Integrand size = 35, antiderivative size = 126 \[ \int \frac {a+b x^2+c x^4}{x^5 \sqrt {d-e x} \sqrt {d+e x}} \, dx=-\frac {a \sqrt {d-e x} \sqrt {d+e x}}{4 d^2 x^4}-\frac {\left (4 b d^2+3 a e^2\right ) \sqrt {d-e x} \sqrt {d+e x}}{8 d^4 x^2}-\frac {\left (8 c d^4+4 b d^2 e^2+3 a e^4\right ) \text {arctanh}\left (\frac {\sqrt {d-e x} \sqrt {d+e x}}{d}\right )}{8 d^5} \]
-1/8*(3*a*e^4+4*b*d^2*e^2+8*c*d^4)*arctanh((-e*x+d)^(1/2)*(e*x+d)^(1/2)/d) /d^5-1/4*a*(-e*x+d)^(1/2)*(e*x+d)^(1/2)/d^2/x^4-1/8*(3*a*e^2+4*b*d^2)*(-e* x+d)^(1/2)*(e*x+d)^(1/2)/d^4/x^2
Time = 0.30 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.81 \[ \int \frac {a+b x^2+c x^4}{x^5 \sqrt {d-e x} \sqrt {d+e x}} \, dx=-\frac {\frac {d \sqrt {d-e x} \sqrt {d+e x} \left (2 a d^2+4 b d^2 x^2+3 a e^2 x^2\right )}{x^4}+2 \left (8 c d^4+4 b d^2 e^2+3 a e^4\right ) \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d-e x}}\right )}{8 d^5} \]
-1/8*((d*Sqrt[d - e*x]*Sqrt[d + e*x]*(2*a*d^2 + 4*b*d^2*x^2 + 3*a*e^2*x^2) )/x^4 + 2*(8*c*d^4 + 4*b*d^2*e^2 + 3*a*e^4)*ArcTanh[Sqrt[d + e*x]/Sqrt[d - e*x]])/d^5
Time = 0.39 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.43, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.229, Rules used = {1905, 1578, 1192, 25, 1471, 25, 298, 219}
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.
| \(\displaystyle \int \frac {a+b x^2+c x^4}{x^5 \sqrt {d-e x} \sqrt {d+e x}} \, dx\) |
| \(\Big \downarrow \) 1905 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^2} \int \frac {c x^4+b x^2+a}{x^5 \sqrt {d^2-e^2 x^2}}dx}{\sqrt {d-e x} \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 1578 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^2} \int \frac {c x^4+b x^2+a}{x^6 \sqrt {d^2-e^2 x^2}}dx^2}{2 \sqrt {d-e x} \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 1192 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^2} \int -\frac {c x^8-\left (2 c d^2+b e^2\right ) x^4+c d^4+a e^4+b d^2 e^2}{\left (d^2-x^4\right )^3}d\sqrt {d^2-e^2 x^2}}{\sqrt {d-e x} \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt {d^2-e^2 x^2} \int \frac {c x^8-\left (2 c d^2+b e^2\right ) x^4+c d^4+a e^4+b d^2 e^2}{\left (d^2-x^4\right )^3}d\sqrt {d^2-e^2 x^2}}{\sqrt {d-e x} \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 1471 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^2} \left (\frac {\int -\frac {4 c d^4-4 c x^4 d^2+4 b e^2 d^2+3 a e^4}{\left (d^2-x^4\right )^2}d\sqrt {d^2-e^2 x^2}}{4 d^2}-\frac {a e^4 \sqrt {d^2-e^2 x^2}}{4 d^2 \left (d^2-x^4\right )^2}\right )}{\sqrt {d-e x} \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^2} \left (-\frac {\int \frac {4 c d^4-4 c x^4 d^2+4 b e^2 d^2+3 a e^4}{\left (d^2-x^4\right )^2}d\sqrt {d^2-e^2 x^2}}{4 d^2}-\frac {a e^4 \sqrt {d^2-e^2 x^2}}{4 d^2 \left (d^2-x^4\right )^2}\right )}{\sqrt {d-e x} \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 298 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^2} \left (-\frac {\frac {\left (3 a e^4+4 b d^2 e^2+8 c d^4\right ) \int \frac {1}{d^2-x^4}d\sqrt {d^2-e^2 x^2}}{2 d^2}+\frac {e^2 \sqrt {d^2-e^2 x^2} \left (\frac {3 a e^2}{d^2}+4 b\right )}{2 \left (d^2-x^4\right )}}{4 d^2}-\frac {a e^4 \sqrt {d^2-e^2 x^2}}{4 d^2 \left (d^2-x^4\right )^2}\right )}{\sqrt {d-e x} \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^2} \left (-\frac {\frac {\text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right ) \left (3 a e^4+4 b d^2 e^2+8 c d^4\right )}{2 d^3}+\frac {e^2 \sqrt {d^2-e^2 x^2} \left (\frac {3 a e^2}{d^2}+4 b\right )}{2 \left (d^2-x^4\right )}}{4 d^2}-\frac {a e^4 \sqrt {d^2-e^2 x^2}}{4 d^2 \left (d^2-x^4\right )^2}\right )}{\sqrt {d-e x} \sqrt {d+e x}}\) |
(Sqrt[d^2 - e^2*x^2]*(-1/4*(a*e^4*Sqrt[d^2 - e^2*x^2])/(d^2*(d^2 - x^4)^2) - ((e^2*(4*b + (3*a*e^2)/d^2)*Sqrt[d^2 - e^2*x^2])/(2*(d^2 - x^4)) + ((8* c*d^4 + 4*b*d^2*e^2 + 3*a*e^4)*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/(2*d^3))/(4 *d^2)))/(Sqrt[d - e*x]*Sqrt[d + e*x])
3.2.37.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[(-( b*c - a*d))*x*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] - Simp[(a*d - b*c*( 2*p + 3))/(2*a*b*(p + 1)) Int[(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/2 + p, 0])
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[2/e^(n + 2*p + 1) Subst[Int[x^( 2*m + 1)*(e*f - d*g + g*x^2)^n*(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4)^p, x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ[p, 0] && ILtQ[n, 0] && IntegerQ[m + 1/2]
Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> With[{Qx = PolynomialQuotient[(a + b*x^2 + c*x^4)^p, d + e*x^2 , x], R = Coeff[PolynomialRemainder[(a + b*x^2 + c*x^4)^p, d + e*x^2, x], x , 0]}, Simp[(-R)*x*((d + e*x^2)^(q + 1)/(2*d*(q + 1))), x] + Simp[1/(2*d*(q + 1)) Int[(d + e*x^2)^(q + 1)*ExpandToSum[2*d*(q + 1)*Qx + R*(2*q + 3), x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^ 2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && LtQ[q, -1]
Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_ )^4)^(p_.), x_Symbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && Int egerQ[(m - 1)/2]
Int[((f_.)*(x_))^(m_.)*((d1_) + (e1_.)*(x_)^(non2_.))^(q_.)*((d2_) + (e2_.) *(x_)^(non2_.))^(q_.)*((a_.) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_))^(p_.), x _Symbol] :> Simp[(d1 + e1*x^(n/2))^FracPart[q]*((d2 + e2*x^(n/2))^FracPart[ q]/(d1*d2 + e1*e2*x^n)^FracPart[q]) Int[(f*x)^m*(d1*d2 + e1*e2*x^n)^q*(a + b*x^n + c*x^(2*n))^p, x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[non2, n/2] && EqQ[d2*e1 + d1*e2, 0]
Time = 0.44 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.13
| method | result | size |
| risch | \(-\frac {\sqrt {e x +d}\, \sqrt {-e x +d}\, \left (3 a \,e^{2} x^{2}+4 b \,d^{2} x^{2}+2 a \,d^{2}\right )}{8 d^{4} x^{4}}-\frac {\left (3 e^{4} a +4 e^{2} d^{2} b +8 d^{4} c \right ) \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right ) \sqrt {\left (e x +d \right ) \left (-e x +d \right )}}{8 d^{4} \sqrt {d^{2}}\, \sqrt {e x +d}\, \sqrt {-e x +d}}\) | \(143\) |
