Integrand size = 22, antiderivative size = 712 \[ \int \frac {x^8}{\left (d+e x^2\right ) \left (a+c x^4\right )^2} \, dx=\frac {d x}{4 c \left (c d^2+a e^2\right )}-\frac {x^3 \left (a e+c d x^2\right )}{4 c \left (c d^2+a e^2\right ) \left (a+c x^4\right )}+\frac {d^{7/2} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e} \left (c d^2+a e^2\right )^2}+\frac {\sqrt [4]{a} d^2 \left (\sqrt {c} d-\sqrt {a} e\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} c^{3/4} \left (c d^2+a e^2\right )^2}+\frac {\sqrt [4]{a} \left (\sqrt {c} d-3 \sqrt {a} e\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} c^{7/4} \left (c d^2+a e^2\right )}-\frac {\sqrt [4]{a} d^2 \left (\sqrt {c} d-\sqrt {a} e\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} c^{3/4} \left (c d^2+a e^2\right )^2}-\frac {\sqrt [4]{a} \left (\sqrt {c} d-3 \sqrt {a} e\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} c^{7/4} \left (c d^2+a e^2\right )}+\frac {\sqrt [4]{a} d^2 \left (\sqrt {c} d+\sqrt {a} e\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} c^{3/4} \left (c d^2+a e^2\right )^2}+\frac {\sqrt [4]{a} \left (\sqrt {c} d+3 \sqrt {a} e\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{16 \sqrt {2} c^{7/4} \left (c d^2+a e^2\right )}-\frac {\sqrt [4]{a} d^2 \left (\sqrt {c} d+\sqrt {a} e\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} c^{3/4} \left (c d^2+a e^2\right )^2}-\frac {\sqrt [4]{a} \left (\sqrt {c} d+3 \sqrt {a} e\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{16 \sqrt {2} c^{7/4} \left (c d^2+a e^2\right )} \]
1/4*d*x/c/(a*e^2+c*d^2)-1/4*x^3*(c*d*x^2+a*e)/c/(a*e^2+c*d^2)/(c*x^4+a)-1/ 16*a^(1/4)*arctan(-1+c^(1/4)*x*2^(1/2)/a^(1/4))*(-3*e*a^(1/2)+d*c^(1/2))/c ^(7/4)/(a*e^2+c*d^2)*2^(1/2)-1/16*a^(1/4)*arctan(1+c^(1/4)*x*2^(1/2)/a^(1/ 4))*(-3*e*a^(1/2)+d*c^(1/2))/c^(7/4)/(a*e^2+c*d^2)*2^(1/2)-1/4*a^(1/4)*d^2 *arctan(-1+c^(1/4)*x*2^(1/2)/a^(1/4))*(-e*a^(1/2)+d*c^(1/2))/c^(3/4)/(a*e^ 2+c*d^2)^2*2^(1/2)-1/4*a^(1/4)*d^2*arctan(1+c^(1/4)*x*2^(1/2)/a^(1/4))*(-e *a^(1/2)+d*c^(1/2))/c^(3/4)/(a*e^2+c*d^2)^2*2^(1/2)+1/8*a^(1/4)*d^2*ln(-a^ (1/4)*c^(1/4)*x*2^(1/2)+a^(1/2)+x^2*c^(1/2))*(e*a^(1/2)+d*c^(1/2))/c^(3/4) /(a*e^2+c*d^2)^2*2^(1/2)-1/8*a^(1/4)*d^2*ln(a^(1/4)*c^(1/4)*x*2^(1/2)+a^(1 /2)+x^2*c^(1/2))*(e*a^(1/2)+d*c^(1/2))/c^(3/4)/(a*e^2+c*d^2)^2*2^(1/2)+1/3 2*a^(1/4)*ln(-a^(1/4)*c^(1/4)*x*2^(1/2)+a^(1/2)+x^2*c^(1/2))*(3*e*a^(1/2)+ d*c^(1/2))/c^(7/4)/(a*e^2+c*d^2)*2^(1/2)-1/32*a^(1/4)*ln(a^(1/4)*c^(1/4)*x *2^(1/2)+a^(1/2)+x^2*c^(1/2))*(3*e*a^(1/2)+d*c^(1/2))/c^(7/4)/(a*e^2+c*d^2 )*2^(1/2)+d^(7/2)*arctan(x*e^(1/2)/d^(1/2))/(a*e^2+c*d^2)^2/e^(1/2)
