3.52 \(\int \frac{1}{\sqrt{a x^2+b x^3+c x^4}} \, dx\)

Optimal. Leaf size=45 \[ -\frac{\tanh ^{-1}\left (\frac{x (2 a+b x)}{2 \sqrt{a} \sqrt{a x^2+b x^3+c x^4}}\right )}{\sqrt{a}} \]

[Out]

-(ArcTanh[(x*(2*a + b*x))/(2*Sqrt[a]*Sqrt[a*x^2 + b*x^3 + c*x^4])]/Sqrt[a])

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Rubi [A]  time = 0.0325963, antiderivative size = 45, 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 \[ -\frac{\tanh ^{-1}\left (\frac{x (2 a+b x)}{2 \sqrt{a} \sqrt{a x^2+b x^3+c x^4}}\right )}{\sqrt{a}} \]

Antiderivative was successfully verified.

[In]  Int[1/Sqrt[a*x^2 + b*x^3 + c*x^4],x]

[Out]

-(ArcTanh[(x*(2*a + b*x))/(2*Sqrt[a]*Sqrt[a*x^2 + b*x^3 + c*x^4])]/Sqrt[a])

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Rubi in Sympy [A]  time = 19.1527, size = 68, normalized size = 1.51 \[ - \frac{x \sqrt{a + b x + c x^{2}} \operatorname{atanh}{\left (\frac{2 a + b x}{2 \sqrt{a} \sqrt{a + b x + c x^{2}}} \right )}}{\sqrt{a} \sqrt{a x^{2} + b x^{3} + c x^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(c*x**4+b*x**3+a*x**2)**(1/2),x)

[Out]

-x*sqrt(a + b*x + c*x**2)*atanh((2*a + b*x)/(2*sqrt(a)*sqrt(a + b*x + c*x**2)))/
(sqrt(a)*sqrt(a*x**2 + b*x**3 + c*x**4))

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Mathematica [A]  time = 0.0701684, size = 70, normalized size = 1.56 \[ \frac{x \sqrt{a+x (b+c x)} \left (\log (x)-\log \left (2 \sqrt{a} \sqrt{a+x (b+c x)}+2 a+b x\right )\right )}{\sqrt{a} \sqrt{x^2 (a+x (b+c x))}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/Sqrt[a*x^2 + b*x^3 + c*x^4],x]

[Out]

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

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Maple [A]  time = 0.008, size = 66, normalized size = 1.5 \[ -{x\sqrt{c{x}^{2}+bx+a}\ln \left ({\frac{1}{x} \left ( 2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a} \right ) } \right ){\frac{1}{\sqrt{c{x}^{4}+b{x}^{3}+a{x}^{2}}}}{\frac{1}{\sqrt{a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(c*x^4+b*x^3+a*x^2)^(1/2),x)

[Out]

-1/(c*x^4+b*x^3+a*x^2)^(1/2)*x*(c*x^2+b*x+a)^(1/2)/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^4 + b*x^3 + a*x^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.277752, size = 1, normalized size = 0.02 \[ \left [\frac{\log \left (\frac{4 \, \sqrt{c x^{4} + b x^{3} + a x^{2}}{\left (a b x + 2 \, a^{2}\right )} -{\left (8 \, a b x^{2} +{\left (b^{2} + 4 \, a c\right )} x^{3} + 8 \, a^{2} x\right )} \sqrt{a}}{x^{3}}\right )}{2 \, \sqrt{a}}, \frac{\sqrt{-a} \arctan \left (\frac{{\left (b x^{2} + 2 \, a x\right )} \sqrt{-a}}{2 \, \sqrt{c x^{4} + b x^{3} + a x^{2}} a}\right )}{a}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(c*x^4 + b*x^3 + a*x^2),x, algorithm="fricas")

[Out]

[1/2*log((4*sqrt(c*x^4 + b*x^3 + a*x^2)*(a*b*x + 2*a^2) - (8*a*b*x^2 + (b^2 + 4*
a*c)*x^3 + 8*a^2*x)*sqrt(a))/x^3)/sqrt(a), sqrt(-a)*arctan(1/2*(b*x^2 + 2*a*x)*s
qrt(-a)/(sqrt(c*x^4 + b*x^3 + a*x^2)*a))/a]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{a x^{2} + b x^{3} + c x^{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(c*x**4+b*x**3+a*x**2)**(1/2),x)

[Out]

Integral(1/sqrt(a*x**2 + b*x**3 + c*x**4), x)

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GIAC/XCAS [A]  time = 0.276477, size = 80, normalized size = 1.78 \[ -\frac{2 \, \arctan \left (\frac{\sqrt{a}}{\sqrt{-a}}\right ){\rm sign}\left (x\right )}{\sqrt{-a}} + \frac{2 \, \arctan \left (-\frac{\sqrt{c} x - \sqrt{c x^{2} + b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a}{\rm sign}\left (x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(c*x^4 + b*x^3 + a*x^2),x, algorithm="giac")

[Out]

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