Integrand size = 22, antiderivative size = 364 \[ \int \frac {1}{(d+e x)^3 \left (a+b x+c x^2\right )^{3/2}} \, dx=-\frac {e}{2 \left (c d^2-b d e+a e^2\right ) (d+e x)^2 \sqrt {a+b x+c x^2}}-\frac {5 e (2 c d-b e)}{4 \left (c d^2-b d e+a e^2\right )^2 (d+e x) \sqrt {a+b x+c x^2}}-\frac {20 a c e (2 c d-b e)^2+\left (b c d-b^2 e+2 a c e\right ) \left (8 c^2 d^2+15 b^2 e^2-4 c e (7 b d+3 a e)\right )+c (2 c d-b e) \left (8 c^2 d^2+15 b^2 e^2-4 c e (2 b d+13 a e)\right ) x}{4 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^3 \sqrt {a+b x+c x^2}}+\frac {3 e^2 \left (16 c^2 d^2+5 b^2 e^2-4 c e (4 b d+a e)\right ) \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{8 \left (c d^2-b d e+a e^2\right )^{7/2}} \] Output:
-1/2*e/(a*e^2-b*d*e+c*d^2)/(e*x+d)^2/(c*x^2+b*x+a)^(1/2)-5/4*e*(-b*e+2*c*d )/(a*e^2-b*d*e+c*d^2)^2/(e*x+d)/(c*x^2+b*x+a)^(1/2)-1/4*(20*a*c*e*(-b*e+2* c*d)^2+(2*a*c*e-b^2*e+b*c*d)*(8*c^2*d^2+15*b^2*e^2-4*c*e*(3*a*e+7*b*d))+c* (-b*e+2*c*d)*(8*c^2*d^2+15*b^2*e^2-4*c*e*(13*a*e+2*b*d))*x)/(-4*a*c+b^2)/( a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^(1/2)+3/8*e^2*(16*c^2*d^2+5*b^2*e^2-4*c *e*(a*e+4*b*d))*arctanh(1/2*(b*d-2*a*e+(-b*e+2*c*d)*x)/(a*e^2-b*d*e+c*d^2) ^(1/2)/(c*x^2+b*x+a)^(1/2))/(a*e^2-b*d*e+c*d^2)^(7/2)
Time = 10.89 (sec) , antiderivative size = 353, normalized size of antiderivative = 0.97 \[ \int \frac {1}{(d+e x)^3 \left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {2 \left (-\frac {e \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) \sqrt {a+x (b+c x)}}{4 \left (c d^2+e (-b d+a e)\right ) (d+e x)^2}+\frac {b^2 e-2 c (a e+c d x)+b c (-d+e x)}{(d+e x)^2 \sqrt {a+x (b+c x)}}+\frac {1}{16} e \left (-\frac {2 (2 c d-b e) \left (8 c^2 d^2+15 b^2 e^2-4 c e (2 b d+13 a e)\right ) \sqrt {a+x (b+c x)}}{\left (c d^2+e (-b d+a e)\right )^2 (d+e x)}-\frac {3 \left (b^2-4 a c\right ) e \left (16 c^2 d^2+5 b^2 e^2-4 c e (4 b d+a e)\right ) \text {arctanh}\left (\frac {-b d+2 a e-2 c d x+b e x}{2 \sqrt {c d^2+e (-b d+a e)} \sqrt {a+x (b+c x)}}\right )}{\left (c d^2+e (-b d+a e)\right )^{5/2}}\right )\right )}{\left (b^2-4 a c\right ) \left (c d^2+e (-b d+a e)\right )} \] Input:
Integrate[1/((d + e*x)^3*(a + b*x + c*x^2)^(3/2)),x]
Output:
(2*(-1/4*(e*(8*c^2*d^2 + 5*b^2*e^2 - 4*c*e*(2*b*d + 3*a*e))*Sqrt[a + x*(b + c*x)])/((c*d^2 + e*(-(b*d) + a*e))*(d + e*x)^2) + (b^2*e - 2*c*(a*e + c* d*x) + b*c*(-d + e*x))/((d + e*x)^2*Sqrt[a + x*(b + c*x)]) + (e*((-2*(2*c* d - b*e)*(8*c^2*d^2 + 15*b^2*e^2 - 4*c*e*(2*b*d + 13*a*e))*Sqrt[a + x*(b + c*x)])/((c*d^2 + e*(-(b*d) + a*e))^2*(d + e*x)) - (3*(b^2 - 4*a*c)*e*(16* c^2*d^2 + 5*b^2*e^2 - 4*c*e*(4*b*d + a*e))*ArcTanh[(-(b*d) + 2*a*e - 2*c*d *x + b*e*x)/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)])])/(c* d^2 + e*(-(b*d) + a*e))^(5/2)))/16))/((b^2 - 4*a*c)*(c*d^2 + e*(-(b*d) + a *e)))
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {1}{(d+e x)^3 \left (a+b x+c x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1165 |
| \(\displaystyle -\frac {2 \int \frac {e \left (-5 e b^2+4 c d b+12 a c e+4 c (2 c d-b e) x\right )}{2 (d+e x)^3 \sqrt {c x^2+b x+a}}dx}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{\left (b^2-4 a c\right ) (d+e x)^2 \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {e \int \frac {-5 e b^2+4 c d b+12 a c e+4 c (2 c d-b e) x}{(d+e x)^3 \sqrt {c x^2+b x+a}}dx}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{\left (b^2-4 a c\right ) (d+e x)^2 \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 1237 |
