Integrand size = 20, antiderivative size = 1324 \[ \int \frac {1}{(d+e x) \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 ) \left (a+b x+c x^2\right )^4}-\frac {7 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^2-4 b^2 e^2-c e (7 b d-16 a e)\right )-2 c (2 c d-b e) \left (7 c^2 d^2-2 b^2 e^2-c e (7 b d-15 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 \left (a+b x+c x^2\right )^3}+\frac {5 a c e (2 c d-b e)^2 \left (7 c^2 d^2-2 b^2 e^2-c e (7 b d-15 a e)\right )-\left (b c d-b^2 e+2 a c e\right ) \left (70 c^4 d^4+6 b^4 e^4+2 b^2 c e^3 (5 b d-24 a e)-15 c^3 d^2 e (7 b d-10 a e)+3 c^2 e^2 \left (5 b^2 d^2-25 a b d e+32 a^2 e^2\right )\right )-2 c (2 c d-b e) \left (35 c^4 d^4+3 b^4 e^4+2 b^2 c e^3 (5 b d-17 a e)-10 c^3 d^2 e (7 b d-11 a e)+c^2 e^2 \left (25 b^2 d^2-110 a b d e+123 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 \left (a+b x+c x^2\right )^2}+\frac {b^7 c d e^6+2 b^8 e^7+256 a^4 c^4 e^7+b^6 c e^5 \left (c d^2-31 a e^2\right )+b^5 c^2 d e^4 \left (c d^2-14 a e^2\right )-b^4 c^2 e^3 \left (125 c^2 d^4+13 a c d^2 e^2-178 a^2 e^4\right )+b^3 c^3 d e^2 \left (295 c^2 d^4+492 a c d^2 e^2+69 a^2 e^4\right )+2 b c^4 d \left (35 c^3 d^6+145 a c^2 d^4 e^2+233 a^2 c d^2 e^4+187 a^3 e^6\right )-b^2 c^3 e \left (245 c^3 d^6+725 a c^2 d^4 e^2+699 a^2 c d^2 e^4+443 a^3 e^6\right )+2 c (2 c d-b e) \left (35 c^6 d^6-b^6 e^6-5 c^5 d^4 e (21 b d-29 a e)-3 b^4 c e^5 (b d-5 a e)-b^2 c^2 e^4 \left (7 b^2 d^2-44 a b d e+82 a^2 e^2\right )+c^4 d^2 e^2 \left (95 b^2 d^2-290 a b d e+233 a^2 e^2\right )-c^3 e^3 \left (15 b^3 d^3-101 a b^2 d^2 e+233 a^2 b d e^2-187 a^3 e^3\right )\right ) x}{2 \left (b^2-4 a c\right )^4 \left (c d^2-b d e+a e^2\right )^4 \left (a+b x+c x^2\right )}-\frac {\left (140 c^9 d^9-b^9 e^9+18 a b^7 c e^9-126 a^2 b^5 c^2 e^9+420 a^3 b^3 c^3 e^9-630 a^4 b c^4 e^9-90 c^8 d^7 e (7 b d-8 a e)+72 c^7 d^5 e^2 \left (15 b^2 d^2-35 a b d e+21 a^2 e^2\right )-84 c^6 d^3 e^3 \left (10 b^3 d^3-36 a b^2 d^2 e+45 a^2 b d e^2-20 a^3 e^3\right )+252 c^5 d e^4 \left (b^4 d^4-5 a b^3 d^3 e+10 a^2 b^2 d^2 e^2-10 a^3 b d e^3+5 a^4 e^4\right )\right ) \text {arctanh}\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 )^5}+\frac {e^9 \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^5}-\frac {e^9 \log \left (a+b x+c x^2\right )}{2 \left (c d^2-b d e+a e^2\right )^5} \]
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)/(c*x^2+b*x+a)^4+1/12*(-7*a*c*e*(-b*e+2*c*d)^2+(2*a*c*e-b^2*e+b*c*d)*(14 *c^2*d^2-4*b^2*e^2-c*e*(-16*a*e+7*b*d))+2*c*(-b*e+2*c*d)*(7*c^2*d^2-2*b^2* e^2-c*e*(-15*a*e+7*b*d))*x)/(-4*a*c+b^2)^2/(a*e^2-b*d*e+c*d^2)^2/(c*x^2+b* x+a)^3+1/12*(5*a*c*e*(-b*e+2*c*d)^2*(7*c^2*d^2-2*b^2*e^2-c*e*(-15*a*e+7*b* d))-(2*a*c*e-b^2*e+b*c*d)*(70*c^4*d^4+6*b^4*e^4+2*b^2*c*e^3*(-24*a*e+5*b*d )-15*c^3*d^2*e*(-10*a*e+7*b*d)+3*c^2*e^2*(32*a^2*e^2-25*a*b*d*e+5*b^2*d^2) )-2*c*(-b*e+2*c*d)*(35*c^4*d^4+3*b^4*e^4+2*b^2*c*e^3*(-17*a*e+5*b*d)-10*c^ 3*d^2*e*(-11*a*e+7*b*d)+c^2*e^2*(123*a^2*e^2-110*a*b*d*e+25*b^2*d^2))*x)/( -4*a*c+b^2)^3/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2+1/2*(b^7*c*d*e^6+2*b^8 *e^7+256*a^4*c^4*e^7+b^6*c*e^5*(-31*a*e^2+c*d^2)+b^5*c^2*d*e^4*(-14*a*e^2+ c*d^2)-b^4*c^2*e^3*(-178*a^2*e^4+13*a*c*d^2*e^2+125*c^2*d^4)+b^3*c^3*d*e^2 *(69*a^2*e^4+492*a*c*d^2*e^2+295*c^2*d^4)+2*b*c^4*d*(187*a^3*e^6+233*a^2*c *d^2*e^4+145*a*c^2*d^4*e^2+35*c^3*d^6)-b^2*c^3*e*(443*a^3*e^6+699*a^2*c*d^ 2*e^4+725*a*c^2*d^4*e^2+245*c^3*d^6)+2*c*(-b*e+2*c*d)*(35*c^6*d^6-b^6*e^6- 5*c^5*d^4*e*(-29*a*e+21*b*d)-3*b^4*c*e^5*(-5*a*e+b*d)-b^2*c^2*e^4*(82*a^2* e^2-44*a*b*d*e+7*b^2*d^2)+c^4*d^2*e^2*(233*a^2*e^2-290*a*b*d*e+95*b^2*d^2) -c^3*e^3*(-187*a^3*e^3+233*a^2*b*d*e^2-101*a*b^2*d^2*e+15*b^3*d^3))*x)/(-4 *a*c+b^2)^4/(a*e^2-b*d*e+c*d^2)^4/(c*x^2+b*x+a)-(140*c^9*d^9-b^9*e^9+18*a* b^7*c*e^9-126*a^2*b^5*c^2*e^9+420*a^3*b^3*c^3*e^9-630*a^4*b*c^4*e^9-90*...
