Integrand size = 26, antiderivative size = 791 \[ \int \frac {d+e x^4}{d^2+b x^4+e^2 x^8} \, dx=-\frac {\arctan \left (\frac {\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {-b+2 d e}}-2 \sqrt {e} x}{\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {-b+2 d e}}}\right )}{4 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {-b+2 d e}}}-\frac {\arctan \left (\frac {\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {-b+2 d e}}-2 \sqrt {e} x}{\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {-b+2 d e}}}\right )}{4 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {-b+2 d e}}}+\frac {\arctan \left (\frac {\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {-b+2 d e}}+2 \sqrt {e} x}{\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {-b+2 d e}}}\right )}{4 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {-b+2 d e}}}+\frac {\arctan \left (\frac {\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {-b+2 d e}}+2 \sqrt {e} x}{\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {-b+2 d e}}}\right )}{4 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {-b+2 d e}}}-\frac {\log \left (\sqrt {d}-\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {-b+2 d e}} x+\sqrt {e} x^2\right )}{8 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {-b+2 d e}}}+\frac {\log \left (\sqrt {d}+\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {-b+2 d e}} x+\sqrt {e} x^2\right )}{8 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {-b+2 d e}}}-\frac {\log \left (\sqrt {d}-\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {-b+2 d e}} x+\sqrt {e} x^2\right )}{8 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {-b+2 d e}}}+\frac {\log \left (\sqrt {d}+\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {-b+2 d e}} x+\sqrt {e} x^2\right )}{8 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {-b+2 d e}}} \]
-1/4*arctan((-2*x*e^(1/2)+(2*d^(1/2)*e^(1/2)+(2*d*e-b)^(1/2))^(1/2))/(2*d^ (1/2)*e^(1/2)-(2*d*e-b)^(1/2))^(1/2))/d^(1/2)/(2*d^(1/2)*e^(1/2)-(2*d*e-b) ^(1/2))^(1/2)+1/4*arctan((2*x*e^(1/2)+(2*d^(1/2)*e^(1/2)+(2*d*e-b)^(1/2))^ (1/2))/(2*d^(1/2)*e^(1/2)-(2*d*e-b)^(1/2))^(1/2))/d^(1/2)/(2*d^(1/2)*e^(1/ 2)-(2*d*e-b)^(1/2))^(1/2)-1/8*ln(d^(1/2)+x^2*e^(1/2)-x*(2*d^(1/2)*e^(1/2)- (2*d*e-b)^(1/2))^(1/2))/d^(1/2)/(2*d^(1/2)*e^(1/2)-(2*d*e-b)^(1/2))^(1/2)+ 1/8*ln(d^(1/2)+x^2*e^(1/2)+x*(2*d^(1/2)*e^(1/2)-(2*d*e-b)^(1/2))^(1/2))/d^ (1/2)/(2*d^(1/2)*e^(1/2)-(2*d*e-b)^(1/2))^(1/2)-1/4*arctan((-2*x*e^(1/2)+( 2*d^(1/2)*e^(1/2)-(2*d*e-b)^(1/2))^(1/2))/(2*d^(1/2)*e^(1/2)+(2*d*e-b)^(1/ 2))^(1/2))/d^(1/2)/(2*d^(1/2)*e^(1/2)+(2*d*e-b)^(1/2))^(1/2)+1/4*arctan((2 *x*e^(1/2)+(2*d^(1/2)*e^(1/2)-(2*d*e-b)^(1/2))^(1/2))/(2*d^(1/2)*e^(1/2)+( 2*d*e-b)^(1/2))^(1/2))/d^(1/2)/(2*d^(1/2)*e^(1/2)+(2*d*e-b)^(1/2))^(1/2)-1 /8*ln(d^(1/2)+x^2*e^(1/2)-x*(2*d^(1/2)*e^(1/2)+(2*d*e-b)^(1/2))^(1/2))/d^( 1/2)/(2*d^(1/2)*e^(1/2)+(2*d*e-b)^(1/2))^(1/2)+1/8*ln(d^(1/2)+x^2*e^(1/2)+ x*(2*d^(1/2)*e^(1/2)+(2*d*e-b)^(1/2))^(1/2))/d^(1/2)/(2*d^(1/2)*e^(1/2)+(2 *d*e-b)^(1/2))^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.04 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.08 \[ \int \frac {d+e x^4}{d^2+b x^4+e^2 x^8} \, dx=\frac {1}{4} \text {RootSum}\left [d^2+b \text {$\#$1}^4+e^2 \text {$\#$1}^8\&,\frac {d \log (x-\text {$\#$1})+e \log (x-\text {$\#$1}) \text {$\#$1}^4}{b \text {$\#$1}^3+2 e^2 \text {$\#$1}^7}\&\right ] \]
RootSum[d^2 + b*#1^4 + e^2*#1^8 & , (d*Log[x - #1] + e*Log[x - #1]*#1^4)/( b*#1^3 + 2*e^2*#1^7) & ]/4
Time = 1.30 (sec) , antiderivative size = 1001, normalized size of antiderivative = 1.27, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {1749, 1407, 27, 1142, 25, 27, 1083, 217, 1103}
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.
