Integrand size = 21, antiderivative size = 262 \[ \int \frac {x^4 (d+e x)}{\left (a+b x+c x^2\right )^2} \, dx=\frac {\left (2 b^2 c d-6 a c^2 d-3 b^3 e+11 a b c e\right ) x}{c^3 \left (b^2-4 a c\right )}-\frac {\left (2 b c d-3 b^2 e+8 a c e\right ) x^2}{2 c^2 \left (b^2-4 a c\right )}+\frac {x^3 \left (a (2 c d-b e)+\left (b c d-b^2 e+2 a c e\right ) x\right )}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {\left (2 b^4 c d-12 a b^2 c^2 d+12 a^2 c^3 d-3 b^5 e+20 a b^3 c e-30 a^2 b c^2 e\right ) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^4 \left (b^2-4 a c\right )^{3/2}}-\frac {\left (2 b c d-3 b^2 e+2 a c e\right ) \log \left (a+b x+c x^2\right )}{2 c^4} \]
(11*a*b*c*e-6*a*c^2*d-3*b^3*e+2*b^2*c*d)*x/c^3/(-4*a*c+b^2)-1/2*(8*a*c*e-3 *b^2*e+2*b*c*d)*x^2/c^2/(-4*a*c+b^2)+x^3*(a*(-b*e+2*c*d)+(2*a*c*e-b^2*e+b* c*d)*x)/c/(-4*a*c+b^2)/(c*x^2+b*x+a)-(-30*a^2*b*c^2*e+12*a^2*c^3*d+20*a*b^ 3*c*e-12*a*b^2*c^2*d-3*b^5*e+2*b^4*c*d)*arctanh((2*c*x+b)/(-4*a*c+b^2)^(1/ 2))/c^4/(-4*a*c+b^2)^(3/2)-1/2*(2*a*c*e-3*b^2*e+2*b*c*d)*ln(c*x^2+b*x+a)/c ^4
Time = 0.23 (sec) , antiderivative size = 249, normalized size of antiderivative = 0.95 \[ \int \frac {x^4 (d+e x)}{\left (a+b x+c x^2\right )^2} \, dx=\frac {2 c (c d-2 b e) x+c^2 e x^2+\frac {2 \left (2 a^3 c^2 e+b^4 (-c d+b e) x+a b^2 \left (b^2 e+4 c^2 d x-b c (d+5 e x)\right )+a^2 c \left (-4 b^2 e-2 c^2 d x+b c (3 d+5 e x)\right )\right )}{\left (b^2-4 a c\right ) (a+x (b+c x))}+\frac {2 \left (-2 b^4 c d+12 a b^2 c^2 d-12 a^2 c^3 d+3 b^5 e-20 a b^3 c e+30 a^2 b c^2 e\right ) \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\left (-b^2+4 a c\right )^{3/2}}+\left (-2 b c d+3 b^2 e-2 a c e\right ) \log (a+x (b+c x))}{2 c^4} \]
(2*c*(c*d - 2*b*e)*x + c^2*e*x^2 + (2*(2*a^3*c^2*e + b^4*(-(c*d) + b*e)*x + a*b^2*(b^2*e + 4*c^2*d*x - b*c*(d + 5*e*x)) + a^2*c*(-4*b^2*e - 2*c^2*d* x + b*c*(3*d + 5*e*x))))/((b^2 - 4*a*c)*(a + x*(b + c*x))) + (2*(-2*b^4*c* d + 12*a*b^2*c^2*d - 12*a^2*c^3*d + 3*b^5*e - 20*a*b^3*c*e + 30*a^2*b*c^2* e)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(3/2) + (-2*b*c* d + 3*b^2*e - 2*a*c*e)*Log[a + x*(b + c*x)])/(2*c^4)
Time = 0.69 (sec) , antiderivative size = 266, normalized size of antiderivative = 1.02, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {1233, 25, 1200, 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.
