Integrand size = 33, antiderivative size = 393 \[ \int \frac {(A+B x) (d+e x)^{11/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {231 e^4 (10 b B d+3 A b e-13 a B e) \sqrt {d+e x}}{128 b^7}+\frac {77 e^4 (10 b B d+3 A b e-13 a B e) (d+e x)^{3/2}}{128 b^6 (b d-a e)}-\frac {231 e^3 (10 b B d+3 A b e-13 a B e) (d+e x)^{5/2}}{640 b^5 (b d-a e) (a+b x)}-\frac {33 e^2 (10 b B d+3 A b e-13 a B e) (d+e x)^{7/2}}{320 b^4 (b d-a e) (a+b x)^2}-\frac {11 e (10 b B d+3 A b e-13 a B e) (d+e x)^{9/2}}{240 b^3 (b d-a e) (a+b x)^3}-\frac {(10 b B d+3 A b e-13 a B e) (d+e x)^{11/2}}{40 b^2 (b d-a e) (a+b x)^4}-\frac {(A b-a B) (d+e x)^{13/2}}{5 b (b d-a e) (a+b x)^5}-\frac {231 e^4 \sqrt {b d-a e} (10 b B d+3 A b e-13 a B e) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 b^{15/2}} \]
77/128*e^4*(3*A*b*e-13*B*a*e+10*B*b*d)*(e*x+d)^(3/2)/b^6/(-a*e+b*d)-231/64 0*e^3*(3*A*b*e-13*B*a*e+10*B*b*d)*(e*x+d)^(5/2)/b^5/(-a*e+b*d)/(b*x+a)-33/ 320*e^2*(3*A*b*e-13*B*a*e+10*B*b*d)*(e*x+d)^(7/2)/b^4/(-a*e+b*d)/(b*x+a)^2 -11/240*e*(3*A*b*e-13*B*a*e+10*B*b*d)*(e*x+d)^(9/2)/b^3/(-a*e+b*d)/(b*x+a) ^3-1/40*(3*A*b*e-13*B*a*e+10*B*b*d)*(e*x+d)^(11/2)/b^2/(-a*e+b*d)/(b*x+a)^ 4-1/5*(A*b-B*a)*(e*x+d)^(13/2)/b/(-a*e+b*d)/(b*x+a)^5-231/128*e^4*(3*A*b*e -13*B*a*e+10*B*b*d)*arctanh(b^(1/2)*(e*x+d)^(1/2)/(-a*e+b*d)^(1/2))*(-a*e+ b*d)^(1/2)/b^(15/2)+231/128*e^4*(3*A*b*e-13*B*a*e+10*B*b*d)*(e*x+d)^(1/2)/ b^7
Time = 2.54 (sec) , antiderivative size = 558, normalized size of antiderivative = 1.42 \[ \int \frac {(A+B x) (d+e x)^{11/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {\sqrt {d+e x} \left (3 A b \left (-3465 a^5 e^5+1155 a^4 b e^4 (d-14 e x)+462 a^3 b^2 e^3 \left (d^2+12 d e x-64 e^2 x^2\right )+66 a^2 b^3 e^2 \left (4 d^3+33 d^2 e x+159 d e^2 x^2-395 e^3 x^3\right )+11 a b^4 e \left (16 d^4+112 d^3 e x+366 d^2 e^2 x^2+880 d e^3 x^3-965 e^4 x^4\right )+b^5 \left (128 d^5+816 d^4 e x+2248 d^3 e^2 x^2+3590 d^2 e^3 x^3+4215 d e^4 x^4-1280 e^5 x^5\right )\right )+B \left (45045 a^6 e^5+1155 a^5 b e^4 (-43 d+182 e x)+924 a^4 b^2 e^3 \left (6 d^2-253 d e x+416 e^2 x^2\right )+66 a^3 b^3 e^2 \left (18 d^3+411 d^2 e x-6547 d e^2 x^2+5135 e^3 x^3\right )+11 a^2 b^4 e \left (32 d^4+524 d^3 e x+4782 d^2 e^2 x^2-35140 d e^3 x^3+12545 e^4 x^4\right )+10 b^6 x \left (48 d^5+328 d^4 e x+1030 d^3 e^2 x^2+2295 d^2 e^3 x^3-2048 d e^4 x^4-128 e^5 x^5\right )+a b^5 \left (96 d^5+1712 d^4 e x+11036 d^3 e^2 x^2+50130 d^2 e^3 x^3-160945 d e^4 x^4+16640 e^5 x^5\right )\right )\right )}{1920 b^7 (a+b x)^5}-\frac {231 e^4 \sqrt {-b d+a e} (10 b B d+3 A b e-13 a B e) \arctan \left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{128 b^{15/2}} \]
-1/1920*(Sqrt[d + e*x]*(3*A*b*(-3465*a^5*e^5 + 1155*a^4*b*e^4*(d - 14*e*x) + 462*a^3*b^2*e^3*(d^2 + 12*d*e*x - 64*e^2*x^2) + 66*a^2*b^3*e^2*(4*d^3 + 33*d^2*e*x + 159*d*e^2*x^2 - 395*e^3*x^3) + 11*a*b^4*e*(16*d^4 + 112*d^3* e*x + 366*d^2*e^2*x^2 + 880*d*e^3*x^3 - 965*e^4*x^4) + b^5*(128*d^5 + 816* d^4*e*x + 2248*d^3*e^2*x^2 + 3590*d^2*e^3*x^3 + 4215*d*e^4*x^4 - 1280*e^5* x^5)) + B*(45045*a^6*e^5 + 1155*a^5*b*e^4*(-43*d + 182*e*x) + 924*a^4*b^2* e^3*(6*d^2 - 253*d*e*x + 416*e^2*x^2) + 66*a^3*b^3*e^2*(18*d^3 + 411*d^2*e *x - 6547*d*e^2*x^2 + 5135*e^3*x^3) + 11*a^2*b^4*e*(32*d^4 + 524*d^3*e*x + 4782*d^2*e^2*x^2 - 35140*d*e^3*x^3 + 12545*e^4*x^4) + 10*b^6*x*(48*d^5 + 328*d^4*e*x + 1030*d^3*e^2*x^2 + 2295*d^2*e^3*x^3 - 2048*d*e^4*x^4 - 128*e ^5*x^5) + a*b^5*(96*d^5 + 1712*d^4*e*x + 11036*d^3*e^2*x^2 + 50130*d^2*e^3 *x^3 - 160945*d*e^4*x^4 + 16640*e^5*x^5))))/(b^7*(a + b*x)^5) - (231*e^4*S qrt[-(b*d) + a*e]*(10*b*B*d + 3*A*b*e - 13*a*B*e)*ArcTan[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[-(b*d) + a*e]])/(128*b^(15/2))
Time = 0.40 (sec) , antiderivative size = 292, normalized size of antiderivative = 0.74, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1184, 27, 87, 51, 51, 51, 51, 60, 60, 73, 221}
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.
