\(\int (d+e x)^4 (a d e+(c d^2+a e^2) x+c d e x^2)^{5/2} \, dx\) [1931]

Optimal result

Integrand size = 37, antiderivative size = 534 \[ \int (d+e x)^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx=\frac {143 \left (c d^2-a e^2\right )^8 \left (c d^2+a e^2+2 c d e x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{131072 c^7 d^7 e^3}-\frac {143 \left (c d^2-a e^2\right )^6 \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{49152 c^6 d^6 e^2}+\frac {143 \left (c d^2-a e^2\right )^4 \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{15360 c^5 d^5 e}+\frac {143 \left (c d^2-a e^2\right )^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{4480 c^4 d^4}+\frac {143 \left (c d^2-a e^2\right )^2 (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{2880 c^3 d^3}+\frac {13 \left (c d^2-a e^2\right ) (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{180 c^2 d^2}+\frac {(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{10 c d}-\frac {143 \left (c d^2-a e^2\right )^{10} \text {arctanh}\left (\frac {c d^2+a e^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{262144 c^{15/2} d^{15/2} e^{7/2}} \]

[Out]

-143/49152*(-a*e^2+c*d^2)^6*(2*c*d*e*x+a*e^2+c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/c^6/d^6/e^2+143/15
360*(-a*e^2+c*d^2)^4*(2*c*d*e*x+a*e^2+c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/c^5/d^5/e+143/4480*(-a*e^
2+c*d^2)^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/c^4/d^4+143/2880*(-a*e^2+c*d^2)^2*(e*x+d)*(a*d*e+(a*e^2+c*d
^2)*x+c*d*e*x^2)^(7/2)/c^3/d^3+13/180*(-a*e^2+c*d^2)*(e*x+d)^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/c^2/d^2
+1/10*(e*x+d)^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/c/d-143/262144*(-a*e^2+c*d^2)^10*arctanh(1/2*(2*c*d*e*
x+a*e^2+c*d^2)/c^(1/2)/d^(1/2)/e^(1/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/c^(15/2)/d^(15/2)/e^(7/2)+143/
131072*(-a*e^2+c*d^2)^8*(2*c*d*e*x+a*e^2+c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/c^7/d^7/e^3

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 534, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.135, Rules used = {684, 654, 626, 635, 212} \[ \int (d+e x)^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx=-\frac {143 \left (c d^2-a e^2\right )^{10} \text {arctanh}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{262144 c^{15/2} d^{15/2} e^{7/2}}+\frac {143 \left (c d^2-a e^2\right )^8 \left (a e^2+c d^2+2 c d e x\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{131072 c^7 d^7 e^3}-\frac {143 \left (c d^2-a e^2\right )^6 \left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{49152 c^6 d^6 e^2}+\frac {143 \left (c d^2-a e^2\right )^4 \left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{15360 c^5 d^5 e}+\frac {143 \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{4480 c^4 d^4}+\frac {143 (d+e x) \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{2880 c^3 d^3}+\frac {13 (d+e x)^2 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{180 c^2 d^2}+\frac {(d+e x)^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{10 c d} \]

[In]

Int[(d + e*x)^4*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2),x]

[Out]

(143*(c*d^2 - a*e^2)^8*(c*d^2 + a*e^2 + 2*c*d*e*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(131072*c^7*d^
7*e^3) - (143*(c*d^2 - a*e^2)^6*(c*d^2 + a*e^2 + 2*c*d*e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(49
152*c^6*d^6*e^2) + (143*(c*d^2 - a*e^2)^4*(c*d^2 + a*e^2 + 2*c*d*e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^
(5/2))/(15360*c^5*d^5*e) + (143*(c*d^2 - a*e^2)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(4480*c^4*d^4
) + (143*(c*d^2 - a*e^2)^2*(d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(2880*c^3*d^3) + (13*(c*d^
2 - a*e^2)*(d + e*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(180*c^2*d^2) + ((d + e*x)^3*(a*d*e + (c
*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(10*c*d) - (143*(c*d^2 - a*e^2)^10*ArcTanh[(c*d^2 + a*e^2 + 2*c*d*e*x)/(2*
Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])])/(262144*c^(15/2)*d^(15/2)*e^(7/2))

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 626

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x)*((a + b*x + c*x^2)^p/(2*c*(2*p + 1
))), x] - Dist[p*((b^2 - 4*a*c)/(2*c*(2*p + 1))), Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x]
&& NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && IntegerQ[4*p]

Rule 635

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 654

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*((a + b*x + c*x^2)^(p +
1)/(2*c*(p + 1))), x] + Dist[(2*c*d - b*e)/(2*c), Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}
, x] && NeQ[2*c*d - b*e, 0] && NeQ[p, -1]

Rule 684

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*(d + e*x)^(m - 1)*
((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 1))), x] + Dist[(m + p)*((2*c*d - b*e)/(c*(m + 2*p + 1))), Int[(d + e
*x)^(m - 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 -
b*d*e + a*e^2, 0] && GtQ[m, 1] && NeQ[m + 2*p + 1, 0] && IntegerQ[2*p]

Rubi steps \begin{align*} \text {integral}& = \frac {(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{10 c d}+\frac {\left (13 \left (d^2-\frac {a e^2}{c}\right )\right ) \int (d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx}{20 d} \\ & = \frac {13 \left (c d^2-a e^2\right ) (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{180 c^2 d^2}+\frac {(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{10 c d}+\frac {\left (143 \left (d^2-\frac {a e^2}{c}\right )^2\right ) \int (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx}{360 d^2} \\ & = \frac {143 \left (c d^2-a e^2\right )^2 (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{2880 c^3 d^3}+\frac {13 \left (c d^2-a e^2\right ) (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{180 c^2 d^2}+\frac {(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{10 c d}+\frac {\left (143 \left (d^2-\frac {a e^2}{c}\right )^3\right ) \int (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx}{640 d^3} \\ & = \frac {143 \left (c d^2-a e^2\right )^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{4480 c^4 d^4}+\frac {143 \left (c d^2-a e^2\right )^2 (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{2880 c^3 d^3}+\frac {13 \left (c d^2-a e^2\right ) (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{180 c^2 d^2}+\frac {(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{10 c d}+\frac {\left (143 \left (d^2-\frac {a e^2}{c}\right )^4\right ) \int \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx}{1280 d^4} \\ & = \frac {143 \left (c d^2-a e^2\right )^4 \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{15360 c^5 d^5 e}+\frac {143 \left (c d^2-a e^2\right )^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{4480 c^4 d^4}+\frac {143 \left (c d^2-a e^2\right )^2 (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{2880 c^3 d^3}+\frac {13 \left (c d^2-a e^2\right ) (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{180 c^2 d^2}+\frac {(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{10 c d}-\frac {\left (143 \left (c d^2-a e^2\right )^6\right ) \int \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2} \, dx}{6144 c^5 d^5 e} \\ & = -\frac {143 \left (c d^2-a e^2\right )^6 \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{49152 c^6 d^6 e^2}+\frac {143 \left (c d^2-a e^2\right )^4 \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{15360 c^5 d^5 e}+\frac {143 \left (c d^2-a e^2\right )^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{4480 c^4 d^4}+\frac {143 \left (c d^2-a e^2\right )^2 (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{2880 c^3 d^3}+\frac {13 \left (c d^2-a e^2\right ) (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{180 c^2 d^2}+\frac {(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{10 c d}+\frac {\left (143 \left (c d^2-a e^2\right )^8\right ) \int \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx}{32768 c^6 d^6 e^2} \\ & = \frac {143 \left (c d^2-a e^2\right )^8 \left (c d^2+a e^2+2 c d e x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{131072 c^7 d^7 e^3}-\frac {143 \left (c d^2-a e^2\right )^6 \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{49152 c^6 d^6 e^2}+\frac {143 \left (c d^2-a e^2\right )^4 \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{15360 c^5 d^5 e}+\frac {143 \left (c d^2-a e^2\right )^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{4480 c^4 d^4}+\frac {143 \left (c d^2-a e^2\right )^2 (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{2880 c^3 d^3}+\frac {13 \left (c d^2-a e^2\right ) (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{180 c^2 d^2}+\frac {(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{10 c d}-\frac {\left (143 \left (c d^2-a e^2\right )^{10}\right ) \int \frac {1}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{262144 c^7 d^7 e^3} \\ & = \frac {143 \left (c d^2-a e^2\right )^8 \left (c d^2+a e^2+2 c d e x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{131072 c^7 d^7 e^3}-\frac {143 \left (c d^2-a e^2\right )^6 \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{49152 c^6 d^6 e^2}+\frac {143 \left (c d^2-a e^2\right )^4 \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{15360 c^5 d^5 e}+\frac {143 \left (c d^2-a e^2\right )^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{4480 c^4 d^4}+\frac {143 \left (c d^2-a e^2\right )^2 (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{2880 c^3 d^3}+\frac {13 \left (c d^2-a e^2\right ) (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{180 c^2 d^2}+\frac {(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{10 c d}-\frac {\left (143 \left (c d^2-a e^2\right )^{10}\right ) \text {Subst}\left (\int \frac {1}{4 c d e-x^2} \, dx,x,\frac {c d^2+a e^2+2 c d e x}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{131072 c^7 d^7 e^3} \\ & = \frac {143 \left (c d^2-a e^2\right )^8 \left (c d^2+a e^2+2 c d e x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{131072 c^7 d^7 e^3}-\frac {143 \left (c d^2-a e^2\right )^6 \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{49152 c^6 d^6 e^2}+\frac {143 \left (c d^2-a e^2\right )^4 \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{15360 c^5 d^5 e}+\frac {143 \left (c d^2-a e^2\right )^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{4480 c^4 d^4}+\frac {143 \left (c d^2-a e^2\right )^2 (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{2880 c^3 d^3}+\frac {13 \left (c d^2-a e^2\right ) (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{180 c^2 d^2}+\frac {(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{10 c d}-\frac {143 \left (c d^2-a e^2\right )^{10} \tanh ^{-1}\left (\frac {c d^2+a e^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{262144 c^{15/2} d^{15/2} e^{7/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.25 (sec) , antiderivative size = 735, normalized size of antiderivative = 1.38 \[ \int (d+e x)^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx=\frac {((a e+c d x) (d+e x))^{5/2} \left (\frac {\sqrt {c} \sqrt {d} \sqrt {e} \left (45045 a^9 e^{18}-15015 a^8 c d e^{16} (29 d+2 e x)+12012 a^7 c^2 d^2 e^{14} \left (157 d^2+24 d e x+2 e^2 x^2\right )-1716 a^6 c^3 d^3 e^{12} \left (2805 d^3+722 d^2 e x+134 d e^2 x^2+12 e^3 x^3\right )+286 a^5 c^4 d^4 e^{10} \left (27985 d^4+10960 d^3 e x+3444 d^2 e^2 x^2+688 d e^3 x^3+64 e^4 x^4\right )-130 a^4 c^5 d^5 e^8 \left (69505 d^5+39690 d^4 e x+19100 d^3 e^2 x^2+6472 d^2 e^3 x^3+1344 d e^4 x^4+128 e^5 x^5\right )+20 a^3 c^6 d^6 e^6 \left (349155 d^6+287800 d^5 e x+203530 d^4 e^2 x^2+105840 d^3 e^3 x^3+37312 d^2 e^4 x^4+7936 d e^5 x^5+768 e^6 x^6\right )+4 a^2 c^7 d^7 e^4 \left (471471 d^7+8537622 d^6 e x+27771366 d^5 e^2 x^2+47303260 d^4 e^3 x^3+47700160 d^3 e^4 x^4+28732032 d^2 e^5 x^5+9597184 d e^6 x^6+1372672 e^7 x^7\right )+a c^8 d^8 e^2 \left (-435435 d^8+288288 d^7 e x+35159496 d^6 e^2 x^2+144375968 d^5 e^3 x^3+275076480 d^4 e^4 x^4+296381440 d^3 e^5 x^5+186500096 d^2 e^6 x^6+64253952 d e^7 x^7+9404416 e^8 x^8\right )+c^9 d^9 \left (45045 d^9-30030 d^8 e x+24024 d^7 e^2 x^2+11775888 d^6 e^3 x^3+53757824 d^5 e^4 x^4+108773120 d^4 e^5 x^5+121912320 d^3 e^6 x^6+78891008 d^2 e^7 x^7+27754496 d e^8 x^8+4128768 e^9 x^9\right )\right )}{(a e+c d x)^2 (d+e x)^2}-\frac {45045 \left (c d^2-a e^2\right )^{10} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {e} \sqrt {a e+c d x}}\right )}{(a e+c d x)^{5/2} (d+e x)^{5/2}}\right )}{41287680 c^{15/2} d^{15/2} e^{7/2}} \]

[In]

Integrate[(d + e*x)^4*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2),x]

[Out]

