3.991 \(\int \frac{1}{\sqrt{a c x+b c x^2}} \, dx\)

Optimal. Leaf size=40 \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c} x}{\sqrt{a c x+b c x^2}}\right )}{\sqrt{b} \sqrt{c}} \]

[Out]

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

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Rubi [A]  time = 0.0169132, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {620, 206} \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c} x}{\sqrt{a c x+b c x^2}}\right )}{\sqrt{b} \sqrt{c}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[a*c*x + b*c*x^2],x]

[Out]

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

Rule 620

Int[1/Sqrt[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(1 - c*x^2), x], x, x/Sqrt[b*x + c*x^2
]], x] /; FreeQ[{b, c}, x]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{a c x+b c x^2}} \, dx &=2 \operatorname{Subst}\left (\int \frac{1}{1-b c x^2} \, dx,x,\frac{x}{\sqrt{a c x+b c x^2}}\right )\\ &=\frac{2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c} x}{\sqrt{a c x+b c x^2}}\right )}{\sqrt{b} \sqrt{c}}\\ \end{align*}

Mathematica [A]  time = 0.0158734, size = 58, normalized size = 1.45 \[ \frac{2 \sqrt{a} \sqrt{x} \sqrt{\frac{b x}{a}+1} \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{\sqrt{b} \sqrt{c x (a+b x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[a*c*x + b*c*x^2],x]

[Out]

(2*Sqrt[a]*Sqrt[x]*Sqrt[1 + (b*x)/a]*ArcSinh[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(Sqrt[b]*Sqrt[c*x*(a + b*x)])

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Maple [A]  time = 0.003, size = 37, normalized size = 0.9 \begin{align*}{\ln \left ({ \left ({\frac{ac}{2}}+bcx \right ){\frac{1}{\sqrt{bc}}}}+\sqrt{bc{x}^{2}+acx} \right ){\frac{1}{\sqrt{bc}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.50532, size = 197, normalized size = 4.92 \begin{align*} \left [\frac{\sqrt{b c} \log \left (2 \, b c x + a c + 2 \, \sqrt{b c x^{2} + a c x} \sqrt{b c}\right )}{b c}, -\frac{2 \, \sqrt{-b c} \arctan \left (\frac{\sqrt{b c x^{2} + a c x} \sqrt{-b c}}{b c x}\right )}{b c}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[sqrt(b*c)*log(2*b*c*x + a*c + 2*sqrt(b*c*x^2 + a*c*x)*sqrt(b*c))/(b*c), -2*sqrt(-b*c)*arctan(sqrt(b*c*x^2 + a
*c*x)*sqrt(-b*c)/(b*c*x))/(b*c)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{a c x + b c x^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/sqrt(a*c*x + b*c*x**2), x)

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Giac [A]  time = 1.25144, size = 68, normalized size = 1.7 \begin{align*} -\frac{\sqrt{b c} \log \left ({\left | -2 \,{\left (\sqrt{b c} x - \sqrt{b c x^{2} + a c x}\right )} b - \sqrt{b c} a \right |}\right )}{b c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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