Integrand size = 25, antiderivative size = 1043 \[ \int \frac {f+g x}{(d+e x)^2 \left (a+b x+c x^2\right )^3} \, dx=\frac {e \left (6 c^4 d^4 f-b^3 e^3 (3 b e f-2 b d g-a e g)-b c e^2 \left (7 a^2 e^2 g-a b e (21 e f-13 d g)-3 b^2 d (e f-d g)\right )+c^3 d^2 (4 a e (6 e f-d g)-3 b d (4 e f+d g))-c^2 e \left (2 a^2 e^2 (15 e f-22 d g)+6 a b d e (4 e f+d g)-b^2 d^2 (3 e f+7 d g)\right )\right )}{\left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^3 (d+e x)}-\frac {b c d f-b^2 e f+2 a c e f-2 a c d g+a b e g+c (2 c d f+2 a e g-b (e f+d g)) x}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x) \left (a+b x+c x^2\right )^2}-\frac {4 a c e (2 c d-b e) (2 c d f+2 a e g-b (e f+d g))-\left (b c d-b^2 e+2 a c e\right ) \left (6 c^2 d^2 f-b e (3 b e f-2 b d g-a e g)+c (2 a e (5 e f-2 d g)-b d (2 e f+3 d g))\right )-c \left (12 c^3 d^3 f+b^2 e^2 (3 b e f-2 b d g-a e g)+2 c^2 d (2 a e (9 e f-2 d g)-3 b d (3 e f+d g))+c e \left (11 b^2 d^2 g+16 a^2 e^2 g-2 a b e (9 e f+5 d g)\right )\right ) x}{2 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 (d+e x) \left (a+b x+c x^2\right )}-\frac {\left (12 c^6 d^6 f+b^5 e^5 (3 b e f-2 b d g-a e g)+b^3 c e^4 \left (10 a^2 e^2 g-b^2 d (6 e f-5 d g)-10 a b e (3 e f-2 d g)\right )-10 a b c^2 e^4 \left (3 a^2 e^2 g-b^2 d (6 e f-5 d g)-3 a b e (3 e f-2 d g)\right )-10 c^3 e^2 \left (2 b^3 d^4 g-8 a b^2 d^3 e g+6 a^3 e^3 (e f-2 d g)+3 a^2 b d e^2 (6 e f-d g)\right )+2 c^5 d^4 (2 a e (15 e f-2 d g)-3 b d (6 e f+d g))+10 c^4 d^2 e \left (2 a^2 e^2 (9 e f-4 d g)-a b d e (12 e f+d g)+b^2 d^2 (3 e f+2 d g)\right )\right ) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2} \left (c d^2-b d e+a e^2\right )^4}+\frac {e^4 (c d (6 e f-5 d g)-e (3 b e f-2 b d g-a e g)) \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^4}-\frac {e^4 (c d (6 e f-5 d g)-e (3 b e f-2 b d g-a e g)) \log \left (a+b x+c x^2\right )}{2 \left (c d^2-b d e+a e^2\right )^4} \]
e*(6*c^4*d^4*f-b^3*e^3*(-a*e*g-2*b*d*g+3*b*e*f)-b*c*e^2*(7*a^2*e^2*g-a*b*e *(-13*d*g+21*e*f)-3*b^2*d*(-d*g+e*f))+c^3*d^2*(4*a*e*(-d*g+6*e*f)-3*b*d*(d *g+4*e*f))-c^2*e*(2*a^2*e^2*(-22*d*g+15*e*f)+6*a*b*d*e*(d*g+4*e*f)-b^2*d^2 *(7*d*g+3*e*f)))/(-4*a*c+b^2)^2/(a*e^2-b*d*e+c*d^2)^3/(e*x+d)+1/2*(-b*c*d* f+b^2*e*f-2*a*c*e*f+2*a*c*d*g-a*b*e*g-c*(2*c*d*f+2*a*e*g-b*(d*g+e*f))*x)/( -4*a*c+b^2)/(a*e^2-b*d*e+c*d^2)/(e*x+d)/(c*x^2+b*x+a)^2+1/2*(-4*a*c*e*(-b* e+2*c*d)*(2*c*d*f+2*a*e*g-b*(d*g+e*f))+(2*a*c*e-b^2*e+b*c*d)*(6*c^2*d^2*f- b*e*(-a*e*g-2*b*d*g+3*b*e*f)+c*(2*a*e*(-2*d*g+5*e*f)-b*d*(3*d*g+2*e*f)))+c *(12*c^3*d^3*f+b^2*e^2*(-a*e*g-2*b*d*g+3*b*e*f)+2*c^2*d*(2*a*e*(-2*d*g+9*e *f)-3*b*d*(d*g+3*e*f))+c*e*(11*b^2*d^2*g+16*a^2*e^2*g-2*a*b*e*(5*d*g+9*e*f )))*x)/(-4*a*c+b^2)^2/(a*e^2-b*d*e+c*d^2)^2/(e*x+d)/(c*x^2+b*x+a)-(12*c^6* d^6*f+b^5*e^5*(-a*e*g-2*b*d*g+3*b*e*f)+b^3*c*e^4*(10*a^2*e^2*g-b^2*d*(-5*d *g+6*e*f)-10*a*b*e*(-2*d*g+3*e*f))-10*a*b*c^2*e^4*(3*a^2*e^2*g-b^2*d*(-5*d *g+6*e*f)-3*a*b*e*(-2*d*g+3*e*f))-10*c^3*e^2*(2*b^3*d^4*g-8*a*b^2*d^3*e*g+ 6*a^3*e^3*(-2*d*g+e*f)+3*a^2*b*d*e^2*(-d*g+6*e*f))+2*c^5*d^4*(2*a*e*(-2*d* g+15*e*f)-3*b*d*(d*g+6*e*f))+10*c^4*d^2*e*(2*a^2*e^2*(-4*d*g+9*e*f)-a*b*d* e*(d*g+12*e*f)+b^2*d^2*(2*d*g+3*e*f)))*arctanh((2*c*x+b)/(-4*a*c+b^2)^(1/2 ))/(-4*a*c+b^2)^(5/2)/(a*e^2-b*d*e+c*d^2)^4+e^4*(c*d*(-5*d*g+6*e*f)-e*(-a* e*g-2*b*d*g+3*b*e*f))*ln(e*x+d)/(a*e^2-b*d*e+c*d^2)^4-1/2*e^4*(c*d*(-5*d*g +6*e*f)-e*(-a*e*g-2*b*d*g+3*b*e*f))*ln(c*x^2+b*x+a)/(a*e^2-b*d*e+c*d^2)...
Time = 5.11 (sec) , antiderivative size = 1050, normalized size of antiderivative = 1.01 \[ \int \frac {f+g x}{(d+e x)^2 \left (a+b x+c x^2\right )^3} \, dx=\frac {1}{2} \left (-\frac {2 e^4 (e f-d g)}{\left (c d^2+e (-b d+a e)\right )^3 (d+e x)}+\frac {-b^3 e^2 f+b^2 e (a e g+c f (2 d-e x))+b c (c d (-d f+2 e f x+d g x)+a e (3 e f-2 d g+e g x))-2 c \left (a^2 e^2 g+c^2 d^2 f x-a c \left (d^2 g+e^2 f x-2 d e (f+g x)\right )\right )}{\left (b^2-4 a c\right ) \left (c d^2+e (-b d+a e)\right )^2 (a+x (b+c x))^2}+\frac {2 b^5 e^3 (2 e f-d g)+b^4 \left (-2 a e^4 g+c e^2 \left (-7 d e f+6 d^2 g+4 e^2 f x-2 d e g x\right )\right )+2 b c^2 \left (3 c^2 d^3 (-d f+4 e f x+d g x)+a^2 e^3 (23 e f-22 d g+7 e g x)+2 a c d e \left (-6 d e f+d^2 g+12 e^2 f x+3 d e g x\right )\right )-b^3 c e \left (a e^2 (29 e f-13 d g+2 e g x)+c d \left (7 d^2 g+6 e^2 f x+3 d e (f-2 g x)\right )\right )+b^2 c \left (15 a^2 e^4 g+c^2 d^2 \left (3 d^2 g-6 e^2 f x+2 d e (6 f-7 g x)\right )-2 a c e^2 \left (9 d^2 g+13 e^2 f x-d e (28 f+5 g x)\right )\right )-4 c^2 \left (4 a^3 e^4 g+3 c^3 d^4 f x-2 a c^2 d^2 e (-6 e f+d g) x+a^2 c e^2 \left (-12 d^2 g-7 e^2 f x+2 d e (8 f+7 g x)\right )\right )}{\left (b^2-4 a c\right )^2 \left (-c d^2+e (b d-a e)\right )^3 (a+x (b+c x))}-\frac {2 \left (-12 c^6 d^6 f+b^5 e^5 (-3 b e f+2 b d g+a e g)+b^3 c e^4 \left (-10 a^2 e^2 g+b^2 d (6 e f-5 d g)+10 a b e (3 e f-2 d g)\right )-10 c^3 e^2 \left (-2 b^3 d^4 g+8 a b^2 d^3 e g-6 a^3 e^3 (e f-2 d g)+3 a^2 b d e^2 (-6 e f+d g)\right )+2 c^5 d^4 (3 b d (6 e f+d g)+2 a e (-15 e f+2 d g))-10 c^4 d^2 e \left (2 a^2 e^2 (9 e f-4 d g)-a b d e (12 e f+d g)+b^2 d^2 (3 e f+2 d g)\right )+10 a b c^2 e^4 \left (3 a^2 e^2 g+3 a b e (-3 e f+2 d g)+b^2 d (-6 e f+5 d g)\right )\right ) \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\left (-b^2+4 a c\right )^{5/2} \left (c d^2+e (-b d+a e)\right )^4}-\frac {2 e^4 (c d (-6 e f+5 d g)+e (3 b e f-2 b d g-a e g)) \log (d+e x)}{\left (c d^2+e (-b d+a e)\right )^4}+\frac {e^4 (c d (-6 e f+5 d g)+e (3 b e f-2 b d g-a e g)) \log (a+x (b+c x))}{\left (c d^2+e (-b d+a e)\right )^4}\right ) \]
((-2*e^4*(e*f - d*g))/((c*d^2 + e*(-(b*d) + a*e))^3*(d + e*x)) + (-(b^3*e^ 2*f) + b^2*e*(a*e*g + c*f*(2*d - e*x)) + b*c*(c*d*(-(d*f) + 2*e*f*x + d*g* x) + a*e*(3*e*f - 2*d*g + e*g*x)) - 2*c*(a^2*e^2*g + c^2*d^2*f*x - a*c*(d^ 2*g + e^2*f*x - 2*d*e*(f + g*x))))/((b^2 - 4*a*c)*(c*d^2 + e*(-(b*d) + a*e ))^2*(a + x*(b + c*x))^2) + (2*b^5*e^3*(2*e*f - d*g) + b^4*(-2*a*e^4*g + c *e^2*(-7*d*e*f + 6*d^2*g + 4*e^2*f*x - 2*d*e*g*x)) + 2*b*c^2*(3*c^2*d^3*(- (d*f) + 4*e*f*x + d*g*x) + a^2*e^3*(23*e*f - 22*d*g + 7*e*g*x) + 2*a*c*d*e *(-6*d*e*f + d^2*g + 12*e^2*f*x + 3*d*e*g*x)) - b^3*c*e*(a*e^2*(29*e*f - 1 3*d*g + 2*e*g*x) + c*d*(7*d^2*g + 6*e^2*f*x + 3*d*e*(f - 2*g*x))) + b^2*c* (15*a^2*e^4*g + c^2*d^2*(3*d^2*g - 6*e^2*f*x + 2*d*e*(6*f - 7*g*x)) - 2*a* c*e^2*(9*d^2*g + 13*e^2*f*x - d*e*(28*f + 5*g*x))) - 4*c^2*(4*a^3*e^4*g + 3*c^3*d^4*f*x - 2*a*c^2*d^2*e*(-6*e*f + d*g)*x + a^2*c*e^2*(-12*d^2*g - 7* e^2*f*x + 2*d*e*(8*f + 7*g*x))))/((b^2 - 4*a*c)^2*(-(c*d^2) + e*(b*d - a*e ))^3*(a + x*(b + c*x))) - (2*(-12*c^6*d^6*f + b^5*e^5*(-3*b*e*f + 2*b*d*g + a*e*g) + b^3*c*e^4*(-10*a^2*e^2*g + b^2*d*(6*e*f - 5*d*g) + 10*a*b*e*(3* e*f - 2*d*g)) - 10*c^3*e^2*(-2*b^3*d^4*g + 8*a*b^2*d^3*e*g - 6*a^3*e^3*(e* f - 2*d*g) + 3*a^2*b*d*e^2*(-6*e*f + d*g)) + 2*c^5*d^4*(3*b*d*(6*e*f + d*g ) + 2*a*e*(-15*e*f + 2*d*g)) - 10*c^4*d^2*e*(2*a^2*e^2*(9*e*f - 4*d*g) - a *b*d*e*(12*e*f + d*g) + b^2*d^2*(3*e*f + 2*d*g)) + 10*a*b*c^2*e^4*(3*a^2*e ^2*g + 3*a*b*e*(-3*e*f + 2*d*g) + b^2*d*(-6*e*f + 5*d*g)))*ArcTan[(b + ...
Time = 3.86 (sec) , antiderivative size = 1117, normalized size of antiderivative = 1.07, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1235, 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.
