Integrand size = 36, antiderivative size = 198 \[ \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{x^6 \sqrt {b^2+a x^2}} \, dx=\frac {\sqrt {b+\sqrt {b^2+a x^2}} \left (-315 a^{5/2} x^4+\sqrt {a} \left (16 b^4-144 b^3 \sqrt {b^2+a x^2}\right )+a^{3/2} \left (-42 b^2 x^2+210 b x^2 \sqrt {b^2+a x^2}\right )\right )}{640 \sqrt {a} b^5 x^5}-\frac {63 a^{5/2} \arctan \left (\frac {\sqrt {a} x}{\sqrt {2} \sqrt {b} \sqrt {b+\sqrt {b^2+a x^2}}}-\frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\sqrt {2} \sqrt {b}}\right )}{64 \sqrt {2} b^{11/2}} \]
1/640*(b+(a*x^2+b^2)^(1/2))^(1/2)*(-315*a^(5/2)*x^4+a^(1/2)*(16*b^4-144*b^ 3*(a*x^2+b^2)^(1/2))+a^(3/2)*(-42*b^2*x^2+210*b*x^2*(a*x^2+b^2)^(1/2)))/a^ (1/2)/b^5/x^5-63/128*a^(5/2)*arctan(1/2*a^(1/2)*x*2^(1/2)/b^(1/2)/(b+(a*x^ 2+b^2)^(1/2))^(1/2)-1/2*(b+(a*x^2+b^2)^(1/2))^(1/2)*2^(1/2)/b^(1/2))*2^(1/ 2)/b^(11/2)
Time = 0.32 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.74 \[ \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{x^6 \sqrt {b^2+a x^2}} \, dx=-\frac {\sqrt {b+\sqrt {b^2+a x^2}} \left (-16 b^4+42 a b^2 x^2+315 a^2 x^4+144 b^3 \sqrt {b^2+a x^2}-210 a b x^2 \sqrt {b^2+a x^2}\right )}{640 b^5 x^5}-\frac {63 a^{5/2} \arctan \left (\frac {\sqrt {a} x}{\sqrt {2} \sqrt {b} \sqrt {b+\sqrt {b^2+a x^2}}}\right )}{128 \sqrt {2} b^{11/2}} \]
-1/640*(Sqrt[b + Sqrt[b^2 + a*x^2]]*(-16*b^4 + 42*a*b^2*x^2 + 315*a^2*x^4 + 144*b^3*Sqrt[b^2 + a*x^2] - 210*a*b*x^2*Sqrt[b^2 + a*x^2]))/(b^5*x^5) - (63*a^(5/2)*ArcTan[(Sqrt[a]*x)/(Sqrt[2]*Sqrt[b]*Sqrt[b + Sqrt[b^2 + a*x^2] ])])/(128*Sqrt[2]*b^(11/2))
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.
\(\displaystyle \int \frac {\sqrt {\sqrt {a x^2+b^2}+b}}{x^6 \sqrt {a x^2+b^2}} \, dx\) |
\(\Big \downarrow \) 7299 |
\(\displaystyle \int \frac {\sqrt {\sqrt {a x^2+b^2}+b}}{x^6 \sqrt {a x^2+b^2}}dx\) |
3.25.47.3.1 Defintions of rubi rules used
\[\int \frac {\sqrt {b +\sqrt {a \,x^{2}+b^{2}}}}{x^{6} \sqrt {a \,x^{2}+b^{2}}}d x\]
Time = 211.13 (sec) , antiderivative size = 330, normalized size of antiderivative = 1.67 \[ \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{x^6 \sqrt {b^2+a x^2}} \, dx=\left [\frac {315 \, \sqrt {\frac {1}{2}} a^{2} x^{5} \sqrt {-\frac {a}{b}} \log \left (-\frac {a^{2} x^{3} + 4 \, a b^{2} x - 4 \, \sqrt {a x^{2} + b^{2}} a b x + 4 \, {\left (2 \, \sqrt {\frac {1}{2}} \sqrt {a x^{2} + b^{2}} b^{2} \sqrt {-\frac {a}{b}} - \sqrt {\frac {1}{2}} {\left (a b x^{2} + 2 \, b^{3}\right )} \sqrt {-\frac {a}{b}}\right )} \sqrt {b + \sqrt {a x^{2} + b^{2}}}}{x^{3}}\right ) - 2 \, {\left (315 \, a^{2} x^{4} + 42 \, a b^{2} x^{2} - 16 \, b^{4} - 6 \, {\left (35 \, a b x^{2} - 24 \, b^{3}\right )} \sqrt {a x^{2} + b^{2}}\right )} \sqrt {b + \sqrt {a x^{2} + b^{2}}}}{1280 \, b^{5} x^{5}}, \frac {315 \, \sqrt {\frac {1}{2}} a^{2} x^{5} \sqrt {\frac {a}{b}} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {b + \sqrt {a x^{2} + b^{2}}} b \sqrt {\frac {a}{b}}}{a x}\right ) - {\left (315 \, a^{2} x^{4} + 42 \, a b^{2} x^{2} - 16 \, b^{4} - 6 \, {\left (35 \, a b x^{2} - 24 \, b^{3}\right )} \sqrt {a x^{2} + b^{2}}\right )} \sqrt {b + \sqrt {a x^{2} + b^{2}}}}{640 \, b^{5} x^{5}}\right ] \]
[1/1280*(315*sqrt(1/2)*a^2*x^5*sqrt(-a/b)*log(-(a^2*x^3 + 4*a*b^2*x - 4*sq rt(a*x^2 + b^2)*a*b*x + 4*(2*sqrt(1/2)*sqrt(a*x^2 + b^2)*b^2*sqrt(-a/b) - sqrt(1/2)*(a*b*x^2 + 2*b^3)*sqrt(-a/b))*sqrt(b + sqrt(a*x^2 + b^2)))/x^3) - 2*(315*a^2*x^4 + 42*a*b^2*x^2 - 16*b^4 - 6*(35*a*b*x^2 - 24*b^3)*sqrt(a* x^2 + b^2))*sqrt(b + sqrt(a*x^2 + b^2)))/(b^5*x^5), 1/640*(315*sqrt(1/2)*a ^2*x^5*sqrt(a/b)*arctan(2*sqrt(1/2)*sqrt(b + sqrt(a*x^2 + b^2))*b*sqrt(a/b )/(a*x)) - (315*a^2*x^4 + 42*a*b^2*x^2 - 16*b^4 - 6*(35*a*b*x^2 - 24*b^3)* sqrt(a*x^2 + b^2))*sqrt(b + sqrt(a*x^2 + b^2)))/(b^5*x^5)]
Result contains complex when optimal does not.
Time = 1.25 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.25 \[ \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{x^6 \sqrt {b^2+a x^2}} \, dx=- \frac {\Gamma \left (\frac {1}{4}\right ) \Gamma \left (\frac {3}{4}\right ) {{}_{3}F_{2}\left (\begin {matrix} - \frac {5}{2}, \frac {1}{4}, \frac {3}{4} \\ - \frac {3}{2}, \frac {1}{2} \end {matrix}\middle | {\frac {a x^{2} e^{i \pi }}{b^{2}}} \right )}}{5 \pi \sqrt {b} x^{5}} \]
-gamma(1/4)*gamma(3/4)*hyper((-5/2, 1/4, 3/4), (-3/2, 1/2), a*x**2*exp_pol ar(I*pi)/b**2)/(5*pi*sqrt(b)*x**5)
\[ \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{x^6 \sqrt {b^2+a x^2}} \, dx=\int { \frac {\sqrt {b + \sqrt {a x^{2} + b^{2}}}}{\sqrt {a x^{2} + b^{2}} x^{6}} \,d x } \]
\[ \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{x^6 \sqrt {b^2+a x^2}} \, dx=\int { \frac {\sqrt {b + \sqrt {a x^{2} + b^{2}}}}{\sqrt {a x^{2} + b^{2}} x^{6}} \,d x } \]
Timed out. \[ \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{x^6 \sqrt {b^2+a x^2}} \, dx=\int \frac {\sqrt {b+\sqrt {b^2+a\,x^2}}}{x^6\,\sqrt {b^2+a\,x^2}} \,d x \]