\(\int (d+e x)^3 (a d e+(c d^2+a e^2) x+c d e x^2)^{5/2} \, dx\) [1932]

Optimal result

Integrand size = 37, antiderivative size = 474 \[ \int (d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx=\frac {55 \left (c d^2-a e^2\right )^7 \left (c d^2+a e^2+2 c d e x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{32768 c^6 d^6 e^3}-\frac {55 \left (c d^2-a e^2\right )^5 \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{12288 c^5 d^5 e^2}+\frac {11 \left (c d^2-a e^2\right )^3 \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{768 c^4 d^4 e}+\frac {11 \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{224 c^3 d^3}+\frac {11 \left (c d^2-a e^2\right ) (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{144 c^2 d^2}+\frac {(d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{9 c d}-\frac {55 \left (c d^2-a e^2\right )^9 \text {arctanh}\left (\frac {c d^2+a e^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{65536 c^{13/2} d^{13/2} e^{7/2}} \]

[Out]

-55/12288*(-a*e^2+c*d^2)^5*(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^5/d^5/e^2+11/768*
(-a*e^2+c*d^2)^3*(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^4/d^4/e+11/224*(-a*e^2+c*d^
2)^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/c^3/d^3+11/144*(-a*e^2+c*d^2)*(e*x+d)*(a*d*e+(a*e^2+c*d^2)*x+c*d*
e*x^2)^(7/2)/c^2/d^2+1/9*(e*x+d)^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/c/d-55/65536*(-a*e^2+c*d^2)^9*arcta
nh(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^(13/2)/d^(13
/2)/e^(7/2)+55/32768*(-a*e^2+c*d^2)^7*(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^6/d^6/
e^3

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 474, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.135, Rules used = {684, 654, 626, 635, 212} \[ \int (d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx=-\frac {55 \left (c d^2-a e^2\right )^9 \text {arctanh}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{65536 c^{13/2} d^{13/2} e^{7/2}}+\frac {55 \left (c d^2-a e^2\right )^7 \left (a e^2+c d^2+2 c d e x\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{32768 c^6 d^6 e^3}-\frac {55 \left (c d^2-a e^2\right )^5 \left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{12288 c^5 d^5 e^2}+\frac {11 \left (c d^2-a e^2\right )^3 \left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{768 c^4 d^4 e}+\frac {11 \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{224 c^3 d^3}+\frac {11 (d+e x) \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{144 c^2 d^2}+\frac {(d+e x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{9 c d} \]

[In]

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

[Out]

(55*(c*d^2 - a*e^2)^7*(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])/(32768*c^6*d^6*
e^3) - (55*(c*d^2 - a*e^2)^5*(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))/(12288
*c^5*d^5*e^2) + (11*(c*d^2 - a*e^2)^3*(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
))/(768*c^4*d^4*e) + (11*(c*d^2 - a*e^2)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(224*c^3*d^3) + (11*
(c*d^2 - a*e^2)*(d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(144*c^2*d^2) + ((d + e*x)^2*(a*d*e +
(c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(9*c*d) - (55*(c*d^2 - a*e^2)^9*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])])/(65536*c^(13/2)*d^(13/2)*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)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{9 c d}+\frac {\left (11 \left (d^2-\frac {a e^2}{c}\right )\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}{18 d} \\ & = \frac {11 \left (c d^2-a e^2\right ) (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{144 c^2 d^2}+\frac {(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}}{9 c d}+\frac {\left (11 \left (d^2-\frac {a e^2}{c}\right )^2\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}{32 d^2} \\ & = \frac {11 \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{224 c^3 d^3}+\frac {11 \left (c d^2-a e^2\right ) (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{144 c^2 d^2}+\frac {(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}}{9 c d}+\frac {\left (11 \left (d^2-\frac {a e^2}{c}\right )^3\right ) \int \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx}{64 d^3} \\ & = \frac {11 \left (c d^2-a e^2\right )^3 \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}}{768 c^4 d^4 e}+\frac {11 \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{224 c^3 d^3}+\frac {11 \left (c d^2-a e^2\right ) (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{144 c^2 d^2}+\frac {(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}}{9 c d}-\frac {\left (55 \left (c d^2-a e^2\right )^5\right ) \int \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2} \, dx}{1536 c^4 d^4 e} \\ & = -\frac {55 \left (c d^2-a e^2\right )^5 \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}}{12288 c^5 d^5 e^2}+\frac {11 \left (c d^2-a e^2\right )^3 \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}}{768 c^4 d^4 e}+\frac {11 \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{224 c^3 d^3}+\frac {11 \left (c d^2-a e^2\right ) (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{144 c^2 d^2}+\frac {(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}}{9 c d}+\frac {\left (55 \left (c d^2-a e^2\right )^7\right ) \int \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx}{8192 c^5 d^5 e^2} \\ & = \frac {55 \left (c d^2-a e^2\right )^7 \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}}{32768 c^6 d^6 e^3}-\frac {55 \left (c d^2-a e^2\right )^5 \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}}{12288 c^5 d^5 e^2}+\frac {11 \left (c d^2-a e^2\right )^3 \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}}{768 c^4 d^4 e}+\frac {11 \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{224 c^3 d^3}+\frac {11 \left (c d^2-a e^2\right ) (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{144 c^2 d^2}+\frac {(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}}{9 c d}-\frac {\left (55 \left (c d^2-a e^2\right )^9\right ) \int \frac {1}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{65536 c^6 d^6 e^3} \\ & = \frac {55 \left (c d^2-a e^2\right )^7 \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}}{32768 c^6 d^6 e^3}-\frac {55 \left (c d^2-a e^2\right )^5 \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}}{12288 c^5 d^5 e^2}+\frac {11 \left (c d^2-a e^2\right )^3 \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}}{768 c^4 d^4 e}+\frac {11 \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{224 c^3 d^3}+\frac {11 \left (c d^2-a e^2\right ) (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{144 c^2 d^2}+\frac {(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}}{9 c d}-\frac {\left (55 \left (c d^2-a e^2\right )^9\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 )}{32768 c^6 d^6 e^3} \\ & = \frac {55 \left (c d^2-a e^2\right )^7 \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}}{32768 c^6 d^6 e^3}-\frac {55 \left (c d^2-a e^2\right )^5 \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}}{12288 c^5 d^5 e^2}+\frac {11 \left (c d^2-a e^2\right )^3 \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}}{768 c^4 d^4 e}+\frac {11 \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{224 c^3 d^3}+\frac {11 \left (c d^2-a e^2\right ) (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{144 c^2 d^2}+\frac {(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}}{9 c d}-\frac {55 \left (c d^2-a e^2\right )^9 \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 )}{65536 c^{13/2} d^{13/2} e^{7/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.66 (sec) , antiderivative size = 625, normalized size of antiderivative = 1.32 \[ \int (d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx=\frac {((a e+c d x) (d+e x))^{5/2} \left (\frac {\sqrt {c} \sqrt {d} \sqrt {e} \left (-3465 a^8 e^{16}+2310 a^7 c d e^{14} (13 d+e x)-462 a^6 c^2 d^2 e^{12} \left (249 d^2+43 d e x+4 e^2 x^2\right )+198 a^5 c^3 d^3 e^{10} \left (1289 d^3+381 d^2 e x+80 d e^2 x^2+8 e^3 x^3\right )-22 a^4 c^4 d^4 e^8 \left (16384 d^4+7531 d^3 e x+2724 d^2 e^2 x^2+616 d e^3 x^3+64 e^4 x^4\right )+2 a^3 c^5 d^5 e^6 \left (167301 d^5+115609 d^4 e x+65536 d^3 e^2 x^2+25584 d^2 e^3 x^3+6016 d e^4 x^4+640 e^5 x^5\right )+6 a^2 c^6 d^6 e^4 \left (19173 d^6+282339 d^5 e x+763652 d^4 e^2 x^2+1040048 d^3 e^3 x^3+786432 d^2 e^4 x^4+315776 d e^5 x^5+52736 e^6 x^6\right )+2 a c^7 d^7 e^2 \left (-15015 d^7+9933 d^6 e x+876816 d^5 e^2 x^2+2988664 d^4 e^3 x^3+4548736 d^3 e^4 x^4+3672960 d^2 e^5 x^5+1540096 d e^6 x^6+265216 e^7 x^7\right )+c^8 d^8 \left (3465 d^8-2310 d^7 e x+1848 d^6 e^2 x^2+588240 d^5 e^3 x^3+2229632 d^4 e^4 x^4+3603200 d^3 e^5 x^5+3025920 d^2 e^6 x^6+1304576 d e^7 x^7+229376 e^8 x^8\right )\right )}{(a e+c d x)^2 (d+e x)^2}-\frac {3465 \left (c d^2-a e^2\right )^9 \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {e} \sqrt {a e+c d x}}\right )}{(a e+c d x)^{5/2} (d+e x)^{5/2}}\right )}{2064384 c^{13/2} d^{13/2} e^{7/2}} \]

[In]