| \(\displaystyle \int \frac {f+g x}{(d+e x)^2 \left (a+b x+c x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 1235 |
| \(\displaystyle -\frac {\int \frac {6 c^2 f d^2-b e (3 b e f-2 b d g-a e g)+c (2 a e (5 e f-2 d g)-b d (2 e f+3 d g))+4 c e (2 c d f+2 a e g-b (e f+d g)) x}{(d+e x)^2 \left (c x^2+b x+a\right )^2}dx}{2 \left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {c x (2 a e g-b (d g+e f)+2 c d f)+a b e g-2 a c d g+2 a c e f+b^2 (-e) f+b c d f}{2 \left (b^2-4 a c\right ) (d+e x) \left (a+b x+c x^2\right )^2 \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 1235 |
| \(\displaystyle -\frac {\frac {-\left (2 a c e+b^2 (-e)+b c d\right ) \left (c (2 a e (5 e f-2 d g)-b d (3 d g+2 e f))-b e (-a e g-2 b d g+3 b e f)+6 c^2 d^2 f\right )+c x \left (4 c e (b d-2 a e) (2 a e g-b (d g+e f)+2 c d f)-(2 c d-b e) \left (c (2 a e (5 e f-2 d g)-b d (3 d g+2 e f))-b e (-a e g-2 b d g+3 b e f)+6 c^2 d^2 f\right )\right )+4 a c e (2 c d-b e) (2 a e g-b (d g+e f)+2 c d f)}{\left (b^2-4 a c\right ) (d+e x) \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )}-\frac {\int -\frac {2 \left (2 c e (b d (c d-b e)+a e (4 c d-b e)) (2 c d f+2 a e g-b (e f+d g))-\frac {1}{2} \left (2 c^2 d^2-2 b^2 e^2+6 a c e^2\right ) \left (6 c^2 f d^2-b e (3 b e f-2 b d g-a e g)+c (2 a e (5 e f-2 d g)-b d (2 e f+3 d g))\right )-c e \left (12 c^3 f d^3+2 c^2 (2 a e (9 e f-2 d g)-3 b d (3 e f+d g)) d+b^2 e^2 (3 b e f-2 b d g-a e g)+c e \left (11 b^2 g d^2+16 a^2 e^2 g-2 a b e (9 e f+5 d g)\right )\right ) x\right )}{(d+e x)^2 \left (c x^2+b x+a\right )}dx}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}}{2 \left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {c x (2 a e g-b (d g+e f)+2 c d f)+a b e g-2 a c d g+2 a c e f+b^2 (-e) f+b c d f}{2 \left (b^2-4 a c\right ) (d+e x) \left (a+b x+c x^2\right )^2 \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {2 \int \frac {2 c e (b d (c d-b e)+a e (4 c d-b e)) (2 c d f+2 a e g-b (e f+d g))-\frac {1}{2} \left (2 c^2 d^2-2 b^2 e^2+6 a c e^2\right ) \left (6 c^2 f d^2-b e (3 b e f-2 b d g-a e g)+c (2 a e (5 e f-2 d g)-b d (2 e f+3 d g))\right )-c e \left (12 c^3 f d^3+2 c^2 (2 a e (9 e f-2 d g)-3 b d (3 e f+d g)) d+b^2 e^2 (3 b e f-2 b d g-a e g)+c e \left (11 b^2 g d^2+16 a^2 e^2 g-2 a b e (9 e f+5 d g)\right )\right ) x}{(d+e x)^2 \left (c x^2+b x+a\right )}dx}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}+\frac {-\left (2 a c e+b^2 (-e)+b c d\right ) \left (c (2 a e (5 e f-2 d g)-b d (3 d g+2 e f))-b e (-a e g-2 b d g+3 b e f)+6 c^2 d^2 f\right )+c x \left (4 c e (b d-2 a e) (2 a e g-b (d g+e f)+2 c d f)-(2 c d-b e) \left (c (2 a e (5 e f-2 d g)-b d (3 d g+2 e f))-b e (-a e g-2 b d g+3 b e f)+6 c^2 d^2 f\right )\right )+4 a c e (2 c d-b e) (2 a e g-b (d g+e f)+2 c d f)}{\left (b^2-4 a c\right ) (d+e x) \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )}}{2 \left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {c x (2 a e g-b (d g+e f)+2 c d f)+a b e g-2 a c d g+2 a c e f+b^2 (-e) f+b c d f}{2 \left (b^2-4 a c\right ) (d+e x) \left (a+b x+c x^2\right )^2 \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 1200 |
