3.2099 \(\int \frac{x}{(a+\frac{b}{x^4})^{5/2}} \, dx\)

Optimal. Leaf size=64 \[ \frac{4 x^2 \sqrt{a+\frac{b}{x^4}}}{3 a^3}-\frac{2 x^2}{3 a^2 \sqrt{a+\frac{b}{x^4}}}-\frac{x^2}{6 a \left (a+\frac{b}{x^4}\right )^{3/2}} \]

[Out]

-x^2/(6*a*(a + b/x^4)^(3/2)) - (2*x^2)/(3*a^2*Sqrt[a + b/x^4]) + (4*Sqrt[a + b/x^4]*x^2)/(3*a^3)

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Rubi [A]  time = 0.019163, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {273, 264} \[ \frac{4 x^2 \sqrt{a+\frac{b}{x^4}}}{3 a^3}-\frac{2 x^2}{3 a^2 \sqrt{a+\frac{b}{x^4}}}-\frac{x^2}{6 a \left (a+\frac{b}{x^4}\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[x/(a + b/x^4)^(5/2),x]

[Out]

-x^2/(6*a*(a + b/x^4)^(3/2)) - (2*x^2)/(3*a^2*Sqrt[a + b/x^4]) + (4*Sqrt[a + b/x^4]*x^2)/(3*a^3)

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 )^{5/2}} \, dx &=-\frac{x^2}{6 a \left (a+\frac{b}{x^4}\right )^{3/2}}+\frac{4 \int \frac{x}{\left (a+\frac{b}{x^4}\right )^{3/2}} \, dx}{3 a}\\ &=-\frac{x^2}{6 a \left (a+\frac{b}{x^4}\right )^{3/2}}-\frac{2 x^2}{3 a^2 \sqrt{a+\frac{b}{x^4}}}+\frac{8 \int \frac{x}{\sqrt{a+\frac{b}{x^4}}} \, dx}{3 a^2}\\ &=-\frac{x^2}{6 a \left (a+\frac{b}{x^4}\right )^{3/2}}-\frac{2 x^2}{3 a^2 \sqrt{a+\frac{b}{x^4}}}+\frac{4 \sqrt{a+\frac{b}{x^4}} x^2}{3 a^3}\\ \end{align*}

Mathematica [A]  time = 0.0217474, size = 51, normalized size = 0.8 \[ \frac{3 a^2 x^8+12 a b x^4+8 b^2}{6 a^3 x^2 \sqrt{a+\frac{b}{x^4}} \left (a x^4+b\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[x/(a + b/x^4)^(5/2),x]

[Out]

(8*b^2 + 12*a*b*x^4 + 3*a^2*x^8)/(6*a^3*Sqrt[a + b/x^4]*x^2*(b + a*x^4))

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Maple [A]  time = 0.006, size = 50, normalized size = 0.8 \begin{align*}{\frac{ \left ( a{x}^{4}+b \right ) \left ( 3\,{x}^{8}{a}^{2}+12\,ab{x}^{4}+8\,{b}^{2} \right ) }{6\,{a}^{3}{x}^{10}} \left ({\frac{a{x}^{4}+b}{{x}^{4}}} \right ) ^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(a+b/x^4)^(5/2),x)

[Out]

1/6*(a*x^4+b)*(3*a^2*x^8+12*a*b*x^4+8*b^2)/a^3/x^10/((a*x^4+b)/x^4)^(5/2)

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Maxima [A]  time = 0.945905, size = 73, normalized size = 1.14 \begin{align*} \frac{\sqrt{a + \frac{b}{x^{4}}} x^{2}}{2 \, a^{3}} + \frac{6 \,{\left (a + \frac{b}{x^{4}}\right )} b x^{4} - b^{2}}{6 \,{\left (a + \frac{b}{x^{4}}\right )}^{\frac{3}{2}} a^{3} x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(a+b/x^4)^(5/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.4919, size = 134, normalized size = 2.09 \begin{align*} \frac{{\left (3 \, a^{2} x^{10} + 12 \, a b x^{6} + 8 \, b^{2} x^{2}\right )} \sqrt{\frac{a x^{4} + b}{x^{4}}}}{6 \,{\left (a^{5} x^{8} + 2 \, a^{4} b x^{4} + a^{3} b^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(a+b/x^4)^(5/2),x, algorithm="fricas")

[Out]

1/6*(3*a^2*x^10 + 12*a*b*x^6 + 8*b^2*x^2)*sqrt((a*x^4 + b)/x^4)/(a^5*x^8 + 2*a^4*b*x^4 + a^3*b^2)

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Sympy [B]  time = 2.50946, size = 163, normalized size = 2.55 \begin{align*} \frac{3 a^{2} b^{\frac{9}{2}} x^{8} \sqrt{\frac{a x^{4}}{b} + 1}}{6 a^{5} b^{4} x^{8} + 12 a^{4} b^{5} x^{4} + 6 a^{3} b^{6}} + \frac{12 a b^{\frac{11}{2}} x^{4} \sqrt{\frac{a x^{4}}{b} + 1}}{6 a^{5} b^{4} x^{8} + 12 a^{4} b^{5} x^{4} + 6 a^{3} b^{6}} + \frac{8 b^{\frac{13}{2}} \sqrt{\frac{a x^{4}}{b} + 1}}{6 a^{5} b^{4} x^{8} + 12 a^{4} b^{5} x^{4} + 6 a^{3} b^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(a+b/x**4)**(5/2),x)

[Out]

3*a**2*b**(9/2)*x**8*sqrt(a*x**4/b + 1)/(6*a**5*b**4*x**8 + 12*a**4*b**5*x**4 + 6*a**3*b**6) + 12*a*b**(11/2)*
x**4*sqrt(a*x**4/b + 1)/(6*a**5*b**4*x**8 + 12*a**4*b**5*x**4 + 6*a**3*b**6) + 8*b**(13/2)*sqrt(a*x**4/b + 1)/
(6*a**5*b**4*x**8 + 12*a**4*b**5*x**4 + 6*a**3*b**6)

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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{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(a+b/x^4)^(5/2),x, algorithm="giac")

[Out]

integrate(x/(a + b/x^4)^(5/2), x)