\(\int (a+b x) (d+e x)^5 (a^2+2 a b x+b^2 x^2)^{3/2} \, dx\) [1970]

Optimal result

Integrand size = 33, antiderivative size = 254 \[ \int (a+b x) (d+e x)^5 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {(b d-a e)^4 (d+e x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{6 e^5 (a+b x)}-\frac {4 b (b d-a e)^3 (d+e x)^7 \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^5 (a+b x)}+\frac {3 b^2 (b d-a e)^2 (d+e x)^8 \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^5 (a+b x)}-\frac {4 b^3 (b d-a e) (d+e x)^9 \sqrt {a^2+2 a b x+b^2 x^2}}{9 e^5 (a+b x)}+\frac {b^4 (d+e x)^{10} \sqrt {a^2+2 a b x+b^2 x^2}}{10 e^5 (a+b x)} \]

[Out]

1/6*(-a*e+b*d)^4*(e*x+d)^6*((b*x+a)^2)^(1/2)/e^5/(b*x+a)-4/7*b*(-a*e+b*d)^3*(e*x+d)^7*((b*x+a)^2)^(1/2)/e^5/(b
*x+a)+3/4*b^2*(-a*e+b*d)^2*(e*x+d)^8*((b*x+a)^2)^(1/2)/e^5/(b*x+a)-4/9*b^3*(-a*e+b*d)*(e*x+d)^9*((b*x+a)^2)^(1
/2)/e^5/(b*x+a)+1/10*b^4*(e*x+d)^10*((b*x+a)^2)^(1/2)/e^5/(b*x+a)

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {784, 21, 45} \[ \int (a+b x) (d+e x)^5 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {3 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^8 (b d-a e)^2}{4 e^5 (a+b x)}-\frac {4 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^7 (b d-a e)^3}{7 e^5 (a+b x)}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^6 (b d-a e)^4}{6 e^5 (a+b x)}+\frac {b^4 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{10}}{10 e^5 (a+b x)}-\frac {4 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^9 (b d-a e)}{9 e^5 (a+b x)} \]

[In]

Int[(a + b*x)*(d + e*x)^5*(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]

[Out]

((b*d - a*e)^4*(d + e*x)^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(6*e^5*(a + b*x)) - (4*b*(b*d - a*e)^3*(d + e*x)^7*S
qrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^5*(a + b*x)) + (3*b^2*(b*d - a*e)^2*(d + e*x)^8*Sqrt[a^2 + 2*a*b*x + b^2*x^
2])/(4*e^5*(a + b*x)) - (4*b^3*(b*d - a*e)*(d + e*x)^9*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*e^5*(a + b*x)) + (b^4
*(d + e*x)^10*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(10*e^5*(a + b*x))

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 45

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, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 784

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dis
t[(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)*(b/2 + c
*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && EqQ[b^2 - 4*a*c, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int (a+b x) \left (a b+b^2 x\right )^3 (d+e x)^5 \, dx}{b^2 \left (a b+b^2 x\right )} \\ & = \frac {\left (b \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int (a+b x)^4 (d+e x)^5 \, dx}{a b+b^2 x} \\ & = \frac {\left (b \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \left (\frac {(-b d+a e)^4 (d+e x)^5}{e^4}-\frac {4 b (b d-a e)^3 (d+e x)^6}{e^4}+\frac {6 b^2 (b d-a e)^2 (d+e x)^7}{e^4}-\frac {4 b^3 (b d-a e) (d+e x)^8}{e^4}+\frac {b^4 (d+e x)^9}{e^4}\right ) \, dx}{a b+b^2 x} \\ & = \frac {(b d-a e)^4 (d+e x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{6 e^5 (a+b x)}-\frac {4 b (b d-a e)^3 (d+e x)^7 \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^5 (a+b x)}+\frac {3 b^2 (b d-a e)^2 (d+e x)^8 \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^5 (a+b x)}-\frac {4 b^3 (b d-a e) (d+e x)^9 \sqrt {a^2+2 a b x+b^2 x^2}}{9 e^5 (a+b x)}+\frac {b^4 (d+e x)^{10} \sqrt {a^2+2 a b x+b^2 x^2}}{10 e^5 (a+b x)} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.07 (sec) , antiderivative size = 322, normalized size of antiderivative = 1.27 \[ \int (a+b x) (d+e x)^5 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {x \sqrt {(a+b x)^2} \left (210 a^4 \left (6 d^5+15 d^4 e x+20 d^3 e^2 x^2+15 d^2 e^3 x^3+6 d e^4 x^4+e^5 x^5\right )+120 a^3 b x \left (21 d^5+70 d^4 e x+105 d^3 e^2 x^2+84 d^2 e^3 x^3+35 d e^4 x^4+6 e^5 x^5\right )+45 a^2 b^2 x^2 \left (56 d^5+210 d^4 e x+336 d^3 e^2 x^2+280 d^2 e^3 x^3+120 d e^4 x^4+21 e^5 x^5\right )+10 a b^3 x^3 \left (126 d^5+504 d^4 e x+840 d^3 e^2 x^2+720 d^2 e^3 x^3+315 d e^4 x^4+56 e^5 x^5\right )+b^4 x^4 \left (252 d^5+1050 d^4 e x+1800 d^3 e^2 x^2+1575 d^2 e^3 x^3+700 d e^4 x^4+126 e^5 x^5\right )\right )}{1260 (a+b x)} \]

[In]

Integrate[(a + b*x)*(d + e*x)^5*(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]

[Out]

(x*Sqrt[(a + b*x)^2]*(210*a^4*(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) +
120*a^3*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) + 45*a^2*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) + 10*a*b^3*x^3*(12
6*d^5 + 504*d^4*e*x + 840*d^3*e^2*x^2 + 720*d^2*e^3*x^3 + 315*d*e^4*x^4 + 56*e^5*x^5) + b^4*x^4*(252*d^5 + 105
0*d^4*e*x + 1800*d^3*e^2*x^2 + 1575*d^2*e^3*x^3 + 700*d*e^4*x^4 + 126*e^5*x^5)))/(1260*(a + b*x))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(413\) vs. \(2(189)=378\).

Time = 0.58 (sec) , antiderivative size = 414, normalized size of antiderivative = 1.63

method	result	size
gosper	\(\frac {x \left (126 b^{4} e^{5} x^{9}+560 x^{8} a \,b^{3} e^{5}+700 x^{8} b^{4} d \,e^{4}+945 x^{7} a^{2} b^{2} e^{5}+3150 x^{7} b^{3} a d \,e^{4}+1575 x^{7} b^{4} d^{2} e^{3}+720 x^{6} b \,a^{3} e^{5}+5400 x^{6} b^{2} a^{2} d \,e^{4}+7200 x^{6} b^{3} a \,d^{2} e^{3}+1800 x^{6} b^{4} d^{3} e^{2}+210 x^{5} a^{4} e^{5}+4200 x^{5} b \,a^{3} d \,e^{4}+12600 x^{5} b^{2} a^{2} d^{2} e^{3}+8400 x^{5} b^{3} a \,d^{3} e^{2}+1050 x^{5} b^{4} d^{4} e +1260 x^{4} a^{4} d \,e^{4}+10080 x^{4} b \,a^{3} d^{2} e^{3}+15120 x^{4} b^{2} a^{2} d^{3} e^{2}+5040 x^{4} b^{3} a \,d^{4} e +252 x^{4} b^{4} d^{5}+3150 x^{3} a^{4} d^{2} e^{3}+12600 x^{3} b \,a^{3} d^{3} e^{2}+9450 x^{3} b^{2} a^{2} d^{4} e +1260 x^{3} b^{3} a \,d^{5}+4200 x^{2} a^{4} d^{3} e^{2}+8400 x^{2} b \,a^{3} d^{4} e +2520 x^{2} b^{2} a^{2} d^{5}+3150 x \,a^{4} d^{4} e +2520 x b \,a^{3} d^{5}+1260 a^{4} d^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{1260 \left (b x +a \right )^{3}}\)	\(414\)
default	\(\frac {x \left (126 b^{4} e^{5} x^{9}+560 x^{8} a \,b^{3} e^{5}+700 x^{8} b^{4} d \,e^{4}+945 x^{7} a^{2} b^{2} e^{5}+3150 x^{7} b^{3} a d \,e^{4}+1575 x^{7} b^{4} d^{2} e^{3}+720 x^{6} b \,a^{3} e^{5}+5400 x^{6} b^{2} a^{2} d \,e^{4}+7200 x^{6} b^{3} a \,d^{2} e^{3}+1800 x^{6} b^{4} d^{3} e^{2}+210 x^{5} a^{4} e^{5}+4200 x^{5} b \,a^{3} d \,e^{4}+12600 x^{5} b^{2} a^{2} d^{2} e^{3}+8400 x^{5} b^{3} a \,d^{3} e^{2}+1050 x^{5} b^{4} d^{4} e +1260 x^{4} a^{4} d \,e^{4}+10080 x^{4} b \,a^{3} d^{2} e^{3}+15120 x^{4} b^{2} a^{2} d^{3} e^{2}+5040 x^{4} b^{3} a \,d^{4} e +252 x^{4} b^{4} d^{5}+3150 x^{3} a^{4} d^{2} e^{3}+12600 x^{3} b \,a^{3} d^{3} e^{2}+9450 x^{3} b^{2} a^{2} d^{4} e +1260 x^{3} b^{3} a \,d^{5}+4200 x^{2} a^{4} d^{3} e^{2}+8400 x^{2} b \,a^{3} d^{4} e +2520 x^{2} b^{2} a^{2} d^{5}+3150 x \,a^{4} d^{4} e +2520 x b \,a^{3} d^{5}+1260 a^{4} d^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{1260 \left (b x +a \right )^{3}}\)	\(414\)
risch	\(\frac {\sqrt {\left (b x +a \right )^{2}}\, b^{4} e^{5} x^{10}}{10 b x +10 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (4 a \,b^{3} e^{5}+5 b^{4} d \,e^{4}\right ) x^{9}}{9 b x +9 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (6 a^{2} b^{2} e^{5}+20 b^{3} a d \,e^{4}+10 b^{4} d^{2} e^{3}\right ) x^{8}}{8 b x +8 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (4 b \,a^{3} e^{5}+30 b^{2} a^{2} d \,e^{4}+40 b^{3} a \,d^{2} e^{3}+10 b^{4} d^{3} e^{2}\right ) x^{7}}{7 b x +7 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (a^{4} e^{5}+20 b \,a^{3} d \,e^{4}+60 b^{2} a^{2} d^{2} e^{3}+40 b^{3} a \,d^{3} e^{2}+5 b^{4} d^{4} e \right ) x^{6}}{6 b x +6 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (5 a^{4} d \,e^{4}+40 b \,a^{3} d^{2} e^{3}+60 b^{2} a^{2} d^{3} e^{2}+20 b^{3} a \,d^{4} e +b^{4} d^{5}\right ) x^{5}}{5 b x +5 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (10 a^{4} d^{2} e^{3}+40 b \,a^{3} d^{3} e^{2}+30 b^{2} a^{2} d^{4} e +4 b^{3} a \,d^{5}\right ) x^{4}}{4 b x +4 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (10 a^{4} d^{3} e^{2}+20 b \,a^{3} d^{4} e +6 b^{2} a^{2} d^{5}\right ) x^{3}}{3 b x +3 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (5 a^{4} d^{4} e +4 b \,a^{3} d^{5}\right ) x^{2}}{2 b x +2 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, a^{4} d^{5} x}{b x +a}\)	\(521\)