| \(\displaystyle -\frac {-e f b^2+c d f b+a e g b+2 a c e f-2 a c d g+c (2 c d f+2 a e g-b (e f+d g)) x}{2 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) (d+e x) \left (c x^2+b x+a\right )^2}-\frac {\frac {4 a c e (2 c d-b e) (2 c d f+2 a e g-b (e f+d g))-\left (-e b^2+c d b+2 a c e\right ) \left (6 c^2 f d^2-b e (3 b e f-2 b d g-a e g)+c (2 a e (5 e f-2 d g)-b d (2 e f+3 d g))\right )+c \left (4 c e (b d-2 a e) (2 c d f+2 a e g-b (e f+d g))-(2 c d-b e) \left (6 c^2 f d^2-b e (3 b e f-2 b d g-a e g)+c (2 a e (5 e f-2 d g)-b d (2 e f+3 d g))\right )\right ) x}{\left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) (d+e x) \left (c x^2+b x+a\right )}+\frac {2 \int \left (\frac {\left (b^2-4 a c\right )^2 (e (3 b e f-2 b d g-a e g)-c d (6 e f-5 d g)) e^5}{\left (c d^2-b e d+a e^2\right )^2 (d+e x)}+\frac {\left (6 c^4 f d^4+c^3 (4 a e (6 e f-d g)-3 b d (4 e f+d g)) d^2-b^3 e^3 (3 b e f-2 b d g-a e g)-b c e^2 \left (-3 d (e f-d g) b^2-a e (21 e f-13 d g) b+7 a^2 e^2 g\right )-c^2 e \left (-b^2 (3 e f+7 d g) d^2+6 a b e (4 e f+d g) d+2 a^2 e^2 (15 e f-22 d g)\right )\right ) e^2}{\left (c d^2-b e d+a e^2\right ) (d+e x)^2}+\frac {-6 c^6 f d^6-c^5 (2 a e (15 e f-2 d g)-3 b d (6 e f+d g)) d^4-5 c^4 e \left (b^2 (3 e f+2 d g) d^2-a b e (12 e f+d g) d+2 a^2 e^2 (9 e f-4 d g)\right ) d^2-b^5 e^5 (3 b e f-2 b d g-a e g)+c^3 e^2 \left (10 b^3 g d^4-40 a b^2 e g d^3+a^2 b e^2 (138 e f-55 d g) d+30 a^3 e^3 (e f-2 d g)\right )+a b c^2 e^4 \left (-9 d (6 e f-5 d g) b^2-23 a e (3 e f-2 d g) b+23 a^2 e^2 g\right )-b^3 c e^4 \left (-d (6 e f-5 d g) b^2-9 a e (3 e f-2 d g) b+9 a^2 e^2 g\right )+c \left (b^2-4 a c\right )^2 e^4 (c d (6 e f-5 d g)-e (3 b e f-2 b d g-a e g)) x}{\left (c d^2-b e d+a e^2\right )^2 \left (c x^2+b x+a\right )}\right )dx}{\left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )}}{2 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {-e f b^2+c d f b+a e g b+2 a c e f-2 a c d g+c (2 c d f+2 a e g-b (e f+d g)) x}{2 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) (d+e x) \left (c x^2+b x+a\right )^2}-\frac {\frac {4 a c e (2 c d-b e) (2 c d f+2 a e g-b (e f+d g))-\left (-e b^2+c d b+2 a c e\right ) \left (6 c^2 f d^2-b e (3 b e f-2 b d g-a e g)+c (2 a e (5 e f-2 d g)-b d (2 e f+3 d g))\right )+c \left (4 c e (b d-2 a e) (2 c d f+2 a e g-b (e f+d g))-(2 c d-b e) \left (6 c^2 f d^2-b e (3 b e f-2 b d g-a e g)+c (2 a e (5 e f-2 d g)-b d (2 e f+3 d g))\right )\right ) x}{\left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) (d+e x) \left (c x^2+b x+a\right )}+\frac {2 \left (-\frac {\left (b^2-4 a c\right )^2 (c d (6 e f-5 d g)-e (3 b e f-2 b d g-a e g)) \log (d+e x) e^4}{\left (c d^2-b e d+a e^2\right )^2}+\frac {\left (b^2-4 a c\right )^2 (c d (6 e f-5 d g)-e (3 b e f-2 b d g-a e g)) \log \left (c x^2+b x+a\right ) e^4}{2 \left (c d^2-b e d+a e^2\right )^2}-\frac {\left (6 c^4 f d^4+c^3 (4 a e (6 e f-d g)-3 b d (4 e f+d g)) d^2-b^3 e^3 (3 b e f-2 b d g-a e g)-b c e^2 \left (-3 d (e f-d g) b^2-a e (21 e f-13 d g) b+7 a^2 e^2 g\right )-c^2 e \left (-b^2 (3 e f+7 d g) d^2+6 a b e (4 e f+d g) d+2 a^2 e^2 (15 e f-22 d g)\right )\right ) e}{\left (c d^2-b e d+a e^2\right ) (d+e x)}+\frac {\left (12 c^6 f d^6+2 c^5 (2 a e (15 e f-2 d g)-3 b d (6 e f+d g)) d^4+10 c^4 e \left (b^2 (3 e f+2 d g) d^2-a b e (12 e f+d g) d+2 a^2 e^2 (9 e f-4 d g)\right ) d^2+b^5 e^5 (3 b e f-2 b d g-a e g)+b^3 c e^4 \left (-d (6 e f-5 d g) b^2-10 a e (3 e f-2 d g) b+10 a^2 e^2 g\right )-10 a b c^2 e^4 \left (-d (6 e f-5 d g) b^2-3 a e (3 e f-2 d g) b+3 a^2 e^2 g\right )-10 c^3 e^2 \left (2 b^3 g d^4-8 a b^2 e g d^3+3 a^2 b e^2 (6 e f-d g) d+6 a^3 e^3 (e f-2 d g)\right )\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-b e d+a e^2\right )^2}\right )}{\left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )}}{2 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )}\) |
