3.1010 \(\int \frac{1}{x^3 \sqrt{a+(2+2 b-2 (1+b)) x^2+c x^4}} \, dx\)

Optimal. Leaf size=21 \[ -\frac{\sqrt{a+c x^4}}{2 a x^2} \]

[Out]

-Sqrt[a + c*x^4]/(2*a*x^2)

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Rubi [A]  time = 0.0046456, antiderivative size = 21, 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 = {4, 264} \[ -\frac{\sqrt{a+c x^4}}{2 a x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-Sqrt[a + c*x^4]/(2*a*x^2)

Rule 4

Int[(u_.)*((a_.) + (c_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[u*(a + c*x^(2*n))^p, x] /; Fre
eQ[{a, b, c, n, p}, x] && EqQ[j, 2*n] && EqQ[b, 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^3 \sqrt{a+(2+2 b-2 (1+b)) x^2+c x^4}} \, dx &=\int \frac{1}{x^3 \sqrt{a+c x^4}} \, dx\\ &=-\frac{\sqrt{a+c x^4}}{2 a x^2}\\ \end{align*}

Mathematica [A]  time = 0.0040815, size = 21, normalized size = 1. \[ -\frac{\sqrt{a+c x^4}}{2 a x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-Sqrt[a + c*x^4]/(2*a*x^2)

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Maple [A]  time = 0.045, size = 18, normalized size = 0.9 \begin{align*} -{\frac{1}{2\,a{x}^{2}}\sqrt{c{x}^{4}+a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.944374, size = 23, normalized size = 1.1 \begin{align*} -\frac{\sqrt{c x^{4} + a}}{2 \, a x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*sqrt(c*x^4 + a)/(a*x^2)

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Fricas [A]  time = 1.4529, size = 41, normalized size = 1.95 \begin{align*} -\frac{\sqrt{c x^{4} + a}}{2 \, a x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*sqrt(c*x^4 + a)/(a*x^2)

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Sympy [A]  time = 0.616605, size = 20, normalized size = 0.95 \begin{align*} - \frac{\sqrt{c} \sqrt{\frac{a}{c x^{4}} + 1}}{2 a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(c)*sqrt(a/(c*x**4) + 1)/(2*a)

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Giac [A]  time = 1.1882, size = 19, normalized size = 0.9 \begin{align*} -\frac{\sqrt{c + \frac{a}{x^{4}}}}{2 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*sqrt(c + a/x^4)/a