Integrand size = 24, antiderivative size = 147 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{13/2}} \, dx=-\frac {2 (B d-A e) (a+b x)^{7/2}}{11 e (b d-a e) (d+e x)^{11/2}}+\frac {2 (7 b B d+4 A b e-11 a B e) (a+b x)^{7/2}}{99 e (b d-a e)^2 (d+e x)^{9/2}}+\frac {4 b (7 b B d+4 A b e-11 a B e) (a+b x)^{7/2}}{693 e (b d-a e)^3 (d+e x)^{7/2}} \]
-2/11*(-A*e+B*d)*(b*x+a)^(7/2)/e/(-a*e+b*d)/(e*x+d)^(11/2)+2/99*(4*A*b*e-1 1*B*a*e+7*B*b*d)*(b*x+a)^(7/2)/e/(-a*e+b*d)^2/(e*x+d)^(9/2)+4/693*b*(4*A*b *e-11*B*a*e+7*B*b*d)*(b*x+a)^(7/2)/e/(-a*e+b*d)^3/(e*x+d)^(7/2)
Time = 0.26 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.91 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{13/2}} \, dx=\frac {2 (a+b x)^{11/2} \left (-63 B d e+63 A e^2+\frac {77 b B d (d+e x)}{a+b x}-\frac {154 A b e (d+e x)}{a+b x}+\frac {77 a B e (d+e x)}{a+b x}+\frac {99 A b^2 (d+e x)^2}{(a+b x)^2}-\frac {99 a b B (d+e x)^2}{(a+b x)^2}\right )}{693 (b d-a e)^3 (d+e x)^{11/2}} \]
(2*(a + b*x)^(11/2)*(-63*B*d*e + 63*A*e^2 + (77*b*B*d*(d + e*x))/(a + b*x) - (154*A*b*e*(d + e*x))/(a + b*x) + (77*a*B*e*(d + e*x))/(a + b*x) + (99* A*b^2*(d + e*x)^2)/(a + b*x)^2 - (99*a*b*B*(d + e*x)^2)/(a + b*x)^2))/(693 *(b*d - a*e)^3*(d + e*x)^(11/2))
Time = 0.23 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.97, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {87, 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.
| \(\displaystyle \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{13/2}} \, dx\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle \frac {(-11 a B e+4 A b e+7 b B d) \int \frac {(a+b x)^{5/2}}{(d+e x)^{11/2}}dx}{11 e (b d-a e)}-\frac {2 (a+b x)^{7/2} (B d-A e)}{11 e (d+e x)^{11/2} (b d-a e)}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle \frac {(-11 a B e+4 A b e+7 b B d) \left (\frac {2 b \int \frac {(a+b x)^{5/2}}{(d+e x)^{9/2}}dx}{9 (b d-a e)}+\frac {2 (a+b x)^{7/2}}{9 (d+e x)^{9/2} (b d-a e)}\right )}{11 e (b d-a e)}-\frac {2 (a+b x)^{7/2} (B d-A e)}{11 e (d+e x)^{11/2} (b d-a e)}\) |
| \(\Big \downarrow \) 48 |