[In]

int((b*x+a)*(e*x+d)^5*(b^2*x^2+2*a*b*x+a^2)^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/1260*x*(126*b^4*e^5*x^9+560*a*b^3*e^5*x^8+700*b^4*d*e^4*x^8+945*a^2*b^2*e^5*x^7+3150*a*b^3*d*e^4*x^7+1575*b^
4*d^2*e^3*x^7+720*a^3*b*e^5*x^6+5400*a^2*b^2*d*e^4*x^6+7200*a*b^3*d^2*e^3*x^6+1800*b^4*d^3*e^2*x^6+210*a^4*e^5
*x^5+4200*a^3*b*d*e^4*x^5+12600*a^2*b^2*d^2*e^3*x^5+8400*a*b^3*d^3*e^2*x^5+1050*b^4*d^4*e*x^5+1260*a^4*d*e^4*x
^4+10080*a^3*b*d^2*e^3*x^4+15120*a^2*b^2*d^3*e^2*x^4+5040*a*b^3*d^4*e*x^4+252*b^4*d^5*x^4+3150*a^4*d^2*e^3*x^3
+12600*a^3*b*d^3*e^2*x^3+9450*a^2*b^2*d^4*e*x^3+1260*a*b^3*d^5*x^3+4200*a^4*d^3*e^2*x^2+8400*a^3*b*d^4*e*x^2+2
520*a^2*b^2*d^5*x^2+3150*a^4*d^4*e*x+2520*a^3*b*d^5*x+1260*a^4*d^5)*((b*x+a)^2)^(3/2)/(b*x+a)^3

Fricas [A] (verification not implemented)

none

Time = 0.43 (sec) , antiderivative size = 360, normalized size of antiderivative = 1.42 \[ \int (a+b x) (d+e x)^5 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {1}{10} \, b^{4} e^{5} x^{10} + a^{4} d^{5} x + \frac {1}{9} \, {\left (5 \, b^{4} d e^{4} + 4 \, a b^{3} e^{5}\right )} x^{9} + \frac {1}{4} \, {\left (5 \, b^{4} d^{2} e^{3} + 10 \, a b^{3} d e^{4} + 3 \, a^{2} b^{2} e^{5}\right )} x^{8} + \frac {2}{7} \, {\left (5 \, b^{4} d^{3} e^{2} + 20 \, a b^{3} d^{2} e^{3} + 15 \, a^{2} b^{2} d e^{4} + 2 \, a^{3} b e^{5}\right )} x^{7} + \frac {1}{6} \, {\left (5 \, b^{4} d^{4} e + 40 \, a b^{3} d^{3} e^{2} + 60 \, a^{2} b^{2} d^{2} e^{3} + 20 \, a^{3} b d e^{4} + a^{4} e^{5}\right )} x^{6} + \frac {1}{5} \, {\left (b^{4} d^{5} + 20 \, a b^{3} d^{4} e + 60 \, a^{2} b^{2} d^{3} e^{2} + 40 \, a^{3} b d^{2} e^{3} + 5 \, a^{4} d e^{4}\right )} x^{5} + \frac {1}{2} \, {\left (2 \, a b^{3} d^{5} + 15 \, a^{2} b^{2} d^{4} e + 20 \, a^{3} b d^{3} e^{2} + 5 \, a^{4} d^{2} e^{3}\right )} x^{4} + \frac {2}{3} \, {\left (3 \, a^{2} b^{2} d^{5} + 10 \, a^{3} b d^{4} e + 5 \, a^{4} d^{3} e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (4 \, a^{3} b d^{5} + 5 \, a^{4} d^{4} e\right )} x^{2} \]

[In]

integrate((b*x+a)*(e*x+d)^5*(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm="fricas")

[Out]

1/10*b^4*e^5*x^10 + a^4*d^5*x + 1/9*(5*b^4*d*e^4 + 4*a*b^3*e^5)*x^9 + 1/4*(5*b^4*d^2*e^3 + 10*a*b^3*d*e^4 + 3*
a^2*b^2*e^5)*x^8 + 2/7*(5*b^4*d^3*e^2 + 20*a*b^3*d^2*e^3 + 15*a^2*b^2*d*e^4 + 2*a^3*b*e^5)*x^7 + 1/6*(5*b^4*d^
4*e + 40*a*b^3*d^3*e^2 + 60*a^2*b^2*d^2*e^3 + 20*a^3*b*d*e^4 + a^4*e^5)*x^6 + 1/5*(b^4*d^5 + 20*a*b^3*d^4*e +
60*a^2*b^2*d^3*e^2 + 40*a^3*b*d^2*e^3 + 5*a^4*d*e^4)*x^5 + 1/2*(2*a*b^3*d^5 + 15*a^2*b^2*d^4*e + 20*a^3*b*d^3*
e^2 + 5*a^4*d^2*e^3)*x^4 + 2/3*(3*a^2*b^2*d^5 + 10*a^3*b*d^4*e + 5*a^4*d^3*e^2)*x^3 + 1/2*(4*a^3*b*d^5 + 5*a^4
*d^4*e)*x^2

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 17473 vs. \(2 (184) = 368\).

Time = 1.58 (sec) , antiderivative size = 17473, normalized size of antiderivative = 68.79 \[ \int (a+b x) (d+e x)^5 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\text {Too large to display} \]

[In]

integrate((b*x+a)*(e*x+d)**5*(b**2*x**2+2*a*b*x+a**2)**(3/2),x)

[Out]

