Integrand size = 33, antiderivative size = 146 \[ \int (a+b x) (d+e x)^5 \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {(b d-a e)^2 (d+e x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{6 e^3 (a+b x)}-\frac {2 b (b d-a e) (d+e x)^7 \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^3 (a+b x)}+\frac {b^2 (d+e x)^8 \sqrt {a^2+2 a b x+b^2 x^2}}{8 e^3 (a+b x)} \]
1/6*(-a*e+b*d)^2*(e*x+d)^6*((b*x+a)^2)^(1/2)/e^3/(b*x+a)-2/7*b*(-a*e+b*d)* (e*x+d)^7*((b*x+a)^2)^(1/2)/e^3/(b*x+a)+1/8*b^2*(e*x+d)^8*((b*x+a)^2)^(1/2 )/e^3/(b*x+a)
Time = 1.04 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.34 \[ \int (a+b x) (d+e x)^5 \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {x \sqrt {(a+b x)^2} \left (28 a^2 \left (6 d^5+15 d^4 e x+20 d^3 e^2 x^2+15 d^2 e^3 x^3+6 d e^4 x^4+e^5 x^5\right )+8 a b x \left (21 d^5+70 d^4 e x+105 d^3 e^2 x^2+84 d^2 e^3 x^3+35 d e^4 x^4+6 e^5 x^5\right )+b^2 x^2 \left (56 d^5+210 d^4 e x+336 d^3 e^2 x^2+280 d^2 e^3 x^3+120 d e^4 x^4+21 e^5 x^5\right )\right )}{168 (a+b x)} \]
(x*Sqrt[(a + b*x)^2]*(28*a^2*(6*d^5 + 15*d^4*e*x + 20*d^3*e^2*x^2 + 15*d^2 *e^3*x^3 + 6*d*e^4*x^4 + e^5*x^5) + 8*a*b*x*(21*d^5 + 70*d^4*e*x + 105*d^3 *e^2*x^2 + 84*d^2*e^3*x^3 + 35*d*e^4*x^4 + 6*e^5*x^5) + b^2*x^2*(56*d^5 + 210*d^4*e*x + 336*d^3*e^2*x^2 + 280*d^2*e^3*x^3 + 120*d*e^4*x^4 + 21*e^5*x ^5)))/(168*(a + b*x))
Time = 0.32 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.64, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {1187, 27, 49, 2009}
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.
| \(\displaystyle \int (a+b x) \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^5 \, dx\) |
| \(\Big \downarrow \) 1187 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int b (a+b x)^2 (d+e x)^5dx}{b (a+b x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int (a+b x)^2 (d+e x)^5dx}{a+b x}\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {b^2 (d+e x)^7}{e^2}-\frac {2 b (b d-a e) (d+e x)^6}{e^2}+\frac {(a e-b d)^2 (d+e x)^5}{e^2}\right )dx}{a+b x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (-\frac {2 b (d+e x)^7 (b d-a e)}{7 e^3}+\frac {(d+e x)^6 (b d-a e)^2}{6 e^3}+\frac {b^2 (d+e x)^8}{8 e^3}\right )}{a+b x}\) |
(Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(((b*d - a*e)^2*(d + e*x)^6)/(6*e^3) - (2*b *(b*d - a*e)*(d + e*x)^7)/(7*e^3) + (b^2*(d + e*x)^8)/(8*e^3)))/(a + b*x)
3.20.54.3.1 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] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (b_.)*(x_ ) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(a + b*x + c*x^2)^FracPart[p]/(c^ IntPart[p]*(b/2 + c*x)^(2*FracPart[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, p}, x] && EqQ[b^2 - 4*a*c, 0] && !IntegerQ[p]
Leaf count of result is larger than twice the leaf count of optimal. \(229\) vs. \(2(107)=214\).
