\(\int \frac {x}{\sqrt {b+a x^4}} \, dx\) [382]

Optimal result

Integrand size = 13, antiderivative size = 31 \[ \int \frac {x}{\sqrt {b+a x^4}} \, dx=\frac {\log \left (\sqrt {a} x^2+\sqrt {b+a x^4}\right )}{2 \sqrt {a}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.97, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {281, 223, 212} \[ \int \frac {x}{\sqrt {b+a x^4}} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {a} x^2}{\sqrt {a x^4+b}}\right )}{2 \sqrt {a}} \]

[In]

Int[x/Sqrt[b + a*x^4],x]

[Out]

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

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 281

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {1}{\sqrt {b+a x^2}} \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \frac {1}{1-a x^2} \, dx,x,\frac {x^2}{\sqrt {b+a x^4}}\right ) \\ & = \frac {\text {arctanh}\left (\frac {\sqrt {a} x^2}{\sqrt {b+a x^4}}\right )}{2 \sqrt {a}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.24 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00 \[ \int \frac {x}{\sqrt {b+a x^4}} \, dx=\frac {\log \left (\sqrt {a} x^2+\sqrt {b+a x^4}\right )}{2 \sqrt {a}} \]

[In]

Integrate[x/Sqrt[b + a*x^4],x]

[Out]

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

Maple [A] (verified)

Time = 1.12 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.74

method	result	size
pseudoelliptic	\(\frac {\operatorname {arctanh}\left (\frac {\sqrt {a \,x^{4}+b}}{x^{2} \sqrt {a}}\right )}{2 \sqrt {a}}\)	\(23\)
default	\(\frac {\ln \left (\sqrt {a}\, x^{2}+\sqrt {a \,x^{4}+b}\right )}{2 \sqrt {a}}\)	\(24\)
elliptic	\(\frac {\ln \left (\sqrt {a}\, x^{2}+\sqrt {a \,x^{4}+b}\right )}{2 \sqrt {a}}\)	\(24\)

[In]

int(x/(a*x^4+b)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.03 \[ \int \frac {x}{\sqrt {b+a x^4}} \, dx=\left [\frac {\log \left (-2 \, a x^{4} - 2 \, \sqrt {a x^{4} + b} \sqrt {a} x^{2} - b\right )}{4 \, \sqrt {a}}, -\frac {\sqrt {-a} \arctan \left (\frac {\sqrt {-a} x^{2}}{\sqrt {a x^{4} + b}}\right )}{2 \, a}\right ] \]

[In]

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

[Out]

[1/4*log(-2*a*x^4 - 2*sqrt(a*x^4 + b)*sqrt(a)*x^2 - b)/sqrt(a), -1/2*sqrt(-a)*arctan(sqrt(-a)*x^2/sqrt(a*x^4 +
b))/a]

Sympy [A] (verification not implemented)

Time = 0.48 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.65 \[ \int \frac {x}{\sqrt {b+a x^4}} \, dx=\frac {\operatorname {asinh}{\left (\frac {\sqrt {a} x^{2}}{\sqrt {b}} \right )}}{2 \sqrt {a}} \]

[In]

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

[Out]

asinh(sqrt(a)*x**2/sqrt(b))/(2*sqrt(a))

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.45 \[ \int \frac {x}{\sqrt {b+a x^4}} \, dx=-\frac {\log \left (-\frac {\sqrt {a} - \frac {\sqrt {a x^{4} + b}}{x^{2}}}{\sqrt {a} + \frac {\sqrt {a x^{4} + b}}{x^{2}}}\right )}{4 \, \sqrt {a}} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.81 \[ \int \frac {x}{\sqrt {b+a x^4}} \, dx=-\frac {\log \left ({\left | -\sqrt {a} x^{2} + \sqrt {a x^{4} + b} \right |}\right )}{2 \, \sqrt {a}} \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {x}{\sqrt {b+a x^4}} \, dx=\int \frac {x}{\sqrt {a\,x^4+b}} \,d x \]

[In]

int(x/(b + a*x^4)^(1/2),x)

[Out]

int(x/(b + a*x^4)^(1/2), x)