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

Optimal. Leaf size=119 \[ -\frac{\left (3 b^2-4 a c\right ) \tanh ^{-1}\left (\frac{x (2 a+b x)}{2 \sqrt{a} \sqrt{a x^2+b x^3+c x^4}}\right )}{8 a^{5/2}}+\frac{3 b \sqrt{a x^2+b x^3+c x^4}}{4 a^2 x^2}-\frac{\sqrt{a x^2+b x^3+c x^4}}{2 a x^3} \]

[Out]

-Sqrt[a*x^2 + b*x^3 + c*x^4]/(2*a*x^3) + (3*b*Sqrt[a*x^2 + b*x^3 + c*x^4])/(4*a^
2*x^2) - ((3*b^2 - 4*a*c)*ArcTanh[(x*(2*a + b*x))/(2*Sqrt[a]*Sqrt[a*x^2 + b*x^3
+ c*x^4])])/(8*a^(5/2))

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Rubi [A]  time = 0.250008, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ -\frac{\left (3 b^2-4 a c\right ) \tanh ^{-1}\left (\frac{x (2 a+b x)}{2 \sqrt{a} \sqrt{a x^2+b x^3+c x^4}}\right )}{8 a^{5/2}}+\frac{3 b \sqrt{a x^2+b x^3+c x^4}}{4 a^2 x^2}-\frac{\sqrt{a x^2+b x^3+c x^4}}{2 a x^3} \]

Antiderivative was successfully verified.

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

[Out]

-Sqrt[a*x^2 + b*x^3 + c*x^4]/(2*a*x^3) + (3*b*Sqrt[a*x^2 + b*x^3 + c*x^4])/(4*a^
2*x^2) - ((3*b^2 - 4*a*c)*ArcTanh[(x*(2*a + b*x))/(2*Sqrt[a]*Sqrt[a*x^2 + b*x^3
+ c*x^4])])/(8*a^(5/2))

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Rubi in Sympy [A]  time = 42.7591, size = 134, normalized size = 1.13 \[ - \frac{\sqrt{a x^{2} + b x^{3} + c x^{4}}}{2 a x^{3}} + \frac{3 b \sqrt{a x^{2} + b x^{3} + c x^{4}}}{4 a^{2} x^{2}} - \frac{x \left (- a c + \frac{3 b^{2}}{4}\right ) \sqrt{a + b x + c x^{2}} \operatorname{atanh}{\left (\frac{2 a + b x}{2 \sqrt{a} \sqrt{a + b x + c x^{2}}} \right )}}{2 a^{\frac{5}{2}} \sqrt{a x^{2} + b x^{3} + c x^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(a*x**2 + b*x**3 + c*x**4)/(2*a*x**3) + 3*b*sqrt(a*x**2 + b*x**3 + c*x**4)/
(4*a**2*x**2) - x*(-a*c + 3*b**2/4)*sqrt(a + b*x + c*x**2)*atanh((2*a + b*x)/(2*
sqrt(a)*sqrt(a + b*x + c*x**2)))/(2*a**(5/2)*sqrt(a*x**2 + b*x**3 + c*x**4))

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Mathematica [A]  time = 0.230935, size = 138, normalized size = 1.16 \[ \frac{x^2 \log (x) \left (3 b^2-4 a c\right ) \sqrt{a+x (b+c x)}+x^2 \left (4 a c-3 b^2\right ) \sqrt{a+x (b+c x)} \log \left (2 \sqrt{a} \sqrt{a+x (b+c x)}+2 a+b x\right )-2 \sqrt{a} (2 a-3 b x) (a+x (b+c x))}{8 a^{5/2} x \sqrt{x^2 (a+x (b+c x))}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[a]*(2*a - 3*b*x)*(a + x*(b + c*x)) + (3*b^2 - 4*a*c)*x^2*Sqrt[a + x*(b
+ c*x)]*Log[x] + (-3*b^2 + 4*a*c)*x^2*Sqrt[a + x*(b + c*x)]*Log[2*a + b*x + 2*Sq
rt[a]*Sqrt[a + x*(b + c*x)]])/(8*a^(5/2)*x*Sqrt[x^2*(a + x*(b + c*x))])

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Maple [A]  time = 0.01, size = 152, normalized size = 1.3 \[ -{\frac{1}{8\,x}\sqrt{c{x}^{2}+bx+a} \left ( 4\,\sqrt{c{x}^{2}+bx+a}{a}^{5/2}-6\,\sqrt{c{x}^{2}+bx+a}{a}^{3/2}xb-4\,c\ln \left ({\frac{2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a}}{x}} \right ){x}^{2}{a}^{2}+3\,\ln \left ({\frac{2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a}}{x}} \right ){x}^{2}a{b}^{2} \right ){\frac{1}{\sqrt{c{x}^{4}+b{x}^{3}+a{x}^{2}}}}{a}^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8*(c*x^2+b*x+a)^(1/2)*(4*(c*x^2+b*x+a)^(1/2)*a^(5/2)-6*(c*x^2+b*x+a)^(1/2)*a^
(3/2)*x*b-4*c*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)*x^2*a^2+3*ln((2*a+b*
x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)*x^2*a*b^2)/x/(c*x^4+b*x^3+a*x^2)^(1/2)/a^(7/
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(c*x^4 + b*x^3 + a*x^2)*x^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.289729, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (3 \, b^{2} - 4 \, a c\right )} \sqrt{a} x^{3} \log \left (-\frac{4 \, \sqrt{c x^{4} + b x^{3} + a x^{2}}{\left (a b x + 2 \, a^{2}\right )} +{\left (8 \, a b x^{2} +{\left (b^{2} + 4 \, a c\right )} x^{3} + 8 \, a^{2} x\right )} \sqrt{a}}{x^{3}}\right ) - 4 \, \sqrt{c x^{4} + b x^{3} + a x^{2}}{\left (3 \, a b x - 2 \, a^{2}\right )}}{16 \, a^{3} x^{3}}, \frac{{\left (3 \, b^{2} - 4 \, a c\right )} \sqrt{-a} x^{3} \arctan \left (\frac{{\left (b x^{2} + 2 \, a x\right )} \sqrt{-a}}{2 \, \sqrt{c x^{4} + b x^{3} + a x^{2}} a}\right ) + 2 \, \sqrt{c x^{4} + b x^{3} + a x^{2}}{\left (3 \, a b x - 2 \, a^{2}\right )}}{8 \, a^{3} x^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/16*((3*b^2 - 4*a*c)*sqrt(a)*x^3*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) - 4*sqrt(c*x^4
+ b*x^3 + a*x^2)*(3*a*b*x - 2*a^2))/(a^3*x^3), 1/8*((3*b^2 - 4*a*c)*sqrt(-a)*x^
3*arctan(1/2*(b*x^2 + 2*a*x)*sqrt(-a)/(sqrt(c*x^4 + b*x^3 + a*x^2)*a)) + 2*sqrt(
c*x^4 + b*x^3 + a*x^2)*(3*a*b*x - 2*a^2))/(a^3*x^3)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: NotImplementedError