Integrand size = 33, antiderivative size = 146 \[ \int (a+b x) (d+e x)^3 \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {(b d-a e)^2 (d+e x)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^3 (a+b x)}-\frac {2 b (b d-a e) (d+e x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^3 (a+b x)}+\frac {b^2 (d+e x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{6 e^3 (a+b x)} \]
1/4*(-a*e+b*d)^2*(e*x+d)^4*((b*x+a)^2)^(1/2)/e^3/(b*x+a)-2/5*b*(-a*e+b*d)* (e*x+d)^5*((b*x+a)^2)^(1/2)/e^3/(b*x+a)+1/6*b^2*(e*x+d)^6*((b*x+a)^2)^(1/2 )/e^3/(b*x+a)
Time = 1.03 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.89 \[ \int (a+b x) (d+e x)^3 \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {x \sqrt {(a+b x)^2} \left (15 a^2 \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )+6 a b x \left (10 d^3+20 d^2 e x+15 d e^2 x^2+4 e^3 x^3\right )+b^2 x^2 \left (20 d^3+45 d^2 e x+36 d e^2 x^2+10 e^3 x^3\right )\right )}{60 (a+b x)} \]
(x*Sqrt[(a + b*x)^2]*(15*a^2*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3) + 6*a*b*x*(10*d^3 + 20*d^2*e*x + 15*d*e^2*x^2 + 4*e^3*x^3) + b^2*x^2*(20*d^ 3 + 45*d^2*e*x + 36*d*e^2*x^2 + 10*e^3*x^3)))/(60*(a + b*x))
Time = 0.28 (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.
| \(\displaystyle \int (a+b x) \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^3 \, dx\) |
| \(\Big \downarrow \) 1187 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int b (a+b x)^2 (d+e x)^3dx}{b (a+b x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int (a+b x)^2 (d+e x)^3dx}{a+b x}\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {b^2 (d+e x)^5}{e^2}-\frac {2 b (b d-a e) (d+e x)^4}{e^2}+\frac {(a e-b d)^2 (d+e x)^3}{e^2}\right )dx}{a+b x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (-\frac {2 b (d+e x)^5 (b d-a e)}{5 e^3}+\frac {(d+e x)^4 (b d-a e)^2}{4 e^3}+\frac {b^2 (d+e x)^6}{6 e^3}\right )}{a+b x}\) |
(Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(((b*d - a*e)^2*(d + e*x)^4)/(4*e^3) - (2*b *(b*d - a*e)*(d + e*x)^5)/(5*e^3) + (b^2*(d + e*x)^6)/(6*e^3)))/(a + b*x)
3.20.56.3.1 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]
Result contains higher order function than in optimal. Order 9 vs. order 2.
Time = 0.37 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.83
| method | result | size |
| default | \(-\frac {\operatorname {csgn}\left (b x +a \right ) \left (b x +a \right )^{3} \left (-10 b^{3} x^{3} e^{3}+6 x^{2} a \,b^{2} e^{3}-36 x^{2} b^{3} d \,e^{2}-3 x \,a^{2} b \,e^{3}+18 x a \,b^{2} d \,e^{2}-45 x \,b^{3} d^{2} e +a^{3} e^{3}-6 a^{2} b d \,e^{2}+15 a \,b^{2} d^{2} e -20 b^{3} d^{3}\right )}{60 b^{4}}\) | \(121\) |
| gosper | \(\frac {x \left (10 x^{5} b^{2} e^{3}+24 x^{4} a b \,e^{3}+36 x^{4} b^{2} d \,e^{2}+15 x^{3} a^{2} e^{3}+90 a b d \,e^{2} x^{3}+45 b^{2} d^{2} e \,x^{3}+60 x^{2} a^{2} d \,e^{2}+120 x^{2} a b \,d^{2} e +20 x^{2} b^{2} d^{3}+90 x \,a^{2} d^{2} e +60 b \,d^{3} a x +60 a^{2} d^{3}\right ) \sqrt {\left (b x +a \right )^{2}}}{60 b x +60 a}\) | \(148\) |
| risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, b^{2} e^{3} x^{6}}{6 b x +6 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (2 a b \,e^{3}+3 b^{2} d \,e^{2}\right ) x^{5}}{5 b x +5 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (a^{2} e^{3}+6 a b d \,e^{2}+3 b^{2} d^{2} e \right ) x^{4}}{4 b x +4 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (3 a^{2} d \,e^{2}+6 a b \,d^{2} e +b^{2} d^{3}\right ) x^{3}}{3 b x +3 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (3 a^{2} d^{2} e +2 b a \,d^{3}\right ) x^{2}}{2 b x +2 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, a^{2} d^{3} x}{b x +a}\) | \(221\) |
