3.23.28 \(\int \frac {1}{(d+e x)^2 (a+b x+c x^2)^5} \, dx\) [2228]

Optimal. Leaf size=1761 \[ \frac {5 e \left (14 c^8 d^8-b^8 e^8-4 c^7 d^6 e (14 b d-19 a e)+b^6 c e^7 (b d+15 a e)+b^4 c^2 e^6 \left (b^2 d^2-16 a b d e-82 a^2 e^2\right )+c^6 d^4 e^2 \left (79 b^2 d^2-228 a b d e+176 a^2 e^2\right )-c^5 d^2 e^3 \left (41 b^3 d^3-197 a b^2 d^2 e+352 a^2 b d e^2-244 a^3 e^3\right )+b^2 c^3 e^5 \left (b^3 d^3-15 a b^2 d^2 e+95 a^2 b d e^2+187 a^3 e^3\right )+c^4 e^4 \left (b^4 d^4-14 a b^3 d^3 e+81 a^2 b^2 d^2 e^2-244 a^3 b d e^3-126 a^4 e^4\right )\right )}{\left (b^2-4 a c\right )^4 \left (c d^2-b d e+a e^2\right )^5 (d+e x)}-\frac {b c d-b^2 e+2 a c e+c (2 c d-b e) x}{4 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x) \left (a+b x+c x^2\right )^4}-\frac {8 a c e (2 c d-b e)^2-\left (b c d-b^2 e+2 a c e\right ) \left (14 c^2 d^2-5 b^2 e^2-6 c e (b d-3 a e)\right )-c (2 c d-b e) \left (14 c^2 d^2-5 b^2 e^2-2 c e (7 b d-17 a e)\right ) x}{12 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 (d+e x) \left (a+b x+c x^2\right )^3}+\frac {3 a c e (2 c d-b e)^2 \left (14 c^2 d^2-5 b^2 e^2-2 c e (7 b d-17 a e)\right )-\left (b c d-b^2 e+2 a c e\right ) \left (70 c^4 d^4+10 b^4 e^4-2 c^3 d^2 e (49 b d-78 a e)+5 b^2 c e^3 (2 b d-15 a e)+3 c^2 e^2 \left (b^2 d^2-18 a b d e+42 a^2 e^2\right )\right )-5 c (2 c d-b e) \left (14 c^4 d^4+2 b^4 e^4+b^2 c e^3 (5 b d-21 a e)-4 c^3 d^2 e (7 b d-12 a e)+3 c^2 e^2 \left (3 b^2 d^2-16 a b d e+22 a^2 e^2\right )\right ) x}{12 \left (b^2-4 a c\right )^3 \left (c d^2-b d e+a e^2\right )^3 (d+e x) \left (a+b x+c x^2\right )^2}-\frac {5 \left (2 a c e (2 c d-b e)^2 \left (14 c^4 d^4+2 b^4 e^4+b^2 c e^3 (5 b d-21 a e)-4 c^3 d^2 e (7 b d-12 a e)+3 c^2 e^2 \left (3 b^2 d^2-16 a b d e+22 a^2 e^2\right )\right )-\left (b c d-b^2 e+2 a c e\right ) \left (42 c^6 d^6-3 b^6 e^6-2 c^5 d^4 e (49 b d-65 a e)-2 b^4 c e^5 (b d-17 a e)+b^2 c^2 e^4 \left (b^2 d^2+16 a b d e-123 a^2 e^2\right )+c^4 d^2 e^2 \left (55 b^2 d^2-164 a b d e+150 a^2 e^2\right )+6 c^3 e^3 \left (b^3 d^3-4 a b^2 d^2 e-3 a^2 b d e^2+21 a^3 e^3\right )\right )-3 c (2 c d-b e) \left (14 c^6 d^6-b^6 e^6-2 c^5 d^4 e (21 b d-31 a e)-2 b^4 c e^5 (b d-7 a e)-b^2 c^2 e^4 \left (3 b^2 d^2-26 a b d e+69 a^2 e^2\right )+c^4 d^2 e^2 \left (37 b^2 d^2-124 a b d e+114 a^2 e^2\right )-2 c^3 e^3 \left (2 b^3 d^3-18 a b^2 d^2 e+57 a^2 b d e^2-65 a^3 e^3\right )\right ) x\right )}{6 \left (b^2-4 a c\right )^4 \left (c d^2-b d e+a e^2\right )^4 (d+e x) \left (a+b x+c x^2\right )}-\frac {5 \left (28 c^{10} d^{10}+b^{10} e^{10}-20 c^9 d^8 e (7 b d-9 a e)-252 a^4 c^5 e^9 (5 b d+a e)+210 a^3 b^2 c^4 e^9 (4 b d+3 a e)-84 a^2 b^4 c^3 e^9 (3 b d+5 a e)+18 a b^6 c^2 e^9 (2 b d+7 a e)-2 b^8 c e^9 (b d+9 a e)+18 c^8 d^6 e^2 \left (15 b^2 d^2-40 a b d e+28 a^2 e^2\right )-24 c^7 d^4 e^3 \left (10 b^3 d^3-42 a b^2 d^2 e+63 a^2 b d e^2-35 a^3 e^3\right )+84 c^6 d^2 e^4 \left (b^4 d^4-6 a b^3 d^3 e+15 a^2 b^2 d^2 e^2-20 a^3 b d e^3+15 a^4 e^4\right )\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{9/2} \left (c d^2-e (b d-a e)\right )^6}+\frac {5 e^9 (2 c d-b e) \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^6}-\frac {5 e^9 (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 )^6} \]

[Out]

5*e*(14*c^8*d^8-b^8*e^8-4*c^7*d^6*e*(-19*a*e+14*b*d)+b^6*c*e^7*(15*a*e+b*d)+b^4*c^2*e^6*(-82*a^2*e^2-16*a*b*d*
e+b^2*d^2)+c^6*d^4*e^2*(176*a^2*e^2-228*a*b*d*e+79*b^2*d^2)-c^5*d^2*e^3*(-244*a^3*e^3+352*a^2*b*d*e^2-197*a*b^
2*d^2*e+41*b^3*d^3)+b^2*c^3*e^5*(187*a^3*e^3+95*a^2*b*d*e^2-15*a*b^2*d^2*e+b^3*d^3)+c^4*e^4*(-126*a^4*e^4-244*
a^3*b*d*e^3+81*a^2*b^2*d^2*e^2-14*a*b^3*d^3*e+b^4*d^4))/(-4*a*c+b^2)^4/(a*e^2-b*d*e+c*d^2)^5/(e*x+d)+1/4*(-b*c
*d+b^2*e-2*a*c*e-c*(-b*e+2*c*d)*x)/(-4*a*c+b^2)/(a*e^2-b*d*e+c*d^2)/(e*x+d)/(c*x^2+b*x+a)^4+1/12*(-8*a*c*e*(-b
*e+2*c*d)^2+(2*a*c*e-b^2*e+b*c*d)*(14*c^2*d^2-5*b^2*e^2-6*c*e*(-3*a*e+b*d))+c*(-b*e+2*c*d)*(14*c^2*d^2-5*b^2*e
^2-2*c*e*(-17*a*e+7*b*d))*x)/(-4*a*c+b^2)^2/(a*e^2-b*d*e+c*d^2)^2/(e*x+d)/(c*x^2+b*x+a)^3+1/12*(3*a*c*e*(-b*e+
2*c*d)^2*(14*c^2*d^2-5*b^2*e^2-2*c*e*(-17*a*e+7*b*d))-(2*a*c*e-b^2*e+b*c*d)*(70*c^4*d^4+10*b^4*e^4-2*c^3*d^2*e
*(-78*a*e+49*b*d)+5*b^2*c*e^3*(-15*a*e+2*b*d)+3*c^2*e^2*(42*a^2*e^2-18*a*b*d*e+b^2*d^2))-5*c*(-b*e+2*c*d)*(14*
c^4*d^4+2*b^4*e^4+b^2*c*e^3*(-21*a*e+5*b*d)-4*c^3*d^2*e*(-12*a*e+7*b*d)+3*c^2*e^2*(22*a^2*e^2-16*a*b*d*e+3*b^2
*d^2))*x)/(-4*a*c+b^2)^3/(a*e^2-b*d*e+c*d^2)^3/(e*x+d)/(c*x^2+b*x+a)^2-5/6*(2*a*c*e*(-b*e+2*c*d)^2*(14*c^4*d^4
+2*b^4*e^4+b^2*c*e^3*(-21*a*e+5*b*d)-4*c^3*d^2*e*(-12*a*e+7*b*d)+3*c^2*e^2*(22*a^2*e^2-16*a*b*d*e+3*b^2*d^2))-
(2*a*c*e-b^2*e+b*c*d)*(42*c^6*d^6-3*b^6*e^6-2*c^5*d^4*e*(-65*a*e+49*b*d)-2*b^4*c*e^5*(-17*a*e+b*d)+b^2*c^2*e^4
*(-123*a^2*e^2+16*a*b*d*e+b^2*d^2)+c^4*d^2*e^2*(150*a^2*e^2-164*a*b*d*e+55*b^2*d^2)+6*c^3*e^3*(21*a^3*e^3-3*a^
2*b*d*e^2-4*a*b^2*d^2*e+b^3*d^3))-3*c*(-b*e+2*c*d)*(14*c^6*d^6-b^6*e^6-2*c^5*d^4*e*(-31*a*e+21*b*d)-2*b^4*c*e^
5*(-7*a*e+b*d)-b^2*c^2*e^4*(69*a^2*e^2-26*a*b*d*e+3*b^2*d^2)+c^4*d^2*e^2*(114*a^2*e^2-124*a*b*d*e+37*b^2*d^2)-
2*c^3*e^3*(-65*a^3*e^3+57*a^2*b*d*e^2-18*a*b^2*d^2*e+2*b^3*d^3))*x)/(-4*a*c+b^2)^4/(a*e^2-b*d*e+c*d^2)^4/(e*x+
d)/(c*x^2+b*x+a)-5*(28*c^10*d^10+b^10*e^10-20*c^9*d^8*e*(-9*a*e+7*b*d)-252*a^4*c^5*e^9*(a*e+5*b*d)+210*a^3*b^2
*c^4*e^9*(3*a*e+4*b*d)-84*a^2*b^4*c^3*e^9*(5*a*e+3*b*d)+18*a*b^6*c^2*e^9*(7*a*e+2*b*d)-2*b^8*c*e^9*(9*a*e+b*d)
+18*c^8*d^6*e^2*(28*a^2*e^2-40*a*b*d*e+15*b^2*d^2)-24*c^7*d^4*e^3*(-35*a^3*e^3+63*a^2*b*d*e^2-42*a*b^2*d^2*e+1
0*b^3*d^3)+84*c^6*d^2*e^4*(15*a^4*e^4-20*a^3*b*d*e^3+15*a^2*b^2*d^2*e^2-6*a*b^3*d^3*e+b^4*d^4))*arctanh((2*c*x
+b)/(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(9/2)/(c*d^2-e*(-a*e+b*d))^6+5*e^9*(-b*e+2*c*d)*ln(e*x+d)/(a*e^2-b*d*e+c*
d^2)^6-5/2*e^9*(-b*e+2*c*d)*ln(c*x^2+b*x+a)/(a*e^2-b*d*e+c*d^2)^6