Time = 4.43 (sec) , antiderivative size = 1289, normalized size of antiderivative = 0.97 \[ \int \frac {1}{(d+e x) \left (a+b x+c x^2\right )^5} \, dx=\frac {1}{12} \left (\frac {-3 b^2 e+6 c (a e+c d x)+3 b c (d-e x)}{\left (b^2-4 a c\right ) \left (-c d^2+e (b d-a e)\right ) (a+x (b+c x))^4}+\frac {4 b^4 e^3+b^3 c e^2 (3 d+4 e x)+4 c^2 \left (8 a^2 e^3+7 c^2 d^3 x+15 a c d e^2 x\right )+2 b c^2 \left (7 c d^2 (d-3 e x)+15 a e^2 (d-e x)\right )+b^2 c e \left (-31 a e^2+3 c d (-7 d+2 e x)\right )}{\left (b^2-4 a c\right )^2 \left (c d^2+e (-b d+a e)\right )^2 (a+x (b+c x))^3}+\frac {-6 b^6 e^5-2 b^5 c e^4 (2 d+3 e x)+4 c^3 \left (48 a^3 e^5+35 c^3 d^5 x+110 a c^2 d^3 e^2 x+123 a^2 c d e^4 x\right )+2 b c^3 \left (35 c^2 d^4 (d-5 e x)+110 a c d^2 e^2 (d-3 e x)+123 a^2 e^4 (d-e x)\right )+b^4 c e^3 \left (70 a e^2-c d (5 d+8 e x)\right )+b^2 c^2 e \left (-267 a^2 e^4-5 c^2 d^3 (35 d-48 e x)+6 a c d e^2 (-55 d+14 e x)\right )+2 b^3 c^2 e^2 \left (5 c d^2 (12 d-e x)+a e^2 (21 d+34 e x)\right )}{\left (b^2-4 a c\right )^3 \left (-c d^2+e (b d-a e)\right )^3 (a+x (b+c x))^2}+\frac {6 \left (2 b^8 e^7+b^7 c e^6 (d+2 e x)+4 c^4 \left (64 a^4 e^7+35 c^4 d^7 x+145 a c^3 d^5 e^2 x+233 a^2 c^2 d^3 e^4 x+187 a^3 c d e^6 x\right )+2 b c^4 \left (35 c^3 d^6 (d-7 e x)+145 a c^2 d^4 e^2 (d-5 e x)+233 a^2 c d^2 e^4 (d-3 e x)+187 a^3 e^6 (d-e x)\right )+b^6 c e^5 \left (-31 a e^2+c d (d+2 e x)\right )+b^5 c^2 e^4 \left (c d^2 (d+2 e x)-2 a e^2 (7 d+15 e x)\right )+b^4 c^2 e^3 \left (178 a^2 e^4+c^2 d^3 (-125 d+2 e x)-a c d e^2 (13 d+28 e x)\right )+b^3 c^3 e^2 \left (5 c^2 d^4 (59 d-50 e x)+2 a c d^2 e^2 (246 d-13 e x)+a^2 e^4 (69 d+164 e x)\right )+b^2 c^3 e \left (-443 a^3 e^6+3 a^2 c d e^4 (-233 d+46 e x)+5 c^3 d^5 (-49 d+118 e x)+a c^2 d^3 e^2 (-725 d+984 e x)\right )\right )}{\left (b^2-4 a c\right )^4 \left (c d^2+e (-b d+a e)\right )^4 (a+x (b+c x))}-\frac {12 \left (140 c^9 d^9-b^9 e^9+18 a b^7 c e^9-126 a^2 b^5 c^2 e^9+420 a^3 b^3 c^3 e^9-630 a^4 b c^4 e^9-90 c^8 d^7 e (7 b d-8 a e)+72 c^7 d^5 e^2 \left (15 b^2 d^2-35 a b d e+21 a^2 e^2\right )-84 c^6 d^3 e^3 \left (10 b^3 d^3-36 a b^2 d^2 e+45 a^2 b d e^2-20 a^3 e^3\right )+252 c^5 d e^4 \left (b^4 d^4-5 a b^3 d^3 e+10 a^2 b^2 d^2 e^2-10 a^3 b d e^3+5 a^4 e^4\right )\right ) \arctan \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 )^5}+\frac {12 e^9 \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^5}-\frac {6 e^9 \log (a+x (b+c x))}{\left (c d^2+e (-b d+a e)\right )^5}\right ) \]
((-3*b^2*e + 6*c*(a*e + c*d*x) + 3*b*c*(d - e*x))/((b^2 - 4*a*c)*(-(c*d^2) + e*(b*d - a*e))*(a + x*(b + c*x))^4) + (4*b^4*e^3 + b^3*c*e^2*(3*d + 4*e *x) + 4*c^2*(8*a^2*e^3 + 7*c^2*d^3*x + 15*a*c*d*e^2*x) + 2*b*c^2*(7*c*d^2* (d - 3*e*x) + 15*a*e^2*(d - e*x)) + b^2*c*e*(-31*a*e^2 + 3*c*d*(-7*d + 2*e *x)))/((b^2 - 4*a*c)^2*(c*d^2 + e*(-(b*d) + a*e))^2*(a + x*(b + c*x))^3) + (-6*b^6*e^5 - 2*b^5*c*e^4*(2*d + 3*e*x) + 4*c^3*(48*a^3*e^5 + 35*c^3*d^5* x + 110*a*c^2*d^3*e^2*x + 123*a^2*c*d*e^4*x) + 2*b*c^3*(35*c^2*d^4*(d - 5* e*x) + 110*a*c*d^2*e^2*(d - 3*e*x) + 123*a^2*e^4*(d - e*x)) + b^4*c*e^3*(7 0*a*e^2 - c*d*(5*d + 8*e*x)) + b^2*c^2*e*(-267*a^2*e^4 - 5*c^2*d^3*(35*d - 48*e*x) + 6*a*c*d*e^2*(-55*d + 14*e*x)) + 2*b^3*c^2*e^2*(5*c*d^2*(12*d - e*x) + a*e^2*(21*d + 34*e*x)))/((b^2 - 4*a*c)^3*(-(c*d^2) + e*(b*d - a*e)) ^3*(a + x*(b + c*x))^2) + (6*(2*b^8*e^7 + b^7*c*e^6*(d + 2*e*x) + 4*c^4*(6 4*a^4*e^7 + 35*c^4*d^7*x + 145*a*c^3*d^5*e^2*x + 233*a^2*c^2*d^3*e^4*x + 1 87*a^3*c*d*e^6*x) + 2*b*c^4*(35*c^3*d^6*(d - 7*e*x) + 145*a*c^2*d^4*e^2*(d - 5*e*x) + 233*a^2*c*d^2*e^4*(d - 3*e*x) + 187*a^3*e^6*(d - e*x)) + b^6*c *e^5*(-31*a*e^2 + c*d*(d + 2*e*x)) + b^5*c^2*e^4*(c*d^2*(d + 2*e*x) - 2*a* e^2*(7*d + 15*e*x)) + b^4*c^2*e^3*(178*a^2*e^4 + c^2*d^3*(-125*d + 2*e*x) - a*c*d*e^2*(13*d + 28*e*x)) + b^3*c^3*e^2*(5*c^2*d^4*(59*d - 50*e*x) + 2* a*c*d^2*e^2*(246*d - 13*e*x) + a^2*e^4*(69*d + 164*e*x)) + b^2*c^3*e*(-443 *a^3*e^6 + 3*a^2*c*d*e^4*(-233*d + 46*e*x) + 5*c^3*d^5*(-49*d + 118*e*x...
