Integrand size = 30, antiderivative size = 318 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{15/2}} \, dx=\frac {2 (b d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{13 e^6 (a+b x) (d+e x)^{13/2}}-\frac {10 b (b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^6 (a+b x) (d+e x)^{11/2}}+\frac {20 b^2 (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{9 e^6 (a+b x) (d+e x)^{9/2}}-\frac {20 b^3 (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^6 (a+b x) (d+e x)^{7/2}}+\frac {2 b^4 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (a+b x) (d+e x)^{5/2}}-\frac {2 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^6 (a+b x) (d+e x)^{3/2}} \]
2/13*(-a*e+b*d)^5*((b*x+a)^2)^(1/2)/e^6/(b*x+a)/(e*x+d)^(13/2)-10/11*b*(-a *e+b*d)^4*((b*x+a)^2)^(1/2)/e^6/(b*x+a)/(e*x+d)^(11/2)+20/9*b^2*(-a*e+b*d) ^3*((b*x+a)^2)^(1/2)/e^6/(b*x+a)/(e*x+d)^(9/2)-20/7*b^3*(-a*e+b*d)^2*((b*x +a)^2)^(1/2)/e^6/(b*x+a)/(e*x+d)^(7/2)+2*b^4*(-a*e+b*d)*((b*x+a)^2)^(1/2)/ e^6/(b*x+a)/(e*x+d)^(5/2)-2/3*b^5*((b*x+a)^2)^(1/2)/e^6/(b*x+a)/(e*x+d)^(3 /2)
Time = 0.13 (sec) , antiderivative size = 235, normalized size of antiderivative = 0.74 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{15/2}} \, dx=-\frac {2 \sqrt {(a+b x)^2} \left (693 a^5 e^5+315 a^4 b e^4 (2 d+13 e x)+70 a^3 b^2 e^3 \left (8 d^2+52 d e x+143 e^2 x^2\right )+30 a^2 b^3 e^2 \left (16 d^3+104 d^2 e x+286 d e^2 x^2+429 e^3 x^3\right )+3 a b^4 e \left (128 d^4+832 d^3 e x+2288 d^2 e^2 x^2+3432 d e^3 x^3+3003 e^4 x^4\right )+b^5 \left (256 d^5+1664 d^4 e x+4576 d^3 e^2 x^2+6864 d^2 e^3 x^3+6006 d e^4 x^4+3003 e^5 x^5\right )\right )}{9009 e^6 (a+b x) (d+e x)^{13/2}} \]
(-2*Sqrt[(a + b*x)^2]*(693*a^5*e^5 + 315*a^4*b*e^4*(2*d + 13*e*x) + 70*a^3 *b^2*e^3*(8*d^2 + 52*d*e*x + 143*e^2*x^2) + 30*a^2*b^3*e^2*(16*d^3 + 104*d ^2*e*x + 286*d*e^2*x^2 + 429*e^3*x^3) + 3*a*b^4*e*(128*d^4 + 832*d^3*e*x + 2288*d^2*e^2*x^2 + 3432*d*e^3*x^3 + 3003*e^4*x^4) + b^5*(256*d^5 + 1664*d ^4*e*x + 4576*d^3*e^2*x^2 + 6864*d^2*e^3*x^3 + 6006*d*e^4*x^4 + 3003*e^5*x ^5)))/(9009*e^6*(a + b*x)*(d + e*x)^(13/2))
Time = 0.30 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.58, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {1102, 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.
