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

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

[Out]

(-2*Sqrt[a + b*x])/(a*Sqrt[x])

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Rubi [A]  time = 0.001619, 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} \[ -\frac{2 \sqrt{a+b x}}{a \sqrt{x}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^(3/2)*Sqrt[a + b*x]),x]

[Out]

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

Mathematica [A]  time = 0.004231, size = 19, normalized size = 1. \[ -\frac{2 \sqrt{a+b x}}{a \sqrt{x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[a + b*x])/(a*Sqrt[x])

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Maple [A]  time = 0.003, size = 16, normalized size = 0.8 \begin{align*} -2\,{\frac{\sqrt{bx+a}}{a\sqrt{x}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.05798, size = 20, normalized size = 1.05 \begin{align*} -\frac{2 \, \sqrt{b x + a}}{a \sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*sqrt(b*x + a)/(a*sqrt(x))

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Fricas [A]  time = 1.9059, size = 41, normalized size = 2.16 \begin{align*} -\frac{2 \, \sqrt{b x + a}}{a \sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*sqrt(b*x + a)/(a*sqrt(x))

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Sympy [A]  time = 1.01257, size = 19, normalized size = 1. \begin{align*} - \frac{2 \sqrt{b} \sqrt{\frac{a}{b x} + 1}}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*sqrt(b)*sqrt(a/(b*x) + 1)/a

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Giac [B]  time = 1.06644, size = 45, normalized size = 2.37 \begin{align*} -\frac{2 \, \sqrt{b x + a} b^{2}}{\sqrt{{\left (b x + a\right )} b - a b} a{\left | b \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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