| \(\displaystyle -\frac {e \left (\frac {\sqrt {a+b x+c x^2} \left (-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right )}{2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}-\frac {\int -\frac {15 e^2 b^3-28 c d e b^2+8 c^2 d^2 b-52 a c e^2 b+80 a c^2 d e+2 c \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) x}{2 (d+e x)^2 \sqrt {c x^2+b x+a}}dx}{2 \left (a e^2-b d e+c d^2\right )}\right )}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{\left (b^2-4 a c\right ) (d+e x)^2 \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {e \left (\frac {\int -\frac {-15 e^2 b^3+28 c d e b^2-4 c \left (2 c d^2-13 a e^2\right ) b-80 a c^2 d e-2 c \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) x}{(d+e x)^2 \sqrt {c x^2+b x+a}}dx}{4 \left (a e^2-b d e+c d^2\right )}+\frac {\sqrt {a+b x+c x^2} \left (-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right )}{2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}\right )}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{\left (b^2-4 a c\right ) (d+e x)^2 \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {e \left (\frac {\sqrt {a+b x+c x^2} \left (-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right )}{2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}-\frac {\int -\frac {15 e^2 b^3-28 c d e b^2+8 c^2 d^2 b-52 a c e^2 b+80 a c^2 d e+2 c \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) x}{(d+e x)^2 \sqrt {c x^2+b x+a}}dx}{4 \left (a e^2-b d e+c d^2\right )}\right )}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{\left (b^2-4 a c\right ) (d+e x)^2 \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {e \left (\frac {\int -\frac {-15 e^2 b^3+28 c d e b^2-4 c \left (2 c d^2-13 a e^2\right ) b-80 a c^2 d e-2 c \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) x}{(d+e x)^2 \sqrt {c x^2+b x+a}}dx}{4 \left (a e^2-b d e+c d^2\right )}+\frac {\sqrt {a+b x+c x^2} \left (-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right )}{2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}\right )}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{\left (b^2-4 a c\right ) (d+e x)^2 \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {e \left (\frac {\sqrt {a+b x+c x^2} \left (-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right )}{2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}-\frac {\int -\frac {15 e^2 b^3-28 c d e b^2+8 c^2 d^2 b-52 a c e^2 b+80 a c^2 d e+2 c \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) x}{(d+e x)^2 \sqrt {c x^2+b x+a}}dx}{4 \left (a e^2-b d e+c d^2\right )}\right )}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{\left (b^2-4 a c\right ) (d+e x)^2 \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {e \left (\frac {\int -\frac {-15 e^2 b^3+28 c d e b^2-4 c \left (2 c d^2-13 a e^2\right ) b-80 a c^2 d e-2 c \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) x}{(d+e x)^2 \sqrt {c x^2+b x+a}}dx}{4 \left (a e^2-b d e+c d^2\right )}+\frac {\sqrt {a+b x+c x^2} \left (-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right )}{2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}\right )}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{\left (b^2-4 a c\right ) (d+e x)^2 \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {e \left (\frac {\sqrt {a+b x+c x^2} \left (-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right )}{2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}-\frac {\int -\frac {15 e^2 b^3-28 c d e b^2+8 c^2 d^2 b-52 a c e^2 b+80 a c^2 d e+2 c \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) x}{(d+e x)^2 \sqrt {c x^2+b x+a}}dx}{4 \left (a e^2-b d e+c d^2\right )}\right )}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{\left (b^2-4 a c\right ) (d+e x)^2 \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {e \left (\frac {\int -\frac {-15 e^2 b^3+28 c d e b^2-4 c \left (2 c d^2-13 a e^2\right ) b-80 a c^2 d e-2 c \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) x}{(d+e x)^2 \sqrt {c x^2+b x+a}}dx}{4 \left (a e^2-b d e+c d^2\right )}+\frac {\sqrt {a+b x+c x^2} \left (-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right )}{2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}\right )}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{\left (b^2-4 a c\right ) (d+e x)^2 \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {e \left (\frac {\sqrt {a+b x+c x^2} \left (-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right )}{2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}-\frac {\int -\frac {15 e^2 b^3-28 c d e b^2+8 c^2 d^2 b-52 a c e^2 b+80 a c^2 d e+2 c \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) x}{(d+e x)^2 \sqrt {c x^2+b x+a}}dx}{4 \left (a e^2-b d e+c d^2\right )}\right )}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{\left (b^2-4 a c\right ) (d+e x)^2 \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {e \left (\frac {\int -\frac {-15 e^2 b^3+28 c d e b^2-4 c \left (2 c d^2-13 a e^2\right ) b-80 a c^2 d e-2 c \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) x}{(d+e x)^2 \sqrt {c x^2+b x+a}}dx}{4 \left (a e^2-b d e+c d^2\right )}+\frac {\sqrt {a+b x+c x^2} \left (-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right )}{2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}\right )}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{\left (b^2-4 a c\right ) (d+e x)^2 \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {e \left (\frac {\sqrt {a+b x+c x^2} \left (-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right )}{2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}-\frac {\int -\frac {15 e^2 b^3-28 c d e b^2+8 c^2 d^2 b-52 a c e^2 b+80 a c^2 d e+2 c \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) x}{(d+e x)^2 \sqrt {c x^2+b x+a}}dx}{4 \left (a e^2-b d e+c d^2\right )}\right )}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{\left (b^2-4 a c\right ) (d+e x)^2 \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {e \left (\frac {\int -\frac {-15 e^2 b^3+28 c d e b^2-4 c \left (2 c d^2-13 a e^2\right ) b-80 a c^2 d e-2 c \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) x}{(d+e x)^2 \sqrt {c x^2+b x+a}}dx}{4 \left (a e^2-b d e+c d^2\right )}+\frac {\sqrt {a+b x+c x^2} \left (-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right )}{2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}\right )}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{\left (b^2-4 a c\right ) (d+e x)^2 \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {e \left (\frac {\sqrt {a+b x+c x^2} \left (-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right )}{2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}-\frac {\int -\frac {15 e^2 b^3-28 c d e b^2+8 c^2 d^2 b-52 a c e^2 b+80 a c^2 d e+2 c \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) x}{(d+e x)^2 \sqrt {c x^2+b x+a}}dx}{4 \left (a e^2-b d e+c d^2\right )}\right )}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{\left (b^2-4 a c\right ) (d+e x)^2 \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {e \left (\frac {\int -\frac {-15 e^2 b^3+28 c d e b^2-4 c \left (2 c d^2-13 a e^2\right ) b-80 a c^2 d e-2 c \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) x}{(d+e x)^2 \sqrt {c x^2+b x+a}}dx}{4 \left (a e^2-b d e+c d^2\right )}+\frac {\sqrt {a+b x+c x^2} \left (-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right )}{2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}\right )}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{\left (b^2-4 a c\right ) (d+e x)^2 \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {e \left (\frac {\sqrt {a+b x+c x^2} \left (-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right )}{2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}-\frac {\int -\frac {15 e^2 b^3-28 c d e b^2+8 c^2 d^2 b-52 a c e^2 b+80 a c^2 d e+2 c \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) x}{(d+e x)^2 \sqrt {c x^2+b x+a}}dx}{4 \left (a e^2-b d e+c d^2\right )}\right )}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{\left (b^2-4 a c\right ) (d+e x)^2 \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {e \left (\frac {\int -\frac {-15 e^2 b^3+28 c d e b^2-4 c \left (2 c d^2-13 a e^2\right ) b-80 a c^2 d e-2 c \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) x}{(d+e x)^2 \sqrt {c x^2+b x+a}}dx}{4 \left (a