| \(\displaystyle \int \frac {x^4 (d+e x)}{\left (a+b x+c x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 1233 |
| \(\displaystyle \frac {\int -\frac {x^2 \left (3 a (2 c d-b e)+\left (-3 e b^2+2 c d b+8 a c e\right ) x\right )}{c x^2+b x+a}dx}{c \left (b^2-4 a c\right )}+\frac {x^3 \left (x \left (2 a c e+b^2 (-e)+b c d\right )+a (2 c d-b e)\right )}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {x^3 \left (x \left (2 a c e+b^2 (-e)+b c d\right )+a (2 c d-b e)\right )}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {\int \frac {x^2 \left (3 a (2 c d-b e)+\left (-3 e b^2+2 c d b+8 a c e\right ) x\right )}{c x^2+b x+a}dx}{c \left (b^2-4 a c\right )}\) |
| \(\Big \downarrow \) 1200 |
| \(\displaystyle \frac {x^3 \left (x \left (2 a c e+b^2 (-e)+b c d\right )+a (2 c d-b e)\right )}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {\int \left (-\frac {-3 e b^3+2 c d b^2+11 a c e b-6 a c^2 d}{c^2}+\frac {\left (-3 e b^2+2 c d b+8 a c e\right ) x}{c}+\frac {a \left (-3 e b^3+2 c d b^2+11 a c e b-6 a c^2 d\right )+\left (b^2-4 a c\right ) \left (-3 e b^2+2 c d b+2 a c e\right ) x}{c^2 \left (c x^2+b x+a\right )}\right )dx}{c \left (b^2-4 a c\right )}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {x^3 \left (x \left (2 a c e+b^2 (-e)+b c d\right )+a (2 c d-b e)\right )}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {\frac {\text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right ) \left (-30 a^2 b c^2 e+12 a^2 c^3 d+20 a b^3 c e-12 a b^2 c^2 d-3 b^5 e+2 b^4 c d\right )}{c^3 \sqrt {b^2-4 a c}}+\frac {\left (b^2-4 a c\right ) \left (2 a c e-3 b^2 e+2 b c d\right ) \log \left (a+b x+c x^2\right )}{2 c^3}+\frac {x^2 \left (8 a c e-3 b^2 e+2 b c d\right )}{2 c}-\frac {x \left (11 a b c e-6 a c^2 d-3 b^3 e+2 b^2 c d\right )}{c^2}}{c \left (b^2-4 a c\right )}\) |
(x^3*(a*(2*c*d - b*e) + (b*c*d - b^2*e + 2*a*c*e)*x))/(c*(b^2 - 4*a*c)*(a + b*x + c*x^2)) - (-(((2*b^2*c*d - 6*a*c^2*d - 3*b^3*e + 11*a*b*c*e)*x)/c^ 2) + ((2*b*c*d - 3*b^2*e + 8*a*c*e)*x^2)/(2*c) + ((2*b^4*c*d - 12*a*b^2*c^ 2*d + 12*a^2*c^3*d - 3*b^5*e + 20*a*b^3*c*e - 30*a^2*b*c^2*e)*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(c^3*Sqrt[b^2 - 4*a*c]) + ((b^2 - 4*a*c)*(2*b* c*d - 3*b^2*e + 2*a*c*e)*Log[a + b*x + c*x^2])/(2*c^3))/(c*(b^2 - 4*a*c))
3.9.90.3.1 Defintions of rubi rules used
