3.714 \(\int \frac{x^7}{(a+b x^2)^{2/3}} \, dx\)
Optimal. Leaf size=80 \[ \frac{9 a^2 \left (a+b x^2\right )^{4/3}}{8 b^4}-\frac{3 a^3 \sqrt [3]{a+b x^2}}{2 b^4}+\frac{3 \left (a+b x^2\right )^{10/3}}{20 b^4}-\frac{9 a \left (a+b x^2\right )^{7/3}}{14 b^4} \]
[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)
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Rubi [A] time = 0.045896, 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} \[ \frac{9 a^2 \left (a+b x^2\right )^{4/3}}{8 b^4}-\frac{3 a^3 \sqrt [3]{a+b x^2}}{2 b^4}+\frac{3 \left (a+b x^2\right )^{10/3}}{20 b^4}-\frac{9 a \left (a+b x^2\right )^{7/3}}{14 b^4} \]
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)
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 \frac{x^7}{\left (a+b x^2\right )^{2/3}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^3}{(a+b x)^{2/3}} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (-\frac{a^3}{b^3 (a+b x)^{2/3}}+\frac{3 a^2 \sqrt [3]{a+b x}}{b^3}-\frac{3 a (a+b x)^{4/3}}{b^3}+\frac{(a+b x)^{7/3}}{b^3}\right ) \, dx,x,x^2\right )\\ &=-\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{9 a \left (a+b x^2\right )^{7/3}}{14 b^4}+\frac{3 \left (a+b x^2\right )^{10/3}}{20 b^4}\\ \end{align*}
Mathematica [A] time = 0.0239664, size = 50, normalized size = 0.62 \[ \frac{3 \sqrt [3]{a+b x^2} \left (27 a^2 b x^2-81 a^3-18 a b^2 x^4+14 b^3 x^6\right )}{280 b^4} \]
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)
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Maple [A] time = 0.006, size = 47, normalized size = 0.6 \begin{align*} -{\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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^7/(b*x^2+a)^(2/3),x)
[Out]
-3/280*(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
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Maxima [A] time = 2.11981, size = 86, normalized size = 1.08 \begin{align*} \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}} \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]
3/20*(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
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Fricas [A] time = 1.69348, size = 109, normalized size = 1.36 \begin{align*} \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}} \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]
3/280*(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
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Sympy [B] time = 2.52817, size = 1690, normalized size = 21.12 \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**(70/3)*(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**1
7*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**1
9*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**(67/3)*b*x**2/(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) - 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) + 3645*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 + 280*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 + 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**1
0 + 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 + 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) - 2583*a*
*(58/3)*b**4*x**8*(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) + 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**15*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)/(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) + 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)
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Giac [A] time = 1.79455, size = 77, normalized size = 0.96 \begin{align*} \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}} \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]
3/280*(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