Integrand size = 26, antiderivative size = 293 \[ \int \frac {(b+2 c x) (d+e x)^5}{\left (a+b x+c x^2\right )^3} \, dx=\frac {5 e^3 \left (3 c^2 d^2+b^2 e^2-c e (2 b d+3 a e)\right ) x}{c^2 \left (b^2-4 a c\right )}+\frac {5 e^4 (2 c d-b e) x^2}{2 c \left (b^2-4 a c\right )}-\frac {(d+e x)^5}{2 \left (a+b x+c x^2\right )^2}-\frac {5 e (d+e x)^3 (b d-2 a e+(2 c d-b e) x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {5 e \left (2 c^4 d^4-b^4 e^4-4 c^3 d^2 e (b d-3 a e)-6 a c^2 e^3 (2 b d+a e)+2 b^2 c e^3 (b d+3 a e)\right ) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^3 \left (b^2-4 a c\right )^{3/2}}+\frac {5 e^4 (2 c d-b e) \log \left (a+b x+c x^2\right )}{2 c^3} \]
5*e^3*(3*c^2*d^2+b^2*e^2-c*e*(3*a*e+2*b*d))*x/c^2/(-4*a*c+b^2)+5/2*e^4*(-b *e+2*c*d)*x^2/c/(-4*a*c+b^2)-1/2*(e*x+d)^5/(c*x^2+b*x+a)^2-5/2*e*(e*x+d)^3 *(b*d-2*a*e+(-b*e+2*c*d)*x)/(-4*a*c+b^2)/(c*x^2+b*x+a)+5*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))/c^3/(-4*a*c+b^2)^(3/2)+5/2*e^4*( -b*e+2*c*d)*ln(c*x^2+b*x+a)/c^3
Time = 0.55 (sec) , antiderivative size = 480, normalized size of antiderivative = 1.64 \[ \int \frac {(b+2 c x) (d+e x)^5}{\left (a+b x+c x^2\right )^3} \, dx=\frac {4 c^2 e^5 x-\frac {b^3 e^5 (a+b x)+c^4 d^4 (d+5 e x)-10 c^3 d^2 e^2 (b d x+a (d+e x))+c^2 e^3 \left (10 b^2 d^2 x+10 a b d (d+e x)+a^2 e (5 d+e x)\right )-b c e^4 \left (2 a^2 e+5 b^2 d x+a b (5 d+3 e x)\right )}{(a+x (b+c x))^2}+\frac {e \left (b^5 e^4-b^4 c e^3 (5 d+8 e x)+b^3 c e^2 \left (-13 a e^2+10 c d (d+3 e x)\right )-2 c^3 \left (5 c^2 d^4 x-10 a c d^2 e (4 d+5 e x)+a^2 e^3 (40 d+9 e x)\right )+b c^2 \left (31 a^2 e^4-5 c^2 d^3 (d-4 e x)-10 a c d e^2 (7 d+10 e x)\right )+2 b^2 c^2 e \left (-5 c d^2 (d+4 e x)+a e^2 (25 d+17 e x)\right )\right )}{\left (b^2-4 a c\right ) (a+x (b+c x))}-\frac {10 c e \left (-2 c^4 d^4+b^4 e^4+4 c^3 d^2 e (b d-3 a e)+6 a c^2 e^3 (2 b d+a e)-2 b^2 c e^3 (b d+3 a e)\right ) \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\left (-b^2+4 a c\right )^{3/2}}+5 c e^4 (2 c d-b e) \log (a+x (b+c x))}{2 c^4} \]
(4*c^2*e^5*x - (b^3*e^5*(a + b*x) + c^4*d^4*(d + 5*e*x) - 10*c^3*d^2*e^2*( b*d*x + a*(d + e*x)) + c^2*e^3*(10*b^2*d^2*x + 10*a*b*d*(d + e*x) + a^2*e* (5*d + e*x)) - b*c*e^4*(2*a^2*e + 5*b^2*d*x + a*b*(5*d + 3*e*x)))/(a + x*( b + c*x))^2 + (e*(b^5*e^4 - b^4*c*e^3*(5*d + 8*e*x) + b^3*c*e^2*(-13*a*e^2 + 10*c*d*(d + 3*e*x)) - 2*c^3*(5*c^2*d^4*x - 10*a*c*d^2*e*(4*d + 5*e*x) + a^2*e^3*(40*d + 9*e*x)) + b*c^2*(31*a^2*e^4 - 5*c^2*d^3*(d - 4*e*x) - 10* a*c*d*e^2*(7*d + 10*e*x)) + 2*b^2*c^2*e*(-5*c*d^2*(d + 4*e*x) + a*e^2*(25* d + 17*e*x))))/((b^2 - 4*a*c)*(a + x*(b + c*x))) - (10*c*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))*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^( 3/2) + 5*c*e^4*(2*c*d - b*e)*Log[a + x*(b + c*x)])/(2*c^4)
Time = 0.69 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.01, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {1222, 1164, 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.