Time = 0.21 (sec) , antiderivative size = 431, normalized size of antiderivative = 0.61 \[ \int \frac {x^8}{\left (d+e x^2\right ) \left (a+c x^4\right )^2} \, dx=\frac {\frac {8 a \left (c d^2+a e^2\right ) x \left (d-e x^2\right )}{c \left (a+c x^4\right )}+\frac {32 d^{7/2} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}-\frac {2 \sqrt {2} \sqrt [4]{a} \left (-5 c^{3/2} d^3+7 \sqrt {a} c d^2 e-a \sqrt {c} d e^2+3 a^{3/2} e^3\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{c^{7/4}}+\frac {2 \sqrt {2} \sqrt [4]{a} \left (-5 c^{3/2} d^3+7 \sqrt {a} c d^2 e-a \sqrt {c} d e^2+3 a^{3/2} e^3\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{c^{7/4}}+\frac {\sqrt {2} \sqrt [4]{a} \left (5 c^{3/2} d^3+7 \sqrt {a} c d^2 e+a \sqrt {c} d e^2+3 a^{3/2} e^3\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{c^{7/4}}-\frac {\sqrt {2} \sqrt [4]{a} \left (5 c^{3/2} d^3+7 \sqrt {a} c d^2 e+a \sqrt {c} d e^2+3 a^{3/2} e^3\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{c^{7/4}}}{32 \left (c d^2+a e^2\right )^2} \]
((8*a*(c*d^2 + a*e^2)*x*(d - e*x^2))/(c*(a + c*x^4)) + (32*d^(7/2)*ArcTan[ (Sqrt[e]*x)/Sqrt[d]])/Sqrt[e] - (2*Sqrt[2]*a^(1/4)*(-5*c^(3/2)*d^3 + 7*Sqr t[a]*c*d^2*e - a*Sqrt[c]*d*e^2 + 3*a^(3/2)*e^3)*ArcTan[1 - (Sqrt[2]*c^(1/4 )*x)/a^(1/4)])/c^(7/4) + (2*Sqrt[2]*a^(1/4)*(-5*c^(3/2)*d^3 + 7*Sqrt[a]*c* d^2*e - a*Sqrt[c]*d*e^2 + 3*a^(3/2)*e^3)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^ (1/4)])/c^(7/4) + (Sqrt[2]*a^(1/4)*(5*c^(3/2)*d^3 + 7*Sqrt[a]*c*d^2*e + a* Sqrt[c]*d*e^2 + 3*a^(3/2)*e^3)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + S qrt[c]*x^2])/c^(7/4) - (Sqrt[2]*a^(1/4)*(5*c^(3/2)*d^3 + 7*Sqrt[a]*c*d^2*e + a*Sqrt[c]*d*e^2 + 3*a^(3/2)*e^3)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)* x + Sqrt[c]*x^2])/c^(7/4))/(32*(c*d^2 + a*e^2)^2)
Time = 1.14 (sec) , antiderivative size = 652, normalized size of antiderivative = 0.92, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.727, Rules used = {1651, 1599, 25, 1603, 27, 1482, 27, 1476, 1082, 217, 1479, 25, 27, 1103, 1611, 2009}
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.