| \(\displaystyle \frac {\left (\frac {4 b (a+b x)^{7/2}}{63 (d+e x)^{7/2} (b d-a e)^2}+\frac {2 (a+b x)^{7/2}}{9 (d+e x)^{9/2} (b d-a e)}\right ) (-11 a B e+4 A b e+7 b B d)}{11 e (b d-a e)}-\frac {2 (a+b x)^{7/2} (B d-A e)}{11 e (d+e x)^{11/2} (b d-a e)}\) |
(-2*(B*d - A*e)*(a + b*x)^(7/2))/(11*e*(b*d - a*e)*(d + e*x)^(11/2)) + ((7 *b*B*d + 4*A*b*e - 11*a*B*e)*((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*e *(b*d - a*e))
3.23.32.3.1 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]))))
Time = 3.57 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.20
| method | result | size |
| gosper | \(-\frac {2 \left (b x +a \right )^{\frac {7}{2}} \left (8 A \,b^{2} e^{2} x^{2}-22 B a b \,e^{2} x^{2}+14 B \,b^{2} d e \,x^{2}-28 A a b \,e^{2} x +44 A \,b^{2} d e x +77 B \,a^{2} e^{2} x -170 B a b d e x +77 b^{2} B \,d^{2} x +63 a^{2} A \,e^{2}-154 A a b d e +99 A \,b^{2} d^{2}+14 B \,a^{2} d e -22 B a b \,d^{2}\right )}{693 \left (e x +d \right )^{\frac {11}{2}} \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}\) | \(177\) |
| default | \(-\frac {2 \left (8 A \,b^{4} e^{2} x^{4}-22 B a \,b^{3} e^{2} x^{4}+14 B \,b^{4} d e \,x^{4}-12 A a \,b^{3} e^{2} x^{3}+44 A \,b^{4} d e \,x^{3}+33 B \,a^{2} b^{2} e^{2} x^{3}-142 B a \,b^{3} d e \,x^{3}+77 B \,b^{4} d^{2} x^{3}+15 A \,a^{2} b^{2} e^{2} x^{2}-66 A a \,b^{3} d e \,x^{2}+99 A \,b^{4} d^{2} x^{2}+132 B \,a^{3} b \,e^{2} x^{2}-312 B \,a^{2} b^{2} d e \,x^{2}+132 B a \,b^{3} d^{2} x^{2}+98 A \,a^{3} b \,e^{2} x -264 A \,a^{2} b^{2} d e x +198 A a \,b^{3} d^{2} x +77 B \,a^{4} e^{2} x -142 B \,a^{3} b d e x +33 B \,a^{2} b^{2} d^{2} x +63 A \,a^{4} e^{2}-154 A \,a^{3} b d e +99 A \,a^{2} b^{2} d^{2}+14 B \,a^{4} d e -22 B \,a^{3} b \,d^{2}\right ) \left (b x +a \right )^{\frac {3}{2}}}{693 \left (e x +d \right )^{\frac {11}{2}} \left (a e -b d \right )^{3}}\) | \(321\) |
-2/693*(b*x+a)^(7/2)*(8*A*b^2*e^2*x^2-22*B*a*b*e^2*x^2+14*B*b^2*d*e*x^2-28 *A*a*b*e^2*x+44*A*b^2*d*e*x+77*B*a^2*e^2*x-170*B*a*b*d*e*x+77*B*b^2*d^2*x+ 63*A*a^2*e^2-154*A*a*b*d*e+99*A*b^2*d^2+14*B*a^2*d*e-22*B*a*b*d^2)/(e*x+d) ^(11/2)/(a^3*e^3-3*a^2*b*d*e^2+3*a*b^2*d^2*e-b^3*d^3)
Leaf count of result is larger than twice the leaf count of optimal. 693 vs. \(2 (129) = 258\).
