Integrand size = 33, antiderivative size = 158 \[ \int (a+b x) (d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=-\frac {2 (b d-a e)^5 (d+e x)^{9/2}}{9 e^6}+\frac {10 b (b d-a e)^4 (d+e x)^{11/2}}{11 e^6}-\frac {20 b^2 (b d-a e)^3 (d+e x)^{13/2}}{13 e^6}+\frac {4 b^3 (b d-a e)^2 (d+e x)^{15/2}}{3 e^6}-\frac {10 b^4 (b d-a e) (d+e x)^{17/2}}{17 e^6}+\frac {2 b^5 (d+e x)^{19/2}}{19 e^6} \] Output:
-2/9*(-a*e+b*d)^5*(e*x+d)^(9/2)/e^6+10/11*b*(-a*e+b*d)^4*(e*x+d)^(11/2)/e^ 6-20/13*b^2*(-a*e+b*d)^3*(e*x+d)^(13/2)/e^6+4/3*b^3*(-a*e+b*d)^2*(e*x+d)^( 15/2)/e^6-10/17*b^4*(-a*e+b*d)*(e*x+d)^(17/2)/e^6+2/19*b^5*(e*x+d)^(19/2)/ e^6
Time = 0.24 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.37 \[ \int (a+b x) (d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {2 (d+e x)^{9/2} \left (46189 a^5 e^5+20995 a^4 b e^4 (-2 d+9 e x)+3230 a^3 b^2 e^3 \left (8 d^2-36 d e x+99 e^2 x^2\right )+646 a^2 b^3 e^2 \left (-16 d^3+72 d^2 e x-198 d e^2 x^2+429 e^3 x^3\right )+19 a b^4 e \left (128 d^4-576 d^3 e x+1584 d^2 e^2 x^2-3432 d e^3 x^3+6435 e^4 x^4\right )+b^5 \left (-256 d^5+1152 d^4 e x-3168 d^3 e^2 x^2+6864 d^2 e^3 x^3-12870 d e^4 x^4+21879 e^5 x^5\right )\right )}{415701 e^6} \] Input:
Integrate[(a + b*x)*(d + e*x)^(7/2)*(a^2 + 2*a*b*x + b^2*x^2)^2,x]
Output:
(2*(d + e*x)^(9/2)*(46189*a^5*e^5 + 20995*a^4*b*e^4*(-2*d + 9*e*x) + 3230* a^3*b^2*e^3*(8*d^2 - 36*d*e*x + 99*e^2*x^2) + 646*a^2*b^3*e^2*(-16*d^3 + 7 2*d^2*e*x - 198*d*e^2*x^2 + 429*e^3*x^3) + 19*a*b^4*e*(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) + b^5*(-256*d^5 + 1 152*d^4*e*x - 3168*d^3*e^2*x^2 + 6864*d^2*e^3*x^3 - 12870*d*e^4*x^4 + 2187 9*e^5*x^5)))/(415701*e^6)
Time = 0.54 (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.
| \(\displaystyle \int (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^2 (d+e x)^{7/2} \, dx\) |
| \(\Big \downarrow \) 1184 |
| \(\displaystyle \frac {\int b^4 (a+b x)^5 (d+e x)^{7/2}dx}{b^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int (a+b x)^5 (d+e x)^{7/2}dx\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle \int \left (-\frac {5 b^4 (d+e x)^{15/2} (b d-a e)}{e^5}+\frac {10 b^3 (d+e x)^{13/2} (b d-a e)^2}{e^5}-\frac {10 b^2 (d+e x)^{11/2} (b d-a e)^3}{e^5}+\frac {5 b (d+e x)^{9/2} (b d-a e)^4}{e^5}+\frac {(d+e x)^{7/2} (a e-b d)^5}{e^5}+\frac {b^5 (d+e x)^{17/2}}{e^5}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {10 b^4 (d+e x)^{17/2} (b d-a e)}{17 e^6}+\frac {4 b^3 (d+e x)^{15/2} (b d-a e)^2}{3 e^6}-\frac {20 b^2 (d+e x)^{13/2} (b d-a e)^3}{13 e^6}+\frac {10 b (d+e x)^{11/2} (b d-a e)^4}{11 e^6}-\frac {2 (d+e x)^{9/2} (b d-a e)^5}{9 e^6}+\frac {2 b^5 (d+e x)^{19/2}}{19 e^6}\) |
Input:
Int[(a + b*x)*(d + e*x)^(7/2)*(a^2 + 2*a*b*x + b^2*x^2)^2,x]
Output:
