Integrand size = 26, antiderivative size = 263 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^2}{(d+e x)^{5/2}} \, dx=\frac {2 d^2 (B d-A e) (c d-b e)^2}{3 e^6 (d+e x)^{3/2}}-\frac {2 d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{e^6 \sqrt {d+e x}}+\frac {2 \left (A e \left (6 c^2 d^2-6 b c d e+b^2 e^2\right )-B d \left (10 c^2 d^2-12 b c d e+3 b^2 e^2\right )\right ) \sqrt {d+e x}}{e^6}-\frac {2 \left (2 A c e (2 c d-b e)-B \left (10 c^2 d^2-8 b c d e+b^2 e^2\right )\right ) (d+e x)^{3/2}}{3 e^6}-\frac {2 c (5 B c d-2 b B e-A c e) (d+e x)^{5/2}}{5 e^6}+\frac {2 B c^2 (d+e x)^{7/2}}{7 e^6} \]
2/3*d^2*(-A*e+B*d)*(-b*e+c*d)^2/e^6/(e*x+d)^(3/2)-2/3*(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)^(3/2)/e^6-2/5*c*(-A*c*e-2*B*b* e+5*B*c*d)*(e*x+d)^(5/2)/e^6+2/7*B*c^2*(e*x+d)^(7/2)/e^6-2*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)^(1/2)+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*x+d)^(1/2)/e^ 6
Time = 0.20 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.03 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^2}{(d+e x)^{5/2}} \, dx=\frac {2 \left (7 A e \left (5 b^2 e^2 \left (8 d^2+12 d e x+3 e^2 x^2\right )+10 b c e \left (-16 d^3-24 d^2 e x-6 d e^2 x^2+e^3 x^3\right )+c^2 \left (128 d^4+192 d^3 e x+48 d^2 e^2 x^2-8 d e^3 x^3+3 e^4 x^4\right )\right )+B \left (35 b^2 e^2 \left (-16 d^3-24 d^2 e x-6 d e^2 x^2+e^3 x^3\right )+14 b c e \left (128 d^4+192 d^3 e x+48 d^2 e^2 x^2-8 d e^3 x^3+3 e^4 x^4\right )-5 c^2 \left (256 d^5+384 d^4 e x+96 d^3 e^2 x^2-16 d^2 e^3 x^3+6 d e^4 x^4-3 e^5 x^5\right )\right )\right )}{105 e^6 (d+e x)^{3/2}} \]
(2*(7*A*e*(5*b^2*e^2*(8*d^2 + 12*d*e*x + 3*e^2*x^2) + 10*b*c*e*(-16*d^3 - 24*d^2*e*x - 6*d*e^2*x^2 + e^3*x^3) + c^2*(128*d^4 + 192*d^3*e*x + 48*d^2* e^2*x^2 - 8*d*e^3*x^3 + 3*e^4*x^4)) + B*(35*b^2*e^2*(-16*d^3 - 24*d^2*e*x - 6*d*e^2*x^2 + e^3*x^3) + 14*b*c*e*(128*d^4 + 192*d^3*e*x + 48*d^2*e^2*x^ 2 - 8*d*e^3*x^3 + 3*e^4*x^4) - 5*c^2*(256*d^5 + 384*d^4*e*x + 96*d^3*e^2*x ^2 - 16*d^2*e^3*x^3 + 6*d*e^4*x^4 - 3*e^5*x^5))))/(105*e^6*(d + e*x)^(3/2) )
Time = 0.42 (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.