Time = 64.93 (sec) , antiderivative size = 693, normalized size of antiderivative = 4.71 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{13/2}} \, dx=\frac {2 \, {\left (63 \, A a^{5} e^{2} + 2 \, {\left (7 \, B b^{5} d e - {\left (11 \, B a b^{4} - 4 \, A b^{5}\right )} e^{2}\right )} x^{5} + {\left (77 \, B b^{5} d^{2} - 4 \, {\left (32 \, B a b^{4} - 11 \, A b^{5}\right )} d e + {\left (11 \, B a^{2} b^{3} - 4 \, A a b^{4}\right )} e^{2}\right )} x^{4} + {\left (11 \, {\left (19 \, B a b^{4} + 9 \, A b^{5}\right )} d^{2} - 2 \, {\left (227 \, B a^{2} b^{3} + 11 \, A a b^{4}\right )} d e + 3 \, {\left (55 \, B a^{3} b^{2} + A a^{2} b^{3}\right )} e^{2}\right )} x^{3} - 11 \, {\left (2 \, B a^{4} b - 9 \, A a^{3} b^{2}\right )} d^{2} + 14 \, {\left (B a^{5} - 11 \, A a^{4} b\right )} d e + {\left (33 \, {\left (5 \, B a^{2} b^{3} + 9 \, A a b^{4}\right )} d^{2} - 2 \, {\left (227 \, B a^{3} b^{2} + 165 \, A a^{2} b^{3}\right )} d e + {\left (209 \, B a^{4} b + 113 \, A a^{3} b^{2}\right )} e^{2}\right )} x^{2} + {\left (11 \, {\left (B a^{3} b^{2} + 27 \, A a^{2} b^{3}\right )} d^{2} - 2 \, {\left (64 \, B a^{4} b + 209 \, A a^{3} b^{2}\right )} d e + 7 \, {\left (11 \, B a^{5} + 23 \, A a^{4} b\right )} e^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}}{693 \, {\left (b^{3} d^{9} - 3 \, a b^{2} d^{8} e + 3 \, a^{2} b d^{7} e^{2} - a^{3} d^{6} e^{3} + {\left (b^{3} d^{3} e^{6} - 3 \, a b^{2} d^{2} e^{7} + 3 \, a^{2} b d e^{8} - a^{3} e^{9}\right )} x^{6} + 6 \, {\left (b^{3} d^{4} e^{5} - 3 \, a b^{2} d^{3} e^{6} + 3 \, a^{2} b d^{2} e^{7} - a^{3} d e^{8}\right )} x^{5} + 15 \, {\left (b^{3} d^{5} e^{4} - 3 \, a b^{2} d^{4} e^{5} + 3 \, a^{2} b d^{3} e^{6} - a^{3} d^{2} e^{7}\right )} x^{4} + 20 \, {\left (b^{3} d^{6} e^{3} - 3 \, a b^{2} d^{5} e^{4} + 3 \, a^{2} b d^{4} e^{5} - a^{3} d^{3} e^{6}\right )} x^{3} + 15 \, {\left (b^{3} d^{7} e^{2} - 3 \, a b^{2} d^{6} e^{3} + 3 \, a^{2} b d^{5} e^{4} - a^{3} d^{4} e^{5}\right )} x^{2} + 6 \, {\left (b^{3} d^{8} e - 3 \, a b^{2} d^{7} e^{2} + 3 \, a^{2} b d^{6} e^{3} - a^{3} d^{5} e^{4}\right )} x\right )}} \]
2/693*(63*A*a^5*e^2 + 2*(7*B*b^5*d*e - (11*B*a*b^4 - 4*A*b^5)*e^2)*x^5 + ( 77*B*b^5*d^2 - 4*(32*B*a*b^4 - 11*A*b^5)*d*e + (11*B*a^2*b^3 - 4*A*a*b^4)* e^2)*x^4 + (11*(19*B*a*b^4 + 9*A*b^5)*d^2 - 2*(227*B*a^2*b^3 + 11*A*a*b^4) *d*e + 3*(55*B*a^3*b^2 + A*a^2*b^3)*e^2)*x^3 - 11*(2*B*a^4*b - 9*A*a^3*b^2 )*d^2 + 14*(B*a^5 - 11*A*a^4*b)*d*e + (33*(5*B*a^2*b^3 + 9*A*a*b^4)*d^2 - 2*(227*B*a^3*b^2 + 165*A*a^2*b^3)*d*e + (209*B*a^4*b + 113*A*a^3*b^2)*e^2) *x^2 + (11*(B*a^3*b^2 + 27*A*a^2*b^3)*d^2 - 2*(64*B*a^4*b + 209*A*a^3*b^2) *d*e + 7*(11*B*a^5 + 23*A*a^4*b)*e^2)*x)*sqrt(b*x + a)*sqrt(e*x + d)/(b^3* d^9 - 3*a*b^2*d^8*e + 3*a^2*b*d^7*e^2 - a^3*d^6*e^3 + (b^3*d^3*e^6 - 3*a*b ^2*d^2*e^7 + 3*a^2*b*d*e^8 - a^3*e^9)*x^6 + 6*(b^3*d^4*e^5 - 3*a*b^2*d^3*e ^6 + 3*a^2*b*d^2*e^7 - a^3*d*e^8)*x^5 + 15*(b^3*d^5*e^4 - 3*a*b^2*d^4*e^5 + 3*a^2*b*d^3*e^6 - a^3*d^2*e^7)*x^4 + 20*(b^3*d^6*e^3 - 3*a*b^2*d^5*e^4 + 3*a^2*b*d^4*e^5 - a^3*d^3*e^6)*x^3 + 15*(b^3*d^7*e^2 - 3*a*b^2*d^6*e^3 + 3*a^2*b*d^5*e^4 - a^3*d^4*e^5)*x^2 + 6*(b^3*d^8*e - 3*a*b^2*d^7*e^2 + 3*a^ 2*b*d^6*e^3 - a^3*d^5*e^4)*x)
Timed out. \[ \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{13/2}} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{13/2}} \, dx=\text {Exception raised: ValueError} \]
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. 663 vs. \(2 (129) = 258\).