Piecewise((sqrt(a**2 + 2*a*b*x + b**2*x**2)*(b**3*e**5*x**9/10 + x**8*(31*a*b**4*e**5/10 + 5*b**5*d*e**4)/(9*b
**2) + x**7*(91*a**2*b**3*e**5/10 + 25*a*b**4*d*e**4 - 17*a*(31*a*b**4*e**5/10 + 5*b**5*d*e**4)/(9*b) + 10*b**
5*d**2*e**3)/(8*b**2) + x**6*(10*a**3*b**2*e**5 + 50*a**2*b**3*d*e**4 - 8*a**2*(31*a*b**4*e**5/10 + 5*b**5*d*e
**4)/(9*b**2) + 50*a*b**4*d**2*e**3 - 15*a*(91*a**2*b**3*e**5/10 + 25*a*b**4*d*e**4 - 17*a*(31*a*b**4*e**5/10
+ 5*b**5*d*e**4)/(9*b) + 10*b**5*d**2*e**3)/(8*b) + 10*b**5*d**3*e**2)/(7*b**2) + x**5*(5*a**4*b*e**5 + 50*a**
3*b**2*d*e**4 + 100*a**2*b**3*d**2*e**3 - 7*a**2*(91*a**2*b**3*e**5/10 + 25*a*b**4*d*e**4 - 17*a*(31*a*b**4*e*
*5/10 + 5*b**5*d*e**4)/(9*b) + 10*b**5*d**2*e**3)/(8*b**2) + 50*a*b**4*d**3*e**2 - 13*a*(10*a**3*b**2*e**5 + 5
0*a**2*b**3*d*e**4 - 8*a**2*(31*a*b**4*e**5/10 + 5*b**5*d*e**4)/(9*b**2) + 50*a*b**4*d**2*e**3 - 15*a*(91*a**2
*b**3*e**5/10 + 25*a*b**4*d*e**4 - 17*a*(31*a*b**4*e**5/10 + 5*b**5*d*e**4)/(9*b) + 10*b**5*d**2*e**3)/(8*b) +
10*b**5*d**3*e**2)/(7*b) + 5*b**5*d**4*e)/(6*b**2) + x**4*(a**5*e**5 + 25*a**4*b*d*e**4 + 100*a**3*b**2*d**2*
e**3 + 100*a**2*b**3*d**3*e**2 - 6*a**2*(10*a**3*b**2*e**5 + 50*a**2*b**3*d*e**4 - 8*a**2*(31*a*b**4*e**5/10 +
5*b**5*d*e**4)/(9*b**2) + 50*a*b**4*d**2*e**3 - 15*a*(91*a**2*b**3*e**5/10 + 25*a*b**4*d*e**4 - 17*a*(31*a*b*
*4*e**5/10 + 5*b**5*d*e**4)/(9*b) + 10*b**5*d**2*e**3)/(8*b) + 10*b**5*d**3*e**2)/(7*b**2) + 25*a*b**4*d**4*e
- 11*a*(5*a**4*b*e**5 + 50*a**3*b**2*d*e**4 + 100*a**2*b**3*d**2*e**3 - 7*a**2*(91*a**2*b**3*e**5/10 + 25*a*b*
*4*d*e**4 - 17*a*(31*a*b**4*e**5/10 + 5*b**5*d*e**4)/(9*b) + 10*b**5*d**2*e**3)/(8*b**2) + 50*a*b**4*d**3*e**2
- 13*a*(10*a**3*b**2*e**5 + 50*a**2*b**3*d*e**4 - 8*a**2*(31*a*b**4*e**5/10 + 5*b**5*d*e**4)/(9*b**2) + 50*a*
b**4*d**2*e**3 - 15*a*(91*a**2*b**3*e**5/10 + 25*a*b**4*d*e**4 - 17*a*(31*a*b**4*e**5/10 + 5*b**5*d*e**4)/(9*b
) + 10*b**5*d**2*e**3)/(8*b) + 10*b**5*d**3*e**2)/(7*b) + 5*b**5*d**4*e)/(6*b) + b**5*d**5)/(5*b**2) + x**3*(5
*a**5*d*e**4 + 50*a**4*b*d**2*e**3 + 100*a**3*b**2*d**3*e**2 + 50*a**2*b**3*d**4*e - 5*a**2*(5*a**4*b*e**5 + 5
0*a**3*b**2*d*e**4 + 100*a**2*b**3*d**2*e**3 - 7*a**2*(91*a**2*b**3*e**5/10 + 25*a*b**4*d*e**4 - 17*a*(31*a*b*
*4*e**5/10 + 5*b**5*d*e**4)/(9*b) + 10*b**5*d**2*e**3)/(8*b**2) + 50*a*b**4*d**3*e**2 - 13*a*(10*a**3*b**2*e**
5 + 50*a**2*b**3*d*e**4 - 8*a**2*(31*a*b**4*e**5/10 + 5*b**5*d*e**4)/(9*b**2) + 50*a*b**4*d**2*e**3 - 15*a*(91
*a**2*b**3*e**5/10 + 25*a*b**4*d*e**4 - 17*a*(31*a*b**4*e**5/10 + 5*b**5*d*e**4)/(9*b) + 10*b**5*d**2*e**3)/(8
*b) + 10*b**5*d**3*e**2)/(7*b) + 5*b**5*d**4*e)/(6*b**2) + 5*a*b**4*d**5 - 9*a*(a**5*e**5 + 25*a**4*b*d*e**4 +
100*a**3*b**2*d**2*e**3 + 100*a**2*b**3*d**3*e**2 - 6*a**2*(10*a**3*b**2*e**5 + 50*a**2*b**3*d*e**4 - 8*a**2*
(31*a*b**4*e**5/10 + 5*b**5*d*e**4)/(9*b**2) + 50*a*b**4*d**2*e**3 - 15*a*(91*a**2*b**3*e**5/10 + 25*a*b**4*d*
e**4 - 17*a*(31*a*b**4*e**5/10 + 5*b**5*d*e**4)/(9*b) + 10*b**5*d**2*e**3)/(8*b) + 10*b**5*d**3*e**2)/(7*b**2)
+ 25*a*b**4*d**4*e - 11*a*(5*a**4*b*e**5 + 50*a**3*b**2*d*e**4 + 100*a**2*b**3*d**2*e**3 - 7*a**2*(91*a**2*b*
*3*e**5/10 + 25*a*b**4*d*e**4 - 17*a*(31*a*b**4*e**5/10 + 5*b**5*d*e**4)/(9*b) + 10*b**5*d**2*e**3)/(8*b**2) +
50*a*b**4*d**3*e**2 - 13*a*(10*a**3*b**2*e**5 + 50*a**2*b**3*d*e**4 - 8*a**2*(31*a*b**4*e**5/10 + 5*b**5*d*e*
*4)/(9*b**2) + 50*a*b**4*d**2*e**3 - 15*a*(91*a**2*b**3*e**5/10 + 25*a*b**4*d*e**4 - 17*a*(31*a*b**4*e**5/10 +
5*b**5*d*e**4)/(9*b) + 10*b**5*d**2*e**3)/(8*b) + 10*b**5*d**3*e**2)/(7*b) + 5*b**5*d**4*e)/(6*b) + b**5*d**5
)/(5*b))/(4*b**2) + x**2*(10*a**5*d**2*e**3 + 50*a**4*b*d**3*e**2 + 50*a**3*b**2*d**4*e + 10*a**2*b**3*d**5 -
4*a**2*(a**5*e**5 + 25*a**4*b*d*e**4 + 100*a**3*b**2*d**2*e**3 + 100*a**2*b**3*d**3*e**2 - 6*a**2*(10*a**3*b**
2*e**5 + 50*a**2*b**3*d*e**4 - 8*a**2*(31*a*b**4*e**5/10 + 5*b**5*d*e**4)/(9*b**2) + 50*a*b**4*d**2*e**3 - 15*
a*(91*a**2*b**3*e**5/10 + 25*a*b**4*d*e**4 - 17*a*(31*a*b**4*e**5/10 + 5*b**5*d*e**4)/(9*b) + 10*b**5*d**2*e**
3)/(8*b) + 10*b**5*d**3*e**2)/(7*b**2) + 25*a*b**4*d**4*e - 11*a*(5*a**4*b*e**5 + 50*a**3*b**2*d*e**4 + 100*a*
*2*b**3*d**2*e**3 - 7*a**2*(91*a**2*b**3*e**5/10 + 25*a*b**4*d*e**4 - 17*a*(31*a*b**4*e**5/10 + 5*b**5*d*e**4)
/(9*b) + 10*b**5*d**2*e**3)/(8*b**2) + 50*a*b**4*d**3*e**2 - 13*a*(10*a**3*b**2*e**5 + 50*a**2*b**3*d*e**4 - 8
*a**2*(31*a*b**4*e**5/10 + 5*b**5*d*e**4)/(9*b**2) + 50*a*b**4*d**2*e**3 - 15*a*(91*a**2*b**3*e**5/10 + 25*a*b
**4*d*e**4 - 17*a*(31*a*b**4*e**5/10 + 5*b**5*d*e**4)/(9*b) + 10*b**5*d**2*e**3)/(8*b) + 10*b**5*d**3*e**2)/(7
*b) + 5*b**5*d**4*e)/(6*b) + b**5*d**5)/(5*b**2) - 7*a*(5*a**5*d*e**4 + 50*a**4*b*d**2*e**3 + 100*a**3*b**2*d*
*3*e**2 + 50*a**2*b**3*d**4*e - 5*a**2*(5*a**4*b*e**5 + 50*a**3*b**2*d*e**4 + 100*a**2*b**3*d**2*e**3 - 7*a**2
*(91*a**2*b**3*e**5/10 + 25*a*b**4*d*e**4 - 17*a*(31*a*b**4*e**5/10 + 5*b**5*d*e**4)/(9*b) + 10*b**5*d**2*e**3
)/(8*b**2) + 50*a*b**4*d**3*e**2 - 13*a*(10*a**3*b**2*e**5 + 50*a**2*b**3*d*e**4 - 8*a**2*(31*a*b**4*e**5/10 +
5*b**5*d*e**4)/(9*b**2) + 50*a*b**4*d**2*e**3 - 15*a*(91*a**2*b**3*e**5/10 + 25*a*b**4*d*e**4 - 17*a*(31*a*b*
*4*e**5/10 + 5*b**5*d*e**4)/(9*b) + 10*b**5*d**2*e**3)/(8*b) + 10*b**5*d**3*e**2)/(7*b) + 5*b**5*d**4*e)/(6*b*
*2) + 5*a*b**4*d**5 - 9*a*(a**5*e**5 + 25*a**4*b*d*e**4 + 100*a**3*b**2*d**2*e**3 + 100*a**2*b**3*d**3*e**2 -
6*a**2*(10*a**3*b**2*e**5 + 50*a**2*b**3*d*e**4 - 8*a**2*(31*a*b**4*e**5/10 + 5*b**5*d*e**4)/(9*b**2) + 50*a*b
**4*d**2*e**3 - 15*a*(91*a**2*b**3*e**5/10 + 25*a*b**4*d*e**4 - 17*a*(31*a*b**4*e**5/10 + 5*b**5*d*e**4)/(9*b)
+ 10*b**5*d**2*e**3)/(8*b) + 10*b**5*d**3*e**2)/(7*b**2) + 25*a*b**4*d**4*e - 11*a*(5*a**4*b*e**5 + 50*a**3*b
**2*d*e**4 + 100*a**2*b**3*d**2*e**3 - 7*a**2*(91*a**2*b**3*e**5/10 + 25*a*b**4*d*e**4 - 17*a*(31*a*b**4*e**5/
10 + 5*b**5*d*e**4)/(9*b) + 10*b**5*d**2*e**3)/(8*b**2) + 50*a*b**4*d**3*e**2 - 13*a*(10*a**3*b**2*e**5 + 50*a
