3.665 \(\int x^7 \sqrt [3]{a+b x^2} \, dx\)
Optimal. Leaf size=80 \[ \frac{9 a^2 \left (a+b x^2\right )^{7/3}}{14 b^4}-\frac{3 a^3 \left (a+b x^2\right )^{4/3}}{8 b^4}+\frac{3 \left (a+b x^2\right )^{13/3}}{26 b^4}-\frac{9 a \left (a+b x^2\right )^{10/3}}{20 b^4} \]
[Out]
(-3*a^3*(a + b*x^2)^(4/3))/(8*b^4) + (9*a^2*(a + b*x^2)^(7/3))/(14*b^4) - (9*a*(a + b*x^2)^(10/3))/(20*b^4) +
(3*(a + b*x^2)^(13/3))/(26*b^4)
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Rubi [A] time = 0.0491851, 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} \[ \frac{9 a^2 \left (a+b x^2\right )^{7/3}}{14 b^4}-\frac{3 a^3 \left (a+b x^2\right )^{4/3}}{8 b^4}+\frac{3 \left (a+b x^2\right )^{13/3}}{26 b^4}-\frac{9 a \left (a+b x^2\right )^{10/3}}{20 b^4} \]
Antiderivative was successfully verified.
[In]
Int[x^7*(a + b*x^2)^(1/3),x]
[Out]
(-3*a^3*(a + b*x^2)^(4/3))/(8*b^4) + (9*a^2*(a + b*x^2)^(7/3))/(14*b^4) - (9*a*(a + b*x^2)^(10/3))/(20*b^4) +
(3*(a + b*x^2)^(13/3))/(26*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 \sqrt [3]{a+b x^2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int x^3 \sqrt [3]{a+b x} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (-\frac{a^3 \sqrt [3]{a+b x}}{b^3}+\frac{3 a^2 (a+b x)^{4/3}}{b^3}-\frac{3 a (a+b x)^{7/3}}{b^3}+\frac{(a+b x)^{10/3}}{b^3}\right ) \, dx,x,x^2\right )\\ &=-\frac{3 a^3 \left (a+b x^2\right )^{4/3}}{8 b^4}+\frac{9 a^2 \left (a+b x^2\right )^{7/3}}{14 b^4}-\frac{9 a \left (a+b x^2\right )^{10/3}}{20 b^4}+\frac{3 \left (a+b x^2\right )^{13/3}}{26 b^4}\\ \end{align*}
Mathematica [A] time = 0.0274937, size = 50, normalized size = 0.62 \[ \frac{3 \left (a+b x^2\right )^{4/3} \left (108 a^2 b x^2-81 a^3-126 a b^2 x^4+140 b^3 x^6\right )}{3640 b^4} \]
Antiderivative was successfully verified.
[In]
Integrate[x^7*(a + b*x^2)^(1/3),x]
[Out]
(3*(a + b*x^2)^(4/3)*(-81*a^3 + 108*a^2*b*x^2 - 126*a*b^2*x^4 + 140*b^3*x^6))/(3640*b^4)
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Maple [A] time = 0.005, size = 47, normalized size = 0.6 \begin{align*} -{\frac{-420\,{b}^{3}{x}^{6}+378\,a{b}^{2}{x}^{4}-324\,{a}^{2}b{x}^{2}+243\,{a}^{3}}{3640\,{b}^{4}} \left ( b{x}^{2}+a \right ) ^{{\frac{4}{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]
-3/3640*(b*x^2+a)^(4/3)*(-140*b^3*x^6+126*a*b^2*x^4-108*a^2*b*x^2+81*a^3)/b^4
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Maxima [A] time = 2.54927, size = 86, normalized size = 1.08 \begin{align*} \frac{3 \,{\left (b x^{2} + a\right )}^{\frac{13}{3}}}{26 \, b^{4}} - \frac{9 \,{\left (b x^{2} + a\right )}^{\frac{10}{3}} a}{20 \, b^{4}} + \frac{9 \,{\left (b x^{2} + a\right )}^{\frac{7}{3}} a^{2}}{14 \, b^{4}} - \frac{3 \,{\left (b x^{2} + a\right )}^{\frac{4}{3}} a^{3}}{8 \, 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]
