Integrand size = 37, antiderivative size = 241 \[ \int \frac {1}{(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\frac {2}{7 \left (c d^2-a e^2\right ) (d+e x)^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {16 c d}{35 \left (c d^2-a e^2\right )^2 (d+e x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {32 c^2 d^2}{35 \left (c d^2-a e^2\right )^3 (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {128 c^3 d^3 \left (c d^2+a e^2+2 c d e x\right )}{35 \left (c d^2-a e^2\right )^5 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \]
2/7/(-a*e^2+c*d^2)/(e*x+d)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+16/35 *c*d/(-a*e^2+c*d^2)^2/(e*x+d)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+32 /35*c^2*d^2/(-a*e^2+c*d^2)^3/(e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/ 2)-128/35*c^3*d^3*(2*c*d*e*x+a*e^2+c*d^2)/(-a*e^2+c*d^2)^5/(a*d*e+(a*e^2+c *d^2)*x+c*d*e*x^2)^(1/2)
Time = 0.23 (sec) , antiderivative size = 193, normalized size of antiderivative = 0.80 \[ \int \frac {1}{(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=-\frac {2 \left (-5 a^4 e^8+4 a^3 c d e^6 (7 d+2 e x)-2 a^2 c^2 d^2 e^4 \left (35 d^2+28 d e x+8 e^2 x^2\right )+4 a c^3 d^3 e^2 \left (35 d^3+70 d^2 e x+56 d e^2 x^2+16 e^3 x^3\right )+c^4 d^4 \left (35 d^4+280 d^3 e x+560 d^2 e^2 x^2+448 d e^3 x^3+128 e^4 x^4\right )\right )}{35 \left (c d^2-a e^2\right )^5 (d+e x)^3 \sqrt {(a e+c d x) (d+e x)}} \]
(-2*(-5*a^4*e^8 + 4*a^3*c*d*e^6*(7*d + 2*e*x) - 2*a^2*c^2*d^2*e^4*(35*d^2 + 28*d*e*x + 8*e^2*x^2) + 4*a*c^3*d^3*e^2*(35*d^3 + 70*d^2*e*x + 56*d*e^2* x^2 + 16*e^3*x^3) + c^4*d^4*(35*d^4 + 280*d^3*e*x + 560*d^2*e^2*x^2 + 448* d*e^3*x^3 + 128*e^4*x^4)))/(35*(c*d^2 - a*e^2)^5*(d + e*x)^3*Sqrt[(a*e + c *d*x)*(d + e*x)])
Time = 0.39 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.12, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.108, Rules used = {1129, 1129, 1129, 1088}
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.
| \(\displaystyle \int \frac {1}{(d+e x)^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1129 |
| \(\displaystyle \frac {8 c d \int \frac {1}{(d+e x)^2 \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}}dx}{7 \left (c d^2-a e^2\right )}+\frac {2}{7 (d+e x)^3 \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\) |
| \(\Big \downarrow \) 1129 |
| \(\displaystyle \frac {8 c d \left (\frac {6 c d \int \frac {1}{(d+e x) \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}}dx}{5 \left (c d^2-a e^2\right )}+\frac {2}{5 (d+e x)^2 \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{7 \left (c d^2-a e^2\right )}+\frac {2}{7 (d+e x)^3 \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\) |
| \(\Big \downarrow \) 1129 |
| \(\displaystyle \frac {8 c d \left (\frac {6 c d \left (\frac {4 c d \int \frac {1}{\left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}}dx}{3 \left (c d^2-a e^2\right )}+\frac {2}{3 (d+e x) \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{5 \left (c d^2-a e^2\right )}+\frac {2}{5 (d+e x)^2 \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{7 \left (c d^2-a e^2\right )}+\frac {2}{7 (d+e x)^3 \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\) |
| \(\Big \downarrow \) 1088 |
| \(\displaystyle \frac {8 c d \left (\frac {6 c d \left (\frac {2}{3 (d+e x) \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {8 c d \left (a e^2+c d^2+2 c d e x\right )}{3 \left (c d^2-a e^2\right )^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{5 \left (c d^2-a e^2\right )}+\frac {2}{5 (d+e x)^2 \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{7 \left (c d^2-a e^2\right )}+\frac {2}{7 (d+e x)^3 \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\) |
