3.2239 \(\int \frac{1}{\sqrt{a+b \sqrt{x}}} \, dx\)

Optimal. Leaf size=40 \[ \frac{4 \left (a+b \sqrt{x}\right )^{3/2}}{3 b^2}-\frac{4 a \sqrt{a+b \sqrt{x}}}{b^2} \]

[Out]

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

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Rubi [A]  time = 0.0471729, antiderivative size = 40, 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 \[ \frac{4 \left (a+b \sqrt{x}\right )^{3/2}}{3 b^2}-\frac{4 a \sqrt{a+b \sqrt{x}}}{b^2} \]

Antiderivative was successfully verified.

[In]  Int[1/Sqrt[a + b*Sqrt[x]],x]

[Out]

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

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Rubi in Sympy [A]  time = 6.03321, size = 36, normalized size = 0.9 \[ - \frac{4 a \sqrt{a + b \sqrt{x}}}{b^{2}} + \frac{4 \left (a + b \sqrt{x}\right )^{\frac{3}{2}}}{3 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0187558, size = 31, normalized size = 0.78 \[ \frac{4 \left (b \sqrt{x}-2 a\right ) \sqrt{a+b \sqrt{x}}}{3 b^2} \]

Antiderivative was successfully verified.

[In]  Integrate[1/Sqrt[a + b*Sqrt[x]],x]

[Out]

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

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Maple [A]  time = 0.006, size = 30, normalized size = 0.8 \[ 4\,{\frac{1/3\, \left ( a+b\sqrt{x} \right ) ^{3/2}-a\sqrt{a+b\sqrt{x}}}{{b}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b*x^(1/2))^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.44255, size = 41, normalized size = 1.02 \[ \frac{4 \,{\left (b \sqrt{x} + a\right )}^{\frac{3}{2}}}{3 \, b^{2}} - \frac{4 \, \sqrt{b \sqrt{x} + a} a}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(b*sqrt(x) + a),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.244414, size = 31, normalized size = 0.78 \[ \frac{4 \, \sqrt{b \sqrt{x} + a}{\left (b \sqrt{x} - 2 \, a\right )}}{3 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(b*sqrt(x) + a),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 3.76088, size = 219, normalized size = 5.48 \[ - \frac{8 a^{\frac{7}{2}} x^{2} \sqrt{1 + \frac{b \sqrt{x}}{a}}}{3 a^{2} b^{2} x^{2} + 3 a b^{3} x^{\frac{5}{2}}} + \frac{8 a^{\frac{7}{2}} x^{2}}{3 a^{2} b^{2} x^{2} + 3 a b^{3} x^{\frac{5}{2}}} - \frac{4 a^{\frac{5}{2}} b x^{\frac{5}{2}} \sqrt{1 + \frac{b \sqrt{x}}{a}}}{3 a^{2} b^{2} x^{2} + 3 a b^{3} x^{\frac{5}{2}}} + \frac{8 a^{\frac{5}{2}} b x^{\frac{5}{2}}}{3 a^{2} b^{2} x^{2} + 3 a b^{3} x^{\frac{5}{2}}} + \frac{4 a^{\frac{3}{2}} b^{2} x^{3} \sqrt{1 + \frac{b \sqrt{x}}{a}}}{3 a^{2} b^{2} x^{2} + 3 a b^{3} x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-8*a**(7/2)*x**2*sqrt(1 + b*sqrt(x)/a)/(3*a**2*b**2*x**2 + 3*a*b**3*x**(5/2)) +
8*a**(7/2)*x**2/(3*a**2*b**2*x**2 + 3*a*b**3*x**(5/2)) - 4*a**(5/2)*b*x**(5/2)*s
qrt(1 + b*sqrt(x)/a)/(3*a**2*b**2*x**2 + 3*a*b**3*x**(5/2)) + 8*a**(5/2)*b*x**(5
/2)/(3*a**2*b**2*x**2 + 3*a*b**3*x**(5/2)) + 4*a**(3/2)*b**2*x**3*sqrt(1 + b*sqr
t(x)/a)/(3*a**2*b**2*x**2 + 3*a*b**3*x**(5/2))

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GIAC/XCAS [A]  time = 0.258369, size = 36, normalized size = 0.9 \[ \frac{4 \,{\left ({\left (b \sqrt{x} + a\right )}^{\frac{3}{2}} - 3 \, \sqrt{b \sqrt{x} + a} a\right )}}{3 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(b*sqrt(x) + a),x, algorithm="giac")

[Out]

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