3.1157 \(\int \frac{1}{(a+b x^4)^{5/4}} \, dx\)

Optimal. Leaf size=16 \[ \frac{x}{a \sqrt [4]{a+b x^4}} \]

[Out]

x/(a*(a + b*x^4)^(1/4))

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Rubi [A]  time = 0.0019039, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {191} \[ \frac{x}{a \sqrt [4]{a+b x^4}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^4)^(-5/4),x]

[Out]

x/(a*(a + b*x^4)^(1/4))

Rule 191

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^(p + 1))/a, x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rubi steps

\begin{align*} \int \frac{1}{\left (a+b x^4\right )^{5/4}} \, dx &=\frac{x}{a \sqrt [4]{a+b x^4}}\\ \end{align*}

Mathematica [A]  time = 0.0053008, size = 16, normalized size = 1. \[ \frac{x}{a \sqrt [4]{a+b x^4}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^4)^(-5/4),x]

[Out]

x/(a*(a + b*x^4)^(1/4))

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Maple [A]  time = 0.002, size = 15, normalized size = 0.9 \begin{align*}{\frac{x}{a}{\frac{1}{\sqrt [4]{b{x}^{4}+a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.04094, size = 19, normalized size = 1.19 \begin{align*} \frac{x}{{\left (b x^{4} + a\right )}^{\frac{1}{4}} a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^4+a)^(5/4),x, algorithm="maxima")

[Out]

x/((b*x^4 + a)^(1/4)*a)

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Fricas [A]  time = 1.4952, size = 50, normalized size = 3.12 \begin{align*} \frac{{\left (b x^{4} + a\right )}^{\frac{3}{4}} x}{a b x^{4} + a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^4+a)^(5/4),x, algorithm="fricas")

[Out]

(b*x^4 + a)^(3/4)*x/(a*b*x^4 + a^2)

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Sympy [B]  time = 0.829706, size = 29, normalized size = 1.81 \begin{align*} \frac{x \Gamma \left (\frac{1}{4}\right )}{4 a^{\frac{5}{4}} \sqrt [4]{1 + \frac{b x^{4}}{a}} \Gamma \left (\frac{5}{4}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x**4+a)**(5/4),x)

[Out]

x*gamma(1/4)/(4*a**(5/4)*(1 + b*x**4/a)**(1/4)*gamma(5/4))

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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{5}{4}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^4+a)^(5/4),x, algorithm="giac")

[Out]

integrate((b*x^4 + a)^(-5/4), x)