Int[(((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.))/((a_.) + (b_.)* (x_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*((f + g* x)^n/(a + b*x + c*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && In tegersQ[n]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(d + e*x)^(m - 1))*(a + b*x + c*x^2) ^(p + 1)*((2*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - (2*c^2*d*f + b^2*e*g - c *(b*e*f + b*d*g + 2*a*e*g))*x)/(c*(p + 1)*(b^2 - 4*a*c))), x] - Simp[1/(c*( p + 1)*(b^2 - 4*a*c)) Int[(d + e*x)^(m - 2)*(a + b*x + c*x^2)^(p + 1)*Sim p[2*c^2*d^2*f*(2*p + 3) + b*e*g*(a*e*(m - 1) + b*d*(p + 2)) - c*(2*a*e*(e*f *(m - 1) + d*g*m) + b*d*(d*g*(2*p + 3) - e*f*(m - 2*p - 4))) + e*(b^2*e*g*( m + p + 1) + 2*c^2*d*f*(m + 2*p + 2) - c*(2*a*e*g*m + b*(e*f + d*g)*(m + 2* p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && LtQ[p, -1] && GtQ[m, 1] && ((EqQ[m, 2] && EqQ[p, -3] && RationalQ[a, b, c, d, e, f, g]) | | !ILtQ[m + 2*p + 3, 0])
Time = 0.29 (sec) , antiderivative size = 349, normalized size of antiderivative = 1.33
| method | result | size |
| default | \(-\frac {-\frac {1}{2} c e \,x^{2}+2 b e x -c d x}{c^{3}}+\frac {\frac {-\frac {\left (5 a^{2} b \,c^{2} e -2 a^{2} c^{3} d -5 a \,b^{3} c e +4 a \,b^{2} c^{2} d +b^{5} e -b^{4} c d \right ) x}{c \left (4 a c -b^{2}\right )}-\frac {a \left (2 a^{2} c^{2} e -4 a \,b^{2} c e +3 a b \,c^{2} d +b^{4} e -b^{3} c d \right )}{c \left (4 a c -b^{2}\right )}}{c \,x^{2}+b x +a}+\frac {\frac {\left (-8 a^{2} c^{2} e +14 a \,b^{2} c e -8 a b \,c^{2} d -3 b^{4} e +2 b^{3} c d \right ) \ln \left (c \,x^{2}+b x +a \right )}{2 c}+\frac {2 \left (11 a^{2} b c e -6 a^{2} c^{2} d -3 a \,b^{3} e +2 a \,b^{2} c d -\frac {\left (-8 a^{2} c^{2} e +14 a \,b^{2} c e -8 a b \,c^{2} d -3 b^{4} e +2 b^{3} c d \right ) b}{2 c}\right ) \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{4 a c -b^{2}}}{c^{3}}\) | \(349\) |
| risch | \(\text {Expression too large to display}\) | \(4104\) |
-1/c^3*(-1/2*c*e*x^2+2*b*e*x-c*d*x)+1/c^3*((-(5*a^2*b*c^2*e-2*a^2*c^3*d-5* a*b^3*c*e+4*a*b^2*c^2*d+b^5*e-b^4*c*d)/c/(4*a*c-b^2)*x-a*(2*a^2*c^2*e-4*a* b^2*c*e+3*a*b*c^2*d+b^4*e-b^3*c*d)/c/(4*a*c-b^2))/(c*x^2+b*x+a)+1/(4*a*c-b ^2)*(1/2*(-8*a^2*c^2*e+14*a*b^2*c*e-8*a*b*c^2*d-3*b^4*e+2*b^3*c*d)/c*ln(c* x^2+b*x+a)+2*(11*a^2*b*c*e-6*a^2*c^2*d-3*a*b^3*e+2*a*b^2*c*d-1/2*(-8*a^2*c ^2*e+14*a*b^2*c*e-8*a*b*c^2*d-3*b^4*e+2*b^3*c*d)*b/c)/(4*a*c-b^2)^(1/2)*ar ctan((2*c*x+b)/(4*a*c-b^2)^(1/2))))
Leaf count of result is larger than twice the leaf count of optimal. 838 vs. \(2 (254) = 508\).
