\(\int \frac {(f+g x)^{3/2} (a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}}{(d+e x)^{5/2}} \, dx\) [751]

Optimal result

Integrand size = 48, antiderivative size = 448 \[ \int \frac {(f+g x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx=-\frac {3 (c d f-a e g)^4 \sqrt {f+g x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{128 c^2 d^2 g^3 \sqrt {d+e x}}-\frac {(c d f-a e g)^3 (f+g x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 c d g^3 \sqrt {d+e x}}+\frac {(c d f-a e g)^2 (f+g x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{16 g^3 \sqrt {d+e x}}-\frac {(c d f-a e g) (f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{8 g^2 (d+e x)^{3/2}}+\frac {(f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 g (d+e x)^{5/2}}-\frac {3 (c d f-a e g)^5 \sqrt {a e+c d x} \sqrt {d+e x} \text {arctanh}\left (\frac {\sqrt {g} \sqrt {a e+c d x}}{\sqrt {c} \sqrt {d} \sqrt {f+g x}}\right )}{128 c^{5/2} d^{5/2} g^{7/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \]

[Out]

-1/8*(-a*e*g+c*d*f)*(g*x+f)^(5/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/g^2/(e*x+d)^(3/2)+1/5*(g*x+f)^(5/2)*
(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/g/(e*x+d)^(5/2)-3/128*(-a*e*g+c*d*f)^5*arctanh(g^(1/2)*(c*d*x+a*e)^(1/
2)/c^(1/2)/d^(1/2)/(g*x+f)^(1/2))*(c*d*x+a*e)^(1/2)*(e*x+d)^(1/2)/c^(5/2)/d^(5/2)/g^(7/2)/(a*d*e+(a*e^2+c*d^2)
*x+c*d*e*x^2)^(1/2)-1/64*(-a*e*g+c*d*f)^3*(g*x+f)^(3/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/c/d/g^3/(e*x+d
)^(1/2)+1/16*(-a*e*g+c*d*f)^2*(g*x+f)^(5/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/g^3/(e*x+d)^(1/2)-3/128*(-
a*e*g+c*d*f)^4*(g*x+f)^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/c^2/d^2/g^3/(e*x+d)^(1/2)

Rubi [A] (verified)

Time = 0.56 (sec) , antiderivative size = 448, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {878, 884, 905, 65, 223, 212} \[ \int \frac {(f+g x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx=-\frac {3 \sqrt {d+e x} \sqrt {a e+c d x} (c d f-a e g)^5 \text {arctanh}\left (\frac {\sqrt {g} \sqrt {a e+c d x}}{\sqrt {c} \sqrt {d} \sqrt {f+g x}}\right )}{128 c^{5/2} d^{5/2} g^{7/2} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {3 \sqrt {f+g x} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^4}{128 c^2 d^2 g^3 \sqrt {d+e x}}-\frac {(f+g x)^{3/2} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^3}{64 c d g^3 \sqrt {d+e x}}+\frac {(f+g x)^{5/2} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^2}{16 g^3 \sqrt {d+e x}}-\frac {(f+g x)^{5/2} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g)}{8 g^2 (d+e x)^{3/2}}+\frac {(f+g x)^{5/2} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{5 g (d+e x)^{5/2}} \]

[In]

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

[Out]

(-3*(c*d*f - a*e*g)^4*Sqrt[f + g*x]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(128*c^2*d^2*g^3*Sqrt[d + e*x
]) - ((c*d*f - a*e*g)^3*(f + g*x)^(3/2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(64*c*d*g^3*Sqrt[d + e*x]
) + ((c*d*f - a*e*g)^2*(f + g*x)^(5/2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(16*g^3*Sqrt[d + e*x]) - (
(c*d*f - a*e*g)*(f + g*x)^(5/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(8*g^2*(d + e*x)^(3/2)) + ((f +
g*x)^(5/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(5*g*(d + e*x)^(5/2)) - (3*(c*d*f - a*e*g)^5*Sqrt[a
*e + c*d*x]*Sqrt[d + e*x]*ArcTanh[(Sqrt[g]*Sqrt[a*e + c*d*x])/(Sqrt[c]*Sqrt[d]*Sqrt[f + g*x])])/(128*c^(5/2)*d
^(5/2)*g^(7/2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

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 223

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

Rule 878

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :>
Simp[(-(d + e*x)^m)*(f + g*x)^(n + 1)*((a + b*x + c*x^2)^p/(g*(m - n - 1))), x] - Dist[m*((c*e*f + c*d*g - b*e
*g)/(e^2*g*(m - n - 1))), Int[(d + e*x)^(m + 1)*(f + g*x)^n*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b,
c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] &&  !Intege
rQ[p] && EqQ[m + p, 0] && GtQ[p, 0] && NeQ[m - n - 1, 0] &&  !IGtQ[n, 0] &&  !(IntegerQ[n + p] && LtQ[n + p +
2, 0]) && RationalQ[n]

