Integrand size = 33, antiderivative size = 146 \[ \int \frac {(a+b x) (d+e x)^{7/2}}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\frac {35 e^2 (b d-a e) \sqrt {d+e x}}{4 b^4}+\frac {35 e^2 (d+e x)^{3/2}}{12 b^3}-\frac {7 e (d+e x)^{5/2}}{4 b^2 (a+b x)}-\frac {(d+e x)^{7/2}}{2 b (a+b x)^2}-\frac {35 e^2 (b d-a e)^{3/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{4 b^{9/2}} \]
35/12*e^2*(e*x+d)^(3/2)/b^3-7/4*e*(e*x+d)^(5/2)/b^2/(b*x+a)-1/2*(e*x+d)^(7 /2)/b/(b*x+a)^2-35/4*e^2*(-a*e+b*d)^(3/2)*arctanh(b^(1/2)*(e*x+d)^(1/2)/(- a*e+b*d)^(1/2))/b^(9/2)+35/4*e^2*(-a*e+b*d)*(e*x+d)^(1/2)/b^4
Time = 0.09 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.11 \[ \int \frac {(a+b x) (d+e x)^{7/2}}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=-\frac {\sqrt {d+e x} \left (105 a^3 e^3+35 a^2 b e^2 (-4 d+5 e x)+7 a b^2 e \left (3 d^2-34 d e x+8 e^2 x^2\right )+b^3 \left (6 d^3+39 d^2 e x-80 d e^2 x^2-8 e^3 x^3\right )\right )}{12 b^4 (a+b x)^2}+\frac {35 e^2 (-b d+a e)^{3/2} \arctan \left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{4 b^{9/2}} \]
-1/12*(Sqrt[d + e*x]*(105*a^3*e^3 + 35*a^2*b*e^2*(-4*d + 5*e*x) + 7*a*b^2* e*(3*d^2 - 34*d*e*x + 8*e^2*x^2) + b^3*(6*d^3 + 39*d^2*e*x - 80*d*e^2*x^2 - 8*e^3*x^3)))/(b^4*(a + b*x)^2) + (35*e^2*(-(b*d) + a*e)^(3/2)*ArcTan[(Sq rt[b]*Sqrt[d + e*x])/Sqrt[-(b*d) + a*e]])/(4*b^(9/2))
Time = 0.27 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.05, 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, 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.
| \(\displaystyle \int \frac {(a+b x) (d+e x)^{7/2}}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 1184 |
| \(\displaystyle b^4 \int \frac {(d+e x)^{7/2}}{b^4 (a+b x)^3}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {(d+e x)^{7/2}}{(a+b x)^3}dx\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {7 e \int \frac {(d+e x)^{5/2}}{(a+b x)^2}dx}{4 b}-\frac {(d+e x)^{7/2}}{2 b (a+b x)^2}\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {7 e \left (\frac {5 e \int \frac {(d+e x)^{3/2}}{a+b x}dx}{2 b}-\frac {(d+e x)^{5/2}}{b (a+b x)}\right )}{4 b}-\frac {(d+e x)^{7/2}}{2 b (a+b x)^2}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {7 e \left (\frac {5 e \left (\frac {(b d-a e) \int \frac {\sqrt {d+e x}}{a+b x}dx}{b}+\frac {2 (d+e x)^{3/2}}{3 b}\right )}{2 b}-\frac {(d+e x)^{5/2}}{b (a+b x)}\right )}{4 b}-\frac {(d+e x)^{7/2}}{2 b (a+b x)^2}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {7 e \left (\frac {5 e \left (\frac {(b d-a e) \left (\frac {(b d-a e) \int \frac {1}{(a+b x) \sqrt {d+e x}}dx}{b}+\frac {2 \sqrt {d+e x}}{b}\right )}{b}+\frac {2 (d+e x)^{3/2}}{3 b}\right )}{2 b}-\frac {(d+e x)^{5/2}}{b (a+b x)}\right )}{4 b}-\frac {(d+e x)^{7/2}}{2 b (a+b x)^2}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {7 e \left (\frac {5 e \left (\frac {(b d-a e) \left (\frac {2 (b d-a e) \int \frac {1}{a+\frac {b (d+e x)}{e}-\frac {b d}{e}}d\sqrt {d+e x}}{b e}+\frac {2 \sqrt {d+e x}}{b}\right )}{b}+\frac {2 (d+e x)^{3/2}}{3 b}\right )}{2 b}-\frac {(d+e x)^{5/2}}{b (a+b x)}\right )}{4 b}-\frac {(d+e x)^{7/2}}{2 b (a+b x)^2}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {7 e \left (\frac {5 e \left (\frac {(b d-a e) \left (\frac {2 \sqrt {d+e x}}{b}-\frac {2 \sqrt {b d-a e} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{3/2}}\right )}{b}+\frac {2 (d+e x)^{3/2}}{3 b}\right )}{2 b}-\frac {(d+e x)^{5/2}}{b (a+b x)}\right )}{4 b}-\frac {(d+e x)^{7/2}}{2 b (a+b x)^2}\) |
