3.392 \(\int \frac{x^3}{\sqrt [3]{a+b x}} \, dx\)

Optimal. Leaf size=72 \[ -\frac{3 a^3 (a+b x)^{2/3}}{2 b^4}+\frac{9 a^2 (a+b x)^{5/3}}{5 b^4}+\frac{3 (a+b x)^{11/3}}{11 b^4}-\frac{9 a (a+b x)^{8/3}}{8 b^4} \]

[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 = 0.0513634, 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 \[ -\frac{3 a^3 (a+b x)^{2/3}}{2 b^4}+\frac{9 a^2 (a+b x)^{5/3}}{5 b^4}+\frac{3 (a+b x)^{11/3}}{11 b^4}-\frac{9 a (a+b x)^{8/3}}{8 b^4} \]

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)

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Rubi in Sympy [A]  time = 10.9982, size = 68, normalized size = 0.94 \[ - \frac{3 a^{3} \left (a + b x\right )^{\frac{2}{3}}}{2 b^{4}} + \frac{9 a^{2} \left (a + b x\right )^{\frac{5}{3}}}{5 b^{4}} - \frac{9 a \left (a + b x\right )^{\frac{8}{3}}}{8 b^{4}} + \frac{3 \left (a + b x\right )^{\frac{11}{3}}}{11 b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3/(b*x+a)**(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)

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Mathematica [A]  time = 0.0245081, size = 46, normalized size = 0.64 \[ \frac{3 (a+b x)^{2/3} \left (-81 a^3+54 a^2 b x-45 a b^2 x^2+40 b^3 x^3\right )}{440 b^4} \]

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 = 0.007, size = 43, normalized size = 0.6 \[ -{\frac{-120\,{b}^{3}{x}^{3}+135\,a{b}^{2}{x}^{2}-162\,{a}^{2}bx+243\,{a}^{3}}{440\,{b}^{4}} \left ( bx+a \right ) ^{{\frac{2}{3}}}} \]

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 = 1.3479, size = 76, normalized size = 1.06 \[ \frac{3 \,{\left (b x + a\right )}^{\frac{11}{3}}}{11 \, b^{4}} - \frac{9 \,{\left (b x + a\right )}^{\frac{8}{3}} a}{8 \, b^{4}} + \frac{9 \,{\left (b x + a\right )}^{\frac{5}{3}} a^{2}}{5 \, b^{4}} - \frac{3 \,{\left (b x + a\right )}^{\frac{2}{3}} a^{3}}{2 \, b^{4}} \]

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 = 0.220721, size = 57, normalized size = 0.79 \[ \frac{3 \,{\left (40 \, b^{3} x^{3} - 45 \, a b^{2} x^{2} + 54 \, a^{2} b x - 81 \, a^{3}\right )}{\left (b x + a\right )}^{\frac{2}{3}}}{440 \, b^{4}} \]

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

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Sympy [A]  time = 8.29501, size = 1640, normalized size = 22.78 \[ \text{result too large to display} \]

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 + 66
00*a**16*b**8*x**4 + 2640*a**15*b**9*x**5 + 440*a**14*b**10*x**6) + 1458*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**16*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) - 3120*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**1
4*b**10*x**6) + 4860*a**(62/3)*b**3*x**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) - 1710*a**(59/3)*b**4*x**4*(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*
*(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 + 6600*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 + 660
0*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 + 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**(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**1
5*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 + 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)

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GIAC/XCAS [A]  time = 0.211391, size = 82, normalized size = 1.14 \[ \frac{3 \,{\left (40 \,{\left (b x + a\right )}^{\frac{11}{3}} b^{30} - 165 \,{\left (b x + a\right )}^{\frac{8}{3}} a b^{30} + 264 \,{\left (b x + a\right )}^{\frac{5}{3}} a^{2} b^{30} - 220 \,{\left (b x + a\right )}^{\frac{2}{3}} a^{3} b^{30}\right )}}{440 \, b^{34}} \]

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)*b^30 - 165*(b*x + a)^(8/3)*a*b^30 + 264*(b*x + a)^(5/
3)*a^2*b^30 - 220*(b*x + a)^(2/3)*a^3*b^30)/b^34