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

Optimal. Leaf size=19 \[ \frac{2 \sqrt{x}}{a \sqrt{a+b x}} \]

[Out]

(2*Sqrt[x])/(a*Sqrt[a + b*x])

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Rubi [A]  time = 0.001838, antiderivative size = 19, 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} \[ \frac{2 \sqrt{x}}{a \sqrt{a+b x}} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[x]*(a + b*x)^(3/2)),x]

[Out]

(2*Sqrt[x])/(a*Sqrt[a + 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} (a+b x)^{3/2}} \, dx &=\frac{2 \sqrt{x}}{a \sqrt{a+b x}}\\ \end{align*}

Mathematica [A]  time = 0.0044999, size = 19, normalized size = 1. \[ \frac{2 \sqrt{x}}{a \sqrt{a+b x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[x])/(a*Sqrt[a + b*x])

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Maple [A]  time = 0.002, size = 16, normalized size = 0.8 \begin{align*} 2\,{\frac{\sqrt{x}}{a\sqrt{bx+a}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.09507, size = 20, normalized size = 1.05 \begin{align*} \frac{2 \, \sqrt{x}}{\sqrt{b x + a} a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(x)/(sqrt(b*x + a)*a)

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Fricas [A]  time = 1.81284, size = 53, normalized size = 2.79 \begin{align*} \frac{2 \, \sqrt{b x + a} \sqrt{x}}{a b x + a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(b*x + a)*sqrt(x)/(a*b*x + a^2)

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Sympy [A]  time = 1.2184, size = 17, normalized size = 0.89 \begin{align*} \frac{2}{a \sqrt{b} \sqrt{\frac{a}{b x} + 1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/(a*sqrt(b)*sqrt(a/(b*x) + 1))

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Giac [B]  time = 1.05505, size = 61, normalized size = 3.21 \begin{align*} \frac{4 \, b^{\frac{3}{2}}}{{\left ({\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2} + a b\right )}{\left | b \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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