Time = 0.82 (sec) , antiderivative size = 663, normalized size of antiderivative = 4.51 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{13/2}} \, dx=\frac {2 \, {\left ({\left (b x + a\right )} {\left (\frac {2 \, {\left (7 \, B b^{14} d^{3} e^{6} {\left | b \right |} - 25 \, B a b^{13} d^{2} e^{7} {\left | b \right |} + 4 \, A b^{14} d^{2} e^{7} {\left | b \right |} + 29 \, B a^{2} b^{12} d e^{8} {\left | b \right |} - 8 \, A a b^{13} d e^{8} {\left | b \right |} - 11 \, B a^{3} b^{11} e^{9} {\left | b \right |} + 4 \, A a^{2} b^{12} e^{9} {\left | b \right |}\right )} {\left (b x + a\right )}}{b^{7} d^{5} e^{5} - 5 \, a b^{6} d^{4} e^{6} + 10 \, a^{2} b^{5} d^{3} e^{7} - 10 \, a^{3} b^{4} d^{2} e^{8} + 5 \, a^{4} b^{3} d e^{9} - a^{5} b^{2} e^{10}} + \frac {11 \, {\left (7 \, B b^{15} d^{4} e^{5} {\left | b \right |} - 32 \, B a b^{14} d^{3} e^{6} {\left | b \right |} + 4 \, A b^{15} d^{3} e^{6} {\left | b \right |} + 54 \, B a^{2} b^{13} d^{2} e^{7} {\left | b \right |} - 12 \, A a b^{14} d^{2} e^{7} {\left | b \right |} - 40 \, B a^{3} b^{12} d e^{8} {\left | b \right |} + 12 \, A a^{2} b^{13} d e^{8} {\left | b \right |} + 11 \, B a^{4} b^{11} e^{9} {\left | b \right |} - 4 \, A a^{3} b^{12} e^{9} {\left | b \right |}\right )}}{b^{7} d^{5} e^{5} - 5 \, a b^{6} d^{4} e^{6} + 10 \, a^{2} b^{5} d^{3} e^{7} - 10 \, a^{3} b^{4} d^{2} e^{8} + 5 \, a^{4} b^{3} d e^{9} - a^{5} b^{2} e^{10}}\right )} - \frac {99 \, {\left (B a b^{15} d^{4} e^{5} {\left | b \right |} - A b^{16} d^{4} e^{5} {\left | b \right |} - 4 \, B a^{2} b^{14} d^{3} e^{6} {\left | b \right |} + 4 \, A a b^{15} d^{3} e^{6} {\left | b \right |} + 6 \, B a^{3} b^{13} d^{2} e^{7} {\left | b \right |} - 6 \, A a^{2} b^{14} d^{2} e^{7} {\left | b \right |} - 4 \, B a^{4} b^{12} d e^{8} {\left | b \right |} + 4 \, A a^{3} b^{13} d e^{8} {\left | b \right |} + B a^{5} b^{11} e^{9} {\left | b \right |} - A