Integrand size = 24, antiderivative size = 309 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{19/2}} \, dx=-\frac {2 (B d-A e) (a+b x)^{7/2}}{17 e (b d-a e) (d+e x)^{17/2}}+\frac {2 (7 b B d+10 A b e-17 a B e) (a+b x)^{7/2}}{255 e (b d-a e)^2 (d+e x)^{15/2}}+\frac {16 b (7 b B d+10 A b e-17 a B e) (a+b x)^{7/2}}{3315 e (b d-a e)^3 (d+e x)^{13/2}}+\frac {32 b^2 (7 b B d+10 A b e-17 a B e) (a+b x)^{7/2}}{12155 e (b d-a e)^4 (d+e x)^{11/2}}+\frac {128 b^3 (7 b B d+10 A b e-17 a B e) (a+b x)^{7/2}}{109395 e (b d-a e)^5 (d+e x)^{9/2}}+\frac {256 b^4 (7 b B d+10 A b e-17 a B e) (a+b x)^{7/2}}{765765 e (b d-a e)^6 (d+e x)^{7/2}} \]
-2/17*(-A*e+B*d)*(b*x+a)^(7/2)/e/(-a*e+b*d)/(e*x+d)^(17/2)+2/255*(10*A*b*e -17*B*a*e+7*B*b*d)*(b*x+a)^(7/2)/e/(-a*e+b*d)^2/(e*x+d)^(15/2)+16/3315*b*( 10*A*b*e-17*B*a*e+7*B*b*d)*(b*x+a)^(7/2)/e/(-a*e+b*d)^3/(e*x+d)^(13/2)+32/ 12155*b^2*(10*A*b*e-17*B*a*e+7*B*b*d)*(b*x+a)^(7/2)/e/(-a*e+b*d)^4/(e*x+d) ^(11/2)+128/109395*b^3*(10*A*b*e-17*B*a*e+7*B*b*d)*(b*x+a)^(7/2)/e/(-a*e+b *d)^5/(e*x+d)^(9/2)+256/765765*b^4*(10*A*b*e-17*B*a*e+7*B*b*d)*(b*x+a)^(7/ 2)/e/(-a*e+b*d)^6/(e*x+d)^(7/2)
Time = 0.65 (sec) , antiderivative size = 341, normalized size of antiderivative = 1.10 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{19/2}} \, dx=\frac {2 (a+b x)^{7/2} \left (45045 B d e^4 (a+b x)^5-45045 A e^5 (a+b x)^5-204204 b B d e^3 (a+b x)^4 (d+e x)+255255 A b e^4 (a+b x)^4 (d+e x)-51051 a B e^4 (a+b x)^4 (d+e x)+353430 b^2 B d e^2 (a+b x)^3 (d+e x)^2-589050 A b^2 e^3 (a+b x)^3 (d+e x)^2+235620 a b B e^3 (a+b x)^3 (d+e x)^2-278460 b^3 B d e (a+b x)^2 (d+e x)^3+696150 A b^3 e^2 (a+b x)^2 (d+e x)^3-417690 a b^2 B e^2 (a+b x)^2 (d+e x)^3+85085 b^4 B d (a+b x) (d+e x)^4-425425 A b^4 e (a+b x) (d+e x)^4+340340 a b^3 B e (a+b x) (d+e x)^4+109395 A b^5 (d+e x)^5-109395 a b^4 B (d+e x)^5\right )}{765765 (b d-a e)^6 (d+e x)^{17/2}} \]
(2*(a + b*x)^(7/2)*(45045*B*d*e^4*(a + b*x)^5 - 45045*A*e^5*(a + b*x)^5 - 204204*b*B*d*e^3*(a + b*x)^4*(d + e*x) + 255255*A*b*e^4*(a + b*x)^4*(d + e *x) - 51051*a*B*e^4*(a + b*x)^4*(d + e*x) + 353430*b^2*B*d*e^2*(a + b*x)^3 *(d + e*x)^2 - 589050*A*b^2*e^3*(a + b*x)^3*(d + e*x)^2 + 235620*a*b*B*e^3 *(a + b*x)^3*(d + e*x)^2 - 278460*b^3*B*d*e*(a + b*x)^2*(d + e*x)^3 + 6961 50*A*b^3*e^2*(a + b*x)^2*(d + e*x)^3 - 417690*a*b^2*B*e^2*(a + b*x)^2*(d + e*x)^3 + 85085*b^4*B*d*(a + b*x)*(d + e*x)^4 - 425425*A*b^4*e*(a + b*x)*( d + e*x)^4 + 340340*a*b^3*B*e*(a + b*x)*(d + e*x)^4 + 109395*A*b^5*(d + e* x)^5 - 109395*a*b^4*B*(d + e*x)^5))/(765765*(b*d - a*e)^6*(d + e*x)^(17/2) )
Time = 0.31 (sec) , antiderivative size = 287, normalized size of antiderivative = 0.93, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {87, 55, 55, 55, 55, 48}
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.
