3.1192 \(\int \frac{\sqrt [4]{a-b x^4}}{x^6} \, dx\)

Optimal. Leaf size=22 \[ -\frac{\left (a-b x^4\right )^{5/4}}{5 a x^5} \]

[Out]

-(a - b*x^4)^(5/4)/(5*a*x^5)

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Rubi [A]  time = 0.0044349, 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} \[ -\frac{\left (a-b x^4\right )^{5/4}}{5 a x^5} \]

Antiderivative was successfully verified.

[In]

Int[(a - b*x^4)^(1/4)/x^6,x]

[Out]

-(a - b*x^4)^(5/4)/(5*a*x^5)

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{\sqrt [4]{a-b x^4}}{x^6} \, dx &=-\frac{\left (a-b x^4\right )^{5/4}}{5 a x^5}\\ \end{align*}

Mathematica [A]  time = 0.0054592, size = 22, normalized size = 1. \[ -\frac{\left (a-b x^4\right )^{5/4}}{5 a x^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(a - b*x^4)^(1/4)/x^6,x]

[Out]

-(a - b*x^4)^(5/4)/(5*a*x^5)

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Maple [A]  time = 0.003, size = 19, normalized size = 0.9 \begin{align*} -{\frac{1}{5\,a{x}^{5}} \left ( -b{x}^{4}+a \right ) ^{{\frac{5}{4}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-b*x^4+a)^(1/4)/x^6,x)

[Out]

-1/5*(-b*x^4+a)^(5/4)/a/x^5

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Maxima [A]  time = 0.964382, size = 24, normalized size = 1.09 \begin{align*} -\frac{{\left (-b x^{4} + a\right )}^{\frac{5}{4}}}{5 \, a x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b*x^4+a)^(1/4)/x^6,x, algorithm="maxima")

[Out]

-1/5*(-b*x^4 + a)^(5/4)/(a*x^5)

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Fricas [A]  time = 1.88859, size = 59, normalized size = 2.68 \begin{align*} \frac{{\left (b x^{4} - a\right )}{\left (-b x^{4} + a\right )}^{\frac{1}{4}}}{5 \, a x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b*x^4+a)^(1/4)/x^6,x, algorithm="fricas")

[Out]

1/5*(b*x^4 - a)*(-b*x^4 + a)^(1/4)/(a*x^5)

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Sympy [B]  time = 1.34896, size = 162, normalized size = 7.36 \begin{align*} \begin{cases} \frac{\sqrt [4]{b} \sqrt [4]{\frac{a}{b x^{4}} - 1} \Gamma \left (- \frac{5}{4}\right )}{4 x^{4} \Gamma \left (- \frac{1}{4}\right )} - \frac{b^{\frac{5}{4}} \sqrt [4]{\frac{a}{b x^{4}} - 1} \Gamma \left (- \frac{5}{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{\sqrt [4]{b} \sqrt [4]{- \frac{a}{b x^{4}} + 1} e^{\frac{i \pi }{4}} \Gamma \left (- \frac{5}{4}\right )}{4 x^{4} \Gamma \left (- \frac{1}{4}\right )} - \frac{b^{\frac{5}{4}} \sqrt [4]{- \frac{a}{b x^{4}} + 1} e^{\frac{i \pi }{4}} \Gamma \left (- \frac{5}{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((-b*x**4+a)**(1/4)/x**6,x)

[Out]

Piecewise((b**(1/4)*(a/(b*x**4) - 1)**(1/4)*gamma(-5/4)/(4*x**4*gamma(-1/4)) - b**(5/4)*(a/(b*x**4) - 1)**(1/4
)*gamma(-5/4)/(4*a*gamma(-1/4)), Abs(a)/(Abs(b)*Abs(x**4)) > 1), (b**(1/4)*(-a/(b*x**4) + 1)**(1/4)*exp(I*pi/4
)*gamma(-5/4)/(4*x**4*gamma(-1/4)) - b**(5/4)*(-a/(b*x**4) + 1)**(1/4)*exp(I*pi/4)*gamma(-5/4)/(4*a*gamma(-1/4
)), True))

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Giac [A]  time = 1.14829, size = 35, normalized size = 1.59 \begin{align*} \frac{{\left (-b x^{4} + a\right )}^{\frac{1}{4}}{\left (b - \frac{a}{x^{4}}\right )}}{5 \, a x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b*x^4+a)^(1/4)/x^6,x, algorithm="giac")

[Out]

1/5*(-b*x^4 + a)^(1/4)*(b - a/x^4)/(a*x)