Integrand size = 26, antiderivative size = 397 \[ \int \frac {b+2 c x}{(d+e x) \left (a+b x+c x^2\right )^3} \, dx=\frac {-\left (\left (b^2-4 a c\right ) (c d-b e)\right )+c \left (b^2-4 a c\right ) e x}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^2}-\frac {e \left (3 b^2 c d e-8 a c^2 d e-2 b^3 e^2-b c \left (c d^2-7 a e^2\right )-2 c \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) x\right )}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 \left (a+b x+c x^2\right )}-\frac {e \left (2 c^4 d^4-b^4 e^4-4 c^3 d^2 e (b d-3 a e)-6 a c^2 e^3 (2 b d+a e)+2 b^2 c e^3 (b d+3 a e)\right ) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2} \left (c d^2-b d e+a e^2\right )^3}-\frac {e^4 (2 c d-b e) \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^3}+\frac {e^4 (2 c d-b e) \log \left (a+b x+c x^2\right )}{2 \left (c d^2-b d e+a e^2\right )^3} \]
1/2*(-(-4*a*c+b^2)*(-b*e+c*d)+c*(-4*a*c+b^2)*e*x)/(-4*a*c+b^2)/(a*e^2-b*d* e+c*d^2)/(c*x^2+b*x+a)^2-1/2*e*(3*b^2*c*d*e-8*a*c^2*d*e-2*b^3*e^2-b*c*(-7* a*e^2+c*d^2)-2*c*(c^2*d^2+b^2*e^2-c*e*(3*a*e+b*d))*x)/(-4*a*c+b^2)/(a*e^2- b*d*e+c*d^2)^2/(c*x^2+b*x+a)-e*(2*c^4*d^4-b^4*e^4-4*c^3*d^2*e*(-3*a*e+b*d) -6*a*c^2*e^3*(a*e+2*b*d)+2*b^2*c*e^3*(3*a*e+b*d))*arctanh((2*c*x+b)/(-4*a* c+b^2)^(1/2))/(-4*a*c+b^2)^(3/2)/(a*e^2-b*d*e+c*d^2)^3-e^4*(-b*e+2*c*d)*ln (e*x+d)/(a*e^2-b*d*e+c*d^2)^3+1/2*e^4*(-b*e+2*c*d)*ln(c*x^2+b*x+a)/(a*e^2- b*d*e+c*d^2)^3
Time = 0.70 (sec) , antiderivative size = 356, normalized size of antiderivative = 0.90 \[ \int \frac {b+2 c x}{(d+e x) \left (a+b x+c x^2\right )^3} \, dx=\frac {1}{2} \left (\frac {-c d+b e+c e x}{\left (c d^2+e (-b d+a e)\right ) (a+x (b+c x))^2}+\frac {e \left (2 b^3 e^2+b^2 c e (-3 d+2 e x)+2 c^2 \left (c d^2 x+a e (4 d-3 e x)\right )+b c \left (-7 a e^2+c d (d-2 e x)\right )\right )}{\left (b^2-4 a c\right ) \left (c d^2+e (-b d+a e)\right )^2 (a+x (b+c x))}-\frac {2 e \left (-2 c^4 d^4+b^4 e^4+4 c^3 d^2 e (b d-3 a e)+6 a c^2 e^3 (2 b d+a e)-2 b^2 c e^3 (b d+3 a e)\right ) \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\left (-b^2+4 a c\right )^{3/2} \left (-c d^2+e (b d-a e)\right )^3}+\frac {2 e^4 (-2 c d+b e) \log (d+e x)}{\left (c d^2+e (-b d+a e)\right )^3}+\frac {e^4 (2 c d-b e) \log (a+x (b+c x))}{\left (c d^2+e (-b d+a e)\right )^3}\right ) \]
((-(c*d) + b*e + c*e*x)/((c*d^2 + e*(-(b*d) + a*e))*(a + x*(b + c*x))^2) + (e*(2*b^3*e^2 + b^2*c*e*(-3*d + 2*e*x) + 2*c^2*(c*d^2*x + a*e*(4*d - 3*e* x)) + b*c*(-7*a*e^2 + c*d*(d - 2*e*x))))/((b^2 - 4*a*c)*(c*d^2 + e*(-(b*d) + a*e))^2*(a + x*(b + c*x))) - (2*e*(-2*c^4*d^4 + b^4*e^4 + 4*c^3*d^2*e*( b*d - 3*a*e) + 6*a*c^2*e^3*(2*b*d + a*e) - 2*b^2*c*e^3*(b*d + 3*a*e))*ArcT an[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/((-b^2 + 4*a*c)^(3/2)*(-(c*d^2) + e*(b *d - a*e))^3) + (2*e^4*(-2*c*d + b*e)*Log[d + e*x])/(c*d^2 + e*(-(b*d) + a *e))^3 + (e^4*(2*c*d - b*e)*Log[a + x*(b + c*x)])/(c*d^2 + e*(-(b*d) + a*e ))^3)/2
Time = 0.97 (sec) , antiderivative size = 463, normalized size of antiderivative = 1.17, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {1235, 27, 1235, 27, 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.