Rule 884

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

Rule 905

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

Rubi steps \begin{align*} \text {integral}& = \frac {(f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 g (d+e x)^{5/2}}-\frac {(c d f-a e g) \int \frac {(f+g x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx}{2 g} \\ & = -\frac {(c d f-a e g) (f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{8 g^2 (d+e x)^{3/2}}+\frac {(f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 g (d+e x)^{5/2}}+\frac {\left (3 (c d f-a e g)^2\right ) \int \frac {(f+g x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x}} \, dx}{16 g^2} \\ & = \frac {(c d f-a e g)^2 (f+g x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{16 g^3 \sqrt {d+e x}}-\frac {(c d f-a e g) (f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{8 g^2 (d+e x)^{3/2}}+\frac {(f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 g (d+e x)^{5/2}}-\frac {(c d f-a e g)^3 \int \frac {\sqrt {d+e x} (f+g x)^{3/2}}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{32 g^3} \\ & = -\frac {(c d f-a e g)^3 (f+g x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 c d g^3 \sqrt {d+e x}}+\frac {(c d f-a e g)^2 (f+g x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{16 g^3 \sqrt {d+e x}}-\frac {(c d f-a e g) (f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{8 g^2 (d+e x)^{3/2}}+\frac {(f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 g (d+e x)^{5/2}}-\frac {\left (3 (c d f-a e g)^4\right ) \int \frac {\sqrt {d+e x} \sqrt {f+g x}}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{128 c d g^3} \\ & = -\frac {3 (c d f-a e g)^4 \sqrt {f+g x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{128 c^2 d^2 g^3 \sqrt {d+e x}}-\frac {(c d f-a e g)^3 (f+g x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 c d g^3 \sqrt {d+e x}}+\frac {(c d f-a e g)^2 (f+g x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{16 g^3 \sqrt {d+e x}}-\frac {(c d f-a e g) (f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{8 g^2 (d+e x)^{3/2}}+\frac {(f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 g (d+e x)^{5/2}}-\frac {\left (3 (c d f-a e g)^5\right ) \int \frac {\sqrt {d+e x}}{\sqrt {f+g x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{256 c^2 d^2 g^3} \\ & = -\frac {3 (c d f-a e g)^4 \sqrt {f+g x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{128 c^2 d^2 g^3 \sqrt {d+e x}}-\frac {(c d f-a e g)^3 (f+g x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 c d g^3 \sqrt {d+e x}}+\frac {(c d f-a e g)^2 (f+g x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{16 g^3 \sqrt {d+e x}}-\frac {(c d f-a e g) (f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{8 g^2 (d+e x)^{3/2}}+\frac {(f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 g (d+e x)^{5/2}}-\frac {\left (3 (c d f-a e g)^5 \sqrt {a e+c d x} \sqrt {d+e x}\right ) \int \frac {1}{\sqrt {a e+c d x} \sqrt {f+g x}} \, dx}{256 c^2 d^2 g^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \\ & = -\frac {3 (c d f-a e g)^4 \sqrt {f+g x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{128 c^2 d^2 g^3 \sqrt {d+e x}}-\frac {(c d f-a e g)^3 (f+g x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 c d g^3 \sqrt {d+e x}}+\frac {(c d f-a e g)^2 (f+g x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{16 g^3 \sqrt {d+e x}}-\frac {(c d f-a e g) (f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{8 g^2 (d+e x)^{3/2}}+\frac {(f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 g (d+e x)^{5/2}}-\frac {\left (3 (c d f-a e g)^5 \sqrt {a e+c d x} \sqrt {d+e x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {f-\frac {a e g}{c d}+\frac {g x^2}{c d}}} \, dx,x,\sqrt {a e+c d x}\right )}{128 c^3 d^3 g^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \\ & = -\frac {3 (c d f-a e g)^4 \sqrt {f+g x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{128 c^2 d^2 g^3 \sqrt {d+e x}}-\frac {(c d f-a e g)^3 (f+g x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 c d g^3 \sqrt {d+e x}}+\frac {(c d f-a e g)^2 (f+g x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{16 g^3 \sqrt {d+e x}}-\frac {(c d f-a e g) (f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{8 g^2 (d+e x)^{3/2}}+\frac {(f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 g (d+e x)^{5/2}}-\frac {\left (3 (c d f-a e g)^5 \sqrt {a e+c d x} \sqrt {d+e x}\right ) \text {Subst}\left (\int \frac {1}{1-\frac {g x^2}{c d}} \, dx,x,\frac {\sqrt {a e+c d x}}{\sqrt {f+g x}}\right )}{128 c^3 d^3 g^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \\ & = -\frac {3 (c d f-a e g)^4 \sqrt {f+g x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{128 c^2 d^2 g^3 \sqrt {d+e x}}-\frac {(c d f-a e g)^3 (f+g x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 c d g^3 \sqrt {d+e x}}+\frac {(c d f-a e g)^2 (f+g x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{16 g^3 \sqrt {d+e x}}-\frac {(c d f-a e g) (f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{8 g^2 (d+e x)^{3/2}}+\frac {(f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 g (d+e x)^{5/2}}-\frac {3 (c d f-a e g)^5 \sqrt {a e+c d x} \sqrt {d+e x} \tanh ^{-1}\left (\frac {\sqrt {g} \sqrt {a e+c d x}}{\sqrt {c} \sqrt {d} \sqrt {f+g x}}\right )}{128 c^{5/2} d^{5/2} g^{7/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.81 (sec) , antiderivative size = 303, normalized size of antiderivative = 0.68 \[ \int \frac {(f+g x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx=\frac {((a e+c d x) (d+e x))^{5/2} \left (\frac {\sqrt {c} \sqrt {d} \sqrt {g} \sqrt {f+g x} \left (-15 a^4 e^4 g^4+10 a^3 c d e^3 g^3 (7 f+g x)+2 a^2 c^2 d^2 e^2 g^2 \left (64 f^2+233 f g x+124 g^2 x^2\right )+2 a c^3 d^3 e g \left (-35 f^3+23 f^2 g x+256 f g^2 x^2+168 g^3 x^3\right )+c^4 d^4 \left (15 f^4-10 f^3 g x+8 f^2 g^2 x^2+176 f g^3 x^3+128 g^4 x^4\right )\right )}{(a e+c d x)^2}-\frac {15 (c d f-a e g)^5 \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {f+g x}}{\sqrt {g} \sqrt {a e+c d x}}\right )}{(a e+c d x)^{5/2}}\right )}{640 c^{5/2} d^{5/2} g^{7/2} (d+e x)^{5/2}} \]

[In]

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

[Out]

(((a*e + c*d*x)*(d + e*x))^(5/2)*((Sqrt[c]*Sqrt[d]*Sqrt[g]*Sqrt[f + g*x]*(-15*a^4*e^4*g^4 + 10*a^3*c*d*e^3*g^3
*(7*f + g*x) + 2*a^2*c^2*d^2*e^2*g^2*(64*f^2 + 233*f*g*x + 124*g^2*x^2) + 2*a*c^3*d^3*e*g*(-35*f^3 + 23*f^2*g*
x + 256*f*g^2*x^2 + 168*g^3*x^3) + c^4*d^4*(15*f^4 - 10*f^3*g*x + 8*f^2*g^2*x^2 + 176*f*g^3*x^3 + 128*g^4*x^4)
))/(a*e + c*d*x)^2 - (15*(c*d*f - a*e*g)^5*ArcTanh[(Sqrt[c]*Sqrt[d]*Sqrt[f + g*x])/(Sqrt[g]*Sqrt[a*e + c*d*x])
])/(a*e + c*d*x)^(5/2)))/(640*c^(5/2)*d^(5/2)*g^(7/2)*(d + e*x)^(5/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1004\) vs. \(2(384)=768\).

Time = 0.62 (sec) , antiderivative size = 1005, normalized size of antiderivative = 2.24

method	result	size
default	\(\text {Expression too large to display}\)	\(1005\)

[In]

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

[Out]

1/1280*(g*x+f)^(1/2)*((c*d*x+a*e)*(e*x+d))^(1/2)*(256*c^4*d^4*g^4*x^4*((g*x+f)*(c*d*x+a*e))^(1/2)*(c*d*g)^(1/2
)+672*a*c^3*d^3*e*g^4*x^3*((g*x+f)*(c*d*x+a*e))^(1/2)*(c*d*g)^(1/2)+352*c^4*d^4*f*g^3*x^3*((g*x+f)*(c*d*x+a*e)
)^(1/2)*(c*d*g)^(1/2)+15*ln(1/2*(2*c*d*g*x+a*e*g+c*d*f+2*((g*x+f)*(c*d*x+a*e))^(1/2)*(c*d*g)^(1/2))/(c*d*g)^(1
/2))*a^5*e^5*g^5-75*ln(1/2*(2*c*d*g*x+a*e*g+c*d*f+2*((g*x+f)*(c*d*x+a*e))^(1/2)*(c*d*g)^(1/2))/(c*d*g)^(1/2))*
a^4*c*d*e^4*f*g^4+150*ln(1/2*(2*c*d*g*x+a*e*g+c*d*f+2*((g*x+f)*(c*d*x+a*e))^(1/2)*(c*d*g)^(1/2))/(c*d*g)^(1/2)
)*a^3*c^2*d^2*e^3*f^2*g^3-150*ln(1/2*(2*c*d*g*x+a*e*g+c*d*f+2*((g*x+f)*(c*d*x+a*e))^(1/2)*(c*d*g)^(1/2))/(c*d*
g)^(1/2))*a^2*c^3*d^3*e^2*f^3*g^2+75*ln(1/2*(2*c*d*g*x+a*e*g+c*d*f+2*((g*x+f)*(c*d*x+a*e))^(1/2)*(c*d*g)^(1/2)
)/(c*d*g)^(1/2))*a*c^4*d^4*e*f^4*g-15*ln(1/2*(2*c*d*g*x+a*e*g+c*d*f+2*((g*x+f)*(c*d*x+a*e))^(1/2)*(c*d*g)^(1/2
))/(c*d*g)^(1/2))*c^5*d^5*f^5+496*a^2*c^2*d^2*e^2*g^4*x^2*((g*x+f)*(c*d*x+a*e))^(1/2)*(c*d*g)^(1/2)+1024*a*c^3
*d^3*e*f*g^3*x^2*((g*x+f)*(c*d*x+a*e))^(1/2)*(c*d*g)^(1/2)+16*c^4*d^4*f^2*g^2*x^2*((g*x+f)*(c*d*x+a*e))^(1/2)*
(c*d*g)^(1/2)+20*((g*x+f)*(c*d*x+a*e))^(1/2)*(c*d*g)^(1/2)*a^3*c*d*e^3*g^4*x+932*((g*x+f)*(c*d*x+a*e))^(1/2)*(
c*d*g)^(1/2)*a^2*c^2*d^2*e^2*f*g^3*x+92*((g*x+f)*(c*d*x+a*e))^(1/2)*(c*d*g)^(1/2)*a*c^3*d^3*e*f^2*g^2*x-20*((g
*x+f)*(c*d*x+a*e))^(1/2)*(c*d*g)^(1/2)*c^4*d^4*f^3*g*x-30*((g*x+f)*(c*d*x+a*e))^(1/2)*(c*d*g)^(1/2)*a^4*e^4*g^
4+140*((g*x+f)*(c*d*x+a*e))^(1/2)*(c*d*g)^(1/2)*a^3*c*d*e^3*f*g^3+256*a^2*c^2*d^2*e^2*f^2*g^2*((g*x+f)*(c*d*x+
a*e))^(1/2)*(c*d*g)^(1/2)-140*((g*x+f)*(c*d*x+a*e))^(1/2)*(c*d*g)^(1/2)*a*c^3*d^3*e*f^3*g+30*((g*x+f)*(c*d*x+a
*e))^(1/2)*(c*d*g)^(1/2)*c^4*d^4*f^4)/(e*x+d)^(1/2)/c^2/d^2/g^3/((g*x+f)*(c*d*x+a*e))^(1/2)/(c*d*g)^(1/2)

