Integrand size = 30, antiderivative size = 1075 \[ \int \frac {A+B x}{\left (a+b x+c x^2\right )^2 \left (d+e x+f x^2\right )} \, dx=-\frac {A c (2 a c e-b (c d+a f))+(A b-a B) \left (2 c^2 d+b^2 f-c (b e+2 a f)\right )+c \left (A b^2 f+2 c (A c d+a B e-a A f)-b (B c d+A c e+a B f)\right ) x}{\left (b^2-4 a c\right ) \left ((c d-a f)^2-(b d-a e) (c e-b f)\right ) \left (a+b x+c x^2\right )}-\frac {\left (b^5 (B d-A e) f^2-2 b^4 f \left (B c d e-A \left (c e^2-c d f+a f^2\right )\right )-4 c^2 \left (A \left (c^3 d^3-3 a^3 f^3-a^2 c f \left (e^2-7 d f\right )+a c^2 d \left (3 e^2-5 d f\right )\right )-a B e \left (c^2 d^2-3 a^2 f^2-a c \left (e^2-2 d f\right )\right )\right )-4 b^2 c \left (B c^2 d^2 e+A f \left (2 c^2 d^2+3 a^2 f^2+3 a c \left (e^2-d f\right )\right )\right )+2 b c \left (B \left (c^3 d^3+3 a^3 f^3+a c^2 d \left (e^2-7 d f\right )+3 a^2 c f \left (e^2+d f\right )\right )+A c e \left (3 c^2 d^2+3 a^2 f^2+a c \left (3 e^2+2 d f\right )\right )\right )-b^3 \left (A c e \left (c e^2-2 c d f-4 a f^2\right )+B \left (4 a c d f^2+a^2 f^3-c^2 d \left (e^2+5 d f\right )\right )\right )\right ) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2} \left (c^2 d^2+f \left (b^2 d-a b e+a^2 f\right )-c \left (b d e-a \left (e^2-2 d f\right )\right )\right )^2}+\frac {\left (B \left (c^2 d e \left (e^2-3 d f\right )-2 c d f \left (b e^2-2 b d f-a e f\right )+f^2 \left (b^2 d e-4 a b d f+a^2 e f\right )\right )-A \left (c^2 \left (e^4-4 d e^2 f+2 d^2 f^2\right )-f^2 \left (2 a b e f-2 a^2 f^2-b^2 \left (e^2-2 d f\right )\right )+2 c f \left (a f \left (e^2-2 d f\right )-b \left (e^3-3 d e f\right )\right )\right )\right ) \text {arctanh}\left (\frac {e+2 f x}{\sqrt {e^2-4 d f}}\right )}{\sqrt {e^2-4 d f} \left (c^2 d^2+f \left (b^2 d-a b e+a^2 f\right )-c \left (b d e-a \left (e^2-2 d f\right )\right )\right )^2}+\frac {\left (A (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )-B \left (2 c d f (b e-a f)-f^2 \left (b^2 d-a^2 f\right )-c^2 d \left (e^2-d f\right )\right )\right ) \log \left (a+b x+c x^2\right )}{2 \left (c^2 d^2+f \left (b^2 d-a b e+a^2 f\right )-c \left (b d e-a \left (e^2-2 d f\right )\right )\right )^2}-\frac {\left (A (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )-B \left (2 c d f (b e-a f)-f^2 \left (b^2 d-a^2 f\right )-c^2 d \left (e^2-d f\right )\right )\right ) \log \left (d+e x+f x^2\right )}{2 \left (c^2 d^2+f \left (b^2 d-a b e+a^2 f\right )-c \left (b d e-a \left (e^2-2 d f\right )\right )\right )^2} \]
(-A*c*(2*a*c*e-b*(a*f+c*d))-(A*b-B*a)*(2*c^2*d+b^2*f-c*(2*a*f+b*e))-c*(A*b ^2*f+2*c*(-A*a*f+A*c*d+B*a*e)-b*(A*c*e+B*a*f+B*c*d))*x)/(-4*a*c+b^2)/((-a* f+c*d)^2-(-a*e+b*d)*(-b*f+c*e))/(c*x^2+b*x+a)-(b^5*(-A*e+B*d)*f^2-2*b^4*f* (B*c*d*e-A*(a*f^2-c*d*f+c*e^2))-4*c^2*(A*(c^3*d^3-3*a^3*f^3-a^2*c*f*(-7*d* f+e^2)+a*c^2*d*(-5*d*f+3*e^2))-a*B*e*(c^2*d^2-3*a^2*f^2-a*c*(-2*d*f+e^2))) -4*b^2*c*(B*c^2*d^2*e+A*f*(2*c^2*d^2+3*a^2*f^2+3*a*c*(-d*f+e^2)))+2*b*c*(B *(c^3*d^3+3*a^3*f^3+a*c^2*d*(-7*d*f+e^2)+3*a^2*c*f*(d*f+e^2))+A*c*e*(3*c^2 *d^2+3*a^2*f^2+a*c*(2*d*f+3*e^2)))-b^3*(A*c*e*(-4*a*f^2-2*c*d*f+c*e^2)+B*( 4*a*c*d*f^2+a^2*f^3-c^2*d*(5*d*f+e^2))))*arctanh((2*c*x+b)/(-4*a*c+b^2)^(1 /2))/(-4*a*c+b^2)^(3/2)/(c^2*d^2+f*(a^2*f-a*b*e+b^2*d)-c*(b*d*e-a*(-2*d*f+ e^2)))^2+1/2*(A*(-b*f+c*e)*(f*(-2*a*f+b*e)-c*(-2*d*f+e^2))-B*(2*c*d*f*(-a* f+b*e)-f^2*(-a^2*f+b^2*d)-c^2*d*(-d*f+e^2)))*ln(c*x^2+b*x+a)/(c^2*d^2+f*(a ^2*f-a*b*e+b^2*d)-c*(b*d*e-a*(-2*d*f+e^2)))^2-1/2*(A*(-b*f+c*e)*(f*(-2*a*f +b*e)-c*(-2*d*f+e^2))-B*(2*c*d*f*(-a*f+b*e)-f^2*(-a^2*f+b^2*d)-c^2*d*(-d*f +e^2)))*ln(f*x^2+e*x+d)/(c^2*d^2+f*(a^2*f-a*b*e+b^2*d)-c*(b*d*e-a*(-2*d*f+ e^2)))^2+(B*(c^2*d*e*(-3*d*f+e^2)-2*c*d*f*(-a*e*f-2*b*d*f+b*e^2)+f^2*(a^2* e*f-4*a*b*d*f+b^2*d*e))-A*(c^2*(2*d^2*f^2-4*d*e^2*f+e^4)-f^2*(2*a*b*e*f-2* a^2*f^2-b^2*(-2*d*f+e^2))+2*c*f*(a*f*(-2*d*f+e^2)-b*(-3*d*e*f+e^3))))*arct anh((2*f*x+e)/(-4*d*f+e^2)^(1/2))/(c^2*d^2+f*(a^2*f-a*b*e+b^2*d)-c*(b*d*e- a*(-2*d*f+e^2)))^2/(-4*d*f+e^2)^(1/2)
Time = 3.87 (sec) , antiderivative size = 952, normalized size of antiderivative = 0.89 \[ \int \frac {A+B x}{\left (a+b x+c x^2\right )^2 \left (d+e x+f x^2\right )} \, dx=\frac {-\frac {2 \left (c^2 d^2-b c d e+f \left (b^2 d-a b e+a^2 f\right )+a c \left (e^2-2 d f\right )\right ) \left (A \left (b^3 f+b^2 c (-e+f x)+b c (-3 a f+c (d-e x))+2 c^2 (c d x+a (e-f x))\right )+B \left (2 a^2 c f-b c^2 d x-a \left (b^2 f+2 c^2 (d-e x)+b c (-e+f x)\right )\right )\right )}{\left (b^2-4 a c\right ) (a+x (b+c x))}-\frac {2 \left (b^5 (B d-A e) f^2+2 b^4 f \left (-B c d e+a A f^2+A c \left (e^2-d f\right )\right )-4 b^2 \left (B c^3 d^2 e+A c f \left (2 c^2 d^2+3 a^2 f^2+3 a c \left (e^2-d f\right )\right )\right )+2 b c \left (B \left (c^3 d^3+3 a^3 f^3+a c^2 d \left (e^2-7 d f\right )+3 a^2 c f \left (e^2+d f\right )\right )+A c e \left (3 c^2 d^2+3 a^2 f^2+a c \left (3 e^2+2 d f\right )\right )\right )+4 c^2 \left (a B e \left (c^2 d^2-3 a^2 f^2-a c \left (e^2-2 d f\right )\right )+A \left (-c^3 d^3+3 a^3 f^3+a^2 c f \left (e^2-7 d f\right )+a c^2 d \left (-3 e^2+5 d f\right )\right )\right )+b^3 \left (A c e \left (-c e^2+2 c d f+4 a f^2\right )+B \left (-4 a c d f^2-a^2 f^3+c^2 d \left (e^2+5 d f\right )\right )\right )\right ) \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\left (-b^2+4 a c\right )^{3/2}}+\frac {2 \left (B \left (c^2 d e \left (-e^2+3 d f\right )-2 c d f \left (-b e^2+2 b d f+a e f\right )+f^2 \left (-b^2 d e+4 a b d f-a^2 e f\right )\right )+A \left (c^2 \left (e^4-4 d e^2 f+2 d^2 f^2\right )+f^2 \left (-2 a b e f+2 a^2 f^2+b^2 \left (e^2-2 d f\right )\right )+2 c f \left (a f \left (e^2-2 d f\right )-b \left (e^3-3 d e f\right )\right )\right )\right ) \arctan \left (\frac {e+2 f x}{\sqrt {-e^2+4 d f}}\right )}{\sqrt {-e^2+4 d f}}-\left (A (c e-b f) \left (f (-b e+2 a f)+c \left (e^2-2 d f\right )\right )+B \left (2 c d f (b e-a f)+f^2 \left (-b^2 d+a^2 f\right )+c^2 d \left (-e^2+d f\right )\right )\right ) \log (a+x (b+c x))+\left (A (c e-b f) \left (f (-b e+2 a f)+c \left (e^2-2 d f\right )\right )+B \left (2 c d f (b e-a f)+f^2 \left (-b^2 d+a^2 f\right )+c^2 d \left (-e^2+d f\right )\right )\right ) \log (d+x (e+f x))}{2 \left (c^2 d^2-b c d e+f \left (b^2 d-a b e+a^2 f\right )+a c \left (e^2-2 d f\right )\right )^2} \]
((-2*(c^2*d^2 - b*c*d*e + f*(b^2*d - a*b*e + a^2*f) + a*c*(e^2 - 2*d*f))*( A*(b^3*f + b^2*c*(-e + f*x) + b*c*(-3*a*f + c*(d - e*x)) + 2*c^2*(c*d*x + a*(e - f*x))) + B*(2*a^2*c*f - b*c^2*d*x - a*(b^2*f + 2*c^2*(d - e*x) + b* c*(-e + f*x)))))/((b^2 - 4*a*c)*(a + x*(b + c*x))) - (2*(b^5*(B*d - A*e)*f ^2 + 2*b^4*f*(-(B*c*d*e) + a*A*f^2 + A*c*(e^2 - d*f)) - 4*b^2*(B*c^3*d^2*e + A*c*f*(2*c^2*d^2 + 3*a^2*f^2 + 3*a*c*(e^2 - d*f))) + 2*b*c*(B*(c^3*d^3 + 3*a^3*f^3 + a*c^2*d*(e^2 - 7*d*f) + 3*a^2*c*f*(e^2 + d*f)) + A*c*e*(3*c^ 2*d^2 + 3*a^2*f^2 + a*c*(3*e^2 + 2*d*f))) + 4*c^2*(a*B*e*(c^2*d^2 - 3*a^2* f^2 - a*c*(e^2 - 2*d*f)) + A*(-(c^3*d^3) + 3*a^3*f^3 + a^2*c*f*(e^2 - 7*d* f) + a*c^2*d*(-3*e^2 + 5*d*f))) + b^3*(A*c*e*(-(c*e^2) + 2*c*d*f + 4*a*f^2 ) + B*(-4*a*c*d*f^2 - a^2*f^3 + c^2*d*(e^2 + 5*d*f))))*ArcTan[(b + 2*c*x)/ Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(3/2) + (2*(B*(c^2*d*e*(-e^2 + 3*d*f) - 2*c*d*f*(-(b*e^2) + 2*b*d*f + a*e*f) + f^2*(-(b^2*d*e) + 4*a*b*d*f - a^2 *e*f)) + A*(c^2*(e^4 - 4*d*e^2*f + 2*d^2*f^2) + f^2*(-2*a*b*e*f + 2*a^2*f^ 2 + b^2*(e^2 - 2*d*f)) + 2*c*f*(a*f*(e^2 - 2*d*f) - b*(e^3 - 3*d*e*f))))*A rcTan[(e + 2*f*x)/Sqrt[-e^2 + 4*d*f]])/Sqrt[-e^2 + 4*d*f] - (A*(c*e - b*f) *(f*(-(b*e) + 2*a*f) + c*(e^2 - 2*d*f)) + B*(2*c*d*f*(b*e - a*f) + f^2*(-( b^2*d) + a^2*f) + c^2*d*(-e^2 + d*f)))*Log[a + x*(b + c*x)] + (A*(c*e - b* f)*(f*(-(b*e) + 2*a*f) + c*(e^2 - 2*d*f)) + B*(2*c*d*f*(b*e - a*f) + f^2*( -(b^2*d) + a^2*f) + c^2*d*(-e^2 + d*f)))*Log[d + x*(e + f*x)])/(2*(c^2*...
