3.838 \(\int \frac{1}{x^3 \sqrt{a-b x^4}} \, dx\)

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

[Out]

-Sqrt[a - b*x^4]/(2*a*x^2)

________________________________________________________________________________________

Rubi [A]  time = 0.0044414, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {264} \[ -\frac{\sqrt{a-b x^4}}{2 a x^2} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^3*Sqrt[a - b*x^4]),x]

[Out]

-Sqrt[a - b*x^4]/(2*a*x^2)

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-b x^4}} \, dx &=-\frac{\sqrt{a-b x^4}}{2 a x^2}\\ \end{align*}

Mathematica [A]  time = 0.0034588, size = 22, normalized size = 1. \[ -\frac{\sqrt{a-b x^4}}{2 a x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^3*Sqrt[a - b*x^4]),x]

[Out]

-Sqrt[a - b*x^4]/(2*a*x^2)

________________________________________________________________________________________

Maple [A]  time = 0.004, size = 19, normalized size = 0.9 \begin{align*} -{\frac{1}{2\,a{x}^{2}}\sqrt{-b{x}^{4}+a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 0.962564, size = 24, normalized size = 1.09 \begin{align*} -\frac{\sqrt{-b x^{4} + a}}{2 \, a x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*sqrt(-b*x^4 + a)/(a*x^2)

________________________________________________________________________________________

Fricas [A]  time = 1.49038, size = 42, normalized size = 1.91 \begin{align*} -\frac{\sqrt{-b x^{4} + a}}{2 \, a x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*sqrt(-b*x^4 + a)/(a*x^2)

________________________________________________________________________________________

Sympy [A]  time = 0.919776, size = 54, normalized size = 2.45 \begin{align*} \begin{cases} - \frac{\sqrt{b} \sqrt{\frac{a}{b x^{4}} - 1}}{2 a} & \text{for}\: \frac{\left |{a}\right |}{\left |{b}\right | \left |{x^{4}}\right |} > 1 \\- \frac{i \sqrt{b} \sqrt{- \frac{a}{b x^{4}} + 1}}{2 a} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-sqrt(b)*sqrt(a/(b*x**4) - 1)/(2*a), Abs(a)/(Abs(b)*Abs(x**4)) > 1), (-I*sqrt(b)*sqrt(-a/(b*x**4) +
 1)/(2*a), True))

________________________________________________________________________________________

Giac [A]  time = 1.09172, size = 22, normalized size = 1. \begin{align*} -\frac{\sqrt{-b + \frac{a}{x^{4}}}}{2 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*sqrt(-b + a/x^4)/a