\(\int \frac {x}{(a+\frac {b}{x^4})^{3/2}} \, dx\) [2091]

Optimal result

Integrand size = 13, antiderivative size = 40 \[ \int \frac {x}{\left (a+\frac {b}{x^4}\right )^{3/2}} \, dx=-\frac {x^2}{2 a \sqrt {a+\frac {b}{x^4}}}+\frac {\sqrt {a+\frac {b}{x^4}} x^2}{a^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {279, 270} \[ \int \frac {x}{\left (a+\frac {b}{x^4}\right )^{3/2}} \, dx=\frac {x^2 \sqrt {a+\frac {b}{x^4}}}{a^2}-\frac {x^2}{2 a \sqrt {a+\frac {b}{x^4}}} \]

[In]

Int[x/(a + b/x^4)^(3/2),x]

[Out]

-1/2*x^2/(a*Sqrt[a + b/x^4]) + (Sqrt[a + b/x^4]*x^2)/a^2

Rule 270

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 279

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] /; Free
Q[{a, b, c, m, n, p}, x] && ILtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[p, -1]

Rubi steps \begin{align*} \text {integral}& = -\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] (verified)

Time = 0.04 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.75 \[ \int \frac {x}{\left (a+\frac {b}{x^4}\right )^{3/2}} \, dx=\frac {2 b+a x^4}{2 a^2 \sqrt {a+\frac {b}{x^4}} x^2} \]

[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)

Maple [A] (verified)

Time = 0.05 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.95

method	result	size
gosper	\(\frac {\left (a \,x^{4}+b \right ) \left (a \,x^{4}+2 b \right )}{2 a^{2} x^{6} \left (\frac {a \,x^{4}+b}{x^{4}}\right )^{\frac {3}{2}}}\)	\(38\)
default	\(\frac {\left (a \,x^{4}+b \right ) \left (a \,x^{4}+2 b \right )}{2 a^{2} x^{6} \left (\frac {a \,x^{4}+b}{x^{4}}\right )^{\frac {3}{2}}}\)	\(38\)
trager	\(\frac {\left (a \,x^{4}+2 b \right ) x^{2} \sqrt {-\frac {-a \,x^{4}-b}{x^{4}}}}{2 \left (a \,x^{4}+b \right ) a^{2}}\)	\(44\)
risch	\(\frac {a \,x^{4}+b}{2 a^{2} \sqrt {\frac {a \,x^{4}+b}{x^{4}}}\, x^{2}}+\frac {b}{2 a^{2} \sqrt {\frac {a \,x^{4}+b}{x^{4}}}\, x^{2}}\)	\(52\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.05 \[ \int \frac {x}{\left (a+\frac {b}{x^4}\right )^{3/2}} \, dx=\frac {{\left (a x^{6} + 2 \, b x^{2}\right )} \sqrt {\frac {a x^{4} + b}{x^{4}}}}{2 \, {\left (a^{3} x^{4} + a^{2} b\right )}} \]

[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)

Sympy [A] (verification not implemented)

Time = 0.52 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.05 \[ \int \frac {x}{\left (a+\frac {b}{x^4}\right )^{3/2}} \, dx=\frac {x^{4}}{2 a \sqrt {b} \sqrt {\frac {a x^{4}}{b} + 1}} + \frac {\sqrt {b}}{a^{2} \sqrt {\frac {a x^{4}}{b} + 1}} \]

[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))

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.90 \[ \int \frac {x}{\left (a+\frac {b}{x^4}\right )^{3/2}} \, dx=\frac {\sqrt {a + \frac {b}{x^{4}}} x^{2}}{2 \, a^{2}} + \frac {b}{2 \, \sqrt {a + \frac {b}{x^{4}}} a^{2} x^{2}} \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.82 \[ \int \frac {x}{\left (a+\frac {b}{x^4}\right )^{3/2}} \, dx=\frac {\frac {\sqrt {a x^{4} + b}}{a} + \frac {b}{\sqrt {a x^{4} + b} a}}{2 \, a} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 6.21 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.60 \[ \int \frac {x}{\left (a+\frac {b}{x^4}\right )^{3/2}} \, dx=\frac {\frac {a\,x^4}{2}+b}{a^2\,x^2\,\sqrt {a+\frac {b}{x^4}}} \]

[In]

int(x/(a + b/x^4)^(3/2),x)

[Out]

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