3.439 \(\int \frac{(a+b x)^2}{\sqrt{x}} \, dx\)
Optimal. Leaf size=34 \[ 2 a^2 \sqrt{x}+\frac{4}{3} a b x^{3/2}+\frac{2}{5} b^2 x^{5/2} \]
[Out]
2*a^2*Sqrt[x] + (4*a*b*x^(3/2))/3 + (2*b^2*x^(5/2))/5
_______________________________________________________________________________________
Rubi [A] time = 0.0212331, antiderivative size = 34, 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
\[ 2 a^2 \sqrt{x}+\frac{4}{3} a b x^{3/2}+\frac{2}{5} b^2 x^{5/2} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x)^2/Sqrt[x],x]
[Out]
2*a^2*Sqrt[x] + (4*a*b*x^(3/2))/3 + (2*b^2*x^(5/2))/5
_______________________________________________________________________________________
Rubi in Sympy [A] time = 3.94715, size = 32, normalized size = 0.94 \[ 2 a^{2} \sqrt{x} + \frac{4 a b x^{\frac{3}{2}}}{3} + \frac{2 b^{2} x^{\frac{5}{2}}}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**2/x**(1/2),x)
[Out]
2*a**2*sqrt(x) + 4*a*b*x**(3/2)/3 + 2*b**2*x**(5/2)/5
_______________________________________________________________________________________
Mathematica [A] time = 0.00859762, size = 28, normalized size = 0.82 \[ \frac{2}{15} \sqrt{x} \left (15 a^2+10 a b x+3 b^2 x^2\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x)^2/Sqrt[x],x]
[Out]
(2*Sqrt[x]*(15*a^2 + 10*a*b*x + 3*b^2*x^2))/15
_______________________________________________________________________________________
Maple [A] time = 0.006, size = 25, normalized size = 0.7 \[{\frac{6\,{b}^{2}{x}^{2}+20\,abx+30\,{a}^{2}}{15}\sqrt{x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^2/x^(1/2),x)
[Out]
2/15*x^(1/2)*(3*b^2*x^2+10*a*b*x+15*a^2)
_______________________________________________________________________________________
Maxima [A] time = 1.33937, size = 32, normalized size = 0.94 \[ \frac{2}{5} \, b^{2} x^{\frac{5}{2}} + \frac{4}{3} \, a b x^{\frac{3}{2}} + 2 \, a^{2} \sqrt{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^2/sqrt(x),x, algorithm="maxima")
[Out]
2/5*b^2*x^(5/2) + 4/3*a*b*x^(3/2) + 2*a^2*sqrt(x)
_______________________________________________________________________________________
Fricas [A] time = 0.206596, size = 32, normalized size = 0.94 \[ \frac{2}{15} \,{\left (3 \, b^{2} x^{2} + 10 \, a b x + 15 \, a^{2}\right )} \sqrt{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^2/sqrt(x),x, algorithm="fricas")
[Out]
2/15*(3*b^2*x^2 + 10*a*b*x + 15*a^2)*sqrt(x)
_______________________________________________________________________________________
Sympy [A] time = 6.08306, size = 1669, normalized size = 49.09 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)**2/x**(1/2),x)
[Out]
Piecewise((-16*a**(21/2)*sqrt(-1 + b*(a/b + x)/a)/(-15*a**8*sqrt(b) + 45*a**7*b*
*(3/2)*(a/b + x) - 45*a**6*b**(5/2)*(a/b + x)**2 + 15*a**5*b**(7/2)*(a/b + x)**3
) + 16*I*a**(21/2)/(-15*a**8*sqrt(b) + 45*a**7*b**(3/2)*(a/b + x) - 45*a**6*b**(
5/2)*(a/b + x)**2 + 15*a**5*b**(7/2)*(a/b + x)**3) + 40*a**(19/2)*b*sqrt(-1 + b*
(a/b + x)/a)*(a/b + x)/(-15*a**8*sqrt(b) + 45*a**7*b**(3/2)*(a/b + x) - 45*a**6*
b**(5/2)*(a/b + x)**2 + 15*a**5*b**(7/2)*(a/b + x)**3) - 48*I*a**(19/2)*b*(a/b +
