Integrand size = 33, antiderivative size = 162 \[ \int \frac {(a+b x) (d+e x)^{5/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {5 e^2 \sqrt {d+e x}}{32 b^3 (a+b x)^2}-\frac {5 e^3 \sqrt {d+e x}}{64 b^3 (b d-a e) (a+b x)}-\frac {5 e (d+e x)^{3/2}}{24 b^2 (a+b x)^3}-\frac {(d+e x)^{5/2}}{4 b (a+b x)^4}+\frac {5 e^4 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 b^{7/2} (b d-a e)^{3/2}} \]
-5/24*e*(e*x+d)^(3/2)/b^2/(b*x+a)^3-1/4*(e*x+d)^(5/2)/b/(b*x+a)^4+5/64*e^4 *arctanh(b^(1/2)*(e*x+d)^(1/2)/(-a*e+b*d)^(1/2))/b^(7/2)/(-a*e+b*d)^(3/2)- 5/32*e^2*(e*x+d)^(1/2)/b^3/(b*x+a)^2-5/64*e^3*(e*x+d)^(1/2)/b^3/(-a*e+b*d) /(b*x+a)
Time = 0.15 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.06 \[ \int \frac {(a+b x) (d+e x)^{5/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {\sqrt {d+e x} \left (15 a^3 e^3+5 a^2 b e^2 (2 d+11 e x)+a b^2 e \left (8 d^2+36 d e x+73 e^2 x^2\right )-b^3 \left (48 d^3+136 d^2 e x+118 d e^2 x^2+15 e^3 x^3\right )\right )}{192 b^3 (b d-a e) (a+b x)^4}+\frac {5 e^4 \arctan \left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{64 b^{7/2} (-b d+a e)^{3/2}} \]
(Sqrt[d + e*x]*(15*a^3*e^3 + 5*a^2*b*e^2*(2*d + 11*e*x) + a*b^2*e*(8*d^2 + 36*d*e*x + 73*e^2*x^2) - b^3*(48*d^3 + 136*d^2*e*x + 118*d*e^2*x^2 + 15*e ^3*x^3)))/(192*b^3*(b*d - a*e)*(a + b*x)^4) + (5*e^4*ArcTan[(Sqrt[b]*Sqrt[ d + e*x])/Sqrt[-(b*d) + a*e]])/(64*b^(7/2)*(-(b*d) + a*e)^(3/2))
Time = 0.29 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.06, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.242, Rules used = {1184, 27, 51, 51, 51, 52, 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.
| \(\displaystyle \int \frac {(a+b x) (d+e x)^{5/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 1184 |
| \(\displaystyle b^6 \int \frac {(d+e x)^{5/2}}{b^6 (a+b x)^5}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {(d+e x)^{5/2}}{(a+b x)^5}dx\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {5 e \int \frac {(d+e x)^{3/2}}{(a+b x)^4}dx}{8 b}-\frac {(d+e x)^{5/2}}{4 b (a+b x)^4}\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {5 e \left (\frac {e \int \frac {\sqrt {d+e x}}{(a+b x)^3}dx}{2 b}-\frac {(d+e x)^{3/2}}{3 b (a+b x)^3}\right )}{8 b}-\frac {(d+e x)^{5/2}}{4 b (a+b x)^4}\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {5 e \left (\frac {e \left (\frac {e \int \frac {1}{(a+b x)^2 \sqrt {d+e x}}dx}{4 b}-\frac {\sqrt {d+e x}}{2 b (a+b x)^2}\right )}{2 b}-\frac {(d+e x)^{3/2}}{3 b (a+b x)^3}\right )}{8 b}-\frac {(d+e x)^{5/2}}{4 b (a+b x)^4}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {5 e \left (\frac {e \left (\frac {e \left (-\frac {e \int \frac {1}{(a+b x) \sqrt {d+e x}}dx}{2 (b d-a e)}-\frac {\sqrt {d+e x}}{(a+b x) (b d-a e)}\right )}{4 b}-\frac {\sqrt {d+e x}}{2 b (a+b x)^2}\right )}{2 b}-\frac {(d+e x)^{3/2}}{3 b (a+b x)^3}\right )}{8 b}-\frac {(d+e x)^{5/2}}{4 b (a+b x)^4}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {5 e \left (\frac {e \left (\frac {e \left (-\frac {\int \frac {1}{a+\frac {b (d+e x)}{e}-\frac {b d}{e}}d\sqrt {d+e x}}{b d-a e}-\frac {\sqrt {d+e x}}{(a+b x) (b d-a e)}\right )}{4 b}-\frac {\sqrt {d+e x}}{2 b (a+b x)^2}\right )}{2 b}-\frac {(d+e x)^{3/2}}{3 b (a+b x)^3}\right )}{8 b}-\frac {(d+e x)^{5/2}}{4 b (a+b x)^4}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {5 e \left (\frac {e \left (\frac {e \left (\frac {e \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{\sqrt {b} (b d-a e)^{3/2}}-\frac {\sqrt {d+e x}}{(a+b x) (b d-a e)}\right )}{4 b}-\frac {\sqrt {d+e x}}{2 b (a+b x)^2}\right )}{2 b}-\frac {(d+e x)^{3/2}}{3 b (a+b x)^3}\right )}{8 b}-\frac {(d+e x)^{5/2}}{4 b (a+b x)^4}\) |
