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

Optimal. Leaf size=49 \[ -\frac{3 a^3}{\sqrt [3]{x}}+\frac{9}{2} a^2 b x^{2/3}+\frac{9}{5} a b^2 x^{5/3}+\frac{3}{8} b^3 x^{8/3} \]

[Out]

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

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Rubi [A]  time = 0.0311833, antiderivative size = 49, 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 \[ -\frac{3 a^3}{\sqrt [3]{x}}+\frac{9}{2} a^2 b x^{2/3}+\frac{9}{5} a b^2 x^{5/3}+\frac{3}{8} b^3 x^{8/3} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^3/x^(4/3),x]

[Out]

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

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Rubi in Sympy [A]  time = 5.62572, size = 48, normalized size = 0.98 \[ - \frac{3 a^{3}}{\sqrt [3]{x}} + \frac{9 a^{2} b x^{\frac{2}{3}}}{2} + \frac{9 a b^{2} x^{\frac{5}{3}}}{5} + \frac{3 b^{3} x^{\frac{8}{3}}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0116151, size = 39, normalized size = 0.8 \[ \frac{3 \left (-40 a^3+60 a^2 b x+24 a b^2 x^2+5 b^3 x^3\right )}{40 \sqrt [3]{x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.007, size = 36, normalized size = 0.7 \[ -{\frac{-15\,{b}^{3}{x}^{3}-72\,a{b}^{2}{x}^{2}-180\,{a}^{2}bx+120\,{a}^{3}}{40}{\frac{1}{\sqrt [3]{x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^3/x^(4/3),x)

[Out]

