Integrand size = 27, antiderivative size = 974 \[ \int \frac {A+B x}{(d+e x) \left (a+b x+c x^2\right )^{7/2}} \, dx=\frac {2 \left (a B (2 c d-b e)-A \left (b c d-b^2 e+2 a c e\right )+c (b B d-2 A c d+A b e-2 a B e) x\right )}{5 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^{5/2}}+\frac {2 \left (8 a c e (2 c d-b e) (b B d-2 A c d+A b e-2 a B e)+\left (b c d-b^2 e+2 a c e\right ) \left (5 b^2 e (B d-A e)-8 b c d (B d+A e)+4 c \left (4 A c d^2-a B d e+5 a A e^2\right )\right )+c \left (8 c e (b d-2 a e) (b B d-2 A c d+A b e-2 a B e)+(2 c d-b e) \left (5 b^2 e (B d-A e)-8 b c d (B d+A e)+4 c \left (4 A c d^2-a B d e+5 a A e^2\right )\right )\right ) x\right )}{15 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 \left (a+b x+c x^2\right )^{3/2}}+\frac {2 \left (4 a c e (2 c d-b e) \left (8 c e (b d-2 a e) (b B d-2 A c d+A b e-2 a B e)+(2 c d-b e) \left (5 b^2 e (B d-A e)-8 b c d (B d+A e)+4 c \left (4 A c d^2-a B d e+5 a A e^2\right )\right )\right )-\left (b c d-b^2 e+2 a c e\right ) \left (8 c d e (b B d-2 A c d+A b e-2 a B e) \left (4 b c d-3 b^2 e+4 a c e\right )+\left (8 c^2 d^2-3 b^2 e^2-4 c e (b d-3 a e)\right ) \left (5 b^2 e (B d-A e)-8 b c d (B d+A e)+4 c \left (4 A c d^2-a B d e+5 a A e^2\right )\right )\right )+c \left (4 c e (b d-2 a e) \left (8 c e (b d-2 a e) (b B d-2 A c d+A b e-2 a B e)+(2 c d-b e) \left (5 b^2 e (B d-A e)-8 b c d (B d+A e)+4 c \left (4 A c d^2-a B d e+5 a A e^2\right )\right )\right )-(2 c d-b e) \left (8 c d e (b B d-2 A c d+A b e-2 a B e) \left (4 b c d-3 b^2 e+4 a c e\right )+\left (8 c^2 d^2-3 b^2 e^2-4 c e (b d-3 a e)\right ) \left (5 b^2 e (B d-A e)-8 b c d (B d+A e)+4 c \left (4 A c d^2-a B d e+5 a A e^2\right )\right )\right )\right ) x\right )}{15 \left (b^2-4 a c\right )^3 \left (c d^2-b d e+a e^2\right )^3 \sqrt {a+b x+c x^2}}-\frac {e^5 (B d-A e) \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{\left (c d^2-b d e+a e^2\right )^{7/2}} \]
2/5*(a*B*(-b*e+2*c*d)-A*(2*a*c*e-b^2*e+b*c*d)+c*(A*b*e-2*A*c*d-2*B*a*e+B*b *d)*x)/(-4*a*c+b^2)/(a*e^2-b*d*e+c*d^2)/(c*x^2+b*x+a)^(5/2)+2/15*(8*a*c*e* (-b*e+2*c*d)*(A*b*e-2*A*c*d-2*B*a*e+B*b*d)+(2*a*c*e-b^2*e+b*c*d)*(5*b^2*e* (-A*e+B*d)-8*b*c*d*(A*e+B*d)+4*c*(5*A*a*e^2+4*A*c*d^2-B*a*d*e))+c*(8*c*e*( -2*a*e+b*d)*(A*b*e-2*A*c*d-2*B*a*e+B*b*d)+(-b*e+2*c*d)*(5*b^2*e*(-A*e+B*d) -8*b*c*d*(A*e+B*d)+4*c*(5*A*a*e^2+4*A*c*d^2-B*a*d*e)))*x)/(-4*a*c+b^2)^2/( a*e^2-b*d*e+c*d^2)^2/(c*x^2+b*x+a)^(3/2)-e^5*(-A*e+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)+2/15*(4*a*c*e*(-b*e+2*c*d)*(8*c*e*(-2*a*e+b*d)*(A*b*e- 2*A*c*d-2*B*a*e+B*b*d)+(-b*e+2*c*d)*(5*b^2*e*(-A*e+B*d)-8*b*c*d*(A*e+B*d)+ 4*c*(5*A*a*e^2+4*A*c*d^2-B*a*d*e)))-(2*a*c*e-b^2*e+b*c*d)*(8*c*d*e*(A*b*e- 2*A*c*d-2*B*a*e+B*b*d)*(4*a*c*e-3*b^2*e+4*b*c*d)+(8*c^2*d^2-3*b^2*e^2-4*c* e*(-3*a*e+b*d))*(5*b^2*e*(-A*e+B*d)-8*b*c*d*(A*e+B*d)+4*c*(5*A*a*e^2+4*A*c *d^2-B*a*d*e)))+c*(4*c*e*(-2*a*e+b*d)*(8*c*e*(-2*a*e+b*d)*(A*b*e-2*A*c*d-2 *B*a*e+B*b*d)+(-b*e+2*c*d)*(5*b^2*e*(-A*e+B*d)-8*b*c*d*(A*e+B*d)+4*c*(5*A* a*e^2+4*A*c*d^2-B*a*d*e)))-(-b*e+2*c*d)*(8*c*d*e*(A*b*e-2*A*c*d-2*B*a*e+B* b*d)*(4*a*c*e-3*b^2*e+4*b*c*d)+(8*c^2*d^2-3*b^2*e^2-4*c*e*(-3*a*e+b*d))*(5 *b^2*e*(-A*e+B*d)-8*b*c*d*(A*e+B*d)+4*c*(5*A*a*e^2+4*A*c*d^2-B*a*d*e))))*x )/(-4*a*c+b^2)^3/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^(1/2)