| default | \(-\frac {\sqrt {-e x +d}\, \sqrt {e x +d}\, \left (3 \ln \left (\frac {2 d \left (\sqrt {-e^{2} x^{2}+d^{2}}\, \operatorname {csgn}\left (d \right )+d \right )}{x}\right ) a \,e^{4} x^{4}+4 \ln \left (\frac {2 d \left (\sqrt {-e^{2} x^{2}+d^{2}}\, \operatorname {csgn}\left (d \right )+d \right )}{x}\right ) b \,d^{2} e^{2} x^{4}+8 \ln \left (\frac {2 d \left (\sqrt {-e^{2} x^{2}+d^{2}}\, \operatorname {csgn}\left (d \right )+d \right )}{x}\right ) c \,d^{4} x^{4}+3 \,\operatorname {csgn}\left (d \right ) a d \,e^{2} x^{2} \sqrt {-e^{2} x^{2}+d^{2}}+4 \,\operatorname {csgn}\left (d \right ) b \,d^{3} x^{2} \sqrt {-e^{2} x^{2}+d^{2}}+2 \,\operatorname {csgn}\left (d \right ) a \,d^{3} \sqrt {-e^{2} x^{2}+d^{2}}\right ) \operatorname {csgn}\left (d \right )}{8 d^{5} \sqrt {-e^{2} x^{2}+d^{2}}\, x^{4}}\) | \(222\) |
-1/8*(e*x+d)^(1/2)*(-e*x+d)^(1/2)*(3*a*e^2*x^2+4*b*d^2*x^2+2*a*d^2)/d^4/x^ 4-1/8/d^4*(3*a*e^4+4*b*d^2*e^2+8*c*d^4)/(d^2)^(1/2)*ln((2*d^2+2*(d^2)^(1/2 )*(-e^2*x^2+d^2)^(1/2))/x)*((e*x+d)*(-e*x+d))^(1/2)/(e*x+d)^(1/2)/(-e*x+d) ^(1/2)
Time = 0.27 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.81 \[ \int \frac {a+b x^2+c x^4}{x^5 \sqrt {d-e x} \sqrt {d+e x}} \, dx=\frac {{\left (8 \, c d^{4} + 4 \, b d^{2} e^{2} + 3 \, a e^{4}\right )} x^{4} \log \left (\frac {\sqrt {e x + d} \sqrt {-e x + d} - d}{x}\right ) - {\left (2 \, a d^{3} + {\left (4 \, b d^{3} + 3 \, a d e^{2}\right )} x^{2}\right )} \sqrt {e x + d} \sqrt {-e x + d}}{8 \, d^{5} x^{4}} \]
1/8*((8*c*d^4 + 4*b*d^2*e^2 + 3*a*e^4)*x^4*log((sqrt(e*x + d)*sqrt(-e*x + d) - d)/x) - (2*a*d^3 + (4*b*d^3 + 3*a*d*e^2)*x^2)*sqrt(e*x + d)*sqrt(-e*x + d))/(d^5*x^4)
Timed out. \[ \int \frac {a+b x^2+c x^4}{x^5 \sqrt {d-e x} \sqrt {d+e x}} \, dx=\text {Timed out} \]
Time = 0.29 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.53 \[ \int \frac {a+b x^2+c x^4}{x^5 \sqrt {d-e x} \sqrt {d+e x}} \, dx=-\frac {c \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right )}{d} - \frac {b e^{2} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right )}{2 \, d^{3}} - \frac {3 \, a e^{4} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right )}{8 \, d^{5}} - \frac {\sqrt {-e^{2} x^{2} + d^{2}} b}{2 \, d^{2} x^{2}} - \frac {3 \, \sqrt {-e^{2} x^{2} + d^{2}} a e^{2}}{8 \, d^{4} x^{2}} - \frac {\sqrt {-e^{2} x^{2} + d^{2}} a}{4 \, d^{2} x^{4}} \]
-c*log(2*d^2/abs(x) + 2*sqrt(-e^2*x^2 + d^2)*d/abs(x))/d - 1/2*b*e^2*log(2 *d^2/abs(x) + 2*sqrt(-e^2*x^2 + d^2)*d/abs(x))/d^3 - 3/8*a*e^4*log(2*d^2/a bs(x) + 2*sqrt(-e^2*x^2 + d^2)*d/abs(x))/d^5 - 1/2*sqrt(-e^2*x^2 + d^2)*b/ (d^2*x^2) - 3/8*sqrt(-e^2*x^2 + d^2)*a*e^2/(d^4*x^2) - 1/4*sqrt(-e^2*x^2 + d^2)*a/(d^2*x^4)
Leaf count of result is larger than twice the leaf count of optimal. 767 vs. \(2 (108) = 216\).