(-2*(b*d - a*e)^5*(d + e*x)^(9/2))/(9*e^6) + (10*b*(b*d - a*e)^4*(d + e*x) ^(11/2))/(11*e^6) - (20*b^2*(b*d - a*e)^3*(d + e*x)^(13/2))/(13*e^6) + (4* b^3*(b*d - a*e)^2*(d + e*x)^(15/2))/(3*e^6) - (10*b^4*(b*d - a*e)*(d + e*x )^(17/2))/(17*e^6) + (2*b^5*(d + e*x)^(19/2))/(19*e^6)
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 = 2.00 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.29
| method | result | size |
| pseudoelliptic | \(\frac {2 \left (e x +d \right )^{\frac {9}{2}} \left (\left (\frac {9}{19} b^{5} x^{5}+\frac {45}{17} a \,b^{4} x^{4}+6 a^{2} b^{3} x^{3}+\frac {90}{13} a^{3} b^{2} x^{2}+\frac {45}{11} a^{4} b x +a^{5}\right ) e^{5}-\frac {10 b d \left (\frac {99}{323} b^{4} x^{4}+\frac {132}{85} a \,b^{3} x^{3}+\frac {198}{65} a^{2} b^{2} x^{2}+\frac {36}{13} a^{3} b x +a^{4}\right ) e^{4}}{11}+\frac {80 b^{2} \left (\frac {429}{1615} b^{3} x^{3}+\frac {99}{85} a \,b^{2} x^{2}+\frac {9}{5} a^{2} b x +a^{3}\right ) d^{2} e^{3}}{143}-\frac {32 b^{3} \left (\frac {99}{323} b^{2} x^{2}+\frac {18}{17} a b x +a^{2}\right ) d^{3} e^{2}}{143}+\frac {128 \left (\frac {9 b x}{19}+a \right ) b^{4} d^{4} e}{2431}-\frac {256 b^{5} d^{5}}{46189}\right )}{9 e^{6}}\) | \(204\) |
| gosper | \(\frac {2 \left (e x +d \right )^{\frac {9}{2}} \left (21879 x^{5} e^{5} b^{5}+122265 x^{4} a \,b^{4} e^{5}-12870 x^{4} b^{5} d \,e^{4}+277134 x^{3} a^{2} b^{3} e^{5}-65208 x^{3} a \,b^{4} d \,e^{4}+6864 x^{3} b^{5} d^{2} e^{3}+319770 x^{2} a^{3} b^{2} e^{5}-127908 x^{2} a^{2} b^{3} d \,e^{4}+30096 x^{2} a \,b^{4} d^{2} e^{3}-3168 x^{2} b^{5} d^{3} e^{2}+188955 a^{4} b \,e^{5} x -116280 a^{3} b^{2} d \,e^{4} x +46512 x \,a^{2} b^{3} d^{2} e^{3}-10944 x a \,b^{4} d^{3} e^{2}+1152 b^{5} d^{4} e x +46189 e^{5} a^{5}-41990 a^{4} b d \,e^{4}+25840 a^{3} b^{2} d^{2} e^{3}-10336 a^{2} b^{3} d^{3} e^{2}+2432 a \,b^{4} d^{4} e -256 b^{5} d^{5}\right )}{415701 e^{6}}\) | \(273\) |
| orering | \(\frac {2 \left (21879 x^{5} e^{5} b^{5}+122265 x^{4} a \,b^{4} e^{5}-12870 x^{4} b^{5} d \,e^{4}+277134 x^{3} a^{2} b^{3} e^{5}-65208 x^{3} a \,b^{4} d \,e^{4}+6864 x^{3} b^{5} d^{2} e^{3}+319770 x^{2} a^{3} b^{2} e^{5}-127908 x^{2} a^{2} b^{3} d \,e^{4}+30096 x^{2} a \,b^{4} d^{2} e^{3}-3168 x^{2} b^{5} d^{3} e^{2}+188955 a^{4} b \,e^{5} x -116280 a^{3} b^{2} d \,e^{4} x +46512 x \,a^{2} b^{3} d^{2} e^{3}-10944 x a \,b^{4} d^{3} e^{2}+1152 b^{5} d^{4} e x +46189 e^{5} a^{5}-41990 a^{4} b d \,e^{4}+25840 a^{3} b^{2} d^{2} e^{3}-10336 a^{2} b^{3} d^{3} e^{2}+2432 a \,b^{4} d^{4} e -256 b^{5} d^{5}\right ) \left (e x +d \right )^{\frac {9}{2}} \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{2}}{415701 e^{6} \left (b x +a \right )^{4}}\) | \(298\) |
