Integrand size = 28, antiderivative size = 187 \[ \int (d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {2 (b d-a e)^6 (d+e x)^{9/2}}{9 e^7}-\frac {12 b (b d-a e)^5 (d+e x)^{11/2}}{11 e^7}+\frac {30 b^2 (b d-a e)^4 (d+e x)^{13/2}}{13 e^7}-\frac {8 b^3 (b d-a e)^3 (d+e x)^{15/2}}{3 e^7}+\frac {30 b^4 (b d-a e)^2 (d+e x)^{17/2}}{17 e^7}-\frac {12 b^5 (b d-a e) (d+e x)^{19/2}}{19 e^7}+\frac {2 b^6 (d+e x)^{21/2}}{21 e^7} \]
2/9*(-a*e+b*d)^6*(e*x+d)^(9/2)/e^7-12/11*b*(-a*e+b*d)^5*(e*x+d)^(11/2)/e^7 +30/13*b^2*(-a*e+b*d)^4*(e*x+d)^(13/2)/e^7-8/3*b^3*(-a*e+b*d)^3*(e*x+d)^(1 5/2)/e^7+30/17*b^4*(-a*e+b*d)^2*(e*x+d)^(17/2)/e^7-12/19*b^5*(-a*e+b*d)*(e *x+d)^(19/2)/e^7+2/21*b^6*(e*x+d)^(21/2)/e^7
Time = 0.18 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.56 \[ \int (d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {2 (d+e x)^{9/2} \left (323323 a^6 e^6+176358 a^5 b e^5 (-2 d+9 e x)+33915 a^4 b^2 e^4 \left (8 d^2-36 d e x+99 e^2 x^2\right )+9044 a^3 b^3 e^3 \left (-16 d^3+72 d^2 e x-198 d e^2 x^2+429 e^3 x^3\right )+399 a^2 b^4 e^2 \left (128 d^4-576 d^3 e x+1584 d^2 e^2 x^2-3432 d e^3 x^3+6435 e^4 x^4\right )+42 a b^5 e \left (-256 d^5+1152 d^4 e x-3168 d^3 e^2 x^2+6864 d^2 e^3 x^3-12870 d e^4 x^4+21879 e^5 x^5\right )+b^6 \left (1024 d^6-4608 d^5 e x+12672 d^4 e^2 x^2-27456 d^3 e^3 x^3+51480 d^2 e^4 x^4-87516 d e^5 x^5+138567 e^6 x^6\right )\right )}{2909907 e^7} \]
(2*(d + e*x)^(9/2)*(323323*a^6*e^6 + 176358*a^5*b*e^5*(-2*d + 9*e*x) + 339 15*a^4*b^2*e^4*(8*d^2 - 36*d*e*x + 99*e^2*x^2) + 9044*a^3*b^3*e^3*(-16*d^3 + 72*d^2*e*x - 198*d*e^2*x^2 + 429*e^3*x^3) + 399*a^2*b^4*e^2*(128*d^4 - 576*d^3*e*x + 1584*d^2*e^2*x^2 - 3432*d*e^3*x^3 + 6435*e^4*x^4) + 42*a*b^5 *e*(-256*d^5 + 1152*d^4*e*x - 3168*d^3*e^2*x^2 + 6864*d^2*e^3*x^3 - 12870* d*e^4*x^4 + 21879*e^5*x^5) + b^6*(1024*d^6 - 4608*d^5*e*x + 12672*d^4*e^2* x^2 - 27456*d^3*e^3*x^3 + 51480*d^2*e^4*x^4 - 87516*d*e^5*x^5 + 138567*e^6 *x^6)))/(2909907*e^7)
Time = 0.34 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {1098, 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.