Time = 0.56 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.58
| method | result | size |
| gosper | \(\frac {x \left (21 b^{2} e^{5} x^{7}+48 x^{6} b a \,e^{5}+120 x^{6} b^{2} d \,e^{4}+28 x^{5} a^{2} e^{5}+280 x^{5} b a d \,e^{4}+280 x^{5} b^{2} d^{2} e^{3}+168 a^{2} d \,e^{4} x^{4}+672 a b \,d^{2} e^{3} x^{4}+336 b^{2} d^{3} e^{2} x^{4}+420 x^{3} a^{2} d^{2} e^{3}+840 x^{3} b a \,d^{3} e^{2}+210 x^{3} b^{2} d^{4} e +560 x^{2} a^{2} d^{3} e^{2}+560 x^{2} b a \,d^{4} e +56 x^{2} b^{2} d^{5}+420 x \,a^{2} d^{4} e +168 x b a \,d^{5}+168 a^{2} d^{5}\right ) \sqrt {\left (b x +a \right )^{2}}}{168 b x +168 a}\) | \(230\) |
| default | \(-\frac {\operatorname {csgn}\left (b x +a \right ) \left (b x +a \right )^{3} \left (-21 x^{5} b^{5} e^{5}+15 x^{4} a \,b^{4} e^{5}-120 x^{4} b^{5} d \,e^{4}-10 x^{3} a^{2} b^{3} e^{5}+80 x^{3} a \,b^{4} d \,e^{4}-280 x^{3} b^{5} d^{2} e^{3}+6 x^{2} a^{3} b^{2} e^{5}-48 x^{2} a^{2} b^{3} d \,e^{4}+168 x^{2} a \,b^{4} d^{2} e^{3}-336 x^{2} b^{5} d^{3} e^{2}-3 x \,a^{4} b \,e^{5}+24 x \,a^{3} b^{2} d \,e^{4}-84 x \,a^{2} b^{3} d^{2} e^{3}+168 x a \,b^{4} d^{3} e^{2}-210 x \,b^{5} d^{4} e +e^{5} a^{5}-8 b d \,e^{4} a^{4}+28 b^{2} d^{2} e^{3} a^{3}-56 b^{3} d^{3} e^{2} a^{2}+70 b^{4} d^{4} e a -56 b^{5} d^{5}\right )}{168 b^{6}}\) | \(278\) |
| risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, b^{2} e^{5} x^{8}}{8 b x +8 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (2 b a \,e^{5}+5 b^{2} d \,e^{4}\right ) x^{7}}{7 b x +7 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (a^{2} e^{5}+10 b a d \,e^{4}+10 b^{2} d^{2} e^{3}\right ) x^{6}}{6 b x +6 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (5 a^{2} d \,e^{4}+20 b a \,d^{2} e^{3}+10 b^{2} d^{3} e^{2}\right ) x^{5}}{5 b x +5 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (10 a^{2} d^{2} e^{3}+20 b a \,d^{3} e^{2}+5 b^{2} d^{4} e \right ) x^{4}}{4 b x +4 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (10 a^{2} d^{3} e^{2}+10 b a \,d^{4} e +b^{2} d^{5}\right ) x^{3}}{3 b x +3 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (5 a^{2} d^{4} e +2 b a \,d^{5}\right ) x^{2}}{2 b x +2 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, a^{2} d^{5} x}{b x +a}\) | \(329\) |
1/168*x*(21*b^2*e^5*x^7+48*a*b*e^5*x^6+120*b^2*d*e^4*x^6+28*a^2*e^5*x^5+28 0*a*b*d*e^4*x^5+280*b^2*d^2*e^3*x^5+168*a^2*d*e^4*x^4+672*a*b*d^2*e^3*x^4+ 336*b^2*d^3*e^2*x^4+420*a^2*d^2*e^3*x^3+840*a*b*d^3*e^2*x^3+210*b^2*d^4*e* x^3+560*a^2*d^3*e^2*x^2+560*a*b*d^4*e*x^2+56*b^2*d^5*x^2+420*a^2*d^4*e*x+1 68*a*b*d^5*x+168*a^2*d^5)*((b*x+a)^2)^(1/2)/(b*x+a)