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

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Maxima [A]  time = 1.35156, size = 47, normalized size = 0.96 \[ \frac{3}{8} \, b^{3} x^{\frac{8}{3}} + \frac{9}{5} \, a b^{2} x^{\frac{5}{3}} + \frac{9}{2} \, a^{2} b x^{\frac{2}{3}} - \frac{3 \, a^{3}}{x^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.20538, size = 47, normalized size = 0.96 \[ \frac{3 \,{\left (5 \, b^{3} x^{3} + 24 \, a b^{2} x^{2} + 60 \, a^{2} b x - 40 \, a^{3}\right )}}{40 \, x^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 10.6286, size = 4004, normalized size = 81.71 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((243*a**(68/3)*b**(1/3)*(-1 + b*(a/b + x)/a)**(2/3)/(40*a**20 - 240*a*
*19*b*(a/b + x) + 600*a**18*b**2*(a/b + x)**2 - 800*a**17*b**3*(a/b + x)**3 + 60
0*a**16*b**4*(a/b + x)**4 - 240*a**15*b**5*(a/b + x)**5 + 40*a**14*b**6*(a/b + x
)**6) - 243*a**(68/3)*b**(1/3)*exp(8*I*pi/3)/(40*a**20 - 240*a**19*b*(a/b + x) +
600*a**18*b**2*(a/b + x)**2 - 800*a**17*b**3*(a/b + x)**3 + 600*a**16*b**4*(a/b
+ x)**4 - 240*a**15*b**5*(a/b + x)**5 + 40*a**14*b**6*(a/b + x)**6) - 1296*a**(
65/3)*b**(4/3)*(-1 + b*(a/b + x)/a)**(2/3)*(a/b + x)/(40*a**20 - 240*a**19*b*(a/
b + x) + 600*a**18*b**2*(a/b + x)**2 - 800*a**17*b**3*(a/b + x)**3 + 600*a**16*b
**4*(a/b + x)**4 - 240*a**15*b**5*(a/b + x)**5 + 40*a**14*b**6*(a/b + x)**6) + 1
458*a**(65/3)*b**(4/3)*(a/b + x)*exp(8*I*pi/3)/(40*a**20 - 240*a**19*b*(a/b + x)
+ 600*a**18*b**2*(a/b + x)**2 - 800*a**17*b**3*(a/b + x)**3 + 600*a**16*b**4*(a
/b + x)**4 - 240*a**15*b**5*(a/b + x)**5 + 40*a**14*b**6*(a/b + x)**6) + 2808*a*
*(62/3)*b**(7/3)*(-1 + b*(a/b + x)/a)**(2/3)*(a/b + x)**2/(40*a**20 - 240*a**19*
b*(a/b + x) + 600*a**18*b**2*(a/b + x)**2 - 800*a**17*b**3*(a/b + x)**3 + 600*a*
*16*b**4*(a/b + x)**4 - 240*a**15*b**5*(a/b + x)**5 + 40*a**14*b**6*(a/b + x)**6
) - 3645*a**(62/3)*b**(7/3)*(a/b + x)**2*exp(8*I*pi/3)/(40*a**20 - 240*a**19*b*(
a/b + x) + 600*a**18*b**2*(a/b + x)**2 - 800*a**17*b**3*(a/b + x)**3 + 600*a**16
*b**4*(a/b + x)**4 - 240*a**15*b**5*(a/b + x)**5 + 40*a**14*b**6*(a/b + x)**6) -
3120*a**(59/3)*b**(10/3)*(-1 + b*(a/b + x)/a)**(2/3)*(a/b + x)**3/(40*a**20 - 2
40*a**19*b*(a/b + x) + 600*a**18*b**2*(a/b + x)**2 - 800*a**17*b**3*(a/b + x)**3
+ 600*a**16*b**4*(a/b + x)**4 - 240*a**15*b**5*(a/b + x)**5 + 40*a**14*b**6*(a/
b + x)**6) + 4860*a**(59/3)*b**(10/3)*(a/b + x)**3*exp(8*I*pi/3)/(40*a**20 - 240
*a**19*b*(a/b + x) + 600*a**18*b**2*(a/b + x)**2 - 800*a**17*b**3*(a/b + x)**3 +
600*a**16*b**4*(a/b + x)**4 - 240*a**15*b**5*(a/b + x)**5 + 40*a**14*b**6*(a/b
+ x)**6) + 1830*a**(56/3)*b**(13/3)*(-1 + b*(a/b + x)/a)**(2/3)*(a/b + x)**4/(40
*a**20 - 240*a**19*b*(a/b + x) + 600*a**18*b**2*(a/b + x)**2 - 800*a**17*b**3*(a
/b + x)**3 + 600*a**16*b**4*(a/b + x)**4 - 240*a**15*b**5*(a/b + x)**5 + 40*a**1
4*b**6*(a/b + x)**6) - 3645*a**(56/3)*b**(13/3)*(a/b + x)**4*exp(8*I*pi/3)/(40*a
**20 - 240*a**19*b*(a/b + x) + 600*a**18*b**2*(a/b + x)**2 - 800*a**17*b**3*(a/b
+ x)**3 + 600*a**16*b**4*(a/b + x)**4 - 240*a**15*b**5*(a/b + x)**5 + 40*a**14*