| \(\displaystyle \int \left (a^2+2 a b x+b^2 x^2\right )^3 (d+e x)^{7/2} \, dx\) |
| \(\Big \downarrow \) 1098 |
| \(\displaystyle \frac {\int b^6 (a+b x)^6 (d+e x)^{7/2}dx}{b^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int (a+b x)^6 (d+e x)^{7/2}dx\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle \int \left (-\frac {6 b^5 (d+e x)^{17/2} (b d-a e)}{e^6}+\frac {15 b^4 (d+e x)^{15/2} (b d-a e)^2}{e^6}-\frac {20 b^3 (d+e x)^{13/2} (b d-a e)^3}{e^6}+\frac {15 b^2 (d+e x)^{11/2} (b d-a e)^4}{e^6}-\frac {6 b (d+e x)^{9/2} (b d-a e)^5}{e^6}+\frac {(d+e x)^{7/2} (a e-b d)^6}{e^6}+\frac {b^6 (d+e x)^{19/2}}{e^6}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {12 b^5 (d+e x)^{19/2} (b d-a e)}{19 e^7}+\frac {30 b^4 (d+e x)^{17/2} (b d-a e)^2}{17 e^7}-\frac {8 b^3 (d+e x)^{15/2} (b d-a e)^3}{3 e^7}+\frac {30 b^2 (d+e x)^{13/2} (b d-a e)^4}{13 e^7}-\frac {12 b (d+e x)^{11/2} (b d-a e)^5}{11 e^7}+\frac {2 (d+e x)^{9/2} (b d-a e)^6}{9 e^7}+\frac {2 b^6 (d+e x)^{21/2}}{21 e^7}\) |
(2*(b*d - a*e)^6*(d + e*x)^(9/2))/(9*e^7) - (12*b*(b*d - a*e)^5*(d + e*x)^ (11/2))/(11*e^7) + (30*b^2*(b*d - a*e)^4*(d + e*x)^(13/2))/(13*e^7) - (8*b ^3*(b*d - a*e)^3*(d + e*x)^(15/2))/(3*e^7) + (30*b^4*(b*d - a*e)^2*(d + e* x)^(17/2))/(17*e^7) - (12*b^5*(b*d - a*e)*(d + e*x)^(19/2))/(19*e^7) + (2* b^6*(d + e*x)^(21/2))/(21*e^7)
3.17.37.3.1 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_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_ Symbol] :> Simp[1/c^p Int[(d + e*x)^m*(b/2 + c*x)^(2*p), x], x] /; FreeQ[ {a, b, c, d, e, m}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
Time = 2.39 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.48
| method | result | size |
| pseudoelliptic | \(\frac {2 \left (e x +d \right )^{\frac {9}{2}} \left (\left (\frac {3}{7} b^{6} x^{6}+a^{6}+\frac {54}{19} a \,x^{5} b^{5}+\frac {135}{17} a^{2} x^{4} b^{4}+12 a^{3} x^{3} b^{3}+\frac {135}{13} a^{4} x^{2} b^{2}+\frac {54}{11} a^{5} x b \right ) e^{6}-\frac {12 b d \left (\frac {33}{133} b^{5} x^{5}+\frac {495}{323} a \,b^{4} x^{4}+\frac {66}{17} a^{2} b^{3} x^{3}+\frac {66}{13} a^{3} b^{2} x^{2}+\frac {45}{13} a^{4} b x +a^{5}\right ) e^{5}}{11}+\frac {120 b^{2} d^{2} \left (\frac {429}{2261} b^{4} x^{4}+\frac {1716}{1615} a \,b^{3} x^{3}+\frac {198}{85} a^{2} b^{2} x^{2}+\frac {12}{5} a^{3} b x +a^{4}\right ) e^{4}}{143}-\frac {64 b^{3} d^{3} \left (\frac {429}{2261} b^{3} x^{3}+\frac {297}{323} a \,b^{2} x^{2}+\frac {27}{17} a^{2} b x +a^{3}\right ) e^{3}}{143}+\frac {384 b^{4} \left (\frac {33}{133} b^{2} x^{2}+\frac {18}{19} a b x +a^{2}\right ) d^{4} e^{2}}{2431}-\frac {1536 b^{5} \left (\frac {3 b x}{7}+a \right ) d^{5} e}{46189}+\frac {1024 b^{6} d^{6}}{323323}\right )}{9 e^{7}}\) | \(276\) |
