Integrand size = 21, antiderivative size = 211 \[ \int \frac {d+e x}{x^2 \left (a+b x+c x^2\right )^2} \, dx=-\frac {2 b^2 d-6 a c d-a b e}{a^2 \left (b^2-4 a c\right ) x}+\frac {b^2 d-2 a c d-a b e+c (b d-2 a e) x}{a \left (b^2-4 a c\right ) x \left (a+b x+c x^2\right )}-\frac {\left (2 b^4 d-12 a b^2 c d+12 a^2 c^2 d-a b^3 e+6 a^2 b c e\right ) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{a^3 \left (b^2-4 a c\right )^{3/2}}-\frac {(2 b d-a e) \log (x)}{a^3}+\frac {(2 b d-a e) \log \left (a+b x+c x^2\right )}{2 a^3} \]
(a*b*e+6*a*c*d-2*b^2*d)/a^2/(-4*a*c+b^2)/x+(b^2*d-2*a*c*d-a*b*e+c*(-2*a*e+ b*d)*x)/a/(-4*a*c+b^2)/x/(c*x^2+b*x+a)-(6*a^2*b*c*e+12*a^2*c^2*d-a*b^3*e-1 2*a*b^2*c*d+2*b^4*d)*arctanh((2*c*x+b)/(-4*a*c+b^2)^(1/2))/a^3/(-4*a*c+b^2 )^(3/2)-(-a*e+2*b*d)*ln(x)/a^3+1/2*(-a*e+2*b*d)*ln(c*x^2+b*x+a)/a^3
Time = 0.22 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.91 \[ \int \frac {d+e x}{x^2 \left (a+b x+c x^2\right )^2} \, dx=\frac {-\frac {2 a d}{x}-\frac {2 a \left (b^3 d+2 a c (a e-c d x)+b^2 (-a e+c d x)-a b c (3 d+e x)\right )}{\left (b^2-4 a c\right ) (a+x (b+c x))}-\frac {2 \left (2 b^4 d-12 a b^2 c d+12 a^2 c^2 d-a b^3 e+6 a^2 b c e\right ) \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\left (-b^2+4 a c\right )^{3/2}}+2 (-2 b d+a e) \log (x)+(2 b d-a e) \log (a+x (b+c x))}{2 a^3} \]
((-2*a*d)/x - (2*a*(b^3*d + 2*a*c*(a*e - c*d*x) + b^2*(-(a*e) + c*d*x) - a *b*c*(3*d + e*x)))/((b^2 - 4*a*c)*(a + x*(b + c*x))) - (2*(2*b^4*d - 12*a* b^2*c*d + 12*a^2*c^2*d - a*b^3*e + 6*a^2*b*c*e)*ArcTan[(b + 2*c*x)/Sqrt[-b ^2 + 4*a*c]])/(-b^2 + 4*a*c)^(3/2) + 2*(-2*b*d + a*e)*Log[x] + (2*b*d - a* e)*Log[a + x*(b + c*x)])/(2*a^3)
Time = 0.55 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.10, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {1235, 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.