Integrate[(d + e*x)^3*(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]*(-3465*a^8*e^16 + 2310*a^7*c*d*e^14*(13*d + e*x) -
462*a^6*c^2*d^2*e^12*(249*d^2 + 43*d*e*x + 4*e^2*x^2) + 198*a^5*c^3*d^3*e^10*(1289*d^3 + 381*d^2*e*x + 80*d*e^
2*x^2 + 8*e^3*x^3) - 22*a^4*c^4*d^4*e^8*(16384*d^4 + 7531*d^3*e*x + 2724*d^2*e^2*x^2 + 616*d*e^3*x^3 + 64*e^4*
x^4) + 2*a^3*c^5*d^5*e^6*(167301*d^5 + 115609*d^4*e*x + 65536*d^3*e^2*x^2 + 25584*d^2*e^3*x^3 + 6016*d*e^4*x^4
+ 640*e^5*x^5) + 6*a^2*c^6*d^6*e^4*(19173*d^6 + 282339*d^5*e*x + 763652*d^4*e^2*x^2 + 1040048*d^3*e^3*x^3 + 7
86432*d^2*e^4*x^4 + 315776*d*e^5*x^5 + 52736*e^6*x^6) + 2*a*c^7*d^7*e^2*(-15015*d^7 + 9933*d^6*e*x + 876816*d^
5*e^2*x^2 + 2988664*d^4*e^3*x^3 + 4548736*d^3*e^4*x^4 + 3672960*d^2*e^5*x^5 + 1540096*d*e^6*x^6 + 265216*e^7*x
^7) + c^8*d^8*(3465*d^8 - 2310*d^7*e*x + 1848*d^6*e^2*x^2 + 588240*d^5*e^3*x^3 + 2229632*d^4*e^4*x^4 + 3603200
*d^3*e^5*x^5 + 3025920*d^2*e^6*x^6 + 1304576*d*e^7*x^7 + 229376*e^8*x^8)))/((a*e + c*d*x)^2*(d + e*x)^2) - (34
65*(c*d^2 - a*e^2)^9*ArcTanh[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/(Sqrt[e]*Sqrt[a*e + c*d*x])])/((a*e + c*d*x)^(5/2
)*(d + e*x)^(5/2))))/(2064384*c^(13/2)*d^(13/2)*e^(7/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(2844\) vs. \(2(432)=864\).

Time = 2.94 (sec) , antiderivative size = 2845, normalized size of antiderivative = 6.00

method	result	size
default	\(\text {Expression too large to display}\)	\(2845\)

[In]

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

[Out]

d^3*(1/12*(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/d/e+5/24*(4*a*c*d^2*e^2-(a*e^2+c*d
^2)^2)/c/d/e*(1/8*(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)^(3/2)/c/d/e+3/16*(4*a*c*d^2*e^2-(a
*e^2+c*d^2)^2)/c/d/e*(1/4*(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)^(1/2)/c/d/e+1/8*(4*a*c*d^2
*e^2-(a*e^2+c*d^2)^2)/c/d/e*ln((1/2*e^2*a+1/2*c*d^2+x*c*d*e)/(c*d*e)^(1/2)+(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(
1/2))/(c*d*e)^(1/2))))+e^3*(1/9*x^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/c/d/e-11/18*(a*e^2+c*d^2)/c/d/e*(1
/8*x*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/c/d/e-9/16*(a*e^2+c*d^2)/c/d/e*(1/7*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*
x^2)^(7/2)/c/d/e-1/2*(a*e^2+c*d^2)/c/d/e*(1/12*(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/d/e+5/24*(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/c/d/e*(1/8*(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)^(3/2)/c/d/e+3/16*(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/c/d/e*(1/4*(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)^(1/2)/c/d/e+1/8*(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/c/d/e*ln((1/2*e^2*a+1/2*c*d^2+x*c*d*e)/(c*d*e)^(1/2
)+(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/(c*d*e)^(1/2)))))-1/8*a/c*(1/12*(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/d/e+5/24*(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/c/d/e*(1/8*(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)^(3/2)/c/d/e+3/16*(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/c/d/e*(1/4*(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)^(1/2)/c/d/e+1/8*(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/c/d/e*ln((1/2*e^2*a+1/2
*c*d^2+x*c*d*e)/(c*d*e)^(1/2)+(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/(c*d*e)^(1/2)))))-2/9*a/c*(1/7*(a*d*e+(
a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/c/d/e-1/2*(a*e^2+c*d^2)/c/d/e*(1/12*(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/d/e+5/24*(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/c/d/e*(1/8*(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)^(3/2)/c/d/e+3/16*(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/c/d/e*(1/4*(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)^(1/2)/c/d/e+1/8*(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/c/d/e*ln((1/2*e^2*a+1/2*c*d^2+
x*c*d*e)/(c*d*e)^(1/2)+(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/(c*d*e)^(1/2))))))+3*d*e^2*(1/8*x*(a*d*e+(a*e^
2+c*d^2)*x+c*d*e*x^2)^(7/2)/c/d/e-9/16*(a*e^2+c*d^2)/c/d/e*(1/7*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/c/d/e-
1/2*(a*e^2+c*d^2)/c/d/e*(1/12*(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/d/e+5/24*(4*a*
c*d^2*e^2-(a*e^2+c*d^2)^2)/c/d/e*(1/8*(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)^(3/2)/c/d/e+3/
16*(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/c/d/e*(1/4*(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)^(1/2)/
c/d/e+1/8*(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/c/d/e*ln((1/2*e^2*a+1/2*c*d^2+x*c*d*e)/(c*d*e)^(1/2)+(a*d*e+(a*e^2+c
*d^2)*x+c*d*e*x^2)^(1/2))/(c*d*e)^(1/2)))))-1/8*a/c*(1/12*(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/d/e+5/24*(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/c/d/e*(1/8*(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)^(3/2)/c/d/e+3/16*(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/c/d/e*(1/4*(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)^(1/2)/c/d/e+1/8*(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/c/d/e*ln((1/2*e^2*a+1/2*c*d^2+x*c*d*e)/(
c*d*e)^(1/2)+(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/(c*d*e)^(1/2)))))+3*d^2*e*(1/7*(a*d*e+(a*e^2+c*d^2)*x+c*
d*e*x^2)^(7/2)/c/d/e-1/2*(a*e^2+c*d^2)/c/d/e*(1/12*(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/d/e+5/24*(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/c/d/e*(1/8*(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)^(3/2)/c/d/e+3/16*(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/c/d/e*(1/4*(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)^(1/2)/c/d/e+1/8*(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/c/d/e*ln((1/2*e^2*a+1/2*c*d^2+x*c*d*e)/(c*d*e)^
(1/2)+(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/(c*d*e)^(1/2)))))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 896 vs. \(2 (432) = 864\).

Time = 0.72 (sec) , antiderivative size = 1806, normalized size of antiderivative = 3.81 \[ \int (d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

[1/8257536*(3465*(c^9*d^18 - 9*a*c^8*d^16*e^2 + 36*a^2*c^7*d^14*e^4 - 84*a^3*c^6*d^12*e^6 + 126*a^4*c^5*d^10*e
^8 - 126*a^5*c^4*d^8*e^10 + 84*a^6*c^3*d^6*e^12 - 36*a^7*c^2*d^4*e^14 + 9*a^8*c*d^2*e^16 - a^9*e^18)*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*(229376*c^9*d^9*e^9*x^8 + 3465*c^9*d
^17*e - 30030*a*c^8*d^15*e^3 + 115038*a^2*c^7*d^13*e^5 + 334602*a^3*c^6*d^11*e^7 - 360448*a^4*c^5*d^9*e^9 + 25
5222*a^5*c^4*d^7*e^11 - 115038*a^6*c^3*d^5*e^13 + 30030*a^7*c^2*d^3*e^15 - 3465*a^8*c*d*e^17 + 14336*(91*c^9*d
^10*e^8 + 37*a*c^8*d^8*e^10)*x^7 + 1024*(2955*c^9*d^11*e^7 + 3008*a*c^8*d^9*e^9 + 309*a^2*c^7*d^7*e^11)*x^6 +
256*(14075*c^9*d^12*e^6 + 28695*a*c^8*d^10*e^8 + 7401*a^2*c^7*d^8*e^10 + 5*a^3*c^6*d^6*e^12)*x^5 + 128*(17419*
c^9*d^13*e^5 + 71074*a*c^8*d^11*e^7 + 36864*a^2*c^7*d^9*e^9 + 94*a^3*c^6*d^7*e^11 - 11*a^4*c^5*d^5*e^13)*x^4 +
16*(36765*c^9*d^14*e^4 + 373583*a*c^8*d^12*e^6 + 390018*a^2*c^7*d^10*e^8 + 3198*a^3*c^6*d^8*e^10 - 847*a^4*c^
5*d^6*e^12 + 99*a^5*c^4*d^4*e^14)*x^3 + 8*(231*c^9*d^15*e^3 + 219204*a*c^8*d^13*e^5 + 572739*a^2*c^7*d^11*e^7
+ 16384*a^3*c^6*d^9*e^9 - 7491*a^4*c^5*d^7*e^11 + 1980*a^5*c^4*d^5*e^13 - 231*a^6*c^3*d^3*e^15)*x^2 - 2*(1155*
c^9*d^16*e^2 - 9933*a*c^8*d^14*e^4 - 847017*a^2*c^7*d^12*e^6 - 115609*a^3*c^6*d^10*e^8 + 82841*a^4*c^5*d^8*e^1
0 - 37719*a^5*c^4*d^6*e^12 + 9933*a^6*c^3*d^4*e^14 - 1155*a^7*c^2*d^2*e^16)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2
+ a*e^2)*x))/(c^7*d^7*e^4), 1/4128768*(3465*(c^9*d^18 - 9*a*c^8*d^16*e^2 + 36*a^2*c^7*d^14*e^4 - 84*a^3*c^6*d
^12*e^6 + 126*a^4*c^5*d^10*e^8 - 126*a^5*c^4*d^8*e^10 + 84*a^6*c^3*d^6*e^12 - 36*a^7*c^2*d^4*e^14 + 9*a^8*c*d^
2*e^16 - a^9*e^18)*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*(229376*c^9*d^9*e^9*x^8 + 3
465*c^9*d^17*e - 30030*a*c^8*d^15*e^3 + 115038*a^2*c^7*d^13*e^5 + 334602*a^3*c^6*d^11*e^7 - 360448*a^4*c^5*d^9
*e^9 + 255222*a^5*c^4*d^7*e^11 - 115038*a^6*c^3*d^5*e^13 + 30030*a^7*c^2*d^3*e^15 - 3465*a^8*c*d*e^17 + 14336*
(91*c^9*d^10*e^8 + 37*a*c^8*d^8*e^10)*x^7 + 1024*(2955*c^9*d^11*e^7 + 3008*a*c^8*d^9*e^9 + 309*a^2*c^7*d^7*e^1
1)*x^6 + 256*(14075*c^9*d^12*e^6 + 28695*a*c^8*d^10*e^8 + 7401*a^2*c^7*d^8*e^10 + 5*a^3*c^6*d^6*e^12)*x^5 + 12
8*(17419*c^9*d^13*e^5 + 71074*a*c^8*d^11*e^7 + 36864*a^2*c^7*d^9*e^9 + 94*a^3*c^6*d^7*e^11 - 11*a^4*c^5*d^5*e^
13)*x^4 + 16*(36765*c^9*d^14*e^4 + 373583*a*c^8*d^12*e^6 + 390018*a^2*c^7*d^10*e^8 + 3198*a^3*c^6*d^8*e^10 - 8
47*a^4*c^5*d^6*e^12 + 99*a^5*c^4*d^4*e^14)*x^3 + 8*(231*c^9*d^15*e^3 + 219204*a*c^8*d^13*e^5 + 572739*a^2*c^7*
d^11*e^7 + 16384*a^3*c^6*d^9*e^9 - 7491*a^4*c^5*d^7*e^11 + 1980*a^5*c^4*d^5*e^13 - 231*a^6*c^3*d^3*e^15)*x^2 -
2*(1155*c^9*d^16*e^2 - 9933*a*c^8*d^14*e^4 - 847017*a^2*c^7*d^12*e^6 - 115609*a^3*c^6*d^10*e^8 + 82841*a^4*c^
5*d^8*e^10 - 37719*a^5*c^4*d^6*e^12 + 9933*a^6*c^3*d^4*e^14 - 1155*a^7*c^2*d^2*e^16)*x)*sqrt(c*d*e*x^2 + a*d*e
+ (c*d^2 + a*e^2)*x))/(c^7*d^7*e^4)]