-1/2*(b*c*d*f - b^2*e*f + 2*a*c*e*f - 2*a*c*d*g + a*b*e*g + c*(2*c*d*f + 2 *a*e*g - b*(e*f + d*g))*x)/((b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*(d + e*x )*(a + b*x + c*x^2)^2) - ((4*a*c*e*(2*c*d - b*e)*(2*c*d*f + 2*a*e*g - b*(e *f + d*g)) - (b*c*d - b^2*e + 2*a*c*e)*(6*c^2*d^2*f - b*e*(3*b*e*f - 2*b*d *g - a*e*g) + c*(2*a*e*(5*e*f - 2*d*g) - b*d*(2*e*f + 3*d*g))) + c*(4*c*e* (b*d - 2*a*e)*(2*c*d*f + 2*a*e*g - b*(e*f + d*g)) - (2*c*d - b*e)*(6*c^2*d ^2*f - b*e*(3*b*e*f - 2*b*d*g - a*e*g) + c*(2*a*e*(5*e*f - 2*d*g) - b*d*(2 *e*f + 3*d*g))))*x)/((b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*(d + e*x)*(a + b*x + c*x^2)) + (2*(-((e*(6*c^4*d^4*f - b^3*e^3*(3*b*e*f - 2*b*d*g - a*e*g ) - b*c*e^2*(7*a^2*e^2*g - a*b*e*(21*e*f - 13*d*g) - 3*b^2*d*(e*f - d*g)) + c^3*d^2*(4*a*e*(6*e*f - d*g) - 3*b*d*(4*e*f + d*g)) - c^2*e*(2*a^2*e^2*( 15*e*f - 22*d*g) + 6*a*b*d*e*(4*e*f + d*g) - b^2*d^2*(3*e*f + 7*d*g))))/(( c*d^2 - b*d*e + a*e^2)*(d + e*x))) + ((12*c^6*d^6*f + b^5*e^5*(3*b*e*f - 2 *b*d*g - a*e*g) + b^3*c*e^4*(10*a^2*e^2*g - b^2*d*(6*e*f - 5*d*g) - 10*a*b *e*(3*e*f - 2*d*g)) - 10*a*b*c^2*e^4*(3*a^2*e^2*g - b^2*d*(6*e*f - 5*d*g) - 3*a*b*e*(3*e*f - 2*d*g)) - 10*c^3*e^2*(2*b^3*d^4*g - 8*a*b^2*d^3*e*g + 6 *a^3*e^3*(e*f - 2*d*g) + 3*a^2*b*d*e^2*(6*e*f - d*g)) + 2*c^5*d^4*(2*a*e*( 15*e*f - 2*d*g) - 3*b*d*(6*e*f + d*g)) + 10*c^4*d^2*e*(2*a^2*e^2*(9*e*f - 4*d*g) - a*b*d*e*(12*e*f + d*g) + b^2*d^2*(3*e*f + 2*d*g)))*ArcTanh[(b + 2 *c*x)/Sqrt[b^2 - 4*a*c]])/(Sqrt[b^2 - 4*a*c]*(c*d^2 - b*d*e + a*e^2)^2)...
3.24.77.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.))/((a_.) + (b_.)* (x_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*((f + g* x)^n/(a + b*x + c*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && In tegersQ[n]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2 *a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^m *(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2*(p + m + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d* m + b*e*m) - b*d*(3*c*d - b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p] )
Leaf count of result is larger than twice the leaf count of optimal. \(3176\) vs. \(2(1033)=2066\).
Time = 1.51 (sec) , antiderivative size = 3177, normalized size of antiderivative = 3.05
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(3177\) |
| risch | \(\text {Expression too large to display}\) | \(1882175\) |