Fricas [A] (verification not implemented)

none

Time = 3.55 (sec) , antiderivative size = 1331, normalized size of antiderivative = 2.97 \[ \int \frac {(f+g x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

[1/2560*(4*(128*c^5*d^5*g^5*x^4 + 15*c^5*d^5*f^4*g - 70*a*c^4*d^4*e*f^3*g^2 + 128*a^2*c^3*d^3*e^2*f^2*g^3 + 70
*a^3*c^2*d^2*e^3*f*g^4 - 15*a^4*c*d*e^4*g^5 + 16*(11*c^5*d^5*f*g^4 + 21*a*c^4*d^4*e*g^5)*x^3 + 8*(c^5*d^5*f^2*
g^3 + 64*a*c^4*d^4*e*f*g^4 + 31*a^2*c^3*d^3*e^2*g^5)*x^2 - 2*(5*c^5*d^5*f^3*g^2 - 23*a*c^4*d^4*e*f^2*g^3 - 233
*a^2*c^3*d^3*e^2*f*g^4 - 5*a^3*c^2*d^2*e^3*g^5)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d)*s
qrt(g*x + f) - 15*(c^5*d^6*f^5 - 5*a*c^4*d^5*e*f^4*g + 10*a^2*c^3*d^4*e^2*f^3*g^2 - 10*a^3*c^2*d^3*e^3*f^2*g^3
+ 5*a^4*c*d^2*e^4*f*g^4 - a^5*d*e^5*g^5 + (c^5*d^5*e*f^5 - 5*a*c^4*d^4*e^2*f^4*g + 10*a^2*c^3*d^3*e^3*f^3*g^2
- 10*a^3*c^2*d^2*e^4*f^2*g^3 + 5*a^4*c*d*e^5*f*g^4 - a^5*e^6*g^5)*x)*sqrt(c*d*g)*log(-(8*c^2*d^2*e*g^2*x^3 +
c^2*d^3*f^2 + 6*a*c*d^2*e*f*g + a^2*d*e^2*g^2 + 4*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(2*c*d*g*x + c*d
*f + a*e*g)*sqrt(c*d*g)*sqrt(e*x + d)*sqrt(g*x + f) + 8*(c^2*d^2*e*f*g + (c^2*d^3 + a*c*d*e^2)*g^2)*x^2 + (c^2
*d^2*e*f^2 + 2*(4*c^2*d^3 + 3*a*c*d*e^2)*f*g + (8*a*c*d^2*e + a^2*e^3)*g^2)*x)/(e*x + d)))/(c^3*d^3*e*g^4*x +
c^3*d^4*g^4), 1/1280*(2*(128*c^5*d^5*g^5*x^4 + 15*c^5*d^5*f^4*g - 70*a*c^4*d^4*e*f^3*g^2 + 128*a^2*c^3*d^3*e^2
*f^2*g^3 + 70*a^3*c^2*d^2*e^3*f*g^4 - 15*a^4*c*d*e^4*g^5 + 16*(11*c^5*d^5*f*g^4 + 21*a*c^4*d^4*e*g^5)*x^3 + 8*
(c^5*d^5*f^2*g^3 + 64*a*c^4*d^4*e*f*g^4 + 31*a^2*c^3*d^3*e^2*g^5)*x^2 - 2*(5*c^5*d^5*f^3*g^2 - 23*a*c^4*d^4*e*
f^2*g^3 - 233*a^2*c^3*d^3*e^2*f*g^4 - 5*a^3*c^2*d^2*e^3*g^5)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sq
rt(e*x + d)*sqrt(g*x + f) + 15*(c^5*d^6*f^5 - 5*a*c^4*d^5*e*f^4*g + 10*a^2*c^3*d^4*e^2*f^3*g^2 - 10*a^3*c^2*d^
3*e^3*f^2*g^3 + 5*a^4*c*d^2*e^4*f*g^4 - a^5*d*e^5*g^5 + (c^5*d^5*e*f^5 - 5*a*c^4*d^4*e^2*f^4*g + 10*a^2*c^3*d^
3*e^3*f^3*g^2 - 10*a^3*c^2*d^2*e^4*f^2*g^3 + 5*a^4*c*d*e^5*f*g^4 - a^5*e^6*g^5)*x)*sqrt(-c*d*g)*arctan(2*sqrt(
c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(-c*d*g)*sqrt(e*x + d)*sqrt(g*x + f)/(2*c*d*e*g*x^2 + c*d^2*f + a*d
*e*g + (c*d*e*f + (2*c*d^2 + a*e^2)*g)*x)))/(c^3*d^3*e*g^4*x + c^3*d^4*g^4)]

Sympy [F(-1)]

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

[In]

integrate((g*x+f)**(3/2)*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2)/(e*x+d)**(5/2),x)

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(5/2)*(g*x + f)^(3/2)/(e*x + d)^(5/2), x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 18597 vs. \(2 (384) = 768\).

