Integrand size = 19, antiderivative size = 158 \[ \int x \left (d+e x^2\right )^3 (a+b \arctan (c x)) \, dx=-\frac {b \left (2 c^2 d-e\right ) \left (2 c^4 d^2-2 c^2 d e+e^2\right ) x}{8 c^7}-\frac {b e \left (6 c^4 d^2-4 c^2 d e+e^2\right ) x^3}{24 c^5}-\frac {b \left (4 c^2 d-e\right ) e^2 x^5}{40 c^3}-\frac {b e^3 x^7}{56 c}-\frac {b \left (c^2 d-e\right )^4 \arctan (c x)}{8 c^8 e}+\frac {\left (d+e x^2\right )^4 (a+b \arctan (c x))}{8 e} \]
-1/8*b*(2*c^2*d-e)*(2*c^4*d^2-2*c^2*d*e+e^2)*x/c^7-1/24*b*e*(6*c^4*d^2-4*c ^2*d*e+e^2)*x^3/c^5-1/40*b*(4*c^2*d-e)*e^2*x^5/c^3-1/56*b*e^3*x^7/c-1/8*b* (c^2*d-e)^4*arctan(c*x)/c^8/e+1/8*(e*x^2+d)^4*(a+b*arctan(c*x))/e
Time = 0.08 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.37 \[ \int x \left (d+e x^2\right )^3 (a+b \arctan (c x)) \, dx=\frac {c x \left (105 b e^3-35 b c^2 e^2 \left (12 d+e x^2\right )+7 b c^4 e \left (90 d^2+20 d e x^2+3 e^2 x^4\right )+105 a c^7 x \left (4 d^3+6 d^2 e x^2+4 d e^2 x^4+e^3 x^6\right )-3 b c^6 \left (140 d^3+70 d^2 e x^2+28 d e^2 x^4+5 e^3 x^6\right )\right )+105 b \left (4 c^6 d^3-6 c^4 d^2 e+4 c^2 d e^2-e^3+c^8 \left (4 d^3 x^2+6 d^2 e x^4+4 d e^2 x^6+e^3 x^8\right )\right ) \arctan (c x)}{840 c^8} \]
(c*x*(105*b*e^3 - 35*b*c^2*e^2*(12*d + e*x^2) + 7*b*c^4*e*(90*d^2 + 20*d*e *x^2 + 3*e^2*x^4) + 105*a*c^7*x*(4*d^3 + 6*d^2*e*x^2 + 4*d*e^2*x^4 + e^3*x ^6) - 3*b*c^6*(140*d^3 + 70*d^2*e*x^2 + 28*d*e^2*x^4 + 5*e^3*x^6)) + 105*b *(4*c^6*d^3 - 6*c^4*d^2*e + 4*c^2*d*e^2 - e^3 + c^8*(4*d^3*x^2 + 6*d^2*e*x ^4 + 4*d*e^2*x^6 + e^3*x^8))*ArcTan[c*x])/(840*c^8)
Time = 0.37 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.99, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {5509, 300, 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.
| \(\displaystyle \int x \left (d+e x^2\right )^3 (a+b \arctan (c x)) \, dx\) |
| \(\Big \downarrow \) 5509 |
| \(\displaystyle \frac {\left (d+e x^2\right )^4 (a+b \arctan (c x))}{8 e}-\frac {b c \int \frac {\left (e x^2+d\right )^4}{c^2 x^2+1}dx}{8 e}\) |
| \(\Big \downarrow \) 300 |
| \(\displaystyle \frac {\left (d+e x^2\right )^4 (a+b \arctan (c x))}{8 e}-\frac {b c \int \left (\frac {e^4 x^6}{c^2}+\frac {\left (4 c^2 d-e\right ) e^3 x^4}{c^4}+\frac {e^2 \left (6 d^2 c^4-4 d e c^2+e^2\right ) x^2}{c^6}+\frac {\left (2 c^2 d-e\right ) e \left (2 d^2 c^4-2 d e c^2+e^2\right )}{c^8}+\frac {d^4 c^8-4 d^3 e c^6+6 d^2 e^2 c^4-4 d e^3 c^2+e^4}{c^8 \left (c^2 x^2+1\right )}\right )dx}{8 e}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\left (d+e x^2\right )^4 (a+b \arctan (c x))}{8 e}-\frac {b c \left (\frac {\arctan (c x) \left (c^2 d-e\right )^4}{c^9}+\frac {e^4 x^7}{7 c^2}+\frac {e^3 x^5 \left (4 c^2 d-e\right )}{5 c^4}+\frac {e x \left (2 c^2 d-e\right ) \left (2 c^4 d^2-2 c^2 d e+e^2\right )}{c^8}+\frac {e^2 x^3 \left (6 c^4 d^2-4 c^2 d e+e^2\right )}{3 c^6}\right )}{8 e}\) |
((d + e*x^2)^4*(a + b*ArcTan[c*x]))/(8*e) - (b*c*(((2*c^2*d - e)*e*(2*c^4* d^2 - 2*c^2*d*e + e^2)*x)/c^8 + (e^2*(6*c^4*d^2 - 4*c^2*d*e + e^2)*x^3)/(3 *c^6) + ((4*c^2*d - e)*e^3*x^5)/(5*c^4) + (e^4*x^7)/(7*c^2) + ((c^2*d - e) ^4*ArcTan[c*x])/c^9))/(8*e)
3.12.39.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Int [PolynomialDivide[(a + b*x^2)^p, (c + d*x^2)^(-q), x], x] /; FreeQ[{a, b, c , d}, x] && NeQ[b*c - a*d, 0] && IGtQ[p, 0] && ILtQ[q, 0] && GeQ[p, -q]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))*(x_)*((d_.) + (e_.)*(x_)^2)^(q_.), x _Symbol] :> Simp[(d + e*x^2)^(q + 1)*((a + b*ArcTan[c*x])/(2*e*(q + 1))), x ] - Simp[b*(c/(2*e*(q + 1))) Int[(d + e*x^2)^(q + 1)/(1 + c^2*x^2), x], x ] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[q, -1]
Time = 0.52 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.67
| method | result | size |
| parts | \(\frac {a \left (e \,x^{2}+d \right )^{4}}{8 e}+\frac {b \left (\frac {\arctan \left (c x \right ) c^{2} e^{3} x^{8}}{8}+\frac {\arctan \left (c x \right ) c^{2} e^{2} x^{6} d}{2}+\frac {3 \arctan \left (c x \right ) c^{2} e \,x^{4} d^{2}}{4}+\frac {\arctan \left (c x \right ) c^{2} x^{2} d^{3}}{2}+\frac {\arctan \left (c x \right ) c^{2} d^{4}}{8 e}-\frac {4 c^{7} d^{3} e x +2 c^{7} d^{2} e^{2} x^{3}+\frac {4 c^{7} d \,e^{3} x^{5}}{5}+\frac {e^{4} c^{7} x^{7}}{7}-6 c^{5} x \,d^{2} e^{2}-\frac {4 c^{5} d \,e^{3} x^{3}}{3}-\frac {e^{4} c^{5} x^{5}}{5}+4 c^{3} x d \,e^{3}+\frac {e^{4} c^{3} x^{3}}{3}-c x \,e^{4}+\left (c^{8} d^{4}-4 c^{6} d^{3} e +6 c^{4} d^{2} e^{2}-4 c^{2} d \,e^{3}+e^{4}\right ) \arctan \left (c x \right )}{8 c^{6} e}\right )}{c^{2}}\) | \(264\) |