Time = 15.29 (sec) , antiderivative size = 924, normalized size of antiderivative = 0.95 \[ \int \frac {A+B x}{(d+e x) \left (a+b x+c x^2\right )^{7/2}} \, dx=\frac {4 \left (4 a c e (2 c d-b e) (b B d-2 A c d+A b e-2 a B e)+\frac {1}{2} \left (b c d-b^2 e+2 a c e\right ) \left (5 b^2 e (B d-A e)-8 b c d (B d+A e)+4 c \left (4 A c d^2-a B d e+5 a A e^2\right )\right )+c \left (4 c e (b d-2 a e) (b B d-2 A c d+A b e-2 a B e)+\frac {1}{2} (2 c d-b e) \left (5 b^2 e (B d-A e)-8 b c d (B d+A e)+4 c \left (4 A c d^2-a B d e+5 a A e^2\right )\right )\right ) x\right )}{15 \left (b^2-4 a c\right )^2 \left (c d^2+e (-b d+a e)\right )^2 (a+x (b+c x))^{3/2}}+\frac {2 \left (A b^2 e+b B c d x-2 A c (a e+c d x)+A b c (-d+e x)+a B (-b e+2 c (d-e x))\right )}{5 \left (b^2-4 a c\right ) \left (c d^2+e (-b d+a e)\right ) (a+x (b+c x))^{5/2}}+\frac {2 \left (15 b^6 e^4 (-B d+A e)+5 b^5 c e^3 (-B d+A e) (d+3 e x)-8 b^3 c^2 e \left (B c d^3 (21 d-31 e x)+2 a B d e^2 (21 d-10 e x)+A c d^2 e (27 d-e x)+2 a A e^3 (3 d+10 e x)\right )-32 c^3 \left (8 A c^3 d^5 x-2 a c^2 d^3 e (B d-13 A e) x+3 a^2 c d e^3 (-3 B d+11 A e) x+a^3 e^4 (-15 B d+15 A e+8 B e x)\right )+16 b^2 c^2 \left (B d \left (-15 a^2 e^4+c^2 d^3 (4 d-21 e x)+3 a c d e^2 (3 d-14 e x)\right )+A e \left (39 a^2 e^4+c^2 d^3 (20 d-27 e x)+3 a c d e^2 (13 d-2 e x)\right )\right )+2 b^4 c e^2 \left (B d \left (85 a e^2+c d (62 d-5 e x)\right )+A e \left (-85 a e^2+c d (2 d+5 e x)\right )\right )-16 b c^2 \left (A c \left (8 c^2 d^4 (d-5 e x)+26 a c d^2 e^2 (d-3 e x)+33 a^2 e^4 (d-e x)\right )-B \left (-8 a^3 e^5+8 c^3 d^5 x+3 a^2 c d e^3 (3 d+5 e x)+2 a c^2 d^3 e (d+9 e x)\right )\right )\right )}{15 \left (b^2-4 a c\right )^3 \left (c d^2+e (-b d+a e)\right )^3 \sqrt {a+x (b+c x)}}+\frac {e^5 (B d-A e) \text {arctanh}\left (\frac {-b d+2 a e-2 c d x+b e x}{2 \sqrt {c d^2+e (-b d+a e)} \sqrt {a+x (b+c x)}}\right )}{\left (c d^2+e (-b d+a e)\right )^{7/2}} \]
(4*(4*a*c*e*(2*c*d - b*e)*(b*B*d - 2*A*c*d + A*b*e - 2*a*B*e) + ((b*c*d - b^2*e + 2*a*c*e)*(5*b^2*e*(B*d - A*e) - 8*b*c*d*(B*d + A*e) + 4*c*(4*A*c*d ^2 - a*B*d*e + 5*a*A*e^2)))/2 + c*(4*c*e*(b*d - 2*a*e)*(b*B*d - 2*A*c*d + A*b*e - 2*a*B*e) + ((2*c*d - b*e)*(5*b^2*e*(B*d - A*e) - 8*b*c*d*(B*d + A* e) + 4*c*(4*A*c*d^2 - a*B*d*e + 5*a*A*e^2)))/2)*x))/(15*(b^2 - 4*a*c)^2*(c *d^2 + e*(-(b*d) + a*e))^2*(a + x*(b + c*x))^(3/2)) + (2*(A*b^2*e + b*B*c* d*x - 2*A*c*(a*e + c*d*x) + A*b*c*(-d + e*x) + a*B*(-(b*e) + 2*c*(d - e*x) )))/(5*(b^2 - 4*a*c)*(c*d^2 + e*(-(b*d) + a*e))*(a + x*(b + c*x))^(5/2)) + (2*(15*b^6*e^4*(-(B*d) + A*e) + 5*b^5*c*e^3*(-(B*d) + A*e)*(d + 3*e*x) - 8*b^3*c^2*e*(B*c*d^3*(21*d - 31*e*x) + 2*a*B*d*e^2*(21*d - 10*e*x) + A*c*d ^2*e*(27*d - e*x) + 2*a*A*e^3*(3*d + 10*e*x)) - 32*c^3*(8*A*c^3*d^5*x - 2* a*c^2*d^3*e*(B*d - 13*A*e)*x + 3*a^2*c*d*e^3*(-3*B*d + 11*A*e)*x + a^3*e^4 *(-15*B*d + 15*A*e + 8*B*e*x)) + 16*b^2*c^2*(B*d*(-15*a^2*e^4 + c^2*d^3*(4 *d - 21*e*x) + 3*a*c*d*e^2*(3*d - 14*e*x)) + A*e*(39*a^2*e^4 + c^2*d^3*(20 *d - 27*e*x) + 3*a*c*d*e^2*(13*d - 2*e*x))) + 2*b^4*c*e^2*(B*d*(85*a*e^2 + c*d*(62*d - 5*e*x)) + A*e*(-85*a*e^2 + c*d*(2*d + 5*e*x))) - 16*b*c^2*(A* c*(8*c^2*d^4*(d - 5*e*x) + 26*a*c*d^2*e^2*(d - 3*e*x) + 33*a^2*e^4*(d - e* x)) - B*(-8*a^3*e^5 + 8*c^3*d^5*x + 3*a^2*c*d*e^3*(3*d + 5*e*x) + 2*a*c^2* d^3*e*(d + 9*e*x)))))/(15*(b^2 - 4*a*c)^3*(c*d^2 + e*(-(b*d) + a*e))^3*Sqr t[a + x*(b + c*x)]) + (e^5*(B*d - A*e)*ArcTanh[(-(b*d) + 2*a*e - 2*c*d*...
