3.413 \(\int \frac{x^3}{(a+b x)^{4/3}} \, dx\)

Optimal. Leaf size=70 \[ \frac{3 a^3}{b^4 \sqrt [3]{a+b x}}+\frac{9 a^2 (a+b x)^{2/3}}{2 b^4}-\frac{9 a (a+b x)^{5/3}}{5 b^4}+\frac{3 (a+b x)^{8/3}}{8 b^4} \]

[Out]

(3*a^3)/(b^4*(a + b*x)^(1/3)) + (9*a^2*(a + b*x)^(2/3))/(2*b^4) - (9*a*(a + b*x)
^(5/3))/(5*b^4) + (3*(a + b*x)^(8/3))/(8*b^4)

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Rubi [A]  time = 0.0516654, antiderivative size = 70, 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 \[ \frac{3 a^3}{b^4 \sqrt [3]{a+b x}}+\frac{9 a^2 (a+b x)^{2/3}}{2 b^4}-\frac{9 a (a+b x)^{5/3}}{5 b^4}+\frac{3 (a+b x)^{8/3}}{8 b^4} \]

Antiderivative was successfully verified.

[In]  Int[x^3/(a + b*x)^(4/3),x]

[Out]

(3*a^3)/(b^4*(a + b*x)^(1/3)) + (9*a^2*(a + b*x)^(2/3))/(2*b^4) - (9*a*(a + b*x)
^(5/3))/(5*b^4) + (3*(a + b*x)^(8/3))/(8*b^4)

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Rubi in Sympy [A]  time = 10.9331, size = 66, normalized size = 0.94 \[ \frac{3 a^{3}}{b^{4} \sqrt [3]{a + b x}} + \frac{9 a^{2} \left (a + b x\right )^{\frac{2}{3}}}{2 b^{4}} - \frac{9 a \left (a + b x\right )^{\frac{5}{3}}}{5 b^{4}} + \frac{3 \left (a + b x\right )^{\frac{8}{3}}}{8 b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3/(b*x+a)**(4/3),x)

[Out]

3*a**3/(b**4*(a + b*x)**(1/3)) + 9*a**2*(a + b*x)**(2/3)/(2*b**4) - 9*a*(a + b*x
)**(5/3)/(5*b**4) + 3*(a + b*x)**(8/3)/(8*b**4)

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Mathematica [A]  time = 0.0264444, size = 46, normalized size = 0.66 \[ \frac{3 \left (81 a^3+27 a^2 b x-9 a b^2 x^2+5 b^3 x^3\right )}{40 b^4 \sqrt [3]{a+b x}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3/(a + b*x)^(4/3),x]

[Out]

(3*(81*a^3 + 27*a^2*b*x - 9*a*b^2*x^2 + 5*b^3*x^3))/(40*b^4*(a + b*x)^(1/3))

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Maple [A]  time = 0.007, size = 43, normalized size = 0.6 \[{\frac{15\,{b}^{3}{x}^{3}-27\,a{b}^{2}{x}^{2}+81\,{a}^{2}bx+243\,{a}^{3}}{40\,{b}^{4}}{\frac{1}{\sqrt [3]{bx+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(b*x+a)^(4/3),x)

[Out]

3/40/(b*x+a)^(1/3)*(5*b^3*x^3-9*a*b^2*x^2+27*a^2*b*x+81*a^3)/b^4

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Maxima [A]  time = 1.32441, size = 76, normalized size = 1.09 \[ \frac{3 \,{\left (b x + a\right )}^{\frac{8}{3}}}{8 \, b^{4}} - \frac{9 \,{\left (b x + a\right )}^{\frac{5}{3}} a}{5 \, b^{4}} + \frac{9 \,{\left (b x + a\right )}^{\frac{2}{3}} a^{2}}{2 \, b^{4}} + \frac{3 \, a^{3}}{{\left (b x + a\right )}^{\frac{1}{3}} b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/(b*x + a)^(4/3),x, algorithm="maxima")

[Out]

3/8*(b*x + a)^(8/3)/b^4 - 9/5*(b*x + a)^(5/3)*a/b^4 + 9/2*(b*x + a)^(2/3)*a^2/b^
4 + 3*a^3/((b*x + a)^(1/3)*b^4)

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Fricas [A]  time = 0.207542, size = 57, normalized size = 0.81 \[ \frac{3 \,{\left (5 \, b^{3} x^{3} - 9 \, a b^{2} x^{2} + 27 \, a^{2} b x + 81 \, a^{3}\right )}}{40 \,{\left (b x + a\right )}^{\frac{1}{3}} b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/(b*x + a)^(4/3),x, algorithm="fricas")

[Out]

3/40*(5*b^3*x^3 - 9*a*b^2*x^2 + 27*a^2*b*x + 81*a^3)/((b*x + a)^(1/3)*b^4)

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Sympy [A]  time = 8.7926, size = 1538, normalized size = 21.97 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3/(b*x+a)**(4/3),x)

[Out]