| gosper | \(\frac {2 \left (e x +d \right )^{\frac {9}{2}} \left (138567 x^{6} b^{6} e^{6}+918918 x^{5} a \,b^{5} e^{6}-87516 x^{5} b^{6} d \,e^{5}+2567565 x^{4} a^{2} b^{4} e^{6}-540540 x^{4} a \,b^{5} d \,e^{5}+51480 x^{4} b^{6} d^{2} e^{4}+3879876 x^{3} a^{3} b^{3} e^{6}-1369368 x^{3} a^{2} b^{4} d \,e^{5}+288288 x^{3} a \,b^{5} d^{2} e^{4}-27456 x^{3} b^{6} d^{3} e^{3}+3357585 x^{2} a^{4} b^{2} e^{6}-1790712 x^{2} a^{3} b^{3} d \,e^{5}+632016 x^{2} a^{2} b^{4} d^{2} e^{4}-133056 x^{2} a \,b^{5} d^{3} e^{3}+12672 x^{2} b^{6} d^{4} e^{2}+1587222 x \,a^{5} b \,e^{6}-1220940 x \,a^{4} b^{2} d \,e^{5}+651168 x \,a^{3} b^{3} d^{2} e^{4}-229824 x \,a^{2} b^{4} d^{3} e^{3}+48384 x a \,b^{5} d^{4} e^{2}-4608 x \,b^{6} d^{5} e +323323 a^{6} e^{6}-352716 a^{5} b d \,e^{5}+271320 a^{4} b^{2} d^{2} e^{4}-144704 a^{3} b^{3} d^{3} e^{3}+51072 a^{2} b^{4} d^{4} e^{2}-10752 a \,b^{5} d^{5} e +1024 b^{6} d^{6}\right )}{2909907 e^{7}}\) | \(377\) |
| derivativedivides | \(\frac {\frac {2 b^{6} \left (e x +d \right )^{\frac {21}{2}}}{21}+\frac {6 \left (2 a e b -2 b^{2} d \right ) b^{4} \left (e x +d \right )^{\frac {19}{2}}}{19}+\frac {2 \left (\left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) b^{4}+2 \left (2 a e b -2 b^{2} d \right )^{2} b^{2}+b^{2} \left (2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a e b -2 b^{2} d \right )^{2}\right )\right ) \left (e x +d \right )^{\frac {17}{2}}}{17}+\frac {2 \left (4 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \left (2 a e b -2 b^{2} d \right ) b^{2}+\left (2 a e b -2 b^{2} d \right ) \left (2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a e b -2 b^{2} d \right )^{2}\right )\right ) \left (e x +d \right )^{\frac {15}{2}}}{15}+\frac {2 \left (\left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \left (2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a e b -2 b^{2} d \right )^{2}\right )+2 \left (2 a e b -2 b^{2} d \right )^{2} \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )+b^{2} \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )^{2}\right ) \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {6 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )^{2} \left (2 a e b -2 b^{2} d \right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )^{3} \left (e x +d \right )^{\frac {9}{2}}}{9}}{e^{7}}\) | \(457\) |
| default | \(\frac {\frac {2 b^{6} \left (e x +d \right )^{\frac {21}{2}}}{21}+\frac {6 \left (2 a e b -2 b^{2} d \right ) b^{4} \left (e x +d \right )^{\frac {19}{2}}}{19}+\frac {2 \left (\left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) b^{4}+2 \left (2 a e b -2 b^{2} d \right )^{2} b^{2}+b^{2} \left (2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a e b -2 b^{2} d \right )^{2}\right )\right ) \left (e x +d \right )^{\frac {17}{2}}}{17}+\frac {2 \left (4 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \left (2 a e b -2 b^{2} d \right ) b^{2}+\left (2 a e b -2 b^{2} d \right ) \left (2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a e b -2 b^{2} d \right )^{2}\right )\right ) \left (e x +d \right )^{\frac {15}{2}}}{15}+\frac {2 \left (\left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \left (2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a e b -2 b^{2} d \right )^{2}\right )+2 \left (2 a e b -2 b^{2} d \right )^{2} \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )+b^{2} \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )^{2}\right ) \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {6 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )^{2} \left (2 a e b -2 b^{2} d \right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )^{3} \left (e x +d \right )^{\frac {9}{2}}}{9}}{e^{7}}\) | \(457\) |
| trager | \(\frac {2 \left (138567 e^{10} b^{6} x^{10}+918918 a \,b^{5} e^{10} x^{9}+466752 b^{6} d \,e^{9} x^{9}+2567565 a^{2} b^{4} e^{10} x^{8}+3135132 a \,b^{5} d \,e^{9} x^{8}+532818 b^{6} d^{2} e^{8} x^{8}+3879876 a^{3} b^{3} e^{10} x^{7}+8900892 a^{2} b^{4} d \,e^{9} x^{7}+3639636 a \,b^{5} d^{2} e^{8} x^{7}+207636 b^{6} d^{3} e^{7} x^{7}+3357585 a^{4} b^{2} e^{10} x^{6}+13728792 a^{3} b^{3} d \,e^{9} x^{6}+10559934 a^{2} b^{4} d^{2} e^{8} x^{6}+1452528 a \,b^{5} d^{3} e^{7} x^{6}+231 b^{6} d^{4} e^{6} x^{6}+1587222 a^{5} b \,e^{10} x^{5}+12209400 a^{4} b^{2} d \,e^{9} x^{5}+16767576 a^{3} b^{3} d^{2} e^{8} x^{5}+4352292 a^{2} b^{4} d^{3} e^{7} x^{5}+2646 a \,b^{5} d^{4} e^{6} x^{5}-252 b^{6} d^{5} e^{5} x^{5}+323323 a^{6} e^{10} x^{4}+5996172 a^{5} b d \,e^{9} x^{4}+15533070 a^{4} b^{2} d^{2} e^{8} x^{4}+7235200 a^{3} b^{3} d^{3} e^{7} x^{4}+13965 a^{2} b^{4} d^{4} e^{6} x^{4}-2940 a \,b^{5} d^{5} e^{5} x^{4}+280 b^{6} d^{6} e^{4} x^{4}+1293292 a^{6} d \,e^{9} x^{3}+8112468 a^{5} b \,d^{2} e^{8} x^{3}+7189980 a^{4} b^{2} d^{3} e^{7} x^{3}+45220 a^{3} b^{3} d^{4} e^{6} x^{3}-15960 a^{2} b^{4} d^{5} e^{5} x^{3}+3360 a \,b^{5} d^{6} e^{4} x^{3}-320 b^{6} d^{7} e^{3} x^{3}+1939938 a^{6} d^{2} e^{8} x^{2}+4232592 a^{5} b \,d^{3} e^{7} x^{2}+101745 a^{4} b^{2} d^{4} e^{6} x^{2}-54264 a^{3} b^{3} d^{5} e^{5} x^{2}+19152 a^{2} b^{4} d^{6} e^{4} x^{2}-4032 a \,b^{5} d^{7} e^{3} x^{2}+384 b^{6} d^{8} e^{2} x^{2}+1293292 a^{6} d^{3} e^{7} x +176358 a^{5} b \,d^{4} e^{6} x -135660 a^{4} b^{2} d^{5} e^{5} x +72352 a^{3} b^{3} d^{6} e^{4} x -25536 a^{2} b^{4} d^{7} e^{3} x +5376 a \,b^{5} d^{8} e^{2} x -512 b^{6} d^{9} e x +323323 a^{6} d^{4} e^{6}-352716 a^{5} b \,d^{5} e^{5}+271320 a^{4} b^{2} d^{6} e^{4}-144704 a^{3} b^{3} d^{7} e^{3}+51072 a^{2} b^{4} d^{8} e^{2}-10752 a \,b^{5} d^{9} e +1024 b^{6} d^{10}\right ) \sqrt {e x +d}}{2909907 e^{7}}\) | \(809\) |