| \(\displaystyle \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{15/2}} \, dx\) |
| \(\Big \downarrow \) 1102 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {b^5 (a+b x)^5}{(d+e x)^{15/2}}dx}{b^5 (a+b x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {(a+b x)^5}{(d+e x)^{15/2}}dx}{a+b x}\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {b^5}{e^5 (d+e x)^{5/2}}-\frac {5 (b d-a e) b^4}{e^5 (d+e x)^{7/2}}+\frac {10 (b d-a e)^2 b^3}{e^5 (d+e x)^{9/2}}-\frac {10 (b d-a e)^3 b^2}{e^5 (d+e x)^{11/2}}+\frac {5 (b d-a e)^4 b}{e^5 (d+e x)^{13/2}}+\frac {(a e-b d)^5}{e^5 (d+e x)^{15/2}}\right )dx}{a+b x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (\frac {2 b^4 (b d-a e)}{e^6 (d+e x)^{5/2}}-\frac {20 b^3 (b d-a e)^2}{7 e^6 (d+e x)^{7/2}}+\frac {20 b^2 (b d-a e)^3}{9 e^6 (d+e x)^{9/2}}-\frac {10 b (b d-a e)^4}{11 e^6 (d+e x)^{11/2}}+\frac {2 (b d-a e)^5}{13 e^6 (d+e x)^{13/2}}-\frac {2 b^5}{3 e^6 (d+e x)^{3/2}}\right )}{a+b x}\) |
(Sqrt[a^2 + 2*a*b*x + b^2*x^2]*((2*(b*d - a*e)^5)/(13*e^6*(d + e*x)^(13/2) ) - (10*b*(b*d - a*e)^4)/(11*e^6*(d + e*x)^(11/2)) + (20*b^2*(b*d - a*e)^3 )/(9*e^6*(d + e*x)^(9/2)) - (20*b^3*(b*d - a*e)^2)/(7*e^6*(d + e*x)^(7/2)) + (2*b^4*(b*d - a*e))/(e^6*(d + e*x)^(5/2)) - (2*b^5)/(3*e^6*(d + e*x)^(3 /2))))/(a + b*x)
3.18.2.3.1 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_S ymbol] :> Simp[(a + b*x + c*x^2)^FracPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*F racPart[p])) Int[(d + e*x)^m*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[b^2 - 4*a*c, 0]
Time = 2.57 (sec) , antiderivative size = 289, normalized size of antiderivative = 0.91
| method | result | size |
| gosper | \(-\frac {2 \left (3003 x^{5} e^{5} b^{5}+9009 x^{4} a \,b^{4} e^{5}+6006 x^{4} b^{5} d \,e^{4}+12870 x^{3} a^{2} b^{3} e^{5}+10296 x^{3} a \,b^{4} d \,e^{4}+6864 x^{3} b^{5} d^{2} e^{3}+10010 x^{2} a^{3} b^{2} e^{5}+8580 x^{2} a^{2} b^{3} d \,e^{4}+6864 x^{2} a \,b^{4} d^{2} e^{3}+4576 x^{2} b^{5} d^{3} e^{2}+4095 a^{4} b \,e^{5} x +3640 a^{3} b^{2} d \,e^{4} x +3120 x \,a^{2} b^{3} d^{2} e^{3}+2496 x a \,b^{4} d^{3} e^{2}+1664 b^{5} d^{4} e x +693 a^{5} e^{5}+630 a^{4} b d \,e^{4}+560 a^{3} b^{2} d^{2} e^{3}+480 a^{2} b^{3} d^{3} e^{2}+384 a \,b^{4} d^{4} e +256 b^{5} d^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{9009 \left (e x +d \right )^{\frac {13}{2}} e^{6} \left (b x +a \right )^{5}}\) | \(289\) |