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.
| \(\displaystyle \int \frac {A+B x}{\left (a+b x+c x^2\right )^2 \left (d+e x+f x^2\right )} \, dx\) |
| \(\Big \downarrow \) 1349 |
| \(\displaystyle -\frac {\int -\frac {(B d f-A e f) b^3-\left (B c d e-A \left (c e^2+a f^2-2 c d f\right )\right ) b^2+c (B d (c d-3 a f)+A e (c d+4 a f)) b-c f \left (A f b^2-(B c d+A c e+a B f) b+2 c (A c d+a B e-a A f)\right ) x^2+2 c \left (a B c d e-A \left (c^2 d^2-3 a c f d+2 a c e^2+2 a^2 f^2\right )\right )-\left (A f^2 b^3+B f (c d-a f) b^2-c \left (A c e^2+B c d e+a B f e+4 a A f^2\right ) b+2 c \left (A c e (c d+a f)+a B \left (c e^2+2 a f^2-2 c d f\right )\right )\right ) x}{\left (c x^2+b x+a\right ) \left (f x^2+e x+d\right )}dx}{\left (b^2-4 a c\right ) \left ((c d-a f)^2-(b d-a e) (c e-b f)\right )}-\frac {(A b-a B) \left (-c (2 a f+b e)+b^2 f+2 c^2 d\right )+c x \left (-b (a B f+A c e+B c d)+2 c (-a A f+a B e+A c d)+A b^2 f\right )+A c (2 a c e-b (a f+c d))}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) \left ((c d-a f)^2-(b d-a e) (c e-b f)\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {(B d-A e) f b^3-\left (-a A f^2+B c d e-A c \left (e^2-2 d f\right )\right ) b^2+c (B d (c d-3 a f)+A e (c d+4 a f)) b-c f \left (A f b^2-(B c d+A c e+a B f) b+2 c (A c d+a B e-a A f)\right ) x^2+2 c \left (a B c d e-A \left (c^2 d^2+2 a^2 f^2+a c \left (2 e^2-3 d f\right )\right )\right )-\left (A f^2 b^3+B f (c d-a f) b^2-c \left (A c e^2+B c d e+a B f e+4 a A f^2\right ) b+2 c \left (A c e (c d+a f)+a B \left (c e^2+2 a f^2-2 c d f\right )\right )\right ) x}{\left (c x^2+b x+a\right ) \left (f x^2+e x+d\right )}dx}{\left (b^2-4 a c\right ) \left ((c d-a f)^2-(b d-a e) (c e-b f)\right )}-\frac {(A b-a B) \left (-c (2 a f+b e)+b^2 f+2 c^2 d\right )+c x \left (-b (a B f+A c e+B c d)+2 c (-a A f+a B e+A c d)+A b^2 f\right )+A c (2 a c e-b (a f+c d))}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) \left ((c d-a f)^2-(b d-a e) (c e-b f)\right )}\) |
| \(\Big \downarrow \) 2141 |
| \(\displaystyle \frac {\frac {\int \frac {(B d-A e) f^2 b^5-2 f \left (-a A f^2+B c d e-A c \left (e^2-d f\right )\right ) b^4-\left (A c e \left (c e^2-3 a f^2-2 c d f\right )+B \left (a^2 f^3+3 a c d f^2-c^2 d \left (e^2+2 d f\right )\right )\right ) b^3-2 c \left (B c d e (c d-2 a f)+A f \left (2 c^2 d^2+5 a^2 f^2+5 a c \left (e^2-d f\right )\right )\right ) b^2+c \left (A c e \left (3 c^2 d^2+7 a^2 f^2+a c \left (5 e^2-2 d f\right )\right )+B \left (c^3 d^3-a c^2 \left (e^2+5 d f\right ) d+5 a^3 f^3+a^2 c f \left (3 e^2-d f\right )\right )\right ) b-2 c^2 \left (A \left (c^3 d^3+a c^2 \left (3 e^2-5 d f\right ) d-3 a^3 f^3-a^2 c f \left (e^2-7 d f\right )\right )-a B e \left (c^2 d^2-3 a^2 f^2-a c \left (e^2-2 d f\right )\right )\right )+c \left (b^2-4 a c\right ) \left (A (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )-B \left (-d \left (e^2-d f\right ) c^2+2 d f (b e-a f) c-f^2 \left (b^2 d-a^2 f\right )\right )\right ) x}{c x^2+b x+a}dx}{c^2 d^2-b c e d+f \left (f a^2-b e a+b^2 d\right )+a c \left (e^2-2 d f\right )}+\frac {\int \frac {\left (b^2-4 a c\right ) \left (B d (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )+A \left (\left (e^4-3 d f e^2+d^2 f^2\right ) c^2+2 f \left (a f \left (e^2-d f\right )-b \left (e^3-2 d e f\right )\right ) c-f^2 \left (-\left (\left (e^2-d f\right ) b^2\right )+2 a e f b-a^2 f^2\right )\right )-f \left (A (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )-B \left (-d \left (e^2-d f\right ) c^2+2 d f (b e-a f) c-f^2 \left (b^2 d-a^2 f\right )\right )\right ) x\right )}{f x^2+e x+d}dx}{c^2 d^2-b c e d+f \left (f a^2-b e a+b^2 d\right )+a c \left (e^2-2 d f\right )}}{\left (b^2-4 a c\right ) \left ((c d-a f)^2-(b d-a e) (c e-b f)\right )}-\frac {A c (2 a c e-b (c d+a f))+(A b-a B) \left (f b^2+2 c^2 d-c (b e+2 a f)\right )+c \left (A f b^2-(B c d+A c e+a B f) b+2 c (A c d+a B e-a A f)\right ) x}{\left (b^2-4 a c\right ) \left ((c d-a f)^2-(b d-a e) (c e-b f)\right ) \left (c x^2+b x+a\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {(B d-A e) f^2 b^5-2 f \left (-a A f^2+B c d e-A c \left (e^2-d f\right )\right ) b^4-\left (A c e \left (c e^2-3 a f^2-2 c d f\right )+B \left (a^2 f^3+3 a c d f^2-c^2 d \left (e^2+2 d f\right )\right )\right ) b^3-2 c \left (B c d e (c d-2 a f)+A f \left (2 c^2 d^2+5 a^2 f^2+5 a c \left (e^2-d f\right )\right )\right ) b^2+c \left (A c e \left (3 c^2 d^2+7 a^2 f^2+a c \left (5 e^2-2 d f\right )\right )+B \left (c^3 d^3-a c^2 \left (e^2+5 d f\right ) d+5 a^3 f^3+a^2 c f \left (3 e^2-d f\right )\right )\right ) b-2 c^2 \left (A \left (c^3 d^3+a c^2 \left (3 e^2-5 d f\right ) d-3 a^3 f^3-a^2 c f \left (e^2-7 d f\right )\right )-a B e \left (c^2 d^2-3 a^2 f^2-a c \left (e^2-2 d f\right )\right )\right )+c \left (b^2-4 a c\right ) \left (A (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )-B \left (-d \left (e^2-d f\right ) c^2+2 d f (b e-a f) c-f^2 \left (b^2 d-a^2 f\right )\right )\right ) x}{c x^2+b x+a}dx}{c^2 d^2-b c e d+f \left (f a^2-b e a+b^2 d\right )+a c \left (e^2-2 d f\right )}+\frac {\left (b^2-4 a c\right ) \int -\frac {B d (c e-b f) \left (c e^2-b f e+2 a f^2-2 c d f\right )-A \left (\left (e^4-3 d f e^2+d^2 f^2\right ) c^2+2 f \left (a f \left (e^2-d f\right )-b \left (e^3-2 d e f\right )\right ) c-f^2 \left (-\left (\left (e^2-d f\right ) b^2\right )+2 a e f b-a^2 f^2\right )\right )-f \left (A (c e-b f) \left (c e^2-b f e+2 a f^2-2 c d f\right )+B \left (-d \left (e^2-d f\right ) c^2+2 d f (b e-a f) c-f^2 \left (b^2 d-a^2 f\right )\right )\right ) x}{f x^2+e x+d}dx}{c^2 d^2-b c e d+f \left (f a^2-b e a+b^2 d\right )+a c \left (e^2-2 d f\right )}}{\left (b^2-4 a c\right ) \left ((c d-a f)^2-(b d-a e) (c e-b f)\right )}-\frac {A c (2 a c e-b (c d+a f))+(A b-a B) \left (f b^2+2 c^2 d-c (b e+2 a f)\right )+c \left (A f b^2-(B c d+A c e+a B f) b+2 c (A c d+a B e-a A f)\right ) x}{\left (b^2-4 a c\right ) \left ((c d-a f)^2-(b d-a e) (c e-b f)\right ) \left (c x^2+b x+a\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {(B d-A e) f^2 b^5-2 f \left (-a A f^2+B c d e-A c \left (e^2-d f\right )\right ) b^4-\left (A c e \left (c e^2-3 a f^2-2 c d f\right )+B \left (a^2 f^3+3 a c d f^2-c^2 d \left (e^2+2 d f\right )\right )\right ) b^3-2 c \left (B c d e (c d-2 a f)+A f \left (2 c^2 d^2+5 a^2 f^2+5 a c \left (e^2-d f\right )\right )\right ) b^2+c \left (A c e \left (3 c^2 d^2+7 a^2 f^2+a c \left (5 e^2-2 d f\right )\right )+B \left (c^3 d^3-a c^2 \left (e^2+5 d f\right ) d+5 a^3 f^3+a^2 c f \left (3 e^2-d f\right )\right )\right ) b-2 c^2 \left (A \left (c^3 d^3+a c^2 \left (3 e^2-5 d f\right ) d-3 a^3 f^3-a^2 c f \left (e^2-7 d f\right )\right )-a B e \left (c^2 d^2-3 a^2 f^2-a c \left (e^2-2 d f\right )\right )\right )+c \left (b^2-4 a c\right ) \left (A (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )-B \left (-d \left (e^2-d f\right ) c^2+2 d f (b e-a f) c-f^2 \left (b^2 d-a^2 f\right )\right )\right ) x}{c x^2+b x+a}dx}{c^2 d^2-b c e d+f \left (f a^2-b e a+b^2 d\right )+a c \left (e^2-2 d f\right )}-\frac {\left (b^2-4 a c\right ) \int -\frac {B d (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )+A \left (\left (e^4-3 d f e^2+d^2 f^2\right ) c^2+2 f \left (a f \left (e^2-d f\right )-b \left (e^3-2 d e f\right )\right ) c-f^2 \left (-\left (\left (e^2-d f\right ) b^2\right )+2 a e f b-a^2 f^2\right )\right )-f \left (A (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )-B \left (-d \left (e^2-d f\right ) c^2+2 d f (b e-a f) c-f^2 \left (b^2 d-a^2 f\right )\right )\right ) x}{f x^2+e x+d}dx}{c^2 d^2-b c e d+f \left (f a^2-b e a+b^2 d\right )+a c \left (e^2-2 d f\right )}}{\left (b^2-4 a c\right ) \left ((c d-a f)^2-(b d-a e) (c e-b f)\right )}-\frac {A c (2 a c e-b (c d+a f))+(A b-a B) \left (f b^2+2 c^2 d-c (b e+2 a f)\right )+c \left (A f b^2-(B c d+A c e+a B f) b+2 c (A c d+a B e-a A f)\right ) x}{\left (b^2-4 a c\right ) \left ((c d-a f)^2-(b d-a e) (c e-b f)\right ) \left (c x^2+b x+a\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {(B d-A e) f^2 b^5-2 f \left (-a A f^2+B c d e-A c \left (e^2-d f\right )\right ) b^4-\left (A c e \left (c e^2-3 a f^2-2 c d f\right )+B \left (a^2 f^3+3 a c d f^2-c^2 d \left (e^2+2 d f\right )\right )\right ) b^3-2 c \left (B c d e (c d-2 a f)+A f \left (2 c^2 d^2+5 a^2 f^2+5 a c \left (e^2-d f\right )\right )\right ) b^2+c \left (A c e \left (3 c^2 d^2+7 a^2 f^2+a c \left (5 e^2-2 d f\right )\right )+B \left (c^3 d^3-a c^2 \left (e^2+5 d f\right ) d+5 a^3 f^3+a^2 c f \left (3 e^2-d f\right )\right )\right ) b-2 c^2 \left (A \left (c^3 d^3+a c^2 \left (3 e^2-5 d f\right ) d-3 a^3 f^3-a^2 c f \left (e^2-7 d f\right )\right )-a B e \left (c^2 d^2-3 a^2 f^2-a c \left (e^2-2 d f\right )\right )\right )+c \left (b^2-4 a c\right ) \left (A (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )-B \left (-d \left (e^2-d f\right ) c^2+2 d f (b e-a f) c-f^2 \left (b^2 d-a^2 f\right )\right )\right ) x}{c x^2+b x+a}dx}{c^2 d^2-b c e d+f \left (f a^2-b e a+b^2 d\right )+a c \left (e^2-2 d f\right )}+\frac {\left (b^2-4 a c\right ) \int -\frac {B d (c e-b