3.619 \(\int \frac{1}{\sqrt{x} (2+b x)^{3/2}} \, dx\)

Optimal. Leaf size=15 \[ \frac{\sqrt{x}}{\sqrt{b x+2}} \]

[Out]

Sqrt[x]/Sqrt[2 + b*x]

_______________________________________________________________________________________

Rubi [A]  time = 0.0110465, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ \frac{\sqrt{x}}{\sqrt{b x+2}} \]

Antiderivative was successfully verified.

[In]  Int[1/(Sqrt[x]*(2 + b*x)^(3/2)),x]

[Out]

Sqrt[x]/Sqrt[2 + b*x]

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 2.04892, size = 12, normalized size = 0.8 \[ \frac{\sqrt{x}}{\sqrt{b x + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x+2)**(3/2)/x**(1/2),x)

[Out]

sqrt(x)/sqrt(b*x + 2)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0107898, size = 15, normalized size = 1. \[ \frac{\sqrt{x}}{\sqrt{b x+2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(Sqrt[x]*(2 + b*x)^(3/2)),x]

[Out]

Sqrt[x]/Sqrt[2 + b*x]

_______________________________________________________________________________________

Maple [A]  time = 0.005, size = 12, normalized size = 0.8 \[{1\sqrt{x}{\frac{1}{\sqrt{bx+2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x+2)^(3/2)/x^(1/2),x)

[Out]

x^(1/2)/(b*x+2)^(1/2)

_______________________________________________________________________________________

Maxima [A]  time = 1.35111, size = 15, normalized size = 1. \[ \frac{\sqrt{x}}{\sqrt{b x + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + 2)^(3/2)*sqrt(x)),x, algorithm="maxima")

[Out]

sqrt(x)/sqrt(b*x + 2)

_______________________________________________________________________________________

Fricas [A]  time = 0.208228, size = 15, normalized size = 1. \[ \frac{\sqrt{x}}{\sqrt{b x + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + 2)^(3/2)*sqrt(x)),x, algorithm="fricas")

[Out]

sqrt(x)/sqrt(b*x + 2)

_______________________________________________________________________________________

Sympy [A]  time = 4.35119, size = 15, normalized size = 1. \[ \frac{1}{\sqrt{b} \sqrt{1 + \frac{2}{b x}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b*x+2)**(3/2)/x**(1/2),x)

[Out]

1/(sqrt(b)*sqrt(1 + 2/(b*x)))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.212357, size = 59, normalized size = 3.93 \[ \frac{4 \, b^{\frac{3}{2}}}{{\left ({\left (\sqrt{b x + 2} \sqrt{b} - \sqrt{{\left (b x + 2\right )} b - 2 \, b}\right )}^{2} + 2 \, b\right )}{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + 2)^(3/2)*sqrt(x)),x, algorithm="giac")

[Out]

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