3/26*(b*x^2 + a)^(13/3)/b^4 - 9/20*(b*x^2 + a)^(10/3)*a/b^4 + 9/14*(b*x^2 + a)^(7/3)*a^2/b^4 - 3/8*(b*x^2 + a)
^(4/3)*a^3/b^4
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Fricas [A] time = 1.49548, size = 135, normalized size = 1.69 \begin{align*} \frac{3 \,{\left (140 \, b^{4} x^{8} + 14 \, a b^{3} x^{6} - 18 \, a^{2} b^{2} x^{4} + 27 \, a^{3} b x^{2} - 81 \, a^{4}\right )}{\left (b x^{2} + a\right )}^{\frac{1}{3}}}{3640 \, 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]
3/3640*(140*b^4*x^8 + 14*a*b^3*x^6 - 18*a^2*b^2*x^4 + 27*a^3*b*x^2 - 81*a^4)*(b*x^2 + a)^(1/3)/b^4
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Sympy [B] time = 2.62751, 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)**(1/3),x)
[Out]
-243*a**(73/3)*(1 + b*x**2/a)**(1/3)/(3640*a**20*b**4 + 21840*a**19*b**5*x**2 + 54600*a**18*b**6*x**4 + 72800*
a**17*b**7*x**6 + 54600*a**16*b**8*x**8 + 21840*a**15*b**9*x**10 + 3640*a**14*b**10*x**12) + 243*a**(73/3)/(36
40*a**20*b**4 + 21840*a**19*b**5*x**2 + 54600*a**18*b**6*x**4 + 72800*a**17*b**7*x**6 + 54600*a**16*b**8*x**8
+ 21840*a**15*b**9*x**10 + 3640*a**14*b**10*x**12) - 1377*a**(70/3)*b*x**2*(1 + b*x**2/a)**(1/3)/(3640*a**20*b
**4 + 21840*a**19*b**5*x**2 + 54600*a**18*b**6*x**4 + 72800*a**17*b**7*x**6 + 54600*a**16*b**8*x**8 + 21840*a*
*15*b**9*x**10 + 3640*a**14*b**10*x**12) + 1458*a**(70/3)*b*x**2/(3640*a**20*b**4 + 21840*a**19*b**5*x**2 + 54
600*a**18*b**6*x**4 + 72800*a**17*b**7*x**6 + 54600*a**16*b**8*x**8 + 21840*a**15*b**9*x**10 + 3640*a**14*b**1
0*x**12) - 3213*a**(67/3)*b**2*x**4*(1 + b*x**2/a)**(1/3)/(3640*a**20*b**4 + 21840*a**19*b**5*x**2 + 54600*a**
18*b**6*x**4 + 72800*a**17*b**7*x**6 + 54600*a**16*b**8*x**8 + 21840*a**15*b**9*x**10 + 3640*a**14*b**10*x**12
) + 3645*a**(67/3)*b**2*x**4/(3640*a**20*b**4 + 21840*a**19*b**5*x**2 + 54600*a**18*b**6*x**4 + 72800*a**17*b*
*7*x**6 + 54600*a**16*b**8*x**8 + 21840*a**15*b**9*x**10 + 3640*a**14*b**10*x**12) - 3927*a**(64/3)*b**3*x**6*
(1 + b*x**2/a)**(1/3)/(3640*a**20*b**4 + 21840*a**19*b**5*x**2 + 54600*a**18*b**6*x**4 + 72800*a**17*b**7*x**6
+ 54600*a**16*b**8*x**8 + 21840*a**15*b**9*x**10 + 3640*a**14*b**10*x**12) + 4860*a**(64/3)*b**3*x**6/(3640*a
**20*b**4 + 21840*a**19*b**5*x**2 + 54600*a**18*b**6*x**4 + 72800*a**17*b**7*x**6 + 54600*a**16*b**8*x**8 + 21
840*a**15*b**9*x**10 + 3640*a**14*b**10*x**12) - 2163*a**(61/3)*b**4*x**8*(1 + b*x**2/a)**(1/3)/(3640*a**20*b*
