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

Optimal. Leaf size=32 \[ -\frac{2 a^2}{\sqrt{x}}+4 a b \sqrt{x}+\frac{2}{3} b^2 x^{3/2} \]

[Out]

(-2*a^2)/Sqrt[x] + 4*a*b*Sqrt[x] + (2*b^2*x^(3/2))/3

_______________________________________________________________________________________

Rubi [A]  time = 0.0214869, antiderivative size = 32, 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 \[ -\frac{2 a^2}{\sqrt{x}}+4 a b \sqrt{x}+\frac{2}{3} b^2 x^{3/2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^2/x^(3/2),x]

[Out]

(-2*a^2)/Sqrt[x] + 4*a*b*Sqrt[x] + (2*b^2*x^(3/2))/3

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 3.88684, size = 31, normalized size = 0.97 \[ - \frac{2 a^{2}}{\sqrt{x}} + 4 a b \sqrt{x} + \frac{2 b^{2} x^{\frac{3}{2}}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*a**2/sqrt(x) + 4*a*b*sqrt(x) + 2*b**2*x**(3/2)/3

_______________________________________________________________________________________

Mathematica [A]  time = 0.0100484, size = 27, normalized size = 0.84 \[ \frac{2 \left (-3 a^2+6 a b x+b^2 x^2\right )}{3 \sqrt{x}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.006, size = 25, normalized size = 0.8 \[ -{\frac{-2\,{b}^{2}{x}^{2}-12\,abx+6\,{a}^{2}}{3}{\frac{1}{\sqrt{x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^2/x^(3/2),x)

[Out]

-2/3*(-b^2*x^2-6*a*b*x+3*a^2)/x^(1/2)

_______________________________________________________________________________________

Maxima [A]  time = 1.36972, size = 32, normalized size = 1. \[ \frac{2}{3} \, b^{2} x^{\frac{3}{2}} + 4 \, a b \sqrt{x} - \frac{2 \, a^{2}}{\sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*b^2*x^(3/2) + 4*a*b*sqrt(x) - 2*a^2/sqrt(x)

