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

Optimal. Leaf size=16 \[ -\frac{\sqrt{b x+2}}{\sqrt{x}} \]

[Out]

-(Sqrt[2 + b*x]/Sqrt[x])

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Rubi [A]  time = 0.0012056, antiderivative size = 16, 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} \[ -\frac{\sqrt{b x+2}}{\sqrt{x}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^(3/2)*Sqrt[2 + b*x]),x]

[Out]

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

Mathematica [A]  time = 0.0034954, size = 16, normalized size = 1. \[ -\frac{\sqrt{b x+2}}{\sqrt{x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[2 + b*x]/Sqrt[x])

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Maple [A]  time = 0.003, size = 13, normalized size = 0.8 \begin{align*} -{\sqrt{bx+2}{\frac{1}{\sqrt{x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.02366, size = 16, normalized size = 1. \begin{align*} -\frac{\sqrt{b x + 2}}{\sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(b*x + 2)/sqrt(x)

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Fricas [A]  time = 1.8098, size = 32, normalized size = 2. \begin{align*} -\frac{\sqrt{b x + 2}}{\sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(b*x + 2)/sqrt(x)

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Sympy [A]  time = 1.12139, size = 15, normalized size = 0.94 \begin{align*} - \sqrt{b} \sqrt{1 + \frac{2}{b x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(b)*sqrt(1 + 2/(b*x))

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Giac [B]  time = 1.07529, size = 39, normalized size = 2.44 \begin{align*} -\frac{\sqrt{b x + 2} b^{2}}{\sqrt{{\left (b x + 2\right )} b - 2 \, b}{\left | b \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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