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

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 = 0.0363187, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c} x}{\sqrt{a c x+b c x^2}}\right )}{\sqrt{b} \sqrt{c}} \]

Antiderivative was successfully verified.

[In]  Int[1/Sqrt[c*(a*x + b*x^2)],x]

[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 in Sympy [A]  time = 2.17579, size = 39, normalized size = 0.98 \[ \frac{2 \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{c} x}{\sqrt{a c x + b c x^{2}}} \right )}}{\sqrt{b} \sqrt{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(c*(b*x**2+a*x))**(1/2),x)

[Out]

2*atanh(sqrt(b)*sqrt(c)*x/sqrt(a*c*x + b*c*x**2))/(sqrt(b)*sqrt(c))

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Mathematica [A]  time = 0.0104209, size = 57, normalized size = 1.42 \[ \frac{2 \sqrt{x} \sqrt{a+b x} \log \left (\sqrt{b} \sqrt{a+b x}+b \sqrt{x}\right )}{\sqrt{b} \sqrt{c x (a+b x)}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.007, size = 37, normalized size = 0.9 \[{1\ln \left ({1 \left ({\frac{ac}{2}}+bcx \right ){\frac{1}{\sqrt{bc}}}}+\sqrt{bc{x}^{2}+acx} \right ){\frac{1}{\sqrt{bc}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(c*(b*x^2+a*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]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.275092, size = 1, normalized size = 0.02 \[ \left [\frac{\log \left (\sqrt{b c}{\left (2 \, b x + a\right )} + 2 \, \sqrt{b c x^{2} + a c x} b\right )}{\sqrt{b c}}, \frac{2 \, \arctan \left (\frac{\sqrt{b c x^{2} + a c x} \sqrt{-b c}}{b c x}\right )}{\sqrt{-b c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{c \left (a x + b x^{2}\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/sqrt(c*(a*x + b*x**2)), x)

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GIAC/XCAS [A]  time = 0.320606, size = 68, normalized size = 1.7 \[ -\frac{\sqrt{b c}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{b c} x - \sqrt{b c x^{2} + a c x}\right )} b - \sqrt{b c} a \right |}\right )}{b c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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