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

Optimal. Leaf size=80 \[ -\frac{3 a^3 \left (a+b x^2\right )^{5/3}}{10 b^4}+\frac{9 a^2 \left (a+b x^2\right )^{8/3}}{16 b^4}+\frac{3 \left (a+b x^2\right )^{14/3}}{28 b^4}-\frac{9 a \left (a+b x^2\right )^{11/3}}{22 b^4} \]

[Out]

(-3*a^3*(a + b*x^2)^(5/3))/(10*b^4) + (9*a^2*(a + b*x^2)^(8/3))/(16*b^4) - (9*a*
(a + b*x^2)^(11/3))/(22*b^4) + (3*(a + b*x^2)^(14/3))/(28*b^4)

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Rubi [A]  time = 0.123728, 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 \[ -\frac{3 a^3 \left (a+b x^2\right )^{5/3}}{10 b^4}+\frac{9 a^2 \left (a+b x^2\right )^{8/3}}{16 b^4}+\frac{3 \left (a+b x^2\right )^{14/3}}{28 b^4}-\frac{9 a \left (a+b x^2\right )^{11/3}}{22 b^4} \]

Antiderivative was successfully verified.

[In]  Int[x^7*(a + b*x^2)^(2/3),x]

[Out]

(-3*a^3*(a + b*x^2)^(5/3))/(10*b^4) + (9*a^2*(a + b*x^2)^(8/3))/(16*b^4) - (9*a*
(a + b*x^2)^(11/3))/(22*b^4) + (3*(a + b*x^2)^(14/3))/(28*b^4)

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Rubi in Sympy [A]  time = 15.4796, size = 75, normalized size = 0.94 \[ - \frac{3 a^{3} \left (a + b x^{2}\right )^{\frac{5}{3}}}{10 b^{4}} + \frac{9 a^{2} \left (a + b x^{2}\right )^{\frac{8}{3}}}{16 b^{4}} - \frac{9 a \left (a + b x^{2}\right )^{\frac{11}{3}}}{22 b^{4}} + \frac{3 \left (a + b x^{2}\right )^{\frac{14}{3}}}{28 b^{4}} \]

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)**(5/3)/(10*b**4) + 9*a**2*(a + b*x**2)**(8/3)/(16*b**4) - 9
*a*(a + b*x**2)**(11/3)/(22*b**4) + 3*(a + b*x**2)**(14/3)/(28*b**4)

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Mathematica [A]  time = 0.0298714, size = 61, normalized size = 0.76 \[ \frac{3 \left (a+b x^2\right )^{2/3} \left (-81 a^4+54 a^3 b x^2-45 a^2 b^2 x^4+40 a b^3 x^6+220 b^4 x^8\right )}{6160 b^4} \]

Antiderivative was successfully verified.

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

[Out]

(3*(a + b*x^2)^(2/3)*(-81*a^4 + 54*a^3*b*x^2 - 45*a^2*b^2*x^4 + 40*a*b^3*x^6 + 2
20*b^4*x^8))/(6160*b^4)

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Maple [A]  time = 0.008, size = 47, normalized size = 0.6 \[ -{\frac{-660\,{b}^{3}{x}^{6}+540\,a{b}^{2}{x}^{4}-405\,{a}^{2}b{x}^{2}+243\,{a}^{3}}{6160\,{b}^{4}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^7*(b*x^2+a)^(2/3),x)

[Out]

-3/6160*(b*x^2+a)^(5/3)*(-220*b^3*x^6+180*a*b^2*x^4-135*a^2*b*x^2+81*a^3)/b^4

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Maxima [A]  time = 1.34686, size = 86, normalized size = 1.08 \[ \frac{3 \,{\left (b x^{2} + a\right )}^{\frac{14}{3}}}{28 \, b^{4}} - \frac{9 \,{\left (b x^{2} + a\right )}^{\frac{11}{3}} a}{22 \, b^{4}} + \frac{9 \,{\left (b x^{2} + a\right )}^{\frac{8}{3}} a^{2}}{16 \, b^{4}} - \frac{3 \,{\left (b x^{2} + a\right )}^{\frac{5}{3}} a^{3}}{10 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/28*(b*x^2 + a)^(14/3)/b^4 - 9/22*(b*x^2 + a)^(11/3)*a/b^4 + 9/16*(b*x^2 + a)^(
8/3)*a^2/b^4 - 3/10*(b*x^2 + a)^(5/3)*a^3/b^4