________________________________________________________________________________________

Rubi [A]
time = 13.95, antiderivative size = 1761, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {754, 836, 814, 648, 632, 212, 642} \begin {gather*} \frac {5 (2 c d-b e) \log (d+e x) e^9}{\left (c d^2-b e d+a e^2\right )^6}-\frac {5 (2 c d-b e) \log \left (c x^2+b x+a\right ) e^9}{2 \left (c d^2-b e d+a e^2\right )^6}+\frac {5 \left (14 c^8 d^8-4 c^7 e (14 b d-19 a e) d^6+c^6 e^2 \left (79 b^2 d^2-228 a b e d+176 a^2 e^2\right ) d^4-c^5 e^3 \left (41 b^3 d^3-197 a b^2 e d^2+352 a^2 b e^2 d-244 a^3 e^3\right ) d^2-b^8 e^8+b^6 c e^7 (b d+15 a e)+b^4 c^2 e^6 \left (b^2 d^2-16 a b e d-82 a^2 e^2\right )+b^2 c^3 e^5 \left (b^3 d^3-15 a b^2 e d^2+95 a^2 b e^2 d+187 a^3 e^3\right )+c^4 e^4 \left (b^4 d^4-14 a b^3 e d^3+81 a^2 b^2 e^2 d^2-244 a^3 b e^3 d-126 a^4 e^4\right )\right ) e}{\left (b^2-4 a c\right )^4 \left (c d^2-b e d+a e^2\right )^5 (d+e x)}-\frac {5 \left (28 c^{10} d^{10}-20 c^9 e (7 b d-9 a e) d^8+18 c^8 e^2 \left (15 b^2 d^2-40 a b e d+28 a^2 e^2\right ) d^6-24 c^7 e^3 \left (10 b^3 d^3-42 a b^2 e d^2+63 a^2 b e^2 d-35 a^3 e^3\right ) d^4+84 c^6 e^4 \left (b^4 d^4-6 a b^3 e d^3+15 a^2 b^2 e^2 d^2-20 a^3 b e^3 d+15 a^4 e^4\right ) d^2+b^{10} e^{10}-252 a^4 c^5 e^9 (5 b d+a e)+210 a^3 b^2 c^4 e^9 (4 b d+3 a e)-84 a^2 b^4 c^3 e^9 (3 b d+5 a e)+18 a b^6 c^2 e^9 (2 b d+7 a e)-2 b^8 c e^9 (b d+9 a e)\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{9/2} \left (c d^2-e (b d-a e)\right )^6}-\frac {5 \left (2 a c e \left (14 c^4 d^4-4 c^3 e (7 b d-12 a e) d^2+2 b^4 e^4+b^2 c e^3 (5 b d-21 a e)+3 c^2 e^2 \left (3 b^2 d^2-16 a b e d+22 a^2 e^2\right )\right ) (2 c d-b e)^2-3 c \left (14 c^6 d^6-2 c^5 e (21 b d-31 a e) d^4+c^4 e^2 \left (37 b^2 d^2-124 a b e d+114 a^2 e^2\right ) d^2-b^6 e^6-2 b^4 c e^5 (b d-7 a e)-b^2 c^2 e^4 \left (3 b^2 d^2-26 a b e d+69 a^2 e^2\right )-2 c^3 e^3 \left (2 b^3 d^3-18 a b^2 e d^2+57 a^2 b e^2 d-65 a^3 e^3\right )\right ) x (2 c d-b e)-\left (-e b^2+c d b+2 a c e\right ) \left (42 c^6 d^6-2 c^5 e (49 b d-65 a e) d^4+c^4 e^2 \left (55 b^2 d^2-164 a b e d+150 a^2 e^2\right ) d^2-3 b^6 e^6-2 b^4 c e^5 (b d-17 a e)+b^2 c^2 e^4 \left (b^2 d^2+16 a b e d-123 a^2 e^2\right )+6 c^3 e^3 \left (b^3 d^3-4 a b^2 e d^2-3 a^2 b e^2 d+21 a^3 e^3\right )\right )\right )}{6 \left (b^2-4 a c\right )^4 \left (c d^2-b e d+a e^2\right )^4 (d+e x) \left (c x^2+b x+a\right )}+\frac {3 a c e \left (14 c^2 d^2-5 b^2 e^2-2 c e (7 b d-17 a e)\right ) (2 c d-b e)^2-5 c \left (14 c^4 d^4-4 c^3 e (7 b d-12 a e) d^2+2 b^4 e^4+b^2 c e^3 (5 b d-21 a e)+3 c^2 e^2 \left (3 b^2 d^2-16 a b e d+22 a^2 e^2\right )\right ) x (2 c d-b e)-\left (-e b^2+c d b+2 a c e\right ) \left (70 c^4 d^4-2 c^3 e (49 b d-78 a e) d^2+10 b^4 e^4+5 b^2 c e^3 (2 b d-15 a e)+3 c^2 e^2 \left (b^2 d^2-18 a b e d+42 a^2 e^2\right )\right )}{12 \left (b^2-4 a c\right )^3 \left (c d^2-b e d+a e^2\right )^3 (d+e x) \left (c x^2+b x+a\right )^2}-\frac {8 a c e (2 c d-b e)^2-c \left (14 c^2 d^2-5 b^2 e^2-2 c e (7 b d-17 a e)\right ) x (2 c d-b e)-\left (-e b^2+c d b+2 a c e\right ) \left (14 c^2 d^2-5 b^2 e^2-6 c e (b d-3 a e)\right )}{12 \left (b^2-4 a c\right )^2 \left (c d^2-b e d+a e^2\right )^2 (d+e x) \left (c x^2+b x+a\right )^3}-\frac {-e b^2+c d b+2 a c e+c (2 c d-b e) x}{4 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) (d+e x) \left (c x^2+b x+a\right )^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x)^2*(a + b*x + c*x^2)^5),x]

[Out]

(5*e*(14*c^8*d^8 - b^8*e^8 - 4*c^7*d^6*e*(14*b*d - 19*a*e) + b^6*c*e^7*(b*d + 15*a*e) + b^4*c^2*e^6*(b^2*d^2 -
16*a*b*d*e - 82*a^2*e^2) + c^6*d^4*e^2*(79*b^2*d^2 - 228*a*b*d*e + 176*a^2*e^2) - c^5*d^2*e^3*(41*b^3*d^3 - 1
97*a*b^2*d^2*e + 352*a^2*b*d*e^2 - 244*a^3*e^3) + b^2*c^3*e^5*(b^3*d^3 - 15*a*b^2*d^2*e + 95*a^2*b*d*e^2 + 187
*a^3*e^3) + c^4*e^4*(b^4*d^4 - 14*a*b^3*d^3*e + 81*a^2*b^2*d^2*e^2 - 244*a^3*b*d*e^3 - 126*a^4*e^4)))/((b^2 -
4*a*c)^4*(c*d^2 - b*d*e + a*e^2)^5*(d + e*x)) - (b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x)/(4*(b^2 - 4*a*c)
*(c*d^2 - b*d*e + a*e^2)*(d + e*x)*(a + b*x + c*x^2)^4) - (8*a*c*e*(2*c*d - b*e)^2 - (b*c*d - b^2*e + 2*a*c*e)
*(14*c^2*d^2 - 5*b^2*e^2 - 6*c*e*(b*d - 3*a*e)) - c*(2*c*d - b*e)*(14*c^2*d^2 - 5*b^2*e^2 - 2*c*e*(7*b*d - 17*
a*e))*x)/(12*(b^2 - 4*a*c)^2*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)*(a + b*x + c*x^2)^3) + (3*a*c*e*(2*c*d - b*e)
^2*(14*c^2*d^2 - 5*b^2*e^2 - 2*c*e*(7*b*d - 17*a*e)) - (b*c*d - b^2*e + 2*a*c*e)*(70*c^4*d^4 + 10*b^4*e^4 - 2*
c^3*d^2*e*(49*b*d - 78*a*e) + 5*b^2*c*e^3*(2*b*d - 15*a*e) + 3*c^2*e^2*(b^2*d^2 - 18*a*b*d*e + 42*a^2*e^2)) -
5*c*(2*c*d - b*e)*(14*c^4*d^4 + 2*b^4*e^4 + b^2*c*e^3*(5*b*d - 21*a*e) - 4*c^3*d^2*e*(7*b*d - 12*a*e) + 3*c^2*
e^2*(3*b^2*d^2 - 16*a*b*d*e + 22*a^2*e^2))*x)/(12*(b^2 - 4*a*c)^3*(c*d^2 - b*d*e + a*e^2)^3*(d + e*x)*(a + b*x
+ c*x^2)^2) - (5*(2*a*c*e*(2*c*d - b*e)^2*(14*c^4*d^4 + 2*b^4*e^4 + b^2*c*e^3*(5*b*d - 21*a*e) - 4*c^3*d^2*e*
(7*b*d - 12*a*e) + 3*c^2*e^2*(3*b^2*d^2 - 16*a*b*d*e + 22*a^2*e^2)) - (b*c*d - b^2*e + 2*a*c*e)*(42*c^6*d^6 -
3*b^6*e^6 - 2*c^5*d^4*e*(49*b*d - 65*a*e) - 2*b^4*c*e^5*(b*d - 17*a*e) + b^2*c^2*e^4*(b^2*d^2 + 16*a*b*d*e - 1
23*a^2*e^2) + c^4*d^2*e^2*(55*b^2*d^2 - 164*a*b*d*e + 150*a^2*e^2) + 6*c^3*e^3*(b^3*d^3 - 4*a*b^2*d^2*e - 3*a^
2*b*d*e^2 + 21*a^3*e^3)) - 3*c*(2*c*d - b*e)*(14*c^6*d^6 - b^6*e^6 - 2*c^5*d^4*e*(21*b*d - 31*a*e) - 2*b^4*c*e
^5*(b*d - 7*a*e) - b^2*c^2*e^4*(3*b^2*d^2 - 26*a*b*d*e + 69*a^2*e^2) + c^4*d^2*e^2*(37*b^2*d^2 - 124*a*b*d*e +
114*a^2*e^2) - 2*c^3*e^3*(2*b^3*d^3 - 18*a*b^2*d^2*e + 57*a^2*b*d*e^2 - 65*a^3*e^3))*x))/(6*(b^2 - 4*a*c)^4*(
c*d^2 - b*d*e + a*e^2)^4*(d + e*x)*(a + b*x + c*x^2)) - (5*(28*c^10*d^10 + b^10*e^10 - 20*c^9*d^8*e*(7*b*d - 9
*a*e) - 252*a^4*c^5*e^9*(5*b*d + a*e) + 210*a^3*b^2*c^4*e^9*(4*b*d + 3*a*e) - 84*a^2*b^4*c^3*e^9*(3*b*d + 5*a*
e) + 18*a*b^6*c^2*e^9*(2*b*d + 7*a*e) - 2*b^8*c*e^9*(b*d + 9*a*e) + 18*c^8*d^6*e^2*(15*b^2*d^2 - 40*a*b*d*e +
28*a^2*e^2) - 24*c^7*d^4*e^3*(10*b^3*d^3 - 42*a*b^2*d^2*e + 63*a^2*b*d*e^2 - 35*a^3*e^3) + 84*c^6*d^2*e^4*(b^4
*d^4 - 6*a*b^3*d^3*e + 15*a^2*b^2*d^2*e^2 - 20*a^3*b*d*e^3 + 15*a^4*e^4))*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c
]])/((b^2 - 4*a*c)^(9/2)*(c*d^2 - e*(b*d - a*e))^6) + (5*e^9*(2*c*d - b*e)*Log[d + e*x])/(c*d^2 - b*d*e + a*e^
2)^6 - (5*e^9*(2*c*d - b*e)*Log[a + b*x + c*x^2])/(2*(c*d^2 - b*d*e + a*e^2)^6)

