Integrand size = 33, antiderivative size = 214 \[ \int (a+b x) \sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=-\frac {2 (b d-a e)^7 (d+e x)^{3/2}}{3 e^8}+\frac {14 b (b d-a e)^6 (d+e x)^{5/2}}{5 e^8}-\frac {6 b^2 (b d-a e)^5 (d+e x)^{7/2}}{e^8}+\frac {70 b^3 (b d-a e)^4 (d+e x)^{9/2}}{9 e^8}-\frac {70 b^4 (b d-a e)^3 (d+e x)^{11/2}}{11 e^8}+\frac {42 b^5 (b d-a e)^2 (d+e x)^{13/2}}{13 e^8}-\frac {14 b^6 (b d-a e) (d+e x)^{15/2}}{15 e^8}+\frac {2 b^7 (d+e x)^{17/2}}{17 e^8} \]
-2/3*(-a*e+b*d)^7*(e*x+d)^(3/2)/e^8+14/5*b*(-a*e+b*d)^6*(e*x+d)^(5/2)/e^8- 6*b^2*(-a*e+b*d)^5*(e*x+d)^(7/2)/e^8+70/9*b^3*(-a*e+b*d)^4*(e*x+d)^(9/2)/e ^8-70/11*b^4*(-a*e+b*d)^3*(e*x+d)^(11/2)/e^8+42/13*b^5*(-a*e+b*d)^2*(e*x+d )^(13/2)/e^8-14/15*b^6*(-a*e+b*d)*(e*x+d)^(15/2)/e^8+2/17*b^7*(e*x+d)^(17/ 2)/e^8
Time = 0.17 (sec) , antiderivative size = 376, normalized size of antiderivative = 1.76 \[ \int (a+b x) \sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {2 (d+e x)^{3/2} \left (36465 a^7 e^7+51051 a^6 b e^6 (-2 d+3 e x)+21879 a^5 b^2 e^5 \left (8 d^2-12 d e x+15 e^2 x^2\right )+12155 a^4 b^3 e^4 \left (-16 d^3+24 d^2 e x-30 d e^2 x^2+35 e^3 x^3\right )+1105 a^3 b^4 e^3 \left (128 d^4-192 d^3 e x+240 d^2 e^2 x^2-280 d e^3 x^3+315 e^4 x^4\right )+255 a^2 b^5 e^2 \left (-256 d^5+384 d^4 e x-480 d^3 e^2 x^2+560 d^2 e^3 x^3-630 d e^4 x^4+693 e^5 x^5\right )+17 a b^6 e \left (1024 d^6-1536 d^5 e x+1920 d^4 e^2 x^2-2240 d^3 e^3 x^3+2520 d^2 e^4 x^4-2772 d e^5 x^5+3003 e^6 x^6\right )+b^7 \left (-2048 d^7+3072 d^6 e x-3840 d^5 e^2 x^2+4480 d^4 e^3 x^3-5040 d^3 e^4 x^4+5544 d^2 e^5 x^5-6006 d e^6 x^6+6435 e^7 x^7\right )\right )}{109395 e^8} \]
(2*(d + e*x)^(3/2)*(36465*a^7*e^7 + 51051*a^6*b*e^6*(-2*d + 3*e*x) + 21879 *a^5*b^2*e^5*(8*d^2 - 12*d*e*x + 15*e^2*x^2) + 12155*a^4*b^3*e^4*(-16*d^3 + 24*d^2*e*x - 30*d*e^2*x^2 + 35*e^3*x^3) + 1105*a^3*b^4*e^3*(128*d^4 - 19 2*d^3*e*x + 240*d^2*e^2*x^2 - 280*d*e^3*x^3 + 315*e^4*x^4) + 255*a^2*b^5*e ^2*(-256*d^5 + 384*d^4*e*x - 480*d^3*e^2*x^2 + 560*d^2*e^3*x^3 - 630*d*e^4 *x^4 + 693*e^5*x^5) + 17*a*b^6*e*(1024*d^6 - 1536*d^5*e*x + 1920*d^4*e^2*x ^2 - 2240*d^3*e^3*x^3 + 2520*d^2*e^4*x^4 - 2772*d*e^5*x^5 + 3003*e^6*x^6) + b^7*(-2048*d^7 + 3072*d^6*e*x - 3840*d^5*e^2*x^2 + 4480*d^4*e^3*x^3 - 50 40*d^3*e^4*x^4 + 5544*d^2*e^5*x^5 - 6006*d*e^6*x^6 + 6435*e^7*x^7)))/(1093 95*e^8)
Time = 0.35 (sec) , antiderivative size = 214, 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.