(((a*e + c*d*x)*(d + e*x))^(5/2)*((Sqrt[c]*Sqrt[d]*Sqrt[e]*(45045*a^9*e^18 - 15015*a^8*c*d*e^16*(29*d + 2*e*x)
+ 12012*a^7*c^2*d^2*e^14*(157*d^2 + 24*d*e*x + 2*e^2*x^2) - 1716*a^6*c^3*d^3*e^12*(2805*d^3 + 722*d^2*e*x + 1
34*d*e^2*x^2 + 12*e^3*x^3) + 286*a^5*c^4*d^4*e^10*(27985*d^4 + 10960*d^3*e*x + 3444*d^2*e^2*x^2 + 688*d*e^3*x^
3 + 64*e^4*x^4) - 130*a^4*c^5*d^5*e^8*(69505*d^5 + 39690*d^4*e*x + 19100*d^3*e^2*x^2 + 6472*d^2*e^3*x^3 + 1344
*d*e^4*x^4 + 128*e^5*x^5) + 20*a^3*c^6*d^6*e^6*(349155*d^6 + 287800*d^5*e*x + 203530*d^4*e^2*x^2 + 105840*d^3*
e^3*x^3 + 37312*d^2*e^4*x^4 + 7936*d*e^5*x^5 + 768*e^6*x^6) + 4*a^2*c^7*d^7*e^4*(471471*d^7 + 8537622*d^6*e*x
+ 27771366*d^5*e^2*x^2 + 47303260*d^4*e^3*x^3 + 47700160*d^3*e^4*x^4 + 28732032*d^2*e^5*x^5 + 9597184*d*e^6*x^
6 + 1372672*e^7*x^7) + a*c^8*d^8*e^2*(-435435*d^8 + 288288*d^7*e*x + 35159496*d^6*e^2*x^2 + 144375968*d^5*e^3*
x^3 + 275076480*d^4*e^4*x^4 + 296381440*d^3*e^5*x^5 + 186500096*d^2*e^6*x^6 + 64253952*d*e^7*x^7 + 9404416*e^8
*x^8) + c^9*d^9*(45045*d^9 - 30030*d^8*e*x + 24024*d^7*e^2*x^2 + 11775888*d^6*e^3*x^3 + 53757824*d^5*e^4*x^4 +
108773120*d^4*e^5*x^5 + 121912320*d^3*e^6*x^6 + 78891008*d^2*e^7*x^7 + 27754496*d*e^8*x^8 + 4128768*e^9*x^9))
)/((a*e + c*d*x)^2*(d + e*x)^2) - (45045*(c*d^2 - a*e^2)^10*ArcTanh[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/(Sqrt[e]*S
qrt[a*e + c*d*x])])/((a*e + c*d*x)^(5/2)*(d + e*x)^(5/2))))/(41287680*c^(15/2)*d^(15/2)*e^(7/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(5006\) vs. \(2(488)=976\).

Time = 3.13 (sec) , antiderivative size = 5007, normalized size of antiderivative = 9.38

method	result	size
default	\(\text {Expression too large to display}\)	\(5007\)

[In]

int((e*x+d)^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x,method=_RETURNVERBOSE)

[Out]

result too large to display

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1050 vs. \(2 (488) = 976\).

Time = 0.70 (sec) , antiderivative size = 2114, normalized size of antiderivative = 3.96 \[ \int (d+e x)^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx=\text {Too large to display} \]

[In]

integrate((e*x+d)^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algorithm="fricas")

[Out]

[1/165150720*(45045*(c^10*d^20 - 10*a*c^9*d^18*e^2 + 45*a^2*c^8*d^16*e^4 - 120*a^3*c^7*d^14*e^6 + 210*a^4*c^6*
d^12*e^8 - 252*a^5*c^5*d^10*e^10 + 210*a^6*c^4*d^8*e^12 - 120*a^7*c^3*d^6*e^14 + 45*a^8*c^2*d^4*e^16 - 10*a^9*
c*d^2*e^18 + a^10*e^20)*sqrt(c*d*e)*log(8*c^2*d^2*e^2*x^2 + c^2*d^4 + 6*a*c*d^2*e^2 + a^2*e^4 - 4*sqrt(c*d*e*x
^2 + a*d*e + (c*d^2 + a*e^2)*x)*(2*c*d*e*x + c*d^2 + a*e^2)*sqrt(c*d*e) + 8*(c^2*d^3*e + a*c*d*e^3)*x) + 4*(41
28768*c^10*d^10*e^10*x^9 + 45045*c^10*d^19*e - 435435*a*c^9*d^17*e^3 + 1885884*a^2*c^8*d^15*e^5 + 6983100*a^3*
c^7*d^13*e^7 - 9035650*a^4*c^6*d^11*e^9 + 8003710*a^5*c^5*d^9*e^11 - 4813380*a^6*c^4*d^7*e^13 + 1885884*a^7*c^
3*d^5*e^15 - 435435*a^8*c^2*d^3*e^17 + 45045*a^9*c*d*e^19 + 229376*(121*c^10*d^11*e^9 + 41*a*c^9*d^9*e^11)*x^8
+ 14336*(5503*c^10*d^12*e^8 + 4482*a*c^9*d^10*e^10 + 383*a^2*c^8*d^8*e^12)*x^7 + 1024*(119055*c^10*d^13*e^7 +
182129*a*c^9*d^11*e^9 + 37489*a^2*c^8*d^9*e^11 + 15*a^3*c^7*d^7*e^13)*x^6 + 256*(424895*c^10*d^14*e^6 + 11577
40*a*c^9*d^12*e^8 + 448938*a^2*c^8*d^10*e^10 + 620*a^3*c^7*d^8*e^12 - 65*a^4*c^6*d^6*e^14)*x^5 + 128*(419983*c
^10*d^15*e^5 + 2149035*a*c^9*d^13*e^7 + 1490630*a^2*c^8*d^11*e^9 + 5830*a^3*c^7*d^9*e^11 - 1365*a^4*c^6*d^7*e^
13 + 143*a^5*c^5*d^5*e^15)*x^4 + 16*(735993*c^10*d^16*e^4 + 9023498*a*c^9*d^14*e^6 + 11825815*a^2*c^8*d^12*e^8
+ 132300*a^3*c^7*d^10*e^10 - 52585*a^4*c^6*d^8*e^12 + 12298*a^5*c^5*d^6*e^14 - 1287*a^6*c^4*d^4*e^16)*x^3 + 8
*(3003*c^10*d^17*e^3 + 4394937*a*c^9*d^15*e^5 + 13885683*a^2*c^8*d^13*e^7 + 508825*a^3*c^7*d^11*e^9 - 310375*a
^4*c^6*d^9*e^11 + 123123*a^5*c^5*d^7*e^13 - 28743*a^6*c^4*d^5*e^15 + 3003*a^7*c^3*d^3*e^17)*x^2 - 2*(15015*c^1
0*d^18*e^2 - 144144*a*c^9*d^16*e^4 - 17075244*a^2*c^8*d^14*e^6 - 2878000*a^3*c^7*d^12*e^8 + 2579850*a^4*c^6*d^
10*e^10 - 1567280*a^5*c^5*d^8*e^12 + 619476*a^6*c^4*d^6*e^14 - 144144*a^7*c^3*d^4*e^16 + 15015*a^8*c^2*d^2*e^1
8)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(c^8*d^8*e^4), 1/82575360*(45045*(c^10*d^20 - 10*a*c^9*d^18
*e^2 + 45*a^2*c^8*d^16*e^4 - 120*a^3*c^7*d^14*e^6 + 210*a^4*c^6*d^12*e^8 - 252*a^5*c^5*d^10*e^10 + 210*a^6*c^4
*d^8*e^12 - 120*a^7*c^3*d^6*e^14 + 45*a^8*c^2*d^4*e^16 - 10*a^9*c*d^2*e^18 + a^10*e^20)*sqrt(-c*d*e)*arctan(1/
2*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(2*c*d*e*x + c*d^2 + a*e^2)*sqrt(-c*d*e)/(c^2*d^2*e^2*x^2 + a*c*
d^2*e^2 + (c^2*d^3*e + a*c*d*e^3)*x)) + 2*(4128768*c^10*d^10*e^10*x^9 + 45045*c^10*d^19*e - 435435*a*c^9*d^17*
e^3 + 1885884*a^2*c^8*d^15*e^5 + 6983100*a^3*c^7*d^13*e^7 - 9035650*a^4*c^6*d^11*e^9 + 8003710*a^5*c^5*d^9*e^1
1 - 4813380*a^6*c^4*d^7*e^13 + 1885884*a^7*c^3*d^5*e^15 - 435435*a^8*c^2*d^3*e^17 + 45045*a^9*c*d*e^19 + 22937
6*(121*c^10*d^11*e^9 + 41*a*c^9*d^9*e^11)*x^8 + 14336*(5503*c^10*d^12*e^8 + 4482*a*c^9*d^10*e^10 + 383*a^2*c^8
*d^8*e^12)*x^7 + 1024*(119055*c^10*d^13*e^7 + 182129*a*c^9*d^11*e^9 + 37489*a^2*c^8*d^9*e^11 + 15*a^3*c^7*d^7*
e^13)*x^6 + 256*(424895*c^10*d^14*e^6 + 1157740*a*c^9*d^12*e^8 + 448938*a^2*c^8*d^10*e^10 + 620*a^3*c^7*d^8*e^
12 - 65*a^4*c^6*d^6*e^14)*x^5 + 128*(419983*c^10*d^15*e^5 + 2149035*a*c^9*d^13*e^7 + 1490630*a^2*c^8*d^11*e^9
+ 5830*a^3*c^7*d^9*e^11 - 1365*a^4*c^6*d^7*e^13 + 143*a^5*c^5*d^5*e^15)*x^4 + 16*(735993*c^10*d^16*e^4 + 90234
98*a*c^9*d^14*e^6 + 11825815*a^2*c^8*d^12*e^8 + 132300*a^3*c^7*d^10*e^10 - 52585*a^4*c^6*d^8*e^12 + 12298*a^5*
c^5*d^6*e^14 - 1287*a^6*c^4*d^4*e^16)*x^3 + 8*(3003*c^10*d^17*e^3 + 4394937*a*c^9*d^15*e^5 + 13885683*a^2*c^8*
d^13*e^7 + 508825*a^3*c^7*d^11*e^9 - 310375*a^4*c^6*d^9*e^11 + 123123*a^5*c^5*d^7*e^13 - 28743*a^6*c^4*d^5*e^1
5 + 3003*a^7*c^3*d^3*e^17)*x^2 - 2*(15015*c^10*d^18*e^2 - 144144*a*c^9*d^16*e^4 - 17075244*a^2*c^8*d^14*e^6 -
2878000*a^3*c^7*d^12*e^8 + 2579850*a^4*c^6*d^10*e^10 - 1567280*a^5*c^5*d^8*e^12 + 619476*a^6*c^4*d^6*e^14 - 14
4144*a^7*c^3*d^4*e^16 + 15015*a^8*c^2*d^2*e^18)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(c^8*d^8*e^4)]

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 24405 vs. \(2 (520) = 1040\).

Time = 34.89 (sec) , antiderivative size = 24405, normalized size of antiderivative = 45.70 \[ \int (d+e x)^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx=\text {Too large to display} \]

[In]

integrate((e*x+d)**4*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2),x)

[Out]