Time = 0.30 (sec) , antiderivative size = 1696, normalized size of antiderivative = 6.47 \[ \int \frac {x^4 (d+e x)}{\left (a+b x+c x^2\right )^2} \, dx=\text {Too large to display} \]
[1/2*((b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*e*x^4 + (2*(b^4*c^3 - 8*a*b^2*c ^4 + 16*a^2*c^5)*d - 3*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*e)*x^3 + (2* (b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*d - (4*b^6*c - 33*a*b^4*c^2 + 72*a^ 2*b^2*c^3 - 16*a^3*c^4)*e)*x^2 + ((2*(b^4*c^2 - 6*a*b^2*c^3 + 6*a^2*c^4)*d - (3*b^5*c - 20*a*b^3*c^2 + 30*a^2*b*c^3)*e)*x^2 + 2*(a*b^4*c - 6*a^2*b^2 *c^2 + 6*a^3*c^3)*d - (3*a*b^5 - 20*a^2*b^3*c + 30*a^3*b*c^2)*e + (2*(b^5* c - 6*a*b^3*c^2 + 6*a^2*b*c^3)*d - (3*b^6 - 20*a*b^4*c + 30*a^2*b^2*c^2)*e )*x)*sqrt(b^2 - 4*a*c)*log((2*c^2*x^2 + 2*b*c*x + b^2 - 2*a*c - sqrt(b^2 - 4*a*c)*(2*c*x + b))/(c*x^2 + b*x + a)) - 2*(a*b^5*c - 7*a^2*b^3*c^2 + 12* a^3*b*c^3)*d + 2*(a*b^6 - 8*a^2*b^4*c + 18*a^3*b^2*c^2 - 8*a^4*c^3)*e - 2* ((b^6*c - 9*a*b^4*c^2 + 26*a^2*b^2*c^3 - 24*a^3*c^4)*d - (b^7 - 11*a*b^5*c + 41*a^2*b^3*c^2 - 52*a^3*b*c^3)*e)*x - ((2*(b^5*c^2 - 8*a*b^3*c^3 + 16*a ^2*b*c^4)*d - (3*b^6*c - 26*a*b^4*c^2 + 64*a^2*b^2*c^3 - 32*a^3*c^4)*e)*x^ 2 + 2*(a*b^5*c - 8*a^2*b^3*c^2 + 16*a^3*b*c^3)*d - (3*a*b^6 - 26*a^2*b^4*c + 64*a^3*b^2*c^2 - 32*a^4*c^3)*e + (2*(b^6*c - 8*a*b^4*c^2 + 16*a^2*b^2*c ^3)*d - (3*b^7 - 26*a*b^5*c + 64*a^2*b^3*c^2 - 32*a^3*b*c^3)*e)*x)*log(c*x ^2 + b*x + a))/(a*b^4*c^4 - 8*a^2*b^2*c^5 + 16*a^3*c^6 + (b^4*c^5 - 8*a*b^ 2*c^6 + 16*a^2*c^7)*x^2 + (b^5*c^4 - 8*a*b^3*c^5 + 16*a^2*b*c^6)*x), 1/2*( (b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*e*x^4 + (2*(b^4*c^3 - 8*a*b^2*c^4 + 1 6*a^2*c^5)*d - 3*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*e)*x^3 + (2*(b^...
Leaf count of result is larger than twice the leaf count of optimal. 1572 vs. \(2 (262) = 524\).