**2*b**3*d*e**4 - 8*a**2*(31*a*b**4*e**5/10 + 5*b**5*d*e**4)/(9*b**2) + 50*a*b**4*d**2*e**3 - 15*a*(91*a**2*b*
*3*e**5/10 + 25*a*b**4*d*e**4 - 17*a*(31*a*b**4*e**5/10 + 5*b**5*d*e**4)/(9*b) + 10*b**5*d**2*e**3)/(8*b) + 10
*b**5*d**3*e**2)/(7*b) + 5*b**5*d**4*e)/(6*b) + b**5*d**5)/(5*b))/(4*b))/(3*b**2) + x*(10*a**5*d**3*e**2 + 25*
a**4*b*d**4*e + 10*a**3*b**2*d**5 - 3*a**2*(5*a**5*d*e**4 + 50*a**4*b*d**2*e**3 + 100*a**3*b**2*d**3*e**2 + 50
*a**2*b**3*d**4*e - 5*a**2*(5*a**4*b*e**5 + 50*a**3*b**2*d*e**4 + 100*a**2*b**3*d**2*e**3 - 7*a**2*(91*a**2*b*
*3*e**5/10 + 25*a*b**4*d*e**4 - 17*a*(31*a*b**4*e**5/10 + 5*b**5*d*e**4)/(9*b) + 10*b**5*d**2*e**3)/(8*b**2) +
50*a*b**4*d**3*e**2 - 13*a*(10*a**3*b**2*e**5 + 50*a**2*b**3*d*e**4 - 8*a**2*(31*a*b**4*e**5/10 + 5*b**5*d*e*
*4)/(9*b**2) + 50*a*b**4*d**2*e**3 - 15*a*(91*a**2*b**3*e**5/10 + 25*a*b**4*d*e**4 - 17*a*(31*a*b**4*e**5/10 +
5*b**5*d*e**4)/(9*b) + 10*b**5*d**2*e**3)/(8*b) + 10*b**5*d**3*e**2)/(7*b) + 5*b**5*d**4*e)/(6*b**2) + 5*a*b*
*4*d**5 - 9*a*(a**5*e**5 + 25*a**4*b*d*e**4 + 100*a**3*b**2*d**2*e**3 + 100*a**2*b**3*d**3*e**2 - 6*a**2*(10*a
**3*b**2*e**5 + 50*a**2*b**3*d*e**4 - 8*a**2*(31*a*b**4*e**5/10 + 5*b**5*d*e**4)/(9*b**2) + 50*a*b**4*d**2*e**
3 - 15*a*(91*a**2*b**3*e**5/10 + 25*a*b**4*d*e**4 - 17*a*(31*a*b**4*e**5/10 + 5*b**5*d*e**4)/(9*b) + 10*b**5*d
**2*e**3)/(8*b) + 10*b**5*d**3*e**2)/(7*b**2) + 25*a*b**4*d**4*e - 11*a*(5*a**4*b*e**5 + 50*a**3*b**2*d*e**4 +
100*a**2*b**3*d**2*e**3 - 7*a**2*(91*a**2*b**3*e**5/10 + 25*a*b**4*d*e**4 - 17*a*(31*a*b**4*e**5/10 + 5*b**5*
d*e**4)/(9*b) + 10*b**5*d**2*e**3)/(8*b**2) + 50*a*b**4*d**3*e**2 - 13*a*(10*a**3*b**2*e**5 + 50*a**2*b**3*d*e
**4 - 8*a**2*(31*a*b**4*e**5/10 + 5*b**5*d*e**4)/(9*b**2) + 50*a*b**4*d**2*e**3 - 15*a*(91*a**2*b**3*e**5/10 +
25*a*b**4*d*e**4 - 17*a*(31*a*b**4*e**5/10 + 5*b**5*d*e**4)/(9*b) + 10*b**5*d**2*e**3)/(8*b) + 10*b**5*d**3*e
**2)/(7*b) + 5*b**5*d**4*e)/(6*b) + b**5*d**5)/(5*b))/(4*b**2) - 5*a*(10*a**5*d**2*e**3 + 50*a**4*b*d**3*e**2
+ 50*a**3*b**2*d**4*e + 10*a**2*b**3*d**5 - 4*a**2*(a**5*e**5 + 25*a**4*b*d*e**4 + 100*a**3*b**2*d**2*e**3 + 1
00*a**2*b**3*d**3*e**2 - 6*a**2*(10*a**3*b**2*e**5 + 50*a**2*b**3*d*e**4 - 8*a**2*(31*a*b**4*e**5/10 + 5*b**5*
d*e**4)/(9*b**2) + 50*a*b**4*d**2*e**3 - 15*a*(91*a**2*b**3*e**5/10 + 25*a*b**4*d*e**4 - 17*a*(31*a*b**4*e**5/
10 + 5*b**5*d*e**4)/(9*b) + 10*b**5*d**2*e**3)/(8*b) + 10*b**5*d**3*e**2)/(7*b**2) + 25*a*b**4*d**4*e - 11*a*(
5*a**4*b*e**5 + 50*a**3*b**2*d*e**4 + 100*a**2*b**3*d**2*e**3 - 7*a**2*(91*a**2*b**3*e**5/10 + 25*a*b**4*d*e**
4 - 17*a*(31*a*b**4*e**5/10 + 5*b**5*d*e**4)/(9*b) + 10*b**5*d**2*e**3)/(8*b**2) + 50*a*b**4*d**3*e**2 - 13*a*
(10*a**3*b**2*e**5 + 50*a**2*b**3*d*e**4 - 8*a**2*(31*a*b**4*e**5/10 + 5*b**5*d*e**4)/(9*b**2) + 50*a*b**4*d**
2*e**3 - 15*a*(91*a**2*b**3*e**5/10 + 25*a*b**4*d*e**4 - 17*a*(31*a*b**4*e**5/10 + 5*b**5*d*e**4)/(9*b) + 10*b
**5*d**2*e**3)/(8*b) + 10*b**5*d**3*e**2)/(7*b) + 5*b**5*d**4*e)/(6*b) + b**5*d**5)/(5*b**2) - 7*a*(5*a**5*d*e
**4 + 50*a**4*b*d**2*e**3 + 100*a**3*b**2*d**3*e**2 + 50*a**2*b**3*d**4*e - 5*a**2*(5*a**4*b*e**5 + 50*a**3*b*
*2*d*e**4 + 100*a**2*b**3*d**2*e**3 - 7*a**2*(91*a**2*b**3*e**5/10 + 25*a*b**4*d*e**4 - 17*a*(31*a*b**4*e**5/1
0 + 5*b**5*d*e**4)/(9*b) + 10*b**5*d**2*e**3)/(8*b**2) + 50*a*b**4*d**3*e**2 - 13*a*(10*a**3*b**2*e**5 + 50*a*
*2*b**3*d*e**4 - 8*a**2*(31*a*b**4*e**5/10 + 5*b**5*d*e**4)/(9*b**2) + 50*a*b**4*d**2*e**3 - 15*a*(91*a**2*b**
3*e**5/10 + 25*a*b**4*d*e**4 - 17*a*(31*a*b**4*e**5/10 + 5*b**5*d*e**4)/(9*b) + 10*b**5*d**2*e**3)/(8*b) + 10*
b**5*d**3*e**2)/(7*b) + 5*b**5*d**4*e)/(6*b**2) + 5*a*b**4*d**5 - 9*a*(a**5*e**5 + 25*a**4*b*d*e**4 + 100*a**3
*b**2*d**2*e**3 + 100*a**2*b**3*d**3*e**2 - 6*a**2*(10*a**3*b**2*e**5 + 50*a**2*b**3*d*e**4 - 8*a**2*(31*a*b**
4*e**5/10 + 5*b**5*d*e**4)/(9*b**2) + 50*a*b**4*d**2*e**3 - 15*a*(91*a**2*b**3*e**5/10 + 25*a*b**4*d*e**4 - 17
*a*(31*a*b**4*e**5/10 + 5*b**5*d*e**4)/(9*b) + 10*b**5*d**2*e**3)/(8*b) + 10*b**5*d**3*e**2)/(7*b**2) + 25*a*b
**4*d**4*e - 11*a*(5*a**4*b*e**5 + 50*a**3*b**2*d*e**4 + 100*a**2*b**3*d**2*e**3 - 7*a**2*(91*a**2*b**3*e**5/1
0 + 25*a*b**4*d*e**4 - 17*a*(31*a*b**4*e**5/10 + 5*b**5*d*e**4)/(9*b) + 10*b**5*d**2*e**3)/(8*b**2) + 50*a*b**
4*d**3*e**2 - 13*a*(10*a**3*b**2*e**5 + 50*a**2*b**3*d*e**4 - 8*a**2*(31*a*b**4*e**5/10 + 5*b**5*d*e**4)/(9*b*
*2) + 50*a*b**4*d**2*e**3 - 15*a*(91*a**2*b**3*e**5/10 + 25*a*b**4*d*e**4 - 17*a*(31*a*b**4*e**5/10 + 5*b**5*d
*e**4)/(9*b) + 10*b**5*d**2*e**3)/(8*b) + 10*b**5*d**3*e**2)/(7*b) + 5*b**5*d**4*e)/(6*b) + b**5*d**5)/(5*b))/
(4*b))/(3*b))/(2*b**2) + (5*a**5*d**4*e + 5*a**4*b*d**5 - 2*a**2*(10*a**5*d**2*e**3 + 50*a**4*b*d**3*e**2 + 50
*a**3*b**2*d**4*e + 10*a**2*b**3*d**5 - 4*a**2*(a**5*e**5 + 25*a**4*b*d*e**4 + 100*a**3*b**2*d**2*e**3 + 100*a
**2*b**3*d**3*e**2 - 6*a**2*(10*a**3*b**2*e**5 + 50*a**2*b**3*d*e**4 - 8*a**2*(31*a*b**4*e**5/10 + 5*b**5*d*e*
*4)/(9*b**2) + 50*a*b**4*d**2*e**3 - 15*a*(91*a**2*b**3*e**5/10 + 25*a*b**4*d*e**4 - 17*a*(31*a*b**4*e**5/10 +
5*b**5*d*e**4)/(9*b) + 10*b**5*d**2*e**3)/(8*b) + 10*b**5*d**3*e**2)/(7*b**2) + 25*a*b**4*d**4*e - 11*a*(5*a*
*4*b*e**5 + 50*a**3*b**2*d*e**4 + 100*a**2*b**3*d**2*e**3 - 7*a**2*(91*a**2*b**3*e**5/10 + 25*a*b**4*d*e**4 -
17*a*(31*a*b**4*e**5/10 + 5*b**5*d*e**4)/(9*b) + 10*b**5*d**2*e**3)/(8*b**2) + 50*a*b**4*d**3*e**2 - 13*a*(10*
a**3*b**2*e**5 + 50*a**2*b**3*d*e**4 - 8*a**2*(31*a*b**4*e**5/10 + 5*b**5*d*e**4)/(9*b**2) + 50*a*b**4*d**2*e*
*3 - 15*a*(91*a**2*b**3*e**5/10 + 25*a*b**4*d*e**4 - 17*a*(31*a*b**4*e**5/10 + 5*b**5*d*e**4)/(9*b) + 10*b**5*
d**2*e**3)/(8*b) + 10*b**5*d**3*e**2)/(7*b) + 5*b**5*d**4*e)/(6*b) + b**5*d**5)/(5*b**2) - 7*a*(5*a**5*d*e**4
+ 50*a**4*b*d**2*e**3 + 100*a**3*b**2*d**3*e**2 + 50*a**2*b**3*d**4*e - 5*a**2*(5*a**4*b*e**5 + 50*a**3*b**2*d
*e**4 + 100*a**2*b**3*d**2*e**3 - 7*a**2*(91*a**2*b**3*e**5/10 + 25*a*b**4*d*e**4 - 17*a*(31*a*b**4*e**5/10 +
5*b**5*d*e**4)/(9*b) + 10*b**5*d**2*e**3)/(8*b**2) + 50*a*b**4*d**3*e**2 - 13*a*(10*a**3*b**2*e**5 + 50*a**2*b
