Integrand size = 33, antiderivative size = 158 \[ \int (a+b x) (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=-\frac {2 (b d-a e)^5 (d+e x)^{5/2}}{5 e^6}+\frac {10 b (b d-a e)^4 (d+e x)^{7/2}}{7 e^6}-\frac {20 b^2 (b d-a e)^3 (d+e x)^{9/2}}{9 e^6}+\frac {20 b^3 (b d-a e)^2 (d+e x)^{11/2}}{11 e^6}-\frac {10 b^4 (b d-a e) (d+e x)^{13/2}}{13 e^6}+\frac {2 b^5 (d+e x)^{15/2}}{15 e^6} \]
-2/5*(-a*e+b*d)^5*(e*x+d)^(5/2)/e^6+10/7*b*(-a*e+b*d)^4*(e*x+d)^(7/2)/e^6- 20/9*b^2*(-a*e+b*d)^3*(e*x+d)^(9/2)/e^6+20/11*b^3*(-a*e+b*d)^2*(e*x+d)^(11 /2)/e^6-10/13*b^4*(-a*e+b*d)*(e*x+d)^(13/2)/e^6+2/15*b^5*(e*x+d)^(15/2)/e^ 6
Time = 0.04 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.37 \[ \int (a+b x) (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {2 (d+e x)^{5/2} \left (9009 a^5 e^5+6435 a^4 b e^4 (-2 d+5 e x)+1430 a^3 b^2 e^3 \left (8 d^2-20 d e x+35 e^2 x^2\right )+390 a^2 b^3 e^2 \left (-16 d^3+40 d^2 e x-70 d e^2 x^2+105 e^3 x^3\right )+15 a b^4 e \left (128 d^4-320 d^3 e x+560 d^2 e^2 x^2-840 d e^3 x^3+1155 e^4 x^4\right )+b^5 \left (-256 d^5+640 d^4 e x-1120 d^3 e^2 x^2+1680 d^2 e^3 x^3-2310 d e^4 x^4+3003 e^5 x^5\right )\right )}{45045 e^6} \]
(2*(d + e*x)^(5/2)*(9009*a^5*e^5 + 6435*a^4*b*e^4*(-2*d + 5*e*x) + 1430*a^ 3*b^2*e^3*(8*d^2 - 20*d*e*x + 35*e^2*x^2) + 390*a^2*b^3*e^2*(-16*d^3 + 40* d^2*e*x - 70*d*e^2*x^2 + 105*e^3*x^3) + 15*a*b^4*e*(128*d^4 - 320*d^3*e*x + 560*d^2*e^2*x^2 - 840*d*e^3*x^3 + 1155*e^4*x^4) + b^5*(-256*d^5 + 640*d^ 4*e*x - 1120*d^3*e^2*x^2 + 1680*d^2*e^3*x^3 - 2310*d*e^4*x^4 + 3003*e^5*x^ 5)))/(45045*e^6)
Time = 0.30 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {1184, 27, 53, 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.