| risch | \(\frac {2 \left (138567 e^{10} b^{6} x^{10}+918918 a \,b^{5} e^{10} x^{9}+466752 b^{6} d \,e^{9} x^{9}+2567565 a^{2} b^{4} e^{10} x^{8}+3135132 a \,b^{5} d \,e^{9} x^{8}+532818 b^{6} d^{2} e^{8} x^{8}+3879876 a^{3} b^{3} e^{10} x^{7}+8900892 a^{2} b^{4} d \,e^{9} x^{7}+3639636 a \,b^{5} d^{2} e^{8} x^{7}+207636 b^{6} d^{3} e^{7} x^{7}+3357585 a^{4} b^{2} e^{10} x^{6}+13728792 a^{3} b^{3} d \,e^{9} x^{6}+10559934 a^{2} b^{4} d^{2} e^{8} x^{6}+1452528 a \,b^{5} d^{3} e^{7} x^{6}+231 b^{6} d^{4} e^{6} x^{6}+1587222 a^{5} b \,e^{10} x^{5}+12209400 a^{4} b^{2} d \,e^{9} x^{5}+16767576 a^{3} b^{3} d^{2} e^{8} x^{5}+4352292 a^{2} b^{4} d^{3} e^{7} x^{5}+2646 a \,b^{5} d^{4} e^{6} x^{5}-252 b^{6} d^{5} e^{5} x^{5}+323323 a^{6} e^{10} x^{4}+5996172 a^{5} b d \,e^{9} x^{4}+15533070 a^{4} b^{2} d^{2} e^{8} x^{4}+7235200 a^{3} b^{3} d^{3} e^{7} x^{4}+13965 a^{2} b^{4} d^{4} e^{6} x^{4}-2940 a \,b^{5} d^{5} e^{5} x^{4}+280 b^{6} d^{6} e^{4} x^{4}+1293292 a^{6} d \,e^{9} x^{3}+8112468 a^{5} b \,d^{2} e^{8} x^{3}+7189980 a^{4} b^{2} d^{3} e^{7} x^{3}+45220 a^{3} b^{3} d^{4} e^{6} x^{3}-15960 a^{2} b^{4} d^{5} e^{5} x^{3}+3360 a \,b^{5} d^{6} e^{4} x^{3}-320 b^{6} d^{7} e^{3} x^{3}+1939938 a^{6} d^{2} e^{8} x^{2}+4232592 a^{5} b \,d^{3} e^{7} x^{2}+101745 a^{4} b^{2} d^{4} e^{6} x^{2}-54264 a^{3} b^{3} d^{5} e^{5} x^{2}+19152 a^{2} b^{4} d^{6} e^{4} x^{2}-4032 a \,b^{5} d^{7} e^{3} x^{2}+384 b^{6} d^{8} e^{2} x^{2}+1293292 a^{6} d^{3} e^{7} x +176358 a^{5} b \,d^{4} e^{6} x -135660 a^{4} b^{2} d^{5} e^{5} x +72352 a^{3} b^{3} d^{6} e^{4} x -25536 a^{2} b^{4} d^{7} e^{3} x +5376 a \,b^{5} d^{8} e^{2} x -512 b^{6} d^{9} e x +323323 a^{6} d^{4} e^{6}-352716 a^{5} b \,d^{5} e^{5}+271320 a^{4} b^{2} d^{6} e^{4}-144704 a^{3} b^{3} d^{7} e^{3}+51072 a^{2} b^{4} d^{8} e^{2}-10752 a \,b^{5} d^{9} e +1024 b^{6} d^{10}\right ) \sqrt {e x +d}}{2909907 e^{7}}\) | \(809\) |
2/9*(e*x+d)^(9/2)*((3/7*b^6*x^6+a^6+54/19*a*x^5*b^5+135/17*a^2*x^4*b^4+12* a^3*x^3*b^3+135/13*a^4*x^2*b^2+54/11*a^5*x*b)*e^6-12/11*b*d*(33/133*b^5*x^ 5+495/323*a*b^4*x^4+66/17*a^2*b^3*x^3+66/13*a^3*b^2*x^2+45/13*a^4*b*x+a^5) *e^5+120/143*b^2*d^2*(429/2261*b^4*x^4+1716/1615*a*b^3*x^3+198/85*a^2*b^2* x^2+12/5*a^3*b*x+a^4)*e^4-64/143*b^3*d^3*(429/2261*b^3*x^3+297/323*a*b^2*x ^2+27/17*a^2*b*x+a^3)*e^3+384/2431*b^4*(33/133*b^2*x^2+18/19*a*b*x+a^2)*d^ 4*e^2-1536/46189*b^5*(3/7*b*x+a)*d^5*e+1024/323323*b^6*d^6)/e^7
Leaf count of result is larger than twice the leaf count of optimal. 729 vs. \(2 (159) = 318\).
Time = 0.38 (sec) , antiderivative size = 729, normalized size of antiderivative = 3.90 \[ \int (d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {2 \, {\left (138567 \, b^{6} e^{10} x^{10} + 1024 \, b^{6} d^{10} - 10752 \, a b^{5} d^{9} e + 51072 \, a^{2} b^{4} d^{8} e^{2} - 144704 \, a^{3} b^{3} d^{7} e^{3} + 271320 \, a^{4} b^{2} d^{6} e^{4} - 352716 \, a^{5} b d^{5} e^{5} + 323323 \, a^{6} d^{4} e^{6} + 14586 \, {\left (32 \, b^{6} d e^{9} + 63 \, a b^{5} e^{10}\right )} x^{9} + 3861 \, {\left (138 \, b^{6} d^{2} e^{8} + 812 \, a b^{5} d e^{9} + 665 \, a^{2} b^{4} e^{10}\right )} x^{8} + 1716 \, {\left (121 \, b^{6} d^{3} e^{7} + 2121 \, a b^{5} d^{2} e^{8} + 5187 \, a^{2} b^{4} d