Time = 1.65 (sec) , antiderivative size = 1054, normalized size of antiderivative = 1.08, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {1235, 27, 1235, 27, 1235, 27, 1154, 219}
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.
| \(\displaystyle \int \frac {A+B x}{(d+e x) \left (a+b x+c x^2\right )^{7/2}} \, dx\) |
| \(\Big \downarrow \) 1235 |
| \(\displaystyle \frac {2 \left (-A \left (2 a c e+b^2 (-e)+b c d\right )+c x (-2 a B e+A b e-2 A c d+b B d)+a B (2 c d-b e)\right )}{5 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2} \left (a e^2-b d e+c d^2\right )}-\frac {2 \int \frac {5 e (B d-A e) b^2-8 c d (B d+A e) b+4 c \left (4 A c d^2-a B e d+5 a A e^2\right )-8 c e (b B d-2 A c d+A b e-2 a B e) x}{2 (d+e x) \left (c x^2+b x+a\right )^{5/2}}dx}{5 \left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \left (-A \left (2 a c e+b^2 (-e)+b c d\right )+c x (-2 a B e+A b e-2 A c d+b B d)+a B (2 c d-b e)\right )}{5 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2} \left (a e^2-b d e+c d^2\right )}-\frac {\int \frac {5 e (B d-A e) b^2-8 c d (B d+A e) b+4 c \left (4 A c d^2-a B e d+5 a A e^2\right )-8 c e (b B d-2 A c d+A b e-2 a B e) x}{(d+e x) \left (c x^2+b x+a\right )^{5/2}}dx}{5 \left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 1235 |
| \(\displaystyle \frac {2 \left (-A \left (2 a c e+b^2 (-e)+b c d\right )+c x (-2 a B e+A b e-2 A c d+b B d)+a B (2 c d-b e)\right )}{5 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2} \left (a e^2-b d e+c d^2\right )}-\frac {-\frac {2 \int \frac {8 c d e (b B d-2 A c d+A b e-2 a B e) \left (-3 e b^2+4 c d b+4 a c e\right )+2 \left (4 c^2 d^2-\frac {3 b^2 e^2}{2}-2 c e (b d-3 a e)\right ) \left (5 e (B d-A e) b^2-8 c d (B d+A e) b+4 c \left (4 A c d^2-a B e d+5 a A e^2\right )\right )+4 c e \left (8 c e (b d-2 a e) (b B d-2 A c d+A b e-2 a B e)+(2 c d-b e) \left (5 e (B d-A e) b^2-8 c d (B d+A e) b+4 c \left (4 A c d^2-a B e d+5 a A e^2\right )\right )\right ) x}{2 (d+e x) \left (c x^2+b x+a\right )^{3/2}}dx}{3 \left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (c x \left ((2 c d-b e) \left (4 c \left (5 a A e^2-a B d e+4 A c d^2\right )+5 b^2 e (B d-A e)-8 b c d (A e+B d)\right )+8 c e (b d-2 a e) (-2 a B e+A b e-2 A c d+b B d)\right )+\left (2 a c e+b^2 (-e)+b c d\right ) \left (4 c \left (5 a A e^2-a B d e+4 A c d^2\right )+5 b^2 e (B d-A e)-8 b c d (A e+B d)\right )+8 a c e (2 c d-b e) (-2 a B e+A b e-2 A c d+b B d)\right )}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2} \left (a e^2-b d e+c d^2\right )}}{5 \left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \left (-A \left (2 a c e+b^2 (-e)+b c d\right )+c x (-2 a B e+A b e-2 A c d+b B d)+a B (2 c d-b e)\right )}{5 