-1/60*csgn(b*x+a)*(b*x+a)^3*(-10*b^3*e^3*x^3+6*a*b^2*e^3*x^2-36*b^3*d*e^2* x^2-3*a^2*b*e^3*x+18*a*b^2*d*e^2*x-45*b^3*d^2*e*x+a^3*e^3-6*a^2*b*d*e^2+15 *a*b^2*d^2*e-20*b^3*d^3)/b^4
Time = 0.29 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.85 \[ \int (a+b x) (d+e x)^3 \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {1}{6} \, b^{2} e^{3} x^{6} + a^{2} d^{3} x + \frac {1}{5} \, {\left (3 \, b^{2} d e^{2} + 2 \, a b e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (3 \, b^{2} d^{2} e + 6 \, a b d e^{2} + a^{2} e^{3}\right )} x^{4} + \frac {1}{3} \, {\left (b^{2} d^{3} + 6 \, a b d^{2} e + 3 \, a^{2} d e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (2 \, a b d^{3} + 3 \, a^{2} d^{2} e\right )} x^{2} \]
1/6*b^2*e^3*x^6 + a^2*d^3*x + 1/5*(3*b^2*d*e^2 + 2*a*b*e^3)*x^5 + 1/4*(3*b ^2*d^2*e + 6*a*b*d*e^2 + a^2*e^3)*x^4 + 1/3*(b^2*d^3 + 6*a*b*d^2*e + 3*a^2 *d*e^2)*x^3 + 1/2*(2*a*b*d^3 + 3*a^2*d^2*e)*x^2
Time = 2.88 (sec) , antiderivative size = 1080, normalized size of antiderivative = 7.40 \[ \int (a+b x) (d+e x)^3 \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\text {Too large to display} \]
a*d**3*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 )) + 3*a*d**2*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)) + 3*a*d*e**2*Piecewise((sqrt(a**2 + 2*a*b*x + b**2*x**2)*(a**3/(1 2*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)) + a*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)) + b*d**3*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)) + 3*b*d**2*e*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...
Leaf count of result is larger than twice the leaf count of optimal. 693 vs. \(2 (107) = 214\).
Time = 0.20 (sec) , antiderivative size = 693, normalized size of antiderivative = 4.75 \[ \int (a+b x) (d+e x)^3 \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} e^{3} x^{3}}{6 \, b} + \frac {1}{2} \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a d^{3} x + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{4} e^{3} x}{2 \, b^{3}} - \frac {3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a e^{3} x^{2}}{10 \, b^{2}} + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{2} d^{3}}{2 \, b} + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{5} e^{3}}{2 \, b^{4}} + \frac {2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{2} e^{3} x}{5 \, b^{3}} - \frac {7 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{3} e^{3}}{15 \, b^{4}} - \frac {{\left (3 \, b d e^{2} + a e^{3}\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{3} x}{2 \, b^{3}} + \frac {3 \, {\left (b d^{2} e + a d e^{2}\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{2} x}{2 \, b^{2}} - \frac {{\left (b d^{3} + 3 \, a d^{2} e\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a x}{2 \, b} + \frac {{\left (3 \, b d e^{2} + a e^{3}\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} x^{2}}{5 \, b^{2}} - \frac {{\left (3 \, b d e^{2} + a e^{3}\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{4}}{2 \, b^{4}} + \frac {3 \, {\left (b d^{2} e + a d e^{2}\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{3}}{2 \, b^{3}} - \frac {{\left (b d^{3} + 3 \, a d^{2} e\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{2}}{2 \, b^{2}} - \frac {7 \, {\left (3 \, b d e^{2} + a e^{3}\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a x}{20 \, b^{3}} + \frac {3 \, {\left (b d^{2} e + a d e^{2}\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} x}{4 \, b^{2}} + \frac {9 \, {\left (3 \, b d e^{2} + a e^{3}\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{2}}{20 \, b^{4}} - \frac {5 \, {\left (b d^{2} e + a d e^{2}\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a}{4 \, b^{3}} + \frac {{\left (b d^{3} + 3 \, a d^{2} e\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}}}{3 \, b^{2}} \]
1/6*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*e^3*x^3/b + 1/2*sqrt(b^2*x^2 + 2*a*b*x + a^2)*a*d^3*x + 1/2*sqrt(b^2*x^2 + 2*a*b*x + a^2)*a^4*e^3*x/b^3 - 3/10*( b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a*e^3*x^2/b^2 + 1/2*sqrt(b^2*x^2 + 2*a*b*x + a^2)*a^2*d^3/b + 1/2*sqrt(b^2*x^2 + 2*a*b*x + a^2)*a^5*e^3/b^4 + 2/5*(b^ 2*x^2 + 2*a*b*x + a^2)^(3/2)*a^2*e^3*x/b^3 - 7/15*(b^2*x^2 + 2*a*b*x + a^2 )^(3/2)*a^3*e^3/b^4 - 1/2*(3*b*d*e^2 + a*e^3)*sqrt(b^2*x^2 + 2*a*b*x + a^2 )*a^3*x/b^3 + 3/2*(b*d^2*e + a*d*e^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2)*a^2*x/ b^2 - 1/2*(b*d^3 + 3*a*d^2*e)*sqrt(b^2*x^2 + 2*a*b*x + a^2)*a*x/b + 1/5*(3 *b*d*e^2 + a*e^3)*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*x^2/b^2 - 1/2*(3*b*d*e^2 + a*e^3)*sqrt(b^2*x^2 + 2*a*b*x + a^2)*a^4/b^4 + 3/2*(b*d^2*e + a*d*e^2)* sqrt(b^2*x^2 + 2*a*b*x + a^2)*a^3/b^3 - 1/2*(b*d^3 + 3*a*d^2*e)*sqrt(b^2*x ^2 + 2*a*b*x + a^2)*a^2/b^2 - 7/20*(3*b*d*e^2 + a*e^3)*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a*x/b^3 + 3/4*(b*d^2*e + a*d*e^2)*(b^2*x^2 + 2*a*b*x + a^2)^( 3/2)*x/b^2 + 9/20*(3*b*d*e^2 + a*e^3)*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^2/ b^4 - 5/4*(b*d^2*e + a*d*e^2)*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a/b^3 + 1/3* (b*d^3 + 3*a*d^2*e)*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)/b^2
Leaf count of result is larger than twice the leaf count of optimal. 255 vs. \(2 (107) = 214\).
Time = 0.26 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.75 \[ \int (a+b x) (d+e x)^3 \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {1}{6} \, b^{2} e^{3} x^{6} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{5} \, b^{2} d e^{2} x^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {2}{5} \, a b e^{3} x^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{4} \, b^{2} d^{2} e x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{2} \, a b d e^{2} x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{4} \, a^{2} e^{3} x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{3} \, b^{2} d^{3} x^{3} \mathrm {sgn}\left (b x + a\right ) + 2 \, a b d^{2} e x^{3} \mathrm {sgn}\left (b x + a\right ) + a^{2} d e^{2} x^{3} \mathrm {sgn}\left (b x + a\right ) + a b d^{3} x^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{2} \, a^{2} d^{2} e x^{2} \mathrm {sgn}\left (b x + a\right ) + a^{2} d^{3} x \mathrm {sgn}\left (b x + a\right ) + \frac {{\left (20 \, a^{3} b^{3} d^{3} - 15 \, a^{4} b^{2} d^{2} e + 6 \, a^{5} b d e^{2} - a^{6} e^{3}\right )} \mathrm {sgn}\left (b x + a\right )}{60 \, b^{4}} \]
1/6*b^2*e^3*x^6*sgn(b*x + a) + 3/5*b^2*d*e^2*x^5*sgn(b*x + a) + 2/5*a*b*e^ 3*x^5*sgn(b*x + a) + 3/4*b^2*d^2*e*x^4*sgn(b*x + a) + 3/2*a*b*d*e^2*x^4*sg n(b*x + a) + 1/4*a^2*e^3*x^4*sgn(b*x + a) + 1/3*b^2*d^3*x^3*sgn(b*x + a) + 2*a*b*d^2*e*x^3*sgn(b*x + a) + a^2*d*e^2*x^3*sgn(b*x + a) + a*b*d^3*x^2*s gn(b*x + a) + 3/2*a^2*d^2*e*x^2*sgn(b*x + a) + a^2*d^3*x*sgn(b*x + a) + 1/ 60*(20*a^3*b^3*d^3 - 15*a^4*b^2*d^2*e + 6*a^5*b*d*e^2 - a^6*e^3)*sgn(b*x + a)/b^4