x)/(-15*a**8*sqrt(b) + 45*a**7*b**(3/2)*(a/b + x) - 45*a**6*b**(5/2)*(a/b + x)*
*2 + 15*a**5*b**(7/2)*(a/b + x)**3) - 30*a**(17/2)*b**2*sqrt(-1 + b*(a/b + x)/a)
*(a/b + x)**2/(-15*a**8*sqrt(b) + 45*a**7*b**(3/2)*(a/b + x) - 45*a**6*b**(5/2)*
(a/b + x)**2 + 15*a**5*b**(7/2)*(a/b + x)**3) + 48*I*a**(17/2)*b**2*(a/b + x)**2
/(-15*a**8*sqrt(b) + 45*a**7*b**(3/2)*(a/b + x) - 45*a**6*b**(5/2)*(a/b + x)**2
+ 15*a**5*b**(7/2)*(a/b + x)**3) + 10*a**(15/2)*b**3*sqrt(-1 + b*(a/b + x)/a)*(a
/b + x)**3/(-15*a**8*sqrt(b) + 45*a**7*b**(3/2)*(a/b + x) - 45*a**6*b**(5/2)*(a/
b + x)**2 + 15*a**5*b**(7/2)*(a/b + x)**3) - 16*I*a**(15/2)*b**3*(a/b + x)**3/(-
15*a**8*sqrt(b) + 45*a**7*b**(3/2)*(a/b + x) - 45*a**6*b**(5/2)*(a/b + x)**2 + 1
5*a**5*b**(7/2)*(a/b + x)**3) - 10*a**(13/2)*b**4*sqrt(-1 + b*(a/b + x)/a)*(a/b
+ x)**4/(-15*a**8*sqrt(b) + 45*a**7*b**(3/2)*(a/b + x) - 45*a**6*b**(5/2)*(a/b +
x)**2 + 15*a**5*b**(7/2)*(a/b + x)**3) + 6*a**(11/2)*b**5*sqrt(-1 + b*(a/b + x)
/a)*(a/b + x)**5/(-15*a**8*sqrt(b) + 45*a**7*b**(3/2)*(a/b + x) - 45*a**6*b**(5/
2)*(a/b + x)**2 + 15*a**5*b**(7/2)*(a/b + x)**3), Abs(b*(a/b + x)/a) > 1), (-16*
I*a**(21/2)*sqrt(1 - b*(a/b + x)/a)/(-15*a**8*sqrt(b) + 45*a**7*b**(3/2)*(a/b +
x) - 45*a**6*b**(5/2)*(a/b + x)**2 + 15*a**5*b**(7/2)*(a/b + x)**3) + 16*I*a**(2
1/2)/(-15*a**8*sqrt(b) + 45*a**7*b**(3/2)*(a/b + x) - 45*a**6*b**(5/2)*(a/b + x)
**2 + 15*a**5*b**(7/2)*(a/b + x)**3) + 40*I*a**(19/2)*b*sqrt(1 - b*(a/b + x)/a)*
(a/b + x)/(-15*a**8*sqrt(b) + 45*a**7*b**(3/2)*(a/b + x) - 45*a**6*b**(5/2)*(a/b
+ x)**2 + 15*a**5*b**(7/2)*(a/b + x)**3) - 48*I*a**(19/2)*b*(a/b + x)/(-15*a**8
*sqrt(b) + 45*a**7*b**(3/2)*(a/b + x) - 45*a**6*b**(5/2)*(a/b + x)**2 + 15*a**5*
b**(7/2)*(a/b + x)**3) - 30*I*a**(17/2)*b**2*sqrt(1 - b*(a/b + x)/a)*(a/b + x)**
2/(-15*a**8*sqrt(b) + 45*a**7*b**(3/2)*(a/b + x) - 45*a**6*b**(5/2)*(a/b + x)**2
+ 15*a**5*b**(7/2)*(a/b + x)**3) + 48*I*a**(17/2)*b**2*(a/b + x)**2/(-15*a**8*s
qrt(b) + 45*a**7*b**(3/2)*(a/b + x) - 45*a**6*b**(5/2)*(a/b + x)**2 + 15*a**5*b*
*(7/2)*(a/b + x)**3) + 10*I*a**(15/2)*b**3*sqrt(1 - b*(a/b + x)/a)*(a/b + x)**3/
(-15*a**8*sqrt(b) + 45*a**7*b**(3/2)*(a/b + x) - 45*a**6*b**(5/2)*(a/b + x)**2 +
15*a**5*b**(7/2)*(a/b + x)**3) - 16*I*a**(15/2)*b**3*(a/b + x)**3/(-15*a**8*sqr
t(b) + 45*a**7*b**(3/2)*(a/b + x) - 45*a**6*b**(5/2)*(a/b + x)**2 + 15*a**5*b**(
7/2)*(a/b + x)**3) - 10*I*a**(13/2)*b**4*sqrt(1 - b*(a/b + x)/a)*(a/b + x)**4/(-
15*a**8*sqrt(b) + 45*a**7*b**(3/2)*(a/b + x) - 45*a**6*b**(5/2)*(a/b + x)**2 + 1
5*a**5*b**(7/2)*(a/b + x)**3) + 6*I*a**(11/2)*b**5*sqrt(1 - b*(a/b + x)/a)*(a/b
+ x)**5/(-15*a**8*sqrt(b) + 45*a**7*b**(3/2)*(a/b + x) - 45*a**6*b**(5/2)*(a/b +
x)**2 + 15*a**5*b**(7/2)*(a/b + x)**3), True))
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.200805, size = 32, normalized size = 0.94 \[ \frac{2}{5} \, b^{2} x^{\frac{5}{2}} + \frac{4}{3} \, a b x^{\frac{3}{2}} + 2 \, a^{2} \sqrt{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^2/sqrt(x),x, algorithm="giac")
[Out]
2/5*b^2*x^(5/2) + 4/3*a*b*x^(3/2) + 2*a^2*sqrt(x)