| \(\displaystyle \int \frac {d+e x}{x^2 \left (a+b x+c x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 1235 |
| \(\displaystyle \frac {c x (b d-2 a e)-a b e-2 a c d+b^2 d}{a x \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {\int -\frac {2 d b^2-a e b-6 a c d+2 c (b d-2 a e) x}{x^2 \left (c x^2+b x+a\right )}dx}{a \left (b^2-4 a c\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {2 d b^2-a e b-6 a c d+2 c (b d-2 a e) x}{x^2 \left (c x^2+b x+a\right )}dx}{a \left (b^2-4 a c\right )}+\frac {c x (b d-2 a e)-a b e-2 a c d+b^2 d}{a x \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\) |
| \(\Big \downarrow \) 1200 |
| \(\displaystyle \frac {\int \left (-\frac {\left (4 a c-b^2\right ) (a e-2 b d)}{a^2 x}+\frac {2 d b^4-a e b^3-10 a c d b^2+5 a^2 c e b+6 a^2 c^2 d+c \left (b^2-4 a c\right ) (2 b d-a e) x}{a^2 \left (c x^2+b x+a\right )}+\frac {2 d b^2-a e b-6 a c d}{a x^2}\right )dx}{a \left (b^2-4 a c\right )}+\frac {c x (b d-2 a e)-a b e-2 a c d+b^2 d}{a x \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {\text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right ) \left (6 a^2 b c e+12 a^2 c^2 d-a b^3 e-12 a b^2 c d+2 b^4 d\right )}{a^2 \sqrt {b^2-4 a c}}+\frac {\left (b^2-4 a c\right ) (2 b d-a e) \log \left (a+b x+c x^2\right )}{2 a^2}-\frac {\log (x) \left (b^2-4 a c\right ) (2 b d-a e)}{a^2}-\frac {-a b e-6 a c d+2 b^2 d}{a x}}{a \left (b^2-4 a c\right )}+\frac {c x (b d-2 a e)-a b e-2 a c d+b^2 d}{a x \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\) |
(b^2*d - 2*a*c*d - a*b*e + c*(b*d - 2*a*e)*x)/(a*(b^2 - 4*a*c)*x*(a + b*x + c*x^2)) + (-((2*b^2*d - 6*a*c*d - a*b*e)/(a*x)) - ((2*b^4*d - 12*a*b^2*c *d + 12*a^2*c^2*d - a*b^3*e + 6*a^2*b*c*e)*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(a^2*Sqrt[b^2 - 4*a*c]) - ((b^2 - 4*a*c)*(2*b*d - a*e)*Log[x])/a^ 2 + ((b^2 - 4*a*c)*(2*b*d - a*e)*Log[a + b*x + c*x^2])/(2*a^2))/(a*(b^2 - 4*a*c))
3.9.96.3.1 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)*(f*(b*c*d - b^2*e + 2 *a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^m *(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2*(p + m + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d* m + b*e*m) - b*d*(3*c*d - b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p] )
Time = 0.21 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.39
| method | result | size |
