3.702 \(\int \frac{x^7}{\sqrt [3]{a+b x^2}} \, dx\)
Optimal. Leaf size=80 \[ \frac{9 a^2 \left (a+b x^2\right )^{5/3}}{10 b^4}-\frac{3 a^3 \left (a+b x^2\right )^{2/3}}{4 b^4}+\frac{3 \left (a+b x^2\right )^{11/3}}{22 b^4}-\frac{9 a \left (a+b x^2\right )^{8/3}}{16 b^4} \]
[Out]
(-3*a^3*(a + b*x^2)^(2/3))/(4*b^4) + (9*a^2*(a + b*x^2)^(5/3))/(10*b^4) - (9*a*(a + b*x^2)^(8/3))/(16*b^4) + (
3*(a + b*x^2)^(11/3))/(22*b^4)
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Rubi [A] time = 0.0463976, 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} \[ \frac{9 a^2 \left (a+b x^2\right )^{5/3}}{10 b^4}-\frac{3 a^3 \left (a+b x^2\right )^{2/3}}{4 b^4}+\frac{3 \left (a+b x^2\right )^{11/3}}{22 b^4}-\frac{9 a \left (a+b x^2\right )^{8/3}}{16 b^4} \]
Antiderivative was successfully verified.
[In]
Int[x^7/(a + b*x^2)^(1/3),x]
[Out]
(-3*a^3*(a + b*x^2)^(2/3))/(4*b^4) + (9*a^2*(a + b*x^2)^(5/3))/(10*b^4) - (9*a*(a + b*x^2)^(8/3))/(16*b^4) + (
3*(a + b*x^2)^(11/3))/(22*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}{\sqrt [3]{a+b x^2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^3}{\sqrt [3]{a+b x}} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (-\frac{a^3}{b^3 \sqrt [3]{a+b x}}+\frac{3 a^2 (a+b x)^{2/3}}{b^3}-\frac{3 a (a+b x)^{5/3}}{b^3}+\frac{(a+b x)^{8/3}}{b^3}\right ) \, dx,x,x^2\right )\\ &=-\frac{3 a^3 \left (a+b x^2\right )^{2/3}}{4 b^4}+\frac{9 a^2 \left (a+b x^2\right )^{5/3}}{10 b^4}-\frac{9 a \left (a+b x^2\right )^{8/3}}{16 b^4}+\frac{3 \left (a+b x^2\right )^{11/3}}{22 b^4}\\ \end{align*}
Mathematica [A] time = 0.0280746, size = 50, normalized size = 0.62 \[ \frac{3 \left (a+b x^2\right )^{2/3} \left (54 a^2 b x^2-81 a^3-45 a b^2 x^4+40 b^3 x^6\right )}{880 b^4} \]
Antiderivative was successfully verified.
[In]
Integrate[x^7/(a + b*x^2)^(1/3),x]
[Out]
(3*(a + b*x^2)^(2/3)*(-81*a^3 + 54*a^2*b*x^2 - 45*a*b^2*x^4 + 40*b^3*x^6))/(880*b^4)
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Maple [A] time = 0.004, size = 47, normalized size = 0.6 \begin{align*} -{\frac{-120\,{b}^{3}{x}^{6}+135\,a{b}^{2}{x}^{4}-162\,{a}^{2}b{x}^{2}+243\,{a}^{3}}{880\,{b}^{4}} \left ( b{x}^{2}+a \right ) ^{{\frac{2}{3}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^7/(b*x^2+a)^(1/3),x)
[Out]
-3/880*(b*x^2+a)^(2/3)*(-40*b^3*x^6+45*a*b^2*x^4-54*a^2*b*x^2+81*a^3)/b^4