b**6*(a/b + x)**6) - 528*a**(53/3)*b**(16/3)*(-1 + b*(a/b + x)/a)**(2/3)*(a/b +
x)**5/(40*a**20 - 240*a**19*b*(a/b + x) + 600*a**18*b**2*(a/b + x)**2 - 800*a**1
7*b**3*(a/b + x)**3 + 600*a**16*b**4*(a/b + x)**4 - 240*a**15*b**5*(a/b + x)**5
+ 40*a**14*b**6*(a/b + x)**6) + 1458*a**(53/3)*b**(16/3)*(a/b + x)**5*exp(8*I*pi
/3)/(40*a**20 - 240*a**19*b*(a/b + x) + 600*a**18*b**2*(a/b + x)**2 - 800*a**17*
b**3*(a/b + x)**3 + 600*a**16*b**4*(a/b + x)**4 - 240*a**15*b**5*(a/b + x)**5 +
40*a**14*b**6*(a/b + x)**6) + 96*a**(50/3)*b**(19/3)*(-1 + b*(a/b + x)/a)**(2/3)
*(a/b + x)**6/(40*a**20 - 240*a**19*b*(a/b + x) + 600*a**18*b**2*(a/b + x)**2 -
800*a**17*b**3*(a/b + x)**3 + 600*a**16*b**4*(a/b + x)**4 - 240*a**15*b**5*(a/b
+ x)**5 + 40*a**14*b**6*(a/b + x)**6) - 243*a**(50/3)*b**(19/3)*(a/b + x)**6*exp
(8*I*pi/3)/(40*a**20 - 240*a**19*b*(a/b + x) + 600*a**18*b**2*(a/b + x)**2 - 800
*a**17*b**3*(a/b + x)**3 + 600*a**16*b**4*(a/b + x)**4 - 240*a**15*b**5*(a/b + x
)**5 + 40*a**14*b**6*(a/b + x)**6) - 48*a**(47/3)*b**(22/3)*(-1 + b*(a/b + x)/a)
**(2/3)*(a/b + x)**7/(40*a**20 - 240*a**19*b*(a/b + x) + 600*a**18*b**2*(a/b + x
)**2 - 800*a**17*b**3*(a/b + x)**3 + 600*a**16*b**4*(a/b + x)**4 - 240*a**15*b**
5*(a/b + x)**5 + 40*a**14*b**6*(a/b + x)**6) + 15*a**(44/3)*b**(25/3)*(-1 + b*(a
/b + x)/a)**(2/3)*(a/b + x)**8/(40*a**20 - 240*a**19*b*(a/b + x) + 600*a**18*b**
2*(a/b + x)**2 - 800*a**17*b**3*(a/b + x)**3 + 600*a**16*b**4*(a/b + x)**4 - 240
*a**15*b**5*(a/b + x)**5 + 40*a**14*b**6*(a/b + x)**6), Abs(b*(a/b + x)/a) > 1),
(243*a**(68/3)*b**(1/3)*(1 - b*(a/b + x)/a)**(2/3)*exp(8*I*pi/3)/(40*a**20 - 24
0*a**19*b*(a/b + x) + 600*a**18*b**2*(a/b + x)**2 - 800*a**17*b**3*(a/b + x)**3
+ 600*a**16*b**4*(a/b + x)**4 - 240*a**15*b**5*(a/b + x)**5 + 40*a**14*b**6*(a/b
+ x)**6) - 243*a**(68/3)*b**(1/3)*exp(8*I*pi/3)/(40*a**20 - 240*a**19*b*(a/b +
x) + 600*a**18*b**2*(a/b + x)**2 - 800*a**17*b**3*(a/b + x)**3 + 600*a**16*b**4*
(a/b + x)**4 - 240*a**15*b**5*(a/b + x)**5 + 40*a**14*b**6*(a/b + x)**6) - 1296*
a**(65/3)*b**(4/3)*(1 - b*(a/b + x)/a)**(2/3)*(a/b + x)*exp(8*I*pi/3)/(40*a**20
- 240*a**19*b*(a/b + x) + 600*a**18*b**2*(a/b + x)**2 - 800*a**17*b**3*(a/b + x)
**3 + 600*a**16*b**4*(a/b + x)**4 - 240*a**15*b**5*(a/b + x)**5 + 40*a**14*b**6*
(a/b + x)**6) + 1458*a**(65/3)*b**(4/3)*(a/b + x)*exp(8*I*pi/3)/(40*a**20 - 240*
a**19*b*(a/b + x) + 600*a**18*b**2*(a/b + x)**2 - 800*a**17*b**3*(a/b + x)**3 +
600*a**16*b**4*(a/b + x)**4 - 240*a**15*b**5*(a/b + x)**5 + 40*a**14*b**6*(a/b +
x)**6) + 2808*a**(62/3)*b**(7/3)*(1 - b*(a/b + x)/a)**(2/3)*(a/b + x)**2*exp(8*
I*pi/3)/(40*a**20 - 240*a**19*b*(a/b + x) + 600*a**18*b**2*(a/b + x)**2 - 800*a*
*17*b**3*(a/b + x)**3 + 600*a**16*b**4*(a/b + x)**4 - 240*a**15*b**5*(a/b + x)**
5 + 40*a**14*b**6*(a/b + x)**6) - 3645*a**(62/3)*b**(7/3)*(a/b + x)**2*exp(8*I*p
i/3)/(40*a**20 - 240*a**19*b*(a/b + x) + 600*a**18*b**2*(a/b + x)**2 - 800*a**17