-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)
3.21.75.3.1 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_)^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 = 0.45 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.99
| method | result | size |
| risch | \(-\frac {2 e^{2} \left (-b e x +9 a e -10 b d \right ) \sqrt {e x +d}}{3 b^{4}}+\frac {\left (2 e^{2} a^{2}-4 a b d e +2 b^{2} d^{2}\right ) e^{2} \left (\frac {-\frac {13 b \left (e x +d \right )^{\frac {3}{2}}}{8}+\left (-\frac {11 a e}{8}+\frac {11 b d}{8}\right ) \sqrt {e x +d}}{\left (b \left (e x +d \right )+a e -b d \right )^{2}}+\frac {35 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{8 \sqrt {\left (a e -b d \right ) b}}\right )}{b^{4}}\) | \(144\) |
| pseudoelliptic | \(\frac {-\frac {35 \left (\left (-\frac {8}{105} x^{3} b^{3}+\frac {8}{15} a \,b^{2} x^{2}+\frac {5}{3} b \,a^{2} x +a^{3}\right ) e^{3}-\frac {4 b \left (\frac {4}{7} b^{2} x^{2}+\frac {17}{10} a b x +a^{2}\right ) d \,e^{2}}{3}+\frac {b^{2} \left (\frac {13 b x}{7}+a \right ) d^{2} e}{5}+\frac {2 b^{3} d^{3}}{35}\right ) \sqrt {\left (a e -b d \right ) b}\, \sqrt {e x +d}}{4}+\frac {35 e^{2} \left (b x +a \right )^{2} \left (a e -b d \right )^{2} \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{4}}{b^{4} \left (b x +a \right )^{2} \sqrt {\left (a e -b d \right ) b}}\) | \(170\) |
| derivativedivides | \(2 e^{2} \left (-\frac {-\frac {b \left (e x +d \right )^{\frac {3}{2}}}{3}+3 a e \sqrt {e x +d}-3 b d \sqrt {e x +d}}{b^{4}}+\frac {\frac {\left (-\frac {13}{8} a^{2} b \,e^{2}+\frac {13}{4} a \,b^{2} d e -\frac {13}{8} b^{3} d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}+\left (-\frac {11}{8} a^{3} e^{3}+\frac {33}{8} a^{2} b d \,e^{2}-\frac {33}{8} a \,b^{2} d^{2} e +\frac {11}{8} b^{3} d^{3}\right ) \sqrt {e x +d}}{\left (b \left (e x +d \right )+a e -b d \right )^{2}}+\frac {35 \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{8 \sqrt {\left (a e -b d \right ) b}}}{b^{4}}\right )\) | \(205\) |
| default | \(2 e^{2} \left (-\frac {-\frac {b \left (e x +d \right )^{\frac {3}{2}}}{3}+3 a e \sqrt {e x +d}-3 b d \sqrt {e x +d}}{b^{4}}+\frac {\frac {\left (-\frac {13}{8} a^{2} b \,e^{2}+\frac {13}{4} a \,b^{2} d e -\frac {13}{8} b^{3} d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}+\left (-\frac {11}{8} a^{3} e^{3}+\frac {33}{8} a^{2} b d \,e^{2}-\frac {33}{8} a \,b^{2} d^{2} e +\frac {11}{8} b^{3} d^{3}\right ) \sqrt {e x +d}}{\left (b \left (e x +d \right )+a e -b d \right )^{2}}+\frac {35 \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{8 \sqrt {\left (a e -b d \right ) b}}}{b^{4}}\right )\) | \(205\) |
-2/3*e^2*(-b*e*x+9*a*e-10*b*d)*(e*x+d)^(1/2)/b^4+1/b^4*(2*a^2*e^2-4*a*b*d* e+2*b^2*d^2)*e^2*((-13/8*b*(e*x+d)^(3/2)+(-11/8*a*e+11/8*b*d)*(e*x+d)^(1/2 ))/(b*(e*x+d)+a*e-b*d)^2+35/8/((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. 255 vs. \(2 (118) = 236\).
