Integrand size = 38, antiderivative size = 528 \[ \int \frac {d+e x+f x^2+g x^3+h x^4+i x^5}{\left (a+b x+c x^2\right )^3} \, dx=-\frac {a b^3 c h+b c^2 \left (c^2 d+a c f-3 a^2 h\right )-a b^4 i-a b^2 c (c g-4 a i)-2 a c^2 \left (c^2 e-a c g+a^2 i\right )+\left (2 c^5 d-c^4 (b e+2 a f)+c^3 \left (b^2 f+3 a b g+2 a^2 h\right )-b^5 i+b^3 c (b h+5 a i)-b c^2 \left (b^2 g+4 a b h+5 a^2 i\right )\right ) x}{2 c^4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac {b^5 c h+b^3 c^2 (c f-8 a h)+2 b c^3 \left (3 c^2 d+a c f+11 a^2 h\right )-b^6 i-b^4 c (c g-11 a i)-16 a^2 c^3 (c g-2 a i)-b^2 c^2 \left (3 c^2 e-5 a c g+39 a^2 i\right )+2 c \left (6 c^5 d-c^4 (3 b e-2 a f)+c^3 \left (b^2 f-3 a b g-10 a^2 h\right )+2 b^5 i-b^3 c (b h+15 a i)+a b c^2 (8 b h+25 a i)\right ) x}{2 c^4 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac {\left (12 c^5 d-c^4 (6 b e-4 a f)+2 c^3 \left (b^2 f-3 a b g+6 a^2 h\right )-b^5 i+10 a b^3 c i-30 a^2 b c^2 i\right ) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^3 \left (b^2-4 a c\right )^{5/2}}+\frac {i \log \left (a+b x+c x^2\right )}{2 c^3} \] Output:
-1/2*(a*b^3*c*h+b*c^2*(-3*a^2*h+a*c*f+c^2*d)-a*b^4*i-a*b^2*c*(-4*a*i+c*g)- 2*a*c^2*(a^2*i-a*c*g+c^2*e)+(2*c^5*d-c^4*(2*a*f+b*e)+c^3*(2*a^2*h+3*a*b*g+ b^2*f)-b^5*i+b^3*c*(5*a*i+b*h)-b*c^2*(5*a^2*i+4*a*b*h+b^2*g))*x)/c^4/(-4*a *c+b^2)/(c*x^2+b*x+a)^2+1/2*(b^5*c*h+b^3*c^2*(-8*a*h+c*f)+2*b*c^3*(11*a^2* h+a*c*f+3*c^2*d)-b^6*i-b^4*c*(-11*a*i+c*g)-16*a^2*c^3*(-2*a*i+c*g)-b^2*c^2 *(39*a^2*i-5*a*c*g+3*c^2*e)+2*c*(6*c^5*d-c^4*(-2*a*f+3*b*e)+c^3*(-10*a^2*h -3*a*b*g+b^2*f)+2*b^5*i-b^3*c*(15*a*i+b*h)+a*b*c^2*(25*a*i+8*b*h))*x)/c^4/ (-4*a*c+b^2)^2/(c*x^2+b*x+a)-(12*c^5*d-c^4*(-4*a*f+6*b*e)+2*c^3*(6*a^2*h-3 *a*b*g+b^2*f)-b^5*i+10*a*b^3*c*i-30*a^2*b*c^2*i)*arctanh((2*c*x+b)/(-4*a*c +b^2)^(1/2))/c^3/(-4*a*c+b^2)^(5/2)+1/2*i*ln(c*x^2+b*x+a)/c^3
Time = 1.21 (sec) , antiderivative size = 488, normalized size of antiderivative = 0.92 \[ \int \frac {d+e x+f x^2+g x^3+h x^4+i x^5}{\left (a+b x+c x^2\right )^3} \, dx=\frac {\frac {b^5 i x+b^4 (a i-c h x)+2 c^2 \left (a^3 i-c^3 d x+a c^2 (e+f x)-a^2 c (g+h x)\right )+b^2 c \left (-4 a^2 i-c^2 f x+a c (g+4 h x)\right )+b^3 c (c g x-a (h+5 i x))+b c^2 \left (c^2 (-d+e x)-a c (f+3 g x)+a^2 (3 h+5 i x)\right )}{\left (b^2-4 a c\right ) (a+x (b+c x))^2}+\frac {-b^6 i+b^5 c (h+4 i x)+b^3 c^2 (c f-8 a h-30 a i x)-b^4 c (-11 a i+c (g+2 h x))+4 c^3 \left (8 a^3 i+3 c^3 d x+a c^2 f x-a^2 c (4 g+5 h x)\right )+b^2 c^2 \left (-39 a^2 i+c^2 (-3 e+2 f x)+a c (5 g+16 h x)\right )+2 b c^3 \left (3 c^2 (d-e x)+a c (f-3 g x)+a^2 (11 h+25 i x)\right )}{\left (b^2-4 a c\right )^2 (a+x (b+c x))}+\frac {2 c \left (12 c^5 d+c^4 (-6 b e+4 a f)+2 c^3 \left (b^2 f-3 a b g+6 a^2 h\right )-b^5 i+10 a b^3 c i-30 a^2 b c^2 i\right ) \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\left (-b^2+4 a c\right )^{5/2}}+c i \log (a+x (b+c x))}{2 c^4} \] Input:
