3.673 \(\int \frac{(a+b x)^3}{x^{5/3}} \, dx\)

Optimal. Leaf size=49 \[ -\frac{3 a^3}{2 x^{2/3}}+9 a^2 b \sqrt [3]{x}+\frac{9}{4} a b^2 x^{4/3}+\frac{3}{7} b^3 x^{7/3} \]

[Out]

(-3*a^3)/(2*x^(2/3)) + 9*a^2*b*x^(1/3) + (9*a*b^2*x^(4/3))/4 + (3*b^3*x^(7/3))/7

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Rubi [A]  time = 0.0316716, 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 \[ -\frac{3 a^3}{2 x^{2/3}}+9 a^2 b \sqrt [3]{x}+\frac{9}{4} a b^2 x^{4/3}+\frac{3}{7} b^3 x^{7/3} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^3/x^(5/3),x]

[Out]

(-3*a^3)/(2*x^(2/3)) + 9*a^2*b*x^(1/3) + (9*a*b^2*x^(4/3))/4 + (3*b^3*x^(7/3))/7

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Rubi in Sympy [A]  time = 5.36094, size = 48, normalized size = 0.98 \[ - \frac{3 a^{3}}{2 x^{\frac{2}{3}}} + 9 a^{2} b \sqrt [3]{x} + \frac{9 a b^{2} x^{\frac{4}{3}}}{4} + \frac{3 b^{3} x^{\frac{7}{3}}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**3/x**(5/3),x)

[Out]

-3*a**3/(2*x**(2/3)) + 9*a**2*b*x**(1/3) + 9*a*b**2*x**(4/3)/4 + 3*b**3*x**(7/3)
/7

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Mathematica [A]  time = 0.0117088, size = 39, normalized size = 0.8 \[ \frac{3 \left (-14 a^3+84 a^2 b x+21 a b^2 x^2+4 b^3 x^3\right )}{28 x^{2/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^3/x^(5/3),x]

[Out]

(3*(-14*a^3 + 84*a^2*b*x + 21*a*b^2*x^2 + 4*b^3*x^3))/(28*x^(2/3))

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Maple [A]  time = 0.007, size = 36, normalized size = 0.7 \[ -{\frac{-12\,{b}^{3}{x}^{3}-63\,a{b}^{2}{x}^{2}-252\,{a}^{2}bx+42\,{a}^{3}}{28}{x}^{-{\frac{2}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^3/x^(5/3),x)

[Out]

-3/28*(-4*b^3*x^3-21*a*b^2*x^2-84*a^2*b*x+14*a^3)/x^(2/3)

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Maxima [A]  time = 1.32241, size = 47, normalized size = 0.96 \[ \frac{3}{7} \, b^{3} x^{\frac{7}{3}} + \frac{9}{4} \, a b^{2} x^{\frac{4}{3}} + 9 \, a^{2} b x^{\frac{1}{3}} - \frac{3 \, a^{3}}{2 \, x^{\frac{2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^3/x^(5/3),x, algorithm="maxima")

[Out]

3/7*b^3*x^(7/3) + 9/4*a*b^2*x^(4/3) + 9*a^2*b*x^(1/3) - 3/2*a^3/x^(2/3)

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Fricas [A]  time = 0.209908, size = 47, normalized size = 0.96 \[ \frac{3 \,{\left (4 \, b^{3} x^{3} + 21 \, a b^{2} x^{2} + 84 \, a^{2} b x - 14 \, a^{3}\right )}}{28 \, x^{\frac{2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^3/x^(5/3),x, algorithm="fricas")

[Out]

3/28*(4*b^3*x^3 + 21*a*b^2*x^2 + 84*a^2*b*x - 14*a^3)/x^(2/3)

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Sympy [A]  time = 10.7685, 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**(5/3),x)

[Out]