| derivativedivides | \(\frac {\frac {2 b^{5} \left (e x +d \right )^{\frac {19}{2}}}{19}+\frac {2 \left (\left (a e -b d \right ) b^{4}+2 b^{3} \left (2 a b e -2 b^{2} d \right )\right ) \left (e x +d \right )^{\frac {17}{2}}}{17}+\frac {2 \left (2 \left (a e -b d \right ) \left (2 a b e -2 b^{2} d \right ) b^{2}+b \left (2 \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a b e -2 b^{2} d \right )^{2}\right )\right ) \left (e x +d \right )^{\frac {15}{2}}}{15}+\frac {2 \left (\left (a e -b d \right ) \left (2 \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a b e -2 b^{2} d \right )^{2}\right )+2 b \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) \left (2 a b e -2 b^{2} d \right )\right ) \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (2 \left (a e -b d \right ) \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) \left (2 a b e -2 b^{2} d \right )+b \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )^{2}\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (a e -b d \right ) \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )^{2} \left (e x +d \right )^{\frac {9}{2}}}{9}}{e^{6}}\) | \(350\) |
| default | \(\frac {\frac {2 b^{5} \left (e x +d \right )^{\frac {19}{2}}}{19}+\frac {2 \left (\left (a e -b d \right ) b^{4}+2 b^{3} \left (2 a b e -2 b^{2} d \right )\right ) \left (e x +d \right )^{\frac {17}{2}}}{17}+\frac {2 \left (2 \left (a e -b d \right ) \left (2 a b e -2 b^{2} d \right ) b^{2}+b \left (2 \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a b e -2 b^{2} d \right )^{2}\right )\right ) \left (e x +d \right )^{\frac {15}{2}}}{15}+\frac {2 \left (\left (a e -b d \right ) \left (2 \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a b e -2 b^{2} d \right )^{2}\right )+2 b \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) \left (2 a b e -2 b^{2} d \right )\right ) \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (2 \left (a e -b d \right ) \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) \left (2 a b e -2 b^{2} d \right )+b \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )^{2}\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (a e -b d \right ) \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )^{2} \left (e x +d \right )^{\frac {9}{2}}}{9}}{e^{6}}\) | \(350\) |
| trager | \(\frac {2 \left (21879 b^{5} e^{9} x^{9}+122265 a \,b^{4} e^{9} x^{8}+74646 b^{5} d \,e^{8} x^{8}+277134 a^{2} b^{3} e^{9} x^{7}+423852 a \,b^{4} d \,e^{8} x^{7}+86658 b^{5} d^{2} e^{7} x^{7}+319770 a^{3} b^{2} e^{9} x^{6}+980628 a^{2} b^{3} d \,e^{8} x^{6}+502854 a \,b^{4} d^{2} e^{7} x^{6}+34584 b^{5} d^{3} e^{6} x^{6}+188955 a^{4} b \,e^{9} x^{5}+1162800 a^{3} b^{2} d \,e^{8} x^{5}+1197684 a^{2} b^{3} d^{2} e^{7} x^{5}+207252 a \,b^{4} d^{3} e^{6} x^{5}+63 b^{5} d^{4} e^{5} x^{5}+46189 a^{5} e^{9} x^{4}+713830 a^{4} b d \,e^{8} x^{4}+1479340 a^{3} b^{2} d^{2} e^{7} x^{4}+516800 a^{2} b^{3} d^{3} e^{6} x^{4}+665 a \,b^{4} d^{4} e^{5} x^{4}-70 b^{5} d^{5} e^{4} x^{4}+184756 