e^{9} + 2261 \, a^{3} b^{3} e^{10}\right )} x^{7} + 231 \, {\left (b^{6} d^{4} e^{6} + 6288 \, a b^{5} d^{3} e^{7} + 45714 \, a^{2} b^{4} d^{2} e^{8} + 59432 \, a^{3} b^{3} d e^{9} + 14535 \, a^{4} b^{2} e^{10}\right )} x^{6} - 126 \, {\left (2 \, b^{6} d^{5} e^{5} - 21 \, a b^{5} d^{4} e^{6} - 34542 \, a^{2} b^{4} d^{3} e^{7} - 133076 \, a^{3} b^{3} d^{2} e^{8} - 96900 \, a^{4} b^{2} d e^{9} - 12597 \, a^{5} b e^{10}\right )} x^{5} + 7 \, {\left (40 \, b^{6} d^{6} e^{4} - 420 \, a b^{5} d^{5} e^{5} + 1995 \, a^{2} b^{4} d^{4} e^{6} + 1033600 \, a^{3} b^{3} d^{3} e^{7} + 2219010 \, a^{4} b^{2} d^{2} e^{8} + 856596 \, a^{5} b d e^{9} + 46189 \, a^{6} e^{10}\right )} x^{4} - 4 \, {\left (80 \, b^{6} d^{7} e^{3} - 840 \, a b^{5} d^{6} e^{4} + 3990 \, a^{2} b^{4} d^{5} e^{5} - 11305 \, a^{3} b^{3} d^{4} e^{6} - 1797495 \, a^{4} b^{2} d^{3} e^{7} - 2028117 \, a^{5} b d^{2} e^{8} - 323323 \, a^{6} d e^{9}\right )} x^{3} + 3 \, {\left (128 \, b^{6} d^{8} e^{2} - 1344 \, a b^{5} d^{7} e^{3} + 6384 \, a^{2} b^{4} d^{6} e^{4} - 18088 \, a^{3} b^{3} d^{5} e^{5} + 33915 \, a^{4} b^{2} d^{4} e^{6} + 1410864 \, a^{5} b d^{3} e^{7} + 646646 \, a^{6} d^{2} e^{8}\right )} x^{2} - 2 \, {\left (256 \, b^{6} d^{9} e - 2688 \, a b^{5} d^{8} e^{2} + 12768 \, a^{2} b^{4} d^{7} e^{3} - 36176 \, a^{3} b^{3} d^{6} e^{4} + 67830 \, a^{4} b^{2} d^{5} e^{5} - 88179 \, a^{5} b d^{4} e^{6} - 646646 \, a^{6} d^{3} e^{7}\right )} x\right )} \sqrt {e x + d}}{2909907 \, e^{7}} \]
2/2909907*(138567*b^6*e^10*x^10 + 1024*b^6*d^10 - 10752*a*b^5*d^9*e + 5107 2*a^2*b^4*d^8*e^2 - 144704*a^3*b^3*d^7*e^3 + 271320*a^4*b^2*d^6*e^4 - 3527 16*a^5*b*d^5*e^5 + 323323*a^6*d^4*e^6 + 14586*(32*b^6*d*e^9 + 63*a*b^5*e^1 0)*x^9 + 3861*(138*b^6*d^2*e^8 + 812*a*b^5*d*e^9 + 665*a^2*b^4*e^10)*x^8 + 1716*(121*b^6*d^3*e^7 + 2121*a*b^5*d^2*e^8 + 5187*a^2*b^4*d*e^9 + 2261*a^ 3*b^3*e^10)*x^7 + 231*(b^6*d^4*e^6 + 6288*a*b^5*d^3*e^7 + 45714*a^2*b^4*d^ 2*e^8 + 59432*a^3*b^3*d*e^9 + 14535*a^4*b^2*e^10)*x^6 - 126*(2*b^6*d^5*e^5 - 21*a*b^5*d^4*e^6 - 34542*a^2*b^4*d^3*e^7 - 133076*a^3*b^3*d^2*e^8 - 969 00*a^4*b^2*d*e^9 - 12597*a^5*b*e^10)*x^5 + 7*(40*b^6*d^6*e^4 - 420*a*b^5*d ^5*e^5 + 1995*a^2*b^4*d^4*e^6 + 1033600*a^3*b^3*d^3*e^7 + 2219010*a^4*b^2* d^2*e^8 + 856596*a^5*b*d*e^9 + 46189*a^6*e^10)*x^4 - 4*(80*b^6*d^7*e^3 - 8 40*a*b^5*d^6*e^4 + 3990*a^2*b^4*d^5*e^5 - 11305*a^3*b^3*d^4*e^6 - 1797495* a^4*b^2*d^3*e^7 - 2028117*a^5*b*d^2*e^8 - 323323*a^6*d*e^9)*x^3 + 3*(128*b ^6*d^8*e^2 - 1344*a*b^5*d^7*e^3 + 6384*a^2*b^4*d^6*e^4 - 18088*a^3*b^3*d^5 *e^5 + 33915*a^4*b^2*d^4*e^6 + 1410864*a^5*b*d^3*e^7 + 646646*a^6*d^2*e^8) *x^2 - 2*(256*b^6*d^9*e - 2688*a*b^5*d^8*e^2 + 12768*a^2*b^4*d^7*e^3 - 361 76*a^3*b^3*d^6*e^4 + 67830*a^4*b^2*d^5*e^5 - 88179*a^5*b*d^4*e^6 - 646646* a^6*d^3*e^7)*x)*sqrt(e*x + d)/e^7
Leaf count of result is larger than twice the leaf count of optimal. 495 vs. \(2 (173) = 346\).
