Integrand size = 22, antiderivative size = 685 \[ \int \frac {x^4}{\left (d+e x^2\right ) \left (a+c x^4\right )^2} \, dx=-\frac {x \left (d-e x^2\right )}{4 \left (c d^2+a e^2\right ) \left (a+c x^4\right )}+\frac {d^{3/2} e^{3/2} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\left (c d^2+a e^2\right )^2}-\frac {\sqrt [4]{c} d^2 \left (\sqrt {c} d-\sqrt {a} e\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} \left (c d^2+a e^2\right )^2}+\frac {\left (3 \sqrt {c} d-\sqrt {a} e\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{3/4} c^{3/4} \left (c d^2+a e^2\right )}+\frac {\sqrt [4]{c} d^2 \left (\sqrt {c} d-\sqrt {a} e\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} \left (c d^2+a e^2\right )^2}-\frac {\left (3 \sqrt {c} d-\sqrt {a} e\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{3/4} c^{3/4} \left (c d^2+a e^2\right )}-\frac {\sqrt [4]{c} d^2 \left (\sqrt {c} d+\sqrt {a} e\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} \left (c d^2+a e^2\right )^2}+\frac {\left (3 \sqrt {c} d+\sqrt {a} e\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{16 \sqrt {2} a^{3/4} c^{3/4} \left (c d^2+a e^2\right )}+\frac {\sqrt [4]{c} d^2 \left (\sqrt {c} d+\sqrt {a} e\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} \left (c d^2+a e^2\right )^2}-\frac {\left (3 \sqrt {c} d+\sqrt {a} e\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{16 \sqrt {2} a^{3/4} c^{3/4} \left (c d^2+a e^2\right )} \]
-1/4*x*(-e*x^2+d)/(a*e^2+c*d^2)/(c*x^4+a)+d^(3/2)*e^(3/2)*arctan(x*e^(1/2) /d^(1/2))/(a*e^2+c*d^2)^2+1/4*c^(1/4)*d^2*arctan(-1+c^(1/4)*x*2^(1/2)/a^(1 /4))*(-e*a^(1/2)+d*c^(1/2))/a^(3/4)/(a*e^2+c*d^2)^2*2^(1/2)+1/4*c^(1/4)*d^ 2*arctan(1+c^(1/4)*x*2^(1/2)/a^(1/4))*(-e*a^(1/2)+d*c^(1/2))/a^(3/4)/(a*e^ 2+c*d^2)^2*2^(1/2)-1/8*c^(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))/a^(3/4)/(a*e^2+c*d^2)^2*2^(1/2)+1/8*c^(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))/a^(3/4)/(a*e^2+c*d^2)^2*2^(1/2)-1/16*arctan(-1+c^(1/4)*x*2^(1/2)/a^( 1/4))*(-e*a^(1/2)+3*d*c^(1/2))/a^(3/4)/c^(3/4)/(a*e^2+c*d^2)*2^(1/2)-1/16* arctan(1+c^(1/4)*x*2^(1/2)/a^(1/4))*(-e*a^(1/2)+3*d*c^(1/2))/a^(3/4)/c^(3/ 4)/(a*e^2+c*d^2)*2^(1/2)+1/32*ln(-a^(1/4)*c^(1/4)*x*2^(1/2)+a^(1/2)+x^2*c^ (1/2))*(e*a^(1/2)+3*d*c^(1/2))/a^(3/4)/c^(3/4)/(a*e^2+c*d^2)*2^(1/2)-1/32* ln(a^(1/4)*c^(1/4)*x*2^(1/2)+a^(1/2)+x^2*c^(1/2))*(e*a^(1/2)+3*d*c^(1/2))/ a^(3/4)/c^(3/4)/(a*e^2+c*d^2)*2^(1/2)
Time = 0.18 (sec) , antiderivative size = 423, normalized size of antiderivative = 0.62 \[ \int \frac {x^4}{\left (d+e x^2\right ) \left (a+c x^4\right )^2} \, dx=\frac {\frac {8 \left (c d^2+a e^2\right ) \left (-d x+e x^3\right )}{a+c x^4}+32 d^{3/2} e^{3/2} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )-\frac {2 \sqrt {2} \left (c^{3/2} d^3-3 \sqrt {a} c d^2 e-3 a \sqrt {c} d e^2+a^{3/2} e^3\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{a^{3/4} c^{3/4}}+\frac {2 \sqrt {2} \left (c^{3/2} d^3-3 \sqrt {a} c d^2 e-3 a \sqrt {c} d e^2+a^{3/2} e^3\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{a^{3/4} c^{3/4}}+\frac {\sqrt {2} \left (-c^{3/2} d^3-3 \sqrt {a} c d^2 e+3 a \sqrt {c} d e^2+a^{3/2} e^3\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{a^{3/4} c^{3/4}}+\frac {\sqrt {2} \left (c^{3/2} d^3+3 \sqrt {a} c d^2 e-3 a \sqrt {c} d e^2-a^{3/2} e^3\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{a^{3/4} c^{3/4}}}{32 \left (c d^2+a e^2\right )^2} \]