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 14834 vs. \(2 (462) = 924\).

Time = 19.60 (sec) , antiderivative size = 14834, normalized size of antiderivative = 31.30 \[ \int (d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx=\text {Too large to display} \]

[In]

integrate((e*x+d)**3*(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**5*x**8/9 + x**7*(3*a*c**2*d**2*e**7 +
6*c**3*d**4*e**5 - c**2*d**2*e**5*(17*a*e**2/2 + 17*c*d**2/2)/9)/(8*c*d*e) + x**6*(3*a**2*c*d*e**8 + 154*a*c**
2*d**3*e**6/9 + 15*c**3*d**5*e**4 - (15*a*e**2/2 + 15*c*d**2/2)*(3*a*c**2*d**2*e**7 + 6*c**3*d**4*e**5 - c**2*
d**2*e**5*(17*a*e**2/2 + 17*c*d**2/2)/9)/(8*c*d*e))/(7*c*d*e) + x**5*(a**3*e**9 + 18*a**2*c*d**2*e**7 + 45*a*c
**2*d**4*e**5 - 7*a*(3*a*c**2*d**2*e**7 + 6*c**3*d**4*e**5 - c**2*d**2*e**5*(17*a*e**2/2 + 17*c*d**2/2)/9)/(8*
c) + 20*c**3*d**6*e**3 - (13*a*e**2/2 + 13*c*d**2/2)*(3*a**2*c*d*e**8 + 154*a*c**2*d**3*e**6/9 + 15*c**3*d**5*
e**4 - (15*a*e**2/2 + 15*c*d**2/2)*(3*a*c**2*d**2*e**7 + 6*c**3*d**4*e**5 - c**2*d**2*e**5*(17*a*e**2/2 + 17*c
*d**2/2)/9)/(8*c*d*e))/(7*c*d*e))/(6*c*d*e) + x**4*(6*a**3*d*e**8 + 45*a**2*c*d**3*e**6 + 60*a*c**2*d**5*e**4
- 6*a*(3*a**2*c*d*e**8 + 154*a*c**2*d**3*e**6/9 + 15*c**3*d**5*e**4 - (15*a*e**2/2 + 15*c*d**2/2)*(3*a*c**2*d*
*2*e**7 + 6*c**3*d**4*e**5 - c**2*d**2*e**5*(17*a*e**2/2 + 17*c*d**2/2)/9)/(8*c*d*e))/(7*c) + 15*c**3*d**7*e**
2 - (11*a*e**2/2 + 11*c*d**2/2)*(a**3*e**9 + 18*a**2*c*d**2*e**7 + 45*a*c**2*d**4*e**5 - 7*a*(3*a*c**2*d**2*e*
*7 + 6*c**3*d**4*e**5 - c**2*d**2*e**5*(17*a*e**2/2 + 17*c*d**2/2)/9)/(8*c) + 20*c**3*d**6*e**3 - (13*a*e**2/2
+ 13*c*d**2/2)*(3*a**2*c*d*e**8 + 154*a*c**2*d**3*e**6/9 + 15*c**3*d**5*e**4 - (15*a*e**2/2 + 15*c*d**2/2)*(3
*a*c**2*d**2*e**7 + 6*c**3*d**4*e**5 - c**2*d**2*e**5*(17*a*e**2/2 + 17*c*d**2/2)/9)/(8*c*d*e))/(7*c*d*e))/(6*
c*d*e))/(5*c*d*e) + x**3*(15*a**3*d**2*e**7 + 60*a**2*c*d**4*e**5 + 45*a*c**2*d**6*e**3 - 5*a*(a**3*e**9 + 18*
a**2*c*d**2*e**7 + 45*a*c**2*d**4*e**5 - 7*a*(3*a*c**2*d**2*e**7 + 6*c**3*d**4*e**5 - c**2*d**2*e**5*(17*a*e**
2/2 + 17*c*d**2/2)/9)/(8*c) + 20*c**3*d**6*e**3 - (13*a*e**2/2 + 13*c*d**2/2)*(3*a**2*c*d*e**8 + 154*a*c**2*d*
*3*e**6/9 + 15*c**3*d**5*e**4 - (15*a*e**2/2 + 15*c*d**2/2)*(3*a*c**2*d**2*e**7 + 6*c**3*d**4*e**5 - c**2*d**2
*e**5*(17*a*e**2/2 + 17*c*d**2/2)/9)/(8*c*d*e))/(7*c*d*e))/(6*c) + 6*c**3*d**8*e - (9*a*e**2/2 + 9*c*d**2/2)*(
6*a**3*d*e**8 + 45*a**2*c*d**3*e**6 + 60*a*c**2*d**5*e**4 - 6*a*(3*a**2*c*d*e**8 + 154*a*c**2*d**3*e**6/9 + 15
*c**3*d**5*e**4 - (15*a*e**2/2 + 15*c*d**2/2)*(3*a*c**2*d**2*e**7 + 6*c**3*d**4*e**5 - c**2*d**2*e**5*(17*a*e*
*2/2 + 17*c*d**2/2)/9)/(8*c*d*e))/(7*c) + 15*c**3*d**7*e**2 - (11*a*e**2/2 + 11*c*d**2/2)*(a**3*e**9 + 18*a**2
*c*d**2*e**7 + 45*a*c**2*d**4*e**5 - 7*a*(3*a*c**2*d**2*e**7 + 6*c**3*d**4*e**5 - c**2*d**2*e**5*(17*a*e**2/2
+ 17*c*d**2/2)/9)/(8*c) + 20*c**3*d**6*e**3 - (13*a*e**2/2 + 13*c*d**2/2)*(3*a**2*c*d*e**8 + 154*a*c**2*d**3*e
**6/9 + 15*c**3*d**5*e**4 - (15*a*e**2/2 + 15*c*d**2/2)*(3*a*c**2*d**2*e**7 + 6*c**3*d**4*e**5 - c**2*d**2*e**
5*(17*a*e**2/2 + 17*c*d**2/2)/9)/(8*c*d*e))/(7*c*d*e))/(6*c*d*e))/(5*c*d*e))/(4*c*d*e) + x**2*(20*a**3*d**3*e*
*6 + 45*a**2*c*d**5*e**4 + 18*a*c**2*d**7*e**2 - 4*a*(6*a**3*d*e**8 + 45*a**2*c*d**3*e**6 + 60*a*c**2*d**5*e**
4 - 6*a*(3*a**2*c*d*e**8 + 154*a*c**2*d**3*e**6/9 + 15*c**3*d**5*e**4 - (15*a*e**2/2 + 15*c*d**2/2)*(3*a*c**2*
d**2*e**7 + 6*c**3*d**4*e**5 - c**2*d**2*e**5*(17*a*e**2/2 + 17*c*d**2/2)/9)/(8*c*d*e))/(7*c) + 15*c**3*d**7*e
**2 - (11*a*e**2/2 + 11*c*d**2/2)*(a**3*e**9 + 18*a**2*c*d**2*e**7 + 45*a*c**2*d**4*e**5 - 7*a*(3*a*c**2*d**2*
e**7 + 6*c**3*d**4*e**5 - c**2*d**2*e**5*(17*a*e**2/2 + 17*c*d**2/2)/9)/(8*c) + 20*c**3*d**6*e**3 - (13*a*e**2
/2 + 13*c*d**2/2)*(3*a**2*c*d*e**8 + 154*a*c**2*d**3*e**6/9 + 15*c**3*d**5*e**4 - (15*a*e**2/2 + 15*c*d**2/2)*
(3*a*c**2*d**2*e**7 + 6*c**3*d**4*e**5 - c**2*d**2*e**5*(17*a*e**2/2 + 17*c*d**2/2)/9)/(8*c*d*e))/(7*c*d*e))/(
6*c*d*e))/(5*c) + c**3*d**9 - (7*a*e**2/2 + 7*c*d**2/2)*(15*a**3*d**2*e**7 + 60*a**2*c*d**4*e**5 + 45*a*c**2*d
**6*e**3 - 5*a*(a**3*e**9 + 18*a**2*c*d**2*e**7 + 45*a*c**2*d**4*e**5 - 7*a*(3*a*c**2*d**2*e**7 + 6*c**3*d**4*
e**5 - c**2*d**2*e**5*(17*a*e**2/2 + 17*c*d**2/2)/9)/(8*c) + 20*c**3*d**6*e**3 - (13*a*e**2/2 + 13*c*d**2/2)*(
3*a**2*c*d*e**8 + 154*a*c**2*d**3*e**6/9 + 15*c**3*d**5*e**4 - (15*a*e**2/2 + 15*c*d**2/2)*(3*a*c**2*d**2*e**7
+ 6*c**3*d**4*e**5 - c**2*d**2*e**5*(17*a*e**2/2 + 17*c*d**2/2)/9)/(8*c*d*e))/(7*c*d*e))/(6*c) + 6*c**3*d**8*
e - (9*a*e**2/2 + 9*c*d**2/2)*(6*a**3*d*e**8 + 45*a**2*c*d**3*e**6 + 60*a*c**2*d**5*e**4 - 6*a*(3*a**2*c*d*e**
8 + 154*a*c**2*d**3*e**6/9 + 15*c**3*d**5*e**4 - (15*a*e**2/2 + 15*c*d**2/2)*(3*a*c**2*d**2*e**7 + 6*c**3*d**4
*e**5 - c**2*d**2*e**5*(17*a*e**2/2 + 17*c*d**2/2)/9)/(8*c*d*e))/(7*c) + 15*c**3*d**7*e**2 - (11*a*e**2/2 + 11
*c*d**2/2)*(a**3*e**9 + 18*a**2*c*d**2*e**7 + 45*a*c**2*d**4*e**5 - 7*a*(3*a*c**2*d**2*e**7 + 6*c**3*d**4*e**5
- c**2*d**2*e**5*(17*a*e**2/2 + 17*c*d**2/2)/9)/(8*c) + 20*c**3*d**6*e**3 - (13*a*e**2/2 + 13*c*d**2/2)*(3*a*
*2*c*d*e**8 + 154*a*c**2*d**3*e**6/9 + 15*c**3*d**5*e**4 - (15*a*e**2/2 + 15*c*d**2/2)*(3*a*c**2*d**2*e**7 + 6
*c**3*d**4*e**5 - c**2*d**2*e**5*(17*a*e**2/2 + 17*c*d**2/2)/9)/(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*(15*a**3*d**4*e**5 + 18*a**2*c*d**6*e**3 + 3*a*c**2*d**8*e - 3*a*(15*a**3*d**2*e**7 + 6
0*a**2*c*d**4*e**5 + 45*a*c**2*d**6*e**3 - 5*a*(a**3*e**9 + 18*a**2*c*d**2*e**7 + 45*a*c**2*d**4*e**5 - 7*a*(3
*a*c**2*d**2*e**7 + 6*c**3*d**4*e**5 - c**2*d**2*e**5*(17*a*e**2/2 + 17*c*d**2/2)/9)/(8*c) + 20*c**3*d**6*e**3
- (13*a*e**2/2 + 13*c*d**2/2)*(3*a**2*c*d*e**8 + 154*a*c**2*d**3*e**6/9 + 15*c**3*d**5*e**4 - (15*a*e**2/2 +
15*c*d**2/2)*(3*a*c**2*d**2*e**7 + 6*c**3*d**4*e**5 - c**2*d**2*e**5*(17*a*e**2/2 + 17*c*d**2/2)/9)/(8*c*d*e))
/(7*c*d*e))/(6*c) + 6*c**3*d**8*e - (9*a*e**2/2 + 9*c*d**2/2)*(6*a**3*d*e**8 + 45*a**2*c*d**3*e**6 + 60*a*c**2
*d**5*e**4 - 6*a*(3*a**2*c*d*e**8 + 154*a*c**2*d**3*e**6/9 + 15*c**3*d**5*e**4 - (15*a*e**2/2 + 15*c*d**2/2)*(