| \(\displaystyle \int \frac {d+e x^4}{b x^4+d^2+e^2 x^8} \, dx\) |
| \(\Big \downarrow \) 1749 |
| \(\displaystyle \frac {\int \frac {1}{x^4-\frac {\sqrt {2 d e-b} x^2}{e}+\frac {d}{e}}dx}{2 e}+\frac {\int \frac {1}{x^4+\frac {\sqrt {2 d e-b} x^2}{e}+\frac {d}{e}}dx}{2 e}\) |
| \(\Big \downarrow \) 1407 |
| \(\displaystyle \frac {\frac {e \int \frac {\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-b}}-\sqrt {e} x}{\sqrt {e} \left (x^2-\frac {\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-b}} x}{\sqrt {e}}+\frac {\sqrt {d}}{\sqrt {e}}\right )}dx}{2 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-b}}}+\frac {e \int \frac {\sqrt {e} x+\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-b}}}{\sqrt {e} \left (x^2+\frac {\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-b}} x}{\sqrt {e}}+\frac {\sqrt {d}}{\sqrt {e}}\right )}dx}{2 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-b}}}}{2 e}+\frac {\frac {e \int \frac {\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-b}}-\sqrt {e} x}{\sqrt {e} \left (x^2-\frac {\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-b}} x}{\sqrt {e}}+\frac {\sqrt {d}}{\sqrt {e}}\right )}dx}{2 \sqrt {d} \sqrt {\sqrt {2 d e-b}+2 \sqrt {d} \sqrt {e}}}+\frac {e \int \frac {\sqrt {e} x+\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-b}}}{\sqrt {e} \left (x^2+\frac {\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-b}} x}{\sqrt {e}}+\frac {\sqrt {d}}{\sqrt {e}}\right )}dx}{2 \sqrt {d} \sqrt {\sqrt {2 d e-b}+2 \sqrt {d} \sqrt {e}}}}{2 e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\sqrt {e} \int \frac {\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-b}}-\sqrt {e} x}{x^2-\frac {\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-b}} x}{\sqrt {e}}+\frac {\sqrt {d}}{\sqrt {e}}}dx}{2 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-b}}}+\frac {\sqrt {e} \int \frac {\sqrt {e} x+\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-b}}}{x^2+\frac {\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-b}} x}{\sqrt {e}}+\frac {\sqrt {d}}{\sqrt {e}}}dx}{2 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-b}}}}{2 e}+\frac {\frac {\sqrt {e} \int \frac {\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-b}}-\sqrt {e} x}{x^2-\frac {\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-b}} x}{\sqrt {e}}+\frac {\sqrt {d}}{\sqrt {e}}}dx}{2 \sqrt {d} \sqrt {\sqrt {2 d e-b}+2 \sqrt {d} \sqrt {e}}}+\frac {\sqrt {e} \int \frac {\sqrt {e} x+\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-b}}}{x^2+\frac {\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-b}} x}{\sqrt {e}}+\frac {\sqrt {d}}{\sqrt {e}}}dx}{2 \sqrt {d} \sqrt {\sqrt {2 d e-b}+2 \sqrt {d} \sqrt {e}}}}{2 e}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {\frac {\sqrt {e} \left (\frac {1}{2} \sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-b}} \int \frac {1}{x^2-\frac {\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-b}} x}{\sqrt {e}}+\frac {\sqrt {d}}{\sqrt {e}}}dx-\frac {1}{2} \sqrt {e} \int -\frac {\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-b}}-2 \sqrt {e} x}{\sqrt {e} \left (x^2-\frac {\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-b}} x}{\sqrt {e}}+\frac {\sqrt {d}}{\sqrt {e}}\right )}dx\right )}{2 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-b}}}+\frac {\sqrt {e} \left (\frac {1}{2} \sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-b}} \int \frac {1}{x^2+\frac {\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-b}} x}{\sqrt {e}}+\frac {\sqrt {d}}{\sqrt {e}}}dx+\frac {1}{2} \sqrt {e} \int \frac {2 \sqrt {e} x+\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-b}}}{\sqrt {e} \left (x^2+\frac {\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-b}} x}{\sqrt {e}}+\frac {\sqrt {d}}{\sqrt {e}}\right )}dx\right )}{2 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-b}}}}{2 e}+\frac {\frac {\sqrt {e} \left (\frac {1}{2} \sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-b}} \int \frac {1}{x^2-\frac {\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-b}} x}{\sqrt {e}}+\frac {\sqrt {d}}{\sqrt {e}}}dx-\frac {1}{2} \sqrt {e} \int -\frac {\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-b}}-2 \sqrt {e} x}{\sqrt {e} \left (x^2-\frac {\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-b}} x}{\sqrt {e}}+\frac {\sqrt {d}}{\sqrt {e}}\right )}dx\right )}{2 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-b}}}+\frac {\sqrt {e} \left (\frac {1}{2} \sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-b}} \int \frac {1}{x^2+\frac {\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-b}} x}{\sqrt {e}}+\frac {\sqrt {d}}{\sqrt {e}}}dx+\frac {1}{2} \sqrt {e} \int \frac {2 \sqrt {e} x+\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-b}}}{\sqrt {e} \left (x^2+\frac {\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-b}} x}{\sqrt {e}}+\frac {\sqrt {d}}{\sqrt {e}}\right )}dx\right )}{2 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-b}}}}{2 e}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\sqrt {e} \left (\frac {1}{2} \sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-b}} \int \frac {1}{x^2-\frac {\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-b}} x}{\sqrt {e}}+\frac {\sqrt {d}}{\sqrt {e}}}dx+\frac {1}{2} \sqrt {e} \int \frac {\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-b}}-2 \sqrt {e} x}{\sqrt {e} \left (x^2-\frac {\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-b}} x}{\sqrt {e}}+\frac {\sqrt {d}}{\sqrt {e}}\right )}dx\right )}{2 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-b}}}+\frac {\sqrt {e} \left (\frac {1}{2} \sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-b}} \int \frac {1}{x^2+\frac {\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-b}} x}{\sqrt {e}}+\frac {\sqrt {d}}{\sqrt {e}}}dx+\frac {1}{2} \sqrt {e} \int \frac {2 \sqrt {e} x+\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-b}}}{\sqrt {e} \left (x^2+\frac {\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-b}} x}{\sqrt {e}}+\frac {\sqrt {d}}{\sqrt {e}}\right )}dx\right )}{2 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-b}}}}{2 e}+\frac {\frac {\sqrt {e} \left (\frac {1}{2} \sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-b}} \int \frac {1}{x^2-\frac {\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-b}} x}{\sqrt {e}}+\frac {\sqrt {d}}{\sqrt {e}}}dx+\frac {1}{2} \sqrt {e} \int \frac {\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-b}}-2 \sqrt {e} x}{\sqrt {e} \left (x^2-\frac {\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-b}} x}{\sqrt {e}}+\frac {\sqrt {d}}{\sqrt {e}}\right )}dx\right )}{2 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-b}}}+\frac {\sqrt {e} \left (\frac {1}{2} \sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-b}} \int \frac {1}{x^2+\frac {\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-b}} x}{\sqrt {e}}+\frac {\sqrt {d}}{\sqrt {e}}}dx+\frac {1}{2} \sqrt {e} \int \frac {2 \sqrt {e} x+\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-b}}}{\sqrt {e} \left (x^2+\frac {\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-b}} x}{\sqrt {e}}+\frac {\sqrt {d}}{\sqrt {e}}\right )}dx\right )}{2 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-b}}}}{2 e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\sqrt {e} \left (\frac {1}{2} \sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-b}} \int \frac {1}{x^2-\frac {\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-b}} x}{\sqrt {e}}+\frac {\sqrt {d}}{\sqrt {e}}}dx+\frac {1}{2} \int \frac {\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-b}}-2 \sqrt {e} x}{x^2-\frac {\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-b}} x}{\sqrt {e}}+\frac {\sqrt {d}}{\sqrt {e}}}dx\right )}{2 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-b}}}+\frac {\sqrt {e} \left (\frac {1}{2} \sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-b}} \int \frac {1}{x^2+\frac {\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-b}} x}{\sqrt {e}}+\frac {\sqrt {d}}{\sqrt {e}}}dx+\frac {1}{2} \int \frac {2 \sqrt {e} x+\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-b}}}{x^2+\frac {\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-b}} x}{\sqrt {e}}+\frac {\sqrt {d}}{\sqrt {e}}}dx\right )}{2 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-b}}}}{2 e}+\frac {\frac {\sqrt {e} \left (\frac {1}{2} \sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-b}} \int \frac {1}{x^2-\frac {\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-b}} x}{\sqrt {e}}+\frac {\sqrt {d}}{\sqrt {e}}}dx+\frac {1}{2} \int \frac {\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-b}}-2 \sqrt {e} x}{x^2-\frac {\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-b}} x}{\sqrt {e}}+\frac {\sqrt {d}}{\sqrt {e}}}dx\right )}{2 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-b}}}+\frac {\sqrt {e} \left (\frac {1}{2} \sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-b}} \int \frac {1}{x^2+\frac {\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-b}} x}{\sqrt {e}}+\frac {\sqrt {d}}{\sqrt {e}}}dx+\frac {1}{2} \int \frac {2 \sqrt {e} x+\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-b}}}{x^2+\frac {\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-b}} x}{\sqrt {e}}+\frac {\sqrt {d}}{\sqrt {e}}}dx\right )}{2 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-b}}}}{2 e}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {\frac {\sqrt {e} \left (\frac {1}{2} \int \frac {\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-b}}-2 \sqrt {e} x}{x^2-\frac {\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-b}} x}{\sqrt {e}}+\frac {\sqrt {d}}{\sqrt {e}}}dx-\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-b}} \int \frac {1}{-\left (2 x-\frac {\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-b}}}{\sqrt {e}}\right )^2-\frac {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-b}}{e}}d\left (2 x-\frac {\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-b}}}{\sqrt {e}}\right )\right )}{2 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-b}}}+\frac {\sqrt {e} \left (\frac {1}{2} \int \frac {2 \sqrt {e} x+\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-b}}}{x^2+\frac {\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-b}} x}{\sqrt {e}}+\frac {\sqrt {d}}{\sqrt {e}}}dx-\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-b}} \int \frac {1}{-\left (2 x+\frac {\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-b}}}{\sqrt {e}}\right )^2-\frac {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-b}}{e}}d\left (2 x+\frac {\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-b}}}{\sqrt {e}}\right )\right )}{2 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-b}}}}{2 e}+\frac {\frac {\sqrt {e} \left (\frac {1}{2} \int \frac {\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-b}}-2 \sqrt {e} x}{x^2-\frac {\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-b}} x}{\sqrt {e}}+\frac {\sqrt {d}}{\sqrt {e}}}dx-\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-b}} \int \frac {1}{-\left (2 x-\frac {\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-b}}}{\sqrt {e}}\right )^2-\frac {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-b}}{e}}d\left (2 x-\frac {\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-b}}}{\sqrt {e}}\right )\right )}{2 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-b}}}+\frac {\sqrt {e} \left (\frac {1}{2} \int \frac {2 \sqrt {e} x+\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-b}}}{x^2+\frac {\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-b}} x}{\sqrt {e}}+\frac {\sqrt {d}}{\sqrt {e}}}dx-\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-b}} \int \frac {1}{-\left (2 x+\frac {\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-b}}}{\sqrt {e}}\right )^2-\frac {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-b}}{e}}d\left (2 x+\frac {\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-b}}}{\sqrt {e}}\right )\right )}{2 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-b}}}}{2 