| \(\displaystyle \int (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^3 \sqrt {d+e x} \, dx\) |
| \(\Big \downarrow \) 1184 |
| \(\displaystyle \frac {\int b^6 (a+b x)^7 \sqrt {d+e x}dx}{b^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int (a+b x)^7 \sqrt {d+e x}dx\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle \int \left (-\frac {7 b^6 (d+e x)^{13/2} (b d-a e)}{e^7}+\frac {21 b^5 (d+e x)^{11/2} (b d-a e)^2}{e^7}-\frac {35 b^4 (d+e x)^{9/2} (b d-a e)^3}{e^7}+\frac {35 b^3 (d+e x)^{7/2} (b d-a e)^4}{e^7}-\frac {21 b^2 (d+e x)^{5/2} (b d-a e)^5}{e^7}+\frac {7 b (d+e x)^{3/2} (b d-a e)^6}{e^7}+\frac {\sqrt {d+e x} (a e-b d)^7}{e^7}+\frac {b^7 (d+e x)^{15/2}}{e^7}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {14 b^6 (d+e x)^{15/2} (b d-a e)}{15 e^8}+\frac {42 b^5 (d+e x)^{13/2} (b d-a e)^2}{13 e^8}-\frac {70 b^4 (d+e x)^{11/2} (b d-a e)^3}{11 e^8}+\frac {70 b^3 (d+e x)^{9/2} (b d-a e)^4}{9 e^8}-\frac {6 b^2 (d+e x)^{7/2} (b d-a e)^5}{e^8}+\frac {14 b (d+e x)^{5/2} (b d-a e)^6}{5 e^8}-\frac {2 (d+e x)^{3/2} (b d-a e)^7}{3 e^8}+\frac {2 b^7 (d+e x)^{17/2}}{17 e^8}\) |
(-2*(b*d - a*e)^7*(d + e*x)^(3/2))/(3*e^8) + (14*b*(b*d - a*e)^6*(d + e*x) ^(5/2))/(5*e^8) - (6*b^2*(b*d - a*e)^5*(d + e*x)^(7/2))/e^8 + (70*b^3*(b*d - a*e)^4*(d + e*x)^(9/2))/(9*e^8) - (70*b^4*(b*d - a*e)^3*(d + e*x)^(11/2 ))/(11*e^8) + (42*b^5*(b*d - a*e)^2*(d + e*x)^(13/2))/(13*e^8) - (14*b^6*( b*d - a*e)*(d + e*x)^(15/2))/(15*e^8) + (2*b^7*(d + e*x)^(17/2))/(17*e^8)
3.21.61.3.1 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 = 0.36 (sec) , antiderivative size = 359, normalized size of antiderivative = 1.68
| method | result | size |
| pseudoelliptic | \(\frac {2 \left (\left (\frac {3}{17} b^{7} x^{7}+a^{7}+\frac {7}{5} a \,b^{6} x^{6}+\frac {63}{13} a^{2} b^{5} x^{5}+\frac {105}{11} a^{3} b^{4} x^{4}+\frac {35}{3} a^{4} b^{3} x^{3}+9 a^{5} b^{2} x^{2}+\frac {21}{5} a^{6} b x \right ) e^{7}-\frac {14 b \left (\frac {1}{17} b^{6} x^{6}+\frac {6}{13} a \,b^{5} x^{5}+\frac {225}{143} a^{2} b^{4} x^{4}+\frac {100}{33} a^{3} b^{3} x^{3}+\frac {25}{7} a^{4} b^{2} x^{2}+\frac {18}{7} a^{5} b x +a^{6}\right ) d \,e^{6}}{5}+\frac {24 b^{2} \left (\frac {7}{221} b^{5} x^{5}+\frac {35}{143} a \,b^{4} x^{4}+\frac {350}{429} a^{2} b^{3} x^{3}+\frac {50}{33} a^{3} b^{2} x^{2}+\frac {5}{3} a^{4} b x +a^{5}\right ) d^{2} e^{5}}{5}-\frac {16 b^{3} \left (\frac {63}{2431} x^{4} b^{4}+\frac {28}{143} a \,b^{3} x^{3}+\frac {90}{143} x^{2} b^{2} a^{2}+\frac {12}{11} b \,a^{3} x +a^{4}\right ) d^{3} e^{4}}{3}+\frac {128 \left (\frac {7}{221} x^{3} b^{3}+\frac {3}{13} a \,b^{2} x^{2}+\frac {9}{13} b \,a^{2} x +a^{3}\right ) b^{4} d^{4} e^{3}}{33}-\frac {256 b^{5} \left (\frac {1}{17} b^{2} x^{2}+\frac {2}{5} a b x +a^{2}\right ) d^{5} e^{2}}{143}+\frac {1024 b^{6} \left (\frac {3 b x}{17}+a \right ) d^{6} e}{2145}-\frac {2048 b^{7} d^{7}}{36465}\right ) \left (e x +d \right )^{\frac {3}{2}}}{3 e^{8}}\) | \(359\) |