Piecewise((sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))*(c**2*d**2*e**6*x**9/10 + x**8*(3*a*c**2*d**2*e**8 +
7*c**3*d**4*e**6 - c**2*d**2*e**6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c*d*e) + x**7*(3*a**2*c*d*e**9 + 201*a*c
**2*d**3*e**7/10 + 21*c**3*d**5*e**5 - (17*a*e**2/2 + 17*c*d**2/2)*(3*a*c**2*d**2*e**8 + 7*c**3*d**4*e**6 - c*
*2*d**2*e**6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c*d*e))/(8*c*d*e) + x**6*(a**3*e**10 + 21*a**2*c*d**2*e**8 + 6
3*a*c**2*d**4*e**6 - 8*a*(3*a*c**2*d**2*e**8 + 7*c**3*d**4*e**6 - c**2*d**2*e**6*(19*a*e**2/2 + 19*c*d**2/2)/1
0)/(9*c) + 35*c**3*d**6*e**4 - (15*a*e**2/2 + 15*c*d**2/2)*(3*a**2*c*d*e**9 + 201*a*c**2*d**3*e**7/10 + 21*c**
3*d**5*e**5 - (17*a*e**2/2 + 17*c*d**2/2)*(3*a*c**2*d**2*e**8 + 7*c**3*d**4*e**6 - c**2*d**2*e**6*(19*a*e**2/2
+ 19*c*d**2/2)/10)/(9*c*d*e))/(8*c*d*e))/(7*c*d*e) + x**5*(7*a**3*d*e**9 + 63*a**2*c*d**3*e**7 + 105*a*c**2*d
**5*e**5 - 7*a*(3*a**2*c*d*e**9 + 201*a*c**2*d**3*e**7/10 + 21*c**3*d**5*e**5 - (17*a*e**2/2 + 17*c*d**2/2)*(3
*a*c**2*d**2*e**8 + 7*c**3*d**4*e**6 - c**2*d**2*e**6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c*d*e))/(8*c) + 35*c*
*3*d**7*e**3 - (13*a*e**2/2 + 13*c*d**2/2)*(a**3*e**10 + 21*a**2*c*d**2*e**8 + 63*a*c**2*d**4*e**6 - 8*a*(3*a*
c**2*d**2*e**8 + 7*c**3*d**4*e**6 - c**2*d**2*e**6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c) + 35*c**3*d**6*e**4 -
(15*a*e**2/2 + 15*c*d**2/2)*(3*a**2*c*d*e**9 + 201*a*c**2*d**3*e**7/10 + 21*c**3*d**5*e**5 - (17*a*e**2/2 + 1
7*c*d**2/2)*(3*a*c**2*d**2*e**8 + 7*c**3*d**4*e**6 - c**2*d**2*e**6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c*d*e))
/(8*c*d*e))/(7*c*d*e))/(6*c*d*e) + x**4*(21*a**3*d**2*e**8 + 105*a**2*c*d**4*e**6 + 105*a*c**2*d**6*e**4 - 6*a
*(a**3*e**10 + 21*a**2*c*d**2*e**8 + 63*a*c**2*d**4*e**6 - 8*a*(3*a*c**2*d**2*e**8 + 7*c**3*d**4*e**6 - c**2*d
**2*e**6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c) + 35*c**3*d**6*e**4 - (15*a*e**2/2 + 15*c*d**2/2)*(3*a**2*c*d*e
**9 + 201*a*c**2*d**3*e**7/10 + 21*c**3*d**5*e**5 - (17*a*e**2/2 + 17*c*d**2/2)*(3*a*c**2*d**2*e**8 + 7*c**3*d
**4*e**6 - c**2*d**2*e**6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c*d*e))/(8*c*d*e))/(7*c) + 21*c**3*d**8*e**2 - (1
1*a*e**2/2 + 11*c*d**2/2)*(7*a**3*d*e**9 + 63*a**2*c*d**3*e**7 + 105*a*c**2*d**5*e**5 - 7*a*(3*a**2*c*d*e**9 +
201*a*c**2*d**3*e**7/10 + 21*c**3*d**5*e**5 - (17*a*e**2/2 + 17*c*d**2/2)*(3*a*c**2*d**2*e**8 + 7*c**3*d**4*e
**6 - c**2*d**2*e**6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c*d*e))/(8*c) + 35*c**3*d**7*e**3 - (13*a*e**2/2 + 13*
c*d**2/2)*(a**3*e**10 + 21*a**2*c*d**2*e**8 + 63*a*c**2*d**4*e**6 - 8*a*(3*a*c**2*d**2*e**8 + 7*c**3*d**4*e**6
- c**2*d**2*e**6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c) + 35*c**3*d**6*e**4 - (15*a*e**2/2 + 15*c*d**2/2)*(3*a
**2*c*d*e**9 + 201*a*c**2*d**3*e**7/10 + 21*c**3*d**5*e**5 - (17*a*e**2/2 + 17*c*d**2/2)*(3*a*c**2*d**2*e**8 +
7*c**3*d**4*e**6 - c**2*d**2*e**6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c*d*e))/(8*c*d*e))/(7*c*d*e))/(6*c*d*e))
/(5*c*d*e) + x**3*(35*a**3*d**3*e**7 + 105*a**2*c*d**5*e**5 + 63*a*c**2*d**7*e**3 - 5*a*(7*a**3*d*e**9 + 63*a*
*2*c*d**3*e**7 + 105*a*c**2*d**5*e**5 - 7*a*(3*a**2*c*d*e**9 + 201*a*c**2*d**3*e**7/10 + 21*c**3*d**5*e**5 - (
17*a*e**2/2 + 17*c*d**2/2)*(3*a*c**2*d**2*e**8 + 7*c**3*d**4*e**6 - c**2*d**2*e**6*(19*a*e**2/2 + 19*c*d**2/2)
/10)/(9*c*d*e))/(8*c) + 35*c**3*d**7*e**3 - (13*a*e**2/2 + 13*c*d**2/2)*(a**3*e**10 + 21*a**2*c*d**2*e**8 + 63
*a*c**2*d**4*e**6 - 8*a*(3*a*c**2*d**2*e**8 + 7*c**3*d**4*e**6 - c**2*d**2*e**6*(19*a*e**2/2 + 19*c*d**2/2)/10
)/(9*c) + 35*c**3*d**6*e**4 - (15*a*e**2/2 + 15*c*d**2/2)*(3*a**2*c*d*e**9 + 201*a*c**2*d**3*e**7/10 + 21*c**3
*d**5*e**5 - (17*a*e**2/2 + 17*c*d**2/2)*(3*a*c**2*d**2*e**8 + 7*c**3*d**4*e**6 - c**2*d**2*e**6*(19*a*e**2/2
+ 19*c*d**2/2)/10)/(9*c*d*e))/(8*c*d*e))/(7*c*d*e))/(6*c) + 7*c**3*d**9*e - (9*a*e**2/2 + 9*c*d**2/2)*(21*a**3
*d**2*e**8 + 105*a**2*c*d**4*e**6 + 105*a*c**2*d**6*e**4 - 6*a*(a**3*e**10 + 21*a**2*c*d**2*e**8 + 63*a*c**2*d
**4*e**6 - 8*a*(3*a*c**2*d**2*e**8 + 7*c**3*d**4*e**6 - c**2*d**2*e**6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c) +
35*c**3*d**6*e**4 - (15*a*e**2/2 + 15*c*d**2/2)*(3*a**2*c*d*e**9 + 201*a*c**2*d**3*e**7/10 + 21*c**3*d**5*e**
5 - (17*a*e**2/2 + 17*c*d**2/2)*(3*a*c**2*d**2*e**8 + 7*c**3*d**4*e**6 - c**2*d**2*e**6*(19*a*e**2/2 + 19*c*d*
*2/2)/10)/(9*c*d*e))/(8*c*d*e))/(7*c) + 21*c**3*d**8*e**2 - (11*a*e**2/2 + 11*c*d**2/2)*(7*a**3*d*e**9 + 63*a*
*2*c*d**3*e**7 + 105*a*c**2*d**5*e**5 - 7*a*(3*a**2*c*d*e**9 + 201*a*c**2*d**3*e**7/10 + 21*c**3*d**5*e**5 - (
17*a*e**2/2 + 17*c*d**2/2)*(3*a*c**2*d**2*e**8 + 7*c**3*d**4*e**6 - c**2*d**2*e**6*(19*a*e**2/2 + 19*c*d**2/2)
/10)/(9*c*d*e))/(8*c) + 35*c**3*d**7*e**3 - (13*a*e**2/2 + 13*c*d**2/2)*(a**3*e**10 + 21*a**2*c*d**2*e**8 + 63
*a*c**2*d**4*e**6 - 8*a*(3*a*c**2*d**2*e**8 + 7*c**3*d**4*e**6 - c**2*d**2*e**6*(19*a*e**2/2 + 19*c*d**2/2)/10
)/(9*c) + 35*c**3*d**6*e**4 - (15*a*e**2/2 + 15*c*d**2/2)*(3*a**2*c*d*e**9 + 201*a*c**2*d**3*e**7/10 + 21*c**3
*d**5*e**5 - (17*a*e**2/2 + 17*c*d**2/2)*(3*a*c**2*d**2*e**8 + 7*c**3*d**4*e**6 - c**2*d**2*e**6*(19*a*e**2/2
+ 19*c*d**2/2)/10)/(9*c*d*e))/(8*c*d*e))/(7*c*d*e))/(6*c*d*e))/(5*c*d*e))/(4*c*d*e) + x**2*(35*a**3*d**4*e**6
+ 63*a**2*c*d**6*e**4 + 21*a*c**2*d**8*e**2 - 4*a*(21*a**3*d**2*e**8 + 105*a**2*c*d**4*e**6 + 105*a*c**2*d**6*
e**4 - 6*a*(a**3*e**10 + 21*a**2*c*d**2*e**8 + 63*a*c**2*d**4*e**6 - 8*a*(3*a*c**2*d**2*e**8 + 7*c**3*d**4*e**
6 - c**2*d**2*e**6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c) + 35*c**3*d**6*e**4 - (15*a*e**2/2 + 15*c*d**2/2)*(3*
a**2*c*d*e**9 + 201*a*c**2*d**3*e**7/10 + 21*c**3*d**5*e**5 - (17*a*e**2/2 + 17*c*d**2/2)*(3*a*c**2*d**2*e**8
+ 7*c**3*d**4*e**6 - c**2*d**2*e**6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c*d*e))/(8*c*d*e))/(7*c) + 21*c**3*d**8
*e**2 - (11*a*e**2/2 + 11*c*d**2/2)*(7*a**3*d*e**9 + 63*a**2*c*d**3*e**7 + 105*a*c**2*d**5*e**5 - 7*a*(3*a**2*
c*d*e**9 + 201*a*c**2*d**3*e**7/10 + 21*c**3*d**5*e**5 - (17*a*e**2/2 + 17*c*d**2/2)*(3*a*c**2*d**2*e**8 + 7*c
**3*d**4*e**6 - c**2*d**2*e**6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c*d*e))/(8*c) + 35*c**3*d**7*e**3 - (13*a*e*
*2/2 + 13*c*d**2/2)*(a**3*e**10 + 21*a**2*c*d**2*e**8 + 63*a*c**2*d**4*e**6 - 8*a*(3*a*c**2*d**2*e**8 + 7*c**3
*d**4*e**6 - c**2*d**2*e**6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c) + 35*c**3*d**6*e**4 - (15*a*e**2/2 + 15*c*d*
*2/2)*(3*a**2*c*d*e**9 + 201*a*c**2*d**3*e**7/10 + 21*c**3*d**5*e**5 - (17*a*e**2/2 + 17*c*d**2/2)*(3*a*c**2*d
**2*e**8 + 7*c**3*d**4*e**6 - c**2*d**2*e**6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c*d*e))/(8*c*d*e))/(7*c*d*e))/
(6*c*d*e))/(5*c) + c**3*d**10 - (7*a*e**2/2 + 7*c*d**2/2)*(35*a**3*d**3*e**7 + 105*a**2*c*d**5*e**5 + 63*a*c**
2*d**7*e**3 - 5*a*(7*a**3*d*e**9 + 63*a**2*c*d**3*e**7 + 105*a*c**2*d**5*e**5 - 7*a*(3*a**2*c*d*e**9 + 201*a*c
**2*d**3*e**7/10 + 21*c**3*d**5*e**5 - (17*a*e**2/2 + 17*c*d**2/2)*(3*a*c**2*d**2*e**8 + 7*c**3*d**4*e**6 - c*
*2*d**2*e**6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c*d*e))/(8*c) + 35*c**3*d**7*e**3 - (13*a*e**2/2 + 13*c*d**2/2
)*(a**3*e**10 + 21*a**2*c*d**2*e**8 + 63*a*c**2*d**4*e**6 - 8*a*(3*a*c**2*d**2*e**8 + 7*c**3*d**4*e**6 - c**2*
d**2*e**6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c) + 35*c**3*d**6*e**4 - (15*a*e**2/2 + 15*c*d**2/2)*(3*a**2*c*d*
e**9 + 201*a*c**2*d**3*e**7/10 + 21*c**3*d**5*e**5 - (17*a*e**2/2 + 17*c*d**2/2)*(3*a*c**2*d**2*e**8 + 7*c**3*
d**4*e**6 - c**2*d**2*e**6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c*d*e))/(8*c*d*e))/(7*c*d*e))/(6*c) + 7*c**3*d**
9*e - (9*a*e**2/2 + 9*c*d**2/2)*(21*a**3*d**2*e**8 + 105*a**2*c*d**4*e**6 + 105*a*c**2*d**6*e**4 - 6*a*(a**3*e
**10 + 21*a**2*c*d**2*e**8 + 63*a*c**2*d**4*e**6 - 8*a*(3*a*c**2*d**2*e**8 + 7*c**3*d**4*e**6 - c**2*d**2*e**6
*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c) + 35*c**3*d**6*e**4 - (15*a*e**2/2 + 15*c*d**2/2)*(3*a**2*c*d*e**9 + 20
1*a*c**2*d**3*e**7/10 + 21*c**3*d**5*e**5 - (17*a*e**2/2 + 17*c*d**2/2)*(3*a*c**2*d**2*e**8 + 7*c**3*d**4*e**6
- c**2*d**2*e**6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c*d*e))/(8*c*d*e))/(7*c) + 21*c**3*d**8*e**2 - (11*a*e**2
/2 + 11*c*d**2/2)*(7*a**3*d*e**9 + 63*a**2*c*d**3*e**7 + 105*a*c**2*d**5*e**5 - 7*a*(3*a**2*c*d*e**9 + 201*a*c
**2*d**3*e**7/10 + 21*c**3*d**5*e**5 - (17*a*e**2/2 + 17*c*d**2/2)*(3*a*c**2*d**2*e**8 + 7*c**3*d**4*e**6 - c*
*2*d**2*e**6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c*d*e))/(8*c) + 35*c**3*d**7*e**3 - (13*a*e**2/2 + 13*c*d**2/2
