Integrand size = 48, antiderivative size = 151 \[ \int \frac {\left (-b+a^2 x^4\right ) \sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\sqrt {b+a^2 x^4}} \, dx=\frac {3 \sqrt {a} b x^2+2 a^{5/2} x^6+2 a^{3/2} x^4 \sqrt {b+a^2 x^4}}{8 \sqrt {a} x \sqrt {a x^2+\sqrt {b+a^2 x^4}}}-\frac {11 b \log \left (a x^2+\sqrt {b+a^2 x^4}+\sqrt {2} \sqrt {a} x \sqrt {a x^2+\sqrt {b+a^2 x^4}}\right )}{8 \sqrt {2} \sqrt {a}} \]
1/8*(3*a^(1/2)*b*x^2+2*a^(5/2)*x^6+2*a^(3/2)*x^4*(a^2*x^4+b)^(1/2))/a^(1/2 )/x/(a*x^2+(a^2*x^4+b)^(1/2))^(1/2)-11/16*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 = 0.77 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.85 \[ \int \frac {\left (-b+a^2 x^4\right ) \sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\sqrt {b+a^2 x^4}} \, dx=\frac {3 b x+2 a x^3 \left (a x^2+\sqrt {b+a^2 x^4}\right )}{8 \sqrt {a x^2+\sqrt {b+a^2 x^4}}}-\frac {11 b \log \left (a x^2+\sqrt {b+a^2 x^4}+\sqrt {2} \sqrt {a} x \sqrt {a x^2+\sqrt {b+a^2 x^4}}\right )}{8 \sqrt {2} \sqrt {a}} \]
(3*b*x + 2*a*x^3*(a*x^2 + Sqrt[b + a^2*x^4]))/(8*Sqrt[a*x^2 + Sqrt[b + a^2 *x^4]]) - (11*b*Log[a*x^2 + Sqrt[b + a^2*x^4] + Sqrt[2]*Sqrt[a]*x*Sqrt[a*x ^2 + Sqrt[b + a^2*x^4]]])/(8*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.
\(\displaystyle \int \frac {\left (a^2 x^4-b\right ) \sqrt {\sqrt {a^2 x^4+b}+a x^2}}{\sqrt {a^2 x^4+b}} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {a^2 x^4 \sqrt {\sqrt {a^2 x^4+b}+a x^2}}{\sqrt {a^2 x^4+b}}-\frac {b \sqrt {\sqrt {a^2 x^4+b}+a x^2}}{\sqrt {a^2 x^4+b}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle a^2 \int \frac {x^4 \sqrt {a x^2+\sqrt {a^2 x^4+b}}}{\sqrt {a^2 x^4+b}}dx-\frac {b \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} x}{\sqrt {\sqrt {a^2 x^4+b}+a x^2}}\right )}{\sqrt {2} \sqrt {a}}\) |
3.21.94.3.1 Defintions of rubi rules used
\[\int \frac {\left (a^{2} x^{4}-b \right ) \sqrt {a \,x^{2}+\sqrt {a^{2} x^{4}+b}}}{\sqrt {a^{2} x^{4}+b}}d x\]
Time = 1.49 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.56 \[ \int \frac {\left (-b+a^2 x^4\right ) \sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\sqrt {b+a^2 x^4}} \, dx=\left [\frac {11 \, \sqrt {2} \sqrt {a} b \log \left (4 \, a^{2} x^{4} + 4 \, \sqrt {a^{2} x^{4} + b} a x^{2} - 2 \, {\left (\sqrt {2} a^{\frac {3}{2}} x^{3} + \sqrt {2} \sqrt {a^{2} x^{4} + b} \sqrt {a} x\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}} + b\right ) - 4 \, {\left (a^{2} x^{3} - 3 \, \sqrt {a^{2} x^{4} + b} a x\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}}}{32 \, a}, \frac {11 \, \sqrt {2} \sqrt {-a} b \arctan \left (\frac {\sqrt {2} \sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}} \sqrt {-a}}{2 \, a x}\right ) - 2 \, {\left (a^{2} x^{3} - 3 \, \sqrt {a^{2} x^{4} + b} a x\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}}}{16 \, a}\right ] \]
integrate((a^2*x^4-b)*(a*x^2+(a^2*x^4+b)^(1/2))^(1/2)/(a^2*x^4+b)^(1/2),x, algorithm="fricas")
[1/32*(11*sqrt(2)*sqrt(a)*b*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 + s qrt(a^2*x^4 + b)) + b) - 4*(a^2*x^3 - 3*sqrt(a^2*x^4 + b)*a*x)*sqrt(a*x^2 + sqrt(a^2*x^4 + b)))/a, 1/16*(11*sqrt(2)*sqrt(-a)*b*arctan(1/2*sqrt(2)*sq rt(a*x^2 + sqrt(a^2*x^4 + b))*sqrt(-a)/(a*x)) - 2*(a^2*x^3 - 3*sqrt(a^2*x^ 4 + b)*a*x)*sqrt(a*x^2 + sqrt(a^2*x^4 + b)))/a]
\[ \int \frac {\left (-b+a^2 x^4\right ) \sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\sqrt {b+a^2 x^4}} \, dx=\int \frac {\sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}} \left (a^{2} x^{4} - b\right )}{\sqrt {a^{2} x^{4} + b}}\, dx \]
\[ \int \frac {\left (-b+a^2 x^4\right ) \sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\sqrt {b+a^2 x^4}} \, dx=\int { \frac {{\left (a^{2} x^{4} - b\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}}}{\sqrt {a^{2} x^{4} + b}} \,d x } \]
integrate((a^2*x^4-b)*(a*x^2+(a^2*x^4+b)^(1/2))^(1/2)/(a^2*x^4+b)^(1/2),x, algorithm="maxima")
\[ \int \frac {\left (-b+a^2 x^4\right ) \sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\sqrt {b+a^2 x^4}} \, dx=\int { \frac {{\left (a^{2} x^{4} - b\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}}}{\sqrt {a^{2} x^{4} + b}} \,d x } \]
Timed out. \[ \int \frac {\left (-b+a^2 x^4\right ) \sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\sqrt {b+a^2 x^4}} \, dx=\int -\frac {\sqrt {\sqrt {a^2\,x^4+b}+a\,x^2}\,\left (b-a^2\,x^4\right )}{\sqrt {a^2\,x^4+b}} \,d x \]