| \(\displaystyle \int \frac {x^8}{\left (a+c x^4\right )^2 \left (d+e x^2\right )} \, dx\) |
| \(\Big \downarrow \) 1651 |
| \(\displaystyle \frac {d^2 \int \frac {x^4}{\left (e x^2+d\right ) \left (c x^4+a\right )}dx}{a e^2+c d^2}-\frac {a \int \frac {x^4 \left (d-e x^2\right )}{\left (c x^4+a\right )^2}dx}{a e^2+c d^2}\) |
| \(\Big \downarrow \) 1599 |
| \(\displaystyle \frac {d^2 \int \frac {x^4}{\left (e x^2+d\right ) \left (c x^4+a\right )}dx}{a e^2+c d^2}-\frac {a \left (\frac {\int -\frac {x^2 \left (c d x^2+3 a e\right )}{c x^4+a}dx}{4 a c}+\frac {x^3 \left (a e+c d x^2\right )}{4 a c \left (a+c x^4\right )}\right )}{a e^2+c d^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {d^2 \int \frac {x^4}{\left (e x^2+d\right ) \left (c x^4+a\right )}dx}{a e^2+c d^2}-\frac {a \left (\frac {x^3 \left (a e+c d x^2\right )}{4 a c \left (a+c x^4\right )}-\frac {\int \frac {x^2 \left (c d x^2+3 a e\right )}{c x^4+a}dx}{4 a c}\right )}{a e^2+c d^2}\) |
| \(\Big \downarrow \) 1603 |
| \(\displaystyle \frac {d^2 \int \frac {x^4}{\left (e x^2+d\right ) \left (c x^4+a\right )}dx}{a e^2+c d^2}-\frac {a \left (\frac {x^3 \left (a e+c d x^2\right )}{4 a c \left (a+c x^4\right )}-\frac {d x-\frac {\int \frac {a c \left (d-3 e x^2\right )}{c x^4+a}dx}{c}}{4 a c}\right )}{a e^2+c d^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {d^2 \int \frac {x^4}{\left (e x^2+d\right ) \left (c x^4+a\right )}dx}{a e^2+c d^2}-\frac {a \left (\frac {x^3 \left (a e+c d x^2\right )}{4 a c \left (a+c x^4\right )}-\frac {d x-a \int \frac {d-3 e x^2}{c x^4+a}dx}{4 a c}\right )}{a e^2+c d^2}\) |
| \(\Big \downarrow \) 1482 |
| \(\displaystyle \frac {d^2 \int \frac {x^4}{\left (e x^2+d\right ) \left (c x^4+a\right )}dx}{a e^2+c d^2}-\frac {a \left (\frac {x^3 \left (a e+c d x^2\right )}{4 a c \left (a+c x^4\right )}-\frac {d x-a \left (\frac {\left (\frac {\sqrt {c} d}{\sqrt {a}}+3 e\right ) \int \frac {\sqrt {c} \left (\sqrt {a}-\sqrt {c} x^2\right )}{c x^4+a}dx}{2 c}+\frac {\left (\frac {\sqrt {c} d}{\sqrt {a}}-3 e\right ) \int \frac {\sqrt {c} \left (\sqrt {c} x^2+\sqrt {a}\right )}{c x^4+a}dx}{2 c}\right )}{4 a c}\right )}{a e^2+c d^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {d^2 \int \frac {x^4}{\left (e x^2+d\right ) \left (c x^4+a\right )}dx}{a e^2+c d^2}-\frac {a \left (\frac {x^3 \left (a e+c d x^2\right )}{4 a c \left (a+c x^4\right )}-\frac {d x-a \left (\frac {\left (\frac {\sqrt {c} d}{\sqrt {a}}+3 e\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{c x^4+a}dx}{2 \sqrt {c}}+\frac {\left (\frac {\sqrt {c} d}{\sqrt {a}}-3 e\right ) \int \frac {\sqrt {c} x^2+\sqrt {a}}{c x^4+a}dx}{2 \sqrt {c}}\right )}{4 a c}\right )}{a e^2+c d^2}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {d^2 \int \frac {x^4}{\left (e x^2+d\right ) \left (c x^4+a\right )}dx}{a e^2+c d^2}-\frac {a \left (\frac {x^3 \left (a e+c d x^2\right )}{4 a c \left (a+c x^4\right )}-\frac {d x-a \left (\frac {\left (\frac {\sqrt {c} d}{\sqrt {a}}-3 e\right ) \left (\frac {\int \frac {1}{x^2-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}}dx}{2 \sqrt {c}}+\frac {\int \frac {1}{x^2+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}}dx}{2 \sqrt {c}}\right )}{2 \sqrt {c}}+\frac {\left (\frac {\sqrt {c} d}{\sqrt {a}}+3 e\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{c x^4+a}dx}{2 \sqrt {c}}\right )}{4 a c}\right )}{a e^2+c d^2}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {d^2 \int \frac {x^4}{\left (e x^2+d\right ) \left (c x^4+a\right )}dx}{a e^2+c d^2}-\frac {a \left (\frac {x^3 \left (a e+c d x^2\right )}{4 a c \left (a+c x^4\right )}-\frac {d x-a \left (\frac {\left (\frac {\sqrt {c} d}{\sqrt {a}}+3 e\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{c x^4+a}dx}{2 \sqrt {c}}+\frac {\left (\frac {\sqrt {c} d}{\sqrt {a}}-3 e\right ) \left (\frac {\int \frac {1}{-\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )^2-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\int \frac {1}{-\left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )^2-1}d\left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )}{2 \sqrt {c}}\right )}{4 a c}\right )}{a e^2+c d^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {d^2 \int \frac {x^4}{\left (e x^2+d\right ) \left (c x^4+a\right )}dx}{a e^2+c d^2}-\frac {a \left (\frac {x^3 \left (a e+c d x^2\right )}{4 a c \left (a+c x^4\right )}-\frac {d x-a \left (\frac {\left (\frac {\sqrt {c} d}{\sqrt {a}}+3 e\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{c x^4+a}dx}{2 \sqrt {c}}+\frac {\left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right ) \left (\frac {\sqrt {c} d}{\sqrt {a}}-3 e\right )}{2 \sqrt {c}}\right )}{4 a c}\right )}{a e^2+c d^2}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {d^2 \int \frac {x^4}{\left (e x^2+d\right ) \left (c x^4+a\right )}dx}{a e^2+c d^2}-\frac {a \left (\frac {x^3 \left (a e+c d x^2\right )}{4 a c \left (a+c x^4\right )}-\frac {d x-a \left (\frac {\left (\frac {\sqrt {c} d}{\sqrt {a}}+3 e\right ) \left (-\frac {\int -\frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{c} x}{\sqrt [4]{c} \left (x^2-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}dx}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{c} x+\sqrt [4]{a}\right )}{\sqrt [4]{c} \left (x^2+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}dx}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )}{2 \sqrt {c}}+\frac {\left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right ) \left (\frac {\sqrt {c} d}{\sqrt {a}}-3 e\right )}{2 \sqrt {c}}\right )}{4 a c}\right )}{a e^2+c d^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {d^2 \int \frac {x^4}{\left (e x^2+d\right ) \left (c x^4+a\right )}dx}{a e^2+c d^2}-\frac {a \left (\frac {x^3 \left (a e+c d x^2\right )}{4 a c \left (a+c x^4\right )}-\frac {d x-a \left (\frac {\left (\frac {\sqrt {c} d}{\sqrt {a}}+3 e\right ) \left (\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{c} x}{\sqrt [4]{c} \left (x^2-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}dx}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{c} x+\sqrt [4]{a}\right )}{\sqrt [4]{c} \left (x^2+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}dx}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )}{2 \sqrt {c}}+\frac {\left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right ) \left (\frac {\sqrt {c} d}{\sqrt {a}}-3 e\right )}{2 \sqrt {c}}\right )}{4 a c}\right )}{a e^2+c d^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {d^2 \int \frac {x^4}{\left (e x^2+d\right ) \left (c x^4+a\right )}dx}{a e^2+c d^2}-\frac {a \left (\frac {x^3 \left (a e+c d x^2\right )}{4 a c \left (a+c x^4\right )}-\frac {d x-a \left (\frac {\left (\frac {\sqrt {c} d}{\sqrt {a}}+3 e\right ) \left (\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{c} x}{x^2-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}}dx}{2 \sqrt {2} \sqrt [4]{a} \sqrt {c}}+\frac {\int \frac {\sqrt {2} \sqrt [4]{c} x+\sqrt [4]{a}}{x^2+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}}dx}{2 \sqrt [4]{a} \sqrt {c}}\right )}{2 \sqrt {c}}+\frac {\left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right ) \left (\frac {\sqrt {c} d}{\sqrt {a}}-3 e\right )}{2 \sqrt {c}}\right )}{4 a c}\right )}{a e^2+c d^2}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {d^2 \int \frac {x^4}{\left (e x^2+d\right ) \left (c x^4+a\right )}dx}{a e^2+c d^2}-\frac {a \left (\frac {x^3 \left (a e+c d x^2\right )}{4 a c \left (a+c x^4\right )}-\frac {d x-a \left (\frac {\left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right ) \left (\frac {\sqrt {c} d}{\sqrt {a}}-3 e\right )}{2 \sqrt {c}}+\frac {\left (\frac {\sqrt {c} d}{\sqrt {a}}+3 e\right ) \left (\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )}{2 \sqrt {c}}\right )}{4 a c}\right )}{a e^2+c d^2}\) |