Time = 6.09 (sec) , antiderivative size = 1470, normalized size of antiderivative = 1.11, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {1165, 1235, 27, 1235, 25, 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.
| \(\displaystyle \int \frac {1}{(d+e x) \left (a+b x+c x^2\right )^5} \, dx\) |
| \(\Big \downarrow \) 1165 |
| \(\displaystyle -\frac {\int \frac {14 c^2 d^2-4 b^2 e^2-c e (7 b d-16 a e)+7 c e (2 c d-b e) x}{(d+e x) \left (c x^2+b x+a\right )^4}dx}{4 \left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^4 \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 1235 |
| \(\displaystyle -\frac {\frac {-2 c x (2 c d-b e) \left (-c e (7 b d-15 a e)-2 b^2 e^2+7 c^2 d^2\right )-\left (2 a c e+b^2 (-e)+b c d\right ) \left (-c e (7 b d-16 a e)-4 b^2 e^2+14 c^2 d^2\right )+7 a c e (2 c d-b e)^2}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3 \left (a e^2-b d e+c d^2\right )}-\frac {\int \frac {2 \left (70 c^4 d^4-15 c^3 e (7 b d-10 a e) d^2+6 b^4 e^4+2 b^2 c e^3 (5 b d-24 a e)+3 c^2 e^2 \left (5 b^2 d^2-25 a b e d+32 a^2 e^2\right )+5 c e (2 c d-b e) \left (7 c^2 d^2-2 b^2 e^2-c e (7 b d-15 a e)\right ) x\right )}{(d+e x) \left (c x^2+b x+a\right )^3}dx}{3 \left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}}{4 \left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^4 \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {-2 c x (2 c d-b e) \left (-c e (7 b d-15 a e)-2 b^2 e^2+7 c^2 d^2\right )-\left (2 a c e+b^2 (-e)+b c d\right ) \left (-c e (7 b d-16 a e)-4 b^2 e^2+14 c^2 d^2\right )+7 a c e (2 c d-b e)^2}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3 \left (a e^2-b d e+c d^2\right )}-\frac {2 \int \frac {70 c^4 d^4-15 c^3 e (7 b d-10 a e) d^2+6 b^4 e^4+2 b^2 c e^3 (5 b d-24 a e)+3 c^2 e^2 \left (5 b^2 d^2-25 a b e d+32 a^2 e^2\right )+5 c e (2 c d-b e) \left (7 c^2 d^2-2 b^2 e^2-c e (7 b d-15 a e)\right ) x}{(d+e x) \left (c x^2+b x+a\right )^3}dx}{3 \left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}}{4 \left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^4 \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 1235 |
| \(\displaystyle -\frac {-e b^2+c d b+2 a c e+c (2 c d-b e) x}{4 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) \left (c x^2+b x+a\right )^4}-\frac {\frac {7 a c e (2 c d-b e)^2-2 c \left (7 c^2 d^2-2 b^2 e^2-c e (7 b d-15 a e)\right ) x (2 c d-b e)-\left (-e b^2+c d b+2 a c e\right ) \left (14 c^2 d^2-4 b^2 e^2-c e (7 b d-16 a e)\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) \left (c x^2+b x+a\right )^3}-\frac {2 \left (\frac {5 a c e \left (7 c^2 d^2-2 b^2 e^2-c e (7 b d-15 a e)\right ) (2 c d-b e)^2-2 c \left (35 c^4 d^4-10 c^3 e (7 b d-11 a e) d^2+3 b^4 e^4+2 b^2 c e^3 (5 b d-17 a e)+c^2 e^2 \left (25 b^2 d^2-110 a b e d+123 a^2 e^2\right )\right ) x (2 c d-b e)-\left (-e b^2+c d b+2 a c e\right ) \left (70 c^4 d^4-15 c^3 e (7 b d-10 a e) d^2+6 b^4 e^4+2 b^2 c e^3 (5 b d-24 a e)+3 c^2 e^2 \left (5 b^2 d^2-25 a b e d+32 a^2 e^2\right )\right )}{2 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) \left (c x^2+b x+a\right )^2}-\frac {\int -\frac {5 c d e (2 c d-b e) \left (-2 e b^2+3 c d b+2 a c e\right ) \left (7 c^2 d^2-2 b^2 e^2-c e (7 b d-15 a e)\right )-\left (6 c^2 d^2-2 b^2 e^2-c e (3 b d-8 a e)\right ) \left (70 c^4 d^4-15 c^3 e (7 b d-10 a e) d^2+6 b^4 e^4+2 b^2 c e^3 (5 b d-24 a e)+3 c^2 e^2 \left (5 b^2 d^2-25 a b e d+32 a^2 e^2\right )\right )-6 c e (2 c d-b e) \left (35 c^4 d^4-10 c^3 e (7 b d-11 a e) d^2+3 b^4 e^4+2 b^2 c e^3 (5 b d-17 a e)+c^2 e^2 \left (25 b^2 d^2-110 a b e d+123 a^2 e^2\right )\right ) x}{(d+e x) \left (c x^2+b x+a\right )^2}dx}{2 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )}\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )}}{4 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {-e b^2+c d b+2 a c e+c (2 c d-b e) x}{4 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) \left (c x^2+b x+a\right )^4}-\frac {\frac {7 a c e (2 c d-b e)^2-2 c \left (7 c^2 d^2-2 b^2 e^2-c e (7 b d-15 a e)\right ) x (2 c d-b e)-\left (-e b^2+c d b+2 a c e\right ) \left (14 c^2 d^2-4 b^2 e^2-c e (7 b d-16 a e)\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) \left (c x^2+b x+a\right )^3}-\frac {2 \left (\frac {5 a c e \left (7 c^2 d^2-2 b^2 e^2-c e (7 b d-15 a e)\right ) (2 c d-b e)^2-2 c \left (35 c^4 d^4-10 c^3 e (7 b d-11 a e) d^2+3 b^4 e^4+2 b^2 c e^3 (5 b d-17 a e)+c^2 e^2 \left (25 b^2 d^2-110 a b e d+123 a^2 e^2\right )\right ) x (2 c d-b e)-\left (-e b^2+c d b+2 a c e\right ) \left (70 c^4 d^4-15 c^3 e (7 b d-10 a e) d^2+6 b^4 e^4+2 b^2 c e^3 (5 b d-24 a e)+3 c^2 e^2 \left (5 b^2 d^2-25 a b e d+32 a^2 e^2\right )\right )}{2 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) \left (c x^2+b x+a\right )^2}+\frac {\int -\frac {6 \left (70 c^6 d^6-5 c^5 e (35 b d-44 a e) d^4+6 c^4 e^2 \left (20 b^2 d^2-55 a b e d+41 a^2 e^2\right ) d^2-2 b^6 e^6-3 b^4 c e^5 (b d-8 a e)-2 b^2 c^2 e^4 \left (2 b^2 d^2-17 a b e d+48 a^2 e^2\right )-c^3 e^3 \left (5 b^3 d^3-42 a b^2 e d^2+123 a^2 b e^2 d-128 a^3 e^3\right )+c e (2 c d-b e) \left (35 c^4 d^4-10 c^3 e (7 b d-11 a e) d^2+3 b^4 e^4+2 b^2 c e^3 (5 b d-17 a e)+c^2 e^2 \left (25 b^2 d^2-110 a b e d+123 a^2 e^2\right )\right ) x\right )}{(d+e x) \left (c x^2+b x+a\right )^2}dx}{2 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )}\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )}}{4 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-e b^2+c d b+2 a c e+c (2 c d-b e) x}{4 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) \left (c x^2+b x+a\right )^4}-\frac {\frac {7 a c e (2 c d-b e)^2-2 c \left (7 c^2 d^2-2 b^2 e^2-c e (7 b d-15 a e)\right ) x (2 c d-b e)-\left (-e b^2+c d b+2 a c e\right ) \left (14 c^2 d^2-4 b^2 e^2-c e (7 b d-16 a e)\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) \left (c x^2+b x+a\right )^3}-\frac {2 \left (\frac {5 a c e \left (7 c^2 d^2-2 b^2 e^2-c e (7 b d-15 a e)\right ) (2 c d-b e)^2-2 c \left (35 c^4 d^4-10 c^3 e (7 b d-11 a e) d^2+3 b^4 e^4+2 b^2 c e^3 (5 b d-17 a e)+c^2 e^2 \left (25 b^2 d^2-110 a b e d+123 a^2 e^2\right )\right ) x (2 c d-b e)-\left (-e b^2+c d b+2 a c e\right ) \left (70 c^4 d^4-15 c^3 e (7 b d-10 a e) d^2+6 b^4 e^4+2 b^2 c e^3 (5 b d-24 a e)+3 c^2 e^2 \left (5 b^2 d^2-25 a b e d+32 a^2 e^2\right )\right )}{2 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) \left (c x^2+b x+a\right )^2}-\frac {3 \int \frac {70 c^6 d^6-5 c^5 e (35 b d-44 a e) d^4+6 c^4 e^2 \left (20 b^2 d^2-55 a b e d+41 a^2 e^2\right ) d^2-2 b^6 e^6-3 b^4 c e^5 (b d-8 a e)-2 b^2 c^2 e^4 \left (2 b^2 d^2-17 a b e d+48 a^2 e^2\right )-c^3 e^3 \left (5 b^3 d^3-42 a b^2 e d^2+123 a^2 b e^2 d-128 a^3 e^3\right )+c e (2 c d-b e) \left (35 c^4 d^4-10 c^3 e (7 b d-11 a e) d^2+3 b^4 e^4+2 b^2 c e^3 (5 b d-17 a e)+c^2 e^2 \left (25 b^2 d^2-110 a b e d+123 a^2 e^2\right )\right ) x}{(d+e x) \left (c x^2+b x+a\right )^2}dx}{\left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )}\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )}}{4 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )}\) |
| \(\Big \downarrow \) 1235 |
| \(\displaystyle -\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 ) \left (c x^2+b x+a\right )^4}-\frac {\frac {7 a c e (2 c d-b e)^2-2 c \left (7 c^2 d^2-2 b^2 e^2-c e (7 b d-15 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^2-4 b^2 e^2-c e (7 b d-16 a e)\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) \left (c x^2+b x+a\right )^3}-\frac {2 \left (\frac {5 a c e \left (7 c^2 d^2-2 b^2 e^2-c e (7 b d-15 a e)\right ) (2 c d-b e)^2-2 c \left (35 c^4 d^4-10 c^3 e (7 b d-11 a e) d^2+3 b^4 e^4+2 b^2 c e^3 (5 b d-17 a e)+c^2 e^2 \left (25 b^2 d^2-110 a b e d+123 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^4-15 c^3 e (7 b d-10 a e) d^2+6 b^4 e^4+2 b^2 c e^3 (5 b d-24 a e)+3 c^2 e^2 \left (5 b^2 d^2-25 a b e d+32 a^2 e^2\right )\right )}{2 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) \left (c x^2+b x+a\right )^2}-\frac {3 \left (-\frac {2 e^7 b^8+c d e^6 b^7+c e^5 \left (c d^2-31 a e^2\right ) b^6+c^2 d e^4 \left (c d^2-14 a e^2\right ) b^5-c^2 e^3 \left (125 c^2 d^4+13 a c e^2 d^2-178 a^2 e^4\right ) b^4+c^3 d e^2 \left (295 c^2 d^4+492 a c e^2 d^2+69 a^2 e^4\right ) b^3-c^3 e \left (245 c^3 d^6+725 a c^2 e^2 d^4+699 a^2 c e^4 d^2+443 a^3 e^6\right ) b^2+2 c^4 d \left (35 c^3 d^6+145 a c^2 e^2 d^4+233 a^2 c e^4 d^2+187 a^3 e^6\right ) b+256 a^4 c^4 e^7+2 c (2 c d-b e) \left (35 c^6 d^6-5 c^5 e (21 b d-29 a e) d^4+c^4 e^2 \left (95 b^2 d^2-290 a b e d+233 a^2 e^2\right ) d^2-b^6 e^6-3 b^4 c e^5 (b d-5 a e)-b^2 c^2 e^4 \left (7 b^2 d^2-44 a b e d+82 a^2 e^2\right )-c^3 e^3 \left (15 b^3 d^3-101 a b^2 e d^2+233 a^2 b e^2 d-187 a^3 e^3\right )\right ) x}{\left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) \left (c x^2+b x+a\right )}-\frac {\int \frac {2 \left (70 c^8 d^8-5 c^7 e (49 b d-58 a e) d^6+c^6 e^2 \left (295 b^2 d^2-725 a b e d+466 a^2 e^2\right ) d^4-c^5 e^3 \left (125 b^3 d^3-492 a b^2 e d^2+699 a^2 b e^2 d-374 a^3 e^3\right ) d^2+b^8 e^8+b^6 c e^7 (b d-16 a e)+b^4 c^2 e^6 \left (b^2 d^2-15 a b e d+96 a^2 e^2\right )+b^2 c^3 e^5 \left (b^3 d^3-14 a b^2 e d^2+82 a^2 b e^2 d-256 a^3 e^3\right )+c^4 e^4 \left (b^4 d^4-13 a b^3 e d^3+69 a^2 b^2 e^2 d^2-187 a^3 b e^3 d+256 a^4 e^4\right )+c e (2 c d-b e) \left (35 c^6 d^6-5 c^5 e (21 b d-29 a e) d^4+c^4 e^2 \left (95 b^2 d^2-290 a b e d+233 a^2 e^2\right ) d^2-b^6 e^6-3 b^4 c e^5 (b d-5 a e)-b^2 c^2 e^4 \left (7 b^2 d^2-44 a b e d+82 a^2 e^2\right )-c^3 e^3 \left (15 b^3 d^3-101 a b^2 e d^2+233 a^2 b e^2 d-187 a^3 e^3\right )\right ) x\right )}{(d+e x) \left (c x^2+b x+a\right )}dx}{\left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )}\right )}{\left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )}\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )}}{4 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\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 ) \left (c x^2+b x+a\right )^4}-\frac {\frac {7 a c e (2 c d-b e)^2-2 c \left (7 c^2 d^2-2 b^2 e^2-c e (7 b d-15 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^2-4 b^2 e^2-c e (7 b d-16 a e)\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) \left (c x^2+b x+a\right )^3}-\frac {2 \left (\frac {5 a c e \left (7 c^2 d^2-2 b^2 e^2-c e (7 b d-15 a e)\right ) (2 c d-b e)^2-2 c \left (35 c^4 d^4-10 c^3 e (7 b d-11 a e) d^2+3 b^4 e^4+2 b^2 c e^3 (5 b d-17 a e)+c^2 e^2 \left (25 b^2 d^2-110 a b e d+123 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^4-15 c^3 e (7 b d-10 a e) d^2+6 b^4 e^4+2 b^2 c e^3 (5 b d-24 a e)+3 c^2 e^2 \left (5 b^2 d^2-25 a b e d+32 a^2 e^2\right )\right )}{2 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) \left (c x^2+b x+a\right )^2}-\frac {3 \left (-\frac {2 e^7 b^8+c d e^6 b^7+c e^5 \left (c d^2-31 a e^2\right ) b^6+c^2 d e^4 \left (c d^2-14 a e^2\right ) b^5-c^2 e^3 \left (125 c^2 d^4+13 a c e^2 d^2-178 a^2 e^4\right ) b^4+c^3 d e^2 \left (295 c^2 d^4+492 a c e^2 d^2+69 a^2 e^4\right ) b^3-c^3 e \left (245 c^3 d^6+725 a c^2 e^2 d^4+699 a^2 c e^4 d^2+443 a^3 e^6\right ) b^2+2 c^4 d \left (35 c^3 d^6+145 a c^2 e^2 d^4+233 a^2 c e^4 d^2+187 a^3 e^6\right ) b+256 a^4 c^4 e^7+2 c (2 c d-b e) \left (35 c^6 d^6-5 c^5 e (21 b d-29 a e) d^4+c^4 e^2 \left (95 b^2 d^2-290 a b e d+233 a^2 e^2\right ) d^2-b^6 e^6-3 b^4 c e^5 (b d-5 a e)-b^2 c^2 e^4 \left (7 b^2 d^2-44 a b e d+82 a^2 e^2\right )-c^3 e^3 \left (15 b^3 d^3-101 a b^2 e d^2+233 a^2 b e^2 d-187 a^3 e^3\right )\right ) x}{\left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) \left (c x^2+b x+a\right )}-\frac {2 \int \frac {70 c^8 d^8-5 c^7 e (49 b d-58 a e) d^6+c^6 e^2 \left (295 b^2 d^2-725 a b e d+466 a^2 e^2\right ) d^4-c^5 e^3 \left (125 b^3 d^3-492 a b^2 e d^2+699 a^2 b e^2 d-374 a^3 e^3\right ) d^2+b^8 e^8+b^6 c e^7 (b d-16 a e)+b^4 c^2 e^6 \left (b^2 d^2-15 a b e d+96 a^2 e^2\right )+b^2 c^3 e^5 \left (b^3 d^3-14 a b^2 e d^2+82 a^2 b e^2 d-256 a^3 e^3\right )+c^4 e^4 \left (b^4 d^4-13 a b^3 e d^3+69 a^2 b^2 e^2 d^2-187 a^3 b e^3 d+256 a^4 e^4\right )+c e (2 c d-b e) \left (35 c^6 d^6-5 c^5 e (21 b d-29 a e) d^4+c^4 e^2 \left (95 b^2 d^2-290 a b e d+233 a^2 e^2\right ) d^2-b^6 e^6-3 b^4 c e^5 (b d-5 a e)-b^2 c^2 e^4 \left (7 b^2 d^2-44 a b e d+82 a^2 e^2\right )-c^3 e^3 \left (15 b^3 d^3-101 a b^2 e d^2+233 a^2 b e^2 d-187 a^3 e^3\right )\right ) x}{(d+e x) \left (c x^2+b x+a\right )}dx}{\left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )}\right )}{\left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )}\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )}}{4 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )}\) |