| default | \(-\frac {d}{a^{2} x}+\frac {\left (a e -2 b d \right ) \ln \left (x \right )}{a^{3}}-\frac {\frac {\frac {a c \left (a b e +2 a c d -b^{2} d \right ) x}{4 a c -b^{2}}-\frac {a \left (2 a^{2} c e -a \,b^{2} e -3 a b c d +d \,b^{3}\right )}{4 a c -b^{2}}}{c \,x^{2}+b x +a}+\frac {\frac {\left (4 a^{2} c^{2} e -a \,b^{2} c e -8 a b \,c^{2} d +2 b^{3} c d \right ) \ln \left (c \,x^{2}+b x +a \right )}{2 c}+\frac {2 \left (5 a^{2} b c e +6 a^{2} c^{2} d -a \,b^{3} e -10 a \,b^{2} c d +2 d \,b^{4}-\frac {\left (4 a^{2} c^{2} e -a \,b^{2} c e -8 a b \,c^{2} d +2 b^{3} c d \right ) b}{2 c}\right ) \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{4 a c -b^{2}}}{a^{3}}\) | \(293\) |
| risch | \(\frac {-\frac {c \left (a b e +6 a c d -2 b^{2} d \right ) x^{2}}{a^{2} \left (4 a c -b^{2}\right )}+\frac {\left (2 a^{2} c e -a \,b^{2} e -7 a b c d +2 d \,b^{3}\right ) x}{\left (4 a c -b^{2}\right ) a^{2}}-\frac {d}{a}}{x \left (c \,x^{2}+b x +a \right )}+\frac {e \ln \left (x \right )}{a^{2}}-\frac {2 \ln \left (x \right ) b d}{a^{3}}+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (64 a^{6} c^{3}-48 a^{5} b^{2} c^{2}+12 a^{4} b^{4} c -a^{3} b^{6}\right ) \textit {\_Z}^{2}+\left (64 c^{3} e \,a^{4}-48 a^{3} b^{2} c^{2} e -128 a^{3} b \,c^{3} d +12 a^{2} b^{4} c e +96 a^{2} b^{3} c^{2} d -a \,b^{6} e -24 a \,b^{5} c d +2 d \,b^{7}\right ) \textit {\_Z} +16 a^{2} c^{3} e^{2}-3 a \,b^{2} c^{2} e^{2}-28 a b \,c^{3} d e +36 a \,c^{4} d^{2}+6 b^{3} c^{2} d e -8 b^{2} c^{3} d^{2}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (96 a^{7} c^{3}-80 a^{6} b^{2} c^{2}+22 a^{5} b^{4} c -2 a^{4} b^{6}\right ) \textit {\_R}^{2}+\left (48 a^{5} c^{3} e -20 a^{4} b^{2} c^{2} e -72 a^{4} b \,c^{3} d +2 a^{3} b^{4} c e +34 a^{3} b^{3} c^{2} d -4 a^{2} b^{5} c d \right ) \textit {\_R} +a^{2} b^{2} c^{2} e^{2}+12 a^{2} b \,c^{3} d e +36 a^{2} c^{4} d^{2}-4 a \,b^{3} c^{2} d e -24 a \,b^{2} c^{3} d^{2}+4 b^{4} c^{2} d^{2}\right ) x +\left (-16 a^{7} b \,c^{2}+8 a^{6} b^{3} c -a^{5} b^{5}\right ) \textit {\_R}^{2}+\left (20 a^{5} b \,c^{2} e +24 a^{5} c^{3} d -9 a^{4} b^{3} c e -46 a^{4} b^{2} c^{2} d +a^{3} b^{5} e +18 a^{3} b^{4} c d -2 a^{2} b^{6} d \right ) \textit {\_R} -4 a^{3} b \,c^{2} e^{2}-24 a^{3} c^{3} d e +a^{2} b^{3} c \,e^{2}+22 a^{2} b^{2} c^{2} d e +48 a^{2} b \,c^{3} d^{2}-4 a \,b^{4} c d e -28 a \,b^{3} c^{2} d^{2}+4 b^{5} c \,d^{2}\right )\right )\) | \(680\) |
-d/a^2/x+(a*e-2*b*d)/a^3*ln(x)-1/a^3*((a*c*(a*b*e+2*a*c*d-b^2*d)/(4*a*c-b^ 2)*x-a*(2*a^2*c*e-a*b^2*e-3*a*b*c*d+b^3*d)/(4*a*c-b^2))/(c*x^2+b*x+a)+1/(4 *a*c-b^2)*(1/2*(4*a^2*c^2*e-a*b^2*c*e-8*a*b*c^2*d+2*b^3*c*d)/c*ln(c*x^2+b* x+a)+2*(5*a^2*b*c*e+6*a^2*c^2*d-a*b^3*e-10*a*b^2*c*d+2*d*b^4-1/2*(4*a^2*c^ 2*e-a*b^2*c*e-8*a*b*c^2*d+2*b^3*c*d)*b/c)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+ b)/(4*a*c-b^2)^(1/2))))
Leaf count of result is larger than twice the leaf count of optimal. 798 vs. \(2 (205) = 410\).