| gosper | \(\frac {2 \left (e x +d \right )^{\frac {3}{2}} \left (6435 x^{7} b^{7} e^{7}+51051 x^{6} a \,b^{6} e^{7}-6006 x^{6} b^{7} d \,e^{6}+176715 x^{5} a^{2} b^{5} e^{7}-47124 x^{5} a \,b^{6} d \,e^{6}+5544 x^{5} b^{7} d^{2} e^{5}+348075 x^{4} a^{3} b^{4} e^{7}-160650 x^{4} a^{2} b^{5} d \,e^{6}+42840 x^{4} a \,b^{6} d^{2} e^{5}-5040 x^{4} b^{7} d^{3} e^{4}+425425 x^{3} a^{4} b^{3} e^{7}-309400 x^{3} a^{3} b^{4} d \,e^{6}+142800 x^{3} a^{2} b^{5} d^{2} e^{5}-38080 x^{3} a \,b^{6} d^{3} e^{4}+4480 x^{3} b^{7} d^{4} e^{3}+328185 x^{2} a^{5} b^{2} e^{7}-364650 x^{2} a^{4} b^{3} d \,e^{6}+265200 x^{2} a^{3} b^{4} d^{2} e^{5}-122400 x^{2} a^{2} b^{5} d^{3} e^{4}+32640 x^{2} a \,b^{6} d^{4} e^{3}-3840 x^{2} b^{7} d^{5} e^{2}+153153 x \,a^{6} b \,e^{7}-262548 x \,a^{5} b^{2} d \,e^{6}+291720 x \,a^{4} b^{3} d^{2} e^{5}-212160 x \,a^{3} b^{4} d^{3} e^{4}+97920 x \,a^{2} b^{5} d^{4} e^{3}-26112 x a \,b^{6} d^{5} e^{2}+3072 x \,b^{7} d^{6} e +36465 e^{7} a^{7}-102102 b d \,e^{6} a^{6}+175032 b^{2} d^{2} e^{5} a^{5}-194480 b^{3} d^{3} e^{4} a^{4}+141440 b^{4} d^{4} e^{3} a^{3}-65280 b^{5} d^{5} e^{2} a^{2}+17408 b^{6} d^{6} e a -2048 b^{7} d^{7}\right )}{109395 e^{8}}\) | \(498\) |
| trager | \(\frac {2 \left (6435 b^{7} e^{8} x^{8}+51051 a \,b^{6} e^{8} x^{7}+429 b^{7} d \,e^{7} x^{7}+176715 a^{2} b^{5} e^{8} x^{6}+3927 a \,b^{6} d \,e^{7} x^{6}-462 b^{7} d^{2} e^{6} x^{6}+348075 a^{3} b^{4} e^{8} x^{5}+16065 a^{2} b^{5} d \,e^{7} x^{5}-4284 a \,b^{6} d^{2} e^{6} x^{5}+504 b^{7} d^{3} e^{5} x^{5}+425425 a^{4} b^{3} e^{8} x^{4}+38675 a^{3} b^{4} d \,e^{7} x^{4}-17850 a^{2} b^{5} d^{2} e^{6} x^{4}+4760 a \,b^{6} d^{3} e^{5} x^{4}-560 b^{7} d^{4} e^{4} x^{4}+328185 a^{5} b^{2} e^{8} x^{3}+60775 a^{4} b^{3} d \,e^{7} x^{3}-44200 a^{3} b^{4} d^{2} e^{6} x^{3}+20400 a^{2} b^{5} d^{3} e^{5} x^{3}-5440 a \,b^{6} d^{4} e^{4} x^{3}+640 b^{7} d^{5} e^{3} x^{3}+153153 a^{6} b \,e^{8} x^{2}+65637 a^{5} b^{2} d \,e^{7} x^{2}-72930 a^{4} b^{3} d^{2} e^{6} x^{2}+53040 a^{3} b^{4} d^{3} e^{5} x^{2}-24480 a^{2} b^{5} d^{4} e^{4} x^{2}+6528 a \,b^{6} d^{5} e^{3} x^{2}-768 b^{7} d^{6} e^{2} x^{2}+36465 a^{7} e^{8} x +51051 a^{6} b d \,e^{7} x -87516 a^{5} b^{2} d^{2} e^{6} x +97240 a^{4} b^{3} d^{3} e^{5} x -70720 a^{3} b^{4} d^{4} e^{4} x +32640 a^{2} b^{5} d^{5} e^{3} x -8704 a \,b^{6} d^{6} e^{2} x +1024 b^{7} d^{7} e x +36465 a^{7} d \,e^{7}-102102 a^{6} b \,d^{2} e^{6}+175032 a^{5} b^{2} d^{3} e^{5}-194480 a^{4} b^{3} d^{4} e^{4}+141440 a^{3} b^{4} d^{5} e^{3}-65280 a^{2} b^{5} d^{6} e^{2}+17408 a \,b^{6} d^{7} e -2048 b^{7} d^{8}\right ) \sqrt {e x +d}}{109395 e^{8}}\) | \(620\) |
| risch | \(\frac {2 \left (6435 b^{7} e^{8} x^{8}+51051 a \,b^{6} e^{8} x^{7}+429 b^{7} d \,e^{7} x^{7}+176715 a^{2} b^{5} e^{8} x^{6}+3927 a \,b^{6} d \,e^{7} x^{6}-462 b^{7} d^{2} e^{6} x^{6}+348075 a^{3} b^{4} e^{8} x^{5}+16065 a^{2} b^{5} d \,e^{7} x^{5}-4284 a \,b^{6} d^{2} e^{6} x^{5}+504 b^{7} d^{3} e^{5} x^{5}+425425 a^{4} b^{3} e^{8} x^{4}+38675 a^{3} b^{4} d \,e^{7} x^{4}-17850 a^{2} b^{5} d^{2} e^{6} x^{4}+4760 a \,b^{6} d^{3} e^{5} x^{4}-560 b^{7} d^{4} e^{4} x^{4}+328185 a^{5} b^{2} e^{8} x^{3}+60775 a^{4} b^{3} d \,e^{7} x^{3}-44200 a^{3} b^{4} d^{2} e^{6} x^{3}+20400 a^{2} b^{5} d^{3} e^{5} x^{3}-5440 a \,b^{6} d^{4} e^{4} x^{3}+640 b^{7} d^{5} e^{3} x^{3}+153153 a^{6} b \,e^{8} x^{2}+65637 a^{5} b^{2} d \,e^{7} x^{2}-72930 a^{4} b^{3} d^{2} e^{6} x^{2}+53040 a^{3} b^{4} d^{3} e^{5} x^{2}-24480 a^{2} b^{5} d^{4} e^{4} x^{2}+6528 a \,b^{6} d^{5} e^{3} x^{2}-768 b^{7} d^{6} e^{2} x^{2}+36465 a^{7} e^{8} x +51051 a^{6} b d \,e^{7} x -87516 a^{5} b^{2} d^{2} e^{6} x +97240 a^{4} b^{3} d^{3} e^{5} x -70720 a^{3} b^{4} d^{4} e^{4} x +32640 a^{2} b^{5} d^{5} e^{3} x -8704 a \,b^{6} d^{6} e^{2} x +1024 b^{7} d^{7} e x +36465 a^{7} d \,e^{7}-102102 a^{6} b \,d^{2} e^{6}+175032 a^{5} b^{2} d^{3} e^{5}-194480 a^{4} b^{3} d^{4} e^{4}+141440 a^{3} b^{4} d^{5} e^{3}-65280 a^{2} b^{5} d^{6} e^{2}+17408 a \,b^{6} d^{7} e -2048 b^{7} d^{8}\right ) \sqrt {e x +d}}{109395 e^{8}}\) | \(620\) |