| derivativedivides | \(\frac {\frac {a \left (e \,c^{2} x^{2}+c^{2} d \right )^{4}}{8 c^{6} e}+\frac {b \left (\frac {\arctan \left (c x \right ) c^{8} d^{4}}{8 e}+\frac {\arctan \left (c x \right ) c^{8} d^{3} x^{2}}{2}+\frac {3 \arctan \left (c x \right ) e \,c^{8} d^{2} x^{4}}{4}+\frac {\arctan \left (c x \right ) e^{2} c^{8} d \,x^{6}}{2}+\frac {\arctan \left (c x \right ) e^{3} c^{8} x^{8}}{8}-\frac {4 c^{7} d^{3} e x +2 c^{7} d^{2} e^{2} x^{3}+\frac {4 c^{7} d \,e^{3} x^{5}}{5}+\frac {e^{4} c^{7} x^{7}}{7}-6 c^{5} x \,d^{2} e^{2}-\frac {4 c^{5} d \,e^{3} x^{3}}{3}-\frac {e^{4} c^{5} x^{5}}{5}+4 c^{3} x d \,e^{3}+\frac {e^{4} c^{3} x^{3}}{3}-c x \,e^{4}+\left (c^{8} d^{4}-4 c^{6} d^{3} e +6 c^{4} d^{2} e^{2}-4 c^{2} d \,e^{3}+e^{4}\right ) \arctan \left (c x \right )}{8 e}\right )}{c^{6}}}{c^{2}}\) | \(275\) |
| default | \(\frac {\frac {a \left (e \,c^{2} x^{2}+c^{2} d \right )^{4}}{8 c^{6} e}+\frac {b \left (\frac {\arctan \left (c x \right ) c^{8} d^{4}}{8 e}+\frac {\arctan \left (c x \right ) c^{8} d^{3} x^{2}}{2}+\frac {3 \arctan \left (c x \right ) e \,c^{8} d^{2} x^{4}}{4}+\frac {\arctan \left (c x \right ) e^{2} c^{8} d \,x^{6}}{2}+\frac {\arctan \left (c x \right ) e^{3} c^{8} x^{8}}{8}-\frac {4 c^{7} d^{3} e x +2 c^{7} d^{2} e^{2} x^{3}+\frac {4 c^{7} d \,e^{3} x^{5}}{5}+\frac {e^{4} c^{7} x^{7}}{7}-6 c^{5} x \,d^{2} e^{2}-\frac {4 c^{5} d \,e^{3} x^{3}}{3}-\frac {e^{4} c^{5} x^{5}}{5}+4 c^{3} x d \,e^{3}+\frac {e^{4} c^{3} x^{3}}{3}-c x \,e^{4}+\left (c^{8} d^{4}-4 c^{6} d^{3} e +6 c^{4} d^{2} e^{2}-4 c^{2} d \,e^{3}+e^{4}\right ) \arctan \left (c x \right )}{8 e}\right )}{c^{6}}}{c^{2}}\) | \(275\) |
| parallelrisch | \(\frac {105 x^{8} \arctan \left (c x \right ) b \,c^{8} e^{3}+105 x^{8} a \,c^{8} e^{3}+420 x^{6} \arctan \left (c x \right ) b \,c^{8} d \,e^{2}-15 b \,c^{7} e^{3} x^{7}+420 x^{6} a \,c^{8} d \,e^{2}+630 x^{4} \arctan \left (c x \right ) b \,c^{8} d^{2} e -84 b \,c^{7} d \,e^{2} x^{5}+630 x^{4} a \,c^{8} d^{2} e +420 x^{2} \arctan \left (c x \right ) b \,c^{8} d^{3}+21 b \,c^{5} e^{3} x^{5}-210 b \,c^{7} d^{2} e \,x^{3}+420 x^{2} a \,c^{8} d^{3}+140 b \,c^{5} d \,e^{2} x^{3}-420 b \,c^{7} d^{3} x +420 b \,c^{6} d^{3} \arctan \left (c x \right )-35 b \,c^{3} e^{3} x^{3}+630 b \,c^{5} d^{2} e x -630 b \,c^{4} d^{2} e \arctan \left (c x \right )-420 b \,c^{3} d \,e^{2} x +420 b \,c^{2} d \,e^{2} \arctan \left (c x \right )+105 b c \,e^{3} x -105 b \,e^{3} \arctan \left (c x \right )}{840 c^{8}}\) | \(289\) |
| risch | \(\text {Expression too large to display}\) | \(1367\) |
1/8*a*(e*x^2+d)^4/e+b/c^2*(1/8*arctan(c*x)*c^2*e^3*x^8+1/2*arctan(c*x)*c^2 *e^2*x^6*d+3/4*arctan(c*x)*c^2*e*x^4*d^2+1/2*arctan(c*x)*c^2*x^2*d^3+1/8*a rctan(c*x)*c^2/e*d^4-1/8/c^6/e*(4*c^7*d^3*e*x+2*c^7*d^2*e^2*x^3+4/5*c^7*d* e^3*x^5+1/7*e^4*c^7*x^7-6*c^5*x*d^2*e^2-4/3*c^5*d*e^3*x^3-1/5*e^4*c^5*x^5+ 4*c^3*x*d*e^3+1/3*e^4*c^3*x^3-c*x*e^4+(c^8*d^4-4*c^6*d^3*e+6*c^4*d^2*e^2-4 *c^2*d*e^3+e^4)*arctan(c*x)))