Integrate[(d + e*x + f*x^2 + g*x^3 + h*x^4 + i*x^5)/(a + b*x + c*x^2)^3,x]
Output:
((b^5*i*x + b^4*(a*i - c*h*x) + 2*c^2*(a^3*i - c^3*d*x + a*c^2*(e + f*x) - a^2*c*(g + h*x)) + b^2*c*(-4*a^2*i - c^2*f*x + a*c*(g + 4*h*x)) + b^3*c*( c*g*x - a*(h + 5*i*x)) + b*c^2*(c^2*(-d + e*x) - a*c*(f + 3*g*x) + a^2*(3* h + 5*i*x)))/((b^2 - 4*a*c)*(a + x*(b + c*x))^2) + (-(b^6*i) + b^5*c*(h + 4*i*x) + b^3*c^2*(c*f - 8*a*h - 30*a*i*x) - b^4*c*(-11*a*i + c*(g + 2*h*x) ) + 4*c^3*(8*a^3*i + 3*c^3*d*x + a*c^2*f*x - a^2*c*(4*g + 5*h*x)) + b^2*c^ 2*(-39*a^2*i + c^2*(-3*e + 2*f*x) + a*c*(5*g + 16*h*x)) + 2*b*c^3*(3*c^2*( d - e*x) + a*c*(f - 3*g*x) + a^2*(11*h + 25*i*x)))/((b^2 - 4*a*c)^2*(a + x *(b + c*x))) + (2*c*(12*c^5*d + c^4*(-6*b*e + 4*a*f) + 2*c^3*(b^2*f - 3*a* b*g + 6*a^2*h) - b^5*i + 10*a*b^3*c*i - 30*a^2*b*c^2*i)*ArcTan[(b + 2*c*x) /Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(5/2) + c*i*Log[a + x*(b + c*x)])/(2* c^4)
Time = 1.22 (sec) , antiderivative size = 567, normalized size of antiderivative = 1.07, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.184, Rules used = {2191, 2191, 27, 1142, 1083, 219, 1103}
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.
| \(\displaystyle \int \frac {d+e x+f x^2+g x^3+h x^4+i x^5}{\left (a+b x+c x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 2191 |
| \(\displaystyle -\frac {\int \frac {-\frac {i b^5}{c^4}+\frac {(b h+3 a i) b^3}{c^3}-3 e b-\frac {\left (-i a^2+2 b h a+b^2 g\right ) b}{c^2}+2 \left (4 a-\frac {b^2}{c}\right ) i x^3-\frac {2 \left (b^2-4 a c\right ) (c h-b i) x^2}{c^2}+6 c d+2 a f+\frac {-2 h a^2+b g a+b^2 f}{c}-\frac {2 \left (b^2-4 a c\right ) \left (i b^2+c^2 g-c (b h+a i)\right ) x}{c^3}}{\left (c x^2+b x+a\right )^2}dx}{2 \left (b^2-4 a c\right )}-\frac {x \left (c^3 \left (2 a^2 h+3 a b g+b^2 f\right )-b c^2 \left (5 a^2 i+4 a b h+b^2 g\right )+b^3 c (5 a i+b h)-c^4 (2 a f+b e)+b^5 (-i)+2 c^5 d\right )+b c^2 \left (-3 a^2 h+a c f+c^2 d\right )-2 a c^2 \left (a^2 i-a c g+c^2 e\right )-a b^4 i+a b^3 c h-a b^2 c (c g-4 a i)}{2 c^4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 2191 |
| \(\displaystyle -\frac {-\frac {\int \frac {2 \left (\left (\frac {a i b^3}{c^2}+f b^2-3 c e b-3 a g b-\frac {7 a^2 i b}{c}+6 c^2 d+2 a c f+6 a^2 h\right ) c^2+\left (b^2-4 a c\right )^2 i x\right )}{c^2 \left (c x^2+b x+a\right )}dx}{b^2-4 a c}-\frac {-b^2 c^2 \left (39 a^2 i-5 a c g+3 c^2 e\right )+2 c x \left (c^3 \left (-10 a^2 h-3 a b g+b^2 f\right )-b^3 c (15 a i+b h)-c^4 (3 b e-2 a f)+a b c^2 (25 a i+8 b h)+2 b^5 i+6 c^5 d\right )+2 b c^3 \left (11 a^2 h+a c f+3 c^2 d\right )-16 a^2 c^3 (c g-2 a i)-b^4 c (c g-11 a i)+b^3 c^2 (c f-8 a h)+b^6 (-i)+b^5 c h}{c^4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}}{2 \left (b^2-4 a c\right )}-\frac {x \left (c^3 \left (2 a^2 h+3 a b g+b^2 f\right )-b c^2 \left (5 a^2 i+4 a b h+b^2 g\right )+b^3 c (5 a i+b h)-c^4 (2 a f+b e)+b^5 (-i)+2 c^5 d\right )+b c^2 \left (-3 a^2 h+a c f+c^2 d\right )-2 a c^2 \left (a^2 i-a c g+c^2 e\right )-a b^4 i+a b^3 c h-a b^2 c (c g-4 a i)}{2 c^4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {2 \int \frac {\left (\frac {a i b^3}{c^2}+f b^2-3 c e b-3 a g b-\frac {7 a^2 i b}{c}+6 c^2 d+2 a c f+6 a^2 h\right ) c^2+\left (b^2-4 a c\right )^2 i x}{c x^2+b x+a}dx}{c^2 \left (b^2-4 a c\right )}-\frac {-b^2 c^2 \left (39 a^2 i-5 a c g+3 c^2 e\right )+2 c x \left (c^3 \left (-10 a^2 h-3 a b g+b^2 f\right )-b^3 c (15 a i+b h)-c^4 (3 b e-2 a f)+a b c^2 (25 a i+8 b h)+2 b^5 i+6 c^5 d\right )+2 b c^3 \left (11 a^2 h+a c f+3 c^2 d\right )-16 a^2 c^3 (c g-2 a i)-b^4 c (c g-11 a i)+b^3 c^2 (c f-8 a h)+b^6 (-i)+b^5 c h}{c^4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}}{2 \left (b^2-4 a c\right )}-\frac {x \left (c^3 \left (2 a^2 h+3 a b g+b^2 f\right )-b c^2 \left (5 a^2 i+4 a b h+b^2 g\right )+b^3 c (5 a i+b h)-c^4 (2 a f+b e)+b^5 (-i)+2 c^5 d\right )+b c^2 \left (-3 a^2 h+a c f+c^2 d\right )-2 a c^2 \left (a^2 i-a c g+c^2 e\right )-a b^4 i+a b^3 c h-a b^2 c (c g-4 a i)}{2 c^4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle -\frac {-\frac {2 \left (\frac {\left (2 c^3 \left (6 a^2 h-3 a b g+b^2 f\right )-30 a^2 b c^2 i+10 a b^3 c i-c^4 (6 b e-4 a f)+b^5 (-i)+12 c^5 d\right ) \int \frac {1}{c x^2+b x+a}dx}{2 c}+\frac {i \left (b^2-4 a c\right )^2 \int \frac {b+2 c x}{c x^2+b x+a}dx}{2 c}\right )}{c^2 \left (b^2-4 a c\right )}-\frac {-b^2 c^2 \left (39 a^2 i-5 a c g+3 c^2 e\right )+2 c x \left (c^3 \left (-10 a^2 h-3 a b g+b^2 f\right )-b^3 c (15 a i+b h)-c^4 (3 b e-2 a f)+a b c^2 (25 a i+8 b h)+2 b^5 i+6 c^5 d\right )+2 b c^3 \left (11 a^2 h+a c f+3 c^2 d\right )-16 a^2 c^3 (c g-2 a i)-b^4 c (c g-11 a i)+b^3 c^2 (c f-8 a h)+b^6 (-i)+b^5 c h}{c^4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}}{2 \left (b^2-4 a c\right )}-\frac {x \left (c^3 \left (2 a^2 h+3 a b g+b^2 f\right )-b c^2 \left (5 a^2 i+4 a b h+b^2 g\right )+b^3 c (5 a i+b h)-c^4 (2 a f+b e)+b^5 (-i)+2 c^5 d\right )+b c^2 \left (-3 a^2 h+a c f+c^2 d\right )-2 a c^2 \left (a^2 i-a c g+c^2 e\right )-a b^4 i+a b^3 c h-a b^2 c (c g-4 a i)}{2 c^4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle -\frac {-\frac {2 \left (\frac {i \left (b^2-4 a c\right )^2 \int \frac {b+2 c x}{c x^2+b x+a}dx}{2 c}-\frac {\left (2 c^3 \left (6 a^2 h-3 a b g+b^2 f\right )-30 a^2 b c^2 i+10 a b^3 c i-c^4 (6 b e-4 a f)+b^5 (-i)+12 c^5 d\right ) \int \frac {1}{b^2-(b+2 c x)^2-4 a c}d(b+2 c x)}{c}\right )}{c^2 \left (b^2-4 a c\right )}-\frac {-b^2 c^2 \left (39 a^2 i-5 a c g+3 c^2 e\right )+2 c x \left (c^3 \left (-10 a^2 h-3 a b g+b^2 f\right )-b^3 c (15 a i+b h)-c^4 (3 b e-2 a f)+a b c^2 (25 a i+8 b h)+2 b^5 i+6 c^5 d\right )+2 b c^3 \left (11 a^2 h+a c f+3 c^2 d\right )-16 a^2 c^3 (c g-2 a i)-b^4 c (c g-11 a i)+b^3 c^2 (c f-8 a h)+b^6 (-i)+b^5 c h}{c^4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}}{2 \left (b^2-4 a c\right )}-\frac {x \left (c^3 \left (2 a^2 h+3 a b