Piecewise((243*a**(67/3)*b**(2/3)*(-1 + b*(a/b + x)/a)**(1/3)/(28*a**20 - 168*a*
*19*b*(a/b + x) + 420*a**18*b**2*(a/b + x)**2 - 560*a**17*b**3*(a/b + x)**3 + 42
0*a**16*b**4*(a/b + x)**4 - 168*a**15*b**5*(a/b + x)**5 + 28*a**14*b**6*(a/b + x
)**6) - 243*a**(67/3)*b**(2/3)*exp(7*I*pi/3)/(28*a**20 - 168*a**19*b*(a/b + x) +
420*a**18*b**2*(a/b + x)**2 - 560*a**17*b**3*(a/b + x)**3 + 420*a**16*b**4*(a/b
+ x)**4 - 168*a**15*b**5*(a/b + x)**5 + 28*a**14*b**6*(a/b + x)**6) - 1377*a**(
64/3)*b**(5/3)*(-1 + b*(a/b + x)/a)**(1/3)*(a/b + x)/(28*a**20 - 168*a**19*b*(a/
b + x) + 420*a**18*b**2*(a/b + x)**2 - 560*a**17*b**3*(a/b + x)**3 + 420*a**16*b
**4*(a/b + x)**4 - 168*a**15*b**5*(a/b + x)**5 + 28*a**14*b**6*(a/b + x)**6) + 1
458*a**(64/3)*b**(5/3)*(a/b + x)*exp(7*I*pi/3)/(28*a**20 - 168*a**19*b*(a/b + x)
+ 420*a**18*b**2*(a/b + x)**2 - 560*a**17*b**3*(a/b + x)**3 + 420*a**16*b**4*(a
/b + x)**4 - 168*a**15*b**5*(a/b + x)**5 + 28*a**14*b**6*(a/b + x)**6) + 3213*a*
*(61/3)*b**(8/3)*(-1 + b*(a/b + x)/a)**(1/3)*(a/b + x)**2/(28*a**20 - 168*a**19*
b*(a/b + x) + 420*a**18*b**2*(a/b + x)**2 - 560*a**17*b**3*(a/b + x)**3 + 420*a*
*16*b**4*(a/b + x)**4 - 168*a**15*b**5*(a/b + x)**5 + 28*a**14*b**6*(a/b + x)**6
) - 3645*a**(61/3)*b**(8/3)*(a/b + x)**2*exp(7*I*pi/3)/(28*a**20 - 168*a**19*b*(
a/b + x) + 420*a**18*b**2*(a/b + x)**2 - 560*a**17*b**3*(a/b + x)**3 + 420*a**16
*b**4*(a/b + x)**4 - 168*a**15*b**5*(a/b + x)**5 + 28*a**14*b**6*(a/b + x)**6) -
3927*a**(58/3)*b**(11/3)*(-1 + b*(a/b + x)/a)**(1/3)*(a/b + x)**3/(28*a**20 - 1
68*a**19*b*(a/b + x) + 420*a**18*b**2*(a/b + x)**2 - 560*a**17*b**3*(a/b + x)**3
+ 420*a**16*b**4*(a/b + x)**4 - 168*a**15*b**5*(a/b + x)**5 + 28*a**14*b**6*(a/
b + x)**6) + 4860*a**(58/3)*b**(11/3)*(a/b + x)**3*exp(7*I*pi/3)/(28*a**20 - 168
*a**19*b*(a/b + x) + 420*a**18*b**2*(a/b + x)**2 - 560*a**17*b**3*(a/b + x)**3 +
420*a**16*b**4*(a/b + x)**4 - 168*a**15*b**5*(a/b + x)**5 + 28*a**14*b**6*(a/b
+ x)**6) + 2625*a**(55/3)*b**(14/3)*(-1 + b*(a/b + x)/a)**(1/3)*(a/b + x)**4/(28
*a**20 - 168*a**19*b*(a/b + x) + 420*a**18*b**2*(a/b + x)**2 - 560*a**17*b**3*(a
