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

Optimal. Leaf size=45 \[ -\frac{2 a^3}{\sqrt{x}}+6 a^2 b \sqrt{x}+2 a b^2 x^{3/2}+\frac{2}{5} b^3 x^{5/2} \]

[Out]

(-2*a^3)/Sqrt[x] + 6*a^2*b*Sqrt[x] + 2*a*b^2*x^(3/2) + (2*b^3*x^(5/2))/5

_______________________________________________________________________________________

Rubi [A]  time = 0.0298119, antiderivative size = 45, 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 \[ -\frac{2 a^3}{\sqrt{x}}+6 a^2 b \sqrt{x}+2 a b^2 x^{3/2}+\frac{2}{5} b^3 x^{5/2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^3/x^(3/2),x]

[Out]

(-2*a^3)/Sqrt[x] + 6*a^2*b*Sqrt[x] + 2*a*b^2*x^(3/2) + (2*b^3*x^(5/2))/5

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 5.16791, size = 44, normalized size = 0.98 \[ - \frac{2 a^{3}}{\sqrt{x}} + 6 a^{2} b \sqrt{x} + 2 a b^{2} x^{\frac{3}{2}} + \frac{2 b^{3} x^{\frac{5}{2}}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*a**3/sqrt(x) + 6*a**2*b*sqrt(x) + 2*a*b**2*x**(3/2) + 2*b**3*x**(5/2)/5

_______________________________________________________________________________________

Mathematica [A]  time = 0.0114352, size = 38, normalized size = 0.84 \[ \frac{2 \left (-5 a^3+15 a^2 b x+5 a b^2 x^2+b^3 x^3\right )}{5 \sqrt{x}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(-5*a^3 + 15*a^2*b*x + 5*a*b^2*x^2 + b^3*x^3))/(5*Sqrt[x])

_______________________________________________________________________________________

Maple [A]  time = 0.006, size = 36, normalized size = 0.8 \[ -{\frac{-2\,{b}^{3}{x}^{3}-10\,a{b}^{2}{x}^{2}-30\,{a}^{2}bx+10\,{a}^{3}}{5}{\frac{1}{\sqrt{x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^3/x^(3/2),x)

[Out]

-2/5*(-b^3*x^3-5*a*b^2*x^2-15*a^2*b*x+5*a^3)/x^(1/2)

_______________________________________________________________________________________

Maxima [A]  time = 1.36576, size = 47, normalized size = 1.04 \[ \frac{2}{5} \, b^{3} x^{\frac{5}{2}} + 2 \, a b^{2} x^{\frac{3}{2}} + 6 \, a^{2} b \sqrt{x} - \frac{2 \, a^{3}}{\sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/5*b^3*x^(5/2) + 2*a*b^2*x^(3/2) + 6*a^2*b*sqrt(x) - 2*a^3/sqrt(x)

_______________________________________________________________________________________

Fricas [A]  time = 0.206739, size = 46, normalized size = 1.02 \[ \frac{2 \,{\left (b^{3} x^{3} + 5 \, a b^{2} x^{2} + 15 \, a^{2} b x - 5 \, a^{3}\right )}}{5 \, \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/5*(b^3*x^3 + 5*a*b^2*x^2 + 15*a^2*b*x - 5*a^3)/sqrt(x)