| \(\displaystyle \int \frac {(A+B x) (d+e x)^{11/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 1184 |
| \(\displaystyle b^6 \int \frac {(A+B x) (d+e x)^{11/2}}{b^6 (a+b x)^6}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {(A+B x) (d+e x)^{11/2}}{(a+b x)^6}dx\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle \frac {(-13 a B e+3 A b e+10 b B d) \int \frac {(d+e x)^{11/2}}{(a+b x)^5}dx}{10 b (b d-a e)}-\frac {(d+e x)^{13/2} (A b-a B)}{5 b (a+b x)^5 (b d-a e)}\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {(-13 a B e+3 A b e+10 b B d) \left (\frac {11 e \int \frac {(d+e x)^{9/2}}{(a+b x)^4}dx}{8 b}-\frac {(d+e x)^{11/2}}{4 b (a+b x)^4}\right )}{10 b (b d-a e)}-\frac {(d+e x)^{13/2} (A b-a B)}{5 b (a+b x)^5 (b d-a e)}\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {(-13 a B e+3 A b e+10 b B d) \left (\frac {11 e \left (\frac {3 e \int \frac {(d+e x)^{7/2}}{(a+b x)^3}dx}{2 b}-\frac {(d+e x)^{9/2}}{3 b (a+b x)^3}\right )}{8 b}-\frac {(d+e x)^{11/2}}{4 b (a+b x)^4}\right )}{10 b (b d-a e)}-\frac {(d+e x)^{13/2} (A b-a B)}{5 b (a+b x)^5 (b d-a e)}\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {(-13 a B e+3 A b e+10 b B d) \left (\frac {11 e \left (\frac {3 e \left (\frac {7 e \int \frac {(d+e x)^{5/2}}{(a+b x)^2}dx}{4 b}-\frac {(d+e x)^{7/2}}{2 b (a+b x)^2}\right )}{2 b}-\frac {(d+e x)^{9/2}}{3 b (a+b x)^3}\right )}{8 b}-\frac {(d+e x)^{11/2}}{4 b (a+b x)^4}\right )}{10 b (b d-a e)}-\frac {(d+e x)^{13/2} (A b-a B)}{5 b (a+b x)^5 (b d-a e)}\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {(-13 a B e+3 A b e+10 b B d) \left (\frac {11 e \left (\frac {3 e \left (\frac {7 e \left (\frac {5 e \int \frac {(d+e x)^{3/2}}{a+b x}dx}{2 b}-\frac {(d+e x)^{5/2}}{b (a+b x)}\right )}{4 b}-\frac {(d+e x)^{7/2}}{2 b (a+b x)^2}\right )}{2 b}-\frac {(d+e x)^{9/2}}{3 b (a+b x)^3}\right )}{8 b}-\frac {(d+e x)^{11/2}}{4 b (a+b x)^4}\right )}{10 b (b d-a e)}-\frac {(d+e x)^{13/2} (A b-a B)}{5 b (a+b x)^5 (b d-a e)}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {(-13 a B e+3 A b e+10 b B d) \left (\frac {11 e \left (\frac {3 e \left (\frac {7 e \left (\frac {5 e \left (\frac {(b d-a e) \int \frac {\sqrt {d+e x}}{a+b x}dx}{b}+\frac {2 (d+e x)^{3/2}}{3 b}\right )}{2 b}-\frac {(d+e x)^{5/2}}{b (a+b x)}\right )}{4 b}-\frac {(d+e x)^{7/2}}{2 b (a+b x)^2}\right )}{2 b}-\frac {(d+e x)^{9/2}}{3 b (a+b x)^3}\right )}{8 b}-\frac {(d+e x)^{11/2}}{4 b (a+b x)^4}\right )}{10 b (b d-a e)}-\frac {(d+e x)^{13/2} (A b-a B)}{5 b (a+b x)^5 (b d-a e)}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {(-13 a B e+3 A b e+10 b B d) \left (\frac {11 e \left (\frac {3 e \left (\frac {7 e \left (\frac {5 e \left (\frac {(b d-a e) \left (\frac {(b d-a e) \int \frac {1}{(a+b x) \sqrt {d+e x}}dx}{b}+\frac {2 \sqrt {d+e x}}{b}\right )}{b}+\frac {2 (d+e x)^{3/2}}{3 b}\right )}{2 b}-\frac {(d+e x)^{5/2}}{b (a+b x)}\right )}{4 b}-\frac {(d+e x)^{7/2}}{2 b (a+b x)^2}\right )}{2 b}-\frac {(d+e x)^{9/2}}{3 b (a+b x)^3}\right )}{8 b}-\frac {(d+e x)^{11/2}}{4 b (a+b x)^4}\right )}{10 b (b d-a e)}-\frac {(d+e x)^{13/2} (A b-a B)}{5 b (a+b x)^5 (b d-a e)}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {(-13 a B e+3 A b e+10 b B d) \left (\frac {11 e \left (\frac {3 e \left (\frac {7 e \left (\frac {5 e \left (\frac {(b d-a e) \left (\frac {2 (b d-a e) \int \frac {1}{a+\frac {b (d+e x)}{e}-\frac {b d}{e}}d\sqrt {d+e x}}{b e}+\frac {2 \sqrt {d+e x}}{b}\right )}{b}+\frac {2 (d+e x)^{3/2}}{3 b}\right )}{2 b}-\frac {(d+e x)^{5/2}}{b (a+b x)}\right )}{4 b}-\frac {(d+e x)^{7/2}}{2 b (a+b x)^2}\right )}{2 b}-\frac {(d+e x)^{9/2}}{3 b (a+b x)^3}\right )}{8 b}-\frac {(d+e x)^{11/2}}{4 b (a+b x)^4}\right )}{10 b (b d-a e)}-\frac {(d+e x)^{13/2} (A b-a B)}{5 b (a+b x)^5 (b d-a e)}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {(-13 a B e+3 A b e+10 b B d) \left (\frac {11 e \left (\frac {3 e \left (\frac {7 e \left (\frac {5 e \left (\frac {(b d-a e) \left (\frac {2 \sqrt {d+e x}}{b}-\frac {2 \sqrt {b d-a e} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{3/2}}\right )}{b}+\frac {2 (d+e x)^{3/2}}{3 b}\right )}{2 b}-\frac {(d+e x)^{5/2}}{b (a+b x)}\right )}{4 b}-\frac {(d+e x)^{7/2}}{2 b (a+b x)^2}\right )}{2 b}-\frac {(d+e x)^{9/2}}{3 b (a+b x)^3}\right )}{8 b}-\frac {(d+e x)^{11/2}}{4 b (a+b x)^4}\right )}{10 b (b d-a e)}-\frac {(d+e x)^{13/2} (A b-a B)}{5 b (a+b x)^5 (b d-a e)}\) |
