3.714 \(\int \frac{x^7}{\left (a+b x^2\right )^{2/3}} \, dx\)

Optimal. Leaf size=80 \[ -\frac{3 a^3 \sqrt [3]{a+b x^2}}{2 b^4}+\frac{9 a^2 \left (a+b x^2\right )^{4/3}}{8 b^4}+\frac{3 \left (a+b x^2\right )^{10/3}}{20 b^4}-\frac{9 a \left (a+b x^2\right )^{7/3}}{14 b^4} \]

[Out]

(-3*a^3*(a + b*x^2)^(1/3))/(2*b^4) + (9*a^2*(a + b*x^2)^(4/3))/(8*b^4) - (9*a*(a
+ b*x^2)^(7/3))/(14*b^4) + (3*(a + b*x^2)^(10/3))/(20*b^4)

_______________________________________________________________________________________

Rubi [A]  time = 0.128401, antiderivative size = 80, 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 \[ -\frac{3 a^3 \sqrt [3]{a+b x^2}}{2 b^4}+\frac{9 a^2 \left (a+b x^2\right )^{4/3}}{8 b^4}+\frac{3 \left (a+b x^2\right )^{10/3}}{20 b^4}-\frac{9 a \left (a+b x^2\right )^{7/3}}{14 b^4} \]

Antiderivative was successfully verified.

[In]  Int[x^7/(a + b*x^2)^(2/3),x]

[Out]

(-3*a^3*(a + b*x^2)^(1/3))/(2*b^4) + (9*a^2*(a + b*x^2)^(4/3))/(8*b^4) - (9*a*(a
+ b*x^2)^(7/3))/(14*b^4) + (3*(a + b*x^2)^(10/3))/(20*b^4)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 15.3842, size = 75, normalized size = 0.94 \[ - \frac{3 a^{3} \sqrt [3]{a + b x^{2}}}{2 b^{4}} + \frac{9 a^{2} \left (a + b x^{2}\right )^{\frac{4}{3}}}{8 b^{4}} - \frac{9 a \left (a + b x^{2}\right )^{\frac{7}{3}}}{14 b^{4}} + \frac{3 \left (a + b x^{2}\right )^{\frac{10}{3}}}{20 b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**7/(b*x**2+a)**(2/3),x)

[Out]

-3*a**3*(a + b*x**2)**(1/3)/(2*b**4) + 9*a**2*(a + b*x**2)**(4/3)/(8*b**4) - 9*a
*(a + b*x**2)**(7/3)/(14*b**4) + 3*(a + b*x**2)**(10/3)/(20*b**4)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0301498, size = 50, normalized size = 0.62 \[ \frac{3 \sqrt [3]{a+b x^2} \left (-81 a^3+27 a^2 b x^2-18 a b^2 x^4+14 b^3 x^6\right )}{280 b^4} \]

Antiderivative was successfully verified.

[In]  Integrate[x^7/(a + b*x^2)^(2/3),x]

[Out]

(3*(a + b*x^2)^(1/3)*(-81*a^3 + 27*a^2*b*x^2 - 18*a*b^2*x^4 + 14*b^3*x^6))/(280*
b^4)

_______________________________________________________________________________________

Maple [A]  time = 0.009, size = 47, normalized size = 0.6 \[ -{\frac{-42\,{b}^{3}{x}^{6}+54\,a{b}^{2}{x}^{4}-81\,{a}^{2}b{x}^{2}+243\,{a}^{3}}{280\,{b}^{4}}\sqrt [3]{b{x}^{2}+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^7/(b*x^2+a)^(2/3),x)

[Out]

-3/280*(b*x^2+a)^(1/3)*(-14*b^3*x^6+18*a*b^2*x^4-27*a^2*b*x^2+81*a^3)/b^4

_______________________________________________________________________________________

Maxima [A]  time = 1.34161, size = 86, normalized size = 1.08 \[ \frac{3 \,{\left (b x^{2} + a\right )}^{\frac{10}{3}}}{20 \, b^{4}} - \frac{9 \,{\left (b x^{2} + a\right )}^{\frac{7}{3}} a}{14 \, b^{4}} + \frac{9 \,{\left (b x^{2} + a\right )}^{\frac{4}{3}} a^{2}}{8 \, b^{4}} - \frac{3 \,{\left (b x^{2} + a\right )}^{\frac{1}{3}} a^{3}}{2 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^7/(b*x^2 + a)^(2/3),x, algorithm="maxima")