)*(a**3*e**10 + 21*a**2*c*d**2*e**8 + 63*a*c**2*d**4*e**6 - 8*a*(3*a*c**2*d**2*e**8 + 7*c**3*d**4*e**6 - c**2*
d**2*e**6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c) + 35*c**3*d**6*e**4 - (15*a*e**2/2 + 15*c*d**2/2)*(3*a**2*c*d*
e**9 + 201*a*c**2*d**3*e**7/10 + 21*c**3*d**5*e**5 - (17*a*e**2/2 + 17*c*d**2/2)*(3*a*c**2*d**2*e**8 + 7*c**3*
d**4*e**6 - c**2*d**2*e**6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c*d*e))/(8*c*d*e))/(7*c*d*e))/(6*c*d*e))/(5*c*d*
e))/(4*c*d*e))/(3*c*d*e) + x*(21*a**3*d**5*e**5 + 21*a**2*c*d**7*e**3 + 3*a*c**2*d**9*e - 3*a*(35*a**3*d**3*e*
*7 + 105*a**2*c*d**5*e**5 + 63*a*c**2*d**7*e**3 - 5*a*(7*a**3*d*e**9 + 63*a**2*c*d**3*e**7 + 105*a*c**2*d**5*e
**5 - 7*a*(3*a**2*c*d*e**9 + 201*a*c**2*d**3*e**7/10 + 21*c**3*d**5*e**5 - (17*a*e**2/2 + 17*c*d**2/2)*(3*a*c*
*2*d**2*e**8 + 7*c**3*d**4*e**6 - c**2*d**2*e**6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c*d*e))/(8*c) + 35*c**3*d*
*7*e**3 - (13*a*e**2/2 + 13*c*d**2/2)*(a**3*e**10 + 21*a**2*c*d**2*e**8 + 63*a*c**2*d**4*e**6 - 8*a*(3*a*c**2*
d**2*e**8 + 7*c**3*d**4*e**6 - c**2*d**2*e**6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c) + 35*c**3*d**6*e**4 - (15*
a*e**2/2 + 15*c*d**2/2)*(3*a**2*c*d*e**9 + 201*a*c**2*d**3*e**7/10 + 21*c**3*d**5*e**5 - (17*a*e**2/2 + 17*c*d
**2/2)*(3*a*c**2*d**2*e**8 + 7*c**3*d**4*e**6 - c**2*d**2*e**6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c*d*e))/(8*c
*d*e))/(7*c*d*e))/(6*c) + 7*c**3*d**9*e - (9*a*e**2/2 + 9*c*d**2/2)*(21*a**3*d**2*e**8 + 105*a**2*c*d**4*e**6
+ 105*a*c**2*d**6*e**4 - 6*a*(a**3*e**10 + 21*a**2*c*d**2*e**8 + 63*a*c**2*d**4*e**6 - 8*a*(3*a*c**2*d**2*e**8
+ 7*c**3*d**4*e**6 - c**2*d**2*e**6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c) + 35*c**3*d**6*e**4 - (15*a*e**2/2
+ 15*c*d**2/2)*(3*a**2*c*d*e**9 + 201*a*c**2*d**3*e**7/10 + 21*c**3*d**5*e**5 - (17*a*e**2/2 + 17*c*d**2/2)*(3
*a*c**2*d**2*e**8 + 7*c**3*d**4*e**6 - c**2*d**2*e**6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c*d*e))/(8*c*d*e))/(7
*c) + 21*c**3*d**8*e**2 - (11*a*e**2/2 + 11*c*d**2/2)*(7*a**3*d*e**9 + 63*a**2*c*d**3*e**7 + 105*a*c**2*d**5*e
**5 - 7*a*(3*a**2*c*d*e**9 + 201*a*c**2*d**3*e**7/10 + 21*c**3*d**5*e**5 - (17*a*e**2/2 + 17*c*d**2/2)*(3*a*c*
*2*d**2*e**8 + 7*c**3*d**4*e**6 - c**2*d**2*e**6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c*d*e))/(8*c) + 35*c**3*d*
*7*e**3 - (13*a*e**2/2 + 13*c*d**2/2)*(a**3*e**10 + 21*a**2*c*d**2*e**8 + 63*a*c**2*d**4*e**6 - 8*a*(3*a*c**2*
d**2*e**8 + 7*c**3*d**4*e**6 - c**2*d**2*e**6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c) + 35*c**3*d**6*e**4 - (15*
a*e**2/2 + 15*c*d**2/2)*(3*a**2*c*d*e**9 + 201*a*c**2*d**3*e**7/10 + 21*c**3*d**5*e**5 - (17*a*e**2/2 + 17*c*d
**2/2)*(3*a*c**2*d**2*e**8 + 7*c**3*d**4*e**6 - c**2*d**2*e**6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c*d*e))/(8*c
*d*e))/(7*c*d*e))/(6*c*d*e))/(5*c*d*e))/(4*c) - (5*a*e**2/2 + 5*c*d**2/2)*(35*a**3*d**4*e**6 + 63*a**2*c*d**6*
e**4 + 21*a*c**2*d**8*e**2 - 4*a*(21*a**3*d**2*e**8 + 105*a**2*c*d**4*e**6 + 105*a*c**2*d**6*e**4 - 6*a*(a**3*
e**10 + 21*a**2*c*d**2*e**8 + 63*a*c**2*d**4*e**6 - 8*a*(3*a*c**2*d**2*e**8 + 7*c**3*d**4*e**6 - c**2*d**2*e**
6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c) + 35*c**3*d**6*e**4 - (15*a*e**2/2 + 15*c*d**2/2)*(3*a**2*c*d*e**9 + 2
01*a*c**2*d**3*e**7/10 + 21*c**3*d**5*e**5 - (17*a*e**2/2 + 17*c*d**2/2)*(3*a*c**2*d**2*e**8 + 7*c**3*d**4*e**
6 - c**2*d**2*e**6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c*d*e))/(8*c*d*e))/(7*c) + 21*c**3*d**8*e**2 - (11*a*e**
2/2 + 11*c*d**2/2)*(7*a**3*d*e**9 + 63*a**2*c*d**3*e**7 + 105*a*c**2*d**5*e**5 - 7*a*(3*a**2*c*d*e**9 + 201*a*
c**2*d**3*e**7/10 + 21*c**3*d**5*e**5 - (17*a*e**2/2 + 17*c*d**2/2)*(3*a*c**2*d**2*e**8 + 7*c**3*d**4*e**6 - c
**2*d**2*e**6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c*d*e))/(8*c) + 35*c**3*d**7*e**3 - (13*a*e**2/2 + 13*c*d**2/
2)*(a**3*e**10 + 21*a**2*c*d**2*e**8 + 63*a*c**2*d**4*e**6 - 8*a*(3*a*c**2*d**2*e**8 + 7*c**3*d**4*e**6 - c**2
*d**2*e**6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c) + 35*c**3*d**6*e**4 - (15*a*e**2/2 + 15*c*d**2/2)*(3*a**2*c*d
*e**9 + 201*a*c**2*d**3*e**7/10 + 21*c**3*d**5*e**5 - (17*a*e**2/2 + 17*c*d**2/2)*(3*a*c**2*d**2*e**8 + 7*c**3
*d**4*e**6 - c**2*d**2*e**6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c*d*e))/(8*c*d*e))/(7*c*d*e))/(6*c*d*e))/(5*c)
+ c**3*d**10 - (7*a*e**2/2 + 7*c*d**2/2)*(35*a**3*d**3*e**7 + 105*a**2*c*d**5*e**5 + 63*a*c**2*d**7*e**3 - 5*a
*(7*a**3*d*e**9 + 63*a**2*c*d**3*e**7 + 105*a*c**2*d**5*e**5 - 7*a*(3*a**2*c*d*e**9 + 201*a*c**2*d**3*e**7/10
+ 21*c**3*d**5*e**5 - (17*a*e**2/2 + 17*c*d**2/2)*(3*a*c**2*d**2*e**8 + 7*c**3*d**4*e**6 - c**2*d**2*e**6*(19*
a*e**2/2 + 19*c*d**2/2)/10)/(9*c*d*e))/(8*c) + 35*c**3*d**7*e**3 - (13*a*e**2/2 + 13*c*d**2/2)*(a**3*e**10 + 2
1*a**2*c*d**2*e**8 + 63*a*c**2*d**4*e**6 - 8*a*(3*a*c**2*d**2*e**8 + 7*c**3*d**4*e**6 - c**2*d**2*e**6*(19*a*e
**2/2 + 19*c*d**2/2)/10)/(9*c) + 35*c**3*d**6*e**4 - (15*a*e**2/2 + 15*c*d**2/2)*(3*a**2*c*d*e**9 + 201*a*c**2
*d**3*e**7/10 + 21*c**3*d**5*e**5 - (17*a*e**2/2 + 17*c*d**2/2)*(3*a*c**2*d**2*e**8 + 7*c**3*d**4*e**6 - c**2*
d**2*e**6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c*d*e))/(8*c*d*e))/(7*c*d*e))/(6*c) + 7*c**3*d**9*e - (9*a*e**2/2
+ 9*c*d**2/2)*(21*a**3*d**2*e**8 + 105*a**2*c*d**4*e**6 + 105*a*c**2*d**6*e**4 - 6*a*(a**3*e**10 + 21*a**2*c*
d**2*e**8 + 63*a*c**2*d**4*e**6 - 8*a*(3*a*c**2*d**2*e**8 + 7*c**3*d**4*e**6 - c**2*d**2*e**6*(19*a*e**2/2 + 1
9*c*d**2/2)/10)/(9*c) + 35*c**3*d**6*e**4 - (15*a*e**2/2 + 15*c*d**2/2)*(3*a**2*c*d*e**9 + 201*a*c**2*d**3*e**
7/10 + 21*c**3*d**5*e**5 - (17*a*e**2/2 + 17*c*d**2/2)*(3*a*c**2*d**2*e**8 + 7*c**3*d**4*e**6 - c**2*d**2*e**6
*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c*d*e))/(8*c*d*e))/(7*c) + 21*c**3*d**8*e**2 - (11*a*e**2/2 + 11*c*d**2/2)
*(7*a**3*d*e**9 + 63*a**2*c*d**3*e**7 + 105*a*c**2*d**5*e**5 - 7*a*(3*a**2*c*d*e**9 + 201*a*c**2*d**3*e**7/10
+ 21*c**3*d**5*e**5 - (17*a*e**2/2 + 17*c*d**2/2)*(3*a*c**2*d**2*e**8 + 7*c**3*d**4*e**6 - c**2*d**2*e**6*(19*
a*e**2/2 + 19*c*d**2/2)/10)/(9*c*d*e))/(8*c) + 35*c**3*d**7*e**3 - (13*a*e**2/2 + 13*c*d**2/2)*(a**3*e**10 + 2
1*a**2*c*d**2*e**8 + 63*a*c**2*d**4*e**6 - 8*a*(3*a*c**2*d**2*e**8 + 7*c**3*d**4*e**6 - c**2*d**2*e**6*(19*a*e
**2/2 + 19*c*d**2/2)/10)/(9*c) + 35*c**3*d**6*e**4 - (15*a*e**2/2 + 15*c*d**2/2)*(3*a**2*c*d*e**9 + 201*a*c**2
*d**3*e**7/10 + 21*c**3*d**5*e**5 - (17*a*e**2/2 + 17*c*d**2/2)*(3*a*c**2*d**2*e**8 + 7*c**3*d**4*e**6 - c**2*
d**2*e**6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c*d*e))/(8*c*d*e))/(7*c*d*e))/(6*c*d*e))/(5*c*d*e))/(4*c*d*e))/(3
*c*d*e))/(2*c*d*e) + (7*a**3*d**6*e**4 + 3*a**2*c*d**8*e**2 - 2*a*(35*a**3*d**4*e**6 + 63*a**2*c*d**6*e**4 + 2
1*a*c**2*d**8*e**2 - 4*a*(21*a**3*d**2*e**8 + 105*a**2*c*d**4*e**6 + 105*a*c**2*d**6*e**4 - 6*a*(a**3*e**10 +
21*a**2*c*d**2*e**8 + 63*a*c**2*d**4*e**6 - 8*a*(3*a*c**2*d**2*e**8 + 7*c**3*d**4*e**6 - c**2*d**2*e**6*(19*a*
e**2/2 + 19*c*d**2/2)/10)/(9*c) + 35*c**3*d**6*e**4 - (15*a*e**2/2 + 15*c*d**2/2)*(3*a**2*c*d*e**9 + 201*a*c**
2*d**3*e**7/10 + 21*c**3*d**5*e**5 - (17*a*e**2/2 + 17*c*d**2/2)*(3*a*c**2*d**2*e**8 + 7*c**3*d**4*e**6 - c**2
*d**2*e**6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c*d*e))/(8*c*d*e))/(7*c) + 21*c**3*d**8*e**2 - (11*a*e**2/2 + 11
*c*d**2/2)*(7*a**3*d*e**9 + 63*a**2*c*d**3*e**7 + 105*a*c**2*d**5*e**5 - 7*a*(3*a**2*c*d*e**9 + 201*a*c**2*d**
3*e**7/10 + 21*c**3*d**5*e**5 - (17*a*e**2/2 + 17*c*d**2/2)*(3*a*c**2*d**2*e**8 + 7*c**3*d**4*e**6 - c**2*d**2
*e**6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c*d*e))/(8*c) + 35*c**3*d**7*e**3 - (13*a*e**2/2 + 13*c*d**2/2)*(a**3
*e**10 + 21*a**2*c*d**2*e**8 + 63*a*c**2*d**4*e**6 - 8*a*(3*a*c**2*d**2*e**8 + 7*c**3*d**4*e**6 - c**2*d**2*e*
*6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c) + 35*c**3*d**6*e**4 - (15*a*e**2/2 + 15*c*d**2/2)*(3*a**2*c*d*e**9 +
201*a*c**2*d**3*e**7/10 + 21*c**3*d**5*e**5 - (17*a*e**2/2 + 17*c*d**2/2)*(3*a*c**2*d**2*e**8 + 7*c**3*d**4*e*
*6 - c**2*d**2*e**6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c*d*e))/(8*c*d*e))/(7*c*d*e))/(6*c*d*e))/(5*c) + c**3*d
**10 - (7*a*e**2/2 + 7*c*d**2/2)*(35*a**3*d**3*e**7 + 105*a**2*c*d**5*e**5 + 63*a*c**2*d**7*e**3 - 5*a*(7*a**3
*d*e**9 + 63*a**2*c*d**3*e**7 + 105*a*c**2*d**5*e**5 - 7*a*(3*a**2*c*d*e**9 + 201*a*c**2*d**3*e**7/10 + 21*c**
3*d**5*e**5 - (17*a*e**2/2 + 17*c*d**2/2)*(3*a*c**2*d**2*e**8 + 7*c**3*d**4*e**6 - c**2*d**2*e**6*(19*a*e**2/2
