Integrand size = 26, antiderivative size = 644 \[ \int \frac {A+B x}{(d+e x)^{5/2} \left (b x+c x^2\right )^3} \, dx=\frac {e \left (72 A c^4 d^4+5 b^4 e^3 (4 B d-7 A e)-9 b^3 c d e^2 (4 B d-5 A e)-36 b c^3 d^3 (B d+4 A e)+3 b^2 c^2 d^2 e (29 B d+9 A e)\right )}{12 b^4 d^3 (c d-b e)^3 (d+e x)^{3/2}}+\frac {e \left (24 A c^5 d^5+8 b^4 c d e^3 (7 B d-10 A e)-5 b^5 e^4 (4 B d-7 A e)-6 b^3 c^2 d^2 e^2 (4 B d-3 A e)+7 b^2 c^3 d^3 e (5 B d+4 A e)-12 b c^4 d^4 (B d+5 A e)\right )}{4 b^4 d^4 (c d-b e)^4 \sqrt {d+e x}}-\frac {A b (c d-b e)+c (2 A c d-b (B d+A e)) x}{2 b^2 d (c d-b e) (d+e x)^{3/2} \left (b x+c x^2\right )^2}+\frac {b (c d-b e) \left (12 A c^2 d^2+b^2 e (4 B d-7 A e)-3 b c d (2 B d+A e)\right )+c \left (24 A c^3 d^3-b^3 e^2 (4 B d-7 A e)+b^2 c d e (23 B d-2 A e)-12 b c^2 d^2 (B d+3 A e)\right ) x}{4 b^4 d^2 (c d-b e)^2 (d+e x)^{3/2} \left (b x+c x^2\right )}-\frac {\left (48 A c^2 d^2-5 b^2 e (4 B d-7 A e)-12 b c d (2 B d-5 A e)\right ) \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{4 b^5 d^{9/2}}+\frac {c^{7/2} \left (48 A c^3 d^2-99 b^3 B e^2-12 b c^2 d (2 B d+13 A e)+11 b^2 c e (8 B d+13 A e)\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{4 b^5 (c d-b e)^{9/2}} \]
1/12*e*(72*A*c^4*d^4+5*b^4*e^3*(-7*A*e+4*B*d)-9*b^3*c*d*e^2*(-5*A*e+4*B*d) -36*b*c^3*d^3*(4*A*e+B*d)+3*b^2*c^2*d^2*e*(9*A*e+29*B*d))/b^4/d^3/(-b*e+c* d)^3/(e*x+d)^(3/2)+1/2*(-A*b*(-b*e+c*d)-c*(2*A*c*d-b*(A*e+B*d))*x)/b^2/d/( -b*e+c*d)/(e*x+d)^(3/2)/(c*x^2+b*x)^2+1/4*(b*(-b*e+c*d)*(12*A*c^2*d^2+b^2* e*(-7*A*e+4*B*d)-3*b*c*d*(A*e+2*B*d))+c*(24*A*c^3*d^3-b^3*e^2*(-7*A*e+4*B* d)+b^2*c*d*e*(-2*A*e+23*B*d)-12*b*c^2*d^2*(3*A*e+B*d))*x)/b^4/d^2/(-b*e+c* d)^2/(e*x+d)^(3/2)/(c*x^2+b*x)-1/4*(48*A*c^2*d^2-5*b^2*e*(-7*A*e+4*B*d)-12 *b*c*d*(-5*A*e+2*B*d))*arctanh((e*x+d)^(1/2)/d^(1/2))/b^5/d^(9/2)+1/4*c^(7 /2)*(48*A*c^3*d^2-99*b^3*B*e^2-12*b*c^2*d*(13*A*e+2*B*d)+11*b^2*c*e*(13*A* e+8*B*d))*arctanh(c^(1/2)*(e*x+d)^(1/2)/(-b*e+c*d)^(1/2))/b^5/(-b*e+c*d)^( 9/2)+1/4*e*(24*A*c^5*d^5+8*b^4*c*d*e^3*(-10*A*e+7*B*d)-5*b^5*e^4*(-7*A*e+4 *B*d)-6*b^3*c^2*d^2*e^2*(-3*A*e+4*B*d)+7*b^2*c^3*d^3*e*(4*A*e+5*B*d)-12*b* c^4*d^4*(5*A*e+B*d))/b^4/d^4/(-b*e+c*d)^4/(e*x+d)^(1/2)
Time = 5.57 (sec) , antiderivative size = 803, normalized size of antiderivative = 1.25 \[ \int \frac {A+B x}{(d+e x)^{5/2} \left (b x+c x^2\right )^3} \, dx=-\frac {\frac {b \left (b B d x \left (36 c^6 d^4 x^2 (d+e x)^2+3 b c^5 d^3 x (18 d-35 e x) (d+e x)^2+4 b^6 e^4 \left (3 d^2+20 d e x+15 e^2 x^2\right )+3 b^2 c^4 d^2 (d+e x)^2 \left (4 d^2-53 d e x+24 e^2 x^2\right )-8 b^5 c e^3 \left (6 d^3+25 d^2 e x+d e^2 x^2-15 e^3 x^3\right )+4 b^4 c^2 e^2 \left (18 d^4+12 d^3 e x-91 d^2 e^2 x^2-64 d e^3 x^3+15 e^4 x^4\right )-8 b^3 c^3 d e \left (6 d^4-6 d^3 e x-24 d^2 e^2 x^2+10 d e^3 x^3+21 e^4 x^4\right )\right )+A \left (-72 c^7 d^5 x^3 (d+e x)^2-36 b c^6 d^4 x^2 (3 d-5 e x) (d+e x)^2-3 b^2 c^5 d^3 x (d+e x)^2 \left (8 d^2-91 d e x+28 e^2 x^2\right )+3 b^3 c^4 d^2 (d+e x)^2 \left (2 d^3+21 d^2 e x-44 d e^2 x^2-18 e^3 x^3\right )-b^7 e^4 \left (-6 d^3+21 d^2 e x+140 d e^2 x^2+105 e^3 x^3\right )-2 b^6 c e^3 \left (12 d^4-30 d^3 e x-139 d^2 e^2 x^2+20 d e^3 x^3+105 e^4 x^4\right )+b^5 c^2 e^2 \left (36 d^5-30 d^4 e x+30 d^3 e^2 x^2+565 d^2 e^3 x^3+340 d e^4 x^4-105 e^5 x^5\right )-4 b^4 c^3 d e \left (6 d^5+15 d^4 e x+45 d^3 e^2 x^2+45 d^2 e^3 x^3-53 d e^4 x^4-60 e^5 x^5\right )\right )\right )}{d^4 (c d-b e)^4 x^2 (b+c x)^2 (d+e x)^{3/2}}+\frac {3 c^{7/2} \left (48 A c^3 d^2-99 b^3 B e^2-12 b c^2 d (2 B d+13 A e)+11 b^2 c e (8 B d+13 A e)\right ) \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {-c d+b e}}\right )}{(-c d+b e)^{9/2}}+\frac {3 \left (48 A c^2 d^2+12 b c d (-2 B d+5 A e)+5 b^2 e (-4 B d+7 A e)\right ) \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{d^{9/2}}}{12 b^5} \]