Time = 0.26 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.63 \[ \int x \left (d+e x^2\right )^3 (a+b \arctan (c x)) \, dx=\frac {105 \, a c^{8} e^{3} x^{8} + 420 \, a c^{8} d e^{2} x^{6} - 15 \, b c^{7} e^{3} x^{7} + 630 \, a c^{8} d^{2} e x^{4} + 420 \, a c^{8} d^{3} x^{2} - 21 \, {\left (4 \, b c^{7} d e^{2} - b c^{5} e^{3}\right )} x^{5} - 35 \, {\left (6 \, b c^{7} d^{2} e - 4 \, b c^{5} d e^{2} + b c^{3} e^{3}\right )} x^{3} - 105 \, {\left (4 \, b c^{7} d^{3} - 6 \, b c^{5} d^{2} e + 4 \, b c^{3} d e^{2} - b c e^{3}\right )} x + 105 \, {\left (b c^{8} e^{3} x^{8} + 4 \, b c^{8} d e^{2} x^{6} + 6 \, b c^{8} d^{2} e x^{4} + 4 \, b c^{8} d^{3} x^{2} + 4 \, b c^{6} d^{3} - 6 \, b c^{4} d^{2} e + 4 \, b c^{2} d e^{2} - b e^{3}\right )} \arctan \left (c x\right )}{840 \, c^{8}} \]
1/840*(105*a*c^8*e^3*x^8 + 420*a*c^8*d*e^2*x^6 - 15*b*c^7*e^3*x^7 + 630*a* c^8*d^2*e*x^4 + 420*a*c^8*d^3*x^2 - 21*(4*b*c^7*d*e^2 - b*c^5*e^3)*x^5 - 3 5*(6*b*c^7*d^2*e - 4*b*c^5*d*e^2 + b*c^3*e^3)*x^3 - 105*(4*b*c^7*d^3 - 6*b *c^5*d^2*e + 4*b*c^3*d*e^2 - b*c*e^3)*x + 105*(b*c^8*e^3*x^8 + 4*b*c^8*d*e ^2*x^6 + 6*b*c^8*d^2*e*x^4 + 4*b*c^8*d^3*x^2 + 4*b*c^6*d^3 - 6*b*c^4*d^2*e + 4*b*c^2*d*e^2 - b*e^3)*arctan(c*x))/c^8
Leaf count of result is larger than twice the leaf count of optimal. 350 vs. \(2 (144) = 288\).
Time = 0.61 (sec) , antiderivative size = 350, normalized size of antiderivative = 2.22 \[ \int x \left (d+e x^2\right )^3 (a+b \arctan (c x)) \, dx=\begin {cases} \frac {a d^{3} x^{2}}{2} + \frac {3 a d^{2} e x^{4}}{4} + \frac {a d e^{2} x^{6}}{2} + \frac {a e^{3} x^{8}}{8} + \frac {b d^{3} x^{2} \operatorname {atan}{\left (c x \right )}}{2} + \frac {3 b d^{2} e x^{4} \operatorname {atan}{\left (c x \right )}}{4} + \frac {b d e^{2} x^{6} \operatorname {atan}{\left (c x \right )}}{2} + \frac {b e^{3} x^{8} \operatorname {atan}{\left (c x \right )}}{8} - \frac {b d^{3} x}{2 c} - \frac {b d^{2} e x^{3}}{4 c} - \frac {b d e^{2} x^{5}}{10 c} - \frac {b e^{3} x^{7}}{56 c} + \frac {b d^{3} \operatorname {atan}{\left (c x \right )}}{2 c^{2}} + \frac {3 b d^{2} e x}{4 c^{3}} + \frac {b d e^{2} x^{3}}{6 c^{3}} + \frac {b e^{3} x^{5}}{40 c^{3}} - \frac {3 b d^{2} e \operatorname {atan}{\left (c x \right )}}{4 c^{4}} - \frac {b d