3.392 \(\int \frac{x^3}{\sqrt [3]{a+b x}} \, dx\)
Optimal. Leaf size=72 \[ \frac{9 a^2 (a+b x)^{5/3}}{5 b^4}-\frac{3 a^3 (a+b x)^{2/3}}{2 b^4}+\frac{3 (a+b x)^{11/3}}{11 b^4}-\frac{9 a (a+b x)^{8/3}}{8 b^4} \]
[Out]
(-3*a^3*(a + b*x)^(2/3))/(2*b^4) + (9*a^2*(a + b*x)^(5/3))/(5*b^4) - (9*a*(a + b*x)^(8/3))/(8*b^4) + (3*(a + b
*x)^(11/3))/(11*b^4)
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Rubi [A] time = 0.0179759, antiderivative size = 72, normalized size of antiderivative = 1.,
number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used =
{43} \[ \frac{9 a^2 (a+b x)^{5/3}}{5 b^4}-\frac{3 a^3 (a+b x)^{2/3}}{2 b^4}+\frac{3 (a+b x)^{11/3}}{11 b^4}-\frac{9 a (a+b x)^{8/3}}{8 b^4} \]
Antiderivative was successfully verified.
[In]
Int[x^3/(a + b*x)^(1/3),x]
[Out]
(-3*a^3*(a + b*x)^(2/3))/(2*b^4) + (9*a^2*(a + b*x)^(5/3))/(5*b^4) - (9*a*(a + b*x)^(8/3))/(8*b^4) + (3*(a + b
*x)^(11/3))/(11*b^4)
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^3}{\sqrt [3]{a+b x}} \, dx &=\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\\ &=-\frac{3 a^3 (a+b x)^{2/3}}{2 b^4}+\frac{9 a^2 (a+b x)^{5/3}}{5 b^4}-\frac{9 a (a+b x)^{8/3}}{8 b^4}+\frac{3 (a+b x)^{11/3}}{11 b^4}\\ \end{align*}
Mathematica [A] time = 0.0449445, size = 46, normalized size = 0.64 \[ \frac{3 (a+b x)^{2/3} \left (54 a^2 b x-81 a^3-45 a b^2 x^2+40 b^3 x^3\right )}{440 b^4} \]
Antiderivative was successfully verified.
[In]
Integrate[x^3/(a + b*x)^(1/3),x]
[Out]
(3*(a + b*x)^(2/3)*(-81*a^3 + 54*a^2*b*x - 45*a*b^2*x^2 + 40*b^3*x^3))/(440*b^4)
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Maple [A] time = 0.004, size = 43, normalized size = 0.6 \begin{align*} -{\frac{-120\,{b}^{3}{x}^{3}+135\,a{b}^{2}{x}^{2}-162\,{a}^{2}bx+243\,{a}^{3}}{440\,{b}^{4}} \left ( bx+a \right ) ^{{\frac{2}{3}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^3/(b*x+a)^(1/3),x)
[Out]
-3/440*(b*x+a)^(2/3)*(-40*b^3*x^3+45*a*b^2*x^2-54*a^2*b*x+81*a^3)/b^4
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Maxima [A] time = 1.07116, size = 76, normalized size = 1.06 \begin{align*} \frac{3 \,{\left (b x + a\right )}^{\frac{11}{3}}}{11 \, b^{4}} - \frac{9 \,{\left (b x + a\right )}^{\frac{8}{3}} a}{8 \, b^{4}} + \frac{9 \,{\left (b x + a\right )}^{\frac{5}{3}} a^{2}}{5 \, b^{4}} - \frac{3 \,{\left (b x + a\right )}^{\frac{2}{3}} a^{3}}{2 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^3/(b*x+a)^(1/3),x, algorithm="maxima")
[Out]
3/11*(b*x + a)^(11/3)/b^4 - 9/8*(b*x + a)^(8/3)*a/b^4 + 9/5*(b*x + a)^(5/3)*a^2/b^4 - 3/2*(b*x + a)^(2/3)*a^3/
b^4
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Fricas [A] time = 1.50877, size = 104, normalized size = 1.44 \begin{align*} \frac{3 \,{\left (40 \, b^{3} x^{3} - 45 \, a b^{2} x^{2} + 54 \, a^{2} b x - 81 \, a^{3}\right )}{\left (b x + a\right )}^{\frac{2}{3}}}{440 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^3/(b*x+a)^(1/3),x, algorithm="fricas")
[Out]
3/440*(40*b^3*x^3 - 45*a*b^2*x^2 + 54*a^2*b*x - 81*a^3)*(b*x + a)^(2/3)/b^4
________________________________________________________________________________________
Sympy [B] time = 3.54217, size = 1640, normalized size = 22.78 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x**3/(b*x+a)**(1/3),x)
[Out]
-243*a**(71/3)*(1 + b*x/a)**(2/3)/(440*a**20*b**4 + 2640*a**19*b**5*x + 6600*a**18*b**6*x**2 + 8800*a**17*b**7
*x**3 + 6600*a**16*b**8*x**4 + 2640*a**15*b**9*x**5 + 440*a**14*b**10*x**6) + 243*a**(71/3)/(440*a**20*b**4 +
2640*a**19*b**5*x + 6600*a**18*b**6*x**2 + 8800*a**17*b**7*x**3 + 6600*a**16*b**8*x**4 + 2640*a**15*b**9*x**5
