Integrand size = 21, antiderivative size = 283 \[ \int \frac {d+e x}{x^3 \left (a+b x+c x^2\right )^2} \, dx=-\frac {3 b^2 d-8 a c d-2 a b e}{2 a^2 \left (b^2-4 a c\right ) x^2}+\frac {3 b^3 d-11 a b c d-2 a b^2 e+6 a^2 c e}{a^3 \left (b^2-4 a c\right ) x}+\frac {b^2 d-2 a c d-a b e+c (b d-2 a e) x}{a \left (b^2-4 a c\right ) x^2 \left (a+b x+c x^2\right )}+\frac {\left (3 b^5 d-20 a b^3 c d+30 a^2 b c^2 d-2 a b^4 e+12 a^2 b^2 c e-12 a^3 c^2 e\right ) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{a^4 \left (b^2-4 a c\right )^{3/2}}+\frac {\left (3 b^2 d-2 a c d-2 a b e\right ) \log (x)}{a^4}-\frac {\left (3 b^2 d-2 a c d-2 a b e\right ) \log \left (a+b x+c x^2\right )}{2 a^4} \]
1/2*(2*a*b*e+8*a*c*d-3*b^2*d)/a^2/(-4*a*c+b^2)/x^2+(6*a^2*c*e-2*a*b^2*e-11 *a*b*c*d+3*b^3*d)/a^3/(-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^2/(c*x^2+b*x+a)+(-12*a^3*c^2*e+12*a^2*b^2*c*e+30*a^2*b* c^2*d-2*a*b^4*e-20*a*b^3*c*d+3*b^5*d)*arctanh((2*c*x+b)/(-4*a*c+b^2)^(1/2) )/a^4/(-4*a*c+b^2)^(3/2)+(-2*a*b*e-2*a*c*d+3*b^2*d)*ln(x)/a^4-1/2*(-2*a*b* e-2*a*c*d+3*b^2*d)*ln(c*x^2+b*x+a)/a^4
Time = 0.28 (sec) , antiderivative size = 253, normalized size of antiderivative = 0.89 \[ \int \frac {d+e x}{x^3 \left (a+b x+c x^2\right )^2} \, dx=\frac {-\frac {a^2 d}{x^2}-\frac {2 a (-2 b d+a e)}{x}+\frac {2 a \left (b^4 d+3 a b c (a e-c d x)+b^3 (-a e+c d x)+2 a^2 c^2 (d+e x)-a b^2 c (4 d+e x)\right )}{\left (b^2-4 a c\right ) (a+x (b+c x))}+\frac {2 \left (3 b^5 d-20 a b^3 c d+30 a^2 b c^2 d-2 a b^4 e+12 a^2 b^2 c e-12 a^3 c^2 e\right ) \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\left (-b^2+4 a c\right )^{3/2}}+2 \left (3 b^2 d-2 a c d-2 a b e\right ) \log (x)+\left (-3 b^2 d+2 a c d+2 a b e\right ) \log (a+x (b+c x))}{2 a^4} \]
(-((a^2*d)/x^2) - (2*a*(-2*b*d + a*e))/x + (2*a*(b^4*d + 3*a*b*c*(a*e - c* d*x) + b^3*(-(a*e) + c*d*x) + 2*a^2*c^2*(d + e*x) - a*b^2*c*(4*d + e*x)))/ ((b^2 - 4*a*c)*(a + x*(b + c*x))) + (2*(3*b^5*d - 20*a*b^3*c*d + 30*a^2*b* c^2*d - 2*a*b^4*e + 12*a^2*b^2*c*e - 12*a^3*c^2*e)*ArcTan[(b + 2*c*x)/Sqrt [-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(3/2) + 2*(3*b^2*d - 2*a*c*d - 2*a*b*e)*Lo g[x] + (-3*b^2*d + 2*a*c*d + 2*a*b*e)*Log[a + x*(b + c*x)])/(2*a^4)
Time = 0.73 (sec) , antiderivative size = 294, normalized size of antiderivative = 1.04, 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.
