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

Optimal. Leaf size=22 \[ \frac{\sqrt{b x^2+c x^4}}{c x} \]

[Out]

Sqrt[b*x^2 + c*x^4]/(c*x)

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Rubi [A]  time = 0.0131807, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071 \[ \frac{\sqrt{b x^2+c x^4}}{c x} \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[b*x^2 + c*x^4]/(c*x)

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Rubi in Sympy [A]  time = 7.59124, size = 15, normalized size = 0.68 \[ \frac{\sqrt{b x^{2} + c x^{4}}}{c x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(b*x**2 + c*x**4)/(c*x)

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Mathematica [A]  time = 0.00949294, size = 22, normalized size = 1. \[ \frac{\sqrt{x^2 \left (b+c x^2\right )}}{c x} \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[x^2*(b + c*x^2)]/(c*x)

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Maple [A]  time = 0.003, size = 26, normalized size = 1.2 \[{\frac{x \left ( c{x}^{2}+b \right ) }{c}{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(c*x^4+b*x^2)^(1/2),x)

[Out]

(c*x^2+b)/c*x/(c*x^4+b*x^2)^(1/2)

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Maxima [A]  time = 0.714677, size = 18, normalized size = 0.82 \[ \frac{\sqrt{c x^{2} + b}}{c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(c*x^2 + b)/c

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Fricas [A]  time = 0.272006, size = 27, normalized size = 1.23 \[ \frac{\sqrt{c x^{4} + b x^{2}}}{c x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(c*x^4 + b*x^2)/(c*x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.278081, size = 42, normalized size = 1.91 \[ -\frac{2 \, \sqrt{b}}{{\left (\sqrt{c + \frac{b}{x^{2}}} - \frac{\sqrt{b}}{x}\right )}^{2} - c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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