-1/12*((b*(b*B*d*x*(36*c^6*d^4*x^2*(d + e*x)^2 + 3*b*c^5*d^3*x*(18*d - 35* e*x)*(d + e*x)^2 + 4*b^6*e^4*(3*d^2 + 20*d*e*x + 15*e^2*x^2) + 3*b^2*c^4*d ^2*(d + e*x)^2*(4*d^2 - 53*d*e*x + 24*e^2*x^2) - 8*b^5*c*e^3*(6*d^3 + 25*d ^2*e*x + d*e^2*x^2 - 15*e^3*x^3) + 4*b^4*c^2*e^2*(18*d^4 + 12*d^3*e*x - 91 *d^2*e^2*x^2 - 64*d*e^3*x^3 + 15*e^4*x^4) - 8*b^3*c^3*d*e*(6*d^4 - 6*d^3*e *x - 24*d^2*e^2*x^2 + 10*d*e^3*x^3 + 21*e^4*x^4)) + A*(-72*c^7*d^5*x^3*(d + e*x)^2 - 36*b*c^6*d^4*x^2*(3*d - 5*e*x)*(d + e*x)^2 - 3*b^2*c^5*d^3*x*(d + e*x)^2*(8*d^2 - 91*d*e*x + 28*e^2*x^2) + 3*b^3*c^4*d^2*(d + e*x)^2*(2*d ^3 + 21*d^2*e*x - 44*d*e^2*x^2 - 18*e^3*x^3) - b^7*e^4*(-6*d^3 + 21*d^2*e* x + 140*d*e^2*x^2 + 105*e^3*x^3) - 2*b^6*c*e^3*(12*d^4 - 30*d^3*e*x - 139* d^2*e^2*x^2 + 20*d*e^3*x^3 + 105*e^4*x^4) + b^5*c^2*e^2*(36*d^5 - 30*d^4*e *x + 30*d^3*e^2*x^2 + 565*d^2*e^3*x^3 + 340*d*e^4*x^4 - 105*e^5*x^5) - 4*b ^4*c^3*d*e*(6*d^5 + 15*d^4*e*x + 45*d^3*e^2*x^2 + 45*d^2*e^3*x^3 - 53*d*e^ 4*x^4 - 60*e^5*x^5))))/(d^4*(c*d - b*e)^4*x^2*(b + c*x)^2*(d + e*x)^(3/2)) + (3*c^(7/2)*(48*A*c^3*d^2 - 99*b^3*B*e^2 - 12*b*c^2*d*(2*B*d + 13*A*e) + 11*b^2*c*e*(8*B*d + 13*A*e))*ArcTan[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[-(c*d) + b*e]])/(-(c*d) + b*e)^(9/2) + (3*(48*A*c^2*d^2 + 12*b*c*d*(-2*B*d + 5*A*e ) + 5*b^2*e*(-4*B*d + 7*A*e))*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/d^(9/2))/b^5
Time = 1.76 (sec) , antiderivative size = 722, normalized size of antiderivative = 1.12, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {1235, 27, 1235, 27, 1198, 1198, 1197, 27, 1480, 221}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {A+B x}{\left (b x+c x^2\right )^3 (d+e x)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 1235 |
| \(\displaystyle -\frac {\int \frac {e (4 B d-7 A e) b^2-3 c d (2 B d+A e) b+12 A c^2 d^2-9 c e (b B d-2 A c d+A b e) x}{2 (d+e x)^{5/2} \left (c x^2+b x\right )^2}dx}{2 b^2 d (c d-b e)}-\frac {c x (2 A c d-b (A e+B d))+A b (c d-b e)}{2 b^2 d \left (b x+c x^2\right )^2 (d+e x)^{3/2} (c d-b e)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {e (4 B d-7 A e) b^2-3 c d (2 B d+A e) b+12 A c^2 d^2-9 c e (b B d-2 A c d+A b e) x}{(d+e x)^{5/2} \left (c x^2+b x\right )^2}dx}{4 b^2 d (c d-b e)}-\frac {c x (2 A c d-b (A e+B d))+A b (c d-b e)}{2 b^2 d \left (b x+c x^2\right )^2 (d+e x)^{3/2} (c d-b e)}\) |
| \(\Big \downarrow \) 1235 |
| \(\displaystyle -\frac {-\frac {\int \frac {\left (-5 e (4 B d-7 A e) b^2-12 c d (2 B d-5 A e) b+48 A c^2 d^2\right ) (c d-b e)^2+5 c e \left (-e^2 (4 B d-7 A e) b^3+c d e (23 B d-2 A e) b^2-12 c^2 d^2 (B d+3 A e) b+24 A c^3 d^3\right ) x}{2 (d+e x)^{5/2} \left (c x^2+b x\right )}dx}{b^2 d (c d-b e)}-\frac {b (c d-b e) \left (b^2 e (4 B d-7 A e)-3 b c d (A e+2 B d)+12 A c^2 d^2\right )+c x \left (b^3 \left (-e^2\right ) (4 B d-7 A e)+b^2 c d e (23 B d-2 A e)-12 b c^2 d^2 (3 A e+B d)+24 A c^3 d^3\right )}{b^2 d \left (b x+c x^2\right ) (d+e x)^{3/2} (c d-b e)}}{4 b^2 d (c d-b e)}-\frac {c x (2 A c d-b (A e+B d))+A b (c d-b e)}{2 b^2 d \left (b x+c x^2\right )^2 (d+e x)^{3/2} (c d-b e)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {\int \frac {\left (-5 e (4 B d-7 A e) b^2-12 c d (2 B d-5 A e) b+48 A c^2 d^2\right ) (c d-b e)^2+5 c e \left (-e^2 (4 B d-7 A e) b^3+c d e (23 B d-2 A e) b^2-12 c^2 d^2 (B d+3 A e) b+24 A c^3 d^3\right ) x}{(d+e x)^{5/2} \left (c x^2+b x\right )}dx}{2 b^2 d (c d-b e)}-\frac {b (c d-b e) \left (b^2 e (4 B d-7 A e)-3 b c d (A e+2 B d)+12 A c^2 d^2\right )+c x \left (b^3 \left (-e^2\right ) (4 B d-7 A e)+b^2 c d e (23 B d-2 A e)-12 b c^2 d^2 (3 A e+B d)+24 A c^3 d^3\right )}{b^2 d \left (b x+c x^2\right ) (d+e x)^{3/2} (c d-b e)}}{4 b^2 d (c d-b e)}-\frac {c x (2 A c d-b (A e+B d))+A b (c d-b e)}{2 b^2 d \left (b x+c x^2\right )^2 (d+e x)^{3/2} (c d-b e)}\) |