Time = 1.35 (sec) , antiderivative size = 1615, normalized size of antiderivative = 7.65 \[ \int \frac {d+e x}{x^2 \left (a+b x+c x^2\right )^2} \, dx=\text {Too large to display} \]
[-1/2*(2*(2*(a*b^4*c - 7*a^2*b^2*c^2 + 12*a^3*c^3)*d - (a^2*b^3*c - 4*a^3* b*c^2)*e)*x^2 + ((2*(b^4*c - 6*a*b^2*c^2 + 6*a^2*c^3)*d - (a*b^3*c - 6*a^2 *b*c^2)*e)*x^3 + (2*(b^5 - 6*a*b^3*c + 6*a^2*b*c^2)*d - (a*b^4 - 6*a^2*b^2 *c)*e)*x^2 + (2*(a*b^4 - 6*a^2*b^2*c + 6*a^3*c^2)*d - (a^2*b^3 - 6*a^3*b*c )*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^2*b^4 - 8*a^3*b^2*c + 16 *a^4*c^2)*d + 2*((2*a*b^5 - 15*a^2*b^3*c + 28*a^3*b*c^2)*d - (a^2*b^4 - 6* a^3*b^2*c + 8*a^4*c^2)*e)*x - ((2*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d - (a*b^4*c - 8*a^2*b^2*c^2 + 16*a^3*c^3)*e)*x^3 + (2*(b^6 - 8*a*b^4*c + 16* a^2*b^2*c^2)*d - (a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*e)*x^2 + (2*(a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*d - (a^2*b^4 - 8*a^3*b^2*c + 16*a^4*c^2)*e)*x )*log(c*x^2 + b*x + a) + 2*((2*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d - (a *b^4*c - 8*a^2*b^2*c^2 + 16*a^3*c^3)*e)*x^3 + (2*(b^6 - 8*a*b^4*c + 16*a^2 *b^2*c^2)*d - (a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*e)*x^2 + (2*(a*b^5 - 8* a^2*b^3*c + 16*a^3*b*c^2)*d - (a^2*b^4 - 8*a^3*b^2*c + 16*a^4*c^2)*e)*x)*l og(x))/((a^3*b^4*c - 8*a^4*b^2*c^2 + 16*a^5*c^3)*x^3 + (a^3*b^5 - 8*a^4*b^ 3*c + 16*a^5*b*c^2)*x^2 + (a^4*b^4 - 8*a^5*b^2*c + 16*a^6*c^2)*x), -1/2*(2 *(2*(a*b^4*c - 7*a^2*b^2*c^2 + 12*a^3*c^3)*d - (a^2*b^3*c - 4*a^3*b*c^2)*e )*x^2 + 2*((2*(b^4*c - 6*a*b^2*c^2 + 6*a^2*c^3)*d - (a*b^3*c - 6*a^2*b*c^2 )*e)*x^3 + (2*(b^5 - 6*a*b^3*c + 6*a^2*b*c^2)*d - (a*b^4 - 6*a^2*b^2*c)...
Timed out. \[ \int \frac {d+e x}{x^2 \left (a+b x+c x^2\right )^2} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {d+e x}{x^2 \left (a+b x+c x^2\right )^2} \, dx=\text {Exception raised: ValueError} \]
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 = 238, normalized size of antiderivative = 1.13 \[ \int \frac {d+e x}{x^2 \left (a+b x+c x^2\right )^2} \, dx=\frac {{\left (2 \, b^{4} d - 12 \, a b^{2} c d + 12 \, a^{2} c^{2} d - a b^{3} e + 6 \, a^{2} b c e\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (a^{3} b^{2} - 4 \, a^{4} c\right )} \sqrt {-b^{2} + 4 \, a c}} - \frac {2 \, b^{2} c d x^{2} - 6 \, a c^{2} d x^{2} - a b c e x^{2} + 2 \, b^{3} d x - 7 \, a b c d x - a b^{2} e x + 2 \, a^{2} c e x + a b^{2} d - 4 \, a^{2} c d}{{\left (a^{2} b^{2} - 4 \, a^{3} c\right )} {\left (c x^{3} + b x^{2} + a x\right )}} + \frac {{\left (2 \, b d - a e\right )} \log \left (c x^{2} + b x + a\right )}{2 \, a^{3}} - \frac {{\left (2 \, b d - a e\right )} \log \left ({\left | x \right |}\right )}{a^{3}} \]