-1/4*(d + e*x)^(5/2)/(b*(a + b*x)^4) + (5*e*(-1/3*(d + e*x)^(3/2)/(b*(a + b*x)^3) + (e*(-1/2*Sqrt[d + e*x]/(b*(a + b*x)^2) + (e*(-(Sqrt[d + e*x]/((b *d - a*e)*(a + b*x))) + (e*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e] ])/(Sqrt[b]*(b*d - a*e)^(3/2))))/(4*b)))/(2*b)))/(8*b)
3.21.86.3.1 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 + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && ILtQ[m, -1] && FractionQ[n] && LtQ[n, 0]
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_)^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 = 0.81 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.98
| method | result | size |
| derivativedivides | \(2 e^{4} \left (\frac {\frac {5 \left (e x +d \right )^{\frac {7}{2}}}{128 \left (a e -b d \right )}-\frac {73 \left (e x +d \right )^{\frac {5}{2}}}{384 b}-\frac {55 \left (a e -b d \right ) \left (e x +d \right )^{\frac {3}{2}}}{384 b^{2}}-\frac {5 \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) \sqrt {e x +d}}{128 b^{3}}}{\left (b \left (e x +d \right )+a e -b d \right )^{4}}+\frac {5 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{128 \left (a e -b d \right ) b^{3} \sqrt {\left (a e -b d \right ) b}}\right )\) | \(159\) |
| default | \(2 e^{4} \left (\frac {\frac {5 \left (e x +d \right )^{\frac {7}{2}}}{128 \left (a e -b d \right )}-\frac {73 \left (e x +d \right )^{\frac {5}{2}}}{384 b}-\frac {55 \left (a e -b d \right ) \left (e x +d \right )^{\frac {3}{2}}}{384 b^{2}}-\frac {5 \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) \sqrt {e x +d}}{128 b^{3}}}{\left (b \left (e x +d \right )+a e -b d \right )^{4}}+\frac {5 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{128 \left (a e -b d \right ) b^{3} \sqrt {\left (a e -b d \right ) b}}\right )\) | \(159\) |
| pseudoelliptic | \(\frac {-\frac {5 \left (\left (-x^{3} b^{3}+\frac {73}{15} a \,b^{2} x^{2}+\frac {11}{3} b \,a^{2} x +a^{3}\right ) e^{3}+\frac {2 \left (-\frac {59}{5} b^{2} x^{2}+\frac {18}{5} a b x +a^{2}\right ) b d \,e^{2}}{3}+\frac {8 b^{2} d^{2} \left (-17 b x +a \right ) e}{15}-\frac {16 b^{3} d^{3}}{5}\right ) \sqrt {\left (a e -b d \right ) b}\, \sqrt {e x +d}}{64}+\frac {5 e^{4} \left (b x +a \right )^{4} \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{64}}{\left (a e -b d \right ) b^{3} \left (b x +a \right )^{4} \sqrt {\left (a e -b d \right ) b}}\) | \(170\) |
2*e^4*((5/128/(a*e-b*d)*(e*x+d)^(7/2)-73/384*(e*x+d)^(5/2)/b-55/384*(a*e-b *d)/b^2*(e*x+d)^(3/2)-5/128*(a^2*e^2-2*a*b*d*e+b^2*d^2)/b^3*(e*x+d)^(1/2)) /(b*(e*x+d)+a*e-b*d)^4+5/128/(a*e-b*d)/b^3/((a*e-b*d)*b)^(1/2)*arctan(b*(e *x+d)^(1/2)/((a*e-b*d)*b)^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 440 vs. \(2 (134) = 268\).
