3.2233 \(\int \sqrt{a+b \sqrt{x}} \, dx\)

Optimal. Leaf size=42 \[ \frac{4 \left (a+b \sqrt{x}\right )^{5/2}}{5 b^2}-\frac{4 a \left (a+b \sqrt{x}\right )^{3/2}}{3 b^2} \]

[Out]

(-4*a*(a + b*Sqrt[x])^(3/2))/(3*b^2) + (4*(a + b*Sqrt[x])^(5/2))/(5*b^2)

_______________________________________________________________________________________

Rubi [A]  time = 0.0474007, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{4 \left (a+b \sqrt{x}\right )^{5/2}}{5 b^2}-\frac{4 a \left (a+b \sqrt{x}\right )^{3/2}}{3 b^2} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b*Sqrt[x]],x]

[Out]

(-4*a*(a + b*Sqrt[x])^(3/2))/(3*b^2) + (4*(a + b*Sqrt[x])^(5/2))/(5*b^2)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 6.04349, size = 37, normalized size = 0.88 \[ - \frac{4 a \left (a + b \sqrt{x}\right )^{\frac{3}{2}}}{3 b^{2}} + \frac{4 \left (a + b \sqrt{x}\right )^{\frac{5}{2}}}{5 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b*x**(1/2))**(1/2),x)

[Out]

-4*a*(a + b*sqrt(x))**(3/2)/(3*b**2) + 4*(a + b*sqrt(x))**(5/2)/(5*b**2)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0193036, size = 40, normalized size = 0.95 \[ \frac{4 \sqrt{a+b \sqrt{x}} \left (-2 a^2+a b \sqrt{x}+3 b^2 x\right )}{15 b^2} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a + b*Sqrt[x]],x]

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.004, size = 30, normalized size = 0.7 \[ 4\,{\frac{1/5\, \left ( a+b\sqrt{x} \right ) ^{5/2}-1/3\, \left ( a+b\sqrt{x} \right ) ^{3/2}a}{{b}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*x^(1/2))^(1/2),x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 1.45126, size = 41, normalized size = 0.98 \[ \frac{4 \,{\left (b \sqrt{x} + a\right )}^{\frac{5}{2}}}{5 \, b^{2}} - \frac{4 \,{\left (b \sqrt{x} + a\right )}^{\frac{3}{2}} a}{3 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*sqrt(x) + a),x, algorithm="maxima")

[Out]

4/5*(b*sqrt(x) + a)^(5/2)/b^2 - 4/3*(b*sqrt(x) + a)^(3/2)*a/b^2

_______________________________________________________________________________________

Fricas [A]  time = 0.24716, size = 43, normalized size = 1.02 \[ \frac{4 \,{\left (3 \, b^{2} x + a b \sqrt{x} - 2 \, a^{2}\right )} \sqrt{b \sqrt{x} + a}}{15 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*sqrt(x) + a),x, algorithm="fricas")

[Out]

4/15*(3*b^2*x + a*b*sqrt(x) - 2*a^2)*sqrt(b*sqrt(x) + a)/b^2

_______________________________________________________________________________________

Sympy [A]  time = 3.9724, size = 272, normalized size = 6.48 \[ - \frac{8 a^{\frac{9}{2}} x^{2} \sqrt{1 + \frac{b \sqrt{x}}{a}}}{15 a^{2} b^{2} x^{2} + 15 a b^{3} x^{\frac{5}{2}}} + \frac{8 a^{\frac{9}{2}} x^{2}}{15 a^{2} b^{2} x^{2} + 15 a b^{3} x^{\frac{5}{2}}} - \frac{4 a^{\frac{7}{2}} b x^{\frac{5}{2}} \sqrt{1 + \frac{b \sqrt{x}}{a}}}{15 a^{2} b^{2} x^{2} + 15 a b^{3} x^{\frac{5}{2}}} + \frac{8 a^{\frac{7}{2}} b x^{\frac{5}{2}}}{15 a^{2} b^{2} x^{2} + 15 a b^{3} x^{\frac{5}{2}}} + \frac{16 a^{\frac{5}{2}} b^{2} x^{3} \sqrt{1 + \frac{b \sqrt{x}}{a}}}{15 a^{2} b^{2} x^{2} + 15 a b^{3} x^{\frac{5}{2}}} + \frac{12 a^{\frac{3}{2}} b^{3} x^{\frac{7}{2}} \sqrt{1 + \frac{b \sqrt{x}}{a}}}{15 a^{2} b^{2} x^{2} + 15 a b^{3} x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b*x**(1/2))**(1/2),x)

[Out]

-8*a**(9/2)*x**2*sqrt(1 + b*sqrt(x)/a)/(15*a**2*b**2*x**2 + 15*a*b**3*x**(5/2))
+ 8*a**(9/2)*x**2/(15*a**2*b**2*x**2 + 15*a*b**3*x**(5/2)) - 4*a**(7/2)*b*x**(5/
2)*sqrt(1 + b*sqrt(x)/a)/(15*a**2*b**2*x**2 + 15*a*b**3*x**(5/2)) + 8*a**(7/2)*b
*x**(5/2)/(15*a**2*b**2*x**2 + 15*a*b**3*x**(5/2)) + 16*a**(5/2)*b**2*x**3*sqrt(
1 + b*sqrt(x)/a)/(15*a**2*b**2*x**2 + 15*a*b**3*x**(5/2)) + 12*a**(3/2)*b**3*x**
(7/2)*sqrt(1 + b*sqrt(x)/a)/(15*a**2*b**2*x**2 + 15*a*b**3*x**(5/2))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.251155, size = 39, normalized size = 0.93 \[ \frac{4 \,{\left (3 \,{\left (b \sqrt{x} + a\right )}^{\frac{5}{2}} - 5 \,{\left (b \sqrt{x} + a\right )}^{\frac{3}{2}} a\right )}}{15 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*sqrt(x) + a),x, algorithm="giac")

[Out]

4/15*(3*(b*sqrt(x) + a)^(5/2) - 5*(b*sqrt(x) + a)^(3/2)*a)/b^2