_______________________________________________________________________________________

Sympy [A]  time = 9.37383, size = 3847, normalized size = 85.49 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((32*a**(45/2)*sqrt(b)*sqrt(-1 + b*(a/b + x)/a)/(5*a**20 - 30*a**19*b*(
a/b + x) + 75*a**18*b**2*(a/b + x)**2 - 100*a**17*b**3*(a/b + x)**3 + 75*a**16*b
**4*(a/b + x)**4 - 30*a**15*b**5*(a/b + x)**5 + 5*a**14*b**6*(a/b + x)**6) - 32*
I*a**(45/2)*sqrt(b)/(5*a**20 - 30*a**19*b*(a/b + x) + 75*a**18*b**2*(a/b + x)**2
- 100*a**17*b**3*(a/b + x)**3 + 75*a**16*b**4*(a/b + x)**4 - 30*a**15*b**5*(a/b
+ x)**5 + 5*a**14*b**6*(a/b + x)**6) - 176*a**(43/2)*b**(3/2)*sqrt(-1 + b*(a/b
+ x)/a)*(a/b + x)/(5*a**20 - 30*a**19*b*(a/b + x) + 75*a**18*b**2*(a/b + x)**2 -
100*a**17*b**3*(a/b + x)**3 + 75*a**16*b**4*(a/b + x)**4 - 30*a**15*b**5*(a/b +
x)**5 + 5*a**14*b**6*(a/b + x)**6) + 192*I*a**(43/2)*b**(3/2)*(a/b + x)/(5*a**2
0 - 30*a**19*b*(a/b + x) + 75*a**18*b**2*(a/b + x)**2 - 100*a**17*b**3*(a/b + x)
**3 + 75*a**16*b**4*(a/b + x)**4 - 30*a**15*b**5*(a/b + x)**5 + 5*a**14*b**6*(a/
b + x)**6) + 396*a**(41/2)*b**(5/2)*sqrt(-1 + b*(a/b + x)/a)*(a/b + x)**2/(5*a**
20 - 30*a**19*b*(a/b + x) + 75*a**18*b**2*(a/b + x)**2 - 100*a**17*b**3*(a/b + x
)**3 + 75*a**16*b**4*(a/b + x)**4 - 30*a**15*b**5*(a/b + x)**5 + 5*a**14*b**6*(a
/b + x)**6) - 480*I*a**(41/2)*b**(5/2)*(a/b + x)**2/(5*a**20 - 30*a**19*b*(a/b +
x) + 75*a**18*b**2*(a/b + x)**2 - 100*a**17*b**3*(a/b + x)**3 + 75*a**16*b**4*(
a/b + x)**4 - 30*a**15*b**5*(a/b + x)**5 + 5*a**14*b**6*(a/b + x)**6) - 462*a**(
39/2)*b**(7/2)*sqrt(-1 + b*(a/b + x)/a)*(a/b + x)**3/(5*a**20 - 30*a**19*b*(a/b
+ x) + 75*a**18*b**2*(a/b + x)**2 - 100*a**17*b**3*(a/b + x)**3 + 75*a**16*b**4*
(a/b + x)**4 - 30*a**15*b**5*(a/b + x)**5 + 5*a**14*b**6*(a/b + x)**6) + 640*I*a
**(39/2)*b**(7/2)*(a/b + x)**3/(5*a**20 - 30*a**19*b*(a/b + x) + 75*a**18*b**2*(
a/b + x)**2 - 100*a**17*b**3*(a/b + x)**3 + 75*a**16*b**4*(a/b + x)**4 - 30*a**1
5*b**5*(a/b + x)**5 + 5*a**14*b**6*(a/b + x)**6) + 290*a**(37/2)*b**(9/2)*sqrt(-
1 + b*(a/b + x)/a)*(a/b + x)**4/(5*a**20 - 30*a**19*b*(a/b + x) + 75*a**18*b**2*
(a/b + x)**2 - 100*a**17*b**3*(a/b + x)**3 + 75*a**16*b**4*(a/b + x)**4 - 30*a**
15*b**5*(a/b + x)**5 + 5*a**14*b**6*(a/b + x)**6) - 480*I*a**(37/2)*b**(9/2)*(a/
b + x)**4/(5*a**20 - 30*a**19*b*(a/b + x) + 75*a**18*b**2*(a/b + x)**2 - 100*a**
17*b**3*(a/b + x)**3 + 75*a**16*b**4*(a/b + x)**4 - 30*a**15*b**5*(a/b + x)**5 +
5*a**14*b**6*(a/b + x)**6) - 92*a**(35/2)*b**(11/2)*sqrt(-1 + b*(a/b + x)/a)*(a
/b + x)**5/(5*a**20 - 30*a**19*b*(a/b + x) + 75*a**18*b**2*(a/b + x)**2 - 100*a*
