3.1252 \(\int \frac{1}{x^2 (a-b x^4)^{3/4}} \, dx\)

Optimal. Leaf size=20 \[ -\frac{\sqrt [4]{a-b x^4}}{a x} \]

[Out]

-((a - b*x^4)^(1/4)/(a*x))

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Rubi [A]  time = 0.004762, antiderivative size = 20, 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} \[ -\frac{\sqrt [4]{a-b x^4}}{a x} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^2*(a - b*x^4)^(3/4)),x]

[Out]

-((a - b*x^4)^(1/4)/(a*x))

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 \left (a-b x^4\right )^{3/4}} \, dx &=-\frac{\sqrt [4]{a-b x^4}}{a x}\\ \end{align*}

Mathematica [A]  time = 0.0042253, size = 20, normalized size = 1. \[ -\frac{\sqrt [4]{a-b x^4}}{a x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((a - b*x^4)^(1/4)/(a*x))

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Maple [A]  time = 0.003, size = 19, normalized size = 1. \begin{align*} -{\frac{1}{ax}\sqrt [4]{-b{x}^{4}+a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.00684, size = 24, normalized size = 1.2 \begin{align*} -\frac{{\left (-b x^{4} + a\right )}^{\frac{1}{4}}}{a x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(-b*x^4 + a)^(1/4)/(a*x)

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Fricas [A]  time = 1.7578, size = 36, normalized size = 1.8 \begin{align*} -\frac{{\left (-b x^{4} + a\right )}^{\frac{1}{4}}}{a x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(-b*x^4 + a)^(1/4)/(a*x)

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Sympy [B]  time = 0.990024, size = 83, normalized size = 4.15 \begin{align*} \begin{cases} \frac{\sqrt [4]{b} \sqrt [4]{\frac{a}{b x^{4}} - 1} \Gamma \left (- \frac{1}{4}\right )}{4 a \Gamma \left (\frac{3}{4}\right )} & \text{for}\: \frac{\left |{a}\right |}{\left |{b}\right | \left |{x^{4}}\right |} > 1 \\- \frac{\sqrt [4]{b} \sqrt [4]{- \frac{a}{b x^{4}} + 1} e^{- \frac{3 i \pi }{4}} \Gamma \left (- \frac{1}{4}\right )}{4 a \Gamma \left (\frac{3}{4}\right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((b**(1/4)*(a/(b*x**4) - 1)**(1/4)*gamma(-1/4)/(4*a*gamma(3/4)), Abs(a)/(Abs(b)*Abs(x**4)) > 1), (-b*
*(1/4)*(-a/(b*x**4) + 1)**(1/4)*exp(-3*I*pi/4)*gamma(-1/4)/(4*a*gamma(3/4)), True))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (-b x^{4} + a\right )}^{\frac{3}{4}} x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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