a^{5} d \,e^{8} x^{3}+965770 a^{4} b \,d^{2} e^{7} x^{3}+684760 a^{3} b^{2} d^{3} e^{6} x^{3}+3230 a^{2} b^{3} d^{4} e^{5} x^{3}-760 a \,b^{4} d^{5} e^{4} x^{3}+80 b^{5} d^{6} e^{3} x^{3}+277134 a^{5} d^{2} e^{7} x^{2}+503880 a^{4} b \,d^{3} e^{6} x^{2}+9690 a^{3} b^{2} d^{4} e^{5} x^{2}-3876 a^{2} b^{3} d^{5} e^{4} x^{2}+912 a \,b^{4} d^{6} e^{3} x^{2}-96 b^{5} d^{7} e^{2} x^{2}+184756 a^{5} d^{3} e^{6} x +20995 a^{4} b \,d^{4} e^{5} x -12920 a^{3} b^{2} d^{5} e^{4} x +5168 a^{2} b^{3} d^{6} e^{3} x -1216 a \,b^{4} d^{7} e^{2} x +128 b^{5} d^{8} e x +46189 a^{5} d^{4} e^{5}-41990 a^{4} b \,d^{5} e^{4}+25840 a^{3} b^{2} d^{6} e^{3}-10336 a^{2} b^{3} d^{7} e^{2}+2432 a \,b^{4} d^{8} e -256 b^{5} d^{9}\right ) \sqrt {e x +d}}{415701 e^{6}}\) | \(637\) |
| risch | \(\frac {2 \left (21879 b^{5} e^{9} x^{9}+122265 a \,b^{4} e^{9} x^{8}+74646 b^{5} d \,e^{8} x^{8}+277134 a^{2} b^{3} e^{9} x^{7}+423852 a \,b^{4} d \,e^{8} x^{7}+86658 b^{5} d^{2} e^{7} x^{7}+319770 a^{3} b^{2} e^{9} x^{6}+980628 a^{2} b^{3} d \,e^{8} x^{6}+502854 a \,b^{4} d^{2} e^{7} x^{6}+34584 b^{5} d^{3} e^{6} x^{6}+188955 a^{4} b \,e^{9} x^{5}+1162800 a^{3} b^{2} d \,e^{8} x^{5}+1197684 a^{2} b^{3} d^{2} e^{7} x^{5}+207252 a \,b^{4} d^{3} e^{6} x^{5}+63 b^{5} d^{4} e^{5} x^{5}+46189 a^{5} e^{9} x^{4}+713830 a^{4} b d \,e^{8} x^{4}+1479340 a^{3} b^{2} d^{2} e^{7} x^{4}+516800 a^{2} b^{3} d^{3} e^{6} x^{4}+665 a \,b^{4} d^{4} e^{5} x^{4}-70 b^{5} d^{5} e^{4} x^{4}+184756 a^{5} d \,e^{8} x^{3}+965770 a^{4} b \,d^{2} e^{7} x^{3}+684760 a^{3} b^{2} d^{3} e^{6} x^{3}+3230 a^{2} b^{3} d^{4} e^{5} x^{3}-760 a \,b^{4} d^{5} e^{4} x^{3}+80 b^{5} d^{6} e^{3} x^{3}+277134 a^{5} d^{2} e^{7} x^{2}+503880 a^{4} b \,d^{3} e^{6} x^{2}+9690 a^{3} b^{2} d^{4} e^{5} x^{2}-3876 a^{2} b^{3} d^{5} e^{4} x^{2}+912 a \,b^{4} d^{6} e^{3} x^{2}-96 b^{5} d^{7} e^{2} x^{2}+184756 a^{5} d^{3} e^{6} x +20995 a^{4} b \,d^{4} e^{5} x -12920 a^{3} b^{2} d^{5} e^{4} x +5168 a^{2} b^{3} d^{6} e^{3} x -1216 a \,b^{4} d^{7} e^{2} x +128 b^{5} d^{8} e x +46189 a^{5} d^{4} e^{5}-41990 a^{4} b \,d^{5} e^{4}+25840 a^{3} b^{2} d^{6} e^{3}-10336 a^{2} b^{3} d^{7} e^{2}+2432 a \,b^{4} d^{8} e -256 b^{5} d^{9}\right ) \sqrt {e x +d}}{415701 e^{6}}\) | \(637\) |
Input:
int((b*x+a)*(e*x+d)^(7/2)*(b^2*x^2+2*a*b*x+a^2)^2,x,method=_RETURNVERBOSE)
Output:
2/9*(e*x+d)^(9/2)*((9/19*b^5*x^5+45/17*a*b^4*x^4+6*a^2*b^3*x^3+90/13*a^3*b ^2*x^2+45/11*a^4*b*x+a^5)*e^5-10/11*b*d*(99/323*b^4*x^4+132/85*a*b^3*x^3+1 98/65*a^2*b^2*x^2+36/13*a^3*b*x+a^4)*e^4+80/143*b^2*(429/1615*b^3*x^3+99/8 5*a*b^2*x^2+9/5*a^2*b*x+a^3)*d^2*e^3-32/143*b^3*(99/323*b^2*x^2+18/17*a*b* x+a^2)*d^3*e^2+128/2431*(9/19*b*x+a)*b^4*d^4*e-256/46189*b^5*d^5)/e^6
Leaf count of result is larger than twice the leaf count of optimal. 579 vs. \(2 (134) = 268\).