| \(\displaystyle \int \frac {(b+2 c x) (d+e x)^5}{\left (a+b x+c x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 1222 |
| \(\displaystyle \frac {5}{2} e \int \frac {(d+e x)^4}{\left (c x^2+b x+a\right )^2}dx-\frac {(d+e x)^5}{2 \left (a+b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 1164 |
| \(\displaystyle \frac {5}{2} e \left (-\frac {\int \frac {2 (d+e x)^2 \left (c d^2-2 b e d+3 a e^2-e (2 c d-b e) x\right )}{c x^2+b x+a}dx}{b^2-4 a c}-\frac {(d+e x)^3 (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\right )-\frac {(d+e x)^5}{2 \left (a+b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{2} e \left (-\frac {2 \int \frac {(d+e x)^2 \left (c d^2-e (2 b d-3 a e)-e (2 c d-b e) x\right )}{c x^2+b x+a}dx}{b^2-4 a c}-\frac {(d+e x)^3 (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\right )-\frac {(d+e x)^5}{2 \left (a+b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 1200 |
| \(\displaystyle \frac {5}{2} e \left (-\frac {2 \int \left (-\frac {(2 c d-b e) x e^3}{c}-\frac {\left (3 c^2 d^2+b^2 e^2-c e (2 b d+3 a e)\right ) e^2}{c^2}+\frac {c^3 d^4-2 c^2 e (b d-3 a e) d^2+a b^2 e^4-a c e^3 (2 b d+3 a e)-\left (b^2-4 a c\right ) e^3 (2 c d-b e) x}{c^2 \left (c x^2+b x+a\right )}\right )dx}{b^2-4 a c}-\frac {(d+e x)^3 (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\right )-\frac {(d+e x)^5}{2 \left (a+b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {5}{2} e \left (-\frac {2 \left (-\frac {\text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right ) \left (2 b^2 c e^3 (3 a e+b d)-4 c^3 d^2 e (b d-3 a e)-6 a c^2 e^3 (a e+2 b d)-b^4 e^4+2 c^4 d^4\right )}{c^3 \sqrt {b^2-4 a c}}-\frac {e^3 \left (b^2-4 a c\right ) (2 c d-b e) \log \left (a+b x+c x^2\right )}{2 c^3}-\frac {e^2 x \left (-c e (3 a e+2 b d)+b^2 e^2+3 c^2 d^2\right )}{c^2}-\frac {e^3 x^2 (2 c d-b e)}{2 c}\right )}{b^2-4 a c}-\frac {(d+e x)^3 (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\right )-\frac {(d+e x)^5}{2 \left (a+b x+c x^2\right )^2}\) |
-1/2*(d + e*x)^5/(a + b*x + c*x^2)^2 + (5*e*(-(((d + e*x)^3*(b*d - 2*a*e + (2*c*d - b*e)*x))/((b^2 - 4*a*c)*(a + b*x + c*x^2))) - (2*(-((e^2*(3*c^2* d^2 + b^2*e^2 - c*e*(2*b*d + 3*a*e))*x)/c^2) - (e^3*(2*c*d - b*e)*x^2)/(2* c) - ((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 ]])/(c^3*Sqrt[b^2 - 4*a*c]) - ((b^2 - 4*a*c)*e^3*(2*c*d - b*e)*Log[a + b*x + c*x^2])/(2*c^3)))/(b^2 - 4*a*c)))/2
3.16.39.3.1 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_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[(d + e*x)^(m - 1)*(d*b - 2*a*e + (2*c*d - b*e)*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] + Simp[1/((p + 1)*(b^2 - 4*a* c)) Int[(d + e*x)^(m - 2)*Simp[e*(2*a*e*(m - 1) + b*d*(2*p - m + 4)) - 2* c*d^2*(2*p + 3) + e*(b*e - 2*d*c)*(m + 2*p + 2)*x, x]*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && LtQ[p, -1] && GtQ[m, 1] && Int QuadraticQ[a, b, c, d, e, m, p, 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[g*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/(2*c*(p + 1))), x] - Simp[e*g*(m/(2*c*(p + 1))) Int[(d + e*x)^(m - 1)* (a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ [2*c*f - b*g, 0] && LtQ[p, -1] && GtQ[m, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(752\) vs. \(2(281)=562\).