f) \left (c e^2-b f e+2 a f^2-2 c d f\right )-A \left (\left (e^4-3 d f e^2+d^2 f^2\right ) c^2+2 f \left (a f \left (e^2-d f\right )-b \left (e^3-2 d e f\right )\right ) c-f^2 \left (-\left (\left (e^2-d f\right ) b^2\right )+2 a e f b-a^2 f^2\right )\right )-f \left (A (c e-b f) \left (c e^2-b f e+2 a f^2-2 c d f\right )+B \left (-d \left (e^2-d f\right ) c^2+2 d f (b e-a f) c-f^2 \left (b^2 d-a^2 f\right )\right )\right ) x}{f x^2+e x+d}dx}{c^2 d^2-b c e d+f \left (f a^2-b e a+b^2 d\right )+a c \left (e^2-2 d f\right )}}{\left (b^2-4 a c\right ) \left ((c d-a f)^2-(b d-a e) (c e-b f)\right )}-\frac {A c (2 a c e-b (c d+a f))+(A b-a B) \left (f b^2+2 c^2 d-c (b e+2 a f)\right )+c \left (A f b^2-(B c d+A c e+a B f) b+2 c (A c d+a B e-a A f)\right ) x}{\left (b^2-4 a c\right ) \left ((c d-a f)^2-(b d-a e) (c e-b f)\right ) \left (c x^2+b x+a\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {(B d-A e) f^2 b^5-2 f \left (-a A f^2+B c d e-A c \left (e^2-d f\right )\right ) b^4-\left (A c e \left (c e^2-3 a f^2-2 c d f\right )+B \left (a^2 f^3+3 a c d f^2-c^2 d \left (e^2+2 d f\right )\right )\right ) b^3-2 c \left (B c d e (c d-2 a f)+A f \left (2 c^2 d^2+5 a^2 f^2+5 a c \left (e^2-d f\right )\right )\right ) b^2+c \left (A c e \left (3 c^2 d^2+7 a^2 f^2+a c \left (5 e^2-2 d f\right )\right )+B \left (c^3 d^3-a c^2 \left (e^2+5 d f\right ) d+5 a^3 f^3+a^2 c f \left (3 e^2-d f\right )\right )\right ) b-2 c^2 \left (A \left (c^3 d^3+a c^2 \left (3 e^2-5 d f\right ) d-3 a^3 f^3-a^2 c f \left (e^2-7 d f\right )\right )-a B e \left (c^2 d^2-3 a^2 f^2-a c \left (e^2-2 d f\right )\right )\right )+c \left (b^2-4 a c\right ) \left (A (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )-B \left (-d \left (e^2-d f\right ) c^2+2 d f (b e-a f) c-f^2 \left (b^2 d-a^2 f\right )\right )\right ) x}{c x^2+b x+a}dx}{c^2 d^2-b c e d+f \left (f a^2-b e a+b^2 d\right )+a c \left (e^2-2 d f\right )}-\frac {\left (b^2-4 a c\right ) \int -\frac {B d (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )+A \left (\left (e^4-3 d f e^2+d^2 f^2\right ) c^2+2 f \left (a f \left (e^2-d f\right )-b \left (e^3-2 d e f\right )\right ) c-f^2 \left (-\left (\left (e^2-d f\right ) b^2\right )+2 a e f b-a^2 f^2\right )\right )-f \left (A (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )-B \left (-d \left (e^2-d f\right ) c^2+2 d f (b e-a f) c-f^2 \left (b^2 d-a^2 f\right )\right )\right ) x}{f x^2+e x+d}dx}{c^2 d^2-b c e d+f \left (f a^2-b e a+b^2 d\right )+a c \left (e^2-2 d f\right )}}{\left (b^2-4 a c\right ) \left ((c d-a f)^2-(b d-a e) (c e-b f)\right )}-\frac {A c (2 a c e-b (c d+a f))+(A b-a B) \left (f b^2+2 c^2 d-c (b e+2 a f)\right )+c \left (A f b^2-(B c d+A c e+a B f) b+2 c (A c d+a B e-a A f)\right ) x}{\left (b^2-4 a c\right ) \left ((c d-a f)^2-(b d-a e) (c e-b f)\right ) \left (c x^2+b x+a\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {(B d-A e) f^2 b^5-2 f \left (-a A f^2+B c d e-A c \left (e^2-d f\right )\right ) b^4-\left (A c e \left (c e^2-3 a f^2-2 c d f\right )+B \left (a^2 f^3+3 a c d f^2-c^2 d \left (e^2+2 d f\right )\right )\right ) b^3-2 c \left (B c d e (c d-2 a f)+A f \left (2 c^2 d^2+5 a^2 f^2+5 a c \left (e^2-d f\right )\right )\right ) b^2+c \left (A c e \left (3 c^2 d^2+7 a^2 f^2+a c \left (5 e^2-2 d f\right )\right )+B \left (c^3 d^3-a c^2 \left (e^2+5 d f\right ) d+5 a^3 f^3+a^2 c f \left (3 e^2-d f\right )\right )\right ) b-2 c^2 \left (A \left (c^3 d^3+a c^2 \left (3 e^2-5 d f\right ) d-3 a^3 f^3-a^2 c f \left (e^2-7 d f\right )\right )-a B e \left (c^2 d^2-3 a^2 f^2-a c \left (e^2-2 d f\right )\right )\right )+c \left (b^2-4 a c\right ) \left (A (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )-B \left (-d \left (e^2-d f\right ) c^2+2 d f (b e-a f) c-f^2 \left (b^2 d-a^2 f\right )\right )\right ) x}{c x^2+b x+a}dx}{c^2 d^2-b c e d+f \left (f a^2-b e a+b^2 d\right )+a c \left (e^2-2 d f\right )}+\frac {\left (b^2-4 a c\right ) \int -\frac {B d (c e-b f) \left (c e^2-b f e+2 a f^2-2 c d f\right )-A \left (\left (e^4-3 d f e^2+d^2 f^2\right ) c^2+2 f \left (a f \left (e^2-d f\right )-b \left (e^3-2 d e f\right )\right ) c-f^2 \left (-\left (\left (e^2-d f\right ) b^2\right )+2 a e f b-a^2 f^2\right )\right )-f \left (A (c e-b f) \left (c e^2-b f e+2 a f^2-2 c d f\right )+B \left (-d \left (e^2-d f\right ) c^2+2 d f (b e-a f) c-f^2 \left (b^2 d-a^2 f\right )\right )\right ) x}{f x^2+e x+d}dx}{c^2 d^2-b c e d+f \left (f a^2-b e a+b^2 d\right )+a c \left (e^2-2 d f\right )}}{\left (b^2-4 a c\right ) \left ((c d-a f)^2-(b d-a e) (c e-b f)\right )}-\frac {A c (2 a c e-b (c d+a f))+(A b-a B) \left (f b^2+2 c^2 d-c (b e+2 a f)\right )+c \left (A f b^2-(B c d+A c e+a B f) b+2 c (A c d+a B e-a A f)\right ) x}{\left (b^2-4 a c\right ) \left ((c d-a f)^2-(b d-a e) (c e-b f)\right ) \left (c x^2+b x+a\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {(B d-A e) f^2 b^5-2 f \left (-a A f^2+B c d e-A c \left (e^2-d f\right )\right ) b^4-\left (A c e \left (c e^2-3 a f^2-2 c d f\right )+B \left (a^2 f^3+3 a c d f^2-c^2 d \left (e^2+2 d f\right )\right )\right ) b^3-2 c \left (B c d e (c d-2 a f)+A f \left (2 c^2 d^2+5 a^2 f^2+5 a c \left (e^2-d f\right )\right )\right ) b^2+c \left (A c e \left (3 c^2 d^2+7 a^2 f^2+a c \left (5 e^2-2 d f\right )\right )+B \left (c^3 d^3-a c^2 \left (e^2+5 d f\right ) d+5 a^3 f^3+a^2 c f \left (3 e^2-d f\right )\right )\right ) b-2 c^2 \left (A \left (c^3 d^3+a c^2 \left (3 e^2-5 d f\right ) d-3 a^3 f^3-a^2 c f \left (e^2-7 d f\right )\right )-a B e \left (c^2 d^2-3 a^2 f^2-a c \left (e^2-2 d f\right )\right )\right )+c \left (b^2-4 a c\right ) \left (A (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )-B \left (-d \left (e^2-d f\right ) c^2+2 d f (b e-a f) c-f^2 \left (b^2 d-a^2 f\right )\right )\right ) x}{c x^2+b x+a}dx}{c^2 d^2-b c e d+f \left (f a^2-b e a+b^2 d\right )+a c \left (e^2-2 d f\right )}-\frac {\left (b^2-4 a c\right ) \int -\frac {B d (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )+A \left (\left (e^4-3 d f e^2+d^2 f^2\right ) c^2+2 f \left (a f \left (e^2-d f\right )-b \left (e^3-2 d e f\right )\right ) c-f^2 \left (-\left (\left (e^2-d f\right ) b^2\right )+2 a e f b-a^2 f^2\right )\right )-f \left (A (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )-B \left (-d \left (e^2-d f\right ) c^2+2 d f (b e-a f) c-f^2 \left (b^2 d-a^2 f\right )\right )\right ) x}{f x^2+e x+d}dx}{c^2 d^2-b c e d+f \left (f a^2-b e a+b^2 d\right )+a c \left (e^2-2 d f\right )}}{\left (b^2-4 a c\right ) \left ((c d-a f)^2-(b d-a e) (c e-b f)\right )}-\frac {A c (2 a c e-b (c d+a f))+(A b-a B) \left (f b^2+2 c^2 d-c (b e+2 a f)\right )+c \left (A f b^2-(B c d+A c e+a B f) b+2 c (A c d+a B e-a A f)\right ) x}{\left (b^2-4 a c\right ) \left ((c d-a f)^2-(b d-a e) (c e-b f)\right ) \left (c x^2+b x+a\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {(B d-A e) f^2 b^5-2 f \left (-a A f^2+B c d e-A c \left (e^2-d f\right )\right ) b^4-\left (A c e \left (c e^2-3 a f^2-2 c d f\right )+B \left (a^2 f^3+3 a c d f^2-c^2 d \left (e^2+2 d f\right )\right )\right ) b^3-2 c \left (B c d e (c d-2 a f)+A f \left (2 c^2 d^2+5 a^2 f^2+5 a c \left (e^2-d f\right )\right )\right ) b^2+c \left (A c e \left (3 c^2 d^2+7 a^2 f^2+a c \left (5 e^2-2 d f\right )\right )+B \left (c^3 d^3-a c^2 \left (e^2+5 d f\right ) d+5 a^3 f^3+a^2 c f \left (3 e^2-d f\right )\right )\right ) b-2 c^2 \left (A \left (c^3 d^3+a c^2 \left (3 e^2-5 d f\right ) d-3 a^3 f^3-a^2 c f \left (e^2-7 d f\right )\right )-a B e \left (c^2 d^2-3 a^2 f^2-a c \left (e^2-2 d f\right )\right )\right )+c \left (b^2-4 a c\right ) \left (A (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )-B \left (-d \left (e^2-d f\right ) c^2+2 d f (b e-a f) c-f^2 \left (b^2 d-a^2 f\right )\right )\right ) x}{c x^2+b x+a}dx}{c^2 d^2-b c e d+f \left (f a^2-b e a+b^2 d\right )+a c \left (e^2-2 d f\right )}+\frac {\left (b^2-4 a c\right ) \int -\frac {B d (c e-b f) \left (c e^2-b f e+2 a f^2-2 c d f\right )-A \left (\left (e^4-3 d f e^2+d^2 f^2\right ) c^2+2 f \left (a f \left (e^2-d f\right )-b \left (e^3-2 d e f\right )\right ) c-f^2 \left (-\left (\left (e^2-d f\right ) b^2\right )+2 a e f b-a^2 f^2\right )\right )-f \left (A (c e-b f) \left (c e^2-b f e+2 a f^2-2 c d f\right )+B \left (-d \left (e^2-d f\right ) c^2+2 d f (b e-a f) c-f^2 \left (b^2 d-a^2 f\right )\right )\right ) x}{f x^2+e x+d}dx}{c^2 d^2-b c e d+f \left (f a^2-b e a+b^2 d\right )+a c \left (e^2-2 d f\right )}}{\left (b^2-4 a c\right ) \left ((c d-a f)^2-(b d-a e) (c e-b f)\right )}-\frac {A c (2 a c e-b (c d+a f))+(A b-a B) \left (f b^2+2 c^2 d-c (b e+2 a f)\right )+c \left (A f b^2-(B c d+A c e+a B f) b+2 c (A c d+a B e-a A f)\right ) x}{\left (b^2-4 a c\right ) \left ((c d-a f)^2-(b d-a e) (c e-b f)\right ) \left (c x^2+b x+a\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {(B d-A e) f^2 b^5-2 f \left (-a A f^2+B c d e-A c \left (e^2-d f\right )\right ) b^4-\left (A c e \left (c e^2-3 a f^2-2 c d f\right )+B \left (a^2 f^3+3 a c d f^2-c^2 d \left (e^2+2 d f\right )\right )\right ) b^3-2 c \left (B c d e (c d-2 a f)+A f \left (2 c^2 d^2+5 