(2*b^4*d - 12*a*b^2*c*d + 12*a^2*c^2*d - a*b^3*e + 6*a^2*b*c*e)*arctan((2* c*x + b)/sqrt(-b^2 + 4*a*c))/((a^3*b^2 - 4*a^4*c)*sqrt(-b^2 + 4*a*c)) - (2 *b^2*c*d*x^2 - 6*a*c^2*d*x^2 - a*b*c*e*x^2 + 2*b^3*d*x - 7*a*b*c*d*x - a*b ^2*e*x + 2*a^2*c*e*x + a*b^2*d - 4*a^2*c*d)/((a^2*b^2 - 4*a^3*c)*(c*x^3 + b*x^2 + a*x)) + 1/2*(2*b*d - a*e)*log(c*x^2 + b*x + a)/a^3 - (2*b*d - a*e) *log(abs(x))/a^3
Time = 11.77 (sec) , antiderivative size = 1366, normalized size of antiderivative = 6.47 \[ \int \frac {d+e x}{x^2 \left (a+b x+c x^2\right )^2} \, dx=\ln \left (96\,a^5\,c^3\,e-2\,a^2\,b^6\,e+4\,a\,b^7\,d+4\,b^8\,d\,x+174\,a^3\,b^3\,c^2\,d-2\,a^2\,b^3\,e\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+6\,a^3\,c^2\,d\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-84\,a^4\,b^2\,c^2\,e-2\,a\,b^7\,e\,x+4\,a\,b^4\,d\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-46\,a^2\,b^5\,c\,d-216\,a^4\,b\,c^3\,d+23\,a^3\,b^4\,c\,e+48\,a^4\,c^4\,d\,x+4\,b^5\,d\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+9\,a^3\,b\,c\,e\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-2\,a\,b^4\,e\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+24\,a^2\,b^5\,c\,e\,x+120\,a^4\,b\,c^3\,e\,x-18\,a^2\,b^2\,c\,d\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+194\,a^2\,b^4\,c^2\,d\,x-276\,a^3\,b^2\,c^3\,d\,x-94\,a^3\,b^3\,c^2\,e\,x-12\,a^3\,c^2\,e\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-48\,a\,b^6\,c\,d\,x-24\,a\,b^3\,c\,d\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+30\,a^2\,b\,c^2\,d\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+12\,a^2\,b^2\,c\,e\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}\right )\,\left (\frac {b^4\,d\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+6\,a^2\,c^2\,d\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-\frac {a\,b^3\,e\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}}{2}-6\,a\,b^2\,c\,d\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+3\,a^2\,b\,c\,e\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}}{-64\,a^6\,c^3+48\,a^5\,b^2\,c^2-12\,a^4\,b^4\,c+a^3\,b^6}-\frac {e}{2\,a^2}+\frac {b\,d}{a^3}\right )-\ln \left (2\,a^2\,b^6\,e-96\,a^5\,c^3\,e-4\,a\,b^7\,d-4\,b^8\,d\,x-174\,a^3\,b^3\,c^2\,d-2\,a^2\,b^3\,e\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+6\,a^3\,c^2\,d\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+84\,a^4\,b^2\,c^2\,e+2\,a\,b^7\,e\,x+4\,a\,b^4\,d\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+46\,a^2\,b^5\,c\,d+216\,a^4\,b\,c^3\,d-23\,a^3\,b^4\,c\,e-48\,a^4\,c^4\,d\,x+4\,b^5\,d\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+9\,a^3\,b\,c\,e\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-2\,a\,b^4\,e\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-24\,a^2\,b^5\,c\,e\,x-120\,a^4\,b\,c^3\,e