e^{2} x}{2 c^{5}} - \frac {b e^{3} x^{3}}{24 c^{5}} + \frac {b d e^{2} \operatorname {atan}{\left (c x \right )}}{2 c^{6}} + \frac {b e^{3} x}{8 c^{7}} - \frac {b e^{3} \operatorname {atan}{\left (c x \right )}}{8 c^{8}} & \text {for}\: c \neq 0 \\a \left (\frac {d^{3} x^{2}}{2} + \frac {3 d^{2} e x^{4}}{4} + \frac {d e^{2} x^{6}}{2} + \frac {e^{3} x^{8}}{8}\right ) & \text {otherwise} \end {cases} \]
Piecewise((a*d**3*x**2/2 + 3*a*d**2*e*x**4/4 + a*d*e**2*x**6/2 + a*e**3*x* *8/8 + b*d**3*x**2*atan(c*x)/2 + 3*b*d**2*e*x**4*atan(c*x)/4 + b*d*e**2*x* *6*atan(c*x)/2 + b*e**3*x**8*atan(c*x)/8 - b*d**3*x/(2*c) - b*d**2*e*x**3/ (4*c) - b*d*e**2*x**5/(10*c) - b*e**3*x**7/(56*c) + b*d**3*atan(c*x)/(2*c* *2) + 3*b*d**2*e*x/(4*c**3) + b*d*e**2*x**3/(6*c**3) + b*e**3*x**5/(40*c** 3) - 3*b*d**2*e*atan(c*x)/(4*c**4) - b*d*e**2*x/(2*c**5) - b*e**3*x**3/(24 *c**5) + b*d*e**2*atan(c*x)/(2*c**6) + b*e**3*x/(8*c**7) - b*e**3*atan(c*x )/(8*c**8), Ne(c, 0)), (a*(d**3*x**2/2 + 3*d**2*e*x**4/4 + d*e**2*x**6/2 + e**3*x**8/8), True))
Time = 0.27 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.47 \[ \int x \left (d+e x^2\right )^3 (a+b \arctan (c x)) \, dx=\frac {1}{8} \, a e^{3} x^{8} + \frac {1}{2} \, a d e^{2} x^{6} + \frac {3}{4} \, a d^{2} e x^{4} + \frac {1}{2} \, a d^{3} x^{2} + \frac {1}{2} \, {\left (x^{2} \arctan \left (c x\right ) - c {\left (\frac {x}{c^{2}} - \frac {\arctan \left (c x\right )}{c^{3}}\right )}\right )} b d^{3} + \frac {1}{4} \, {\left (3 \, x^{4} \arctan \left (c x\right ) - c {\left (\frac {c^{2} x^{3} - 3 \, x}{c^{4}} + \frac {3 \, \arctan \left (c x\right )}{c^{5}}\right )}\right )} b d^{2} e + \frac {1}{30} \, {\left (15 \, x^{6} \arctan \left (c x\right ) - c {\left (\frac {3 \, c^{4} x^{5} - 5 \, c^{2} x^{3} + 15 \, x}{c^{6}} - \frac {15 \, \arctan \left (c x\right )}{c^{7}}\right )}\right )} b d e^{2} + \frac {1}{840} \, {\left (105 \, x^{8} \arctan \left (c x\right ) - c {\left (\frac {15 \, c^{6} x^{7} - 21 \, c^{4} x^{5} + 35 \, c^{2} x^{3} - 105 \, x}{c^{8}} + \frac {105 \, \arctan \left (c x\right )}{c^{9}}\right )}\right )} b e^{3} \]
1/8*a*e^3*x^8 + 1/2*a*d*e^2*x^6 + 3/4*a*d^2*e*x^4 + 1/2*a*d^3*x^2 + 1/2*(x ^2*arctan(c*x) - c*(x/c^2 - arctan(c*x)/c^3))*b*d^3 + 1/4*(3*x^4*arctan(c* x) - c*((c^2*x^3 - 3*x)/c^4 + 3*arctan(c*x)/c^5))*b*d^2*e + 1/30*(15*x^6*a rctan(c*x) - c*((3*c^4*x^5 - 5*c^2*x^3 + 15*x)/c^6 - 15*arctan(c*x)/c^7))* b*d*e^2 + 1/840*(105*x^8*arctan(c*x) - c*((15*c^6*x^7 - 21*c^4*x^5 + 35*c^ 2*x^3 - 105*x)/c^8 + 105*arctan(c*x)/c^9))*b*e^3