((8*(c*d^2 + a*e^2)*(-(d*x) + e*x^3))/(a + c*x^4) + 32*d^(3/2)*e^(3/2)*Arc Tan[(Sqrt[e]*x)/Sqrt[d]] - (2*Sqrt[2]*(c^(3/2)*d^3 - 3*Sqrt[a]*c*d^2*e - 3 *a*Sqrt[c]*d*e^2 + a^(3/2)*e^3)*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/( a^(3/4)*c^(3/4)) + (2*Sqrt[2]*(c^(3/2)*d^3 - 3*Sqrt[a]*c*d^2*e - 3*a*Sqrt[ c]*d*e^2 + a^(3/2)*e^3)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(a^(3/4)* c^(3/4)) + (Sqrt[2]*(-(c^(3/2)*d^3) - 3*Sqrt[a]*c*d^2*e + 3*a*Sqrt[c]*d*e^ 2 + a^(3/2)*e^3)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/( a^(3/4)*c^(3/4)) + (Sqrt[2]*(c^(3/2)*d^3 + 3*Sqrt[a]*c*d^2*e - 3*a*Sqrt[c] *d*e^2 - a^(3/2)*e^3)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^ 2])/(a^(3/4)*c^(3/4)))/(32*(c*d^2 + a*e^2)^2)
Time = 0.92 (sec) , antiderivative size = 635, normalized size of antiderivative = 0.93, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.636, Rules used = {1651, 1485, 1493, 25, 1482, 27, 1476, 1082, 217, 1479, 25, 27, 1103, 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.
| \(\displaystyle \int \frac {x^4}{\left (a+c x^4\right )^2 \left (d+e x^2\right )} \, dx\) |
| \(\Big \downarrow \) 1651 |
| \(\displaystyle \frac {d^2 \int \frac {1}{\left (e x^2+d\right ) \left (c x^4+a\right )}dx}{a e^2+c d^2}-\frac {a \int \frac {d-e x^2}{\left (c x^4+a\right )^2}dx}{a e^2+c d^2}\) |
| \(\Big \downarrow \) 1485 |
| \(\displaystyle \frac {d^2 \int \left (\frac {e^2}{\left (c d^2+a e^2\right ) \left (e x^2+d\right )}+\frac {c \left (d-e x^2\right )}{\left (c d^2+a e^2\right ) \left (c x^4+a\right )}\right )dx}{a e^2+c d^2}-\frac {a \int \frac {d-e x^2}{\left (c x^4+a\right )^2}dx}{a e^2+c d^2}\) |
| \(\Big \downarrow \) 1493 |
| \(\displaystyle \frac {d^2 \int \left (\frac {e^2}{\left (c d^2+a e^2\right ) \left (e x^2+d\right )}+\frac {c \left (d-e x^2\right )}{\left (c d^2+a e^2\right ) \left (c x^4+a\right )}\right )dx}{a e^2+c d^2}-\frac {a \left (\frac {x \left (d-e x^2\right )}{4 a \left (a+c x^4\right )}-\frac {\int -\frac {3 d-e x^2}{c x^4+a}dx}{4 a}\right )}{a e^2+c d^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {d^2 \int \left (\frac {e^2}{\left (c d^2+a e^2\right ) \left (e x^2+d\right )}+\frac {c \left (d-e x^2\right )}{\left (c d^2+a e^2\right ) \left (c x^4+a\right )}\right )dx}{a e^2+c d^2}-\frac {a \left (\frac {\int \frac {3 d-e x^2}{c x^4+a}dx}{4 a}+\frac {x \left (d-e x^2\right )}{4 a \left (a+c x^4\right )}\right )}{a e^2+c d^2}\) |
| \(\Big \downarrow \) 1482 |
| \(\displaystyle \frac {d^2 \int \left (\frac {e^2}{\left (c d^2+a e^2\right ) \left (e x^2+d\right )}+\frac {c \left (d-e x^2\right )}{\left (c d^2+a e^2\right ) \left (c x^4+a\right )}\right )dx}{a e^2+c d^2}-\frac {a \left (\frac {\frac {\left (\frac {3 \sqrt {c} d}{\sqrt {a}}+e\right ) \int \frac {\sqrt {c} \left (\sqrt {a}-\sqrt {c} x^2\right )}{c x^4+a}dx}{2 c}+\frac {\left (\frac {3 \sqrt {c} d}{\sqrt {a}}-e\right ) \int \frac {\sqrt {c} \left (\sqrt {c} x^2+\sqrt {a}\right )}{c x^4+a}dx}{2 c}}{4 a}+\frac {x \left (d-e x^2\right )}{4 a \left (a+c x^4\right )}\right )}{a e^2+c d^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {d^2 \int \left (\frac {e^2}{\left (c d^2+a e^2\right ) \left (e x^2+d\right )}+\frac {c \left (d-e x^2\right )}{\left (c d^2+a e^2\right ) \left (c x^4+a\right )}\right )dx}{a e^2+c d^2}-\frac {a \left (\frac {\frac {\left (\frac {3 \sqrt {c} d}{\sqrt {a}}+e\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{c x^4+a}dx}{2 \sqrt {c}}+\frac {\left (\frac {3 \sqrt {c} d}{\sqrt {a}}-e\right ) \int \frac {\sqrt {c} x^2+\sqrt {a}}{c x^4+a}dx}{2 \sqrt {c}}}{4 a}+\frac {x \left (d-e x^2\right )}{4 a \left (a+c x^4\right )}\right )}{a e^2+c d^2}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {d^2 \int \left (\frac {e^2}{\left (c d^2+a e^2\right ) \left (e x^2+d\right )}+\frac {c \left (d-e x^2\right )}{\left (c d^2+a e^2\right ) \left (c x^4+a\right )}\right )dx}{a e^2+c d^2}-\frac {a \left (\frac {\frac {\left (\frac {3 \sqrt {c} d}{\sqrt {a}}-e\right ) \left (\frac {\int \frac {1}{x^2-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}}dx}{2 \sqrt {c}}+\frac {\int \frac {1}{x^2+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}}dx}{2 \sqrt {c}}\right )}{2 \sqrt {c}}+\frac {\left (\frac {3 \sqrt {c} d}{\sqrt {a}}+e\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{c x^4+a}dx}{2 \sqrt {c}}}{4 a}+\frac {x \left (d-e x^2\right )}{4 a \left (a+c x^4\right )}\right )}{a e^2+c d^2}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {d^2 \int \left (\frac {e^2}{\left (c d^2+a e^2\right ) \left (e x^2+d\right )}+\frac {c \left (d-e x^2\right )}{\left (c d^2+a e^2\right ) \left (c x^4+a\right )}\right )dx}{a e^2+c d^2}-\frac {a \left (\frac {\frac {\left (\frac {3 \sqrt {c} d}{\sqrt {a}}+e\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{c x^4+a}dx}{2 \sqrt {c}}+\frac {\left (\frac {3 \sqrt {c} d}{\sqrt {a}}-e\right ) \left (\frac {\int \frac {1}{-\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )^2-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\int \frac {1}{-\left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )^2-1}d\left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )}{2 \sqrt {c}}}{4 a}+\frac {x \left (d-e x^2\right )}{4 a \left (a+c x^4\right )}\right )}{a e^2+c d^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {d^2 \int \left (\frac {e^2}{\left (c d^2+a e^2\right ) \left (e x^2+d\right )}+\frac {c \left (d-e x^2\right )}{\left (c d^2+a e^2\right ) \left (c x^4+a\right )}\right )dx}{a e^2+c d^2}-\frac {a \left (\frac {\frac {\left (\frac {3 \sqrt {c} d}{\sqrt {a}}+e\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{c x^4+a}dx}{2 \sqrt {c}}+\frac {\left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right ) \left (\frac {3 \sqrt {c} d}{\sqrt {a}}-e\right )}{2 \sqrt {c}}}{4 a}+\frac {x \left (d-e x^2\right )}{4 a \left (a+c x^4\right )}\right )}{a e^2+c d^2}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {d^2 \int \left (\frac {e^2}{\left (c d^2+a e^2\right ) \left (e x^2+d\right )}+\frac {c \left (d-e x^2\right )}{\left (c d^2+a e^2\right ) \left (c x^4+a\right )}\right )dx}{a e^2+c d^2}-\frac {a \left (\frac {\frac {\left (\frac {3 \sqrt {c} d}{\sqrt {a}}+e\right ) \left (-\frac {\int -\frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{c} x}{\sqrt [4]{c} \left (x^2-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}dx}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{c} x+\sqrt [4]{a}\right )}{\sqrt [4]{c} \left (x^2+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}dx}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )}{2 \sqrt {c}}+\frac {\left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right ) \left (\frac {3 \sqrt {c} d}{\sqrt {a}}-e\right )}{2 \sqrt {c}}}{4 a}+\frac {x \left (d-e x^2\right )}{4 a \left (a+c x^4\right )}\right )}{a e^2+c d^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {d^2 \int \left (\frac {e^2}{\left (c d^2+a e^2\right ) \left (e x^2+d\right )}+\frac {c \left (d-e x^2\right )}{\left (c d^2+a e^2\right ) \left (c x^4+a\right )}\right )dx}{a e^2+c d^2}-\frac {a \left (\frac {\frac {\left (\frac {3 \sqrt {c} d}{\sqrt {a}}+e\right ) \left (\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{c} x}{\sqrt [4]{c} \left (x^2-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}dx}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{c} x+\sqrt [4]{a}\right )}{\sqrt [4]{c} \left (x^2+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}dx}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )}{2 \sqrt {c}}+\frac {\left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right ) \left (\frac {3 \sqrt {c} d}{\sqrt {a}}-e\right )}{2 \sqrt {c}}}{4 a}+\frac {x \left (d-e x^2\right )}{4 a \left (a+c x^4\right )}\right )}{a e^2+c d^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {d^2 \int \left (\frac {e^2}{\left (c d^2+a e^2\right ) \left (e x^2+d\right )}+\frac {c \left (d-e x^2\right )}{\left (c d^2+a e^2\right ) \left (c x^4+a\right )}\right )dx}{a e^2+c d^2}-\frac {a \left (\frac {\frac {\left (\frac {3 \sqrt {c} d}{\sqrt {a}}+e\right ) \left (\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{c} x}{x^2-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}}dx}{2 \sqrt {2} \sqrt [4]{a} \sqrt {c}}+\frac {\int \frac {\sqrt {2} \sqrt [4]{c} x+\sqrt [4]{a}}{x^2+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}}dx}{2 \sqrt [4]{a} \sqrt {c}}\right )}{2 \sqrt {c}}+\frac {\left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right ) \left (\frac {3 \sqrt {c} d}{\sqrt {a}}-e\right )}{2 \sqrt {c}}}{4 a}+\frac {x \left (d-e x^2\right )}{4 a \left (a+c x^4\right )}\right )}{a e^2+c d^2}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {d^2 \int \left (\frac {e^2}{\left (c d^2+a e^2\right ) \left (e x^2+d\right )}+\frac {c \left (d-e x^2\right )}{\left (c d^2+a e^2\right ) \left (c x^4+a\right )}\right )dx}{a e^2+c d^2}-\frac {a \left (\frac {\frac {\left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right ) \left (\frac {3 \sqrt {c} d}{\sqrt {a}}-e\right )}{2 \sqrt {c}}+\frac {\left (\frac {3 \sqrt {c} d}{\sqrt {a}}+e\right ) \left (\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )}{2 \sqrt {c}}}{4 a}+\frac {x \left (d-e x^2\right )}{4 a \left (a+c x^4\right )}\right )}{a e^2+c d^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {d^2 \left (-\frac {\sqrt [4]{c} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right ) \left (\sqrt {c} d-\sqrt {a} e\right )}{2 \sqrt {2} a^{3/4} \left (a e^2+c d^2\right )}+\frac {\sqrt [4]{c} \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right ) \left (\sqrt {c} d-\sqrt {a} e\right )}{2 \sqrt {2} a^{3/4} \left (a e^2+c d^2\right )}-\frac {\sqrt [4]{c} \left (\sqrt {a} e+\sqrt {c} d\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} \left (a e^2+c d^2\right )}+\frac {\sqrt [4]{c} \left (\sqrt {a} e+\sqrt {c} d\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} \left (a e^2+c d^2\right )}+\frac {e^{3/2} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} \left (a e^2+c d^2\right )}\right )}{a e^2+c d^2}-\frac {a \left (\frac {\frac {\left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right ) \left (\frac {3 \sqrt {c} d}{\sqrt {a}}-e\right )}{2 \sqrt {c}}+\frac {\left (\frac {3 \sqrt {c} d}{\sqrt {a}}+e\right ) \left (\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )}{2 \sqrt {c}}}{4 a}+\frac {x \left (d-e x^2\right )}{4 a \left (a+c x^4\right )}\right )}{a e^2+c d^2}\) |