a^{4} b^{12} e^{9} {\left | b \right |}\right )}}{b^{7} d^{5} e^{5} - 5 \, a b^{6} d^{4} e^{6} + 10 \, a^{2} b^{5} d^{3} e^{7} - 10 \, a^{3} b^{4} d^{2} e^{8} + 5 \, a^{4} b^{3} d e^{9} - a^{5} b^{2} e^{10}}\right )} {\left (b x + a\right )}^{\frac {7}{2}}}{693 \, {\left (b^{2} d + {\left (b x + a\right )} b e - a b e\right )}^{\frac {11}{2}}} \]
2/693*((b*x + a)*(2*(7*B*b^14*d^3*e^6*abs(b) - 25*B*a*b^13*d^2*e^7*abs(b) + 4*A*b^14*d^2*e^7*abs(b) + 29*B*a^2*b^12*d*e^8*abs(b) - 8*A*a*b^13*d*e^8* abs(b) - 11*B*a^3*b^11*e^9*abs(b) + 4*A*a^2*b^12*e^9*abs(b))*(b*x + a)/(b^ 7*d^5*e^5 - 5*a*b^6*d^4*e^6 + 10*a^2*b^5*d^3*e^7 - 10*a^3*b^4*d^2*e^8 + 5* a^4*b^3*d*e^9 - a^5*b^2*e^10) + 11*(7*B*b^15*d^4*e^5*abs(b) - 32*B*a*b^14* d^3*e^6*abs(b) + 4*A*b^15*d^3*e^6*abs(b) + 54*B*a^2*b^13*d^2*e^7*abs(b) - 12*A*a*b^14*d^2*e^7*abs(b) - 40*B*a^3*b^12*d*e^8*abs(b) + 12*A*a^2*b^13*d* e^8*abs(b) + 11*B*a^4*b^11*e^9*abs(b) - 4*A*a^3*b^12*e^9*abs(b))/(b^7*d^5* e^5 - 5*a*b^6*d^4*e^6 + 10*a^2*b^5*d^3*e^7 - 10*a^3*b^4*d^2*e^8 + 5*a^4*b^ 3*d*e^9 - a^5*b^2*e^10)) - 99*(B*a*b^15*d^4*e^5*abs(b) - A*b^16*d^4*e^5*ab s(b) - 4*B*a^2*b^14*d^3*e^6*abs(b) + 4*A*a*b^15*d^3*e^6*abs(b) + 6*B*a^3*b ^13*d^2*e^7*abs(b) - 6*A*a^2*b^14*d^2*e^7*abs(b) - 4*B*a^4*b^12*d*e^8*abs( b) + 4*A*a^3*b^13*d*e^8*abs(b) + B*a^5*b^11*e^9*abs(b) - A*a^4*b^12*e^9*ab s(b))/(b^7*d^5*e^5 - 5*a*b^6*d^4*e^6 + 10*a^2*b^5*d^3*e^7 - 10*a^3*b^4*d^2 *e^8 + 5*a^4*b^3*d*e^9 - a^5*b^2*e^10))*(b*x + a)^(7/2)/(b^2*d + (b*x + a) *b*e - a*b*e)^(11/2)
Time = 3.13 (sec) , antiderivative size = 509, normalized size of antiderivative = 3.46 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{13/2}} \, dx=-\frac {\sqrt {d+e\,x}\,\left (\frac {\sqrt {a+b\,x}\,\left (28\,B\,a^5\,d\,e+126\,A\,a^5\,e^2-44\,B\,a^4\,b\,d^2-308\,A\,a^4\,b\,d\,e+198\,A\,a^3\,b^2\,d^2\right )}{693\,e^6\,{\left (a\,e-b\,d\right )}^3}+\frac {x\,\sqrt {a+b\,x}\,\left (154\,B\,a^5\,e^2-256\,B\,a^4\,b\,d\,e+322\,A\,a^4\,b\,e^2+22\,B\,a^3\,b^2\,d^2-836\,A\,a^3\,b^2\,d\,e+594\,A\,a^2\,b^3\,d^2\right )}{693\,e^6\,{\left (a\,e-b\,d\right )}^3}+\frac {x^2\,\sqrt {a+b\,x}\,\left (418\,B\,a^4\,b\,e^2-908\,B\,a^3\,b^2\,d\,e+226\,A\,a^3\,b^2\,e^2+330\,B\,a^2\,b^3\,d^2-660\,A\,a^2\,b^3\,d\,e+594\,A\,a\,b^4\,d^2\right )}{693\,e^6\,{\left (a\,e-b\,d\right )}^3}+\frac {x^3\,\sqrt {a+b\,x}\,\left (330\,B\,a^3\,b^2\,e^2-908\,B\,a^2\,b^3\,d\,e+6\,A\,a^2\,b^3\,e^2+418\,B\,a\,b^4\,d^2-44\,A\,a\,b^4\,d\,e+198\,A\,b^5\,d^2\right )}{693\,e^6\,{\left (a\,e-b\,d\right )}^3}+\frac {4\,b^4\,x^5\,\sqrt {a+b\,x}\,\left (4\,A\,b\,e-11\,B\,a\,e+7\,B\,b\,d\right )}{693\,e^5\,{\left (a\,e-b\,d\right )}^3}-\frac {2\,b^3\,x^4\,\left (a\,e-11\,b\,d\right )\,\sqrt {a+b\,x}\,\left (4\,A\,b\,e-11\,B\,a\,e+7\,B\,b\,d\right )}{693\,e^6\,{\left (a\,e-b\,d\right )}^3}\right )}{x^6+\frac {d^6}{e^6}+\frac {6\,d\,x^5}{e}+\frac {6\,d^5\,x}{e^5}+\frac {15\,d^2\,x^4}{e^2}+\frac {20\,d^3\,x^3}{e^3}+\frac {15\,d^4\,x^2}{e^4}} \]
-((d + e*x)^(1/2)*(((a + b*x)^(1/2)*(126*A*a^5*e^2 + 28*B*a^5*d*e - 44*B*a ^4*b*d^2 + 198*A*a^3*b^2*d^2 - 308*A*a^4*b*d*e))/(693*e^6*(a*e - b*d)^3) + (x*(a + b*x)^(1/2)*(154*B*a^5*e^2 + 322*A*a^4*b*e^2 + 594*A*a^2*b^3*d^2 + 22*B*a^3*b^2*d^2 - 256*B*a^4*b*d*e - 836*A*a^3*b^2*d*e))/(693*e^6*(a*e - b*d)^3) + (x^2*(a + b*x)^(1/2)*(594*A*a*b^4*d^2 + 418*B*a^4*b*e^2 + 226*A* a^3*b^2*e^2 + 330*B*a^2*b^3*d^2 - 660*A*a^2*b^3*d*e - 908*B*a^3*b^2*d*e))/ (693*e^6*(a*e - b*d)^3) + (x^3*(a + b*x)^(1/2)*(198*A*b^5*d^2 + 418*B*a*b^ 4*d^2 + 6*A*a^2*b^3*e^2 + 330*B*a^3*b^2*e^2 - 44*A*a*b^4*d*e - 908*B*a^2*b ^3*d*e))/(693*e^6*(a*e - b*d)^3) + (4*b^4*x^5*(a + b*x)^(1/2)*(4*A*b*e - 1 1*B*a*e + 7*B*b*d))/(693*e^5*(a*e - b*d)^3) - (2*b^3*x^4*(a*e - 11*b*d)*(a + b*x)^(1/2)*(4*A*b*e - 11*B*a*e + 7*B*b*d))/(693*e^6*(a*e - b*d)^3)))/(x ^6 + d^6/e^6 + (6*d*x^5)/e + (6*d^5*x)/e^5 + (15*d^2*x^4)/e^2 + (20*d^3*x^ 3)/e^3 + (15*d^4*x^2)/e^4)