a^2 f^2+5 a c \left (e^2-d f\right )\right )\right ) b^2+c \left (A c e \left (3 c^2 d^2+7 a^2 f^2+a c \left (5 e^2-2 d f\right )\right )+B \left (c^3 d^3-a c^2 \left (e^2+5 d f\right ) d+5 a^3 f^3+a^2 c f \left (3 e^2-d f\right )\right )\right ) b-2 c^2 \left (A \left (c^3 d^3+a c^2 \left (3 e^2-5 d f\right ) d-3 a^3 f^3-a^2 c f \left (e^2-7 d f\right )\right )-a B e \left (c^2 d^2-3 a^2 f^2-a c \left (e^2-2 d f\right )\right )\right )+c \left (b^2-4 a c\right ) \left (A (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )-B \left (-d \left (e^2-d f\right ) c^2+2 d f (b e-a f) c-f^2 \left (b^2 d-a^2 f\right )\right )\right ) x}{c x^2+b x+a}dx}{c^2 d^2-b c e d+f \left (f a^2-b e a+b^2 d\right )+a c \left (e^2-2 d f\right )}-\frac {\left (b^2-4 a c\right ) \int -\frac {B d (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )+A \left (\left (e^4-3 d f e^2+d^2 f^2\right ) c^2+2 f \left (a f \left (e^2-d f\right )-b \left (e^3-2 d e f\right )\right ) c-f^2 \left (-\left (\left (e^2-d f\right ) b^2\right )+2 a e f b-a^2 f^2\right )\right )-f \left (A (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )-B \left (-d \left (e^2-d f\right ) c^2+2 d f (b e-a f) c-f^2 \left (b^2 d-a^2 f\right )\right )\right ) x}{f x^2+e x+d}dx}{c^2 d^2-b c e d+f \left (f a^2-b e a+b^2 d\right )+a c \left (e^2-2 d f\right )}}{\left (b^2-4 a c\right ) \left ((c d-a f)^2-(b d-a e) (c e-b f)\right )}-\frac {A c (2 a c e-b (c d+a f))+(A b-a B) \left (f b^2+2 c^2 d-c (b e+2 a f)\right )+c \left (A f b^2-(B c d+A c e+a B f) b+2 c (A c d+a B e-a A f)\right ) x}{\left (b^2-4 a c\right ) \left ((c d-a f)^2-(b d-a e) (c e-b f)\right ) \left (c x^2+b x+a\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {(B d-A e) f^2 b^5-2 f \left (-a A f^2+B c d e-A c \left (e^2-d f\right )\right ) b^4-\left (A c e \left (c e^2-3 a f^2-2 c d f\right )+B \left (a^2 f^3+3 a c d f^2-c^2 d \left (e^2+2 d f\right )\right )\right ) b^3-2 c \left (B c d e (c d-2 a f)+A f \left (2 c^2 d^2+5 a^2 f^2+5 a c \left (e^2-d f\right )\right )\right ) b^2+c \left (A c e \left (3 c^2 d^2+7 a^2 f^2+a c \left (5 e^2-2 d f\right )\right )+B \left (c^3 d^3-a c^2 \left (e^2+5 d f\right ) d+5 a^3 f^3+a^2 c f \left (3 e^2-d f\right )\right )\right ) b-2 c^2 \left (A \left (c^3 d^3+a c^2 \left (3 e^2-5 d f\right ) d-3 a^3 f^3-a^2 c f \left (e^2-7 d f\right )\right )-a B e \left (c^2 d^2-3 a^2 f^2-a c \left (e^2-2 d f\right )\right )\right )+c \left (b^2-4 a c\right ) \left (A (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )-B \left (-d \left (e^2-d f\right ) c^2+2 d f (b e-a f) c-f^2 \left (b^2 d-a^2 f\right )\right )\right ) x}{c x^2+b x+a}dx}{c^2 d^2-b c e d+f \left (f a^2-b e a+b^2 d\right )+a c \left (e^2-2 d f\right )}+\frac {\left (b^2-4 a c\right ) \int -\frac {B d (c e-b f) \left (c e^2-b f e+2 a f^2-2 c d f\right )-A \left (\left (e^4-3 d f e^2+d^2 f^2\right ) c^2+2 f \left (a f \left (e^2-d f\right )-b \left (e^3-2 d e f\right )\right ) c-f^2 \left (-\left (\left (e^2-d f\right ) b^2\right )+2 a e f b-a^2 f^2\right )\right )-f \left (A (c e-b f) \left (c e^2-b f e+2 a f^2-2 c d f\right )+B \left (-d \left (e^2-d f\right ) c^2+2 d f (b e-a f) c-f^2 \left (b^2 d-a^2 f\right )\right )\right ) x}{f x^2+e x+d}dx}{c^2 d^2-b c e d+f \left (f a^2-b e a+b^2 d\right )+a c \left (e^2-2 d f\right )}}{\left (b^2-4 a c\right ) \left ((c d-a f)^2-(b d-a e) (c e-b f)\right )}-\frac {A c (2 a c e-b (c d+a f))+(A b-a B) \left (f b^2+2 c^2 d-c (b e+2 a f)\right )+c \left (A f b^2-(B c d+A c e+a B f) b+2 c (A c d+a B e-a A f)\right ) x}{\left (b^2-4 a c\right ) \left ((c d-a f)^2-(b d-a e) (c e-b f)\right ) \left (c x^2+b x+a\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {(B d-A e) f^2 b^5-2 f \left (-a A f^2+B c d e-A c \left (e^2-d f\right )\right ) b^4-\left (A c e \left (c e^2-3 a f^2-2 c d f\right )+B \left (a^2 f^3+3 a c d f^2-c^2 d \left (e^2+2 d f\right )\right )\right ) b^3-2 c \left (B c d e (c d-2 a f)+A f \left (2 c^2 d^2+5 a^2 f^2+5 a c \left (e^2-d f\right )\right )\right ) b^2+c \left (A c e \left (3 c^2 d^2+7 a^2 f^2+a c \left (5 e^2-2 d f\right )\right )+B \left (c^3 d^3-a c^2 \left (e^2+5 d f\right ) d+5 a^3 f^3+a^2 c f \left (3 e^2-d f\right )\right )\right ) b-2 c^2 \left (A \left (c^3 d^3+a c^2 \left (3 e^2-5 d f\right ) d-3 a^3 f^3-a^2 c f \left (e^2-7 d f\right )\right )-a B e \left (c^2 d^2-3 a^2 f^2-a c \left (e^2-2 d f\right )\right )\right )+c \left (b^2-4 a c\right ) \left (A (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )-B \left (-d \left (e^2-d f\right ) c^2+2 d f (b e-a f) c-f^2 \left (b^2 d-a^2 f\right )\right )\right ) x}{c x^2+b x+a}dx}{c^2 d^2-b c e d+f \left (f a^2-b e a+b^2 d\right )+a c \left (e^2-2 d f\right )}-\frac {\left (b^2-4 a c\right ) \int -\frac {B d (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )+A \left (\left (e^4-3 d f e^2+d^2 f^2\right ) c^2+2 f \left (a f \left (e^2-d f\right )-b \left (e^3-2 d e f\right )\right ) c-f^2 \left (-\left (\left (e^2-d f\right ) b^2\right )+2 a e f b-a^2 f^2\right )\right )-f \left (A (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )-B \left (-d \left (e^2-d f\right ) c^2+2 d f (b e-a f) c-f^2 \left (b^2 d-a^2 f\right )\right )\right ) x}{f x^2+e x+d}dx}{c^2 d^2-b c e d+f \left (f a^2-b e a+b^2 d\right )+a c \left (e^2-2 d f\right )}}{\left (b^2-4 a c\right ) \left ((c d-a f)^2-(b d-a e) (c e-b f)\right )}-\frac {A c (2 a c e-b (c d+a f))+(A b-a B) \left (f b^2+2 c^2 d-c (b e+2 a f)\right )+c \left (A f b^2-(B c d+A c e+a B f) b+2 c (A c d+a B e-a A f)\right ) x}{\left (b^2-4 a c\right ) \left ((c d-a f)^2-(b d-a e) (c e-b f)\right ) \left (c x^2+b x+a\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {(B d-A e) f^2 b^5-2 f \left (-a A f^2+B c d e-A c \left (e^2-d f\right )\right ) b^4-\left (A c e \left (c e^2-3 a f^2-2 c d f\right )+B \left (a^2 f^3+3 a c d f^2-c^2 d \left (e^2+2 d f\right )\right )\right ) b^3-2 c \left (B c d e (c d-2 a f)+A f \left (2 c^2 d^2+5 a^2 f^2+5 a c \left (e^2-d f\right )\right )\right ) b^2+c \left (A c e \left (3 c^2 d^2+7 a^2 f^2+a c \left (5 e^2-2 d f\right )\right )+B \left (c^3 d^3-a c^2 \left (e^2+5 d f\right ) d+5 a^3 f^3+a^2 c f \left (3 e^2-d f\right )\right )\right ) b-2 c^2 \left (A \left (c^3 d^3+a c^2 \left (3 e^2-5 d f\right ) d-3 a^3 f^3-a^2 c f \left (e^2-7 d f\right )\right )-a B e \left (c^2 d^2-3 a^2 f^2-a c \left (e^2-2 d f\right )\right )\right )+c \left (b^2-4 a c\right ) \left (A (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )-B \left (-d \left (e^2-d f\right ) c^2+2 d f (b e-a f) c-f^2 \left (b^2 d-a^2 f\right )\right )\right ) x}{c x^2+b x+a}dx}{c^2 d^2-b c e d+f \left (f a^2-b e a+b^2 d\right )+a c \left (e^2-2 d f\right )}+\frac {\left (b^2-4 a c\right ) \int -\frac {B d (c e-b f) \left (c e^2-b f e+2 a f^2-2 c d f\right )-A \left (\left (e^4-3 d f e^2+d^2 f^2\right ) c^2+2 f \left (a f \left (e^2-d f\right )-b \left (e^3-2 d e f\right )\right ) c-f^2 \left (-\left (\left (e^2-d f\right ) b^2\right )+2 a e f b-a^2 f^2\right )\right )-f \left (A (c e-b f) \left (c e^2-b f e+2 a f^2-2 c d f\right )+B \left (-d \left (e^2-d f\right ) c^2+2 d f (b e-a f) c-f^2 \left (b^2 d-a^2 f\right )\right )\right ) x}{f x^2+e x+d}dx}{c^2 d^2-b c e d+f \left (f a^2-b e a+b^2 d\right )+a c \left (e^2-2 d f\right )}}{\left (b^2-4 a c\right ) \left ((c d-a f)^2-(b d-a e) (c e-b f)\right )}-\frac {A c (2 a c e-b (c d+a f))+(A b-a B) \left (f b^2+2 c^2 d-c (b e+2 a f)\right )+c \left (A f b^2-(B c d+A c e+a B f) b+2 c (A c d+a B e-a A f)\right ) x}{\left (b^2-4 a c\right ) \left ((c d-a f)^2-(b d-a e) (c e-b f)\right ) \left (c x^2+b x+a\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {(B d-A e) f^2 b^5-2 f \left (-a A f^2+B c d e-A c \left (e^2-d f\right )\right ) b^4-\left (A c e \left (c e^2-3 a f^2-2 c d f\right )+B \left (a^2 f^3+3 a c d f^2-c^2 d \left (e^2+2 d f\right )\right )\right ) b^3-2 c \left (B c d e (c d-2 a f)+A f \left (2 c^2 d^2+5 a^2 f^2+5 a c \left (e^2-d f\right )\right )\right ) b^2+c \left (A c e \left (3 c^2 d^2+7 a^2 f^2+a c \left (5 e^2-2 d f\right )\right )+B \left (c^3 d^3-a c^2 \left (e^2+5 d f\right ) d+5 a^3 f^3+a^2 c f \left (3 e^2-d f\right )\right )\right ) b-2 c^2 \left (A \left (c^3 d^3+a c^2 \left (3 e^2-5 d f\right ) d-3 a^3 f^3-a^2 c f \left (e^2-7 d f\right )\right )-a B e \left (c^2 d^2-3 a^2 f^2-a c \left (e^2-2 d f\right )\right )\right )+c \left (b^2-4 a c\right ) \left (A (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )-B \left (-d \left (e^2-d f\right ) c^2+2 d f (b e-a f) c-f^2 \left (b^2 d-a^2 f\right )\right )\right ) x}{c x^2+b x+a}dx}{c^2 d^2-b c e d+f \left (f a^2-b e a+b^2 d\right )+a c \left (e^2-2 d f\right )}-\frac {\left (b^2-4 a c\right ) \int -\frac {B d (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )+A \left (\left (e^4-3 d f e^2+d^2 f^2\right ) c^2+2 f \left (a f \left (e^2-d f\right )-b \left (e^3-2 d e f\right )\right ) c-f^2 \left (-\left (\left (e^2-d f\right ) b^2\right )+2 a e f b-a^2 f^2\right )\right )-f \left (A (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )-B \left (-d \left (e^2-d f\right ) c^2+2 d f (b e-a f) c-f^2 \left (b^2 d-a^2 f\right )\right )\right ) x}{f x^2+e x+d}dx}{c^2 d^2-b c e d+f \left (f a^2-b e a+b^2 d\right )+a c \left (e^2-2 d f\right )}}{\left (b^2-4 a c\right ) \left ((c d-a f)^2-(b d-a e) (c e-b f)\right )}-\frac {A c (2 a c e-b (c d+a f))+(A b-a B) \left (f b^2+2 c^2 d-c (b e+2 a f)\right )+c \left (A f b^2-(B c d+A c e+a B f) b+2 c (A c d+a B e-a A f)\right ) x}{\left (b^2-4 a c\right ) \left ((c d-a f)^2-(b d-a e) (c e-b f)\right ) \left (c x^2+b x+a\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {(B d-A e) f^2 b^5-2 f \left (-a A f^2+B c d e-A c \left (e^2-d f\right )\right ) b^4-\left (A c e \left (c e^2-3 a f^2-2 c d f\right )+B \left (a^2 f^3+3 a c d f^2-c^2 d \left (e^2+2 d f\right )\right )\right ) b^3-2 c \left (B c d e (c d-2 a f)+A f \left (2 c^2 d^2+5 a^2 f^2+5 a c \left (e^2-d f\right )\right )\right ) b^2+c \left (A c e \left (3 c^2 d^2+7 a^2 f^2+a c \left (5 e^2-2 d f\right )\right )+B \left (c^3 d^3-a c^2 \left (e^2+5 d f\right ) d+5 a^3 f^3+a^2 c f \left (3 e^2-d f\right )\right )\right ) b-2 c^2 \left (A \left (c^3 d^3+a c^2 \left (3 e^2-5 d f\right ) d-3 a^3 f^3-a^2 c f \left (e^2-7 d f\right )\right )-a B e \left (c^2 d^2-3 a^2 f^2-a c \left (e^2-2 d f\right )\right )\right )+c \left (b^2-4 a c\right ) \left (A (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )-B \left (-d \left (e^2-d f\right ) c^2+2 d f (b e-a f) c-f^2 \left (b^2 d-a^2 f\right )\right )\right ) x}{c x^2+b x+a}dx}{c^2 d^2-b c e d+f \left (f a^2-b e a+b^2 d\right )+a c \left (e^2-2 d f\right )}+\frac {\left (b^2-4 a c\right ) \int -\frac {B d (c e-b f) \left (c e^2-b f e+2 a f^2-2 c d f\right )-A \left (\left (e^4-3 d f e^2+d^2 f^2\right ) c^2+2 f \left (a f \left (e^2-d f\right )-b \left (e^3-2 d e f\right )\right ) c-f^2 \left (-\left (\left (e^2-d f\right ) b^2\right )+2 a e f b-a^2 f^2\right )\right )-f \left (A (c e-b f) \left (c e^2-b f e+2 a f^2-2 c d f\right )+B \left (-d \left (e^2-d f\right ) c^2+2 d f (b e-a f) c-f^2 \left (b^2 d-a^2 f\right )\right )\right ) x}{f x^2+e x+d}dx}{c^2 d^2-b c e d+f \left (f a^2-b e a+b^2 d\right )+a c \left (e^2-2 d f\right )}}{\left (b^2-4 a c\right ) \left ((c d-a f)^2-(b d-a e) (c e-b f)\right )}-\frac {A c (2 a c e-b (c d+a f))+(A b-a B) \left (f b^2+2 c^2 d-c (b e+2 a f)\right )+c \left (A f b^2-(B c d+A c e+a B f) b+2 c (A c d+a B e-a A f)\right ) x}{\left (b^2-4 a c\right ) \left ((c d-a f)^2-(b d-a e) (c e-b f)\right ) \left (c x^2+b x+a\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {(B d-A e) f^2 b^5-2 f \left (-a A f^2+B c d e-A c \left (e^2-d f\right )\right ) b^4-\left (A c e \left (c e^2-3 a f^2-2 c d f\right )+B \left (a^2 f^3+3 a c d f^2-c^2 d \left (e^2+2 d f\right )\right )\right ) b^3-2 c \left (B c d e (c d-2 a f)+A f \left (2 c^2 d^2+5 a^2 f^2+5 a c \left (e^2-d f\right )\right )\right ) b^2+c \left (A c e \left (3 c^2 d^2+7 a^2 f^2+a c \left (5 e^2-2 d f\right )\right )+B \left (c^3 d^3-a c^2 \left (e^2+5 d f\right ) d+5 a^3 f^3+a^2 c f \left (3 e^2-d f\right )\right )\right ) b-2 c^2 \left (A \left (c^3 d^3+a c^2 \left (3 e^2-5 d f\right ) d-3 a^3 f^3-a^2 c f \left (e^2-7 d f\right )\right )-a B e \left (c^2 d^2-3 a^2 f^2-a c \left (e^2-2 d f\right )\right )\right )+c \left (b^2-4 a c\right ) \left (A (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )-B \left (-d \left (e^2-d f\right ) c^2+2 d f (b e-a f) c-f^2 \left (b^2 d-a^2 f\right )\right )\right ) x}{c x^2+b x+a}dx}{c^2 d^2-b c e d+f \left (f a^2-b e a+b^2 d\right )+a c \left (e^2-2 d f\right )}-\frac {\left (b^2-4 a c\right ) \int -\frac {B d (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )+A \left (\left (e^4-3 d f e^2+d^2 f^2\right ) c^2+2 f \left (a f \left (e^2-d f\right )-b \left (e^3-2 d e f\right )\right ) c-f^2 \left (-\left (\left (e^2-d f\right ) b^2\right )+2 a e f b-a^2 f^2\right )\right )-f \left (A (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )-B \left (-d \left (e^2-d f\right ) c^2+2 d f (b e-a f) c-f^2 \left (b^2 d-a^2 f\right )\right )\right ) x}{f x^2+e x+d}dx}{c^2 d^2-b c e d+f \left (f a^2-b e a+b^2 d\right )+a c \left (e^2-2 d f\right )}}{\left (b^2-4 a c\right ) \left ((c d-a f)^2-(b d-a e) (c e-b f)\right )}-\frac {A c (2 a c e-b (c d+a f))+(A b-a B) \left (f b^2+2 c^2 d-c (b e+2 a f)\right )+c \left (A f b^2-(B c d+A c e+a B f) b+2 c (A c d+a B e-a A f)\right ) x}{\left (b^2-4 a c\right ) \left ((c d-a f)^2-(b d-a e) (c e-b f)\right ) \left (c x^2+b x+a\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {(B d-A e) f^2 b^5-2 f \left (-a A f^2+B c d e-A c \left (e^2-d f\right )\right ) b^4-\left (A c e \left (c e^2-3 a f^2-2 c d f\right )+B \left (a^2 f^3+3 a c d f^2-c^2 d \left (e^2+2 d f\right )\right )\right ) b^3-2 c \left (B c d e (c d-2 a f)+A f \left (2 c^2 d^2+5 a^2 f^2+5 a c \left (e^2-d f\right )\right )\right ) b^2+c \left (A c e \left (3 c^2 d^2+7 a^2 f^2+a c \left (5 e^2-2 d f\right )\right )+B \left (c^3 d^3-a c^2 \left (e^2+5 d f\right ) d+5 a^3 f^3+a^2 c f \left (3 e^2-d f\right )\right )\right ) b-2 c^2 \left (A \left (c^3 d^3+a c^2 \left (3 e^2-5 d f\right ) d-3 a^3 f^3-a^2 c f \left (e^2-7 d f\right )\right )-a B e \left (c^2 d^2-3 a^2 f^2-a c \left (e^2-2 d f\right )\right )\right )+c \left (b^2-4 a c\right ) \left (A (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )-B \left (-d \left (e^2-d f\right ) c^2+2 d f (b e-a f) c-f^2 \left (b^2 d-a^2 f\right )\right )\right ) x}{c x^2+b x+a}dx}{c^2 d^2-b c e d+f \left (f a^2-b e a+b^2 d\right )+a c \left (e^2-2 d f\right )}+\frac {\left (b^2-4 a c\right ) \int -\frac {B d (c e-b f) \left (c e^2-b f e+2 a f^2-2 c d f\right )-A \left (\left (e^4-3 d f e^2+d^2 f^2\right ) c^2+2 f \left (a f \left (e^2-d f\right )-b \left (e^3-2 d e f\right )\right ) c-f^2 \left (-\left (\left (e^2-d f\right ) b^2\right )+2 a e f b-a^2 f^2\right )\right )-f \left (A (c e-b f) \left (c e^2-b f e+2 a f^2-2 c d f\right )+B \left (-d \left (e^2-d f\right ) c^2+2 d f (b e-a f) c-f^2 \left (b^2 d-a^2 f\right )\right )\right ) x}{f x^2+e x+d}dx}{c^2 d^2-b c e d+f \left (f a^2-b e a+b^2 d\right )+a c \left (e^2-2 d f\right )}}{\left (b^2-4 a c\right ) \left ((c d-a f)^2-(b d-a e) (c e-b f)\right )}-\frac {A c (2 a c e-b (c d+a f))+(A b-a B) \left (f b^2+2 c^2 d-c (b e+2 a f)\right )+c \left (A f b^2-(B c d+A c e+a B f) b+2 c (A c d+a B e-a A f)\right ) x}{\left (b^2-4 a c\right ) \left ((c d-a f)^2-(b d-a e) (c e-b f)\right ) \left (c x^2+b x+a\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {(B d-A e) f^2 b^5-2 f \left (-a A f^2+B c d e-A c \left (e^2-d f\right )\right ) b^4-\left (A c e \left (c e^2-3 a f^2-2 c d f\right )+B \left (a^2 f^3+3 a c d f^2-c^2 d \left (e^2+2 d f\right )\right )\right ) b^3-2 c \left (B c d e (c d-2 a f)+A f \left (2 c^2 d^2+5 a^2 f^2+5 a c \left (e^2-d f\right )\right )\right ) b^2+c \left (A c e \left (3 c^2 d^2+7 a^2 f^2+a c \left (5 e^2-2 d f\right )\right )+B \left (c^3 d^3-a c^2 \left (e^2+5 d f\right ) d+5 a^3 f^3+a^2 c f \left (3 e^2-d f\right )\right )\right ) b-2 c^2 \left (A \left (c^3 d^3+a c^2 \left (3 e^2-5 d f\right ) d-3 a^3 f^3-a^2 c f \left (e^2-7 d f\right )\right )-a B e \left (c^2 d^2-3 a^2 f^2-a c \left (e^2-2 d f\right )\right )\right )+c \left (b^2-4 a c\right ) \left (A (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )-B \left (-d \left (e^2-d f\right ) c^2+2 d f (b e-a f) c-f^2 \left (b^2 d-a^2 f\right )\right )\right ) x}{c x^2+b x+a}dx}{c^2 d^2-b c e d+f \left (f a^2-b e a+b^2 d\right )+a c \left (e^2-2 d f\right )}-\frac {\left (b^2-4 a c\right ) \int -\frac {B d (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )+A \left (\left (e^4-3 d f e^2+d^2 f^2\right ) c^2+2 f \left (a f \left (e^2-d f\right )-b \left (e^3-2 d e f\right )\right ) c-f^2 \left (-\left (\left (e^2-d f\right ) b^2\right )+2 a e f b-a^2 f^2\right )\right )-f \left (A (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )-B \left (-d \left (e^2-d f\right ) c^2+2 d f (b e-a f) c-f^2 \left (b^2 d-a^2 f\right )\right )\right ) x}{f x^2+e x+d}dx}{c^2 d^2-b c e d+f \left (f a^2-b e a+b^2 d\right )+a c \left (e^2-2 d f\right )}}{\left (b^2-4 a c\right ) \left ((c d-a f)^2-(b d-a e) (c e-b f)\right )}-\frac {A c (2 a c e-b (c d+a f))+(A b-a B) \left (f b^2+2 c^2 d-c (b e+2 a f)\right )+c \left (A f b^2-(B c d+A c e+a B f) b+2 c (A c d+a B e-a A f)\right ) x}{\left (b^2-4 a c\right ) \left ((c d-a f)^2-(b d-a e) (c e-b f)\right ) \left (c x^2+b x+a\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {(B d-A e) f^2 b^5-2 f \left (-a A f^2+B c d e-A c \left (e^2-d f\right )\right ) b^4-\left (A c e \left (c e^2-3 a f^2-2 c d f\right )+B \left (a^2 f^3+3 a c d f^2-c^2 d \left (e^2+2 d f\right )\right )\right ) b^3-2 c \left (B c d e (c d-2 a f)+A f \left (2 c^2 d^2+5 a^2 f^2+5 a c \left (e^2-d f\right )\right )\right ) b^2+c \left (A c e \left (3 c^2 d^2+7 a^2 f^2+a c \left (5 e^2-2 d f\right )\right )+B \left (c^3 d^3-a c^2 \left (e^2+5 d f\right ) d+5 a^3 f^3+a^2 c f \left (3 e^2-d f\right )\right )\right ) b-2 c^2 \left (A \left (c^3 d^3+a c^2 \left (3 e^2-5 d f\right ) d-3 a^3 f^3-a^2 c f \left (e^2-7 d f\right )\right )-a