e^2-b d e+c d^2\right )}+\frac {\sqrt {a+b x+c x^2} \left (-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right )}{2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}\right )}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{\left (b^2-4 a c\right ) (d+e x)^2 \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {e \left (\frac {\sqrt {a+b x+c x^2} \left (-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right )}{2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}-\frac {\int -\frac {15 e^2 b^3-28 c d e b^2+8 c^2 d^2 b-52 a c e^2 b+80 a c^2 d e+2 c \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) x}{(d+e x)^2 \sqrt {c x^2+b x+a}}dx}{4 \left (a e^2-b d e+c d^2\right )}\right )}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{\left (b^2-4 a c\right ) (d+e x)^2 \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {e \left (\frac {\int -\frac {-15 e^2 b^3+28 c d e b^2-4 c \left (2 c d^2-13 a e^2\right ) b-80 a c^2 d e-2 c \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) x}{(d+e x)^2 \sqrt {c x^2+b x+a}}dx}{4 \left (a e^2-b d e+c d^2\right )}+\frac {\sqrt {a+b x+c x^2} \left (-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right )}{2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}\right )}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{\left (b^2-4 a c\right ) (d+e x)^2 \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {e \left (\frac {\sqrt {a+b x+c x^2} \left (-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right )}{2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}-\frac {\int -\frac {15 e^2 b^3-28 c d e b^2+8 c^2 d^2 b-52 a c e^2 b+80 a c^2 d e+2 c \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) x}{(d+e x)^2 \sqrt {c x^2+b x+a}}dx}{4 \left (a e^2-b d e+c d^2\right )}\right )}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{\left (b^2-4 a c\right ) (d+e x)^2 \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {e \left (\frac {\int -\frac {-15 e^2 b^3+28 c d e b^2-4 c \left (2 c d^2-13 a e^2\right ) b-80 a c^2 d e-2 c \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) x}{(d+e x)^2 \sqrt {c x^2+b x+a}}dx}{4 \left (a e^2-b d e+c d^2\right )}+\frac {\sqrt {a+b x+c x^2} \left (-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right )}{2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}\right )}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{\left (b^2-4 a c\right ) (d+e x)^2 \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {e \left (\frac {\sqrt {a+b x+c x^2} \left (-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right )}{2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}-\frac {\int -\frac {15 e^2 b^3-28 c d e b^2+8 c^2 d^2 b-52 a c e^2 b+80 a c^2 d e+2 c \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) x}{(d+e x)^2 \sqrt {c x^2+b x+a}}dx}{4 \left (a e^2-b d e+c d^2\right )}\right )}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{\left (b^2-4 a c\right ) (d+e x)^2 \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {e \left (\frac {\int -\frac {-15 e^2 b^3+28 c d e b^2-4 c \left (2 c d^2-13 a e^2\right ) b-80 a c^2 d e-2 c \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) x}{(d+e x)^2 \sqrt {c x^2+b x+a}}dx}{4 \left (a e^2-b d e+c d^2\right )}+\frac {\sqrt {a+b x+c x^2} \left (-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right )}{2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}\right )}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{\left (b^2-4 a c\right ) (d+e x)^2 \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {e \left (\frac {\sqrt {a+b x+c x^2} \left (-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right )}{2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}-\frac {\int -\frac {15 e^2 b^3-28 c d e b^2+8 c^2 d^2 b-52 a c e^2 b+80 a c^2 d e+2 c \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) x}{(d+e x)^2 \sqrt {c x^2+b x+a}}dx}{4 \left (a e^2-b d e+c d^2\right )}\right )}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{\left (b^2-4 a c\right ) (d+e x)^2 \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {e \left (\frac {\int -\frac {-15 e^2 b^3+28 c d e b^2-4 c \left (2 c d^2-13 a e^2\right ) b-80 a c^2 d e-2 c \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) x}{(d+e x)^2 \sqrt {c x^2+b x+a}}dx}{4 \left (a e^2-b d e+c d^2\right )}+\frac {\sqrt {a+b x+c x^2} \left (-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right )}{2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}\right )}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{\left (b^2-4 a c\right ) (d+e x)^2 \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {e \left (\frac {\sqrt {a+b x+c x^2} \left (-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right )}{2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}-\frac {\int -\frac {15 e^2 b^3-28 c d e b^2+8 c^2 d^2 b-52 a c e^2 b+80 a c^2 d e+2 c \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) x}{(d+e x)^2 \sqrt {c x^2+b x+a}}dx}{4 \left (a e^2-b d e+c d^2\right )}\right )}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{\left (b^2-4 a c\right ) (d+e x)^2 \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {e \left (\frac {\int -\frac {-15 e^2 b^3+28 c d e b^2-4 c \left (2 c d^2-13 a e^2\right ) b-80 a c^2 d e-2 c \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) x}{(d+e x)^2 \sqrt {c x^2+b x+a}}dx}{4 \left (a e^2-b d e+c d^2\right )}+\frac {\sqrt {a+b x+c x^2} \left (-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right )}{2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}\right )}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{\left (b^2-4 a c\right ) (d+e x)^2 \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {e \left (\frac {\sqrt {a+b x+c x^2} \left (-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right )}{2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}-\frac {\int -\frac {15 e^2 b^3-28 c d e b^2+8 c^2 d^2 b-52 a c e^2 b+80 a c^2 d e+2 c \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) x}{(d+e x)^2 \sqrt {c x^2+b x+a}}dx}{4 \left (a e^2-b d e+c d^2\right )}\right )}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{\left (b^2-4 a c\right ) (d+e x)^2 \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {e \left (\frac {\int -\frac {-15 e^2 b^3+28 c d e b^2-4 c \left (2 c d^2-13 a e^2\right ) b-80 a c^2 d e-2 c \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) x}{(d+e x)^2 \sqrt {c x^2+b x+a}}dx}{4 \left (a e^2-b d e+c d^2\right )}+\frac {\sqrt {a+b x+c x^2} \left (-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right )}{2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}\right )}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{\left (b^2-4 a c\right ) (d+e x)^2 \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}\) |
Input:
Int[1/((d + e*x)^3*(a + b*x + c*x^2)^(3/2)),x]
Output:
$Aborted
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[(d + e*x)^(m + 1)*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e) *x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^ 2))), x] + Simp[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^m*Simp[b*c*d*e*(2*p - m + 2) + b^2*e^2*(m + p + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3) - c*e*(2*c*d - b*e)*(m + 2*p + 4)*x, x]*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && LtQ[p, -1] && IntQuadraticQ[a, b, c, d, e, m, p, x]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(e*f - d*g)*(d + e*x)^(m + 1)*((a + b* x + c*x^2)^(p + 1)/((m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[1/((m + 1) *(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*Simp[ (c*d*f - f*b*e + a*e*g)*(m + 1) + b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && LtQ[m, -1 ] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Leaf count of result is larger than twice the leaf count of optimal. \(1188\) vs. \(2(342)=684\).
