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]

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Rubi [A]  time = 0.0013378, 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, Rules used = {37} \[ \frac{\sqrt{x}}{\sqrt{b x+2}} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[x]*(2 + b*x)^(3/2)),x]

[Out]

Sqrt[x]/Sqrt[2 + b*x]

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{x} (2+b x)^{3/2}} \, dx &=\frac{\sqrt{x}}{\sqrt{2+b x}}\\ \end{align*}

Mathematica [A]  time = 0.0032298, 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]

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Maple [A]  time = 0.003, size = 12, normalized size = 0.8 \begin{align*}{\sqrt{x}{\frac{1}{\sqrt{bx+2}}}} \end{align*}

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)

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Maxima [A]  time = 1.08694, size = 15, normalized size = 1. \begin{align*} \frac{\sqrt{x}}{\sqrt{b x + 2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(x)/sqrt(b*x + 2)

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Fricas [A]  time = 1.48699, size = 31, normalized size = 2.07 \begin{align*} \frac{\sqrt{x}}{\sqrt{b x + 2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(x)/sqrt(b*x + 2)

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Sympy [A]  time = 1.11237, size = 15, normalized size = 1. \begin{align*} \frac{1}{\sqrt{b} \sqrt{1 + \frac{2}{b x}}} \end{align*}

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)))

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Giac [B]  time = 1.07288, size = 59, normalized size = 3.93 \begin{align*} \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 |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+2)^(3/2)/x^(1/2),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))