Time = 5.44 (sec) , antiderivative size = 18597, normalized size of antiderivative = 41.51 \[ \int \frac {(f+g x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/1920*(480*a^2*f*((4*((c*d*e^2*f*g - a*e^3*g^2)*log(abs(-sqrt(e^2*f + (e*x + d)*e*g - d*e*g)*sqrt(c*d*g) + sq
rt(-c*d*e^2*f*g + a*e^3*g^2 + (e^2*f + (e*x + d)*e*g - d*e*g)*c*d*g)))/sqrt(c*d*g) + sqrt(-c*d*e^2*f*g + a*e^3
*g^2 + (e^2*f + (e*x + d)*e*g - d*e*g)*c*d*g)*sqrt(e^2*f + (e*x + d)*e*g - d*e*g))*e*f*abs(g)/g^2 - 4*((c*d*e^
2*f*g - a*e^3*g^2)*log(abs(-sqrt(e^2*f + (e*x + d)*e*g - d*e*g)*sqrt(c*d*g) + sqrt(-c*d*e^2*f*g + a*e^3*g^2 +
(e^2*f + (e*x + d)*e*g - d*e*g)*c*d*g)))/sqrt(c*d*g) + sqrt(-c*d*e^2*f*g + a*e^3*g^2 + (e^2*f + (e*x + d)*e*g
- d*e*g)*c*d*g)*sqrt(e^2*f + (e*x + d)*e*g - d*e*g))*d*abs(g)/g + (sqrt(-c*d*e^2*f*g + a*e^3*g^2 + (e^2*f + (e
*x + d)*e*g - d*e*g)*c*d*g)*(2*e^2*f + 2*(e*x + d)*e*g - 2*d*e*g - (5*c^2*d^2*e^2*f - 4*c^2*d^3*e*g - a*c*d*e^
3*g)/(c^2*d^2))*sqrt(e^2*f + (e*x + d)*e*g - d*e*g) - (3*c^2*d^2*e^4*f^2*g - 4*c^2*d^3*e^3*f*g^2 - 2*a*c*d*e^5
*f*g^2 + 4*a*c*d^2*e^4*g^3 - a^2*e^6*g^3)*log(abs(-sqrt(e^2*f + (e*x + d)*e*g - d*e*g)*sqrt(c*d*g) + sqrt(-c*d
*e^2*f*g + a*e^3*g^2 + (e^2*f + (e*x + d)*e*g - d*e*g)*c*d*g)))/(sqrt(c*d*g)*c*d))*abs(g)/(e*g^2))/g - (c^2*d^
2*e^3*f^2*g*abs(g)*log(abs(-sqrt(e^2*f - d*e*g)*sqrt(c*d*g) + sqrt(-c*d^2*e*g^2 + a*e^3*g^2))) - 2*a*c*d*e^4*f
*g^2*abs(g)*log(abs(-sqrt(e^2*f - d*e*g)*sqrt(c*d*g) + sqrt(-c*d^2*e*g^2 + a*e^3*g^2))) + a^2*e^5*g^3*abs(g)*l
og(abs(-sqrt(e^2*f - d*e*g)*sqrt(c*d*g) + sqrt(-c*d^2*e*g^2 + a*e^3*g^2))) + sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sq
rt(e^2*f - d*e*g)*sqrt(c*d*g)*c*d*e*f*abs(g) - 2*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(e^2*f - d*e*g)*sqrt(c*d*g
)*c*d^2*g*abs(g) + sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*a*e^2*g*abs(g))/(sqrt(c*d*g)
*c*d*g^3))*abs(e)^2/e^2 + 10*c^2*d^2*f*((192*((c*d*e^2*f*g - a*e^3*g^2)*log(abs(-sqrt(e^2*f + (e*x + d)*e*g -
d*e*g)*sqrt(c*d*g) + sqrt(-c*d*e^2*f*g + a*e^3*g^2 + (e^2*f + (e*x + d)*e*g - d*e*g)*c*d*g)))/sqrt(c*d*g) + sq
rt(-c*d*e^2*f*g + a*e^3*g^2 + (e^2*f + (e*x + d)*e*g - d*e*g)*c*d*g)*sqrt(e^2*f + (e*x + d)*e*g - d*e*g))*d^2*
e*f*abs(g)/g^2 - 192*((c*d*e^2*f*g - a*e^3*g^2)*log(abs(-sqrt(e^2*f + (e*x + d)*e*g - d*e*g)*sqrt(c*d*g) + sqr
t(-c*d*e^2*f*g + a*e^3*g^2 + (e^2*f + (e*x + d)*e*g - d*e*g)*c*d*g)))/sqrt(c*d*g) + sqrt(-c*d*e^2*f*g + a*e^3*
g^2 + (e^2*f + (e*x + d)*e*g - d*e*g)*c*d*g)*sqrt(e^2*f + (e*x + d)*e*g - d*e*g))*d^3*abs(g)/g + 8*(sqrt(-c*d*
e^2*f*g + a*e^3*g^2 + (e^2*f + (e*x + d)*e*g - d*e*g)*c*d*g)*sqrt(e^2*f + (e*x + d)*e*g - d*e*g)*(2*(e^2*f + (
e*x + d)*e*g - d*e*g)*(4*(e^2*f + (e*x + d)*e*g - d*e*g)/(e^2*g^2) - (13*c^4*d^4*e^3*f*g^5 - 12*c^4*d^5*e^2*g^
6 - a*c^3*d^3*e^4*g^6)/(c^4*d^4*e^3*g^7)) + 3*(11*c^4*d^4*e^5*f^2*g^5 - 20*c^4*d^5*e^4*f*g^6 - 2*a*c^3*d^3*e^6
*f*g^6 + 8*c^4*d^6*e^3*g^7 + 4*a*c^3*d^4*e^5*g^7 - a^2*c^2*d^2*e^7*g^7)/(c^4*d^4*e^3*g^7)) + 3*(5*c^3*d^3*e^4*
f^3 - 12*c^3*d^4*e^3*f^2*g - 3*a*c^2*d^2*e^5*f^2*g + 8*c^3*d^5*e^2*f*g^2 + 8*a*c^2*d^3*e^4*f*g^2 - a^2*c*d*e^6
*f*g^2 - 8*a*c^2*d^4*e^3*g^3 + 4*a^2*c*d^2*e^5*g^3 - a^3*e^7*g^3)*log(abs(-sqrt(e^2*f + (e*x + d)*e*g - d*e*g)
*sqrt(c*d*g) + sqrt(-c*d*e^2*f*g + a*e^3*g^2 + (e^2*f + (e*x + d)*e*g - d*e*g)*c*d*g)))/(sqrt(c*d*g)*c^2*d^2*g
))*e*f*abs(g)/g^2 - 24*(sqrt(-c*d*e^2*f*g + a*e^3*g^2 + (e^2*f + (e*x + d)*e*g - d*e*g)*c*d*g)*sqrt(e^2*f + (e
*x + d)*e*g - d*e*g)*(2*(e^2*f + (e*x + d)*e*g - d*e*g)*(4*(e^2*f + (e*x + d)*e*g - d*e*g)/(e^2*g^2) - (13*c^4
*d^4*e^3*f*g^5 - 12*c^4*d^5*e^2*g^6 - a*c^3*d^3*e^4*g^6)/(c^4*d^4*e^3*g^7)) + 3*(11*c^4*d^4*e^5*f^2*g^5 - 20*c
^4*d^5*e^4*f*g^6 - 2*a*c^3*d^3*e^6*f*g^6 + 8*c^4*d^6*e^3*g^7 + 4*a*c^3*d^4*e^5*g^7 - a^2*c^2*d^2*e^7*g^7)/(c^4
*d^4*e^3*g^7)) + 3*(5*c^3*d^3*e^4*f^3 - 12*c^3*d^4*e^3*f^2*g - 3*a*c^2*d^2*e^5*f^2*g + 8*c^3*d^5*e^2*f*g^2 + 8
*a*c^2*d^3*e^4*f*g^2 - a^2*c*d*e^6*f*g^2 - 8*a*c^2*d^4*e^3*g^3 + 4*a^2*c*d^2*e^5*g^3 - a^3*e^7*g^3)*log(abs(-s