| \(\displaystyle \int \frac {(A+B x) \left (b x+c x^2\right )^2}{(d+e x)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 1195 |
| \(\displaystyle \int \left (\frac {\sqrt {d+e x} \left (B \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )-2 A c e (2 c d-b e)\right )}{e^5}+\frac {A e \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )-B d \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )}{e^5 \sqrt {d+e x}}-\frac {d^2 (B d-A e) (c d-b e)^2}{e^5 (d+e x)^{5/2}}+\frac {c (d+e x)^{3/2} (A c e+2 b B e-5 B c d)}{e^5}+\frac {d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{e^5 (d+e x)^{3/2}}+\frac {B c^2 (d+e x)^{5/2}}{e^5}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 (d+e x)^{3/2} \left (2 A c e (2 c d-b e)-B \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )\right )}{3 e^6}+\frac {2 \sqrt {d+e x} \left (A e \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )-B d \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )\right )}{e^6}+\frac {2 d^2 (B d-A e) (c d-b e)^2}{3 e^6 (d+e x)^{3/2}}-\frac {2 c (d+e x)^{5/2} (-A c e-2 b B e+5 B c d)}{5 e^6}-\frac {2 d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{e^6 \sqrt {d+e x}}+\frac {2 B c^2 (d+e x)^{7/2}}{7 e^6}\) |
(2*d^2*(B*d - A*e)*(c*d - b*e)^2)/(3*e^6*(d + e*x)^(3/2)) - (2*d*(c*d - b* e)*(B*d*(5*c*d - 3*b*e) - 2*A*e*(2*c*d - b*e)))/(e^6*Sqrt[d + e*x]) + (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))*Sqrt[d + e*x])/e^6 - (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))*(d + e*x)^(3/2))/(3*e^6) - (2*c*(5*B*c*d - 2*b*B*e - A*c*e)*(d + e*x)^(5/2))/(5*e^6) + (2*B*c^2*(d + e*x)^(7/2))/(7*e^6)
3.13.27.3.1 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 = 0.37 (sec) , antiderivative size = 253, normalized size of antiderivative = 0.96
| method | result | size |
| pseudoelliptic | \(\frac {\frac {2 \left (5 \left (3 e^{5} x^{5}-6 d \,e^{4} x^{4}+16 d^{2} e^{3} x^{3}-96 d^{3} e^{2} x^{2}-384 d^{4} e x -256 d^{5}\right ) B +896 \left (\frac {3}{128} e^{4} x^{4}-\frac {1}{16} d \,e^{3} x^{3}+\frac {3}{8} d^{2} e^{2} x^{2}+\frac {3}{2} d^{3} e x +d^{4}\right ) A e \right ) c^{2}}{105}-\frac {64 \left (\frac {\left (-\frac {3}{16} e^{4} x^{4}+\frac {1}{2} d \,e^{3} x^{3}-3 d^{2} e^{2} x^{2}-12 d^{3} e x -8 d^{4}\right ) B}{5}+\left (-\frac {1}{8} e^{2} x^{2}+d e x +d^{2}\right ) \left (\frac {e x}{2}+d \right ) A e \right ) e b c}{3}+\frac {16 e^{2} b^{2} \left (\left (\frac {1}{8} e^{3} x^{3}-\frac {3}{4} d \,e^{2} x^{2}-3 d^{2} e x -2 d^{3}\right ) B +A e \left (\frac {3}{8} e^{2} x^{2}+\frac {3}{2} d e x +d^{2}\right )\right )}{3}}{\left (e x +d \right )^{\frac {3}{2}} e^{6}}\) | \(253\) |