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Maxima [A] time = 1.72394, size = 86, normalized size = 1.08 \begin{align*} \frac{3 \,{\left (b x^{2} + a\right )}^{\frac{11}{3}}}{22 \, b^{4}} - \frac{9 \,{\left (b x^{2} + a\right )}^{\frac{8}{3}} a}{16 \, b^{4}} + \frac{9 \,{\left (b x^{2} + a\right )}^{\frac{5}{3}} a^{2}}{10 \, b^{4}} - \frac{3 \,{\left (b x^{2} + a\right )}^{\frac{2}{3}} a^{3}}{4 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^7/(b*x^2+a)^(1/3),x, algorithm="maxima")
[Out]
3/22*(b*x^2 + a)^(11/3)/b^4 - 9/16*(b*x^2 + a)^(8/3)*a/b^4 + 9/10*(b*x^2 + a)^(5/3)*a^2/b^4 - 3/4*(b*x^2 + a)^
(2/3)*a^3/b^4
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Fricas [A] time = 1.65236, size = 109, normalized size = 1.36 \begin{align*} \frac{3 \,{\left (40 \, b^{3} x^{6} - 45 \, a b^{2} x^{4} + 54 \, a^{2} b x^{2} - 81 \, a^{3}\right )}{\left (b x^{2} + a\right )}^{\frac{2}{3}}}{880 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^7/(b*x^2+a)^(1/3),x, algorithm="fricas")
[Out]
3/880*(40*b^3*x^6 - 45*a*b^2*x^4 + 54*a^2*b*x^2 - 81*a^3)*(b*x^2 + a)^(2/3)/b^4
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Sympy [B] time = 2.49541, 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)**(1/3),x)
[Out]
-243*a**(71/3)*(1 + b*x**2/a)**(2/3)/(880*a**20*b**4 + 5280*a**19*b**5*x**2 + 13200*a**18*b**6*x**4 + 17600*a*
*17*b**7*x**6 + 13200*a**16*b**8*x**8 + 5280*a**15*b**9*x**10 + 880*a**14*b**10*x**12) + 243*a**(71/3)/(880*a*
*20*b**4 + 5280*a**19*b**5*x**2 + 13200*a**18*b**6*x**4 + 17600*a**17*b**7*x**6 + 13200*a**16*b**8*x**8 + 5280
*a**15*b**9*x**10 + 880*a**14*b**10*x**12) - 1296*a**(68/3)*b*x**2*(1 + b*x**2/a)**(2/3)/(880*a**20*b**4 + 528
0*a**19*b**5*x**2 + 13200*a**18*b**6*x**4 + 17600*a**17*b**7*x**6 + 13200*a**16*b**8*x**8 + 5280*a**15*b**9*x*
*10 + 880*a**14*b**10*x**12) + 1458*a**(68/3)*b*x**2/(880*a**20*b**4 + 5280*a**19*b**5*x**2 + 13200*a**18*b**6
*x**4 + 17600*a**17*b**7*x**6 + 13200*a**16*b**8*x**8 + 5280*a**15*b**9*x**10 + 880*a**14*b**10*x**12) - 2808*
a**(65/3)*b**2*x**4*(1 + b*x**2/a)**(2/3)/(880*a**20*b**4 + 5280*a**19*b**5*x**2 + 13200*a**18*b**6*x**4 + 176
00*a**17*b**7*x**6 + 13200*a**16*b**8*x**8 + 5280*a**15*b**9*x**10 + 880*a**14*b**10*x**12) + 3645*a**(65/3)*b
**2*x**4/(880*a**20*b**4 + 5280*a**19*b**5*x**2 + 13200*a**18*b**6*x**4 + 17600*a**17*b**7*x**6 + 13200*a**16*
b**8*x**8 + 5280*a**15*b**9*x**10 + 880*a**14*b**10*x**12) - 3120*a**(62/3)*b**3*x**6*(1 + b*x**2/a)**(2/3)/(8
80*a**20*b**4 + 5280*a**19*b**5*x**2 + 13200*a**18*b**6*x**4 + 17600*a**17*b**7*x**6 + 13200*a**16*b**8*x**8 +
5280*a**15*b**9*x**10 + 880*a**14*b**10*x**12) + 4860*a**(62/3)*b**3*x**6/(880*a**20*b**4 + 5280*a**19*b**5*x