+ 440*a**14*b**10*x**6) - 1296*a**(68/3)*b*x*(1 + b*x/a)**(2/3)/(440*a**20*b**4 + 2640*a**19*b**5*x + 6600*a**
18*b**6*x**2 + 8800*a**17*b**7*x**3 + 6600*a**16*b**8*x**4 + 2640*a**15*b**9*x**5 + 440*a**14*b**10*x**6) + 14
58*a**(68/3)*b*x/(440*a**20*b**4 + 2640*a**19*b**5*x + 6600*a**18*b**6*x**2 + 8800*a**17*b**7*x**3 + 6600*a**1
6*b**8*x**4 + 2640*a**15*b**9*x**5 + 440*a**14*b**10*x**6) - 2808*a**(65/3)*b**2*x**2*(1 + b*x/a)**(2/3)/(440*
a**20*b**4 + 2640*a**19*b**5*x + 6600*a**18*b**6*x**2 + 8800*a**17*b**7*x**3 + 6600*a**16*b**8*x**4 + 2640*a**
15*b**9*x**5 + 440*a**14*b**10*x**6) + 3645*a**(65/3)*b**2*x**2/(440*a**20*b**4 + 2640*a**19*b**5*x + 6600*a**
18*b**6*x**2 + 8800*a**17*b**7*x**3 + 6600*a**16*b**8*x**4 + 2640*a**15*b**9*x**5 + 440*a**14*b**10*x**6) - 31
20*a**(62/3)*b**3*x**3*(1 + b*x/a)**(2/3)/(440*a**20*b**4 + 2640*a**19*b**5*x + 6600*a**18*b**6*x**2 + 8800*a*
*17*b**7*x**3 + 6600*a**16*b**8*x**4 + 2640*a**15*b**9*x**5 + 440*a**14*b**10*x**6) + 4860*a**(62/3)*b**3*x**3
/(440*a**20*b**4 + 2640*a**19*b**5*x + 6600*a**18*b**6*x**2 + 8800*a**17*b**7*x**3 + 6600*a**16*b**8*x**4 + 26
40*a**15*b**9*x**5 + 440*a**14*b**10*x**6) - 1710*a**(59/3)*b**4*x**4*(1 + b*x/a)**(2/3)/(440*a**20*b**4 + 264
0*a**19*b**5*x + 6600*a**18*b**6*x**2 + 8800*a**17*b**7*x**3 + 6600*a**16*b**8*x**4 + 2640*a**15*b**9*x**5 + 4
40*a**14*b**10*x**6) + 3645*a**(59/3)*b**4*x**4/(440*a**20*b**4 + 2640*a**19*b**5*x + 6600*a**18*b**6*x**2 + 8
800*a**17*b**7*x**3 + 6600*a**16*b**8*x**4 + 2640*a**15*b**9*x**5 + 440*a**14*b**10*x**6) + 72*a**(56/3)*b**5*
x**5*(1 + b*x/a)**(2/3)/(440*a**20*b**4 + 2640*a**19*b**5*x + 6600*a**18*b**6*x**2 + 8800*a**17*b**7*x**3 + 66
00*a**16*b**8*x**4 + 2640*a**15*b**9*x**5 + 440*a**14*b**10*x**6) + 1458*a**(56/3)*b**5*x**5/(440*a**20*b**4 +
2640*a**19*b**5*x + 6600*a**18*b**6*x**2 + 8800*a**17*b**7*x**3 + 6600*a**16*b**8*x**4 + 2640*a**15*b**9*x**5
+ 440*a**14*b**10*x**6) + 1104*a**(53/3)*b**6*x**6*(1 + b*x/a)**(2/3)/(440*a**20*b**4 + 2640*a**19*b**5*x + 6
600*a**18*b**6*x**2 + 8800*a**17*b**7*x**3 + 6600*a**16*b**8*x**4 + 2640*a**15*b**9*x**5 + 440*a**14*b**10*x**
6) + 243*a**(53/3)*b**6*x**6/(440*a**20*b**4 + 2640*a**19*b**5*x + 6600*a**18*b**6*x**2 + 8800*a**17*b**7*x**3
+ 6600*a**16*b**8*x**4 + 2640*a**15*b**9*x**5 + 440*a**14*b**10*x**6) + 1152*a**(50/3)*b**7*x**7*(1 + b*x/a)*
*(2/3)/(440*a**20*b**4 + 2640*a**19*b**5*x + 6600*a**18*b**6*x**2 + 8800*a**17*b**7*x**3 + 6600*a**16*b**8*x**
4 + 2640*a**15*b**9*x**5 + 440*a**14*b**10*x**6) + 585*a**(47/3)*b**8*x**8*(1 + b*x/a)**(2/3)/(440*a**20*b**4
+ 2640*a**19*b**5*x + 6600*a**18*b**6*x**2 + 8800*a**17*b**7*x**3 + 6600*a**16*b**8*x**4 + 2640*a**15*b**9*x**
5 + 440*a**14*b**10*x**6) + 120*a**(44/3)*b**9*x**9*(1 + b*x/a)**(2/3)/(440*a**20*b**4 + 2640*a**19*b**5*x + 6
600*a**18*b**6*x**2 + 8800*a**17*b**7*x**3 + 6600*a**16*b**8*x**4 + 2640*a**15*b**9*x**5 + 440*a**14*b**10*x**
6)
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Giac [A] time = 1.22544, size = 66, normalized size = 0.92 \begin{align*} \frac{3 \,{\left (40 \,{\left (b x + a\right )}^{\frac{11}{3}} - 165 \,{\left (b x + a\right )}^{\frac{8}{3}} a + 264 \,{\left (b x + a\right )}^{\frac{5}{3}} a^{2} - 220 \,{\left (b x + a\right )}^{\frac{2}{3}} a^{3}\right )}}{440 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^3/(b*x+a)^(1/3),x, algorithm="giac")
[Out]
3/440*(40*(b*x + a)^(11/3) - 165*(b*x + a)^(8/3)*a + 264*(b*x + a)^(5/3)*a^2 - 220*(b*x + a)^(2/3)*a^3)/b^4