3.678 \(\int x^5 (a+b x^2)^{2/3} \, dx\)

Optimal. Leaf size=59 \[ \frac {3 a^2 \left (a+b x^2\right )^{5/3}}{10 b^3}+\frac {3 \left (a+b x^2\right )^{11/3}}{22 b^3}-\frac {3 a \left (a+b x^2\right )^{8/3}}{8 b^3} \]

[Out]

3/10*a^2*(b*x^2+a)^(5/3)/b^3-3/8*a*(b*x^2+a)^(8/3)/b^3+3/22*(b*x^2+a)^(11/3)/b^3

________________________________________________________________________________________

Rubi [A]  time = 0.04, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {266, 43} \[ \frac {3 a^2 \left (a+b x^2\right )^{5/3}}{10 b^3}+\frac {3 \left (a+b x^2\right )^{11/3}}{22 b^3}-\frac {3 a \left (a+b x^2\right )^{8/3}}{8 b^3} \]

Antiderivative was successfully verified.

[In]

Int[x^5*(a + b*x^2)^(2/3),x]

[Out]

(3*a^2*(a + b*x^2)^(5/3))/(10*b^3) - (3*a*(a + b*x^2)^(8/3))/(8*b^3) + (3*(a + b*x^2)^(11/3))/(22*b^3)

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps

\begin {align*} \int x^5 \left (a+b x^2\right )^{2/3} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int x^2 (a+b x)^{2/3} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {a^2 (a+b x)^{2/3}}{b^2}-\frac {2 a (a+b x)^{5/3}}{b^2}+\frac {(a+b x)^{8/3}}{b^2}\right ) \, dx,x,x^2\right )\\ &=\frac {3 a^2 \left (a+b x^2\right )^{5/3}}{10 b^3}-\frac {3 a \left (a+b x^2\right )^{8/3}}{8 b^3}+\frac {3 \left (a+b x^2\right )^{11/3}}{22 b^3}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.02, size = 39, normalized size = 0.66 \[ \frac {3 \left (a+b x^2\right )^{5/3} \left (9 a^2-15 a b x^2+20 b^2 x^4\right )}{440 b^3} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5*(a + b*x^2)^(2/3),x]

[Out]

(3*(a + b*x^2)^(5/3)*(9*a^2 - 15*a*b*x^2 + 20*b^2*x^4))/(440*b^3)

________________________________________________________________________________________

fricas [A]  time = 0.81, size = 46, normalized size = 0.78 \[ \frac {3 \, {\left (20 \, b^{3} x^{6} + 5 \, a b^{2} x^{4} - 6 \, a^{2} b x^{2} + 9 \, a^{3}\right )} {\left (b x^{2} + a\right )}^{\frac {2}{3}}}{440 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/440*(20*b^3*x^6 + 5*a*b^2*x^4 - 6*a^2*b*x^2 + 9*a^3)*(b*x^2 + a)^(2/3)/b^3

________________________________________________________________________________________

giac [A]  time = 0.97, size = 43, normalized size = 0.73 \[ \frac {3 \, {\left (20 \, {\left (b x^{2} + a\right )}^{\frac {11}{3}} - 55 \, {\left (b x^{2} + a\right )}^{\frac {8}{3}} a + 44 \, {\left (b x^{2} + a\right )}^{\frac {5}{3}} a^{2}\right )}}{440 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/440*(20*(b*x^2 + a)^(11/3) - 55*(b*x^2 + a)^(8/3)*a + 44*(b*x^2 + a)^(5/3)*a^2)/b^3

________________________________________________________________________________________

maple [A]  time = 0.01, size = 36, normalized size = 0.61 \[ \frac {3 \left (b \,x^{2}+a \right )^{\frac {5}{3}} \left (20 b^{2} x^{4}-15 a b \,x^{2}+9 a^{2}\right )}{440 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(b*x^2+a)^(2/3),x)

[Out]

3/440*(b*x^2+a)^(5/3)*(20*b^2*x^4-15*a*b*x^2+9*a^2)/b^3

________________________________________________________________________________________

maxima [A]  time = 1.35, size = 47, normalized size = 0.80 \[ \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {11}{3}}}{22 \, b^{3}} - \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {8}{3}} a}{8 \, b^{3}} + \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {5}{3}} a^{2}}{10 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/22*(b*x^2 + a)^(11/3)/b^3 - 3/8*(b*x^2 + a)^(8/3)*a/b^3 + 3/10*(b*x^2 + a)^(5/3)*a^2/b^3

________________________________________________________________________________________

mupad [B]  time = 4.66, size = 44, normalized size = 0.75 \[ {\left (b\,x^2+a\right )}^{2/3}\,\left (\frac {3\,x^6}{22}+\frac {27\,a^3}{440\,b^3}+\frac {3\,a\,x^4}{88\,b}-\frac {9\,a^2\,x^2}{220\,b^2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(a + b*x^2)^(2/3),x)

[Out]

(a + b*x^2)^(2/3)*((3*x^6)/22 + (27*a^3)/(440*b^3) + (3*a*x^4)/(88*b) - (9*a^2*x^2)/(220*b^2))