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2} \left (a e^2-b d e+c d^2\right )}-\frac {-\frac {\int \frac {8 c d e (b B d-2 A c d+A b e-2 a B e) \left (-3 e b^2+4 c d b+4 a c e\right )+2 \left (4 c^2 d^2-\frac {3 b^2 e^2}{2}-2 c e (b d-3 a e)\right ) \left (5 e (B d-A e) b^2-8 c d (B d+A e) b+4 c \left (4 A c d^2-a B e d+5 a A e^2\right )\right )+4 c e \left (8 c e (b d-2 a e) (b B d-2 A c d+A b e-2 a B e)+(2 c d-b e) \left (5 e (B d-A e) b^2-8 c d (B d+A e) b+4 c \left (4 A c d^2-a B e d+5 a A e^2\right )\right )\right ) x}{(d+e x) \left (c x^2+b x+a\right )^{3/2}}dx}{3 \left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (c x \left ((2 c d-b e) \left (4 c \left (5 a A e^2-a B d e+4 A c d^2\right )+5 b^2 e (B d-A e)-8 b c d (A e+B d)\right )+8 c e (b d-2 a e) (-2 a B e+A b e-2 A c d+b B d)\right )+\left (2 a c e+b^2 (-e)+b c d\right ) \left (4 c \left (5 a A e^2-a B d e+4 A c d^2\right )+5 b^2 e (B d-A e)-8 b c d (A e+B d)\right )+8 a c e (2 c d-b e) (-2 a B e+A b e-2 A c d+b B d)\right )}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2} \left (a e^2-b d e+c d^2\right )}}{5 \left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 1235 |
| \(\displaystyle \frac {2 \left (a B (2 c d-b e)-A \left (-e b^2+c d b+2 a c e\right )+c (b B d-2 A c d+A b e-2 a B e) x\right )}{5 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) \left (c x^2+b x+a\right )^{5/2}}-\frac {-\frac {2 \left (8 a c e (2 c d-b e) (b B d-2 A c d+A b e-2 a B e)+\left (-e b^2+c d b+2 a c e\right ) \left (5 e (B d-A e) b^2-8 c d (B d+A e) b+4 c \left (4 A c d^2-a B e d+5 a A e^2\right )\right )+c \left (8 c e (b d-2 a e) (b B d-2 A c d+A b e-2 a B e)+(2 c d-b e) \left (5 e (B d-A e) b^2-8 c d (B d+A e) b+4 c \left (4 A c d^2-a B e d+5 a A e^2\right )\right )\right ) x\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) \left (c x^2+b x+a\right )^{3/2}}-\frac {\frac {2 \left (4 a c e (2 c d-b e) \left (8 c e (b d-2 a e) (b B d-2 A c d+A b e-2 a B e)+(2 c d-b e) \left (5 e (B d-A e) b^2-8 c d (B d+A e) b+4 c \left (4 A c d^2-a B e d+5 a A e^2\right )\right )\right )-\left (-e b^2+c d b+2 a c e\right ) \left (8 c d e (b B d-2 A c d+A b e-2 a B e) \left (-3 e b^2+4 c d b+4 a c e\right )+2 \left (4 c^2 d^2-\frac {3 b^2 e^2}{2}-2 c e (b d-3 a e)\right ) \left (5 e (B d-A e) b^2-8 c d (B d+A e) b+4 c \left (4 A c d^2-a B e d+5 a A e^2\right )\right )\right )+c \left (4 c e (b d-2 a e) \left (8 c e (b d-2 a e) (b B d-2 A c d+A b e-2 a B e)+(2 c d-b e) \left (5 e (B d-A e) b^2-8 c d (B d+A e) b+4 c \left (4 A c d^2-a B e d+5 a A e^2\right )\right )\right )-(2 c d-b e) \left (8 c d e (b B d-2 A c d+A b e-2 a B e) \left (-3 e b^2+4 c d b+4 a c e\right )+2 \left (4 c^2 d^2-\frac {3 b^2 e^2}{2}-2 c e (b d-3 a e)\right ) \left (5 e (B d-A e) b^2-8 c d (B d+A e) b+4 c \left (4 A c d^2-a B e d+5 a A e^2\right )\right )\right )\right ) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) \sqrt {c x^2+b x+a}}-\frac {2 \int \frac {15 \left (b^2-4 a c\right )^3 e^5 (B d-A e)}{2 (d+e x) \sqrt {c x^2+b x+a}}dx}{\left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )}}{3 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )}}{5 