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 648

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 754

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(b
*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*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] + Dist[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^m*Simp[b*c*d*e*(2*p - m
+ 2) + b^2*e^2*(m + p + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3) - c*e*(2*c*d - b*e)*(m + 2*p + 4)*x
, x]*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b
*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && LtQ[p, -1] && IntQuadraticQ[a, b, c, d, e, m, p, x]

Rule 814

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[Exp
andIntegrand[(d + e*x)^m*((f + g*x)/(a + b*x + c*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 -
4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[m]

Rule 836

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] + Dist[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] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] ||
IntegerQ[p] || IntegersQ[2*m, 2*p])

Rubi steps

\begin {align*} \int \frac {1}{(d+e x)^2 \left (a+b x+c x^2\right )^5} \, dx &=-\frac {b c d-b^2 e+2 a c e+c (2 c d-b e) x}{4 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x) \left (a+b x+c x^2\right )^4}-\frac {\int \frac {14 c^2 d^2-5 b^2 e^2-6 c e (b d-3 a e)+8 c e (2 c d-b e) x}{(d+e x)^2 \left (a+b x+c x^2\right )^4} \, dx}{4 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )}\\ &=-\frac {b c d-b^2 e+2 a c e+c (2 c d-b e) x}{4 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x) \left (a+b x+c x^2\right )^4}-\frac {8 a c e (2 c d-b e)^2-\left (b c d-b^2 e+2 a c e\right ) \left (14 c^2 d^2-5 b^2 e^2-6 c e (b d-3 a e)\right )-c (2 c d-b e) \left (14 c^2 d^2-5 b^2 e^2-2 c e (7 b d-17 a e)\right ) x}{12 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 (d+e x) \left (a+b x+c x^2\right )^3}+\frac {\int \frac {2 \left (70 c^4 d^4+10 b^4 e^4-2 c^3 d^2 e (49 b d-78 a e)+5 b^2 c e^3 (2 b d-15 a e)+3 c^2 e^2 \left (b^2 d^2-18 a b d e+42 a^2 e^2\right )\right )+6 c e (2 c d-b e) \left (14 c^2 d^2-5 b^2 e^2-2 c e (7 b d-17 a e)\right ) x}{(d+e x)^2 \left (a+b x+c x^2\right )^3} \, dx}{12 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2}\\ &=-\frac {b c d-b^2 e+2 a c e+c (2 c d-b e) x}{4 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x) \left (a+b x+c x^2\right )^4}-\frac {8 a c e (2 c d-b e)^2-\left (b c d-b^2 e+2 a c e\right ) \left (14 c^2 d^2-5 b^2 e^2-6 c e (b d-3 a e)\right )-c (2 c d-b e) \left (14 c^2 d^2-5 b^2 e^2-2 c e (7 b d-17 a e)\right ) x}{12 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 (d+e x) \left (a+b x+c x^2\right )^3}+\frac {3 a c e (2 c d-b e)^2 \left (14 c^2 d^2-5 b^2 e^2-2 c e (7 b d-17 a e)\right )-\left (b c d-b^2 e+2 a c e\right ) \left (70 c^4 d^4+10 b^4 e^4-2 c^3 d^2 e (49 b d-78 a e)+5 b^2 c e^3 (2 b d-15 a e)+3 c^2 e^2 \left (b^2 d^2-18 a b d e+42 a^2 e^2\right )\right )-5 c (2 c d-b e) \left (14 c^4 d^4+2 b^4 e^4+b^2 c e^3 (5 b d-21 a e)-4 c^3 d^2 e (7 b d-12 a e)+3 c^2 e^2 \left (3 b^2 d^2-16 a b d e+22 a^2 e^2\right )\right ) x}{12 \left (b^2-4 a c\right )^3 \left (c d^2-b d e+a e^2\right )^3 (d+e x) \left (a+b x+c x^2\right )^2}-\frac {\int \frac {20 \left (42 c^6 d^6-3 b^6 e^6-2 c^5 d^4 e (49 b d-65 a e)-2 b^4 c e^5 (b d-17 a e)+b^2 c^2 e^4 \left (b^2 d^2+16 a b d e-123 a^2 e^2\right )+c^4 d^2 e^2 \left (55 b^2 d^2-164 a b d e+150 a^2 e^2\right )+6 c^3 e^3 \left (b^3 d^3-4 a b^2 d^2 e-3 a^2 b d e^2+21 a^3 e^3\right )\right )+40 c e (2 c d-b e) \left (14 c^4 d^4+2 b^4 e^4+b^2 c e^3 (5 b d-21 a e)-4 c^3 d^2 e (7 b d-12 a e)+3 c^2 e^2 \left (3 b^2 d^2-16 a b d e+22 a^2 e^2\right )\right ) x}{(d+e x)^2 \left (a+b x+c x^2\right )^2} \, dx}{24 \left (b^2-4 a c\right )^3 \left (c d^2-b d e+a e^2\right )^3}\\ &=-\frac {b c d-b^2 e+2 a c e+c (2 c d-b e) x}{4 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x) \left (a+b x+c x^2\right )^4}-\frac {8 a c e (2 c d-b e)^2-\left (b c d-b^2 e+2 a c e\right ) \left (14 c^2 d^2-5 b^2 e^2-6 c e (b d-3 a e)\right )-c (2 c d-b e) \left (14 c^2 d^2-5 b^2 e^2-2 c e (7 b d-17 a e)\right ) x}{12 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 (d+e x) \left (a+b x+c x^2\right )^3}+\frac {3 a c e (2 c d-b e)^2 \left (14 c^2 d^2-5 b^2 e^2-2 c e (7 b d-17 a e)\right )-\left (b c d-b^2 e+2 a c e\right ) \left (70 c^4 d^4+10 b^4 e^4-2 c^3 d^2 e (49 b d-78 a e)+5 b^2 c e^3 (2 b d-15 a e)+3 c^2 e^2 \left (b^2 d^2-18 a b d e+42 a^2 e^2\right )\right )-5 c (2 c d-b e) \left (14 c^4 d^4+2 b^4 e^4+b^2 c e^3 (5 b d-21 a e)-4 c^3 d^2 e (7 b d-12 a e)+3 c^2 e^2 \left (3 b^2 d^2-16 a b d e+22 a^2 e^2\right )\right ) x}{12 \left (b^2-4 a c\right )^3 \left (c d^2-b d e+a e^2\right )^3 (d+e x) \left (a+b x+c x^2\right )^2}-\frac {5 \left (2 a c e (2 c d-b e)^2 \left (14 c^4 d^4+2 b^4 e^4+b^2 c e^3 (5 b d-21 a e)-4 c^3 d^2 e (7 b d-12 a e)+3 c^2 e^2 \left (3 b^2 d^2-16 a b d e+22 a^2 e^2\right )\right )-\left (b c d-b^2 e+2 a c e\right ) \left (42 c^6 d^6-3 b^6 e^6-2 c^5 d^4 e (49 b d-65 a e)-2 b^4 c e^5 (b d-17 a e)+b^2 c^2 e^4 \left (b^2 d^2+16 a b d e-123 a^2 e^2\right )+c^4 d^2 e^2 \left (55 b^2 d^2-164 a b d e+150 a^2 e^2\right )+6 c^3 e^3 \left (b^3 d^3-4 a b^2 d^2 e-3 a^2 b d e^2+21 a^3 e^3\right )\right )-3 c (2 c d-b e) \left (14 c^6 d^6-b^6 e^6-2 c^5 d^4 e (21 b d-31 a e)-2 b^4 c e^5 (b d-7 a e)-b^2 c^2 e^4 \left (3 b^2 d^2-26 a b d e+69 a^2 e^2\right )+c^4 d^2 e^2 \left (37 b^2 d^2-124 a b d e+114 a^2 e^2\right )-2 c^3 e^3 \left (2 b^3 d^3-18 a b^2 d^2 e+57 a^2 b d e^2-65 a^3 e^3\right )\right ) x\right )}{6 \left (b^2-4 a c\right )^4 \left (c d^2-b d e+a e^2\right )^4 (d+e x) \left (a+b x+c x^2\right )}+\frac {\int \frac {120 \left (14 c^8 d^8+b^8 e^8-15 a b^6 c e^8-6 c^7 d^6 e (7 b d-8 a e)-b^4 c^2 e^6 \left (b^2 d^2-2 a b d e-82 a^2 e^2\right )+c^6 d^4 e^2 \left (37 b^2 d^2-82 a b d e+52 a^2 e^2\right )-c^5 d^2 e^3 \left (4 b^3 d^3+a b^2 d^2 e-10 a^2 b d e^2-16 a^3 e^3\right )-b^2 c^3 e^5 \left (2 b^3 d^3-17 a b^2 d^2 e+26 a^2 b d e^2+187 a^3 e^3\right )-3 c^4 e^4 \left (b^4 d^4-10 a b^3 d^3 e+35 a^2 b^2 d^2 e^2-38 a^3 b d e^3-42 a^4 e^4\right )\right )+120 c e (2 c d-b e) \left (14 c^6 d^6-b^6 e^6-2 c^5 d^4 e (21 b d-31 a e)-2 b^4 c e^5 (b d-7 a e)-b^2 c^2 e^4 \left (3 b^2 d^2-26 a b d e+69 a^2 e^2\right )+c^4 d^2 e^2 \left (37 b^2 d^2-124 a b d e+114 a^2 e^2\right )-2 c^3 e^3 \left (2 b^3 d^3-18 a b^2 d^2 e+57 a^2 b d e^2-65 a^3 e^3\right )\right ) x}{(d+e x)^2 \left (a+b x+c x^2\right )} \, dx}{24 \left (b^2-4 a c\right )^4 \left (c d^2-b d e+a e^2\right )^4}\\ &=-\frac {b c d-b^2 e+2 a c e+c (2 c d-b e) x}{4 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x) \left (a+b x+c x^2\right )^4}-\frac {8 a c e (2 c d-b e)^2-\left (b c d-b^2 e+2 a c e\right ) \left (14 c^2 d^2-5 b^2 e^2-6 c e (b d-3 a e)\right )-c (2 c d-b e) \left (14 c^2 d^2-5 b^2 e^2-2 c e (7 b d-17 a e)\right ) x}{12 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 (d+e x) \left (a+b x+c x^2\right )^3}+\frac {3 a c e (2 c d-b e)^2 \left (14 c^2 d^2-5 b^2 e^2-2 c e (7 b d-17 a e)\right )-\left (b c d-b^2 e+2 a c e\right ) \left (70 c^4 d^4+10 b^4 e^4-2 c^3 d^2 e (49 b d-78 a e)+5 b^2 c e^3 (2 b d-15 a e)+3 c^2 e^2 \left (b^2 d^2-18 a b d e+42 a^2 e^2\right )\right )-5 c (2 c d-b e) \left (14 c^4 d^4+2 b^4 e^4+b^2 c e^3 (5 b d-21 a e)-4 c^3 d^2 e (7 b d-12 a e)+3 c^2 e^2 \left (3 b^2 d^2-16 a b d e+22 a^2 e^2\right )\right ) x}{12 \left (b^2-4 a c\right )^3 \left (c d^2-b d e+a e^2\right )^3 (d+e x) \left (a+b x+c x^2\right )^2}-\frac {5 \left (2 a c e (2 c d-b e)^2 \left (14 c^4 d^4+2 b^4 e^4+b^2 c e^3 (5 b d-21 a e)-4 c^3 d^2 e (7 b d-12 a e)+3 c^2 e^2 \left (3 b^2 d^2-16 a b d e+22 a^2 e^2\right )\right )-\left (b c d-b^2 e+2 a c e\right ) \left (42 c^6 d^6-3 b^6 e^6-2 c^5 d^4 e (49 b d-65 a e)-2 b^4 c e^5 (b d-17 a e)+b^2 c^2 e^4 \left (b^2 d^2+16 a b d e-123 a^2 e^2\right )+c^4 d^2 e^2 \left (55 b^2 d^2-164 a b d e+150 a^2 e^2\right )+6 c^3 e^3 \left (b^3 d^3-4 a b^2 d^2 e-3 a^2 b d e^2+21 a^3 e^3\right )\right )-3 c (2 c d-b e) \left (14 c^6 d^6-b^6 e^6-2 c^5 d^4 e (21 b d-31 a e)-2 b^4 c e^5 (b d-7 a e)-b^2 c^2 e^4 \left (3 b^2 d^2-26 a b d e+69 a^2 e^2\right )+c^4 d^2 e^2 \left (37 b^2 d^2-124 a b d e+114 a^2 e^2\right )-2 c^3 e^3 \left (2 b^3 d^3-18 a b^2 d^2 e+57 a^2 b d e^2-65 a^3 e^3\right )\right ) x\right )}{6 \left (b^2-4 a c\right )^4 \left (c d^2-b d e+a e^2\right )^4 (d+e x) \left (a+b x+c x^2\right )}+\frac {\int \left (\frac {120 e^2 \left (-14 c^8 d^8+b^8 e^8+4 c^7 d^6 e (14 b d-19 a e)-b^6 c e^7 (b d+15 a e)-b^4 c^2 e^6 \left (b^2 d^2-16 a b d e-82 a^2 e^2\right )-c^6 d^4 e^2 \left (79 b^2 d^2-228 a b d e+176 a^2 e^2\right )+c^5 d^2 e^3 \left (41 b^3 d^3-197 a b^2 d^2 e+352 a^2 b d e^2-244 a^3 e^3\right )-b^2 c^3 e^5 \left (b^3 d^3-15 a b^2 d^2 e+95 a^2 b d e^2+187 a^3 e^3\right )-c^4 e^4 \left (b^4 d^4-14 a b^3 d^3 e+81 a^2 b^2 d^2 e^2-244 a^3 b d e^3-126 a^4 e^4\right )\right )}{\left (c d^2-b d e+a e^2\right ) (d+e x)^2}-\frac {120 \left (b^2-4 a c\right )^4 e^{10} (-2 c d+b e)}{\left (c d^2-b d e+a e^2\right )^2 (d+e x)}+\frac {120 \left (14 c^{10} d^{10}+b^{10} e^{10}-10 c^9 d^8 e (7 b d-9 a e)-b^8 c e^9 (2 b d+17 a e)-2 a^4 c^5 e^9 (443 b d+63 a e)+a b^6 c^2 e^9 (34 b d+111 a e)-2 a^2 b^4 c^3 e^9 (111 b d+169 a e)+a^3 b^2 c^4 e^9 (676 b d+443 a e)+9 c^8 d^6 e^2 \left (15 b^2 d^2-40 a b d e+28 a^2 e^2\right )-12 c^7 d^4 e^3 \left (10 b^3 d^3-42 a b^2 d^2 e+63 a^2 b d e^2-35 a^3 e^3\right )+42 c^6 d^2 e^4 \left (b^4 d^4-6 a b^3 d^3 e+15 a^2 b^2 d^2 e^2-20 a^3 b d e^3+15 a^4 e^4\right )-c \left (b^2-4 a c\right )^4 e^9 (2 c d-b e) x\right )}{\left (c d^2-b d e+a e^2\right )^2 \left (a+b x+c x^2\right )}\right ) \, dx}{24 \left (b^2-4 a c\right )^4 \left (c d^2-b d e+a e^2\right )^4}\\ &=\frac {5 e \left (14 c^8 d^8-b^8 e^8-4 c^7 d^6 e (14 b d-19 a e)+b^6 c e^7 (b d+15 a e)+b^4 c^2 e^6 \left (b^2 d^2-16 a b d e-82 a^2 e^2\right )+c^6 d^4 e^2 \left (79 b^2 d^2-228 a b d e+176 a^2 e^2\right )-c^5 d^2 e^3 \left (41 b^3 d^3-197 a b^2 d^2 e+352 a^2 b d e^2-244 a^3 e^3\right )+b^2 c^3 e^5 \left (b^3 d^3-15 a b^2 d^2 e+95 a^2 b d e^2+187 a^3 e^3\right )+c^4 e^4 \left (b^4 d^4-14 a b^3 d^3 e+81 a^2 b^2 d^2 e^2-244 a^3 b d e^3-126 a^4 e^4\right )\right )}{\left (b^2-4 a c\right )^4 \left (c d^2-b d e+a e^2\right )^5 (d+e x)}-\frac {b c d-b^2 e+2 a c e+c (2 c d-b e) x}{4 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x) \left (a+b x+c x^2\right )^4}-\frac {8 a c e (2 c d-b e)^2-\left (b c d-b^2 e+2 a c e\right ) \left (14 c^2 d^2-5 b^2 e^2-6 c e (b d-3 a e)\right )-c (2 c d-b e) \left (14 c^2 d^2-5 b^2 e^2-2 c e (7 b d-17 a e)\right ) x}{12 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 (d+e x) \left (a+b x+c x^2\right )^3}+\frac {3 a c e (2 c d-b e)^2 \left (14 c^2 d^2-5 b^2 e^2-2 c e (7 b d-17 a e)\right )-\left (b c d-b^2 e+2 a c e\right ) \left (70 c^4 d^4+10 b^4 e^4-2 c^3 d^2 e (49 b d-78 a e)+5 b^2 c e^3 (2 b d-15 a e)+3 c^2 e^2 \left (b^2 d^2-18 a b d e+42 a^2 e^2\right )\right )-5 c (2 c d-b e) \left (14 c^4 d^4+2 b^4 e^4+b^2 c e^3 (5 b d-21 a e)-4 c^3 d^2 e (7 b d-12 a e)+3 c^2 e^2 \left (3 b^2 d^2-16 a b d e+22 a^2 e^2\right )\right ) x}{12 \left (b^2-4 a c\right )^3 \left (c d^2-b d e+a e^2\right )^3 (d+e x) \left (a+b x+c x^2\right )^2}-\frac {5 \left (2 a c e (2 c d-b e)^2 \left (14 c^4 d^4+2 b^4 e^4+b^2 c e^3 (5 b d-21 a e)-4 c^3 d^2 e (7 b d-12 a e)+3 c^2 e^2 \left (3 b^2 d^2-16 a b d e+22 a^2 e^2\right )\right )-\left (b c d-b^2 e+2 a c e\right ) \left (42 c^6 d^6-3 b^6 e^6-2 c^5 d^4 e (49 b d-65 a e)-2 b^4 c e^5 (b d-17 a e)+b^2 c^2 e^4 \left (b^2 d^2+16 a b d e-123 a^2 e^2\right )+c^4 d^2 e^2 \left (55 b^2 d^2-164 a b d e+150 a^2 e^2\right )+6 c^3 e^3 \left (b^3 d^3-4 a b^2 d^2 e-3 a^2 b d e^2+21 a^3 e^3\right )\right )-3 c (2 c d-b e) \left (14 c^6 d^6-b^6 e^6-2 c^5 d^4 e (21 b d-31 a e)-2 b^4 c e^5 (b d-7 a e)-b^2 c^2 e^4 \left (3 b^2 d^2-26 a b d e+69 a^2 e^2\right )+c^4 d^2 e^2 \left (37 b^2 d^2-124 a b d e+114 a^2 e^2\right )-2 c^3 e^3 \left (2 b^3 d^3-18 a b^2 d^2 e+57 a^2 b d e^2-65 a^3 e^3\right )\right ) x\right )}{6 \left (b^2-4 a c\right )^4 \left (c d^2-b d e+a e^2\right )^4 (d+e x) \left (a+b x+c x^2\right )}+\frac {5 e^9 (2 c d-b e) \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^6}+\frac {5 \int \frac {14 c^{10} d^{10}+b^{10} e^{10}-10 c^9 d^8 e (7 b d-9 a e)-b^8 c e^9 (2 b d+17 a e)-2 a^4 c^5 e^9 (443 b d+63 a e)+a b^6 c^2 e^9 (34 b d+111 a e)-2 a^2 b^4 c^3 e^9 (111 b d+169 a e)+a^3 b^2 c^4 e^9 (676 b d+443 a e)+9 c^8 d^6 e^2 \left (15 b^2 d^2-40 a b d e+28 a^2 e^2\right )-12 c^7 d^4 e^3 \left (10 b^3 d^3-42 a b^2 d^2 e+63 a^2 b d e^2-35 a^3 e^3\right )+42 c^6 d^2 e^4 \left (b^4 d^4-6 a b^3 d^3 e+15 a^2 b^2 d^2 e^2-20 a^3 b d e^3+15 a^4 e^4\right )-c \left (b^2-4 a c\right )^4 e^9 (2 c d-b e) x}{a+b x+c x^2} \, dx}{\left (b^2-4 a c\right )^4 \left (c d^2-b d e+a e^2\right )^6}\\ &=\frac {5 e \left (14 c^8 d^8-b^8 e^8-4 c^7 d^6 e (14 b d-19 a e)+b^6 c e^7 (b d+15 a e)+b^4 c^2 e^6 \left (b^2 d^2-16 a b d e-82 a^2 e^2\right )+c^6 d^4 e^2 \left (79 b^2 d^2-228 a b d e+176 a^2 e^2\right )-c^5 d^2 e^3 \left (41 b^3 d^3-197 a b^2 d^2 e+352 a^2 b d e^2-244 a^3 e^3\right )+b^2 c^3 e^5 \left (b^3 d^3-15 a b^2 d^2 e+95 a^2 b d e^2+187 a^3 e^3\right )+c^4 e^4 \left (b^4 d^4-14 a b^3 d^3 e+81 a^2 b^2 d^2 e^2-244 a^3 b d e^3-126 a^4 e^4\right )\right )}{\left (b^2-4 a c\right )^4 \left (c d^2-b d e+a e^2\right )^5 (d+e x)}-\frac {b c d-b^2 e+2 a c e+c (2 c d-b e) x}{4 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x) \left (a+b x+c x^2\right )^4}-\frac {8 a c e (2 c d-b e)^2-\left (b c d-b^2 e+2 a c e\right ) \left (14 c^2 d^2-5 b^2 e^2-6 c e (b d-3 a e)\right )-c (2 c d-b e) \left (14 c^2 d^2-5 b^2 e^2-2 c e (7 b d-17 a e)\right ) x}{12 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 (d+e x) \left (a+b x+c x^2\right )^3}+\frac {3 a c e (2 c d-b e)^2 \left (14 c^2 d^2-5 b^2 e^2-2 c e (7 b d-17 a e)\right )-\left (b c d-b^2 e+2 a c e\right ) \left (70 c^4 d^4+10 b^4 e^4-2 c^3 d^2 e (49 b d-78 a e)+5 b^2 c e^3 (2 b d-15 a e)+3 c^2 e^2 \left (b^2 d^2-18 a b d e+42 a^2 e^2\right )\right )-5 c (2 c d-b e) \left (14 c^4 d^4+2 b^4 e^4+b^2 c e^3 (5 b d-21 a e)-4 c^3 d^2 e (7 b d-12 a e)+3 c^2 e^2 \left (3 b^2 d^2-16 a b d e+22 a^2 e^2\right )\right ) x}{12 \left (b^2-4 a c\right )^3 \left (c d^2-b d e+a e^2\right )^3 (d+e x) \left (a+b x+c x^2\right )^2}-\frac {5 \left (2 a c e (2 c d-b e)^2 \left (14 c^4 d^4+2 b^4 e^4+b^2 c e^3 (5 b d-21 a e)-4 c^3 d^2 e (7 b d-12 a e)+3 c^2 e^2 \left (3 b^2 d^2-16 a b d e+22 a^2 e^2\right )\right )-\left (b c d-b^2 e+2 a c e\right ) \left (42 c^6 d^6-3 b^6 e^6-2 c^5 d^4 e (49 b d-65 a e)-2 b^4 c e^5 (b d-17 a e)+b^2 c^2 e^4 \left (b^2 d^2+16 a b d e-123 a^2 e^2\right )+c^4 d^2 e^2 \left (55 b^2 d^2-164 a b d e+150 a^2 e^2\right )+6 c^3 e^3 \left (b^3 d^3-4 a b^2 d^2 e-3 a^2 b d e^2+21 a^3 e^3\right )\right )-3 c (2 c d-b e) \left (14 c^6 d^6-b^6 e^6-2 c^5 d^4 e (21 b d-31 a e)-2 b^4 c e^5 (b d-7 a e)-b^2 c^2 e^4 \left (3 b^2 d^2-26 a b d e+69 a^2 e^2\right )+c^4 d^2 e^2 \left (37 b^2 d^2-124 a b d e+114 a^2 