| \(\displaystyle \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{19/2}} \, dx\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle \frac {(-17 a B e+10 A b e+7 b B d) \int \frac {(a+b x)^{5/2}}{(d+e x)^{17/2}}dx}{17 e (b d-a e)}-\frac {2 (a+b x)^{7/2} (B d-A e)}{17 e (d+e x)^{17/2} (b d-a e)}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle \frac {(-17 a B e+10 A b e+7 b B d) \left (\frac {8 b \int \frac {(a+b x)^{5/2}}{(d+e x)^{15/2}}dx}{15 (b d-a e)}+\frac {2 (a+b x)^{7/2}}{15 (d+e x)^{15/2} (b d-a e)}\right )}{17 e (b d-a e)}-\frac {2 (a+b x)^{7/2} (B d-A e)}{17 e (d+e x)^{17/2} (b d-a e)}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle \frac {(-17 a B e+10 A b e+7 b B d) \left (\frac {8 b \left (\frac {6 b \int \frac {(a+b x)^{5/2}}{(d+e x)^{13/2}}dx}{13 (b d-a e)}+\frac {2 (a+b x)^{7/2}}{13 (d+e x)^{13/2} (b d-a e)}\right )}{15 (b d-a e)}+\frac {2 (a+b x)^{7/2}}{15 (d+e x)^{15/2} (b d-a e)}\right )}{17 e (b d-a e)}-\frac {2 (a+b x)^{7/2} (B d-A e)}{17 e (d+e x)^{17/2} (b d-a e)}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle \frac {(-17 a B e+10 A b e+7 b B d) \left (\frac {8 b \left (\frac {6 b \left (\frac {4 b \int \frac {(a+b x)^{5/2}}{(d+e x)^{11/2}}dx}{11 (b d-a e)}+\frac {2 (a+b x)^{7/2}}{11 (d+e x)^{11/2} (b d-a e)}\right )}{13 (b d-a e)}+\frac {2 (a+b x)^{7/2}}{13 (d+e x)^{13/2} (b d-a e)}\right )}{15 (b d-a e)}+\frac {2 (a+b x)^{7/2}}{15 (d+e x)^{15/2} (b d-a e)}\right )}{17 e (b d-a e)}-\frac {2 (a+b x)^{7/2} (B d-A e)}{17 e (d+e x)^{17/2} (b d-a e)}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle \frac {(-17 a B e+10 A b e+7 b B d) \left (\frac {8 b \left (\frac {6 b \left (\frac {4 b \left (\frac {2 b \int \frac {(a+b x)^{5/2}}{(d+e x)^{9/2}}dx}{9 (b d-a e)}+\frac {2 (a+b x)^{7/2}}{9 (d+e x)^{9/2} (b d-a e)}\right )}{11 (b d-a e)}+\frac {2 (a+b x)^{7/2}}{11 (d+e x)^{11/2} (b d-a e)}\right )}{13 (b d-a e)}+\frac {2 (a+b x)^{7/2}}{13 (d+e x)^{13/2} (b d-a e)}\right )}{15 (b d-a e)}+\frac {2 (a+b x)^{7/2}}{15 (d+e x)^{15/2} (b d-a e)}\right )}{17 e (b d-a e)}-\frac {2 (a+b x)^{7/2} (B d-A e)}{17 e (d+e x)^{17/2} (b d-a e)}\) |
| \(\Big \downarrow \) 48 |
| \(\displaystyle \frac {\left (\frac {2 (a+b x)^{7/2}}{15 (d+e x)^{15/2} (b d-a e)}+\frac {8 b \left (\frac {2 (a+b x)^{7/2}}{13 (d+e x)^{13/2} (b d-a e)}+\frac {6 b \left (\frac {2 (a+b x)^{7/2}}{11 (d+e x)^{11/2} (b d-a e)}+\frac {4 b \left (\frac {4 b (a+b x)^{7/2}}{63 (d+e x)^{7/2} (b d-a e)^2}+\frac {2 (a+b x)^{7/2}}{9 (d+e x)^{9/2} (b d-a e)}\right )}{11 (b d-a e)}\right )}{13 (b d-a e)}\right )}{15 (b d-a e)}\right ) (-17 a B e+10 A b e+7 b B d)}{17 e (b d-a e)}-\frac {2 (a+b x)^{7/2} (B d-A e)}{17 e (d+e x)^{17/2} (b d-a e)}\) |