Time = 0.09 (sec) , antiderivative size = 579, normalized size of antiderivative = 3.66 \[ \int (a+b x) (d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {2 \, {\left (21879 \, b^{5} e^{9} x^{9} - 256 \, b^{5} d^{9} + 2432 \, a b^{4} d^{8} e - 10336 \, a^{2} b^{3} d^{7} e^{2} + 25840 \, a^{3} b^{2} d^{6} e^{3} - 41990 \, a^{4} b d^{5} e^{4} + 46189 \, a^{5} d^{4} e^{5} + 1287 \, {\left (58 \, b^{5} d e^{8} + 95 \, a b^{4} e^{9}\right )} x^{8} + 858 \, {\left (101 \, b^{5} d^{2} e^{7} + 494 \, a b^{4} d e^{8} + 323 \, a^{2} b^{3} e^{9}\right )} x^{7} + 66 \, {\left (524 \, b^{5} d^{3} e^{6} + 7619 \, a b^{4} d^{2} e^{7} + 14858 \, a^{2} b^{3} d e^{8} + 4845 \, a^{3} b^{2} e^{9}\right )} x^{6} + 9 \, {\left (7 \, b^{5} d^{4} e^{5} + 23028 \, a b^{4} d^{3} e^{6} + 133076 \, a^{2} b^{3} d^{2} e^{7} + 129200 \, a^{3} b^{2} d e^{8} + 20995 \, a^{4} b e^{9}\right )} x^{5} - {\left (70 \, b^{5} d^{5} e^{4} - 665 \, a b^{4} d^{4} e^{5} - 516800 \, a^{2} b^{3} d^{3} e^{6} - 1479340 \, a^{3} b^{2} d^{2} e^{7} - 713830 \, a^{4} b d e^{8} - 46189 \, a^{5} e^{9}\right )} x^{4} + 2 \, {\left (40 \, b^{5} d^{6} e^{3} - 380 \, a b^{4} d^{5} e^{4} + 1615 \, a^{2} b^{3} d^{4} e^{5} + 342380 \, a^{3} b^{2} d^{3} e^{6} + 482885 \, a^{4} b d^{2} e^{7} + 92378 \, a^{5} d e^{8}\right )} x^{3} - 6 \, {\left (16 \, b^{5} d^{7} e^{2} - 152 \, a b^{4} d^{6} e^{3} + 646 \, a^{2} b^{3} d^{5} e^{4} - 1615 \, a^{3} b^{2} d^{4} e^{5} - 83980 \, a^{4} b d^{3} e^{6} - 46189 \, a^{5} d^{2} e^{7}\right )} x^{2} + {\left (128 \, b^{5} d^{8} e - 1216 \, a b^{4} d^{7} e^{2} + 5168 \, a^{2} b^{3} d^{6} e^{3} - 12920 \, a^{3} b^{2} d^{5} e^{4} + 20995 \, a^{4} b d^{4} e^{5} + 184756 \, a^{5} d^{3} e^{6}\right )} x\right )} \sqrt {e x + d}}{415701 \, e^{6}} \] Input:
integrate((b*x+a)*(e*x+d)^(7/2)*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="fric as")
Output:
2/415701*(21879*b^5*e^9*x^9 - 256*b^5*d^9 + 2432*a*b^4*d^8*e - 10336*a^2*b ^3*d^7*e^2 + 25840*a^3*b^2*d^6*e^3 - 41990*a^4*b*d^5*e^4 + 46189*a^5*d^4*e ^5 + 1287*(58*b^5*d*e^8 + 95*a*b^4*e^9)*x^8 + 858*(101*b^5*d^2*e^7 + 494*a *b^4*d*e^8 + 323*a^2*b^3*e^9)*x^7 + 66*(524*b^5*d^3*e^6 + 7619*a*b^4*d^2*e ^7 + 14858*a^2*b^3*d*e^8 + 4845*a^3*b^2*e^9)*x^6 + 9*(7*b^5*d^4*e^5 + 2302 8*a*b^4*d^3*e^6 + 133076*a^2*b^3*d^2*e^7 + 129200*a^3*b^2*d*e^8 + 20995*a^ 4*b*e^9)*x^5 - (70*b^5*d^5*e^4 - 665*a*b^4*d^4*e^5 - 516800*a^2*b^3*d^3*e^ 6 - 1479340*a^3*b^2*d^2*e^7 - 713830*a^4*b*d*e^8 - 46189*a^5*e^9)*x^4 + 2* (40*b^5*d^6*e^3 - 380*a*b^4*d^5*e^4 + 1615*a^2*b^3*d^4*e^5 + 342380*a^3*b^ 2*d^3*e^6 + 482885*a^4*b*d^2*e^7 + 92378*a^5*d*e^8)*x^3 - 6*(16*b^5*d^7*e^ 2 - 152*a*b^4*d^6*e^3 + 646*a^2*b^3*d^5*e^4 - 1615*a^3*b^2*d^4*e^5 - 83980 *a^4*b*d^3*e^6 - 46189*a^5*d^2*e^7)*x^2 + (128*b^5*d^8*e - 1216*a*b^4*d^7* e^2 + 5168*a^2*b^3*d^6*e^3 - 12920*a^3*b^2*d^5*e^4 + 20995*a^4*b*d^4*e^5 + 184756*a^5*d^3*e^6)*x)*sqrt(e*x + d)/e^6
Leaf count of result is larger than twice the leaf count of optimal. 1187 vs. \(2 (146) = 292\).
