Integrand size = 48, antiderivative size = 186 \[ \int \frac {\left (-b+a^2 x^2\right ) \sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\sqrt {b+a^2 x^4}} \, dx=\frac {1}{2} a x \sqrt {a x^2+\sqrt {b+a^2 x^4}}-\frac {\sqrt {a} \sqrt {b} \arctan \left (\frac {a x^2}{\sqrt {b}}+\frac {\sqrt {b+a^2 x^4}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt {a} x \sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\sqrt {b}}\right )}{\sqrt {2}}-\frac {b \log \left (a x^2+\sqrt {b+a^2 x^4}+\sqrt {2} \sqrt {a} x \sqrt {a x^2+\sqrt {b+a^2 x^4}}\right )}{\sqrt {2} \sqrt {a}} \]
1/2*a*x*(a*x^2+(a^2*x^4+b)^(1/2))^(1/2)-1/2*a^(1/2)*b^(1/2)*arctan(a*x^2/b ^(1/2)+(a^2*x^4+b)^(1/2)/b^(1/2)+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)-1/2*b*ln(a*x^2+(a^2*x^4+b)^(1/2)+2^(1/2)*a^(1/2) *x*(a*x^2+(a^2*x^4+b)^(1/2))^(1/2))*2^(1/2)/a^(1/2)
Time = 0.44 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.95 \[ \int \frac {\left (-b+a^2 x^2\right ) \sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\sqrt {b+a^2 x^4}} \, dx=\frac {a^{3/2} x \sqrt {a x^2+\sqrt {b+a^2 x^4}}+\sqrt {2} a \sqrt {b} \arctan \left (\frac {a x^2+\sqrt {b+a^2 x^4}-\sqrt {2} \sqrt {a} x \sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\sqrt {b}}\right )+\sqrt {2} b \log \left (a x^2+\sqrt {b+a^2 x^4}-\sqrt {2} \sqrt {a} x \sqrt {a x^2+\sqrt {b+a^2 x^4}}\right )}{2 \sqrt {a}} \]
(a^(3/2)*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]] + Sqrt[2]*a*Sqrt[b]*ArcTan[(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[b]] + Sqrt[2]*b*Log[a*x^2 + Sqrt[b + a^2*x^4] - Sqrt[2]*Sqrt[a]*x*S qrt[a*x^2 + Sqrt[b + a^2*x^4]]])/(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.
\(\displaystyle \int \frac {\left (a^2 x^2-b\right ) \sqrt {\sqrt {a^2 x^4+b}+a x^2}}{\sqrt {a^2 x^4+b}} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {a^2 x^2 \sqrt {\sqrt {a^2 x^4+b}+a x^2}}{\sqrt {a^2 x^4+b}}-\frac {b \sqrt {\sqrt {a^2 x^4+b}+a x^2}}{\sqrt {a^2 x^4+b}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle a^2 \int \frac {x^2 \sqrt {a x^2+\sqrt {a^2 x^4+b}}}{\sqrt {a^2 x^4+b}}dx-\frac {b \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} x}{\sqrt {\sqrt {a^2 x^4+b}+a x^2}}\right )}{\sqrt {2} \sqrt {a}}\) |
3.24.52.3.1 Defintions of rubi rules used
\[\int \frac {\left (a^{2} x^{2}-b \right ) \sqrt {a \,x^{2}+\sqrt {a^{2} x^{4}+b}}}{\sqrt {a^{2} x^{4}+b}}d x\]
Timed out. \[ \int \frac {\left (-b+a^2 x^2\right ) \sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\sqrt {b+a^2 x^4}} \, dx=\text {Timed out} \]
integrate((a^2*x^2-b)*(a*x^2+(a^2*x^4+b)^(1/2))^(1/2)/(a^2*x^4+b)^(1/2),x, algorithm="fricas")
\[ \int \frac {\left (-b+a^2 x^2\right ) \sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\sqrt {b+a^2 x^4}} \, dx=\int \frac {\sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}} \left (a^{2} x^{2} - b\right )}{\sqrt {a^{2} x^{4} + b}}\, dx \]
\[ \int \frac {\left (-b+a^2 x^2\right ) \sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\sqrt {b+a^2 x^4}} \, dx=\int { \frac {{\left (a^{2} x^{2} - b\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}}}{\sqrt {a^{2} x^{4} + b}} \,d x } \]
integrate((a^2*x^2-b)*(a*x^2+(a^2*x^4+b)^(1/2))^(1/2)/(a^2*x^4+b)^(1/2),x, algorithm="maxima")
\[ \int \frac {\left (-b+a^2 x^2\right ) \sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\sqrt {b+a^2 x^4}} \, dx=\int { \frac {{\left (a^{2} x^{2} - b\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}}}{\sqrt {a^{2} x^{4} + b}} \,d x } \]
Timed out. \[ \int \frac {\left (-b+a^2 x^2\right ) \sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\sqrt {b+a^2 x^4}} \, dx=\int -\frac {\sqrt {\sqrt {a^2\,x^4+b}+a\,x^2}\,\left (b-a^2\,x^2\right )}{\sqrt {a^2\,x^4+b}} \,d x \]