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]

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

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Rubi [A]  time = 0.02, antiderivative size = 64, normalized size of antiderivative = 1.00, 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 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]

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]

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*}

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Mathematica [A]  time = 0.02, size = 51, normalized size = 0.80 \[ \frac {3 a^2 x^8+12 a b x^4+8 b^2}{6 a^3 x^2 \sqrt {a+\frac {b}{x^4}} \left (a x^4+b\right )} \]

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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fricas [A]  time = 0.67, size = 65, normalized size = 1.02 \[ \frac {{\left (3 \, a^{2} x^{10} + 12 \, a b x^{6} + 8 \, b^{2} x^{2}\right )} \sqrt {\frac {a x^{4} + b}{x^{4}}}}{6 \, {\left (a^{5} x^{8} + 2 \, a^{4} b x^{4} + a^{3} b^{2}\right )}} \]

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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giac [A]  time = 0.19, size = 53, normalized size = 0.83 \[ \frac {\sqrt {a x^{4} + b}}{2 \, a^{3}} - \frac {4 \, \sqrt {b}}{3 \, a^{3}} + \frac {6 \, {\left (a x^{4} + b\right )} b - b^{2}}{6 \, {\left (a x^{4} + b\right )}^{\frac {3}{2}} a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.01, size = 50, normalized size = 0.78 \[ \frac {\left (a \,x^{4}+b \right ) \left (3 x^{8} a^{2}+12 a b \,x^{4}+8 b^{2}\right )}{6 \left (\frac {a \,x^{4}+b}{x^{4}}\right )^{\frac {5}{2}} a^{3} x^{10}} \]

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 = 0.88, size = 54, normalized size = 0.84 \[ \frac {\sqrt {a + \frac {b}{x^{4}}} x^{2}}{2 \, a^{3}} + \frac {6 \, {\left (a + \frac {b}{x^{4}}\right )} b x^{4} - b^{2}}{6 \, {\left (a + \frac {b}{x^{4}}\right )}^{\frac {3}{2}} a^{3} x^{6}} \]

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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mupad [B]  time = 1.48, size = 38, normalized size = 0.59 \[ \frac {3\,a^2\,x^8+12\,a\,b\,x^4+8\,b^2}{6\,a^3\,x^6\,{\left (a+\frac {b}{x^4}\right )}^{3/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(a + b/x^4)^(5/2),x)

[Out]

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

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sympy [B]  time = 1.66, size = 163, normalized size = 2.55 \[ \frac {3 a^{2} b^{\frac {9}{2}} x^{8} \sqrt {\frac {a x^{4}}{b} + 1}}{6 a^{5} b^{4} x^{8} + 12 a^{4} b^{5} x^{4} + 6 a^{3} b^{6}} + \frac {12 a b^{\frac {11}{2}} x^{4} \sqrt {\frac {a x^{4}}{b} + 1}}{6 a^{5} b^{4} x^{8} + 12 a^{4} b^{5} x^{4} + 6 a^{3} b^{6}} + \frac {8 b^{\frac {13}{2}} \sqrt {\frac {a x^{4}}{b} + 1}}{6 a^{5} b^{4} x^{8} + 12 a^{4} b^{5} x^{4} + 6 a^{3} b^{6}} \]

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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