| \(\Big \downarrow \) 1200 |
| \(\displaystyle -\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 ) \left (c x^2+b x+a\right )^4}-\frac {\frac {7 a c e (2 c d-b e)^2-2 c \left (7 c^2 d^2-2 b^2 e^2-c e (7 b d-15 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^2-4 b^2 e^2-c e (7 b d-16 a e)\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) \left (c x^2+b x+a\right )^3}-\frac {2 \left (\frac {5 a c e \left (7 c^2 d^2-2 b^2 e^2-c e (7 b d-15 a e)\right ) (2 c d-b e)^2-2 c \left (35 c^4 d^4-10 c^3 e (7 b d-11 a e) d^2+3 b^4 e^4+2 b^2 c e^3 (5 b d-17 a e)+c^2 e^2 \left (25 b^2 d^2-110 a b e d+123 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^4-15 c^3 e (7 b d-10 a e) d^2+6 b^4 e^4+2 b^2 c e^3 (5 b d-24 a e)+3 c^2 e^2 \left (5 b^2 d^2-25 a b e d+32 a^2 e^2\right )\right )}{2 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) \left (c x^2+b x+a\right )^2}-\frac {3 \left (-\frac {2 e^7 b^8+c d e^6 b^7+c e^5 \left (c d^2-31 a e^2\right ) b^6+c^2 d e^4 \left (c d^2-14 a e^2\right ) b^5-c^2 e^3 \left (125 c^2 d^4+13 a c e^2 d^2-178 a^2 e^4\right ) b^4+c^3 d e^2 \left (295 c^2 d^4+492 a c e^2 d^2+69 a^2 e^4\right ) b^3-c^3 e \left (245 c^3 d^6+725 a c^2 e^2 d^4+699 a^2 c e^4 d^2+443 a^3 e^6\right ) b^2+2 c^4 d \left (35 c^3 d^6+145 a c^2 e^2 d^4+233 a^2 c e^4 d^2+187 a^3 e^6\right ) b+256 a^4 c^4 e^7+2 c (2 c d-b e) \left (35 c^6 d^6-5 c^5 e (21 b d-29 a e) d^4+c^4 e^2 \left (95 b^2 d^2-290 a b e d+233 a^2 e^2\right ) d^2-b^6 e^6-3 b^4 c e^5 (b d-5 a e)-b^2 c^2 e^4 \left (7 b^2 d^2-44 a b e d+82 a^2 e^2\right )-c^3 e^3 \left (15 b^3 d^3-101 a b^2 e d^2+233 a^2 b e^2 d-187 a^3 e^3\right )\right ) x}{\left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) \left (c x^2+b x+a\right )}-\frac {2 \int \left (\frac {\left (b^2-4 a c\right )^4 e^{10}}{\left (c d^2-b e d+a e^2\right ) (d+e x)}+\frac {70 c^9 d^9-45 c^8 e (7 b d-8 a e) d^7+36 c^7 e^2 \left (15 b^2 d^2-35 a b e d+21 a^2 e^2\right ) d^5-42 c^6 e^3 \left (10 b^3 d^3-36 a b^2 e d^2+45 a^2 b e^2 d-20 a^3 e^3\right ) d^3+126 c^5 e^4 \left (b^4 d^4-5 a b^3 e d^3+10 a^2 b^2 e^2 d^2-10 a^3 b e^3 d+5 a^4 e^4\right ) d-b^9 e^9-443 a^4 b c^4 e^9+338 a^3 b^3 c^3 e^9-111 a^2 b^5 c^2 e^9+17 a b^7 c e^9-c \left (b^2-4 a c\right )^4 e^9 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 (c d^2-b e d+a e^2\right )}\right )}{\left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )}\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )}}{4 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\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 ) \left (c x^2+b x+a\right )^4}-\frac {\frac {7 a c e (2 c d-b e)^2-2 c \left (7 c^2 d^2-2 b^2 e^2-c e (7 b d-15 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^2-4 b^2 e^2-c e (7 b d-16 a e)\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) \left (c x^2+b x+a\right )^3}-\frac {2 \left (\frac {5 a c e \left (7 c^2 d^2-2 b^2 e^2-c e (7 b d-15 a e)\right ) (2 c d-b e)^2-2 c \left (35 c^4 d^4-10 c^3 e (7 b d-11 a e) d^2+3 b^4 e^4+2 b^2 c e^3 (5 b d-17 a e)+c^2 e^2 \left (25 b^2 d^2-110 a b e d+123 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^4-15 c^3 e (7 b d-10 a e) d^2+6 b^4 e^4+2 b^2 c e^3 (5 b d-24 a e)+3 c^2 e^2 \left (5 b^2 d^2-25 a b e d+32 a^2 e^2\right )\right )}{2 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) \left (c x^2+b x+a\right )^2}-\frac {3 \left (-\frac {2 e^7 b^8+c d e^6 b^7+c e^5 \left (c d^2-31 a e^2\right ) b^6+c^2 d e^4 \left (c d^2-14 a e^2\right ) b^5-c^2 e^3 \left (125 c^2 d^4+13 a c e^2 d^2-178 a^2 e^4\right ) b^4+c^3 d e^2 \left (295 c^2 d^4+492 