e^4*(a*e^2*g+2*b*d*e*g-3*b*e^2*f-5*c*d^2*g+6*c*d*e*f)/(a*e^2-b*d*e+c*d^2)^ 4*ln(e*x+d)+(d*g-e*f)*e^4/(a*e^2-b*d*e+c*d^2)^3/(e*x+d)-1/(a*e^2-b*d*e+c*d ^2)^4*((c^2*(7*a^3*b*c*e^6*g-28*a^3*c^2*d*e^5*g+14*a^3*c^2*e^6*f-a^2*b^3*e ^6*g-2*a^2*b^2*c*d*e^5*g-13*a^2*b^2*c*e^6*f+41*a^2*b*c^2*d^2*e^4*g+10*a^2* b*c^2*d*e^5*f-24*a^2*c^3*d^3*e^3*g-10*a^2*c^3*d^2*e^4*f+2*a*b^4*e^6*f-3*a* b^3*c*d^2*e^4*g+10*a*b^3*c*d*e^5*f-8*a*b^2*c^2*d^3*e^3*g-40*a*b^2*c^2*d^2* e^4*f+5*a*b*c^3*d^4*e^2*g+60*a*b*c^3*d^3*e^3*f+4*a*c^4*d^5*e*g-30*a*c^4*d^ 4*e^2*f+b^5*d^2*e^4*g-2*b^5*d*e^5*f-4*b^4*c*d^3*e^3*g+5*b^4*c*d^2*e^4*f+10 *b^3*c^2*d^4*e^2*g-10*b^2*c^3*d^5*e*g-15*b^2*c^3*d^4*e^2*f+3*b*c^4*d^6*g+1 8*b*c^4*d^5*e*f-6*c^5*d^6*f)/(16*a^2*c^2-8*a*b^2*c+b^4)*x^3-1/2*c*(16*a^4* c^2*e^6*g-29*a^3*b^2*c*e^6*g+84*a^3*b*c^2*d*e^5*g-74*a^3*b*c^2*e^6*f-32*a^ 3*c^3*d^2*e^4*g+64*a^3*c^3*d*e^5*f+4*a^2*b^4*e^6*g+6*a^2*b^3*c*d*e^5*g+55* a^2*b^3*c*e^6*f-123*a^2*b^2*c^2*d^2*e^4*g-30*a^2*b^2*c^2*d*e^5*f+136*a^2*b *c^3*d^3*e^3*g-66*a^2*b*c^3*d^2*e^4*f-48*a^2*c^4*d^4*e^2*g+64*a^2*c^4*d^3* e^3*f-8*a*b^5*e^6*f+15*a*b^4*c*d^2*e^4*g-42*a*b^4*c*d*e^5*f-8*a*b^3*c^2*d^ 3*e^3*g+168*a*b^3*c^2*d^2*e^4*f+9*a*b^2*c^3*d^4*e^2*g-212*a*b^2*c^3*d^3*e^ 3*f-12*a*b*c^4*d^5*e*g+90*a*b*c^4*d^4*e^2*f-4*b^6*d^2*e^4*g+8*b^6*d*e^5*f+ 16*b^5*c*d^3*e^3*g-21*b^5*c*d^2*e^4*f-33*b^4*c^2*d^4*e^2*g+4*b^4*c^2*d^3*e ^3*f+30*b^3*c^3*d^5*e*g+45*b^3*c^3*d^4*e^2*f-9*b^2*c^4*d^6*g-54*b^2*c^4*d^ 5*e*f+18*b*c^5*d^6*f)/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2+(a^4*b*c^2*e^6*g-3...
Timed out. \[ \int \frac {f+g x}{(d+e x)^2 \left (a+b x+c x^2\right )^3} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {f+g x}{(d+e x)^2 \left (a+b x+c x^2\right )^3} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {f+g x}{(d+e x)^2 \left (a+b x+c x^2\right )^3} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` for more deta
Leaf count of result is larger than twice the leaf count of optimal. 3633 vs. \(2 (1032) = 2064\).
Time = 0.76 (sec) , antiderivative size = 3633, normalized size of antiderivative = 3.48 \[ \int \frac {f+g x}{(d+e x)^2 \left (a+b x+c x^2\right )^3} \, dx=\text {Too large to display} \]
-1/2*(6*c*d*e^5*f - 3*b*e^6*f - 5*c*d^2*e^4*g + 2*b*d*e^5*g + a*e^6*g)*log (c - 2*c*d/(e*x + d) + c*d^2/(e*x + d)^2 + b*e/(e*x + d) - b*d*e/(e*x + d) ^2 + a*e^2/(e*x + d)^2)/(c^4*d^8 - 4*b*c^3*d^7*e + 6*b^2*c^2*d^6*e^2 + 4*a *c^3*d^6*e^2 - 4*b^3*c*d^5*e^3 - 12*a*b*c^2*d^5*e^3 + b^4*d^4*e^4 + 12*a*b ^2*c*d^4*e^4 + 6*a^2*c^2*d^4*e^4 - 4*a*b^3*d^3*e^5 - 12*a^2*b*c*d^3*e^5 + 6*a^2*b^2*d^2*e^6 + 4*a^3*c*d^2*e^6 - 4*a^3*b*d*e^7 + a^4*e^8) - (e^11*f/( e*x + d) - d*e^10*g/(e*x + d))/(c^3*d^6*e^6 - 3*b*c^2*d^5*e^7 + 3*b^2*c*d^ 4*e^8 + 3*a*c^2*d^4*e^8 - b^3*d^3*e^9 - 6*a*b*c*d^3*e^9 + 3*a*b^2*d^2*e^10 + 3*a^2*c*d^2*e^10 - 3*a^2*b*d*e^11 + a^3*e^12) + (12*c^6*d^6*e^2*f - 36* b*c^5*d^5*e^3*f + 30*b^2*c^4*d^4*e^4*f + 60*a*c^5*d^4*e^4*f - 120*a*b*c^4* d^3*e^5*f + 180*a^2*c^4*d^2*e^6*f - 6*b^5*c*d*e^7*f + 60*a*b^3*c^2*d*e^7*f - 180*a^2*b*c^3*d*e^7*f + 3*b^6*e^8*f - 30*a*b^4*c*e^8*f + 90*a^2*b^2*c^2 *e^8*f - 60*a^3*c^3*e^8*f - 6*b*c^5*d^6*e^2*g + 20*b^2*c^4*d^5*e^3*g - 8*a *c^5*d^5*e^3*g - 20*b^3*c^3*d^4*e^4*g - 10*a*b*c^4*d^4*e^4*g + 80*a*b^2*c^ 3*d^3*e^5*g - 80*a^2*c^4*d^3*e^5*g + 5*b^5*c*d^2*e^6*g - 50*a*b^3*c^2*d^2* e^6*g + 30*a^2*b*c^3*d^2*e^6*g - 2*b^6*d*e^7*g + 20*a*b^4*c*d*e^7*g - 60*a ^2*b^2*c^2*d*e^7*g + 120*a^3*c^3*d*e^7*g - a*b^5*e^8*g + 10*a^2*b^3*c*e^8* g - 30*a^3*b*c^2*e^8*g)*arctan((2*c*d - 2*c*d^2/(e*x + d) - b*e + 2*b*d*e/ (e*x + d) - 2*a*e^2/(e*x + d))/(sqrt(-b^2 + 4*a*c)*e))/((b^4*c^4*d^8 - 8*a *b^2*c^5*d^8 + 16*a^2*c^6*d^8 - 4*b^5*c^3*d^7*e + 32*a*b^3*c^4*d^7*e - ...