| \(\Big \downarrow \) 1611 |
| \(\displaystyle \frac {d^2 \int \left (\frac {d^2}{\left (c d^2+a e^2\right ) \left (e x^2+d\right )}-\frac {a \left (d-e x^2\right )}{\left (c d^2+a e^2\right ) \left (c x^4+a\right )}\right )dx}{a e^2+c d^2}-\frac {a \left (\frac {x^3 \left (a e+c d x^2\right )}{4 a c \left (a+c x^4\right )}-\frac {d x-a \left (\frac {\left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right ) \left (\frac {\sqrt {c} d}{\sqrt {a}}-3 e\right )}{2 \sqrt {c}}+\frac {\left (\frac {\sqrt {c} d}{\sqrt {a}}+3 e\right ) \left (\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )}{2 \sqrt {c}}\right )}{4 a c}\right )}{a e^2+c d^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {d^2 \left (\frac {\sqrt [4]{a} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right ) \left (\sqrt {c} d-\sqrt {a} e\right )}{2 \sqrt {2} c^{3/4} \left (a e^2+c d^2\right )}-\frac {\sqrt [4]{a} \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right ) \left (\sqrt {c} d-\sqrt {a} e\right )}{2 \sqrt {2} c^{3/4} \left (a e^2+c d^2\right )}+\frac {d^{3/2} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e} \left (a e^2+c d^2\right )}+\frac {\sqrt [4]{a} \left (\sqrt {a} e+\sqrt {c} d\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{4 \sqrt {2} c^{3/4} \left (a e^2+c d^2\right )}-\frac {\sqrt [4]{a} \left (\sqrt {a} e+\sqrt {c} d\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{4 \sqrt {2} c^{3/4} \left (a e^2+c d^2\right )}\right )}{a e^2+c d^2}-\frac {a \left (\frac {x^3 \left (a e+c d x^2\right )}{4 a c \left (a+c x^4\right )}-\frac {d x-a \left (\frac {\left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right ) \left (\frac {\sqrt {c} d}{\sqrt {a}}-3 e\right )}{2 \sqrt {c}}+\frac {\left (\frac {\sqrt {c} d}{\sqrt {a}}+3 e\right ) \left (\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )}{2 \sqrt {c}}\right )}{4 a c}\right )}{a e^2+c d^2}\) |
(d^2*((d^(3/2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(Sqrt[e]*(c*d^2 + a*e^2)) + (a ^(1/4)*(Sqrt[c]*d - Sqrt[a]*e)*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(2 *Sqrt[2]*c^(3/4)*(c*d^2 + a*e^2)) - (a^(1/4)*(Sqrt[c]*d - Sqrt[a]*e)*ArcTa n[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*c^(3/4)*(c*d^2 + a*e^2)) + (a^(1/4)*(Sqrt[c]*d + Sqrt[a]*e)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(4*Sqrt[2]*c^(3/4)*(c*d^2 + a*e^2)) - (a^(1/4)*(Sqrt[c]*d + Sqrt[a]*e)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(4*Sqr t[2]*c^(3/4)*(c*d^2 + a*e^2))))/(c*d^2 + a*e^2) - (a*((x^3*(a*e + c*d*x^2) )/(4*a*c*(a + c*x^4)) - (d*x - a*((((Sqrt[c]*d)/Sqrt[a] - 3*e)*(-(ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)]/(Sqrt[2]*a^(1/4)*c^(1/4))) + ArcTan[1 + (S qrt[2]*c^(1/4)*x)/a^(1/4)]/(Sqrt[2]*a^(1/4)*c^(1/4))))/(2*Sqrt[c]) + (((Sq rt[c]*d)/Sqrt[a] + 3*e)*(-1/2*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sq rt[c]*x^2]/(Sqrt[2]*a^(1/4)*c^(1/4)) + Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/ 4)*x + Sqrt[c]*x^2]/(2*Sqrt[2]*a^(1/4)*c^(1/4))))/(2*Sqrt[c])))/(4*a*c)))/ (c*d^2 + a*e^2)