Time = 0.46 (sec) , antiderivative size = 894, normalized size of antiderivative = 5.52 \[ \int \frac {(a+b x) (d+e x)^{5/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\left [-\frac {15 \, {\left (b^{4} e^{4} x^{4} + 4 \, a b^{3} e^{4} x^{3} + 6 \, a^{2} b^{2} e^{4} x^{2} + 4 \, a^{3} b e^{4} x + a^{4} e^{4}\right )} \sqrt {b^{2} d - a b e} \log \left (\frac {b e x + 2 \, b d - a e - 2 \, \sqrt {b^{2} d - a b e} \sqrt {e x + d}}{b x + a}\right ) + 2 \, {\left (48 \, b^{5} d^{4} - 56 \, a b^{4} d^{3} e - 2 \, a^{2} b^{3} d^{2} e^{2} - 5 \, a^{3} b^{2} d e^{3} + 15 \, a^{4} b e^{4} + 15 \, {\left (b^{5} d e^{3} - a b^{4} e^{4}\right )} x^{3} + {\left (118 \, b^{5} d^{2} e^{2} - 191 \, a b^{4} d e^{3} + 73 \, a^{2} b^{3} e^{4}\right )} x^{2} + {\left (136 \, b^{5} d^{3} e - 172 \, a b^{4} d^{2} e^{2} - 19 \, a^{2} b^{3} d e^{3} + 55 \, a^{3} b^{2} e^{4}\right )} x\right )} \sqrt {e x + d}}{384 \, {\left (a^{4} b^{6} d^{2} - 2 \, a^{5} b^{5} d e + a^{6} b^{4} e^{2} + {\left (b^{10} d^{2} - 2 \, a b^{9} d e + a^{2} b^{8} e^{2}\right )} x^{4} + 4 \, {\left (a b^{9} d^{2} - 2 \, a^{2} b^{8} d e + a^{3} b^{7} e^{2}\right )} x^{3} + 6 \, {\left (a^{2} b^{8} d^{2} - 2 \, a^{3} b^{7} d e + a^{4} b^{6} e^{2}\right )} x^{2} + 4 \, {\left (a^{3} b^{7} d^{2} - 2 \, a^{4} b^{6} d e + a^{5} b^{5} e^{2}\right )} x\right )}}, -\frac {15 \, {\left (b^{4} e^{4} x^{4} + 4 \, a b^{3} e^{4} x^{3} + 6 \, a^{2} b^{2} e^{4} x^{2} + 4 \, a^{3} b e^{4} x + a^{4} e^{4}\right )} \sqrt {-b^{2} d + a b e} \arctan \left (\frac {\sqrt {-b^{2} d + a b e} \sqrt {e x + d}}{b e x + b d}\right ) + {\left (48 \, b^{5} d^{4} - 56 \, a b^{4} d^{3} e - 2 \, a^{2} b^{3} d^{2} e^{2} - 5 \, a^{3} b^{2} d e^{3} + 15 \, a^{4} b e^{4} + 15 \, {\left (b^{5} d e^{3} - a b^{4} e^{4}\right )} x^{3} + {\left (118 \, b^{5} d^{2} e^{2} - 191 \, a b^{4} d e^{3} + 73 \, a^{2} b^{3} e^{4}\right )} x^{2} + {\left (136 \, b^{5} d^{3} e - 172 \, a b^{4} d^{2} e^{2} - 19 \, a^{2} b^{3} d e^{3} + 55 \, a^{3} b^{2} e^{4}\right )} x\right )} \sqrt {e x + d}}{192 \, {\left (a^{4} b^{6} d^{2} - 2 \, a^{5} b^{5} d e + a^{6} b^{4} e^{2} + {\left (b^{10} d^{2} - 2 \, a b^{9} d e + a^{2} b^{8} e^{2}\right )} x^{4} + 4 \, {\left (a b^{9} d^{2} - 2 \, a^{2} b^{8} d e + a^{3} b^{7} e^{2}\right )} x^{3} + 6 \, {\left (a^{2} b^{8} d^{2} - 2 \, a^{3} b^{7} d e + a^{4} b^{6} e^{2}\right )} x^{2} + 4 \, {\left (a^{3} b^{7} d^{2} - 2 \, a^{4} b^{6} d e + a^{5} b^{5} e^{2}\right )} x\right )}}\right ] \]