| \(\displaystyle \int (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^2 (d+e x)^{3/2} \, dx\) |
| \(\Big \downarrow \) 1184 |
| \(\displaystyle \frac {\int b^4 (a+b x)^5 (d+e x)^{3/2}dx}{b^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int (a+b x)^5 (d+e x)^{3/2}dx\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle \int \left (-\frac {5 b^4 (d+e x)^{11/2} (b d-a e)}{e^5}+\frac {10 b^3 (d+e x)^{9/2} (b d-a e)^2}{e^5}-\frac {10 b^2 (d+e x)^{7/2} (b d-a e)^3}{e^5}+\frac {5 b (d+e x)^{5/2} (b d-a e)^4}{e^5}+\frac {(d+e x)^{3/2} (a e-b d)^5}{e^5}+\frac {b^5 (d+e x)^{13/2}}{e^5}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {10 b^4 (d+e x)^{13/2} (b d-a e)}{13 e^6}+\frac {20 b^3 (d+e x)^{11/2} (b d-a e)^2}{11 e^6}-\frac {20 b^2 (d+e x)^{9/2} (b d-a e)^3}{9 e^6}+\frac {10 b (d+e x)^{7/2} (b d-a e)^4}{7 e^6}-\frac {2 (d+e x)^{5/2} (b d-a e)^5}{5 e^6}+\frac {2 b^5 (d+e x)^{15/2}}{15 e^6}\) |
(-2*(b*d - a*e)^5*(d + e*x)^(5/2))/(5*e^6) + (10*b*(b*d - a*e)^4*(d + e*x) ^(7/2))/(7*e^6) - (20*b^2*(b*d - a*e)^3*(d + e*x)^(9/2))/(9*e^6) + (20*b^3 *(b*d - a*e)^2*(d + e*x)^(11/2))/(11*e^6) - (10*b^4*(b*d - a*e)*(d + e*x)^ (13/2))/(13*e^6) + (2*b^5*(d + e*x)^(15/2))/(15*e^6)
3.21.52.3.1 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_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (b_.)*(x_ ) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[1/c^p Int[(d + e*x)^m*(f + g*x )^n*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x] && E qQ[b^2 - 4*a*c, 0] && IntegerQ[p]
Time = 0.32 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.29
| method | result | size |
| pseudoelliptic | \(\frac {2 \left (\left (\frac {1}{3} b^{5} x^{5}+\frac {25}{13} a \,b^{4} x^{4}+\frac {50}{11} a^{2} b^{3} x^{3}+\frac {50}{9} a^{3} b^{2} x^{2}+\frac {25}{7} a^{4} b x +a^{5}\right ) e^{5}-\frac {10 b \left (\frac {7}{39} x^{4} b^{4}+\frac {140}{143} a \,b^{3} x^{3}+\frac {70}{33} x^{2} b^{2} a^{2}+\frac {20}{9} b \,a^{3} x +a^{4}\right ) d \,e^{4}}{7}+\frac {80 b^{2} \left (\frac {21}{143} x^{3} b^{3}+\frac {105}{143} a \,b^{2} x^{2}+\frac {15}{11} b \,a^{2} x +a^{3}\right ) d^{2} e^{3}}{63}-\frac {160 b^{3} \left (\frac {7}{39} b^{2} x^{2}+\frac {10}{13} a b x +a^{2}\right ) d^{3} e^{2}}{231}+\frac {640 b^{4} \left (\frac {b x}{3}+a \right ) d^{4} e}{3003}-\frac {256 b^{5} d^{5}}{9009}\right ) \left (e x +d \right )^{\frac {5}{2}}}{5 e^{6}}\) | \(204\) |