qrt(e^2*f + (e*x + d)*e*g - d*e*g)*sqrt(c*d*g) + sqrt(-c*d*e^2*f*g + a*e^3*g^2 + (e^2*f + (e*x + d)*e*g - d*e*
g)*c*d*g)))/(sqrt(c*d*g)*c^2*d^2*g))*d*abs(g)/g - 96*(sqrt(-c*d*e^2*f*g + a*e^3*g^2 + (e^2*f + (e*x + d)*e*g -
d*e*g)*c*d*g)*(2*e^2*f + 2*(e*x + d)*e*g - 2*d*e*g - (5*c^2*d^2*e^2*f - 4*c^2*d^3*e*g - a*c*d*e^3*g)/(c^2*d^2
))*sqrt(e^2*f + (e*x + d)*e*g - d*e*g) - (3*c^2*d^2*e^4*f^2*g - 4*c^2*d^3*e^3*f*g^2 - 2*a*c*d*e^5*f*g^2 + 4*a*
c*d^2*e^4*g^3 - a^2*e^6*g^3)*log(abs(-sqrt(e^2*f + (e*x + d)*e*g - d*e*g)*sqrt(c*d*g) + sqrt(-c*d*e^2*f*g + a*
e^3*g^2 + (e^2*f + (e*x + d)*e*g - d*e*g)*c*d*g)))/(sqrt(c*d*g)*c*d))*d*f*abs(g)/g^3 + 144*(sqrt(-c*d*e^2*f*g
+ a*e^3*g^2 + (e^2*f + (e*x + d)*e*g - d*e*g)*c*d*g)*(2*e^2*f + 2*(e*x + d)*e*g - 2*d*e*g - (5*c^2*d^2*e^2*f -
4*c^2*d^3*e*g - a*c*d*e^3*g)/(c^2*d^2))*sqrt(e^2*f + (e*x + d)*e*g - d*e*g) - (3*c^2*d^2*e^4*f^2*g - 4*c^2*d^
3*e^3*f*g^2 - 2*a*c*d*e^5*f*g^2 + 4*a*c*d^2*e^4*g^3 - a^2*e^6*g^3)*log(abs(-sqrt(e^2*f + (e*x + d)*e*g - d*e*g
)*sqrt(c*d*g) + sqrt(-c*d*e^2*f*g + a*e^3*g^2 + (e^2*f + (e*x + d)*e*g - d*e*g)*c*d*g)))/(sqrt(c*d*g)*c*d))*d^
2*abs(g)/(e*g^2) + (sqrt(-c*d*e^2*f*g + a*e^3*g^2 + (e^2*f + (e*x + d)*e*g - d*e*g)*c*d*g)*sqrt(e^2*f + (e*x +
d)*e*g - d*e*g)*(2*(e^2*f + (e*x + d)*e*g - d*e*g)*(4*(e^2*f + (e*x + d)*e*g - d*e*g)*(6*(e^2*f + (e*x + d)*e
*g - d*e*g)/(e^3*g^3) - (25*c^6*d^6*e^5*f*g^11 - 24*c^6*d^7*e^4*g^12 - a*c^5*d^5*e^6*g^12)/(c^6*d^6*e^6*g^14))
+ (163*c^6*d^6*e^7*f^2*g^11 - 312*c^6*d^7*e^6*f*g^12 - 14*a*c^5*d^5*e^8*f*g^12 + 144*c^6*d^8*e^5*g^13 + 24*a*
c^5*d^6*e^7*g^13 - 5*a^2*c^4*d^4*e^9*g^13)/(c^6*d^6*e^6*g^14)) - 3*(93*c^6*d^6*e^9*f^3*g^11 - 264*c^6*d^7*e^8*
f^2*g^12 - 15*a*c^5*d^5*e^10*f^2*g^12 + 240*c^6*d^8*e^7*f*g^13 + 48*a*c^5*d^6*e^9*f*g^13 - 9*a^2*c^4*d^4*e^11*
f*g^13 - 64*c^6*d^9*e^6*g^14 - 48*a*c^5*d^7*e^8*g^14 + 24*a^2*c^4*d^5*e^10*g^14 - 5*a^3*c^3*d^3*e^12*g^14)/(c^
6*d^6*e^6*g^14)) - 3*(35*c^4*d^4*e^5*f^4 - 120*c^4*d^5*e^4*f^3*g - 20*a*c^3*d^3*e^6*f^3*g + 144*c^4*d^6*e^3*f^
2*g^2 + 72*a*c^3*d^4*e^5*f^2*g^2 - 6*a^2*c^2*d^2*e^7*f^2*g^2 - 64*c^4*d^7*e^2*f*g^3 - 96*a*c^3*d^5*e^4*f*g^3 +
24*a^2*c^2*d^3*e^6*f*g^3 - 4*a^3*c*d*e^8*f*g^3 + 64*a*c^3*d^6*e^3*g^4 - 48*a^2*c^2*d^4*e^5*g^4 + 24*a^3*c*d^2
*e^7*g^4 - 5*a^4*e^9*g^4)*log(abs(-sqrt(e^2*f + (e*x + d)*e*g - d*e*g)*sqrt(c*d*g) + sqrt(-c*d*e^2*f*g + a*e^3
*g^2 + (e^2*f + (e*x + d)*e*g - d*e*g)*c*d*g)))/(sqrt(c*d*g)*c^3*d^3*g^2))*abs(g)/g)/(e^2*g) - (15*c^4*d^4*e^5
*f^4*g*abs(g)*log(abs(-sqrt(e^2*f - d*e*g)*sqrt(c*d*g) + sqrt(-c*d^2*e*g^2 + a*e^3*g^2))) - 12*a*c^3*d^3*e^6*f
^3*g^2*abs(g)*log(abs(-sqrt(e^2*f - d*e*g)*sqrt(c*d*g) + sqrt(-c*d^2*e*g^2 + a*e^3*g^2))) - 6*a^2*c^2*d^2*e^7*
f^2*g^3*abs(g)*log(abs(-sqrt(e^2*f - d*e*g)*sqrt(c*d*g) + sqrt(-c*d^2*e*g^2 + a*e^3*g^2))) - 12*a^3*c*d*e^8*f*
g^4*abs(g)*log(abs(-sqrt(e^2*f - d*e*g)*sqrt(c*d*g) + sqrt(-c*d^2*e*g^2 + a*e^3*g^2))) + 15*a^4*e^9*g^5*abs(g)
*log(abs(-sqrt(e^2*f - d*e*g)*sqrt(c*d*g) + sqrt(-c*d^2*e*g^2 + a*e^3*g^2))) + 15*sqrt(-c*d^2*e*g^2 + a*e^3*g^
2)*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*c^3*d^3*e^3*f^3*abs(g) + 10*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(e^2*f - d*e
*g)*sqrt(c*d*g)*c^3*d^4*e^2*f^2*g*abs(g) - 7*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*a*
c^2*d^2*e^4*f^2*g*abs(g) + 8*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*c^3*d^5*e*f*g^2*ab
s(g) - 4*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*a*c^2*d^3*e^3*f*g^2*abs(g) - 7*sqrt(-c
*d^2*e*g^2 + a*e^3*g^2)*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*a^2*c*d*e^5*f*g^2*abs(g) - 48*sqrt(-c*d^2*e*g^2 + a*e^
3*g^2)*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*c^3*d^6*g^3*abs(g) + 8*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(e^2*f - d*e*
g)*sqrt(c*d*g)*a*c^2*d^4*e^2*g^3*abs(g) + 10*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*a^
2*c*d^2*e^4*g^3*abs(g) + 15*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*a^3*e^6*g^3*abs(g))