e}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {\sqrt {e} \left (\frac {\sqrt {e} \sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-b}} \arctan \left (\frac {\sqrt {e} \left (2 x-\frac {\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-b}}}{\sqrt {e}}\right )}{\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-b}}}\right )}{\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-b}}}+\frac {1}{2} \int \frac {\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-b}}-2 \sqrt {e} x}{x^2-\frac {\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-b}} x}{\sqrt {e}}+\frac {\sqrt {d}}{\sqrt {e}}}dx\right )}{2 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-b}}}+\frac {\sqrt {e} \left (\frac {\sqrt {e} \sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-b}} \arctan \left (\frac {\sqrt {e} \left (2 x+\frac {\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-b}}}{\sqrt {e}}\right )}{\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-b}}}\right )}{\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-b}}}+\frac {1}{2} \int \frac {2 \sqrt {e} x+\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-b}}}{x^2+\frac {\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-b}} x}{\sqrt {e}}+\frac {\sqrt {d}}{\sqrt {e}}}dx\right )}{2 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-b}}}}{2 e}+\frac {\frac {\sqrt {e} \left (\frac {\sqrt {e} \sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-b}} \arctan \left (\frac {\sqrt {e} \left (2 x-\frac {\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-b}}}{\sqrt {e}}\right )}{\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-b}}}\right )}{\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-b}}}+\frac {1}{2} \int \frac {\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-b}}-2 \sqrt {e} x}{x^2-\frac {\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-b}} x}{\sqrt {e}}+\frac {\sqrt {d}}{\sqrt {e}}}dx\right )}{2 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-b}}}+\frac {\sqrt {e} \left (\frac {\sqrt {e} \sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-b}} \arctan \left (\frac {\sqrt {e} \left (2 x+\frac {\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-b}}}{\sqrt {e}}\right )}{\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-b}}}\right )}{\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-b}}}+\frac {1}{2} \int \frac {2 \sqrt {e} x+\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-b}}}{x^2+\frac {\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-b}} x}{\sqrt {e}}+\frac {\sqrt {d}}{\sqrt {e}}}dx\right )}{2 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-b}}}}{2 e}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\frac {\sqrt {e} \left (\frac {\sqrt {e} \sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-b}} \arctan \left (\frac {\sqrt {e} \left (2 x-\frac {\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-b}}}{\sqrt {e}}\right )}{\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-b}}}\right )}{\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-b}}}-\frac {1}{2} \sqrt {e} \log \left (\sqrt {e} x^2-\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-b}} x+\sqrt {d}\right )\right )}{2 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-b}}}+\frac {\sqrt {e} \left (\frac {\sqrt {e} \sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-b}} \arctan \left (\frac {\sqrt {e} \left (2 x+\frac {\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-b}}}{\sqrt {e}}\right )}{\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-b}}}\right )}{\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-b}}}+\frac {1}{2} \sqrt {e} \log \left (\sqrt {e} x^2+\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-b}} x+\sqrt {d}\right )\right )}{2 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-b}}}}{2 e}+\frac {\frac {\sqrt {e} \left (\frac {\sqrt {e} \sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-b}} \arctan \left (\frac {\sqrt {e} \left (2 x-\frac {\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-b}}}{\sqrt {e}}\right )}{\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-b}}}\right )}{\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-b}}}-\frac {1}{2} \sqrt {e} \log \left (\sqrt {e} x^2-\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-b}} x+\sqrt {d}\right )\right )}{2 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-b}}}+\frac {\sqrt {e} \left (\frac {\sqrt {e} \sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-b}} \arctan \left (\frac {\sqrt {e} \left (2 x+\frac {\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-b}}}{\sqrt {e}}\right )}{\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-b}}}\right )}{\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-b}}}+\frac {1}{2} \sqrt {e} \log \left (\sqrt {e} x^2+\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-b}} x+\sqrt {d}\right )\right )}{2 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-b}}}}{2 e}\) |
((Sqrt[e]*((Sqrt[e]*Sqrt[2*Sqrt[d]*Sqrt[e] - Sqrt[-b + 2*d*e]]*ArcTan[(Sqr t[e]*(-(Sqrt[2*Sqrt[d]*Sqrt[e] - Sqrt[-b + 2*d*e]]/Sqrt[e]) + 2*x))/Sqrt[2 *Sqrt[d]*Sqrt[e] + Sqrt[-b + 2*d*e]]])/Sqrt[2*Sqrt[d]*Sqrt[e] + Sqrt[-b + 2*d*e]] - (Sqrt[e]*Log[Sqrt[d] - Sqrt[2*Sqrt[d]*Sqrt[e] - Sqrt[-b + 2*d*e] ]*x + Sqrt[e]*x^2])/2))/(2*Sqrt[d]*Sqrt[2*Sqrt[d]*Sqrt[e] - Sqrt[-b + 2*d* e]]) + (Sqrt[e]*((Sqrt[e]*Sqrt[2*Sqrt[d]*Sqrt[e] - Sqrt[-b + 2*d*e]]*ArcTa n[(Sqrt[e]*(Sqrt[2*Sqrt[d]*Sqrt[e] - Sqrt[-b + 2*d*e]]/Sqrt[e] + 2*x))/Sqr t[2*Sqrt[d]*Sqrt[e] + Sqrt[-b + 2*d*e]]])/Sqrt[2*Sqrt[d]*Sqrt[e] + Sqrt[-b + 2*d*e]] + (Sqrt[e]*Log[Sqrt[d] + Sqrt[2*Sqrt[d]*Sqrt[e] - Sqrt[-b + 2*d *e]]*x + Sqrt[e]*x^2])/2))/(2*Sqrt[d]*Sqrt[2*Sqrt[d]*Sqrt[e] - Sqrt[-b + 2 *d*e]]))/(2*e) + ((Sqrt[e]*((Sqrt[e]*Sqrt[2*Sqrt[d]*Sqrt[e] + Sqrt[-b + 2* d*e]]*ArcTan[(Sqrt[e]*(-(Sqrt[2*Sqrt[d]*Sqrt[e] + Sqrt[-b + 2*d*e]]/Sqrt[e ]) + 2*x))/Sqrt[2*Sqrt[d]*Sqrt[e] - Sqrt[-b + 2*d*e]]])/Sqrt[2*Sqrt[d]*Sqr t[e] - Sqrt[-b + 2*d*e]] - (Sqrt[e]*Log[Sqrt[d] - Sqrt[2*Sqrt[d]*Sqrt[e] + Sqrt[-b + 2*d*e]]*x + Sqrt[e]*x^2])/2))/(2*Sqrt[d]*Sqrt[2*Sqrt[d]*Sqrt[e] + Sqrt[-b + 2*d*e]]) + (Sqrt[e]*((Sqrt[e]*Sqrt[2*Sqrt[d]*Sqrt[e] + Sqrt[- b + 2*d*e]]*ArcTan[(Sqrt[e]*(Sqrt[2*Sqrt[d]*Sqrt[e] + Sqrt[-b + 2*d*e]]/Sq rt[e] + 2*x))/Sqrt[2*Sqrt[d]*Sqrt[e] - Sqrt[-b + 2*d*e]]])/Sqrt[2*Sqrt[d]* Sqrt[e] - Sqrt[-b + 2*d*e]] + (Sqrt[e]*Log[Sqrt[d] + Sqrt[2*Sqrt[d]*Sqrt[e ] + Sqrt[-b + 2*d*e]]*x + Sqrt[e]*x^2])/2))/(2*Sqrt[d]*Sqrt[2*Sqrt[d]*S...
3.1.5.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(-1), x_Symbol] :> With[{q = Rt[a/ c, 2]}, With[{r = Rt[2*q - b/c, 2]}, Simp[1/(2*c*q*r) Int[(r - x)/(q - r* x + x^2), x], x] + Simp[1/(2*c*q*r) Int[(r + x)/(q + r*x + x^2), x], x]]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && NegQ[b^2 - 4*a*c]
Int[((d_) + (e_.)*(x_)^(n_))/((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_)), x _Symbol] :> With[{q = Rt[2*(d/e) - b/c, 2]}, Simp[e/(2*c) Int[1/Simp[d/e + q*x^(n/2) + x^n, x], x], x] + Simp[e/(2*c) Int[1/Simp[d/e - q*x^(n/2) + x^n, x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - a*e^2, 0] && IGtQ[n/2, 0] && (GtQ[2*(d/e) - b/c, 0] || ( !LtQ[2*(d/e) - b/c, 0] && EqQ[d, e*Rt[a/c, 2]]))
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.17 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.07
| method | result | size |
| default | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (e^{2} \textit {\_Z}^{8}+\textit {\_Z}^{4} b +d^{2}\right )}{\sum }\frac {\left (\textit {\_R}^{4} e +d \right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{7} e^{2}+\textit {\_R}^{3} b}\right )}{4}\) | \(53\) |
| risch | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (e^{2} \textit {\_Z}^{8}+\textit {\_Z}^{4} b +d^{2}\right )}{\sum }\frac {\left (\textit {\_R}^{4} e +d \right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{7} e^{2}+\textit {\_R}^{3} b}\right )}{4}\) | \(53\) |
Leaf count of result is larger than twice the leaf count of optimal. 2461 vs. \(2 (581) = 1162\).