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Fricas [A]  time = 0.211101, size = 77, normalized size = 0.96 \[ \frac{3 \,{\left (220 \, b^{4} x^{8} + 40 \, a b^{3} x^{6} - 45 \, a^{2} b^{2} x^{4} + 54 \, a^{3} b x^{2} - 81 \, a^{4}\right )}{\left (b x^{2} + a\right )}^{\frac{2}{3}}}{6160 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/6160*(220*b^4*x^8 + 40*a*b^3*x^6 - 45*a^2*b^2*x^4 + 54*a^3*b*x^2 - 81*a^4)*(b*
x^2 + a)^(2/3)/b^4

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Sympy [A]  time = 11.0237, size = 1795, normalized size = 22.44 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-243*a**(74/3)*(1 + b*x**2/a)**(2/3)/(6160*a**20*b**4 + 36960*a**19*b**5*x**2 +
92400*a**18*b**6*x**4 + 123200*a**17*b**7*x**6 + 92400*a**16*b**8*x**8 + 36960*a
**15*b**9*x**10 + 6160*a**14*b**10*x**12) + 243*a**(74/3)/(6160*a**20*b**4 + 369
60*a**19*b**5*x**2 + 92400*a**18*b**6*x**4 + 123200*a**17*b**7*x**6 + 92400*a**1
6*b**8*x**8 + 36960*a**15*b**9*x**10 + 6160*a**14*b**10*x**12) - 1296*a**(71/3)*
b*x**2*(1 + b*x**2/a)**(2/3)/(6160*a**20*b**4 + 36960*a**19*b**5*x**2 + 92400*a*
*18*b**6*x**4 + 123200*a**17*b**7*x**6 + 92400*a**16*b**8*x**8 + 36960*a**15*b**
9*x**10 + 6160*a**14*b**10*x**12) + 1458*a**(71/3)*b*x**2/(6160*a**20*b**4 + 369
60*a**19*b**5*x**2 + 92400*a**18*b**6*x**4 + 123200*a**17*b**7*x**6 + 92400*a**1
6*b**8*x**8 + 36960*a**15*b**9*x**10 + 6160*a**14*b**10*x**12) - 2808*a**(68/3)*
b**2*x**4*(1 + b*x**2/a)**(2/3)/(6160*a**20*b**4 + 36960*a**19*b**5*x**2 + 92400
*a**18*b**6*x**4 + 123200*a**17*b**7*x**6 + 92400*a**16*b**8*x**8 + 36960*a**15*
b**9*x**10 + 6160*a**14*b**10*x**12) + 3645*a**(68/3)*b**2*x**4/(6160*a**20*b**4
+ 36960*a**19*b**5*x**2 + 92400*a**18*b**6*x**4 + 123200*a**17*b**7*x**6 + 9240
0*a**16*b**8*x**8 + 36960*a**15*b**9*x**10 + 6160*a**14*b**10*x**12) - 3120*a**(
65/3)*b**3*x**6*(1 + b*x**2/a)**(2/3)/(6160*a**20*b**4 + 36960*a**19*b**5*x**2 +
92400*a**18*b**6*x**4 + 123200*a**17*b**7*x**6 + 92400*a**16*b**8*x**8 + 36960*
a**15*b**9*x**10 + 6160*a**14*b**10*x**12) + 4860*a**(65/3)*b**3*x**6/(6160*a**2
0*b**4 + 36960*a**19*b**5*x**2 + 92400*a**18*b**6*x**4 + 123200*a**17*b**7*x**6
+ 92400*a**16*b**8*x**8 + 36960*a**15*b**9*x**10 + 6160*a**14*b**10*x**12) - 105
0*a**(62/3)*b**4*x**8*(1 + b*x**2/a)**(2/3)/(6160*a**20*b**4 + 36960*a**19*b**5*
x**2 + 92400*a**18*b**6*x**4 + 123200*a**17*b**7*x**6 + 92400*a**16*b**8*x**8 +
36960*a**15*b**9*x**10 + 6160*a**14*b**10*x**12) + 3645*a**(62/3)*b**4*x**8/(616