| \(\displaystyle \int \frac {b+2 c x}{(d+e x) \left (a+b x+c x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 1235 |
| \(\displaystyle -\frac {\int \frac {\left (b^2-4 a c\right ) e (c d-2 b e-3 c e x)}{(d+e x) \left (c x^2+b x+a\right )^2}dx}{2 \left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {\left (b^2-4 a c\right ) (c d-b e)-c e x \left (b^2-4 a c\right )}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2 \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {e \int \frac {c d-2 b e-3 c e x}{(d+e x) \left (c x^2+b x+a\right )^2}dx}{2 \left (a e^2-b d e+c d^2\right )}-\frac {\left (b^2-4 a c\right ) (c d-b e)-c e x \left (b^2-4 a c\right )}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2 \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 1235 |
| \(\displaystyle -\frac {e \left (\frac {-2 c x \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right )-b c \left (c d^2-7 a e^2\right )-8 a c^2 d e-2 b^3 e^2+3 b^2 c d e}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )}-\frac {\int \frac {2 \left (c^3 d^3-c^2 e (b d-5 a e) d+b^3 e^3-b c e^2 (b d+4 a e)+c e \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) x\right )}{(d+e x) \left (c x^2+b x+a\right )}dx}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}\right )}{2 \left (a e^2-b d e+c d^2\right )}-\frac {\left (b^2-4 a c\right ) (c d-b e)-c e x \left (b^2-4 a c\right )}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2 \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {e \left (\frac {-2 c x \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right )-b c \left (c d^2-7 a e^2\right )-8 a c^2 d e-2 b^3 e^2+3 b^2 c d e}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )}-\frac {2 \int \frac {c^3 d^3-c^2 e (b d-5 a e) d+b^3 e^3-b c e^2 (b d+4 a e)+c e \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) x}{(d+e x) \left (c x^2+b x+a\right )}dx}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}\right )}{2 \left (a e^2-b d e+c d^2\right )}-\frac {\left (b^2-4 a c\right ) (c d-b e)-c e x \left (b^2-4 a c\right )}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2 \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 1200 |
| \(\displaystyle -\frac {e \left (\frac {-2 c x \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right )-b c \left (c d^2-7 a e^2\right )-8 a c^2 d e-2 b^3 e^2+3 b^2 c d e}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )}-\frac {2 \int \left (\frac {\left (b^2-4 a c\right ) (b e-2 c d) e^4}{\left (c d^2-b e d+a e^2\right ) (d+e x)}+\frac {c^4 d^4-2 c^3 e (b d-3 a e) d^2-b^4 e^4-a c^2 e^3 (10 b d+3 a e)+b^2 c e^3 (2 b d+5 a e)+c \left (b^2-4 a c\right ) e^3 (2 c d-b e) x}{\left (c d^2-b e d+a e^2\right ) \left (c x^2+b x+a\right )}\right )dx}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}\right )}{2 \left (a e^2-b d e+c d^2\right )}-\frac {\left (b^2-4 a c\right ) (c d-b e)-c e x \left (b^2-4 a c\right )}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2 \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {e \left (\frac {-2 c x \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right )-b c \left (c d^2-7 a e^2\right )-8 a c^2 d e-2 b^3 e^2+3 b^2 c d e}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (-\frac {\text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right ) \left (2 b^2 c e^3 (3 a e+b d)-4 c^3 d^2 e (b d-3 a e)-6 a c^2 e^3 (a e+2 b d)-b^4 e^4+2 c^4 d^4\right )}{\sqrt {b^2-4 a c} \left (a e^2-b d e+c d^2\right )}+\frac {e^3 \left (b^2-4 a c\right ) (2 c d-b e) \log \left (a+b x+c x^2\right )}{2 \left (a e^2-b d e+c d^2\right )}-\frac {e^3 \left (b^2-4 a c\right ) (2 c d-b e) \log (d+e x)}{a e^2-b d e+c d^2}\right )}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}\right )}{2 \left (a e^2-b d e+c d^2\right )}-\frac {\left (b^2-4 a c\right ) (c d-b e)-c e x \left (b^2-4 a c\right )}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2 \left (a e^2-b d e+c d^2\right )}\) |
-1/2*((b^2 - 4*a*c)*(c*d - b*e) - c*(b^2 - 4*a*c)*e*x)/((b^2 - 4*a*c)*(c*d ^2 - b*d*e + a*e^2)*(a + b*x + c*x^2)^2) - (e*((3*b^2*c*d*e - 8*a*c^2*d*e - 2*b^3*e^2 - b*c*(c*d^2 - 7*a*e^2) - 2*c*(c^2*d^2 + b^2*e^2 - c*e*(b*d + 3*a*e))*x)/((b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*(a + b*x + c*x^2)) - (2* (-(((2*c^4*d^4 - b^4*e^4 - 4*c^3*d^2*e*(b*d - 3*a*e) - 6*a*c^2*e^3*(2*b*d + a*e) + 2*b^2*c*e^3*(b*d + 3*a*e))*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]] )/(Sqrt[b^2 - 4*a*c]*(c*d^2 - b*d*e + a*e^2))) - ((b^2 - 4*a*c)*e^3*(2*c*d - b*e)*Log[d + e*x])/(c*d^2 - b*d*e + a*e^2) + ((b^2 - 4*a*c)*e^3*(2*c*d - b*e)*Log[a + b*x + c*x^2])/(2*(c*d^2 - b*d*e + a*e^2))))/((b^2 - 4*a*c)* (c*d^2 - b*d*e + a*e^2))))/(2*(c*d^2 - b*d*e + a*e^2))
3.16.45.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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] )
Leaf count of result is larger than twice the leaf count of optimal. \(874\) vs. \(2(387)=774\).
Time = 0.74 (sec) , antiderivative size = 875, normalized size of antiderivative = 2.20
| method | result | size |
| default | \(\frac {\left (b e -2 c d \right ) e^{4} \ln \left (e x +d \right )}{\left (e^{2} a -b d e +c \,d^{2}\right )^{3}}+\frac {\frac {\frac {c^{2} e \left (3 c \,e^{4} a^{2}-a \,b^{2} e^{4}-2 a b c d \,e^{3}+2 a \,c^{2} d^{2} e^{2}+b^{3} d \,e^{3}-2 b^{2} c \,d^{2} e^{2}+2 b \,c^{2} d^{3} e -c^{3} d^{4}\right ) x^{3}}{4 a c -b^{2}}+\frac {c e \left (13 c \,e^{4} b \,a^{2}-8 a^{2} c^{2} d \,e^{3}-4 a \,b^{3} e^{4}-8 a \,b^{2} c d \,e^{3}+18 a b \,c^{2} d^{2} e^{2}-8 a \,c^{3} d^{3} e +4 b^{4} d \,e^{3}-9 b^{3} c \,d^{2} e^{2}+8 b^{2} c^{2} d^{3} e -3 b \,d^{4} c^{3}\right ) x^{2}}{8 a c -2 b^{2}}+\frac {e \left (5 a^{3} c^{2} e^{4}+2 a^{2} b^{2} c \,e^{4}-10 a^{2} b \,c^{2} d \,e^{3}+6 a^{2} c^{3} d^{2} e^{2}-a \,b^{4} e^{4}+6 a \,b^{2} c^{2} d^{2} e^{2}-6 a b \,c^{3} d^{3} e +a \,c^{4} d^{4}+b^{5} d \,e^{3}-3 b^{4} c \,d^{2} e^{2}+3 b^{3} c^{2} d^{3} e -b^{2} c^{3} d^{4}\right ) x}{4 a c -b^{2}}+\frac {11 a^{3} b c \,e^{5}-12 a^{3} c^{2} d \,e^{4}-3 a^{2} b^{3} e^{5}-11 a^{2} b^{2} c d \,e^{4}+30 a^{2} b \,c^{2} d^{2} e^{3}-16 a^{2} c^{3} d^{3} e^{2}+4 a \,b^{4} d \,e^{4}-5 a \,b^{3} c \,d^{2} e^{3}-6 a \,b^{2} c^{2} d^{3} e^{2}+11 a b \,c^{3} d^{4} e -4 a \,c^{4} d^{5}-b^{5} d^{2} e^{3}+3 b^{4} c \,d^{3} e^{2}-3 b^{3} c^{2} d^{4} e +b^{2} c^{3} d^{5}}{8 a c -2 b^{2}}}{\left (c \,x^{2}+b x +a \right )^{2}}+\frac {e \left (\frac {\left (-4 a b \,c^{2} e^{4}+8 a \,c^{3} d \,e^{3}+b^{3} c \,e^{4}-2 b^{2} c^{2} d \,e^{3}\right ) \ln \left (c \,x^{2}+b x +a \right )}{2 c}+\frac {2 \left (3 a^{2} c^{2} e^{4}-5 a \,b^{2} c \,e^{4}+10 a b \,c^{2} d \,e^{3}-6 a \,c^{3} d^{2} e^{2}+b^{4} e^{4}-2 b^{3} c d \,e^{3}+2 b \,c^{3} d^{3} e -c^{4} d^{4}-\frac {\left (-4 a b \,c^{2} e^{4}+8 a \,c^{3} d \,e^{3}+b^{3} c \,e^{4}-2 b^{2} c^{2} d \,e^{3}\right ) b}{2 c}\right ) \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}\right )}{4 a c -b^{2}}}{\left (e^{2} a -b d e +c \,d^{2}\right )^{3}}\) | \(875\) |
| risch | \(\text {Expression too large to display}\) | \(3962\) |
(b*e-2*c*d)*e^4/(a*e^2-b*d*e+c*d^2)^3*ln(e*x+d)+1/(a*e^2-b*d*e+c*d^2)^3*(( c^2*e*(3*a^2*c*e^4-a*b^2*e^4-2*a*b*c*d*e^3+2*a*c^2*d^2*e^2+b^3*d*e^3-2*b^2 *c*d^2*e^2+2*b*c^2*d^3*e-c^3*d^4)/(4*a*c-b^2)*x^3+1/2*c*e*(13*a^2*b*c*e^4- 8*a^2*c^2*d*e^3-4*a*b^3*e^4-8*a*b^2*c*d*e^3+18*a*b*c^2*d^2*e^2-8*a*c^3*d^3 *e+4*b^4*d*e^3-9*b^3*c*d^2*e^2+8*b^2*c^2*d^3*e-3*b*c^3*d^4)/(4*a*c-b^2)*x^ 2+e*(5*a^3*c^2*e^4+2*a^2*b^2*c*e^4-10*a^2*b*c^2*d*e^3+6*a^2*c^3*d^2*e^2-a* b^4*e^4+6*a*b^2*c^2*d^2*e^2-6*a*b*c^3*d^3*e+a*c^4*d^4+b^5*d*e^3-3*b^4*c*d^ 2*e^2+3*b^3*c^2*d^3*e-b^2*c^3*d^4)/(4*a*c-b^2)*x+1/2*(11*a^3*b*c*e^5-12*a^ 3*c^2*d*e^4-3*a^2*b^3*e^5-11*a^2*b^2*c*d*e^4+30*a^2*b*c^2*d^2*e^3-16*a^2*c ^3*d^3*e^2+4*a*b^4*d*e^4-5*a*b^3*c*d^2*e^3-6*a*b^2*c^2*d^3*e^2+11*a*b*c^3* d^4*e-4*a*c^4*d^5-b^5*d^2*e^3+3*b^4*c*d^3*e^2-3*b^3*c^2*d^4*e+b^2*c^3*d^5) /(4*a*c-b^2))/(c*x^2+b*x+a)^2+e/(4*a*c-b^2)*(1/2*(-4*a*b*c^2*e^4+8*a*c^3*d *e^3+b^3*c*e^4-2*b^2*c^2*d*e^3)/c*ln(c*x^2+b*x+a)+2*(3*a^2*c^2*e^4-5*a*b^2 *c*e^4+10*a*b*c^2*d*e^3-6*a*c^3*d^2*e^2+b^4*e^4-2*b^3*c*d*e^3+2*b*c^3*d^3* e-c^4*d^4-1/2*(-4*a*b*c^2*e^4+8*a*c^3*d*e^3+b^3*c*e^4-2*b^2*c^2*d*e^3)*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. 3323 vs. \(2 (387) = 774\).