a c e^2 d^2+69 a^2 e^4\right ) b^3-c^3 e \left (245 c^3 d^6+725 a c^2 e^2 d^4+699 a^2 c e^4 d^2+443 a^3 e^6\right ) b^2+2 c^4 d \left (35 c^3 d^6+145 a c^2 e^2 d^4+233 a^2 c e^4 d^2+187 a^3 e^6\right ) b+256 a^4 c^4 e^7+2 c (2 c d-b e) \left (35 c^6 d^6-5 c^5 e (21 b d-29 a e) d^4+c^4 e^2 \left (95 b^2 d^2-290 a b e d+233 a^2 e^2\right ) d^2-b^6 e^6-3 b^4 c e^5 (b d-5 a e)-b^2 c^2 e^4 \left (7 b^2 d^2-44 a b e d+82 a^2 e^2\right )-c^3 e^3 \left (15 b^3 d^3-101 a b^2 e d^2+233 a^2 b e^2 d-187 a^3 e^3\right )\right ) x}{\left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) \left (c x^2+b x+a\right )}-\frac {2 \left (\frac {\left (b^2-4 a c\right )^4 \log (d+e x) e^9}{c d^2-b e d+a e^2}-\frac {\left (b^2-4 a c\right )^4 \log \left (c x^2+b x+a\right ) e^9}{2 \left (c d^2-b e d+a e^2\right )}-\frac {\left (140 c^9 d^9-90 c^8 e (7 b d-8 a e) d^7+72 c^7 e^2 \left (15 b^2 d^2-35 a b e d+21 a^2 e^2\right ) d^5-84 c^6 e^3 \left (10 b^3 d^3-36 a b^2 e d^2+45 a^2 b e^2 d-20 a^3 e^3\right ) d^3+252 c^5 e^4 \left (b^4 d^4-5 a b^3 e d^3+10 a^2 b^2 e^2 d^2-10 a^3 b e^3 d+5 a^4 e^4\right ) d-b^9 e^9-630 a^4 b c^4 e^9+420 a^3 b^3 c^3 e^9-126 a^2 b^5 c^2 e^9+18 a b^7 c e^9\right ) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c} \left (c d^2-e (b d-a e)\right )}\right )}{\left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )}\right )}{\left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )}\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )}}{4 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )}\) |
-1/4*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x)/((b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*(a + b*x + c*x^2)^4) - ((7*a*c*e*(2*c*d - b*e)^2 - (b*c*d - b^2*e + 2*a*c*e)*(14*c^2*d^2 - 4*b^2*e^2 - c*e*(7*b*d - 16*a*e)) - 2*c*( 2*c*d - b*e)*(7*c^2*d^2 - 2*b^2*e^2 - c*e*(7*b*d - 15*a*e))*x)/(3*(b^2 - 4 *a*c)*(c*d^2 - b*d*e + a*e^2)*(a + b*x + c*x^2)^3) - (2*((5*a*c*e*(2*c*d - b*e)^2*(7*c^2*d^2 - 2*b^2*e^2 - c*e*(7*b*d - 15*a*e)) - (b*c*d - b^2*e + 2*a*c*e)*(70*c^4*d^4 + 6*b^4*e^4 + 2*b^2*c*e^3*(5*b*d - 24*a*e) - 15*c^3*d ^2*e*(7*b*d - 10*a*e) + 3*c^2*e^2*(5*b^2*d^2 - 25*a*b*d*e + 32*a^2*e^2)) - 2*c*(2*c*d - b*e)*(35*c^4*d^4 + 3*b^4*e^4 + 2*b^2*c*e^3*(5*b*d - 17*a*e) - 10*c^3*d^2*e*(7*b*d - 11*a*e) + c^2*e^2*(25*b^2*d^2 - 110*a*b*d*e + 123* a^2*e^2))*x)/(2*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*(a + b*x + c*x^2)^2) - (3*(-((b^7*c*d*e^6 + 2*b^8*e^7 + 256*a^4*c^4*e^7 + b^6*c*e^5*(c*d^2 - 3 1*a*e^2) + b^5*c^2*d*e^4*(c*d^2 - 14*a*e^2) - b^4*c^2*e^3*(125*c^2*d^4 + 1 3*a*c*d^2*e^2 - 178*a^2*e^4) + b^3*c^3*d*e^2*(295*c^2*d^4 + 492*a*c*d^2*e^ 2 + 69*a^2*e^4) + 2*b*c^4*d*(35*c^3*d^6 + 145*a*c^2*d^4*e^2 + 233*a^2*c*d^ 2*e^4 + 187*a^3*e^6) - b^2*c^3*e*(245*c^3*d^6 + 725*a*c^2*d^4*e^2 + 699*a^ 2*c*d^2*e^4 + 443*a^3*e^6) + 2*c*(2*c*d - b*e)*(35*c^6*d^6 - b^6*e^6 - 5*c ^5*d^4*e*(21*b*d - 29*a*e) - 3*b^4*c*e^5*(b*d - 5*a*e) - b^2*c^2*e^4*(7*b^ 2*d^2 - 44*a*b*d*e + 82*a^2*e^2) + c^4*d^2*e^2*(95*b^2*d^2 - 290*a*b*d*e + 233*a^2*e^2) - c^3*e^3*(15*b^3*d^3 - 101*a*b^2*d^2*e + 233*a^2*b*d*e^2...
3.23.27.3.1 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)*(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] + Simp[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] && LtQ[p, -1] && IntQuadraticQ[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[(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. \(5747\) vs. \(2(1310)=2620\).
Time = 22.75 (sec) , antiderivative size = 5748, normalized size of antiderivative = 4.34
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(5748\) |
| risch | \(\text {Expression too large to display}\) | \(406145\) |
Timed out. \[ \int \frac {1}{(d+e x) \left (a+b x+c x^2\right )^5} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {1}{(d+e x) \left (a+b x+c x^2\right )^5} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {1}{(d+e x) \left (a+b x+c x^2\right )^5} \, dx=\text {Exception raised: ValueError} \]
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. 6749 vs. \(2 (1310) = 2620\).