Time = 0.61 (sec) , antiderivative size = 767, normalized size of antiderivative = 6.09 \[ \int \frac {a+b x^2+c x^4}{x^5 \sqrt {d-e x} \sqrt {d+e x}} \, dx=-\frac {\frac {{\left (8 \, c d^{4} e + 4 \, b d^{2} e^{3} + 3 \, a e^{5}\right )} \log \left ({\left | -\frac {\sqrt {2} \sqrt {d} - \sqrt {-e x + d}}{\sqrt {e x + d}} + \frac {\sqrt {e x + d}}{\sqrt {2} \sqrt {d} - \sqrt {-e x + d}} + 2 \right |}\right )}{d^{5}} - \frac {{\left (8 \, c d^{4} e + 4 \, b d^{2} e^{3} + 3 \, a e^{5}\right )} \log \left ({\left | -\frac {\sqrt {2} \sqrt {d} - \sqrt {-e x + d}}{\sqrt {e x + d}} + \frac {\sqrt {e x + d}}{\sqrt {2} \sqrt {d} - \sqrt {-e x + d}} - 2 \right |}\right )}{d^{5}} - \frac {4 \, {\left (4 \, b d^{2} e^{3} {\left (\frac {\sqrt {2} \sqrt {d} - \sqrt {-e x + d}}{\sqrt {e x + d}} - \frac {\sqrt {e x + d}}{\sqrt {2} \sqrt {d} - \sqrt {-e x + d}}\right )}^{7} + 5 \, a e^{5} {\left (\frac {\sqrt {2} \sqrt {d} - \sqrt {-e x + d}}{\sqrt {e x + d}} - \frac {\sqrt {e x + d}}{\sqrt {2} \sqrt {d} - \sqrt {-e x + d}}\right )}^{7} - 16 \, b d^{2} e^{3} {\left (\frac {\sqrt {2} \sqrt {d} - \sqrt {-e x + d}}{\sqrt {e x + d}} - \frac {\sqrt {e x + d}}{\sqrt {2} \sqrt {d} - \sqrt {-e x + d}}\right )}^{5} + 12 \, a e^{5} {\left (\frac {\sqrt {2} \sqrt {d} - \sqrt {-e x + d}}{\sqrt {e x + d}} - \frac {\sqrt {e x + d}}{\sqrt {2} \sqrt {d} - \sqrt {-e x + d}}\right )}^{5} - 64 \, b d^{2} e^{3} {\left (\frac {\sqrt {2} \sqrt {d} - \sqrt {-e x + d}}{\sqrt {e x + d}} - \frac {\sqrt {e x + d}}{\sqrt {2} \sqrt {d} - \sqrt {-e x + d}}\right )}^{3} + 48 \, a e^{5} {\left (\frac {\sqrt {2} \sqrt {d} - \sqrt {-e x + d}}{\sqrt {e x + d}} - \frac {\sqrt {e x + d}}{\sqrt {2} \sqrt {d} - \sqrt {-e x + d}}\right )}^{3} + 256 \, b d^{2} e^{3} {\left (\frac {\sqrt {2} \sqrt {d} - \sqrt {-e x + d}}{\sqrt {e x + d}} - \frac {\sqrt {e x + d}}{\sqrt {2} \sqrt {d} - \sqrt {-e x + d}}\right )} + 320 \, a e^{5} {\left (\frac {\sqrt {2} \sqrt {d} - \sqrt {-e x + d}}{\sqrt {e x + d}} - \frac {\sqrt {e x + d}}{\sqrt {2} \sqrt {d} - \sqrt {-e x + d}}\right )}\right )}}{{\left ({\left (\frac {\sqrt {2} \sqrt {d} - \sqrt {-e x + d}}{\sqrt {e x + d}} - \frac {\sqrt {e x + d}}{\sqrt {2} \sqrt {d} - \sqrt {-e x + d}}\right )}^{2} - 4\right )}^{4} d^{5}}}{8 \, e} \]
-1/8*((8*c*d^4*e + 4*b*d^2*e^3 + 3*a*e^5)*log(abs(-(sqrt(2)*sqrt(d) - sqrt (-e*x + d))/sqrt(e*x + d) + sqrt(e*x + d)/(sqrt(2)*sqrt(d) - sqrt(-e*x + d )) + 2))/d^5 - (8*c*d^4*e + 4*b*d^2*e^3 + 3*a*e^5)*log(abs(-(sqrt(2)*sqrt( d) - sqrt(-e*x + d))/sqrt(e*x + d) + sqrt(e*x + d)/(sqrt(2)*sqrt(d) - sqrt (-e*x + d)) - 2))/d^5 - 4*(4*b*d^2*e^3*((sqrt(2)*sqrt(d) - sqrt(-e*x + d)) /sqrt(e*x + d) - sqrt(e*x + d)/(sqrt(2)*sqrt(d) - sqrt(-e*x + d)))^7 + 5*a *e^5*((sqrt(2)*sqrt(d) - sqrt(-e*x + d))/sqrt(e*x + d) - sqrt(e*x + d)/(sq rt(2)*sqrt(d) - sqrt(-e*x + d)))^7 - 16*b*d^2*e^3*((sqrt(2)*sqrt(d) - sqrt (-e*x + d))/sqrt(e*x + d) - sqrt(e*x + d)/(sqrt(2)*sqrt(d) - sqrt(-e*x + d )))^5 + 12*a*e^5*((sqrt(2)*sqrt(d) - sqrt(-e*x + d))/sqrt(e*x + d) - sqrt( e*x + d)/(sqrt(2)*sqrt(d) - sqrt(-e*x + d)))^5 - 64*b*d^2*e^3*((sqrt(2)*sq rt(d) - sqrt(-e*x + d))/sqrt(e*x + d) - sqrt(e*x + d)/(sqrt(2)*sqrt(d) - s qrt(-e*x + d)))^3 + 48*a*e^5*((sqrt(2)*sqrt(d) - sqrt(-e*x + d))/sqrt(e*x + d) - sqrt(e*x + d)/(sqrt(2)*sqrt(d) - sqrt(-e*x + d)))^3 + 256*b*d^2*e^3 *((sqrt(2)*sqrt(d) - sqrt(-e*x + d))/sqrt(e*x + d) - sqrt(e*x + d)/(sqrt(2 )*sqrt(d) - sqrt(-e*x + d))) + 320*a*e^5*((sqrt(2)*sqrt(d) - sqrt(-e*x + d ))/sqrt(e*x + d) - sqrt(e*x + d)/(sqrt(2)*sqrt(d) - sqrt(-e*x + d))))/(((( sqrt(2)*sqrt(d) - sqrt(-e*x + d))/sqrt(e*x + d) - sqrt(e*x + d)/(sqrt(2)*s qrt(d) - sqrt(-e*x + d)))^2 - 4)^4*d^5))/e
Time = 15.50 (sec) , antiderivative size = 932, normalized size of antiderivative = 7.40 \[ \int \frac {a+b x^2+c x^4}{x^5 \sqrt {d-e x} \sqrt {d+e x}} \, dx=\frac {\frac {a\,e^4}{4}+\frac {6\,a\,e^4\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^2}{{\left (\sqrt {d-e\,x}-\sqrt {d}\right )}^2}-\frac {53\,a\,e^4\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^4}{2\,{\left (\sqrt {d-e\,x}-\sqrt {d}\right )}^4}-\frac {87\,a\,e^4\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^6}{{\left (\sqrt {d-e\,x}-\sqrt {d}\right )}^6}+\frac {657\,a\,e^4\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^8}{4\,{\left (\sqrt {d-e\,x}-\sqrt {d}\right )}^8}-\frac {121\,a\,e^4\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^{10}}{{\left (\sqrt {d-e\,x}-\sqrt {d}\right )}^{10}}}{\frac {256\,d^5\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^4}{{\left (\sqrt {d-e\,x}-\sqrt {d}\right )}^4}-\frac {1024\,d^5\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^6}{{\left (\sqrt {d-e\,x}-\sqrt {d}\right )}^6}+\frac {1536\,d^5\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^8}{{\left (\sqrt {d-e\,x}-\sqrt {d}\right )}^8}-\frac {1024\,d^5\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^{10}}{{\left (\sqrt {d-e\,x}-\sqrt {d}\right )}^{10}}+\frac {256\,d^5\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^{12}}{{\left (\sqrt {d-e\,x}-\sqrt {d}\right )}^{12}}}-\frac {\frac {b\,e^2\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^2}{{\left (\sqrt {d-e\,x}-\sqrt {d}\right )}^2}-\frac {b\,e^2}{2}+\frac {15\,b\,e^2\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^4}{2\,{\left (\sqrt {d-e\,x}-\sqrt {d}\right )}^4}}{\frac {16\,d^3\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^2}{{\left (\sqrt {d-e\,x}-\sqrt {d}\right )}^2}-\frac {32\,d^3\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^4}{{\left (\sqrt {d-e\,x}-\sqrt {d}\right )}^4}+\frac {16\,d^3\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^6}{{\left (\sqrt {d-e\,x}-\sqrt {d}\right )}^6}}+\frac {c\,\left (\ln \left (\frac {{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^2}{{\left (\sqrt {d-e\,x}-\sqrt {d}\right )}^2}-1\right )-\ln \left (\frac {\sqrt {d+e\,x}-\sqrt {d}}{\sqrt {d-e\,x}-\sqrt {d}}\right )\right )}{d}-\frac {3\,a\,e^4\,\ln \left (\frac {\sqrt {d+e\,x}-\sqrt {d}}{\sqrt {d-e\,x}-\sqrt {d}}\right )}{8\,d^5}-\frac {b\,e^2\,\ln \left (\frac {\sqrt {d+e\,x}-\sqrt {d}}{\sqrt {d-e\,x}-\sqrt {d}}\right )}{2\,d^3}+\frac {3\,a\,e^4\,\ln \left (\frac {{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^2}{{\left (\sqrt {d-e\,x}-\sqrt {d}\right )}^2}-1\right )}{8\,d^5}+\frac {b\,e^2\,\ln \left (\frac {{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^2}{{\left (\sqrt {d-e\,x}-\sqrt {d}\right )}^2}-1\right )}{2\,d^3}+\frac {7\,a\,e^4\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^2}{256\,d^5\,{\left (\sqrt {d-e\,x}-\sqrt {d}\right )}^2}+\frac {a\,e^4\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^4}{1024\,d^5\,{\left (\sqrt {d-e\,x}-\sqrt {d}\right )}^4}+\frac {b\,e^2\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^2}{32\,d^3\,{\left (\sqrt {d-e\,x}-\sqrt {d}\right )}^2} \]
((a*e^4)/4 + (6*a*e^4*((d + e*x)^(1/2) - d^(1/2))^2)/((d - e*x)^(1/2) - d^ (1/2))^2 - (53*a*e^4*((d + e*x)^(1/2) - d^(1/2))^4)/(2*((d - e*x)^(1/2) - d^(1/2))^4) - (87*a*e^4*((d + e*x)^(1/2) - d^(1/2))^6)/((d - e*x)^(1/2) - d^(1/2))^6 + (657*a*e^4*((d + e*x)^(1/2) - d^(1/2))^8)/(4*((d - e*x)^(1/2) - d^(1/2))^8) - (121*a*e^4*((d + e*x)^(1/2) - d^(1/2))^10)/((d - e*x)^(1/ 2) - d^(1/2))^10)/((256*d^5*((d + e*x)^(1/2) - d^(1/2))^4)/((d - e*x)^(1/2 ) - d^(1/2))^4 - (1024*d^5*((d + e*x)^(1/2) - d^(1/2))^6)/((d - e*x)^(1/2) - d^(1/2))^6 + (1536*d^5*((d + e*x)^(1/2) - d^(1/2))^8)/((d - e*x)^(1/2) - d^(1/2))^8 - (1024*d^5*((d + e*x)^(1/2) - d^(1/2))^10)/((d - e*x)^(1/2) - d^(1/2))^10 + (256*d^5*((d + e*x)^(1/2) - d^(1/2))^12)/((d - e*x)^(1/2) - d^(1/2))^12) - ((b*e^2*((d + e*x)^(1/2) - d^(1/2))^2)/((d - e*x)^(1/2) - d^(1/2))^2 - (b*e^2)/2 + (15*b*e^2*((d + e*x)^(1/2) - d^(1/2))^4)/(2*((d - e*x)^(1/2) - d^(1/2))^4))/((16*d^3*((d + e*x)^(1/2) - d^(1/2))^2)/((d - e*x)^(1/2) - d^(1/2))^2 - (32*d^3*((d + e*x)^(1/2) - d^(1/2))^4)/((d - e*x )^(1/2) - d^(1/2))^4 + (16*d^3*((d + e*x)^(1/2) - d^(1/2))^6)/((d - e*x)^( 1/2) - d^(1/2))^6) + (c*(log(((d + e*x)^(1/2) - d^(1/2))^2/((d - e*x)^(1/2 ) - d^(1/2))^2 - 1) - log(((d + e*x)^(1/2) - d^(1/2))/((d - e*x)^(1/2) - d ^(1/2)))))/d - (3*a*e^4*log(((d + e*x)^(1/2) - d^(1/2))/((d - e*x)^(1/2) - d^(1/2))))/(8*d^5) - (b*e^2*log(((d + e*x)^(1/2) - d^(1/2))/((d - e*x)^(1 /2) - d^(1/2))))/(2*d^3) + (3*a*e^4*log(((d + e*x)^(1/2) - d^(1/2))^2/(...