Time = 0.31 (sec) , antiderivative size = 520, normalized size of antiderivative = 3.56 \[ \int \frac {(a+b x) (d+e x)^{7/2}}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\left [-\frac {105 \, {\left (a^{2} b d e^{2} - a^{3} e^{3} + {\left (b^{3} d e^{2} - a b^{2} e^{3}\right )} x^{2} + 2 \, {\left (a b^{2} d e^{2} - a^{2} b e^{3}\right )} x\right )} \sqrt {\frac {b d - a e}{b}} \log \left (\frac {b e x + 2 \, b d - a e + 2 \, \sqrt {e x + d} b \sqrt {\frac {b d - a e}{b}}}{b x + a}\right ) - 2 \, {\left (8 \, b^{3} e^{3} x^{3} - 6 \, b^{3} d^{3} - 21 \, a b^{2} d^{2} e + 140 \, a^{2} b d e^{2} - 105 \, a^{3} e^{3} + 8 \, {\left (10 \, b^{3} d e^{2} - 7 \, a b^{2} e^{3}\right )} x^{2} - {\left (39 \, b^{3} d^{2} e - 238 \, a b^{2} d e^{2} + 175 \, a^{2} b e^{3}\right )} x\right )} \sqrt {e x + d}}{24 \, {\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\right )}}, -\frac {105 \, {\left (a^{2} b d e^{2} - a^{3} e^{3} + {\left (b^{3} d e^{2} - a b^{2} e^{3}\right )} x^{2} + 2 \, {\left (a b^{2} d e^{2} - a^{2} b e^{3}\right )} x\right )} \sqrt {-\frac {b d - a e}{b}} \arctan \left (-\frac {\sqrt {e x + d} b \sqrt {-\frac {b d - a e}{b}}}{b d - a e}\right ) - {\left (8 \, b^{3} e^{3} x^{3} - 6 \, b^{3} d^{3} - 21 \, a b^{2} d^{2} e + 140 \, a^{2} b d e^{2} - 105 \, a^{3} e^{3} + 8 \, {\left (10 \, b^{3} d e^{2} - 7 \, a b^{2} e^{3}\right )} x^{2} - {\left (39 \, b^{3} d^{2} e - 238 \, a b^{2} d e^{2} + 175 \, a^{2} b e^{3}\right )} x\right )} \sqrt {e x + d}}{12 \, {\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\right )}}\right ] \]
[-1/24*(105*(a^2*b*d*e^2 - a^3*e^3 + (b^3*d*e^2 - a*b^2*e^3)*x^2 + 2*(a*b^ 2*d*e^2 - a^2*b*e^3)*x)*sqrt((b*d - a*e)/b)*log((b*e*x + 2*b*d - a*e + 2*s qrt(e*x + d)*b*sqrt((b*d - a*e)/b))/(b*x + a)) - 2*(8*b^3*e^3*x^3 - 6*b^3* d^3 - 21*a*b^2*d^2*e + 140*a^2*b*d*e^2 - 105*a^3*e^3 + 8*(10*b^3*d*e^2 - 7 *a*b^2*e^3)*x^2 - (39*b^3*d^2*e - 238*a*b^2*d*e^2 + 175*a^2*b*e^3)*x)*sqrt (e*x + d))/(b^6*x^2 + 2*a*b^5*x + a^2*b^4), -1/12*(105*(a^2*b*d*e^2 - a^3* e^3 + (b^3*d*e^2 - a*b^2*e^3)*x^2 + 2*(a*b^2*d*e^2 - a^2*b*e^3)*x)*sqrt(-( b*d - a*e)/b)*arctan(-sqrt(e*x + d)*b*sqrt(-(b*d - a*e)/b)/(b*d - a*e)) - (8*b^3*e^3*x^3 - 6*b^3*d^3 - 21*a*b^2*d^2*e + 140*a^2*b*d*e^2 - 105*a^3*e^ 3 + 8*(10*b^3*d*e^2 - 7*a*b^2*e^3)*x^2 - (39*b^3*d^2*e - 238*a*b^2*d*e^2 + 175*a^2*b*e^3)*x)*sqrt(e*x + d))/(b^6*x^2 + 2*a*b^5*x + a^2*b^4)]
Timed out. \[ \int \frac {(a+b x) (d+e x)^{7/2}}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {(a+b x) (d+e x)^{7/2}}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(a*e-b*d>0)', see `assume?` for m ore detail
Leaf count of result is larger than twice the leaf count of optimal. 263 vs. \(2 (118) = 236\).