/(sqrt(c*d*g)*c^3*d^3*e^2*g^5))*abs(e)^2/e^4 + 20*a*c*d*g*((192*((c*d*e^2*f*g - a*e^3*g^2)*log(abs(-sqrt(e^2*f
+ (e*x + d)*e*g - d*e*g)*sqrt(c*d*g) + sqrt(-c*d*e^2*f*g + a*e^3*g^2 + (e^2*f + (e*x + d)*e*g - d*e*g)*c*d*g)
))/sqrt(c*d*g) + sqrt(-c*d*e^2*f*g + a*e^3*g^2 + (e^2*f + (e*x + d)*e*g - d*e*g)*c*d*g)*sqrt(e^2*f + (e*x + d)
*e*g - d*e*g))*d^2*e*f*abs(g)/g^2 - 192*((c*d*e^2*f*g - a*e^3*g^2)*log(abs(-sqrt(e^2*f + (e*x + d)*e*g - d*e*g
)*sqrt(c*d*g) + sqrt(-c*d*e^2*f*g + a*e^3*g^2 + (e^2*f + (e*x + d)*e*g - d*e*g)*c*d*g)))/sqrt(c*d*g) + sqrt(-c
*d*e^2*f*g + a*e^3*g^2 + (e^2*f + (e*x + d)*e*g - d*e*g)*c*d*g)*sqrt(e^2*f + (e*x + d)*e*g - d*e*g))*d^3*abs(g
)/g + 8*(sqrt(-c*d*e^2*f*g + a*e^3*g^2 + (e^2*f + (e*x + d)*e*g - d*e*g)*c*d*g)*sqrt(e^2*f + (e*x + d)*e*g - d
*e*g)*(2*(e^2*f + (e*x + d)*e*g - d*e*g)*(4*(e^2*f + (e*x + d)*e*g - d*e*g)/(e^2*g^2) - (13*c^4*d^4*e^3*f*g^5
- 12*c^4*d^5*e^2*g^6 - a*c^3*d^3*e^4*g^6)/(c^4*d^4*e^3*g^7)) + 3*(11*c^4*d^4*e^5*f^2*g^5 - 20*c^4*d^5*e^4*f*g^
6 - 2*a*c^3*d^3*e^6*f*g^6 + 8*c^4*d^6*e^3*g^7 + 4*a*c^3*d^4*e^5*g^7 - a^2*c^2*d^2*e^7*g^7)/(c^4*d^4*e^3*g^7))
+ 3*(5*c^3*d^3*e^4*f^3 - 12*c^3*d^4*e^3*f^2*g - 3*a*c^2*d^2*e^5*f^2*g + 8*c^3*d^5*e^2*f*g^2 + 8*a*c^2*d^3*e^4*
f*g^2 - a^2*c*d*e^6*f*g^2 - 8*a*c^2*d^4*e^3*g^3 + 4*a^2*c*d^2*e^5*g^3 - a^3*e^7*g^3)*log(abs(-sqrt(e^2*f + (e*
x + d)*e*g - d*e*g)*sqrt(c*d*g) + sqrt(-c*d*e^2*f*g + a*e^3*g^2 + (e^2*f + (e*x + d)*e*g - d*e*g)*c*d*g)))/(sq
rt(c*d*g)*c^2*d^2*g))*e*f*abs(g)/g^2 - 24*(sqrt(-c*d*e^2*f*g + a*e^3*g^2 + (e^2*f + (e*x + d)*e*g - d*e*g)*c*d
*g)*sqrt(e^2*f + (e*x + d)*e*g - d*e*g)*(2*(e^2*f + (e*x + d)*e*g - d*e*g)*(4*(e^2*f + (e*x + d)*e*g - d*e*g)/
(e^2*g^2) - (13*c^4*d^4*e^3*f*g^5 - 12*c^4*d^5*e^2*g^6 - a*c^3*d^3*e^4*g^6)/(c^4*d^4*e^3*g^7)) + 3*(11*c^4*d^4
*e^5*f^2*g^5 - 20*c^4*d^5*e^4*f*g^6 - 2*a*c^3*d^3*e^6*f*g^6 + 8*c^4*d^6*e^3*g^7 + 4*a*c^3*d^4*e^5*g^7 - a^2*c^
2*d^2*e^7*g^7)/(c^4*d^4*e^3*g^7)) + 3*(5*c^3*d^3*e^4*f^3 - 12*c^3*d^4*e^3*f^2*g - 3*a*c^2*d^2*e^5*f^2*g + 8*c^
3*d^5*e^2*f*g^2 + 8*a*c^2*d^3*e^4*f*g^2 - a^2*c*d*e^6*f*g^2 - 8*a*c^2*d^4*e^3*g^3 + 4*a^2*c*d^2*e^5*g^3 - a^3*
e^7*g^3)*log(abs(-sqrt(e^2*f + (e*x + d)*e*g - d*e*g)*sqrt(c*d*g) + sqrt(-c*d*e^2*f*g + a*e^3*g^2 + (e^2*f + (
e*x + d)*e*g - d*e*g)*c*d*g)))/(sqrt(c*d*g)*c^2*d^2*g))*d*abs(g)/g - 96*(sqrt(-c*d*e^2*f*g + a*e^3*g^2 + (e^2*
f + (e*x + d)*e*g - d*e*g)*c*d*g)*(2*e^2*f + 2*(e*x + d)*e*g - 2*d*e*g - (5*c^2*d^2*e^2*f - 4*c^2*d^3*e*g - a*
c*d*e^3*g)/(c^2*d^2))*sqrt(e^2*f + (e*x + d)*e*g - d*e*g) - (3*c^2*d^2*e^4*f^2*g - 4*c^2*d^3*e^3*f*g^2 - 2*a*c
*d*e^5*f*g^2 + 4*a*c*d^2*e^4*g^3 - a^2*e^6*g^3)*log(abs(-sqrt(e^2*f + (e*x + d)*e*g - d*e*g)*sqrt(c*d*g) + sqr
t(-c*d*e^2*f*g + a*e^3*g^2 + (e^2*f + (e*x + d)*e*g - d*e*g)*c*d*g)))/(sqrt(c*d*g)*c*d))*d*f*abs(g)/g^3 + 144*
(sqrt(-c*d*e^2*f*g + a*e^3*g^2 + (e^2*f + (e*x + d)*e*g - d*e*g)*c*d*g)*(2*e^2*f + 2*(e*x + d)*e*g - 2*d*e*g -
(5*c^2*d^2*e^2*f - 4*c^2*d^3*e*g - a*c*d*e^3*g)/(c^2*d^2))*sqrt(e^2*f + (e*x + d)*e*g - d*e*g) - (3*c^2*d^2*e
^4*f^2*g - 4*c^2*d^3*e^3*f*g^2 - 2*a*c*d*e^5*f*g^2 + 4*a*c*d^2*e^4*g^3 - a^2*e^6*g^3)*log(abs(-sqrt(e^2*f + (e
*x + d)*e*g - d*e*g)*sqrt(c*d*g) + sqrt(-c*d*e^2*f*g + a*e^3*g^2 + (e^2*f + (e*x + d)*e*g - d*e*g)*c*d*g)))/(s
qrt(c*d*g)*c*d))*d^2*abs(g)/(e*g^2) + (sqrt(-c*d*e^2*f*g + a*e^3*g^2 + (e^2*f + (e*x + d)*e*g - d*e*g)*c*d*g)*
sqrt(e^2*f + (e*x + d)*e*g - d*e*g)*(2*(e^2*f + (e*x + d)*e*g - d*e*g)*(4*(e^2*f + (e*x + d)*e*g - d*e*g)*(6*(
e^2*f + (e*x + d)*e*g - d*e*g)/(e^3*g^3) - (25*c^6*d^6*e^5*f*g^11 - 24*c^6*d^7*e^4*g^12 - a*c^5*d^5*e^6*g^12)/
(c^6*d^6*e^6*g^14)) + (163*c^6*d^6*e^7*f^2*g^11 - 312*c^6*d^7*e^6*f*g^12 - 14*a*c^5*d^5*e^8*f*g^12 + 144*c^6*d
^8*e^5*g^13 + 24*a*c^5*d^6*e^7*g^13 - 5*a^2*c^4*d^4*e^9*g^13)/(c^6*d^6*e^6*g^14)) - 3*(93*c^6*d^6*e^9*f^3*g^11
- 264*c^6*d^7*e^8*f^2*g^12 - 15*a*c^5*d^5*e^10*f^2*g^12 + 240*c^6*d^8*e^7*f*g^13 + 48*a*c^5*d^6*e^9*f*g^13 -