*4 + 21840*a**19*b**5*x**2 + 54600*a**18*b**6*x**4 + 72800*a**17*b**7*x**6 + 54600*a**16*b**8*x**8 + 21840*a**
15*b**9*x**10 + 3640*a**14*b**10*x**12) + 3645*a**(61/3)*b**4*x**8/(3640*a**20*b**4 + 21840*a**19*b**5*x**2 +
54600*a**18*b**6*x**4 + 72800*a**17*b**7*x**6 + 54600*a**16*b**8*x**8 + 21840*a**15*b**9*x**10 + 3640*a**14*b*
*10*x**12) + 1827*a**(58/3)*b**5*x**10*(1 + b*x**2/a)**(1/3)/(3640*a**20*b**4 + 21840*a**19*b**5*x**2 + 54600*
a**18*b**6*x**4 + 72800*a**17*b**7*x**6 + 54600*a**16*b**8*x**8 + 21840*a**15*b**9*x**10 + 3640*a**14*b**10*x*
*12) + 1458*a**(58/3)*b**5*x**10/(3640*a**20*b**4 + 21840*a**19*b**5*x**2 + 54600*a**18*b**6*x**4 + 72800*a**1
7*b**7*x**6 + 54600*a**16*b**8*x**8 + 21840*a**15*b**9*x**10 + 3640*a**14*b**10*x**12) + 6573*a**(55/3)*b**6*x
**12*(1 + b*x**2/a)**(1/3)/(3640*a**20*b**4 + 21840*a**19*b**5*x**2 + 54600*a**18*b**6*x**4 + 72800*a**17*b**7
*x**6 + 54600*a**16*b**8*x**8 + 21840*a**15*b**9*x**10 + 3640*a**14*b**10*x**12) + 243*a**(55/3)*b**6*x**12/(3
640*a**20*b**4 + 21840*a**19*b**5*x**2 + 54600*a**18*b**6*x**4 + 72800*a**17*b**7*x**6 + 54600*a**16*b**8*x**8
+ 21840*a**15*b**9*x**10 + 3640*a**14*b**10*x**12) + 8787*a**(52/3)*b**7*x**14*(1 + b*x**2/a)**(1/3)/(3640*a*
*20*b**4 + 21840*a**19*b**5*x**2 + 54600*a**18*b**6*x**4 + 72800*a**17*b**7*x**6 + 54600*a**16*b**8*x**8 + 218
40*a**15*b**9*x**10 + 3640*a**14*b**10*x**12) + 6498*a**(49/3)*b**8*x**16*(1 + b*x**2/a)**(1/3)/(3640*a**20*b*
*4 + 21840*a**19*b**5*x**2 + 54600*a**18*b**6*x**4 + 72800*a**17*b**7*x**6 + 54600*a**16*b**8*x**8 + 21840*a**
15*b**9*x**10 + 3640*a**14*b**10*x**12) + 2562*a**(46/3)*b**9*x**18*(1 + b*x**2/a)**(1/3)/(3640*a**20*b**4 + 2
1840*a**19*b**5*x**2 + 54600*a**18*b**6*x**4 + 72800*a**17*b**7*x**6 + 54600*a**16*b**8*x**8 + 21840*a**15*b**
9*x**10 + 3640*a**14*b**10*x**12) + 420*a**(43/3)*b**10*x**20*(1 + b*x**2/a)**(1/3)/(3640*a**20*b**4 + 21840*a
**19*b**5*x**2 + 54600*a**18*b**6*x**4 + 72800*a**17*b**7*x**6 + 54600*a**16*b**8*x**8 + 21840*a**15*b**9*x**1
0 + 3640*a**14*b**10*x**12)
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Giac [A] time = 1.84312, size = 77, normalized size = 0.96 \begin{align*} \frac{3 \,{\left (140 \,{\left (b x^{2} + a\right )}^{\frac{13}{3}} - 546 \,{\left (b x^{2} + a\right )}^{\frac{10}{3}} a + 780 \,{\left (b x^{2} + a\right )}^{\frac{7}{3}} a^{2} - 455 \,{\left (b x^{2} + a\right )}^{\frac{4}{3}} a^{3}\right )}}{3640 \, 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]
3/3640*(140*(b*x^2 + a)^(13/3) - 546*(b*x^2 + a)^(10/3)*a + 780*(b*x^2 + a)^(7/3)*a^2 - 455*(b*x^2 + a)^(4/3)*
a^3)/b^4