| \(\Big \downarrow \) 1198 |
| \(\displaystyle -\frac {-\frac {\frac {\int \frac {\left (-5 e (4 B d-7 A e) b^2-12 c d (2 B d-5 A e) b+48 A c^2 d^2\right ) (c d-b e)^3+c e \left (5 e^3 (4 B d-7 A e) b^4-9 c d e^2 (4 B d-5 A e) b^3+3 c^2 d^2 e (29 B d+9 A e) b^2-36 c^3 d^3 (B d+4 A e) b+72 A c^4 d^4\right ) x}{(d+e x)^{3/2} \left (c x^2+b x\right )}dx}{d (c d-b e)}+\frac {2 e \left (5 b^4 e^3 (4 B d-7 A e)-9 b^3 c d e^2 (4 B d-5 A e)+3 b^2 c^2 d^2 e (9 A e+29 B d)-36 b c^3 d^3 (4 A e+B d)+72 A c^4 d^4\right )}{3 d (d+e x)^{3/2} (c d-b e)}}{2 b^2 d (c d-b e)}-\frac {b (c d-b e) \left (b^2 e (4 B d-7 A e)-3 b c d (A e+2 B d)+12 A c^2 d^2\right )+c x \left (b^3 \left (-e^2\right ) (4 B d-7 A e)+b^2 c d e (23 B d-2 A e)-12 b c^2 d^2 (3 A e+B d)+24 A c^3 d^3\right )}{b^2 d \left (b x+c x^2\right ) (d+e x)^{3/2} (c d-b e)}}{4 b^2 d (c d-b e)}-\frac {c x (2 A c d-b (A e+B d))+A b (c d-b e)}{2 b^2 d \left (b x+c x^2\right )^2 (d+e x)^{3/2} (c d-b e)}\) |
| \(\Big \downarrow \) 1198 |
| \(\displaystyle -\frac {-\frac {\frac {\frac {\int \frac {\left (-5 e (4 B d-7 A e) b^2-12 c d (2 B d-5 A e) b+48 A c^2 d^2\right ) (c d-b e)^4+c e \left (-5 e^4 (4 B d-7 A e) b^5+8 c d e^3 (7 B d-10 A e) b^4-6 c^2 d^2 e^2 (4 B d-3 A e) b^3+7 c^3 d^3 e (5 B d+4 A e) b^2-12 c^4 d^4 (B d+5 A e) b+24 A c^5 d^5\right ) x}{\sqrt {d+e x} \left (c x^2+b x\right )}dx}{d (c d-b e)}+\frac {2 e \left (-5 b^5 e^4 (4 B d-7 A e)+8 b^4 c d e^3 (7 B d-10 A e)-6 b^3 c^2 d^2 e^2 (4 B d-3 A e)+7 b^2 c^3 d^3 e (4 A e+5 B d)-12 b c^4 d^4 (5 A e+B d)+24 A c^5 d^5\right )}{d \sqrt {d+e x} (c d-b e)}}{d (c d-b e)}+\frac {2 e \left (5 b^4 e^3 (4 B d-7 A e)-9 b^3 c d e^2 (4 B d-5 A e)+3 b^2 c^2 d^2 e (9 A e+29 B d)-36 b c^3 d^3 (4 A e+B d)+72 A c^4 d^4\right )}{3 d (d+e x)^{3/2} (c d-b e)}}{2 b^2 d (c d-b e)}-\frac {b (c d-b e) \left (b^2 e (4 B d-7 A e)-3 b c d (A e+2 B d)+12 A c^2 d^2\right )+c x \left (b^3 \left (-e^2\right ) (4 B d-7 A e)+b^2 c d e (23 B d-2 A e)-12 b c^2 d^2 (3 A e+B d)+24 A c^3 d^3\right )}{b^2 d \left (b x+c x^2\right ) (d+e x)^{3/2} (c d-b e)}}{4 b^2 d (c d-b e)}-\frac {c x (2 A c d-b (A e+B d))+A b (c d-b e)}{2 b^2 d \left (b x+c x^2\right )^2 (d+e x)^{3/2} (c d-b e)}\) |
| \(\Big \downarrow \) 1197 |
| \(\displaystyle -\frac {A b (c d-b e)+c (2 A c d-b (B d+A e)) x}{2 b^2 d (c d-b e) (d+e x)^{3/2} \left (c x^2+b x\right )^2}-\frac {-\frac {b (c d-b e) \left (e (4 B d-7 A e) b^2-3 c d (2 B d+A e) b+12 A c^2 d^2\right )+c \left (-e^2 (4 B d-7 A e) b^3+c d e (23 B d-2 A e) b^2-12 c^2 d^2 (B d+3 A e) b+24 A c^3 d^3\right ) x}{b^2 d (c d-b e) (d+e x)^{3/2} \left (c x^2+b x\right )}-\frac {\frac {2 e \left (5 e^3 (4 B d-7 A e) b^4-9 c d e^2 (4 B d-5 A e) b^3+3 c^2 d^2 e (29 B d+9 A e) b^2-36 c^3 d^3 (B d+4 A e) b+72 A c^4 d^4\right )}{3 d (c d-b e) (d+e x)^{3/2}}+\frac {\frac {2 e \left (-5 e^4 (4 B d-7 A e) b^5+8 c d e^3 (7 B d-10 A e) b^4-6 c^2 d^2 e^2 (4 B d-3 A e) b^3+7 c^3 d^3 e (5 B d+4 A e) b^2-12 c^4 d^4 (B d+5 A e) b+24 A c^5 d^5\right )}{d (c d-b e) \sqrt {d+e x}}+\frac {2 \int \frac {e \left (-5 e^5 (4 B d-7 A e) b^6+c d e^4 (76 B d-115 A e) b^5-2 c^2 d^2 e^3 (40 B d-49 A e) b^4-10 c^3 d^3 e^2 (4 B d-A e) b^3+c^4 d^4 e (41 B d+55 A e) b^2-12 c^5 d^5 (B d+6 A e) b+24 A c^6 d^6+c \left (-5 e^4 (4 B d-7 A e) b^5+8 c d e^3 (7 B d-10 A e) b^4-6 c^2 