Time = 1.13 (sec) , antiderivative size = 1187, normalized size of antiderivative = 7.51 \[ \int (a+b x) (d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx =\text {Too large to display} \] Input:
integrate((b*x+a)*(e*x+d)**(7/2)*(b**2*x**2+2*a*b*x+a**2)**2,x)
Output:
Piecewise((2*a**5*d**4*sqrt(d + e*x)/(9*e) + 8*a**5*d**3*x*sqrt(d + e*x)/9 + 4*a**5*d**2*e*x**2*sqrt(d + e*x)/3 + 8*a**5*d*e**2*x**3*sqrt(d + e*x)/9 + 2*a**5*e**3*x**4*sqrt(d + e*x)/9 - 20*a**4*b*d**5*sqrt(d + e*x)/(99*e** 2) + 10*a**4*b*d**4*x*sqrt(d + e*x)/(99*e) + 80*a**4*b*d**3*x**2*sqrt(d + e*x)/33 + 460*a**4*b*d**2*e*x**3*sqrt(d + e*x)/99 + 340*a**4*b*d*e**2*x**4 *sqrt(d + e*x)/99 + 10*a**4*b*e**3*x**5*sqrt(d + e*x)/11 + 160*a**3*b**2*d **6*sqrt(d + e*x)/(1287*e**3) - 80*a**3*b**2*d**5*x*sqrt(d + e*x)/(1287*e* *2) + 20*a**3*b**2*d**4*x**2*sqrt(d + e*x)/(429*e) + 4240*a**3*b**2*d**3*x **3*sqrt(d + e*x)/1287 + 9160*a**3*b**2*d**2*e*x**4*sqrt(d + e*x)/1287 + 8 00*a**3*b**2*d*e**2*x**5*sqrt(d + e*x)/143 + 20*a**3*b**2*e**3*x**6*sqrt(d + e*x)/13 - 64*a**2*b**3*d**7*sqrt(d + e*x)/(1287*e**4) + 32*a**2*b**3*d* *6*x*sqrt(d + e*x)/(1287*e**3) - 8*a**2*b**3*d**5*x**2*sqrt(d + e*x)/(429* e**2) + 20*a**2*b**3*d**4*x**3*sqrt(d + e*x)/(1287*e) + 3200*a**2*b**3*d** 3*x**4*sqrt(d + e*x)/1287 + 824*a**2*b**3*d**2*e*x**5*sqrt(d + e*x)/143 + 184*a**2*b**3*d*e**2*x**6*sqrt(d + e*x)/39 + 4*a**2*b**3*e**3*x**7*sqrt(d + e*x)/3 + 256*a*b**4*d**8*sqrt(d + e*x)/(21879*e**5) - 128*a*b**4*d**7*x* sqrt(d + e*x)/(21879*e**4) + 32*a*b**4*d**6*x**2*sqrt(d + e*x)/(7293*e**3) - 80*a*b**4*d**5*x**3*sqrt(d + e*x)/(21879*e**2) + 70*a*b**4*d**4*x**4*sq rt(d + e*x)/(21879*e) + 2424*a*b**4*d**3*x**5*sqrt(d + e*x)/2431 + 1604*a* b**4*d**2*e*x**6*sqrt(d + e*x)/663 + 104*a*b**4*d*e**2*x**7*sqrt(d + e*...