e^2\right )-2 c^3 e^3 \left (2 b^3 d^3-18 a b^2 d^2 e+57 a^2 b d e^2-65 a^3 e^3\right )\right ) x\right )}{6 \left (b^2-4 a c\right )^4 \left (c d^2-b d e+a e^2\right )^4 (d+e x) \left (a+b x+c x^2\right )}+\frac {5 e^9 (2 c d-b e) \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^6}-\frac {\left (5 e^9 (2 c d-b e)\right ) \int \frac {b+2 c x}{a+b x+c x^2} \, dx}{2 \left (c d^2-b d e+a e^2\right )^6}+\frac {\left (5 \left (28 c^{10} d^{10}+b^{10} e^{10}-20 c^9 d^8 e (7 b d-9 a e)-252 a^4 c^5 e^9 (5 b d+a e)+210 a^3 b^2 c^4 e^9 (4 b d+3 a e)-84 a^2 b^4 c^3 e^9 (3 b d+5 a e)+18 a b^6 c^2 e^9 (2 b d+7 a e)-2 b^8 c e^9 (b d+9 a e)+18 c^8 d^6 e^2 \left (15 b^2 d^2-40 a b d e+28 a^2 e^2\right )-24 c^7 d^4 e^3 \left (10 b^3 d^3-42 a b^2 d^2 e+63 a^2 b d e^2-35 a^3 e^3\right )+84 c^6 d^2 e^4 \left (b^4 d^4-6 a b^3 d^3 e+15 a^2 b^2 d^2 e^2-20 a^3 b d e^3+15 a^4 e^4\right )\right )\right ) \int \frac {1}{a+b x+c x^2} \, dx}{2 \left (b^2-4 a c\right )^4 \left (c d^2-b d e+a e^2\right )^6}\\ &=\frac {5 e \left (14 c^8 d^8-b^8 e^8-4 c^7 d^6 e (14 b d-19 a e)+b^6 c e^7 (b d+15 a e)+b^4 c^2 e^6 \left (b^2 d^2-16 a b d e-82 a^2 e^2\right )+c^6 d^4 e^2 \left (79 b^2 d^2-228 a b d e+176 a^2 e^2\right )-c^5 d^2 e^3 \left (41 b^3 d^3-197 a b^2 d^2 e+352 a^2 b d e^2-244 a^3 e^3\right )+b^2 c^3 e^5 \left (b^3 d^3-15 a b^2 d^2 e+95 a^2 b d e^2+187 a^3 e^3\right )+c^4 e^4 \left (b^4 d^4-14 a b^3 d^3 e+81 a^2 b^2 d^2 e^2-244 a^3 b d e^3-126 a^4 e^4\right )\right )}{\left (b^2-4 a c\right )^4 \left (c d^2-b d e+a e^2\right )^5 (d+e x)}-\frac {b c d-b^2 e+2 a c e+c (2 c d-b e) x}{4 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x) \left (a+b x+c x^2\right )^4}-\frac {8 a c e (2 c d-b e)^2-\left (b c d-b^2 e+2 a c e\right ) \left (14 c^2 d^2-5 b^2 e^2-6 c e (b d-3 a e)\right )-c (2 c d-b e) \left (14 c^2 d^2-5 b^2 e^2-2 c e (7 b d-17 a e)\right ) x}{12 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 (d+e x) \left (a+b x+c x^2\right )^3}+\frac {3 a c e (2 c d-b e)^2 \left (14 c^2 d^2-5 b^2 e^2-2 c e (7 b d-17 a e)\right )-\left (b c d-b^2 e+2 a c e\right ) \left (70 c^4 d^4+10 b^4 e^4-2 c^3 d^2 e (49 b d-78 a e)+5 b^2 c e^3 (2 b d-15 a e)+3 c^2 e^2 \left (b^2 d^2-18 a b d e+42 a^2 e^2\right )\right )-5 c (2 c d-b e) \left (14 c^4 d^4+2 b^4 e^4+b^2 c e^3 (5 b d-21 a e)-4 c^3 d^2 e (7 b d-12 a e)+3 c^2 e^2 \left (3 b^2 d^2-16 a b d e+22 a^2 e^2\right )\right ) x}{12 \left (b^2-4 a c\right )^3 \left (c d^2-b d e+a e^2\right )^3 (d+e x) \left (a+b x+c x^2\right )^2}-\frac {5 \left (2 a c e (2 c d-b e)^2 \left (14 c^4 d^4+2 b^4 e^4+b^2 c e^3 (5 b d-21 a e)-4 c^3 d^2 e (7 b d-12 a e)+3 c^2 e^2 \left (3 b^2 d^2-16 a b d e+22 a^2 e^2\right )\right )-\left (b c d-b^2 e+2 a c e\right ) \left (42 c^6 d^6-3 b^6 e^6-2 c^5 d^4 e (49 b d-65 a e)-2 b^4 c e^5 (b d-17 a e)+b^2 c^2 e^4 \left (b^2 d^2+16 a b d e-123 a^2 e^2\right )+c^4 d^2 e^2 \left (55 b^2 d^2-164 a b d e+150 a^2 e^2\right )+6 c^3 e^3 \left (b^3 d^3-4 a b^2 d^2 e-3 a^2 b d e^2+21 a^3 e^3\right )\right )-3 c (2 c d-b e) \left (14 c^6 d^6-b^6 e^6-2 c^5 d^4 e (21 b d-31 a e)-2 b^4 c e^5 (b d-7 a e)-b^2 c^2 e^4 \left (3 b^2 d^2-26 a b d e+69 a^2 e^2\right )+c^4 d^2 e^2 \left (37 b^2 d^2-124 a b d e+114 a^2 e^2\right )-2 c^3 e^3 \left (2 b^3 d^3-18 a b^2 d^2 e+57 a^2 b d e^2-65 a^3 e^3\right )\right ) x\right )}{6 \left (b^2-4 a c\right )^4 \left (c d^2-b d e+a e^2\right )^4 (d+e x) \left (a+b x+c x^2\right )}+\frac {5 e^9 (2 c d-b e) \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^6}-\frac {5 e^9 (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 )^6}-\frac {\left (5 \left (28 c^{10} d^{10}+b^{10} e^{10}-20 c^9 d^8 e (7 b d-9 a e)-252 a^4 c^5 e^9 (5 b d+a e)+210 a^3 b^2 c^4 e^9 (4 b d+3 a e)-84 a^2 b^4 c^3 e^9 (3 b d+5 a e)+18 a b^6 c^2 e^9 (2 b d+7 a e)-2 b^8 c e^9 (b d+9 a e)+18 c^8 d^6 e^2 \left (15 b^2 d^2-40 a b d e+28 a^2 e^2\right )-24 c^7 d^4 e^3 \left (10 b^3 d^3-42 a b^2 d^2 e+63 a^2 b d e^2-35 a^3 e^3\right )+84 c^6 d^2 e^4 \left (b^4 d^4-6 a b^3 d^3 e+15 a^2 b^2 d^2 e^2-20 a^3 b d e^3+15 a^4 e^4\right )\right )\right ) \text {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{\left (b^2-4 a c\right )^4 \left (c d^2-b d e+a e^2\right )^6}\\ &=\frac {5 e \left (14 c^8 d^8-b^8 e^8-4 c^7 d^6 e (14 b d-19 a e)+b^6 c e^7 (b d+15 a e)+b^4 c^2 e^6 \left (b^2 d^2-16 a b d e-82 a^2 e^2\right )+c^6 d^4 e^2 \left (79 b^2 d^2-228 a b d e+176 a^2 e^2\right )-c^5 d^2 e^3 \left (41 b^3 d^3-197 a b^2 d^2 e+352 a^2 b d e^2-244 a^3 e^3\right )+b^2 c^3 e^5 \left (b^3 d^3-15 a b^2 d^2 e+95 a^2 b d e^2+187 a^3 e^3\right )+c^4 e^4 \left (b^4 d^4-14 a b^3 d^3 e+81 a^2 b^2 d^2 e^2-244 a^3 b d e^3-126 a^4 e^4\right )\right )}{\left (b^2-4 a c\right )^4 \left (c d^2-b d e+a e^2\right )^5 (d+e x)}-\frac {b c d-b^2 e+2 a c e+c (2 c d-b e) x}{4 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x) \left (a+b x+c x^2\right )^4}-\frac {8 a c e (2 c d-b e)^2-\left (b c d-b^2 e+2 a c e\right ) \left (14 c^2 d^2-5 b^2 e^2-6 c e (b d-3 a e)\right )-c (2 c d-b e) \left (14 c^2 d^2-5 b^2 e^2-2 c e (7 b d-17 a e)\right ) x}{12 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 (d+e x) \left (a+b x+c x^2\right )^3}+\frac {3 a c e (2 c d-b e)^2 \left (14 c^2 d^2-5 b^2 e^2-2 c e (7 b d-17 a e)\right )-\left (b c d-b^2 e+2 a c e\right ) \left (70 c^4 d^4+10 b^4 e^4-2 c^3 d^2 e (49 b d-78 a e)+5 b^2 c e^3 (2 b d-15 a e)+3 c^2 e^2 \left (b^2 d^2-18 a b d e+42 a^2 e^2\right )\right )-5 c (2 c d-b e) \left (14 c^4 d^4+2 b^4 e^4+b^2 c e^3 (5 b d-21 a e)-4 c^3 d^2 e (7 b d-12 a e)+3 c^2 e^2 \left (3 b^2 d^2-16 a b d e+22 a^2 e^2\right )\right ) x}{12 \left (b^2-4 a c\right )^3 \left (c d^2-b d e+a e^2\right )^3 (d+e x) \left (a+b x+c x^2\right )^2}-\frac {5 \left (2 a c e (2 c d-b e)^2 \left (14 c^4 d^4+2 b^4 e^4+b^2 c e^3 (5 b d-21 a e)-4 c^3 d^2 e (7 b d-12 a e)+3 c^2 e^2 \left (3 b^2 d^2-16 a b d e+22 a^2 e^2\right )\right )-\left (b c d-b^2 e+2 a c e\right ) \left (42 c^6 d^6-3 b^6 e^6-2 c^5 d^4 e (49 b d-65 a e)-2 b^4 c e^5 (b d-17 a e)+b^2 c^2 e^4 \left (b^2 d^2+16 a b d e-123 a^2 e^2\right )+c^4 d^2 e^2 \left (55 b^2 d^2-164 a b d e+150 a^2 e^2\right )+6 c^3 e^3 \left (b^3 d^3-4 a b^2 d^2 e-3 a^2 b d e^2+21 a^3 e^3\right )\right )-3 c (2 c d-b e) \left (14 c^6 d^6-b^6 e^6-2 c^5 d^4 e (21 b d-31 a e)-2 b^4 c e^5 (b d-7 a e)-b^2 c^2 e^4 \left (3 b^2 d^2-26 a b d e+69 a^2 e^2\right )+c^4 d^2 e^2 \left (37 b^2 d^2-124 a b d e+114 a^2 e^2\right )-2 c^3 e^3 \left (2 b^3 d^3-18 a b^2 d^2 e+57 a^2 b d e^2-65 a^3 e^3\right )\right ) x\right )}{6 \left (b^2-4 a c\right )^4 \left (c d^2-b d e+a e^2\right )^4 (d+e x) \left (a+b x+c x^2\right )}-\frac {5 \left (28 c^{10} d^{10}+b^{10} e^{10}-20 c^9 d^8 e (7 b d-9 a e)-252 a^4 c^5 e^9 (5 b d+a e)+210 a^3 b^2 c^4 e^9 (4 b d+3 a e)-84 a^2 b^4 c^3 e^9 (3 b d+5 a e)+18 a b^6 c^2 e^9 (2 b d+7 a e)-2 b^8 c e^9 (b d+9 a e)+18 c^8 d^6 e^2 \left (15 b^2 d^2-40 a b d e+28 a^2 e^2\right )-24 c^7 d^4 e^3 \left (10 b^3 d^3-42 a b^2 d^2 e+63 a^2 b d e^2-35 a^3 e^3\right )+84 c^6 d^2 e^4 \left (b^4 d^4-6 a b^3 d^3 e+15 a^2 b^2 d^2 e^2-20 a^3 b d e^3+15 a^4 e^4\right )\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{9/2} \left (c d^2-e (b d-a e)\right )^6}+\frac {5 e^9 (2 c d-b e) \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^6}-\frac {5 e^9 (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 )^6}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 6.66, size = 2147, normalized size = 1.22 \begin {gather*} \text {Result too large to show} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((d + e*x)^2*(a + b*x + c*x^2)^5),x]