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \left (a B (2 c d-b e)-A \left (-e b^2+c d b+2 a c e\right )+c (b B d-2 A c d+A b e-2 a B e) x\right )}{5 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) \left (c x^2+b x+a\right )^{5/2}}-\frac {-\frac {2 \left (8 a c e (2 c d-b e) (b B d-2 A c d+A b e-2 a B e)+\left (-e b^2+c d b+2 a c e\right ) \left (5 e (B d-A e) b^2-8 c d (B d+A e) b+4 c \left (4 A c d^2-a B e d+5 a A e^2\right )\right )+c \left (8 c e (b d-2 a e) (b B d-2 A c d+A b e-2 a B e)+(2 c d-b e) \left (5 e (B d-A e) b^2-8 c d (B d+A e) b+4 c \left (4 A c d^2-a B e d+5 a A e^2\right )\right )\right ) x\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) \left (c x^2+b x+a\right )^{3/2}}-\frac {\frac {2 \left (4 a c e (2 c d-b e) \left (8 c e (b d-2 a e) (b B d-2 A c d+A b e-2 a B e)+(2 c d-b e) \left (5 e (B d-A e) b^2-8 c d (B d+A e) b+4 c \left (4 A c d^2-a B e d+5 a A e^2\right )\right )\right )-\left (-e b^2+c d b+2 a c e\right ) \left (8 c d e (b B d-2 A c d+A b e-2 a B e) \left (-3 e b^2+4 c d b+4 a c e\right )+2 \left (4 c^2 d^2-\frac {3 b^2 e^2}{2}-2 c e (b d-3 a e)\right ) \left (5 e (B d-A e) b^2-8 c d (B d+A e) b+4 c \left (4 A c d^2-a B e d+5 a A e^2\right )\right )\right )+c \left (4 c e (b d-2 a e) \left (8 c e (b d-2 a e) (b B d-2 A c d+A b e-2 a B e)+(2 c d-b e) \left (5 e (B d-A e) b^2-8 c d (B d+A e) b+4 c \left (4 A c d^2-a B e d+5 a A e^2\right )\right )\right )-(2 c d-b e) \left (8 c d e (b B d-2 A c d+A b e-2 a B e) \left (-3 e b^2+4 c d b+4 a c e\right )+2 \left (4 c^2 d^2-\frac {3 b^2 e^2}{2}-2 c e (b d-3 a e)\right ) \left (5 e (B d-A e) b^2-8 c d (B d+A e) b+4 c \left (4 A c d^2-a B e d+5 a A e^2\right )\right )\right )\right ) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) \sqrt {c x^2+b x+a}}-\frac {15 \left (b^2-4 a c\right )^2 e^5 (B d-A e) \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{c d^2-b e d+a e^2}}{3 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )}}{5 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {2 \left (a B (2 c d-b e)-A \left (-e b^2+c d b+2 a c e\right )+c (b B d-2 A c d+A b e-2 a B e) x\right )}{5 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) \left (c x^2+b x+a\right )^{5/2}}-\frac {-\frac {2 \left (8 a c e (2 c d-b e) (b B d-2 A c d+A b e-2 a B e)+\left (-e b^2+c d b+2 a c e\right ) \left (5 e (B d-A e) b^2-8 c d (B d+A e) b+4 c \left (4 A c d^2-a B e d+5 a A e^2\right )\right )+c \left (8 c e (b d-2 a e) (b B d-2 A c d+A b e-2 a B e)+(2 c d-b e) \left (5 e (B d-A e) b^2-8 c d (B d+A e) b+4 c \left (4 A c d^2-a B e d+5 a A e^2\right )\right )\right ) x\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) \left (c x^2+b x+a\right )^{3/2}}-\frac {\frac {30 \left (b^2-4 a c\right )^2 (B d-A e) \int \frac {1}{4 \left (c d^2-b e d+a e^2\right )-\frac {(b d-2 a e+(2 c d-b e) x)^2}{c x^2+b x+a}}d\left (-\frac {b d-2 a e+(2 c d-b e) x}{\sqrt {c x^2+b x+a}}\right ) e^5}{c d^2-b e d+a e^2}+\frac {2 \left (4 a c e (2 c d-b e) \left (8 c e (b d-2 a e) (b B d-2 A c d+A b e-2 a B e)+(2 c d-b e) \left (5 e (B d-A e) b^2-8 c d (B d+A e) b+4 c \left (4 A c d^2-a B e d+5 a A e^2\right )\right )\right )-\left (-e b^2+c d b+2 a c e\right ) \left (8 c d e (b B d-2 A c d+A b e-2 a B e) \left (-3 e b^2+4 c d b+4 a c e\right )+2 \left (4 c^2 d^2-\frac {3 b^2 e^2}{2}-2 c e (b d-3 a e)\right ) \left (5 e (B d-A e) b^2-8 c d (B d+A e) b+4 c \left (4 A c d^2-a B e d+5 a A e^2\right )\right )\right )+c \left (4 c e (b d-2 a e) \left (8 c e (b d-2 a e) (b B d-2 A c d+A b e-2 a B e)+(2 c d-b e) \left (5 e (B d-A e) b^2-8 c d (B d+A e) b+4 c \left (4 A c d^2-a B e d+5 a A e^2\right )\right )\right )-(2 c d-b e) \left (8 c d e (b B d-2 A c d+A b e-2 a B e) \left (-3 e b^2+4 c d b+4 a c e\right )+2 \left (4 c^2 d^2-\frac {3 b^2 e^2}{2}-2 c e (b d-3 a e)\right ) \left (5 e (B d-A e) b^2-8 c d (B d+A e) b+4 c \left (4 A c d^2-a B e d+5 a A e^2\right )\right )\right )\right ) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) \sqrt {c x^2+b x+a}}}{3 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )}}{5 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {2 \left (a B (2 c d-b e)-A \left (-e b^2+c d b+2 a c e\right )+c (b B d-2 A c d+A b e-2 a B e) x\right )}{5 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) \left (c x^2+b x+a\right )^{5/2}}-\frac {-\frac {2 \left (8 a c e (2 c d-b e) (b B d-2 A c d+A b e-2 a B e)+\left (-e b^2+c d b+2 a c e\right ) \left (5 e (B d-A e) b^2-8 c d (B d+A e) b+4 c \left (4 A c d^2-a B e d+5 a A e^2\right )\right )+c \left (8 c e (b d-2 a e) (b B d-2 A c d+A b e-2 a B e)+(2 c d-b e) \left (5 e (B d-A e) b^2-8 c d (B d+A e) b+4 c \left (4 A c d^2-a B e d+5 a A e^2\right )\right )\right ) x\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) \left (c x^2+b x+a\right )^{3/2}}-\frac {\frac {2 \left (4 a c e (2 c d-b e) \left (8 c e (b d-2 a e) (b B d-2 A c d+A b e-2 a B e)+(2 c d-b e) \left (5 e (B d-A e) b^2-8 c d (B d+A e) b+4 c \left (4 A c d^2-a B e d+5 a A e^2\right )\right )\right )-\left (-e b^2+c d b+2 a c e\right ) \left (8 c d e (b B d-2 A c d+A b e-2 a B e) \left (-3 e b^2+4 c d b+4 a c e\right )+2 \left (4 c^2 d^2-\frac {3 b^2 e^2}{2}-2 c e (b d-3 a e)\right ) \left (5 e (B d-A e) b^2-8 c d (B d+A e) b+4 c \left (4 A c d^2-a B e d+5 a A e^2\right )\right )\right )+c \left (4 c e (b d-2 a e) \left (8 c e (b d-2 a e) (b B d-2 A c d+A b e-2 a B e)+(2 c d-b e) \left (5 e (B d-A e) b^2-8 c d (B d+A e) b+4 c \left (4 A c d^2-a B e d+5 a A e^2\right )\right )\right )-(2 c d-b e) \left (8 c d e (b B d-2 A c d+A b e-2 a B e) \left (-3 e b^2+4 c d b+4 a c e\right )+2 \left (4 c^2 d^2-\frac {3 b^2 e^2}{2}-2 c e (b d-3 a e)\right ) \left (5 e (B d-A e) b^2-8 c d (B d+A e) b+4 c \left (4 A c d^2-a B e d+5 a A e^2\right )\right )\right )\right ) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) \sqrt {c x^2+b x+a}}-\frac {15 \left (b^2-4 a c\right )^2 e^5 (B d-A e) \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b e d+a e^2} \sqrt {c x^2+b x+a}}\right )}{\left (c d^2-b e d+a e^2\right )^{3/2}}}{3 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )}}{5 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )}\) |