-1/5*((A*b - a*B)*(d + e*x)^(13/2))/(b*(b*d - a*e)*(a + b*x)^5) + ((10*b*B *d + 3*A*b*e - 13*a*B*e)*(-1/4*(d + e*x)^(11/2)/(b*(a + b*x)^4) + (11*e*(- 1/3*(d + e*x)^(9/2)/(b*(a + b*x)^3) + (3*e*(-1/2*(d + e*x)^(7/2)/(b*(a + b *x)^2) + (7*e*(-((d + e*x)^(5/2)/(b*(a + b*x))) + (5*e*((2*(d + e*x)^(3/2) )/(3*b) + ((b*d - a*e)*((2*Sqrt[d + e*x])/b - (2*Sqrt[b*d - a*e]*ArcTanh[( Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/b^(3/2)))/b))/(2*b)))/(4*b)))/(2* b)))/(8*b)))/(10*b*(b*d - a*e))
3.19.25.3.1 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] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n}, x ] && ILtQ[m, -1] && FractionQ[n] && GtQ[n, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
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]))))
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
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 = 1.45 (sec) , antiderivative size = 487, normalized size of antiderivative = 1.24
| method | result | size |
| pseudoelliptic | \(\frac {-\frac {693 \left (a e -b d \right ) \left (\left (A e +\frac {10 B d}{3}\right ) b -\frac {13 B a e}{3}\right ) \left (b x +a \right )^{5} e^{4} \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{128}+\frac {693 \sqrt {\left (a e -b d \right ) b}\, \left (\left (\frac {256 \left (\frac {B x}{3}+A \right ) x^{5} e^{5}}{693}-\frac {281 \left (-\frac {4096 B x}{2529}+A \right ) x^{4} d \,e^{4}}{231}-\frac {718 \left (\frac {765 B x}{359}+A \right ) x^{3} d^{2} e^{3}}{693}-\frac {2248 \left (\frac {2575 B x}{1686}+A \right ) x^{2} d^{3} e^{2}}{3465}-\frac {272 \left (\frac {205 B x}{153}+A \right ) x \,d^{4} e}{1155}-\frac {128 d^{5} \left (\frac {5 B x}{4}+A \right )}{3465}\right ) b^{6}-\frac {16 \left (\left (\frac {1040}{33} B \,x^{5}-\frac {965}{16} A \,x^{4}\right ) e^{5}+55 x^{3} d \left (-\frac {32189 B x}{5808}+A \right ) e^{4}+\frac {183 \left (\frac {2785 B x}{671}+A \right ) x^{2} d^{2} e^{3}}{8}+7 x \,d^{3} \left (\frac {2759 B x}{924}+A \right ) e^{2}+d^{4} \left (\frac {107 B x}{33}+A \right ) e +\frac {2 B \,d^{5}}{11}\right ) a \,b^{5}}{315}-\frac {8 \left (\left (\frac {12545}{72} B \,x^{4}-\frac {395}{4} A \,x^{3}\right ) e^{4}+\frac {159 \left (-\frac {17570 B x}{1431}+A \right ) x^{2} d \,e^{3}}{4}+\frac {33 x \,d^{2} \left (\frac {797 B x}{99}+A \right ) e^{2}}{4}+d^{3} \left (\frac {131 B x}{18}+A \right ) e +\frac {4 B \,d^{4}}{9}\right ) e \,a^{2} b^{4}}{105}-\frac {2 \left (\left (\frac {5135}{21} B \,x^{3}-64 A \,x^{2}\right ) e^{3}+12 \left (-\frac {6547 B x}{252}+A \right ) x d \,e^{2}+d^{2} \left (\frac {137 B x}{7}+A \right ) e +\frac {6 B \,d^{3}}{7}\right ) e^{2} a^{3} b^{3}}{15}-\frac {\left (-14 x \left (-\frac {832 B x}{105}+A \right ) e^{2}+d \left (-\frac {1012 B x}{15}+A \right ) e +\frac {8 B \,d^{2}}{5}\right ) e^{3} a^{4} b^{2}}{3}+\left (\left (A -\frac {182 B x}{9}\right ) e +\frac {43 B d}{9}\right ) e^{4} a^{5} b -\frac {13 B \,a^{6} e^{5}}{3}\right ) \sqrt {e x +d}}{128}}{\left (b x +a \right )^{5} b^{7} \sqrt {\left (a e -b d \right ) b}}\) | \(487\) |
