3.378 \(\int x^3 (a+b x)^{2/3} \, dx\)

Optimal. Leaf size=72 \[ -\frac{3 a^3 (a+b x)^{5/3}}{5 b^4}+\frac{9 a^2 (a+b x)^{8/3}}{8 b^4}+\frac{3 (a+b x)^{14/3}}{14 b^4}-\frac{9 a (a+b x)^{11/3}}{11 b^4} \]

[Out]

(-3*a^3*(a + b*x)^(5/3))/(5*b^4) + (9*a^2*(a + b*x)^(8/3))/(8*b^4) - (9*a*(a + b
*x)^(11/3))/(11*b^4) + (3*(a + b*x)^(14/3))/(14*b^4)

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Rubi [A]  time = 0.0526865, 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 \[ -\frac{3 a^3 (a+b x)^{5/3}}{5 b^4}+\frac{9 a^2 (a+b x)^{8/3}}{8 b^4}+\frac{3 (a+b x)^{14/3}}{14 b^4}-\frac{9 a (a+b x)^{11/3}}{11 b^4} \]

Antiderivative was successfully verified.

[In]  Int[x^3*(a + b*x)^(2/3),x]

[Out]

(-3*a^3*(a + b*x)^(5/3))/(5*b^4) + (9*a^2*(a + b*x)^(8/3))/(8*b^4) - (9*a*(a + b
*x)^(11/3))/(11*b^4) + (3*(a + b*x)^(14/3))/(14*b^4)

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Rubi in Sympy [A]  time = 11.1939, size = 68, normalized size = 0.94 \[ - \frac{3 a^{3} \left (a + b x\right )^{\frac{5}{3}}}{5 b^{4}} + \frac{9 a^{2} \left (a + b x\right )^{\frac{8}{3}}}{8 b^{4}} - \frac{9 a \left (a + b x\right )^{\frac{11}{3}}}{11 b^{4}} + \frac{3 \left (a + b x\right )^{\frac{14}{3}}}{14 b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3*(b*x+a)**(2/3),x)

[Out]

-3*a**3*(a + b*x)**(5/3)/(5*b**4) + 9*a**2*(a + b*x)**(8/3)/(8*b**4) - 9*a*(a +
b*x)**(11/3)/(11*b**4) + 3*(a + b*x)**(14/3)/(14*b**4)

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Mathematica [A]  time = 0.0226996, size = 57, normalized size = 0.79 \[ \frac{3 (a+b x)^{2/3} \left (-81 a^4+54 a^3 b x-45 a^2 b^2 x^2+40 a b^3 x^3+220 b^4 x^4\right )}{3080 b^4} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3*(a + b*x)^(2/3),x]

[Out]

(3*(a + b*x)^(2/3)*(-81*a^4 + 54*a^3*b*x - 45*a^2*b^2*x^2 + 40*a*b^3*x^3 + 220*b
^4*x^4))/(3080*b^4)

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Maple [A]  time = 0.007, size = 43, normalized size = 0.6 \[ -{\frac{-660\,{b}^{3}{x}^{3}+540\,a{b}^{2}{x}^{2}-405\,{a}^{2}bx+243\,{a}^{3}}{3080\,{b}^{4}} \left ( bx+a \right ) ^{{\frac{5}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(b*x+a)^(2/3),x)

[Out]

-3/3080*(b*x+a)^(5/3)*(-220*b^3*x^3+180*a*b^2*x^2-135*a^2*b*x+81*a^3)/b^4

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Maxima [A]  time = 1.34062, size = 76, normalized size = 1.06 \[ \frac{3 \,{\left (b x + a\right )}^{\frac{14}{3}}}{14 \, b^{4}} - \frac{9 \,{\left (b x + a\right )}^{\frac{11}{3}} a}{11 \, b^{4}} + \frac{9 \,{\left (b x + a\right )}^{\frac{8}{3}} a^{2}}{8 \, b^{4}} - \frac{3 \,{\left (b x + a\right )}^{\frac{5}{3}} a^{3}}{5 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(2/3)*x^3,x, algorithm="maxima")

[Out]