d^2 e^2 (4 B d-3 A e) b^3+7 c^3 d^3 e (5 B d+4 A e) b^2-12 c^4 d^4 (B d+5 A e) b+24 A c^5 d^5\right ) (d+e x)\right )}{c (d+e x)^2-(2 c d-b e) (d+e x)+d (c d-b e)}d\sqrt {d+e x}}{d (c d-b e)}}{d (c d-b e)}}{2 b^2 d (c d-b e)}}{4 b^2 d (c d-b e)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {A b (c d-b e)+c (2 A c d-b (B d+A e)) x}{2 b^2 d (c d-b e) (d+e x)^{3/2} \left (c x^2+b x\right )^2}-\frac {-\frac {b (c d-b e) \left (e (4 B d-7 A e) b^2-3 c d (2 B d+A e) b+12 A c^2 d^2\right )+c \left (-e^2 (4 B d-7 A e) b^3+c d e (23 B d-2 A e) b^2-12 c^2 d^2 (B d+3 A e) b+24 A c^3 d^3\right ) x}{b^2 d (c d-b e) (d+e x)^{3/2} \left (c x^2+b x\right )}-\frac {\frac {2 e \left (5 e^3 (4 B d-7 A e) b^4-9 c d e^2 (4 B d-5 A e) b^3+3 c^2 d^2 e (29 B d+9 A e) b^2-36 c^3 d^3 (B d+4 A e) b+72 A c^4 d^4\right )}{3 d (c d-b e) (d+e x)^{3/2}}+\frac {\frac {2 e \left (-5 e^4 (4 B d-7 A e) b^5+8 c d e^3 (7 B d-10 A e) b^4-6 c^2 d^2 e^2 (4 B d-3 A e) b^3+7 c^3 d^3 e (5 B d+4 A e) b^2-12 c^4 d^4 (B d+5 A e) b+24 A c^5 d^5\right )}{d (c d-b e) \sqrt {d+e x}}+\frac {2 e \int \frac {-5 e^5 (4 B d-7 A e) b^6+c d e^4 (76 B d-115 A e) b^5-2 c^2 d^2 e^3 (40 B d-49 A e) b^4-10 c^3 d^3 e^2 (4 B d-A e) b^3+c^4 d^4 e (41 B d+55 A e) b^2-12 c^5 d^5 (B d+6 A e) b+24 A c^6 d^6+c \left (-5 e^4 (4 B d-7 A e) b^5+8 c d e^3 (7 B d-10 A e) b^4-6 c^2 d^2 e^2 (4 B d-3 A e) b^3+7 c^3 d^3 e (5 B d+4 A e) b^2-12 c^4 d^4 (B d+5 A e) b+24 A c^5 d^5\right ) (d+e x)}{c (d+e x)^2-(2 c d-b e) (d+e x)+d (c d-b e)}d\sqrt {d+e x}}{d (c d-b e)}}{d (c d-b e)}}{2 b^2 d (c d-b e)}}{4 b^2 d (c d-b e)}\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle -\frac {-\frac {\frac {\frac {2 e \left (\frac {c (c d-b e)^4 \left (-5 b^2 e (4 B d-7 A e)-12 b c d (2 B d-5 A e)+48 A c^2 d^2\right ) \int \frac {1}{c (d+e x)-c d}d\sqrt {d+e x}}{b e}-\frac {c^4 d^4 \left (11 b^2 c e (13 A e+8 B d)-12 b c^2 d (13 A e+2 B d)+48 A c^3 d^2-99 b^3 B e^2\right ) \int \frac {1}{-c d+b e+c (d+e x)}d\sqrt {d+e x}}{b e}\right )}{d (c d-b e)}+\frac {2 e \left (-5 b^5 e^4 (4 B d-7 A e)+8 b^4 c d e^3 (7 B d-10 A e)-6 b^3 c^2 d^2 e^2 (4 B d-3 A e)+7 b^2 c^3 d^3 e (4 A e+5 B d)-12 b c^4 d^4 (5 A e+B d)+24 A c^5 d^5\right )}{d \sqrt {d+e x} (c d-b e)}}{d (c d-b e)}+\frac {2 e \left (5 b^4 e^3 (4 B d-7 A e)-9 b^3 c d e^2 (4 B d-5 A e)+3 b^2 c^2 d^2 e (9 A e+29 B d)-36 b c^3 d^3 (4 A e+B d)+72 A c^4 d^4\right )}{3 d (d+e x)^{3/2} (c d-b e)}}{2 b^2 d (c d-b e)}-\frac {b (c d-b e) \left (b^2 e (4 B d-7 A e)-3 b c d (A e+2 B d)+12 A c^2 d^2\right )+c x \left (b^3 \left (-e^2\right ) (4 B d-7 A e)+b^2 c d e (23 B d-2 A e)-12 b c^2 d^2 (3 A e+B d)+24 A c^3 d^3\right )}{b^2 d \left (b x+c x^2\right ) (d+e x)^{3/2} (c d-b e)}}{4 b^2 d (c d-b e)}-\frac {c x (2 A c d-b (A e+B d))+A b (c d-b e)}{2 b^2 d \left (b x+c x^2\right )^2 (d+e x)^{3/2} (c d-b e)}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {-\frac {\frac {\frac {2 e \left (\frac {c^{7/2} d^4 \left (11 b^2 c e (13 A e+8 B d)-12 b c^2 d (13 A e+2 B d)+48 A c^3 d^2-99 b^3 B e^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b e \sqrt {c d-b e}}-\frac {\text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) (c d-b e)^4 \left (-5 b^2 e (4 B d-7 A e)-12 b c d (2 B d-5 A e)+48 A c^2 d^2\right )}{b \sqrt {d} e}\right )}{d (c d-b e)}+\frac {2 e \left (-5 b^5 e^4 (4 B d-7 A e)+8 b^4 c d e^3 (7 B d-10 A e)-6 b^3 c^2 d^2 e^2 (4 B d-3 A e)+7 b^2 c^3 d^3 e (4 A e+5 B d)-12 b c^4 d^4 (5 A e+B d)+24 A c^5 d^5\right )}{d \sqrt {d+e x} (c d-b e)}}{d (c d-b e)}+\frac {2 e \left (5 b^4 e^3 (4 B d-7 A e)-9 b^3 c d e^2 (4 B d-5 A e)+3 b^2 c^2 d^2 e (9 A e+29 B d)-36 b c^3 d^3 (4 A e+B d)+72 A c^4 d^4\right )}{3 d (d+e x)^{3/2} (c d-b e)}}{2 b^2 d (c d-b e)}-\frac {b (c d-b e) \left (b^2 e (4 B d-7 A e)-3 b c d (A e+2 B d)+12 A c^2 d^2\right )+c x \left (b^3 \left (-e^2\right ) (4 B d-7 A e)+b^2 c d e (23 B d-2 A e)-12 b c^2 d^2 (3 A e+B d)+24 A c^3 d^3\right )}{b^2 d \left (b x+c x^2\right ) (d+e x)^{3/2} (c d-b e)}}{4 b^2 d (c d-b e)}-\frac {c x (2 A c d-b (A e+B d))+A b (c d-b e)}{2 b^2 d \left (b x+c x^2\right )^2 (d+e x)^{3/2} (c d-b e)}\) |