| risch | \(\frac {2 e^{4} \left (B b e x +3 A b e -18 B a e +16 B b d \right ) \sqrt {e x +d}}{3 b^{7}}-\frac {\left (2 a e -2 b d \right ) e^{4} \left (\frac {\left (-\frac {843}{256} A e \,b^{5}+\frac {2373}{256} B e \,b^{4} a -\frac {765}{128} B d \,b^{5}\right ) \left (e x +d \right )^{\frac {9}{2}}-\frac {b^{3} \left (3981 A a b \,e^{2}-3981 A \,b^{2} d e -12131 a^{2} B \,e^{2}+20281 B a b d e -8150 B \,b^{2} d^{2}\right ) \left (e x +d \right )^{\frac {7}{2}}}{384}+\left (-\frac {131}{10} A \,a^{2} b^{3} e^{3}+\frac {131}{5} A a \,b^{4} d \,e^{2}-\frac {131}{10} A \,b^{5} d^{2} e +\frac {1253}{30} B \,e^{3} a^{3} b^{2}-\frac {561}{5} B \,a^{2} b^{3} d \,e^{2}+\frac {991}{10} B a \,b^{4} d^{2} e -\frac {86}{3} B \,b^{5} d^{3}\right ) \left (e x +d \right )^{\frac {5}{2}}+\left (-\frac {977}{128} A \,a^{3} b^{2} e^{4}+\frac {2931}{128} A \,a^{2} b^{3} d \,e^{3}-\frac {2931}{128} A a \,b^{4} d^{2} e^{2}+\frac {977}{128} A \,b^{5} d^{3} e +\frac {9629}{384} B \,e^{4} a^{4} b -\frac {35585}{384} B \,a^{3} b^{2} d \,e^{3}+\frac {16327}{128} B \,a^{2} b^{3} d^{2} e^{2}-\frac {29723}{384} B a \,b^{4} d^{3} e +\frac {3349}{192} B \,b^{5} d^{4}\right ) \left (e x +d \right )^{\frac {3}{2}}+\left (-\frac {437}{256} A \,a^{4} b \,e^{5}+\frac {437}{64} A \,a^{3} b^{2} d \,e^{4}-\frac {1311}{128} A \,a^{2} b^{3} d^{2} e^{3}+\frac {437}{64} A a \,b^{4} d^{3} e^{2}-\frac {437}{256} A \,b^{5} d^{4} e +\frac {1467}{256} B \,a^{5} e^{5}-\frac {3449}{128} B \,a^{4} b d \,e^{4}+\frac {6461}{128} B \,a^{3} b^{2} d^{2} e^{3}-\frac {753}{16} B \,a^{2} b^{3} d^{3} e^{2}+\frac {5587}{256} B a \,b^{4} d^{4} e -\frac {515}{128} B \,b^{5} d^{5}\right ) \sqrt {e x +d}}{\left (b \left (e x +d \right )+a e -b d \right )^{5}}+\frac {231 \left (3 A b e -13 B a e +10 B b d \right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{256 \sqrt {\left (a e -b d \right ) b}}\right )}{b^{7}}\) | \(554\) |
| derivativedivides | \(2 e^{4} \left (\frac {\frac {B \left (e x +d \right )^{\frac {3}{2}} b}{3}+A b e \sqrt {e x +d}-6 B a e \sqrt {e x +d}+5 B b d \sqrt {e x +d}}{b^{7}}-\frac {\frac {\left (-\frac {843}{256} A a \,b^{5} e^{2}+\frac {843}{256} A \,b^{6} d e +\frac {2373}{256} B \,a^{2} b^{4} e^{2}-\frac {3903}{256} B a \,b^{5} d e +\frac {765}{128} B \,b^{6} d^{2}\right ) \left (e x +d \right )^{\frac {9}{2}}-\frac {b^{3} \left (3981 A \,a^{2} b \,e^{3}-7962 A a \,b^{2} d \,e^{2}+3981 A \,b^{3} d^{2} e -12131 B \,e^{3} a^{3}+32412 B \,a^{2} b d \,e^{2}-28431 B a \,b^{2} d^{2} e +8150 B \,b^{3} d^{3}\right ) \left (e x +d \right )^{\frac {7}{2}}}{384}+\left (-\frac {131}{10} A \,a^{3} b^{3} e^{4}+\frac {393}{10} A \,a^{2} b^{4} d \,e^{3}-\frac {393}{10} A a \,b^{5} d^{2} e^{2}+\frac {131}{10} A \,b^{6} d^{3} e +\frac {1253}{30} B \,a^{4} b^{2} e^{4}-\frac {4619}{30} B \,a^{3} b^{3} d \,e^{3}+\frac {2113}{10} B \,a^{2} b^{4} d^{2} e^{2}-\frac {3833}{30} B a \,b^{5} d^{3} e +\frac {86}{3} B \,b^{6} d^{4}\right ) \left (e x +d \right )^{\frac {5}{2}}+\left (-\frac {977}{128} A \,a^{4} b^{2} e^{5}+\frac {977}{32} A \,a^{3} b^{3} d \,e^{4}-\frac {2931}{64} A \,a^{2} b^{4} d^{2} e^{3}+\frac {977}{32} A a \,b^{5} d^{3} e^{2}-\frac {977}{128} A \,b^{6} d^{4} e +\frac {9629}{384} B \,a^{5} b \,e^{5}-\frac {22607}{192} B \,a^{4} b^{2} d \,e^{4}+\frac {42283}{192} B \,a^{3} b^{3} d^{2} e^{3}-\frac {4919}{24} B \,a^{2} b^{4} d^{3} e^{2}+\frac {36421}{384} B a \,b^{5} d^{4} e -\frac {3349}{192} B \,b^{6} d^{5}\right ) \left (e x +d \right )^{\frac {3}{2}}+\left (\frac {437}{256} A \,b^{6} d^{5} e -\frac {437}{256} A \,a^{5} b \,e^{6}-\frac {8365}{256} B \,a^{5} b d \,e^{5}+\frac {4955}{64} B \,a^{4} b^{2} d^{2} e^{4}-\frac {12485}{128} B \,a^{3} b^{3} d^{3} e^{3}+\frac {2185}{128} A \,a^{2} b^{4} d^{3} e^{3}-\frac {2185}{256} A a \,b^{5} d^{4} e^{2}+\frac {1467}{256} B \,a^{6} e^{6}+\frac {515}{128} B \,b^{6} d^{6}-\frac {2185}{128} A \,a^{3} b^{3} d^{2} e^{4}+\frac {17635}{256} B \,a^{2} b^{4} d^{4} e^{2}-\frac {6617}{256} B a \,b^{5} d^{5} e +\frac {2185}{256} A \,a^{4} b^{2} d \,e^{5}\right ) \sqrt {e x +d}}{\left (b \left (e x +d \right )+a e -b d \right )^{5}}+\frac {231 \left (3 A a b \,e^{2}-3 A \,b^{2} d e -13 a^{2} B \,e^{2}+23 B a b d e -10 B \,b^{2} d^{2}\right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{256 \sqrt {\left (a e -b d \right ) b}}}{b^{7}}\right )\) | \(734\) |
