3.971 \(\int \frac{x}{\sqrt{a+b x^2-c x^4}} \, dx\)

Optimal. Leaf size=44 \[ -\frac{\tan ^{-1}\left (\frac{b-2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2-c x^4}}\right )}{2 \sqrt{c}} \]

[Out]

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

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Rubi [A]  time = 0.063748, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ -\frac{\tan ^{-1}\left (\frac{b-2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2-c x^4}}\right )}{2 \sqrt{c}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 7.31274, size = 39, normalized size = 0.89 \[ - \frac{\operatorname{atan}{\left (\frac{b - 2 c x^{2}}{2 \sqrt{c} \sqrt{a + b x^{2} - c x^{4}}} \right )}}{2 \sqrt{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.0280529, size = 51, normalized size = 1.16 \[ \frac{i \log \left (2 \sqrt{a+b x^2-c x^4}-\frac{i \left (2 c x^2-b\right )}{\sqrt{c}}\right )}{2 \sqrt{c}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.011, size = 36, normalized size = 0.8 \[{\frac{1}{2}\arctan \left ({1\sqrt{c} \left ({x}^{2}-{\frac{b}{2\,c}} \right ){\frac{1}{\sqrt{-c{x}^{4}+b{x}^{2}+a}}}} \right ){\frac{1}{\sqrt{c}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(-c*x^4+b*x^2+a)^(1/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.288194, size = 1, normalized size = 0.02 \[ \left [\frac{\log \left (4 \, \sqrt{-c x^{4} + b x^{2} + a}{\left (2 \, c^{2} x^{2} - b c\right )} +{\left (8 \, c^{2} x^{4} - 8 \, b c x^{2} + b^{2} - 4 \, a c\right )} \sqrt{-c}\right )}{4 \, \sqrt{-c}}, \frac{\arctan \left (\frac{2 \, c x^{2} - b}{2 \, \sqrt{-c x^{4} + b x^{2} + a} \sqrt{c}}\right )}{2 \, \sqrt{c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.292851, size = 61, normalized size = 1.39 \[ -\frac{{\rm ln}\left ({\left | 2 \,{\left (\sqrt{-c} x^{2} - \sqrt{-c x^{4} + b x^{2} + a}\right )} \sqrt{-c} + b \right |}\right )}{2 \, \sqrt{-c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*ln(abs(2*(sqrt(-c)*x^2 - sqrt(-c*x^4 + b*x^2 + a))*sqrt(-c) + b))/sqrt(-c)