g+b^2 f\right )-b c^2 \left (5 a^2 i+4 a b h+b^2 g\right )+b^3 c (5 a i+b h)-c^4 (2 a f+b e)+b^5 (-i)+2 c^5 d\right )+b c^2 \left (-3 a^2 h+a c f+c^2 d\right )-2 a c^2 \left (a^2 i-a c g+c^2 e\right )-a b^4 i+a b^3 c h-a b^2 c (c g-4 a i)}{2 c^4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {-\frac {2 \left (\frac {i \left (b^2-4 a c\right )^2 \int \frac {b+2 c x}{c x^2+b x+a}dx}{2 c}-\frac {\text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right ) \left (2 c^3 \left (6 a^2 h-3 a b g+b^2 f\right )-30 a^2 b c^2 i+10 a b^3 c i-c^4 (6 b e-4 a f)+b^5 (-i)+12 c^5 d\right )}{c \sqrt {b^2-4 a c}}\right )}{c^2 \left (b^2-4 a c\right )}-\frac {-b^2 c^2 \left (39 a^2 i-5 a c g+3 c^2 e\right )+2 c x \left (c^3 \left (-10 a^2 h-3 a b g+b^2 f\right )-b^3 c (15 a i+b h)-c^4 (3 b e-2 a f)+a b c^2 (25 a i+8 b h)+2 b^5 i+6 c^5 d\right )+2 b c^3 \left (11 a^2 h+a c f+3 c^2 d\right )-16 a^2 c^3 (c g-2 a i)-b^4 c (c g-11 a i)+b^3 c^2 (c f-8 a h)+b^6 (-i)+b^5 c h}{c^4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}}{2 \left (b^2-4 a c\right )}-\frac {x \left (c^3 \left (2 a^2 h+3 a b g+b^2 f\right )-b c^2 \left (5 a^2 i+4 a b h+b^2 g\right )+b^3 c (5 a i+b h)-c^4 (2 a f+b e)+b^5 (-i)+2 c^5 d\right )+b c^2 \left (-3 a^2 h+a c f+c^2 d\right )-2 a c^2 \left (a^2 i-a c g+c^2 e\right )-a b^4 i+a b^3 c h-a b^2 c (c g-4 a i)}{2 c^4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle -\frac {-\frac {2 \left (\frac {i \left (b^2-4 a c\right )^2 \log \left (a+b x+c x^2\right )}{2 c}-\frac {\text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right ) \left (2 c^3 \left (6 a^2 h-3 a b g+b^2 f\right )-30 a^2 b c^2 i+10 a b^3 c i-c^4 (6 b e-4 a f)+b^5 (-i)+12 c^5 d\right )}{c \sqrt {b^2-4 a c}}\right )}{c^2 \left (b^2-4 a c\right )}-\frac {-b^2 c^2 \left (39 a^2 i-5 a c g+3 c^2 e\right )+2 c x \left (c^3 \left (-10 a^2 h-3 a b g+b^2 f\right )-b^3 c (15 a i+b h)-c^4 (3 b e-2 a f)+a b c^2 (25 a i+8 b h)+2 b^5 i+6 c^5 d\right )+2 b c^3 \left (11 a^2 h+a c f+3 c^2 d\right )-16 a^2 c^3 (c g-2 a i)-b^4 c (c g-11 a i)+b^3 c^2 (c f-8 a h)+b^6 (-i)+b^5 c h}{c^4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}}{2 \left (b^2-4 a c\right )}-\frac {x \left (c^3 \left (2 a^2 h+3 a b g+b^2 f\right )-b c^2 \left (5 a^2 i+4 a b h+b^2 g\right )+b^3 c (5 a i+b h)-c^4 (2 a f+b e)+b^5 (-i)+2 c^5 d\right )+b c^2 \left (-3 a^2 h+a c f+c^2 d\right )-2 a c^2 \left (a^2 i-a c g+c^2 e\right )-a b^4 i+a b^3 c h-a b^2 c (c g-4 a i)}{2 c^4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\) |
Input:
Int[(d + e*x + f*x^2 + g*x^3 + h*x^4 + i*x^5)/(a + b*x + c*x^2)^3,x]
Output:
-1/2*(a*b^3*c*h + b*c^2*(c^2*d + a*c*f - 3*a^2*h) - a*b^4*i - a*b^2*c*(c*g - 4*a*i) - 2*a*c^2*(c^2*e - a*c*g + a^2*i) + (2*c^5*d - c^4*(b*e + 2*a*f) + c^3*(b^2*f + 3*a*b*g + 2*a^2*h) - b^5*i + b^3*c*(b*h + 5*a*i) - b*c^2*( b^2*g + 4*a*b*h + 5*a^2*i))*x)/(c^4*(b^2 - 4*a*c)*(a + b*x + c*x^2)^2) - ( -((b^5*c*h + b^3*c^2*(c*f - 8*a*h) + 2*b*c^3*(3*c^2*d + a*c*f + 11*a^2*h) - b^6*i - b^4*c*(c*g - 11*a*i) - 16*a^2*c^3*(c*g - 2*a*i) - b^2*c^2*(3*c^2 *e - 5*a*c*g + 39*a^2*i) + 2*c*(6*c^5*d - c^4*(3*b*e - 2*a*f) + c^3*(b^2*f - 3*a*b*g - 10*a^2*h) + 2*b^5*i - b^3*c*(b*h + 15*a*i) + a*b*c^2*(8*b*h + 25*a*i))*x)/(c^4*(b^2 - 4*a*c)*(a + b*x + c*x^2))) - (2*(-(((12*c^5*d - c ^4*(6*b*e - 4*a*f) + 2*c^3*(b^2*f - 3*a*b*g + 6*a^2*h) - b^5*i + 10*a*b^3* c*i - 30*a^2*b*c^2*i)*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(c*Sqrt[b^2 - 4*a*c])) + ((b^2 - 4*a*c)^2*i*Log[a + b*x + c*x^2])/(2*c)))/(c^2*(b^2 - 4*a*c)))/(2*(b^2 - 4*a*c))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Int[(Pq_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, a + b*x + c*x^2, x], f = Coeff[PolynomialRemainder[P q, a + b*x + c*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x + c*x^2, x], x, 1]}, Simp[(b*f - 2*a*g + (2*c*f - b*g)*x)*((a + b*x + c*x^2)^ (p + 1)/((p + 1)*(b^2 - 4*a*c))), x] + Simp[1/((p + 1)*(b^2 - 4*a*c)) Int [(a + b*x + c*x^2)^(p + 1)*ExpandToSum[(p + 1)*(b^2 - 4*a*c)*Q - (2*p + 3)* (2*c*f - b*g), x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && NeQ[b^ 2 - 4*a*c, 0] && LtQ[p, -1]
Time = 1.79 (sec) , antiderivative size = 769, normalized size of antiderivative = 1.46
| method | result | size |
| default | \(\frac {\frac {\left (25 a^{2} b \,c^{2} i -10 a^{2} c^{3} h -15 a \,b^{3} c i +8 a \,b^{2} c^{2} h -3 a b \,c^{3} g +2 a \,c^{4} f +2 b^{5} i -b^{4} c h +b^{2} c^{3} f -3 b \,c^{4} e +6 c^{5} d \right ) x^{3}}{c^{2} \left (16 a^{2} c^{2}-8 c a \,b^{2}+b^{4}\right )}+\frac {\left (32 a^{3} c^{3} i +11 a^{2} b^{2} c^{2} i +2 a^{2} b \,c^{3} h -16 a^{2} c^{4} g -19 a \,b^{4} c i +8 a \,b^{3} c^{2} h -a \,b^{2} c^{3} g +6 a b \,c^{4} f +3 b^{6} i -b^{5} c h -b^{4} c^{2} g +3 b^{3} c^{3} f -9 b^{2} c^{4} e +18 b \,c^{5} d \right ) x^{2}}{2 \left (16 a^{2} c^{2}-8 c a \,b^{2}+b^{4}\right ) c^{3}}+\frac {\left (31 a^{3} b \,c^{2} i -6 a^{3} c^{3} h -22 a^{2} b^{3} c i +10 a^{2} b^{2} c^{2} h -5 a^{2} b \,c^{3} g -2 a^{2} c^{4} f +3 a \,b^{5} i -a \,b^{4} c h -a \,b^{3} c^{2} g +5 a \,b^{2} c^{3} f -5 a b \,c^{4} e +10 a \,c^{5} d -b^{3} c^{3} e +2 b^{2} c^{4} d \right ) x}{\left (16 a^{2} c^{2}-8 c a \,b^{2}+b^{4}\right ) c^{3}}+\frac {24 a^{4} c^{2} i -21 a^{3} b^{2} c i +10 a^{3} b \,c^{2} h -8 a^{3} c^{3} g +3 a^{2} b^{4} i -a^{2} b^{3} c h -a^{2} b^{2} c^{2} g +6 a^{2} b \,c^{3} f -8 a^{2} c^{4} e -a \,b^{2} c^{3} e +10 a b \,c^{4} d -b^{3} c^{3} d}{2 c^{3} \left (16 a^{2} c^{2}-8 c a \,b^{2}+b^{4}\right )}}{\left (c \,x^{2}+b x +a \right )^{2}}+\frac {\frac {\left (16 a^{2} c^{2} i -8 a \,b^{2} i c +b^{4} i \right ) \ln \left (c \,x^{2}+b x +a \right )}{2 c}+\frac {2 \left (-7 a^{2} b i c +6 a^{2} c^{2} h +a \,b^{3} i -3 a b \,c^{2} g +2 a \,c^{3} f +b^{2} c^{2} f -3 b e \,c^{3}+6 c^{4} d -\frac {\left (16 a^{2} c^{2} i -8 a \,b^{2} i c +b^{4} i \right ) b}{2 c}\right ) \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{c^{2} \left (16 a^{2} c^{2}-8 c a \,b^{2}+b^{4}\right )}\) | \(769\) |