2/(7*(c*d^2 - a*e^2)*(d + e*x)^3*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^ 2]) + (8*c*d*(2/(5*(c*d^2 - a*e^2)*(d + e*x)^2*Sqrt[a*d*e + (c*d^2 + a*e^2 )*x + c*d*e*x^2]) + (6*c*d*(2/(3*(c*d^2 - a*e^2)*(d + e*x)*Sqrt[a*d*e + (c *d^2 + a*e^2)*x + c*d*e*x^2]) - (8*c*d*(c*d^2 + a*e^2 + 2*c*d*e*x))/(3*(c* d^2 - a*e^2)^3*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])))/(5*(c*d^2 - a*e^2))))/(7*(c*d^2 - a*e^2))
3.20.62.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[-2*((b + 2*c*x)/((b^2 - 4*a*c)*Sqrt[a + b*x + c*x^2])), x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[(-e)*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/((m + p + 1)*(2* c*d - b*e))), x] + Simp[c*(Simplify[m + 2*p + 2]/((m + p + 1)*(2*c*d - b*e) )) Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d , e, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && ILtQ[Simplify[m + 2*p + 2], 0]
Time = 2.80 (sec) , antiderivative size = 307, normalized size of antiderivative = 1.27
| method | result | size |
| gosper | \(-\frac {2 \left (c d x +a e \right ) \left (-128 c^{4} d^{4} e^{4} x^{4}-64 a \,c^{3} d^{3} e^{5} x^{3}-448 c^{4} d^{5} e^{3} x^{3}+16 a^{2} c^{2} d^{2} e^{6} x^{2}-224 a \,c^{3} d^{4} e^{4} x^{2}-560 c^{4} d^{6} e^{2} x^{2}-8 a^{3} c d \,e^{7} x +56 a^{2} c^{2} d^{3} e^{5} x -280 a \,c^{3} d^{5} e^{3} x -280 c^{4} d^{7} e x +5 a^{4} e^{8}-28 a^{3} c \,d^{2} e^{6}+70 a^{2} c^{2} d^{4} e^{4}-140 a \,c^{3} d^{6} e^{2}-35 c^{4} d^{8}\right )}{35 \left (e x +d \right )^{2} \left (a^{5} e^{10}-5 a^{4} c \,d^{2} e^{8}+10 a^{3} c^{2} d^{4} e^{6}-10 a^{2} c^{3} d^{6} e^{4}+5 a \,c^{4} d^{8} e^{2}-c^{5} d^{10}\right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {3}{2}}}\) | \(307\) |
| default | \(\frac {-\frac {2}{7 \left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{3} \sqrt {c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}-\frac {8 c d e \left (-\frac {2}{5 \left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{2} \sqrt {c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}-\frac {6 c d e \left (-\frac {2}{3 \left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right ) \sqrt {c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}+\frac {8 c d e \left (2 c d e \left (x +\frac {d}{e}\right )+e^{2} a -c \,d^{2}\right )}{3 \left (e^{2} a -c \,d^{2}\right )^{3} \sqrt {c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}\right )}{5 \left (e^{2} a -c \,d^{2}\right )}\right )}{7 \left (e^{2} a -c \,d^{2}\right )}}{e^{3}}\) | \(308\) |
| trager | \(-\frac {2 \left (-128 c^{4} d^{4} e^{4} x^{4}-64 a \,c^{3} d^{3} e^{5} x^{3}-448 c^{4} d^{5} e^{3} x^{3}+16 a^{2} c^{2} d^{2} e^{6} x^{2}-224 a \,c^{3} d^{4} e^{4} x^{2}-560 c^{4} d^{6} e^{2} x^{2}-8 a^{3} c d \,e^{7} x +56 a^{2} c^{2} d^{3} e^{5} x -280 a \,c^{3} d^{5} e^{3} x -280 c^{4} d^{7} e x +5 a^{4} e^{8}-28 a^{3} c \,d^{2} e^{6}+70 a^{2} c^{2} d^{4} e^{4}-140 a \,c^{3} d^{6} e^{2}-35 c^{4} d^{8}\right ) \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}}{35 \left (c d x +a e \right ) \left (e^{2} a -c \,d^{2}\right ) \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right ) \left (e x +d \right )^{4}}\) | \(308\) |
-2/35*(c*d*x+a*e)*(-128*c^4*d^4*e^4*x^4-64*a*c^3*d^3*e^5*x^3-448*c^4*d^5*e ^3*x^3+16*a^2*c^2*d^2*e^6*x^2-224*a*c^3*d^4*e^4*x^2-560*c^4*d^6*e^2*x^2-8* a^3*c*d*e^7*x+56*a^2*c^2*d^3*e^5*x-280*a*c^3*d^5*e^3*x-280*c^4*d^7*e*x+5*a ^4*e^8-28*a^3*c*d^2*e^6+70*a^2*c^2*d^4*e^4-140*a*c^3*d^6*e^2-35*c^4*d^8)/( e*x+d)^2/(a^5*e^10-5*a^4*c*d^2*e^8+10*a^3*c^2*d^4*e^6-10*a^2*c^3*d^6*e^4+5 *a*c^4*d^8*e^2-c^5*d^10)/(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(3/2)
Leaf count of result is larger than twice the leaf count of optimal. 733 vs. \(2 (225) = 450\).