[Out]

-(e^9/((c*d^2 - b*d*e + a*e^2)^5*(d + e*x))) + (b*c^2*d^2 - 2*b^2*c*d*e + 4*a*c^2*d*e + b^3*e^2 - 3*a*b*c*e^2
+ 2*c^3*d^2*x - 2*b*c^2*d*e*x + b^2*c*e^2*x - 2*a*c^2*e^2*x)/(4*(-b^2 + 4*a*c)*(c*d^2 - b*d*e + a*e^2)^2*(a +
b*x + c*x^2)^4) + (14*b*c^4*d^4 - 28*b^2*c^3*d^3*e + 9*b^3*c^2*d^2*e^2 + 48*a*b*c^3*d^2*e^2 + 13*b^4*c*d*e^3 -
112*a*b^2*c^2*d*e^3 + 128*a^2*c^3*d*e^3 - 8*b^5*e^4 + 59*a*b^3*c*e^4 - 94*a^2*b*c^2*e^4 + 28*c^5*d^4*x - 56*b
*c^4*d^3*e*x + 18*b^2*c^3*d^2*e^2*x + 96*a*c^4*d^2*e^2*x + 10*b^3*c^2*d*e^3*x - 96*a*b*c^3*d*e^3*x - 8*b^4*c*e
^4*x + 54*a*b^2*c^2*e^4*x - 60*a^2*c^3*e^4*x)/(12*(-b^2 + 4*a*c)^2*(c*d^2 - b*d*e + a*e^2)^3*(a + b*x + c*x^2)
^3) + (70*b*c^6*d^6 - 210*b^2*c^5*d^5*e + 185*b^3*c^4*d^4*e^2 + 310*a*b*c^5*d^4*e^2 - 20*b^4*c^3*d^3*e^3 - 620
*a*b^2*c^4*d^3*e^3 - 15*b^5*c^2*d^2*e^4 + 180*a*b^3*c^3*d^2*e^4 + 570*a^2*b*c^4*d^2*e^4 - 28*b^6*c*d*e^5 + 346
*a*b^4*c^2*d*e^5 - 1434*a^2*b^2*c^3*d*e^5 + 1152*a^3*c^4*d*e^5 + 18*b^7*e^6 - 206*a*b^5*c*e^6 + 759*a^2*b^3*c^
2*e^6 - 822*a^3*b*c^3*e^6 + 140*c^7*d^6*x - 420*b*c^6*d^5*e*x + 370*b^2*c^5*d^4*e^2*x + 620*a*c^6*d^4*e^2*x -
40*b^3*c^4*d^3*e^3*x - 1240*a*b*c^5*d^3*e^3*x - 30*b^4*c^3*d^2*e^4*x + 360*a*b^2*c^4*d^2*e^4*x + 1140*a^2*c^5*
d^2*e^4*x - 20*b^5*c^2*d*e^5*x + 260*a*b^3*c^3*d*e^5*x - 1140*a^2*b*c^4*d*e^5*x + 18*b^6*c*e^6*x - 196*a*b^4*c
^2*e^6*x + 654*a^2*b^2*c^3*e^6*x - 492*a^3*c^4*e^6*x)/(12*(-b^2 + 4*a*c)^3*(c*d^2 - b*d*e + a*e^2)^4*(a + b*x
+ c*x^2)^2) + (70*b*c^8*d^8 - 280*b^2*c^7*d^7*e + 395*b^3*c^6*d^6*e^2 + 380*a*b*c^7*d^6*e^2 - 205*b^4*c^5*d^5*
e^3 - 1140*a*b^2*c^6*d^5*e^3 + 5*b^5*c^4*d^4*e^4 + 985*a*b^3*c^5*d^4*e^4 + 880*a^2*b*c^6*d^4*e^4 + 5*b^6*c^3*d
^3*e^5 - 70*a*b^4*c^4*d^3*e^5 - 1760*a^2*b^2*c^5*d^3*e^5 + 5*b^7*c^2*d^2*e^6 - 75*a*b^5*c^3*d^2*e^6 + 405*a^2*
b^3*c^4*d^2*e^6 + 1220*a^3*b*c^5*d^2*e^6 + 13*b^8*c*d*e^7 - 208*a*b^6*c^2*d*e^7 + 1243*a^2*b^4*c^3*d*e^7 - 326
8*a^3*b^2*c^4*d*e^7 + 2048*a^4*c^5*d*e^7 - 8*b^9*e^8 + 123*a*b^7*c*e^8 - 698*a^2*b^5*c^2*e^8 + 1703*a^3*b^3*c^
3*e^8 - 1398*a^4*b*c^4*e^8 + 140*c^9*d^8*x - 560*b*c^8*d^7*e*x + 790*b^2*c^7*d^6*e^2*x + 760*a*c^8*d^6*e^2*x -
410*b^3*c^6*d^5*e^3*x - 2280*a*b*c^7*d^5*e^3*x + 10*b^4*c^5*d^4*e^4*x + 1970*a*b^2*c^6*d^4*e^4*x + 1760*a^2*c
^7*d^4*e^4*x + 10*b^5*c^4*d^3*e^5*x - 140*a*b^3*c^5*d^3*e^5*x - 3520*a^2*b*c^6*d^3*e^5*x + 10*b^6*c^3*d^2*e^6*
x - 150*a*b^4*c^4*d^2*e^6*x + 810*a^2*b^2*c^5*d^2*e^6*x + 2440*a^3*c^6*d^2*e^6*x + 10*b^7*c^2*d*e^7*x - 160*a*
b^5*c^3*d*e^7*x + 950*a^2*b^3*c^4*d*e^7*x - 2440*a^3*b*c^5*d*e^7*x - 8*b^8*c*e^8*x + 118*a*b^6*c^2*e^8*x - 628
*a^2*b^4*c^3*e^8*x + 1358*a^3*b^2*c^4*e^8*x - 748*a^4*c^5*e^8*x)/(2*(-b^2 + 4*a*c)^4*(c*d^2 - b*d*e + a*e^2)^5
*(a + b*x + c*x^2)) + (5*(28*c^10*d^10 - 140*b*c^9*d^9*e + 270*b^2*c^8*d^8*e^2 + 180*a*c^9*d^8*e^2 - 240*b^3*c
^7*d^7*e^3 - 720*a*b*c^8*d^7*e^3 + 84*b^4*c^6*d^6*e^4 + 1008*a*b^2*c^7*d^6*e^4 + 504*a^2*c^8*d^6*e^4 - 504*a*b
^3*c^6*d^5*e^5 - 1512*a^2*b*c^7*d^5*e^5 + 1260*a^2*b^2*c^6*d^4*e^6 + 840*a^3*c^7*d^4*e^6 - 1680*a^3*b*c^6*d^3*
e^7 + 1260*a^4*c^6*d^2*e^8 - 2*b^9*c*d*e^9 + 36*a*b^7*c^2*d*e^9 - 252*a^2*b^5*c^3*d*e^9 + 840*a^3*b^3*c^4*d*e^
9 - 1260*a^4*b*c^5*d*e^9 + b^10*e^10 - 18*a*b^8*c*e^10 + 126*a^2*b^6*c^2*e^10 - 420*a^3*b^4*c^3*e^10 + 630*a^4
*b^2*c^4*e^10 - 252*a^5*c^5*e^10)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/((b^2 - 4*a*c)^4*Sqrt[-b^2 + 4*a*c]*
(-(c*d^2) + b*d*e - a*e^2)^6) + (5*(2*c*d*e^9 - b*e^10)*Log[d + e*x])/(c*d^2 - b*d*e + a*e^2)^6 - (5*(2*c*d*e^
9 - b*e^10)*Log[a + b*x + c*x^2])/(2*(c*d^2 - b*d*e + a*e^2)^6)

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(6979\) vs. \(2(1746)=3492\).
time = 1.72, size = 6980, normalized size = 3.96

method	result	size
default	\(\text {Expression too large to display}\)	\(6980\)
risch	\(\text {Expression too large to display}\)	\(34856\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)^2/(c*x^2+b*x+a)^5,x,method=_RETURNVERBOSE)

[Out]

result too large to display

________________________________________________________________________________________

Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^2/(c*x^2+b*x+a)^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` f
or more deta

________________________________________________________________________________________

Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^2/(c*x^2+b*x+a)^5,x, algorithm="fricas")

[Out]