| default | \(-\frac {2 \left (3003 x^{5} e^{5} b^{5}+9009 x^{4} a \,b^{4} e^{5}+6006 x^{4} b^{5} d \,e^{4}+12870 x^{3} a^{2} b^{3} e^{5}+10296 x^{3} a \,b^{4} d \,e^{4}+6864 x^{3} b^{5} d^{2} e^{3}+10010 x^{2} a^{3} b^{2} e^{5}+8580 x^{2} a^{2} b^{3} d \,e^{4}+6864 x^{2} a \,b^{4} d^{2} e^{3}+4576 x^{2} b^{5} d^{3} e^{2}+4095 a^{4} b \,e^{5} x +3640 a^{3} b^{2} d \,e^{4} x +3120 x \,a^{2} b^{3} d^{2} e^{3}+2496 x a \,b^{4} d^{3} e^{2}+1664 b^{5} d^{4} e x +693 a^{5} e^{5}+630 a^{4} b d \,e^{4}+560 a^{3} b^{2} d^{2} e^{3}+480 a^{2} b^{3} d^{3} e^{2}+384 a \,b^{4} d^{4} e +256 b^{5} d^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{9009 \left (e x +d \right )^{\frac {13}{2}} e^{6} \left (b x +a \right )^{5}}\) | \(289\) |
-2/9009/(e*x+d)^(13/2)*(3003*b^5*e^5*x^5+9009*a*b^4*e^5*x^4+6006*b^5*d*e^4 *x^4+12870*a^2*b^3*e^5*x^3+10296*a*b^4*d*e^4*x^3+6864*b^5*d^2*e^3*x^3+1001 0*a^3*b^2*e^5*x^2+8580*a^2*b^3*d*e^4*x^2+6864*a*b^4*d^2*e^3*x^2+4576*b^5*d ^3*e^2*x^2+4095*a^4*b*e^5*x+3640*a^3*b^2*d*e^4*x+3120*a^2*b^3*d^2*e^3*x+24 96*a*b^4*d^3*e^2*x+1664*b^5*d^4*e*x+693*a^5*e^5+630*a^4*b*d*e^4+560*a^3*b^ 2*d^2*e^3+480*a^2*b^3*d^3*e^2+384*a*b^4*d^4*e+256*b^5*d^5)*((b*x+a)^2)^(5/ 2)/e^6/(b*x+a)^5
Time = 0.41 (sec) , antiderivative size = 338, normalized size of antiderivative = 1.06 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{15/2}} \, dx=-\frac {2 \, {\left (3003 \, b^{5} e^{5} x^{5} + 256 \, b^{5} d^{5} + 384 \, a b^{4} d^{4} e + 480 \, a^{2} b^{3} d^{3} e^{2} + 560 \, a^{3} b^{2} d^{2} e^{3} + 630 \, a^{4} b d e^{4} + 693 \, a^{5} e^{5} + 3003 \, {\left (2 \, b^{5} d e^{4} + 3 \, a b^{4} e^{5}\right )} x^{4} + 858 \, {\left (8 \, b^{5} d^{2} e^{3} + 12 \, a b^{4} d e^{4} + 15 \, a^{2} b^{3} e^{5}\right )} x^{3} + 286 \, {\left (16 \, b^{5} d^{3} e^{2} + 24 \, a b^{4} d^{2} e^{3} + 30 \, a^{2} b^{3} d e^{4} + 35 \, a^{3} b^{2} e^{5}\right )} x^{2} + 13 \, {\left (128 \, b^{5} d^{4} e + 192 \, a b^{4} d^{3} e^{2} + 240 \, a^{2} b^{3} d^{2} e^{3} + 280 \, a^{3} b^{2} d e^{4} + 315 \, a^{4} b e^{5}\right )} x\right )} \sqrt {e x + d}}{9009 \, {\left (e^{13} x^{7} + 7 \, d e^{12} x^{6} + 21 \, d^{2} e^{11} x^{5} + 35 \, d^{3} e^{10} x^{4} + 35 \, d^{4} e^{9} x^{3} + 21 \, d^{5} e^{8} x^{2} + 7 \, d^{6} e^{7} x + d^{7} e^{6}\right )}} \]
