Integrand size = 26, antiderivative size = 263 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^2}{(d+e x)^{7/2}} \, dx=\frac {2 d^2 (B d-A e) (c d-b e)^2}{5 e^6 (d+e x)^{5/2}}-\frac {2 d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{3 e^6 (d+e x)^{3/2}}-\frac {2 \left (A e \left (6 c^2 d^2-6 b c d e+b^2 e^2\right )-B d \left (10 c^2 d^2-12 b c d e+3 b^2 e^2\right )\right )}{e^6 \sqrt {d+e x}}-\frac {2 \left (2 A c e (2 c d-b e)-B \left (10 c^2 d^2-8 b c d e+b^2 e^2\right )\right ) \sqrt {d+e x}}{e^6}-\frac {2 c (5 B c d-2 b B e-A c e) (d+e x)^{3/2}}{3 e^6}+\frac {2 B c^2 (d+e x)^{5/2}}{5 e^6} \]
2/5*d^2*(-A*e+B*d)*(-b*e+c*d)^2/e^6/(e*x+d)^(5/2)-2/3*d*(-b*e+c*d)*(B*d*(- 3*b*e+5*c*d)-2*A*e*(-b*e+2*c*d))/e^6/(e*x+d)^(3/2)-2/3*c*(-A*c*e-2*B*b*e+5 *B*c*d)*(e*x+d)^(3/2)/e^6+2/5*B*c^2*(e*x+d)^(5/2)/e^6-2*(A*e*(b^2*e^2-6*b* c*d*e+6*c^2*d^2)-B*d*(3*b^2*e^2-12*b*c*d*e+10*c^2*d^2))/e^6/(e*x+d)^(1/2)- 2*(2*A*c*e*(-b*e+2*c*d)-B*(b^2*e^2-8*b*c*d*e+10*c^2*d^2))*(e*x+d)^(1/2)/e^ 6
Time = 0.20 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.03 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^2}{(d+e x)^{7/2}} \, dx=\frac {2 \left (A e \left (-b^2 e^2 \left (8 d^2+20 d e x+15 e^2 x^2\right )+6 b c e \left (16 d^3+40 d^2 e x+30 d e^2 x^2+5 e^3 x^3\right )-c^2 \left (128 d^4+320 d^3 e x+240 d^2 e^2 x^2+40 d e^3 x^3-5 e^4 x^4\right )\right )+B \left (3 b^2 e^2 \left (16 d^3+40 d^2 e x+30 d e^2 x^2+5 e^3 x^3\right )-2 b c e \left (128 d^4+320 d^3 e x+240 d^2 e^2 x^2+40 d e^3 x^3-5 e^4 x^4\right )+c^2 \left (256 d^5+640 d^4 e x+480 d^3 e^2 x^2+80 d^2 e^3 x^3-10 d e^4 x^4+3 e^5 x^5\right )\right )\right )}{15 e^6 (d+e x)^{5/2}} \]
(2*(A*e*(-(b^2*e^2*(8*d^2 + 20*d*e*x + 15*e^2*x^2)) + 6*b*c*e*(16*d^3 + 40 *d^2*e*x + 30*d*e^2*x^2 + 5*e^3*x^3) - c^2*(128*d^4 + 320*d^3*e*x + 240*d^ 2*e^2*x^2 + 40*d*e^3*x^3 - 5*e^4*x^4)) + B*(3*b^2*e^2*(16*d^3 + 40*d^2*e*x + 30*d*e^2*x^2 + 5*e^3*x^3) - 2*b*c*e*(128*d^4 + 320*d^3*e*x + 240*d^2*e^ 2*x^2 + 40*d*e^3*x^3 - 5*e^4*x^4) + c^2*(256*d^5 + 640*d^4*e*x + 480*d^3*e ^2*x^2 + 80*d^2*e^3*x^3 - 10*d*e^4*x^4 + 3*e^5*x^5))))/(15*e^6*(d + e*x)^( 5/2))
Time = 0.41 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {1195, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {(A+B x) \left (b x+c x^2\right )^2}{(d+e x)^{7/2}} \, dx\) |