-1/2*(A*b*(c*d - b*e) + c*(2*A*c*d - b*(B*d + A*e))*x)/(b^2*d*(c*d - b*e)* (d + e*x)^(3/2)*(b*x + c*x^2)^2) - (-((b*(c*d - b*e)*(12*A*c^2*d^2 + b^2*e *(4*B*d - 7*A*e) - 3*b*c*d*(2*B*d + A*e)) + c*(24*A*c^3*d^3 - b^3*e^2*(4*B *d - 7*A*e) + b^2*c*d*e*(23*B*d - 2*A*e) - 12*b*c^2*d^2*(B*d + 3*A*e))*x)/ (b^2*d*(c*d - b*e)*(d + e*x)^(3/2)*(b*x + c*x^2))) - ((2*e*(72*A*c^4*d^4 + 5*b^4*e^3*(4*B*d - 7*A*e) - 9*b^3*c*d*e^2*(4*B*d - 5*A*e) - 36*b*c^3*d^3* (B*d + 4*A*e) + 3*b^2*c^2*d^2*e*(29*B*d + 9*A*e)))/(3*d*(c*d - b*e)*(d + e *x)^(3/2)) + ((2*e*(24*A*c^5*d^5 + 8*b^4*c*d*e^3*(7*B*d - 10*A*e) - 5*b^5* e^4*(4*B*d - 7*A*e) - 6*b^3*c^2*d^2*e^2*(4*B*d - 3*A*e) + 7*b^2*c^3*d^3*e* (5*B*d + 4*A*e) - 12*b*c^4*d^4*(B*d + 5*A*e)))/(d*(c*d - b*e)*Sqrt[d + e*x ]) + (2*e*(-(((c*d - b*e)^4*(48*A*c^2*d^2 - 5*b^2*e*(4*B*d - 7*A*e) - 12*b *c*d*(2*B*d - 5*A*e))*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(b*Sqrt[d]*e)) + (c^ (7/2)*d^4*(48*A*c^3*d^2 - 99*b^3*B*e^2 - 12*b*c^2*d*(2*B*d + 13*A*e) + 11* b^2*c*e*(8*B*d + 13*A*e))*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]] )/(b*e*Sqrt[c*d - b*e])))/(d*(c*d - b*e)))/(d*(c*d - b*e)))/(2*b^2*d*(c*d - b*e)))/(4*b^2*d*(c*d - b*e))
3.13.54.3.1 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[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)), x_Symbol] :> Simp[2 Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; Fr eeQ[{a, b, c, d, e, f, g}, x]
Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(e*f - d*g)*((d + e*x)^(m + 1)/((m + 1)*(c *d^2 - b*d*e + a*e^2))), x] + Simp[1/(c*d^2 - b*d*e + a*e^2) Int[(d + e*x )^(m + 1)*(Simp[c*d*f - f*b*e + a*e*g - c*(e*f - d*g)*x, x]/(a + b*x + c*x^ 2)), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && FractionQ[m] && LtQ[m, -1 ]
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] )
Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : > With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e/2 + (2*c*d - b*e)/(2*q)) Int[1/( b/2 - q/2 + c*x^2), x], x] + Simp[(e/2 - (2*c*d - b*e)/(2*q)) Int[1/(b/2 + q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^2 - 4*a*c]
Time = 2.92 (sec) , antiderivative size = 460, normalized size of antiderivative = 0.71
| method | result | size |
| risch | \(-\frac {\sqrt {e x +d}\, \left (-11 A b e x -12 A c d x +4 B b d x +2 A b d \right )}{4 d^{4} b^{4} x^{2}}+\frac {e \left (-\frac {8 c^{4} d^{4} \left (\frac {\left (\frac {23}{8} A \,b^{2} c^{2} e^{2}-\frac {3}{2} A b \,c^{3} d e -\frac {19}{8} B \,b^{3} c \,e^{2}+B \,b^{2} c^{2} d e \right ) \left (e x +d \right )^{\frac {3}{2}}+\frac {b e \left (25 A \,b^{2} c \,e^{2}-37 A b \,c^{2} d e +12 A \,c^{3} d^{2}-21 b^{3} B \,e^{2}+29 B \,b^{2} c d e -8 B b \,c^{2} d^{2}\right ) \sqrt {e x +d}}{8}}{\left (c \left (e x +d \right )+b e -c d \right )^{2}}+\frac {\left (143 A \,b^{2} c \,e^{2}-156 A b \,c^{2} d e +48 A \,c^{3} d^{2}-99 b^{3} B \,e^{2}+88 B \,b^{2} c d e -24 B b \,c^{2} d^{2}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{8 \sqrt {\left (b e -c d \right ) c}}\right )}{\left (b e -c d \right )^{4} b e}-\frac {\left (35 A \,b^{2} e^{2}+60 A b c d e +48 A \,c^{2} d^{2}-20 B \,b^{2} d e -24 c \,d^{2} B b \right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{b e \sqrt {d}}+\frac {8 b^{4} e^{3} \left (3 A b \,e^{2}-6 A c d e -2 B b d e +5 B c \,d^{2}\right )}{\left (b e -c d \right )^{4} \sqrt {e x +d}}+\frac {8 b^{4} d \,e^{3} \left (A e -B d \right )}{3 \left (b e -c d \right )^{3} \left (e x +d \right )^{\frac {3}{2}}}\right )}{4 b^{4} d^{4}}\) | \(460\) |
