Integrand size = 37, antiderivative size = 81 \[ \int \frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\sqrt {b+a^2 x^4}} \, dx=\frac {\log \left (i a^{3/2} x^2+i \sqrt {a} \sqrt {b+a^2 x^4}+i \sqrt {2} a x \sqrt {a x^2+\sqrt {b+a^2 x^4}}\right )}{\sqrt {2} \sqrt {a}} \] Output:
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)
Time = 0.34 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.93 \[ \int \frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\sqrt {b+a^2 x^4}} \, dx=\frac {\log \left (i \sqrt {a} \left (a x^2+\sqrt {b+a^2 x^4}+\sqrt {2} \sqrt {a} x \sqrt {a x^2+\sqrt {b+a^2 x^4}}\right )\right )}{\sqrt {2} \sqrt {a}} \] Input:
Integrate[Sqrt[a*x^2 + Sqrt[b + a^2*x^4]]/Sqrt[b + a^2*x^4],x]
Output:
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])
Time = 0.26 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.58, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, Rules used = {2557, 219}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\sqrt {\sqrt {a^2 x^4+b}+a x^2}}{\sqrt {a^2 x^4+b}} \, dx\) |
\(\Big \downarrow \) 2557 |
\(\displaystyle \int \frac {1}{1-\frac {2 a x^2}{\sqrt {a^2 x^4+b}+a x^2}}d\frac {x}{\sqrt {\sqrt {a^2 x^4+b}+a x^2}}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} x}{\sqrt {\sqrt {a^2 x^4+b}+a x^2}}\right )}{\sqrt {2} \sqrt {a}}\) |
Input:
Int[Sqrt[a*x^2 + Sqrt[b + a^2*x^4]]/Sqrt[b + a^2*x^4],x]
Output:
ArcTanh[(Sqrt[2]*Sqrt[a]*x)/Sqrt[a*x^2 + Sqrt[b + a^2*x^4]]]/(Sqrt[2]*Sqrt [a])
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] && (Gt Q[a, 0] || LtQ[b, 0])
Int[Sqrt[(c_.)*(x_)^2 + (d_.)*Sqrt[(a_) + (b_.)*(x_)^4]]/Sqrt[(a_) + (b_.)* (x_)^4], x_Symbol] :> Simp[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]
\[\int \frac {\sqrt {a \,x^{2}+\sqrt {a^{2} x^{4}+b}}}{\sqrt {a^{2} x^{4}+b}}d x\]
Input:
int((a*x^2+(a^2*x^4+b)^(1/2))^(1/2)/(a^2*x^4+b)^(1/2),x)
Output:
int((a*x^2+(a^2*x^4+b)^(1/2))^(1/2)/(a^2*x^4+b)^(1/2),x)
Time = 1.13 (sec) , antiderivative size = 168, normalized size of antiderivative = 2.07 \[ \int \frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\sqrt {b+a^2 x^4}} \, dx=\left [\frac {\sqrt {2} \log \left (4 \, a^{2} x^{4} + 4 \, \sqrt {a^{2} x^{4} + b} a x^{2} + 2 \, {\left (\sqrt {2} a^{\frac {3}{2}} x^{3} + \sqrt {2} \sqrt {a^{2} x^{4} + b} \sqrt {a} x\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}} + b\right )}{4 \, \sqrt {a}}, -\frac {1}{2} \, \sqrt {2} \sqrt {-\frac {1}{a}} \arctan \left (-\frac {{\left (\sqrt {2} a^{2} x^{3} \sqrt {-\frac {1}{a}} - \sqrt {2} \sqrt {a^{2} x^{4} + b} a x \sqrt {-\frac {1}{a}}\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}}}{b}\right )\right ] \] Input:
integrate((a*x^2+(a^2*x^4+b)^(1/2))^(1/2)/(a^2*x^4+b)^(1/2),x, algorithm=" fricas")
Output:
[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)*sqrt(a)*x)*sqrt(a*x^2 + sqrt(a^2*x^4 + b )) + b)/sqrt(a), -1/2*sqrt(2)*sqrt(-1/a)*arctan(-(sqrt(2)*a^2*x^3*sqrt(-1/ a) - sqrt(2)*sqrt(a^2*x^4 + b)*a*x*sqrt(-1/a))*sqrt(a*x^2 + sqrt(a^2*x^4 + b))/b)]
\[ \int \frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\sqrt {b+a^2 x^4}} \, dx=\int \frac {\sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}}}{\sqrt {a^{2} x^{4} + b}}\, dx \] Input:
integrate((a*x**2+(a**2*x**4+b)**(1/2))**(1/2)/(a**2*x**4+b)**(1/2),x)
Output:
Integral(sqrt(a*x**2 + sqrt(a**2*x**4 + b))/sqrt(a**2*x**4 + b), x)
\[ \int \frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\sqrt {b+a^2 x^4}} \, dx=\int { \frac {\sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}}}{\sqrt {a^{2} x^{4} + b}} \,d x } \] Input:
integrate((a*x^2+(a^2*x^4+b)^(1/2))^(1/2)/(a^2*x^4+b)^(1/2),x, algorithm=" maxima")
Output:
integrate(sqrt(a*x^2 + sqrt(a^2*x^4 + b))/sqrt(a^2*x^4 + b), x)
\[ \int \frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\sqrt {b+a^2 x^4}} \, dx=\int { \frac {\sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}}}{\sqrt {a^{2} x^{4} + b}} \,d x } \] Input:
integrate((a*x^2+(a^2*x^4+b)^(1/2))^(1/2)/(a^2*x^4+b)^(1/2),x, algorithm=" giac")
Output:
integrate(sqrt(a*x^2 + sqrt(a^2*x^4 + b))/sqrt(a^2*x^4 + b), x)
Timed out. \[ \int \frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\sqrt {b+a^2 x^4}} \, dx=\int \frac {\sqrt {\sqrt {a^2\,x^4+b}+a\,x^2}}{\sqrt {a^2\,x^4+b}} \,d x \] Input:
int(((b + a^2*x^4)^(1/2) + a*x^2)^(1/2)/(b + a^2*x^4)^(1/2),x)
Output:
int(((b + a^2*x^4)^(1/2) + a*x^2)^(1/2)/(b + a^2*x^4)^(1/2), x)
Time = 0.16 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.78 \[ \int \frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\sqrt {b+a^2 x^4}} \, dx=\frac {\sqrt {a}\, \sqrt {2}\, \left (-\mathrm {log}\left (\sqrt {\sqrt {a^{2} x^{4}+b}+a \,x^{2}}-\sqrt {a}\, \sqrt {2}\, x \right )+\mathrm {log}\left (\sqrt {\sqrt {a^{2} x^{4}+b}+a \,x^{2}}+\sqrt {a}\, \sqrt {2}\, x \right )\right )}{4 a} \] Input:
int((a*x^2+(a^2*x^4+b)^(1/2))^(1/2)/(a^2*x^4+b)^(1/2),x)
Output:
(sqrt(a)*sqrt(2)*( - log(sqrt(sqrt(a**2*x**4 + b) + a*x**2) - sqrt(a)*sqrt (2)*x) + log(sqrt(sqrt(a**2*x**4 + b) + a*x**2) + sqrt(a)*sqrt(2)*x)))/(4* a)