Time = 1.69 (sec) , antiderivative size = 495, normalized size of antiderivative = 2.65 \[ \int (d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\begin {cases} \frac {2 \left (\frac {b^{6} \left (d + e x\right )^{\frac {21}{2}}}{21 e^{6}} + \frac {\left (d + e x\right )^{\frac {19}{2}} \cdot \left (6 a b^{5} e - 6 b^{6} d\right )}{19 e^{6}} + \frac {\left (d + e x\right )^{\frac {17}{2}} \cdot \left (15 a^{2} b^{4} e^{2} - 30 a b^{5} d e + 15 b^{6} d^{2}\right )}{17 e^{6}} + \frac {\left (d + e x\right )^{\frac {15}{2}} \cdot \left (20 a^{3} b^{3} e^{3} - 60 a^{2} b^{4} d e^{2} + 60 a b^{5} d^{2} e - 20 b^{6} d^{3}\right )}{15 e^{6}} + \frac {\left (d + e x\right )^{\frac {13}{2}} \cdot \left (15 a^{4} b^{2} e^{4} - 60 a^{3} b^{3} d e^{3} + 90 a^{2} b^{4} d^{2} e^{2} - 60 a b^{5} d^{3} e + 15 b^{6} d^{4}\right )}{13 e^{6}} + \frac {\left (d + e x\right )^{\frac {11}{2}} \cdot \left (6 a^{5} b e^{5} - 30 a^{4} b^{2} d e^{4} + 60 a^{3} b^{3} d^{2} e^{3} - 60 a^{2} b^{4} d^{3} e^{2} + 30 a b^{5} d^{4} e - 6 b^{6} d^{5}\right )}{11 e^{6}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \left (a^{6} e^{6} - 6 a^{5} b d e^{5} + 15 a^{4} b^{2} d^{2} e^{4} - 20 a^{3} b^{3} d^{3} e^{3} + 15 a^{2} b^{4} d^{4} e^{2} - 6 a b^{5} d^{5} e + b^{6} d^{6}\right )}{9 e^{6}}\right )}{e} & \text {for}\: e \neq 0 \\d^{\frac {7}{2}} \left (a^{6} x + 3 a^{5} b x^{2} + 5 a^{4} b^{2} x^{3} + 5 a^{3} b^{3} x^{4} + 3 a^{2} b^{4} x^{5} + a b^{5} x^{6} + \frac {b^{6} x^{7}}{7}\right ) & \text {otherwise} \end {cases} \]
Piecewise((2*(b**6*(d + e*x)**(21/2)/(21*e**6) + (d + e*x)**(19/2)*(6*a*b* *5*e - 6*b**6*d)/(19*e**6) + (d + e*x)**(17/2)*(15*a**2*b**4*e**2 - 30*a*b **5*d*e + 15*b**6*d**2)/(17*e**6) + (d + e*x)**(15/2)*(20*a**3*b**3*e**3 - 60*a**2*b**4*d*e**2 + 60*a*b**5*d**2*e - 20*b**6*d**3)/(15*e**6) + (d + e *x)**(13/2)*(15*a**4*b**2*e**4 - 60*a**3*b**3*d*e**3 + 90*a**2*b**4*d**2*e **2 - 60*a*b**5*d**3*e + 15*b**6*d**4)/(13*e**6) + (d + e*x)**(11/2)*(6*a* *5*b*e**5 - 30*a**4*b**2*d*e**4 + 60*a**3*b**3*d**2*e**3 - 60*a**2*b**4*d* *3*e**2 + 30*a*b**5*d**4*e - 6*b**6*d**5)/(11*e**6) + (d + e*x)**(9/2)*(a* *6*e**6 - 6*a**5*b*d*e**5 + 15*a**4*b**2*d**2*e**4 - 20*a**3*b**3*d**3*e** 3 + 15*a**2*b**4*d**4*e**2 - 6*a*b**5*d**5*e + b**6*d**6)/(9*e**6))/e, Ne( e, 0)), (d**(7/2)*(a**6*x + 3*a**5*b*x**2 + 5*a**4*b**2*x**3 + 5*a**3*b**3 *x**4 + 3*a**2*b**4*x**5 + a*b**5*x**6 + b**6*x**7/7), True))
Leaf count of result is larger than twice the leaf count of optimal. 350 vs. \(2 (159) = 318\).
