\(\int \frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\sqrt {b+a^2 x^4}} \, dx\) [1097]

Optimal result

Integrand size = 37, antiderivative size = 81 \[ \int \frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\sqrt {b+a^2 x^4}} \, dx=\frac {\log \left (i a^{3/2} x^2+i \sqrt {a} \sqrt {b+a^2 x^4}+i \sqrt {2} a x \sqrt {a x^2+\sqrt {b+a^2 x^4}}\right )}{\sqrt {2} \sqrt {a}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.58, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, Rules used = {2157, 212} \[ \int \frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\sqrt {b+a^2 x^4}} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} x}{\sqrt {\sqrt {a^2 x^4+b}+a x^2}}\right )}{\sqrt {2} \sqrt {a}} \]

[In]

Int[Sqrt[a*x^2 + Sqrt[b + a^2*x^4]]/Sqrt[b + a^2*x^4],x]

[Out]

ArcTanh[(Sqrt[2]*Sqrt[a]*x)/Sqrt[a*x^2 + Sqrt[b + a^2*x^4]]]/(Sqrt[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 2157

Int[Sqrt[(c_.)*(x_)^2 + (d_.)*Sqrt[(a_) + (b_.)*(x_)^4]]/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Dist[d, Subst
[Int[1/(1 - 2*c*x^2), x], x, x/Sqrt[c*x^2 + d*Sqrt[a + b*x^4]]], x] /; FreeQ[{a, b, c, d}, x] && EqQ[c^2 - b*d
^2, 0]

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

Mathematica [A] (verified)

Time = 0.31 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.93 \[ \int \frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\sqrt {b+a^2 x^4}} \, dx=\frac {\log \left (i \sqrt {a} \left (a x^2+\sqrt {b+a^2 x^4}+\sqrt {2} \sqrt {a} x \sqrt {a x^2+\sqrt {b+a^2 x^4}}\right )\right )}{\sqrt {2} \sqrt {a}} \]

[In]

Integrate[Sqrt[a*x^2 + Sqrt[b + a^2*x^4]]/Sqrt[b + a^2*x^4],x]

[Out]

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

Maple [F]

\[\int \frac {\sqrt {a \,x^{2}+\sqrt {x^{4} a^{2}+b}}}{\sqrt {x^{4} a^{2}+b}}d x\]

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 1.19 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.67 \[ \int \frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\sqrt {b+a^2 x^4}} \, dx=\left [\frac {\sqrt {2} \log \left (4 \, a^{2} x^{4} + 4 \, \sqrt {a^{2} x^{4} + b} a x^{2} + 2 \, {\left (\sqrt {2} a^{\frac {3}{2}} x^{3} + \sqrt {2} \sqrt {a^{2} x^{4} + b} \sqrt {a} x\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}} + b\right )}{4 \, \sqrt {a}}, -\frac {1}{2} \, \sqrt {2} \sqrt {-\frac {1}{a}} \arctan \left (\frac {\sqrt {2} \sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}} \sqrt {-\frac {1}{a}}}{2 \, x}\right )\right ] \]

[In]

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

[Out]

[1/4*sqrt(2)*log(4*a^2*x^4 + 4*sqrt(a^2*x^4 + b)*a*x^2 + 2*(sqrt(2)*a^(3/2)*x^3 + sqrt(2)*sqrt(a^2*x^4 + b)*sq
rt(a)*x)*sqrt(a*x^2 + sqrt(a^2*x^4 + b)) + b)/sqrt(a), -1/2*sqrt(2)*sqrt(-1/a)*arctan(1/2*sqrt(2)*sqrt(a*x^2 +
sqrt(a^2*x^4 + b))*sqrt(-1/a)/x)]

Sympy [F]

\[ \int \frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\sqrt {b+a^2 x^4}} \, dx=\int \frac {\sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}}}{\sqrt {a^{2} x^{4} + b}}\, dx \]

[In]

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

[Out]

Integral(sqrt(a*x**2 + sqrt(a**2*x**4 + b))/sqrt(a**2*x**4 + b), x)

Maxima [F]

\[ \int \frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\sqrt {b+a^2 x^4}} \, dx=\int { \frac {\sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}}}{\sqrt {a^{2} x^{4} + b}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\sqrt {b+a^2 x^4}} \, dx=\int { \frac {\sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}}}{\sqrt {a^{2} x^{4} + b}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\sqrt {b+a^2 x^4}} \, dx=\int \frac {\sqrt {\sqrt {a^2\,x^4+b}+a\,x^2}}{\sqrt {a^2\,x^4+b}} \,d x \]

[In]

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

[Out]

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