Time = 4.00 (sec) , antiderivative size = 1572, normalized size of antiderivative = 6.00 \[ \int \frac {x^4 (d+e x)}{\left (a+b x+c x^2\right )^2} \, dx=\text {Too large to display} \]
x*(-2*b*e/c**3 + d/c**2) + (-sqrt(-(4*a*c - b**2)**3)*(30*a**2*b*c**2*e - 12*a**2*c**3*d - 20*a*b**3*c*e + 12*a*b**2*c**2*d + 3*b**5*e - 2*b**4*c*d) /(2*c**4*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) - (2*a*c *e - 3*b**2*e + 2*b*c*d)/(2*c**4))*log(x + (16*a**3*c**2*e - 17*a**2*b**2* c*e + 10*a**2*b*c**2*d + 16*a**2*c**5*(-sqrt(-(4*a*c - b**2)**3)*(30*a**2* b*c**2*e - 12*a**2*c**3*d - 20*a*b**3*c*e + 12*a*b**2*c**2*d + 3*b**5*e - 2*b**4*c*d)/(2*c**4*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6 )) - (2*a*c*e - 3*b**2*e + 2*b*c*d)/(2*c**4)) + 3*a*b**4*e - 2*a*b**3*c*d - 8*a*b**2*c**4*(-sqrt(-(4*a*c - b**2)**3)*(30*a**2*b*c**2*e - 12*a**2*c** 3*d - 20*a*b**3*c*e + 12*a*b**2*c**2*d + 3*b**5*e - 2*b**4*c*d)/(2*c**4*(6 4*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) - (2*a*c*e - 3*b**2 *e + 2*b*c*d)/(2*c**4)) + b**4*c**3*(-sqrt(-(4*a*c - b**2)**3)*(30*a**2*b* c**2*e - 12*a**2*c**3*d - 20*a*b**3*c*e + 12*a*b**2*c**2*d + 3*b**5*e - 2* b**4*c*d)/(2*c**4*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) - (2*a*c*e - 3*b**2*e + 2*b*c*d)/(2*c**4)))/(30*a**2*b*c**2*e - 12*a**2*c **3*d - 20*a*b**3*c*e + 12*a*b**2*c**2*d + 3*b**5*e - 2*b**4*c*d)) + (sqrt (-(4*a*c - b**2)**3)*(30*a**2*b*c**2*e - 12*a**2*c**3*d - 20*a*b**3*c*e + 12*a*b**2*c**2*d + 3*b**5*e - 2*b**4*c*d)/(2*c**4*(64*a**3*c**3 - 48*a**2* b**2*c**2 + 12*a*b**4*c - b**6)) - (2*a*c*e - 3*b**2*e + 2*b*c*d)/(2*c**4) )*log(x + (16*a**3*c**2*e - 17*a**2*b**2*c*e + 10*a**2*b*c**2*d + 16*a*...
Exception generated. \[ \int \frac {x^4 (d+e x)}{\left (a+b x+c x^2\right )^2} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` for more deta
Time = 0.27 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.08 \[ \int \frac {x^4 (d+e x)}{\left (a+b x+c x^2\right )^2} \, dx=\frac {{\left (2 \, b^{4} c d - 12 \, a b^{2} c^{2} d + 12 \, a^{2} c^{3} d - 3 \, b^{5} e + 20 \, a b^{3} c e - 30 \, a^{2} b c^{2} e\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (b^{2} c^{4} - 4 \, a c^{5}\right )} \sqrt {-b^{2} + 4 \, a c}} - \frac {{\left (2 \, b c d - 3 \, b^{2} e + 2 \, a c e\right )} \log \left (c x^{2} + b x + a\right )}{2 \, c^{4}} + \frac {c^{2} e x^{2} + 2 \, c^{2} d x - 4 \, b c e x}{2 \, c^{4}} - \frac {a b^{3} c d - 3 \, a^{2} b c^{2} d - a b^{4} e + 4 \, a^{2} b^{2} c e - 2 \, a^{3} c^{2} e + {\left (b^{4} c d - 4 \, a b^{2} c^{2} d + 2 \, a^{2} c^{3} d - b^{5} e + 5 \, a b^{3} c e - 5 \, a^{2} b c^{2} e\right )} x}{{\left (c x^{2} + b x + a\right )} {\left (b^{2} - 4 \, a c\right )} c^{4}} \]