Time = 1.32 (sec) , antiderivative size = 1189, normalized size of antiderivative = 3.27
Input:
int(1/(e*x+d)^3/(c*x^2+b*x+a)^(3/2),x,method=_RETURNVERBOSE)
Output:
1/e^3*(-1/2/(a*e^2-b*d*e+c*d^2)*e^2/(x+d/e)^2/(c*(x+d/e)^2+(b*e-2*c*d)/e*( x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)-5/4*(b*e-2*c*d)*e/(a*e^2-b*d*e+c*d^2 )*(-1/(a*e^2-b*d*e+c*d^2)*e^2/(x+d/e)/(c*(x+d/e)^2+(b*e-2*c*d)/e*(x+d/e)+( a*e^2-b*d*e+c*d^2)/e^2)^(1/2)-3/2*(b*e-2*c*d)*e/(a*e^2-b*d*e+c*d^2)*(1/(a* e^2-b*d*e+c*d^2)*e^2/(c*(x+d/e)^2+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2 )/e^2)^(1/2)-(b*e-2*c*d)*e/(a*e^2-b*d*e+c*d^2)*(2*c*(x+d/e)+(b*e-2*c*d)/e) /(4*c*(a*e^2-b*d*e+c*d^2)/e^2-(b*e-2*c*d)^2/e^2)/(c*(x+d/e)^2+(b*e-2*c*d)/ e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)-1/(a*e^2-b*d*e+c*d^2)*e^2/((a*e^2 -b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(x+d/ e)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*(c*(x+d/e)^2+(b*e-2*c*d)/e*(x+d/e)+(a *e^2-b*d*e+c*d^2)/e^2)^(1/2))/(x+d/e)))-4*c/(a*e^2-b*d*e+c*d^2)*e^2*(2*c*( x+d/e)+(b*e-2*c*d)/e)/(4*c*(a*e^2-b*d*e+c*d^2)/e^2-(b*e-2*c*d)^2/e^2)/(c*( x+d/e)^2+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))-3/2*c/(a*e^ 2-b*d*e+c*d^2)*e^2*(1/(a*e^2-b*d*e+c*d^2)*e^2/(c*(x+d/e)^2+(b*e-2*c*d)/e*( x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)-(b*e-2*c*d)*e/(a*e^2-b*d*e+c*d^2)*(2 *c*(x+d/e)+(b*e-2*c*d)/e)/(4*c*(a*e^2-b*d*e+c*d^2)/e^2-(b*e-2*c*d)^2/e^2)/ (c*(x+d/e)^2+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)-1/(a*e^2 -b*d*e+c*d^2)*e^2/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2 )/e^2+(b*e-2*c*d)/e*(x+d/e)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*(c*(x+d/e)^2 +(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(x+d/e))))
Leaf count of result is larger than twice the leaf count of optimal. 2845 vs. \(2 (342) = 684\).