Time = 22.09 (sec) , antiderivative size = 40079, normalized size of antiderivative = 38.43 \[ \int \frac {f+g x}{(d+e x)^2 \left (a+b x+c x^2\right )^3} \, dx=\text {Too large to display} \]
symsum(log(root(286720*a^9*b*c^8*d^7*e^9*z^3 + 286720*a^8*b*c^9*d^9*e^7*z^ 3 + 172032*a^10*b*c^7*d^5*e^11*z^3 + 172032*a^7*b*c^10*d^11*e^5*z^3 + 5734 4*a^11*b*c^6*d^3*e^13*z^3 + 57344*a^6*b*c^11*d^13*e^3*z^3 - 10240*a^11*b^3 *c^4*d*e^15*z^3 - 10240*a^4*b^3*c^11*d^15*e*z^3 + 5120*a^10*b^5*c^3*d*e^15 *z^3 + 5120*a^3*b^5*c^10*d^15*e*z^3 - 1280*a^9*b^7*c^2*d*e^15*z^3 - 1280*a ^2*b^7*c^9*d^15*e*z^3 - 1232*a^5*b^12*c*d^4*e^12*z^3 - 1232*a*b^12*c^5*d^1 2*e^4*z^3 + 1064*a^6*b^11*c*d^3*e^13*z^3 + 1064*a*b^11*c^6*d^13*e^3*z^3 + 840*a^4*b^13*c*d^5*e^11*z^3 + 840*a*b^13*c^4*d^11*e^5*z^3 - 552*a^7*b^10*c *d^2*e^14*z^3 - 552*a*b^10*c^7*d^14*e^2*z^3 - 280*a^3*b^14*c*d^6*e^10*z^3 - 280*a*b^14*c^3*d^10*e^6*z^3 - 8*a^2*b^15*c*d^7*e^9*z^3 - 8*a*b^15*c^2*d^ 9*e^7*z^3 + 8192*a^12*b*c^5*d*e^15*z^3 + 8192*a^5*b*c^12*d^15*e*z^3 + 160* a^8*b^9*c*d*e^15*z^3 + 160*a*b^9*c^8*d^15*e*z^3 + 36*a*b^16*c*d^8*e^8*z^3 - 483840*a^8*b^2*c^8*d^8*e^8*z^3 - 365568*a^7*b^5*c^6*d^7*e^9*z^3 - 365568 *a^6*b^5*c^7*d^9*e^7*z^3 - 358400*a^9*b^2*c^7*d^6*e^10*z^3 - 358400*a^7*b^ 2*c^9*d^10*e^6*z^3 + 241920*a^7*b^4*c^7*d^8*e^8*z^3 + 215040*a^8*b^4*c^6*d ^6*e^10*z^3 + 215040*a^8*b^3*c^7*d^7*e^9*z^3 + 215040*a^7*b^3*c^8*d^9*e^7* z^3 + 215040*a^6*b^4*c^8*d^10*e^6*z^3 - 193536*a^8*b^5*c^5*d^5*e^11*z^3 - 193536*a^5*b^5*c^8*d^11*e^5*z^3 - 136192*a^10*b^2*c^6*d^4*e^12*z^3 - 13619 2*a^6*b^2*c^10*d^12*e^4*z^3 + 133056*a^6*b^6*c^6*d^8*e^8*z^3 + 125440*a^9* b^4*c^5*d^4*e^12*z^3 + 125440*a^5*b^4*c^9*d^12*e^4*z^3 - 109944*a^5*b^8...