3.3.52.3.1 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] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{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_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 2*(d/e), 2]}, Simp[e/(2*c) Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ e/(2*c) Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ -2*(d/e), 2]}, Simp[e/(2*c*q) Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Simp[e/(2*c*q) Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ a*c, 2]}, Simp[(d*q + a*e)/(2*a*c) Int[(q + c*x^2)/(a + c*x^4), x], x] + Simp[(d*q - a*e)/(2*a*c) Int[(q - c*x^2)/(a + c*x^4), x], x]] /; FreeQ[{a , c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && NegQ[(- a)*c]
Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)*((a_) + (c_.)*(x_)^4)^(p_.), x _Symbol] :> Simp[f*(f*x)^(m - 1)*(a + c*x^4)^(p + 1)*((a*e - c*d*x^2)/(4*a* c*(p + 1))), x] - Simp[f^2/(4*a*c*(p + 1)) Int[(f*x)^(m - 2)*(a + c*x^4)^ (p + 1)*(a*e*(m - 1) - c*d*(4*p + 4 + m + 1)*x^2), x], x] /; FreeQ[{a, c, d , e, f}, x] && LtQ[p, -1] && GtQ[m, 1] && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])
Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)*((a_) + (c_.)*(x_)^4)^(p_), x_ Symbol] :> Simp[e*f*(f*x)^(m - 1)*((a + c*x^4)^(p + 1)/(c*(m + 4*p + 3))), x] - Simp[f^2/(c*(m + 4*p + 3)) Int[(f*x)^(m - 2)*(a + c*x^4)^p*(a*e*(m - 1) - c*d*(m + 4*p + 3)*x^2), x], x] /; FreeQ[{a, c, d, e, f, p}, x] && GtQ [m, 1] && NeQ[m + 4*p + 3, 0] && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[ m])
Int[(((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.))/((a_) + (c_.)*(x_)^4), x_Symbol] :> Int[ExpandIntegrand[(f*x)^m*((d + e*x^2)^q/(a + c*x^4)), x], x] /; FreeQ[{a, c, d, e, f, m}, x] && IntegerQ[q] && IntegerQ[m]
Int[(((f_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^4)^(p_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(-a)*(f^4/(c*d^2 + a*e^2)) Int[(f*x)^(m - 4)*(d - e*x^2 )*(a + c*x^4)^p, x], x] + Simp[d^2*(f^4/(c*d^2 + a*e^2)) Int[(f*x)^(m - 4 )*((a + c*x^4)^(p + 1)/(d + e*x^2)), x], x] /; FreeQ[{a, c, d, e, f}, x] && LtQ[p, -1] && GtQ[m, 2]
Time = 0.46 (sec) , antiderivative size = 334, normalized size of antiderivative = 0.47
| method | result | size |
| default | \(-\frac {a \left (\frac {\frac {e \left (a \,e^{2}+c \,d^{2}\right ) x^{3}}{4 c}-\frac {d \left (a \,e^{2}+c \,d^{2}\right ) x}{4 c}}{c \,x^{4}+a}+\frac {\frac {\left (d \,e^{2} a +5 d^{3} c \right ) \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}{x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8 a}+\frac {\left (-3 a \,e^{3}-7 c \,d^{2} e \right ) \sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}{x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8 c \left (\frac {a}{c}\right )^{\frac {1}{4}}}}{4 c}\right )}{\left (a \,e^{2}+c \,d^{2}\right )^{2}}+\frac {d^{4} \arctan \left (\frac {e x}{\sqrt {e d}}\right )}{\left (a \,e^{2}+c \,d^{2}\right )^{2} \sqrt {e d}}\) | \(334\) |
| risch | \(\text {Expression too large to display}\) | \(1581\) |
-a/(a*e^2+c*d^2)^2*((1/4*e*(a*e^2+c*d^2)/c*x^3-1/4*d*(a*e^2+c*d^2)/c*x)/(c *x^4+a)+1/4/c*(1/8*(a*d*e^2+5*c*d^3)*(a/c)^(1/4)/a*2^(1/2)*(ln((x^2+(a/c)^ (1/4)*x*2^(1/2)+(a/c)^(1/2))/(x^2-(a/c)^(1/4)*x*2^(1/2)+(a/c)^(1/2)))+2*ar ctan(2^(1/2)/(a/c)^(1/4)*x+1)+2*arctan(2^(1/2)/(a/c)^(1/4)*x-1))+1/8*(-3*a *e^3-7*c*d^2*e)/c/(a/c)^(1/4)*2^(1/2)*(ln((x^2-(a/c)^(1/4)*x*2^(1/2)+(a/c) ^(1/2))/(x^2+(a/c)^(1/4)*x*2^(1/2)+(a/c)^(1/2)))+2*arctan(2^(1/2)/(a/c)^(1 /4)*x+1)+2*arctan(2^(1/2)/(a/c)^(1/4)*x-1))))+1/(a*e^2+c*d^2)^2*d^4/(e*d)^ (1/2)*arctan(e*x/(e*d)^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 4918 vs. \(2 (538) = 1076\).