| derivativedivides | \(2 e^{4} \left (-\frac {-A e +B d}{3 d^{3} \left (b e -c d \right )^{3} \left (e x +d \right )^{\frac {3}{2}}}-\frac {-3 A b \,e^{2}+6 A c d e +2 B b d e -5 B c \,d^{2}}{d^{4} \left (b e -c d \right )^{4} \sqrt {e x +d}}-\frac {c^{4} \left (\frac {\left (\frac {23}{8} A \,b^{2} c^{2} e^{2}-\frac {3}{2} A b \,c^{3} d e -\frac {19}{8} B \,b^{3} c \,e^{2}+B \,b^{2} c^{2} d e \right ) \left (e x +d \right )^{\frac {3}{2}}+\frac {b e \left (25 A \,b^{2} c \,e^{2}-37 A b \,c^{2} d e +12 A \,c^{3} d^{2}-21 b^{3} B \,e^{2}+29 B \,b^{2} c d e -8 B b \,c^{2} d^{2}\right ) \sqrt {e x +d}}{8}}{\left (c \left (e x +d \right )+b e -c d \right )^{2}}+\frac {\left (143 A \,b^{2} c \,e^{2}-156 A b \,c^{2} d e +48 A \,c^{3} d^{2}-99 b^{3} B \,e^{2}+88 B \,b^{2} c d e -24 B b \,c^{2} d^{2}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{8 \sqrt {\left (b e -c d \right ) c}}\right )}{\left (b e -c d \right )^{4} e^{4} b^{5}}-\frac {\frac {-\frac {b e \left (11 A b e +12 A c d -4 B b d \right ) \left (e x +d \right )^{\frac {3}{2}}}{8}+\left (\frac {13}{8} A \,b^{2} d \,e^{2}+\frac {3}{2} A b c \,d^{2} e -\frac {1}{2} B \,b^{2} d^{2} e \right ) \sqrt {e x +d}}{e^{2} x^{2}}+\frac {\left (35 A \,b^{2} e^{2}+60 A b c d e +48 A \,c^{2} d^{2}-20 B \,b^{2} d e -24 c \,d^{2} B b \right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{8 \sqrt {d}}}{b^{5} d^{4} e^{4}}\right )\) | \(482\) |
| default | \(2 e^{4} \left (-\frac {-A e +B d}{3 d^{3} \left (b e -c d \right )^{3} \left (e x +d \right )^{\frac {3}{2}}}-\frac {-3 A b \,e^{2}+6 A c d e +2 B b d e -5 B c \,d^{2}}{d^{4} \left (b e -c d \right )^{4} \sqrt {e x +d}}-\frac {c^{4} \left (\frac {\left (\frac {23}{8} A \,b^{2} c^{2} e^{2}-\frac {3}{2} A b \,c^{3} d e -\frac {19}{8} B \,b^{3} c \,e^{2}+B \,b^{2} c^{2} d e \right ) \left (e x +d \right )^{\frac {3}{2}}+\frac {b e \left (25 A \,b^{2} c \,e^{2}-37 A b \,c^{2} d e +12 A \,c^{3} d^{2}-21 b^{3} B \,e^{2}+29 B \,b^{2} c d e -8 B b \,c^{2} d^{2}\right ) \sqrt {e x +d}}{8}}{\left (c \left (e x +d \right )+b e -c d \right )^{2}}+\frac {\left (143 A \,b^{2} c \,e^{2}-156 A b \,c^{2} d e +48 A \,c^{3} d^{2}-99 b^{3} B \,e^{2}+88 B \,b^{2} c d e -24 B b \,c^{2} d^{2}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{8 \sqrt {\left (b e -c d \right ) c}}\right )}{\left (b e -c d \right )^{4} e^{4} b^{5}}-\frac {\frac {-\frac {b e \left (11 A b e +12 A c d -4 B b d \right ) \left (e x +d \right )^{\frac {3}{2}}}{8}+\left (\frac {13}{8} A \,b^{2} d \,e^{2}+\frac {3}{2} A b c \,d^{2} e -\frac {1}{2} B \,b^{2} d^{2} e \right ) \sqrt {e x +d}}{e^{2} x^{2}}+\frac {\left (35 A \,b^{2} e^{2}+60 A b c d e +48 A \,c^{2} d^{2}-20 B \,b^{2} d e -24 c \,d^{2} B b \right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{8 \sqrt {d}}}{b^{5} d^{4} e^{4}}\right )\) | \(482\) |