-2/9009*(3003*b^5*e^5*x^5 + 256*b^5*d^5 + 384*a*b^4*d^4*e + 480*a^2*b^3*d^ 3*e^2 + 560*a^3*b^2*d^2*e^3 + 630*a^4*b*d*e^4 + 693*a^5*e^5 + 3003*(2*b^5* d*e^4 + 3*a*b^4*e^5)*x^4 + 858*(8*b^5*d^2*e^3 + 12*a*b^4*d*e^4 + 15*a^2*b^ 3*e^5)*x^3 + 286*(16*b^5*d^3*e^2 + 24*a*b^4*d^2*e^3 + 30*a^2*b^3*d*e^4 + 3 5*a^3*b^2*e^5)*x^2 + 13*(128*b^5*d^4*e + 192*a*b^4*d^3*e^2 + 240*a^2*b^3*d ^2*e^3 + 280*a^3*b^2*d*e^4 + 315*a^4*b*e^5)*x)*sqrt(e*x + d)/(e^13*x^7 + 7 *d*e^12*x^6 + 21*d^2*e^11*x^5 + 35*d^3*e^10*x^4 + 35*d^4*e^9*x^3 + 21*d^5* e^8*x^2 + 7*d^6*e^7*x + d^7*e^6)
Timed out. \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{15/2}} \, dx=\text {Timed out} \]
Time = 0.21 (sec) , antiderivative size = 327, normalized size of antiderivative = 1.03 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{15/2}} \, dx=-\frac {2 \, {\left (3003 \, b^{5} e^{5} x^{5} + 256 \, b^{5} d^{5} + 384 \, a b^{4} d^{4} e + 480 \, a^{2} b^{3} d^{3} e^{2} + 560 \, a^{3} b^{2} d^{2} e^{3} + 630 \, a^{4} b d e^{4} + 693 \, a^{5} e^{5} + 3003 \, {\left (2 \, b^{5} d e^{4} + 3 \, a b^{4} e^{5}\right )} x^{4} + 858 \, {\left (8 \, b^{5} d^{2} e^{3} + 12 \, a b^{4} d e^{4} + 15 \, a^{2} b^{3} e^{5}\right )} x^{3} + 286 \, {\left (16 \, b^{5} d^{3} e^{2} + 24 \, a b^{4} d^{2} e^{3} + 30 \, a^{2} b^{3} d e^{4} + 35 \, a^{3} b^{2} e^{5}\right )} x^{2} + 13 \, {\left (128 \, b^{5} d^{4} e + 192 \, a b^{4} d^{3} e^{2} + 240 \, a^{2} b^{3} d^{2} e^{3} + 280 \, a^{3} b^{2} d e^{4} + 315 \, a^{4} b e^{5}\right )} x\right )}}{9009 \, {\left (e^{12} x^{6} + 6 \, d e^{11} x^{5} + 15 \, d^{2} e^{10} x^{4} + 20 \, d^{3} e^{9} x^{3} + 15 \, d^{4} e^{8} x^{2} + 6 \, d^{5} e^{7} x + d^{6} e^{6}\right )} \sqrt {e x + d}} \]
-2/9009*(3003*b^5*e^5*x^5 + 256*b^5*d^5 + 384*a*b^4*d^4*e + 480*a^2*b^3*d^ 3*e^2 + 560*a^3*b^2*d^2*e^3 + 630*a^4*b*d*e^4 + 693*a^5*e^5 + 3003*(2*b^5* d*e^4 + 3*a*b^4*e^5)*x^4 + 858*(8*b^5*d^2*e^3 + 12*a*b^4*d*e^4 + 15*a^2*b^ 3*e^5)*x^3 + 286*(16*b^5*d^3*e^2 + 24*a*b^4*d^2*e^3 + 30*a^2*b^3*d*e^4 + 3 5*a^3*b^2*e^5)*x^2 + 13*(128*b^5*d^4*e + 192*a*b^4*d^3*e^2 + 240*a^2*b^3*d ^2*e^3 + 280*a^3*b^2*d*e^4 + 315*a^4*b*e^5)*x)/((e^12*x^6 + 6*d*e^11*x^5 + 15*d^2*e^10*x^4 + 20*d^3*e^9*x^3 + 15*d^4*e^8*x^2 + 6*d^5*e^7*x + d^6*e^6 )*sqrt(e*x + d))