[Out]

3/20*(b*x^2 + a)^(10/3)/b^4 - 9/14*(b*x^2 + a)^(7/3)*a/b^4 + 9/8*(b*x^2 + a)^(4/
3)*a^2/b^4 - 3/2*(b*x^2 + a)^(1/3)*a^3/b^4

_______________________________________________________________________________________

Fricas [A]  time = 0.214285, size = 62, normalized size = 0.78 \[ \frac{3 \,{\left (14 \, b^{3} x^{6} - 18 \, a b^{2} x^{4} + 27 \, a^{2} b x^{2} - 81 \, a^{3}\right )}{\left (b x^{2} + a\right )}^{\frac{1}{3}}}{280 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^7/(b*x^2 + a)^(2/3),x, algorithm="fricas")

[Out]

3/280*(14*b^3*x^6 - 18*a*b^2*x^4 + 27*a^2*b*x^2 - 81*a^3)*(b*x^2 + a)^(1/3)/b^4

_______________________________________________________________________________________

Sympy [A]  time = 9.4416, size = 1690, normalized size = 21.12 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**7/(b*x**2+a)**(2/3),x)

[Out]

-243*a**(70/3)*(1 + b*x**2/a)**(1/3)/(280*a**20*b**4 + 1680*a**19*b**5*x**2 + 42
00*a**18*b**6*x**4 + 5600*a**17*b**7*x**6 + 4200*a**16*b**8*x**8 + 1680*a**15*b*
*9*x**10 + 280*a**14*b**10*x**12) + 243*a**(70/3)/(280*a**20*b**4 + 1680*a**19*b
**5*x**2 + 4200*a**18*b**6*x**4 + 5600*a**17*b**7*x**6 + 4200*a**16*b**8*x**8 +
1680*a**15*b**9*x**10 + 280*a**14*b**10*x**12) - 1377*a**(67/3)*b*x**2*(1 + b*x*
*2/a)**(1/3)/(280*a**20*b**4 + 1680*a**19*b**5*x**2 + 4200*a**18*b**6*x**4 + 560
0*a**17*b**7*x**6 + 4200*a**16*b**8*x**8 + 1680*a**15*b**9*x**10 + 280*a**14*b**
10*x**12) + 1458*a**(67/3)*b*x**2/(280*a**20*b**4 + 1680*a**19*b**5*x**2 + 4200*
a**18*b**6*x**4 + 5600*a**17*b**7*x**6 + 4200*a**16*b**8*x**8 + 1680*a**15*b**9*
x**10 + 280*a**14*b**10*x**12) - 3213*a**(64/3)*b**2*x**4*(1 + b*x**2/a)**(1/3)/
(280*a**20*b**4 + 1680*a**19*b**5*x**2 + 4200*a**18*b**6*x**4 + 5600*a**17*b**7*
x**6 + 4200*a**16*b**8*x**8 + 1680*a**15*b**9*x**10 + 280*a**14*b**10*x**12) + 3
645*a**(64/3)*b**2*x**4/(280*a**20*b**4 + 1680*a**19*b**5*x**2 + 4200*a**18*b**6
*x**4 + 5600*a**17*b**7*x**6 + 4200*a**16*b**8*x**8 + 1680*a**15*b**9*x**10 + 28
0*a**14*b**10*x**12) - 3927*a**(61/3)*b**3*x**6*(1 + b*x**2/a)**(1/3)/(280*a**20
*b**4 + 1680*a**19*b**5*x**2 + 4200*a**18*b**6*x**4 + 5600*a**17*b**7*x**6 + 420
0*a**16*b**8*x**8 + 1680*a**15*b**9*x**10 + 280*a**14*b**10*x**12) + 4860*a**(61
/3)*b**3*x**6/(280*a**20*b**4 + 1680*a**19*b**5*x**2 + 4200*a**18*b**6*x**4 + 56
00*a**17*b**7*x**6 + 4200*a**16*b**8*x**8 + 1680*a**15*b**9*x**10 + 280*a**14*b*