Time = 56.93 (sec) , antiderivative size = 6667, normalized size of antiderivative = 16.79 \[ \int \frac {b+2 c x}{(d+e x) \left (a+b x+c x^2\right )^3} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {b+2 c x}{(d+e x) \left (a+b x+c x^2\right )^3} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {b+2 c x}{(d+e x) \left (a+b x+c x^2\right )^3} \, dx=\text {Exception raised: ValueError} \]
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
Leaf count of result is larger than twice the leaf count of optimal. 1183 vs. \(2 (387) = 774\).
Time = 0.28 (sec) , antiderivative size = 1183, normalized size of antiderivative = 2.98 \[ \int \frac {b+2 c x}{(d+e x) \left (a+b x+c x^2\right )^3} \, dx=\frac {{\left (2 \, c d e^{4} - b e^{5}\right )} \log \left (c x^{2} + b x + a\right )}{2 \, {\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} + 3 \, a c^{2} d^{4} e^{2} - b^{3} d^{3} e^{3} - 6 \, a b c d^{3} e^{3} + 3 \, a b^{2} d^{2} e^{4} + 3 \, a^{2} c d^{2} e^{4} - 3 \, a^{2} b d e^{5} + a^{3} e^{6}\right )}} - \frac {{\left (2 \, c d e^{5} - b e^{6}\right )} \log \left ({\left | e x + d \right |}\right )}{c^{3} d^{6} e - 3 \, b c^{2} d^{5} e^{2} + 3 \, b^{2} c d^{4} e^{3} + 3 \, a c^{2} d^{4} e^{3} - b^{3} d^{3} e^{4} - 6 \, a b c d^{3} e^{4} + 3 \, a b^{2} d^{2} e^{5} + 3 \, a^{2} c d^{2} e^{5} - 3 \, a^{2} b d e^{6} + a^{3} e^{7}} + \frac {{\left (2 \, c^{4} d^{4} e - 4 \, b c^{3} d^{3} e^{2} + 12 \, a c^{3} d^{2} e^{3} + 2 \, b^{3} c d e^{4} - 12 \, a b c^{2} d e^{4} - b^{4} e^{5} + 6 \, a b^{2} c e^{5} - 6 \, a^{2} c^{2} e^{5}\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (b^{2} c^{3} d^{6} - 4 \, a c^{4} d^{6} - 3 \, b^{3} c^{2} d^{5} e + 12 \, a b c^{3} d^{5} e + 3 \, b^{4} c d^{4} e^{2} - 9 \, a b^{2} c^{2} d^{4} e^{2} - 12 \, a^{2} c^{3} d^{4} e^{2} - b^{5} d^{3} e^{3} - 2 \, a b^{3} c d^{3} e^{3} + 24 \, a^{2} b c^{2} d^{3} e^{3} + 3 \, a b^{4} d^{2} e^{4} - 9 \, a^{2} b^{2} c d^{2} e^{4} - 12 \, a^{3} c^{2} d^{2} e^{4} - 3 \, a^{2} b^{3} d e^{5} + 12 \, a^{3} b c d e^{5} + a^{3} b^{2} e^{6} - 4 \, a^{4} c e^{6}\right )} \sqrt {-b^{2} + 4 \, a c}} - \frac {b^{2} c^{3} d^{5} - 4 \, a c^{4} d^{5} - 3 \, b^{3} c^{2} d^{4} e + 11 \, a b c^{3} d^{4} e + 3 \, b^{4} c d^{3} e^{2} - 6 \, a b^{2} c^{2} d^{3} e^{2} - 16 \, a^{2} c^{3} d^{3} e^{2} - b^{5} d^{2} e^{3} - 5 \, a b^{3} c d^{2} e^{3} + 30 \, a^{2} b c^{2} d^{2} e^{3} + 4 \, a b^{4} d e^{4} - 11 \, a^{2} b^{2} c d e^{4} - 12 \, a^{3} c^{2} d e^{4} - 3 \, a^{2} b^{3} e^{5} + 11 \, a^{3} b