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

Optimal. Leaf size=23 \[ -\frac{\sqrt{b x^2+c x^4}}{b x^2} \]

[Out]

-(Sqrt[b*x^2 + c*x^4]/(b*x^2))

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Rubi [A]  time = 0.0400943, antiderivative size = 23, 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, Rules used = {3, 2014} \[ -\frac{\sqrt{b x^2+c x^4}}{b x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[b*x^2 + c*x^4]/(b*x^2))

Rule 3

Int[(u_.)*((a_) + (c_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[u*(b*x^n + c*x^(2*n))^p, x] /;
FreeQ[{a, b, c, n, p}, x] && EqQ[j, 2*n] && EqQ[a, 0]

Rule 2014

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> -Simp[(c^(j - 1)*(c*x)^(m - j
+ 1)*(a*x^j + b*x^n)^(p + 1))/(a*(n - j)*(p + 1)), x] /; FreeQ[{a, b, c, j, m, n, p}, x] &&  !IntegerQ[p] && N
eQ[n, j] && EqQ[m + n*p + n - j + 1, 0] && (IntegerQ[j] || GtQ[c, 0])

Rubi steps

\begin{align*} \int \frac{1}{x \sqrt{2+2 a-2 (1+a)+b x^2+c x^4}} \, dx &=\int \frac{1}{x \sqrt{b x^2+c x^4}} \, dx\\ &=-\frac{\sqrt{b x^2+c x^4}}{b x^2}\\ \end{align*}

Mathematica [A]  time = 0.0070438, size = 23, normalized size = 1. \[ -\frac{\sqrt{x^2 \left (b+c x^2\right )}}{b x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[x^2*(b + c*x^2)]/(b*x^2))

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Maple [A]  time = 0.043, size = 26, normalized size = 1.1 \begin{align*} -{\frac{c{x}^{2}+b}{b}{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.45414, size = 41, normalized size = 1.78 \begin{align*} -\frac{\sqrt{c x^{4} + b x^{2}}}{b x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(c*x^4+b*x^2)^(1/2),x, algorithm="fricas")

[Out]

-sqrt(c*x^4 + b*x^2)/(b*x^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.13326, size = 19, normalized size = 0.83 \begin{align*} -\frac{\sqrt{c + \frac{b}{x^{2}}}}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(c*x^4+b*x^2)^(1/2),x, algorithm="giac")

[Out]

-sqrt(c + b/x^2)/b