Integrand size = 23, antiderivative size = 89 \[ \int \sqrt {a x^2+\sqrt {b+a^2 x^4}} \, dx=\frac {1}{2} x \sqrt {a x^2+\sqrt {b+a^2 x^4}}+\frac {\sqrt {b} \arctan \left (\frac {\sqrt {2} \sqrt {a} x \sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\sqrt {b}}\right )}{2 \sqrt {2} \sqrt {a}} \] Output:
1/2*x*(a*x^2+(a^2*x^4+b)^(1/2))^(1/2)+1/4*b^(1/2)*arctan(2^(1/2)*a^(1/2)*x *(a*x^2+(a^2*x^4+b)^(1/2))^(1/2)/b^(1/2))*2^(1/2)/a^(1/2)
Time = 0.14 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00 \[ \int \sqrt {a x^2+\sqrt {b+a^2 x^4}} \, dx=\frac {1}{2} x \sqrt {a x^2+\sqrt {b+a^2 x^4}}+\frac {\sqrt {b} \arctan \left (\frac {\sqrt {2} \sqrt {a} x \sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\sqrt {b}}\right )}{2 \sqrt {2} \sqrt {a}} \] Input:
Integrate[Sqrt[a*x^2 + Sqrt[b + a^2*x^4]],x]
Output:
(x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]])/2 + (Sqrt[b]*ArcTan[(Sqrt[2]*Sqrt[a]*x *Sqrt[a*x^2 + Sqrt[b + a^2*x^4]])/Sqrt[b]])/(2*Sqrt[2]*Sqrt[a])
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.
\(\displaystyle \int \sqrt {\sqrt {a^2 x^4+b}+a x^2} \, dx\) |
\(\Big \downarrow \) 7299 |
\(\displaystyle \int \sqrt {\sqrt {a^2 x^4+b}+a x^2}dx\) |
Input:
Int[Sqrt[a*x^2 + Sqrt[b + a^2*x^4]],x]
Output:
$Aborted
\[\int \sqrt {a \,x^{2}+\sqrt {a^{2} x^{4}+b}}d x\]
Input:
int((a*x^2+(a^2*x^4+b)^(1/2))^(1/2),x)
Output:
int((a*x^2+(a^2*x^4+b)^(1/2))^(1/2),x)
Time = 2.17 (sec) , antiderivative size = 249, normalized size of antiderivative = 2.80 \[ \int \sqrt {a x^2+\sqrt {b+a^2 x^4}} \, dx=\left [\frac {1}{8} \, \sqrt {2} \sqrt {-\frac {b}{a}} \log \left (4 \, a^{2} b x^{4} - 4 \, \sqrt {a^{2} x^{4} + b} a b x^{2} + b^{2} + 2 \, {\left (2 \, \sqrt {2} \sqrt {a^{2} x^{4} + b} a^{2} x^{3} \sqrt {-\frac {b}{a}} - \sqrt {2} {\left (2 \, a^{3} x^{5} + a b x\right )} \sqrt {-\frac {b}{a}}\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}}\right ) + \frac {1}{2} \, \sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}} x, -\frac {1}{4} \, \sqrt {2} \sqrt {\frac {b}{a}} \arctan \left (-\frac {{\left (\sqrt {2} a x^{2} \sqrt {\frac {b}{a}} - \sqrt {2} \sqrt {a^{2} x^{4} + b} \sqrt {\frac {b}{a}}\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}}}{2 \, b x}\right ) + \frac {1}{2} \, \sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}} x\right ] \] Input:
integrate((a*x^2+(a^2*x^4+b)^(1/2))^(1/2),x, algorithm="fricas")
Output:
[1/8*sqrt(2)*sqrt(-b/a)*log(4*a^2*b*x^4 - 4*sqrt(a^2*x^4 + b)*a*b*x^2 + b^ 2 + 2*(2*sqrt(2)*sqrt(a^2*x^4 + b)*a^2*x^3*sqrt(-b/a) - sqrt(2)*(2*a^3*x^5 + a*b*x)*sqrt(-b/a))*sqrt(a*x^2 + sqrt(a^2*x^4 + b))) + 1/2*sqrt(a*x^2 + sqrt(a^2*x^4 + b))*x, -1/4*sqrt(2)*sqrt(b/a)*arctan(-1/2*(sqrt(2)*a*x^2*sq rt(b/a) - sqrt(2)*sqrt(a^2*x^4 + b)*sqrt(b/a))*sqrt(a*x^2 + sqrt(a^2*x^4 + b))/(b*x)) + 1/2*sqrt(a*x^2 + sqrt(a^2*x^4 + b))*x]
\[ \int \sqrt {a x^2+\sqrt {b+a^2 x^4}} \, dx=\int \sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}}\, dx \] Input:
integrate((a*x**2+(a**2*x**4+b)**(1/2))**(1/2),x)
Output:
Integral(sqrt(a*x**2 + sqrt(a**2*x**4 + b)), x)
\[ \int \sqrt {a x^2+\sqrt {b+a^2 x^4}} \, dx=\int { \sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}} \,d x } \] Input:
integrate((a*x^2+(a^2*x^4+b)^(1/2))^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(a*x^2 + sqrt(a^2*x^4 + b)), x)
\[ \int \sqrt {a x^2+\sqrt {b+a^2 x^4}} \, dx=\int { \sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}} \,d x } \] Input:
integrate((a*x^2+(a^2*x^4+b)^(1/2))^(1/2),x, algorithm="giac")
Output:
integrate(sqrt(a*x^2 + sqrt(a^2*x^4 + b)), x)
Timed out. \[ \int \sqrt {a x^2+\sqrt {b+a^2 x^4}} \, dx=\int \sqrt {\sqrt {a^2\,x^4+b}+a\,x^2} \,d x \] Input:
int(((b + a^2*x^4)^(1/2) + a*x^2)^(1/2),x)
Output:
int(((b + a^2*x^4)^(1/2) + a*x^2)^(1/2), x)
\[ \int \sqrt {a x^2+\sqrt {b+a^2 x^4}} \, dx=\int \sqrt {\sqrt {a^{2} x^{4}+b}+a \,x^{2}}d x \] Input:
int((a*x^2+(a^2*x^4+b)^(1/2))^(1/2),x)
Output:
int(sqrt(sqrt(a**2*x**4 + b) + a*x**2),x)