| \(\displaystyle \int \frac {d+e x}{x^3 \left (a+b x+c x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 1235 |
| \(\displaystyle \frac {c x (b d-2 a e)-a b e-2 a c d+b^2 d}{a x^2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {\int -\frac {3 d b^2-2 a e b-8 a c d+3 c (b d-2 a e) x}{x^3 \left (c x^2+b x+a\right )}dx}{a \left (b^2-4 a c\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {3 d b^2-2 a e b-8 a c d+3 c (b d-2 a e) x}{x^3 \left (c x^2+b x+a\right )}dx}{a \left (b^2-4 a c\right )}+\frac {c x (b d-2 a e)-a b e-2 a c d+b^2 d}{a x^2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\) |
| \(\Big \downarrow \) 1200 |
| \(\displaystyle \frac {\int \left (\frac {3 d b^2-2 a e b-8 a c d}{a x^3}+\frac {\left (4 a c-b^2\right ) \left (-3 d b^2+2 a e b+2 a c d\right )}{a^3 x}+\frac {-3 d b^5+2 a e b^4+17 a c d b^3-10 a^2 c e b^2-19 a^2 c^2 d b+6 a^3 c^2 e-c \left (b^2-4 a c\right ) \left (3 d b^2-2 a e b-2 a c d\right ) x}{a^3 \left (c x^2+b x+a\right )}+\frac {-3 d b^3+2 a e b^2+11 a c d b-6 a^2 c e}{a^2 x^2}\right )dx}{a \left (b^2-4 a c\right )}+\frac {c x (b d-2 a e)-a b e-2 a c d+b^2 d}{a x^2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {\left (b^2-4 a c\right ) \left (-2 a b e-2 a c d+3 b^2 d\right ) \log \left (a+b x+c x^2\right )}{2 a^3}+\frac {\log (x) \left (b^2-4 a c\right ) \left (-2 a b e-2 a c d+3 b^2 d\right )}{a^3}+\frac {6 a^2 c e-2 a b^2 e-11 a b c d+3 b^3 d}{a^2 x}+\frac {\text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right ) \left (-12 a^3 c^2 e+12 a^2 b^2 c e+30 a^2 b c^2 d-2 a b^4 e-20 a b^3 c d+3 b^5 d\right )}{a^3 \sqrt {b^2-4 a c}}-\frac {-2 a b e-8 a c d+3 b^2 d}{2 a x^2}}{a \left (b^2-4 a c\right )}+\frac {c x (b d-2 a e)-a b e-2 a c d+b^2 d}{a x^2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\) |
(b^2*d - 2*a*c*d - a*b*e + c*(b*d - 2*a*e)*x)/(a*(b^2 - 4*a*c)*x^2*(a + b* x + c*x^2)) + (-1/2*(3*b^2*d - 8*a*c*d - 2*a*b*e)/(a*x^2) + (3*b^3*d - 11* a*b*c*d - 2*a*b^2*e + 6*a^2*c*e)/(a^2*x) + ((3*b^5*d - 20*a*b^3*c*d + 30*a ^2*b*c^2*d - 2*a*b^4*e + 12*a^2*b^2*c*e - 12*a^3*c^2*e)*ArcTanh[(b + 2*c*x )/Sqrt[b^2 - 4*a*c]])/(a^3*Sqrt[b^2 - 4*a*c]) + ((b^2 - 4*a*c)*(3*b^2*d - 2*a*c*d - 2*a*b*e)*Log[x])/a^3 - ((b^2 - 4*a*c)*(3*b^2*d - 2*a*c*d - 2*a*b *e)*Log[a + b*x + c*x^2])/(2*a^3))/(a*(b^2 - 4*a*c))