*10*x**12) - 2583*a**(58/3)*b**4*x**8*(1 + b*x**2/a)**(1/3)/(280*a**20*b**4 + 16
80*a**19*b**5*x**2 + 4200*a**18*b**6*x**4 + 5600*a**17*b**7*x**6 + 4200*a**16*b*
*8*x**8 + 1680*a**15*b**9*x**10 + 280*a**14*b**10*x**12) + 3645*a**(58/3)*b**4*x
**8/(280*a**20*b**4 + 1680*a**19*b**5*x**2 + 4200*a**18*b**6*x**4 + 5600*a**17*b
**7*x**6 + 4200*a**16*b**8*x**8 + 1680*a**15*b**9*x**10 + 280*a**14*b**10*x**12)
- 693*a**(55/3)*b**5*x**10*(1 + b*x**2/a)**(1/3)/(280*a**20*b**4 + 1680*a**19*b
**5*x**2 + 4200*a**18*b**6*x**4 + 5600*a**17*b**7*x**6 + 4200*a**16*b**8*x**8 +
1680*a**15*b**9*x**10 + 280*a**14*b**10*x**12) + 1458*a**(55/3)*b**5*x**10/(280*
a**20*b**4 + 1680*a**19*b**5*x**2 + 4200*a**18*b**6*x**4 + 5600*a**17*b**7*x**6
+ 4200*a**16*b**8*x**8 + 1680*a**15*b**9*x**10 + 280*a**14*b**10*x**12) + 273*a*
*(52/3)*b**6*x**12*(1 + b*x**2/a)**(1/3)/(280*a**20*b**4 + 1680*a**19*b**5*x**2
+ 4200*a**18*b**6*x**4 + 5600*a**17*b**7*x**6 + 4200*a**16*b**8*x**8 + 1680*a**1
5*b**9*x**10 + 280*a**14*b**10*x**12) + 243*a**(52/3)*b**6*x**12/(280*a**20*b**4
+ 1680*a**19*b**5*x**2 + 4200*a**18*b**6*x**4 + 5600*a**17*b**7*x**6 + 4200*a**
16*b**8*x**8 + 1680*a**15*b**9*x**10 + 280*a**14*b**10*x**12) + 387*a**(49/3)*b*
*7*x**14*(1 + b*x**2/a)**(1/3)/(280*a**20*b**4 + 1680*a**19*b**5*x**2 + 4200*a**
18*b**6*x**4 + 5600*a**17*b**7*x**6 + 4200*a**16*b**8*x**8 + 1680*a**15*b**9*x**
10 + 280*a**14*b**10*x**12) + 198*a**(46/3)*b**8*x**16*(1 + b*x**2/a)**(1/3)/(28
0*a**20*b**4 + 1680*a**19*b**5*x**2 + 4200*a**18*b**6*x**4 + 5600*a**17*b**7*x**
6 + 4200*a**16*b**8*x**8 + 1680*a**15*b**9*x**10 + 280*a**14*b**10*x**12) + 42*a
**(43/3)*b**9*x**18*(1 + b*x**2/a)**(1/3)/(280*a**20*b**4 + 1680*a**19*b**5*x**2
+ 4200*a**18*b**6*x**4 + 5600*a**17*b**7*x**6 + 4200*a**16*b**8*x**8 + 1680*a**
15*b**9*x**10 + 280*a**14*b**10*x**12)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.217219, size = 77, normalized size = 0.96 \[ \frac{3 \,{\left (14 \,{\left (b x^{2} + a\right )}^{\frac{10}{3}} - 60 \,{\left (b x^{2} + a\right )}^{\frac{7}{3}} a + 105 \,{\left (b x^{2} + a\right )}^{\frac{4}{3}} a^{2} - 140 \,{\left (b x^{2} + a\right )}^{\frac{1}{3}} a^{3}\right )}}{280 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^7/(b*x^2 + a)^(2/3),x, algorithm="giac")

[Out]

3/280*(14*(b*x^2 + a)^(10/3) - 60*(b*x^2 + a)^(7/3)*a + 105*(b*x^2 + a)^(4/3)*a^
2 - 140*(b*x^2 + a)^(1/3)*a^3)/b^4