Time = 0.32 (sec) , antiderivative size = 2461, normalized size of antiderivative = 3.11 \[ \int \frac {d+e x^4}{d^2+b x^4+e^2 x^8} \, dx=\text {Too large to display} \]
1/4*sqrt(sqrt(1/2)*sqrt(-((4*d^4*e^2 + 4*b*d^3*e + b^2*d^2)*sqrt(-(2*d*e - b)/(8*d^7*e^3 + 12*b*d^6*e^2 + 6*b^2*d^5*e + b^3*d^4)) + b)/(4*d^4*e^2 + 4*b*d^3*e + b^2*d^2)))*log(e*x + 1/2*(2*d*e - (4*d^4*e^2 + 4*b*d^3*e + b^2 *d^2)*sqrt(-(2*d*e - b)/(8*d^7*e^3 + 12*b*d^6*e^2 + 6*b^2*d^5*e + b^3*d^4) ) + b)*sqrt(sqrt(1/2)*sqrt(-((4*d^4*e^2 + 4*b*d^3*e + b^2*d^2)*sqrt(-(2*d* e - b)/(8*d^7*e^3 + 12*b*d^6*e^2 + 6*b^2*d^5*e + b^3*d^4)) + b)/(4*d^4*e^2 + 4*b*d^3*e + b^2*d^2)))) - 1/4*sqrt(sqrt(1/2)*sqrt(-((4*d^4*e^2 + 4*b*d^ 3*e + b^2*d^2)*sqrt(-(2*d*e - b)/(8*d^7*e^3 + 12*b*d^6*e^2 + 6*b^2*d^5*e + b^3*d^4)) + b)/(4*d^4*e^2 + 4*b*d^3*e + b^2*d^2)))*log(e*x - 1/2*(2*d*e - (4*d^4*e^2 + 4*b*d^3*e + b^2*d^2)*sqrt(-(2*d*e - b)/(8*d^7*e^3 + 12*b*d^6 *e^2 + 6*b^2*d^5*e + b^3*d^4)) + b)*sqrt(sqrt(1/2)*sqrt(-((4*d^4*e^2 + 4*b *d^3*e + b^2*d^2)*sqrt(-(2*d*e - b)/(8*d^7*e^3 + 12*b*d^6*e^2 + 6*b^2*d^5* e + b^3*d^4)) + b)/(4*d^4*e^2 + 4*b*d^3*e + b^2*d^2)))) + 1/4*sqrt(-sqrt(1 /2)*sqrt(-((4*d^4*e^2 + 4*b*d^3*e + b^2*d^2)*sqrt(-(2*d*e - b)/(8*d^7*e^3 + 12*b*d^6*e^2 + 6*b^2*d^5*e + b^3*d^4)) + b)/(4*d^4*e^2 + 4*b*d^3*e + b^2 *d^2)))*log(e*x + 1/2*(2*d*e - (4*d^4*e^2 + 4*b*d^3*e + b^2*d^2)*sqrt(-(2* d*e - b)/(8*d^7*e^3 + 12*b*d^6*e^2 + 6*b^2*d^5*e + b^3*d^4)) + b)*sqrt(-sq rt(1/2)*sqrt(-((4*d^4*e^2 + 4*b*d^3*e + b^2*d^2)*sqrt(-(2*d*e - b)/(8*d^7* e^3 + 12*b*d^6*e^2 + 6*b^2*d^5*e + b^3*d^4)) + b)/(4*d^4*e^2 + 4*b*d^3*e + b^2*d^2)))) - 1/4*sqrt(-sqrt(1/2)*sqrt(-((4*d^4*e^2 + 4*b*d^3*e + b^2*...