| gosper | \(\frac {2 \left (e x +d \right )^{\frac {5}{2}} \left (3003 x^{5} b^{5} e^{5}+17325 x^{4} a \,b^{4} e^{5}-2310 x^{4} b^{5} d \,e^{4}+40950 x^{3} a^{2} b^{3} e^{5}-12600 x^{3} a \,b^{4} d \,e^{4}+1680 x^{3} b^{5} d^{2} e^{3}+50050 x^{2} a^{3} b^{2} e^{5}-27300 x^{2} a^{2} b^{3} d \,e^{4}+8400 x^{2} a \,b^{4} d^{2} e^{3}-1120 x^{2} b^{5} d^{3} e^{2}+32175 x \,a^{4} b \,e^{5}-28600 x \,a^{3} b^{2} d \,e^{4}+15600 x \,a^{2} b^{3} d^{2} e^{3}-4800 x a \,b^{4} d^{3} e^{2}+640 x \,b^{5} d^{4} e +9009 e^{5} a^{5}-12870 b d \,e^{4} a^{4}+11440 b^{2} d^{2} e^{3} a^{3}-6240 b^{3} d^{3} e^{2} a^{2}+1920 b^{4} d^{4} e a -256 b^{5} d^{5}\right )}{45045 e^{6}}\) | \(273\) |
| derivativedivides | \(\frac {\frac {2 b^{5} \left (e x +d \right )^{\frac {15}{2}}}{15}+\frac {2 \left (\left (a e -b d \right ) b^{4}+2 b^{3} \left (2 a b e -2 b^{2} d \right )\right ) \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (2 \left (a e -b d \right ) \left (2 a b e -2 b^{2} d \right ) b^{2}+b \left (2 \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a b e -2 b^{2} d \right )^{2}\right )\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (\left (a e -b d \right ) \left (2 \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a b e -2 b^{2} d \right )^{2}\right )+2 b \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) \left (2 a b e -2 b^{2} d \right )\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (2 \left (a e -b d \right ) \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) \left (2 a b e -2 b^{2} d \right )+b \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )^{2}\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (a e -b d \right ) \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}}{e^{6}}\) | \(350\) |
| default | \(\frac {\frac {2 b^{5} \left (e x +d \right )^{\frac {15}{2}}}{15}+\frac {2 \left (\left (a e -b d \right ) b^{4}+2 b^{3} \left (2 a b e -2 b^{2} d \right )\right ) \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (2 \left (a e -b d \right ) \left (2 a b e -2 b^{2} d \right ) b^{2}+b \left (2 \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a b e -2 b^{2} d \right )^{2}\right )\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (\left (a e -b d \right ) \left (2 \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a b e -2 b^{2} d \right )^{2}\right )+2 b \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) \left (2 a b e -2 b^{2} d \right )\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (2 \left (a e -b d \right ) \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) \left (2 a b e -2 b^{2} d \right )+b \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )^{2}\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (a e -b d \right ) \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}}{e^{6}}\) | \(350\) |