Time = 0.27 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.80 \[ \int \frac {(a+b x) (d+e x)^{7/2}}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\frac {35 \, {\left (b^{2} d^{2} e^{2} - 2 \, a b d e^{3} + a^{2} e^{4}\right )} \arctan \left (\frac {\sqrt {e x + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{4 \, \sqrt {-b^{2} d + a b e} b^{4}} - \frac {13 \, {\left (e x + d\right )}^{\frac {3}{2}} b^{3} d^{2} e^{2} - 11 \, \sqrt {e x + d} b^{3} d^{3} e^{2} - 26 \, {\left (e x + d\right )}^{\frac {3}{2}} a b^{2} d e^{3} + 33 \, \sqrt {e x + d} a b^{2} d^{2} e^{3} + 13 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{2} b e^{4} - 33 \, \sqrt {e x + d} a^{2} b d e^{4} + 11 \, \sqrt {e x + d} a^{3} e^{5}}{4 \, {\left ({\left (e x + d\right )} b - b d + a e\right )}^{2} b^{4}} + \frac {2 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} b^{6} e^{2} + 9 \, \sqrt {e x + d} b^{6} d e^{2} - 9 \, \sqrt {e x + d} a b^{5} e^{3}\right )}}{3 \, b^{9}} \]
35/4*(b^2*d^2*e^2 - 2*a*b*d*e^3 + a^2*e^4)*arctan(sqrt(e*x + d)*b/sqrt(-b^ 2*d + a*b*e))/(sqrt(-b^2*d + a*b*e)*b^4) - 1/4*(13*(e*x + d)^(3/2)*b^3*d^2 *e^2 - 11*sqrt(e*x + d)*b^3*d^3*e^2 - 26*(e*x + d)^(3/2)*a*b^2*d*e^3 + 33* sqrt(e*x + d)*a*b^2*d^2*e^3 + 13*(e*x + d)^(3/2)*a^2*b*e^4 - 33*sqrt(e*x + d)*a^2*b*d*e^4 + 11*sqrt(e*x + d)*a^3*e^5)/(((e*x + d)*b - b*d + a*e)^2*b ^4) + 2/3*((e*x + d)^(3/2)*b^6*e^2 + 9*sqrt(e*x + d)*b^6*d*e^2 - 9*sqrt(e* x + d)*a*b^5*e^3)/b^9
Time = 0.14 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.84 \[ \int \frac {(a+b x) (d+e x)^{7/2}}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\frac {2\,e^2\,{\left (d+e\,x\right )}^{3/2}}{3\,b^3}-\frac {\sqrt {d+e\,x}\,\left (\frac {11\,a^3\,e^5}{4}-\frac {33\,a^2\,b\,d\,e^4}{4}+\frac {33\,a\,b^2\,d^2\,e^3}{4}-\frac {11\,b^3\,d^3\,e^2}{4}\right )+{\left (d+e\,x\right )}^{3/2}\,\left (\frac {13\,a^2\,b\,e^4}{4}-\frac {13\,a\,b^2\,d\,e^3}{2}+\frac {13\,b^3\,d^2\,e^2}{4}\right )}{b^6\,{\left (d+e\,x\right )}^2-\left (2\,b^6\,d-2\,a\,b^5\,e\right )\,\left (d+e\,x\right )+b^6\,d^2+a^2\,b^4\,e^2-2\,a\,b^5\,d\,e}+\frac {2\,e^2\,\left (3\,b^3\,d-3\,a\,b^2\,e\right )\,\sqrt {d+e\,x}}{b^6}+\frac {35\,e^2\,\mathrm {atan}\left (\frac {\sqrt {b}\,e^2\,{\left (a\,e-b\,d\right )}^{3/2}\,\sqrt {d+e\,x}}{a^2\,e^4-2\,a\,b\,d\,e^3+b^2\,d^2\,e^2}\right )\,{\left (a\,e-b\,d\right )}^{3/2}}{4\,b^{9/2}} \]
(2*e^2*(d + e*x)^(3/2))/(3*b^3) - ((d + e*x)^(1/2)*((11*a^3*e^5)/4 - (11*b ^3*d^3*e^2)/4 + (33*a*b^2*d^2*e^3)/4 - (33*a^2*b*d*e^4)/4) + (d + e*x)^(3/ 2)*((13*a^2*b*e^4)/4 + (13*b^3*d^2*e^2)/4 - (13*a*b^2*d*e^3)/2))/(b^6*(d + e*x)^2 - (2*b^6*d - 2*a*b^5*e)*(d + e*x) + b^6*d^2 + a^2*b^4*e^2 - 2*a*b^ 5*d*e) + (2*e^2*(3*b^3*d - 3*a*b^2*e)*(d + e*x)^(1/2))/b^6 + (35*e^2*atan( (b^(1/2)*e^2*(a*e - b*d)^(3/2)*(d + e*x)^(1/2))/(a^2*e^4 + b^2*d^2*e^2 - 2 *a*b*d*e^3))*(a*e - b*d)^(3/2))/(4*b^(9/2))