[-1/384*(15*(b^4*e^4*x^4 + 4*a*b^3*e^4*x^3 + 6*a^2*b^2*e^4*x^2 + 4*a^3*b*e ^4*x + a^4*e^4)*sqrt(b^2*d - a*b*e)*log((b*e*x + 2*b*d - a*e - 2*sqrt(b^2* d - a*b*e)*sqrt(e*x + d))/(b*x + a)) + 2*(48*b^5*d^4 - 56*a*b^4*d^3*e - 2* a^2*b^3*d^2*e^2 - 5*a^3*b^2*d*e^3 + 15*a^4*b*e^4 + 15*(b^5*d*e^3 - a*b^4*e ^4)*x^3 + (118*b^5*d^2*e^2 - 191*a*b^4*d*e^3 + 73*a^2*b^3*e^4)*x^2 + (136* b^5*d^3*e - 172*a*b^4*d^2*e^2 - 19*a^2*b^3*d*e^3 + 55*a^3*b^2*e^4)*x)*sqrt (e*x + d))/(a^4*b^6*d^2 - 2*a^5*b^5*d*e + a^6*b^4*e^2 + (b^10*d^2 - 2*a*b^ 9*d*e + a^2*b^8*e^2)*x^4 + 4*(a*b^9*d^2 - 2*a^2*b^8*d*e + a^3*b^7*e^2)*x^3 + 6*(a^2*b^8*d^2 - 2*a^3*b^7*d*e + a^4*b^6*e^2)*x^2 + 4*(a^3*b^7*d^2 - 2* a^4*b^6*d*e + a^5*b^5*e^2)*x), -1/192*(15*(b^4*e^4*x^4 + 4*a*b^3*e^4*x^3 + 6*a^2*b^2*e^4*x^2 + 4*a^3*b*e^4*x + a^4*e^4)*sqrt(-b^2*d + a*b*e)*arctan( sqrt(-b^2*d + a*b*e)*sqrt(e*x + d)/(b*e*x + b*d)) + (48*b^5*d^4 - 56*a*b^4 *d^3*e - 2*a^2*b^3*d^2*e^2 - 5*a^3*b^2*d*e^3 + 15*a^4*b*e^4 + 15*(b^5*d*e^ 3 - a*b^4*e^4)*x^3 + (118*b^5*d^2*e^2 - 191*a*b^4*d*e^3 + 73*a^2*b^3*e^4)* x^2 + (136*b^5*d^3*e - 172*a*b^4*d^2*e^2 - 19*a^2*b^3*d*e^3 + 55*a^3*b^2*e ^4)*x)*sqrt(e*x + d))/(a^4*b^6*d^2 - 2*a^5*b^5*d*e + a^6*b^4*e^2 + (b^10*d ^2 - 2*a*b^9*d*e + a^2*b^8*e^2)*x^4 + 4*(a*b^9*d^2 - 2*a^2*b^8*d*e + a^3*b ^7*e^2)*x^3 + 6*(a^2*b^8*d^2 - 2*a^3*b^7*d*e + a^4*b^6*e^2)*x^2 + 4*(a^3*b ^7*d^2 - 2*a^4*b^6*d*e + a^5*b^5*e^2)*x)]
Timed out. \[ \int \frac {(a+b x) (d+e x)^{5/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {(a+b x) (d+e x)^{5/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\text {Exception raised: ValueError} \]
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
Time = 0.29 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.60 \[ \int \frac {(a+b x) (d+e x)^{5/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {5 \, e^{4} \arctan \left (\frac {\sqrt {e x + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{64 \, {\left (b^{4} d - a b^{3} e\right )} \sqrt {-b^{2} d + a b e}} - \frac {15 \, {\left (e x + d\right )}^{\frac {7}{2}} b^{3} e^{4} + 73 \, {\left (e x + d\right )}^{\frac {5}{2}} b^{3} d e^{4} - 55 \, {\left (e x + d\right )}^{\frac {3}{2}} b^{3} d^{2} e^{4} + 15 \, \sqrt {e x + d} b^{3} d^{3} e^{4} - 73 \, {\left (e x + d\right )}^{\frac {5}{2}} a b^{2} e^{5} + 