Time = 0.31 (sec) , antiderivative size = 437, normalized size of antiderivative = 1.37 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{15/2}} \, dx=-\frac {2 \, {\left (3003 \, {\left (e x + d\right )}^{5} b^{5} \mathrm {sgn}\left (b x + a\right ) - 9009 \, {\left (e x + d\right )}^{4} b^{5} d \mathrm {sgn}\left (b x + a\right ) + 12870 \, {\left (e x + d\right )}^{3} b^{5} d^{2} \mathrm {sgn}\left (b x + a\right ) - 10010 \, {\left (e x + d\right )}^{2} b^{5} d^{3} \mathrm {sgn}\left (b x + a\right ) + 4095 \, {\left (e x + d\right )} b^{5} d^{4} \mathrm {sgn}\left (b x + a\right ) - 693 \, b^{5} d^{5} \mathrm {sgn}\left (b x + a\right ) + 9009 \, {\left (e x + d\right )}^{4} a b^{4} e \mathrm {sgn}\left (b x + a\right ) - 25740 \, {\left (e x + d\right )}^{3} a b^{4} d e \mathrm {sgn}\left (b x + a\right ) + 30030 \, {\left (e x + d\right )}^{2} a b^{4} d^{2} e \mathrm {sgn}\left (b x + a\right ) - 16380 \, {\left (e x + d\right )} a b^{4} d^{3} e \mathrm {sgn}\left (b x + a\right ) + 3465 \, a b^{4} d^{4} e \mathrm {sgn}\left (b x + a\right ) + 12870 \, {\left (e x + d\right )}^{3} a^{2} b^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) - 30030 \, {\left (e x + d\right )}^{2} a^{2} b^{3} d e^{2} \mathrm {sgn}\left (b x + a\right ) + 24570 \, {\left (e x + d\right )} a^{2} b^{3} d^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) - 6930 \, a^{2} b^{3} d^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) + 10010 \, {\left (e x + d\right )}^{2} a^{3} b^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) - 16380 \, {\left (e x + d\right )} a^{3} b^{2} d e^{3} \mathrm {sgn}\left (b x + a\right ) + 6930 \, a^{3} b^{2} d^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + 4095 \, {\left (e x + d\right )} a^{4} b e^{4} \mathrm {sgn}\left (b x + a\right ) - 3465 \, a^{4} b d e^{4} \mathrm {sgn}\left (b x + a\right ) + 693 \, a^{5} e^{5} \mathrm {sgn}\left (b x + a\right )\right )}}{9009 \, {\left (e x + d\right )}^{\frac {13}{2}} e^{6}} \]
-2/9009*(3003*(e*x + d)^5*b^5*sgn(b*x + a) - 9009*(e*x + d)^4*b^5*d*sgn(b* x + a) + 12870*(e*x + d)^3*b^5*d^2*sgn(b*x + a) - 10010*(e*x + d)^2*b^5*d^ 3*sgn(b*x + a) + 4095*(e*x + d)*b^5*d^4*sgn(b*x + a) - 693*b^5*d^5*sgn(b*x + a) + 9009*(e*x + d)^4*a*b^4*e*sgn(b*x + a) - 25740*(e*x + d)^3*a*b^4*d* e*sgn(b*x + a) + 30030*(e*x + d)^2*a*b^4*d^2*e*sgn(b*x + a) - 16380*(e*x + d)*a*b^4*d^3*e*sgn(b*x + a) + 3465*a*b^4*d^4*e*sgn(b*x + a) + 12870*(e*x + d)^3*a^2*b^3*e^2*sgn(b*x + a) - 30030*(e*x + d)^2*a^2*b^3*d*e^2*sgn(b*x + a) + 24570*(e*x + d)*a^2*b^3*d^2*e^2*sgn(b*x + a) - 6930*a^2*b^3*d^3*e^2 *sgn(b*x + a) + 10010*(e*x + d)^2*a^3*b^2*e^3*sgn(b*x + a) - 16380*(e*x + d)*a^3*b^2*d*e^3*sgn(b*x + a) + 6930*a^3*b^2*d^2*e^3*sgn(b*x + a) + 4095*( e*x + d)*a^4*b*e^4*sgn(b*x + a) - 3465*a^4*b*d*e^4*sgn(b*x + a) + 693*a^5* e^5*sgn(b*x + a))/((e*x + d)^(13/2)*e^6)