(-2*(B*d - A*e)*(a + b*x)^(7/2))/(17*e*(b*d - a*e)*(d + e*x)^(17/2)) + ((7 *b*B*d + 10*A*b*e - 17*a*B*e)*((2*(a + b*x)^(7/2))/(15*(b*d - a*e)*(d + e* x)^(15/2)) + (8*b*((2*(a + b*x)^(7/2))/(13*(b*d - a*e)*(d + e*x)^(13/2)) + (6*b*((2*(a + b*x)^(7/2))/(11*(b*d - a*e)*(d + e*x)^(11/2)) + (4*b*((2*(a + b*x)^(7/2))/(9*(b*d - a*e)*(d + e*x)^(9/2)) + (4*b*(a + b*x)^(7/2))/(63 *(b*d - a*e)^2*(d + e*x)^(7/2))))/(11*(b*d - a*e))))/(13*(b*d - a*e))))/(1 5*(b*d - a*e))))/(17*e*(b*d - a*e))
3.23.35.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp [(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{ a, b, c, d, m, n}, x] && EqQ[m + n + 2, 0] && NeQ[m, -1]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(S implify[m + n + 2]/((b*c - a*d)*(m + 1))) Int[(a + b*x)^Simplify[m + 1]*( c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && ILtQ[Simplify[m + n + 2], 0] && NeQ[m, -1] && !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[ c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (SumSimplerQ[m, 1] || !SumSimp lerQ[n, 1])
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
Leaf count of result is larger than twice the leaf count of optimal. \(721\) vs. \(2(273)=546\).
Time = 1.10 (sec) , antiderivative size = 722, normalized size of antiderivative = 2.34
| method | result | size |
| gosper | \(-\frac {2 \left (b x +a \right )^{\frac {7}{2}} \left (-1280 A \,b^{5} e^{5} x^{5}+2176 B a \,b^{4} e^{5} x^{5}-896 B \,b^{5} d \,e^{4} x^{5}+4480 A a \,b^{4} e^{5} x^{4}-10880 A \,b^{5} d \,e^{4} x^{4}-7616 B \,a^{2} b^{3} e^{5} x^{4}+21632 B a \,b^{4} d \,e^{4} x^{4}-7616 B \,b^{5} d^{2} e^{3} x^{4}-10080 A \,a^{2} b^{3} e^{5} x^{3}+38080 A a \,b^{4} d \,e^{4} x^{3}-40800 A \,b^{5} d^{2} e^{3} x^{3}+17136 B \,a^{3} b^{2} e^{5} x^{3}-71792 B \,a^{2} b^{3} d \,e^{4} x^{3}+96016 B a \,b^{4} d^{2} e^{3} x^{3}-28560 B \,b^{5} d^{3} e^{2} x^{3}+18480 A \,a^{3} b^{2} e^{5} x^{2}-85680 A \,a^{2} b^{3} d \,e^{4} x^{2}+142800 A a \,b^{4} d^{2} e^{3} x^{2}-88400 A \,b^{5} d^{3} e^{2} x^{2}-31416 B \,a^{4} b \,e^{5} x^{2}+158592 B \,a^{3} b^{2} d \,e^{4} x^{2}-302736 B \,a^{2} b^{3} d^{2} e^{3} x^{2}+250240 B a \,b^{4} d^{3} e^{2} x^{2}-61880 B \,b^{5} d^{4} e \,x^{2}-30030 A \,a^{4} b \,e^{5} x +157080 A \,a^{3} b^{2} d \,e^{4} x -321300 A \,a^{2} b^{3} d^{2} e^{3} x +309400 A a \,b^{4} d^{3} e^{2} x -121550 A \,b^{5} d^{4} e x +51051 B \,a^{5} e^{5} x -288057 B \,a^{4} b d \,e^{4} x +656166 B \,a^{3} b^{2} d^{2} e^{3} x -750890 B \,a^{2} b^{3} d^{3} e^{2} x +423215 B a \,b^{4} d^{4} e x -85085 B \,b^{5} d^{5} x +45045 A \,a^{5} e^{5}-255255 A \,a^{4} b d \,e^{4}+589050 A \,a^{3} b^{2} d^{2} e^{3}-696150 A \,a^{2} b^{3} d^{3} e^{2}+425425 A a \,b^{4} d^{4} e -109395 A \,b^{5} d^{5}+6006 B \,a^{5} d \,e^{4}-31416 B \,a^{4} b \,d^{2} e^{3}+64260 B \,a^{3} b^{2} d^{3} e^{2}-61880 B \,a^{2} b^{3} d^{4} e +24310 B a \,b^{4} d^{5}\right )}{765765 \left (e x +d \right )^{\frac {17}{2}} \left (a^{6} e^{6}-6 a^{5} b d \,e^{5}+15 a^{4} b^{2} d^{2} e^{4}-20 a^{3} b^{3} d^{3} e^{3}+15 a^{2} b^{4} d^{4} e^{2}-6 a \,b^{5} d^{5} e +b^{6} d^{6}\right )}\) | \(722\) |
| default | \(\text {Expression too large to display}\) | \(1041\) |
-2/765765*(b*x+a)^(7/2)*(-1280*A*b^5*e^5*x^5+2176*B*a*b^4*e^5*x^5-896*B*b^ 5*d*e^4*x^5+4480*A*a*b^4*e^5*x^4-10880*A*b^5*d*e^4*x^4-7616*B*a^2*b^3*e^5* x^4+21632*B*a*b^4*d*e^4*x^4-7616*B*b^5*d^2*e^3*x^4-10080*A*a^2*b^3*e^5*x^3 +38080*A*a*b^4*d*e^4*x^3-40800*A*b^5*d^2*e^3*x^3+17136*B*a^3*b^2*e^5*x^3-7 1792*B*a^2*b^3*d*e^4*x^3+96016*B*a*b^4*d^2*e^3*x^3-28560*B*b^5*d^3*e^2*x^3 +18480*A*a^3*b^2*e^5*x^2-85680*A*a^2*b^3*d*e^4*x^2+142800*A*a*b^4*d^2*e^3* x^2-88400*A*b^5*d^3*e^2*x^2-31416*B*a^4*b*e^5*x^2+158592*B*a^3*b^2*d*e^4*x ^2-302736*B*a^2*b^3*d^2*e^3*x^2+250240*B*a*b^4*d^3*e^2*x^2-61880*B*b^5*d^4 *e*x^2-30030*A*a^4*b*e^5*x+157080*A*a^3*b^2*d*e^4*x-321300*A*a^2*b^3*d^2*e ^3*x+309400*A*a*b^4*d^3*e^2*x-121550*A*b^5*d^4*e*x+51051*B*a^5*e^5*x-28805 7*B*a^4*b*d*e^4*x+656166*B*a^3*b^2*d^2*e^3*x-750890*B*a^2*b^3*d^3*e^2*x+42 3215*B*a*b^4*d^4*e*x-85085*B*b^5*d^5*x+45045*A*a^5*e^5-255255*A*a^4*b*d*e^ 4+589050*A*a^3*b^2*d^2*e^3-696150*A*a^2*b^3*d^3*e^2+425425*A*a*b^4*d^4*e-1 09395*A*b^5*d^5+6006*B*a^5*d*e^4-31416*B*a^4*b*d^2*e^3+64260*B*a^3*b^2*d^3 *e^2-61880*B*a^2*b^3*d^4*e+24310*B*a*b^4*d^5)/(e*x+d)^(17/2)/(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)
Timed out. \[ \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{19/2}} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{19/2}} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{19/2}} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e*(a*e-b*d)>0)', see `assume?` f or more de
Leaf count of result is larger than twice the leaf count of optimal. 1875 vs. \(2 (273) = 546\).