| trager | \(\frac {2 \left (3003 b^{5} e^{7} x^{7}+17325 a \,b^{4} e^{7} x^{6}+3696 b^{5} d \,e^{6} x^{6}+40950 a^{2} b^{3} e^{7} x^{5}+22050 a \,b^{4} d \,e^{6} x^{5}+63 b^{5} d^{2} e^{5} x^{5}+50050 a^{3} b^{2} e^{7} x^{4}+54600 a^{2} b^{3} d \,e^{6} x^{4}+525 a \,b^{4} d^{2} e^{5} x^{4}-70 b^{5} d^{3} e^{4} x^{4}+32175 a^{4} b \,e^{7} x^{3}+71500 a^{3} b^{2} d \,e^{6} x^{3}+1950 a^{2} b^{3} d^{2} e^{5} x^{3}-600 a \,b^{4} d^{3} e^{4} x^{3}+80 b^{5} d^{4} e^{3} x^{3}+9009 a^{5} e^{7} x^{2}+51480 a^{4} b d \,e^{6} x^{2}+4290 a^{3} b^{2} d^{2} e^{5} x^{2}-2340 a^{2} b^{3} d^{3} e^{4} x^{2}+720 a \,b^{4} d^{4} e^{3} x^{2}-96 b^{5} d^{5} e^{2} x^{2}+18018 a^{5} d \,e^{6} x +6435 a^{4} b \,d^{2} e^{5} x -5720 a^{3} b^{2} d^{3} e^{4} x +3120 a^{2} b^{3} d^{4} e^{3} x -960 a \,b^{4} d^{5} e^{2} x +128 b^{5} d^{6} e x +9009 a^{5} d^{2} e^{5}-12870 a^{4} b \,d^{3} e^{4}+11440 a^{3} b^{2} d^{4} e^{3}-6240 a^{2} b^{3} d^{5} e^{2}+1920 a \,b^{4} d^{6} e -256 b^{5} d^{7}\right ) \sqrt {e x +d}}{45045 e^{6}}\) | \(453\) |
| risch | \(\frac {2 \left (3003 b^{5} e^{7} x^{7}+17325 a \,b^{4} e^{7} x^{6}+3696 b^{5} d \,e^{6} x^{6}+40950 a^{2} b^{3} e^{7} x^{5}+22050 a \,b^{4} d \,e^{6} x^{5}+63 b^{5} d^{2} e^{5} x^{5}+50050 a^{3} b^{2} e^{7} x^{4}+54600 a^{2} b^{3} d \,e^{6} x^{4}+525 a \,b^{4} d^{2} e^{5} x^{4}-70 b^{5} d^{3} e^{4} x^{4}+32175 a^{4} b \,e^{7} x^{3}+71500 a^{3} b^{2} d \,e^{6} x^{3}+1950 a^{2} b^{3} d^{2} e^{5} x^{3}-600 a \,b^{4} d^{3} e^{4} x^{3}+80 b^{5} d^{4} e^{3} x^{3}+9009 a^{5} e^{7} x^{2}+51480 a^{4} b d \,e^{6} x^{2}+4290 a^{3} b^{2} d^{2} e^{5} x^{2}-2340 a^{2} b^{3} d^{3} e^{4} x^{2}+720 a \,b^{4} d^{4} e^{3} x^{2}-96 b^{5} d^{5} e^{2} x^{2}+18018 a^{5} d \,e^{6} x +6435 a^{4} b \,d^{2} e^{5} x -5720 a^{3} b^{2} d^{3} e^{4} x +3120 a^{2} b^{3} d^{4} e^{3} x -960 a \,b^{4} d^{5} e^{2} x +128 b^{5} d^{6} e x +9009 a^{5} d^{2} e^{5}-12870 a^{4} b \,d^{3} e^{4}+11440 a^{3} b^{2} d^{4} e^{3}-6240 a^{2} b^{3} d^{5} e^{2}+1920 a \,b^{4} d^{6} e -256 b^{5} d^{7}\right ) \sqrt {e x +d}}{45045 e^{6}}\) | \(453\) |
2/5*((1/3*b^5*x^5+25/13*a*b^4*x^4+50/11*a^2*b^3*x^3+50/9*a^3*b^2*x^2+25/7* a^4*b*x+a^5)*e^5-10/7*b*(7/39*x^4*b^4+140/143*a*b^3*x^3+70/33*x^2*b^2*a^2+ 20/9*b*a^3*x+a^4)*d*e^4+80/63*b^2*(21/143*x^3*b^3+105/143*a*b^2*x^2+15/11* b*a^2*x+a^3)*d^2*e^3-160/231*b^3*(7/39*b^2*x^2+10/13*a*b*x+a^2)*d^3*e^2+64 0/3003*b^4*(1/3*b*x+a)*d^4*e-256/9009*b^5*d^5)*(e*x+d)^(5/2)/e^6
Leaf count of result is larger than twice the leaf count of optimal. 418 vs. \(2 (134) = 268\).