| default | \(2 e^{4} \left (\frac {\frac {B \left (e x +d \right )^{\frac {3}{2}} b}{3}+A b e \sqrt {e x +d}-6 B a e \sqrt {e x +d}+5 B b d \sqrt {e x +d}}{b^{7}}-\frac {\frac {\left (-\frac {843}{256} A a \,b^{5} e^{2}+\frac {843}{256} A \,b^{6} d e +\frac {2373}{256} B \,a^{2} b^{4} e^{2}-\frac {3903}{256} B a \,b^{5} d e +\frac {765}{128} B \,b^{6} d^{2}\right ) \left (e x +d \right )^{\frac {9}{2}}-\frac {b^{3} \left (3981 A \,a^{2} b \,e^{3}-7962 A a \,b^{2} d \,e^{2}+3981 A \,b^{3} d^{2} e -12131 B \,e^{3} a^{3}+32412 B \,a^{2} b d \,e^{2}-28431 B a \,b^{2} d^{2} e +8150 B \,b^{3} d^{3}\right ) \left (e x +d \right )^{\frac {7}{2}}}{384}+\left (-\frac {131}{10} A \,a^{3} b^{3} e^{4}+\frac {393}{10} A \,a^{2} b^{4} d \,e^{3}-\frac {393}{10} A a \,b^{5} d^{2} e^{2}+\frac {131}{10} A \,b^{6} d^{3} e +\frac {1253}{30} B \,a^{4} b^{2} e^{4}-\frac {4619}{30} B \,a^{3} b^{3} d \,e^{3}+\frac {2113}{10} B \,a^{2} b^{4} d^{2} e^{2}-\frac {3833}{30} B a \,b^{5} d^{3} e +\frac {86}{3} B \,b^{6} d^{4}\right ) \left (e x +d \right )^{\frac {5}{2}}+\left (-\frac {977}{128} A \,a^{4} b^{2} e^{5}+\frac {977}{32} A \,a^{3} b^{3} d \,e^{4}-\frac {2931}{64} A \,a^{2} b^{4} d^{2} e^{3}+\frac {977}{32} A a \,b^{5} d^{3} e^{2}-\frac {977}{128} A \,b^{6} d^{4} e +\frac {9629}{384} B \,a^{5} b \,e^{5}-\frac {22607}{192} B \,a^{4} b^{2} d \,e^{4}+\frac {42283}{192} B \,a^{3} b^{3} d^{2} e^{3}-\frac {4919}{24} B \,a^{2} b^{4} d^{3} e^{2}+\frac {36421}{384} B a \,b^{5} d^{4} e -\frac {3349}{192} B \,b^{6} d^{5}\right ) \left (e x +d \right )^{\frac {3}{2}}+\left (\frac {437}{256} A \,b^{6} d^{5} e -\frac {437}{256} A \,a^{5} b \,e^{6}-\frac {8365}{256} B \,a^{5} b d \,e^{5}+\frac {4955}{64} B \,a^{4} b^{2} d^{2} e^{4}-\frac {12485}{128} B \,a^{3} b^{3} d^{3} e^{3}+\frac {2185}{128} A \,a^{2} b^{4} d^{3} e^{3}-\frac {2185}{256} A a \,b^{5} d^{4} e^{2}+\frac {1467}{256} B \,a^{6} e^{6}+\frac {515}{128} B \,b^{6} d^{6}-\frac {2185}{128} A \,a^{3} b^{3} d^{2} e^{4}+\frac {17635}{256} B \,a^{2} b^{4} d^{4} e^{2}-\frac {6617}{256} B a \,b^{5} d^{5} e +\frac {2185}{256} A \,a^{4} b^{2} d \,e^{5}\right ) \sqrt {e x +d}}{\left (b \left (e x +d \right )+a e -b d \right )^{5}}+\frac {231 \left (3 A a b \,e^{2}-3 A \,b^{2} d e -13 a^{2} B \,e^{2}+23 B a b d e -10 B \,b^{2} d^{2}\right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{256 \sqrt {\left (a e -b d \right ) b}}}{b^{7}}\right )\) | \(734\) |
693/128*(-(a*e-b*d)*((A*e+10/3*B*d)*b-13/3*B*a*e)*(b*x+a)^5*e^4*arctan(b*( e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2))+((a*e-b*d)*b)^(1/2)*((256/693*(1/3*B*x+A )*x^5*e^5-281/231*(-4096/2529*B*x+A)*x^4*d*e^4-718/693*(765/359*B*x+A)*x^3 *d^2*e^3-2248/3465*(2575/1686*B*x+A)*x^2*d^3*e^2-272/1155*(205/153*B*x+A)* x*d^4*e-128/3465*d^5*(5/4*B*x+A))*b^6-16/315*((1040/33*B*x^5-965/16*A*x^4) *e^5+55*x^3*d*(-32189/5808*B*x+A)*e^4+183/8*(2785/671*B*x+A)*x^2*d^2*e^3+7 *x*d^3*(2759/924*B*x+A)*e^2+d^4*(107/33*B*x+A)*e+2/11*B*d^5)*a*b^5-8/105*( (12545/72*B*x^4-395/4*A*x^3)*e^4+159/4*(-17570/1431*B*x+A)*x^2*d*e^3+33/4* x*d^2*(797/99*B*x+A)*e^2+d^3*(131/18*B*x+A)*e+4/9*B*d^4)*e*a^2*b^4-2/15*(( 5135/21*B*x^3-64*A*x^2)*e^3+12*(-6547/252*B*x+A)*x*d*e^2+d^2*(137/7*B*x+A) *e+6/7*B*d^3)*e^2*a^3*b^3-1/3*(-14*x*(-832/105*B*x+A)*e^2+d*(-1012/15*B*x+ A)*e+8/5*B*d^2)*e^3*a^4*b^2+((A-182/9*B*x)*e+43/9*B*d)*e^4*a^5*b-13/3*B*a^ 6*e^5)*(e*x+d)^(1/2))/((a*e-b*d)*b)^(1/2)/(b*x+a)^5/b^7
Leaf count of result is larger than twice the leaf count of optimal. 926 vs. \(2 (353) = 706\).