(2*b^4*c*d - 12*a*b^2*c^2*d + 12*a^2*c^3*d - 3*b^5*e + 20*a*b^3*c*e - 30*a ^2*b*c^2*e)*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/((b^2*c^4 - 4*a*c^5)*sq rt(-b^2 + 4*a*c)) - 1/2*(2*b*c*d - 3*b^2*e + 2*a*c*e)*log(c*x^2 + b*x + a) /c^4 + 1/2*(c^2*e*x^2 + 2*c^2*d*x - 4*b*c*e*x)/c^4 - (a*b^3*c*d - 3*a^2*b* c^2*d - a*b^4*e + 4*a^2*b^2*c*e - 2*a^3*c^2*e + (b^4*c*d - 4*a*b^2*c^2*d + 2*a^2*c^3*d - b^5*e + 5*a*b^3*c*e - 5*a^2*b*c^2*e)*x)/((c*x^2 + b*x + a)* (b^2 - 4*a*c)*c^4)
Time = 10.60 (sec) , antiderivative size = 427, normalized size of antiderivative = 1.63 \[ \int \frac {x^4 (d+e x)}{\left (a+b x+c x^2\right )^2} \, dx=x\,\left (\frac {d}{c^2}-\frac {2\,b\,e}{c^3}\right )-\frac {\frac {a\,\left (2\,e\,a^2\,c^2-4\,e\,a\,b^2\,c+3\,d\,a\,b\,c^2+e\,b^4-d\,b^3\,c\right )}{c\,\left (4\,a\,c-b^2\right )}+\frac {x\,\left (5\,e\,a^2\,b\,c^2-2\,d\,a^2\,c^3-5\,e\,a\,b^3\,c+4\,d\,a\,b^2\,c^2+e\,b^5-d\,b^4\,c\right )}{c\,\left (4\,a\,c-b^2\right )}}{c^4\,x^2+b\,c^3\,x+a\,c^3}+\frac {e\,x^2}{2\,c^2}-\frac {\ln \left (c\,x^2+b\,x+a\right )\,\left (128\,e\,a^4\,c^4-288\,e\,a^3\,b^2\,c^3+128\,d\,a^3\,b\,c^4+168\,e\,a^2\,b^4\,c^2-96\,d\,a^2\,b^3\,c^3-38\,e\,a\,b^6\,c+24\,d\,a\,b^5\,c^2+3\,e\,b^8-2\,d\,b^7\,c\right )}{2\,\left (64\,a^3\,c^7-48\,a^2\,b^2\,c^6+12\,a\,b^4\,c^5-b^6\,c^4\right )}+\frac {\mathrm {atan}\left (\frac {2\,c\,x}{\sqrt {4\,a\,c-b^2}}-\frac {b^3\,c^3-4\,a\,b\,c^4}{c^3\,{\left (4\,a\,c-b^2\right )}^{3/2}}\right )\,\left (30\,e\,a^2\,b\,c^2-12\,d\,a^2\,c^3-20\,e\,a\,b^3\,c+12\,d\,a\,b^2\,c^2+3\,e\,b^5-2\,d\,b^4\,c\right )}{c^4\,{\left (4\,a\,c-b^2\right )}^{3/2}} \]
x*(d/c^2 - (2*b*e)/c^3) - ((a*(b^4*e + 2*a^2*c^2*e - b^3*c*d + 3*a*b*c^2*d - 4*a*b^2*c*e))/(c*(4*a*c - b^2)) + (x*(b^5*e - 2*a^2*c^3*d - b^4*c*d - 5 *a*b^3*c*e + 4*a*b^2*c^2*d + 5*a^2*b*c^2*e))/(c*(4*a*c - b^2)))/(a*c^3 + c ^4*x^2 + b*c^3*x) + (e*x^2)/(2*c^2) - (log(a + b*x + c*x^2)*(3*b^8*e + 128 *a^4*c^4*e - 2*b^7*c*d - 96*a^2*b^3*c^3*d + 168*a^2*b^4*c^2*e - 288*a^3*b^ 2*c^3*e - 38*a*b^6*c*e + 24*a*b^5*c^2*d + 128*a^3*b*c^4*d))/(2*(64*a^3*c^7 - b^6*c^4 + 12*a*b^4*c^5 - 48*a^2*b^2*c^6)) + (atan((2*c*x)/(4*a*c - b^2) ^(1/2) - (b^3*c^3 - 4*a*b*c^4)/(c^3*(4*a*c - b^2)^(3/2)))*(3*b^5*e - 12*a^ 2*c^3*d - 2*b^4*c*d - 20*a*b^3*c*e + 12*a*b^2*c^2*d + 30*a^2*b*c^2*e))/(c^ 4*(4*a*c - b^2)^(3/2))