B e \left (c^2 d^2-3 a^2 f^2-a c \left (e^2-2 d f\right )\right )\right )+c \left (b^2-4 a c\right ) \left (A (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )-B \left (-d \left (e^2-d f\right ) c^2+2 d f (b e-a f) c-f^2 \left (b^2 d-a^2 f\right )\right )\right ) x}{c x^2+b x+a}dx}{c^2 d^2-b c e d+f \left (f a^2-b e a+b^2 d\right )+a c \left (e^2-2 d f\right )}+\frac {\left (b^2-4 a c\right ) \int -\frac {B d (c e-b f) \left (c e^2-b f e+2 a f^2-2 c d f\right )-A \left (\left (e^4-3 d f e^2+d^2 f^2\right ) c^2+2 f \left (a f \left (e^2-d f\right )-b \left (e^3-2 d e f\right )\right ) c-f^2 \left (-\left (\left (e^2-d f\right ) b^2\right )+2 a e f b-a^2 f^2\right )\right )-f \left (A (c e-b f) \left (c e^2-b f e+2 a f^2-2 c d f\right )+B \left (-d \left (e^2-d f\right ) c^2+2 d f (b e-a f) c-f^2 \left (b^2 d-a^2 f\right )\right )\right ) x}{f x^2+e x+d}dx}{c^2 d^2-b c e d+f \left (f a^2-b e a+b^2 d\right )+a c \left (e^2-2 d f\right )}}{\left (b^2-4 a c\right ) \left ((c d-a f)^2-(b d-a e) (c e-b f)\right )}-\frac {A c (2 a c e-b (c d+a f))+(A b-a B) \left (f b^2+2 c^2 d-c (b e+2 a f)\right )+c \left (A f b^2-(B c d+A c e+a B f) b+2 c (A c d+a B e-a A f)\right ) x}{\left (b^2-4 a c\right ) \left ((c d-a f)^2-(b d-a e) (c e-b f)\right ) \left (c x^2+b x+a\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {(B d-A e) f^2 b^5-2 f \left (-a A f^2+B c d e-A c \left (e^2-d f\right )\right ) b^4-\left (A c e \left (c e^2-3 a f^2-2 c d f\right )+B \left (a^2 f^3+3 a c d f^2-c^2 d \left (e^2+2 d f\right )\right )\right ) b^3-2 c \left (B c d e (c d-2 a f)+A f \left (2 c^2 d^2+5 a^2 f^2+5 a c \left (e^2-d f\right )\right )\right ) b^2+c \left (A c e \left (3 c^2 d^2+7 a^2 f^2+a c \left (5 e^2-2 d f\right )\right )+B \left (c^3 d^3-a c^2 \left (e^2+5 d f\right ) d+5 a^3 f^3+a^2 c f \left (3 e^2-d f\right )\right )\right ) b-2 c^2 \left (A \left (c^3 d^3+a c^2 \left (3 e^2-5 d f\right ) d-3 a^3 f^3-a^2 c f \left (e^2-7 d f\right )\right )-a B e \left (c^2 d^2-3 a^2 f^2-a c \left (e^2-2 d f\right )\right )\right )+c \left (b^2-4 a c\right ) \left (A (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )-B \left (-d \left (e^2-d f\right ) c^2+2 d f (b e-a f) c-f^2 \left (b^2 d-a^2 f\right )\right )\right ) x}{c x^2+b x+a}dx}{c^2 d^2-b c e d+f \left (f a^2-b e a+b^2 d\right )+a c \left (e^2-2 d f\right )}-\frac {\left (b^2-4 a c\right ) \int -\frac {B d (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )+A \left (\left (e^4-3 d f e^2+d^2 f^2\right ) c^2+2 f \left (a f \left (e^2-d f\right )-b \left (e^3-2 d e f\right )\right ) c-f^2 \left (-\left (\left (e^2-d f\right ) b^2\right )+2 a e f b-a^2 f^2\right )\right )-f \left (A (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )-B \left (-d \left (e^2-d f\right ) c^2+2 d f (b e-a f) c-f^2 \left (b^2 d-a^2 f\right )\right )\right ) x}{f x^2+e x+d}dx}{c^2 d^2-b c e d+f \left (f a^2-b e a+b^2 d\right )+a c \left (e^2-2 d f\right )}}{\left (b^2-4 a c\right ) \left ((c d-a f)^2-(b d-a e) (c e-b f)\right )}-\frac {A c (2 a c e-b (c d+a f))+(A b-a B) \left (f b^2+2 c^2 d-c (b e+2 a f)\right )+c \left (A f b^2-(B c d+A c e+a B f) b+2 c (A c d+a B e-a A f)\right ) x}{\left (b^2-4 a c\right ) \left ((c d-a f)^2-(b d-a e) (c e-b f)\right ) \left (c x^2+b x+a\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {(B d-A e) f^2 b^5-2 f \left (-a A f^2+B c d e-A c \left (e^2-d f\right )\right ) b^4-\left (A c e \left (c e^2-3 a f^2-2 c d f\right )+B \left (a^2 f^3+3 a c d f^2-c^2 d \left (e^2+2 d f\right )\right )\right ) b^3-2 c \left (B c d e (c d-2 a f)+A f \left (2 c^2 d^2+5 a^2 f^2+5 a c \left (e^2-d f\right )\right )\right ) b^2+c \left (A c e \left (3 c^2 d^2+7 a^2 f^2+a c \left (5 e^2-2 d f\right )\right )+B \left (c^3 d^3-a c^2 \left (e^2+5 d f\right ) d+5 a^3 f^3+a^2 c f \left (3 e^2-d f\right )\right )\right ) b-2 c^2 \left (A \left (c^3 d^3+a c^2 \left (3 e^2-5 d f\right ) d-3 a^3 f^3-a^2 c f \left (e^2-7 d f\right )\right )-a B e \left (c^2 d^2-3 a^2 f^2-a c \left (e^2-2 d f\right )\right )\right )+c \left (b^2-4 a c\right ) \left (A (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )-B \left (-d \left (e^2-d f\right ) c^2+2 d f (b e-a f) c-f^2 \left (b^2 d-a^2 f\right )\right )\right ) x}{c x^2+b x+a}dx}{c^2 d^2-b c e d+f \left (f a^2-b e a+b^2 d\right )+a c \left (e^2-2 d f\right )}+\frac {\left (b^2-4 a c\right ) \int -\frac {B d (c e-b f) \left (c e^2-b f e+2 a f^2-2 c d f\right )-A \left (\left (e^4-3 d f e^2+d^2 f^2\right ) c^2+2 f \left (a f \left (e^2-d f\right )-b \left (e^3-2 d e f\right )\right ) c-f^2 \left (-\left (\left (e^2-d f\right ) b^2\right )+2 a e f b-a^2 f^2\right )\right )-f \left (A (c e-b f) \left (c e^2-b f e+2 a f^2-2 c d f\right )+B \left (-d \left (e^2-d f\right ) c^2+2 d f (b e-a f) c-f^2 \left (b^2 d-a^2 f\right )\right )\right ) x}{f x^2+e x+d}dx}{c^2 d^2-b c e d+f \left (f a^2-b e a+b^2 d\right )+a c \left (e^2-2 d f\right )}}{\left (b^2-4 a c\right ) \left ((c d-a f)^2-(b d-a e) (c e-b f)\right )}-\frac {A c (2 a c e-b (c d+a f))+(A b-a B) \left (f b^2+2 c^2 d-c (b e+2 a f)\right )+c \left (A f b^2-(B c d+A c e+a B f) b+2 c (A c d+a B e-a A f)\right ) x}{\left (b^2-4 a c\right ) \left ((c d-a f)^2-(b d-a e) (c e-b f)\right ) \left (c x^2+b x+a\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {(B d-A e) f^2 b^5-2 f \left (-a A f^2+B c d e-A c \left (e^2-d f\right )\right ) b^4-\left (A c e \left (c e^2-3 a f^2-2 c d f\right )+B \left (a^2 f^3+3 a c d f^2-c^2 d \left (e^2+2 d f\right )\right )\right ) b^3-2 c \left (B c d e (c d-2 a f)+A f \left (2 c^2 d^2+5 a^2 f^2+5 a c \left (e^2-d f\right )\right )\right ) b^2+c \left (A c e \left (3 c^2 d^2+7 a^2 f^2+a c \left (5 e^2-2 d f\right )\right )+B \left (c^3 d^3-a c^2 \left (e^2+5 d f\right ) d+5 a^3 f^3+a^2 c f \left (3 e^2-d f\right )\right )\right ) b-2 c^2 \left (A \left (c^3 d^3+a c^2 \left (3 e^2-5 d f\right ) d-3 a^3 f^3-a^2 c f \left (e^2-7 d f\right )\right )-a B e \left (c^2 d^2-3 a^2 f^2-a c \left (e^2-2 d f\right )\right )\right )+c \left (b^2-4 a c\right ) \left (A (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )-B \left (-d \left (e^2-d f\right ) c^2+2 d f (b e-a f) c-f^2 \left (b^2 d-a^2 f\right )\right )\right ) x}{c x^2+b x+a}dx}{c^2 d^2-b c e d+f \left (f a^2-b e a+b^2 d\right )+a c \left (e^2-2 d f\right )}-\frac {\left (b^2-4 a c\right ) \int -\frac {B d (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )+A \left (\left (e^4-3 d f e^2+d^2 f^2\right ) c^2+2 f \left (a f \left (e^2-d f\right )-b \left (e^3-2 d e f\right )\right ) c-f^2 \left (-\left (\left (e^2-d f\right ) b^2\right )+2 a e f b-a^2 f^2\right )\right )-f \left (A (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )-B \left (-d \left (e^2-d f\right ) c^2+2 d f (b e-a f) c-f^2 \left (b^2 d-a^2 f\right )\right )\right ) x}{f x^2+e x+d}dx}{c^2 d^2-b c e d+f \left (f a^2-b e a+b^2 d\right )+a c \left (e^2-2 d f\right )}}{\left (b^2-4 a c\right ) \left ((c d-a f)^2-(b d-a e) (c e-b f)\right )}-\frac {A c (2 a c e-b (c d+a f))+(A b-a B) \left (f b^2+2 c^2 d-c (b e+2 a f)\right )+c \left (A f b^2-(B c d+A c e+a B f) b+2 c (A c d+a B e-a A f)\right ) x}{\left (b^2-4 a c\right ) \left ((c d-a f)^2-(b d-a e) (c e-b f)\right ) \left (c x^2+b x+a\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {(B d-A e) f^2 b^5-2 f \left (-a A f^2+B c d e-A c \left (e^2-d f\right )\right ) b^4-\left (A c e \left (c e^2-3 a f^2-2 c d f\right )+B \left (a^2 f^3+3 a c d f^2-c^2 d \left (e^2+2 d f\right )\right )\right ) b^3-2 c \left (B c d e (c d-2 a f)+A f \left (2 c^2 d^2+5 a^2 f^2+5 a c \left (e^2-d f\right )\right )\right ) b^2+c \left (A c e \left (3 c^2 d^2+7 a^2 f^2+a c \left (5 e^2-2 d f\right )\right )+B \left (c^3 d^3-a c^2 \left (e^2+5 d f\right ) d+5 a^3 f^3+a^2 c f \left (3 e^2-d f\right )\right )\right ) b-2 c^2 \left (A \left (c^3 d^3+a c^2 \left (3 e^2-5 d f\right ) d-3 a^3 f^3-a^2 c f \left (e^2-7 d f\right )\right )-a B e \left (c^2 d^2-3 a^2 f^2-a c \left (e^2-2 d f\right )\right )\right )+c \left (b^2-4 a c\right ) \left (A (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )-B \left (-d \left (e^2-d f\right ) c^2+2 d f (b e-a f) c-f^2 \left (b^2 d-a^2 f\right )\right )\right ) x}{c x^2+b x+a}dx}{c^2 d^2-b c e d+f \left (f a^2-b e a+b^2 d\right )+a c \left (e^2-2 d f\right )}+\frac {\left (b^2-4 a c\right ) \int -\frac {B d (c e-b f) \left (c e^2-b f e+2 a f^2-2 c d f\right )-A \left (\left (e^4-3 d f e^2+d^2 f^2\right ) c^2+2 f \left (a f \left (e^2-d f\right )-b \left (e^3-2 d e f\right )\right ) c-f^2 \left (-\left (\left (e^2-d f\right ) b^2\right )+2 a e f b-a^2 f^2\right )\right )-f \left (A (c e-b f) \left (c e^2-b f e+2 a f^2-2 c d f\right )+B \left (-d \left (e^2-d f\right ) c^2+2 d f (b e-a f) c-f^2 \left (b^2 d-a^2 f\right )\right )\right ) x}{f x^2+e x+d}dx}{c^2 d^2-b c e d+f \left (f a^2-b e a+b^2 d\right )+a c \left (e^2-2 d f\right )}}{\left (b^2-4 a c\right ) \left ((c d-a f)^2-(b d-a e) (c e-b f)\right )}-\frac {A c (2 a c e-b (c d+a f))+(A b-a B) \left (f b^2+2 c^2 d-c (b e+2 a f)\right )+c \left (A f b^2-(B c d+A c e+a B f) b+2 c (A c d+a B e-a