Time = 0.45 (sec) , antiderivative size = 1862, normalized size of antiderivative = 4.74 \[ \int \frac {(A+B x) (d+e x)^{11/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\text {Too large to display} \]
[-1/3840*(3465*(10*B*a^5*b*d*e^4 - (13*B*a^6 - 3*A*a^5*b)*e^5 + (10*B*b^6* d*e^4 - (13*B*a*b^5 - 3*A*b^6)*e^5)*x^5 + 5*(10*B*a*b^5*d*e^4 - (13*B*a^2* b^4 - 3*A*a*b^5)*e^5)*x^4 + 10*(10*B*a^2*b^4*d*e^4 - (13*B*a^3*b^3 - 3*A*a ^2*b^4)*e^5)*x^3 + 10*(10*B*a^3*b^3*d*e^4 - (13*B*a^4*b^2 - 3*A*a^3*b^3)*e ^5)*x^2 + 5*(10*B*a^4*b^2*d*e^4 - (13*B*a^5*b - 3*A*a^4*b^2)*e^5)*x)*sqrt( (b*d - a*e)/b)*log((b*e*x + 2*b*d - a*e + 2*sqrt(e*x + d)*b*sqrt((b*d - a* e)/b))/(b*x + a)) - 2*(1280*B*b^6*e^5*x^6 - 96*(B*a*b^5 + 4*A*b^6)*d^5 - 1 76*(2*B*a^2*b^4 + 3*A*a*b^5)*d^4*e - 396*(3*B*a^3*b^3 + 2*A*a^2*b^4)*d^3*e ^2 - 1386*(4*B*a^4*b^2 + A*a^3*b^3)*d^2*e^3 + 1155*(43*B*a^5*b - 3*A*a^4*b ^2)*d*e^4 - 3465*(13*B*a^6 - 3*A*a^5*b)*e^5 + 1280*(16*B*b^6*d*e^4 - (13*B *a*b^5 - 3*A*b^6)*e^5)*x^5 - 5*(4590*B*b^6*d^2*e^3 - (32189*B*a*b^5 - 2529 *A*b^6)*d*e^4 + 2123*(13*B*a^2*b^4 - 3*A*a*b^5)*e^5)*x^4 - 10*(1030*B*b^6* d^3*e^2 + 3*(1671*B*a*b^5 + 359*A*b^6)*d^2*e^3 - 22*(1757*B*a^2*b^4 - 132* A*a*b^5)*d*e^4 + 2607*(13*B*a^3*b^3 - 3*A*a^2*b^4)*e^5)*x^3 - 2*(1640*B*b^ 6*d^4*e + 2*(2759*B*a*b^5 + 1686*A*b^6)*d^3*e^2 + 33*(797*B*a^2*b^4 + 183* A*a*b^5)*d^2*e^3 - 33*(6547*B*a^3*b^3 - 477*A*a^2*b^4)*d*e^4 + 14784*(13*B *a^4*b^2 - 3*A*a^3*b^3)*e^5)*x^2 - 2*(240*B*b^6*d^5 + 8*(107*B*a*b^5 + 153 *A*b^6)*d^4*e + 22*(131*B*a^2*b^4 + 84*A*a*b^5)*d^3*e^2 + 99*(137*B*a^3*b^ 3 + 33*A*a^2*b^4)*d^2*e^3 - 462*(253*B*a^4*b^2 - 18*A*a^3*b^3)*d*e^4 + 808 5*(13*B*a^5*b - 3*A*a^4*b^2)*e^5)*x)*sqrt(e*x + d))/(b^12*x^5 + 5*a*b^1...
Timed out. \[ \int \frac {(A+B x) (d+e x)^{11/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {(A+B x) (d+e x)^{11/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(a*e-b*d>0)', see `assume?` for m ore detail
Leaf count of result is larger than twice the leaf count of optimal. 1066 vs. \(2 (353) = 706\).