| risch | \(\frac {2 \left (15 B \,c^{2} x^{3} e^{3}+21 A \,c^{2} e^{3} x^{2}+42 B b c \,e^{3} x^{2}-60 B \,c^{2} d \,e^{2} x^{2}+70 A b c \,e^{3} x -98 A \,c^{2} d \,e^{2} x +35 B \,b^{2} e^{3} x -196 B b c d \,e^{2} x +185 B \,c^{2} d^{2} e x +105 A \,b^{2} e^{3}-560 A b c d \,e^{2}+511 A \,c^{2} d^{2} e -280 B \,b^{2} d \,e^{2}+1022 B b c \,d^{2} e -790 B \,c^{2} d^{3}\right ) \sqrt {e x +d}}{105 e^{6}}+\frac {2 \left (6 A b \,e^{3} x -12 A c d \,e^{2} x -9 B x b d \,e^{2}+15 B c \,d^{2} e x +5 A b d \,e^{2}-11 A c \,d^{2} e -8 B b \,d^{2} e +14 B c \,d^{3}\right ) d \left (b e -c d \right )}{3 e^{6} \left (e x +d \right )^{\frac {3}{2}}}\) | \(258\) |
| gosper | \(\frac {-\frac {512}{21} B \,c^{2} d^{5}+\frac {2}{5} A \,x^{4} c^{2} e^{5}+\frac {2}{3} B \,x^{3} b^{2} e^{5}+2 A \,x^{2} b^{2} e^{5}+\frac {2}{7} B \,x^{5} c^{2} e^{5}+\frac {256}{5} B x b c \,d^{3} e^{2}-32 A x b c \,d^{2} e^{3}+\frac {64}{5} B \,x^{2} b c \,d^{2} e^{3}-8 A \,x^{2} b c d \,e^{4}-\frac {32}{15} B \,x^{3} b c d \,e^{4}+\frac {16}{3} A \,b^{2} d^{2} e^{3}+\frac {256}{15} A \,c^{2} d^{4} e -\frac {32}{3} B \,b^{2} d^{3} e^{2}-\frac {64}{3} A b c \,d^{3} e^{2}+\frac {512}{15} B b c \,d^{4} e +\frac {4}{3} A \,x^{3} b c \,e^{5}-\frac {16}{15} A \,x^{3} c^{2} d \,e^{4}+\frac {32}{21} B \,x^{3} c^{2} d^{2} e^{3}+\frac {32}{5} A \,x^{2} c^{2} d^{2} e^{3}-4 B \,x^{2} b^{2} d \,e^{4}-\frac {64}{7} B \,x^{2} c^{2} d^{3} e^{2}+8 A x \,b^{2} d \,e^{4}+\frac {128}{5} A x \,c^{2} d^{3} e^{2}-16 B x \,b^{2} d^{2} e^{3}-\frac {256}{7} B x \,c^{2} d^{4} e +\frac {4}{5} B \,x^{4} b c \,e^{5}-\frac {4}{7} B \,x^{4} c^{2} d \,e^{4}}{\left (e x +d \right )^{\frac {3}{2}} e^{6}}\) | \(341\) |
| trager | \(\frac {-\frac {512}{21} B \,c^{2} d^{5}+\frac {2}{5} A \,x^{4} c^{2} e^{5}+\frac {2}{3} B \,x^{3} b^{2} e^{5}+2 A \,x^{2} b^{2} e^{5}+\frac {2}{7} B \,x^{5} c^{2} e^{5}+\frac {256}{5} B x b c \,d^{3} e^{2}-32 A x b c \,d^{2} e^{3}+\frac {64}{5} B \,x^{2} b c \,d^{2} e^{3}-8 A \,x^{2} b c d \,e^{4}-\frac {32}{15} B \,x^{3} b c d \,e^{4}+\frac {16}{3} A \,b^{2} d^{2} e^{3}+\frac {256}{15} A \,c^{2} d^{4} e -\frac {32}{3} B \,b^{2} d^{3} e^{2}-\frac {64}{3} A b c \,d^{3} e^{2}+\frac {512}{15} B b c \,d^{4} e +\frac {4}{3} A \,x^{3} b