Time = 0.71 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.35 \[ \int (a+b x) (d+e x)^5 \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {1}{8} \, b^{2} e^{5} x^{8} + a^{2} d^{5} x + \frac {1}{7} \, {\left (5 \, b^{2} d e^{4} + 2 \, a b e^{5}\right )} x^{7} + \frac {1}{6} \, {\left (10 \, b^{2} d^{2} e^{3} + 10 \, a b d e^{4} + a^{2} e^{5}\right )} x^{6} + {\left (2 \, b^{2} d^{3} e^{2} + 4 \, a b d^{2} e^{3} + a^{2} d e^{4}\right )} x^{5} + \frac {5}{4} \, {\left (b^{2} d^{4} e + 4 \, a b d^{3} e^{2} + 2 \, a^{2} d^{2} e^{3}\right )} x^{4} + \frac {1}{3} \, {\left (b^{2} d^{5} + 10 \, a b d^{4} e + 10 \, a^{2} d^{3} e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (2 \, a b d^{5} + 5 \, a^{2} d^{4} e\right )} x^{2} \]
1/8*b^2*e^5*x^8 + a^2*d^5*x + 1/7*(5*b^2*d*e^4 + 2*a*b*e^5)*x^7 + 1/6*(10* b^2*d^2*e^3 + 10*a*b*d*e^4 + a^2*e^5)*x^6 + (2*b^2*d^3*e^2 + 4*a*b*d^2*e^3 + a^2*d*e^4)*x^5 + 5/4*(b^2*d^4*e + 4*a*b*d^3*e^2 + 2*a^2*d^2*e^3)*x^4 + 1/3*(b^2*d^5 + 10*a*b*d^4*e + 10*a^2*d^3*e^2)*x^3 + 1/2*(2*a*b*d^5 + 5*a^2 *d^4*e)*x^2
Time = 5.19 (sec) , antiderivative size = 2018, normalized size of antiderivative = 13.82 \[ \int (a+b x) (d+e x)^5 \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\text {Too large to display} \]
a*d**5*Piecewise(((a/(2*b) + x/2)*sqrt(a**2 + 2*a*b*x + b**2*x**2), Ne(b** 2, 0)), ((a**2 + 2*a*b*x)**(3/2)/(3*a*b), Ne(a*b, 0)), (x*sqrt(a**2), True )) + 5*a*d**4*e*Piecewise((sqrt(a**2 + 2*a*b*x + b**2*x**2)*(-a**2/(6*b**2 ) + a*x/(6*b) + x**2/3), Ne(b**2, 0)), ((-a**2*(a**2 + 2*a*b*x)**(3/2)/3 + (a**2 + 2*a*b*x)**(5/2)/5)/(2*a**2*b**2), Ne(a*b, 0)), (x**2*sqrt(a**2)/2 , True)) + 10*a*d**3*e**2*Piecewise((sqrt(a**2 + 2*a*b*x + b**2*x**2)*(a** 3/(12*b**3) - a**2*x/(12*b**2) + a*x**2/(12*b) + x**3/4), Ne(b**2, 0)), (( a**4*(a**2 + 2*a*b*x)**(3/2)/3 - 2*a**2*(a**2 + 2*a*b*x)**(5/2)/5 + (a**2 + 2*a*b*x)**(7/2)/7)/(4*a**3*b**3), Ne(a*b, 0)), (x**3*sqrt(a**2)/3, True) ) + 10*a*d**2*e**3*Piecewise((sqrt(a**2 + 2*a*b*x + b**2*x**2)*(-a**4/(20* b**4) + a**3*x/(20*b**3) - a**2*x**2/(20*b**2) + a*x**3/(20*b) + x**4/5), Ne(b**2, 0)), ((-a**6*(a**2 + 2*a*b*x)**(3/2)/3 + 3*a**4*(a**2 + 2*a*b*x)* *(5/2)/5 - 3*a**2*(a**2 + 2*a*b*x)**(7/2)/7 + (a**2 + 2*a*b*x)**(9/2)/9)/( 8*a**4*b**4), Ne(a*b, 0)), (x**4*sqrt(a**2)/4, True)) + 5*a*d*e**4*Piecewi se((sqrt(a**2 + 2*a*b*x + b**2*x**2)*(a**5/(30*b**5) - a**4*x/(30*b**4) + a**3*x**2/(30*b**3) - a**2*x**3/(30*b**2) + a*x**4/(30*b) + x**5/6), Ne(b* *2, 0)), ((a**8*(a**2 + 2*a*b*x)**(3/2)/3 - 4*a**6*(a**2 + 2*a*b*x)**(5/2) /5 + 6*a**4*(a**2 + 2*a*b*x)**(7/2)/7 - 4*a**2*(a**2 + 2*a*b*x)**(9/2)/9 + (a**2 + 2*a*b*x)**(11/2)/11)/(16*a**5*b**5), Ne(a*b, 0)), (x**5*sqrt(a**2 )/5, True)) + a*e**5*Piecewise((sqrt(a**2 + 2*a*b*x + b**2*x**2)*(-a**6...
Leaf count of result is larger than twice the leaf count of optimal. 1323 vs. \(2 (107) = 214\).