Time = 0.20 (sec) , antiderivative size = 350, normalized size of antiderivative = 1.87 \[ \int (d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {2 \, {\left (138567 \, {\left (e x + d\right )}^{\frac {21}{2}} b^{6} - 918918 \, {\left (b^{6} d - a b^{5} e\right )} {\left (e x + d\right )}^{\frac {19}{2}} + 2567565 \, {\left (b^{6} d^{2} - 2 \, a b^{5} d e + a^{2} b^{4} e^{2}\right )} {\left (e x + d\right )}^{\frac {17}{2}} - 3879876 \, {\left (b^{6} d^{3} - 3 \, a b^{5} d^{2} e + 3 \, a^{2} b^{4} d e^{2} - a^{3} b^{3} e^{3}\right )} {\left (e x + d\right )}^{\frac {15}{2}} + 3357585 \, {\left (b^{6} d^{4} - 4 \, a b^{5} d^{3} e + 6 \, a^{2} b^{4} d^{2} e^{2} - 4 \, a^{3} b^{3} d e^{3} + a^{4} b^{2} e^{4}\right )} {\left (e x + d\right )}^{\frac {13}{2}} - 1587222 \, {\left (b^{6} d^{5} - 5 \, a b^{5} d^{4} e + 10 \, a^{2} b^{4} d^{3} e^{2} - 10 \, a^{3} b^{3} d^{2} e^{3} + 5 \, a^{4} b^{2} d e^{4} - a^{5} b e^{5}\right )} {\left (e x + d\right )}^{\frac {11}{2}} + 323323 \, {\left (b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} + a^{6} e^{6}\right )} {\left (e x + d\right )}^{\frac {9}{2}}\right )}}{2909907 \, e^{7}} \]
2/2909907*(138567*(e*x + d)^(21/2)*b^6 - 918918*(b^6*d - a*b^5*e)*(e*x + d )^(19/2) + 2567565*(b^6*d^2 - 2*a*b^5*d*e + a^2*b^4*e^2)*(e*x + d)^(17/2) - 3879876*(b^6*d^3 - 3*a*b^5*d^2*e + 3*a^2*b^4*d*e^2 - a^3*b^3*e^3)*(e*x + d)^(15/2) + 3357585*(b^6*d^4 - 4*a*b^5*d^3*e + 6*a^2*b^4*d^2*e^2 - 4*a^3* b^3*d*e^3 + a^4*b^2*e^4)*(e*x + d)^(13/2) - 1587222*(b^6*d^5 - 5*a*b^5*d^4 *e + 10*a^2*b^4*d^3*e^2 - 10*a^3*b^3*d^2*e^3 + 5*a^4*b^2*d*e^4 - a^5*b*e^5 )*(e*x + d)^(11/2) + 323323*(b^6*d^6 - 6*a*b^5*d^5*e + 15*a^2*b^4*d^4*e^2 - 20*a^3*b^3*d^3*e^3 + 15*a^4*b^2*d^2*e^4 - 6*a^5*b*d*e^5 + a^6*e^6)*(e*x + d)^(9/2))/e^7
Leaf count of result is larger than twice the leaf count of optimal. 2771 vs. \(2 (159) = 318\).
Time = 0.33 (sec) , antiderivative size = 2771, normalized size of antiderivative = 14.82 \[ \int (d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\text {Too large to display} \]
2/14549535*(14549535*sqrt(e*x + d)*a^6*d^4 + 19399380*((e*x + d)^(3/2) - 3 *sqrt(e*x + d)*d)*a^6*d^3 + 29099070*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d) *a^5*b*d^4/e + 5819814*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt (e*x + d)*d^2)*a^6*d^2 + 14549535*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)* d + 15*sqrt(e*x + d)*d^2)*a^4*b^2*d^4/e^2 + 23279256*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*a^5*b*d^3/e + 1662804*(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^6*d + 8314020*(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^3*d^4/e^3 + 24942060*(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^2*d^3/e^2 + 14965236*(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*d^2/e + 461 89*(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^6 + 692835*(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^4*d^4/e^4 + 3695120*(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^3*d^3/e^3 + 4157010*(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^2*d^2/e^2 + 1108536*(35*(e*x + ...
Time = 9.59 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.87 \[ \int (d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {2\,b^6\,{\left (d+e\,x\right )}^{21/2}}{21\,e^7}-\frac {\left (12\,b^6\,d-12\,a\,b^5\,e\right )\,{\left (d+e\,x\right )}^{19/2}}{19\,e^7}+\frac {2\,{\left (a\,e-b\,d\right )}^6\,{\left (d+e\,x\right )}^{9/2}}{9\,e^7}+\frac {30\,b^2\,{\left (a\,e-b\,d\right )}^4\,{\left (d+e\,x\right )}^{13/2}}{13\,e^7}+\frac {8\,b^3\,{\left (a\,e-b\,d\right )}^3\,{\left (d+e\,x\right )}^{15/2}}{3\,e^7}+\frac {30\,b^4\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{17/2}}{17\,e^7}+\frac {12\,b\,{\left (a\,e-b\,d\right )}^5\,{\left (d+e\,x\right )}^{11/2}}{11\,e^7} \]
(2*b^6*(d + e*x)^(21/2))/(21*e^7) - ((12*b^6*d - 12*a*b^5*e)*(d + e*x)^(19 /2))/(19*e^7) + (2*(a*e - b*d)^6*(d + e*x)^(9/2))/(9*e^7) + (30*b^2*(a*e - b*d)^4*(d + e*x)^(13/2))/(13*e^7) + (8*b^3*(a*e - b*d)^3*(d + e*x)^(15/2) )/(3*e^7) + (30*b^4*(a*e - b*d)^2*(d + e*x)^(17/2))/(17*e^7) + (12*b*(a*e - b*d)^5*(d + e*x)^(11/2))/(11*e^7)