**3*d*e**4 - 8*a**2*(31*a*b**4*e**5/10 + 5*b**5*d*e**4)/(9*b**2) + 50*a*b**4*d**2*e**3 - 15*a*(91*a**2*b**3*e*
*5/10 + 25*a*b**4*d*e**4 - 17*a*(31*a*b**4*e**5/10 + 5*b**5*d*e**4)/(9*b) + 10*b**5*d**2*e**3)/(8*b) + 10*b**5
*d**3*e**2)/(7*b) + 5*b**5*d**4*e)/(6*b**2) + 5*a*b**4*d**5 - 9*a*(a**5*e**5 + 25*a**4*b*d*e**4 + 100*a**3*b**
2*d**2*e**3 + 100*a**2*b**3*d**3*e**2 - 6*a**2*(10*a**3*b**2*e**5 + 50*a**2*b**3*d*e**4 - 8*a**2*(31*a*b**4*e*
*5/10 + 5*b**5*d*e**4)/(9*b**2) + 50*a*b**4*d**2*e**3 - 15*a*(91*a**2*b**3*e**5/10 + 25*a*b**4*d*e**4 - 17*a*(
31*a*b**4*e**5/10 + 5*b**5*d*e**4)/(9*b) + 10*b**5*d**2*e**3)/(8*b) + 10*b**5*d**3*e**2)/(7*b**2) + 25*a*b**4*
d**4*e - 11*a*(5*a**4*b*e**5 + 50*a**3*b**2*d*e**4 + 100*a**2*b**3*d**2*e**3 - 7*a**2*(91*a**2*b**3*e**5/10 +
25*a*b**4*d*e**4 - 17*a*(31*a*b**4*e**5/10 + 5*b**5*d*e**4)/(9*b) + 10*b**5*d**2*e**3)/(8*b**2) + 50*a*b**4*d*
*3*e**2 - 13*a*(10*a**3*b**2*e**5 + 50*a**2*b**3*d*e**4 - 8*a**2*(31*a*b**4*e**5/10 + 5*b**5*d*e**4)/(9*b**2)
+ 50*a*b**4*d**2*e**3 - 15*a*(91*a**2*b**3*e**5/10 + 25*a*b**4*d*e**4 - 17*a*(31*a*b**4*e**5/10 + 5*b**5*d*e**
4)/(9*b) + 10*b**5*d**2*e**3)/(8*b) + 10*b**5*d**3*e**2)/(7*b) + 5*b**5*d**4*e)/(6*b) + b**5*d**5)/(5*b))/(4*b
))/(3*b**2) - 3*a*(10*a**5*d**3*e**2 + 25*a**4*b*d**4*e + 10*a**3*b**2*d**5 - 3*a**2*(5*a**5*d*e**4 + 50*a**4*
b*d**2*e**3 + 100*a**3*b**2*d**3*e**2 + 50*a**2*b**3*d**4*e - 5*a**2*(5*a**4*b*e**5 + 50*a**3*b**2*d*e**4 + 10
0*a**2*b**3*d**2*e**3 - 7*a**2*(91*a**2*b**3*e**5/10 + 25*a*b**4*d*e**4 - 17*a*(31*a*b**4*e**5/10 + 5*b**5*d*e
**4)/(9*b) + 10*b**5*d**2*e**3)/(8*b**2) + 50*a*b**4*d**3*e**2 - 13*a*(10*a**3*b**2*e**5 + 50*a**2*b**3*d*e**4
- 8*a**2*(31*a*b**4*e**5/10 + 5*b**5*d*e**4)/(9*b**2) + 50*a*b**4*d**2*e**3 - 15*a*(91*a**2*b**3*e**5/10 + 25
*a*b**4*d*e**4 - 17*a*(31*a*b**4*e**5/10 + 5*b**5*d*e**4)/(9*b) + 10*b**5*d**2*e**3)/(8*b) + 10*b**5*d**3*e**2
)/(7*b) + 5*b**5*d**4*e)/(6*b**2) + 5*a*b**4*d**5 - 9*a*(a**5*e**5 + 25*a**4*b*d*e**4 + 100*a**3*b**2*d**2*e**
3 + 100*a**2*b**3*d**3*e**2 - 6*a**2*(10*a**3*b**2*e**5 + 50*a**2*b**3*d*e**4 - 8*a**2*(31*a*b**4*e**5/10 + 5*
b**5*d*e**4)/(9*b**2) + 50*a*b**4*d**2*e**3 - 15*a*(91*a**2*b**3*e**5/10 + 25*a*b**4*d*e**4 - 17*a*(31*a*b**4*
e**5/10 + 5*b**5*d*e**4)/(9*b) + 10*b**5*d**2*e**3)/(8*b) + 10*b**5*d**3*e**2)/(7*b**2) + 25*a*b**4*d**4*e - 1
1*a*(5*a**4*b*e**5 + 50*a**3*b**2*d*e**4 + 100*a**2*b**3*d**2*e**3 - 7*a**2*(91*a**2*b**3*e**5/10 + 25*a*b**4*
d*e**4 - 17*a*(31*a*b**4*e**5/10 + 5*b**5*d*e**4)/(9*b) + 10*b**5*d**2*e**3)/(8*b**2) + 50*a*b**4*d**3*e**2 -
13*a*(10*a**3*b**2*e**5 + 50*a**2*b**3*d*e**4 - 8*a**2*(31*a*b**4*e**5/10 + 5*b**5*d*e**4)/(9*b**2) + 50*a*b**
4*d**2*e**3 - 15*a*(91*a**2*b**3*e**5/10 + 25*a*b**4*d*e**4 - 17*a*(31*a*b**4*e**5/10 + 5*b**5*d*e**4)/(9*b) +
10*b**5*d**2*e**3)/(8*b) + 10*b**5*d**3*e**2)/(7*b) + 5*b**5*d**4*e)/(6*b) + b**5*d**5)/(5*b))/(4*b**2) - 5*a
*(10*a**5*d**2*e**3 + 50*a**4*b*d**3*e**2 + 50*a**3*b**2*d**4*e + 10*a**2*b**3*d**5 - 4*a**2*(a**5*e**5 + 25*a
**4*b*d*e**4 + 100*a**3*b**2*d**2*e**3 + 100*a**2*b**3*d**3*e**2 - 6*a**2*(10*a**3*b**2*e**5 + 50*a**2*b**3*d*
e**4 - 8*a**2*(31*a*b**4*e**5/10 + 5*b**5*d*e**4)/(9*b**2) + 50*a*b**4*d**2*e**3 - 15*a*(91*a**2*b**3*e**5/10
+ 25*a*b**4*d*e**4 - 17*a*(31*a*b**4*e**5/10 + 5*b**5*d*e**4)/(9*b) + 10*b**5*d**2*e**3)/(8*b) + 10*b**5*d**3*
e**2)/(7*b**2) + 25*a*b**4*d**4*e - 11*a*(5*a**4*b*e**5 + 50*a**3*b**2*d*e**4 + 100*a**2*b**3*d**2*e**3 - 7*a*
*2*(91*a**2*b**3*e**5/10 + 25*a*b**4*d*e**4 - 17*a*(31*a*b**4*e**5/10 + 5*b**5*d*e**4)/(9*b) + 10*b**5*d**2*e*
*3)/(8*b**2) + 50*a*b**4*d**3*e**2 - 13*a*(10*a**3*b**2*e**5 + 50*a**2*b**3*d*e**4 - 8*a**2*(31*a*b**4*e**5/10
+ 5*b**5*d*e**4)/(9*b**2) + 50*a*b**4*d**2*e**3 - 15*a*(91*a**2*b**3*e**5/10 + 25*a*b**4*d*e**4 - 17*a*(31*a*
b**4*e**5/10 + 5*b**5*d*e**4)/(9*b) + 10*b**5*d**2*e**3)/(8*b) + 10*b**5*d**3*e**2)/(7*b) + 5*b**5*d**4*e)/(6*
b) + b**5*d**5)/(5*b**2) - 7*a*(5*a**5*d*e**4 + 50*a**4*b*d**2*e**3 + 100*a**3*b**2*d**3*e**2 + 50*a**2*b**3*d
**4*e - 5*a**2*(5*a**4*b*e**5 + 50*a**3*b**2*d*e**4 + 100*a**2*b**3*d**2*e**3 - 7*a**2*(91*a**2*b**3*e**5/10 +
25*a*b**4*d*e**4 - 17*a*(31*a*b**4*e**5/10 + 5*b**5*d*e**4)/(9*b) + 10*b**5*d**2*e**3)/(8*b**2) + 50*a*b**4*d
**3*e**2 - 13*a*(10*a**3*b**2*e**5 + 50*a**2*b**3*d*e**4 - 8*a**2*(31*a*b**4*e**5/10 + 5*b**5*d*e**4)/(9*b**2)
+ 50*a*b**4*d**2*e**3 - 15*a*(91*a**2*b**3*e**5/10 + 25*a*b**4*d*e**4 - 17*a*(31*a*b**4*e**5/10 + 5*b**5*d*e*
*4)/(9*b) + 10*b**5*d**2*e**3)/(8*b) + 10*b**5*d**3*e**2)/(7*b) + 5*b**5*d**4*e)/(6*b**2) + 5*a*b**4*d**5 - 9*
a*(a**5*e**5 + 25*a**4*b*d*e**4 + 100*a**3*b**2*d**2*e**3 + 100*a**2*b**3*d**3*e**2 - 6*a**2*(10*a**3*b**2*e**
5 + 50*a**2*b**3*d*e**4 - 8*a**2*(31*a*b**4*e**5/10 + 5*b**5*d*e**4)/(9*b**2) + 50*a*b**4*d**2*e**3 - 15*a*(91
*a**2*b**3*e**5/10 + 25*a*b**4*d*e**4 - 17*a*(31*a*b**4*e**5/10 + 5*b**5*d*e**4)/(9*b) + 10*b**5*d**2*e**3)/(8
*b) + 10*b**5*d**3*e**2)/(7*b**2) + 25*a*b**4*d**4*e - 11*a*(5*a**4*b*e**5 + 50*a**3*b**2*d*e**4 + 100*a**2*b*
*3*d**2*e**3 - 7*a**2*(91*a**2*b**3*e**5/10 + 25*a*b**4*d*e**4 - 17*a*(31*a*b**4*e**5/10 + 5*b**5*d*e**4)/(9*b
) + 10*b**5*d**2*e**3)/(8*b**2) + 50*a*b**4*d**3*e**2 - 13*a*(10*a**3*b**2*e**5 + 50*a**2*b**3*d*e**4 - 8*a**2
*(31*a*b**4*e**5/10 + 5*b**5*d*e**4)/(9*b**2) + 50*a*b**4*d**2*e**3 - 15*a*(91*a**2*b**3*e**5/10 + 25*a*b**4*d
*e**4 - 17*a*(31*a*b**4*e**5/10 + 5*b**5*d*e**4)/(9*b) + 10*b**5*d**2*e**3)/(8*b) + 10*b**5*d**3*e**2)/(7*b) +
5*b**5*d**4*e)/(6*b) + b**5*d**5)/(5*b))/(4*b))/(3*b))/(2*b))/b**2) + (a/b + x)*(a**5*d**5 - a**2*(10*a**5*d*
*3*e**2 + 25*a**4*b*d**4*e + 10*a**3*b**2*d**5 - 3*a**2*(5*a**5*d*e**4 + 50*a**4*b*d**2*e**3 + 100*a**3*b**2*d
**3*e**2 + 50*a**2*b**3*d**4*e - 5*a**2*(5*a**4*b*e**5 + 50*a**3*b**2*d*e**4 + 100*a**2*b**3*d**2*e**3 - 7*a**
2*(91*a**2*b**3*e**5/10 + 25*a*b**4*d*e**4 - 17*a*(31*a*b**4*e**5/10 + 5*b**5*d*e**4)/(9*b) + 10*b**5*d**2*e**
3)/(8*b**2) + 50*a*b**4*d**3*e**2 - 13*a*(10*a**3*b**2*e**5 + 50*a**2*b**3*d*e**4 - 8*a**2*(31*a*b**4*e**5/10