Time = 0.57 (sec) , antiderivative size = 753, normalized size of antiderivative = 2.57
| method | result | size |
| default | \(\frac {2 e^{5} x}{c^{2}}-\frac {\frac {-\frac {e \left (9 a^{2} c^{2} e^{4}-17 a \,b^{2} c \,e^{4}+50 a b \,c^{2} d \,e^{3}-50 a \,c^{3} d^{2} e^{2}+4 b^{4} e^{4}-15 b^{3} c d \,e^{3}+20 b^{2} c^{2} d^{2} e^{2}-10 b \,c^{3} d^{3} e +5 c^{4} d^{4}\right ) x^{3}}{4 a c -b^{2}}+\frac {e \left (13 a^{2} b \,c^{2} e^{4}-80 a^{2} c^{3} d \,e^{3}+21 a \,b^{3} c \,e^{4}-50 a \,b^{2} c^{2} d \,e^{3}+30 a b \,c^{3} d^{2} e^{2}+80 a \,c^{4} d^{3} e -7 b^{5} e^{4}+25 b^{4} c d \,e^{3}-30 b^{3} c^{2} d^{2} e^{2}+10 b^{2} c^{3} d^{3} e -15 b \,c^{4} d^{4}\right ) x^{2}}{2 \left (4 a c -b^{2}\right ) c}-\frac {e \left (7 a^{3} c^{2} e^{4}-26 a^{2} b^{2} c \,e^{4}+70 a^{2} b \,c^{2} d \,e^{3}-30 a^{2} c^{3} d^{2} e^{2}+7 a \,b^{4} e^{4}-25 a \,b^{3} c d \,e^{3}+30 a \,b^{2} c^{2} d^{2} e^{2}-30 a b \,c^{3} d^{3} e -5 a \,c^{4} d^{4}+5 b^{2} c^{3} d^{4}\right ) x}{\left (4 a c -b^{2}\right ) c}+\frac {23 a^{3} b c \,e^{5}-60 a^{3} c^{2} d \,e^{4}-7 a^{2} b^{3} e^{5}+25 a^{2} b^{2} c d \,e^{4}-30 a^{2} b \,c^{2} d^{2} e^{3}+40 a^{2} c^{3} d^{3} e^{2}-5 a b \,c^{3} d^{4} e +4 a \,c^{4} d^{5}-b^{2} c^{3} d^{5}}{2 \left (4 a c -b^{2}\right ) c}}{\left (c \,x^{2}+b x +a \right )^{2}}+\frac {5 e \left (\frac {\left (4 c \,e^{4} b a -8 a \,c^{2} d \,e^{3}-b^{3} e^{4}+2 b^{2} c d \,e^{3}\right ) \ln \left (c \,x^{2}+b x +a \right )}{2 c}+\frac {2 \left (3 c \,e^{4} a^{2}-a \,b^{2} e^{4}+2 a b c d \,e^{3}-6 a \,c^{2} d^{2} e^{2}+2 b \,c^{2} d^{3} e -c^{3} d^{4}-\frac {\left (4 c \,e^{4} b a -8 a \,c^{2} d \,e^{3}-b^{3} e^{4}+2 b^{2} c d \,e^{3}\right ) b}{2 c}\right ) \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}\right )}{4 a c -b^{2}}}{c^{2}}\) | \(753\) |
| risch | \(\text {Expression too large to display}\) | \(5961\) |