Time = 4.33 (sec) , antiderivative size = 5732, normalized size of antiderivative = 15.75 \[ \int \frac {1}{(d+e x)^3 \left (a+b x+c x^2\right )^{3/2}} \, dx=\text {Too large to display} \] Input:
integrate(1/(e*x+d)^3/(c*x^2+b*x+a)^(3/2),x, algorithm="fricas")
Output:
Too large to include
\[ \int \frac {1}{(d+e x)^3 \left (a+b x+c x^2\right )^{3/2}} \, dx=\int \frac {1}{\left (d + e x\right )^{3} \left (a + b x + c x^{2}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(1/(e*x+d)**3/(c*x**2+b*x+a)**(3/2),x)
Output:
Integral(1/((d + e*x)**3*(a + b*x + c*x**2)**(3/2)), x)
Exception generated. \[ \int \frac {1}{(d+e x)^3 \left (a+b x+c x^2\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(1/(e*x+d)^3/(c*x^2+b*x+a)^(3/2),x, algorithm="maxima")
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(a*e^2-b*d*e>0)', see `assume?` f or more de
Leaf count of result is larger than twice the leaf count of optimal. 2718 vs. \(2 (342) = 684\).
Time = 0.29 (sec) , antiderivative size = 2718, normalized size of antiderivative = 7.47 \[ \int \frac {1}{(d+e x)^3 \left (a+b x+c x^2\right )^{3/2}} \, dx=\text {Too large to display} \] Input:
integrate(1/(e*x+d)^3/(c*x^2+b*x+a)^(3/2),x, algorithm="giac")
Output:
-2*((2*c^7*d^9 - 9*b*c^6*d^8*e + 18*b^2*c^5*d^7*e^2 - 21*b^3*c^4*d^6*e^3 + 15*b^4*c^3*d^5*e^4 + 6*a*b^2*c^4*d^5*e^4 - 12*a^2*c^5*d^5*e^4 - 6*b^5*c^2 *d^4*e^5 - 15*a*b^3*c^3*d^4*e^5 + 30*a^2*b*c^4*d^4*e^5 + b^6*c*d^3*e^6 + 1 2*a*b^4*c^2*d^3*e^6 - 18*a^2*b^2*c^3*d^3*e^6 - 16*a^3*c^4*d^3*e^6 - 3*a*b^ 5*c*d^2*e^7 - 3*a^2*b^3*c^2*d^2*e^7 + 24*a^3*b*c^3*d^2*e^7 + 3*a^2*b^4*c*d *e^8 - 6*a^3*b^2*c^2*d*e^8 - 6*a^4*c^3*d*e^8 - a^3*b^3*c*e^9 + 3*a^4*b*c^2 *e^9)*x/(b^2*c^6*d^12 - 4*a*c^7*d^12 - 6*b^3*c^5*d^11*e + 24*a*b*c^6*d^11* e + 15*b^4*c^4*d^10*e^2 - 54*a*b^2*c^5*d^10*e^2 - 24*a^2*c^6*d^10*e^2 - 20 *b^5*c^3*d^9*e^3 + 50*a*b^3*c^4*d^9*e^3 + 120*a^2*b*c^5*d^9*e^3 + 15*b^6*c ^2*d^8*e^4 - 225*a^2*b^2*c^4*d^8*e^4 - 60*a^3*c^5*d^8*e^4 - 6*b^7*c*d^7*e^ 5 - 36*a*b^5*c^2*d^7*e^5 + 180*a^2*b^3*c^3*d^7*e^5 + 240*a^3*b*c^4*d^7*e^5 + b^8*d^6*e^6 + 26*a*b^6*c*d^6*e^6 - 30*a^2*b^4*c^2*d^6*e^6 - 340*a^3*b^2 *c^3*d^6*e^6 - 80*a^4*c^4*d^6*e^6 - 6*a*b^7*d^5*e^7 - 36*a^2*b^5*c*d^5*e^7 + 180*a^3*b^3*c^2*d^5*e^7 + 240*a^4*b*c^3*d^5*e^7 + 15*a^2*b^6*d^4*e^8 - 225*a^4*b^2*c^2*d^4*e^8 - 60*a^5*c^3*d^4*e^8 - 20*a^3*b^5*d^3*e^9 + 50*a^4 *b^3*c*d^3*e^9 + 120*a^5*b*c^2*d^3*e^9 + 15*a^4*b^4*d^2*e^10 - 54*a^5*b^2* c*d^2*e^10 - 24*a^6*c^2*d^2*e^10 - 6*a^5*b^3*d*e^11 + 24*a^6*b*c*d*e^11 + a^6*b^2*e^12 - 4*a^7*c*e^12) + (b*c^6*d^9 - 6*b^2*c^5*d^8*e + 6*a*c^6*d^8* e + 15*b^3*c^4*d^7*e^2 - 24*a*b*c^5*d^7*e^2 - 20*b^4*c^3*d^6*e^3 + 34*a*b^ 2*c^4*d^6*e^3 + 16*a^2*c^5*d^6*e^3 + 15*b^5*c^2*d^5*e^4 - 15*a*b^3*c^3*...