Time = 10.71 (sec) , antiderivative size = 472, normalized size of antiderivative = 1.48 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{15/2}} \, dx=-\frac {\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}\,\left (\frac {1386\,a^5\,e^5+1260\,a^4\,b\,d\,e^4+1120\,a^3\,b^2\,d^2\,e^3+960\,a^2\,b^3\,d^3\,e^2+768\,a\,b^4\,d^4\,e+512\,b^5\,d^5}{9009\,b\,e^{12}}+\frac {2\,b^4\,x^5}{3\,e^7}+\frac {2\,b^3\,x^4\,\left (3\,a\,e+2\,b\,d\right )}{3\,e^8}+\frac {x\,\left (8190\,a^4\,b\,e^5+7280\,a^3\,b^2\,d\,e^4+6240\,a^2\,b^3\,d^2\,e^3+4992\,a\,b^4\,d^3\,e^2+3328\,b^5\,d^4\,e\right )}{9009\,b\,e^{12}}+\frac {4\,b^2\,x^3\,\left (15\,a^2\,e^2+12\,a\,b\,d\,e+8\,b^2\,d^2\right )}{21\,e^9}+\frac {4\,b\,x^2\,\left (35\,a^3\,e^3+30\,a^2\,b\,d\,e^2+24\,a\,b^2\,d^2\,e+16\,b^3\,d^3\right )}{63\,e^{10}}\right )}{x^7\,\sqrt {d+e\,x}+\frac {a\,d^6\,\sqrt {d+e\,x}}{b\,e^6}+\frac {x^6\,\left (a\,e+6\,b\,d\right )\,\sqrt {d+e\,x}}{b\,e}+\frac {3\,d\,x^5\,\left (2\,a\,e+5\,b\,d\right )\,\sqrt {d+e\,x}}{b\,e^2}+\frac {d^5\,x\,\left (6\,a\,e+b\,d\right )\,\sqrt {d+e\,x}}{b\,e^6}+\frac {5\,d^2\,x^4\,\left (3\,a\,e+4\,b\,d\right )\,\sqrt {d+e\,x}}{b\,e^3}+\frac {5\,d^3\,x^3\,\left (4\,a\,e+3\,b\,d\right )\,\sqrt {d+e\,x}}{b\,e^4}+\frac {3\,d^4\,x^2\,\left (5\,a\,e+2\,b\,d\right )\,\sqrt {d+e\,x}}{b\,e^5}} \]
-((a^2 + b^2*x^2 + 2*a*b*x)^(1/2)*((1386*a^5*e^5 + 512*b^5*d^5 + 960*a^2*b ^3*d^3*e^2 + 1120*a^3*b^2*d^2*e^3 + 768*a*b^4*d^4*e + 1260*a^4*b*d*e^4)/(9 009*b*e^12) + (2*b^4*x^5)/(3*e^7) + (2*b^3*x^4*(3*a*e + 2*b*d))/(3*e^8) + (x*(8190*a^4*b*e^5 + 3328*b^5*d^4*e + 4992*a*b^4*d^3*e^2 + 7280*a^3*b^2*d* e^4 + 6240*a^2*b^3*d^2*e^3))/(9009*b*e^12) + (4*b^2*x^3*(15*a^2*e^2 + 8*b^ 2*d^2 + 12*a*b*d*e))/(21*e^9) + (4*b*x^2*(35*a^3*e^3 + 16*b^3*d^3 + 24*a*b ^2*d^2*e + 30*a^2*b*d*e^2))/(63*e^10)))/(x^7*(d + e*x)^(1/2) + (a*d^6*(d + e*x)^(1/2))/(b*e^6) + (x^6*(a*e + 6*b*d)*(d + e*x)^(1/2))/(b*e) + (3*d*x^ 5*(2*a*e + 5*b*d)*(d + e*x)^(1/2))/(b*e^2) + (d^5*x*(6*a*e + b*d)*(d + e*x )^(1/2))/(b*e^6) + (5*d^2*x^4*(3*a*e + 4*b*d)*(d + e*x)^(1/2))/(b*e^3) + ( 5*d^3*x^3*(4*a*e + 3*b*d)*(d + e*x)^(1/2))/(b*e^4) + (3*d^4*x^2*(5*a*e + 2 *b*d)*(d + e*x)^(1/2))/(b*e^5))