Time = 9.62 (sec) , antiderivative size = 9856, normalized size of antiderivative = 13.84 \[ \int \frac {x^8}{\left (d+e x^2\right ) \left (a+c x^4\right )^2} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {x^8}{\left (d+e x^2\right ) \left (a+c x^4\right )^2} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {x^8}{\left (d+e x^2\right ) \left (a+c x^4\right )^2} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
Time = 0.29 (sec) , antiderivative size = 599, normalized size of antiderivative = 0.84 \[ \int \frac {x^8}{\left (d+e x^2\right ) \left (a+c x^4\right )^2} \, dx=\frac {d^{4} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{{\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \sqrt {d e}} - \frac {{\left (5 \, \left (a c^{3}\right )^{\frac {1}{4}} c^{3} d^{3} + \left (a c^{3}\right )^{\frac {1}{4}} a c^{2} d e^{2} - 7 \, \left (a c^{3}\right )^{\frac {3}{4}} c d^{2} e - 3 \, \left (a c^{3}\right )^{\frac {3}{4}} a e^{3}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{8 \, {\left (\sqrt {2} c^{6} d^{4} + 2 \, \sqrt {2} a c^{5} d^{2} e^{2} + \sqrt {2} a^{2} c^{4} e^{4}\right )}} - \frac {{\left (5 \, \left (a c^{3}\right )^{\frac {1}{4}} c^{3} d^{3} + \left (a c^{3}\right )^{\frac {1}{4}} a c^{2} d e^{2} - 7 \, \left (a c^{3}\right )^{\frac {3}{4}} c d^{2} e - 3 \, \left (a c^{3}\right )^{\frac {3}{4}} a e^{3}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{8 \, {\left (\sqrt {2} c^{6} d^{4} + 2 \, \sqrt {2} a c^{5} d^{2} e^{2} + \sqrt {2} a^{2} c^{4} e^{4}\right )}} - \frac {{\left (5 \, \left (a c^{3}\right )^{\frac {1}{4}} c^{3} d^{3} + \left (a c^{3}\right )^{\frac {1}{4}} a c^{2} d e^{2} + 7 \, \left (a c^{3}\right )^{\frac {3}{4}} c d^{2} e + 3 \, \left (a c^{3}\right )^{\frac {3}{4}} a e^{3}\right )} \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{16 \, {\left (\sqrt {2} c^{6} d^{4} + 2 \, \sqrt {2} a c^{5} d^{2} e^{2} + \sqrt {2} a^{2} c^{4} e^{4}\right )}} + \frac {{\left (5 \, \left (a c^{3}\right )^{\frac {1}{4}} c^{3} d^{3} + \left (a c^{3}\right )^{\frac {1}{4}} a c^{2} d e^{2} + 7 \, \left (a c^{3}\right )^{\frac {3}{4}} c d^{2} e + 3 \, \left (a c^{3}\right )^{\frac {3}{4}} a e^{3}\right )} \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{16 \, {\left (\sqrt {2} c^{6} d^{4} + 2 \, \sqrt {2} a c^{5} d^{2} e^{2} + \sqrt {2} a^{2} c^{4} e^{4}\right )}} - \frac {a e x^{3} - a d x}{4 \, {\left (c x^{4} + a\right )} {\left (c^{2} d^{2} + a c e^{2}\right )}} \]