Timed out

________________________________________________________________________________________

Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)**2/(c*x**2+b*x+a)**5,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 8036 vs. \(2 (1758) = 3516\).
time = 1.65, size = 8036, normalized size = 4.56 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^2/(c*x^2+b*x+a)^5,x, algorithm="giac")

[Out]

5*(28*c^10*d^10*e^2 - 140*b*c^9*d^9*e^3 + 270*b^2*c^8*d^8*e^4 + 180*a*c^9*d^8*e^4 - 240*b^3*c^7*d^7*e^5 - 720*
a*b*c^8*d^7*e^5 + 84*b^4*c^6*d^6*e^6 + 1008*a*b^2*c^7*d^6*e^6 + 504*a^2*c^8*d^6*e^6 - 504*a*b^3*c^6*d^5*e^7 -
1512*a^2*b*c^7*d^5*e^7 + 1260*a^2*b^2*c^6*d^4*e^8 + 840*a^3*c^7*d^4*e^8 - 1680*a^3*b*c^6*d^3*e^9 + 1260*a^4*c^
6*d^2*e^10 - 2*b^9*c*d*e^11 + 36*a*b^7*c^2*d*e^11 - 252*a^2*b^5*c^3*d*e^11 + 840*a^3*b^3*c^4*d*e^11 - 1260*a^4
*b*c^5*d*e^11 + b^10*e^12 - 18*a*b^8*c*e^12 + 126*a^2*b^6*c^2*e^12 - 420*a^3*b^4*c^3*e^12 + 630*a^4*b^2*c^4*e^
12 - 252*a^5*c^5*e^12)*arctan((2*c*d - 2*c*d^2/(x*e + d) - b*e + 2*b*d*e/(x*e + d) - 2*a*e^2/(x*e + d))*e^(-1)
/sqrt(-b^2 + 4*a*c))*e^(-2)/((b^8*c^6*d^12 - 16*a*b^6*c^7*d^12 + 96*a^2*b^4*c^8*d^12 - 256*a^3*b^2*c^9*d^12 +
256*a^4*c^10*d^12 - 6*b^9*c^5*d^11*e + 96*a*b^7*c^6*d^11*e - 576*a^2*b^5*c^7*d^11*e + 1536*a^3*b^3*c^8*d^11*e
- 1536*a^4*b*c^9*d^11*e + 15*b^10*c^4*d^10*e^2 - 234*a*b^8*c^5*d^10*e^2 + 1344*a^2*b^6*c^6*d^10*e^2 - 3264*a^3
*b^4*c^7*d^10*e^2 + 2304*a^4*b^2*c^8*d^10*e^2 + 1536*a^5*c^9*d^10*e^2 - 20*b^11*c^3*d^9*e^3 + 290*a*b^9*c^4*d^
9*e^3 - 1440*a^2*b^7*c^5*d^9*e^3 + 2240*a^3*b^5*c^6*d^9*e^3 + 2560*a^4*b^3*c^7*d^9*e^3 - 7680*a^5*b*c^8*d^9*e^
3 + 15*b^12*c^2*d^8*e^4 - 180*a*b^10*c^3*d^8*e^4 + 495*a^2*b^8*c^4*d^8*e^4 + 1680*a^3*b^6*c^5*d^8*e^4 - 10080*
a^4*b^4*c^6*d^8*e^4 + 11520*a^5*b^2*c^7*d^8*e^4 + 3840*a^6*c^8*d^8*e^4 - 6*b^13*c*d^7*e^5 + 36*a*b^11*c^2*d^7*
e^5 + 324*a^2*b^9*c^3*d^7*e^5 - 3264*a^3*b^7*c^4*d^7*e^5 + 8064*a^4*b^5*c^5*d^7*e^5 - 15360*a^6*b*c^7*d^7*e^5
+ b^14*d^6*e^6 + 14*a*b^12*c*d^6*e^6 - 294*a^2*b^10*c^2*d^6*e^6 + 1204*a^3*b^8*c^3*d^6*e^6 + 896*a^4*b^6*c^4*d
^6*e^6 - 13440*a^5*b^4*c^5*d^6*e^6 + 17920*a^6*b^2*c^6*d^6*e^6 + 5120*a^7*c^7*d^6*e^6 - 6*a*b^13*d^5*e^7 + 36*
a^2*b^11*c*d^5*e^7 + 324*a^3*b^9*c^2*d^5*e^7 - 3264*a^4*b^7*c^3*d^5*e^7 + 8064*a^5*b^5*c^4*d^5*e^7 - 15360*a^7
*b*c^6*d^5*e^7 + 15*a^2*b^12*d^4*e^8 - 180*a^3*b^10*c*d^4*e^8 + 495*a^4*b^8*c^2*d^4*e^8 + 1680*a^5*b^6*c^3*d^4
*e^8 - 10080*a^6*b^4*c^4*d^4*e^8 + 11520*a^7*b^2*c^5*d^4*e^8 + 3840*a^8*c^6*d^4*e^8 - 20*a^3*b^11*d^3*e^9 + 29
0*a^4*b^9*c*d^3*e^9 - 1440*a^5*b^7*c^2*d^3*e^9 + 2240*a^6*b^5*c^3*d^3*e^9 + 2560*a^7*b^3*c^4*d^3*e^9 - 7680*a^
8*b*c^5*d^3*e^9 + 15*a^4*b^10*d^2*e^10 - 234*a^5*b^8*c*d^2*e^10 + 1344*a^6*b^6*c^2*d^2*e^10 - 3264*a^7*b^4*c^3
*d^2*e^10 + 2304*a^8*b^2*c^4*d^2*e^10 + 1536*a^9*c^5*d^2*e^10 - 6*a^5*b^9*d*e^11 + 96*a^6*b^7*c*d*e^11 - 576*a
^7*b^5*c^2*d*e^11 + 1536*a^8*b^3*c^3*d*e^11 - 1536*a^9*b*c^4*d*e^11 + a^6*b^8*e^12 - 16*a^7*b^6*c*e^12 + 96*a^
8*b^4*c^2*e^12 - 256*a^9*b^2*c^3*e^12 + 256*a^10*c^4*e^12)*sqrt(-b^2 + 4*a*c)) - 5/2*(2*c*d*e^9 - b*e^10)*log(
c - 2*c*d/(x*e + d) + c*d^2/(x*e + d)^2 + b*e/(x*e + d) - b*d*e/(x*e + d)^2 + a*e^2/(x*e + d)^2)/(c^6*d^12 - 6
*b*c^5*d^11*e + 15*b^2*c^4*d^10*e^2 + 6*a*c^5*d^10*e^2 - 20*b^3*c^3*d^9*e^3 - 30*a*b*c^4*d^9*e^3 + 15*b^4*c^2*
d^8*e^4 + 60*a*b^2*c^3*d^8*e^4 + 15*a^2*c^4*d^8*e^4 - 6*b^5*c*d^7*e^5 - 60*a*b^3*c^2*d^7*e^5 - 60*a^2*b*c^3*d^
7*e^5 + b^6*d^6*e^6 + 30*a*b^4*c*d^6*e^6 + 90*a^2*b^2*c^2*d^6*e^6 + 20*a^3*c^3*d^6*e^6 - 6*a*b^5*d^5*e^7 - 60*
a^2*b^3*c*d^5*e^7 - 60*a^3*b*c^2*d^5*e^7 + 15*a^2*b^4*d^4*e^8 + 60*a^3*b^2*c*d^4*e^8 + 15*a^4*c^2*d^4*e^8 - 20
*a^3*b^3*d^3*e^9 - 30*a^4*b*c*d^3*e^9 + 15*a^4*b^2*d^2*e^10 + 6*a^5*c*d^2*e^10 - 6*a^5*b*d*e^11 + a^6*e^12) -
e^19/((c^5*d^10*e^10 - 5*b*c^4*d^9*e^11 + 10*b^2*c^3*d^8*e^12 + 5*a*c^4*d^8*e^12 - 10*b^3*c^2*d^7*e^13 - 20*a*
b*c^3*d^7*e^13 + 5*b^4*c*d^6*e^14 + 30*a*b^2*c^2*d^6*e^14 + 10*a^2*c^3*d^6*e^14 - b^5*d^5*e^15 - 20*a*b^3*c*d^
5*e^15 - 30*a^2*b*c^2*d^5*e^15 + 5*a*b^4*d^4*e^16 + 30*a^2*b^2*c*d^4*e^16 + 10*a^3*c^2*d^4*e^16 - 10*a^2*b^3*d
^3*e^17 - 20*a^3*b*c*d^3*e^17 + 10*a^3*b^2*d^2*e^18 + 5*a^4*c*d^2*e^18 - 5*a^4*b*d*e^19 + a^5*e^20)*(x*e + d))
+ 1/12*(840*c^13*d^9*e - 3780*b*c^12*d^8*e^2 + 6280*b^2*c^11*d^7*e^3 + 5120*a*c^12*d^7*e^3 - 4340*b^3*c^10*d^
6*e^4 - 17920*a*b*c^11*d^6*e^4 + 738*b^4*c^9*d^5*e^5 + 20136*a*b^2*c^10*d^5*e^5 + 13488*a^2*c^11*d^5*e^5 + 185
*b^5*c^8*d^4*e^6 - 5540*a*b^3*c^9*d^4*e^6 - 33720*a^2*b*c^10*d^4*e^6 + 80*b^6*c^7*d^3*e^7 - 1700*a*b^4*c^8*d^3
*e^7 + 17880*a^2*b^2*c^9*d^3*e^7 + 21120*a^3*c^10*d^3*e^7 + 45*b^7*c^6*d^2*e^8 - 870*a*b^5*c^7*d^2*e^8 + 6900*
a^2*b^3*c^8*d^2*e^8 - 31680*a^3*b*c^9*d^2*e^8 - 202*b^8*c^5*d*e^9 + 3142*a*b^6*c^6*d*e^9 - 17982*a^2*b^4*c^7*d
*e^9 + 43352*a^3*b^2*c^8*d*e^9 - 27512*a^4*c^9*d*e^9 + 77*b^9*c^4*e^10 - 1184*a*b^7*c^5*e^10 + 6717*a^2*b^5*c^
6*e^10 - 16396*a^3*b^3*c^7*e^10 + 13756*a^4*b*c^8*e^10 - 4*(1470*c^13*d^10*e^2 - 7350*b*c^12*d^9*e^3 + 14315*b
^2*c^11*d^8*e^4 + 8890*a*c^12*d^8*e^4 - 13160*b^3*c^10*d^7*e^5 - 35560*a*b*c^11*d^7*e^5 + 5186*b^4*c^9*d^6*e^6
+ 50632*a*b^2*c^10*d^6*e^6 + 23196*a^2*c^11*d^6*e^6 - 368*b^5*c^8*d^5*e^7 - 27436*a*b^3*c^9*d^5*e^7 - 69588*a
^2*b*c^10*d^5*e^7 - 25*b^6*c^7*d^4*e^8 + 2140*a*b^4*c^8*d^4*e^8 + 60030*a^2*b^2*c^9*d^4*e^8 + 35940*a^3*c^10*d
^4*e^8 + 10*b^7*c^6*d^3*e^9 - 40*a*b^5*c^7*d^3*e^9 - 4080*a^2*b^3*c^8*d^3*e^9 - 71880*a^3*b*c^9*d^3*e^9 - 422*
b^8*c^5*d^2*e^10 + 6722*a*b^6*c^6*d^2*e^10 - 40...