Time = 19.01 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.17 \[ \int \frac {d+e x^4}{d^2+b x^4+e^2 x^8} \, dx=\operatorname {RootSum} {\left (t^{8} \cdot \left (65536 b^{4} d^{2} + 524288 b^{3} d^{3} e + 1572864 b^{2} d^{4} e^{2} + 2097152 b d^{5} e^{3} + 1048576 d^{6} e^{4}\right ) + t^{4} \cdot \left (256 b^{3} + 1024 b^{2} d e + 1024 b d^{2} e^{2}\right ) + e^{2}, \left ( t \mapsto t \log {\left (x + \frac {1024 t^{5} b^{2} d^{2} + 4096 t^{5} b d^{3} e + 4096 t^{5} d^{4} e^{2} + 4 t b + 4 t d e}{e} \right )} \right )\right )} \]
RootSum(_t**8*(65536*b**4*d**2 + 524288*b**3*d**3*e + 1572864*b**2*d**4*e* *2 + 2097152*b*d**5*e**3 + 1048576*d**6*e**4) + _t**4*(256*b**3 + 1024*b** 2*d*e + 1024*b*d**2*e**2) + e**2, Lambda(_t, _t*log(x + (1024*_t**5*b**2*d **2 + 4096*_t**5*b*d**3*e + 4096*_t**5*d**4*e**2 + 4*_t*b + 4*_t*d*e)/e)))
\[ \int \frac {d+e x^4}{d^2+b x^4+e^2 x^8} \, dx=\int { \frac {e x^{4} + d}{e^{2} x^{8} + b x^{4} + d^{2}} \,d x } \]
\[ \int \frac {d+e x^4}{d^2+b x^4+e^2 x^8} \, dx=\int { \frac {e x^{4} + d}{e^{2} x^{8} + b x^{4} + d^{2}} \,d x } \]
Time = 9.72 (sec) , antiderivative size = 10409, normalized size of antiderivative = 13.16 \[ \int \frac {d+e x^4}{d^2+b x^4+e^2 x^8} \, dx=\text {Too large to display} \]
2*atan(((x*(32*b*d^5*e^13 - 4*b^4*d^2*e^10 + 24*b^3*d^3*e^11 - 48*b^2*d^4* e^12) + (-(b^3 + ((b - 2*d*e)*(b + 2*d*e)^5)^(1/2) + 4*b*d^2*e^2 + 4*b^2*d *e)/(512*(b^4*d^2 + 16*d^6*e^4 + 8*b^3*d^3*e + 32*b*d^5*e^3 + 24*b^2*d^4*e ^2)))^(1/4)*((x*(65536*d^9*e^15 - 32768*b*d^8*e^14 + 1024*b^7*d^2*e^8 - 20 48*b^6*d^3*e^9 - 10240*b^5*d^4*e^10 + 20480*b^4*d^5*e^11 + 32768*b^3*d^6*e ^12 - 65536*b^2*d^7*e^13) - (-(b^3 + ((b - 2*d*e)*(b + 2*d*e)^5)^(1/2) + 4 *b*d^2*e^2 + 4*b^2*d*e)/(512*(b^4*d^2 + 16*d^6*e^4 + 8*b^3*d^3*e + 32*b*d^ 5*e^3 + 24*b^2*d^4*e^2)))^(1/4)*(262144*d^10*e^15 - 262144*b*d^9*e^14 + 40 96*b^7*d^3*e^8 - 4096*b^6*d^4*e^9 - 49152*b^5*d^5*e^10 + 49152*b^4*d^6*e^1 1 + 196608*b^3*d^7*e^12 - 196608*b^2*d^8*e^13)*1i)*(-(b^3 + ((b - 2*d*e)*( b + 2*d*e)^5)^(1/2) + 4*b*d^2*e^2 + 4*b^2*d*e)/(512*(b^4*d^2 + 16*d^6*e^4 + 8*b^3*d^3*e + 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(3/4)*1i - 256*d^7*e^14 + 256*b*d^6*e^13 + 16*b^4*d^3*e^10 - 64*b^3*d^4*e^11)*1i)*(-(b^3 + ((b - 2* d*e)*(b + 2*d*e)^5)^(1/2) + 4*b*d^2*e^2 + 4*b^2*d*e)/(512*(b^4*d^2 + 16*d^ 6*e^4 + 8*b^3*d^3*e + 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(1/4) + (x*(32*b*d^ 5*e^13 - 4*b^4*d^2*e^10 + 24*b^3*d^3*e^11 - 48*b^2*d^4*e^12) + (-(b^3 + (( b - 2*d*e)*(b + 2*d*e)^5)^(1/2) + 4*b*d^2*e^2 + 4*b^2*d*e)/(512*(b^4*d^2 + 16*d^6*e^4 + 8*b^3*d^3*e + 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(1/4)*((x*(65 536*d^9*e^15 - 32768*b*d^8*e^14 + 1024*b^7*d^2*e^8 - 2048*b^6*d^3*e^9 - 10 240*b^5*d^4*e^10 + 20480*b^4*d^5*e^11 + 32768*b^3*d^6*e^12 - 65536*b^2*...