(2*(a*B*(2*c*d - b*e) - A*(b*c*d - b^2*e + 2*a*c*e) + c*(b*B*d - 2*A*c*d + A*b*e - 2*a*B*e)*x))/(5*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*(a + b*x + c*x^2)^(5/2)) - ((-2*(8*a*c*e*(2*c*d - b*e)*(b*B*d - 2*A*c*d + A*b*e - 2*a *B*e) + (b*c*d - b^2*e + 2*a*c*e)*(5*b^2*e*(B*d - A*e) - 8*b*c*d*(B*d + A* e) + 4*c*(4*A*c*d^2 - a*B*d*e + 5*a*A*e^2)) + c*(8*c*e*(b*d - 2*a*e)*(b*B* d - 2*A*c*d + A*b*e - 2*a*B*e) + (2*c*d - b*e)*(5*b^2*e*(B*d - A*e) - 8*b* c*d*(B*d + A*e) + 4*c*(4*A*c*d^2 - a*B*d*e + 5*a*A*e^2)))*x))/(3*(b^2 - 4* a*c)*(c*d^2 - b*d*e + a*e^2)*(a + b*x + c*x^2)^(3/2)) - ((2*(4*a*c*e*(2*c* d - b*e)*(8*c*e*(b*d - 2*a*e)*(b*B*d - 2*A*c*d + A*b*e - 2*a*B*e) + (2*c*d - b*e)*(5*b^2*e*(B*d - A*e) - 8*b*c*d*(B*d + A*e) + 4*c*(4*A*c*d^2 - a*B* d*e + 5*a*A*e^2))) - (b*c*d - b^2*e + 2*a*c*e)*(8*c*d*e*(b*B*d - 2*A*c*d + A*b*e - 2*a*B*e)*(4*b*c*d - 3*b^2*e + 4*a*c*e) + 2*(4*c^2*d^2 - (3*b^2*e^ 2)/2 - 2*c*e*(b*d - 3*a*e))*(5*b^2*e*(B*d - A*e) - 8*b*c*d*(B*d + A*e) + 4 *c*(4*A*c*d^2 - a*B*d*e + 5*a*A*e^2))) + c*(4*c*e*(b*d - 2*a*e)*(8*c*e*(b* d - 2*a*e)*(b*B*d - 2*A*c*d + A*b*e - 2*a*B*e) + (2*c*d - b*e)*(5*b^2*e*(B *d - A*e) - 8*b*c*d*(B*d + A*e) + 4*c*(4*A*c*d^2 - a*B*d*e + 5*a*A*e^2))) - (2*c*d - b*e)*(8*c*d*e*(b*B*d - 2*A*c*d + A*b*e - 2*a*B*e)*(4*b*c*d - 3* b^2*e + 4*a*c*e) + 2*(4*c^2*d^2 - (3*b^2*e^2)/2 - 2*c*e*(b*d - 3*a*e))*(5* b^2*e*(B*d - A*e) - 8*b*c*d*(B*d + A*e) + 4*c*(4*A*c*d^2 - a*B*d*e + 5*a*A *e^2))))*x))/((b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*Sqrt[a + b*x + c*x^...
3.25.94.3.1 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[((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[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym bol] :> Simp[-2 Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c , d, e}, x]
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] )
Time = 0.52 (sec) , antiderivative size = 1422, normalized size of antiderivative = 1.46
B/e*(2/5*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(5/2)+16/5*c/(4*a*c-b^2)*(2/3 *(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)+16/3*c/(4*a*c-b^2)^2*(2*c*x+b)/ (c*x^2+b*x+a)^(1/2)))+(A*e-B*d)/e^2*(1/5/(a*e^2-b*d*e+c*d^2)*e^2/((x+d/e)^ 2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(5/2)-1/2*(b*e-2*c*d)*e /(a*e^2-b*d*e+c*d^2)*(2/5*(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)/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d* e+c*d^2)/e^2)^(5/2)+16/5*c/(4*c*(a*e^2-b*d*e+c*d^2)/e^2-(b*e-2*c*d)^2/e^2) *(2/3*(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)/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)+ 16/3*c/(4*c*(a*e^2-b*d*e+c*d^2)/e^2-(b*e-2*c*d)^2/e^2)^2*(2*c*(x+d/e)+(b*e -2*c*d)/e)/((x+d/e)^2*c+(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*(1/3/(a*e^2-b*d*e+c*d^2)*e^2/((x+d/e)^2*c+( b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)-1/2*(b*e-2*c*d)*e/(a*e ^2-b*d*e+c*d^2)*(2/3*(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)/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d ^2)/e^2)^(3/2)+16/3*c/(4*c*(a*e^2-b*d*e+c*d^2)/e^2-(b*e-2*c*d)^2/e^2)^2*(2 *c*(x+d/e)+(b*e-2*c*d)/e)/((x+d/e)^2*c+(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*(1/(a*e^2-b*d*e+c*d^2)*e^2/(( x+d/e)^2*c+(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...
