\(\int \sqrt {a+\frac {b}{x^4}} x \, dx\) [2059]

Optimal result

Integrand size = 13, antiderivative size = 49 \[ \int \sqrt {a+\frac {b}{x^4}} x \, dx=\frac {1}{2} \sqrt {a+\frac {b}{x^4}} x^2-\frac {1}{2} \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b}}{\sqrt {a+\frac {b}{x^4}} x^2}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {342, 281, 283, 223, 212} \[ \int \sqrt {a+\frac {b}{x^4}} x \, dx=\frac {1}{2} x^2 \sqrt {a+\frac {b}{x^4}}-\frac {1}{2} \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b}}{x^2 \sqrt {a+\frac {b}{x^4}}}\right ) \]

[In]

Int[Sqrt[a + b/x^4]*x,x]

[Out]

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

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]

Rule 283

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

Rule 342

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Subst[Int[(a + b/x^n)^p/x^(m + 2), x], x, 1/x] /;
FreeQ[{a, b, p}, x] && ILtQ[n, 0] && IntegerQ[m]

Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {\sqrt {a+b x^4}}{x^3} \, dx,x,\frac {1}{x}\right ) \\ & = -\left (\frac {1}{2} \text {Subst}\left (\int \frac {\sqrt {a+b x^2}}{x^2} \, dx,x,\frac {1}{x^2}\right )\right ) \\ & = \frac {1}{2} \sqrt {a+\frac {b}{x^4}} x^2-\frac {1}{2} b \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\frac {1}{x^2}\right ) \\ & = \frac {1}{2} \sqrt {a+\frac {b}{x^4}} x^2-\frac {1}{2} b \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {1}{\sqrt {a+\frac {b}{x^4}} x^2}\right ) \\ & = \frac {1}{2} \sqrt {a+\frac {b}{x^4}} x^2-\frac {1}{2} \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b}}{\sqrt {a+\frac {b}{x^4}} x^2}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.35 \[ \int \sqrt {a+\frac {b}{x^4}} x \, dx=\frac {\sqrt {a+\frac {b}{x^4}} x^2 \left (\sqrt {b+a x^4}-\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b+a x^4}}{\sqrt {b}}\right )\right )}{2 \sqrt {b+a x^4}} \]

[In]

Integrate[Sqrt[a + b/x^4]*x,x]

[Out]

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

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.33

method	result	size
default	\(-\frac {\sqrt {\frac {a \,x^{4}+b}{x^{4}}}\, x^{2} \left (\sqrt {b}\, \ln \left (\frac {2 b +2 \sqrt {b}\, \sqrt {a \,x^{4}+b}}{x^{2}}\right )-\sqrt {a \,x^{4}+b}\right )}{2 \sqrt {a \,x^{4}+b}}\)	\(65\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 112, normalized size of antiderivative = 2.29 \[ \int \sqrt {a+\frac {b}{x^4}} x \, dx=\left [\frac {1}{2} \, x^{2} \sqrt {\frac {a x^{4} + b}{x^{4}}} + \frac {1}{4} \, \sqrt {b} \log \left (\frac {a x^{4} - 2 \, \sqrt {b} x^{2} \sqrt {\frac {a x^{4} + b}{x^{4}}} + 2 \, b}{x^{4}}\right ), \frac {1}{2} \, x^{2} \sqrt {\frac {a x^{4} + b}{x^{4}}} + \frac {1}{2} \, \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x^{2} \sqrt {\frac {a x^{4} + b}{x^{4}}}}{b}\right )\right ] \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.89 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.35 \[ \int \sqrt {a+\frac {b}{x^4}} x \, dx=\frac {\sqrt {a} x^{2}}{2 \sqrt {1 + \frac {b}{a x^{4}}}} - \frac {\sqrt {b} \operatorname {asinh}{\left (\frac {\sqrt {b}}{\sqrt {a} x^{2}} \right )}}{2} + \frac {b}{2 \sqrt {a} x^{2} \sqrt {1 + \frac {b}{a x^{4}}}} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.22 \[ \int \sqrt {a+\frac {b}{x^4}} x \, dx=\frac {1}{2} \, \sqrt {a + \frac {b}{x^{4}}} x^{2} + \frac {1}{4} \, \sqrt {b} \log \left (\frac {\sqrt {a + \frac {b}{x^{4}}} x^{2} - \sqrt {b}}{\sqrt {a + \frac {b}{x^{4}}} x^{2} + \sqrt {b}}\right ) \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.73 \[ \int \sqrt {a+\frac {b}{x^4}} x \, dx=\frac {b \arctan \left (\frac {\sqrt {a x^{4} + b}}{\sqrt {-b}}\right )}{2 \, \sqrt {-b}} + \frac {1}{2} \, \sqrt {a x^{4} + b} \]

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int(x*(a + b/x^4)^(1/2),x)

[Out]

int(x*(a + b/x^4)^(1/2), x)