+ 19*c*d**2/2)/10)/(9*c*d*e))/(8*c) + 35*c**3*d**7*e**3 - (13*a*e**2/2 + 13*c*d**2/2)*(a**3*e**10 + 21*a**2*c
*d**2*e**8 + 63*a*c**2*d**4*e**6 - 8*a*(3*a*c**2*d**2*e**8 + 7*c**3*d**4*e**6 - c**2*d**2*e**6*(19*a*e**2/2 +
19*c*d**2/2)/10)/(9*c) + 35*c**3*d**6*e**4 - (15*a*e**2/2 + 15*c*d**2/2)*(3*a**2*c*d*e**9 + 201*a*c**2*d**3*e*
*7/10 + 21*c**3*d**5*e**5 - (17*a*e**2/2 + 17*c*d**2/2)*(3*a*c**2*d**2*e**8 + 7*c**3*d**4*e**6 - c**2*d**2*e**
6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c*d*e))/(8*c*d*e))/(7*c*d*e))/(6*c) + 7*c**3*d**9*e - (9*a*e**2/2 + 9*c*d
**2/2)*(21*a**3*d**2*e**8 + 105*a**2*c*d**4*e**6 + 105*a*c**2*d**6*e**4 - 6*a*(a**3*e**10 + 21*a**2*c*d**2*e**
8 + 63*a*c**2*d**4*e**6 - 8*a*(3*a*c**2*d**2*e**8 + 7*c**3*d**4*e**6 - c**2*d**2*e**6*(19*a*e**2/2 + 19*c*d**2
/2)/10)/(9*c) + 35*c**3*d**6*e**4 - (15*a*e**2/2 + 15*c*d**2/2)*(3*a**2*c*d*e**9 + 201*a*c**2*d**3*e**7/10 + 2
1*c**3*d**5*e**5 - (17*a*e**2/2 + 17*c*d**2/2)*(3*a*c**2*d**2*e**8 + 7*c**3*d**4*e**6 - c**2*d**2*e**6*(19*a*e
**2/2 + 19*c*d**2/2)/10)/(9*c*d*e))/(8*c*d*e))/(7*c) + 21*c**3*d**8*e**2 - (11*a*e**2/2 + 11*c*d**2/2)*(7*a**3
*d*e**9 + 63*a**2*c*d**3*e**7 + 105*a*c**2*d**5*e**5 - 7*a*(3*a**2*c*d*e**9 + 201*a*c**2*d**3*e**7/10 + 21*c**
3*d**5*e**5 - (17*a*e**2/2 + 17*c*d**2/2)*(3*a*c**2*d**2*e**8 + 7*c**3*d**4*e**6 - c**2*d**2*e**6*(19*a*e**2/2
+ 19*c*d**2/2)/10)/(9*c*d*e))/(8*c) + 35*c**3*d**7*e**3 - (13*a*e**2/2 + 13*c*d**2/2)*(a**3*e**10 + 21*a**2*c
*d**2*e**8 + 63*a*c**2*d**4*e**6 - 8*a*(3*a*c**2*d**2*e**8 + 7*c**3*d**4*e**6 - c**2*d**2*e**6*(19*a*e**2/2 +
19*c*d**2/2)/10)/(9*c) + 35*c**3*d**6*e**4 - (15*a*e**2/2 + 15*c*d**2/2)*(3*a**2*c*d*e**9 + 201*a*c**2*d**3*e*
*7/10 + 21*c**3*d**5*e**5 - (17*a*e**2/2 + 17*c*d**2/2)*(3*a*c**2*d**2*e**8 + 7*c**3*d**4*e**6 - c**2*d**2*e**
6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c*d*e))/(8*c*d*e))/(7*c*d*e))/(6*c*d*e))/(5*c*d*e))/(4*c*d*e))/(3*c) - (3
*a*e**2/2 + 3*c*d**2/2)*(21*a**3*d**5*e**5 + 21*a**2*c*d**7*e**3 + 3*a*c**2*d**9*e - 3*a*(35*a**3*d**3*e**7 +
105*a**2*c*d**5*e**5 + 63*a*c**2*d**7*e**3 - 5*a*(7*a**3*d*e**9 + 63*a**2*c*d**3*e**7 + 105*a*c**2*d**5*e**5 -
7*a*(3*a**2*c*d*e**9 + 201*a*c**2*d**3*e**7/10 + 21*c**3*d**5*e**5 - (17*a*e**2/2 + 17*c*d**2/2)*(3*a*c**2*d*
*2*e**8 + 7*c**3*d**4*e**6 - c**2*d**2*e**6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c*d*e))/(8*c) + 35*c**3*d**7*e*
*3 - (13*a*e**2/2 + 13*c*d**2/2)*(a**3*e**10 + 21*a**2*c*d**2*e**8 + 63*a*c**2*d**4*e**6 - 8*a*(3*a*c**2*d**2*
e**8 + 7*c**3*d**4*e**6 - c**2*d**2*e**6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c) + 35*c**3*d**6*e**4 - (15*a*e**
2/2 + 15*c*d**2/2)*(3*a**2*c*d*e**9 + 201*a*c**2*d**3*e**7/10 + 21*c**3*d**5*e**5 - (17*a*e**2/2 + 17*c*d**2/2
)*(3*a*c**2*d**2*e**8 + 7*c**3*d**4*e**6 - c**2*d**2*e**6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c*d*e))/(8*c*d*e)
)/(7*c*d*e))/(6*c) + 7*c**3*d**9*e - (9*a*e**2/2 + 9*c*d**2/2)*(21*a**3*d**2*e**8 + 105*a**2*c*d**4*e**6 + 105
*a*c**2*d**6*e**4 - 6*a*(a**3*e**10 + 21*a**2*c*d**2*e**8 + 63*a*c**2*d**4*e**6 - 8*a*(3*a*c**2*d**2*e**8 + 7*
c**3*d**4*e**6 - c**2*d**2*e**6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c) + 35*c**3*d**6*e**4 - (15*a*e**2/2 + 15*
c*d**2/2)*(3*a**2*c*d*e**9 + 201*a*c**2*d**3*e**7/10 + 21*c**3*d**5*e**5 - (17*a*e**2/2 + 17*c*d**2/2)*(3*a*c*
*2*d**2*e**8 + 7*c**3*d**4*e**6 - c**2*d**2*e**6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c*d*e))/(8*c*d*e))/(7*c) +
21*c**3*d**8*e**2 - (11*a*e**2/2 + 11*c*d**2/2)*(7*a**3*d*e**9 + 63*a**2*c*d**3*e**7 + 105*a*c**2*d**5*e**5 -
7*a*(3*a**2*c*d*e**9 + 201*a*c**2*d**3*e**7/10 + 21*c**3*d**5*e**5 - (17*a*e**2/2 + 17*c*d**2/2)*(3*a*c**2*d*
*2*e**8 + 7*c**3*d**4*e**6 - c**2*d**2*e**6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c*d*e))/(8*c) + 35*c**3*d**7*e*
*3 - (13*a*e**2/2 + 13*c*d**2/2)*(a**3*e**10 + 21*a**2*c*d**2*e**8 + 63*a*c**2*d**4*e**6 - 8*a*(3*a*c**2*d**2*
e**8 + 7*c**3*d**4*e**6 - c**2*d**2*e**6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c) + 35*c**3*d**6*e**4 - (15*a*e**
2/2 + 15*c*d**2/2)*(3*a**2*c*d*e**9 + 201*a*c**2*d**3*e**7/10 + 21*c**3*d**5*e**5 - (17*a*e**2/2 + 17*c*d**2/2
)*(3*a*c**2*d**2*e**8 + 7*c**3*d**4*e**6 - c**2*d**2*e**6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c*d*e))/(8*c*d*e)
)/(7*c*d*e))/(6*c*d*e))/(5*c*d*e))/(4*c) - (5*a*e**2/2 + 5*c*d**2/2)*(35*a**3*d**4*e**6 + 63*a**2*c*d**6*e**4
+ 21*a*c**2*d**8*e**2 - 4*a*(21*a**3*d**2*e**8 + 105*a**2*c*d**4*e**6 + 105*a*c**2*d**6*e**4 - 6*a*(a**3*e**10
+ 21*a**2*c*d**2*e**8 + 63*a*c**2*d**4*e**6 - 8*a*(3*a*c**2*d**2*e**8 + 7*c**3*d**4*e**6 - c**2*d**2*e**6*(19
*a*e**2/2 + 19*c*d**2/2)/10)/(9*c) + 35*c**3*d**6*e**4 - (15*a*e**2/2 + 15*c*d**2/2)*(3*a**2*c*d*e**9 + 201*a*
c**2*d**3*e**7/10 + 21*c**3*d**5*e**5 - (17*a*e**2/2 + 17*c*d**2/2)*(3*a*c**2*d**2*e**8 + 7*c**3*d**4*e**6 - c
**2*d**2*e**6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c*d*e))/(8*c*d*e))/(7*c) + 21*c**3*d**8*e**2 - (11*a*e**2/2 +
11*c*d**2/2)*(7*a**3*d*e**9 + 63*a**2*c*d**3*e**7 + 105*a*c**2*d**5*e**5 - 7*a*(3*a**2*c*d*e**9 + 201*a*c**2*
d**3*e**7/10 + 21*c**3*d**5*e**5 - (17*a*e**2/2 + 17*c*d**2/2)*(3*a*c**2*d**2*e**8 + 7*c**3*d**4*e**6 - c**2*d
**2*e**6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c*d*e))/(8*c) + 35*c**3*d**7*e**3 - (13*a*e**2/2 + 13*c*d**2/2)*(a
**3*e**10 + 21*a**2*c*d**2*e**8 + 63*a*c**2*d**4*e**6 - 8*a*(3*a*c**2*d**2*e**8 + 7*c**3*d**4*e**6 - c**2*d**2
*e**6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c) + 35*c**3*d**6*e**4 - (15*a*e**2/2 + 15*c*d**2/2)*(3*a**2*c*d*e**9
+ 201*a*c**2*d**3*e**7/10 + 21*c**3*d**5*e**5 - (17*a*e**2/2 + 17*c*d**2/2)*(3*a*c**2*d**2*e**8 + 7*c**3*d**4
*e**6 - c**2*d**2*e**6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c*d*e))/(8*c*d*e))/(7*c*d*e))/(6*c*d*e))/(5*c) + c**
3*d**10 - (7*a*e**2/2 + 7*c*d**2/2)*(35*a**3*d**3*e**7 + 105*a**2*c*d**5*e**5 + 63*a*c**2*d**7*e**3 - 5*a*(7*a
**3*d*e**9 + 63*a**2*c*d**3*e**7 + 105*a*c**2*d**5*e**5 - 7*a*(3*a**2*c*d*e**9 + 201*a*c**2*d**3*e**7/10 + 21*
c**3*d**5*e**5 - (17*a*e**2/2 + 17*c*d**2/2)*(3*a*c**2*d**2*e**8 + 7*c**3*d**4*e**6 - c**2*d**2*e**6*(19*a*e**
2/2 + 19*c*d**2/2)/10)/(9*c*d*e))/(8*c) + 35*c**3*d**7*e**3 - (13*a*e**2/2 + 13*c*d**2/2)*(a**3*e**10 + 21*a**
2*c*d**2*e**8 + 63*a*c**2*d**4*e**6 - 8*a*(3*a*c**2*d**2*e**8 + 7*c**3*d**4*e**6 - c**2*d**2*e**6*(19*a*e**2/2
+ 19*c*d**2/2)/10)/(9*c) + 35*c**3*d**6*e**4 - (15*a*e**2/2 + 15*c*d**2/2)*(3*a**2*c*d*e**9 + 201*a*c**2*d**3
*e**7/10 + 21*c**3*d**5*e**5 - (17*a*e**2/2 + 17*c*d**2/2)*(3*a*c**2*d**2*e**8 + 7*c**3*d**4*e**6 - c**2*d**2*
e**6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c*d*e))/(8*c*d*e))/(7*c*d*e))/(6*c) + 7*c**3*d**9*e - (9*a*e**2/2 + 9*
c*d**2/2)*(21*a**3*d**2*e**8 + 105*a**2*c*d**4*e**6 + 105*a*c**2*d**6*e**4 - 6*a*(a**3*e**10 + 21*a**2*c*d**2*
e**8 + 63*a*c**2*d**4*e**6 - 8*a*(3*a*c**2*d**2*e**8 + 7*c**3*d**4*e**6 - c**2*d**2*e**6*(19*a*e**2/2 + 19*c*d
**2/2)/10)/(9*c) + 35*c**3*d**6*e**4 - (15*a*e**2/2 + 15*c*d**2/2)*(3*a**2*c*d*e**9 + 201*a*c**2*d**3*e**7/10
+ 21*c**3*d**5*e**5 - (17*a*e**2/2 + 17*c*d**2/2)*(3*a*c**2*d**2*e**8 + 7*c**3*d**4*e**6 - c**2*d**2*e**6*(19*
a*e**2/2 + 19*c*d**2/2)/10)/(9*c*d*e))/(8*c*d*e))/(7*c) + 21*c**3*d**8*e**2 - (11*a*e**2/2 + 11*c*d**2/2)*(7*a
**3*d*e**9 + 63*a**2*c*d**3*e**7 + 105*a*c**2*d**5*e**5 - 7*a*(3*a**2*c*d*e**9 + 201*a*c**2*d**3*e**7/10 + 21*
c**3*d**5*e**5 - (17*a*e**2/2 + 17*c*d**2/2)*(3*a*c**2*d**2*e**8 + 7*c**3*d**4*e**6 - c**2*d**2*e**6*(19*a*e**
2/2 + 19*c*d**2/2)/10)/(9*c*d*e))/(8*c) + 35*c**3*d**7*e**3 - (13*a*e**2/2 + 13*c*d**2/2)*(a**3*e**10 + 21*a**
2*c*d**2*e**8 + 63*a*c**2*d**4*e**6 - 8*a*(3*a*c**2*d**2*e**8 + 7*c**3*d**4*e**6 - c**2*d**2*e**6*(19*a*e**2/2
+ 19*c*d**2/2)/10)/(9*c) + 35*c**3*d**6*e**4 - (15*a*e**2/2 + 15*c*d**2/2)*(3*a**2*c*d*e**9 + 201*a*c**2*d**3
*e**7/10 + 21*c**3*d**5*e**5 - (17*a*e**2/2 + 17*c*d**2/2)*(3*a*c**2*d**2*e**8 + 7*c**3*d**4*e**6 - c**2*d**2*
e**6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c*d*e))/(8*c*d*e))/(7*c*d*e))/(6*c*d*e))/(5*c*d*e))/(4*c*d*e))/(3*c*d*
e))/(2*c*d*e))/(c*d*e)) + (a**3*d**7*e**3 - a*(21*a**3*d**5*e**5 + 21*a**2*c*d**7*e**3 + 3*a*c**2*d**9*e - 3*a
*(35*a**3*d**3*e**7 + 105*a**2*c*d**5*e**5 + 63*a*c**2*d**7*e**3 - 5*a*(7*a**3*d*e**9 + 63*a**2*c*d**3*e**7 +
105*a*c**2*d**5*e**5 - 7*a*(3*a**2*c*d*e**9 + 201*a*c**2*d**3*e**7/10 + 21*c**3*d**5*e**5 - (17*a*e**2/2 + 17*
c*d**2/2)*(3*a*c**2*d**2*e**8 + 7*c**3*d**4*e**6 - c**2*d**2*e**6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c*d*e))/(