________________________________________________________________________________________

Mupad [B]
time = 26.54, size = 2500, normalized size = 1.42 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((d + e*x)^2*(a + b*x + c*x^2)^5),x)

[Out]

((3*b^12*d^4*e^5 - 3072*a^8*c^4*e^9 - 3*b^7*c^5*d^9 - 12*a^4*b^8*e^9 + 50*a*b^5*c^6*d^9 + 1116*a^3*b*c^8*d^9 +
192*a^5*b^6*c*e^9 - 17*a*b^11*d^3*e^6 - 77*a^3*b^9*d*e^8 + 768*a^4*c^8*d^8*e + 15*b^8*c^4*d^8*e - 15*b^11*c*d
^5*e^4 - 326*a^2*b^3*c^7*d^9 - 1152*a^6*b^4*c^2*e^9 + 3072*a^7*b^2*c^3*e^9 + 43*a^2*b^10*d^2*e^7 + 4352*a^5*c^
7*d^6*e^3 + 11008*a^6*c^6*d^4*e^5 + 19712*a^7*c^5*d^2*e^7 - 30*b^9*c^3*d^7*e^2 + 30*b^10*c^2*d^6*e^3 - 2823*a^
2*b^5*c^5*d^7*e^2 + 1905*a^2*b^6*c^4*d^6*e^3 + 150*a^2*b^7*c^3*d^5*e^4 - 684*a^2*b^8*c^2*d^4*e^5 + 7874*a^3*b^
3*c^6*d^7*e^2 - 1758*a^3*b^4*c^5*d^6*e^3 - 5655*a^3*b^5*c^4*d^5*e^4 + 3864*a^3*b^6*c^3*d^4*e^5 + 402*a^3*b^7*c
^2*d^3*e^6 - 17096*a^4*b^2*c^6*d^6*e^3 + 22650*a^4*b^3*c^5*d^5*e^4 - 4572*a^4*b^4*c^4*d^4*e^5 - 7415*a^4*b^5*c
^3*d^3*e^6 + 1721*a^4*b^6*c^2*d^2*e^7 - 22464*a^5*b^2*c^5*d^4*e^5 + 23486*a^5*b^3*c^4*d^3*e^6 + 2818*a^5*b^4*c
^3*d^2*e^7 - 22248*a^6*b^2*c^4*d^2*e^7 - 248*a*b^6*c^5*d^8*e + 20*a*b^10*c*d^4*e^5 + 1184*a^4*b^7*c*d*e^8 - 13
756*a^7*b*c^4*d*e^8 + 475*a*b^7*c^4*d^7*e^2 - 420*a*b^8*c^3*d^6*e^3 + 140*a*b^9*c^2*d^5*e^4 + 1592*a^2*b^4*c^6
*d^8*e + 143*a^2*b^9*c*d^3*e^6 - 5232*a^3*b^2*c^7*d^8*e - 534*a^3*b^8*c*d^2*e^7 + 2712*a^4*b*c^7*d^7*e^2 - 800
*a^5*b*c^6*d^5*e^4 - 6717*a^5*b^5*c^2*d*e^8 - 8472*a^6*b*c^5*d^3*e^6 + 16396*a^6*b^3*c^3*d*e^8)/(12*(256*a^4*c
^9*d^10 + a^5*b^8*e^10 + 256*a^9*c^4*e^10 + b^8*c^5*d^10 - b^13*d^5*e^5 - 16*a*b^6*c^6*d^10 - 16*a^6*b^6*c*e^1
0 + 5*a*b^12*d^4*e^6 - 5*a^4*b^9*d*e^9 - 5*b^9*c^4*d^9*e + 5*b^12*c*d^6*e^4 + 96*a^2*b^4*c^7*d^10 - 256*a^3*b^
2*c^8*d^10 + 96*a^7*b^4*c^2*e^10 - 256*a^8*b^2*c^3*e^10 - 10*a^2*b^11*d^3*e^7 + 10*a^3*b^10*d^2*e^8 + 1280*a^5
*c^8*d^8*e^2 + 2560*a^6*c^7*d^6*e^4 + 2560*a^7*c^6*d^4*e^6 + 1280*a^8*c^5*d^2*e^8 + 10*b^10*c^3*d^8*e^2 - 10*b
^11*c^2*d^7*e^3 + 880*a^2*b^6*c^5*d^8*e^2 - 640*a^2*b^7*c^4*d^7*e^3 + 10*a^2*b^8*c^3*d^6*e^4 + 194*a^2*b^9*c^2
*d^5*e^5 - 2080*a^3*b^4*c^6*d^8*e^2 + 640*a^3*b^5*c^5*d^7*e^3 + 1440*a^3*b^6*c^4*d^6*e^4 - 1184*a^3*b^7*c^3*d^
5*e^5 + 10*a^3*b^8*c^2*d^4*e^6 + 1280*a^4*b^2*c^7*d^8*e^2 + 2560*a^4*b^3*c^6*d^7*e^3 - 5440*a^4*b^4*c^5*d^6*e^
4 + 1984*a^4*b^5*c^4*d^5*e^5 + 1440*a^4*b^6*c^3*d^4*e^6 - 640*a^4*b^7*c^2*d^3*e^7 + 5120*a^5*b^2*c^6*d^6*e^4 +
2560*a^5*b^3*c^5*d^5*e^5 - 5440*a^5*b^4*c^4*d^4*e^6 + 640*a^5*b^5*c^3*d^3*e^7 + 880*a^5*b^6*c^2*d^2*e^8 + 512
0*a^6*b^2*c^5*d^4*e^6 + 2560*a^6*b^3*c^4*d^3*e^7 - 2080*a^6*b^4*c^3*d^2*e^8 + 1280*a^7*b^2*c^4*d^2*e^8 + 80*a*
b^7*c^5*d^9*e - 4*a*b^11*c*d^5*e^5 - 1280*a^4*b*c^8*d^9*e + 80*a^5*b^7*c*d*e^9 - 1280*a^8*b*c^4*d*e^9 - 155*a*
b^8*c^4*d^8*e^2 + 140*a*b^9*c^3*d^7*e^3 - 50*a*b^10*c^2*d^6*e^4 - 480*a^2*b^5*c^6*d^9*e - 50*a^2*b^10*c*d^4*e^
6 + 1280*a^3*b^3*c^7*d^9*e + 140*a^3*b^9*c*d^3*e^7 - 155*a^4*b^8*c*d^2*e^8 - 5120*a^5*b*c^7*d^7*e^3 - 7680*a^6
*b*c^6*d^5*e^5 - 480*a^6*b^5*c^2*d*e^9 - 5120*a^7*b*c^5*d^3*e^7 + 1280*a^7*b^3*c^3*d*e^9)) + (5*x^8*(14*c^12*d
^8*e - 126*a^4*c^8*e^9 - b^8*c^4*e^9 + 15*a*b^6*c^5*e^9 + 76*a*c^11*d^6*e^3 - 56*b*c^11*d^7*e^2 + b^7*c^5*d*e^
8 - 82*a^2*b^4*c^6*e^9 + 187*a^3*b^2*c^7*e^9 + 176*a^2*c^10*d^4*e^5 + 244*a^3*c^9*d^2*e^7 + 79*b^2*c^10*d^6*e^
3 - 41*b^3*c^9*d^5*e^4 + b^4*c^8*d^4*e^5 + b^5*c^7*d^3*e^6 + b^6*c^6*d^2*e^7 + 81*a^2*b^2*c^8*d^2*e^7 - 228*a*
b*c^10*d^5*e^4 - 16*a*b^5*c^6*d*e^8 - 244*a^3*b*c^8*d*e^8 + 197*a*b^2*c^9*d^4*e^5 - 14*a*b^3*c^8*d^3*e^6 - 15*
a*b^4*c^7*d^2*e^7 - 352*a^2*b*c^9*d^3*e^6 + 95*a^2*b^3*c^7*d*e^8))/(256*a^4*c^9*d^10 + a^5*b^8*e^10 + 256*a^9*
c^4*e^10 + b^8*c^5*d^10 - b^13*d^5*e^5 - 16*a*b^6*c^6*d^10 - 16*a^6*b^6*c*e^10 + 5*a*b^12*d^4*e^6 - 5*a^4*b^9*
d*e^9 - 5*b^9*c^4*d^9*e + 5*b^12*c*d^6*e^4 + 96*a^2*b^4*c^7*d^10 - 256*a^3*b^2*c^8*d^10 + 96*a^7*b^4*c^2*e^10
- 256*a^8*b^2*c^3*e^10 - 10*a^2*b^11*d^3*e^7 + 10*a^3*b^10*d^2*e^8 + 1280*a^5*c^8*d^8*e^2 + 2560*a^6*c^7*d^6*e
^4 + 2560*a^7*c^6*d^4*e^6 + 1280*a^8*c^5*d^2*e^8 + 10*b^10*c^3*d^8*e^2 - 10*b^11*c^2*d^7*e^3 + 880*a^2*b^6*c^5
*d^8*e^2 - 640*a^2*b^7*c^4*d^7*e^3 + 10*a^2*b^8*c^3*d^6*e^4 + 194*a^2*b^9*c^2*d^5*e^5 - 2080*a^3*b^4*c^6*d^8*e
^2 + 640*a^3*b^5*c^5*d^7*e^3 + 1440*a^3*b^6*c^4*d^6*e^4 - 1184*a^3*b^7*c^3*d^5*e^5 + 10*a^3*b^8*c^2*d^4*e^6 +
1280*a^4*b^2*c^7*d^8*e^2 + 2560*a^4*b^3*c^6*d^7*e^3 - 5440*a^4*b^4*c^5*d^6*e^4 + 1984*a^4*b^5*c^4*d^5*e^5 + 14
40*a^4*b^6*c^3*d^4*e^6 - 640*a^4*b^7*c^2*d^3*e^7 + 5120*a^5*b^2*c^6*d^6*e^4 + 2560*a^5*b^3*c^5*d^5*e^5 - 5440*
a^5*b^4*c^4*d^4*e^6 + 640*a^5*b^5*c^3*d^3*e^7 + 880*a^5*b^6*c^2*d^2*e^8 + 5120*a^6*b^2*c^5*d^4*e^6 + 2560*a^6*
b^3*c^4*d^3*e^7 - 2080*a^6*b^4*c^3*d^2*e^8 + 1280*a^7*b^2*c^4*d^2*e^8 + 80*a*b^7*c^5*d^9*e - 4*a*b^11*c*d^5*e^
5 - 1280*a^4*b*c^8*d^9*e + 80*a^5*b^7*c*d*e^9 - 1280*a^8*b*c^4*d*e^9 - 155*a*b^8*c^4*d^8*e^2 + 140*a*b^9*c^3*d
^7*e^3 - 50*a*b^10*c^2*d^6*e^4 - 480*a^2*b^5*c^6*d^9*e - 50*a^2*b^10*c*d^4*e^6 + 1280*a^3*b^3*c^7*d^9*e + 140*
a^3*b^9*c*d^3*e^7 - 155*a^4*b^8*c*d^2*e^8 - 5120*a^5*b*c^7*d^7*e^3 - 7680*a^6*b*c^6*d^5*e^5 - 480*a^6*b^5*c^2*
d*e^9 - 5120*a^7*b*c^5*d^3*e^7 + 1280*a^7*b^3*c^3*d*e^9) + (x^3*(2044*a^2*c^10*d^9 - 15*b^12*d*e^8 - 105*a*b^1
1*e^9 + 84*b^4*c^8*d^9 + 2828*a*b^2*c^9*d^9 + 1...

________________________________________________________________________________________