c \,e^{5}-\frac {16}{15} A \,x^{3} c^{2} d \,e^{4}+\frac {32}{21} B \,x^{3} c^{2} d^{2} e^{3}+\frac {32}{5} A \,x^{2} c^{2} d^{2} e^{3}-4 B \,x^{2} b^{2} d \,e^{4}-\frac {64}{7} B \,x^{2} c^{2} d^{3} e^{2}+8 A x \,b^{2} d \,e^{4}+\frac {128}{5} A x \,c^{2} d^{3} e^{2}-16 B x \,b^{2} d^{2} e^{3}-\frac {256}{7} B x \,c^{2} d^{4} e +\frac {4}{5} B \,x^{4} b c \,e^{5}-\frac {4}{7} B \,x^{4} c^{2} d \,e^{4}}{\left (e x +d \right )^{\frac {3}{2}} e^{6}}\) | \(341\) |
| derivativedivides | \(\frac {\frac {2 B \,c^{2} \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 A \,c^{2} e \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {4 B b c e \left (e x +d \right )^{\frac {5}{2}}}{5}-2 B \,c^{2} d \left (e x +d \right )^{\frac {5}{2}}+\frac {4 A b c \,e^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}-\frac {8 A \,c^{2} d e \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {2 B \,b^{2} e^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}-\frac {16 B b c d e \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {20 B \,c^{2} d^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}+2 A \,b^{2} e^{3} \sqrt {e x +d}-12 A b c d \,e^{2} \sqrt {e x +d}+12 A \,c^{2} d^{2} e \sqrt {e x +d}-6 B \,b^{2} d \,e^{2} \sqrt {e x +d}+24 B b c \,d^{2} e \sqrt {e x +d}-20 B \,c^{2} d^{3} \sqrt {e x +d}-\frac {2 d^{2} \left (A \,b^{2} e^{3}-2 A b c d \,e^{2}+A \,c^{2} d^{2} e -B \,b^{2} d \,e^{2}+2 B b c \,d^{2} e -B \,c^{2} d^{3}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}+\frac {2 d \left (2 A \,b^{2} e^{3}-6 A b c d \,e^{2}+4 A \,c^{2} d^{2} e -3 B \,b^{2} d \,e^{2}+8 B b c \,d^{2} e -5 B \,c^{2} d^{3}\right )}{\sqrt {e x +d}}}{e^{6}}\) | \(367\) |
| default | \(\frac {\frac {2 B \,c^{2} \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 A \,c^{2} e \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {4 B b c e \left (e x +d \right )^{\frac {5}{2}}}{5}-2 B \,c^{2} d \left (e x +d \right )^{\frac {5}{2}}+\frac {4 A b c \,e^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}-\frac {8 A \,c^{2} d e \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {2 B \,b^{2} e^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}-\frac {16 B b c d e \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {20 B \,c^{2} d^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}+2 A \,b^{2} e^{3} \sqrt {e x +d}-12 A b c d \,e^{2} \sqrt {e x +d}+12 A \,c^{2} d^{2} e \sqrt {e x +d}-6 B \,b^{2} d \,e^{2} \sqrt {e x +d}+24 B b c \,d^{2} e \sqrt {e x +d}-20 B \,c^{2} d^{3} \sqrt {e x +d}-\frac {2 d^{2} \left (A \,b^{2} e^{3}-2 A b c d \,e^{2}+A \,c^{2} d^{2} e -B \,b^{2} d \,e^{2}+2 B b c \,d^{2} e -B \,c^{2} d^{3}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}+\frac {2 d \left (2 A \,b^{2} e^{3}-6 A b c d \,e^{2}+4 A \,c^{2} d^{2} e -3 B \,b^{2} d \,e^{2}+8 B b c \,d^{2} e -5 B \,c^{2} d^{3}\right )}{\sqrt {e x +d}}}{e^{6}}\) | \(367\) |
2/105*((5*(3*e^5*x^5-6*d*e^4*x^4+16*d^2*e^3*x^3-96*d^3*e^2*x^2-384*d^4*e*x -256*d^5)*B+896*(3/128*e^4*x^4-1/16*d*e^3*x^3+3/8*d^2*e^2*x^2+3/2*d^3*e*x+ d^4)*A*e)*c^2-1120*(1/5*(-3/16*e^4*x^4+1/2*d*e^3*x^3-3*d^2*e^2*x^2-12*d^3* e*x-8*d^4)*B+(-1/8*e^2*x^2+d*e*x+d^2)*(1/2*e*x+d)*A*e)*e*b*c+280*e^2*b^2*( (1/8*e^3*x^3-3/4*d*e^2*x^2-3*d^2*e*x-2*d^3)*B+A*e*(3/8*e^2*x^2+3/2*d*e*x+d ^2)))/(e*x+d)^(3/2)/e^6
Time = 0.33 (sec) , antiderivative size = 310, normalized size of antiderivative = 1.18 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^2}{(d+e x)^{5/2}} \, dx=\frac {2 \, {\left (15 \, B c^{2} e^{5} x^{5} - 1280 \, B c^{2} d^{5} + 280 \, A b^{2} d^{2} e^{3} + 896 \, {\left (2 \, B b c + A c^{2}\right )} d^{4} e - 560 \, {\left (B b^{2} + 2 \, A b c\right )} d^{3} e^{2} - 3 \, {\left (10 \, B c^{2} d e^{4} - 7 \, {\left (2 \, B b c + A c^{2}\right )} e^{5}\right )} x^{4} + {\left (80 \, B c^{2} d^{2} e^{3} - 56 \, {\left (2 \, B b c + A c^{2}\right )} d e^{4} + 35 \, {\left (B b^{2} + 2 \, A b c\right )} e^{5}\right )} x^{3} - 3 \, {\left (160 \, B c^{2} d^{3} e^{2} - 35 \, A b^{2} e^{5} - 112 \, {\left (2 \, B b c + A c^{2}\right )} d^{2} e^{3} + 70 \, {\left (B b^{2} + 2 \, A b c\right )} d e^{4}\right )} x^{2} - 12 \, {\left (160 \, B c^{2} d^{4} e - 35 \, A b^{2} d e^{4} - 112 \, {\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} + 70 \, {\left (B b^{2} + 2 \, A b c\right )} d^{2} e^{3}\right )} x\right )} \sqrt {e x + d}}{105 \, {\left (e^{8} x^{2} + 2 \, d e^{7} x + d^{2} e^{6}\right )}} \]
2/105*(15*B*c^2*e^5*x^5 - 1280*B*c^2*d^5 + 280*A*b^2*d^2*e^3 + 896*(2*B*b* c + A*c^2)*d^4*e - 560*(B*b^2 + 2*A*b*c)*d^3*e^2 - 3*(10*B*c^2*d*e^4 - 7*( 2*B*b*c + A*c^2)*e^5)*x^4 + (80*B*c^2*d^2*e^3 - 56*(2*B*b*c + A*c^2)*d*e^4 + 35*(B*b^2 + 2*A*b*c)*e^5)*x^3 - 3*(160*B*c^2*d^3*e^2 - 35*A*b^2*e^5 - 1 12*(2*B*b*c + A*c^2)*d^2*e^3 + 70*(B*b^2 + 2*A*b*c)*d*e^4)*x^2 - 12*(160*B *c^2*d^4*e - 35*A*b^2*d*e^4 - 112*(2*B*b*c + A*c^2)*d^3*e^2 + 70*(B*b^2 + 2*A*b*c)*d^2*e^3)*x)*sqrt(e*x + d)/(e^8*x^2 + 2*d*e^7*x + d^2*e^6)