| \(\Big \downarrow \) 1195 |
| \(\displaystyle \int \left (\frac {B \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )-2 A c e (2 c d-b e)}{e^5 \sqrt {d+e x}}+\frac {A e \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )-B d \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )}{e^5 (d+e x)^{3/2}}-\frac {d^2 (B d-A e) (c d-b e)^2}{e^5 (d+e x)^{7/2}}+\frac {c \sqrt {d+e x} (A c e+2 b B e-5 B c d)}{e^5}+\frac {d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{e^5 (d+e x)^{5/2}}+\frac {B c^2 (d+e x)^{3/2}}{e^5}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 \sqrt {d+e x} \left (2 A c e (2 c d-b e)-B \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )\right )}{e^6}-\frac {2 \left (A e \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )-B d \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )\right )}{e^6 \sqrt {d+e x}}+\frac {2 d^2 (B d-A e) (c d-b e)^2}{5 e^6 (d+e x)^{5/2}}-\frac {2 c (d+e x)^{3/2} (-A c e-2 b B e+5 B c d)}{3 e^6}-\frac {2 d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{3 e^6 (d+e x)^{3/2}}+\frac {2 B c^2 (d+e x)^{5/2}}{5 e^6}\) |
(2*d^2*(B*d - A*e)*(c*d - b*e)^2)/(5*e^6*(d + e*x)^(5/2)) - (2*d*(c*d - b* e)*(B*d*(5*c*d - 3*b*e) - 2*A*e*(2*c*d - b*e)))/(3*e^6*(d + e*x)^(3/2)) - (2*(A*e*(6*c^2*d^2 - 6*b*c*d*e + b^2*e^2) - B*d*(10*c^2*d^2 - 12*b*c*d*e + 3*b^2*e^2)))/(e^6*Sqrt[d + e*x]) - (2*(2*A*c*e*(2*c*d - b*e) - B*(10*c^2* d^2 - 8*b*c*d*e + b^2*e^2))*Sqrt[d + e*x])/e^6 - (2*c*(5*B*c*d - 2*b*B*e - A*c*e)*(d + e*x)^(3/2))/(3*e^6) + (2*B*c^2*(d + e*x)^(5/2))/(5*e^6)
3.13.28.3.1 Defintions of rubi rules used
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x _) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x ] && IGtQ[p, 0]
Time = 0.36 (sec) , antiderivative size = 257, normalized size of antiderivative = 0.98
| method | result | size |
| pseudoelliptic | \(\frac {\left (\left (6 e^{5} x^{5}-20 d \,e^{4} x^{4}+160 d^{2} e^{3} x^{3}+960 d^{3} e^{2} x^{2}+1280 d^{4} e x +512 d^{5}\right ) B -256 A e \left (-\frac {5}{128} e^{4} x^{4}+\frac {5}{16} d \,e^{3} x^{3}+\frac {15}{8} d^{2} e^{2} x^{2}+\frac {5}{2} d^{3} e x +d^{4}\right )\right ) c^{2}+192 \left (\left (\frac {5}{48} e^{4} x^{4}-\frac {5}{6} d \,e^{3} x^{3}-5 d^{2} e^{2} x^{2}-\frac {20}{3} d^{3} e x -\frac {8}{3} d^{4}\right ) B +A e \left (\frac {5}{16} e^{3} x^{3}+\frac {15}{8} d \,e^{2} x^{2}+\frac {5}{2} d^{2} e x +d^{3}\right )\right ) e b c -16 \left (\left (-\frac {15}{8} e^{3} x^{3}-\frac {45}{4} d \,e^{2} x^{2}-15 d^{2} e x -6 d^{3}\right ) B +A e \left (\frac {15}{8} e^{2} x^{2}+\frac {5}{2} d e x +d^{2}\right )\right ) e^{2} b^{2}}{15 \left (e x +d \right )^{\frac {5}{2}} e^{6}}\) | \(257\) |
| derivativedivides | \(\frac {\frac {2 B \,c^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 A \,c^{2} e \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {4 B b c e \left (e x +d \right )^{\frac {3}{2}}}{3}-\frac {10 B \,c^{2} d \left (e x +d \right )^{\frac {3}{2}}}{3}+4 A b c \,e^{2} \sqrt {e x +d}-8 A \,c^{2} d e \sqrt {e x +d}+2 B \,b^{2} e^{2} \sqrt {e x +d}-16 B b c d e \sqrt {e x +d}+20 B \,c^{2} d^{2} \sqrt {e x +d}-\frac {2 d^{2} \left (A \,b^{2} e^{3}-2 A b c d \,e^{2}+A \,c^{2} d^{2} e -B \,b^{2} d \,e^{2}+2 B b c \,d^{2} e -B \,c^{2} d^{3}\right )}{5 \left (e x +d \right )^{\frac {5}{2}}}+\frac {2 d \left (2 A \,b^{2} e^{3}-6 A b c d \,e^{2}+4 A \,c^{2} d^{2} e -3 B \,b^{2} d \,e^{2}+8 B b c \,d^{2} e -5 B \,c^{2} d^{3}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}-\frac {2 \left (A \,b^{2} e^{3}-6 A b c d \,e^{2}+6 A \,c^{2} d^{2} e -3 B \,b^{2} d \,e^{2}+12 B b c \,d^{2} e -10 B \,c^{2} d^{3}\right )}{\sqrt {e x +d}}}{e^{6}}\) | \(335\) |
| default | \(\frac {\frac {2 B \,c^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 A \,c^{2} e \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {4 B b c e \left (e x +d \right )^{\frac {3}{2}}}{3}-\frac {10 B \,c^{2} d \left (e x +d \right )^{\frac {3}{2}}}{3}+4 A b c \,e^{2} \sqrt {e x +d}-8 A \,c^{2} d e \sqrt {e x +d}+2 B \,b^{2} e^{2} \sqrt {e x +d}-16 B b c d e \sqrt {e x +d}+20 B \,c^{2} d^{2} \sqrt {e x +d}-\frac {2 d^{2} \left (A \,b^{2} e^{3}-2 A b c d \,e^{2}+A \,c^{2} d^{2} e -B \,b^{2} d \,e^{2}+2 B b c \,d^{2} e -B \,c^{2} d^{3}\right )}{5 \left (e x +d \right )^{\frac {5}{2}}}+\frac {2 d \left (2 A \,b^{2} e^{3}-6 A b c d \,e^{2}+4 A \,c^{2} d^{2} e -3 B \,b^{2} d \,e^{2}+8 B b c \,d^{2} e -5 B \,c^{2} d^{3}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}-\frac {2 \left (A \,b^{2} e^{3}-6 A b c d \,e^{2}+6 A \,c^{2} d^{2} e -3 B \,b^{2} d \,e^{2}+12 B b c \,d^{2} e -10 B \,c^{2} d^{3}\right )}{\sqrt {e x +d}}}{e^{6}}\) | \(335\) |