________________________________________________________________________________________

sympy [B]  time = 2.00, size = 700, normalized size = 11.86 \[ \frac {27 a^{\frac {35}{3}} \left (1 + \frac {b x^{2}}{a}\right )^{\frac {2}{3}}}{440 a^{8} b^{3} + 1320 a^{7} b^{4} x^{2} + 1320 a^{6} b^{5} x^{4} + 440 a^{5} b^{6} x^{6}} - \frac {27 a^{\frac {35}{3}}}{440 a^{8} b^{3} + 1320 a^{7} b^{4} x^{2} + 1320 a^{6} b^{5} x^{4} + 440 a^{5} b^{6} x^{6}} + \frac {63 a^{\frac {32}{3}} b x^{2} \left (1 + \frac {b x^{2}}{a}\right )^{\frac {2}{3}}}{440 a^{8} b^{3} + 1320 a^{7} b^{4} x^{2} + 1320 a^{6} b^{5} x^{4} + 440 a^{5} b^{6} x^{6}} - \frac {81 a^{\frac {32}{3}} b x^{2}}{440 a^{8} b^{3} + 1320 a^{7} b^{4} x^{2} + 1320 a^{6} b^{5} x^{4} + 440 a^{5} b^{6} x^{6}} + \frac {42 a^{\frac {29}{3}} b^{2} x^{4} \left (1 + \frac {b x^{2}}{a}\right )^{\frac {2}{3}}}{440 a^{8} b^{3} + 1320 a^{7} b^{4} x^{2} + 1320 a^{6} b^{5} x^{4} + 440 a^{5} b^{6} x^{6}} - \frac {81 a^{\frac {29}{3}} b^{2} x^{4}}{440 a^{8} b^{3} + 1320 a^{7} b^{4} x^{2} + 1320 a^{6} b^{5} x^{4} + 440 a^{5} b^{6} x^{6}} + \frac {78 a^{\frac {26}{3}} b^{3} x^{6} \left (1 + \frac {b x^{2}}{a}\right )^{\frac {2}{3}}}{440 a^{8} b^{3} + 1320 a^{7} b^{4} x^{2} + 1320 a^{6} b^{5} x^{4} + 440 a^{5} b^{6} x^{6}} - \frac {27 a^{\frac {26}{3}} b^{3} x^{6}}{440 a^{8} b^{3} + 1320 a^{7} b^{4} x^{2} + 1320 a^{6} b^{5} x^{4} + 440 a^{5} b^{6} x^{6}} + \frac {207 a^{\frac {23}{3}} b^{4} x^{8} \left (1 + \frac {b x^{2}}{a}\right )^{\frac {2}{3}}}{440 a^{8} b^{3} + 1320 a^{7} b^{4} x^{2} + 1320 a^{6} b^{5} x^{4} + 440 a^{5} b^{6} x^{6}} + \frac {195 a^{\frac {20}{3}} b^{5} x^{10} \left (1 + \frac {b x^{2}}{a}\right )^{\frac {2}{3}}}{440 a^{8} b^{3} + 1320 a^{7} b^{4} x^{2} + 1320 a^{6} b^{5} x^{4} + 440 a^{5} b^{6} x^{6}} + \frac {60 a^{\frac {17}{3}} b^{6} x^{12} \left (1 + \frac {b x^{2}}{a}\right )^{\frac {2}{3}}}{440 a^{8} b^{3} + 1320 a^{7} b^{4} x^{2} + 1320 a^{6} b^{5} x^{4} + 440 a^{5} b^{6} x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

27*a**(35/3)*(1 + b*x**2/a)**(2/3)/(440*a**8*b**3 + 1320*a**7*b**4*x**2 + 1320*a**6*b**5*x**4 + 440*a**5*b**6*
x**6) - 27*a**(35/3)/(440*a**8*b**3 + 1320*a**7*b**4*x**2 + 1320*a**6*b**5*x**4 + 440*a**5*b**6*x**6) + 63*a**
(32/3)*b*x**2*(1 + b*x**2/a)**(2/3)/(440*a**8*b**3 + 1320*a**7*b**4*x**2 + 1320*a**6*b**5*x**4 + 440*a**5*b**6
*x**6) - 81*a**(32/3)*b*x**2/(440*a**8*b**3 + 1320*a**7*b**4*x**2 + 1320*a**6*b**5*x**4 + 440*a**5*b**6*x**6)
+ 42*a**(29/3)*b**2*x**4*(1 + b*x**2/a)**(2/3)/(440*a**8*b**3 + 1320*a**7*b**4*x**2 + 1320*a**6*b**5*x**4 + 44
0*a**5*b**6*x**6) - 81*a**(29/3)*b**2*x**4/(440*a**8*b**3 + 1320*a**7*b**4*x**2 + 1320*a**6*b**5*x**4 + 440*a*
*5*b**6*x**6) + 78*a**(26/3)*b**3*x**6*(1 + b*x**2/a)**(2/3)/(440*a**8*b**3 + 1320*a**7*b**4*x**2 + 1320*a**6*
b**5*x**4 + 440*a**5*b**6*x**6) - 27*a**(26/3)*b**3*x**6/(440*a**8*b**3 + 1320*a**7*b**4*x**2 + 1320*a**6*b**5
*x**4 + 440*a**5*b**6*x**6) + 207*a**(23/3)*b**4*x**8*(1 + b*x**2/a)**(2/3)/(440*a**8*b**3 + 1320*a**7*b**4*x*
*2 + 1320*a**6*b**5*x**4 + 440*a**5*b**6*x**6) + 195*a**(20/3)*b**5*x**10*(1 + b*x**2/a)**(2/3)/(440*a**8*b**3
+ 1320*a**7*b**4*x**2 + 1320*a**6*b**5*x**4 + 440*a**5*b**6*x**6) + 60*a**(17/3)*b**6*x**12*(1 + b*x**2/a)**(
2/3)/(440*a**8*b**3 + 1320*a**7*b**4*x**2 + 1320*a**6*b**5*x**4 + 440*a**5*b**6*x**6)

________________________________________________________________________________________