\right )}^{\frac{4}{3}} a^{2}\right )}}{280 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(1/3)*x^5,x, algorithm="giac")

[Out]

3/280*(14*(b*x^2 + a)^(10/3) - 40*(b*x^2 + a)^(7/3)*a + 35*(b*x^2 + a)^(4/3)*a^2
)/b^3