8*c) + 35*c**3*d**7*e**3 - (13*a*e**2/2 + 13*c*d**2/2)*(a**3*e**10 + 21*a**2*c*d**2*e**8 + 63*a*c**2*d**4*e**6
- 8*a*(3*a*c**2*d**2*e**8 + 7*c**3*d**4*e**6 - c**2*d**2*e**6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c) + 35*c**3
*d**6*e**4 - (15*a*e**2/2 + 15*c*d**2/2)*(3*a**2*c*d*e**9 + 201*a*c**2*d**3*e**7/10 + 21*c**3*d**5*e**5 - (17*
a*e**2/2 + 17*c*d**2/2)*(3*a*c**2*d**2*e**8 + 7*c**3*d**4*e**6 - c**2*d**2*e**6*(19*a*e**2/2 + 19*c*d**2/2)/10
)/(9*c*d*e))/(8*c*d*e))/(7*c*d*e))/(6*c) + 7*c**3*d**9*e - (9*a*e**2/2 + 9*c*d**2/2)*(21*a**3*d**2*e**8 + 105*
a**2*c*d**4*e**6 + 105*a*c**2*d**6*e**4 - 6*a*(a**3*e**10 + 21*a**2*c*d**2*e**8 + 63*a*c**2*d**4*e**6 - 8*a*(3
*a*c**2*d**2*e**8 + 7*c**3*d**4*e**6 - c**2*d**2*e**6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c) + 35*c**3*d**6*e**
4 - (15*a*e**2/2 + 15*c*d**2/2)*(3*a**2*c*d*e**9 + 201*a*c**2*d**3*e**7/10 + 21*c**3*d**5*e**5 - (17*a*e**2/2
+ 17*c*d**2/2)*(3*a*c**2*d**2*e**8 + 7*c**3*d**4*e**6 - c**2*d**2*e**6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c*d*
e))/(8*c*d*e))/(7*c) + 21*c**3*d**8*e**2 - (11*a*e**2/2 + 11*c*d**2/2)*(7*a**3*d*e**9 + 63*a**2*c*d**3*e**7 +
105*a*c**2*d**5*e**5 - 7*a*(3*a**2*c*d*e**9 + 201*a*c**2*d**3*e**7/10 + 21*c**3*d**5*e**5 - (17*a*e**2/2 + 17*
c*d**2/2)*(3*a*c**2*d**2*e**8 + 7*c**3*d**4*e**6 - c**2*d**2*e**6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c*d*e))/(
8*c) + 35*c**3*d**7*e**3 - (13*a*e**2/2 + 13*c*d**2/2)*(a**3*e**10 + 21*a**2*c*d**2*e**8 + 63*a*c**2*d**4*e**6
- 8*a*(3*a*c**2*d**2*e**8 + 7*c**3*d**4*e**6 - c**2*d**2*e**6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c) + 35*c**3
*d**6*e**4 - (15*a*e**2/2 + 15*c*d**2/2)*(3*a**2*c*d*e**9 + 201*a*c**2*d**3*e**7/10 + 21*c**3*d**5*e**5 - (17*
a*e**2/2 + 17*c*d**2/2)*(3*a*c**2*d**2*e**8 + 7*c**3*d**4*e**6 - c**2*d**2*e**6*(19*a*e**2/2 + 19*c*d**2/2)/10
)/(9*c*d*e))/(8*c*d*e))/(7*c*d*e))/(6*c*d*e))/(5*c*d*e))/(4*c) - (5*a*e**2/2 + 5*c*d**2/2)*(35*a**3*d**4*e**6
+ 63*a**2*c*d**6*e**4 + 21*a*c**2*d**8*e**2 - 4*a*(21*a**3*d**2*e**8 + 105*a**2*c*d**4*e**6 + 105*a*c**2*d**6*
e**4 - 6*a*(a**3*e**10 + 21*a**2*c*d**2*e**8 + 63*a*c**2*d**4*e**6 - 8*a*(3*a*c**2*d**2*e**8 + 7*c**3*d**4*e**
6 - c**2*d**2*e**6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c) + 35*c**3*d**6*e**4 - (15*a*e**2/2 + 15*c*d**2/2)*(3*
a**2*c*d*e**9 + 201*a*c**2*d**3*e**7/10 + 21*c**3*d**5*e**5 - (17*a*e**2/2 + 17*c*d**2/2)*(3*a*c**2*d**2*e**8
+ 7*c**3*d**4*e**6 - c**2*d**2*e**6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c*d*e))/(8*c*d*e))/(7*c) + 21*c**3*d**8
*e**2 - (11*a*e**2/2 + 11*c*d**2/2)*(7*a**3*d*e**9 + 63*a**2*c*d**3*e**7 + 105*a*c**2*d**5*e**5 - 7*a*(3*a**2*
c*d*e**9 + 201*a*c**2*d**3*e**7/10 + 21*c**3*d**5*e**5 - (17*a*e**2/2 + 17*c*d**2/2)*(3*a*c**2*d**2*e**8 + 7*c
**3*d**4*e**6 - c**2*d**2*e**6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c*d*e))/(8*c) + 35*c**3*d**7*e**3 - (13*a*e*
*2/2 + 13*c*d**2/2)*(a**3*e**10 + 21*a**2*c*d**2*e**8 + 63*a*c**2*d**4*e**6 - 8*a*(3*a*c**2*d**2*e**8 + 7*c**3
*d**4*e**6 - c**2*d**2*e**6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c) + 35*c**3*d**6*e**4 - (15*a*e**2/2 + 15*c*d*
*2/2)*(3*a**2*c*d*e**9 + 201*a*c**2*d**3*e**7/10 + 21*c**3*d**5*e**5 - (17*a*e**2/2 + 17*c*d**2/2)*(3*a*c**2*d
**2*e**8 + 7*c**3*d**4*e**6 - c**2*d**2*e**6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c*d*e))/(8*c*d*e))/(7*c*d*e))/
(6*c*d*e))/(5*c) + c**3*d**10 - (7*a*e**2/2 + 7*c*d**2/2)*(35*a**3*d**3*e**7 + 105*a**2*c*d**5*e**5 + 63*a*c**
2*d**7*e**3 - 5*a*(7*a**3*d*e**9 + 63*a**2*c*d**3*e**7 + 105*a*c**2*d**5*e**5 - 7*a*(3*a**2*c*d*e**9 + 201*a*c
**2*d**3*e**7/10 + 21*c**3*d**5*e**5 - (17*a*e**2/2 + 17*c*d**2/2)*(3*a*c**2*d**2*e**8 + 7*c**3*d**4*e**6 - c*
*2*d**2*e**6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c*d*e))/(8*c) + 35*c**3*d**7*e**3 - (13*a*e**2/2 + 13*c*d**2/2
)*(a**3*e**10 + 21*a**2*c*d**2*e**8 + 63*a*c**2*d**4*e**6 - 8*a*(3*a*c**2*d**2*e**8 + 7*c**3*d**4*e**6 - c**2*
d**2*e**6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c) + 35*c**3*d**6*e**4 - (15*a*e**2/2 + 15*c*d**2/2)*(3*a**2*c*d*
e**9 + 201*a*c**2*d**3*e**7/10 + 21*c**3*d**5*e**5 - (17*a*e**2/2 + 17*c*d**2/2)*(3*a*c**2*d**2*e**8 + 7*c**3*
d**4*e**6 - c**2*d**2*e**6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c*d*e))/(8*c*d*e))/(7*c*d*e))/(6*c) + 7*c**3*d**
9*e - (9*a*e**2/2 + 9*c*d**2/2)*(21*a**3*d**2*e**8 + 105*a**2*c*d**4*e**6 + 105*a*c**2*d**6*e**4 - 6*a*(a**3*e
**10 + 21*a**2*c*d**2*e**8 + 63*a*c**2*d**4*e**6 - 8*a*(3*a*c**2*d**2*e**8 + 7*c**3*d**4*e**6 - c**2*d**2*e**6
*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c) + 35*c**3*d**6*e**4 - (15*a*e**2/2 + 15*c*d**2/2)*(3*a**2*c*d*e**9 + 20
1*a*c**2*d**3*e**7/10 + 21*c**3*d**5*e**5 - (17*a*e**2/2 + 17*c*d**2/2)*(3*a*c**2*d**2*e**8 + 7*c**3*d**4*e**6
- c**2*d**2*e**6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c*d*e))/(8*c*d*e))/(7*c) + 21*c**3*d**8*e**2 - (11*a*e**2
/2 + 11*c*d**2/2)*(7*a**3*d*e**9 + 63*a**2*c*d**3*e**7 + 105*a*c**2*d**5*e**5 - 7*a*(3*a**2*c*d*e**9 + 201*a*c
**2*d**3*e**7/10 + 21*c**3*d**5*e**5 - (17*a*e**2/2 + 17*c*d**2/2)*(3*a*c**2*d**2*e**8 + 7*c**3*d**4*e**6 - c*
*2*d**2*e**6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c*d*e))/(8*c) + 35*c**3*d**7*e**3 - (13*a*e**2/2 + 13*c*d**2/2
)*(a**3*e**10 + 21*a**2*c*d**2*e**8 + 63*a*c**2*d**4*e**6 - 8*a*(3*a*c**2*d**2*e**8 + 7*c**3*d**4*e**6 - c**2*
d**2*e**6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c) + 35*c**3*d**6*e**4 - (15*a*e**2/2 + 15*c*d**2/2)*(3*a**2*c*d*
e**9 + 201*a*c**2*d**3*e**7/10 + 21*c**3*d**5*e**5 - (17*a*e**2/2 + 17*c*d**2/2)*(3*a*c**2*d**2*e**8 + 7*c**3*
d**4*e**6 - c**2*d**2*e**6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c*d*e))/(8*c*d*e))/(7*c*d*e))/(6*c*d*e))/(5*c*d*
e))/(4*c*d*e))/(3*c*d*e))/(2*c) - (a*e**2 + c*d**2)*(7*a**3*d**6*e**4 + 3*a**2*c*d**8*e**2 - 2*a*(35*a**3*d**4
*e**6 + 63*a**2*c*d**6*e**4 + 21*a*c**2*d**8*e**2 - 4*a*(21*a**3*d**2*e**8 + 105*a**2*c*d**4*e**6 + 105*a*c**2
*d**6*e**4 - 6*a*(a**3*e**10 + 21*a**2*c*d**2*e**8 + 63*a*c**2*d**4*e**6 - 8*a*(3*a*c**2*d**2*e**8 + 7*c**3*d*
*4*e**6 - c**2*d**2*e**6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c) + 35*c**3*d**6*e**4 - (15*a*e**2/2 + 15*c*d**2/
2)*(3*a**2*c*d*e**9 + 201*a*c**2*d**3*e**7/10 + 21*c**3*d**5*e**5 - (17*a*e**2/2 + 17*c*d**2/2)*(3*a*c**2*d**2
*e**8 + 7*c**3*d**4*e**6 - c**2*d**2*e**6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c*d*e))/(8*c*d*e))/(7*c) + 21*c**
3*d**8*e**2 - (11*a*e**2/2 + 11*c*d**2/2)*(7*a**3*d*e**9 + 63*a**2*c*d**3*e**7 + 105*a*c**2*d**5*e**5 - 7*a*(3
*a**2*c*d*e**9 + 201*a*c**2*d**3*e**7/10 + 21*c**3*d**5*e**5 - (17*a*e**2/2 + 17*c*d**2/2)*(3*a*c**2*d**2*e**8
+ 7*c**3*d**4*e**6 - c**2*d**2*e**6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c*d*e))/(8*c) + 35*c**3*d**7*e**3 - (1
3*a*e**2/2 + 13*c*d**2/2)*(a**3*e**10 + 21*a**2*c*d**2*e**8 + 63*a*c**2*d**4*e**6 - 8*a*(3*a*c**2*d**2*e**8 +
7*c**3*d**4*e**6 - c**2*d**2*e**6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c) + 35*c**3*d**6*e**4 - (15*a*e**2/2 + 1
5*c*d**2/2)*(3*a**2*c*d*e**9 + 201*a*c**2*d**3*e**7/10 + 21*c**3*d**5*e**5 - (17*a*e**2/2 + 17*c*d**2/2)*(3*a*
c**2*d**2*e**8 + 7*c**3*d**4*e**6 - c**2*d**2*e**6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c*d*e))/(8*c*d*e))/(7*c*
d*e))/(6*c*d*e))/(5*c) + c**3*d**10 - (7*a*e**2/2 + 7*c*d**2/2)*(35*a**3*d**3*e**7 + 105*a**2*c*d**5*e**5 + 63
*a*c**2*d**7*e**3 - 5*a*(7*a**3*d*e**9 + 63*a**2*c*d**3*e**7 + 105*a*c**2*d**5*e**5 - 7*a*(3*a**2*c*d*e**9 + 2
01*a*c**2*d**3*e**7/10 + 21*c**3*d**5*e**5 - (17*a*e**2/2 + 17*c*d**2/2)*(3*a*c**2*d**2*e**8 + 7*c**3*d**4*e**
6 - c**2*d**2*e**6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c*d*e))/(8*c) + 35*c**3*d**7*e**3 - (13*a*e**2/2 + 13*c*
d**2/2)*(a**3*e**10 + 21*a**2*c*d**2*e**8 + 63*a*c**2*d**4*e**6 - 8*a*(3*a*c**2*d**2*e**8 + 7*c**3*d**4*e**6 -
c**2*d**2*e**6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c) + 35*c**3*d**6*e**4 - (15*a*e**2/2 + 15*c*d**2/2)*(3*a**
2*c*d*e**9 + 201*a*c**2*d**3*e**7/10 + 21*c**3*d**5*e**5 - (17*a*e**2/2 + 17*c*d**2/2)*(3*a*c**2*d**2*e**8 + 7
*c**3*d**4*e**6 - c**2*d**2*e**6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c*d*e))/(8*c*d*e))/(7*c*d*e))/(6*c) + 7*c*
*3*d**9*e - (9*a*e**2/2 + 9*c*d**2/2)*(21*a**3*d**2*e**8 + 105*a**2*c*d**4*e**6 + 105*a*c**2*d**6*e**4 - 6*a*(
a**3*e**10 + 21*a**2*c*d**2*e**8 + 63*a*c**2*d**4*e**6 - 8*a*(3*a*c**2*d**2*e**8 + 7*c**3*d**4*e**6 - c**2*d**
2*e**6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c) + 35*c**3*d**6*e**4 - (15*a*e**2/2 + 15*c*d**2/2)*(3*a**2*c*d*e**
