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

Optimal. Leaf size=38 \[ -\frac{\tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{\sqrt{a}} \]

[Out]

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

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Rubi [A]  time = 0.0428515, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ -\frac{\tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{\sqrt{a}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 6.56539, size = 34, normalized size = 0.89 \[ - \frac{\operatorname{atanh}{\left (\frac{2 a + b x}{2 \sqrt{a} \sqrt{a + b x + c x^{2}}} \right )}}{\sqrt{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.055654, size = 39, normalized size = 1.03 \[ \frac{\log (x)-\log \left (2 \sqrt{a} \sqrt{a+x (b+c x)}+2 a+b x\right )}{\sqrt{a}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.005, size = 35, normalized size = 0.9 \[ -{1\ln \left ({\frac{1}{x} \left ( 2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a} \right ) } \right ){\frac{1}{\sqrt{a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(c*x^2+b*x+a)^(1/2),x)

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.29034, size = 1, normalized size = 0.03 \[ \left [\frac{\log \left (\frac{4 \,{\left (a b x + 2 \, a^{2}\right )} \sqrt{c x^{2} + b x + a} -{\left (8 \, a b x +{\left (b^{2} + 4 \, a c\right )} x^{2} + 8 \, a^{2}\right )} \sqrt{a}}{x^{2}}\right )}{2 \, \sqrt{a}}, -\frac{\arctan \left (\frac{{\left (b x + 2 \, a\right )} \sqrt{-a}}{2 \, \sqrt{c x^{2} + b x + a} a}\right )}{\sqrt{-a}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/2*log((4*(a*b*x + 2*a^2)*sqrt(c*x^2 + b*x + a) - (8*a*b*x + (b^2 + 4*a*c)*x^2
+ 8*a^2)*sqrt(a))/x^2)/sqrt(a), -arctan(1/2*(b*x + 2*a)*sqrt(-a)/(sqrt(c*x^2 +
b*x + a)*a))/sqrt(-a)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.270821, size = 47, normalized size = 1.24 \[ \frac{2 \, \arctan \left (-\frac{\sqrt{c} x - \sqrt{c x^{2} + b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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