Time = 9.66 (sec) , antiderivative size = 733, normalized size of antiderivative = 3.04 \[ \int \frac {1}{(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=-\frac {2 \, {\left (128 \, c^{4} d^{4} e^{4} x^{4} + 35 \, c^{4} d^{8} + 140 \, a c^{3} d^{6} e^{2} - 70 \, a^{2} c^{2} d^{4} e^{4} + 28 \, a^{3} c d^{2} e^{6} - 5 \, a^{4} e^{8} + 64 \, {\left (7 \, c^{4} d^{5} e^{3} + a c^{3} d^{3} e^{5}\right )} x^{3} + 16 \, {\left (35 \, c^{4} d^{6} e^{2} + 14 \, a c^{3} d^{4} e^{4} - a^{2} c^{2} d^{2} e^{6}\right )} x^{2} + 8 \, {\left (35 \, c^{4} d^{7} e + 35 \, a c^{3} d^{5} e^{3} - 7 \, a^{2} c^{2} d^{3} e^{5} + a^{3} c d e^{7}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{35 \, {\left (a c^{5} d^{14} e - 5 \, a^{2} c^{4} d^{12} e^{3} + 10 \, a^{3} c^{3} d^{10} e^{5} - 10 \, a^{4} c^{2} d^{8} e^{7} + 5 \, a^{5} c d^{6} e^{9} - a^{6} d^{4} e^{11} + {\left (c^{6} d^{11} e^{4} - 5 \, a c^{5} d^{9} e^{6} + 10 \, a^{2} c^{4} d^{7} e^{8} - 10 \, a^{3} c^{3} d^{5} e^{10} + 5 \, a^{4} c^{2} d^{3} e^{12} - a^{5} c d e^{14}\right )} x^{5} + {\left (4 \, c^{6} d^{12} e^{3} - 19 \, a c^{5} d^{10} e^{5} + 35 \, a^{2} c^{4} d^{8} e^{7} - 30 \, a^{3} c^{3} d^{6} e^{9} + 10 \, a^{4} c^{2} d^{4} e^{11} + a^{5} c d^{2} e^{13} - a^{6} e^{15}\right )} x^{4} + 2 \, {\left (3 \, c^{6} d^{13} e^{2} - 13 \, a c^{5} d^{11} e^{4} + 20 \, a^{2} c^{4} d^{9} e^{6} - 10 \, a^{3} c^{3} d^{7} e^{8} - 5 \, a^{4} c^{2} d^{5} e^{10} + 7 \, a^{5} c d^{3} e^{12} - 2 \, a^{6} d e^{14}\right )} x^{3} + 2 \, {\left (2 \, c^{6} d^{14} e - 7 \, a c^{5} d^{12} e^{3} + 5 \, a^{2} c^{4} d^{10} e^{5} + 10 \, a^{3} c^{3} d^{8} e^{7} - 20 \, a^{4} c^{2} d^{6} e^{9} + 13 \, a^{5} c d^{4} e^{11} - 3 \, a^{6} d^{2} e^{13}\right )} x^{2} + {\left (c^{6} d^{15} - a c^{5} d^{13} e^{2} - 10 \, a^{2} c^{4} d^{11} e^{4} + 30 \, a^{3} c^{3} d^{9} e^{6} - 35 \, a^{4} c^{2} d^{7} e^{8} + 19 \, a^{5} c d^{5} e^{10} - 4 \, a^{6} d^{3} e^{12}\right )} x\right )}} \]
-2/35*(128*c^4*d^4*e^4*x^4 + 35*c^4*d^8 + 140*a*c^3*d^6*e^2 - 70*a^2*c^2*d ^4*e^4 + 28*a^3*c*d^2*e^6 - 5*a^4*e^8 + 64*(7*c^4*d^5*e^3 + a*c^3*d^3*e^5) *x^3 + 16*(35*c^4*d^6*e^2 + 14*a*c^3*d^4*e^4 - a^2*c^2*d^2*e^6)*x^2 + 8*(3 5*c^4*d^7*e + 35*a*c^3*d^5*e^3 - 7*a^2*c^2*d^3*e^5 + a^3*c*d*e^7)*x)*sqrt( c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)/(a*c^5*d^14*e - 5*a^2*c^4*d^12*e^3 + 10*a^3*c^3*d^10*e^5 - 10*a^4*c^2*d^8*e^7 + 5*a^5*c*d^6*e^9 - a^6*d^4*e^1 1 + (c^6*d^11*e^4 - 5*a*c^5*d^9*e^6 + 10*a^2*c^4*d^7*e^8 - 10*a^3*c^3*d^5* e^10 + 5*a^4*c^2*d^3*e^12 - a^5*c*d*e^14)*x^5 + (4*c^6*d^12*e^3 - 19*a*c^5 *d^10*e^5 + 35*a^2*c^4*d^8*e^7 - 30*a^3*c^3*d^6*e^9 + 10*a^4*c^2*d^4*e^11 + a^5*c*d^2*e^13 - a^6*e^15)*x^4 + 2*(3*c^6*d^13*e^2 - 13*a*c^5*d^11*e^4 + 20*a^2*c^4*d^9*e^6 - 10*a^3*c^3*d^7*e^8 - 5*a^4*c^2*d^5*e^10 + 7*a^5*c*d^ 3*e^12 - 2*a^6*d*e^14)*x^3 + 2*(2*c^6*d^14*e - 7*a*c^5*d^12*e^3 + 5*a^2*c^ 4*d^10*e^5 + 10*a^3*c^3*d^8*e^7 - 20*a^4*c^2*d^6*e^9 + 13*a^5*c*d^4*e^11 - 3*a^6*d^2*e^13)*x^2 + (c^6*d^15 - a*c^5*d^13*e^2 - 10*a^2*c^4*d^11*e^4 + 30*a^3*c^3*d^9*e^6 - 35*a^4*c^2*d^7*e^8 + 19*a^5*c*d^5*e^10 - 4*a^6*d^3*e^ 12)*x)
Timed out. \[ \int \frac {1}{(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {1}{(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \]
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*(a*e^2-c*d^2)>0)', see `assume ?` for mor
\[ \int \frac {1}{(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {3}{2}} {\left (e x + d\right )}^{3}} \,d x } \]
Time = 11.79 (sec) , antiderivative size = 2121, normalized size of antiderivative = 8.80 \[ \int \frac {1}{(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\text {Too large to display} \]
(((d*((24*c^4*d^5*e^3)/(35*(a*e^2 - c*d^2)^4*(3*a^2*e^5 + 3*c^2*d^4*e - 6* a*c*d^2*e^3)) + (4*c^3*d^3*e^3*(17*a*e^2 - 29*c*d^2))/(35*(a*e^2 - c*d^2)^ 4*(3*a^2*e^5 + 3*c^2*d^4*e - 6*a*c*d^2*e^3))))/e - (e^2*(70*c^4*d^6 - 256* a*c^3*d^4*e^2 + 162*a^2*c^2*d^2*e^4))/(35*(a*e^2 - c*d^2)^4*(3*a^2*e^5 + 3 *c^2*d^4*e - 6*a*c*d^2*e^3)))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2 ))/(d + e*x)^2 - (((e^2*(14*c^2*d^3 - 26*a*c*d*e^2))/(7*(a*e^2 - c*d^2)^2* (5*a^2*e^5 + 5*c^2*d^4*e - 10*a*c*d^2*e^3)) + (12*c^2*d^3*e^2)/(7*(a*e^2 - c*d^2)^2*(5*a^2*e^5 + 5*c^2*d^4*e - 10*a*c*d^2*e^3)))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x)^3 - (((d*((16*c^5*d^6*e^2)/(35*(a*e^ 2 - c*d^2)^7) + (16*c^4*d^4*e^2*(7*a*e^2 - 13*c*d^2))/(105*(a*e^2 - c*d^2) ^7)))/e + (4*c^3*d^3*e*(23*c^2*d^4 - 17*a^2*e^4 + 6*a*c*d^2*e^2))/(105*(a* e^2 - c*d^2)^7))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x) + (((24*c^3*d^4*e^2)/(35*(a*e^2 - c*d^2)^4*(3*a*e^3 - 3*c*d^2*e)) - (12*c^ 2*d^2*e^2*(a*e^2 + c*d^2))/(35*(a*e^2 - c*d^2)^4*(3*a*e^3 - 3*c*d^2*e)))*( x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x)^2 - (2*e^2*(x*(a*e ^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/((d + e*x)^4*(7*a^2*e^5 + 7*c^2*d^ 4*e - 14*a*c*d^2*e^3)) - (((a*(((a*e^2 + c*d^2)*((16*c^7*d^7*e^4*(a*e^2 + c*d^2))/(35*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) + (32*c^7*d^7*e^4*(4*a*e^2 - 13*c*d^2))/(105*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/(c*d*e) + (16*c^6*d^6*e^3*(27*a^2*...