c e^{5} - 2 \, {\left (c^{5} d^{4} e - 2 \, b c^{4} d^{3} e^{2} + 2 \, b^{2} c^{3} d^{2} e^{3} - 2 \, a c^{4} d^{2} e^{3} - b^{3} c^{2} d e^{4} + 2 \, a b c^{3} d e^{4} + a b^{2} c^{2} e^{5} - 3 \, a^{2} c^{3} e^{5}\right )} x^{3} - {\left (3 \, b c^{4} d^{4} e - 8 \, b^{2} c^{3} d^{3} e^{2} + 8 \, a c^{4} d^{3} e^{2} + 9 \, b^{3} c^{2} d^{2} e^{3} - 18 \, a b c^{3} d^{2} e^{3} - 4 \, b^{4} c d e^{4} + 8 \, a b^{2} c^{2} d e^{4} + 8 \, a^{2} c^{3} d e^{4} + 4 \, a b^{3} c e^{5} - 13 \, a^{2} b c^{2} e^{5}\right )} x^{2} - 2 \, {\left (b^{2} c^{3} d^{4} e - a c^{4} d^{4} e - 3 \, b^{3} c^{2} d^{3} e^{2} + 6 \, a b c^{3} d^{3} e^{2} + 3 \, b^{4} c d^{2} e^{3} - 6 \, a b^{2} c^{2} d^{2} e^{3} - 6 \, a^{2} c^{3} d^{2} e^{3} - b^{5} d e^{4} + 10 \, a^{2} b c^{2} d e^{4} + a b^{4} e^{5} - 2 \, a^{2} b^{2} c e^{5} - 5 \, a^{3} c^{2} e^{5}\right )} x}{2 \, {\left (c d^{2} - b d e + a e^{2}\right )}^{3} {\left (c x^{2} + b x + a\right )}^{2} {\left (b^{2} - 4 \, a c\right )}} \]
1/2*(2*c*d*e^4 - b*e^5)*log(c*x^2 + b*x + a)/(c^3*d^6 - 3*b*c^2*d^5*e + 3* b^2*c*d^4*e^2 + 3*a*c^2*d^4*e^2 - b^3*d^3*e^3 - 6*a*b*c*d^3*e^3 + 3*a*b^2* d^2*e^4 + 3*a^2*c*d^2*e^4 - 3*a^2*b*d*e^5 + a^3*e^6) - (2*c*d*e^5 - b*e^6) *log(abs(e*x + d))/(c^3*d^6*e - 3*b*c^2*d^5*e^2 + 3*b^2*c*d^4*e^3 + 3*a*c^ 2*d^4*e^3 - b^3*d^3*e^4 - 6*a*b*c*d^3*e^4 + 3*a*b^2*d^2*e^5 + 3*a^2*c*d^2* e^5 - 3*a^2*b*d*e^6 + a^3*e^7) + (2*c^4*d^4*e - 4*b*c^3*d^3*e^2 + 12*a*c^3 *d^2*e^3 + 2*b^3*c*d*e^4 - 12*a*b*c^2*d*e^4 - b^4*e^5 + 6*a*b^2*c*e^5 - 6* a^2*c^2*e^5)*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/((b^2*c^3*d^6 - 4*a*c^ 4*d^6 - 3*b^3*c^2*d^5*e + 12*a*b*c^3*d^5*e + 3*b^4*c*d^4*e^2 - 9*a*b^2*c^2 *d^4*e^2 - 12*a^2*c^3*d^4*e^2 - b^5*d^3*e^3 - 2*a*b^3*c*d^3*e^3 + 24*a^2*b *c^2*d^3*e^3 + 3*a*b^4*d^2*e^4 - 9*a^2*b^2*c*d^2*e^4 - 12*a^3*c^2*d^2*e^4 - 3*a^2*b^3*d*e^5 + 12*a^3*b*c*d*e^5 + a^3*b^2*e^6 - 4*a^4*c*e^6)*sqrt(-b^ 2 + 4*a*c)) - 1/2*(b^2*c^3*d^5 - 4*a*c^4*d^5 - 3*b^3*c^2*d^4*e + 11*a*b*c^ 3*d^4*e + 3*b^4*c*d^3*e^2 - 6*a*b^2*c^2*d^3*e^2 - 16*a^2*c^3*d^3*e^2 - b^5 *d^2*e^3 - 5*a*b^3*c*d^2*e^3 + 30*a^2*b*c^2*d^2*e^3 + 4*a*b^4*d*e^4 - 11*a ^2*b^2*c*d*e^4 - 12*a^3*c^2*d*e^4 - 3*a^2*b^3*e^5 + 11*a^3*b*c*e^5 - 2*(c^ 5*d^4*e - 2*b*c^4*d^3*e^2 + 2*b^2*c^3*d^2*e^3 - 2*a*c^4*d^2*e^3 - b^3*c^2* d*e^4 + 2*a*b*c^3*d*e^4 + a*b^2*c^2*e^5 - 3*a^2*c^3*e^5)*x^3 - (3*b*c^4*d^ 4*e - 8*b^2*c^3*d^3*e^2 + 8*a*c^4*d^3*e^2 + 9*b^3*c^2*d^2*e^3 - 18*a*b*c^3 *d^2*e^3 - 4*b^4*c*d*e^4 + 8*a*b^2*c^2*d*e^4 + 8*a^2*c^3*d*e^4 + 4*a*b^...
