3.2083 \(\int \frac{x}{\sqrt{a+\frac{b}{x^4}}} \, dx\)

Optimal. Leaf size=21 \[ \frac{x^2 \sqrt{a+\frac{b}{x^4}}}{2 a} \]

[Out]

(Sqrt[a + b/x^4]*x^2)/(2*a)

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Rubi [A]  time = 0.0046884, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {264} \[ \frac{x^2 \sqrt{a+\frac{b}{x^4}}}{2 a} \]

Antiderivative was successfully verified.

[In]

Int[x/Sqrt[a + b/x^4],x]

[Out]

(Sqrt[a + b/x^4]*x^2)/(2*a)

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{x}{\sqrt{a+\frac{b}{x^4}}} \, dx &=\frac{\sqrt{a+\frac{b}{x^4}} x^2}{2 a}\\ \end{align*}

Mathematica [A]  time = 0.0092191, size = 21, normalized size = 1. \[ \frac{x^2 \sqrt{a+\frac{b}{x^4}}}{2 a} \]

Antiderivative was successfully verified.

[In]

Integrate[x/Sqrt[a + b/x^4],x]

[Out]

(Sqrt[a + b/x^4]*x^2)/(2*a)

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Maple [A]  time = 0.004, size = 29, normalized size = 1.4 \begin{align*}{\frac{a{x}^{4}+b}{2\,a{x}^{2}}{\frac{1}{\sqrt{{\frac{a{x}^{4}+b}{{x}^{4}}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(a*x^4+b)/a/x^2/((a*x^4+b)/x^4)^(1/2)

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Maxima [A]  time = 0.964356, size = 23, normalized size = 1.1 \begin{align*} \frac{\sqrt{a + \frac{b}{x^{4}}} x^{2}}{2 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(a + b/x^4)*x^2/a

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Fricas [A]  time = 1.45134, size = 45, normalized size = 2.14 \begin{align*} \frac{x^{2} \sqrt{\frac{a x^{4} + b}{x^{4}}}}{2 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^2*sqrt((a*x^4 + b)/x^4)/a

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Sympy [A]  time = 0.729963, size = 19, normalized size = 0.9 \begin{align*} \frac{\sqrt{b} \sqrt{\frac{a x^{4}}{b} + 1}}{2 a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(b)*sqrt(a*x**4/b + 1)/(2*a)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{\sqrt{a + \frac{b}{x^{4}}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x/sqrt(a + b/x^4), x)