| pseudoelliptic | \(-\frac {\frac {c^{4} d^{\frac {9}{2}} x^{2} \left (c x +b \right )^{2} \left (143 A \,b^{2} c \,e^{2}-156 A b \,c^{2} d e +48 A \,c^{3} d^{2}-99 b^{3} B \,e^{2}+88 B \,b^{2} c d e -24 B b \,c^{2} d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}} \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{2}+\sqrt {\left (b e -c d \right ) c}\, \left (\frac {\left (e x +d \right )^{\frac {3}{2}} x^{2} \left (c x +b \right )^{2} \left (b e -c d \right )^{4} \left (35 A \,b^{2} e^{2}+60 A b c d e +48 A \,c^{2} d^{2}-20 B \,b^{2} d e -24 c \,d^{2} B b \right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{2}+\left (\left (\left (2 B x +A \right ) d^{3}-\frac {7 x \left (-\frac {80 B x}{21}+A \right ) e \,d^{2}}{2}-\frac {70 x^{2} \left (-\frac {3 B x}{7}+A \right ) e^{2} d}{3}-\frac {35 A \,e^{3} x^{3}}{2}\right ) e^{4} b^{7}-4 c \left (\left (2 B x +A \right ) d^{4}-\frac {5 \left (-\frac {10 B x}{3}+A \right ) x e \,d^{3}}{2}-\frac {139 x^{2} \left (-\frac {4 B x}{139}+A \right ) e^{2} d^{2}}{12}+\frac {5 e^{3} x^{3} \left (-3 B x +A \right ) d}{3}+\frac {35 A \,e^{4} x^{4}}{4}\right ) e^{3} b^{6}+6 c^{2} \left (\left (2 B x +A \right ) d^{5}-\frac {5 x \left (-\frac {8 B x}{5}+A \right ) e \,d^{4}}{6}+\frac {5 \left (-\frac {182 B x}{15}+A \right ) x^{2} e^{2} d^{3}}{6}+\frac {565 \left (-\frac {256 B x}{565}+A \right ) x^{3} e^{3} d^{2}}{36}+\frac {85 \left (\frac {3 B x}{17}+A \right ) x^{4} e^{4} d}{9}-\frac {35 A \,e^{5} x^{5}}{12}\right ) e^{2} b^{5}-4 c^{3} \left (\left (2 B x +A \right ) d^{5}+\frac {5 x e \left (-\frac {4 B x}{5}+A \right ) d^{4}}{2}+\frac {15 x^{2} \left (-\frac {16 B x}{15}+A \right ) e^{2} d^{3}}{2}+\frac {15 \left (\frac {4 B x}{9}+A \right ) x^{3} e^{3} d^{2}}{2}-\frac {53 x^{4} \left (-\frac {42 B x}{53}+A \right ) e^{4} d}{6}-10 A \,e^{5} x^{5}\right ) d e \,b^{4}+c^{4} \left (\left (2 B x +A \right ) d^{3}+\frac {21 x e \left (-\frac {53 B x}{21}+A \right ) d^{2}}{2}-22 x^{2} \left (-\frac {6 B x}{11}+A \right ) e^{2} d -9 A \,e^{3} x^{3}\right ) d^{2} \left (e x +d \right )^{2} b^{3}-4 c^{5} x \,d^{3} \left (\left (-\frac {9 B x}{4}+A \right ) d^{2}-\frac {91 \left (-\frac {5 B x}{13}+A \right ) x e d}{8}+\frac {7 A \,e^{2} x^{2}}{2}\right ) \left (e x +d \right )^{2} b^{2}-18 c^{6} \left (\left (-\frac {B x}{3}+A \right ) d -\frac {5 A e x}{3}\right ) x^{2} d^{4} \left (e x +d \right )^{2} b -12 d^{5} A \,c^{7} x^{3} \left (e x +d \right )^{2}\right ) \sqrt {d}\, b \right )}{2 \left (e x +d \right )^{\frac {3}{2}} \sqrt {\left (b e -c d \right ) c}\, \left (c x +b \right )^{2} b^{5} \left (b e -c d \right )^{4} x^{2} d^{\frac {9}{2}}}\) | \(737\) |
-1/4*(e*x+d)^(1/2)*(-11*A*b*e*x-12*A*c*d*x+4*B*b*d*x+2*A*b*d)/d^4/b^4/x^2+ 1/4/b^4/d^4*e*(-8*c^4*d^4/(b*e-c*d)^4/b/e*(((23/8*A*b^2*c^2*e^2-3/2*A*b*c^ 3*d*e-19/8*B*b^3*c*e^2+B*b^2*c^2*d*e)*(e*x+d)^(3/2)+1/8*b*e*(25*A*b^2*c*e^ 2-37*A*b*c^2*d*e+12*A*c^3*d^2-21*B*b^3*e^2+29*B*b^2*c*d*e-8*B*b*c^2*d^2)*( e*x+d)^(1/2))/(c*(e*x+d)+b*e-c*d)^2+1/8*(143*A*b^2*c*e^2-156*A*b*c^2*d*e+4 8*A*c^3*d^2-99*B*b^3*e^2+88*B*b^2*c*d*e-24*B*b*c^2*d^2)/((b*e-c*d)*c)^(1/2 )*arctan(c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)))-(35*A*b^2*e^2+60*A*b*c*d*e+ 48*A*c^2*d^2-20*B*b^2*d*e-24*B*b*c*d^2)/b/e/d^(1/2)*arctanh((e*x+d)^(1/2)/ d^(1/2))+8*b^4*e^3*(3*A*b*e^2-6*A*c*d*e-2*B*b*d*e+5*B*c*d^2)/(b*e-c*d)^4/( e*x+d)^(1/2)+8/3*b^4*d*e^3*(A*e-B*d)/(b*e-c*d)^3/(e*x+d)^(3/2))
Leaf count of result is larger than twice the leaf count of optimal. 2738 vs. \(2 (608) = 1216\).
Time = 263.17 (sec) , antiderivative size = 10984, normalized size of antiderivative = 17.06 \[ \int \frac {A+B x}{(d+e x)^{5/2} \left (b x+c x^2\right )^3} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {A+B x}{(d+e x)^{5/2} \left (b x+c x^2\right )^3} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {A+B x}{(d+e x)^{5/2} \left (b x+c x^2\right )^3} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(b*e-c*d>0)', see `assume?` for m ore detail
Leaf count of result is larger than twice the leaf count of optimal. 1583 vs. \(2 (608) = 1216\).