| derivativedivides | \(\frac {\frac {2 b^{7} \left (e x +d \right )^{\frac {17}{2}}}{17}+\frac {2 \left (\left (a e -b d \right ) b^{6}+3 b^{5} \left (2 a b e -2 b^{2} d \right )\right ) \left (e x +d \right )^{\frac {15}{2}}}{15}+\frac {2 \left (3 \left (a e -b d \right ) \left (2 a b e -2 b^{2} d \right ) b^{4}+b \left (\left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) b^{4}+2 \left (2 a b e -2 b^{2} d \right )^{2} b^{2}+b^{2} \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 )\right ) \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (\left (a e -b d \right ) \left (\left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) b^{4}+2 \left (2 a b e -2 b^{2} d \right )^{2} b^{2}+b^{2} \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 )+b \left (4 \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^{2}+\left (2 a b e -2 b^{2} 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 )\right )\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (\left (a e -b d \right ) \left (4 \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^{2}+\left (2 a b e -2 b^{2} 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 )\right )+b \left (\left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\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 \left (2 a b e -2 b^{2} d \right )^{2} \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )+b^{2} \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )^{2}\right )\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (\left (a e -b d \right ) \left (\left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\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 \left (2 a b e -2 b^{2} d \right )^{2} \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )+b^{2} \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )^{2}\right )+3 b \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )^{2} \left (2 a b e -2 b^{2} d \right )\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (3 \left (a e -b d \right ) \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )^{2} \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 )^{3}\right ) \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 \left (a e -b d \right ) \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )^{3} \left (e x +d \right )^{\frac {3}{2}}}{3}}{e^{8}}\) | \(936\) |
| default | \(\frac {\frac {2 b^{7} \left (e x +d \right )^{\frac {17}{2}}}{17}+\frac {2 \left (\left (a e -b d \right ) b^{6}+3 b^{5} \left (2 a b e -2 b^{2} d \right )\right ) \left (e x +d \right )^{\frac {15}{2}}}{15}+\frac {2 \left (3 \left (a e -b d \right ) \left (2 a b e -2 b^{2} d \right ) b^{4}+b \left (\left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) b^{4}+2 \left (2 a b e -2 b^{2} d \right )^{2} b^{2}+b^{2} \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 )\right ) \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (\left (a e -b d \right ) \left (\left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) b^{4}+2 \left (2 a b e -2 b^{2} d \right )^{2} b^{2}+b^{2} \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 )+b \left (4 \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^{2}+\left (2 a b e -2 b^{2} 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 )\right )\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (\left (a e -b d \right ) \left (4 \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^{2}+\left (2 a b e -2 b^{2} 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 )\right )+b \left (\left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\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 \left (2 a b e -2 b^{2} d \right )^{2} \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )+b^{2} \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )^{2}\right )\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (\left (a e -b d \right ) \left (\left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\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 \left (2 a b e -2 b^{2} d \right )^{2} \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )+b^{2} \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )^{2}\right )+3 b \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )^{2} \left (2 a b e -2 b^{2} d \right )\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (3 \left (a e -b d \right ) \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )^{2} \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 )^{3}\right ) \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 \left (a e -b d \right ) \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )^{3} \left (e x +d \right )^{\frac {3}{2}}}{3}}{e^{8}}\) | \(936\) |
2/3*((3/17*b^7*x^7+a^7+7/5*a*b^6*x^6+63/13*a^2*b^5*x^5+105/11*a^3*b^4*x^4+ 35/3*a^4*b^3*x^3+9*a^5*b^2*x^2+21/5*a^6*b*x)*e^7-14/5*b*(1/17*b^6*x^6+6/13 *a*b^5*x^5+225/143*a^2*b^4*x^4+100/33*a^3*b^3*x^3+25/7*a^4*b^2*x^2+18/7*a^ 5*b*x+a^6)*d*e^6+24/5*b^2*(7/221*b^5*x^5+35/143*a*b^4*x^4+350/429*a^2*b^3* x^3+50/33*a^3*b^2*x^2+5/3*a^4*b*x+a^5)*d^2*e^5-16/3*b^3*(63/2431*x^4*b^4+2 8/143*a*b^3*x^3+90/143*x^2*b^2*a^2+12/11*b*a^3*x+a^4)*d^3*e^4+128/33*(7/22 1*x^3*b^3+3/13*a*b^2*x^2+9/13*b*a^2*x+a^3)*b^4*d^4*e^3-256/143*b^5*(1/17*b ^2*x^2+2/5*a*b*x+a^2)*d^5*e^2+1024/2145*b^6*(3/17*b*x+a)*d^6*e-2048/36465* b^7*d^7)*(e*x+d)^(3/2)/e^8
Leaf count of result is larger than twice the leaf count of optimal. 568 vs. \(2 (184) = 368\).
Time = 0.31 (sec) , antiderivative size = 568, normalized size of antiderivative = 2.65 \[ \int (a+b x) \sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {2 \, {\left (6435 \, b^{7} e^{8} x^{8} - 2048 \, b^{7} d^{8} + 17408 \, a b^{6} d^{7} e - 65280 \, a^{2} b^{5} d^{6} e^{2} + 141440 \, a^{3} b^{4} d^{5} e^{3} - 194480 \, a^{4} b^{3} d^{4} e^{4} + 175032 \, a^{5} b^{2} d^{3} e^{5} - 102102 \, a^{6} b d^{2} e^{6} + 36465 \, a^{7} d e^{7} + 429 \, {\left (b^{7} d e^{7} + 119 \, a b^{6} e^{8}\right )} x^{7} - 231 \, {\left (2 \, b^{7} d^{2} e^{6} - 17 \, a b^{6} d e^{7} - 765 \, a^{2} b^{5} e^{8}\right )} x^{6} + 63 \, {\left (8 \, b^{7} d^{3} e^{5} - 68 \, a b^{6} d^{2} e^{6} + 255 \, a^{2} b^{5} d e^{7} + 5525 \, a^{3} b^{4} e^{8}\right )} x^{5} - 35 \, {\left (16 \, b^{7} d^{4} e^{4} - 136 \, a b^{6} d^{3} e^{5} + 510 \, a^{2} b^{5} d^{2} e^{6} - 1105 \, a^{3} b^{4} d e^{7} - 12155 \, a^{4} b^{3} e^{8}\right )} x^{4} + 5 \, {\left (128 \, b^{7} d^{5} e^{3} - 1088 \, a b^{6} d^{4} e^{4} + 4080 \, a^{2} b^{5} d^{3} e^{5} - 8840 \, a^{3} b^{4} d^{2} e^{6} + 12155 \, a^{4} b^{3} d e^{7} + 65637 \, a^{5} b^{2} e^{8}\right )} x^{3} - 3 \, {\left (256 \, b^{7} d^{6} e^{2} - 2176 \, a b^{6} d^{5} e^{3} + 8160 \, a^{2} b^{5} d^{4} e^{4} - 17680 \, a^{3} b^{4} d^{3} e^{5} + 24310 \, a^{4} b^{3} d^{2} e^{6} - 21879 \, a^{5} b^{2} d e^{7} - 51051 \, a^{6} b e^{8}\right )} x^{2} + {\left (1024 \, b^{7} d^{7} e - 8704 \, a b^{6} d^{6} e^{2} + 32640 \, a^{2} b^{5} d^{5} e^{3} - 70720 \, a^{3} b^{4} d^{4} e^{4} + 97240 \, a^{4} b^{3} d^{3} e^{5} - 87516 \, a^{5} b^{2} d^{2} e^{6} + 51051 \, a^{6} b d e^{7} + 36465 \, a^{7} e^{8}\right )} x\right )} \sqrt {e x + d}}{109395 \, e^{8}} \]
2/109395*(6435*b^7*e^8*x^8 - 2048*b^7*d^8 + 17408*a*b^6*d^7*e - 65280*a^2* b^5*d^6*e^2 + 141440*a^3*b^4*d^5*e^3 - 194480*a^4*b^3*d^4*e^4 + 175032*a^5 *b^2*d^3*e^5 - 102102*a^6*b*d^2*e^6 + 36465*a^7*d*e^7 + 429*(b^7*d*e^7 + 1 19*a*b^6*e^8)*x^7 - 231*(2*b^7*d^2*e^6 - 17*a*b^6*d*e^7 - 765*a^2*b^5*e^8) *x^6 + 63*(8*b^7*d^3*e^5 - 68*a*b^6*d^2*e^6 + 255*a^2*b^5*d*e^7 + 5525*a^3 *b^4*e^8)*x^5 - 35*(16*b^7*d^4*e^4 - 136*a*b^6*d^3*e^5 + 510*a^2*b^5*d^2*e ^6 - 1105*a^3*b^4*d*e^7 - 12155*a^4*b^3*e^8)*x^4 + 5*(128*b^7*d^5*e^3 - 10 88*a*b^6*d^4*e^4 + 4080*a^2*b^5*d^3*e^5 - 8840*a^3*b^4*d^2*e^6 + 12155*a^4 *b^3*d*e^7 + 65637*a^5*b^2*e^8)*x^3 - 3*(256*b^7*d^6*e^2 - 2176*a*b^6*d^5* e^3 + 8160*a^2*b^5*d^4*e^4 - 17680*a^3*b^4*d^3*e^5 + 24310*a^4*b^3*d^2*e^6 - 21879*a^5*b^2*d*e^7 - 51051*a^6*b*e^8)*x^2 + (1024*b^7*d^7*e - 8704*a*b ^6*d^6*e^2 + 32640*a^2*b^5*d^5*e^3 - 70720*a^3*b^4*d^4*e^4 + 97240*a^4*b^3 *d^3*e^5 - 87516*a^5*b^2*d^2*e^6 + 51051*a^6*b*d*e^7 + 36465*a^7*e^8)*x)*s qrt(e*x + d)/e^8
Leaf count of result is larger than twice the leaf count of optimal. 576 vs. \(2 (199) = 398\).
Time = 1.76 (sec) , antiderivative size = 576, normalized size of antiderivative = 2.69 \[ \int (a+b x) \sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\begin {cases} \frac {2 \left (\frac {b^{7} \left (d + e x\right )^{\frac {17}{2}}}{17 e^{7}} + \frac {\left (d + e x\right )^{\frac {15}{2}} \cdot \left (7 a b^{6} e - 7 b^{7} d\right )}{15 e^{7}} + \frac {\left (d + e x\right )^{\frac {13}{2}} \cdot \left (21 a^{2} b^{5} e^{2} - 42 a b^{6} d e + 21 b^{7} d^{2}\right )}{13 e^{7}} + \frac {\left (d + e x\right )^{\frac {11}{2}} \cdot \left (35 a^{3} b^{4} e^{3} - 105 a^{2} b^{5} d e^{2} + 105 a b^{6} d^{2} e - 35 b^{7} d^{3}\right )}{11 e^{7}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \cdot \left (35 a^{4} b^{3} e^{4} - 140 a^{3} b^{4} d e^{3} + 210 a^{2} b^{5} d^{2} e^{2} - 140 a b^{6} d^{3} e + 35 b^{7} d^{4}\right )}{9 e^{7}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \cdot \left (21 a^{5} b^{2} e^{5} - 105 a^{4} b^{3} d e^{4} + 210 a^{3} b^{4} d^{2} e^{3} - 210 a^{2} b^{5} d^{3} e^{2} + 105 a b^{6} d^{4} e - 21 b^{7} d^{5}\right )}{7 e^{7}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \cdot \left (7 a^{6} b e^{6} - 42 a^{5} b^{2} d e^{5} + 105 a^{4} b^{3} d^{2} e^{4} - 140 a^{3} b^{4} d^{3} e^{3} + 105 a^{2} b^{5} d^{4} e^{2} - 