3.1018 \(\int \frac{1}{x^2 \sqrt{a+b x^2+(2+2 c-2 (1+c)) x^4}} \, dx\)

Optimal. Leaf size=19 \[ -\frac{\sqrt{a+b x^2}}{a x} \]

[Out]

-(Sqrt[a + b*x^2]/(a*x))

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Rubi [A]  time = 0.0044502, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069, Rules used = {5, 264} \[ -\frac{\sqrt{a+b x^2}}{a x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[a + b*x^2]/(a*x))

Rule 5

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

Rule 264

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

Rubi steps

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

Mathematica [A]  time = 0.003643, size = 19, normalized size = 1. \[ -\frac{\sqrt{a+b x^2}}{a x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[a + b*x^2]/(a*x))

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Maple [A]  time = 0.043, size = 18, normalized size = 1. \begin{align*} -{\frac{1}{ax}\sqrt{b{x}^{2}+a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(b*x^2+a)^(1/2)/a/x

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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^2/(b*x^2+a)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.52835, size = 32, normalized size = 1.68 \begin{align*} -\frac{\sqrt{b x^{2} + a}}{a x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(b*x^2 + a)/(a*x)

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Sympy [A]  time = 0.57357, size = 19, normalized size = 1. \begin{align*} - \frac{\sqrt{b} \sqrt{\frac{a}{b x^{2}} + 1}}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**2/(b*x**2+a)**(1/2),x)

[Out]

-sqrt(b)*sqrt(a/(b*x**2) + 1)/a

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Giac [A]  time = 1.18817, size = 41, normalized size = 2.16 \begin{align*} \frac{2 \, \sqrt{b}}{{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} - a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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