9 + 201*a*c**2*d**3*e**7/10 + 21*c**3*d**5*e**5 - (17*a*e**2/2 + 17*c*d**2/2)*(3*a*c**2*d**2*e**8 + 7*c**3*d**
4*e**6 - c**2*d**2*e**6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c*d*e))/(8*c*d*e))/(7*c) + 21*c**3*d**8*e**2 - (11*
a*e**2/2 + 11*c*d**2/2)*(7*a**3*d*e**9 + 63*a**2*c*d**3*e**7 + 105*a*c**2*d**5*e**5 - 7*a*(3*a**2*c*d*e**9 + 2
01*a*c**2*d**3*e**7/10 + 21*c**3*d**5*e**5 - (17*a*e**2/2 + 17*c*d**2/2)*(3*a*c**2*d**2*e**8 + 7*c**3*d**4*e**
6 - c**2*d**2*e**6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c*d*e))/(8*c) + 35*c**3*d**7*e**3 - (13*a*e**2/2 + 13*c*
d**2/2)*(a**3*e**10 + 21*a**2*c*d**2*e**8 + 63*a*c**2*d**4*e**6 - 8*a*(3*a*c**2*d**2*e**8 + 7*c**3*d**4*e**6 -
c**2*d**2*e**6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c) + 35*c**3*d**6*e**4 - (15*a*e**2/2 + 15*c*d**2/2)*(3*a**
2*c*d*e**9 + 201*a*c**2*d**3*e**7/10 + 21*c**3*d**5*e**5 - (17*a*e**2/2 + 17*c*d**2/2)*(3*a*c**2*d**2*e**8 + 7
*c**3*d**4*e**6 - c**2*d**2*e**6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c*d*e))/(8*c*d*e))/(7*c*d*e))/(6*c*d*e))/(
5*c*d*e))/(4*c*d*e))/(3*c) - (3*a*e**2/2 + 3*c*d**2/2)*(21*a**3*d**5*e**5 + 21*a**2*c*d**7*e**3 + 3*a*c**2*d**
9*e - 3*a*(35*a**3*d**3*e**7 + 105*a**2*c*d**5*e**5 + 63*a*c**2*d**7*e**3 - 5*a*(7*a**3*d*e**9 + 63*a**2*c*d**
3*e**7 + 105*a*c**2*d**5*e**5 - 7*a*(3*a**2*c*d*e**9 + 201*a*c**2*d**3*e**7/10 + 21*c**3*d**5*e**5 - (17*a*e**
2/2 + 17*c*d**2/2)*(3*a*c**2*d**2*e**8 + 7*c**3*d**4*e**6 - c**2*d**2*e**6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*
c*d*e))/(8*c) + 35*c**3*d**7*e**3 - (13*a*e**2/2 + 13*c*d**2/2)*(a**3*e**10 + 21*a**2*c*d**2*e**8 + 63*a*c**2*
d**4*e**6 - 8*a*(3*a*c**2*d**2*e**8 + 7*c**3*d**4*e**6 - c**2*d**2*e**6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c)
+ 35*c**3*d**6*e**4 - (15*a*e**2/2 + 15*c*d**2/2)*(3*a**2*c*d*e**9 + 201*a*c**2*d**3*e**7/10 + 21*c**3*d**5*e*
*5 - (17*a*e**2/2 + 17*c*d**2/2)*(3*a*c**2*d**2*e**8 + 7*c**3*d**4*e**6 - c**2*d**2*e**6*(19*a*e**2/2 + 19*c*d
**2/2)/10)/(9*c*d*e))/(8*c*d*e))/(7*c*d*e))/(6*c) + 7*c**3*d**9*e - (9*a*e**2/2 + 9*c*d**2/2)*(21*a**3*d**2*e*
*8 + 105*a**2*c*d**4*e**6 + 105*a*c**2*d**6*e**4 - 6*a*(a**3*e**10 + 21*a**2*c*d**2*e**8 + 63*a*c**2*d**4*e**6
- 8*a*(3*a*c**2*d**2*e**8 + 7*c**3*d**4*e**6 - c**2*d**2*e**6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c) + 35*c**3
*d**6*e**4 - (15*a*e**2/2 + 15*c*d**2/2)*(3*a**2*c*d*e**9 + 201*a*c**2*d**3*e**7/10 + 21*c**3*d**5*e**5 - (17*
a*e**2/2 + 17*c*d**2/2)*(3*a*c**2*d**2*e**8 + 7*c**3*d**4*e**6 - c**2*d**2*e**6*(19*a*e**2/2 + 19*c*d**2/2)/10
)/(9*c*d*e))/(8*c*d*e))/(7*c) + 21*c**3*d**8*e**2 - (11*a*e**2/2 + 11*c*d**2/2)*(7*a**3*d*e**9 + 63*a**2*c*d**
3*e**7 + 105*a*c**2*d**5*e**5 - 7*a*(3*a**2*c*d*e**9 + 201*a*c**2*d**3*e**7/10 + 21*c**3*d**5*e**5 - (17*a*e**
2/2 + 17*c*d**2/2)*(3*a*c**2*d**2*e**8 + 7*c**3*d**4*e**6 - c**2*d**2*e**6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*
c*d*e))/(8*c) + 35*c**3*d**7*e**3 - (13*a*e**2/2 + 13*c*d**2/2)*(a**3*e**10 + 21*a**2*c*d**2*e**8 + 63*a*c**2*
d**4*e**6 - 8*a*(3*a*c**2*d**2*e**8 + 7*c**3*d**4*e**6 - c**2*d**2*e**6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c)
+ 35*c**3*d**6*e**4 - (15*a*e**2/2 + 15*c*d**2/2)*(3*a**2*c*d*e**9 + 201*a*c**2*d**3*e**7/10 + 21*c**3*d**5*e*
*5 - (17*a*e**2/2 + 17*c*d**2/2)*(3*a*c**2*d**2*e**8 + 7*c**3*d**4*e**6 - c**2*d**2*e**6*(19*a*e**2/2 + 19*c*d
**2/2)/10)/(9*c*d*e))/(8*c*d*e))/(7*c*d*e))/(6*c*d*e))/(5*c*d*e))/(4*c) - (5*a*e**2/2 + 5*c*d**2/2)*(35*a**3*d
**4*e**6 + 63*a**2*c*d**6*e**4 + 21*a*c**2*d**8*e**2 - 4*a*(21*a**3*d**2*e**8 + 105*a**2*c*d**4*e**6 + 105*a*c
**2*d**6*e**4 - 6*a*(a**3*e**10 + 21*a**2*c*d**2*e**8 + 63*a*c**2*d**4*e**6 - 8*a*(3*a*c**2*d**2*e**8 + 7*c**3
*d**4*e**6 - c**2*d**2*e**6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c) + 35*c**3*d**6*e**4 - (15*a*e**2/2 + 15*c*d*
*2/2)*(3*a**2*c*d*e**9 + 201*a*c**2*d**3*e**7/10 + 21*c**3*d**5*e**5 - (17*a*e**2/2 + 17*c*d**2/2)*(3*a*c**2*d
**2*e**8 + 7*c**3*d**4*e**6 - c**2*d**2*e**6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c*d*e))/(8*c*d*e))/(7*c) + 21*
c**3*d**8*e**2 - (11*a*e**2/2 + 11*c*d**2/2)*(7*a**3*d*e**9 + 63*a**2*c*d**3*e**7 + 105*a*c**2*d**5*e**5 - 7*a
*(3*a**2*c*d*e**9 + 201*a*c**2*d**3*e**7/10 + 21*c**3*d**5*e**5 - (17*a*e**2/2 + 17*c*d**2/2)*(3*a*c**2*d**2*e
**8 + 7*c**3*d**4*e**6 - c**2*d**2*e**6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c*d*e))/(8*c) + 35*c**3*d**7*e**3 -
(13*a*e**2/2 + 13*c*d**2/2)*(a**3*e**10 + 21*a**2*c*d**2*e**8 + 63*a*c**2*d**4*e**6 - 8*a*(3*a*c**2*d**2*e**8
+ 7*c**3*d**4*e**6 - c**2*d**2*e**6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c) + 35*c**3*d**6*e**4 - (15*a*e**2/2
+ 15*c*d**2/2)*(3*a**2*c*d*e**9 + 201*a*c**2*d**3*e**7/10 + 21*c**3*d**5*e**5 - (17*a*e**2/2 + 17*c*d**2/2)*(3
*a*c**2*d**2*e**8 + 7*c**3*d**4*e**6 - c**2*d**2*e**6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c*d*e))/(8*c*d*e))/(7
*c*d*e))/(6*c*d*e))/(5*c) + c**3*d**10 - (7*a*e**2/2 + 7*c*d**2/2)*(35*a**3*d**3*e**7 + 105*a**2*c*d**5*e**5 +
63*a*c**2*d**7*e**3 - 5*a*(7*a**3*d*e**9 + 63*a**2*c*d**3*e**7 + 105*a*c**2*d**5*e**5 - 7*a*(3*a**2*c*d*e**9
+ 201*a*c**2*d**3*e**7/10 + 21*c**3*d**5*e**5 - (17*a*e**2/2 + 17*c*d**2/2)*(3*a*c**2*d**2*e**8 + 7*c**3*d**4*
e**6 - c**2*d**2*e**6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c*d*e))/(8*c) + 35*c**3*d**7*e**3 - (13*a*e**2/2 + 13
*c*d**2/2)*(a**3*e**10 + 21*a**2*c*d**2*e**8 + 63*a*c**2*d**4*e**6 - 8*a*(3*a*c**2*d**2*e**8 + 7*c**3*d**4*e**
6 - c**2*d**2*e**6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c) + 35*c**3*d**6*e**4 - (15*a*e**2/2 + 15*c*d**2/2)*(3*
a**2*c*d*e**9 + 201*a*c**2*d**3*e**7/10 + 21*c**3*d**5*e**5 - (17*a*e**2/2 + 17*c*d**2/2)*(3*a*c**2*d**2*e**8
+ 7*c**3*d**4*e**6 - c**2*d**2*e**6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c*d*e))/(8*c*d*e))/(7*c*d*e))/(6*c) + 7
*c**3*d**9*e - (9*a*e**2/2 + 9*c*d**2/2)*(21*a**3*d**2*e**8 + 105*a**2*c*d**4*e**6 + 105*a*c**2*d**6*e**4 - 6*
a*(a**3*e**10 + 21*a**2*c*d**2*e**8 + 63*a*c**2*d**4*e**6 - 8*a*(3*a*c**2*d**2*e**8 + 7*c**3*d**4*e**6 - c**2*
d**2*e**6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c) + 35*c**3*d**6*e**4 - (15*a*e**2/2 + 15*c*d**2/2)*(3*a**2*c*d*
e**9 + 201*a*c**2*d**3*e**7/10 + 21*c**3*d**5*e**5 - (17*a*e**2/2 + 17*c*d**2/2)*(3*a*c**2*d**2*e**8 + 7*c**3*
d**4*e**6 - c**2*d**2*e**6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c*d*e))/(8*c*d*e))/(7*c) + 21*c**3*d**8*e**2 - (
11*a*e**2/2 + 11*c*d**2/2)*(7*a**3*d*e**9 + 63*a**2*c*d**3*e**7 + 105*a*c**2*d**5*e**5 - 7*a*(3*a**2*c*d*e**9
+ 201*a*c**2*d**3*e**7/10 + 21*c**3*d**5*e**5 - (17*a*e**2/2 + 17*c*d**2/2)*(3*a*c**2*d**2*e**8 + 7*c**3*d**4*
e**6 - c**2*d**2*e**6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c*d*e))/(8*c) + 35*c**3*d**7*e**3 - (13*a*e**2/2 + 13
*c*d**2/2)*(a**3*e**10 + 21*a**2*c*d**2*e**8 + 63*a*c**2*d**4*e**6 - 8*a*(3*a*c**2*d**2*e**8 + 7*c**3*d**4*e**
6 - c**2*d**2*e**6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c) + 35*c**3*d**6*e**4 - (15*a*e**2/2 + 15*c*d**2/2)*(3*
a**2*c*d*e**9 + 201*a*c**2*d**3*e**7/10 + 21*c**3*d**5*e**5 - (17*a*e**2/2 + 17*c*d**2/2)*(3*a*c**2*d**2*e**8
+ 7*c**3*d**4*e**6 - c**2*d**2*e**6*(19*a*e**2/2 + 19*c*d**2/2)/10)/(9*c*d*e))/(8*c*d*e))/(7*c*d*e))/(6*c*d*e)
)/(5*c*d*e))/(4*c*d*e))/(3*c*d*e))/(2*c*d*e))/(2*c*d*e))*Piecewise((log(a*e**2 + c*d**2 + 2*c*d*e*x + 2*sqrt(c
*d*e)*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2)))/sqrt(c*d*e), Ne(a*d*e - (a*e**2 + c*d**2)**2/(4*c*d*e),
0)), ((x - (-a*e**2 - c*d**2)/(2*c*d*e))*log(x - (-a*e**2 - c*d**2)/(2*c*d*e))/sqrt(c*d*e*(x - (-a*e**2 - c*d*
*2)/(2*c*d*e))**2), True)), Ne(c*d*e, 0)), (2*(c**4*d**12*(a*d*e + x*(a*e**2 + c*d**2))**(7/2)/(7*(a**4*e**8 +
4*a**3*c*d**2*e**6 + 6*a**2*c**2*d**4*e**4 + 4*a*c**3*d**6*e**2 + c**4*d**8)) + 4*c**3*d**9*e*(a*d*e + x*(a*e
**2 + c*d**2))**(9/2)/(9*(a**4*e**8 + 4*a**3*c*d**2*e**6 + 6*a**2*c**2*d**4*e**4 + 4*a*c**3*d**6*e**2 + c**4*d
**8)) + 6*c**2*d**6*e**2*(a*d*e + x*(a*e**2 + c*d**2))**(11/2)/(11*(a**4*e**8 + 4*a**3*c*d**2*e**6 + 6*a**2*c*
*2*d**4*e**4 + 4*a*c**3*d**6*e**2 + c**4*d**8)) + 4*c*d**3*e**3*(a*d*e + x*(a*e**2 + c*d**2))**(13/2)/(13*(a**
4*e**8 + 4*a**3*c*d**2*e**6 + 6*a**2*c**2*d**4*e**4 + 4*a*c**3*d**6*e**2 + c**4*d**8)) + e**4*(a*d*e + x*(a*e*
*2 + c*d**2))**(15/2)/(15*(a**4*e**8 + 4*a**3*c*d**2*e**6 + 6*a**2*c**2*d**4*e**4 + 4*a*c**3*d**6*e**2 + c**4*
d**8)))/(a*e**2 + c*d**2), Ne(a*e**2 + c*d**2, 0)), ((a*d*e)**(5/2)*Piecewise((d**4*x, Eq(e, 0)), ((d + e*x)**
5/(5*e), True)), True))