| risch | \(\text {Expression too large to display}\) | \(4132\) |
Input:
int((i*x^5+h*x^4+g*x^3+f*x^2+e*x+d)/(c*x^2+b*x+a)^3,x,method=_RETURNVERBOS E)
Output:
((25*a^2*b*c^2*i-10*a^2*c^3*h-15*a*b^3*c*i+8*a*b^2*c^2*h-3*a*b*c^3*g+2*a*c ^4*f+2*b^5*i-b^4*c*h+b^2*c^3*f-3*b*c^4*e+6*c^5*d)/c^2/(16*a^2*c^2-8*a*b^2* c+b^4)*x^3+1/2*(32*a^3*c^3*i+11*a^2*b^2*c^2*i+2*a^2*b*c^3*h-16*a^2*c^4*g-1 9*a*b^4*c*i+8*a*b^3*c^2*h-a*b^2*c^3*g+6*a*b*c^4*f+3*b^6*i-b^5*c*h-b^4*c^2* g+3*b^3*c^3*f-9*b^2*c^4*e+18*b*c^5*d)/(16*a^2*c^2-8*a*b^2*c+b^4)/c^3*x^2+( 31*a^3*b*c^2*i-6*a^3*c^3*h-22*a^2*b^3*c*i+10*a^2*b^2*c^2*h-5*a^2*b*c^3*g-2 *a^2*c^4*f+3*a*b^5*i-a*b^4*c*h-a*b^3*c^2*g+5*a*b^2*c^3*f-5*a*b*c^4*e+10*a* c^5*d-b^3*c^3*e+2*b^2*c^4*d)/(16*a^2*c^2-8*a*b^2*c+b^4)/c^3*x+1/2/c^3*(24* a^4*c^2*i-21*a^3*b^2*c*i+10*a^3*b*c^2*h-8*a^3*c^3*g+3*a^2*b^4*i-a^2*b^3*c* h-a^2*b^2*c^2*g+6*a^2*b*c^3*f-8*a^2*c^4*e-a*b^2*c^3*e+10*a*b*c^4*d-b^3*c^3 *d)/(16*a^2*c^2-8*a*b^2*c+b^4))/(c*x^2+b*x+a)^2+1/c^2/(16*a^2*c^2-8*a*b^2* c+b^4)*(1/2*(16*a^2*c^2*i-8*a*b^2*c*i+b^4*i)/c*ln(c*x^2+b*x+a)+2*(-7*a^2*b *i*c+6*a^2*c^2*h+a*b^3*i-3*a*b*c^2*g+2*a*c^3*f+b^2*c^2*f-3*b*e*c^3+6*c^4*d -1/2*(16*a^2*c^2*i-8*a*b^2*c*i+b^4*i)*b/c)/(4*a*c-b^2)^(1/2)*arctan((2*c*x +b)/(4*a*c-b^2)^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 1730 vs. \(2 (518) = 1036\).
Time = 0.16 (sec) , antiderivative size = 3480, normalized size of antiderivative = 6.59 \[ \int \frac {d+e x+f x^2+g x^3+h x^4+i x^5}{\left (a+b x+c x^2\right )^3} \, dx=\text {Too large to display} \] Input:
integrate((i*x^5+h*x^4+g*x^3+f*x^2+e*x+d)/(c*x^2+b*x+a)^3,x, algorithm="fr icas")
Output:
Too large to include
Timed out. \[ \int \frac {d+e x+f x^2+g x^3+h x^4+i x^5}{\left (a+b x+c x^2\right )^3} \, dx=\text {Timed out} \] Input:
integrate((i*x**5+h*x**4+g*x**3+f*x**2+e*x+d)/(c*x**2+b*x+a)**3,x)
Output:
Timed out
Exception generated. \[ \int \frac {d+e x+f x^2+g x^3+h x^4+i x^5}{\left (a+b x+c x^2\right )^3} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((i*x^5+h*x^4+g*x^3+f*x^2+e*x+d)/(c*x^2+b*x+a)^3,x, algorithm="ma xima")
Output:
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
Time = 0.32 (sec) , antiderivative size = 646, normalized size of antiderivative = 1.22 \[ \int \frac {d+e x+f x^2+g x^3+h x^4+i x^5}{\left (a+b x+c x^2\right )^3} \, dx=\frac {{\left (12 \, c^{5} d - 6 \, b c^{4} e + 2 \, b^{2} c^{3} f + 4 \, a c^{4} f - 6 \, a b c^{3} g + 12 \, a^{2} c^{3} h - b^{5} i + 10 \, a b^{3} c i - 30 \, a^{2} b c^{2} i\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}\right )} \sqrt {-b^{2} + 4 \, a c}} + \frac {i \log \left (c x^{2} + b x + a\right )}{2 \, c^{3}} - \frac {b^{3} c^{3} d - 10 \, a b c^{4} d + a b^{2} c^{3} e + 8 \, a^{2} c^{4} e - 6 \, a^{2} b c^{3} f + a^{2} b^{2} c^{2} g + 8 \, a^{3} c^{3} g + a^{2} b^{3} c h - 10 \, a^{3} b c^{2} h - 3 \, a^{2} b^{4} i + 21 \, a^{3} b^{2} c i - 24 \, a^{4} c^{2} i - 2 \, {\left (6 \, c^{6} d - 3 \, b c^{5} e + b^{2} c^{4} f + 2 \, a c^{5} f - 3 \, a b c^{4} g - b^{4} c^{2} h + 8 \, a b^{2} c^{3} h - 10 \, a^{2} c^{4} h + 2 \, b^{5} c i - 15 \, a b^{3} c^{2} i + 25 \, a^{2} b c^{3} i\right )} x^{3} - {\left (18 \, b c^{5} d - 9 \, b^{2} c^{4} e + 3 \, b^{3} c^{3} f + 6 \, a b c^{4} f - b^{4} c^{2} g - a b^{2} c^{3} g - 16 \, a^{2} c^{4} g - b^{5} c h + 8 \, a b^{3} c^{2} h + 2 \, a^{2} b c^{3} h + 3 \, b^{6} i - 19 \, a b^{4} c i + 11 \, a^{2} b^{2} c^{2} i + 32 \, a^{3} c^{3} i\right )} x^{2} - 2 \, {\left (2 \, b^{2} c^{4} d + 10 \, a c^{5} d - b^{3} c^{3} e - 5 \, a b c^{4} e + 5 \, a b^{2} c^{3} f - 2 \, a^{2} c^{4} f - a b^{3} c^{2} g - 5 \, a^{2} b c^{3} g - a b^{4} c h + 10 \, a^{2} b^{2} c^{2} h - 6 \, a^{3} c^{3} h + 3 \, a b^{5} i - 22 \, a^{2} b^{3} c i + 31 \, a^{3} b c^{2} i\right )} x}{2 \, {\left (c x^{2} + b x + a\right )}^{2} {\left (b^{2} - 4 \, a c\right )}^{2} c^{3}} \] Input:
integrate((i*x^5+h*x^4+g*x^3+f*x^2+e*x+d)/(c*x^2+b*x+a)^3,x, algorithm="gi ac")
Output:
(12*c^5*d - 6*b*c^4*e + 2*b^2*c^3*f + 4*a*c^4*f - 6*a*b*c^3*g + 12*a^2*c^3 *h - b^5*i + 10*a*b^3*c*i - 30*a^2*b*c^2*i)*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/((b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*sqrt(-b^2 + 4*a*c)) + 1/2*i *log(c*x^2 + b*x + a)/c^3 - 1/2*(b^3*c^3*d - 10*a*b*c^4*d + a*b^2*c^3*e + 8*a^2*c^4*e - 6*a^2*b*c^3*f + a^2*b^2*c^2*g + 8*a^3*c^3*g + a^2*b^3*c*h - 10*a^3*b*c^2*h - 3*a^2*b^4*i + 21*a^3*b^2*c*i - 24*a^4*c^2*i - 2*(6*c^6*d - 3*b*c^5*e + b^2*c^4*f + 2*a*c^5*f - 3*a*b*c^4*g - b^4*c^2*h + 8*a*b^2*c^ 3*h - 10*a^2*c^4*h + 2*b^5*c*i - 15*a*b^3*c^2*i + 25*a^2*b*c^3*i)*x^3 - (1 8*b*c^5*d - 9*b^2*c^4*e + 3*b^3*c^3*f + 6*a*b*c^4*f - b^4*c^2*g - a*b^2*c^ 3*g - 16*a^2*c^4*g - b^5*c*h + 8*a*b^3*c^2*h + 2*a^2*b*c^3*h + 3*b^6*i - 1 9*a*b^4*c*i + 11*a^2*b^2*c^2*i + 32*a^3*c^3*i)*x^2 - 2*(2*b^2*c^4*d + 10*a *c^5*d - b^3*c^3*e - 5*a*b*c^4*e + 5*a*b^2*c^3*f - 2*a^2*c^4*f - a*b^3*c^2 *g - 5*a^2*b*c^3*g - a*b^4*c*h + 10*a^2*b^2*c^2*h - 6*a^3*c^3*h + 3*a*b^5* i - 22*a^2*b^3*c*i + 31*a^3*b*c^2*i)*x)/((c*x^2 + b*x + a)^2*(b^2 - 4*a*c) ^2*c^3)
Time = 19.93 (sec) , antiderivative size = 1027, normalized size of antiderivative = 1.95 \[ \int \frac {d+e x+f x^2+g x^3+h x^4+i x^5}{\left (a+b x+c x^2\right )^3} \, dx =\text {Too large to display} \] Input:
int((d + e*x + f*x^2 + g*x^3 + h*x^4 + i*x^5)/(a + b*x + c*x^2)^3,x)
Output:
(atan((x*(32*a^2*c^5*(4*a*c - b^2)^(5/2) + 2*b^4*c^3*(4*a*c - b^2)^(5/2) - 16*a*b^2*c^4*(4*a*c - b^2)^(5/2)))/(c^2*(4*a*c - b^2)^5) + ((32*a^2*c^5*( 4*a*c - b^2)^(5/2) + 2*b^4*c^3*(4*a*c - b^2)^(5/2) - 16*a*b^2*c^4*(4*a*c - b^2)^(5/2))*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4))/(2*c^5*(4*a*c - b^2)^ 5*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)))*(12*c^5*d - b^5*i + 2*b^2*c^3*f + 12*a^ 2*c^3*h + 4*a*c^4*f - 6*b*c^4*e - 