(d^2*((e^(3/2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(Sqrt[d]*(c*d^2 + a*e^2)) - (c ^(1/4)*(Sqrt[c]*d - Sqrt[a]*e)*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(2 *Sqrt[2]*a^(3/4)*(c*d^2 + a*e^2)) + (c^(1/4)*(Sqrt[c]*d - Sqrt[a]*e)*ArcTa n[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(3/4)*(c*d^2 + a*e^2)) - (c^(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]*a^(3/4)*(c*d^2 + a*e^2)) + (c^(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]*a^(3/4)*(c*d^2 + a*e^2))))/(c*d^2 + a*e^2) - (a*((x*(d - e*x^2))/(4*a *(a + c*x^4)) + ((((3*Sqrt[c]*d)/Sqrt[a] - e)*(-(ArcTan[1 - (Sqrt[2]*c^(1/ 4)*x)/a^(1/4)]/(Sqrt[2]*a^(1/4)*c^(1/4))) + ArcTan[1 + (Sqrt[2]*c^(1/4)*x) /a^(1/4)]/(Sqrt[2]*a^(1/4)*c^(1/4))))/(2*Sqrt[c]) + (((3*Sqrt[c]*d)/Sqrt[a ] + e)*(-1/2*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[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*d^2 + a*e^2)
3.3.54.3.1 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[((d_) + (e_.)*(x_)^2)^(q_)/((a_) + (c_.)*(x_)^4), x_Symbol] :> Int[Expa ndIntegrand[(d + e*x^2)^q/(a + c*x^4), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && IntegerQ[q]
Int[((d_) + (e_.)*(x_)^2)*((a_) + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[(-x )*(d + e*x^2)*((a + c*x^4)^(p + 1)/(4*a*(p + 1))), x] + Simp[1/(4*a*(p + 1) ) Int[Simp[d*(4*p + 5) + e*(4*p + 7)*x^2, x]*(a + c*x^4)^(p + 1), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && Integer Q[2*p]
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 = 0.48 (sec) , antiderivative size = 327, normalized size of antiderivative = 0.48
| method | result | size |
| default | \(-\frac {\frac {\left (-\frac {1}{4} a \,e^{3}-\frac {1}{4} c \,d^{2} e \right ) x^{3}+\left (\frac {1}{4} d \,e^{2} a +\frac {1}{4} d^{3} c \right ) x}{c \,x^{4}+a}+\frac {\left (3 d \,e^{2} a -d^{3} c \right ) \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}{x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{32 a}+\frac {\left (-a \,e^{3}+3 c \,d^{2} e \right ) \sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}{x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{32 c \left (\frac {a}{c}\right )^{\frac {1}{4}}}}{\left (a \,e^{2}+c \,d^{2}\right )^{2}}+\frac {d^{2} e^{2} \arctan \left (\frac {e x}{\sqrt {e d}}\right )}{\left (a \,e^{2}+c \,d^{2}\right )^{2} \sqrt {e d}}\) | \(327\) |
| risch | \(\text {Expression too large to display}\) | \(1401\) |
-1/(a*e^2+c*d^2)^2*(((-1/4*a*e^3-1/4*c*d^2*e)*x^3+(1/4*d*e^2*a+1/4*d^3*c)* x)/(c*x^4+a)+1/32*(3*a*d*e^2-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*arc tan(2^(1/2)/(a/c)^(1/4)*x+1)+2*arctan(2^(1/2)/(a/c)^(1/4)*x-1))+1/32*(-a*e ^3+3*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)))+d^2*e^2/(a*e^2+c*d^2)^2/(e*d)^( 1/2)*arctan(e*x/(e*d)^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 4829 vs. \(2 (514) = 1028\).