_______________________________________________________________________________________

Fricas [A]  time = 0.206447, size = 31, normalized size = 0.97 \[ \frac{2 \,{\left (b^{2} x^{2} + 6 \, a b x - 3 \, a^{2}\right )}}{3 \, \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 5.87073, size = 1324, normalized size = 41.38 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-16*a**(19/2)*sqrt(b)*sqrt(-1 + b*(a/b + x)/a)/(-3*a**8 + 9*a**7*b*(a
/b + x) - 9*a**6*b**2*(a/b + x)**2 + 3*a**5*b**3*(a/b + x)**3) + 16*I*a**(19/2)*
sqrt(b)/(-3*a**8 + 9*a**7*b*(a/b + x) - 9*a**6*b**2*(a/b + x)**2 + 3*a**5*b**3*(
a/b + x)**3) + 40*a**(17/2)*b**(3/2)*sqrt(-1 + b*(a/b + x)/a)*(a/b + x)/(-3*a**8
+ 9*a**7*b*(a/b + x) - 9*a**6*b**2*(a/b + x)**2 + 3*a**5*b**3*(a/b + x)**3) - 4
8*I*a**(17/2)*b**(3/2)*(a/b + x)/(-3*a**8 + 9*a**7*b*(a/b + x) - 9*a**6*b**2*(a/
b + x)**2 + 3*a**5*b**3*(a/b + x)**3) - 30*a**(15/2)*b**(5/2)*sqrt(-1 + b*(a/b +
x)/a)*(a/b + x)**2/(-3*a**8 + 9*a**7*b*(a/b + x) - 9*a**6*b**2*(a/b + x)**2 + 3
*a**5*b**3*(a/b + x)**3) + 48*I*a**(15/2)*b**(5/2)*(a/b + x)**2/(-3*a**8 + 9*a**
7*b*(a/b + x) - 9*a**6*b**2*(a/b + x)**2 + 3*a**5*b**3*(a/b + x)**3) + 4*a**(13/
2)*b**(7/2)*sqrt(-1 + b*(a/b + x)/a)*(a/b + x)**3/(-3*a**8 + 9*a**7*b*(a/b + x)
- 9*a**6*b**2*(a/b + x)**2 + 3*a**5*b**3*(a/b + x)**3) - 16*I*a**(13/2)*b**(7/2)
*(a/b + x)**3/(-3*a**8 + 9*a**7*b*(a/b + x) - 9*a**6*b**2*(a/b + x)**2 + 3*a**5*
b**3*(a/b + x)**3) + 2*a**(11/2)*b**(9/2)*sqrt(-1 + b*(a/b + x)/a)*(a/b + x)**4/
(-3*a**8 + 9*a**7*b*(a/b + x) - 9*a**6*b**2*(a/b + x)**2 + 3*a**5*b**3*(a/b + x)
**3), Abs(b*(a/b + x)/a) > 1), (-16*I*a**(19/2)*sqrt(b)*sqrt(1 - b*(a/b + x)/a)/
(-3*a**8 + 9*a**7*b*(a/b + x) - 9*a**6*b**2*(a/b + x)**2 + 3*a**5*b**3*(a/b + x)
**3) + 16*I*a**(19/2)*sqrt(b)/(-3*a**8 + 9*a**7*b*(a/b + x) - 9*a**6*b**2*(a/b +
x)**2 + 3*a**5*b**3*(a/b + x)**3) + 40*I*a**(17/2)*b**(3/2)*sqrt(1 - b*(a/b + x
)/a)*(a/b + x)/(-3*a**8 + 9*a**7*b*(a/b + x) - 9*a**6*b**2*(a/b + x)**2 + 3*a**5
*b**3*(a/b + x)**3) - 48*I*a**(17/2)*b**(3/2)*(a/b + x)/(-3*a**8 + 9*a**7*b*(a/b
+ x) - 9*a**6*b**2*(a/b + x)**2 + 3*a**5*b**3*(a/b + x)**3) - 30*I*a**(15/2)*b*
*(5/2)*sqrt(1 - b*(a/b + x)/a)*(a/b + x)**2/(-3*a**8 + 9*a**7*b*(a/b + x) - 9*a*
*6*b**2*(a/b + x)**2 + 3*a**5*b**3*(a/b + x)**3) + 48*I*a**(15/2)*b**(5/2)*(a/b
+ x)**2/(-3*a**8 + 9*a**7*b*(a/b + x) - 9*a**6*b**2*(a/b + x)**2 + 3*a**5*b**3*(
a/b + x)**3) + 4*I*a**(13/2)*b**(7/2)*sqrt(1 - b*(a/b + x)/a)*(a/b + x)**3/(-3*a
**8 + 9*a**7*b*(a/b + x) - 9*a**6*b**2*(a/b + x)**2 + 3*a**5*b**3*(a/b + x)**3)
- 16*I*a**(13/2)*b**(7/2)*(a/b + x)**3/(-3*a**8 + 9*a**7*b*(a/b + x) - 9*a**6*b*
*2*(a/b + x)**2 + 3*a**5*b**3*(a/b + x)**3) + 2*I*a**(11/2)*b**(9/2)*sqrt(1 - b*
(a/b + x)/a)*(a/b + x)**4/(-3*a**8 + 9*a**7*b*(a/b + x) - 9*a**6*b**2*(a/b + x)*
*2 + 3*a**5*b**3*(a/b + x)**3), True))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.203367, size = 32, normalized size = 1. \[ \frac{2}{3} \, b^{2} x^{\frac{3}{2}} + 4 \, a b \sqrt{x} - \frac{2 \, a^{2}}{\sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*b^2*x^(3/2) + 4*a*b*sqrt(x) - 2*a^2/sqrt(x)