\,x-18\,a^2\,b^2\,c\,d\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-194\,a^2\,b^4\,c^2\,d\,x+276\,a^3\,b^2\,c^3\,d\,x+94\,a^3\,b^3\,c^2\,e\,x-12\,a^3\,c^2\,e\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+48\,a\,b^6\,c\,d\,x-24\,a\,b^3\,c\,d\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+30\,a^2\,b\,c^2\,d\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+12\,a^2\,b^2\,c\,e\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}\right )\,\left (\frac {b^4\,d\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+6\,a^2\,c^2\,d\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-\frac {a\,b^3\,e\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}}{2}-6\,a\,b^2\,c\,d\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+3\,a^2\,b\,c\,e\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}}{-64\,a^6\,c^3+48\,a^5\,b^2\,c^2-12\,a^4\,b^4\,c+a^3\,b^6}+\frac {e}{2\,a^2}-\frac {b\,d}{a^3}\right )-\frac {\frac {d}{a}-\frac {x\,\left (2\,c\,e\,a^2-e\,a\,b^2-7\,c\,d\,a\,b+2\,d\,b^3\right )}{a^2\,\left (4\,a\,c-b^2\right )}+\frac {c\,x^2\,\left (-2\,d\,b^2+a\,e\,b+6\,a\,c\,d\right )}{a^2\,\left (4\,a\,c-b^2\right )}}{c\,x^3+b\,x^2+a\,x}+\frac {\ln \left (x\right )\,\left (a\,e-2\,b\,d\right )}{a^3} \]
log(96*a^5*c^3*e - 2*a^2*b^6*e + 4*a*b^7*d + 4*b^8*d*x + 174*a^3*b^3*c^2*d - 2*a^2*b^3*e*(-(4*a*c - b^2)^3)^(1/2) + 6*a^3*c^2*d*(-(4*a*c - b^2)^3)^( 1/2) - 84*a^4*b^2*c^2*e - 2*a*b^7*e*x + 4*a*b^4*d*(-(4*a*c - b^2)^3)^(1/2) - 46*a^2*b^5*c*d - 216*a^4*b*c^3*d + 23*a^3*b^4*c*e + 48*a^4*c^4*d*x + 4* b^5*d*x*(-(4*a*c - b^2)^3)^(1/2) + 9*a^3*b*c*e*(-(4*a*c - b^2)^3)^(1/2) - 2*a*b^4*e*x*(-(4*a*c - b^2)^3)^(1/2) + 24*a^2*b^5*c*e*x + 120*a^4*b*c^3*e* x - 18*a^2*b^2*c*d*(-(4*a*c - b^2)^3)^(1/2) + 194*a^2*b^4*c^2*d*x - 276*a^ 3*b^2*c^3*d*x - 94*a^3*b^3*c^2*e*x - 12*a^3*c^2*e*x*(-(4*a*c - b^2)^3)^(1/ 2) - 48*a*b^6*c*d*x - 24*a*b^3*c*d*x*(-(4*a*c - b^2)^3)^(1/2) + 30*a^2*b*c ^2*d*x*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b^2*c*e*x*(-(4*a*c - b^2)^3)^(1/2 ))*((b^4*d*(-(4*a*c - b^2)^3)^(1/2) + 6*a^2*c^2*d*(-(4*a*c - b^2)^3)^(1/2) - (a*b^3*e*(-(4*a*c - b^2)^3)^(1/2))/2 - 6*a*b^2*c*d*(-(4*a*c - b^2)^3)^( 1/2) + 3*a^2*b*c*e*(-(4*a*c - b^2)^3)^(1/2))/(a^3*b^6 - 64*a^6*c^3 - 12*a^ 4*b^4*c + 48*a^5*b^2*c^2) - e/(2*a^2) + (b*d)/a^3) - log(2*a^2*b^6*e - 96* a^5*c^3*e - 4*a*b^7*d - 4*b^8*d*x - 174*a^3*b^3*c^2*d - 2*a^2*b^3*e*(-(4*a *c - b^2)^3)^(1/2) + 6*a^3*c^2*d*(-(4*a*c - b^2)^3)^(1/2) + 84*a^4*b^2*c^2 *e + 2*a*b^7*e*x + 4*a*b^4*d*(-(4*a*c - b^2)^3)^(1/2) + 46*a^2*b^5*c*d + 2 16*a^4*b*c^3*d - 23*a^3*b^4*c*e - 48*a^4*c^4*d*x + 4*b^5*d*x*(-(4*a*c - b^ 2)^3)^(1/2) + 9*a^3*b*c*e*(-(4*a*c - b^2)^3)^(1/2) - 2*a*b^4*e*x*(-(4*a*c - b^2)^3)^(1/2) - 24*a^2*b^5*c*e*x - 120*a^4*b*c^3*e*x - 18*a^2*b^2*c*d...