Maxima [F(-2)]

Exception generated. \[ \int (d+e x)^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((e*x+d)^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more
details)Is e

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1099 vs. \(2 (488) = 976\).

Time = 0.41 (sec) , antiderivative size = 1099, normalized size of antiderivative = 2.06 \[ \int (d+e x)^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx=\text {Too large to display} \]

[In]

integrate((e*x+d)^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algorithm="giac")

[Out]

1/41287680*sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e)*(2*(4*(2*(8*(2*(4*(14*(16*(18*c^2*d^2*e^6*x + (121*c^11
*d^12*e^14 + 41*a*c^10*d^10*e^16)/(c^9*d^9*e^9))*x + (5503*c^11*d^13*e^13 + 4482*a*c^10*d^11*e^15 + 383*a^2*c^
9*d^9*e^17)/(c^9*d^9*e^9))*x + (119055*c^11*d^14*e^12 + 182129*a*c^10*d^12*e^14 + 37489*a^2*c^9*d^10*e^16 + 15
*a^3*c^8*d^8*e^18)/(c^9*d^9*e^9))*x + (424895*c^11*d^15*e^11 + 1157740*a*c^10*d^13*e^13 + 448938*a^2*c^9*d^11*
e^15 + 620*a^3*c^8*d^9*e^17 - 65*a^4*c^7*d^7*e^19)/(c^9*d^9*e^9))*x + (419983*c^11*d^16*e^10 + 2149035*a*c^10*
d^14*e^12 + 1490630*a^2*c^9*d^12*e^14 + 5830*a^3*c^8*d^10*e^16 - 1365*a^4*c^7*d^8*e^18 + 143*a^5*c^6*d^6*e^20)
/(c^9*d^9*e^9))*x + (735993*c^11*d^17*e^9 + 9023498*a*c^10*d^15*e^11 + 11825815*a^2*c^9*d^13*e^13 + 132300*a^3
*c^8*d^11*e^15 - 52585*a^4*c^7*d^9*e^17 + 12298*a^5*c^6*d^7*e^19 - 1287*a^6*c^5*d^5*e^21)/(c^9*d^9*e^9))*x + (
3003*c^11*d^18*e^8 + 4394937*a*c^10*d^16*e^10 + 13885683*a^2*c^9*d^14*e^12 + 508825*a^3*c^8*d^12*e^14 - 310375
*a^4*c^7*d^10*e^16 + 123123*a^5*c^6*d^8*e^18 - 28743*a^6*c^5*d^6*e^20 + 3003*a^7*c^4*d^4*e^22)/(c^9*d^9*e^9))*
x - (15015*c^11*d^19*e^7 - 144144*a*c^10*d^17*e^9 - 17075244*a^2*c^9*d^15*e^11 - 2878000*a^3*c^8*d^13*e^13 + 2
579850*a^4*c^7*d^11*e^15 - 1567280*a^5*c^6*d^9*e^17 + 619476*a^6*c^5*d^7*e^19 - 144144*a^7*c^4*d^5*e^21 + 1501
5*a^8*c^3*d^3*e^23)/(c^9*d^9*e^9))*x + (45045*c^11*d^20*e^6 - 435435*a*c^10*d^18*e^8 + 1885884*a^2*c^9*d^16*e^
10 + 6983100*a^3*c^8*d^14*e^12 - 9035650*a^4*c^7*d^12*e^14 + 8003710*a^5*c^6*d^10*e^16 - 4813380*a^6*c^5*d^8*e
^18 + 1885884*a^7*c^4*d^6*e^20 - 435435*a^8*c^3*d^4*e^22 + 45045*a^9*c^2*d^2*e^24)/(c^9*d^9*e^9)) + 143/262144
*(c^10*d^20 - 10*a*c^9*d^18*e^2 + 45*a^2*c^8*d^16*e^4 - 120*a^3*c^7*d^14*e^6 + 210*a^4*c^6*d^12*e^8 - 252*a^5*
c^5*d^10*e^10 + 210*a^6*c^4*d^8*e^12 - 120*a^7*c^3*d^6*e^14 + 45*a^8*c^2*d^4*e^16 - 10*a^9*c*d^2*e^18 + a^10*e
^20)*log(abs(-c*d^2 - a*e^2 - 2*sqrt(c*d*e)*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))))/(s
qrt(c*d*e)*c^7*d^7*e^3)

Mupad [F(-1)]

Timed out. \[ \int (d+e x)^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx=\int {\left (d+e\,x\right )}^4\,{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{5/2} \,d x \]

[In]

int((d + e*x)^4*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2),x)

[Out]

int((d + e*x)^4*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2), x)