3/14*(b*x + a)^(14/3)/b^4 - 9/11*(b*x + a)^(11/3)*a/b^4 + 9/8*(b*x + a)^(8/3)*a^
2/b^4 - 3/5*(b*x + a)^(5/3)*a^3/b^4

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Fricas [A]  time = 0.208001, size = 72, normalized size = 1. \[ \frac{3 \,{\left (220 \, b^{4} x^{4} + 40 \, a b^{3} x^{3} - 45 \, a^{2} b^{2} x^{2} + 54 \, a^{3} b x - 81 \, a^{4}\right )}{\left (b x + a\right )}^{\frac{2}{3}}}{3080 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(2/3)*x^3,x, algorithm="fricas")

[Out]

3/3080*(220*b^4*x^4 + 40*a*b^3*x^3 - 45*a^2*b^2*x^2 + 54*a^3*b*x - 81*a^4)*(b*x
+ a)^(2/3)/b^4

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Sympy [A]  time = 8.92595, size = 1742, normalized size = 24.19 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3*(b*x+a)**(2/3),x)

[Out]

-243*a**(74/3)*(1 + b*x/a)**(2/3)/(3080*a**20*b**4 + 18480*a**19*b**5*x + 46200*
a**18*b**6*x**2 + 61600*a**17*b**7*x**3 + 46200*a**16*b**8*x**4 + 18480*a**15*b*
*9*x**5 + 3080*a**14*b**10*x**6) + 243*a**(74/3)/(3080*a**20*b**4 + 18480*a**19*
b**5*x + 46200*a**18*b**6*x**2 + 61600*a**17*b**7*x**3 + 46200*a**16*b**8*x**4 +
18480*a**15*b**9*x**5 + 3080*a**14*b**10*x**6) - 1296*a**(71/3)*b*x*(1 + b*x/a)
**(2/3)/(3080*a**20*b**4 + 18480*a**19*b**5*x + 46200*a**18*b**6*x**2 + 61600*a*
*17*b**7*x**3 + 46200*a**16*b**8*x**4 + 18480*a**15*b**9*x**5 + 3080*a**14*b**10
*x**6) + 1458*a**(71/3)*b*x/(3080*a**20*b**4 + 18480*a**19*b**5*x + 46200*a**18*
b**6*x**2 + 61600*a**17*b**7*x**3 + 46200*a**16*b**8*x**4 + 18480*a**15*b**9*x**
5 + 3080*a**14*b**10*x**6) - 2808*a**(68/3)*b**2*x**2*(1 + b*x/a)**(2/3)/(3080*a
**20*b**4 + 18480*a**19*b**5*x + 46200*a**18*b**6*x**2 + 61600*a**17*b**7*x**3 +
46200*a**16*b**8*x**4 + 18480*a**15*b**9*x**5 + 3080*a**14*b**10*x**6) + 3645*a
**(68/3)*b**2*x**2/(3080*a**20*b**4 + 18480*a**19*b**5*x + 46200*a**18*b**6*x**2
+ 61600*a**17*b**7*x**3 + 46200*a**16*b**8*x**4 + 18480*a**15*b**9*x**5 + 3080*
a**14*b**10*x**6) - 3120*a**(65/3)*b**3*x**3*(1 + b*x/a)**(2/3)/(3080*a**20*b**4
+ 18480*a**19*b**5*x + 46200*a**18*b**6*x**2 + 61600*a**17*b**7*x**3 + 46200*a*
*16*b**8*x**4 + 18480*a**15*b**9*x**5 + 3080*a**14*b**10*x**6) + 4860*a**(65/3)*
b**3*x**3/(3080*a**20*b**4 + 18480*a**19*b**5*x + 46200*a**18*b**6*x**2 + 61600*
a**17*b**7*x**3 + 46200*a**16*b**8*x**4 + 18480*a**15*b**9*x**5 + 3080*a**14*b**
10*x**6) - 1050*a**(62/3)*b**4*x**4*(1 + b*x/a)**(2/3)/(3080*a**20*b**4 + 18480*
a**19*b**5*x + 46200*a**18*b**6*x**2 + 61600*a**17*b**7*x**3 + 46200*a**16*b**8*