**2 + 13200*a**18*b**6*x**4 + 17600*a**17*b**7*x**6 + 13200*a**16*b**8*x**8 + 5280*a**15*b**9*x**10 + 880*a**1
4*b**10*x**12) - 1710*a**(59/3)*b**4*x**8*(1 + b*x**2/a)**(2/3)/(880*a**20*b**4 + 5280*a**19*b**5*x**2 + 13200
*a**18*b**6*x**4 + 17600*a**17*b**7*x**6 + 13200*a**16*b**8*x**8 + 5280*a**15*b**9*x**10 + 880*a**14*b**10*x**
12) + 3645*a**(59/3)*b**4*x**8/(880*a**20*b**4 + 5280*a**19*b**5*x**2 + 13200*a**18*b**6*x**4 + 17600*a**17*b*
*7*x**6 + 13200*a**16*b**8*x**8 + 5280*a**15*b**9*x**10 + 880*a**14*b**10*x**12) + 72*a**(56/3)*b**5*x**10*(1
+ b*x**2/a)**(2/3)/(880*a**20*b**4 + 5280*a**19*b**5*x**2 + 13200*a**18*b**6*x**4 + 17600*a**17*b**7*x**6 + 13
200*a**16*b**8*x**8 + 5280*a**15*b**9*x**10 + 880*a**14*b**10*x**12) + 1458*a**(56/3)*b**5*x**10/(880*a**20*b*
*4 + 5280*a**19*b**5*x**2 + 13200*a**18*b**6*x**4 + 17600*a**17*b**7*x**6 + 13200*a**16*b**8*x**8 + 5280*a**15
*b**9*x**10 + 880*a**14*b**10*x**12) + 1104*a**(53/3)*b**6*x**12*(1 + b*x**2/a)**(2/3)/(880*a**20*b**4 + 5280*
a**19*b**5*x**2 + 13200*a**18*b**6*x**4 + 17600*a**17*b**7*x**6 + 13200*a**16*b**8*x**8 + 5280*a**15*b**9*x**1
0 + 880*a**14*b**10*x**12) + 243*a**(53/3)*b**6*x**12/(880*a**20*b**4 + 5280*a**19*b**5*x**2 + 13200*a**18*b**
6*x**4 + 17600*a**17*b**7*x**6 + 13200*a**16*b**8*x**8 + 5280*a**15*b**9*x**10 + 880*a**14*b**10*x**12) + 1152
*a**(50/3)*b**7*x**14*(1 + b*x**2/a)**(2/3)/(880*a**20*b**4 + 5280*a**19*b**5*x**2 + 13200*a**18*b**6*x**4 + 1
7600*a**17*b**7*x**6 + 13200*a**16*b**8*x**8 + 5280*a**15*b**9*x**10 + 880*a**14*b**10*x**12) + 585*a**(47/3)*
b**8*x**16*(1 + b*x**2/a)**(2/3)/(880*a**20*b**4 + 5280*a**19*b**5*x**2 + 13200*a**18*b**6*x**4 + 17600*a**17*
b**7*x**6 + 13200*a**16*b**8*x**8 + 5280*a**15*b**9*x**10 + 880*a**14*b**10*x**12) + 120*a**(44/3)*b**9*x**18*
(1 + b*x**2/a)**(2/3)/(880*a**20*b**4 + 5280*a**19*b**5*x**2 + 13200*a**18*b**6*x**4 + 17600*a**17*b**7*x**6 +
13200*a**16*b**8*x**8 + 5280*a**15*b**9*x**10 + 880*a**14*b**10*x**12)
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Giac [A] time = 2.48188, size = 77, normalized size = 0.96 \begin{align*} \frac{3 \,{\left (40 \,{\left (b x^{2} + a\right )}^{\frac{11}{3}} - 165 \,{\left (b x^{2} + a\right )}^{\frac{8}{3}} a + 264 \,{\left (b x^{2} + a\right )}^{\frac{5}{3}} a^{2} - 220 \,{\left (b x^{2} + a\right )}^{\frac{2}{3}} a^{3}\right )}}{880 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^7/(b*x^2+a)^(1/3),x, algorithm="giac")
[Out]
3/880*(40*(b*x^2 + a)^(11/3) - 165*(b*x^2 + a)^(8/3)*a + 264*(b*x^2 + a)^(5/3)*a^2 - 220*(b*x^2 + a)^(2/3)*a^3
)/b^4