9*a^2*c^4*d^4*e^11*f*g^13 - 64*c^6*d^9*e^6*g^14 - 48*a*c^5*d^7*e^8*g^14 + 24*a^2*c^4*d^5*e^10*g^14 - 5*a^3*c^3
*d^3*e^12*g^14)/(c^6*d^6*e^6*g^14)) - 3*(35*c^4*d^4*e^5*f^4 - 120*c^4*d^5*e^4*f^3*g - 20*a*c^3*d^3*e^6*f^3*g +
144*c^4*d^6*e^3*f^2*g^2 + 72*a*c^3*d^4*e^5*f^2*g^2 - 6*a^2*c^2*d^2*e^7*f^2*g^2 - 64*c^4*d^7*e^2*f*g^3 - 96*a*
c^3*d^5*e^4*f*g^3 + 24*a^2*c^2*d^3*e^6*f*g^3 - 4*a^3*c*d*e^8*f*g^3 + 64*a*c^3*d^6*e^3*g^4 - 48*a^2*c^2*d^4*e^5
*g^4 + 24*a^3*c*d^2*e^7*g^4 - 5*a^4*e^9*g^4)*log(abs(-sqrt(e^2*f + (e*x + d)*e*g - d*e*g)*sqrt(c*d*g) + sqrt(-
c*d*e^2*f*g + a*e^3*g^2 + (e^2*f + (e*x + d)*e*g - d*e*g)*c*d*g)))/(sqrt(c*d*g)*c^3*d^3*g^2))*abs(g)/g)/(e^2*g
) - (15*c^4*d^4*e^5*f^4*g*abs(g)*log(abs(-sqrt(e^2*f - d*e*g)*sqrt(c*d*g) + sqrt(-c*d^2*e*g^2 + a*e^3*g^2))) -
12*a*c^3*d^3*e^6*f^3*g^2*abs(g)*log(abs(-sqrt(e^2*f - d*e*g)*sqrt(c*d*g) + sqrt(-c*d^2*e*g^2 + a*e^3*g^2))) -
6*a^2*c^2*d^2*e^7*f^2*g^3*abs(g)*log(abs(-sqrt(e^2*f - d*e*g)*sqrt(c*d*g) + sqrt(-c*d^2*e*g^2 + a*e^3*g^2)))
- 12*a^3*c*d*e^8*f*g^4*abs(g)*log(abs(-sqrt(e^2*f - d*e*g)*sqrt(c*d*g) + sqrt(-c*d^2*e*g^2 + a*e^3*g^2))) + 15
*a^4*e^9*g^5*abs(g)*log(abs(-sqrt(e^2*f - d*e*g)*sqrt(c*d*g) + sqrt(-c*d^2*e*g^2 + a*e^3*g^2))) + 15*sqrt(-c*d
^2*e*g^2 + a*e^3*g^2)*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*c^3*d^3*e^3*f^3*abs(g) + 10*sqrt(-c*d^2*e*g^2 + a*e^3*g^
2)*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*c^3*d^4*e^2*f^2*g*abs(g) - 7*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(e^2*f - d*
e*g)*sqrt(c*d*g)*a*c^2*d^2*e^4*f^2*g*abs(g) + 8*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)
*c^3*d^5*e*f*g^2*abs(g) - 4*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*a*c^2*d^3*e^3*f*g^2
*abs(g) - 7*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*a^2*c*d*e^5*f*g^2*abs(g) - 48*sqrt(
-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*c^3*d^6*g^3*abs(g) + 8*sqrt(-c*d^2*e*g^2 + a*e^3*g^2
)*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*a*c^2*d^4*e^2*g^3*abs(g) + 10*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(e^2*f - d*
e*g)*sqrt(c*d*g)*a^2*c*d^2*e^4*g^3*abs(g) + 15*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*
a^3*e^6*g^3*abs(g))/(sqrt(c*d*g)*c^3*d^3*e^2*g^5))*abs(e)^2/e^3 - c^2*d^2*g*((1920*((c*d*e^2*f*g - a*e^3*g^2)*
log(abs(-sqrt(e^2*f + (e*x + d)*e*g - d*e*g)*sqrt(c*d*g) + sqrt(-c*d*e^2*f*g + a*e^3*g^2 + (e^2*f + (e*x + d)*
e*g - d*e*g)*c*d*g)))/sqrt(c*d*g) + sqrt(-c*d*e^2*f*g + a*e^3*g^2 + (e^2*f + (e*x + d)*e*g - d*e*g)*c*d*g)*sqr
t(e^2*f + (e*x + d)*e*g - d*e*g))*d^3*e*f*abs(g)/g^2 - 1920*((c*d*e^2*f*g - a*e^3*g^2)*log(abs(-sqrt(e^2*f + (
e*x + d)*e*g - d*e*g)*sqrt(c*d*g) + sqrt(-c*d*e^2*f*g + a*e^3*g^2 + (e^2*f + (e*x + d)*e*g - d*e*g)*c*d*g)))/s
qrt(c*d*g) + sqrt(-c*d*e^2*f*g + a*e^3*g^2 + (e^2*f + (e*x + d)*e*g - d*e*g)*c*d*g)*sqrt(e^2*f + (e*x + d)*e*g
- d*e*g))*d^4*abs(g)/g + 240*(sqrt(-c*d*e^2*f*g + a*e^3*g^2 + (e^2*f + (e*x + d)*e*g - d*e*g)*c*d*g)*sqrt(e^2
*f + (e*x + d)*e*g - d*e*g)*(2*(e^2*f + (e*x + d)*e*g - d*e*g)*(4*(e^2*f + (e*x + d)*e*g - d*e*g)/(e^2*g^2) -
(13*c^4*d^4*e^3*f*g^5 - 12*c^4*d^5*e^2*g^6 - a*c^3*d^3*e^4*g^6)/(c^4*d^4*e^3*g^7)) + 3*(11*c^4*d^4*e^5*f^2*g^5
- 20*c^4*d^5*e^4*f*g^6 - 2*a*c^3*d^3*e^6*f*g^6 + 8*c^4*d^6*e^3*g^7 + 4*a*c^3*d^4*e^5*g^7 - a^2*c^2*d^2*e^7*g^
7)/(c^4*d^4*e^3*g^7)) + 3*(5*c^3*d^3*e^4*f^3 - 12*c^3*d^4*e^3*f^2*g - 3*a*c^2*d^2*e^5*f^2*g + 8*c^3*d^5*e^2*f*
g^2 + 8*a*c^2*d^3*e^4*f*g^2 - a^2*c*d*e^6*f*g^2 - 8*a*c^2*d^4*e^3*g^3 + 4*a^2*c*d^2*e^5*g^3 - a^3*e^7*g^3)*log
(abs(-sqrt(e^2*f + (e*x + d)*e*g - d*e*g)*sqrt(c*d*g) + sqrt(-c*d*e^2*f*g + a*e^3*g^2 + (e^2*f + (e*x + d)*e*g
- d*e*g)*c*d*g)))/(sqrt(c*d*g)*c^2*d^2*g))*d*e*f*abs(g)/g^2 - 480*(sqrt(-c*d*e^2*f*g + a*e^3*g^2 + (e^2*f + (
e*x + d)*e*g - d*e*g)*c*d*g)*sqrt(e^2*f + (e*x + d)*e*g - d*e*g)*(2*(e^2*f + (e*x + d)*e*g - d*e*g)*(4*(e^2*f
+ (e*x + d)*e*g - d*e*g)/(e^2*g^2) - (13*c^4*d^4*e^3*f*g^5 - 12*c^4*d^5*e^2*g^6 - a*c^3*d^3*e^4*g^6)/(c^4*d^4*