| gosper | \(-\frac {2 \left (-3 B \,x^{5} c^{2} e^{5}-5 A \,x^{4} c^{2} e^{5}-10 B \,x^{4} b c \,e^{5}+10 B \,x^{4} c^{2} d \,e^{4}-30 A \,x^{3} b c \,e^{5}+40 A \,x^{3} c^{2} d \,e^{4}-15 B \,x^{3} b^{2} e^{5}+80 B \,x^{3} b c d \,e^{4}-80 B \,x^{3} c^{2} d^{2} e^{3}+15 A \,x^{2} b^{2} e^{5}-180 A \,x^{2} b c d \,e^{4}+240 A \,x^{2} c^{2} d^{2} e^{3}-90 B \,x^{2} b^{2} d \,e^{4}+480 B \,x^{2} b c \,d^{2} e^{3}-480 B \,x^{2} c^{2} d^{3} e^{2}+20 A x \,b^{2} d \,e^{4}-240 A x b c \,d^{2} e^{3}+320 A x \,c^{2} d^{3} e^{2}-120 B x \,b^{2} d^{2} e^{3}+640 B x b c \,d^{3} e^{2}-640 B x \,c^{2} d^{4} e +8 A \,b^{2} d^{2} e^{3}-96 A b c \,d^{3} e^{2}+128 A \,c^{2} d^{4} e -48 B \,b^{2} d^{3} e^{2}+256 B b c \,d^{4} e -256 B \,c^{2} d^{5}\right )}{15 \left (e x +d \right )^{\frac {5}{2}} e^{6}}\) | \(341\) |
| trager | \(-\frac {2 \left (-3 B \,x^{5} c^{2} e^{5}-5 A \,x^{4} c^{2} e^{5}-10 B \,x^{4} b c \,e^{5}+10 B \,x^{4} c^{2} d \,e^{4}-30 A \,x^{3} b c \,e^{5}+40 A \,x^{3} c^{2} d \,e^{4}-15 B \,x^{3} b^{2} e^{5}+80 B \,x^{3} b c d \,e^{4}-80 B \,x^{3} c^{2} d^{2} e^{3}+15 A \,x^{2} b^{2} e^{5}-180 A \,x^{2} b c d \,e^{4}+240 A \,x^{2} c^{2} d^{2} e^{3}-90 B \,x^{2} b^{2} d \,e^{4}+480 B \,x^{2} b c \,d^{2} e^{3}-480 B \,x^{2} c^{2} d^{3} e^{2}+20 A x \,b^{2} d \,e^{4}-240 A x b c \,d^{2} e^{3}+320 A x \,c^{2} d^{3} e^{2}-120 B x \,b^{2} d^{2} e^{3}+640 B x b c \,d^{3} e^{2}-640 B x \,c^{2} d^{4} e +8 A \,b^{2} d^{2} e^{3}-96 A b c \,d^{3} e^{2}+128 A \,c^{2} d^{4} e -48 B \,b^{2} d^{3} e^{2}+256 B b c \,d^{4} e -256 B \,c^{2} d^{5}\right )}{15 \left (e x +d \right )^{\frac {5}{2}} e^{6}}\) | \(341\) |
| risch | \(\frac {2 \left (3 B \,c^{2} e^{2} x^{2}+5 A \,c^{2} e^{2} x +10 e^{2} B b c x -19 B \,c^{2} d e x +30 A b c \,e^{2}-55 A \,c^{2} d e +15 B \,b^{2} e^{2}-110 B b c d e +128 B \,c^{2} d^{2}\right ) \sqrt {e x +d}}{15 e^{6}}-\frac {2 \left (15 A \,x^{2} b^{2} e^{5}-90 A \,x^{2} b c d \,e^{4}+90 A \,x^{2} c^{2} d^{2} e^{3}-45 B \,x^{2} b^{2} d \,e^{4}+180 B \,x^{2} b c \,d^{2} e^{3}-150 B \,x^{2} c^{2} d^{3} e^{2}+20 A x \,b^{2} d \,e^{4}-150 A x b c \,d^{2} e^{3}+160 A x \,c^{2} d^{3} e^{2}-75 B x \,b^{2} d^{2} e^{3}+320 B x b c \,d^{3} e^{2}-275 B x \,c^{2} d^{4} e +8 A \,b^{2} d^{2} e^{3}-66 A b c \,d^{3} e^{2}+73 A \,c^{2} d^{4} e -33 B \,b^{2} d^{3} e^{2}+146 B b c \,d^{4} e -128 B \,c^{2} d^{5}\right )}{15 e^{6} \sqrt {e x +d}\, \left (e^{2} x^{2}+2 d e x +d^{2}\right )}\) | \(343\) |