/b + x)**3 + 420*a**16*b**4*(a/b + x)**4 - 168*a**15*b**5*(a/b + x)**5 + 28*a**1
4*b**6*(a/b + x)**6) - 3645*a**(55/3)*b**(14/3)*(a/b + x)**4*exp(7*I*pi/3)/(28*a
**20 - 168*a**19*b*(a/b + x) + 420*a**18*b**2*(a/b + x)**2 - 560*a**17*b**3*(a/b
+ x)**3 + 420*a**16*b**4*(a/b + x)**4 - 168*a**15*b**5*(a/b + x)**5 + 28*a**14*
b**6*(a/b + x)**6) - 903*a**(52/3)*b**(17/3)*(-1 + b*(a/b + x)/a)**(1/3)*(a/b +
x)**5/(28*a**20 - 168*a**19*b*(a/b + x) + 420*a**18*b**2*(a/b + x)**2 - 560*a**1
7*b**3*(a/b + x)**3 + 420*a**16*b**4*(a/b + x)**4 - 168*a**15*b**5*(a/b + x)**5
+ 28*a**14*b**6*(a/b + x)**6) + 1458*a**(52/3)*b**(17/3)*(a/b + x)**5*exp(7*I*pi
/3)/(28*a**20 - 168*a**19*b*(a/b + x) + 420*a**18*b**2*(a/b + x)**2 - 560*a**17*
b**3*(a/b + x)**3 + 420*a**16*b**4*(a/b + x)**4 - 168*a**15*b**5*(a/b + x)**5 +
28*a**14*b**6*(a/b + x)**6) + 147*a**(49/3)*b**(20/3)*(-1 + b*(a/b + x)/a)**(1/3
)*(a/b + x)**6/(28*a**20 - 168*a**19*b*(a/b + x) + 420*a**18*b**2*(a/b + x)**2 -
560*a**17*b**3*(a/b + x)**3 + 420*a**16*b**4*(a/b + x)**4 - 168*a**15*b**5*(a/b
+ x)**5 + 28*a**14*b**6*(a/b + x)**6) - 243*a**(49/3)*b**(20/3)*(a/b + x)**6*ex
p(7*I*pi/3)/(28*a**20 - 168*a**19*b*(a/b + x) + 420*a**18*b**2*(a/b + x)**2 - 56
0*a**17*b**3*(a/b + x)**3 + 420*a**16*b**4*(a/b + x)**4 - 168*a**15*b**5*(a/b +
x)**5 + 28*a**14*b**6*(a/b + x)**6) - 33*a**(46/3)*b**(23/3)*(-1 + b*(a/b + x)/a
)**(1/3)*(a/b + x)**7/(28*a**20 - 168*a**19*b*(a/b + x) + 420*a**18*b**2*(a/b +
x)**2 - 560*a**17*b**3*(a/b + x)**3 + 420*a**16*b**4*(a/b + x)**4 - 168*a**15*b*
*5*(a/b + x)**5 + 28*a**14*b**6*(a/b + x)**6) + 12*a**(43/3)*b**(26/3)*(-1 + b*(
a/b + x)/a)**(1/3)*(a/b + x)**8/(28*a**20 - 168*a**19*b*(a/b + x) + 420*a**18*b*
*2*(a/b + x)**2 - 560*a**17*b**3*(a/b + x)**3 + 420*a**16*b**4*(a/b + x)**4 - 16
8*a**15*b**5*(a/b + x)**5 + 28*a**14*b**6*(a/b + x)**6), Abs(b*(a/b + x)/a) > 1)
, (243*a**(67/3)*b**(2/3)*(1 - b*(a/b + x)/a)**(1/3)*exp(7*I*pi/3)/(28*a**20 - 1
68*a**19*b*(a/b + x) + 420*a**18*b**2*(a/b + x)**2 - 560*a**17*b**3*(a/b + x)**3
+ 420*a**16*b**4*(a/b + x)**4 - 168*a**15*b**5*(a/b + x)**5 + 28*a**14*b**6*(a/