243*a**(68/3)*(1 + b*x/a)**(2/3)/(40*a**20*b**4 + 240*a**19*b**5*x + 600*a**18*b
**6*x**2 + 800*a**17*b**7*x**3 + 600*a**16*b**8*x**4 + 240*a**15*b**9*x**5 + 40*
a**14*b**10*x**6) - 243*a**(68/3)/(40*a**20*b**4 + 240*a**19*b**5*x + 600*a**18*
b**6*x**2 + 800*a**17*b**7*x**3 + 600*a**16*b**8*x**4 + 240*a**15*b**9*x**5 + 40
*a**14*b**10*x**6) + 1296*a**(65/3)*b*x*(1 + b*x/a)**(2/3)/(40*a**20*b**4 + 240*
a**19*b**5*x + 600*a**18*b**6*x**2 + 800*a**17*b**7*x**3 + 600*a**16*b**8*x**4 +
240*a**15*b**9*x**5 + 40*a**14*b**10*x**6) - 1458*a**(65/3)*b*x/(40*a**20*b**4
+ 240*a**19*b**5*x + 600*a**18*b**6*x**2 + 800*a**17*b**7*x**3 + 600*a**16*b**8*
x**4 + 240*a**15*b**9*x**5 + 40*a**14*b**10*x**6) + 2808*a**(62/3)*b**2*x**2*(1
+ b*x/a)**(2/3)/(40*a**20*b**4 + 240*a**19*b**5*x + 600*a**18*b**6*x**2 + 800*a*
*17*b**7*x**3 + 600*a**16*b**8*x**4 + 240*a**15*b**9*x**5 + 40*a**14*b**10*x**6)
- 3645*a**(62/3)*b**2*x**2/(40*a**20*b**4 + 240*a**19*b**5*x + 600*a**18*b**6*x
**2 + 800*a**17*b**7*x**3 + 600*a**16*b**8*x**4 + 240*a**15*b**9*x**5 + 40*a**14
*b**10*x**6) + 3120*a**(59/3)*b**3*x**3*(1 + b*x/a)**(2/3)/(40*a**20*b**4 + 240*
a**19*b**5*x + 600*a**18*b**6*x**2 + 800*a**17*b**7*x**3 + 600*a**16*b**8*x**4 +
240*a**15*b**9*x**5 + 40*a**14*b**10*x**6) - 4860*a**(59/3)*b**3*x**3/(40*a**20
*b**4 + 240*a**19*b**5*x + 600*a**18*b**6*x**2 + 800*a**17*b**7*x**3 + 600*a**16
*b**8*x**4 + 240*a**15*b**9*x**5 + 40*a**14*b**10*x**6) + 1830*a**(56/3)*b**4*x*
*4*(1 + b*x/a)**(2/3)/(40*a**20*b**4 + 240*a**19*b**5*x + 600*a**18*b**6*x**2 +
800*a**17*b**7*x**3 + 600*a**16*b**8*x**4 + 240*a**15*b**9*x**5 + 40*a**14*b**10
*x**6) - 3645*a**(56/3)*b**4*x**4/(40*a**20*b**4 + 240*a**19*b**5*x + 600*a**18*
b**6*x**2 + 800*a**17*b**7*x**3 + 600*a**16*b**8*x**4 + 240*a**15*b**9*x**5 + 40
*a**14*b**10*x**6) + 528*a**(53/3)*b**5*x**5*(1 + b*x/a)**(2/3)/(40*a**20*b**4 +
240*a**19*b**5*x + 600*a**18*b**6*x**2 + 800*a**17*b**7*x**3 + 600*a**16*b**8*x
**4 + 240*a**15*b**9*x**5 + 40*a**14*b**10*x**6) - 1458*a**(53/3)*b**5*x**5/(40*
a**20*b**4 + 240*a**19*b**5*x + 600*a**18*b**6*x**2 + 800*a**17*b**7*x**3 + 600*
a**16*b**8*x**4 + 240*a**15*b**9*x**5 + 40*a**14*b**10*x**6) + 96*a**(50/3)*b**6
*x**6*(1 + b*x/a)**(2/3)/(40*a**20*b**4 + 240*a**19*b**5*x + 600*a**18*b**6*x**2
+ 800*a**17*b**7*x**3 + 600*a**16*b**8*x**4 + 240*a**15*b**9*x**5 + 40*a**14*b*
*10*x**6) - 243*a**(50/3)*b**6*x**6/(40*a**20*b**4 + 240*a**19*b**5*x + 600*a**1
8*b**6*x**2 + 800*a**17*b**7*x**3 + 600*a**16*b**8*x**4 + 240*a**15*b**9*x**5 +
40*a**14*b**10*x**6) + 48*a**(47/3)*b**7*x**7*(1 + b*x/a)**(2/3)/(40*a**20*b**4
+ 240*a**19*b**5*x + 600*a**18*b**6*x**2 + 800*a**17*b**7*x**3 + 600*a**16*b**8*
x**4 + 240*a**15*b**9*x**5 + 40*a**14*b**10*x**6) + 15*a**(44/3)*b**8*x**8*(1 +
b*x/a)**(2/3)/(40*a**20*b**4 + 240*a**19*b**5*x + 600*a**18*b**6*x**2 + 800*a**1
7*b**7*x**3 + 600*a**16*b**8*x**4 + 240*a**15*b**9*x**5 + 40*a**14*b**10*x**6)

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GIAC/XCAS [A]  time = 0.2417, size = 84, normalized size = 1.2 \[ \frac{3 \, a^{3}}{{\left (b x + a\right )}^{\frac{1}{3}} b^{4}} + \frac{3 \,{\left (5 \,{\left (b x + a\right )}^{\frac{8}{3}} b^{28} - 24 \,{\left (b x + a\right )}^{\frac{5}{3}} a b^{28} + 60 \,{\left (b x + a\right )}^{\frac{2}{3}} a^{2} b^{28}\right )}}{40 \, b^{32}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/(b*x + a)^(4/3),x, algorithm="giac")

[Out]

3*a^3/((b*x + a)^(1/3)*b^4) + 3/40*(5*(b*x + a)^(8/3)*b^28 - 24*(b*x + a)^(5/3)*
a*b^28 + 60*(b*x + a)^(2/3)*a^2*b^28)/b^32