3.1005 \(\int \frac{\sqrt [4]{a+b x^4}}{x^6} \, dx\)

Optimal. Leaf size=21 \[ -\frac{\left (a+b x^4\right )^{5/4}}{5 a x^5} \]

[Out]

-(a + b*x^4)^(5/4)/(5*a*x^5)

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Rubi [A]  time = 0.0047634, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, 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 = 0.0044163, size = 21, 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 = 18, 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 = 0.990606, size = 23, normalized size = 1.1 \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 = 1.75156, size = 43, normalized size = 2.05 \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="fricas")

[Out]

-1/5*(b*x^4 + a)^(5/4)/(a*x^5)

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Sympy [B]  time = 1.24901, size = 68, normalized size = 3.24 \begin{align*} \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 )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**4+a)**(1/4)/x**6,x)

[Out]

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))

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Giac [A]  time = 1.16983, size = 32, normalized size = 1.52 \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)