3.677 \(\int x^7 (a+b x^2)^{2/3} \, dx\)
Optimal. Leaf size=80 \[ \frac{9 a^2 \left (a+b x^2\right )^{8/3}}{16 b^4}-\frac{3 a^3 \left (a+b x^2\right )^{5/3}}{10 b^4}+\frac{3 \left (a+b x^2\right )^{14/3}}{28 b^4}-\frac{9 a \left (a+b x^2\right )^{11/3}}{22 b^4} \]
[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 = 0.0473386, 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, Rules used =
{266, 43} \[ \frac{9 a^2 \left (a+b x^2\right )^{8/3}}{16 b^4}-\frac{3 a^3 \left (a+b x^2\right )^{5/3}}{10 b^4}+\frac{3 \left (a+b x^2\right )^{14/3}}{28 b^4}-\frac{9 a \left (a+b x^2\right )^{11/3}}{22 b^4} \]
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)
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]]
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])
Rubi steps
\begin{align*} \int x^7 \left (a+b x^2\right )^{2/3} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int x^3 (a+b x)^{2/3} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (-\frac{a^3 (a+b x)^{2/3}}{b^3}+\frac{3 a^2 (a+b x)^{5/3}}{b^3}-\frac{3 a (a+b x)^{8/3}}{b^3}+\frac{(a+b x)^{11/3}}{b^3}\right ) \, dx,x,x^2\right )\\ &=-\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{9 a \left (a+b x^2\right )^{11/3}}{22 b^4}+\frac{3 \left (a+b x^2\right )^{14/3}}{28 b^4}\\ \end{align*}
Mathematica [A] time = 0.0246458, size = 50, normalized size = 0.62 \[ \frac{3 \left (a+b x^2\right )^{5/3} \left (135 a^2 b x^2-81 a^3-180 a b^2 x^4+220 b^3 x^6\right )}{6160 b^4} \]
Antiderivative was successfully verified.
[In]
Integrate[x^7*(a + b*x^2)^(2/3),x]
[Out]
(3*(a + b*x^2)^(5/3)*(-81*a^3 + 135*a^2*b*x^2 - 180*a*b^2*x^4 + 220*b^3*x^6))/(6160*b^4)
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Maple [A] time = 0.005, size = 47, normalized size = 0.6 \begin{align*} -{\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}}}} \end{align*}
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 = 1.59909, size = 86, normalized size = 1.08 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^7*(b*x^2+a)^(2/3),x, algorithm="maxima")
[Out]
3/28*(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 = 1.71919, size = 135, normalized size = 1.69 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^7*(b*x^2+a)^(2/3),x, algorithm="fricas")
[Out]
3/6160*(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 [B] time = 2.93274, size = 1795, normalized size = 22.44 \begin{align*} \text{result too large to display} \end{align*}
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)/(6
160*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) - 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 + 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) - 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 + 92400*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**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) - 1050*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/(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 + 616
0*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**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**(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**10*x**12) + 11004*a**
(56/3)*b**6*x**12*(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 + 123
200*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 + 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) + 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**1
6*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 [A] time = 2.97849, size = 77, normalized size = 0.96 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^7*(b*x^2+a)^(2/3),x, algorithm="giac")
[Out]
3/6160*(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