3*a*c**2*d**2*e**7 + 6*c**3*d**4*e**5 - c**2*d**2*e**5*(17*a*e**2/2 + 17*c*d**2/2)/9)/(8*c*d*e))/(7*c) + 15*c*
*3*d**7*e**2 - (11*a*e**2/2 + 11*c*d**2/2)*(a**3*e**9 + 18*a**2*c*d**2*e**7 + 45*a*c**2*d**4*e**5 - 7*a*(3*a*c
**2*d**2*e**7 + 6*c**3*d**4*e**5 - c**2*d**2*e**5*(17*a*e**2/2 + 17*c*d**2/2)/9)/(8*c) + 20*c**3*d**6*e**3 - (
13*a*e**2/2 + 13*c*d**2/2)*(3*a**2*c*d*e**8 + 154*a*c**2*d**3*e**6/9 + 15*c**3*d**5*e**4 - (15*a*e**2/2 + 15*c
*d**2/2)*(3*a*c**2*d**2*e**7 + 6*c**3*d**4*e**5 - c**2*d**2*e**5*(17*a*e**2/2 + 17*c*d**2/2)/9)/(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)*(20*a**3*d**3*e**6 + 45*a**2*c*d**5*e**4 + 18*
a*c**2*d**7*e**2 - 4*a*(6*a**3*d*e**8 + 45*a**2*c*d**3*e**6 + 60*a*c**2*d**5*e**4 - 6*a*(3*a**2*c*d*e**8 + 154
*a*c**2*d**3*e**6/9 + 15*c**3*d**5*e**4 - (15*a*e**2/2 + 15*c*d**2/2)*(3*a*c**2*d**2*e**7 + 6*c**3*d**4*e**5 -
c**2*d**2*e**5*(17*a*e**2/2 + 17*c*d**2/2)/9)/(8*c*d*e))/(7*c) + 15*c**3*d**7*e**2 - (11*a*e**2/2 + 11*c*d**2
/2)*(a**3*e**9 + 18*a**2*c*d**2*e**7 + 45*a*c**2*d**4*e**5 - 7*a*(3*a*c**2*d**2*e**7 + 6*c**3*d**4*e**5 - c**2
*d**2*e**5*(17*a*e**2/2 + 17*c*d**2/2)/9)/(8*c) + 20*c**3*d**6*e**3 - (13*a*e**2/2 + 13*c*d**2/2)*(3*a**2*c*d*
e**8 + 154*a*c**2*d**3*e**6/9 + 15*c**3*d**5*e**4 - (15*a*e**2/2 + 15*c*d**2/2)*(3*a*c**2*d**2*e**7 + 6*c**3*d
**4*e**5 - c**2*d**2*e**5*(17*a*e**2/2 + 17*c*d**2/2)/9)/(8*c*d*e))/(7*c*d*e))/(6*c*d*e))/(5*c) + c**3*d**9 -
(7*a*e**2/2 + 7*c*d**2/2)*(15*a**3*d**2*e**7 + 60*a**2*c*d**4*e**5 + 45*a*c**2*d**6*e**3 - 5*a*(a**3*e**9 + 18
*a**2*c*d**2*e**7 + 45*a*c**2*d**4*e**5 - 7*a*(3*a*c**2*d**2*e**7 + 6*c**3*d**4*e**5 - c**2*d**2*e**5*(17*a*e*
*2/2 + 17*c*d**2/2)/9)/(8*c) + 20*c**3*d**6*e**3 - (13*a*e**2/2 + 13*c*d**2/2)*(3*a**2*c*d*e**8 + 154*a*c**2*d
**3*e**6/9 + 15*c**3*d**5*e**4 - (15*a*e**2/2 + 15*c*d**2/2)*(3*a*c**2*d**2*e**7 + 6*c**3*d**4*e**5 - c**2*d**
2*e**5*(17*a*e**2/2 + 17*c*d**2/2)/9)/(8*c*d*e))/(7*c*d*e))/(6*c) + 6*c**3*d**8*e - (9*a*e**2/2 + 9*c*d**2/2)*
(6*a**3*d*e**8 + 45*a**2*c*d**3*e**6 + 60*a*c**2*d**5*e**4 - 6*a*(3*a**2*c*d*e**8 + 154*a*c**2*d**3*e**6/9 + 1
5*c**3*d**5*e**4 - (15*a*e**2/2 + 15*c*d**2/2)*(3*a*c**2*d**2*e**7 + 6*c**3*d**4*e**5 - c**2*d**2*e**5*(17*a*e
**2/2 + 17*c*d**2/2)/9)/(8*c*d*e))/(7*c) + 15*c**3*d**7*e**2 - (11*a*e**2/2 + 11*c*d**2/2)*(a**3*e**9 + 18*a**
2*c*d**2*e**7 + 45*a*c**2*d**4*e**5 - 7*a*(3*a*c**2*d**2*e**7 + 6*c**3*d**4*e**5 - c**2*d**2*e**5*(17*a*e**2/2
+ 17*c*d**2/2)/9)/(8*c) + 20*c**3*d**6*e**3 - (13*a*e**2/2 + 13*c*d**2/2)*(3*a**2*c*d*e**8 + 154*a*c**2*d**3*
e**6/9 + 15*c**3*d**5*e**4 - (15*a*e**2/2 + 15*c*d**2/2)*(3*a*c**2*d**2*e**7 + 6*c**3*d**4*e**5 - c**2*d**2*e*
*5*(17*a*e**2/2 + 17*c*d**2/2)/9)/(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)
+ (6*a**3*d**5*e**4 + 3*a**2*c*d**7*e**2 - 2*a*(20*a**3*d**3*e**6 + 45*a**2*c*d**5*e**4 + 18*a*c**2*d**7*e**2
- 4*a*(6*a**3*d*e**8 + 45*a**2*c*d**3*e**6 + 60*a*c**2*d**5*e**4 - 6*a*(3*a**2*c*d*e**8 + 154*a*c**2*d**3*e**6
/9 + 15*c**3*d**5*e**4 - (15*a*e**2/2 + 15*c*d**2/2)*(3*a*c**2*d**2*e**7 + 6*c**3*d**4*e**5 - c**2*d**2*e**5*(
17*a*e**2/2 + 17*c*d**2/2)/9)/(8*c*d*e))/(7*c) + 15*c**3*d**7*e**2 - (11*a*e**2/2 + 11*c*d**2/2)*(a**3*e**9 +
18*a**2*c*d**2*e**7 + 45*a*c**2*d**4*e**5 - 7*a*(3*a*c**2*d**2*e**7 + 6*c**3*d**4*e**5 - c**2*d**2*e**5*(17*a*
e**2/2 + 17*c*d**2/2)/9)/(8*c) + 20*c**3*d**6*e**3 - (13*a*e**2/2 + 13*c*d**2/2)*(3*a**2*c*d*e**8 + 154*a*c**2
*d**3*e**6/9 + 15*c**3*d**5*e**4 - (15*a*e**2/2 + 15*c*d**2/2)*(3*a*c**2*d**2*e**7 + 6*c**3*d**4*e**5 - c**2*d
**2*e**5*(17*a*e**2/2 + 17*c*d**2/2)/9)/(8*c*d*e))/(7*c*d*e))/(6*c*d*e))/(5*c) + c**3*d**9 - (7*a*e**2/2 + 7*c
*d**2/2)*(15*a**3*d**2*e**7 + 60*a**2*c*d**4*e**5 + 45*a*c**2*d**6*e**3 - 5*a*(a**3*e**9 + 18*a**2*c*d**2*e**7
+ 45*a*c**2*d**4*e**5 - 7*a*(3*a*c**2*d**2*e**7 + 6*c**3*d**4*e**5 - c**2*d**2*e**5*(17*a*e**2/2 + 17*c*d**2/
2)/9)/(8*c) + 20*c**3*d**6*e**3 - (13*a*e**2/2 + 13*c*d**2/2)*(3*a**2*c*d*e**8 + 154*a*c**2*d**3*e**6/9 + 15*c
**3*d**5*e**4 - (15*a*e**2/2 + 15*c*d**2/2)*(3*a*c**2*d**2*e**7 + 6*c**3*d**4*e**5 - c**2*d**2*e**5*(17*a*e**2
/2 + 17*c*d**2/2)/9)/(8*c*d*e))/(7*c*d*e))/(6*c) + 6*c**3*d**8*e - (9*a*e**2/2 + 9*c*d**2/2)*(6*a**3*d*e**8 +
45*a**2*c*d**3*e**6 + 60*a*c**2*d**5*e**4 - 6*a*(3*a**2*c*d*e**8 + 154*a*c**2*d**3*e**6/9 + 15*c**3*d**5*e**4
- (15*a*e**2/2 + 15*c*d**2/2)*(3*a*c**2*d**2*e**7 + 6*c**3*d**4*e**5 - c**2*d**2*e**5*(17*a*e**2/2 + 17*c*d**2
/2)/9)/(8*c*d*e))/(7*c) + 15*c**3*d**7*e**2 - (11*a*e**2/2 + 11*c*d**2/2)*(a**3*e**9 + 18*a**2*c*d**2*e**7 + 4
5*a*c**2*d**4*e**5 - 7*a*(3*a*c**2*d**2*e**7 + 6*c**3*d**4*e**5 - c**2*d**2*e**5*(17*a*e**2/2 + 17*c*d**2/2)/9
)/(8*c) + 20*c**3*d**6*e**3 - (13*a*e**2/2 + 13*c*d**2/2)*(3*a**2*c*d*e**8 + 154*a*c**2*d**3*e**6/9 + 15*c**3*
d**5*e**4 - (15*a*e**2/2 + 15*c*d**2/2)*(3*a*c**2*d**2*e**7 + 6*c**3*d**4*e**5 - c**2*d**2*e**5*(17*a*e**2/2 +
17*c*d**2/2)/9)/(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)*(15*
a**3*d**4*e**5 + 18*a**2*c*d**6*e**3 + 3*a*c**2*d**8*e - 3*a*(15*a**3*d**2*e**7 + 60*a**2*c*d**4*e**5 + 45*a*c
**2*d**6*e**3 - 5*a*(a**3*e**9 + 18*a**2*c*d**2*e**7 + 45*a*c**2*d**4*e**5 - 7*a*(3*a*c**2*d**2*e**7 + 6*c**3*
d**4*e**5 - c**2*d**2*e**5*(17*a*e**2/2 + 17*c*d**2/2)/9)/(8*c) + 20*c**3*d**6*e**3 - (13*a*e**2/2 + 13*c*d**2
/2)*(3*a**2*c*d*e**8 + 154*a*c**2*d**3*e**6/9 + 15*c**3*d**5*e**4 - (15*a*e**2/2 + 15*c*d**2/2)*(3*a*c**2*d**2
*e**7 + 6*c**3*d**4*e**5 - c**2*d**2*e**5*(17*a*e**2/2 + 17*c*d**2/2)/9)/(8*c*d*e))/(7*c*d*e))/(6*c) + 6*c**3*
d**8*e - (9*a*e**2/2 + 9*c*d**2/2)*(6*a**3*d*e**8 + 45*a**2*c*d**3*e**6 + 60*a*c**2*d**5*e**4 - 6*a*(3*a**2*c*
d*e**8 + 154*a*c**2*d**3*e**6/9 + 15*c**3*d**5*e**4 - (15*a*e**2/2 + 15*c*d**2/2)*(3*a*c**2*d**2*e**7 + 6*c**3
*d**4*e**5 - c**2*d**2*e**5*(17*a*e**2/2 + 17*c*d**2/2)/9)/(8*c*d*e))/(7*c) + 15*c**3*d**7*e**2 - (11*a*e**2/2
+ 11*c*d**2/2)*(a**3*e**9 + 18*a**2*c*d**2*e**7 + 45*a*c**2*d**4*e**5 - 7*a*(3*a*c**2*d**2*e**7 + 6*c**3*d**4
*e**5 - c**2*d**2*e**5*(17*a*e**2/2 + 17*c*d**2/2)/9)/(8*c) + 20*c**3*d**6*e**3 - (13*a*e**2/2 + 13*c*d**2/2)*
