3.2091 \(\int \frac{x}{(a+\frac{b}{x^4})^{3/2}} \, dx\)

Optimal. Leaf size=40 \[ \frac{x^2 \sqrt{a+\frac{b}{x^4}}}{a^2}-\frac{x^2}{2 a \sqrt{a+\frac{b}{x^4}}} \]

[Out]

-x^2/(2*a*Sqrt[a + b/x^4]) + (Sqrt[a + b/x^4]*x^2)/a^2

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Rubi [A]  time = 0.0109285, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {273, 264} \[ \frac{x^2 \sqrt{a+\frac{b}{x^4}}}{a^2}-\frac{x^2}{2 a \sqrt{a+\frac{b}{x^4}}} \]

Antiderivative was successfully verified.

[In]

Int[x/(a + b/x^4)^(3/2),x]

[Out]

-x^2/(2*a*Sqrt[a + b/x^4]) + (Sqrt[a + b/x^4]*x^2)/a^2

Rule 273

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(
a*c*n*(p + 1)), x] + Dist[(m + n*(p + 1) + 1)/(a*n*(p + 1)), Int[(c*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[
{a, b, c, m, n, p}, x] && ILtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[p, -1]

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}{\left (a+\frac{b}{x^4}\right )^{3/2}} \, dx &=-\frac{x^2}{2 a \sqrt{a+\frac{b}{x^4}}}+\frac{2 \int \frac{x}{\sqrt{a+\frac{b}{x^4}}} \, dx}{a}\\ &=-\frac{x^2}{2 a \sqrt{a+\frac{b}{x^4}}}+\frac{\sqrt{a+\frac{b}{x^4}} x^2}{a^2}\\ \end{align*}

Mathematica [A]  time = 0.0154149, size = 30, normalized size = 0.75 \[ \frac{a x^4+2 b}{2 a^2 x^2 \sqrt{a+\frac{b}{x^4}}} \]

Antiderivative was successfully verified.

[In]

Integrate[x/(a + b/x^4)^(3/2),x]

[Out]

(2*b + a*x^4)/(2*a^2*Sqrt[a + b/x^4]*x^2)

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Maple [A]  time = 0.004, size = 38, normalized size = 1. \begin{align*}{\frac{ \left ( a{x}^{4}+b \right ) \left ( a{x}^{4}+2\,b \right ) }{2\,{a}^{2}{x}^{6}} \left ({\frac{a{x}^{4}+b}{{x}^{4}}} \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(a+b/x^4)^(3/2),x)

[Out]

1/2*(a*x^4+b)*(a*x^4+2*b)/a^2/x^6/((a*x^4+b)/x^4)^(3/2)

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Maxima [A]  time = 0.957835, size = 49, normalized size = 1.22 \begin{align*} \frac{\sqrt{a + \frac{b}{x^{4}}} x^{2}}{2 \, a^{2}} + \frac{b}{2 \, \sqrt{a + \frac{b}{x^{4}}} a^{2} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(a+b/x^4)^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.5176, size = 85, normalized size = 2.12 \begin{align*} \frac{{\left (a x^{6} + 2 \, b x^{2}\right )} \sqrt{\frac{a x^{4} + b}{x^{4}}}}{2 \,{\left (a^{3} x^{4} + a^{2} b\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(a+b/x^4)^(3/2),x, algorithm="fricas")

[Out]

1/2*(a*x^6 + 2*b*x^2)*sqrt((a*x^4 + b)/x^4)/(a^3*x^4 + a^2*b)

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Sympy [A]  time = 1.48567, size = 42, normalized size = 1.05 \begin{align*} \frac{x^{4}}{2 a \sqrt{b} \sqrt{\frac{a x^{4}}{b} + 1}} + \frac{\sqrt{b}}{a^{2} \sqrt{\frac{a x^{4}}{b} + 1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(a+b/x**4)**(3/2),x)

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{{\left (a + \frac{b}{x^{4}}\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(a+b/x^4)^(3/2),x, algorithm="giac")

[Out]

integrate(x/(a + b/x^4)^(3/2), x)