42 a b^{6} d^{5} e + 7 b^{7} d^{6}\right )}{5 e^{7}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \left (a^{7} e^{7} - 7 a^{6} b d e^{6} + 21 a^{5} b^{2} d^{2} e^{5} - 35 a^{4} b^{3} d^{3} e^{4} + 35 a^{3} b^{4} d^{4} e^{3} - 21 a^{2} b^{5} d^{5} e^{2} + 7 a b^{6} d^{6} e - b^{7} d^{7}\right )}{3 e^{7}}\right )}{e} & \text {for}\: e \neq 0 \\\sqrt {d} \left (\begin {cases} a^{7} x & \text {for}\: b = 0 \\\frac {\left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{4}}{8 b} & \text {otherwise} \end {cases}\right ) & \text {otherwise} \end {cases} \]
Piecewise((2*(b**7*(d + e*x)**(17/2)/(17*e**7) + (d + e*x)**(15/2)*(7*a*b* *6*e - 7*b**7*d)/(15*e**7) + (d + e*x)**(13/2)*(21*a**2*b**5*e**2 - 42*a*b **6*d*e + 21*b**7*d**2)/(13*e**7) + (d + e*x)**(11/2)*(35*a**3*b**4*e**3 - 105*a**2*b**5*d*e**2 + 105*a*b**6*d**2*e - 35*b**7*d**3)/(11*e**7) + (d + e*x)**(9/2)*(35*a**4*b**3*e**4 - 140*a**3*b**4*d*e**3 + 210*a**2*b**5*d** 2*e**2 - 140*a*b**6*d**3*e + 35*b**7*d**4)/(9*e**7) + (d + e*x)**(7/2)*(21 *a**5*b**2*e**5 - 105*a**4*b**3*d*e**4 + 210*a**3*b**4*d**2*e**3 - 210*a** 2*b**5*d**3*e**2 + 105*a*b**6*d**4*e - 21*b**7*d**5)/(7*e**7) + (d + e*x)* *(5/2)*(7*a**6*b*e**6 - 42*a**5*b**2*d*e**5 + 105*a**4*b**3*d**2*e**4 - 14 0*a**3*b**4*d**3*e**3 + 105*a**2*b**5*d**4*e**2 - 42*a*b**6*d**5*e + 7*b** 7*d**6)/(5*e**7) + (d + e*x)**(3/2)*(a**7*e**7 - 7*a**6*b*d*e**6 + 21*a**5 *b**2*d**2*e**5 - 35*a**4*b**3*d**3*e**4 + 35*a**3*b**4*d**4*e**3 - 21*a** 2*b**5*d**5*e**2 + 7*a*b**6*d**6*e - b**7*d**7)/(3*e**7))/e, Ne(e, 0)), (s qrt(d)*Piecewise((a**7*x, Eq(b, 0)), ((a**2 + 2*a*b*x + b**2*x**2)**4/(8*b ), True)), True))
Leaf count of result is larger than twice the leaf count of optimal. 456 vs. \(2 (184) = 368\).
Time = 0.20 (sec) , antiderivative size = 456, normalized size of antiderivative = 2.13 \[ \int (a+b x) \sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {2 \, {\left (6435 \, {\left (e x + d\right )}^{\frac {17}{2}} b^{7} - 51051 \, {\left (b^{7} d - a b^{6} e\right )} {\left (e x + d\right )}^{\frac {15}{2}} + 176715 \, {\left (b^{7} d^{2} - 2 \, a b^{6} d e + a^{2} b^{5} e^{2}\right )} {\left (e x + d\right )}^{\frac {13}{2}} - 348075 \, {\left (b^{7} d^{3} - 3 \, a b^{6} d^{2} e + 3 \, a^{2} b^{5} d e^{2} - a^{3} b^{4} e^{3}\right )} {\left (e x + d\right )}^{\frac {11}{2}} + 425425 \, {\left (b^{7} d^{4} - 4 \, a b^{6} d^{3} e + 6 \, a^{2} b^{5} d^{2} e^{2} - 4 \, a^{3} b^{4} d e^{3} + a^{4} b^{3} e^{4}\right )} {\left (e x + d\right )}^{\frac {9}{2}} - 328185 \, {\left (b^{7} d^{5} - 5 \, a b^{6} d^{4} e + 10 \, a^{2} b^{5} d^{3} e^{2} - 10 \, a^{3} b^{4} d^{2} e^{3} + 5 \, a^{4} b^{3} d e^{4} - a^{5} b^{2} e^{5}\right )} {\left (e x + d\right )}^{\frac {7}{2}} + 153153 \, {\left (b^{7} d^{6} - 6 \, a b^{6} d^{5} e + 15 \, a^{2} b^{5} d^{4} e^{2} - 20 \, a^{3} b^{4} d^{3} e^{3} + 15 \, a^{4} b^{3} d^{2} e^{4} - 6 \, a^{5} b^{2} d e^{5} + a^{6} b e^{6}\right )} {\left (e x + d\right )}^{\frac {5}{2}} - 36465 \, {\left (b^{7} d^{7} - 7 \, a b^{6} d^{6} e + 21 \, a^{2} b^{5} d^{5} e^{2} - 35 \, a^{3} b^{4} d^{4} e^{3} + 35 \, a^{4} b^{3} d^{3} e^{4} - 21 \, a^{5} b^{2} d^{2} e^{5} + 7 \, a^{6} b d e^{6} - a^{7} e^{7}\right )} {\left (e x + d\right )}^{\frac {3}{2}}\right )}}{109395 \, e^{8}} \]