b + x)**6) - 243*a**(67/3)*b**(2/3)*exp(7*I*pi/3)/(28*a**20 - 168*a**19*b*(a/b +
x) + 420*a**18*b**2*(a/b + x)**2 - 560*a**17*b**3*(a/b + x)**3 + 420*a**16*b**4
*(a/b + x)**4 - 168*a**15*b**5*(a/b + x)**5 + 28*a**14*b**6*(a/b + x)**6) - 1377
*a**(64/3)*b**(5/3)*(1 - b*(a/b + x)/a)**(1/3)*(a/b + x)*exp(7*I*pi/3)/(28*a**20
- 168*a**19*b*(a/b + x) + 420*a**18*b**2*(a/b + x)**2 - 560*a**17*b**3*(a/b + x
)**3 + 420*a**16*b**4*(a/b + x)**4 - 168*a**15*b**5*(a/b + x)**5 + 28*a**14*b**6
*(a/b + x)**6) + 1458*a**(64/3)*b**(5/3)*(a/b + x)*exp(7*I*pi/3)/(28*a**20 - 168
*a**19*b*(a/b + x) + 420*a**18*b**2*(a/b + x)**2 - 560*a**17*b**3*(a/b + x)**3 +
420*a**16*b**4*(a/b + x)**4 - 168*a**15*b**5*(a/b + x)**5 + 28*a**14*b**6*(a/b
+ x)**6) + 3213*a**(61/3)*b**(8/3)*(1 - b*(a/b + x)/a)**(1/3)*(a/b + x)**2*exp(7
*I*pi/3)/(28*a**20 - 168*a**19*b*(a/b + x) + 420*a**18*b**2*(a/b + x)**2 - 560*a
**17*b**3*(a/b + x)**3 + 420*a**16*b**4*(a/b + x)**4 - 168*a**15*b**5*(a/b + x)*
*5 + 28*a**14*b**6*(a/b + x)**6) - 3645*a**(61/3)*b**(8/3)*(a/b + x)**2*exp(7*I*
pi/3)/(28*a**20 - 168*a**19*b*(a/b + x) + 420*a**18*b**2*(a/b + x)**2 - 560*a**1
7*b**3*(a/b + x)**3 + 420*a**16*b**4*(a/b + x)**4 - 168*a**15*b**5*(a/b + x)**5
+ 28*a**14*b**6*(a/b + x)**6) - 3927*a**(58/3)*b**(11/3)*(1 - b*(a/b + x)/a)**(1
/3)*(a/b + x)**3*exp(7*I*pi/3)/(28*a**20 - 168*a**19*b*(a/b + x) + 420*a**18*b**
2*(a/b + x)**2 - 560*a**17*b**3*(a/b + x)**3 + 420*a**16*b**4*(a/b + x)**4 - 168
*a**15*b**5*(a/b + x)**5 + 28*a**14*b**6*(a/b + x)**6) + 4860*a**(58/3)*b**(11/3
)*(a/b + x)**3*exp(7*I*pi/3)/(28*a**20 - 168*a**19*b*(a/b + x) + 420*a**18*b**2*
(a/b + x)**2 - 560*a**17*b**3*(a/b + x)**3 + 420*a**16*b**4*(a/b + x)**4 - 168*a
**15*b**5*(a/b + x)**5 + 28*a**14*b**6*(a/b + x)**6) + 2625*a**(55/3)*b**(14/3)*
(1 - b*(a/b + x)/a)**(1/3)*(a/b + x)**4*exp(7*I*pi/3)/(28*a**20 - 168*a**19*b*(a
/b + x) + 420*a**18*b**2*(a/b + x)**2 - 560*a**17*b**3*(a/b + x)**3 + 420*a**16*
b**4*(a/b + x)**4 - 168*a**15*b**5*(a/b + x)**5 + 28*a**14*b**6*(a/b + x)**6) -
3645*a**(55/3)*b**(14/3)*(a/b + x)**4*exp(7*I*pi/3)/(28*a**20 - 168*a**19*b*(a/b