Time = 0.38 (sec) , antiderivative size = 1583, normalized size of antiderivative = 2.46 \[ \int \frac {A+B x}{(d+e x)^{5/2} \left (b x+c x^2\right )^3} \, dx=\text {Too large to display} \]
1/4*(24*B*b*c^6*d^2 - 48*A*c^7*d^2 - 88*B*b^2*c^5*d*e + 156*A*b*c^6*d*e + 99*B*b^3*c^4*e^2 - 143*A*b^2*c^5*e^2)*arctan(sqrt(e*x + d)*c/sqrt(-c^2*d + b*c*e))/((b^5*c^4*d^4 - 4*b^6*c^3*d^3*e + 6*b^7*c^2*d^2*e^2 - 4*b^8*c*d*e ^3 + b^9*e^4)*sqrt(-c^2*d + b*c*e)) + 2/3*(15*(e*x + d)*B*c*d^2*e^4 + B*c* d^3*e^4 - 6*(e*x + d)*B*b*d*e^5 - 18*(e*x + d)*A*c*d*e^5 - B*b*d^2*e^5 - A *c*d^2*e^5 + 9*(e*x + d)*A*b*e^6 + A*b*d*e^6)/((c^4*d^8 - 4*b*c^3*d^7*e + 6*b^2*c^2*d^6*e^2 - 4*b^3*c*d^5*e^3 + b^4*d^4*e^4)*(e*x + d)^(3/2)) - 1/4* (12*(e*x + d)^(7/2)*B*b*c^6*d^5*e - 24*(e*x + d)^(7/2)*A*c^7*d^5*e - 36*(e *x + d)^(5/2)*B*b*c^6*d^6*e + 72*(e*x + d)^(5/2)*A*c^7*d^6*e + 36*(e*x + d )^(3/2)*B*b*c^6*d^7*e - 72*(e*x + d)^(3/2)*A*c^7*d^7*e - 12*sqrt(e*x + d)* B*b*c^6*d^8*e + 24*sqrt(e*x + d)*A*c^7*d^8*e - 35*(e*x + d)^(7/2)*B*b^2*c^ 5*d^4*e^2 + 60*(e*x + d)^(7/2)*A*b*c^6*d^4*e^2 + 123*(e*x + d)^(5/2)*B*b^2 *c^5*d^5*e^2 - 216*(e*x + d)^(5/2)*A*b*c^6*d^5*e^2 - 141*(e*x + d)^(3/2)*B *b^2*c^5*d^6*e^2 + 252*(e*x + d)^(3/2)*A*b*c^6*d^6*e^2 + 53*sqrt(e*x + d)* B*b^2*c^5*d^7*e^2 - 96*sqrt(e*x + d)*A*b*c^6*d^7*e^2 + 24*(e*x + d)^(7/2)* B*b^3*c^4*d^3*e^3 - 28*(e*x + d)^(7/2)*A*b^2*c^5*d^3*e^3 - 125*(e*x + d)^( 5/2)*B*b^3*c^4*d^4*e^3 + 175*(e*x + d)^(5/2)*A*b^2*c^5*d^4*e^3 + 182*(e*x + d)^(3/2)*B*b^3*c^4*d^5*e^3 - 274*(e*x + d)^(3/2)*A*b^2*c^5*d^5*e^3 - 81* sqrt(e*x + d)*B*b^3*c^4*d^6*e^3 + 127*sqrt(e*x + d)*A*b^2*c^5*d^6*e^3 - 16 *(e*x + d)^(7/2)*B*b^4*c^3*d^2*e^4 - 18*(e*x + d)^(7/2)*A*b^3*c^4*d^2*e...
Time = 16.74 (sec) , antiderivative size = 24572, normalized size of antiderivative = 38.16 \[ \int \frac {A+B x}{(d+e x)^{5/2} \left (b x+c x^2\right )^3} \, dx=\text {Too large to display} \]
log(14598144*A^3*b^9*c^27*d^32*e^4 - 884736*A^3*b^8*c^28*d^33*e^3 - ((d + e*x)^(1/2)*(589824*A^2*b^12*c^27*d^36*e^2 - 10616832*A^2*b^13*c^26*d^35*e^ 3 + 89518080*A^2*b^14*c^25*d^34*e^4 - 468971520*A^2*b^15*c^24*d^33*e^5 + 1 707439360*A^2*b^16*c^23*d^32*e^6 - 4579446784*A^2*b^17*c^22*d^31*e^7 + 936 4822016*A^2*b^18*c^21*d^30*e^8 - 14937190400*A^2*b^19*c^20*d^29*e^9 + 1893 6107520*A^2*b^20*c^19*d^28*e^10 - 19535324160*A^2*b^21*c^18*d^27*e^11 + 17 074641408*A^2*b^22*c^17*d^26*e^12 - 13484230656*A^2*b^23*c^16*d^25*e^13 + 10265639040*A^2*b^24*c^15*d^24*e^14 - 7643066880*A^2*b^25*c^14*d^23*e^15 + 5421597440*A^2*b^26*c^13*d^22*e^16 - 3708136960*A^2*b^27*c^12*d^21*e^17 + 2608529792*A^2*b^28*c^11*d^20*e^18 - 1894041600*A^2*b^29*c^10*d^19*e^19 + 1274465280*A^2*b^30*c^9*d^18*e^20 - 707773440*A^2*b^31*c^8*d^17*e^21 + 30 1648512*A^2*b^32*c^7*d^16*e^22 - 93688320*A^2*b^33*c^6*d^15*e^23 + 1993088 0*A^2*b^34*c^5*d^14*e^24 - 2598400*A^2*b^35*c^4*d^13*e^25 + 156800*A^2*b^3 6*c^3*d^12*e^26 + 147456*B^2*b^14*c^25*d^36*e^2 - 2777088*B^2*b^15*c^24*d^ 35*e^3 + 24555520*B^2*b^16*c^23*d^34*e^4 - 135055360*B^2*b^17*c^22*d^33*e^ 5 + 515884160*B^2*b^18*c^21*d^32*e^6 - 1446258176*B^2*b^19*c^20*d^31*e^7 + 3062171904*B^2*b^20*c^19*d^30*e^8 - 4951119360*B^2*b^21*c^18*d^29*e^9 + 6 076371840*B^2*b^22*c^17*d^28*e^10 - 5478190080*B^2*b^23*c^16*d^27*e^11 + 3 273549312*B^2*b^24*c^15*d^26*e^12 - 766116864*B^2*b^25*c^14*d^25*e^13 - 66 8122240*B^2*b^26*c^13*d^24*e^14 + 721318400*B^2*b^27*c^12*d^23*e^15 - 1...