2/109395*(6435*(e*x + d)^(17/2)*b^7 - 51051*(b^7*d - a*b^6*e)*(e*x + d)^(1 5/2) + 176715*(b^7*d^2 - 2*a*b^6*d*e + a^2*b^5*e^2)*(e*x + d)^(13/2) - 348 075*(b^7*d^3 - 3*a*b^6*d^2*e + 3*a^2*b^5*d*e^2 - a^3*b^4*e^3)*(e*x + d)^(1 1/2) + 425425*(b^7*d^4 - 4*a*b^6*d^3*e + 6*a^2*b^5*d^2*e^2 - 4*a^3*b^4*d*e ^3 + a^4*b^3*e^4)*(e*x + d)^(9/2) - 328185*(b^7*d^5 - 5*a*b^6*d^4*e + 10*a ^2*b^5*d^3*e^2 - 10*a^3*b^4*d^2*e^3 + 5*a^4*b^3*d*e^4 - a^5*b^2*e^5)*(e*x + d)^(7/2) + 153153*(b^7*d^6 - 6*a*b^6*d^5*e + 15*a^2*b^5*d^4*e^2 - 20*a^3 *b^4*d^3*e^3 + 15*a^4*b^3*d^2*e^4 - 6*a^5*b^2*d*e^5 + a^6*b*e^6)*(e*x + d) ^(5/2) - 36465*(b^7*d^7 - 7*a*b^6*d^6*e + 21*a^2*b^5*d^5*e^2 - 35*a^3*b^4* d^4*e^3 + 35*a^4*b^3*d^3*e^4 - 21*a^5*b^2*d^2*e^5 + 7*a^6*b*d*e^6 - a^7*e^ 7)*(e*x + d)^(3/2))/e^8
Leaf count of result is larger than twice the leaf count of optimal. 1054 vs. \(2 (184) = 368\).
Time = 0.29 (sec) , antiderivative size = 1054, normalized size of antiderivative = 4.93 \[ \int (a+b x) \sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\text {Too large to display} \]
2/109395*(109395*sqrt(e*x + d)*a^7*d + 36465*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*a^7 + 255255*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*a^6*b*d/e + 15 3153*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*a^5 *b^2*d/e^2 + 51051*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*a^6*b/e + 109395*(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^3*d/e^3 + 65637*(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^5*b^2/e^2 + 12155*(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^4*d/e^4 + 12155*(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^ 4*b^3/e^3 + 3315*(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)*a^2*b^5*d/e^5 + 5525*(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)*a^3*b^4/e^4 + 255*(231*(e*x + d) ^(13/2) - 1638*(e*x + d)^(11/2)*d + 5005*(e*x + d)^(9/2)*d^2 - 8580*(e*x + d)^(7/2)*d^3 + 9009*(e*x + d)^(5/2)*d^4 - 6006*(e*x + d)^(3/2)*d^5 + 3003 *sqrt(e*x + d)*d^6)*a*b^6*d/e^6 + 765*(231*(e*x + d)^(13/2) - 1638*(e*x + d)^(11/2)*d + 5005*(e*x + d)^(9/2)*d^2 - 8580*(e*x + d)^(7/2)*d^3 + 900...
Time = 10.95 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.87 \[ \int (a+b x) \sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {2\,b^7\,{\left (d+e\,x\right )}^{17/2}}{17\,e^8}-\frac {\left (14\,b^7\,d-14\,a\,b^6\,e\right )\,{\left (d+e\,x\right )}^{15/2}}{15\,e^8}+\frac {2\,{\left (a\,e-b\,d\right )}^7\,{\left (d+e\,x\right )}^{3/2}}{3\,e^8}+\frac {6\,b^2\,{\left (a\,e-b\,d\right )}^5\,{\left (d+e\,x\right )}^{7/2}}{e^8}+\frac {70\,b^3\,{\left (a\,e-b\,d\right )}^4\,{\left (d+e\,x\right )}^{9/2}}{9\,e^8}+\frac {70\,b^4\,{\left (a\,e-b\,d\right )}^3\,{\left (d+e\,x\right )}^{11/2}}{11\,e^8}+\frac {42\,b^5\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{13/2}}{13\,e^8}+\frac {14\,b\,{\left (a\,e-b\,d\right )}^6\,{\left (d+e\,x\right )}^{5/2}}{5\,e^8} \]
(2*b^7*(d + e*x)^(17/2))/(17*e^8) - ((14*b^7*d - 14*a*b^6*e)*(d + e*x)^(15 /2))/(15*e^8) + (2*(a*e - b*d)^7*(d + e*x)^(3/2))/(3*e^8) + (6*b^2*(a*e - b*d)^5*(d + e*x)^(7/2))/e^8 + (70*b^3*(a*e - b*d)^4*(d + e*x)^(9/2))/(9*e^ 8) + (70*b^4*(a*e - b*d)^3*(d + e*x)^(11/2))/(11*e^8) + (42*b^5*(a*e - b*d )^2*(d + e*x)^(13/2))/(13*e^8) + (14*b*(a*e - b*d)^6*(d + e*x)^(5/2))/(5*e ^8)