(3*a**2*c*d*e**8 + 154*a*c**2*d**3*e**6/9 + 15*c**3*d**5*e**4 - (15*a*e**2/2 + 15*c*d**2/2)*(3*a*c**2*d**2*e**
7 + 6*c**3*d**4*e**5 - c**2*d**2*e**5*(17*a*e**2/2 + 17*c*d**2/2)/9)/(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)*(20*a**3*d**3*e**6 + 45*a**2*c*d**5*e**4 + 18*a*c**2*d**7*e**2 - 4*a*(6*a
**3*d*e**8 + 45*a**2*c*d**3*e**6 + 60*a*c**2*d**5*e**4 - 6*a*(3*a**2*c*d*e**8 + 154*a*c**2*d**3*e**6/9 + 15*c*
*3*d**5*e**4 - (15*a*e**2/2 + 15*c*d**2/2)*(3*a*c**2*d**2*e**7 + 6*c**3*d**4*e**5 - c**2*d**2*e**5*(17*a*e**2/
2 + 17*c*d**2/2)/9)/(8*c*d*e))/(7*c) + 15*c**3*d**7*e**2 - (11*a*e**2/2 + 11*c*d**2/2)*(a**3*e**9 + 18*a**2*c*
d**2*e**7 + 45*a*c**2*d**4*e**5 - 7*a*(3*a*c**2*d**2*e**7 + 6*c**3*d**4*e**5 - c**2*d**2*e**5*(17*a*e**2/2 + 1
7*c*d**2/2)/9)/(8*c) + 20*c**3*d**6*e**3 - (13*a*e**2/2 + 13*c*d**2/2)*(3*a**2*c*d*e**8 + 154*a*c**2*d**3*e**6
/9 + 15*c**3*d**5*e**4 - (15*a*e**2/2 + 15*c*d**2/2)*(3*a*c**2*d**2*e**7 + 6*c**3*d**4*e**5 - c**2*d**2*e**5*(
17*a*e**2/2 + 17*c*d**2/2)/9)/(8*c*d*e))/(7*c*d*e))/(6*c*d*e))/(5*c) + c**3*d**9 - (7*a*e**2/2 + 7*c*d**2/2)*(
15*a**3*d**2*e**7 + 60*a**2*c*d**4*e**5 + 45*a*c**2*d**6*e**3 - 5*a*(a**3*e**9 + 18*a**2*c*d**2*e**7 + 45*a*c*
*2*d**4*e**5 - 7*a*(3*a*c**2*d**2*e**7 + 6*c**3*d**4*e**5 - c**2*d**2*e**5*(17*a*e**2/2 + 17*c*d**2/2)/9)/(8*c
) + 20*c**3*d**6*e**3 - (13*a*e**2/2 + 13*c*d**2/2)*(3*a**2*c*d*e**8 + 154*a*c**2*d**3*e**6/9 + 15*c**3*d**5*e
**4 - (15*a*e**2/2 + 15*c*d**2/2)*(3*a*c**2*d**2*e**7 + 6*c**3*d**4*e**5 - c**2*d**2*e**5*(17*a*e**2/2 + 17*c*
d**2/2)/9)/(8*c*d*e))/(7*c*d*e))/(6*c) + 6*c**3*d**8*e - (9*a*e**2/2 + 9*c*d**2/2)*(6*a**3*d*e**8 + 45*a**2*c*
d**3*e**6 + 60*a*c**2*d**5*e**4 - 6*a*(3*a**2*c*d*e**8 + 154*a*c**2*d**3*e**6/9 + 15*c**3*d**5*e**4 - (15*a*e*
*2/2 + 15*c*d**2/2)*(3*a*c**2*d**2*e**7 + 6*c**3*d**4*e**5 - c**2*d**2*e**5*(17*a*e**2/2 + 17*c*d**2/2)/9)/(8*
c*d*e))/(7*c) + 15*c**3*d**7*e**2 - (11*a*e**2/2 + 11*c*d**2/2)*(a**3*e**9 + 18*a**2*c*d**2*e**7 + 45*a*c**2*d
**4*e**5 - 7*a*(3*a*c**2*d**2*e**7 + 6*c**3*d**4*e**5 - c**2*d**2*e**5*(17*a*e**2/2 + 17*c*d**2/2)/9)/(8*c) +
20*c**3*d**6*e**3 - (13*a*e**2/2 + 13*c*d**2/2)*(3*a**2*c*d*e**8 + 154*a*c**2*d**3*e**6/9 + 15*c**3*d**5*e**4
- (15*a*e**2/2 + 15*c*d**2/2)*(3*a*c**2*d**2*e**7 + 6*c**3*d**4*e**5 - c**2*d**2*e**5*(17*a*e**2/2 + 17*c*d**2
/2)/9)/(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**6*e**3
- a*(15*a**3*d**4*e**5 + 18*a**2*c*d**6*e**3 + 3*a*c**2*d**8*e - 3*a*(15*a**3*d**2*e**7 + 60*a**2*c*d**4*e**5
+ 45*a*c**2*d**6*e**3 - 5*a*(a**3*e**9 + 18*a**2*c*d**2*e**7 + 45*a*c**2*d**4*e**5 - 7*a*(3*a*c**2*d**2*e**7
+ 6*c**3*d**4*e**5 - c**2*d**2*e**5*(17*a*e**2/2 + 17*c*d**2/2)/9)/(8*c) + 20*c**3*d**6*e**3 - (13*a*e**2/2 +
13*c*d**2/2)*(3*a**2*c*d*e**8 + 154*a*c**2*d**3*e**6/9 + 15*c**3*d**5*e**4 - (15*a*e**2/2 + 15*c*d**2/2)*(3*a*
c**2*d**2*e**7 + 6*c**3*d**4*e**5 - c**2*d**2*e**5*(17*a*e**2/2 + 17*c*d**2/2)/9)/(8*c*d*e))/(7*c*d*e))/(6*c)
+ 6*c**3*d**8*e - (9*a*e**2/2 + 9*c*d**2/2)*(6*a**3*d*e**8 + 45*a**2*c*d**3*e**6 + 60*a*c**2*d**5*e**4 - 6*a*(
3*a**2*c*d*e**8 + 154*a*c**2*d**3*e**6/9 + 15*c**3*d**5*e**4 - (15*a*e**2/2 + 15*c*d**2/2)*(3*a*c**2*d**2*e**7
+ 6*c**3*d**4*e**5 - c**2*d**2*e**5*(17*a*e**2/2 + 17*c*d**2/2)/9)/(8*c*d*e))/(7*c) + 15*c**3*d**7*e**2 - (11
*a*e**2/2 + 11*c*d**2/2)*(a**3*e**9 + 18*a**2*c*d**2*e**7 + 45*a*c**2*d**4*e**5 - 7*a*(3*a*c**2*d**2*e**7 + 6*
c**3*d**4*e**5 - c**2*d**2*e**5*(17*a*e**2/2 + 17*c*d**2/2)/9)/(8*c) + 20*c**3*d**6*e**3 - (13*a*e**2/2 + 13*c
*d**2/2)*(3*a**2*c*d*e**8 + 154*a*c**2*d**3*e**6/9 + 15*c**3*d**5*e**4 - (15*a*e**2/2 + 15*c*d**2/2)*(3*a*c**2
*d**2*e**7 + 6*c**3*d**4*e**5 - c**2*d**2*e**5*(17*a*e**2/2 + 17*c*d**2/2)/9)/(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)*(20*a**3*d**3*e**6 + 45*a**2*c*d**5*e**4 + 18*a*c**2*d**7*e**2 -
4*a*(6*a**3*d*e**8 + 45*a**2*c*d**3*e**6 + 60*a*c**2*d**5*e**4 - 6*a*(3*a**2*c*d*e**8 + 154*a*c**2*d**3*e**6/
9 + 15*c**3*d**5*e**4 - (15*a*e**2/2 + 15*c*d**2/2)*(3*a*c**2*d**2*e**7 + 6*c**3*d**4*e**5 - c**2*d**2*e**5*(1
7*a*e**2/2 + 17*c*d**2/2)/9)/(8*c*d*e))/(7*c) + 15*c**3*d**7*e**2 - (11*a*e**2/2 + 11*c*d**2/2)*(a**3*e**9 + 1
8*a**2*c*d**2*e**7 + 45*a*c**2*d**4*e**5 - 7*a*(3*a*c**2*d**2*e**7 + 6*c**3*d**4*e**5 - c**2*d**2*e**5*(17*a*e
**2/2 + 17*c*d**2/2)/9)/(8*c) + 20*c**3*d**6*e**3 - (13*a*e**2/2 + 13*c*d**2/2)*(3*a**2*c*d*e**8 + 154*a*c**2*
d**3*e**6/9 + 15*c**3*d**5*e**4 - (15*a*e**2/2 + 15*c*d**2/2)*(3*a*c**2*d**2*e**7 + 6*c**3*d**4*e**5 - c**2*d*
*2*e**5*(17*a*e**2/2 + 17*c*d**2/2)/9)/(8*c*d*e))/(7*c*d*e))/(6*c*d*e))/(5*c) + c**3*d**9 - (7*a*e**2/2 + 7*c*
d**2/2)*(15*a**3*d**2*e**7 + 60*a**2*c*d**4*e**5 + 45*a*c**2*d**6*e**3 - 5*a*(a**3*e**9 + 18*a**2*c*d**2*e**7
+ 45*a*c**2*d**4*e**5 - 7*a*(3*a*c**2*d**2*e**7 + 6*c**3*d**4*e**5 - c**2*d**2*e**5*(17*a*e**2/2 + 17*c*d**2/2
)/9)/(8*c) + 20*c**3*d**6*e**3 - (13*a*e**2/2 + 13*c*d**2/2)*(3*a**2*c*d*e**8 + 154*a*c**2*d**3*e**6/9 + 15*c*
*3*d**5*e**4 - (15*a*e**2/2 + 15*c*d**2/2)*(3*a*c**2*d**2*e**7 + 6*c**3*d**4*e**5 - c**2*d**2*e**5*(17*a*e**2/
2 + 17*c*d**2/2)/9)/(8*c*d*e))/(7*c*d*e))/(6*c) + 6*c**3*d**8*e - (9*a*e**2/2 + 9*c*d**2/2)*(6*a**3*d*e**8 + 4
5*a**2*c*d**3*e**6 + 60*a*c**2*d**5*e**4 - 6*a*(3*a**2*c*d*e**8 + 154*a*c**2*d**3*e**6/9 + 15*c**3*d**5*e**4 -
(15*a*e**2/2 + 15*c*d**2/2)*(3*a*c**2*d**2*e**7 + 6*c**3*d**4*e**5 - c**2*d**2*e**5*(17*a*e**2/2 + 17*c*d**2/
2)/9)/(8*c*d*e))/(7*c) + 15*c**3*d**7*e**2 - (11*a*e**2/2 + 11*c*d**2/2)*(a**3*e**9 + 18*a**2*c*d**2*e**7 + 45
*a*c**2*d**4*e**5 - 7*a*(3*a*c**2*d**2*e**7 + 6*c**3*d**4*e**5 - c**2*d**2*e**5*(17*a*e**2/2 + 17*c*d**2/2)/9)
/(8*c) + 20*c**3*d**6*e**3 - (13*a*e**2/2 + 13*c*d**2/2)*(3*a**2*c*d*e**8 + 154*a*c**2*d**3*e**6/9 + 15*c**3*d
**5*e**4 - (15*a*e**2/2 + 15*c*d**2/2)*(3*a*c**2*d**2*e**7 + 6*c**3*d**4*e**5 - c**2*d**2*e**5*(17*a*e**2/2 +
17*c*d**2/2)/9)/(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)*(6
*a**3*d**5*e**4 + 3*a**2*c*d**7*e**2 - 2*a*(20*a**3*d**3*e**6 + 45*a**2*c*d**5*e**4 + 18*a*c**2*d**7*e**2 - 4*
a*(6*a**3*d*e**8 + 45*a**2*c*d**3*e**6 + 60*a*c**2*d**5*e**4 - 6*a*(3*a**2*c*d*e**8 + 154*a*c**2*d**3*e**6/9 +
