3.1222 \(\int \frac{1}{x^4 \sqrt [4]{a-b x^4}} \, dx\)

Optimal. Leaf size=22 \[ -\frac{\left (a-b x^4\right )^{3/4}}{3 a x^3} \]

[Out]

-(a - b*x^4)^(3/4)/(3*a*x^3)

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Rubi [A]  time = 0.0046507, 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} \[ -\frac{\left (a-b x^4\right )^{3/4}}{3 a x^3} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^4*(a - b*x^4)^(1/4)),x]

[Out]

-(a - b*x^4)^(3/4)/(3*a*x^3)

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

Mathematica [A]  time = 0.0041998, size = 22, normalized size = 1. \[ -\frac{\left (a-b x^4\right )^{3/4}}{3 a x^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a - b*x^4)^(3/4)/(3*a*x^3)

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Maple [A]  time = 0.003, size = 19, normalized size = 0.9 \begin{align*} -{\frac{1}{3\,a{x}^{3}} \left ( -b{x}^{4}+a \right ) ^{{\frac{3}{4}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.982862, size = 24, normalized size = 1.09 \begin{align*} -\frac{{\left (-b x^{4} + a\right )}^{\frac{3}{4}}}{3 \, a x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(-b*x^4 + a)^(3/4)/(a*x^3)

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Fricas [A]  time = 1.78086, size = 45, normalized size = 2.05 \begin{align*} -\frac{{\left (-b x^{4} + a\right )}^{\frac{3}{4}}}{3 \, a x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(-b*x^4 + a)^(3/4)/(a*x^3)

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Sympy [A]  time = 1.00889, size = 82, normalized size = 3.73 \begin{align*} \begin{cases} \frac{b^{\frac{3}{4}} \left (\frac{a}{b x^{4}} - 1\right )^{\frac{3}{4}} \Gamma \left (- \frac{3}{4}\right )}{4 a \Gamma \left (\frac{1}{4}\right )} & \text{for}\: \frac{\left |{a}\right |}{\left |{b}\right | \left |{x^{4}}\right |} > 1 \\- \frac{b^{\frac{3}{4}} \left (- \frac{a}{b x^{4}} + 1\right )^{\frac{3}{4}} e^{- \frac{i \pi }{4}} \Gamma \left (- \frac{3}{4}\right )}{4 a \Gamma \left (\frac{1}{4}\right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((b**(3/4)*(a/(b*x**4) - 1)**(3/4)*gamma(-3/4)/(4*a*gamma(1/4)), Abs(a)/(Abs(b)*Abs(x**4)) > 1), (-b*
*(3/4)*(-a/(b*x**4) + 1)**(3/4)*exp(-I*pi/4)*gamma(-3/4)/(4*a*gamma(1/4)), 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{1}{4}} x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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