Leaf count of result is larger than twice the leaf count of optimal. 8020 vs. \(2 (960) = 1920\).
Time = 219.23 (sec) , antiderivative size = 16082, normalized size of antiderivative = 16.51 \[ \int \frac {A+B x}{(d+e x) \left (a+b x+c x^2\right )^{7/2}} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {A+B x}{(d+e x) \left (a+b x+c x^2\right )^{7/2}} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {A+B x}{(d+e x) \left (a+b x+c x^2\right )^{7/2}} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume((b/e-(2*c*d)/e^2)^2>0)', see `as sume?` for
Leaf count of result is larger than twice the leaf count of optimal. 81273 vs. \(2 (960) = 1920\).
Time = 2.54 (sec) , antiderivative size = 81273, normalized size of antiderivative = 83.44 \[ \int \frac {A+B x}{(d+e x) \left (a+b x+c x^2\right )^{7/2}} \, dx=\text {Too large to display} \]
-2*(B*d*e^5 - A*e^6)*arctan(-((sqrt(c)*x - sqrt(c*x^2 + b*x + a))*e + sqrt (c)*d)/sqrt(-c*d^2 + b*d*e - a*e^2))/((c^3*d^6 - 3*b*c^2*d^5*e + 3*b^2*c*d ^4*e^2 + 3*a*c^2*d^4*e^2 - b^3*d^3*e^3 - 6*a*b*c*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a^2*c*d^2*e^4 - 3*a^2*b*d*e^5 + a^3*e^6)*sqrt(-c*d^2 + b*d*e - a*e^2) ) + 2/15*((((((128*B*b*c^22*d^35 - 256*A*c^23*d^35 - 2256*B*b^2*c^21*d^34* e + 64*B*a*c^22*d^34*e + 4480*A*b*c^22*d^34*e + 18728*B*b^3*c^20*d^33*e^2 + 1248*B*a*b*c^21*d^33*e^2 - 36912*A*b^2*c^21*d^33*e^2 - 4672*A*a*c^22*d^3 3*e^2 - 97250*B*b^4*c^19*d^32*e^3 - 30192*B*a*b^2*c^20*d^32*e^3 + 190168*A *b^3*c^20*d^32*e^3 + 1248*B*a^2*c^21*d^32*e^3 + 77088*A*a*b*c^21*d^32*e^3 + 353775*B*b^5*c^18*d^31*e^4 + 260360*B*a*b^3*c^19*d^31*e^4 - 686110*A*b^4 *c^19*d^31*e^4 + 240*B*a^2*b*c^20*d^31*e^4 - 596496*A*a*b^2*c^20*d^31*e^4 - 40416*A*a^2*c^21*d^31*e^4 - 956689*B*b^6*c^17*d^30*e^5 - 1326390*B*a*b^4 *c^18*d^30*e^5 + 1839633*A*b^5*c^18*d^30*e^5 - 166560*B*a^2*b^2*c^19*d^30* e^5 + 2873080*A*a*b^3*c^19*d^30*e^5 + 10784*B*a^3*c^20*d^30*e^5 + 626448*A *a^2*b*c^20*d^30*e^5 + 1991143*B*b^7*c^16*d^29*e^6 + 4620363*B*a*b^5*c^17* d^29*e^6 - 3795695*A*b^6*c^17*d^29*e^6 + 1614360*B*a^2*b^3*c^18*d^29*e^6 - 9640650*A*a*b^4*c^18*d^29*e^6 - 51920*B*a^3*b*c^19*d^29*e^6 - 4533600*A*a ^2*b^2*c^19*d^29*e^6 - 219680*A*a^3*c^20*d^29*e^6 - 3256929*B*b^8*c^15*d^2 8*e^7 - 11788084*B*a*b^6*c^16*d^28*e^7 + 6155801*A*b^7*c^16*d^28*e^7 - 821 3730*B*a^2*b^4*c^17*d^28*e^7 + 23893941*A*a*b^5*c^17*d^28*e^7 - 475440*...
Timed out. \[ \int \frac {A+B x}{(d+e x) \left (a+b x+c x^2\right )^{7/2}} \, dx=\int \frac {A+B\,x}{\left (d+e\,x\right )\,{\left (c\,x^2+b\,x+a\right )}^{7/2}} \,d x \]