Time = 0.31 (sec) , antiderivative size = 1066, normalized size of antiderivative = 2.71 \[ \int \frac {(A+B x) (d+e x)^{11/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\text {Too large to display} \]
231/128*(10*B*b^2*d^2*e^4 - 23*B*a*b*d*e^5 + 3*A*b^2*d*e^5 + 13*B*a^2*e^6 - 3*A*a*b*e^6)*arctan(sqrt(e*x + d)*b/sqrt(-b^2*d + a*b*e))/(sqrt(-b^2*d + a*b*e)*b^7) - 1/1920*(22950*(e*x + d)^(9/2)*B*b^6*d^2*e^4 - 81500*(e*x + d)^(7/2)*B*b^6*d^3*e^4 + 110080*(e*x + d)^(5/2)*B*b^6*d^4*e^4 - 66980*(e*x + d)^(3/2)*B*b^6*d^5*e^4 + 15450*sqrt(e*x + d)*B*b^6*d^6*e^4 - 58545*(e*x + d)^(9/2)*B*a*b^5*d*e^5 + 12645*(e*x + d)^(9/2)*A*b^6*d*e^5 + 284310*(e* x + d)^(7/2)*B*a*b^5*d^2*e^5 - 39810*(e*x + d)^(7/2)*A*b^6*d^2*e^5 - 49062 4*(e*x + d)^(5/2)*B*a*b^5*d^3*e^5 + 50304*(e*x + d)^(5/2)*A*b^6*d^3*e^5 + 364210*(e*x + d)^(3/2)*B*a*b^5*d^4*e^5 - 29310*(e*x + d)^(3/2)*A*b^6*d^4*e ^5 - 99255*sqrt(e*x + d)*B*a*b^5*d^5*e^5 + 6555*sqrt(e*x + d)*A*b^6*d^5*e^ 5 + 35595*(e*x + d)^(9/2)*B*a^2*b^4*e^6 - 12645*(e*x + d)^(9/2)*A*a*b^5*e^ 6 - 324120*(e*x + d)^(7/2)*B*a^2*b^4*d*e^6 + 79620*(e*x + d)^(7/2)*A*a*b^5 *d*e^6 + 811392*(e*x + d)^(5/2)*B*a^2*b^4*d^2*e^6 - 150912*(e*x + d)^(5/2) *A*a*b^5*d^2*e^6 - 787040*(e*x + d)^(3/2)*B*a^2*b^4*d^3*e^6 + 117240*(e*x + d)^(3/2)*A*a*b^5*d^3*e^6 + 264525*sqrt(e*x + d)*B*a^2*b^4*d^4*e^6 - 3277 5*sqrt(e*x + d)*A*a*b^5*d^4*e^6 + 121310*(e*x + d)^(7/2)*B*a^3*b^3*e^7 - 3 9810*(e*x + d)^(7/2)*A*a^2*b^4*e^7 - 591232*(e*x + d)^(5/2)*B*a^3*b^3*d*e^ 7 + 150912*(e*x + d)^(5/2)*A*a^2*b^4*d*e^7 + 845660*(e*x + d)^(3/2)*B*a^3* b^3*d^2*e^7 - 175860*(e*x + d)^(3/2)*A*a^2*b^4*d^2*e^7 - 374550*sqrt(e*x + d)*B*a^3*b^3*d^3*e^7 + 65550*sqrt(e*x + d)*A*a^2*b^4*d^3*e^7 + 160384*...
Time = 0.49 (sec) , antiderivative size = 996, normalized size of antiderivative = 2.53 \[ \int \frac {(A+B x) (d+e x)^{11/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\left (\frac {2\,A\,e^5-2\,B\,d\,e^4}{b^6}+\frac {2\,B\,e^4\,\left (6\,b^6\,d-6\,a\,b^5\,e\right )}{b^{12}}\right )\,\sqrt {d+e\,x}-\frac {{\left (d+e\,x\right )}^{5/2}\,\left (\frac {1253\,B\,a^4\,b^2\,e^8}{15}-\frac {4619\,B\,a^3\,b^3\,d\,e^7}{15}-\frac {131\,A\,a^3\,b^3\,e^8}{5}+\frac {2113\,B\,a^2\,b^4\,d^2\,e^6}{5}+\frac {393\,A\,a^2\,b^4\,d\,e^7}{5}-\frac {3833\,B\,a\,b^5\,d^3\,e^5}{15}-\frac {393\,A\,a\,b^5\,d^2\,e^6}{5}+\frac {172\,B\,b^6\,d^4\,e^4}{3}+\frac {131\,A\,b^6\,d^3\,e^5}{5}\right )-{\left (d+e\,x\right )}^{3/2}\,\left (-\frac {9629\,B\,a^5\,b\,e^9}{192}+\frac {22607\,B\,a^4\,b^2\,d\,e^8}{96}+\frac {977\,A\,a^4\,b^2\,e^9}{64}-\frac {42283\,B\,a^3\,b^3\,d^2\,e^7}{96}-\frac {977\,A\,a^3\,b^3\,d\,e^8}{16}+\frac {4919\,B\,a^2\,b^4\,d^3\,e^6}{12}+\frac {2931\,A\,a^2\,b^4\,d^2\,e^7}{32}-\frac {36421\,B\,a\,b^5\,d^4\,e^5}{192}-\frac {977\,A\,a\,b^5\,d^3\,e^6}{16}+\frac {3349\,B\,b^6\,d^5\,e^4}{96}+\frac {977\,A\,b^6\,d^4\,e^5}{64}\right )-{\left (d+e\,x\right )}^{7/2}\,\left (-\frac {12131\,B\,a^3\,b^3\,e^7}{192}+\frac {2701\,B\,a^2\,b^4\,d\,e^6}{16}+\frac {1327\,A\,a^2\,b^4\,e^7}{64}-\frac {9477\,B\,a\,b^5\,d^2\,e^5}{64}-\frac {1327\,A\,a\,b^5\,d\,e^6}{32}+\frac {4075\,B\,b^6\,d^3\,e^4}{96}+\frac {1327\,A\,b^6\,d^2\,e^5}{64}\right )+\sqrt {d+e\,x}\,\left (\frac {1467\,B\,a^6\,e^{10}}{128}-\frac {8365\,B\,a^5\,b\,d\,e^9}{128}-\frac {437\,A\,a^5\,b\,e^{10}}{128}+\frac {4955\,B\,a^4\,b^2\,d^2\,e^8}{32}+\frac {2185\,A\,a^4\,b^2\,d\,e^9}{128}-\frac {12485\,B\,a^3\,b^3\,d^3\,e^7}{64}-\frac {2185\,A\,a^3\,b^3\,d^2\,e^8}{64}+\frac {17635\,B\,a^2\,b^4\,d^4\,e^6}{128}+\frac {2185\,A\,a^2\,b^4\,d^3\,e^7}{64}-\frac {6617\,B\,a\,b^5\,d^5\,e^5}{128}-\frac {2185\,A\,a\,b^5\,d^4\,e^6}{128}+\frac {515\,B\,b^6\,d^6\,e^4}{64}+\frac {437\,A\,b^6\,d^5\,e^5}{128}\right )+{\left (d+e\,x\right )}^{9/2}\,\left (\frac {2373\,B\,a^2\,b^4\,e^6}{128}-\frac {3903\,B\,a\,b^5\,d\,e^5}{128}-\frac {843\,A\,a\,b^5\,e^6}{128}+\frac {765\,B\,b^6\,d^2\,e^4}{64}+\frac {843\,A\,b^6\,d\,e^5}{128}\right )}{\left (d+e\,x\right )\,\left (5\,a^4\,b^8\,e^4-20\,a^3\,b^9\,d\,e^3+30\,a^2\,b^{10}\,d^2\,e^2-20\,a\,b^{11}\,d^3\,e+5\,b^{12}\,d^4\right )-{\left (d+e\,x\right )}^2\,\left (-10\,a^3\,b^9\,e^3+30\,a^2\,b^{10}\,d\,e^2-30\,a\,b^{11}\,d^2\,e+10\,b^{12}\,d^3\right )+b^{12}\,{\left (d+e\,x\right )}^5-\left (5\,b^{12}\,d-5\,a\,b^{11}\,e\right )\,{\left (d+e\,x\right )}^4-b^{12}\,d^5+{\left (d+e\,x\right )}^3\,\left (10\,a^2\,b^{10}\,e^2-20\,a\,b^{11}\,d\,e+10\,b^{12}\,d^2\right )+a^5\,b^7\,e^5-5\,a^4\,b^8\,d\,e^4-10\,a^2\,b^{10}\,d^3\,e^2+10\,a^3\,b^9\,d^2\,e^3+5\,a\,b^{11}\,d^4\,e}+\frac {2\,B\,e^4\,{\left (d+e\,x\right )}^{3/2}}{3\,b^6}+\frac {e^4\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d+e\,x}\,1{}\mathrm {i}}{\sqrt {b\,d-a\,e}}\right )\,\sqrt {b\,d-a\,e}\,\left (3\,A\,b\,e-13\,B\,a\,e+10\,B\,b\,d\right )\,231{}\mathrm {i}}{128\,b^{15/2}} \]
((2*A*e^5 - 2*B*d*e^4)/b^6 + (2*B*e^4*(6*b^6*d - 6*a*b^5*e))/b^12)*(d + e* x)^(1/2) - ((d + e*x)^(5/2)*((1253*B*a^4*b^2*e^8)/15 - (131*A*a^3*b^3*e^8) /5 + (131*A*b^6*d^3*e^5)/5 + (172*B*b^6*d^4*e^4)/3 - (393*A*a*b^5*d^2*e^6) /5 + (393*A*a^2*b^4*d*e^7)/5 - (3833*B*a*b^5*d^3*e^5)/15 - (4619*B*a^3*b^3 *d*e^7)/15 + (2113*B*a^2*b^4*d^2*e^6)/5) - (d + e*x)^(3/2)*((977*A*a^4*b^2 *e^9)/64 - (9629*B*a^5*b*e^9)/192 + (977*A*b^6*d^4*e^5)/64 + (3349*B*b^6*d ^5*e^4)/96 - (977*A*a*b^5*d^3*e^6)/16 - (977*A*a^3*b^3*d*e^8)/16 - (36421* B*a*b^5*d^4*e^5)/192 + (22607*B*a^4*b^2*d*e^8)/96 + (2931*A*a^2*b^4*d^2*e^ 7)/32 + (4919*B*a^2*b^4*d^3*e^6)/12 - (42283*B*a^3*b^3*d^2*e^7)/96) - (d + e*x)^(7/2)*((1327*A*a^2*b^4*e^7)/64 - (12131*B*a^3*b^3*e^7)/192 + (1327*A *b^6*d^2*e^5)/64 + (4075*B*b^6*d^3*e^4)/96 - (9477*B*a*b^5*d^2*e^5)/64 + ( 2701*B*a^2*b^4*d*e^6)/16 - (1327*A*a*b^5*d*e^6)/32) + (d + e*x)^(1/2)*((14 67*B*a^6*e^10)/128 - (437*A*a^5*b*e^10)/128 + (437*A*b^6*d^5*e^5)/128 + (5 15*B*b^6*d^6*e^4)/64 - (2185*A*a*b^5*d^4*e^6)/128 + (2185*A*a^4*b^2*d*e^9) /128 - (6617*B*a*b^5*d^5*e^5)/128 + (2185*A*a^2*b^4*d^3*e^7)/64 - (2185*A* a^3*b^3*d^2*e^8)/64 + (17635*B*a^2*b^4*d^4*e^6)/128 - (12485*B*a^3*b^3*d^3 *e^7)/64 + (4955*B*a^4*b^2*d^2*e^8)/32 - (8365*B*a^5*b*d*e^9)/128) + (d + e*x)^(9/2)*((843*A*b^6*d*e^5)/128 - (843*A*a*b^5*e^6)/128 + (2373*B*a^2*b^ 4*e^6)/128 + (765*B*b^6*d^2*e^4)/64 - (3903*B*a*b^5*d*e^5)/128))/((d + e*x )*(5*b^12*d^4 + 5*a^4*b^8*e^4 - 20*a^3*b^9*d*e^3 + 30*a^2*b^10*d^2*e^2 ...