*17*b**3*(a/b + x)**3 + 75*a**16*b**4*(a/b + x)**4 - 30*a**15*b**5*(a/b + x)**5
+ 5*a**14*b**6*(a/b + x)**6) + 192*I*a**(35/2)*b**(11/2)*(a/b + x)**5/(5*a**20 -
30*a**19*b*(a/b + x) + 75*a**18*b**2*(a/b + x)**2 - 100*a**17*b**3*(a/b + x)**3
+ 75*a**16*b**4*(a/b + x)**4 - 30*a**15*b**5*(a/b + x)**5 + 5*a**14*b**6*(a/b +
x)**6) + 16*a**(33/2)*b**(13/2)*sqrt(-1 + b*(a/b + x)/a)*(a/b + x)**6/(5*a**20
- 30*a**19*b*(a/b + x) + 75*a**18*b**2*(a/b + x)**2 - 100*a**17*b**3*(a/b + x)**
3 + 75*a**16*b**4*(a/b + x)**4 - 30*a**15*b**5*(a/b + x)**5 + 5*a**14*b**6*(a/b
+ x)**6) - 32*I*a**(33/2)*b**(13/2)*(a/b + x)**6/(5*a**20 - 30*a**19*b*(a/b + x)
+ 75*a**18*b**2*(a/b + x)**2 - 100*a**17*b**3*(a/b + x)**3 + 75*a**16*b**4*(a/b
+ x)**4 - 30*a**15*b**5*(a/b + x)**5 + 5*a**14*b**6*(a/b + x)**6) - 6*a**(31/2)
*b**(15/2)*sqrt(-1 + b*(a/b + x)/a)*(a/b + x)**7/(5*a**20 - 30*a**19*b*(a/b + x)
+ 75*a**18*b**2*(a/b + x)**2 - 100*a**17*b**3*(a/b + x)**3 + 75*a**16*b**4*(a/b
+ x)**4 - 30*a**15*b**5*(a/b + x)**5 + 5*a**14*b**6*(a/b + x)**6) + 2*a**(29/2)
*b**(17/2)*sqrt(-1 + b*(a/b + x)/a)*(a/b + x)**8/(5*a**20 - 30*a**19*b*(a/b + x)
+ 75*a**18*b**2*(a/b + x)**2 - 100*a**17*b**3*(a/b + x)**3 + 75*a**16*b**4*(a/b
+ x)**4 - 30*a**15*b**5*(a/b + x)**5 + 5*a**14*b**6*(a/b + x)**6), Abs(b*(a/b +
x)/a) > 1), (32*I*a**(45/2)*sqrt(b)*sqrt(1 - b*(a/b + x)/a)/(5*a**20 - 30*a**19
*b*(a/b + x) + 75*a**18*b**2*(a/b + x)**2 - 100*a**17*b**3*(a/b + x)**3 + 75*a**
16*b**4*(a/b + x)**4 - 30*a**15*b**5*(a/b + x)**5 + 5*a**14*b**6*(a/b + x)**6) -
32*I*a**(45/2)*sqrt(b)/(5*a**20 - 30*a**19*b*(a/b + x) + 75*a**18*b**2*(a/b + x
)**2 - 100*a**17*b**3*(a/b + x)**3 + 75*a**16*b**4*(a/b + x)**4 - 30*a**15*b**5*
(a/b + x)**5 + 5*a**14*b**6*(a/b + x)**6) - 176*I*a**(43/2)*b**(3/2)*sqrt(1 - b*
(a/b + x)/a)*(a/b + x)/(5*a**20 - 30*a**19*b*(a/b + x) + 75*a**18*b**2*(a/b + x)
**2 - 100*a**17*b**3*(a/b + x)**3 + 75*a**16*b**4*(a/b + x)**4 - 30*a**15*b**5*(
a/b + x)**5 + 5*a**14*b**6*(a/b + x)**6) + 192*I*a**(43/2)*b**(3/2)*(a/b + x)/(5
*a**20 - 30*a**19*b*(a/b + x) + 75*a**18*b**2*(a/b + x)**2 - 100*a**17*b**3*(a/b
+ x)**3 + 75*a**16*b**4*(a/b + x)**4 - 30*a**15*b**5*(a/b + x)**5 + 5*a**14*b**
6*(a/b + x)**6) + 396*I*a**(41/2)*b**(5/2)*sqrt(1 - b*(a/b + x)/a)*(a/b + x)**2/
(5*a**20 - 30*a**19*b*(a/b + x) + 75*a**18*b**2*(a/b + x)**2 - 100*a**17*b**3*(a
/b + x)**3 + 75*a**16*b**4*(a/b + x)**4 - 30*a**15*b**5*(a/b + x)**5 + 5*a**14*b
**6*(a/b + x)**6) - 480*I*a**(41/2)*b**(5/2)*(a/b + x)**2/(5*a**20 - 30*a**19*b*
(a/b + x) + 75*a**18*b**2*(a/b + x)**2 - 100*a**17*b**3*(a/b + x)**3 + 75*a**16*
b**4*(a/b + x)**4 - 30*a**15*b**5*(a/b + x)**5 + 5*a**14*b**6*(a/b + x)**6) - 46
