3.986 \(\int \frac{1}{x \left (a+b x^2+c x^4\right )^{3/2}} \, dx\)

Optimal. Leaf size=89 \[ \frac{-2 a c+b^2+b c x^2}{a \left (b^2-4 a c\right ) \sqrt{a+b x^2+c x^4}}-\frac{\tanh ^{-1}\left (\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right )}{2 a^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.170805, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{-2 a c+b^2+b c x^2}{a \left (b^2-4 a c\right ) \sqrt{a+b x^2+c x^4}}-\frac{\tanh ^{-1}\left (\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right )}{2 a^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 19.3991, size = 78, normalized size = 0.88 \[ \frac{- 2 a c + b^{2} + b c x^{2}}{a \left (- 4 a c + b^{2}\right ) \sqrt{a + b x^{2} + c x^{4}}} - \frac{\operatorname{atanh}{\left (\frac{2 a + b x^{2}}{2 \sqrt{a} \sqrt{a + b x^{2} + c x^{4}}} \right )}}{2 a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.274109, size = 100, normalized size = 1.12 \[ \frac{\log \left (x^2\right )-\log \left (2 \sqrt{a} \sqrt{a+x^2 \left (b+c x^2\right )}+2 a+b x^2\right )}{2 a^{3/2}}+\frac{2 a c-b^2-b c x^2}{a \left (4 a c-b^2\right ) \sqrt{a+b x^2+c x^4}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.018, size = 99, normalized size = 1.1 \[{\frac{1}{2\,a}{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}+a}}}}-{\frac{b \left ( 2\,c{x}^{2}+b \right ) }{2\,a \left ( 4\,ac-{b}^{2} \right ) }{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}+a}}}}-{\frac{1}{2}\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,a+b{x}^{2}+2\,\sqrt{a}\sqrt{c{x}^{4}+b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.317143, size = 1, normalized size = 0.01 \[ \left [\frac{4 \, \sqrt{c x^{4} + b x^{2} + a}{\left (b c x^{2} + b^{2} - 2 \, a c\right )} \sqrt{a} +{\left ({\left (b^{2} c - 4 \, a c^{2}\right )} x^{4} + a b^{2} - 4 \, a^{2} c +{\left (b^{3} - 4 \, a b c\right )} x^{2}\right )} \log \left (\frac{4 \, \sqrt{c x^{4} + b x^{2} + a}{\left (a b x^{2} + 2 \, a^{2}\right )} -{\left ({\left (b^{2} + 4 \, a c\right )} x^{4} + 8 \, a b x^{2} + 8 \, a^{2}\right )} \sqrt{a}}{x^{4}}\right )}{4 \,{\left ({\left (a b^{2} c - 4 \, a^{2} c^{2}\right )} x^{4} + a^{2} b^{2} - 4 \, a^{3} c +{\left (a b^{3} - 4 \, a^{2} b c\right )} x^{2}\right )} \sqrt{a}}, \frac{2 \, \sqrt{c x^{4} + b x^{2} + a}{\left (b c x^{2} + b^{2} - 2 \, a c\right )} \sqrt{-a} -{\left ({\left (b^{2} c - 4 \, a c^{2}\right )} x^{4} + a b^{2} - 4 \, a^{2} c +{\left (b^{3} - 4 \, a b c\right )} x^{2}\right )} \arctan \left (\frac{{\left (b x^{2} + 2 \, a\right )} \sqrt{-a}}{2 \, \sqrt{c x^{4} + b x^{2} + a} a}\right )}{2 \,{\left ({\left (a b^{2} c - 4 \, a^{2} c^{2}\right )} x^{4} + a^{2} b^{2} - 4 \, a^{3} c +{\left (a b^{3} - 4 \, a^{2} b c\right )} x^{2}\right )} \sqrt{-a}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (c x^{4} + b x^{2} + a\right )}^{\frac{3}{2}} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/((c*x^4 + b*x^2 + a)^(3/2)*x), x)