3.9.97.3.1 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 = 0.22 (sec) , antiderivative size = 375, normalized size of antiderivative = 1.33
| method | result | size |
| default | \(-\frac {d}{2 a^{2} x^{2}}-\frac {a e -2 b d}{a^{3} x}+\frac {\left (-2 a b e -2 a c d +3 b^{2} d \right ) \ln \left (x \right )}{a^{4}}-\frac {\frac {\frac {a c \left (2 a^{2} c e -a \,b^{2} e -3 a b c d +d \,b^{3}\right ) x}{4 a c -b^{2}}+\frac {a \left (3 a^{2} b c e +2 a^{2} c^{2} d -a \,b^{3} e -4 a \,b^{2} c d +d \,b^{4}\right )}{4 a c -b^{2}}}{c \,x^{2}+b x +a}+\frac {\frac {\left (-8 a^{2} b \,c^{2} e -8 a^{2} c^{3} d +2 a \,b^{3} c e +14 a \,b^{2} c^{2} d -3 b^{4} c d \right ) \ln \left (c \,x^{2}+b x +a \right )}{2 c}+\frac {2 \left (6 a^{3} c^{2} e -10 c e \,b^{2} a^{2}-19 a^{2} b \,c^{2} d +2 e \,b^{4} a +17 a \,b^{3} c d -3 b^{5} d -\frac {\left (-8 a^{2} b \,c^{2} e -8 a^{2} c^{3} d +2 a \,b^{3} c e +14 a \,b^{2} c^{2} d -3 b^{4} c d \right ) b}{2 c}\right ) \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{4 a c -b^{2}}}{a^{4}}\) | \(375\) |
| risch | \(\frac {-\frac {c \left (6 a^{2} c e -2 a \,b^{2} e -11 a b c d +3 d \,b^{3}\right ) x^{3}}{\left (4 a c -b^{2}\right ) a^{3}}-\frac {\left (14 a^{2} b c e +8 a^{2} c^{2} d -4 a \,b^{3} e -25 a \,b^{2} c d +6 d \,b^{4}\right ) x^{2}}{2 a^{3} \left (4 a c -b^{2}\right )}-\frac {\left (2 a e -3 b d \right ) x}{2 a^{2}}-\frac {d}{2 a}}{x^{2} \left (c \,x^{2}+b x +a \right )}-\frac {2 \ln \left (x \right ) b e}{a^{3}}-\frac {2 c d \ln \left (x \right )}{a^{3}}+\frac {3 \ln \left (x \right ) b^{2} d}{a^{4}}+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (64 a^{7} c^{3}-48 a^{6} b^{2} c^{2}+12 a^{5} b^{4} c -a^{4} b^{6}\right ) \textit {\_Z}^{2}+\left (-128 c^{3} e \,a^{4} b -128 a^{4} c^{4} d +96 c^{2} e \,a^{3} b^{3}+288 a^{3} b^{2} c^{3} d -24 c e \,a^{2} b^{5}-168 a^{2} b^{4} c^{2} d +2 e a \,b^{7}+38 a \,b^{6} c d -3 b^{8} d \right ) \textit {\_Z} +36 a^{2} c^{4} e^{2}-8 a \,b^{2} c^{3} e^{2}-52 a b \,c^{4} d e +64 a \,c^{5} d^{2}+12 b^{3} c^{3} d e -15 b^{2} c^{4} d^{2}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (96 a^{9} c^{3}-80 a^{8} b^{2} c^{2}+22 a^{7} b^{4} c -2 a^{6} b^{6}\right ) \textit {\_R}^{2}+\left (-72 a^{6} b \,c^{3} e -96 a^{6} c^{4} d +34 a^{5} b^{3} c^{2} e +148 a^{5} b^{2} c^{3} d -4 a^{4} b^{5} c e -55 a^{4} b^{4} c^{2} d +6 a^{3} b^{6} c d \right ) \textit {\_R} +36 a^{4} c^{4} e^{2}-24 a^{3} b^{2} c^{3} e^{2}-132 a^{3} b \,c^{4} d e +4 a^{2} b^{4} c^{2} e^{2}+80 a^{2} b^{3} c^{3} d e +121 a^{2} b^{2} c^{4} d^{2}-12 a \,b^{5} c^{2} d e -66 a \,b^{4} c^{3} d^{2}+9 b^{6} c^{2} d^{2}\right ) x +\left (-16 a^{9} b \,c^{2}+8 a^{8} b^{3} c -a^{7} b^{5}\right ) \textit {\_R}^{2}+\left (24 a^{7} c^{3} e -46 a^{6} b^{2} c^{2} e -76 a^{6} b \,c^{3} d +18 a^{5} b^{4} c e +87 a^{5} b^{3} c^{2} d -2 a^{4} b^{6} e -29 a^{4} b^{5} c d +3 a^{3} b^{7} d \right ) \textit {\_R} +48 a^{4} b \,c^{3} e^{2}+48 a^{4} c^{4} d e -28 a^{3} b^{3} c^{2} e^{2}-188 a^{3} b^{2} c^{3} d e -88 a^{3} b \,c^{4} d^{2}+4 a^{2} b^{5} c \,e^{2}+92 a^{2} b^{4} c^{2} d e +178 a^{2} b^{3} c^{3} d^{2}-12 a \,b^{6} c d e -75 a \,b^{5} c^{2} d^{2}+9 b^{7} c \,d^{2}\right )\right )\) | \(855\) |
-1/2*d/a^2/x^2-(a*e-2*b*d)/a^3/x+(-2*a*b*e-2*a*c*d+3*b^2*d)*ln(x)/a^4-1/a^ 4*((a*c*(2*a^2*c*e-a*b^2*e-3*a*b*c*d+b^3*d)/(4*a*c-b^2)*x+a*(3*a^2*b*c*e+2 *a^2*c^2*d-a*b^3*e-4*a*b^2*c*d+b^4*d)/(4*a*c-b^2))/(c*x^2+b*x+a)+1/(4*a*c- b^2)*(1/2*(-8*a^2*b*c^2*e-8*a^2*c^3*d+2*a*b^3*c*e+14*a*b^2*c^2*d-3*b^4*c*d )/c*ln(c*x^2+b*x+a)+2*(6*a^3*c^2*e-10*c*e*b^2*a^2-19*a^2*b*c^2*d+2*e*b^4*a +17*a*b^3*c*d-3*b^5*d-1/2*(-8*a^2*b*c^2*e-8*a^2*c^3*d+2*a*b^3*c*e+14*a*b^2 *c^2*d-3*b^4*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. 992 vs. \(2 (275) = 550\).
Time = 2.50 (sec) , antiderivative size = 2003, normalized size of antiderivative = 7.08 \[ \int \frac {d+e x}{x^3 \left (a+b x+c x^2\right )^2} \, dx=\text {Too large to display} \]
[1/2*(2*((3*a*b^5*c - 23*a^2*b^3*c^2 + 44*a^3*b*c^3)*d - 2*(a^2*b^4*c - 7* a^3*b^2*c^2 + 12*a^4*c^3)*e)*x^3 + ((6*a*b^6 - 49*a^2*b^4*c + 108*a^3*b^2* c^2 - 32*a^4*c^3)*d - 2*(2*a^2*b^5 - 15*a^3*b^3*c + 28*a^4*b*c^2)*e)*x^2 + (((3*b^5*c - 20*a*b^3*c^2 + 30*a^2*b*c^3)*d - 2*(a*b^4*c - 6*a^2*b^2*c^2 + 6*a^3*c^3)*e)*x^4 + ((3*b^6 - 20*a*b^4*c + 30*a^2*b^2*c^2)*d - 2*(a*b^5 - 6*a^2*b^3*c + 6*a^3*b*c^2)*e)*x^3 + ((3*a*b^5 - 20*a^2*b^3*c + 30*a^3*b* c^2)*d - 2*(a^2*b^4 - 6*a^3*b^2*c + 6*a^4*c^2)*e)*x^2)*sqrt(b^2 - 4*a*c)*l og((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)) - (a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2)*d + (3*(a^2*b^5 - 8 *a^3*b^3*c + 16*a^4*b*c^2)*d - 2*(a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2)*e)*x - (((3*b^6*c - 26*a*b^4*c^2 + 64*a^2*b^2*c^3 - 32*a^3*c^4)*d - 2*(a*b^5*c - 8*a^2*b^3*c^2 + 16*a^3*b*c^3)*e)*x^4 + ((3*b^7 - 26*a*b^5*c + 64*a^2*b^ 3*c^2 - 32*a^3*b*c^3)*d - 2*(a*b^6 - 8*a^2*b^4*c + 16*a^3*b^2*c^2)*e)*x^3 + ((3*a*b^6 - 26*a^2*b^4*c + 64*a^3*b^2*c^2 - 32*a^4*c^3)*d - 2*(a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2)*e)*x^2)*log(c*x^2 + b*x + a) + 2*(((3*b^6*c - 26*a*b^4*c^2 + 64*a^2*b^2*c^3 - 32*a^3*c^4)*d - 2*(a*b^5*c - 8*a^2*b^3*c^ 2 + 16*a^3*b*c^3)*e)*x^4 + ((3*b^7 - 26*a*b^5*c + 64*a^2*b^3*c^2 - 32*a^3* b*c^3)*d - 2*(a*b^6 - 8*a^2*b^4*c + 16*a^3*b^2*c^2)*e)*x^3 + ((3*a*b^6 - 2 6*a^2*b^4*c + 64*a^3*b^2*c^2 - 32*a^4*c^3)*d - 2*(a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2)*e)*x^2)*log(x))/((a^4*b^4*c - 8*a^5*b^2*c^2 + 16*a^6*c^3)...
Timed out. \[ \int \frac {d+e x}{x^3 \left (a+b x+c x^2\right )^2} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {d+e x}{x^3 \left (a+b x+c x^2\right )^2} \, dx=\text {Exception raised: ValueError} \]
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 = 0.29 (sec) , antiderivative size = 334, normalized size of antiderivative = 1.18 \[ \int \frac {d+e x}{x^3 \left (a+b x+c x^2\right )^2} \, dx=-\frac {{\left (3 \, b^{5} d - 20 \, a b^{3} c d + 30 \, a^{2} b c^{2} d - 2 \, a b^{4} e + 12 \, a^{2} b^{2} c e - 12 \, a^{3} c^{2} e\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (a^{4} b^{2} - 4 \, a^{5} c\right )} \sqrt {-b^{2} + 4 \, a c}} - \frac {{\left (3 \, b^{2} d - 2 \, a c d - 2 \, a b e\right )} \log \left (c x^{2} + b x + a\right )}{2 \, a^{4}} + \frac {{\left (3 \, b^{2} d - 2 \, a c d - 2 \, a b e\right )} \log \left ({\left | x \right |}\right )}{a^{4}} - \frac {a^{3} b^{2} d - 4 \, a^{4} c d - 2 \, {\left (3 \, a b^{3} c d - 11 \, a^{2} b c^{2} d - 2 \, a^{2} b^{2} c e + 6 \, a^{3} c^{2} e\right )} x^{3} - {\left (6 \, a b^{4} d - 25 \, a^{2} b^{2} c d + 8 \, a^{3} c^{2} d - 4 \, a^{2} b^{3} e + 14 \, a^{3} b c e\right )} x^{2} - {\left (3 \, a^{2} b^{3} d - 12 \, a^{3} b c d - 2 \, a^{3} b^{2} e + 8 \, a^{4} c e\right )} x}{2 \, {\left (c x^{2} + b x + a\right )} {\left (b^{2} - 4 \, a c\right )} a^{4} x^{2}} \]