1/15*(((6*e^5*x^5-20*d*e^4*x^4+160*d^2*e^3*x^3+960*d^3*e^2*x^2+1280*d^4*e* x+512*d^5)*B-256*A*e*(-5/128*e^4*x^4+5/16*d*e^3*x^3+15/8*d^2*e^2*x^2+5/2*d ^3*e*x+d^4))*c^2+192*((5/48*e^4*x^4-5/6*d*e^3*x^3-5*d^2*e^2*x^2-20/3*d^3*e *x-8/3*d^4)*B+A*e*(5/16*e^3*x^3+15/8*d*e^2*x^2+5/2*d^2*e*x+d^3))*e*b*c-16* ((-15/8*e^3*x^3-45/4*d*e^2*x^2-15*d^2*e*x-6*d^3)*B+A*e*(15/8*e^2*x^2+5/2*d *e*x+d^2))*e^2*b^2)/(e*x+d)^(5/2)/e^6
Time = 0.31 (sec) , antiderivative size = 322, normalized size of antiderivative = 1.22 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^2}{(d+e x)^{7/2}} \, dx=\frac {2 \, {\left (3 \, B c^{2} e^{5} x^{5} + 256 \, B c^{2} d^{5} - 8 \, A b^{2} d^{2} e^{3} - 128 \, {\left (2 \, B b c + A c^{2}\right )} d^{4} e + 48 \, {\left (B b^{2} + 2 \, A b c\right )} d^{3} e^{2} - 5 \, {\left (2 \, B c^{2} d e^{4} - {\left (2 \, B b c + A c^{2}\right )} e^{5}\right )} x^{4} + 5 \, {\left (16 \, B c^{2} d^{2} e^{3} - 8 \, {\left (2 \, B b c + A c^{2}\right )} d e^{4} + 3 \, {\left (B b^{2} + 2 \, A b c\right )} e^{5}\right )} x^{3} + 15 \, {\left (32 \, B c^{2} d^{3} e^{2} - A b^{2} e^{5} - 16 \, {\left (2 \, B b c + A c^{2}\right )} d^{2} e^{3} + 6 \, {\left (B b^{2} + 2 \, A b c\right )} d e^{4}\right )} x^{2} + 20 \, {\left (32 \, B c^{2} d^{4} e - A b^{2} d e^{4} - 16 \, {\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} + 6 \, {\left (B b^{2} + 2 \, A b c\right )} d^{2} e^{3}\right )} x\right )} \sqrt {e x + d}}{15 \, {\left (e^{9} x^{3} + 3 \, d e^{8} x^{2} + 3 \, d^{2} e^{7} x + d^{3} e^{6}\right )}} \]
2/15*(3*B*c^2*e^5*x^5 + 256*B*c^2*d^5 - 8*A*b^2*d^2*e^3 - 128*(2*B*b*c + A *c^2)*d^4*e + 48*(B*b^2 + 2*A*b*c)*d^3*e^2 - 5*(2*B*c^2*d*e^4 - (2*B*b*c + A*c^2)*e^5)*x^4 + 5*(16*B*c^2*d^2*e^3 - 8*(2*B*b*c + A*c^2)*d*e^4 + 3*(B* b^2 + 2*A*b*c)*e^5)*x^3 + 15*(32*B*c^2*d^3*e^2 - A*b^2*e^5 - 16*(2*B*b*c + A*c^2)*d^2*e^3 + 6*(B*b^2 + 2*A*b*c)*d*e^4)*x^2 + 20*(32*B*c^2*d^4*e - A* b^2*d*e^4 - 16*(2*B*b*c + A*c^2)*d^3*e^2 + 6*(B*b^2 + 2*A*b*c)*d^2*e^3)*x) *sqrt(e*x + d)/(e^9*x^3 + 3*d*e^8*x^2 + 3*d^2*e^7*x + d^3*e^6)
Leaf count of result is larger than twice the leaf count of optimal. 1833 vs. \(2 (265) = 530\).