\[ \int x \left (d+e x^2\right )^3 (a+b \arctan (c x)) \, dx=\int { {\left (e x^{2} + d\right )}^{3} {\left (b \arctan \left (c x\right ) + a\right )} x \,d x } \]
Time = 0.63 (sec) , antiderivative size = 442, normalized size of antiderivative = 2.80 \[ \int x \left (d+e x^2\right )^3 (a+b \arctan (c x)) \, dx=x\,\left (\frac {\frac {\frac {b\,e^3}{8\,c^3}-\frac {b\,d\,e^2}{2\,c}}{c^2}+\frac {3\,b\,d^2\,e}{4\,c}}{c^2}-\frac {b\,d^3}{2\,c}\right )-x^6\,\left (\frac {a\,e^3}{6\,c^2}-\frac {a\,e^2\,\left (3\,d\,c^2+e\right )}{6\,c^2}\right )+x^4\,\left (\frac {\frac {a\,e^3}{c^2}-\frac {a\,e^2\,\left (3\,d\,c^2+e\right )}{c^2}}{4\,c^2}+\frac {3\,a\,d\,e\,\left (d\,c^2+e\right )}{4\,c^2}\right )+x^5\,\left (\frac {b\,e^3}{40\,c^3}-\frac {b\,d\,e^2}{10\,c}\right )+\mathrm {atan}\left (c\,x\right )\,\left (\frac {b\,d^3\,x^2}{2}+\frac {3\,b\,d^2\,e\,x^4}{4}+\frac {b\,d\,e^2\,x^6}{2}+\frac {b\,e^3\,x^8}{8}\right )-x^3\,\left (\frac {\frac {b\,e^3}{8\,c^3}-\frac {b\,d\,e^2}{2\,c}}{3\,c^2}+\frac {b\,d^2\,e}{4\,c}\right )-x^2\,\left (\frac {\frac {\frac {a\,e^3}{c^2}-\frac {a\,e^2\,\left (3\,d\,c^2+e\right )}{c^2}}{c^2}+\frac {3\,a\,d\,e\,\left (d\,c^2+e\right )}{c^2}}{2\,c^2}-\frac {a\,d^2\,\left (d\,c^2+3\,e\right )}{2\,c^2}\right )+\frac {a\,e^3\,x^8}{8}-\frac {b\,e^3\,x^7}{56\,c}-\frac {b\,\mathrm {atan}\left (\frac {b\,c\,x\,\left (e-2\,c^2\,d\right )\,\left (2\,c^4\,d^2-2\,c^2\,d\,e+e^2\right )}{-4\,b\,c^6\,d^3+6\,b\,c^4\,d^2\,e-4\,b\,c^2\,d\,e^2+b\,e^3}\right )\,\left (e-2\,c^2\,d\right )\,\left (2\,c^4\,d^2-2\,c^2\,d\,e+e^2\right )}{8\,c^8} \]
x*((((b*e^3)/(8*c^3) - (b*d*e^2)/(2*c))/c^2 + (3*b*d^2*e)/(4*c))/c^2 - (b* d^3)/(2*c)) - x^6*((a*e^3)/(6*c^2) - (a*e^2*(e + 3*c^2*d))/(6*c^2)) + x^4* (((a*e^3)/c^2 - (a*e^2*(e + 3*c^2*d))/c^2)/(4*c^2) + (3*a*d*e*(e + c^2*d)) /(4*c^2)) + x^5*((b*e^3)/(40*c^3) - (b*d*e^2)/(10*c)) + atan(c*x)*((b*d^3* x^2)/2 + (b*e^3*x^8)/8 + (3*b*d^2*e*x^4)/4 + (b*d*e^2*x^6)/2) - x^3*(((b*e ^3)/(8*c^3) - (b*d*e^2)/(2*c))/(3*c^2) + (b*d^2*e)/(4*c)) - x^2*((((a*e^3) /c^2 - (a*e^2*(e + 3*c^2*d))/c^2)/c^2 + (3*a*d*e*(e + c^2*d))/c^2)/(2*c^2) - (a*d^2*(3*e + c^2*d))/(2*c^2)) + (a*e^3*x^8)/8 - (b*e^3*x^7)/(56*c) - ( b*atan((b*c*x*(e - 2*c^2*d)*(e^2 + 2*c^4*d^2 - 2*c^2*d*e))/(b*e^3 - 4*b*c^ 6*d^3 - 4*b*c^2*d*e^2 + 6*b*c^4*d^2*e))*(e - 2*c^2*d)*(e^2 + 2*c^4*d^2 - 2 *c^2*d*e))/(8*c^8)