Time = 1.74 (sec) , antiderivative size = 1875, normalized size of antiderivative = 6.07 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{19/2}} \, dx=\text {Too large to display} \]
2/765765*((8*(2*(4*(b*x + a)*(2*(7*B*b^20*d^3*e^12*abs(b) - 31*B*a*b^19*d^ 2*e^13*abs(b) + 10*A*b^20*d^2*e^13*abs(b) + 41*B*a^2*b^18*d*e^14*abs(b) - 20*A*a*b^19*d*e^14*abs(b) - 17*B*a^3*b^17*e^15*abs(b) + 10*A*a^2*b^18*e^15 *abs(b))*(b*x + a)/(b^10*d^8*e^8 - 8*a*b^9*d^7*e^9 + 28*a^2*b^8*d^6*e^10 - 56*a^3*b^7*d^5*e^11 + 70*a^4*b^6*d^4*e^12 - 56*a^5*b^5*d^3*e^13 + 28*a^6* b^4*d^2*e^14 - 8*a^7*b^3*d*e^15 + a^8*b^2*e^16) + 17*(7*B*b^21*d^4*e^11*ab s(b) - 38*B*a*b^20*d^3*e^12*abs(b) + 10*A*b^21*d^3*e^12*abs(b) + 72*B*a^2* b^19*d^2*e^13*abs(b) - 30*A*a*b^20*d^2*e^13*abs(b) - 58*B*a^3*b^18*d*e^14* abs(b) + 30*A*a^2*b^19*d*e^14*abs(b) + 17*B*a^4*b^17*e^15*abs(b) - 10*A*a^ 3*b^18*e^15*abs(b))/(b^10*d^8*e^8 - 8*a*b^9*d^7*e^9 + 28*a^2*b^8*d^6*e^10 - 56*a^3*b^7*d^5*e^11 + 70*a^4*b^6*d^4*e^12 - 56*a^5*b^5*d^3*e^13 + 28*a^6 *b^4*d^2*e^14 - 8*a^7*b^3*d*e^15 + a^8*b^2*e^16)) + 255*(7*B*b^22*d^5*e^10 *abs(b) - 45*B*a*b^21*d^4*e^11*abs(b) + 10*A*b^22*d^4*e^11*abs(b) + 110*B* a^2*b^20*d^3*e^12*abs(b) - 40*A*a*b^21*d^3*e^12*abs(b) - 130*B*a^3*b^19*d^ 2*e^13*abs(b) + 60*A*a^2*b^20*d^2*e^13*abs(b) + 75*B*a^4*b^18*d*e^14*abs(b ) - 40*A*a^3*b^19*d*e^14*abs(b) - 17*B*a^5*b^17*e^15*abs(b) + 10*A*a^4*b^1 8*e^15*abs(b))/(b^10*d^8*e^8 - 8*a*b^9*d^7*e^9 + 28*a^2*b^8*d^6*e^10 - 56* a^3*b^7*d^5*e^11 + 70*a^4*b^6*d^4*e^12 - 56*a^5*b^5*d^3*e^13 + 28*a^6*b^4* d^2*e^14 - 8*a^7*b^3*d*e^15 + a^8*b^2*e^16))*(b*x + a) + 1105*(7*B*b^23*d^ 6*e^9*abs(b) - 52*B*a*b^22*d^5*e^10*abs(b) + 10*A*b^23*d^5*e^10*abs(b) ...