Time = 5.78 (sec) , antiderivative size = 9678, normalized size of antiderivative = 14.13 \[ \int \frac {x^4}{\left (d+e x^2\right ) \left (a+c x^4\right )^2} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {x^4}{\left (d+e x^2\right ) \left (a+c x^4\right )^2} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {x^4}{\left (d+e x^2\right ) \left (a+c x^4\right )^2} \, dx=\text {Exception raised: ValueError} \]
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 = 607, normalized size of antiderivative = 0.89 \[ \int \frac {x^4}{\left (d+e x^2\right ) \left (a+c x^4\right )^2} \, dx=\frac {d^{2} e^{2} \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 (\left (a c^{3}\right )^{\frac {1}{4}} c^{3} d^{3} - 3 \, \left (a c^{3}\right )^{\frac {1}{4}} a c^{2} d e^{2} - 3 \, \left (a c^{3}\right )^{\frac {3}{4}} c d^{2} e + \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} a c^{5} d^{4} + 2 \, \sqrt {2} a^{2} c^{4} d^{2} e^{2} + \sqrt {2} a^{3} c^{3} e^{4}\right )}} + \frac {{\left (\left (a c^{3}\right )^{\frac {1}{4}} c^{3} d^{3} - 3 \, \left (a c^{3}\right )^{\frac {1}{4}} a c^{2} d e^{2} - 3 \, \left (a c^{3}\right )^{\frac {3}{4}} c d^{2} e + \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} a c^{5} d^{4} + 2 \, \sqrt {2} a^{2} c^{4} d^{2} e^{2} + \sqrt {2} a^{3} c^{3} e^{4}\right )}} + \frac {{\left (\left (a c^{3}\right )^{\frac {1}{4}} c^{3} d^{3} - 3 \, \left (a c^{3}\right )^{\frac {1}{4}} a c^{2} d e^{2} + 3 \, \left (a c^{3}\right )^{\frac {3}{4}} c d^{2} e - \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} a c^{5} d^{4} + 2 \, \sqrt {2} a^{2} c^{4} d^{2} e^{2} + \sqrt {2} a^{3} c^{3} e^{4}\right )}} - \frac {{\left (\left (a c^{3}\right )^{\frac {1}{4}} c^{3} d^{3} - 3 \, \left (a c^{3}\right )^{\frac {1}{4}} a c^{2} d e^{2} + 3 \, \left (a c^{3}\right )^{\frac {3}{4}} c d^{2} e - \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} a c^{5} d^{4} + 2 \, \sqrt {2} a^{2} c^{4} d^{2} e^{2} + \sqrt {2} a^{3} c^{3} e^{4}\right )}} + \frac {e x^{3} - d x}{4 \, {\left (c x^{4} + a\right )} {\left (c d^{2} + a e^{2}\right )}} \]
