3.756 \(\int \frac{1}{x \sqrt{c x} \sqrt{a+b x}} \, dx\)

Optimal. Leaf size=21 \[ -\frac{2 \sqrt{a+b x}}{a \sqrt{c x}} \]

[Out]

(-2*Sqrt[a + b*x])/(a*Sqrt[c*x])

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Rubi [A]  time = 0.0047832, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {16, 37} \[ -\frac{2 \sqrt{a+b x}}{a \sqrt{c x}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x*Sqrt[c*x]*Sqrt[a + b*x]),x]

[Out]

(-2*Sqrt[a + b*x])/(a*Sqrt[c*x])

Rule 16

Int[(u_.)*(v_)^(m_.)*((b_)*(v_))^(n_), x_Symbol] :> Dist[1/b^m, Int[u*(b*v)^(m + n), x], x] /; FreeQ[{b, n}, x
] && IntegerQ[m]

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

Mathematica [A]  time = 0.0092778, size = 23, normalized size = 1.1 \[ -\frac{2 c x \sqrt{a+b x}}{a (c x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x*Sqrt[c*x]*Sqrt[a + b*x]),x]

[Out]

(-2*c*x*Sqrt[a + b*x])/(a*(c*x)^(3/2))

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Maple [A]  time = 0.005, size = 18, normalized size = 0.9 \begin{align*} -2\,{\frac{\sqrt{bx+a}}{a\sqrt{cx}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.95213, size = 49, normalized size = 2.33 \begin{align*} -\frac{2 \, \sqrt{b x + a} \sqrt{c x}}{a c x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*sqrt(b*x + a)*sqrt(c*x)/(a*c*x)

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Sympy [A]  time = 1.71165, size = 24, normalized size = 1.14 \begin{align*} - \frac{2 \sqrt{b} \sqrt{\frac{a}{b x} + 1}}{a \sqrt{c}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*sqrt(b)*sqrt(a/(b*x) + 1)/(a*sqrt(c))

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Giac [B]  time = 1.38457, size = 47, normalized size = 2.24 \begin{align*} -\frac{2 \, \sqrt{b x + a} b^{2}}{\sqrt{{\left (b x + a\right )} b c - a b c} a{\left | b \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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