Time = 14.76 (sec) , antiderivative size = 2461, normalized size of antiderivative = 6.20 \[ \int \frac {b+2 c x}{(d+e x) \left (a+b x+c x^2\right )^3} \, dx=\text {Too large to display} \]
(log((-(4*a*c - b^2)^3)^(1/2) - b^3 + 4*a*b*c + 8*a*c^2*x - 2*b^2*c*x)*(b^ 7*e^5 + b^4*e^5*(-(4*a*c - b^2)^3)^(1/2) - 64*a^3*b*c^3*e^5 + 128*a^3*c^4* d*e^4 - 2*c^4*d^4*e*(-(4*a*c - b^2)^3)^(1/2) + 48*a^2*b^3*c^2*e^5 + 6*a^2* c^2*e^5*(-(4*a*c - b^2)^3)^(1/2) - 12*a*b^5*c*e^5 - 2*b^6*c*d*e^4 - 6*a*b^ 2*c*e^5*(-(4*a*c - b^2)^3)^(1/2) + 24*a*b^4*c^2*d*e^4 - 2*b^3*c*d*e^4*(-(4 *a*c - b^2)^3)^(1/2) - 96*a^2*b^2*c^3*d*e^4 - 12*a*c^3*d^2*e^3*(-(4*a*c - b^2)^3)^(1/2) + 4*b*c^3*d^3*e^2*(-(4*a*c - b^2)^3)^(1/2) + 12*a*b*c^2*d*e^ 4*(-(4*a*c - b^2)^3)^(1/2)))/(2*(64*a^3*c^6*d^6 - a^3*b^6*e^6 + 64*a^6*c^3 *e^6 - b^6*c^3*d^6 + b^9*d^3*e^3 + 12*a*b^4*c^4*d^6 + 12*a^4*b^4*c*e^6 - 3 *a*b^8*d^2*e^4 + 3*a^2*b^7*d*e^5 + 3*b^7*c^2*d^5*e - 3*b^8*c*d^4*e^2 - 48* a^2*b^2*c^5*d^6 - 48*a^5*b^2*c^2*e^6 + 192*a^4*c^5*d^4*e^2 + 192*a^5*c^4*d ^2*e^4 - 108*a^2*b^4*c^3*d^4*e^2 - 24*a^2*b^5*c^2*d^3*e^3 + 48*a^3*b^2*c^4 *d^4*e^2 + 224*a^3*b^3*c^3*d^3*e^3 - 108*a^3*b^4*c^2*d^2*e^4 + 48*a^4*b^2* c^3*d^2*e^4 - 36*a*b^5*c^3*d^5*e - 6*a*b^7*c*d^3*e^3 - 192*a^3*b*c^5*d^5*e - 36*a^3*b^5*c*d*e^5 - 192*a^5*b*c^3*d*e^5 + 33*a*b^6*c^2*d^4*e^2 + 144*a ^2*b^3*c^4*d^5*e + 33*a^2*b^6*c*d^2*e^4 - 384*a^4*b*c^4*d^3*e^3 + 144*a^4* b^3*c^2*d*e^5)) - ((3*a*b^3*e^3 + 4*a*c^3*d^3 - b^4*d*e^2 - b^2*c^2*d^3 + 12*a^2*c^2*d*e^2 - 11*a^2*b*c*e^3 + 2*b^3*c*d^2*e - 7*a*b*c^2*d^2*e)/(2*(4 *a*c^3*d^4 + 4*a^3*c*e^4 - a^2*b^2*e^4 - b^2*c^2*d^4 - b^4*d^2*e^2 + 8*a^2 *c^2*d^2*e^2 + 2*a*b^3*d*e^3 + 2*b^3*c*d^3*e - 8*a*b*c^2*d^3*e - 8*a^2*...