Time = 0.03 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.64 \[ \int (a+b x) (d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {2 \, {\left (21879 \, {\left (e x + d\right )}^{\frac {19}{2}} b^{5} - 122265 \, {\left (b^{5} d - a b^{4} e\right )} {\left (e x + d\right )}^{\frac {17}{2}} + 277134 \, {\left (b^{5} d^{2} - 2 \, a b^{4} d e + a^{2} b^{3} e^{2}\right )} {\left (e x + d\right )}^{\frac {15}{2}} - 319770 \, {\left (b^{5} d^{3} - 3 \, a b^{4} d^{2} e + 3 \, a^{2} b^{3} d e^{2} - a^{3} b^{2} e^{3}\right )} {\left (e x + d\right )}^{\frac {13}{2}} + 188955 \, {\left (b^{5} d^{4} - 4 \, a b^{4} d^{3} e + 6 \, a^{2} b^{3} d^{2} e^{2} - 4 \, a^{3} b^{2} d e^{3} + a^{4} b e^{4}\right )} {\left (e x + d\right )}^{\frac {11}{2}} - 46189 \, {\left (b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}\right )} {\left (e x + d\right )}^{\frac {9}{2}}\right )}}{415701 \, e^{6}} \] Input:
integrate((b*x+a)*(e*x+d)^(7/2)*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="maxi ma")
Output:
2/415701*(21879*(e*x + d)^(19/2)*b^5 - 122265*(b^5*d - a*b^4*e)*(e*x + d)^ (17/2) + 277134*(b^5*d^2 - 2*a*b^4*d*e + a^2*b^3*e^2)*(e*x + d)^(15/2) - 3 19770*(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)^ (13/2) + 188955*(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)^(11/2) - 46189*(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)^( 9/2))/e^6
Leaf count of result is larger than twice the leaf count of optimal. 2186 vs. \(2 (134) = 268\).
Time = 0.18 (sec) , antiderivative size = 2186, normalized size of antiderivative = 13.84 \[ \int (a+b x) (d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\text {Too large to display} \] Input:
integrate((b*x+a)*(e*x+d)^(7/2)*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="giac ")
Output:
2/14549535*(14549535*sqrt(e*x + d)*a^5*d^4 + 19399380*((e*x + d)^(3/2) - 3 *sqrt(e*x + d)*d)*a^5*d^3 + 24249225*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d) *a^4*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^5*d^2 + 9699690*(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^4/e^2 + 19399380*(3*(e*x + d)^(5/2) - 1 0*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*a^4*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^5*d + 4157010*(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^4/e^3 + 16628040*(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^3/e^2 + 12471030*(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*d^2/e + 4618 9*(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^5 + 230945*(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*b^4*d^4/e^4 + 1847560*(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^3/e^3 + 2771340*(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*d^2/e^2 + 923780*(35*(e*x + d)^(...
Time = 11.10 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.87 \[ \int (a+b x) (d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {2\,b^5\,{\left (d+e\,x\right )}^{19/2}}{19\,e^6}-\frac {\left (10\,b^5\,d-10\,a\,b^4\,e\right )\,{\left (d+e\,x\right )}^{17/2}}{17\,e^6}+\frac {2\,{\left (a\,e-b\,d\right )}^5\,{\left (d+e\,x\right )}^{9/2}}{9\,e^6}+\frac {20\,b^2\,{\left (a\,e-b\,d\right )}^3\,{\left (d+e\,x\right )}^{13/2}}{13\,e^6}+\frac {4\,b^3\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{15/2}}{3\,e^6}+\frac {10\,b\,{\left (a\,e-b\,d\right )}^4\,{\left (d+e\,x\right )}^{11/2}}{11\,e^6} \] Input:
int((a + b*x)*(d + e*x)^(7/2)*(a^2 + b^2*x^2 + 2*a*b*x)^2,x)
Output:
(2*b^5*(d + e*x)^(19/2))/(19*e^6) - ((10*b^5*d - 10*a*b^4*e)*(d + e*x)^(17 /2))/(17*e^6) + (2*(a*e - b*d)^5*(d + e*x)^(9/2))/(9*e^6) + (20*b^2*(a*e - b*d)^3*(d + e*x)^(13/2))/(13*e^6) + (4*b^3*(a*e - b*d)^2*(d + e*x)^(15/2) )/(3*e^6) + (10*b*(a*e - b*d)^4*(d + e*x)^(11/2))/(11*e^6)
Time = 0.21 (sec) , antiderivative size = 635, normalized size of antiderivative = 4.02 \[ \int (a+b x) (d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {2 \sqrt {e x +d}\, \left (21879 b^{5} e^{9} x^{9}+122265 a \,b^{4} e^{9} x^{8}+74646 b^{5} d \,e^{8} x^{8}+277134 a^{2} b^{3} e^{9} x^{7}+423852 a \,b^{4} d \,e^{8} x^{7}+86658 b^{5} d^{2} e^{7} x^{7}+319770 a^{3} b^{2} e^{9} x^{6}+980628 a^{2} b^{3} d \,e^{8} x^{6}+502854 a \,b^{4} d^{2} e^{7} x^{6}+34584 b^{5} d^{3} e^{6} x^{6}+188955 a^{4} b \,e^{9} x^{5}+1162800 a^{3} b^{2} d \,e^{8} x^{5}+1197684 a^{2} b^{3} d^{2} e^{7} x^{5}+207252 a \,b^{4} d^{3} e^{6} x^{5}+63 b^{5} d^{4} e^{5} x^{5}+46189 a^{5} e^{9} x^{4}+713830 a^{4} b d \,e^{8} x^{4}+1479340 a^{3} b^{2} d^{2} e^{7} x^{4}+516800 a^{2} b^{3} d^{3} e^{6} x^{4}+665 a \,b^{4} d^{4} e^{5} x^{4}-70 b^{5} d^{5} e^{4} x^{4}+184756 a^{5} d \,e^{8} x^{3}+965770 a^{4} b \,d^{2} e^{7} x^{3}+684760 a^{3} b^{2} d^{3} e^{6} x^{3}+3230 a^{2} b^{3} d^{4} e^{5} x^{3}-760 a \,b^{4} d^{5} e^{4} x^{3}+80 b^{5} d^{6} e^{3} x^{3}+277134 a^{5} d^{2} e^{7} x^{2}+503880 a^{4} b \,d^{3} e^{6} x^{2}+9690 a^{3} b^{2} d^{4} e^{5} x^{2}-3876 a^{2} b^{3} d^{5} e^{4} x^{2}+912 a \,b^{4} d^{6} e^{3} x^{2}-96 b^{5} d^{7} e^{2} x^{2}+184756 a^{5} d^{3} e^{6} x +20995 a^{4} b \,d^{4} e^{5} x -12920 a^{3} b^{2} d^{5} e^{4} x +5168 a^{2} b^{3} d^{6} e^{3} x -1216 a \,b^{4} d^{7} e^{2} x +128 b^{5} d^{8} e x +46189 a^{5} d^{4} e^{5}-41990 a^{4} b \,d^{5} e^{4}+25840 a^{3} b^{2} d^{6} e^{3}-10336 a^{2} b^{3} d^{7} e^{2}+2432 a \,b^{4} d^{8} e -256 b^{5} d^{9}\right )}{415701 e^{6}} \] Input:
int((b*x+a)*(e*x+d)^(7/2)*(b^2*x^2+2*a*b*x+a^2)^2,x)
Output:
(2*sqrt(d + e*x)*(46189*a**5*d**4*e**5 + 184756*a**5*d**3*e**6*x + 277134* a**5*d**2*e**7*x**2 + 184756*a**5*d*e**8*x**3 + 46189*a**5*e**9*x**4 - 419 90*a**4*b*d**5*e**4 + 20995*a**4*b*d**4*e**5*x + 503880*a**4*b*d**3*e**6*x **2 + 965770*a**4*b*d**2*e**7*x**3 + 713830*a**4*b*d*e**8*x**4 + 188955*a* *4*b*e**9*x**5 + 25840*a**3*b**2*d**6*e**3 - 12920*a**3*b**2*d**5*e**4*x + 9690*a**3*b**2*d**4*e**5*x**2 + 684760*a**3*b**2*d**3*e**6*x**3 + 1479340 *a**3*b**2*d**2*e**7*x**4 + 1162800*a**3*b**2*d*e**8*x**5 + 319770*a**3*b* *2*e**9*x**6 - 10336*a**2*b**3*d**7*e**2 + 5168*a**2*b**3*d**6*e**3*x - 38 76*a**2*b**3*d**5*e**4*x**2 + 3230*a**2*b**3*d**4*e**5*x**3 + 516800*a**2* b**3*d**3*e**6*x**4 + 1197684*a**2*b**3*d**2*e**7*x**5 + 980628*a**2*b**3* d*e**8*x**6 + 277134*a**2*b**3*e**9*x**7 + 2432*a*b**4*d**8*e - 1216*a*b** 4*d**7*e**2*x + 912*a*b**4*d**6*e**3*x**2 - 760*a*b**4*d**5*e**4*x**3 + 66 5*a*b**4*d**4*e**5*x**4 + 207252*a*b**4*d**3*e**6*x**5 + 502854*a*b**4*d** 2*e**7*x**6 + 423852*a*b**4*d*e**8*x**7 + 122265*a*b**4*e**9*x**8 - 256*b* *5*d**9 + 128*b**5*d**8*e*x - 96*b**5*d**7*e**2*x**2 + 80*b**5*d**6*e**3*x **3 - 70*b**5*d**5*e**4*x**4 + 63*b**5*d**4*e**5*x**5 + 34584*b**5*d**3*e* *6*x**6 + 86658*b**5*d**2*e**7*x**7 + 74646*b**5*d*e**8*x**8 + 21879*b**5* e**9*x**9))/(415701*e**6)