Time = 0.34 (sec) , antiderivative size = 6749, normalized size of antiderivative = 5.10 \[ \int \frac {1}{(d+e x) \left (a+b x+c x^2\right )^5} \, dx=\text {Too large to display} \]
e^10*log(abs(e*x + d))/(c^5*d^10*e - 5*b*c^4*d^9*e^2 + 10*b^2*c^3*d^8*e^3 + 5*a*c^4*d^8*e^3 - 10*b^3*c^2*d^7*e^4 - 20*a*b*c^3*d^7*e^4 + 5*b^4*c*d^6* e^5 + 30*a*b^2*c^2*d^6*e^5 + 10*a^2*c^3*d^6*e^5 - b^5*d^5*e^6 - 20*a*b^3*c *d^5*e^6 - 30*a^2*b*c^2*d^5*e^6 + 5*a*b^4*d^4*e^7 + 30*a^2*b^2*c*d^4*e^7 + 10*a^3*c^2*d^4*e^7 - 10*a^2*b^3*d^3*e^8 - 20*a^3*b*c*d^3*e^8 + 10*a^3*b^2 *d^2*e^9 + 5*a^4*c*d^2*e^9 - 5*a^4*b*d*e^10 + a^5*e^11) - 1/2*e^9*log(c*x^ 2 + b*x + a)/(c^5*d^10 - 5*b*c^4*d^9*e + 10*b^2*c^3*d^8*e^2 + 5*a*c^4*d^8* e^2 - 10*b^3*c^2*d^7*e^3 - 20*a*b*c^3*d^7*e^3 + 5*b^4*c*d^6*e^4 + 30*a*b^2 *c^2*d^6*e^4 + 10*a^2*c^3*d^6*e^4 - b^5*d^5*e^5 - 20*a*b^3*c*d^5*e^5 - 30* a^2*b*c^2*d^5*e^5 + 5*a*b^4*d^4*e^6 + 30*a^2*b^2*c*d^4*e^6 + 10*a^3*c^2*d^ 4*e^6 - 10*a^2*b^3*d^3*e^7 - 20*a^3*b*c*d^3*e^7 + 10*a^3*b^2*d^2*e^8 + 5*a ^4*c*d^2*e^8 - 5*a^4*b*d*e^9 + a^5*e^10) + (140*c^9*d^9 - 630*b*c^8*d^8*e + 1080*b^2*c^7*d^7*e^2 + 720*a*c^8*d^7*e^2 - 840*b^3*c^6*d^6*e^3 - 2520*a* b*c^7*d^6*e^3 + 252*b^4*c^5*d^5*e^4 + 3024*a*b^2*c^6*d^5*e^4 + 1512*a^2*c^ 7*d^5*e^4 - 1260*a*b^3*c^5*d^4*e^5 - 3780*a^2*b*c^6*d^4*e^5 + 2520*a^2*b^2 *c^5*d^3*e^6 + 1680*a^3*c^6*d^3*e^6 - 2520*a^3*b*c^5*d^2*e^7 + 1260*a^4*c^ 5*d*e^8 - b^9*e^9 + 18*a*b^7*c*e^9 - 126*a^2*b^5*c^2*e^9 + 420*a^3*b^3*c^3 *e^9 - 630*a^4*b*c^4*e^9)*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/((b^8*c^5 *d^10 - 16*a*b^6*c^6*d^10 + 96*a^2*b^4*c^7*d^10 - 256*a^3*b^2*c^8*d^10 + 2 56*a^4*c^9*d^10 - 5*b^9*c^4*d^9*e + 80*a*b^7*c^5*d^9*e - 480*a^2*b^5*c^...
Time = 34.81 (sec) , antiderivative size = 59814, normalized size of antiderivative = 45.18 \[ \int \frac {1}{(d+e x) \left (a+b x+c x^2\right )^5} \, dx=\text {Too large to display} \]
((25*a^3*b^8*e^7 + 3200*a^7*c^4*e^7 - 3*b^7*c^4*d^7 - 3*b^11*d^3*e^4 + 50* a*b^5*c^5*d^7 + 1116*a^3*b*c^7*d^7 - 385*a^4*b^6*c*e^7 + 13*a*b^10*d^2*e^5 - 23*a^2*b^9*d*e^6 + 384*a^4*c^7*d^6*e + 12*b^8*c^3*d^6*e + 12*b^10*c*d^4 *e^3 - 326*a^2*b^3*c^6*d^7 + 2175*a^5*b^4*c^2*e^7 - 5150*a^6*b^2*c^3*e^7 + 1664*a^5*c^6*d^4*e^3 + 2944*a^6*c^5*d^2*e^5 - 18*b^9*c^2*d^5*e^2 - 1681*a ^2*b^5*c^4*d^5*e^2 + 598*a^2*b^6*c^3*d^4*e^3 + 314*a^2*b^7*c^2*d^3*e^4 + 4 724*a^3*b^3*c^5*d^5*e^2 + 307*a^3*b^4*c^4*d^4*e^3 - 2812*a^3*b^5*c^3*d^3*e ^4 + 574*a^3*b^6*c^2*d^2*e^5 - 11522*a^4*b^2*c^5*d^4*e^3 + 9254*a^4*b^3*c^ 4*d^3*e^4 + 985*a^4*b^4*c^3*d^2*e^5 - 10910*a^5*b^2*c^4*d^2*e^5 - 199*a*b^ 6*c^4*d^6*e + 10*a*b^9*c*d^3*e^4 + 356*a^3*b^7*c*d*e^6 + 956*a^6*b*c^4*d*e ^6 + 284*a*b^7*c^3*d^5*e^2 - 158*a*b^8*c^2*d^4*e^3 + 1285*a^2*b^4*c^5*d^6* e - 167*a^2*b^8*c*d^2*e^5 - 4290*a^3*b^2*c^6*d^6*e + 3252*a^4*b*c^6*d^5*e^ 2 - 1983*a^4*b^5*c^2*d*e^6 + 3092*a^5*b*c^5*d^3*e^4 + 4204*a^5*b^3*c^3*d*e ^6)/(12*(a^4*b^8*e^8 + 256*a^4*c^8*d^8 + 256*a^8*c^4*e^8 + b^8*c^4*d^8 + b ^12*d^4*e^4 - 16*a*b^6*c^5*d^8 - 16*a^5*b^6*c*e^8 - 4*a*b^11*d^3*e^5 - 4*a ^3*b^9*d*e^7 - 4*b^9*c^3*d^7*e - 4*b^11*c*d^5*e^3 + 96*a^2*b^4*c^6*d^8 - 2 56*a^3*b^2*c^7*d^8 + 96*a^6*b^4*c^2*e^8 - 256*a^7*b^2*c^3*e^8 + 6*a^2*b^10 *d^2*e^6 + 1024*a^5*c^7*d^6*e^2 + 1536*a^6*c^6*d^4*e^4 + 1024*a^7*c^5*d^2* e^6 + 6*b^10*c^2*d^6*e^2 + 512*a^2*b^6*c^4*d^6*e^2 - 192*a^2*b^7*c^3*d^5*e ^3 - 90*a^2*b^8*c^2*d^4*e^4 - 1152*a^3*b^4*c^5*d^6*e^2 - 128*a^3*b^5*c^...