x**4 + 18480*a**15*b**9*x**5 + 3080*a**14*b**10*x**6) + 3645*a**(62/3)*b**4*x**4
/(3080*a**20*b**4 + 18480*a**19*b**5*x + 46200*a**18*b**6*x**2 + 61600*a**17*b**
7*x**3 + 46200*a**16*b**8*x**4 + 18480*a**15*b**9*x**5 + 3080*a**14*b**10*x**6)
+ 4032*a**(59/3)*b**5*x**5*(1 + b*x/a)**(2/3)/(3080*a**20*b**4 + 18480*a**19*b**
5*x + 46200*a**18*b**6*x**2 + 61600*a**17*b**7*x**3 + 46200*a**16*b**8*x**4 + 18
480*a**15*b**9*x**5 + 3080*a**14*b**10*x**6) + 1458*a**(59/3)*b**5*x**5/(3080*a*
*20*b**4 + 18480*a**19*b**5*x + 46200*a**18*b**6*x**2 + 61600*a**17*b**7*x**3 +
46200*a**16*b**8*x**4 + 18480*a**15*b**9*x**5 + 3080*a**14*b**10*x**6) + 11004*a
**(56/3)*b**6*x**6*(1 + b*x/a)**(2/3)/(3080*a**20*b**4 + 18480*a**19*b**5*x + 46
200*a**18*b**6*x**2 + 61600*a**17*b**7*x**3 + 46200*a**16*b**8*x**4 + 18480*a**1
5*b**9*x**5 + 3080*a**14*b**10*x**6) + 243*a**(56/3)*b**6*x**6/(3080*a**20*b**4
+ 18480*a**19*b**5*x + 46200*a**18*b**6*x**2 + 61600*a**17*b**7*x**3 + 46200*a**
16*b**8*x**4 + 18480*a**15*b**9*x**5 + 3080*a**14*b**10*x**6) + 14352*a**(53/3)*
b**7*x**7*(1 + b*x/a)**(2/3)/(3080*a**20*b**4 + 18480*a**19*b**5*x + 46200*a**18
*b**6*x**2 + 61600*a**17*b**7*x**3 + 46200*a**16*b**8*x**4 + 18480*a**15*b**9*x*
*5 + 3080*a**14*b**10*x**6) + 10485*a**(50/3)*b**8*x**8*(1 + b*x/a)**(2/3)/(3080
*a**20*b**4 + 18480*a**19*b**5*x + 46200*a**18*b**6*x**2 + 61600*a**17*b**7*x**3
+ 46200*a**16*b**8*x**4 + 18480*a**15*b**9*x**5 + 3080*a**14*b**10*x**6) + 4080
*a**(47/3)*b**9*x**9*(1 + b*x/a)**(2/3)/(3080*a**20*b**4 + 18480*a**19*b**5*x +
46200*a**18*b**6*x**2 + 61600*a**17*b**7*x**3 + 46200*a**16*b**8*x**4 + 18480*a*
*15*b**9*x**5 + 3080*a**14*b**10*x**6) + 660*a**(44/3)*b**10*x**10*(1 + b*x/a)**
(2/3)/(3080*a**20*b**4 + 18480*a**19*b**5*x + 46200*a**18*b**6*x**2 + 61600*a**1
7*b**7*x**3 + 46200*a**16*b**8*x**4 + 18480*a**15*b**9*x**5 + 3080*a**14*b**10*x
**6)

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GIAC/XCAS [A]  time = 0.220525, size = 82, normalized size = 1.14 \[ \frac{3 \,{\left (220 \,{\left (b x + a\right )}^{\frac{14}{3}} b^{39} - 840 \,{\left (b x + a\right )}^{\frac{11}{3}} a b^{39} + 1155 \,{\left (b x + a\right )}^{\frac{8}{3}} a^{2} b^{39} - 616 \,{\left (b x + a\right )}^{\frac{5}{3}} a^{3} b^{39}\right )}}{3080 \, b^{43}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(2/3)*x^3,x, algorithm="giac")

[Out]

3/3080*(220*(b*x + a)^(14/3)*b^39 - 840*(b*x + a)^(11/3)*a*b^39 + 1155*(b*x + a)
^(8/3)*a^2*b^39 - 616*(b*x + a)^(5/3)*a^3*b^39)/b^43