2*e^5/c^2*x-1/c^2*((-e*(9*a^2*c^2*e^4-17*a*b^2*c*e^4+50*a*b*c^2*d*e^3-50*a *c^3*d^2*e^2+4*b^4*e^4-15*b^3*c*d*e^3+20*b^2*c^2*d^2*e^2-10*b*c^3*d^3*e+5* c^4*d^4)/(4*a*c-b^2)*x^3+1/2*e*(13*a^2*b*c^2*e^4-80*a^2*c^3*d*e^3+21*a*b^3 *c*e^4-50*a*b^2*c^2*d*e^3+30*a*b*c^3*d^2*e^2+80*a*c^4*d^3*e-7*b^5*e^4+25*b ^4*c*d*e^3-30*b^3*c^2*d^2*e^2+10*b^2*c^3*d^3*e-15*b*c^4*d^4)/(4*a*c-b^2)/c *x^2-e*(7*a^3*c^2*e^4-26*a^2*b^2*c*e^4+70*a^2*b*c^2*d*e^3-30*a^2*c^3*d^2*e ^2+7*a*b^4*e^4-25*a*b^3*c*d*e^3+30*a*b^2*c^2*d^2*e^2-30*a*b*c^3*d^3*e-5*a* c^4*d^4+5*b^2*c^3*d^4)/(4*a*c-b^2)/c*x+1/2*(23*a^3*b*c*e^5-60*a^3*c^2*d*e^ 4-7*a^2*b^3*e^5+25*a^2*b^2*c*d*e^4-30*a^2*b*c^2*d^2*e^3+40*a^2*c^3*d^3*e^2 -5*a*b*c^3*d^4*e+4*a*c^4*d^5-b^2*c^3*d^5)/(4*a*c-b^2)/c)/(c*x^2+b*x+a)^2+5 *e/(4*a*c-b^2)*(1/2*(4*a*b*c*e^4-8*a*c^2*d*e^3-b^3*e^4+2*b^2*c*d*e^3)/c*ln (c*x^2+b*x+a)+2*(3*c*e^4*a^2-a*b^2*e^4+2*a*b*c*d*e^3-6*a*c^2*d^2*e^2+2*b*c ^2*d^3*e-c^3*d^4-1/2*(4*a*b*c*e^4-8*a*c^2*d*e^3-b^3*e^4+2*b^2*c*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. 1799 vs. \(2 (281) = 562\).
Time = 0.40 (sec) , antiderivative size = 3618, normalized size of antiderivative = 12.35 \[ \int \frac {(b+2 c x) (d+e x)^5}{\left (a+b x+c x^2\right )^3} \, dx=\text {Too large to display} \]
[1/2*(4*(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*e^5*x^5 + 8*(b^5*c^2 - 8*a*b^ 3*c^3 + 16*a^2*b*c^4)*e^5*x^4 - (b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*d^5 - 5*(a*b^3*c^3 - 4*a^2*b*c^4)*d^4*e + 40*(a^2*b^2*c^3 - 4*a^3*c^4)*d^3*e^2 - 30*(a^2*b^3*c^2 - 4*a^3*b*c^3)*d^2*e^3 + 5*(5*a^2*b^4*c - 32*a^3*b^2*c^2 + 48*a^4*c^3)*d*e^4 - (7*a^2*b^5 - 51*a^3*b^3*c + 92*a^4*b*c^2)*e^5 - 2*( 5*(b^2*c^5 - 4*a*c^6)*d^4*e - 10*(b^3*c^4 - 4*a*b*c^5)*d^3*e^2 + 10*(2*b^4 *c^3 - 13*a*b^2*c^4 + 20*a^2*c^5)*d^2*e^3 - 5*(3*b^5*c^2 - 22*a*b^3*c^3 + 40*a^2*b*c^4)*d*e^4 + (2*b^6*c - 21*a*b^4*c^2 + 77*a^2*b^2*c^3 - 100*a^3*c ^4)*e^5)*x^3 - (15*(b^3*c^4 - 4*a*b*c^5)*d^4*e - 10*(b^4*c^3 + 4*a*b^2*c^4 - 32*a^2*c^5)*d^3*e^2 + 30*(b^5*c^2 - 5*a*b^3*c^3 + 4*a^2*b*c^4)*d^2*e^3 - 5*(5*b^6*c - 30*a*b^4*c^2 + 24*a^2*b^2*c^3 + 64*a^3*c^4)*d*e^4 + (7*b^7 - 57*a*b^5*c + 135*a^2*b^3*c^2 - 76*a^3*b*c^3)*e^5)*x^2 + 5*(2*a^2*c^4*d^4 *e - 4*a^2*b*c^3*d^3*e^2 + 12*a^3*c^3*d^2*e^3 + 2*(a^2*b^3*c - 6*a^3*b*c^2 )*d*e^4 - (a^2*b^4 - 6*a^3*b^2*c + 6*a^4*c^2)*e^5 + (2*c^6*d^4*e - 4*b*c^5 *d^3*e^2 + 12*a*c^5*d^2*e^3 + 2*(b^3*c^3 - 6*a*b*c^4)*d*e^4 - (b^4*c^2 - 6 *a*b^2*c^3 + 6*a^2*c^4)*e^5)*x^4 + 2*(2*b*c^5*d^4*e - 4*b^2*c^4*d^3*e^2 + 12*a*b*c^4*d^2*e^3 + 2*(b^4*c^2 - 6*a*b^2*c^3)*d*e^4 - (b^5*c - 6*a*b^3*c^ 2 + 6*a^2*b*c^3)*e^5)*x^3 + (2*(b^2*c^4 + 2*a*c^5)*d^4*e - 4*(b^3*c^3 + 2* a*b*c^4)*d^3*e^2 + 12*(a*b^2*c^3 + 2*a^2*c^4)*d^2*e^3 + 2*(b^5*c - 4*a*b^3 *c^2 - 12*a^2*b*c^3)*d*e^4 - (b^6 - 4*a*b^4*c - 6*a^2*b^2*c^2 + 12*a^3*...
Timed out. \[ \int \frac {(b+2 c x) (d+e x)^5}{\left (a+b x+c x^2\right )^3} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {(b+2 c x) (d+e x)^5}{\left (a+b x+c x^2\right )^3} \, dx=\text {Exception raised: ValueError} \]
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. 672 vs. \(2 (281) = 562\).
Time = 0.28 (sec) , antiderivative size = 672, normalized size of antiderivative = 2.29 \[ \int \frac {(b+2 c x) (d+e x)^5}{\left (a+b x+c x^2\right )^3} \, dx=\frac {2 \, e^{5} x}{c^{2}} - \frac {5 \, {\left (2 \, c^{4} d^{4} e - 4 \, b c^{3} d^{3} e^{2} + 12 \, a c^{3} d^{2} e^{3} + 2 \, b^{3} c d e^{4} - 12 \, a b c^{2} d e^{4} - b^{4} e^{5} + 6 \, a b^{2} c e^{5} - 6 \, a^{2} c^{2} e^{5}\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (b^{2} c^{3} - 4 \, a c^{4}\right )} \sqrt {-b^{2} + 4 \, a c}} + \frac {5 \, {\left (2 \, c d e^{4} - b e^{5}\right )} \log \left (c x^{2} + b x + a\right )}{2 \, c^{3}} - \frac {b^{2} c^{3} d^{5} - 4 \, a c^{4} d^{5} + 5 \, a b c^{3} d^{4} e - 40 \, a^{2} c^{3} d^{3} e^{2} + 30 \, a^{2} b c^{2} d^{2} e^{3} - 25 \, a^{2} b^{2} c d e^{4} + 60 \, a^{3} c^{2} d e^{4} + 7 \, a^{2} b^{3} e^{5} - 23 \, a^{3} b c e^{5} + 2 \, {\left (5 \, c^{5} d^{4} e - 10 \, b c^{4} d^{3} e^{2} + 