Time = 0.44 (sec) , antiderivative size = 418, normalized size of antiderivative = 2.65 \[ \int (a+b x) (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {2 \, {\left (3003 \, b^{5} e^{7} x^{7} - 256 \, b^{5} d^{7} + 1920 \, a b^{4} d^{6} e - 6240 \, a^{2} b^{3} d^{5} e^{2} + 11440 \, a^{3} b^{2} d^{4} e^{3} - 12870 \, a^{4} b d^{3} e^{4} + 9009 \, a^{5} d^{2} e^{5} + 231 \, {\left (16 \, b^{5} d e^{6} + 75 \, a b^{4} e^{7}\right )} x^{6} + 63 \, {\left (b^{5} d^{2} e^{5} + 350 \, a b^{4} d e^{6} + 650 \, a^{2} b^{3} e^{7}\right )} x^{5} - 35 \, {\left (2 \, b^{5} d^{3} e^{4} - 15 \, a b^{4} d^{2} e^{5} - 1560 \, a^{2} b^{3} d e^{6} - 1430 \, a^{3} b^{2} e^{7}\right )} x^{4} + 5 \, {\left (16 \, b^{5} d^{4} e^{3} - 120 \, a b^{4} d^{3} e^{4} + 390 \, a^{2} b^{3} d^{2} e^{5} + 14300 \, a^{3} b^{2} d e^{6} + 6435 \, a^{4} b e^{7}\right )} x^{3} - 3 \, {\left (32 \, b^{5} d^{5} e^{2} - 240 \, a b^{4} d^{4} e^{3} + 780 \, a^{2} b^{3} d^{3} e^{4} - 1430 \, a^{3} b^{2} d^{2} e^{5} - 17160 \, a^{4} b d e^{6} - 3003 \, a^{5} e^{7}\right )} x^{2} + {\left (128 \, b^{5} d^{6} e - 960 \, a b^{4} d^{5} e^{2} + 3120 \, a^{2} b^{3} d^{4} e^{3} - 5720 \, a^{3} b^{2} d^{3} e^{4} + 6435 \, a^{4} b d^{2} e^{5} + 18018 \, a^{5} d e^{6}\right )} x\right )} \sqrt {e x + d}}{45045 \, e^{6}} \]
2/45045*(3003*b^5*e^7*x^7 - 256*b^5*d^7 + 1920*a*b^4*d^6*e - 6240*a^2*b^3* d^5*e^2 + 11440*a^3*b^2*d^4*e^3 - 12870*a^4*b*d^3*e^4 + 9009*a^5*d^2*e^5 + 231*(16*b^5*d*e^6 + 75*a*b^4*e^7)*x^6 + 63*(b^5*d^2*e^5 + 350*a*b^4*d*e^6 + 650*a^2*b^3*e^7)*x^5 - 35*(2*b^5*d^3*e^4 - 15*a*b^4*d^2*e^5 - 1560*a^2* b^3*d*e^6 - 1430*a^3*b^2*e^7)*x^4 + 5*(16*b^5*d^4*e^3 - 120*a*b^4*d^3*e^4 + 390*a^2*b^3*d^2*e^5 + 14300*a^3*b^2*d*e^6 + 6435*a^4*b*e^7)*x^3 - 3*(32* b^5*d^5*e^2 - 240*a*b^4*d^4*e^3 + 780*a^2*b^3*d^3*e^4 - 1430*a^3*b^2*d^2*e ^5 - 17160*a^4*b*d*e^6 - 3003*a^5*e^7)*x^2 + (128*b^5*d^6*e - 960*a*b^4*d^ 5*e^2 + 3120*a^2*b^3*d^4*e^3 - 5720*a^3*b^2*d^3*e^4 + 6435*a^4*b*d^2*e^5 + 18018*a^5*d*e^6)*x)*sqrt(e*x + d)/e^6
Leaf count of result is larger than twice the leaf count of optimal. 347 vs. \(2 (146) = 292\).
Time = 1.63 (sec) , antiderivative size = 347, normalized size of antiderivative = 2.20 \[ \int (a+b x) (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\begin {cases} \frac {2 \left (\frac {b^{5} \left (d + e x\right )^{\frac {15}{2}}}{15 e^{5}} + \frac {\left (d + e x\right )^{\frac {13}{2}} \cdot \left (5 a b^{4} e - 5 b^{5} d\right )}{13 e^{5}} + \frac {\left (d + e x\right )^{\frac {11}{2}} \cdot \left (10 a^{2} b^{3} e^{2} - 20 a b^{4} d e + 10 b^{5} d^{2}\right )}{11 e^{5}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \cdot \left (10 a^{3} b^{2} e^{3} - 30 a^{2} b^{3} d e^{2} + 30 a b^{4} d^{2} e - 10 b^{5} d^{3}\right )}{9 e^{5}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \cdot \left (5 a^{4} b e^{4} - 20 a^{3} b^{2} d e^{3} + 