Time = 0.21 (sec) , antiderivative size = 1323, normalized size of antiderivative = 9.06 \[ \int (a+b x) (d+e x)^5 \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\text {Too large to display} \]
1/8*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*e^5*x^5/b - 13/56*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a*e^5*x^4/b^2 + 9/28*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^2*e^5*x ^3/b^3 + 1/2*sqrt(b^2*x^2 + 2*a*b*x + a^2)*a*d^5*x + 1/2*sqrt(b^2*x^2 + 2* a*b*x + a^2)*a^6*e^5*x/b^5 - 11/28*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^3*e^5 *x^2/b^4 + 1/2*sqrt(b^2*x^2 + 2*a*b*x + a^2)*a^2*d^5/b + 1/2*sqrt(b^2*x^2 + 2*a*b*x + a^2)*a^7*e^5/b^6 + 25/56*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^4*e ^5*x/b^5 - 27/56*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^5*e^5/b^6 + 1/7*(5*b*d* e^4 + a*e^5)*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*x^4/b^2 - 11/42*(5*b*d*e^4 + a*e^5)*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a*x^3/b^3 + 5/6*(2*b*d^2*e^3 + a*d* e^4)*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*x^3/b^2 - 1/2*(5*b*d*e^4 + a*e^5)*sqr t(b^2*x^2 + 2*a*b*x + a^2)*a^5*x/b^5 + 5/2*(2*b*d^2*e^3 + a*d*e^4)*sqrt(b^ 2*x^2 + 2*a*b*x + a^2)*a^4*x/b^4 - 5*(b*d^3*e^2 + a*d^2*e^3)*sqrt(b^2*x^2 + 2*a*b*x + a^2)*a^3*x/b^3 + 5/2*(b*d^4*e + 2*a*d^3*e^2)*sqrt(b^2*x^2 + 2* a*b*x + a^2)*a^2*x/b^2 - 1/2*(b*d^5 + 5*a*d^4*e)*sqrt(b^2*x^2 + 2*a*b*x + a^2)*a*x/b + 5/14*(5*b*d*e^4 + a*e^5)*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^2* x^2/b^4 - 3/2*(2*b*d^2*e^3 + a*d*e^4)*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a*x^ 2/b^3 + 2*(b*d^3*e^2 + a*d^2*e^3)*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*x^2/b^2 - 1/2*(5*b*d*e^4 + a*e^5)*sqrt(b^2*x^2 + 2*a*b*x + a^2)*a^6/b^6 + 5/2*(2*b *d^2*e^3 + a*d*e^4)*sqrt(b^2*x^2 + 2*a*b*x + a^2)*a^5/b^5 - 5*(b*d^3*e^2 + a*d^2*e^3)*sqrt(b^2*x^2 + 2*a*b*x + a^2)*a^4/b^4 + 5/2*(b*d^4*e + 2*a*...
Leaf count of result is larger than twice the leaf count of optimal. 401 vs. \(2 (107) = 214\).
Time = 0.27 (sec) , antiderivative size = 401, normalized size of antiderivative = 2.75 \[ \int (a+b x) (d+e x)^5 \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {1}{8} \, b^{2} e^{5} x^{8} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{7} \, b^{2} d e^{4} x^{7} \mathrm {sgn}\left (b x + a\right ) + \frac {2}{7} \, a b e^{5} x^{7} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{3} \, b^{2} d^{2} e^{3} x^{6} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{3} \, a b d e^{4} x^{6} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{6} \, a^{2} e^{5} x^{6} \mathrm {sgn}\left (b x + a\right ) + 2 \, b^{2} d^{3} e^{2} x^{5} \mathrm {sgn}\left (b x + a\right ) + 4 \, a b d^{2} e^{3} x^{5} \mathrm {sgn}\left (b x + a\right ) + a^{2} d e^{4} x^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{4} \, b^{2} d^{4} e x^{4} \mathrm {sgn}\left (b x + a\right ) + 5 \, a b d^{3} e^{2} x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{2} \, a^{2} d^{2} e^{3} x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{3} \, b^{2} d^{5} x^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {10}{3} \, a