Time = 0.58 (sec) , antiderivative size = 1833, normalized size of antiderivative = 6.97 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^2}{(d+e x)^{7/2}} \, dx=\text {Too large to display} \]
Piecewise((-16*A*b**2*d**2*e**3/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x* sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)) - 40*A*b**2*d*e**4*x/(15*d**2* e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x )) - 30*A*b**2*e**5*x**2/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)) + 192*A*b*c*d**3*e**2/(15*d**2*e**6*s qrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)) + 4 80*A*b*c*d**2*e**3*x/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e* x) + 15*e**8*x**2*sqrt(d + e*x)) + 360*A*b*c*d*e**4*x**2/(15*d**2*e**6*sqr t(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)) + 60* A*b*c*e**5*x**3/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)) - 256*A*c**2*d**4*e/(15*d**2*e**6*sqrt(d + e*x ) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)) - 640*A*c**2*d **3*e**2*x/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e* *8*x**2*sqrt(d + e*x)) - 480*A*c**2*d**2*e**3*x**2/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)) - 80*A*c**2 *d*e**4*x**3/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15* e**8*x**2*sqrt(d + e*x)) + 10*A*c**2*e**5*x**4/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)) + 96*B*b**2*d** 3*e**2/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x **2*sqrt(d + e*x)) + 240*B*b**2*d**2*e**3*x/(15*d**2*e**6*sqrt(d + e*x)...
Time = 0.21 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.13 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^2}{(d+e x)^{7/2}} \, dx=\frac {2 \, {\left (\frac {3 \, {\left (e x + d\right )}^{\frac {5}{2}} B c^{2} - 5 \, {\left (5 \, B c^{2} d - {\left (2 \, B b c + A c^{2}\right )} e\right )} {\left (e x + d\right )}^{\frac {3}{2}} + 15 \, {\left (10 \, B c^{2} d^{2} - 4 \, {\left (2 \, B b c + A c^{2}\right )} d e + {\left (B b^{2} + 2 \, A b c\right )} e^{2}\right )} \sqrt {e x + d}}{e^{5}} + \frac {3 \, B c^{2} d^{5} - 3 \, A b^{2} d^{2} e^{3} - 3 \, {\left (2 \, B b c + A c^{2}\right )} d^{4} e + 3 \, {\left (B b^{2} + 2 \, A b c\right )} d^{3} e^{2} + 15 \, {\left (10 \, B c^{2} d^{3} - A b^{2} e^{3} - 6 \, {\left (2 \, B b c + A c^{2}\right )} d^{2} e + 3 \, {\left (B b^{2} + 2 \, A b c\right )} d e^{2}\right )} {\left (e x + d\right )}^{2} - 5 \, {\left (5 \, B c^{2} d^{4} - 2 \, A b^{2} d e^{3} - 4 \, {\left (2 \, B b c + A c^{2}\right )} d^{3} e + 3 \, {\left (B b^{2} + 2 \, A b c\right )} d^{2} e^{2}\right )} {\left (e x + d\right )}}{{\left (e x + d\right )}^{\frac {5}{2}} e^{5}}\right )}}{15 \, e} \]