+ 5*b**5*d*e**4)/(9*b**2) + 50*a*b**4*d**2*e**3 - 15*a*(91*a**2*b**3*e**5/10 + 25*a*b**4*d*e**4 - 17*a*(31*a*b
**4*e**5/10 + 5*b**5*d*e**4)/(9*b) + 10*b**5*d**2*e**3)/(8*b) + 10*b**5*d**3*e**2)/(7*b) + 5*b**5*d**4*e)/(6*b
**2) + 5*a*b**4*d**5 - 9*a*(a**5*e**5 + 25*a**4*b*d*e**4 + 100*a**3*b**2*d**2*e**3 + 100*a**2*b**3*d**3*e**2 -
6*a**2*(10*a**3*b**2*e**5 + 50*a**2*b**3*d*e**4 - 8*a**2*(31*a*b**4*e**5/10 + 5*b**5*d*e**4)/(9*b**2) + 50*a*
b**4*d**2*e**3 - 15*a*(91*a**2*b**3*e**5/10 + 25*a*b**4*d*e**4 - 17*a*(31*a*b**4*e**5/10 + 5*b**5*d*e**4)/(9*b
) + 10*b**5*d**2*e**3)/(8*b) + 10*b**5*d**3*e**2)/(7*b**2) + 25*a*b**4*d**4*e - 11*a*(5*a**4*b*e**5 + 50*a**3*
b**2*d*e**4 + 100*a**2*b**3*d**2*e**3 - 7*a**2*(91*a**2*b**3*e**5/10 + 25*a*b**4*d*e**4 - 17*a*(31*a*b**4*e**5
/10 + 5*b**5*d*e**4)/(9*b) + 10*b**5*d**2*e**3)/(8*b**2) + 50*a*b**4*d**3*e**2 - 13*a*(10*a**3*b**2*e**5 + 50*
a**2*b**3*d*e**4 - 8*a**2*(31*a*b**4*e**5/10 + 5*b**5*d*e**4)/(9*b**2) + 50*a*b**4*d**2*e**3 - 15*a*(91*a**2*b
**3*e**5/10 + 25*a*b**4*d*e**4 - 17*a*(31*a*b**4*e**5/10 + 5*b**5*d*e**4)/(9*b) + 10*b**5*d**2*e**3)/(8*b) + 1
0*b**5*d**3*e**2)/(7*b) + 5*b**5*d**4*e)/(6*b) + b**5*d**5)/(5*b))/(4*b**2) - 5*a*(10*a**5*d**2*e**3 + 50*a**4
*b*d**3*e**2 + 50*a**3*b**2*d**4*e + 10*a**2*b**3*d**5 - 4*a**2*(a**5*e**5 + 25*a**4*b*d*e**4 + 100*a**3*b**2*
d**2*e**3 + 100*a**2*b**3*d**3*e**2 - 6*a**2*(10*a**3*b**2*e**5 + 50*a**2*b**3*d*e**4 - 8*a**2*(31*a*b**4*e**5
/10 + 5*b**5*d*e**4)/(9*b**2) + 50*a*b**4*d**2*e**3 - 15*a*(91*a**2*b**3*e**5/10 + 25*a*b**4*d*e**4 - 17*a*(31
*a*b**4*e**5/10 + 5*b**5*d*e**4)/(9*b) + 10*b**5*d**2*e**3)/(8*b) + 10*b**5*d**3*e**2)/(7*b**2) + 25*a*b**4*d*
*4*e - 11*a*(5*a**4*b*e**5 + 50*a**3*b**2*d*e**4 + 100*a**2*b**3*d**2*e**3 - 7*a**2*(91*a**2*b**3*e**5/10 + 25
*a*b**4*d*e**4 - 17*a*(31*a*b**4*e**5/10 + 5*b**5*d*e**4)/(9*b) + 10*b**5*d**2*e**3)/(8*b**2) + 50*a*b**4*d**3
*e**2 - 13*a*(10*a**3*b**2*e**5 + 50*a**2*b**3*d*e**4 - 8*a**2*(31*a*b**4*e**5/10 + 5*b**5*d*e**4)/(9*b**2) +
50*a*b**4*d**2*e**3 - 15*a*(91*a**2*b**3*e**5/10 + 25*a*b**4*d*e**4 - 17*a*(31*a*b**4*e**5/10 + 5*b**5*d*e**4)
/(9*b) + 10*b**5*d**2*e**3)/(8*b) + 10*b**5*d**3*e**2)/(7*b) + 5*b**5*d**4*e)/(6*b) + b**5*d**5)/(5*b**2) - 7*
a*(5*a**5*d*e**4 + 50*a**4*b*d**2*e**3 + 100*a**3*b**2*d**3*e**2 + 50*a**2*b**3*d**4*e - 5*a**2*(5*a**4*b*e**5
+ 50*a**3*b**2*d*e**4 + 100*a**2*b**3*d**2*e**3 - 7*a**2*(91*a**2*b**3*e**5/10 + 25*a*b**4*d*e**4 - 17*a*(31*
a*b**4*e**5/10 + 5*b**5*d*e**4)/(9*b) + 10*b**5*d**2*e**3)/(8*b**2) + 50*a*b**4*d**3*e**2 - 13*a*(10*a**3*b**2
*e**5 + 50*a**2*b**3*d*e**4 - 8*a**2*(31*a*b**4*e**5/10 + 5*b**5*d*e**4)/(9*b**2) + 50*a*b**4*d**2*e**3 - 15*a
*(91*a**2*b**3*e**5/10 + 25*a*b**4*d*e**4 - 17*a*(31*a*b**4*e**5/10 + 5*b**5*d*e**4)/(9*b) + 10*b**5*d**2*e**3
)/(8*b) + 10*b**5*d**3*e**2)/(7*b) + 5*b**5*d**4*e)/(6*b**2) + 5*a*b**4*d**5 - 9*a*(a**5*e**5 + 25*a**4*b*d*e*
*4 + 100*a**3*b**2*d**2*e**3 + 100*a**2*b**3*d**3*e**2 - 6*a**2*(10*a**3*b**2*e**5 + 50*a**2*b**3*d*e**4 - 8*a
**2*(31*a*b**4*e**5/10 + 5*b**5*d*e**4)/(9*b**2) + 50*a*b**4*d**2*e**3 - 15*a*(91*a**2*b**3*e**5/10 + 25*a*b**
4*d*e**4 - 17*a*(31*a*b**4*e**5/10 + 5*b**5*d*e**4)/(9*b) + 10*b**5*d**2*e**3)/(8*b) + 10*b**5*d**3*e**2)/(7*b
**2) + 25*a*b**4*d**4*e - 11*a*(5*a**4*b*e**5 + 50*a**3*b**2*d*e**4 + 100*a**2*b**3*d**2*e**3 - 7*a**2*(91*a**
2*b**3*e**5/10 + 25*a*b**4*d*e**4 - 17*a*(31*a*b**4*e**5/10 + 5*b**5*d*e**4)/(9*b) + 10*b**5*d**2*e**3)/(8*b**
2) + 50*a*b**4*d**3*e**2 - 13*a*(10*a**3*b**2*e**5 + 50*a**2*b**3*d*e**4 - 8*a**2*(31*a*b**4*e**5/10 + 5*b**5*
d*e**4)/(9*b**2) + 50*a*b**4*d**2*e**3 - 15*a*(91*a**2*b**3*e**5/10 + 25*a*b**4*d*e**4 - 17*a*(31*a*b**4*e**5/
10 + 5*b**5*d*e**4)/(9*b) + 10*b**5*d**2*e**3)/(8*b) + 10*b**5*d**3*e**2)/(7*b) + 5*b**5*d**4*e)/(6*b) + b**5*
d**5)/(5*b))/(4*b))/(3*b))/(2*b**2) - a*(5*a**5*d**4*e + 5*a**4*b*d**5 - 2*a**2*(10*a**5*d**2*e**3 + 50*a**4*b
*d**3*e**2 + 50*a**3*b**2*d**4*e + 10*a**2*b**3*d**5 - 4*a**2*(a**5*e**5 + 25*a**4*b*d*e**4 + 100*a**3*b**2*d*
*2*e**3 + 100*a**2*b**3*d**3*e**2 - 6*a**2*(10*a**3*b**2*e**5 + 50*a**2*b**3*d*e**4 - 8*a**2*(31*a*b**4*e**5/1
0 + 5*b**5*d*e**4)/(9*b**2) + 50*a*b**4*d**2*e**3 - 15*a*(91*a**2*b**3*e**5/10 + 25*a*b**4*d*e**4 - 17*a*(31*a
*b**4*e**5/10 + 5*b**5*d*e**4)/(9*b) + 10*b**5*d**2*e**3)/(8*b) + 10*b**5*d**3*e**2)/(7*b**2) + 25*a*b**4*d**4
*e - 11*a*(5*a**4*b*e**5 + 50*a**3*b**2*d*e**4 + 100*a**2*b**3*d**2*e**3 - 7*a**2*(91*a**2*b**3*e**5/10 + 25*a
*b**4*d*e**4 - 17*a*(31*a*b**4*e**5/10 + 5*b**5*d*e**4)/(9*b) + 10*b**5*d**2*e**3)/(8*b**2) + 50*a*b**4*d**3*e
**2 - 13*a*(10*a**3*b**2*e**5 + 50*a**2*b**3*d*e**4 - 8*a**2*(31*a*b**4*e**5/10 + 5*b**5*d*e**4)/(9*b**2) + 50
*a*b**4*d**2*e**3 - 15*a*(91*a**2*b**3*e**5/10 + 25*a*b**4*d*e**4 - 17*a*(31*a*b**4*e**5/10 + 5*b**5*d*e**4)/(
9*b) + 10*b**5*d**2*e**3)/(8*b) + 10*b**5*d**3*e**2)/(7*b) + 5*b**5*d**4*e)/(6*b) + b**5*d**5)/(5*b**2) - 7*a*
(5*a**5*d*e**4 + 50*a**4*b*d**2*e**3 + 100*a**3*b**2*d**3*e**2 + 50*a**2*b**3*d**4*e - 5*a**2*(5*a**4*b*e**5 +
50*a**3*b**2*d*e**4 + 100*a**2*b**3*d**2*e**3 - 7*a**2*(91*a**2*b**3*e**5/10 + 25*a*b**4*d*e**4 - 17*a*(31*a*
b**4*e**5/10 + 5*b**5*d*e**4)/(9*b) + 10*b**5*d**2*e**3)/(8*b**2) + 50*a*b**4*d**3*e**2 - 13*a*(10*a**3*b**2*e
**5 + 50*a**2*b**3*d*e**4 - 8*a**2*(31*a*b**4*e**5/10 + 5*b**5*d*e**4)/(9*b**2) + 50*a*b**4*d**2*e**3 - 15*a*(
91*a**2*b**3*e**5/10 + 25*a*b**4*d*e**4 - 17*a*(31*a*b**4*e**5/10 + 5*b**5*d*e**4)/(9*b) + 10*b**5*d**2*e**3)/
(8*b) + 10*b**5*d**3*e**2)/(7*b) + 5*b**5*d**4*e)/(6*b**2) + 5*a*b**4*d**5 - 9*a*(a**5*e**5 + 25*a**4*b*d*e**4
+ 100*a**3*b**2*d**2*e**3 + 100*a**2*b**3*d**3*e**2 - 6*a**2*(10*a**3*b**2*e**5 + 50*a**2*b**3*d*e**4 - 8*a**
2*(31*a*b**4*e**5/10 + 5*b**5*d*e**4)/(9*b**2) + 50*a*b**4*d**2*e**3 - 15*a*(91*a**2*b**3*e**5/10 + 25*a*b**4*
d*e**4 - 17*a*(31*a*b**4*e**5/10 + 5*b**5*d*e**4)/(9*b) + 10*b**5*d**2*e**3)/(8*b) + 10*b**5*d**3*e**2)/(7*b**