Time = 11.46 (sec) , antiderivative size = 734, normalized size of antiderivative = 5.03 \[ \int (a+b x) (d+e x)^3 \sqrt {a^2+2 a b x+b^2 x^2} \, dx=a\,d^3\,\left (\frac {x}{2}+\frac {a}{2\,b}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}+\frac {d^3\,\left (8\,b^2\,\left (a^2+b^2\,x^2\right )-12\,a^2\,b^2+4\,a\,b^3\,x\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{24\,b^3}+\frac {e^3\,x^3\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}}{6\,b}-\frac {19\,a^2\,e^3\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}\,\left (a^3-5\,a\,b^2\,x^2+3\,b\,x\,\left (a^2+2\,a\,b\,x+b^2\,x^2\right )-4\,a^2\,b\,x\right )}{120\,b^4}-\frac {a^3\,e^3\,\left (8\,b^2\,\left (a^2+b^2\,x^2\right )-12\,a^2\,b^2+4\,a\,b^3\,x\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{60\,b^6}+\frac {a\,e^3\,x^2\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}}{5\,b^2}+\frac {3\,d\,e^2\,x^2\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}}{5\,b}-\frac {3\,a\,e^3\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}\,\left (4\,b^2\,x^2\,\left (a^2+2\,a\,b\,x+b^2\,x^2\right )-a^4+9\,a^2\,b^2\,x^2+8\,a^3\,b\,x-7\,a\,b\,x\,\left (a^2+2\,a\,b\,x+b^2\,x^2\right )\right )}{40\,b^4}+\frac {3\,d^2\,e\,x\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}}{4\,b}-\frac {7\,a\,d\,e^2\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}\,\left (a^3-5\,a\,b^2\,x^2+3\,b\,x\,\left (a^2+2\,a\,b\,x+b^2\,x^2\right )-4\,a^2\,b\,x\right )}{20\,b^3}-\frac {a\,d^2\,e\,\left (8\,b^2\,\left (a^2+b^2\,x^2\right )-12\,a^2\,b^2+4\,a\,b^3\,x\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{32\,b^4}+\frac {3\,a\,d\,e^2\,x\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}}{4\,b^2}-\frac {33\,a^2\,d\,e^2\,\left (8\,b^2\,\left (a^2+b^2\,x^2\right )-12\,a^2\,b^2+4\,a\,b^3\,x\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{160\,b^5}-\frac {3\,a^2\,d^2\,e\,\left (\frac {x}{2}+\frac {a}{2\,b}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{4\,b}-\frac {3\,a^3\,d\,e^2\,\left (\frac {x}{2}+\frac {a}{2\,b}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{4\,b^2} \]
a*d^3*(x/2 + a/(2*b))*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2) + (d^3*(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^3*x^3*(a^2 + b^2*x^2 + 2*a*b*x)^(3/2))/(6*b) - (19*a^2*e^3*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2)*(a^3 - 5*a*b^2*x^2 + 3*b*x*(a^2 + b^2*x^2 + 2*a* b*x) - 4*a^2*b*x))/(120*b^4) - (a^3*e^3*(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))/(60*b^6) + (a*e^3*x^2*(a^2 + b^2*x^2 + 2*a*b*x)^(3/2))/(5*b^2) + (3*d*e^2*x^2*(a^2 + b^2*x^2 + 2*a*b *x)^(3/2))/(5*b) - (3*a*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)))/(40*b^4) + (3*d^2*e*x*(a^2 + b^2*x^2 + 2*a*b*x)^(3/2) )/(4*b) - (7*a*d*e^2*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2)*(a^3 - 5*a*b^2*x^2 + 3*b*x*(a^2 + b^2*x^2 + 2*a*b*x) - 4*a^2*b*x))/(20*b^3) - (a*d^2*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))/ (32*b^4) + (3*a*d*e^2*x*(a^2 + b^2*x^2 + 2*a*b*x)^(3/2))/(4*b^2) - (33*a^2 *d*e^2*(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))/(160*b^5) - (3*a^2*d^2*e*(x/2 + a/(2*b))*(a^2 + b^2*x^2 + 2 *a*b*x)^(1/2))/(4*b) - (3*a^3*d*e^2*(x/2 + a/(2*b))*(a^2 + b^2*x^2 + 2*a*b *x)^(1/2))/(4*b^2)