2/15*((3*(e*x + d)^(5/2)*B*c^2 - 5*(5*B*c^2*d - (2*B*b*c + A*c^2)*e)*(e*x + d)^(3/2) + 15*(10*B*c^2*d^2 - 4*(2*B*b*c + A*c^2)*d*e + (B*b^2 + 2*A*b*c )*e^2)*sqrt(e*x + d))/e^5 + (3*B*c^2*d^5 - 3*A*b^2*d^2*e^3 - 3*(2*B*b*c + A*c^2)*d^4*e + 3*(B*b^2 + 2*A*b*c)*d^3*e^2 + 15*(10*B*c^2*d^3 - A*b^2*e^3 - 6*(2*B*b*c + A*c^2)*d^2*e + 3*(B*b^2 + 2*A*b*c)*d*e^2)*(e*x + d)^2 - 5*( 5*B*c^2*d^4 - 2*A*b^2*d*e^3 - 4*(2*B*b*c + A*c^2)*d^3*e + 3*(B*b^2 + 2*A*b *c)*d^2*e^2)*(e*x + d))/((e*x + d)^(5/2)*e^5))/e
Time = 0.26 (sec) , antiderivative size = 419, normalized size of antiderivative = 1.59 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^2}{(d+e x)^{7/2}} \, dx=\frac {2 \, {\left (150 \, {\left (e x + d\right )}^{2} B c^{2} d^{3} - 25 \, {\left (e x + d\right )} B c^{2} d^{4} + 3 \, B c^{2} d^{5} - 180 \, {\left (e x + d\right )}^{2} B b c d^{2} e - 90 \, {\left (e x + d\right )}^{2} A c^{2} d^{2} e + 40 \, {\left (e x + d\right )} B b c d^{3} e + 20 \, {\left (e x + d\right )} A c^{2} d^{3} e - 6 \, B b c d^{4} e - 3 \, A c^{2} d^{4} e + 45 \, {\left (e x + d\right )}^{2} B b^{2} d e^{2} + 90 \, {\left (e x + d\right )}^{2} A b c d e^{2} - 15 \, {\left (e x + d\right )} B b^{2} d^{2} e^{2} - 30 \, {\left (e x + d\right )} A b c d^{2} e^{2} + 3 \, B b^{2} d^{3} e^{2} + 6 \, A b c d^{3} e^{2} - 15 \, {\left (e x + d\right )}^{2} A b^{2} e^{3} + 10 \, {\left (e x + d\right )} A b^{2} d e^{3} - 3 \, A b^{2} d^{2} e^{3}\right )}}{15 \, {\left (e x + d\right )}^{\frac {5}{2}} e^{6}} + \frac {2 \, {\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} B c^{2} e^{24} - 25 \, {\left (e x + d\right )}^{\frac {3}{2}} B c^{2} d e^{24} + 150 \, \sqrt {e x + d} B c^{2} d^{2} e^{24} + 10 \, {\left (e x + d\right )}^{\frac {3}{2}} B b c e^{25} + 5 \, {\left (e x + d\right )}^{\frac {3}{2}} A c^{2} e^{25} - 120 \, \sqrt {e x + d} B b c d e^{25} - 60 \, \sqrt {e x + d} A c^{2} d e^{25} + 15 \, \sqrt {e x + d} B b^{2} e^{26} + 30 \, \sqrt {e x + d} A b c e^{26}\right )}}{15 \, e^{30}} \]