+ x) + 420*a**18*b**2*(a/b + x)**2 - 560*a**17*b**3*(a/b + x)**3 + 420*a**16*b*
*4*(a/b + x)**4 - 168*a**15*b**5*(a/b + x)**5 + 28*a**14*b**6*(a/b + x)**6) - 90
3*a**(52/3)*b**(17/3)*(1 - b*(a/b + x)/a)**(1/3)*(a/b + x)**5*exp(7*I*pi/3)/(28*
a**20 - 168*a**19*b*(a/b + x) + 420*a**18*b**2*(a/b + x)**2 - 560*a**17*b**3*(a/
b + x)**3 + 420*a**16*b**4*(a/b + x)**4 - 168*a**15*b**5*(a/b + x)**5 + 28*a**14
*b**6*(a/b + x)**6) + 1458*a**(52/3)*b**(17/3)*(a/b + x)**5*exp(7*I*pi/3)/(28*a*
*20 - 168*a**19*b*(a/b + x) + 420*a**18*b**2*(a/b + x)**2 - 560*a**17*b**3*(a/b
+ x)**3 + 420*a**16*b**4*(a/b + x)**4 - 168*a**15*b**5*(a/b + x)**5 + 28*a**14*b
**6*(a/b + x)**6) + 147*a**(49/3)*b**(20/3)*(1 - b*(a/b + x)/a)**(1/3)*(a/b + x)
**6*exp(7*I*pi/3)/(28*a**20 - 168*a**19*b*(a/b + x) + 420*a**18*b**2*(a/b + x)**
2 - 560*a**17*b**3*(a/b + x)**3 + 420*a**16*b**4*(a/b + x)**4 - 168*a**15*b**5*(
a/b + x)**5 + 28*a**14*b**6*(a/b + x)**6) - 243*a**(49/3)*b**(20/3)*(a/b + x)**6
*exp(7*I*pi/3)/(28*a**20 - 168*a**19*b*(a/b + x) + 420*a**18*b**2*(a/b + x)**2 -
560*a**17*b**3*(a/b + x)**3 + 420*a**16*b**4*(a/b + x)**4 - 168*a**15*b**5*(a/b
+ x)**5 + 28*a**14*b**6*(a/b + x)**6) - 33*a**(46/3)*b**(23/3)*(1 - b*(a/b + x)
/a)**(1/3)*(a/b + x)**7*exp(7*I*pi/3)/(28*a**20 - 168*a**19*b*(a/b + x) + 420*a*
*18*b**2*(a/b + x)**2 - 560*a**17*b**3*(a/b + x)**3 + 420*a**16*b**4*(a/b + x)**
4 - 168*a**15*b**5*(a/b + x)**5 + 28*a**14*b**6*(a/b + x)**6) + 12*a**(43/3)*b**
(26/3)*(1 - b*(a/b + x)/a)**(1/3)*(a/b + x)**8*exp(7*I*pi/3)/(28*a**20 - 168*a**
19*b*(a/b + x) + 420*a**18*b**2*(a/b + x)**2 - 560*a**17*b**3*(a/b + x)**3 + 420
*a**16*b**4*(a/b + x)**4 - 168*a**15*b**5*(a/b + x)**5 + 28*a**14*b**6*(a/b + x)
**6), True))

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GIAC/XCAS [A]  time = 0.204165, size = 47, normalized size = 0.96 \[ \frac{3}{7} \, b^{3} x^{\frac{7}{3}} + \frac{9}{4} \, a b^{2} x^{\frac{4}{3}} + 9 \, a^{2} b x^{\frac{1}{3}} - \frac{3 \, a^{3}}{2 \, x^{\frac{2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^3/x^(5/3),x, algorithm="giac")

[Out]

3/7*b^3*x^(7/3) + 9/4*a*b^2*x^(4/3) + 9*a^2*b*x^(1/3) - 3/2*a^3/x^(2/3)