0*a**20*b**4 + 36960*a**19*b**5*x**2 + 92400*a**18*b**6*x**4 + 123200*a**17*b**7
*x**6 + 92400*a**16*b**8*x**8 + 36960*a**15*b**9*x**10 + 6160*a**14*b**10*x**12)
+ 4032*a**(59/3)*b**5*x**10*(1 + b*x**2/a)**(2/3)/(6160*a**20*b**4 + 36960*a**1
9*b**5*x**2 + 92400*a**18*b**6*x**4 + 123200*a**17*b**7*x**6 + 92400*a**16*b**8*
x**8 + 36960*a**15*b**9*x**10 + 6160*a**14*b**10*x**12) + 1458*a**(59/3)*b**5*x*
*10/(6160*a**20*b**4 + 36960*a**19*b**5*x**2 + 92400*a**18*b**6*x**4 + 123200*a*
*17*b**7*x**6 + 92400*a**16*b**8*x**8 + 36960*a**15*b**9*x**10 + 6160*a**14*b**1
0*x**12) + 11004*a**(56/3)*b**6*x**12*(1 + b*x**2/a)**(2/3)/(6160*a**20*b**4 + 3
6960*a**19*b**5*x**2 + 92400*a**18*b**6*x**4 + 123200*a**17*b**7*x**6 + 92400*a*
*16*b**8*x**8 + 36960*a**15*b**9*x**10 + 6160*a**14*b**10*x**12) + 243*a**(56/3)
*b**6*x**12/(6160*a**20*b**4 + 36960*a**19*b**5*x**2 + 92400*a**18*b**6*x**4 + 1
23200*a**17*b**7*x**6 + 92400*a**16*b**8*x**8 + 36960*a**15*b**9*x**10 + 6160*a*
*14*b**10*x**12) + 14352*a**(53/3)*b**7*x**14*(1 + b*x**2/a)**(2/3)/(6160*a**20*
b**4 + 36960*a**19*b**5*x**2 + 92400*a**18*b**6*x**4 + 123200*a**17*b**7*x**6 +
92400*a**16*b**8*x**8 + 36960*a**15*b**9*x**10 + 6160*a**14*b**10*x**12) + 10485
*a**(50/3)*b**8*x**16*(1 + b*x**2/a)**(2/3)/(6160*a**20*b**4 + 36960*a**19*b**5*
x**2 + 92400*a**18*b**6*x**4 + 123200*a**17*b**7*x**6 + 92400*a**16*b**8*x**8 +
36960*a**15*b**9*x**10 + 6160*a**14*b**10*x**12) + 4080*a**(47/3)*b**9*x**18*(1
+ b*x**2/a)**(2/3)/(6160*a**20*b**4 + 36960*a**19*b**5*x**2 + 92400*a**18*b**6*x
**4 + 123200*a**17*b**7*x**6 + 92400*a**16*b**8*x**8 + 36960*a**15*b**9*x**10 +
6160*a**14*b**10*x**12) + 660*a**(44/3)*b**10*x**20*(1 + b*x**2/a)**(2/3)/(6160*
a**20*b**4 + 36960*a**19*b**5*x**2 + 92400*a**18*b**6*x**4 + 123200*a**17*b**7*x
**6 + 92400*a**16*b**8*x**8 + 36960*a**15*b**9*x**10 + 6160*a**14*b**10*x**12)

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GIAC/XCAS [A]  time = 0.217977, size = 77, normalized size = 0.96 \[ \frac{3 \,{\left (220 \,{\left (b x^{2} + a\right )}^{\frac{14}{3}} - 840 \,{\left (b x^{2} + a\right )}^{\frac{11}{3}} a + 1155 \,{\left (b x^{2} + a\right )}^{\frac{8}{3}} a^{2} - 616 \,{\left (b x^{2} + a\right )}^{\frac{5}{3}} a^{3}\right )}}{6160 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/6160*(220*(b*x^2 + a)^(14/3) - 840*(b*x^2 + a)^(11/3)*a + 1155*(b*x^2 + a)^(8/
3)*a^2 - 616*(b*x^2 + a)^(5/3)*a^3)/b^4