d^4*arctan(e*x/sqrt(d*e))/((c^2*d^4 + 2*a*c*d^2*e^2 + a^2*e^4)*sqrt(d*e)) - 1/8*(5*(a*c^3)^(1/4)*c^3*d^3 + (a*c^3)^(1/4)*a*c^2*d*e^2 - 7*(a*c^3)^(3/ 4)*c*d^2*e - 3*(a*c^3)^(3/4)*a*e^3)*arctan(1/2*sqrt(2)*(2*x + sqrt(2)*(a/c )^(1/4))/(a/c)^(1/4))/(sqrt(2)*c^6*d^4 + 2*sqrt(2)*a*c^5*d^2*e^2 + sqrt(2) *a^2*c^4*e^4) - 1/8*(5*(a*c^3)^(1/4)*c^3*d^3 + (a*c^3)^(1/4)*a*c^2*d*e^2 - 7*(a*c^3)^(3/4)*c*d^2*e - 3*(a*c^3)^(3/4)*a*e^3)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(a/c)^(1/4))/(a/c)^(1/4))/(sqrt(2)*c^6*d^4 + 2*sqrt(2)*a*c^5*d^2 *e^2 + sqrt(2)*a^2*c^4*e^4) - 1/16*(5*(a*c^3)^(1/4)*c^3*d^3 + (a*c^3)^(1/4 )*a*c^2*d*e^2 + 7*(a*c^3)^(3/4)*c*d^2*e + 3*(a*c^3)^(3/4)*a*e^3)*log(x^2 + sqrt(2)*x*(a/c)^(1/4) + sqrt(a/c))/(sqrt(2)*c^6*d^4 + 2*sqrt(2)*a*c^5*d^2 *e^2 + sqrt(2)*a^2*c^4*e^4) + 1/16*(5*(a*c^3)^(1/4)*c^3*d^3 + (a*c^3)^(1/4 )*a*c^2*d*e^2 + 7*(a*c^3)^(3/4)*c*d^2*e + 3*(a*c^3)^(3/4)*a*e^3)*log(x^2 - sqrt(2)*x*(a/c)^(1/4) + sqrt(a/c))/(sqrt(2)*c^6*d^4 + 2*sqrt(2)*a*c^5*d^2 *e^2 + sqrt(2)*a^2*c^4*e^4) - 1/4*(a*e*x^3 - a*d*x)/((c*x^4 + a)*(c^2*d^2 + a*c*e^2))
Time = 9.06 (sec) , antiderivative size = 18343, normalized size of antiderivative = 25.76 \[ \int \frac {x^8}{\left (d+e x^2\right ) \left (a+c x^4\right )^2} \, dx=\text {Too large to display} \]
((a*d*x)/(4*c*(a*e^2 + c*d^2)) - (a*e*x^3)/(4*c*(a*e^2 + c*d^2)))/(a + c*x ^4) + atan(((((5120*a^2*c^8*d^13*e + 432*a^8*c^2*d*e^13 - 17232*a^3*c^7*d^ 11*e^3 - 37776*a^4*c^6*d^9*e^5 - 13600*a^5*c^5*d^7*e^7 + 4320*a^6*c^4*d^5* e^9 + 2928*a^7*c^3*d^3*e^11)/(256*(c^7*d^8 + a^4*c^3*e^8 + 4*a*c^6*d^6*e^2 + 6*a^2*c^5*d^4*e^4 + 4*a^3*c^4*d^2*e^6)) - (((81920*a^5*c^9*d^8*e^8 - 73 728*a^3*c^11*d^12*e^4 - 61440*a^4*c^10*d^10*e^6 - 20480*a^2*c^12*d^14*e^2 + 184320*a^6*c^8*d^6*e^10 + 122880*a^7*c^7*d^4*e^12 + 28672*a^8*c^6*d^2*e^ 14)/(256*(c^7*d^8 + a^4*c^3*e^8 + 4*a*c^6*d^6*e^2 + 6*a^2*c^5*d^4*e^4 + 4* a^3*c^4*d^2*e^6)) - (x*((25*c^3*d^6*(-a*c^7)^(1/2) - 9*a^3*e^6*(-a*c^7)^(1 /2) + 6*a^3*c^4*d*e^5 + 44*a^2*c^5*d^3*e^3 + 70*a*c^6*d^5*e - 39*a*c^2*d^4 *e^2*(-a*c^7)^(1/2) - 41*a^2*c*d^2*e^4*(-a*c^7)^(1/2))/(256*(c^11*d^8 + a^ 4*c^7*e^8 + 4*a*c^10*d^6*e^2 + 6*a^2*c^9*d^4*e^4 + 4*a^3*c^8*d^2*e^6)))^(1 /2)*(65536*a^9*c^7*e^17 - 65536*a^2*c^14*d^14*e^3 - 327680*a^3*c^13*d^12*e ^5 - 589824*a^4*c^12*d^10*e^7 - 327680*a^5*c^11*d^8*e^9 + 327680*a^6*c^10* d^6*e^11 + 589824*a^7*c^9*d^4*e^13 + 327680*a^8*c^8*d^2*e^15))/(128*(c^7*d ^8 + a^4*c^3*e^8 + 4*a*c^6*d^6*e^2 + 6*a^2*c^5*d^4*e^4 + 4*a^3*c^4*d^2*e^6 )))*((25*c^3*d^6*(-a*c^7)^(1/2) - 9*a^3*e^6*(-a*c^7)^(1/2) + 6*a^3*c^4*d*e ^5 + 44*a^2*c^5*d^3*e^3 + 70*a*c^6*d^5*e - 39*a*c^2*d^4*e^2*(-a*c^7)^(1/2) - 41*a^2*c*d^2*e^4*(-a*c^7)^(1/2))/(256*(c^11*d^8 + a^4*c^7*e^8 + 4*a*c^1 0*d^6*e^2 + 6*a^2*c^9*d^4*e^4 + 4*a^3*c^8*d^2*e^6)))^(1/2) + (x*(1920*a...