2) + 25*a*b**4*d**4*e - 11*a*(5*a**4*b*e**5 + 50*a**3*b**2*d*e**4 + 100*a**2*b**3*d**2*e**3 - 7*a**2*(91*a**2*
b**3*e**5/10 + 25*a*b**4*d*e**4 - 17*a*(31*a*b**4*e**5/10 + 5*b**5*d*e**4)/(9*b) + 10*b**5*d**2*e**3)/(8*b**2)
+ 50*a*b**4*d**3*e**2 - 13*a*(10*a**3*b**2*e**5 + 50*a**2*b**3*d*e**4 - 8*a**2*(31*a*b**4*e**5/10 + 5*b**5*d*
e**4)/(9*b**2) + 50*a*b**4*d**2*e**3 - 15*a*(91*a**2*b**3*e**5/10 + 25*a*b**4*d*e**4 - 17*a*(31*a*b**4*e**5/10
+ 5*b**5*d*e**4)/(9*b) + 10*b**5*d**2*e**3)/(8*b) + 10*b**5*d**3*e**2)/(7*b) + 5*b**5*d**4*e)/(6*b) + b**5*d*
*5)/(5*b))/(4*b))/(3*b**2) - 3*a*(10*a**5*d**3*e**2 + 25*a**4*b*d**4*e + 10*a**3*b**2*d**5 - 3*a**2*(5*a**5*d*
e**4 + 50*a**4*b*d**2*e**3 + 100*a**3*b**2*d**3*e**2 + 50*a**2*b**3*d**4*e - 5*a**2*(5*a**4*b*e**5 + 50*a**3*b
**2*d*e**4 + 100*a**2*b**3*d**2*e**3 - 7*a**2*(91*a**2*b**3*e**5/10 + 25*a*b**4*d*e**4 - 17*a*(31*a*b**4*e**5/
10 + 5*b**5*d*e**4)/(9*b) + 10*b**5*d**2*e**3)/(8*b**2) + 50*a*b**4*d**3*e**2 - 13*a*(10*a**3*b**2*e**5 + 50*a
**2*b**3*d*e**4 - 8*a**2*(31*a*b**4*e**5/10 + 5*b**5*d*e**4)/(9*b**2) + 50*a*b**4*d**2*e**3 - 15*a*(91*a**2*b*
*3*e**5/10 + 25*a*b**4*d*e**4 - 17*a*(31*a*b**4*e**5/10 + 5*b**5*d*e**4)/(9*b) + 10*b**5*d**2*e**3)/(8*b) + 10
*b**5*d**3*e**2)/(7*b) + 5*b**5*d**4*e)/(6*b**2) + 5*a*b**4*d**5 - 9*a*(a**5*e**5 + 25*a**4*b*d*e**4 + 100*a**
3*b**2*d**2*e**3 + 100*a**2*b**3*d**3*e**2 - 6*a**2*(10*a**3*b**2*e**5 + 50*a**2*b**3*d*e**4 - 8*a**2*(31*a*b*
*4*e**5/10 + 5*b**5*d*e**4)/(9*b**2) + 50*a*b**4*d**2*e**3 - 15*a*(91*a**2*b**3*e**5/10 + 25*a*b**4*d*e**4 - 1
7*a*(31*a*b**4*e**5/10 + 5*b**5*d*e**4)/(9*b) + 10*b**5*d**2*e**3)/(8*b) + 10*b**5*d**3*e**2)/(7*b**2) + 25*a*
b**4*d**4*e - 11*a*(5*a**4*b*e**5 + 50*a**3*b**2*d*e**4 + 100*a**2*b**3*d**2*e**3 - 7*a**2*(91*a**2*b**3*e**5/
10 + 25*a*b**4*d*e**4 - 17*a*(31*a*b**4*e**5/10 + 5*b**5*d*e**4)/(9*b) + 10*b**5*d**2*e**3)/(8*b**2) + 50*a*b*
*4*d**3*e**2 - 13*a*(10*a**3*b**2*e**5 + 50*a**2*b**3*d*e**4 - 8*a**2*(31*a*b**4*e**5/10 + 5*b**5*d*e**4)/(9*b
**2) + 50*a*b**4*d**2*e**3 - 15*a*(91*a**2*b**3*e**5/10 + 25*a*b**4*d*e**4 - 17*a*(31*a*b**4*e**5/10 + 5*b**5*
d*e**4)/(9*b) + 10*b**5*d**2*e**3)/(8*b) + 10*b**5*d**3*e**2)/(7*b) + 5*b**5*d**4*e)/(6*b) + b**5*d**5)/(5*b))
/(4*b**2) - 5*a*(10*a**5*d**2*e**3 + 50*a**4*b*d**3*e**2 + 50*a**3*b**2*d**4*e + 10*a**2*b**3*d**5 - 4*a**2*(a
**5*e**5 + 25*a**4*b*d*e**4 + 100*a**3*b**2*d**2*e**3 + 100*a**2*b**3*d**3*e**2 - 6*a**2*(10*a**3*b**2*e**5 +
50*a**2*b**3*d*e**4 - 8*a**2*(31*a*b**4*e**5/10 + 5*b**5*d*e**4)/(9*b**2) + 50*a*b**4*d**2*e**3 - 15*a*(91*a**
2*b**3*e**5/10 + 25*a*b**4*d*e**4 - 17*a*(31*a*b**4*e**5/10 + 5*b**5*d*e**4)/(9*b) + 10*b**5*d**2*e**3)/(8*b)
+ 10*b**5*d**3*e**2)/(7*b**2) + 25*a*b**4*d**4*e - 11*a*(5*a**4*b*e**5 + 50*a**3*b**2*d*e**4 + 100*a**2*b**3*d
**2*e**3 - 7*a**2*(91*a**2*b**3*e**5/10 + 25*a*b**4*d*e**4 - 17*a*(31*a*b**4*e**5/10 + 5*b**5*d*e**4)/(9*b) +
10*b**5*d**2*e**3)/(8*b**2) + 50*a*b**4*d**3*e**2 - 13*a*(10*a**3*b**2*e**5 + 50*a**2*b**3*d*e**4 - 8*a**2*(31
*a*b**4*e**5/10 + 5*b**5*d*e**4)/(9*b**2) + 50*a*b**4*d**2*e**3 - 15*a*(91*a**2*b**3*e**5/10 + 25*a*b**4*d*e**
4 - 17*a*(31*a*b**4*e**5/10 + 5*b**5*d*e**4)/(9*b) + 10*b**5*d**2*e**3)/(8*b) + 10*b**5*d**3*e**2)/(7*b) + 5*b
**5*d**4*e)/(6*b) + b**5*d**5)/(5*b**2) - 7*a*(5*a**5*d*e**4 + 50*a**4*b*d**2*e**3 + 100*a**3*b**2*d**3*e**2 +
50*a**2*b**3*d**4*e - 5*a**2*(5*a**4*b*e**5 + 50*a**3*b**2*d*e**4 + 100*a**2*b**3*d**2*e**3 - 7*a**2*(91*a**2
*b**3*e**5/10 + 25*a*b**4*d*e**4 - 17*a*(31*a*b**4*e**5/10 + 5*b**5*d*e**4)/(9*b) + 10*b**5*d**2*e**3)/(8*b**2
) + 50*a*b**4*d**3*e**2 - 13*a*(10*a**3*b**2*e**5 + 50*a**2*b**3*d*e**4 - 8*a**2*(31*a*b**4*e**5/10 + 5*b**5*d
*e**4)/(9*b**2) + 50*a*b**4*d**2*e**3 - 15*a*(91*a**2*b**3*e**5/10 + 25*a*b**4*d*e**4 - 17*a*(31*a*b**4*e**5/1
0 + 5*b**5*d*e**4)/(9*b) + 10*b**5*d**2*e**3)/(8*b) + 10*b**5*d**3*e**2)/(7*b) + 5*b**5*d**4*e)/(6*b**2) + 5*a
*b**4*d**5 - 9*a*(a**5*e**5 + 25*a**4*b*d*e**4 + 100*a**3*b**2*d**2*e**3 + 100*a**2*b**3*d**3*e**2 - 6*a**2*(1
0*a**3*b**2*e**5 + 50*a**2*b**3*d*e**4 - 8*a**2*(31*a*b**4*e**5/10 + 5*b**5*d*e**4)/(9*b**2) + 50*a*b**4*d**2*
e**3 - 15*a*(91*a**2*b**3*e**5/10 + 25*a*b**4*d*e**4 - 17*a*(31*a*b**4*e**5/10 + 5*b**5*d*e**4)/(9*b) + 10*b**
5*d**2*e**3)/(8*b) + 10*b**5*d**3*e**2)/(7*b**2) + 25*a*b**4*d**4*e - 11*a*(5*a**4*b*e**5 + 50*a**3*b**2*d*e**
4 + 100*a**2*b**3*d**2*e**3 - 7*a**2*(91*a**2*b**3*e**5/10 + 25*a*b**4*d*e**4 - 17*a*(31*a*b**4*e**5/10 + 5*b*
*5*d*e**4)/(9*b) + 10*b**5*d**2*e**3)/(8*b**2) + 50*a*b**4*d**3*e**2 - 13*a*(10*a**3*b**2*e**5 + 50*a**2*b**3*
d*e**4 - 8*a**2*(31*a*b**4*e**5/10 + 5*b**5*d*e**4)/(9*b**2) + 50*a*b**4*d**2*e**3 - 15*a*(91*a**2*b**3*e**5/1
0 + 25*a*b**4*d*e**4 - 17*a*(31*a*b**4*e**5/10 + 5*b**5*d*e**4)/(9*b) + 10*b**5*d**2*e**3)/(8*b) + 10*b**5*d**
3*e**2)/(7*b) + 5*b**5*d**4*e)/(6*b) + b**5*d**5)/(5*b))/(4*b))/(3*b))/(2*b))/b)*log(a/b + x)/sqrt(b**2*(a/b +
x)**2), Ne(b**2, 0)), (((a**2 + 2*a*b*x)**(5/2)*(-a**6*e**5 + 10*a**5*b*d*e**4 - 40*a**4*b**2*d**2*e**3 + 80*
a**3*b**3*d**3*e**2 - 80*a**2*b**4*d**4*e + 32*a*b**5*d**5)/(320*b**5) + (a**2 + 2*a*b*x)**(7/2)*(2*a**5*e**5
- 15*a**4*b*d*e**4 + 40*a**3*b**2*d**2*e**3 - 40*a**2*b**3*d**3*e**2 + 16*b**5*d**5)/(224*a*b**5) + (a**2 + 2*
a*b*x)**(9/2)*(-5*a**4*e**5 + 20*a**3*b*d*e**4 - 80*a*b**3*d**3*e**2 + 80*b**4*d**4*e)/(576*a**2*b**5) + (a**2
+ 2*a*b*x)**(11/2)*(5*a**2*d*e**4 - 20*a*b*d**2*e**3 + 20*b**2*d**3*e**2)/(176*a**3*b**4) + (a**2 + 2*a*b*x)*
*(13/2)*(5*a**2*e**5 - 30*a*b*d*e**4 + 40*b**2*d**2*e**3)/(832*a**4*b**5) + (a**2 + 2*a*b*x)**(15/2)*(-2*a*e**
5 + 5*b*d*e**4)/(480*a**5*b**5) + e**5*(a**2 + 2*a*b*x)**(17/2)/(1088*a**6*b**5))/(a*b), Ne(a*b, 0)), ((a*d**5
*x + b*e**5*x**7/7 + x**6*(a*e**5 + 5*b*d*e**4)/6 + x**5*(5*a*d*e**4 + 10*b*d**2*e**3)/5 + x**4*(10*a*d**2*e**
3 + 10*b*d**3*e**2)/4 + x**3*(10*a*d**3*e**2 + 5*b*d**4*e)/3 + x**2*(5*a*d**4*e + b*d**5)/2)*(a**2)**(3/2), Tr
ue))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1323 vs. \(2 (189) = 378\).