30 a^{2} b^{3} d^{2} e^{2} - 20 a b^{4} d^{3} e + 5 b^{5} d^{4}\right )}{7 e^{5}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \left (a^{5} e^{5} - 5 a^{4} b d e^{4} + 10 a^{3} b^{2} d^{2} e^{3} - 10 a^{2} b^{3} d^{3} e^{2} + 5 a b^{4} d^{4} e - b^{5} d^{5}\right )}{5 e^{5}}\right )}{e} & \text {for}\: e \neq 0 \\d^{\frac {3}{2}} \left (\begin {cases} a^{5} x & \text {for}\: b = 0 \\\frac {\left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{3}}{6 b} & \text {otherwise} \end {cases}\right ) & \text {otherwise} \end {cases} \]
Piecewise((2*(b**5*(d + e*x)**(15/2)/(15*e**5) + (d + e*x)**(13/2)*(5*a*b* *4*e - 5*b**5*d)/(13*e**5) + (d + e*x)**(11/2)*(10*a**2*b**3*e**2 - 20*a*b **4*d*e + 10*b**5*d**2)/(11*e**5) + (d + e*x)**(9/2)*(10*a**3*b**2*e**3 - 30*a**2*b**3*d*e**2 + 30*a*b**4*d**2*e - 10*b**5*d**3)/(9*e**5) + (d + e*x )**(7/2)*(5*a**4*b*e**4 - 20*a**3*b**2*d*e**3 + 30*a**2*b**3*d**2*e**2 - 2 0*a*b**4*d**3*e + 5*b**5*d**4)/(7*e**5) + (d + e*x)**(5/2)*(a**5*e**5 - 5* a**4*b*d*e**4 + 10*a**3*b**2*d**2*e**3 - 10*a**2*b**3*d**3*e**2 + 5*a*b**4 *d**4*e - b**5*d**5)/(5*e**5))/e, Ne(e, 0)), (d**(3/2)*Piecewise((a**5*x, Eq(b, 0)), ((a**2 + 2*a*b*x + b**2*x**2)**3/(6*b), True)), True))
Time = 0.19 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.64 \[ \int (a+b x) (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {2 \, {\left (3003 \, {\left (e x + d\right )}^{\frac {15}{2}} b^{5} - 17325 \, {\left (b^{5} d - a b^{4} e\right )} {\left (e x + d\right )}^{\frac {13}{2}} + 40950 \, {\left (b^{5} d^{2} - 2 \, a b^{4} d e + a^{2} b^{3} e^{2}\right )} {\left (e x + d\right )}^{\frac {11}{2}} - 50050 \, {\left (b^{5} d^{3} - 3 \, a b^{4} d^{2} e + 3 \, a^{2} b^{3} d e^{2} - a^{3} b^{2} e^{3}\right )} {\left (e x + d\right )}^{\frac {9}{2}} + 32175 \, {\left (b^{5} d^{4} - 4 \, a b^{4} d^{3} e + 6 \, a^{2} b^{3} d^{2} e^{2} - 4 \, a^{3} b^{2} d e^{3} + a^{4} b e^{4}\right )} {\left (e x + d\right )}^{\frac {7}{2}} - 9009 \, {\left (b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}\right )} {\left (e x + d\right )}^{\frac {5}{2}}\right )}}{45045 \, e^{6}} \]
2/45045*(3003*(e*x + d)^(15/2)*b^5 - 17325*(b^5*d - a*b^4*e)*(e*x + d)^(13 /2) + 40950*(b^5*d^2 - 2*a*b^4*d*e + a^2*b^3*e^2)*(e*x + d)^(11/2) - 50050 *(b^5*d^3 - 3*a*b^4*d^2*e + 3*a^2*b^3*d*e^2 - a^3*b^2*e^3)*(e*x + d)^(9/2) + 32175*(b^5*d^4 - 4*a*b^4*d^3*e + 6*a^2*b^3*d^2*e^2 - 4*a^3*b^2*d*e^3 + a^4*b*e^4)*(e*x + d)^(7/2) - 9009*(b^5*d^5 - 5*a*b^4*d^4*e + 10*a^2*b^3*d^ 3*e^2 - 10*a^3*b^2*d^2*e^3 + 5*a^4*b*d*e^4 - a^5*e^5)*(e*x + d)^(5/2))/e^6
Leaf count of result is larger than twice the leaf count of optimal. 1084 vs. \(2 (134) = 268\).