Time = 8.83 (sec) , antiderivative size = 354, normalized size of antiderivative = 1.35 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^2}{(d+e x)^{5/2}} \, dx=\begin {cases} \frac {2 \left (\frac {B c^{2} \left (d + e x\right )^{\frac {7}{2}}}{7 e^{5}} + \frac {d^{2} \left (- A e + B d\right ) \left (b e - c d\right )^{2}}{3 e^{5} \left (d + e x\right )^{\frac {3}{2}}} - \frac {d \left (b e - c d\right ) \left (- 2 A b e^{2} + 4 A c d e + 3 B b d e - 5 B c d^{2}\right )}{e^{5} \sqrt {d + e x}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \left (A c^{2} e + 2 B b c e - 5 B c^{2} d\right )}{5 e^{5}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \cdot \left (2 A b c e^{2} - 4 A c^{2} d e + B b^{2} e^{2} - 8 B b c d e + 10 B c^{2} d^{2}\right )}{3 e^{5}} + \frac {\sqrt {d + e x} \left (A b^{2} e^{3} - 6 A b c d e^{2} + 6 A c^{2} d^{2} e - 3 B b^{2} d e^{2} + 12 B b c d^{2} e - 10 B c^{2} d^{3}\right )}{e^{5}}\right )}{e} & \text {for}\: e \neq 0 \\\frac {\frac {A b^{2} x^{3}}{3} + \frac {B c^{2} x^{6}}{6} + \frac {x^{5} \left (A c^{2} + 2 B b c\right )}{5} + \frac {x^{4} \cdot \left (2 A b c + B b^{2}\right )}{4}}{d^{\frac {5}{2}}} & \text {otherwise} \end {cases} \]
Piecewise((2*(B*c**2*(d + e*x)**(7/2)/(7*e**5) + d**2*(-A*e + B*d)*(b*e - c*d)**2/(3*e**5*(d + e*x)**(3/2)) - d*(b*e - c*d)*(-2*A*b*e**2 + 4*A*c*d*e + 3*B*b*d*e - 5*B*c*d**2)/(e**5*sqrt(d + e*x)) + (d + e*x)**(5/2)*(A*c**2 *e + 2*B*b*c*e - 5*B*c**2*d)/(5*e**5) + (d + e*x)**(3/2)*(2*A*b*c*e**2 - 4 *A*c**2*d*e + B*b**2*e**2 - 8*B*b*c*d*e + 10*B*c**2*d**2)/(3*e**5) + sqrt( d + e*x)*(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)/e**5)/e, Ne(e, 0)), ((A*b**2*x**3/3 + B*c**2*x**6/6 + x**5*(A*c**2 + 2*B*b*c)/5 + x**4*(2*A*b*c + B*b**2)/4)/d* *(5/2), True))
Time = 0.21 (sec) , antiderivative size = 297, normalized size of antiderivative = 1.13 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^2}{(d+e x)^{5/2}} \, dx=\frac {2 \, {\left (\frac {15 \, {\left (e x + d\right )}^{\frac {7}{2}} B c^{2} - 21 \, {\left (5 \, B c^{2} d - {\left (2 \, B b c + A c^{2}\right )} e\right )} {\left (e x + d\right )}^{\frac {5}{2}} + 35 \, {\left (10 \, B c^{2} d^{2} - 4 \, {\left (2 \, B b c + A c^{2}\right )} d e + {\left (B b^{2} + 2 \, A b c\right )} e^{2}\right )} {\left (e x + d\right )}^{\frac {3}{2}} - 105 \, {\left (10 \, B c^{2} d^{3} - A b^{2} e^{3} - 6 \, {\left (2 \, B b c + A c^{2}\right )} d^{2} e + 3 \, {\left (B b^{2} + 2 \, A b c\right )} d e^{2}\right )} \sqrt {e x + d}}{e^{5}} + \frac {35 \, {\left (B c^{2} d^{5} - A b^{2} d^{2} e^{3} - {\left (2 \, B b c + A c^{2}\right )} d^{4} e + {\left (B b^{2} + 2 \, A b c\right )} d^{3} e^{2} - 3 \, {\left (5 \, B c^{2} d^{4} - 2 \, A b^{2} d e^{3} - 4 \, {\left (2 \, B b c + A c^{2}\right )} d^{3} e + 3 \, {\left (B b^{2} + 2 \, A b c\right )} d^{2} e^{2}\right )} {\left (e x + d\right )}\right )}}{{\left (e x + d\right )}^{\frac {3}{2}} e^{5}}\right )}}{105 \, e} \]
2/105*((15*(e*x + d)^(7/2)*B*c^2 - 21*(5*B*c^2*d - (2*B*b*c + A*c^2)*e)*(e *x + d)^(5/2) + 35*(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)*(e*x + d)^(3/2) - 105*(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)*sqrt(e*x + d))/e^5 + 35*(B*c^2*d^ 5 - A*b^2*d^2*e^3 - (2*B*b*c + A*c^2)*d^4*e + (B*b^2 + 2*A*b*c)*d^3*e^2 - 3*(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)^(3/2)*e^5))/e
Time = 0.28 (sec) , antiderivative size = 424, normalized size of antiderivative = 1.61 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^2}{(d+e x)^{5/2}} \, dx=-\frac {2 \, {\left (15 \, {\left (e x + d\right )} B c^{2} d^{4} - B c^{2} d^{5} - 24 \, {\left (e x + d\right )} B b c d^{3} e - 12 \, {\left (e x + d\right )} A c^{2} d^{3} e + 2 \, B b c d^{4} e + A c^{2} d^{4} e + 9 \, {\left (e x + d\right )} B b^{2} d^{2} e^{2} + 18 \, {\left (e x + d\right )} A b c d^{2} e^{2} - B b^{2} d^{3} e^{2} - 2 \, A b c d^{3} e^{2} - 6 \, {\left (e x + d\right )} A b^{2} d e^{3} + A b^{2} d^{2} e^{3}\right )}}{3 \, {\left (e x + d\right )}^{\frac {3}{2}} e^{6}} + \frac {2 \, {\left (15 \, {\left (e x + d\right )}^{\frac {7}{2}} B c^{2} e^{36} - 105 \, {\left (e x + d\right )}^{\frac {5}{2}} B c^{2} d e^{36} + 350 \, {\left (e x + d\right )}^{\frac {3}{2}} B c^{2} d^{2} e^{36} - 1050 \, \sqrt {e x + d} B c^{2} d^{3} e^{36} + 42 \, {\left (e x + d\right )}^{\frac {5}{2}} B b c e^{37} + 21 \, {\left (e x + d\right )}^{\frac {5}{2}} A c^{2} e^{37} - 280 \, {\left (e x + d\right )}^{\frac {3}{2}} B b c d e^{37} - 140 \, {\left (e x + d\right )}^{\frac {3}{2}} A c^{2} d e^{37} + 1260 \, \sqrt {e x + d} B b c d^{2} e^{37} + 630 \, \sqrt {e x + d} A c^{2} d^{2} e^{37} + 35 \, {\left (e x + d\right )}^{\frac {3}{2}} B b^{2} e^{38} + 70 \, {\left (e x + d\right )}^{\frac {3}{2}} A b c e^{38} - 315 \, \sqrt {e x + d} B b^{2} d e^{38} - 630 \, \sqrt {e x + d} A b c d e^{38} + 105 \, \sqrt {e x + d} A b^{2} e^{39}\right )}}{105 \, e^{42}} \]