15*c**3*d**5*e**4 - (15*a*e**2/2 + 15*c*d**2/2)*(3*a*c**2*d**2*e**7 + 6*c**3*d**4*e**5 - c**2*d**2*e**5*(17*a
*e**2/2 + 17*c*d**2/2)/9)/(8*c*d*e))/(7*c) + 15*c**3*d**7*e**2 - (11*a*e**2/2 + 11*c*d**2/2)*(a**3*e**9 + 18*a
**2*c*d**2*e**7 + 45*a*c**2*d**4*e**5 - 7*a*(3*a*c**2*d**2*e**7 + 6*c**3*d**4*e**5 - c**2*d**2*e**5*(17*a*e**2
/2 + 17*c*d**2/2)/9)/(8*c) + 20*c**3*d**6*e**3 - (13*a*e**2/2 + 13*c*d**2/2)*(3*a**2*c*d*e**8 + 154*a*c**2*d**
3*e**6/9 + 15*c**3*d**5*e**4 - (15*a*e**2/2 + 15*c*d**2/2)*(3*a*c**2*d**2*e**7 + 6*c**3*d**4*e**5 - c**2*d**2*
e**5*(17*a*e**2/2 + 17*c*d**2/2)/9)/(8*c*d*e))/(7*c*d*e))/(6*c*d*e))/(5*c) + c**3*d**9 - (7*a*e**2/2 + 7*c*d**
2/2)*(15*a**3*d**2*e**7 + 60*a**2*c*d**4*e**5 + 45*a*c**2*d**6*e**3 - 5*a*(a**3*e**9 + 18*a**2*c*d**2*e**7 + 4
5*a*c**2*d**4*e**5 - 7*a*(3*a*c**2*d**2*e**7 + 6*c**3*d**4*e**5 - c**2*d**2*e**5*(17*a*e**2/2 + 17*c*d**2/2)/9
)/(8*c) + 20*c**3*d**6*e**3 - (13*a*e**2/2 + 13*c*d**2/2)*(3*a**2*c*d*e**8 + 154*a*c**2*d**3*e**6/9 + 15*c**3*
d**5*e**4 - (15*a*e**2/2 + 15*c*d**2/2)*(3*a*c**2*d**2*e**7 + 6*c**3*d**4*e**5 - c**2*d**2*e**5*(17*a*e**2/2 +
17*c*d**2/2)/9)/(8*c*d*e))/(7*c*d*e))/(6*c) + 6*c**3*d**8*e - (9*a*e**2/2 + 9*c*d**2/2)*(6*a**3*d*e**8 + 45*a
**2*c*d**3*e**6 + 60*a*c**2*d**5*e**4 - 6*a*(3*a**2*c*d*e**8 + 154*a*c**2*d**3*e**6/9 + 15*c**3*d**5*e**4 - (1
5*a*e**2/2 + 15*c*d**2/2)*(3*a*c**2*d**2*e**7 + 6*c**3*d**4*e**5 - c**2*d**2*e**5*(17*a*e**2/2 + 17*c*d**2/2)/
9)/(8*c*d*e))/(7*c) + 15*c**3*d**7*e**2 - (11*a*e**2/2 + 11*c*d**2/2)*(a**3*e**9 + 18*a**2*c*d**2*e**7 + 45*a*
c**2*d**4*e**5 - 7*a*(3*a*c**2*d**2*e**7 + 6*c**3*d**4*e**5 - c**2*d**2*e**5*(17*a*e**2/2 + 17*c*d**2/2)/9)/(8
*c) + 20*c**3*d**6*e**3 - (13*a*e**2/2 + 13*c*d**2/2)*(3*a**2*c*d*e**8 + 154*a*c**2*d**3*e**6/9 + 15*c**3*d**5
*e**4 - (15*a*e**2/2 + 15*c*d**2/2)*(3*a*c**2*d**2*e**7 + 6*c**3*d**4*e**5 - c**2*d**2*e**5*(17*a*e**2/2 + 17*
c*d**2/2)/9)/(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)*(15*a**3
*d**4*e**5 + 18*a**2*c*d**6*e**3 + 3*a*c**2*d**8*e - 3*a*(15*a**3*d**2*e**7 + 60*a**2*c*d**4*e**5 + 45*a*c**2*
d**6*e**3 - 5*a*(a**3*e**9 + 18*a**2*c*d**2*e**7 + 45*a*c**2*d**4*e**5 - 7*a*(3*a*c**2*d**2*e**7 + 6*c**3*d**4
*e**5 - c**2*d**2*e**5*(17*a*e**2/2 + 17*c*d**2/2)/9)/(8*c) + 20*c**3*d**6*e**3 - (13*a*e**2/2 + 13*c*d**2/2)*
(3*a**2*c*d*e**8 + 154*a*c**2*d**3*e**6/9 + 15*c**3*d**5*e**4 - (15*a*e**2/2 + 15*c*d**2/2)*(3*a*c**2*d**2*e**
7 + 6*c**3*d**4*e**5 - c**2*d**2*e**5*(17*a*e**2/2 + 17*c*d**2/2)/9)/(8*c*d*e))/(7*c*d*e))/(6*c) + 6*c**3*d**8
*e - (9*a*e**2/2 + 9*c*d**2/2)*(6*a**3*d*e**8 + 45*a**2*c*d**3*e**6 + 60*a*c**2*d**5*e**4 - 6*a*(3*a**2*c*d*e*
*8 + 154*a*c**2*d**3*e**6/9 + 15*c**3*d**5*e**4 - (15*a*e**2/2 + 15*c*d**2/2)*(3*a*c**2*d**2*e**7 + 6*c**3*d**
4*e**5 - c**2*d**2*e**5*(17*a*e**2/2 + 17*c*d**2/2)/9)/(8*c*d*e))/(7*c) + 15*c**3*d**7*e**2 - (11*a*e**2/2 + 1
1*c*d**2/2)*(a**3*e**9 + 18*a**2*c*d**2*e**7 + 45*a*c**2*d**4*e**5 - 7*a*(3*a*c**2*d**2*e**7 + 6*c**3*d**4*e**
5 - c**2*d**2*e**5*(17*a*e**2/2 + 17*c*d**2/2)/9)/(8*c) + 20*c**3*d**6*e**3 - (13*a*e**2/2 + 13*c*d**2/2)*(3*a
**2*c*d*e**8 + 154*a*c**2*d**3*e**6/9 + 15*c**3*d**5*e**4 - (15*a*e**2/2 + 15*c*d**2/2)*(3*a*c**2*d**2*e**7 +
6*c**3*d**4*e**5 - c**2*d**2*e**5*(17*a*e**2/2 + 17*c*d**2/2)/9)/(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)*(20*a**3*d**3*e**6 + 45*a**2*c*d**5*e**4 + 18*a*c**2*d**7*e**2 - 4*a*(6*a**3*
d*e**8 + 45*a**2*c*d**3*e**6 + 60*a*c**2*d**5*e**4 - 6*a*(3*a**2*c*d*e**8 + 154*a*c**2*d**3*e**6/9 + 15*c**3*d
**5*e**4 - (15*a*e**2/2 + 15*c*d**2/2)*(3*a*c**2*d**2*e**7 + 6*c**3*d**4*e**5 - c**2*d**2*e**5*(17*a*e**2/2 +
17*c*d**2/2)/9)/(8*c*d*e))/(7*c) + 15*c**3*d**7*e**2 - (11*a*e**2/2 + 11*c*d**2/2)*(a**3*e**9 + 18*a**2*c*d**2
*e**7 + 45*a*c**2*d**4*e**5 - 7*a*(3*a*c**2*d**2*e**7 + 6*c**3*d**4*e**5 - c**2*d**2*e**5*(17*a*e**2/2 + 17*c*
d**2/2)/9)/(8*c) + 20*c**3*d**6*e**3 - (13*a*e**2/2 + 13*c*d**2/2)*(3*a**2*c*d*e**8 + 154*a*c**2*d**3*e**6/9 +
15*c**3*d**5*e**4 - (15*a*e**2/2 + 15*c*d**2/2)*(3*a*c**2*d**2*e**7 + 6*c**3*d**4*e**5 - c**2*d**2*e**5*(17*a
*e**2/2 + 17*c*d**2/2)/9)/(8*c*d*e))/(7*c*d*e))/(6*c*d*e))/(5*c) + c**3*d**9 - (7*a*e**2/2 + 7*c*d**2/2)*(15*a
**3*d**2*e**7 + 60*a**2*c*d**4*e**5 + 45*a*c**2*d**6*e**3 - 5*a*(a**3*e**9 + 18*a**2*c*d**2*e**7 + 45*a*c**2*d
**4*e**5 - 7*a*(3*a*c**2*d**2*e**7 + 6*c**3*d**4*e**5 - c**2*d**2*e**5*(17*a*e**2/2 + 17*c*d**2/2)/9)/(8*c) +
20*c**3*d**6*e**3 - (13*a*e**2/2 + 13*c*d**2/2)*(3*a**2*c*d*e**8 + 154*a*c**2*d**3*e**6/9 + 15*c**3*d**5*e**4
- (15*a*e**2/2 + 15*c*d**2/2)*(3*a*c**2*d**2*e**7 + 6*c**3*d**4*e**5 - c**2*d**2*e**5*(17*a*e**2/2 + 17*c*d**2
/2)/9)/(8*c*d*e))/(7*c*d*e))/(6*c) + 6*c**3*d**8*e - (9*a*e**2/2 + 9*c*d**2/2)*(6*a**3*d*e**8 + 45*a**2*c*d**3
*e**6 + 60*a*c**2*d**5*e**4 - 6*a*(3*a**2*c*d*e**8 + 154*a*c**2*d**3*e**6/9 + 15*c**3*d**5*e**4 - (15*a*e**2/2
+ 15*c*d**2/2)*(3*a*c**2*d**2*e**7 + 6*c**3*d**4*e**5 - c**2*d**2*e**5*(17*a*e**2/2 + 17*c*d**2/2)/9)/(8*c*d*
e))/(7*c) + 15*c**3*d**7*e**2 - (11*a*e**2/2 + 11*c*d**2/2)*(a**3*e**9 + 18*a**2*c*d**2*e**7 + 45*a*c**2*d**4*
e**5 - 7*a*(3*a*c**2*d**2*e**7 + 6*c**3*d**4*e**5 - c**2*d**2*e**5*(17*a*e**2/2 + 17*c*d**2/2)/9)/(8*c) + 20*c
**3*d**6*e**3 - (13*a*e**2/2 + 13*c*d**2/2)*(3*a**2*c*d*e**8 + 154*a*c**2*d**3*e**6/9 + 15*c**3*d**5*e**4 - (1
5*a*e**2/2 + 15*c*d**2/2)*(3*a*c**2*d**2*e**7 + 6*c**3*d**4*e**5 - c**2*d**2*e**5*(17*a*e**2/2 + 17*c*d**2/2)/
9)/(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**3*d**9*(a*d*e + x*(a*e**2
+ c*d**2))**(7/2)/(7*(a**3*e**6 + 3*a**2*c*d**2*e**4 + 3*a*c**2*d**4*e**2 + c**3*d**6)) + c**2*d**6*e*(a*d*e +
x*(a*e**2 + c*d**2))**(9/2)/(3*(a**3*e**6 + 3*a**2*c*d**2*e**4 + 3*a*c**2*d**4*e**2 + c**3*d**6)) + 3*c*d**3*
e**2*(a*d*e + x*(a*e**2 + c*d**2))**(11/2)/(11*(a**3*e**6 + 3*a**2*c*d**2*e**4 + 3*a*c**2*d**4*e**2 + c**3*d**
6)) + e**3*(a*d*e + x*(a*e**2 + c*d**2))**(13/2)/(13*(a**3*e**6 + 3*a**2*c*d**2*e**4 + 3*a*c**2*d**4*e**2 + c*
*3*d**6)))/(a*e**2 + c*d**2), Ne(a*e**2 + c*d**2, 0)), ((a*d*e)**(5/2)*Piecewise((d**3*x, Eq(e, 0)), ((d + e*x
)**4/(4*e), True)), True))