110 \, {\left (e x + d\right )}^{\frac {3}{2}} a b^{2} d e^{5} - 45 \, \sqrt {e x + d} a b^{2} d^{2} e^{5} - 55 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{2} b e^{6} + 45 \, \sqrt {e x + d} a^{2} b d e^{6} - 15 \, \sqrt {e x + d} a^{3} e^{7}}{192 \, {\left (b^{4} d - a b^{3} e\right )} {\left ({\left (e x + d\right )} b - b d + a e\right )}^{4}} \]
-5/64*e^4*arctan(sqrt(e*x + d)*b/sqrt(-b^2*d + a*b*e))/((b^4*d - a*b^3*e)* sqrt(-b^2*d + a*b*e)) - 1/192*(15*(e*x + d)^(7/2)*b^3*e^4 + 73*(e*x + d)^( 5/2)*b^3*d*e^4 - 55*(e*x + d)^(3/2)*b^3*d^2*e^4 + 15*sqrt(e*x + d)*b^3*d^3 *e^4 - 73*(e*x + d)^(5/2)*a*b^2*e^5 + 110*(e*x + d)^(3/2)*a*b^2*d*e^5 - 45 *sqrt(e*x + d)*a*b^2*d^2*e^5 - 55*(e*x + d)^(3/2)*a^2*b*e^6 + 45*sqrt(e*x + d)*a^2*b*d*e^6 - 15*sqrt(e*x + d)*a^3*e^7)/((b^4*d - a*b^3*e)*((e*x + d) *b - b*d + a*e)^4)
Time = 0.14 (sec) , antiderivative size = 309, normalized size of antiderivative = 1.91 \[ \int \frac {(a+b x) (d+e x)^{5/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {5\,e^4\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d+e\,x}}{\sqrt {a\,e-b\,d}}\right )}{64\,b^{7/2}\,{\left (a\,e-b\,d\right )}^{3/2}}-\frac {\frac {73\,e^4\,{\left (d+e\,x\right )}^{5/2}}{192\,b}-\frac {5\,e^4\,{\left (d+e\,x\right )}^{7/2}}{64\,\left (a\,e-b\,d\right )}+\frac {5\,e^4\,\sqrt {d+e\,x}\,\left (a^2\,e^2-2\,a\,b\,d\,e+b^2\,d^2\right )}{64\,b^3}+\frac {55\,e^4\,\left (a\,e-b\,d\right )\,{\left (d+e\,x\right )}^{3/2}}{192\,b^2}}{b^4\,{\left (d+e\,x\right )}^4-\left (4\,b^4\,d-4\,a\,b^3\,e\right )\,{\left (d+e\,x\right )}^3-\left (d+e\,x\right )\,\left (-4\,a^3\,b\,e^3+12\,a^2\,b^2\,d\,e^2-12\,a\,b^3\,d^2\,e+4\,b^4\,d^3\right )+a^4\,e^4+b^4\,d^4+{\left (d+e\,x\right )}^2\,\left (6\,a^2\,b^2\,e^2-12\,a\,b^3\,d\,e+6\,b^4\,d^2\right )+6\,a^2\,b^2\,d^2\,e^2-4\,a\,b^3\,d^3\,e-4\,a^3\,b\,d\,e^3} \]
(5*e^4*atan((b^(1/2)*(d + e*x)^(1/2))/(a*e - b*d)^(1/2)))/(64*b^(7/2)*(a*e - b*d)^(3/2)) - ((73*e^4*(d + e*x)^(5/2))/(192*b) - (5*e^4*(d + e*x)^(7/2 ))/(64*(a*e - b*d)) + (5*e^4*(d + e*x)^(1/2)*(a^2*e^2 + b^2*d^2 - 2*a*b*d* e))/(64*b^3) + (55*e^4*(a*e - b*d)*(d + e*x)^(3/2))/(192*b^2))/(b^4*(d + e *x)^4 - (4*b^4*d - 4*a*b^3*e)*(d + e*x)^3 - (d + e*x)*(4*b^4*d^3 - 4*a^3*b *e^3 + 12*a^2*b^2*d*e^2 - 12*a*b^3*d^2*e) + a^4*e^4 + b^4*d^4 + (d + e*x)^ 2*(6*b^4*d^2 + 6*a^2*b^2*e^2 - 12*a*b^3*d*e) + 6*a^2*b^2*d^2*e^2 - 4*a*b^3 *d^3*e - 4*a^3*b*d*e^3)