Timed out. \[ \int \frac {1}{(d+e x)^3 \left (a+b x+c x^2\right )^{3/2}} \, dx=\int \frac {1}{{\left (d+e\,x\right )}^3\,{\left (c\,x^2+b\,x+a\right )}^{3/2}} \,d x \] Input:
int(1/((d + e*x)^3*(a + b*x + c*x^2)^(3/2)),x)
Output:
int(1/((d + e*x)^3*(a + b*x + c*x^2)^(3/2)), x)
Time = 0.92 (sec) , antiderivative size = 9369, normalized size of antiderivative = 25.74 \[ \int \frac {1}{(d+e x)^3 \left (a+b x+c x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:
int(1/(e*x+d)^3/(c*x^2+b*x+a)^(3/2),x)
Output:
(48*sqrt(a*e**2 - b*d*e + c*d**2)*log( - 2*sqrt(a + b*x + c*x**2)*sqrt(a*e **2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*a**3*c**2*d**2*e**4 + 96*sqrt(a*e**2 - b*d*e + c*d**2)*log( - 2*sqrt(a + b*x + c*x**2)*sqrt(a *e**2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*a**3*c**2*d*e**5* x + 48*sqrt(a*e**2 - b*d*e + c*d**2)*log( - 2*sqrt(a + b*x + c*x**2)*sqrt( a*e**2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*a**3*c**2*e**6*x **2 - 72*sqrt(a*e**2 - b*d*e + c*d**2)*log( - 2*sqrt(a + b*x + c*x**2)*sqr t(a*e**2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*a**2*b**2*c*d* *2*e**4 - 144*sqrt(a*e**2 - b*d*e + c*d**2)*log( - 2*sqrt(a + b*x + c*x**2 )*sqrt(a*e**2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*a**2*b**2 *c*d*e**5*x - 72*sqrt(a*e**2 - b*d*e + c*d**2)*log( - 2*sqrt(a + b*x + c*x **2)*sqrt(a*e**2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*a**2*b **2*c*e**6*x**2 + 192*sqrt(a*e**2 - b*d*e + c*d**2)*log( - 2*sqrt(a + b*x + c*x**2)*sqrt(a*e**2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*a **2*b*c**2*d**3*e**3 + 432*sqrt(a*e**2 - b*d*e + c*d**2)*log( - 2*sqrt(a + b*x + c*x**2)*sqrt(a*e**2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d *x)*a**2*b*c**2*d**2*e**4*x + 288*sqrt(a*e**2 - b*d*e + c*d**2)*log( - 2*s qrt(a + b*x + c*x**2)*sqrt(a*e**2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*a**2*b*c**2*d*e**5*x**2 + 48*sqrt(a*e**2 - b*d*e + c*d**2)*log( - 2*sqrt(a + b*x + c*x**2)*sqrt(a*e**2 - b*d*e + c*d**2) - 2*a*e + b*d...