d^2*e^2*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*((a*c^3)^(1/4)*c^3*d^3 - 3*(a*c^3)^(1/4)*a*c^2*d*e^2 - 3*(a*c^3) ^(3/4)*c*d^2*e + (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)*a*c^5*d^4 + 2*sqrt(2)*a^2*c^4*d^2*e^2 + s qrt(2)*a^3*c^3*e^4) + 1/8*((a*c^3)^(1/4)*c^3*d^3 - 3*(a*c^3)^(1/4)*a*c^2*d *e^2 - 3*(a*c^3)^(3/4)*c*d^2*e + (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)*a*c^5*d^4 + 2*sqrt(2)*a^2 *c^4*d^2*e^2 + sqrt(2)*a^3*c^3*e^4) + 1/16*((a*c^3)^(1/4)*c^3*d^3 - 3*(a*c ^3)^(1/4)*a*c^2*d*e^2 + 3*(a*c^3)^(3/4)*c*d^2*e - (a*c^3)^(3/4)*a*e^3)*log (x^2 + sqrt(2)*x*(a/c)^(1/4) + sqrt(a/c))/(sqrt(2)*a*c^5*d^4 + 2*sqrt(2)*a ^2*c^4*d^2*e^2 + sqrt(2)*a^3*c^3*e^4) - 1/16*((a*c^3)^(1/4)*c^3*d^3 - 3*(a *c^3)^(1/4)*a*c^2*d*e^2 + 3*(a*c^3)^(3/4)*c*d^2*e - (a*c^3)^(3/4)*a*e^3)*l og(x^2 - sqrt(2)*x*(a/c)^(1/4) + sqrt(a/c))/(sqrt(2)*a*c^5*d^4 + 2*sqrt(2) *a^2*c^4*d^2*e^2 + sqrt(2)*a^3*c^3*e^4) + 1/4*(e*x^3 - d*x)/((c*x^4 + a)*( c*d^2 + a*e^2))
Time = 10.46 (sec) , antiderivative size = 17180, normalized size of antiderivative = 25.08 \[ \int \frac {x^4}{\left (d+e x^2\right ) \left (a+c x^4\right )^2} \, dx=\text {Too large to display} \]
- atan(((((((28672*a^2*c^8*d^10*e^4 - 4096*a*c^9*d^12*e^2 + 155648*a^3*c^7 *d^8*e^6 + 253952*a^4*c^6*d^6*e^8 + 176128*a^5*c^5*d^4*e^10 + 45056*a^6*c^ 4*d^2*e^12)/(256*(a^3*e^6 + c^3*d^6 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4)) - (x*((a^3*e^6*(-a^3*c^3)^(1/2) - c^3*d^6*(-a^3*c^3)^(1/2) + 6*a^2*c^4*d^5 *e + 6*a^4*c^2*d*e^5 - 20*a^3*c^3*d^3*e^3 + 15*a*c^2*d^4*e^2*(-a^3*c^3)^(1 /2) - 15*a^2*c*d^2*e^4*(-a^3*c^3)^(1/2))/(256*(a^3*c^7*d^8 + a^7*c^3*e^8 + 4*a^4*c^6*d^6*e^2 + 6*a^5*c^5*d^4*e^4 + 4*a^6*c^4*d^2*e^6)))^(1/2)*(65536 *a^9*c^4*e^17 - 65536*a^2*c^11*d^14*e^3 - 327680*a^3*c^10*d^12*e^5 - 58982 4*a^4*c^9*d^10*e^7 - 327680*a^5*c^8*d^8*e^9 + 327680*a^6*c^7*d^6*e^11 + 58 9824*a^7*c^6*d^4*e^13 + 327680*a^8*c^5*d^2*e^15))/(128*(a^4*e^8 + c^4*d^8 + 4*a*c^3*d^6*e^2 + 4*a^3*c*d^2*e^6 + 6*a^2*c^2*d^4*e^4)))*((a^3*e^6*(-a^3 *c^3)^(1/2) - c^3*d^6*(-a^3*c^3)^(1/2) + 6*a^2*c^4*d^5*e + 6*a^4*c^2*d*e^5 - 20*a^3*c^3*d^3*e^3 + 15*a*c^2*d^4*e^2*(-a^3*c^3)^(1/2) - 15*a^2*c*d^2*e ^4*(-a^3*c^3)^(1/2))/(256*(a^3*c^7*d^8 + a^7*c^3*e^8 + 4*a^4*c^6*d^6*e^2 + 6*a^5*c^5*d^4*e^4 + 4*a^6*c^4*d^2*e^6)))^(1/2) + (x*(256*a*c^8*d^11*e^4 - 128*c^9*d^13*e^2 + 2944*a^6*c^3*d*e^14 + 21632*a^2*c^7*d^9*e^6 + 32256*a^ 3*c^6*d^7*e^8 + 4224*a^4*c^5*d^5*e^10 - 3840*a^5*c^4*d^3*e^12))/(128*(a^4* e^8 + c^4*d^8 + 4*a*c^3*d^6*e^2 + 4*a^3*c*d^2*e^6 + 6*a^2*c^2*d^4*e^4)))*( (a^3*e^6*(-a^3*c^3)^(1/2) - c^3*d^6*(-a^3*c^3)^(1/2) + 6*a^2*c^4*d^5*e + 6 *a^4*c^2*d*e^5 - 20*a^3*c^3*d^3*e^3 + 15*a*c^2*d^4*e^2*(-a^3*c^3)^(1/2)...