20 \, b^{2} c^{3} d^{2} e^{3} - 50 \, a c^{4} d^{2} e^{3} - 15 \, b^{3} c^{2} d e^{4} + 50 \, a b c^{3} d e^{4} + 4 \, b^{4} c e^{5} - 17 \, a b^{2} c^{2} e^{5} + 9 \, a^{2} c^{3} e^{5}\right )} x^{3} + {\left (15 \, b c^{4} d^{4} e - 10 \, b^{2} c^{3} d^{3} e^{2} - 80 \, a c^{4} d^{3} e^{2} + 30 \, b^{3} c^{2} d^{2} e^{3} - 30 \, a b c^{3} d^{2} e^{3} - 25 \, b^{4} c d e^{4} + 50 \, a b^{2} c^{2} d e^{4} + 80 \, a^{2} c^{3} d e^{4} + 7 \, b^{5} e^{5} - 21 \, a b^{3} c e^{5} - 13 \, a^{2} b c^{2} e^{5}\right )} x^{2} + 2 \, {\left (5 \, b^{2} c^{3} d^{4} e - 5 \, a c^{4} d^{4} e - 30 \, a b c^{3} d^{3} e^{2} + 30 \, a b^{2} c^{2} d^{2} e^{3} - 30 \, a^{2} c^{3} d^{2} e^{3} - 25 \, a b^{3} c d e^{4} + 70 \, a^{2} b c^{2} d e^{4} + 7 \, a b^{4} e^{5} - 26 \, a^{2} b^{2} c e^{5} + 7 \, a^{3} c^{2} e^{5}\right )} x}{2 \, {\left (c x^{2} + b x + a\right )}^{2} {\left (b^{2} - 4 \, a c\right )} c^{3}} \]
2*e^5*x/c^2 - 5*(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)*arct an((2*c*x + b)/sqrt(-b^2 + 4*a*c))/((b^2*c^3 - 4*a*c^4)*sqrt(-b^2 + 4*a*c) ) + 5/2*(2*c*d*e^4 - b*e^5)*log(c*x^2 + b*x + a)/c^3 - 1/2*(b^2*c^3*d^5 - 4*a*c^4*d^5 + 5*a*b*c^3*d^4*e - 40*a^2*c^3*d^3*e^2 + 30*a^2*b*c^2*d^2*e^3 - 25*a^2*b^2*c*d*e^4 + 60*a^3*c^2*d*e^4 + 7*a^2*b^3*e^5 - 23*a^3*b*c*e^5 + 2*(5*c^5*d^4*e - 10*b*c^4*d^3*e^2 + 20*b^2*c^3*d^2*e^3 - 50*a*c^4*d^2*e^3 - 15*b^3*c^2*d*e^4 + 50*a*b*c^3*d*e^4 + 4*b^4*c*e^5 - 17*a*b^2*c^2*e^5 + 9*a^2*c^3*e^5)*x^3 + (15*b*c^4*d^4*e - 10*b^2*c^3*d^3*e^2 - 80*a*c^4*d^3*e ^2 + 30*b^3*c^2*d^2*e^3 - 30*a*b*c^3*d^2*e^3 - 25*b^4*c*d*e^4 + 50*a*b^2*c ^2*d*e^4 + 80*a^2*c^3*d*e^4 + 7*b^5*e^5 - 21*a*b^3*c*e^5 - 13*a^2*b*c^2*e^ 5)*x^2 + 2*(5*b^2*c^3*d^4*e - 5*a*c^4*d^4*e - 30*a*b*c^3*d^3*e^2 + 30*a*b^ 2*c^2*d^2*e^3 - 30*a^2*c^3*d^2*e^3 - 25*a*b^3*c*d*e^4 + 70*a^2*b*c^2*d*e^4 + 7*a*b^4*e^5 - 26*a^2*b^2*c*e^5 + 7*a^3*c^2*e^5)*x)/((c*x^2 + b*x + a)^2 *(b^2 - 4*a*c)*c^3)