-2/3*(15*(e*x + d)*B*c^2*d^4 - B*c^2*d^5 - 24*(e*x + d)*B*b*c*d^3*e - 12*( e*x + d)*A*c^2*d^3*e + 2*B*b*c*d^4*e + A*c^2*d^4*e + 9*(e*x + d)*B*b^2*d^2 *e^2 + 18*(e*x + d)*A*b*c*d^2*e^2 - B*b^2*d^3*e^2 - 2*A*b*c*d^3*e^2 - 6*(e *x + d)*A*b^2*d*e^3 + A*b^2*d^2*e^3)/((e*x + d)^(3/2)*e^6) + 2/105*(15*(e* x + d)^(7/2)*B*c^2*e^36 - 105*(e*x + d)^(5/2)*B*c^2*d*e^36 + 350*(e*x + d) ^(3/2)*B*c^2*d^2*e^36 - 1050*sqrt(e*x + d)*B*c^2*d^3*e^36 + 42*(e*x + d)^( 5/2)*B*b*c*e^37 + 21*(e*x + d)^(5/2)*A*c^2*e^37 - 280*(e*x + d)^(3/2)*B*b* c*d*e^37 - 140*(e*x + d)^(3/2)*A*c^2*d*e^37 + 1260*sqrt(e*x + d)*B*b*c*d^2 *e^37 + 630*sqrt(e*x + d)*A*c^2*d^2*e^37 + 35*(e*x + d)^(3/2)*B*b^2*e^38 + 70*(e*x + d)^(3/2)*A*b*c*e^38 - 315*sqrt(e*x + d)*B*b^2*d*e^38 - 630*sqrt (e*x + d)*A*b*c*d*e^38 + 105*sqrt(e*x + d)*A*b^2*e^39)/e^42
Time = 0.08 (sec) , antiderivative size = 316, normalized size of antiderivative = 1.20 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^2}{(d+e x)^{5/2}} \, dx=\frac {{\left (d+e\,x\right )}^{5/2}\,\left (2\,A\,c^2\,e-10\,B\,c^2\,d+4\,B\,b\,c\,e\right )}{5\,e^6}-\frac {\left (d+e\,x\right )\,\left (6\,B\,b^2\,d^2\,e^2-4\,A\,b^2\,d\,e^3-16\,B\,b\,c\,d^3\,e+12\,A\,b\,c\,d^2\,e^2+10\,B\,c^2\,d^4-8\,A\,c^2\,d^3\,e\right )-\frac {2\,B\,c^2\,d^5}{3}+\frac {2\,A\,c^2\,d^4\,e}{3}+\frac {2\,A\,b^2\,d^2\,e^3}{3}-\frac {2\,B\,b^2\,d^3\,e^2}{3}+\frac {4\,B\,b\,c\,d^4\,e}{3}-\frac {4\,A\,b\,c\,d^3\,e^2}{3}}{e^6\,{\left (d+e\,x\right )}^{3/2}}+\frac {\sqrt {d+e\,x}\,\left (-6\,B\,b^2\,d\,e^2+2\,A\,b^2\,e^3+24\,B\,b\,c\,d^2\,e-12\,A\,b\,c\,d\,e^2-20\,B\,c^2\,d^3+12\,A\,c^2\,d^2\,e\right )}{e^6}+\frac {{\left (d+e\,x\right )}^{3/2}\,\left (2\,B\,b^2\,e^2-16\,B\,b\,c\,d\,e+4\,A\,b\,c\,e^2+20\,B\,c^2\,d^2-8\,A\,c^2\,d\,e\right )}{3\,e^6}+\frac {2\,B\,c^2\,{\left (d+e\,x\right )}^{7/2}}{7\,e^6} \]
((d + e*x)^(5/2)*(2*A*c^2*e - 10*B*c^2*d + 4*B*b*c*e))/(5*e^6) - ((d + e*x )*(10*B*c^2*d^4 - 4*A*b^2*d*e^3 - 8*A*c^2*d^3*e + 6*B*b^2*d^2*e^2 - 16*B*b *c*d^3*e + 12*A*b*c*d^2*e^2) - (2*B*c^2*d^5)/3 + (2*A*c^2*d^4*e)/3 + (2*A* b^2*d^2*e^3)/3 - (2*B*b^2*d^3*e^2)/3 + (4*B*b*c*d^4*e)/3 - (4*A*b*c*d^3*e^ 2)/3)/(e^6*(d + e*x)^(3/2)) + ((d + e*x)^(1/2)*(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))/e^6 + ((d + e*x)^(3/2)*(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))/(3*e^6) + (2*B*c^2*(d + e*x)^(7/2))/(7*e^6)