-(3*b^5*d - 20*a*b^3*c*d + 30*a^2*b*c^2*d - 2*a*b^4*e + 12*a^2*b^2*c*e - 1 2*a^3*c^2*e)*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/((a^4*b^2 - 4*a^5*c)*s qrt(-b^2 + 4*a*c)) - 1/2*(3*b^2*d - 2*a*c*d - 2*a*b*e)*log(c*x^2 + b*x + a )/a^4 + (3*b^2*d - 2*a*c*d - 2*a*b*e)*log(abs(x))/a^4 - 1/2*(a^3*b^2*d - 4 *a^4*c*d - 2*(3*a*b^3*c*d - 11*a^2*b*c^2*d - 2*a^2*b^2*c*e + 6*a^3*c^2*e)* x^3 - (6*a*b^4*d - 25*a^2*b^2*c*d + 8*a^3*c^2*d - 4*a^2*b^3*e + 14*a^3*b*c *e)*x^2 - (3*a^2*b^3*d - 12*a^3*b*c*d - 2*a^3*b^2*e + 8*a^4*c*e)*x)/((c*x^ 2 + b*x + a)*(b^2 - 4*a*c)*a^4*x^2)
Time = 11.73 (sec) , antiderivative size = 1661, normalized size of antiderivative = 5.87 \[ \int \frac {d+e x}{x^3 \left (a+b x+c x^2\right )^2} \, dx=\text {Too large to display} \]
log(192*a^5*c^4*d - 4*a^2*b^7*e + 6*a*b^8*d + 6*b^9*d*x + 307*a^3*b^4*c^2* d - 492*a^4*b^2*c^3*d + 4*a^2*b^4*e*(-(4*a*c - b^2)^3)^(1/2) - 174*a^4*b^3 *c^2*e + 6*a^4*c^2*e*(-(4*a*c - b^2)^3)^(1/2) - 4*a*b^8*e*x - 6*a*b^5*d*(- (4*a*c - b^2)^3)^(1/2) - 73*a^2*b^6*c*d + 46*a^3*b^5*c*e + 216*a^5*b*c^3*e - 6*b^6*d*x*(-(4*a*c - b^2)^3)^(1/2) - 48*a^5*c^4*e*x + 312*a^4*b*c^4*d*x + 4*a*b^5*e*x*(-(4*a*c - b^2)^3)^(1/2) + 48*a^2*b^6*c*e*x + 31*a^2*b^3*c* d*(-(4*a*c - b^2)^3)^(1/2) - 27*a^3*b*c^2*d*(-(4*a*c - b^2)^3)^(1/2) - 18* a^3*b^2*c*e*(-(4*a*c - b^2)^3)^(1/2) + 339*a^2*b^5*c^2*d*x - 602*a^3*b^3*c ^3*d*x + 24*a^3*c^3*d*x*(-(4*a*c - b^2)^3)^(1/2) - 194*a^3*b^4*c^2*e*x + 2 76*a^4*b^2*c^3*e*x - 76*a*b^7*c*d*x - 69*a^2*b^2*c^2*d*x*(-(4*a*c - b^2)^3 )^(1/2) + 40*a*b^4*c*d*x*(-(4*a*c - b^2)^3)^(1/2) - 24*a^2*b^3*c*e*x*(-(4* a*c - b^2)^3)^(1/2) + 30*a^3*b*c^2*e*x*(-(4*a*c - b^2)^3)^(1/2))*(((3*b^5* d*(-(4*a*c - b^2)^3)^(1/2))/2 - 6*a^3*c^2*e*(-(4*a*c - b^2)^3)^(1/2) - a*b ^4*e*(-(4*a*c - b^2)^3)^(1/2) - 10*a*b^3*c*d*(-(4*a*c - b^2)^3)^(1/2) + 15 *a^2*b*c^2*d*(-(4*a*c - b^2)^3)^(1/2) + 6*a^2*b^2*c*e*(-(4*a*c - b^2)^3)^( 1/2))/(a^4*b^6 - 64*a^7*c^3 - 12*a^5*b^4*c + 48*a^6*b^2*c^2) - (3*b^2*d)/( 2*a^4) + (b*e)/a^3 + (c*d)/a^3) - log(4*a^2*b^7*e - 192*a^5*c^4*d - 6*a*b^ 8*d - 6*b^9*d*x - 307*a^3*b^4*c^2*d + 492*a^4*b^2*c^3*d + 4*a^2*b^4*e*(-(4 *a*c - b^2)^3)^(1/2) + 174*a^4*b^3*c^2*e + 6*a^4*c^2*e*(-(4*a*c - b^2)^3)^ (1/2) + 4*a*b^8*e*x - 6*a*b^5*d*(-(4*a*c - b^2)^3)^(1/2) + 73*a^2*b^6*c...