Maxima [F(-2)]

Exception generated. \[ \int (d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((e*x+d)^3*(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. 936 vs. \(2 (432) = 864\).

Time = 0.38 (sec) , antiderivative size = 936, normalized size of antiderivative = 1.97 \[ \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=\frac {1}{2064384} \, \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (2 \, {\left (4 \, {\left (14 \, {\left (16 \, c^{2} d^{2} e^{5} x + \frac {91 \, c^{10} d^{11} e^{12} + 37 \, a c^{9} d^{9} e^{14}}{c^{8} d^{8} e^{8}}\right )} x + \frac {2955 \, c^{10} d^{12} e^{11} + 3008 \, a c^{9} d^{10} e^{13} + 309 \, a^{2} c^{8} d^{8} e^{15}}{c^{8} d^{8} e^{8}}\right )} x + \frac {14075 \, c^{10} d^{13} e^{10} + 28695 \, a c^{9} d^{11} e^{12} + 7401 \, a^{2} c^{8} d^{9} e^{14} + 5 \, a^{3} c^{7} d^{7} e^{16}}{c^{8} d^{8} e^{8}}\right )} x + \frac {17419 \, c^{10} d^{14} e^{9} + 71074 \, a c^{9} d^{12} e^{11} + 36864 \, a^{2} c^{8} d^{10} e^{13} + 94 \, a^{3} c^{7} d^{8} e^{15} - 11 \, a^{4} c^{6} d^{6} e^{17}}{c^{8} d^{8} e^{8}}\right )} x + \frac {36765 \, c^{10} d^{15} e^{8} + 373583 \, a c^{9} d^{13} e^{10} + 390018 \, a^{2} c^{8} d^{11} e^{12} + 3198 \, a^{3} c^{7} d^{9} e^{14} - 847 \, a^{4} c^{6} d^{7} e^{16} + 99 \, a^{5} c^{5} d^{5} e^{18}}{c^{8} d^{8} e^{8}}\right )} x + \frac {231 \, c^{10} d^{16} e^{7} + 219204 \, a c^{9} d^{14} e^{9} + 572739 \, a^{2} c^{8} d^{12} e^{11} + 16384 \, a^{3} c^{7} d^{10} e^{13} - 7491 \, a^{4} c^{6} d^{8} e^{15} + 1980 \, a^{5} c^{5} d^{6} e^{17} - 231 \, a^{6} c^{4} d^{4} e^{19}}{c^{8} d^{8} e^{8}}\right )} x - \frac {1155 \, c^{10} d^{17} e^{6} - 9933 \, a c^{9} d^{15} e^{8} - 847017 \, a^{2} c^{8} d^{13} e^{10} - 115609 \, a^{3} c^{7} d^{11} e^{12} + 82841 \, a^{4} c^{6} d^{9} e^{14} - 37719 \, a^{5} c^{5} d^{7} e^{16} + 9933 \, a^{6} c^{4} d^{5} e^{18} - 1155 \, a^{7} c^{3} d^{3} e^{20}}{c^{8} d^{8} e^{8}}\right )} x + \frac {3465 \, c^{10} d^{18} e^{5} - 30030 \, a c^{9} d^{16} e^{7} + 115038 \, a^{2} c^{8} d^{14} e^{9} + 334602 \, a^{3} c^{7} d^{12} e^{11} - 360448 \, a^{4} c^{6} d^{10} e^{13} + 255222 \, a^{5} c^{5} d^{8} e^{15} - 115038 \, a^{6} c^{4} d^{6} e^{17} + 30030 \, a^{7} c^{3} d^{4} e^{19} - 3465 \, a^{8} c^{2} d^{2} e^{21}}{c^{8} d^{8} e^{8}}\right )} + \frac {55 \, {\left (c^{9} d^{18} - 9 \, a c^{8} d^{16} e^{2} + 36 \, a^{2} c^{7} d^{14} e^{4} - 84 \, a^{3} c^{6} d^{12} e^{6} + 126 \, a^{4} c^{5} d^{10} e^{8} - 126 \, a^{5} c^{4} d^{8} e^{10} + 84 \, a^{6} c^{3} d^{6} e^{12} - 36 \, a^{7} c^{2} d^{4} e^{14} + 9 \, a^{8} c d^{2} e^{16} - a^{9} e^{18}\right )} \log \left ({\left | -c d^{2} - a e^{2} - 2 \, \sqrt {c d e} {\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )} \right |}\right )}{65536 \, \sqrt {c d e} c^{6} d^{6} e^{3}} \]