Time = 0.29 (sec) , antiderivative size = 1084, normalized size of antiderivative = 6.86 \[ \int (a+b x) (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\text {Too large to display} \]
2/45045*(45045*sqrt(e*x + d)*a^5*d^2 + 30030*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*a^5*d + 75075*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*a^4*b*d^2/e + 3003*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*a^ 5 + 30030*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2 )*a^3*b^2*d^2/e^2 + 30030*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*s qrt(e*x + d)*d^2)*a^4*b*d/e + 12870*(5*(e*x + d)^(7/2) - 21*(e*x + d)^(5/2 )*d + 35*(e*x + d)^(3/2)*d^2 - 35*sqrt(e*x + d)*d^3)*a^2*b^3*d^2/e^3 + 257 40*(5*(e*x + d)^(7/2) - 21*(e*x + d)^(5/2)*d + 35*(e*x + d)^(3/2)*d^2 - 35 *sqrt(e*x + d)*d^3)*a^3*b^2*d/e^2 + 6435*(5*(e*x + d)^(7/2) - 21*(e*x + d) ^(5/2)*d + 35*(e*x + d)^(3/2)*d^2 - 35*sqrt(e*x + d)*d^3)*a^4*b/e + 715*(3 5*(e*x + d)^(9/2) - 180*(e*x + d)^(7/2)*d + 378*(e*x + d)^(5/2)*d^2 - 420* (e*x + d)^(3/2)*d^3 + 315*sqrt(e*x + d)*d^4)*a*b^4*d^2/e^4 + 2860*(35*(e*x + d)^(9/2) - 180*(e*x + d)^(7/2)*d + 378*(e*x + d)^(5/2)*d^2 - 420*(e*x + d)^(3/2)*d^3 + 315*sqrt(e*x + d)*d^4)*a^2*b^3*d/e^3 + 1430*(35*(e*x + d)^ (9/2) - 180*(e*x + d)^(7/2)*d + 378*(e*x + d)^(5/2)*d^2 - 420*(e*x + d)^(3 /2)*d^3 + 315*sqrt(e*x + d)*d^4)*a^3*b^2/e^2 + 65*(63*(e*x + d)^(11/2) - 3 85*(e*x + d)^(9/2)*d + 990*(e*x + d)^(7/2)*d^2 - 1386*(e*x + d)^(5/2)*d^3 + 1155*(e*x + d)^(3/2)*d^4 - 693*sqrt(e*x + d)*d^5)*b^5*d^2/e^5 + 650*(63* (e*x + d)^(11/2) - 385*(e*x + d)^(9/2)*d + 990*(e*x + d)^(7/2)*d^2 - 1386* (e*x + d)^(5/2)*d^3 + 1155*(e*x + d)^(3/2)*d^4 - 693*sqrt(e*x + d)*d^5)...
Time = 0.05 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.87 \[ \int (a+b x) (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {2\,b^5\,{\left (d+e\,x\right )}^{15/2}}{15\,e^6}-\frac {\left (10\,b^5\,d-10\,a\,b^4\,e\right )\,{\left (d+e\,x\right )}^{13/2}}{13\,e^6}+\frac {2\,{\left (a\,e-b\,d\right )}^5\,{\left (d+e\,x\right )}^{5/2}}{5\,e^6}+\frac {20\,b^2\,{\left (a\,e-b\,d\right )}^3\,{\left (d+e\,x\right )}^{9/2}}{9\,e^6}+\frac {20\,b^3\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{11/2}}{11\,e^6}+\frac {10\,b\,{\left (a\,e-b\,d\right )}^4\,{\left (d+e\,x\right )}^{7/2}}{7\,e^6} \]