6*a*b*c^3*g + 10*a*b^3*c*i - 30*a^2*b*c^ 2*i))/(c^3*(4*a*c - b^2)^(5/2)) - (log(a + b*x + c*x^2)*(b^10*i - 1024*a^5 *c^5*i + 160*a^2*b^6*c^2*i - 640*a^3*b^4*c^3*i + 1280*a^4*b^2*c^4*i - 20*a *b^8*c*i))/(2*(1024*a^5*c^8 - b^10*c^3 + 20*a*b^8*c^4 - 160*a^2*b^6*c^5 + 640*a^3*b^4*c^6 - 1280*a^4*b^2*c^7)) - ((8*a^2*c^4*e + b^3*c^3*d + 8*a^3*c ^3*g - 3*a^2*b^4*i - 24*a^4*c^2*i + a^2*b^2*c^2*g - 10*a*b*c^4*d + a*b^2*c ^3*e - 6*a^2*b*c^3*f + a^2*b^3*c*h - 10*a^3*b*c^2*h + 21*a^3*b^2*c*i)/(2*c ^3*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) - (x^2*(3*b^6*i - 9*b^2*c^4*e - 16*a^2* c^4*g + 3*b^3*c^3*f - b^4*c^2*g + 32*a^3*c^3*i + 18*b*c^5*d - b^5*c*h + 11 *a^2*b^2*c^2*i + 6*a*b*c^4*f - 19*a*b^4*c*i - a*b^2*c^3*g + 8*a*b^3*c^2*h + 2*a^2*b*c^3*h))/(2*c^3*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + (x*(2*a^2*c^4*f - 2*b^2*c^4*d + b^3*c^3*e + 6*a^3*c^3*h - 10*a*c^5*d - 3*a*b^5*i - 10*a^2 *b^2*c^2*h + 5*a*b*c^4*e + a*b^4*c*h - 5*a*b^2*c^3*f + a*b^3*c^2*g + 5*a^2 *b*c^3*g + 22*a^2*b^3*c*i - 31*a^3*b*c^2*i))/(c^3*(b^4 + 16*a^2*c^2 - 8*a* b^2*c)) - (x^3*(6*c^5*d + 2*b^5*i + b^2*c^3*f - 10*a^2*c^3*h + 2*a*c^4*...
Time = 0.17 (sec) , antiderivative size = 3913, normalized size of antiderivative = 7.41 \[ \int \frac {d+e x+f x^2+g x^3+h x^4+i x^5}{\left (a+b x+c x^2\right )^3} \, dx =\text {Too large to display} \] Input:
int((i*x^5+h*x^4+g*x^3+f*x^2+e*x+d)/(c*x^2+b*x+a)^3,x)
Output:
( - 60*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**4*b**2*c **2*i + 24*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**4*b* c**3*h + 20*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**3*b **4*c*i - 120*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**3 *b**3*c**2*i*x - 12*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2) )*a**3*b**2*c**3*g + 48*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b **2))*a**3*b**2*c**3*h*x - 120*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4* a*c - b**2))*a**3*b**2*c**3*i*x**2 + 8*sqrt(4*a*c - b**2)*atan((b + 2*c*x) /sqrt(4*a*c - b**2))*a**3*b*c**4*f + 48*sqrt(4*a*c - b**2)*atan((b + 2*c*x )/sqrt(4*a*c - b**2))*a**3*b*c**4*h*x**2 - 2*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**2*b**6*i + 40*sqrt(4*a*c - b**2)*atan((b + 2 *c*x)/sqrt(4*a*c - b**2))*a**2*b**5*c*i*x - 20*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**2*b**4*c**2*i*x**2 + 4*sqrt(4*a*c - b**2)* atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**2*b**3*c**3*f - 24*sqrt(4*a*c - b* *2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**2*b**3*c**3*g*x + 24*sqrt(4*a* c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**2*b**3*c**3*h*x**2 - 120 *sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**2*b**3*c**3*i* x**3 - 12*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**2*b** 2*c**4*e + 16*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**2 *b**2*c**4*f*x - 24*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b*...