*b**3*(a/b + x)**3 + 600*a**16*b**4*(a/b + x)**4 - 240*a**15*b**5*(a/b + x)**5 +
40*a**14*b**6*(a/b + x)**6) - 3120*a**(59/3)*b**(10/3)*(1 - b*(a/b + x)/a)**(2/
3)*(a/b + x)**3*exp(8*I*pi/3)/(40*a**20 - 240*a**19*b*(a/b + x) + 600*a**18*b**2
*(a/b + x)**2 - 800*a**17*b**3*(a/b + x)**3 + 600*a**16*b**4*(a/b + x)**4 - 240*
a**15*b**5*(a/b + x)**5 + 40*a**14*b**6*(a/b + x)**6) + 4860*a**(59/3)*b**(10/3)
*(a/b + x)**3*exp(8*I*pi/3)/(40*a**20 - 240*a**19*b*(a/b + x) + 600*a**18*b**2*(
a/b + x)**2 - 800*a**17*b**3*(a/b + x)**3 + 600*a**16*b**4*(a/b + x)**4 - 240*a*
*15*b**5*(a/b + x)**5 + 40*a**14*b**6*(a/b + x)**6) + 1830*a**(56/3)*b**(13/3)*(
1 - b*(a/b + x)/a)**(2/3)*(a/b + x)**4*exp(8*I*pi/3)/(40*a**20 - 240*a**19*b*(a/
b + x) + 600*a**18*b**2*(a/b + x)**2 - 800*a**17*b**3*(a/b + x)**3 + 600*a**16*b
**4*(a/b + x)**4 - 240*a**15*b**5*(a/b + x)**5 + 40*a**14*b**6*(a/b + x)**6) - 3
645*a**(56/3)*b**(13/3)*(a/b + x)**4*exp(8*I*pi/3)/(40*a**20 - 240*a**19*b*(a/b
+ x) + 600*a**18*b**2*(a/b + x)**2 - 800*a**17*b**3*(a/b + x)**3 + 600*a**16*b**
4*(a/b + x)**4 - 240*a**15*b**5*(a/b + x)**5 + 40*a**14*b**6*(a/b + x)**6) - 528
*a**(53/3)*b**(16/3)*(1 - b*(a/b + x)/a)**(2/3)*(a/b + x)**5*exp(8*I*pi/3)/(40*a
**20 - 240*a**19*b*(a/b + x) + 600*a**18*b**2*(a/b + x)**2 - 800*a**17*b**3*(a/b
+ x)**3 + 600*a**16*b**4*(a/b + x)**4 - 240*a**15*b**5*(a/b + x)**5 + 40*a**14*
b**6*(a/b + x)**6) + 1458*a**(53/3)*b**(16/3)*(a/b + x)**5*exp(8*I*pi/3)/(40*a**
20 - 240*a**19*b*(a/b + x) + 600*a**18*b**2*(a/b + x)**2 - 800*a**17*b**3*(a/b +
x)**3 + 600*a**16*b**4*(a/b + x)**4 - 240*a**15*b**5*(a/b + x)**5 + 40*a**14*b*
*6*(a/b + x)**6) + 96*a**(50/3)*b**(19/3)*(1 - b*(a/b + x)/a)**(2/3)*(a/b + x)**
6*exp(8*I*pi/3)/(40*a**20 - 240*a**19*b*(a/b + x) + 600*a**18*b**2*(a/b + x)**2
- 800*a**17*b**3*(a/b + x)**3 + 600*a**16*b**4*(a/b + x)**4 - 240*a**15*b**5*(a/
b + x)**5 + 40*a**14*b**6*(a/b + x)**6) - 243*a**(50/3)*b**(19/3)*(a/b + x)**6*e
xp(8*I*pi/3)/(40*a**20 - 240*a**19*b*(a/b + x) + 600*a**18*b**2*(a/b + x)**2 - 8
00*a**17*b**3*(a/b + x)**3 + 600*a**16*b**4*(a/b + x)**4 - 240*a**15*b**5*(a/b +
x)**5 + 40*a**14*b**6*(a/b + x)**6) - 48*a**(47/3)*b**(22/3)*(1 - b*(a/b + x)/a
)**(2/3)*(a/b + x)**7*exp(8*I*pi/3)/(40*a**20 - 240*a**19*b*(a/b + x) + 600*a**1
8*b**2*(a/b + x)**2 - 800*a**17*b**3*(a/b + x)**3 + 600*a**16*b**4*(a/b + x)**4
- 240*a**15*b**5*(a/b + x)**5 + 40*a**14*b**6*(a/b + x)**6) + 15*a**(44/3)*b**(2
5/3)*(1 - b*(a/b + x)/a)**(2/3)*(a/b + x)**8*exp(8*I*pi/3)/(40*a**20 - 240*a**19
*b*(a/b + x) + 600*a**18*b**2*(a/b + x)**2 - 800*a**17*b**3*(a/b + x)**3 + 600*a
**16*b**4*(a/b + x)**4 - 240*a**15*b**5*(a/b + x)**5 + 40*a**14*b**6*(a/b + x)**
6), True))

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GIAC/XCAS [A]  time = 0.207048, size = 47, normalized size = 0.96 \[ \frac{3}{8} \, b^{3} x^{\frac{8}{3}} + \frac{9}{5} \, a b^{2} x^{\frac{5}{3}} + \frac{9}{2} \, a^{2} b x^{\frac{2}{3}} - \frac{3 \, a^{3}}{x^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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