2/15*(150*(e*x + d)^2*B*c^2*d^3 - 25*(e*x + d)*B*c^2*d^4 + 3*B*c^2*d^5 - 1 80*(e*x + d)^2*B*b*c*d^2*e - 90*(e*x + d)^2*A*c^2*d^2*e + 40*(e*x + d)*B*b *c*d^3*e + 20*(e*x + d)*A*c^2*d^3*e - 6*B*b*c*d^4*e - 3*A*c^2*d^4*e + 45*( e*x + d)^2*B*b^2*d*e^2 + 90*(e*x + d)^2*A*b*c*d*e^2 - 15*(e*x + d)*B*b^2*d ^2*e^2 - 30*(e*x + d)*A*b*c*d^2*e^2 + 3*B*b^2*d^3*e^2 + 6*A*b*c*d^3*e^2 - 15*(e*x + d)^2*A*b^2*e^3 + 10*(e*x + d)*A*b^2*d*e^3 - 3*A*b^2*d^2*e^3)/((e *x + d)^(5/2)*e^6) + 2/15*(3*(e*x + d)^(5/2)*B*c^2*e^24 - 25*(e*x + d)^(3/ 2)*B*c^2*d*e^24 + 150*sqrt(e*x + d)*B*c^2*d^2*e^24 + 10*(e*x + d)^(3/2)*B* b*c*e^25 + 5*(e*x + d)^(3/2)*A*c^2*e^25 - 120*sqrt(e*x + d)*B*b*c*d*e^25 - 60*sqrt(e*x + d)*A*c^2*d*e^25 + 15*sqrt(e*x + d)*B*b^2*e^26 + 30*sqrt(e*x + d)*A*b*c*e^26)/e^30
Time = 10.44 (sec) , antiderivative size = 312, normalized size of antiderivative = 1.19 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^2}{(d+e x)^{7/2}} \, dx=\frac {{\left (d+e\,x\right )}^{3/2}\,\left (2\,A\,c^2\,e-10\,B\,c^2\,d+4\,B\,b\,c\,e\right )}{3\,e^6}+\frac {\sqrt {d+e\,x}\,\left (2\,B\,b^2\,e^2-16\,B\,b\,c\,d\,e+4\,A\,b\,c\,e^2+20\,B\,c^2\,d^2-8\,A\,c^2\,d\,e\right )}{e^6}-\frac {\left (d+e\,x\right )\,\left (2\,B\,b^2\,d^2\,e^2-\frac {4\,A\,b^2\,d\,e^3}{3}-\frac {16\,B\,b\,c\,d^3\,e}{3}+4\,A\,b\,c\,d^2\,e^2+\frac {10\,B\,c^2\,d^4}{3}-\frac {8\,A\,c^2\,d^3\,e}{3}\right )+{\left (d+e\,x\right )}^2\,\left (-6\,B\,b^2\,d\,e^2+2\,A\,b^2\,e^3+24\,B\,b\,c\,d^2\,e-12\,A\,b\,c\,d\,e^2-20\,B\,c^2\,d^3+12\,A\,c^2\,d^2\,e\right )-\frac {2\,B\,c^2\,d^5}{5}+\frac {2\,A\,c^2\,d^4\,e}{5}+\frac {2\,A\,b^2\,d^2\,e^3}{5}-\frac {2\,B\,b^2\,d^3\,e^2}{5}+\frac {4\,B\,b\,c\,d^4\,e}{5}-\frac {4\,A\,b\,c\,d^3\,e^2}{5}}{e^6\,{\left (d+e\,x\right )}^{5/2}}+\frac {2\,B\,c^2\,{\left (d+e\,x\right )}^{5/2}}{5\,e^6} \]
((d + e*x)^(3/2)*(2*A*c^2*e - 10*B*c^2*d + 4*B*b*c*e))/(3*e^6) + ((d + e*x )^(1/2)*(2*B*b^2*e^2 + 20*B*c^2*d^2 + 4*A*b*c*e^2 - 8*A*c^2*d*e - 16*B*b*c *d*e))/e^6 - ((d + e*x)*((10*B*c^2*d^4)/3 - (4*A*b^2*d*e^3)/3 - (8*A*c^2*d ^3*e)/3 + 2*B*b^2*d^2*e^2 - (16*B*b*c*d^3*e)/3 + 4*A*b*c*d^2*e^2) + (d + e *x)^2*(2*A*b^2*e^3 - 20*B*c^2*d^3 + 12*A*c^2*d^2*e - 6*B*b^2*d*e^2 - 12*A* b*c*d*e^2 + 24*B*b*c*d^2*e) - (2*B*c^2*d^5)/5 + (2*A*c^2*d^4*e)/5 + (2*A*b ^2*d^2*e^3)/5 - (2*B*b^2*d^3*e^2)/5 + (4*B*b*c*d^4*e)/5 - (4*A*b*c*d^3*e^2 )/5)/(e^6*(d + e*x)^(5/2)) + (2*B*c^2*(d + e*x)^(5/2))/(5*e^6)