[In]

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

[Out]

1/2064384*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*c^2*d^2*e^5*x + (91*c^10*d^11*
e^12 + 37*a*c^9*d^9*e^14)/(c^8*d^8*e^8))*x + (2955*c^10*d^12*e^11 + 3008*a*c^9*d^10*e^13 + 309*a^2*c^8*d^8*e^1
5)/(c^8*d^8*e^8))*x + (14075*c^10*d^13*e^10 + 28695*a*c^9*d^11*e^12 + 7401*a^2*c^8*d^9*e^14 + 5*a^3*c^7*d^7*e^
16)/(c^8*d^8*e^8))*x + (17419*c^10*d^14*e^9 + 71074*a*c^9*d^12*e^11 + 36864*a^2*c^8*d^10*e^13 + 94*a^3*c^7*d^8
*e^15 - 11*a^4*c^6*d^6*e^17)/(c^8*d^8*e^8))*x + (36765*c^10*d^15*e^8 + 373583*a*c^9*d^13*e^10 + 390018*a^2*c^8
*d^11*e^12 + 3198*a^3*c^7*d^9*e^14 - 847*a^4*c^6*d^7*e^16 + 99*a^5*c^5*d^5*e^18)/(c^8*d^8*e^8))*x + (231*c^10*
d^16*e^7 + 219204*a*c^9*d^14*e^9 + 572739*a^2*c^8*d^12*e^11 + 16384*a^3*c^7*d^10*e^13 - 7491*a^4*c^6*d^8*e^15
+ 1980*a^5*c^5*d^6*e^17 - 231*a^6*c^4*d^4*e^19)/(c^8*d^8*e^8))*x - (1155*c^10*d^17*e^6 - 9933*a*c^9*d^15*e^8 -
847017*a^2*c^8*d^13*e^10 - 115609*a^3*c^7*d^11*e^12 + 82841*a^4*c^6*d^9*e^14 - 37719*a^5*c^5*d^7*e^16 + 9933*
a^6*c^4*d^5*e^18 - 1155*a^7*c^3*d^3*e^20)/(c^8*d^8*e^8))*x + (3465*c^10*d^18*e^5 - 30030*a*c^9*d^16*e^7 + 1150
38*a^2*c^8*d^14*e^9 + 334602*a^3*c^7*d^12*e^11 - 360448*a^4*c^6*d^10*e^13 + 255222*a^5*c^5*d^8*e^15 - 115038*a
^6*c^4*d^6*e^17 + 30030*a^7*c^3*d^4*e^19 - 3465*a^8*c^2*d^2*e^21)/(c^8*d^8*e^8)) + 55/65536*(c^9*d^18 - 9*a*c^
8*d^16*e^2 + 36*a^2*c^7*d^14*e^4 - 84*a^3*c^6*d^12*e^6 + 126*a^4*c^5*d^10*e^8 - 126*a^5*c^4*d^8*e^10 + 84*a^6*
c^3*d^6*e^12 - 36*a^7*c^2*d^4*e^14 + 9*a^8*c*d^2*e^16 - a^9*e^18)*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))))/(sqrt(c*d*e)*c^6*d^6*e^3)

Mupad [F(-1)]

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

[In]

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

[Out]

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