Time = 11.90 (sec) , antiderivative size = 1148, normalized size of antiderivative = 3.92 \[ \int \frac {(b+2 c x) (d+e x)^5}{\left (a+b x+c x^2\right )^3} \, dx=\frac {\frac {x^3\,\left (9\,a^2\,c^2\,e^5-17\,a\,b^2\,c\,e^5+50\,a\,b\,c^2\,d\,e^4-50\,a\,c^3\,d^2\,e^3+4\,b^4\,e^5-15\,b^3\,c\,d\,e^4+20\,b^2\,c^2\,d^2\,e^3-10\,b\,c^3\,d^3\,e^2+5\,c^4\,d^4\,e\right )}{4\,a\,c-b^2}+\frac {-23\,a^3\,b\,c\,e^5+60\,a^3\,c^2\,d\,e^4+7\,a^2\,b^3\,e^5-25\,a^2\,b^2\,c\,d\,e^4+30\,a^2\,b\,c^2\,d^2\,e^3-40\,a^2\,c^3\,d^3\,e^2+5\,a\,b\,c^3\,d^4\,e-4\,a\,c^4\,d^5+b^2\,c^3\,d^5}{2\,c\,\left (4\,a\,c-b^2\right )}-\frac {x^2\,\left (13\,a^2\,b\,c^2\,e^5-80\,a^2\,c^3\,d\,e^4+21\,a\,b^3\,c\,e^5-50\,a\,b^2\,c^2\,d\,e^4+30\,a\,b\,c^3\,d^2\,e^3+80\,a\,c^4\,d^3\,e^2-7\,b^5\,e^5+25\,b^4\,c\,d\,e^4-30\,b^3\,c^2\,d^2\,e^3+10\,b^2\,c^3\,d^3\,e^2-15\,b\,c^4\,d^4\,e\right )}{2\,c\,\left (4\,a\,c-b^2\right )}+\frac {x\,\left (7\,a^3\,c^2\,e^5-26\,a^2\,b^2\,c\,e^5+70\,a^2\,b\,c^2\,d\,e^4-30\,a^2\,c^3\,d^2\,e^3+7\,a\,b^4\,e^5-25\,a\,b^3\,c\,d\,e^4+30\,a\,b^2\,c^2\,d^2\,e^3-30\,a\,b\,c^3\,d^3\,e^2-5\,a\,c^4\,d^4\,e+5\,b^2\,c^3\,d^4\,e\right )}{c\,\left (4\,a\,c-b^2\right )}}{a^2\,c^2+c^4\,x^4+x^2\,\left (b^2\,c^2+2\,a\,c^3\right )+2\,b\,c^3\,x^3+2\,a\,b\,c^2\,x}+\frac {\ln \left (c\,x^2+b\,x+a\right )\,\left (-320\,a^3\,b\,c^3\,e^5+640\,d\,a^3\,c^4\,e^4+240\,a^2\,b^3\,c^2\,e^5-480\,d\,a^2\,b^2\,c^3\,e^4-60\,a\,b^5\,c\,e^5+120\,d\,a\,b^4\,c^2\,e^4+5\,b^7\,e^5-10\,d\,b^6\,c\,e^4\right )}{2\,\left (64\,a^3\,c^6-48\,a^2\,b^2\,c^5+12\,a\,b^4\,c^4-b^6\,c^3\right )}+\frac {2\,e^5\,x}{c^2}+\frac {5\,e\,\mathrm {atan}\left (\frac {c^3\,{\left (4\,a\,c-b^2\right )}^{5/2}\,\left (\frac {5\,e\,\left (b^3\,c^2-4\,a\,b\,c^3\right )\,\left (6\,a^2\,c^2\,e^4-6\,a\,b^2\,c\,e^4+12\,a\,b\,c^2\,d\,e^3-12\,a\,c^3\,d^2\,e^2+b^4\,e^4-2\,b^3\,c\,d\,e^3+4\,b\,c^3\,d^3\,e-2\,c^4\,d^4\right )}{c^5\,{\left (4\,a\,c-b^2\right )}^4}-\frac {10\,e\,x\,\left (6\,a^2\,c^2\,e^4-6\,a\,b^2\,c\,e^4+12\,a\,b\,c^2\,d\,e^3-12\,a\,c^3\,d^2\,e^2+b^4\,e^4-2\,b^3\,c\,d\,e^3+4\,b\,c^3\,d^3\,e-2\,c^4\,d^4\right )}{c^2\,{\left (4\,a\,c-b^2\right )}^3}\right )}{30\,a^2\,c^2\,e^5-30\,a\,b^2\,c\,e^5+60\,a\,b\,c^2\,d\,e^4-60\,a\,c^3\,d^2\,e^3+5\,b^4\,e^5-10\,b^3\,c\,d\,e^4+20\,b\,c^3\,d^3\,e^2-10\,c^4\,d^4\,e}\right )\,\left (6\,a^2\,c^2\,e^4-6\,a\,b^2\,c\,e^4+12\,a\,b\,c^2\,d\,e^3-12\,a\,c^3\,d^2\,e^2+b^4\,e^4-2\,b^3\,c\,d\,e^3+4\,b\,c^3\,d^3\,e-2\,c^4\,d^4\right )}{c^3\,{\left (4\,a\,c-b^2\right )}^{3/2}} \]
((x^3*(4*b^4*e^5 + 5*c^4*d^4*e + 9*a^2*c^2*e^5 - 50*a*c^3*d^2*e^3 - 10*b*c ^3*d^3*e^2 + 20*b^2*c^2*d^2*e^3 - 17*a*b^2*c*e^5 - 15*b^3*c*d*e^4 + 50*a*b *c^2*d*e^4))/(4*a*c - b^2) + (7*a^2*b^3*e^5 - 4*a*c^4*d^5 + b^2*c^3*d^5 + 60*a^3*c^2*d*e^4 - 40*a^2*c^3*d^3*e^2 - 23*a^3*b*c*e^5 + 5*a*b*c^3*d^4*e - 25*a^2*b^2*c*d*e^4 + 30*a^2*b*c^2*d^2*e^3)/(2*c*(4*a*c - b^2)) - (x^2*(13 *a^2*b*c^2*e^5 - 7*b^5*e^5 + 80*a*c^4*d^3*e^2 - 80*a^2*c^3*d*e^4 + 10*b^2* c^3*d^3*e^2 - 30*b^3*c^2*d^2*e^3 + 21*a*b^3*c*e^5 - 15*b*c^4*d^4*e + 25*b^ 4*c*d*e^4 + 30*a*b*c^3*d^2*e^3 - 50*a*b^2*c^2*d*e^4))/(2*c*(4*a*c - b^2)) + (x*(7*a*b^4*e^5 + 7*a^3*c^2*e^5 - 26*a^2*b^2*c*e^5 + 5*b^2*c^3*d^4*e - 3 0*a^2*c^3*d^2*e^3 - 5*a*c^4*d^4*e - 25*a*b^3*c*d*e^4 - 30*a*b*c^3*d^3*e^2 + 70*a^2*b*c^2*d*e^4 + 30*a*b^2*c^2*d^2*e^3))/(c*(4*a*c - b^2)))/(a^2*c^2 + c^4*x^4 + x^2*(2*a*c^3 + b^2*c^2) + 2*b*c^3*x^3 + 2*a*b*c^2*x) + (log(a + b*x + c*x^2)*(5*b^7*e^5 - 320*a^3*b*c^3*e^5 + 640*a^3*c^4*d*e^4 + 240*a^ 2*b^3*c^2*e^5 - 60*a*b^5*c*e^5 - 10*b^6*c*d*e^4 + 120*a*b^4*c^2*d*e^4 - 48 0*a^2*b^2*c^3*d*e^4))/(2*(64*a^3*c^6 - b^6*c^3 + 12*a*b^4*c^4 - 48*a^2*b^2 *c^5)) + (2*e^5*x)/c^2 + (5*e*atan((c^3*(4*a*c - b^2)^(5/2)*((5*e*(b^3*c^2 - 4*a*b*c^3)*(b^4*e^4 - 2*c^4*d^4 + 6*a^2*c^2*e^4 - 12*a*c^3*d^2*e^2 - 6* a*b^2*c*e^4 + 4*b*c^3*d^3*e - 2*b^3*c*d*e^3 + 12*a*b*c^2*d*e^3))/(c^5*(4*a *c - b^2)^4) - (10*e*x*(b^4*e^4 - 2*c^4*d^4 + 6*a^2*c^2*e^4 - 12*a*c^3*d^2 *e^2 - 6*a*b^2*c*e^4 + 4*b*c^3*d^3*e - 2*b^3*c*d*e^3 + 12*a*b*c^2*d*e^3...