A f)\right ) x}{\left (b^2-4 a c\right ) \left ((c d-a f)^2-(b d-a e) (c e-b f)\right ) \left (c x^2+b x+a\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {(B d-A e) f^2 b^5-2 f \left (-a A f^2+B c d e-A c \left (e^2-d f\right )\right ) b^4-\left (A c e \left (c e^2-3 a f^2-2 c d f\right )+B \left (a^2 f^3+3 a c d f^2-c^2 d \left (e^2+2 d f\right )\right )\right ) b^3-2 c \left (B c d e (c d-2 a f)+A f \left (2 c^2 d^2+5 a^2 f^2+5 a c \left (e^2-d f\right )\right )\right ) b^2+c \left (A c e \left (3 c^2 d^2+7 a^2 f^2+a c \left (5 e^2-2 d f\right )\right )+B \left (c^3 d^3-a c^2 \left (e^2+5 d f\right ) d+5 a^3 f^3+a^2 c f \left (3 e^2-d f\right )\right )\right ) b-2 c^2 \left (A \left (c^3 d^3+a c^2 \left (3 e^2-5 d f\right ) d-3 a^3 f^3-a^2 c f \left (e^2-7 d f\right )\right )-a B e \left (c^2 d^2-3 a^2 f^2-a c \left (e^2-2 d f\right )\right )\right )+c \left (b^2-4 a c\right ) \left (A (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )-B \left (-d \left (e^2-d f\right ) c^2+2 d f (b e-a f) c-f^2 \left (b^2 d-a^2 f\right )\right )\right ) x}{c x^2+b x+a}dx}{c^2 d^2-b c e d+f \left (f a^2-b e a+b^2 d\right )+a c \left (e^2-2 d f\right )}-\frac {\left (b^2-4 a c\right ) \int -\frac {B d (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )+A \left (\left (e^4-3 d f e^2+d^2 f^2\right ) c^2+2 f \left (a f \left (e^2-d f\right )-b \left (e^3-2 d e f\right )\right ) c-f^2 \left (-\left (\left (e^2-d f\right ) b^2\right )+2 a e f b-a^2 f^2\right )\right )-f \left (A (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )-B \left (-d \left (e^2-d f\right ) c^2+2 d f (b e-a f) c-f^2 \left (b^2 d-a^2 f\right )\right )\right ) x}{f x^2+e x+d}dx}{c^2 d^2-b c e d+f \left (f a^2-b e a+b^2 d\right )+a c \left (e^2-2 d f\right )}}{\left (b^2-4 a c\right ) \left ((c d-a f)^2-(b d-a e) (c e-b f)\right )}-\frac {A c (2 a c e-b (c d+a f))+(A b-a B) \left (f b^2+2 c^2 d-c (b e+2 a f)\right )+c \left (A f b^2-(B c d+A c e+a B f) b+2 c (A c d+a B e-a A f)\right ) x}{\left (b^2-4 a c\right ) \left ((c d-a f)^2-(b d-a e) (c e-b f)\right ) \left (c x^2+b x+a\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {(B d-A e) f^2 b^5-2 f \left (-a A f^2+B c d e-A c \left (e^2-d f\right )\right ) b^4-\left (A c e \left (c e^2-3 a f^2-2 c d f\right )+B \left (a^2 f^3+3 a c d f^2-c^2 d \left (e^2+2 d f\right )\right )\right ) b^3-2 c \left (B c d e (c d-2 a f)+A f \left (2 c^2 d^2+5 a^2 f^2+5 a c \left (e^2-d f\right )\right )\right ) b^2+c \left (A c e \left (3 c^2 d^2+7 a^2 f^2+a c \left (5 e^2-2 d f\right )\right )+B \left (c^3 d^3-a c^2 \left (e^2+5 d f\right ) d+5 a^3 f^3+a^2 c f \left (3 e^2-d f\right )\right )\right ) b-2 c^2 \left (A \left (c^3 d^3+a c^2 \left (3 e^2-5 d f\right ) d-3 a^3 f^3-a^2 c f \left (e^2-7 d f\right )\right )-a B e \left (c^2 d^2-3 a^2 f^2-a c \left (e^2-2 d f\right )\right )\right )+c \left (b^2-4 a c\right ) \left (A (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )-B \left (-d \left (e^2-d f\right ) c^2+2 d f (b e-a f) c-f^2 \left (b^2 d-a^2 f\right )\right )\right ) x}{c x^2+b x+a}dx}{c^2 d^2-b c e d+f \left (f a^2-b e a+b^2 d\right )+a c \left (e^2-2 d f\right )}+\frac {\left (b^2-4 a c\right ) \int -\frac {B d (c e-b f) \left (c e^2-b f e+2 a f^2-2 c d f\right )-A \left (\left (e^4-3 d f e^2+d^2 f^2\right ) c^2+2 f \left (a f \left (e^2-d f\right )-b \left (e^3-2 d e f\right )\right ) c-f^2 \left (-\left (\left (e^2-d f\right ) b^2\right )+2 a e f b-a^2 f^2\right )\right )-f \left (A (c e-b f) \left (c e^2-b f e+2 a f^2-2 c d f\right )+B \left (-d \left (e^2-d f\right ) c^2+2 d f (b e-a f) c-f^2 \left (b^2 d-a^2 f\right )\right )\right ) x}{f x^2+e x+d}dx}{c^2 d^2-b c e d+f \left (f a^2-b e a+b^2 d\right )+a c \left (e^2-2 d f\right )}}{\left (b^2-4 a c\right ) \left ((c d-a f)^2-(b d-a e) (c e-b f)\right )}-\frac {A c (2 a c e-b (c d+a f))+(A b-a B) \left (f b^2+2 c^2 d-c (b e+2 a f)\right )+c \left (A f b^2-(B c d+A c e+a B f) b+2 c (A c d+a B e-a A f)\right ) x}{\left (b^2-4 a c\right ) \left ((c d-a f)^2-(b d-a e) (c e-b f)\right ) \left (c x^2+b x+a\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {(B d-A e) f^2 b^5-2 f \left (-a A f^2+B c d e-A c \left (e^2-d f\right )\right ) b^4-\left (A c e \left (c e^2-3 a f^2-2 c d f\right )+B \left (a^2 f^3+3 a c d f^2-c^2 d \left (e^2+2 d f\right )\right )\right ) b^3-2 c \left (B c d e (c d-2 a f)+A f \left (2 c^2 d^2+5 a^2 f^2+5 a c \left (e^2-d f\right )\right )\right ) b^2+c \left (A c e \left (3 c^2 d^2+7 a^2 f^2+a c \left (5 e^2-2 d f\right )\right )+B \left (c^3 d^3-a c^2 \left (e^2+5 d f\right ) d+5 a^3 f^3+a^2 c f \left (3 e^2-d f\right )\right )\right ) b-2 c^2 \left (A \left (c^3 d^3+a c^2 \left (3 e^2-5 d f\right ) d-3 a^3 f^3-a^2 c f \left (e^2-7 d f\right )\right )-a B e \left (c^2 d^2-3 a^2 f^2-a c \left (e^2-2 d f\right )\right )\right )+c \left (b^2-4 a c\right ) \left (A (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )-B \left (-d \left (e^2-d f\right ) c^2+2 d f (b e-a f) c-f^2 \left (b^2 d-a^2 f\right )\right )\right ) x}{c x^2+b x+a}dx}{c^2 d^2-b c e d+f \left (f a^2-b e a+b^2 d\right )+a c \left (e^2-2 d f\right )}-\frac {\left (b^2-4 a c\right ) \int -\frac {B d (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )+A \left (\left (e^4-3 d f e^2+d^2 f^2\right ) c^2+2 f \left (a f \left (e^2-d f\right )-b \left (e^3-2 d e f\right )\right ) c-f^2 \left (-\left (\left (e^2-d f\right ) b^2\right )+2 a e f b-a^2 f^2\right )\right )-f \left (A (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )-B \left (-d \left (e^2-d f\right ) c^2+2 d f (b e-a f) c-f^2 \left (b^2 d-a^2 f\right )\right )\right ) x}{f x^2+e x+d}dx}{c^2 d^2-b c e d+f \left (f a^2-b e a+b^2 d\right )+a c \left (e^2-2 d f\right )}}{\left (b^2-4 a c\right ) \left ((c d-a f)^2-(b d-a e) (c e-b f)\right )}-\frac {A c (2 a c e-b (c d+a f))+(A b-a B) \left (f b^2+2 c^2 d-c (b e+2 a f)\right )+c \left (A f b^2-(B c d+A c e+a B f) b+2 c (A c d+a B e-a A f)\right ) x}{\left (b^2-4 a c\right ) \left ((c d-a f)^2-(b d-a e) (c e-b f)\right ) \left (c x^2+b x+a\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {(B d-A e) f^2 b^5-2 f \left (-a A f^2+B c d e-A c \left (e^2-d f\right )\right ) b^4-\left (A c e \left (c e^2-3 a f^2-2 c d f\right )+B \left (a^2 f^3+3 a c d f^2-c^2 d \left (e^2+2 d f\right )\right )\right ) b^3-2 c \left (B c d e (c d-2 a f)+A f \left (2 c^2 d^2+5 a^2 f^2+5 a c \left (e^2-d f\right )\right )\right ) b^2+c \left (A c e \left (3 c^2 d^2+7 a^2 f^2+a c \left (5 e^2-2 d f\right )\right )+B \left (c^3 d^3-a c^2 \left (e^2+5 d f\right ) d+5 a^3 f^3+a^2 c f \left (3 e^2-d f\right )\right )\right ) b-2 c^2 \left (A \left (c^3 d^3+a c^2 \left (3 e^2-5 d f\right ) d-3 a^3 f^3-a^2 c f \left (e^2-7 d f\right )\right )-a B e \left (c^2 d^2-3 a^2 f^2-a c \left (e^2-2 d f\right )\right )\right )+c \left (b^2-4 a c\right ) \left (A (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )-B \left (-d \left (e^2-d f\right ) c^2+2 d f (b e-a f) c-f^2 \left (b^2 d-a^2 f\right )\right )\right ) x}{c x^2+b x+a}dx}{c^2 d^2-b c e d+f \left (f a^2-b e a+b^2 d\right )+a c \left (e^2-2 d f\right )}+\frac {\left (b^2-4 a c\right ) \int -\frac {B d (c e-b f) \left (c e^2-b f e+2 a f^2-2 c d f\right )-A \left (\left (e^4-3 d f e^2+d^2 f^2\right ) c^2+2 f \left (a f \left (e^2-d f\right )-b \left (e^3-2 d e f\right )\right ) c-f^2 \left (-\left (\left (e^2-d f\right ) b^2\right )+2 a e f b-a^2 f^2\right )\right )-f \left (A (c e-b f) \left (c e^2-b f e+2 a f^2-2 c d f\right )+B \left (-d \left (e^2-d f\right ) c^2+2 d f (b e-a f) c-f^2 \left (b^2 d-a^2 f\right )\right )\right ) x}{f x^2+e x+d}dx}{c^2 d^2-b c e d+f \left (f a^2-b e a+b^2 d\right )+a c \left (e^2-2 d f\right )}}{\left (b^2-4 a c\right ) \left ((c d-a f)^2-(b d-a e) (c e-b f)\right )}-\frac {A c (2 a c e-b (c d+a f))+(A b-a B) \left (f b^2+2 c^2 d-c (b e+2 a f)\right )+c \left (A f b^2-(B c d+A c e+a B f) b+2 c (A c d+a B e-a A f)\right ) x}{\left (b^2-4 a c\right ) \left ((c d-a f)^2-(b d-a e) (c e-b f)\right ) \left (c x^2+b x+a\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {(B d-A e) f^2 b^5-2 f \left (-a A f^2+B c d e-A c \left (e^2-d f\right )\right ) b^4-\left (A c e \left (c e^2-3 a f^2-2 c d f\right )+B \left (a^2 f^3+3 a c d f^2-c^2 d \left (e^2+2 d f\right )\right )\right ) b^3-2 c \left (B c d e (c d-2 a f)+A f \left (2 c^2 d^2+5 a^2 f^2+5 a c \left (e^2-d f\right )\right )\right ) b^2+c \left (A c e \left (3 c^2 d^2+7 a^2 f^2+a c \left (5 e^2-2 d f\right )\right )+B \left (c^3 d^3-a c^2 \left (e^2+5 d f\right ) d+5 a^3 f^3+a^2 c f \left (3 e^2-d f\right )\right )\right ) b-2 c^2 \left (A \left (c^3 d^3+a c^2 \left (3 e^2-5 d f\right ) d-3 a^3 f^3-a^2 c f \left (e^2-7 d f\right )\right )-a B e \left (c^2 d^2-3 a^2 f^2-a c \left (e^2-2 d f\right )\right )\right )+c \left (b^2-4 a c\right ) \left (A (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )-B \left (-d \left (e^2-d f\right ) c^2+2 d f (b e-a f) c-f^2 \left (b^2 d-a^2 f\right )\right )\right ) x}{c x^2+b x+a}dx}{c^2 d^2-b c e d+f \left (f a^2-b e a+b^2 d\right )+a c \left (e^2-2 d f\right )}-\frac {\left (b^2-4 a c\right ) \int -\frac {B d (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )+A \left (\left (e^4-3 d f e^2+d^2 f^2\right ) c^2+2 f \left (a f \left (e^2-d f\right )-b \left (e^3-2 d e f\right )\right ) c-f^2 \left (-\left (\left (e^2-d f\right ) b^2\right )+2 a e f b-a^2 f^2\right )\right )-f \left (A (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )-B \left (-d \left (e^2-d f\right ) c^2+2 d f (b e-a f) c-f^2 \left (b^2 d-a^2 f\right )\right )\right ) x}{f x^2+e x+d}dx}{c^2 d^2-b c e d+f \left (f a^2-b e a+b^2 d\right )+a c \left (e^2-2 d f\right )}}{\left (b^2-4 a c\right ) \left ((c d-a f)^2-(b d-a e) (c e-b f)\right )}-\frac {A c (2 a c e-b (c d+a f))+(A b-a B) \left (f b^2+2 c^2 d-c (b e+2 a f)\right )+c \left (A f b^2-(B c d+A c e+a B f) b+2 c (A c d+a B e-a A f)\right ) x}{\left (b^2-4 a c\right ) \left ((c d-a f)^2-(b d-a e) (c e-b f)\right ) \left (c x^2+b x+a\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {(B d-A e) f^2 b^5-2 f \left (-a A f^2+B c d e-A c \left (e^2-d f\right )\right ) b^4-\left (A c e \left (c e^2-3 a f^2-2 c d f\right )+B \left (a^2 f^3+3 a c d f^2-c^2 d \left (e^2+2 d f\right )\right )\right ) b^3-2 c \left (B c d e (c d-2 a f)+A f \left (2 c^2 d^2+5 a^2 f^2+5 a c \left (e^2-d f\right )\right )\right ) b^2+c \left (A c e \left (3 c^2 d^2+7 a^2 f^2+a c \left (5 e^2-2 d f\right )\right )+B \left (c^3 d^3-a c^2 \left (e^2+5 d f\right ) d+5 a^3 f^3+a^2 c f \left (3 e^2-d f\right )\right )\right ) b-2 c^2 \left (A \left (c^3 d^3+a c^2 \left (3 e^2-5 d f\right ) d-3 a^3 f^3-a^2 c f \left (e^2-7 d f\right )\right )-a B e \left (c^2 d^2-3 a^2 f^2-a c \left (e^2-2 d f\right )\right )\right )+c \left (b^2-4 a c\right ) \left (A (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )-B \left (-d \left (e^2-d f\right ) c^2+2 d f (b e-a f) c-f^2 \left (b^2 d-a^2 f\right )\right )\right ) x}{c x^2+b x+a}dx}{c^2 d^2-b c e d+f \left (f a^2-b e a+b^2 d\right )+a c \left (e^2-2 d f\right )}+\frac {\left (b^2-4 a c\right ) \int -\frac {B d (c e-b f) \left (c e^2-b f e+2 a f^2-2 c d f\right )-A \left (\left (e^4-3 d f e^2+d^2 f^2\right ) c^2+2 f \left (a f \left (e^2-d f\right )-b \left (e^3-2 d e f\right )\right ) c-f^2 \left (-\left (\left (e^2-d f\right ) b^2\right )+2 a e f b-a^2 f^2\right )\right )-f \left (A (c e-b f) \left (c e^2-b f e+2 a f^2-2 c d f\right )+B \left (-d \left (e^2-d f\right ) c^2+2 d f (b e-a f) c-f^2 \left (b^2 d-a^2 f\right )\right )\right ) x}{f x^2+e x+d}dx}{c^2 d^2-b c e d+f \left (f a^2-b e a+b^2 d\right )+a c \left (e^2-2 d f\right )}}{\left (b^2-4 a c\right ) \left ((c d-a f)^2-(b d-a e) (c e-b f)\right )}-\frac {A c (2 a c e-b (c d+a f))+(A b-a B) \left (f b^2+2 c^2 d-c (b e+2 a f)\right )+c \left (A f b^2-(B c d+A c e+a B f) b+2 c (A c d+a B e-a A f)\right ) x}{\left (b^2-4 a c\right ) \left ((c d-a f)^2-(b d-a e) (c e-b f)\right ) \left (c x^2+b x+a\right )}\) |
3.1.16.3.1 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[((g_.) + (h_.)*(x_))*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_) + (e _.)*(x_) + (f_.)*(x_)^2)^(q_), x_Symbol] :> Simp[(a + b*x + c*x^2)^(p + 1)* ((d + e*x + f*x^2)^(q + 1)/((b^2 - 4*a*c)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1)))*(g*c*(2*a*c*e - b*(c*d + a*f)) + (g*b - a*h)*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)) + c*(g*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)) - h*(b* c*d - 2*a*c*e + a*b*f))*x), x] + Simp[1/((b^2 - 4*a*c)*((c*d - a*f)^2 - (b* d - a*e)*(c*e - b*f))*(p + 1)) Int[(a + b*x + c*x^2)^(p + 1)*(d + e*x + f *x^2)^q*Simp[(b*h - 2*g*c)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1 ) + (b^2*(g*f) - b*(h*c*d + g*c*e + a*h*f) + 2*(g*c*(c*d - a*f) - a*((-h)*c *e)))*(a*f*(p + 1) - c*d*(p + 2)) - e*((g*c)*(2*a*c*e - b*(c*d + a*f)) + (g *b - a*h)*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)))*(p + q + 2) - (2*f*((g*c)*(2 *a*c*e - b*(c*d + a*f)) + (g*b - a*h)*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)))* (p + q + 2) - (b^2*g*f - b*(h*c*d + g*c*e + a*h*f) + 2*(g*c*(c*d - a*f) - a *((-h)*c*e)))*(b*f*(p + 1) - c*e*(2*p + q + 4)))*x - c*f*(b^2*(g*f) - b*(h* c*d + g*c*e + a*h*f) + 2*(g*c*(c*d - a*f) + a*h*c*e))*(2*p + 2*q + 5)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, q}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && LtQ[p, -1] && NeQ[(c*d - a*f)^2 - (b*d - a*e)*(c* e - b*f), 0] && !( !IntegerQ[p] && ILtQ[q, -1])
Int[(Px_)/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*((d_) + (e_.)*(x_) + (f_.)*(x _)^2)), x_Symbol] :> With[{A = Coeff[Px, x, 0], B = Coeff[Px, x, 1], C = Co eff[Px, x, 2], q = c^2*d^2 - b*c*d*e + a*c*e^2 + b^2*d*f - 2*a*c*d*f - a*b* e*f + a^2*f^2}, Simp[1/q Int[(A*c^2*d - a*c*C*d - A*b*c*e + a*B*c*e + A*b ^2*f - a*b*B*f - a*A*c*f + a^2*C*f + c*(B*c*d - b*C*d - A*c*e + a*C*e + A*b *f - a*B*f)*x)/(a + b*x + c*x^2), x], x] + Simp[1/q Int[(c*C*d^2 - B*c*d* e + A*c*e^2 + b*B*d*f - A*c*d*f - a*C*d*f - A*b*e*f + a*A*f^2 - f*(B*c*d - b*C*d - A*c*e + a*C*e + A*b*f - a*B*f)*x)/(d + e*x + f*x^2), x], x] /; NeQ[ q, 0]] /; FreeQ[{a, b, c, d, e, f}, x] && PolyQ[Px, x, 2]
Timed out.
hanged
Timed out. \[ \int \frac {A+B x}{\left (a+b x+c x^2\right )^2 \left (d+e x+f x^2\right )} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {A+B x}{\left (a+b x+c x^2\right )^2 \left (d+e x+f x^2\right )} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {A+B x}{\left (a+b x+c x^2\right )^2 \left (d+e x+f x^2\right )} \, dx=\text {Exception raised: ValueError} \]
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*d*f-e^2>0)', see `assume?` for more deta
Leaf count of result is larger than twice the leaf count of optimal. 3236 vs. \(2 (1061) = 2122\).
Time = 0.33 (sec) , antiderivative size = 3236, normalized size of antiderivative = 3.01 \[ \int \frac {A+B x}{\left (a+b x+c x^2\right )^2 \left (d+e x+f x^2\right )} \, dx=\text {Too large to display} \]
1/2*(B*c^2*d*e^2 - A*c^2*e^3 - B*c^2*d^2*f - 2*B*b*c*d*e*f + 2*A*c^2*d*e*f + 2*A*b*c*e^2*f + B*b^2*d*f^2 + 2*B*a*c*d*f^2 - 2*A*b*c*d*f^2 - A*b^2*e*f ^2 - 2*A*a*c*e*f^2 - B*a^2*f^3 + 2*A*a*b*f^3)*log(c*x^2 + b*x + a)/(c^4*d^ 4 - 2*b*c^3*d^3*e + b^2*c^2*d^2*e^2 + 2*a*c^3*d^2*e^2 - 2*a*b*c^2*d*e^3 + a^2*c^2*e^4 + 2*b^2*c^2*d^3*f - 4*a*c^3*d^3*f - 2*b^3*c*d^2*e*f + 2*a*b*c^ 2*d^2*e*f + 4*a*b^2*c*d*e^2*f - 4*a^2*c^2*d*e^2*f - 2*a^2*b*c*e^3*f + b^4* d^2*f^2 - 4*a*b^2*c*d^2*f^2 + 6*a^2*c^2*d^2*f^2 - 2*a*b^3*d*e*f^2 + 2*a^2* b*c*d*e*f^2 + a^2*b^2*e^2*f^2 + 2*a^3*c*e^2*f^2 + 2*a^2*b^2*d*f^3 - 4*a^3* c*d*f^3 - 2*a^3*b*e*f^3 + a^4*f^4) - 1/2*(B*c^2*d*e^2 - A*c^2*e^3 - B*c^2* d^2*f - 2*B*b*c*d*e*f + 2*A*c^2*d*e*f + 2*A*b*c*e^2*f + B*b^2*d*f^2 + 2*B* a*c*d*f^2 - 2*A*b*c*d*f^2 - A*b^2*e*f^2 - 2*A*a*c*e*f^2 - B*a^2*f^3 + 2*A* a*b*f^3)*log(f*x^2 + e*x + d)/(c^4*d^4 - 2*b*c^3*d^3*e + b^2*c^2*d^2*e^2 + 2*a*c^3*d^2*e^2 - 2*a*b*c^2*d*e^3 + a^2*c^2*e^4 + 2*b^2*c^2*d^3*f - 4*a*c ^3*d^3*f - 2*b^3*c*d^2*e*f + 2*a*b*c^2*d^2*e*f + 4*a*b^2*c*d*e^2*f - 4*a^2 *c^2*d*e^2*f - 2*a^2*b*c*e^3*f + b^4*d^2*f^2 - 4*a*b^2*c*d^2*f^2 + 6*a^2*c ^2*d^2*f^2 - 2*a*b^3*d*e*f^2 + 2*a^2*b*c*d*e*f^2 + a^2*b^2*e^2*f^2 + 2*a^3 *c*e^2*f^2 + 2*a^2*b^2*d*f^3 - 4*a^3*c*d*f^3 - 2*a^3*b*e*f^3 + a^4*f^4) + (2*B*b*c^4*d^3 - 4*A*c^5*d^3 - 4*B*b^2*c^3*d^2*e + 4*B*a*c^4*d^2*e + 6*A*b *c^4*d^2*e + B*b^3*c^2*d*e^2 + 2*B*a*b*c^3*d*e^2 - 12*A*a*c^4*d*e^2 - A*b^ 3*c^2*e^3 - 4*B*a^2*c^3*e^3 + 6*A*a*b*c^3*e^3 + 5*B*b^3*c^2*d^2*f - 14*...
Time = 50.27 (sec) , antiderivative size = 118429, normalized size of antiderivative = 110.17 \[ \int \frac {A+B x}{\left (a+b x+c x^2\right )^2 \left (d+e x+f x^2\right )} \, dx=\text {Too large to display} \]
symsum(log((x*(4*A^3*b^3*c^4*f^6 + 16*B^3*a^3*c^4*f^6 - 3*B^3*a^2*b^2*c^3* f^6 + 4*B^3*a^2*c^5*e^2*f^4 + B^3*b^2*c^5*d^2*f^4 - 16*A^3*a*b*c^5*f^6 + 1 6*A^3*a*c^6*e*f^5 + 20*A^2*B*a^2*c^5*f^6 - 3*A^2*B*b^4*c^3*f^6 + 4*A^2*B*c ^7*d^2*f^4 - 16*B^3*a^2*c^5*d*f^5 - 4*A^3*b^2*c^5*e*f^5 + 6*B^3*a*b^2*c^4* d*f^5 - 4*B^3*a^2*b*c^4*e*f^5 + A^2*B*b^2*c^5*e^2*f^4 - 24*A^2*B*a*c^6*d*f ^5 + 6*A*B^2*a*b^3*c^3*f^6 - 28*A*B^2*a^2*b*c^4*f^6 + 8*A^2*B*a*b^2*c^4*f^ 6 - 4*A*B^2*b*c^6*d^2*f^4 + 8*A*B^2*a^2*c^5*e*f^5 - 6*A*B^2*b^3*c^4*d*f^5 + 8*A^2*B*b^2*c^5*d*f^5 + 2*A^2*B*b^3*c^4*e*f^5 - 4*B^3*a*b*c^5*d*e*f^4 - 4*A*B^2*a*b*c^5*e^2*f^4 + 2*A*B^2*a*b^2*c^4*e*f^5 + 2*A*B^2*b^2*c^5*d*e*f^ 4 + 16*A*B^2*a*b*c^5*d*f^5 - 12*A^2*B*a*b*c^5*e*f^5 + 8*A*B^2*a*c^6*d*e*f^ 4 - 4*A^2*B*b*c^6*d*e*f^4))/(16*a^2*c^6*d^4 + a^4*b^4*f^4 + 16*a^4*c^4*e^4 + b^4*c^4*d^4 + 16*a^6*c^2*f^4 + b^8*d^2*f^2 - 8*a*b^2*c^5*d^4 - 8*a^5*b^ 2*c*f^4 + 2*a^2*b^6*d*f^3 - 2*a^3*b^5*e*f^3 - 64*a^3*c^5*d^3*f - 64*a^5*c^ 3*d*f^3 - 2*b^5*c^3*d^3*e + 2*b^6*c^2*d^3*f + a^2*b^4*c^2*e^4 - 8*a^3*b^2* c^3*e^4 + 32*a^3*c^5*d^2*e^2 + a^2*b^6*e^2*f^2 + 96*a^4*c^4*d^2*f^2 + b^6* c^2*d^2*e^2 + 32*a^5*c^3*e^2*f^2 - 2*a*b^7*d*e*f^2 - 2*b^7*c*d^2*e*f + 54* a^2*b^4*c^2*d^2*f^2 - 112*a^3*b^2*c^3*d^2*f^2 + 16*a*b^3*c^4*d^3*e - 2*a*b ^5*c^2*d*e^3 - 32*a^2*b*c^5*d^3*e - 32*a^3*b*c^4*d*e^3 - 20*a*b^4*c^3*d^3* f - 12*a*b^6*c*d^2*f^2 - 20*a^3*b^4*c*d*f^3 - 2*a^2*b^5*c*e^3*f - 32*a^4*b *c^3*e^3*f + 16*a^4*b^3*c*e*f^3 - 32*a^5*b*c^2*e*f^3 - 64*a^4*c^4*d*e^2...