2*I*a**(39/2)*b**(7/2)*sqrt(1 - b*(a/b + x)/a)*(a/b + x)**3/(5*a**20 - 30*a**19*
b*(a/b + x) + 75*a**18*b**2*(a/b + x)**2 - 100*a**17*b**3*(a/b + x)**3 + 75*a**1
6*b**4*(a/b + x)**4 - 30*a**15*b**5*(a/b + x)**5 + 5*a**14*b**6*(a/b + x)**6) +
640*I*a**(39/2)*b**(7/2)*(a/b + x)**3/(5*a**20 - 30*a**19*b*(a/b + x) + 75*a**18
*b**2*(a/b + x)**2 - 100*a**17*b**3*(a/b + x)**3 + 75*a**16*b**4*(a/b + x)**4 -
30*a**15*b**5*(a/b + x)**5 + 5*a**14*b**6*(a/b + x)**6) + 290*I*a**(37/2)*b**(9/
2)*sqrt(1 - b*(a/b + x)/a)*(a/b + x)**4/(5*a**20 - 30*a**19*b*(a/b + x) + 75*a**
18*b**2*(a/b + x)**2 - 100*a**17*b**3*(a/b + x)**3 + 75*a**16*b**4*(a/b + x)**4
- 30*a**15*b**5*(a/b + x)**5 + 5*a**14*b**6*(a/b + x)**6) - 480*I*a**(37/2)*b**(
9/2)*(a/b + x)**4/(5*a**20 - 30*a**19*b*(a/b + x) + 75*a**18*b**2*(a/b + x)**2 -
100*a**17*b**3*(a/b + x)**3 + 75*a**16*b**4*(a/b + x)**4 - 30*a**15*b**5*(a/b +
x)**5 + 5*a**14*b**6*(a/b + x)**6) - 92*I*a**(35/2)*b**(11/2)*sqrt(1 - b*(a/b +
x)/a)*(a/b + x)**5/(5*a**20 - 30*a**19*b*(a/b + x) + 75*a**18*b**2*(a/b + x)**2
- 100*a**17*b**3*(a/b + x)**3 + 75*a**16*b**4*(a/b + x)**4 - 30*a**15*b**5*(a/b
+ x)**5 + 5*a**14*b**6*(a/b + x)**6) + 192*I*a**(35/2)*b**(11/2)*(a/b + x)**5/(
5*a**20 - 30*a**19*b*(a/b + x) + 75*a**18*b**2*(a/b + x)**2 - 100*a**17*b**3*(a/
b + x)**3 + 75*a**16*b**4*(a/b + x)**4 - 30*a**15*b**5*(a/b + x)**5 + 5*a**14*b*
*6*(a/b + x)**6) + 16*I*a**(33/2)*b**(13/2)*sqrt(1 - b*(a/b + x)/a)*(a/b + x)**6
/(5*a**20 - 30*a**19*b*(a/b + x) + 75*a**18*b**2*(a/b + x)**2 - 100*a**17*b**3*(
a/b + x)**3 + 75*a**16*b**4*(a/b + x)**4 - 30*a**15*b**5*(a/b + x)**5 + 5*a**14*
b**6*(a/b + x)**6) - 32*I*a**(33/2)*b**(13/2)*(a/b + x)**6/(5*a**20 - 30*a**19*b
*(a/b + x) + 75*a**18*b**2*(a/b + x)**2 - 100*a**17*b**3*(a/b + x)**3 + 75*a**16
*b**4*(a/b + x)**4 - 30*a**15*b**5*(a/b + x)**5 + 5*a**14*b**6*(a/b + x)**6) - 6
*I*a**(31/2)*b**(15/2)*sqrt(1 - b*(a/b + x)/a)*(a/b + x)**7/(5*a**20 - 30*a**19*
b*(a/b + x) + 75*a**18*b**2*(a/b + x)**2 - 100*a**17*b**3*(a/b + x)**3 + 75*a**1
6*b**4*(a/b + x)**4 - 30*a**15*b**5*(a/b + x)**5 + 5*a**14*b**6*(a/b + x)**6) +
2*I*a**(29/2)*b**(17/2)*sqrt(1 - b*(a/b + x)/a)*(a/b + x)**8/(5*a**20 - 30*a**19
*b*(a/b + x) + 75*a**18*b**2*(a/b + x)**2 - 100*a**17*b**3*(a/b + x)**3 + 75*a**
16*b**4*(a/b + x)**4 - 30*a**15*b**5*(a/b + x)**5 + 5*a**14*b**6*(a/b + x)**6),
True))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.203137, size = 47, normalized size = 1.04 \[ \frac{2}{5} \, b^{3} x^{\frac{5}{2}} + 2 \, a b^{2} x^{\frac{3}{2}} + 6 \, a^{2} b \sqrt{x} - \frac{2 \, a^{3}}{\sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/5*b^3*x^(5/2) + 2*a*b^2*x^(3/2) + 6*a^2*b*sqrt(x) - 2*a^3/sqrt(x)