Time = 5.66 (sec) , antiderivative size = 1144, normalized size of antiderivative = 3.70 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{19/2}} \, dx=\frac {\sqrt {d+e\,x}\,\left (\frac {x^2\,\sqrt {a+b\,x}\,\left (-243474\,B\,a^7\,b\,e^5+1375122\,B\,a^6\,b^2\,d\,e^4-127050\,A\,a^6\,b^2\,e^5-3143028\,B\,a^5\,b^3\,d^2\,e^3+760410\,A\,a^5\,b^3\,d\,e^4+3619300\,B\,a^4\,b^4\,d^3\,e^2-1892100\,A\,a^4\,b^4\,d^2\,e^3-2044250\,B\,a^3\,b^5\,d^4\,e+2497300\,A\,a^3\,b^5\,d^3\,e^2+364650\,B\,a^2\,b^6\,d^5-1823250\,A\,a^2\,b^6\,d^4\,e+656370\,A\,a\,b^7\,d^5\right )}{765765\,e^9\,{\left (a\,e-b\,d\right )}^6}-\frac {x\,\sqrt {a+b\,x}\,\left (102102\,B\,a^8\,e^5-540078\,B\,a^7\,b\,d\,e^4+210210\,A\,a^7\,b\,e^5+1123836\,B\,a^6\,b^2\,d^2\,e^3-1217370\,A\,a^6\,b^2\,d\,e^4-1116220\,B\,a^5\,b^3\,d^3\,e^2+2891700\,A\,a^5\,b^3\,d^2\,e^3+475150\,B\,a^4\,b^4\,d^4\,e-3558100\,A\,a^4\,b^4\,d^3\,e^2-24310\,B\,a^3\,b^5\,d^5+2309450\,A\,a^3\,b^5\,d^4\,e-656370\,A\,a^2\,b^6\,d^5\right )}{765765\,e^9\,{\left (a\,e-b\,d\right )}^6}-\frac {\sqrt {a+b\,x}\,\left (12012\,B\,a^8\,d\,e^4+90090\,A\,a^8\,e^5-62832\,B\,a^7\,b\,d^2\,e^3-510510\,A\,a^7\,b\,d\,e^4+128520\,B\,a^6\,b^2\,d^3\,e^2+1178100\,A\,a^6\,b^2\,d^2\,e^3-123760\,B\,a^5\,b^3\,d^4\,e-1392300\,A\,a^5\,b^3\,d^3\,e^2+48620\,B\,a^4\,b^4\,d^5+850850\,A\,a^4\,b^4\,d^4\,e-218790\,A\,a^3\,b^5\,d^5\right )}{765765\,e^9\,{\left (a\,e-b\,d\right )}^6}+\frac {x^3\,\sqrt {a+b\,x}\,\left (-152082\,B\,a^6\,b^2\,e^5+908362\,B\,a^5\,b^3\,d\,e^4-630\,A\,a^5\,b^3\,e^5-2249780\,B\,a^4\,b^4\,d^2\,e^3+5950\,A\,a^4\,b^4\,d\,e^4+2932500\,B\,a^3\,b^5\,d^3\,e^2-25500\,A\,a^3\,b^5\,d^2\,e^3-2044250\,B\,a^2\,b^6\,d^4\,e+66300\,A\,a^2\,b^6\,d^3\,e^2+461890\,B\,a\,b^7\,d^5-121550\,A\,a\,b^7\,d^4\,e+218790\,A\,b^8\,d^5\right )}{765765\,e^9\,{\left (a\,e-b\,d\right )}^6}+\frac {256\,b^7\,x^8\,\sqrt {a+b\,x}\,\left (10\,A\,b\,e-17\,B\,a\,e+7\,B\,b\,d\right )}{765765\,e^5\,{\left (a\,e-b\,d\right )}^6}-\frac {16\,b^4\,x^5\,\sqrt {a+b\,x}\,\left (10\,A\,b\,e-17\,B\,a\,e+7\,B\,b\,d\right )\,\left (5\,a^3\,e^3-51\,a^2\,b\,d\,e^2+255\,a\,b^2\,d^2\,e-1105\,b^3\,d^3\right )}{765765\,e^8\,{\left (a\,e-b\,d\right )}^6}-\frac {128\,b^6\,x^7\,\left (a\,e-17\,b\,d\right )\,\sqrt {a+b\,x}\,\left (10\,A\,b\,e-17\,B\,a\,e+7\,B\,b\,d\right )}{765765\,e^6\,{\left (a\,e-b\,d\right )}^6}+\frac {2\,b^3\,x^4\,\sqrt {a+b\,x}\,\left (10\,A\,b\,e-17\,B\,a\,e+7\,B\,b\,d\right )\,\left (7\,a^4\,e^4-68\,a^3\,b\,d\,e^3+306\,a^2\,b^2\,d^2\,e^2-884\,a\,b^3\,d^3\,e+2431\,b^4\,d^4\right )}{153153\,e^9\,{\left (a\,e-b\,d\right )}^6}+\frac {32\,b^5\,x^6\,\sqrt {a+b\,x}\,\left (3\,a^2\,e^2-34\,a\,b\,d\,e+255\,b^2\,d^2\right )\,\left (10\,A\,b\,e-17\,B\,a\,e+7\,B\,b\,d\right )}{765765\,e^7\,{\left (a\,e-b\,d\right )}^6}\right )}{x^9+\frac {d^9}{e^9}+\frac {9\,d\,x^8}{e}+\frac {9\,d^8\,x}{e^8}+\frac {36\,d^2\,x^7}{e^2}+\frac {84\,d^3\,x^6}{e^3}+\frac {126\,d^4\,x^5}{e^4}+\frac {126\,d^5\,x^4}{e^5}+\frac {84\,d^6\,x^3}{e^6}+\frac {36\,d^7\,x^2}{e^7}} \]
((d + e*x)^(1/2)*((x^2*(a + b*x)^(1/2)*(656370*A*a*b^7*d^5 - 243474*B*a^7* b*e^5 - 127050*A*a^6*b^2*e^5 + 364650*B*a^2*b^6*d^5 - 1823250*A*a^2*b^6*d^ 4*e + 760410*A*a^5*b^3*d*e^4 - 2044250*B*a^3*b^5*d^4*e + 1375122*B*a^6*b^2 *d*e^4 + 2497300*A*a^3*b^5*d^3*e^2 - 1892100*A*a^4*b^4*d^2*e^3 + 3619300*B *a^4*b^4*d^3*e^2 - 3143028*B*a^5*b^3*d^2*e^3))/(765765*e^9*(a*e - b*d)^6) - (x*(a + b*x)^(1/2)*(102102*B*a^8*e^5 + 210210*A*a^7*b*e^5 - 656370*A*a^2 *b^6*d^5 - 24310*B*a^3*b^5*d^5 + 2309450*A*a^3*b^5*d^4*e - 1217370*A*a^6*b ^2*d*e^4 + 475150*B*a^4*b^4*d^4*e - 3558100*A*a^4*b^4*d^3*e^2 + 2891700*A* a^5*b^3*d^2*e^3 - 1116220*B*a^5*b^3*d^3*e^2 + 1123836*B*a^6*b^2*d^2*e^3 - 540078*B*a^7*b*d*e^4))/(765765*e^9*(a*e - b*d)^6) - ((a + b*x)^(1/2)*(9009 0*A*a^8*e^5 + 12012*B*a^8*d*e^4 - 218790*A*a^3*b^5*d^5 + 48620*B*a^4*b^4*d ^5 + 850850*A*a^4*b^4*d^4*e - 123760*B*a^5*b^3*d^4*e - 62832*B*a^7*b*d^2*e ^3 - 1392300*A*a^5*b^3*d^3*e^2 + 1178100*A*a^6*b^2*d^2*e^3 + 128520*B*a^6* b^2*d^3*e^2 - 510510*A*a^7*b*d*e^4))/(765765*e^9*(a*e - b*d)^6) + (x^3*(a + b*x)^(1/2)*(218790*A*b^8*d^5 + 461890*B*a*b^7*d^5 - 630*A*a^5*b^3*e^5 - 152082*B*a^6*b^2*e^5 + 5950*A*a^4*b^4*d*e^4 - 2044250*B*a^2*b^6*d^4*e + 90 8362*B*a^5*b^3*d*e^4 + 66300*A*a^2*b^6*d^3*e^2 - 25500*A*a^3*b^5*d^2*e^3 + 2932500*B*a^3*b^5*d^3*e^2 - 2249780*B*a^4*b^4*d^2*e^3 - 121550*A*a*b^7*d^ 4*e))/(765765*e^9*(a*e - b*d)^6) + (256*b^7*x^8*(a + b*x)^(1/2)*(10*A*b*e - 17*B*a*e + 7*B*b*d))/(765765*e^5*(a*e - b*d)^6) - (16*b^4*x^5*(a + b*...