Time = 0.22 (sec) , antiderivative size = 1323, normalized size of antiderivative = 5.21 \[ \int (a+b x) (d+e x)^5 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\text {Too large to display} \]

[In]

integrate((b*x+a)*(e*x+d)^5*(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm="maxima")

[Out]

1/10*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*e^5*x^5/b - 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a*e^5*x^4/b^2 + 5/24*(b^2
*x^2 + 2*a*b*x + a^2)^(5/2)*a^2*e^5*x^3/b^3 + 1/4*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a*d^5*x + 1/4*(b^2*x^2 + 2*a
*b*x + a^2)^(3/2)*a^6*e^5*x/b^5 - 13/56*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^3*e^5*x^2/b^4 + 1/4*(b^2*x^2 + 2*a*b
*x + a^2)^(3/2)*a^2*d^5/b + 1/4*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^7*e^5/b^6 + 41/168*(b^2*x^2 + 2*a*b*x + a^2)
^(5/2)*a^4*e^5*x/b^5 - 209/840*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^5*e^5/b^6 + 1/9*(5*b*d*e^4 + a*e^5)*(b^2*x^2
+ 2*a*b*x + a^2)^(5/2)*x^4/b^2 - 13/72*(5*b*d*e^4 + a*e^5)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a*x^3/b^3 + 5/8*(2*
b*d^2*e^3 + a*d*e^4)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*x^3/b^2 - 1/4*(5*b*d*e^4 + a*e^5)*(b^2*x^2 + 2*a*b*x + a^
2)^(3/2)*a^5*x/b^5 + 5/4*(2*b*d^2*e^3 + a*d*e^4)*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^4*x/b^4 - 5/2*(b*d^3*e^2 +
a*d^2*e^3)*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^3*x/b^3 + 5/4*(b*d^4*e + 2*a*d^3*e^2)*(b^2*x^2 + 2*a*b*x + a^2)^(
3/2)*a^2*x/b^2 - 1/4*(b*d^5 + 5*a*d^4*e)*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a*x/b + 37/168*(5*b*d*e^4 + a*e^5)*(b
^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^2*x^2/b^4 - 55/56*(2*b*d^2*e^3 + a*d*e^4)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a*x^
2/b^3 + 10/7*(b*d^3*e^2 + a*d^2*e^3)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*x^2/b^2 - 1/4*(5*b*d*e^4 + a*e^5)*(b^2*x^
2 + 2*a*b*x + a^2)^(3/2)*a^6/b^6 + 5/4*(2*b*d^2*e^3 + a*d*e^4)*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^5/b^5 - 5/2*(
b*d^3*e^2 + a*d^2*e^3)*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^4/b^4 + 5/4*(b*d^4*e + 2*a*d^3*e^2)*(b^2*x^2 + 2*a*b*
x + a^2)^(3/2)*a^3/b^3 - 1/4*(b*d^5 + 5*a*d^4*e)*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^2/b^2 - 121/504*(5*b*d*e^4
+ a*e^5)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^3*x/b^5 + 65/56*(2*b*d^2*e^3 + a*d*e^4)*(b^2*x^2 + 2*a*b*x + a^2)^(
5/2)*a^2*x/b^4 - 15/7*(b*d^3*e^2 + a*d^2*e^3)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a*x/b^3 + 5/6*(b*d^4*e + 2*a*d^3
*e^2)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*x/b^2 + 125/504*(5*b*d*e^4 + a*e^5)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^4/
b^6 - 69/56*(2*b*d^2*e^3 + a*d*e^4)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^3/b^5 + 17/7*(b*d^3*e^2 + a*d^2*e^3)*(b^
2*x^2 + 2*a*b*x + a^2)^(5/2)*a^2/b^4 - 7/6*(b*d^4*e + 2*a*d^3*e^2)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a/b^3 + 1/5
*(b*d^5 + 5*a*d^4*e)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)/b^2

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 657 vs. \(2 (189) = 378\).

Time = 0.28 (sec) , antiderivative size = 657, normalized size of antiderivative = 2.59 \[ \int (a+b x) (d+e x)^5 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {1}{10} \, b^{4} e^{5} x^{10} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{9} \, b^{4} d e^{4} x^{9} \mathrm {sgn}\left (b x + a\right ) + \frac {4}{9} \, a b^{3} e^{5} x^{9} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{4} \, b^{4} d^{2} e^{3} x^{8} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{2} \, a b^{3} d e^{4} x^{8} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{4} \, a^{2} b^{2} e^{5} x^{8} \mathrm {sgn}\left (b x + a\right ) + \frac {10}{7} \, b^{4} d^{3} e^{2} x^{7} \mathrm {sgn}\left (b x + a\right ) + \frac {40}{7} \, a b^{3} d^{2} e^{3} x^{7} \mathrm {sgn}\left (b x + a\right ) + \frac {30}{7} \, a^{2} b^{2} d e^{4} x^{7} \mathrm {sgn}\left (b x + a\right ) + \frac {4}{7} \, a^{3} b e^{5} x^{7} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{6} \, b^{4} d^{4} e x^{6} \mathrm {sgn}\left (b x + a\right ) + \frac {20}{3} \, a b^{3} d^{3} e^{2} x^{6} \mathrm {sgn}\left (b x + a\right ) + 10 \, a^{2} b^{2} d^{2} e^{3} x^{6} \mathrm {sgn}\left (b x + a\right ) + \frac {10}{3} \, a^{3} b d e^{4} x^{6} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{6} \, a^{4} e^{5} x^{6} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{5} \, b^{4} d^{5} x^{5} \mathrm {sgn}\left (b x + a\right ) + 4 \, a b^{3} d^{4} e x^{5} \mathrm {sgn}\left (b x + a\right ) + 12 \, a^{2} b^{2} d^{3} e^{2} x^{5} \mathrm {sgn}\left (b x + a\right ) + 8 \, a^{3} b d^{2} e^{3} x^{5} \mathrm {sgn}\left (b x + a\right ) + a^{4} d e^{4} x^{5} \mathrm {sgn}\left (b x + a\right ) + a b^{3} d^{5} x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {15}{2} \, a^{2} b^{2} d^{4} e x^{4} \mathrm {sgn}\left (b x + a\right ) + 10 \, a^{3} b d^{3} e^{2} x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{2} \, a^{4} d^{2} e^{3} x^{4} \mathrm {sgn}\left (b x + a\right ) + 2 \, a^{2} b^{2} d^{5} x^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {20}{3} \, a^{3} b d^{4} e x^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {10}{3} \, a^{4} d^{3} e^{2} x^{3} \mathrm {sgn}\left (b x + a\right ) + 2 \, a^{3} b d^{5} x^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{2} \, a^{4} d^{4} e x^{2} \mathrm {sgn}\left (b x + a\right ) + a^{4} d^{5} x \mathrm {sgn}\left (b x + a\right ) + \frac {{\left (252 \, a^{5} b^{5} d^{5} - 210 \, a^{6} b^{4} d^{4} e + 120 \, a^{7} b^{3} d^{3} e^{2} - 45 \, a^{8} b^{2} d^{2} e^{3} + 10 \, a^{9} b d e^{4} - a^{10} e^{5}\right )} \mathrm {sgn}\left (b x + a\right )}{1260 \, b^{6}} \]

[In]

integrate((b*x+a)*(e*x+d)^5*(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm="giac")

[Out]

1/10*b^4*e^5*x^10*sgn(b*x + a) + 5/9*b^4*d*e^4*x^9*sgn(b*x + a) + 4/9*a*b^3*e^5*x^9*sgn(b*x + a) + 5/4*b^4*d^2
*e^3*x^8*sgn(b*x + a) + 5/2*a*b^3*d*e^4*x^8*sgn(b*x + a) + 3/4*a^2*b^2*e^5*x^8*sgn(b*x + a) + 10/7*b^4*d^3*e^2
*x^7*sgn(b*x + a) + 40/7*a*b^3*d^2*e^3*x^7*sgn(b*x + a) + 30/7*a^2*b^2*d*e^4*x^7*sgn(b*x + a) + 4/7*a^3*b*e^5*
x^7*sgn(b*x + a) + 5/6*b^4*d^4*e*x^6*sgn(b*x + a) + 20/3*a*b^3*d^3*e^2*x^6*sgn(b*x + a) + 10*a^2*b^2*d^2*e^3*x
^6*sgn(b*x + a) + 10/3*a^3*b*d*e^4*x^6*sgn(b*x + a) + 1/6*a^4*e^5*x^6*sgn(b*x + a) + 1/5*b^4*d^5*x^5*sgn(b*x +
a) + 4*a*b^3*d^4*e*x^5*sgn(b*x + a) + 12*a^2*b^2*d^3*e^2*x^5*sgn(b*x + a) + 8*a^3*b*d^2*e^3*x^5*sgn(b*x + a)
+ a^4*d*e^4*x^5*sgn(b*x + a) + a*b^3*d^5*x^4*sgn(b*x + a) + 15/2*a^2*b^2*d^4*e*x^4*sgn(b*x + a) + 10*a^3*b*d^3
*e^2*x^4*sgn(b*x + a) + 5/2*a^4*d^2*e^3*x^4*sgn(b*x + a) + 2*a^2*b^2*d^5*x^3*sgn(b*x + a) + 20/3*a^3*b*d^4*e*x
^3*sgn(b*x + a) + 10/3*a^4*d^3*e^2*x^3*sgn(b*x + a) + 2*a^3*b*d^5*x^2*sgn(b*x + a) + 5/2*a^4*d^4*e*x^2*sgn(b*x
+ a) + a^4*d^5*x*sgn(b*x + a) + 1/1260*(252*a^5*b^5*d^5 - 210*a^6*b^4*d^4*e + 120*a^7*b^3*d^3*e^2 - 45*a^8*b^
2*d^2*e^3 + 10*a^9*b*d*e^4 - a^10*e^5)*sgn(b*x + a)/b^6

Mupad [F(-1)]

Timed out. \[ \int (a+b x) (d+e x)^5 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\int \left (a+b\,x\right )\,{\left (d+e\,x\right )}^5\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2} \,d x \]

[In]

int((a + b*x)*(d + e*x)^5*(a^2 + b^2*x^2 + 2*a*b*x)^(3/2),x)

[Out]

int((a + b*x)*(d + e*x)^5*(a^2 + b^2*x^2 + 2*a*b*x)^(3/2), x)