b d^{4} e x^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {10}{3} \, a^{2} d^{3} e^{2} x^{3} \mathrm {sgn}\left (b x + a\right ) + a b d^{5} x^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{2} \, a^{2} d^{4} e x^{2} \mathrm {sgn}\left (b x + a\right ) + a^{2} d^{5} x \mathrm {sgn}\left (b x + a\right ) + \frac {{\left (56 \, a^{3} b^{5} d^{5} - 70 \, a^{4} b^{4} d^{4} e + 56 \, a^{5} b^{3} d^{3} e^{2} - 28 \, a^{6} b^{2} d^{2} e^{3} + 8 \, a^{7} b d e^{4} - a^{8} e^{5}\right )} \mathrm {sgn}\left (b x + a\right )}{168 \, b^{6}} \]
1/8*b^2*e^5*x^8*sgn(b*x + a) + 5/7*b^2*d*e^4*x^7*sgn(b*x + a) + 2/7*a*b*e^ 5*x^7*sgn(b*x + a) + 5/3*b^2*d^2*e^3*x^6*sgn(b*x + a) + 5/3*a*b*d*e^4*x^6* sgn(b*x + a) + 1/6*a^2*e^5*x^6*sgn(b*x + a) + 2*b^2*d^3*e^2*x^5*sgn(b*x + a) + 4*a*b*d^2*e^3*x^5*sgn(b*x + a) + a^2*d*e^4*x^5*sgn(b*x + a) + 5/4*b^2 *d^4*e*x^4*sgn(b*x + a) + 5*a*b*d^3*e^2*x^4*sgn(b*x + a) + 5/2*a^2*d^2*e^3 *x^4*sgn(b*x + a) + 1/3*b^2*d^5*x^3*sgn(b*x + a) + 10/3*a*b*d^4*e*x^3*sgn( b*x + a) + 10/3*a^2*d^3*e^2*x^3*sgn(b*x + a) + a*b*d^5*x^2*sgn(b*x + a) + 5/2*a^2*d^4*e*x^2*sgn(b*x + a) + a^2*d^5*x*sgn(b*x + a) + 1/168*(56*a^3*b^ 5*d^5 - 70*a^4*b^4*d^4*e + 56*a^5*b^3*d^3*e^2 - 28*a^6*b^2*d^2*e^3 + 8*a^7 *b*d*e^4 - a^8*e^5)*sgn(b*x + a)/b^6
Time = 13.23 (sec) , antiderivative size = 1541, normalized size of antiderivative = 10.55 \[ \int (a+b x) (d+e x)^5 \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\text {Too large to display} \]
a*d^5*(x/2 + a/(2*b))*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2) + (d^5*(8*b^2*(a^2 + b^2*x^2) - 12*a^2*b^2 + 4*a*b^3*x)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(24*b ^3) + (e^5*x^5*(a^2 + b^2*x^2 + 2*a*b*x)^(3/2))/(8*b) + (a*e^5*x^4*(a^2 + b^2*x^2 + 2*a*b*x)^(3/2))/(7*b^2) + (5*d*e^4*x^4*(a^2 + b^2*x^2 + 2*a*b*x) ^(3/2))/(7*b) - (13*a*e^5*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2)*(6*b^4*x^4*(a^2 + b^2*x^2 + 2*a*b*x) - a^6 + 20*a^4*b^2*x^2 + 19*a^5*b*x - 11*a*b^3*x^3*(a ^2 + b^2*x^2 + 2*a*b*x) + 15*a^2*b^2*x^2*(a^2 + b^2*x^2 + 2*a*b*x) - 18*a^ 3*b*x*(a^2 + b^2*x^2 + 2*a*b*x)))/(336*b^6) + (2*d^3*e^2*x^2*(a^2 + b^2*x^ 2 + 2*a*b*x)^(3/2))/b + (5*d^2*e^3*x^3*(a^2 + b^2*x^2 + 2*a*b*x)^(3/2))/(3 *b) + (5*d^4*e*x*(a^2 + b^2*x^2 + 2*a*b*x)^(3/2))/(4*b) - (a^3*e^5*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2)*(4*b^2*x^2*(a^2 + b^2*x^2 + 2*a*b*x) - a^4 + 9*a^ 2*b^2*x^2 + 8*a^3*b*x - 7*a*b*x*(a^2 + b^2*x^2 + 2*a*b*x)))/(35*b^6) - (41 *a^2*e^5*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2)*(a^5 + 5*b^3*x^3*(a^2 + b^2*x^2 + 2*a*b*x) - 14*a^3*b^2*x^2 - 13*a^4*b*x - 9*a*b^2*x^2*(a^2 + b^2*x^2 + 2*a *b*x) + 12*a^2*b*x*(a^2 + b^2*x^2 + 2*a*b*x)))/(560*b^6) - (5*a*d^4*e*(8*b ^2*(a^2 + b^2*x^2) - 12*a^2*b^2 + 4*a*b^3*x)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/ 2))/(96*b^4) - (5*a^3*d^3*e^2*(x/2 + a/(2*b))*(a^2 + b^2*x^2 + 2*a*b*x)^(1 /2))/(2*b^2) - (3*a*d^2*e^3*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2)*(4*b^2*x^2*(a^ 2 + b^2*x^2 + 2*a*b*x) - a^4 + 9*a^2*b^2*x^2 + 8*a^3*b*x - 7*a*b*x*(a^2 + b^2*x^2 + 2*a*b*x)))/(4*b^4) - (29*a^2*d*e^4*(a^2 + b^2*x^2 + 2*a*b*x)^...