e^3*g^7)) + 3*(11*c^4*d^4*e^5*f^2*g^5 - 20*c^4*d^5*e^4*f*g^6 - 2*a*c^3*d^3*e^6*f*g^6 + 8*c^4*d^6*e^3*g^7 + 4*a
*c^3*d^4*e^5*g^7 - a^2*c^2*d^2*e^7*g^7)/(c^4*d^4*e^3*g^7)) + 3*(5*c^3*d^3*e^4*f^3 - 12*c^3*d^4*e^3*f^2*g - 3*a
*c^2*d^2*e^5*f^2*g + 8*c^3*d^5*e^2*f*g^2 + 8*a*c^2*d^3*e^4*f*g^2 - a^2*c*d*e^6*f*g^2 - 8*a*c^2*d^4*e^3*g^3 + 4
*a^2*c*d^2*e^5*g^3 - a^3*e^7*g^3)*log(abs(-sqrt(e^2*f + (e*x + d)*e*g - d*e*g)*sqrt(c*d*g) + sqrt(-c*d*e^2*f*g
+ a*e^3*g^2 + (e^2*f + (e*x + d)*e*g - d*e*g)*c*d*g)))/(sqrt(c*d*g)*c^2*d^2*g))*d^2*abs(g)/g - 1440*(sqrt(-c*
d*e^2*f*g + a*e^3*g^2 + (e^2*f + (e*x + d)*e*g - d*e*g)*c*d*g)*(2*e^2*f + 2*(e*x + d)*e*g - 2*d*e*g - (5*c^2*d
^2*e^2*f - 4*c^2*d^3*e*g - a*c*d*e^3*g)/(c^2*d^2))*sqrt(e^2*f + (e*x + d)*e*g - d*e*g) - (3*c^2*d^2*e^4*f^2*g
- 4*c^2*d^3*e^3*f*g^2 - 2*a*c*d*e^5*f*g^2 + 4*a*c*d^2*e^4*g^3 - a^2*e^6*g^3)*log(abs(-sqrt(e^2*f + (e*x + d)*e
*g - d*e*g)*sqrt(c*d*g) + sqrt(-c*d*e^2*f*g + a*e^3*g^2 + (e^2*f + (e*x + d)*e*g - d*e*g)*c*d*g)))/(sqrt(c*d*g
)*c*d))*d^2*f*abs(g)/g^3 + 1920*(sqrt(-c*d*e^2*f*g + a*e^3*g^2 + (e^2*f + (e*x + d)*e*g - d*e*g)*c*d*g)*(2*e^2
*f + 2*(e*x + d)*e*g - 2*d*e*g - (5*c^2*d^2*e^2*f - 4*c^2*d^3*e*g - a*c*d*e^3*g)/(c^2*d^2))*sqrt(e^2*f + (e*x
+ d)*e*g - d*e*g) - (3*c^2*d^2*e^4*f^2*g - 4*c^2*d^3*e^3*f*g^2 - 2*a*c*d*e^5*f*g^2 + 4*a*c*d^2*e^4*g^3 - a^2*e
^6*g^3)*log(abs(-sqrt(e^2*f + (e*x + d)*e*g - d*e*g)*sqrt(c*d*g) + sqrt(-c*d*e^2*f*g + a*e^3*g^2 + (e^2*f + (e
*x + d)*e*g - d*e*g)*c*d*g)))/(sqrt(c*d*g)*c*d))*d^3*abs(g)/(e*g^2) - 10*(sqrt(-c*d*e^2*f*g + a*e^3*g^2 + (e^2
*f + (e*x + d)*e*g - d*e*g)*c*d*g)*sqrt(e^2*f + (e*x + d)*e*g - d*e*g)*(2*(e^2*f + (e*x + d)*e*g - d*e*g)*(4*(
e^2*f + (e*x + d)*e*g - d*e*g)*(6*(e^2*f + (e*x + d)*e*g - d*e*g)/(e^3*g^3) - (25*c^6*d^6*e^5*f*g^11 - 24*c^6*
d^7*e^4*g^12 - a*c^5*d^5*e^6*g^12)/(c^6*d^6*e^6*g^14)) + (163*c^6*d^6*e^7*f^2*g^11 - 312*c^6*d^7*e^6*f*g^12 -
14*a*c^5*d^5*e^8*f*g^12 + 144*c^6*d^8*e^5*g^13 + 24*a*c^5*d^6*e^7*g^13 - 5*a^2*c^4*d^4*e^9*g^13)/(c^6*d^6*e^6*
g^14)) - 3*(93*c^6*d^6*e^9*f^3*g^11 - 264*c^6*d^7*e^8*f^2*g^12 - 15*a*c^5*d^5*e^10*f^2*g^12 + 240*c^6*d^8*e^7*
f*g^13 + 48*a*c^5*d^6*e^9*f*g^13 - 9*a^2*c^4*d^4*e^11*f*g^13 - 64*c^6*d^9*e^6*g^14 - 48*a*c^5*d^7*e^8*g^14 + 2
4*a^2*c^4*d^5*e^10*g^14 - 5*a^3*c^3*d^3*e^12*g^14)/(c^6*d^6*e^6*g^14)) - 3*(35*c^4*d^4*e^5*f^4 - 120*c^4*d^5*e
^4*f^3*g - 20*a*c^3*d^3*e^6*f^3*g + 144*c^4*d^6*e^3*f^2*g^2 + 72*a*c^3*d^4*e^5*f^2*g^2 - 6*a^2*c^2*d^2*e^7*f^2
*g^2 - 64*c^4*d^7*e^2*f*g^3 - 96*a*c^3*d^5*e^4*f*g^3 + 24*a^2*c^2*d^3*e^6*f*g^3 - 4*a^3*c*d*e^8*f*g^3 + 64*a*c
^3*d^6*e^3*g^4 - 48*a^2*c^2*d^4*e^5*g^4 + 24*a^3*c*d^2*e^7*g^4 - 5*a^4*e^9*g^4)*log(abs(-sqrt(e^2*f + (e*x + d
)*e*g - d*e*g)*sqrt(c*d*g) + sqrt(-c*d*e^2*f*g + a*e^3*g^2 + (e^2*f + (e*x + d)*e*g - d*e*g)*c*d*g)))/(sqrt(c*
d*g)*c^3*d^3*g^2))*e*f*abs(g)/g^2 + 40*(sqrt(-c*d*e^2*f*g + a*e^3*g^2 + (e^2*f + (e*x + d)*e*g - d*e*g)*c*d*g)
*sqrt(e^2*f + (e*x + d)*e*g - d*e*g)*(2*(e^2*f + (e*x + d)*e*g - d*e*g)*(4*(e^2*f + (e*x + d)*e*g - d*e*g)*(6*
(e^2*f + (e*x + d)*e*g - d*e*g)/(e^3*g^3) - (25*c^6*d^6*e^5*f*g^11 - 24*c^6*d^7*e^4*g^12 - a*c^5*d^5*e^6*g^12)
/(c^6*d^6*e^6*g^14)) + (163*c^6*d^6*e^7*f^2*g^11 - 312*c^6*d^7*e^6*f*g^12 - 14*a*c^5*d^5*e^8*f*g^12 + 144*c^6*
d^8*e^5*g^13 + 24*a*c^5*d^6*e^7*g^13 - 5*a^2*c^4*d^4*e^9*g^13)/(c^6*d^6*e^6*g^14)) - 3*(93*c^6*d^6*e^9*f^3*g^1
1 - 264*c^6*d^7*e^8*f^2*g^12 - 15*a*c^5*d^5*e^10*f^2*g^12 + 240*c^6*d^8*e^7*f*g^13 + 48*a*c^5*d^6*e^9*f*g^13 -
9*a^2*c^4*d^4*e^11*f*g^13 - 64*c^6*d^9*e^6*g^14 - 48*a*c^5*d^7*e^8*g^14 + 24*a^2*c^4*d^5*e^10*g^14 - 5*a^3*c^
3*d^3*e^12*g^14)/(c^6*d^6*e^6*g^14)) - 3*(35*c^4*d^4*e^5*f^4 - 120*c^4*d^5*e^4*f^3*g - 20*a*c^3*d^3*e^6*f^3*g
+ 144*c^4*d^6*e^3*f^2*g^2 + 72*a*c^3*d^4*e^5*f^2*g^2 - 6*a^2*c^2*d^2*e^7*f^2*g^2 - 64*c^4*d^7*e^2*f*g^3 - 96*a
*c^3*d^5*e^4*f*g^3 + 24*a^2*c^2*d^3*e^6*f*g^3 - 4*a^3*c*d*e^8*f*g^3 + 64*a*c^3*d^6*e^3*g^4 - 48*a^2*c^2*d^4*e^
5*g^4 + 24*a^3*c*d^2*e^7*g^4 - 5*a^4*e^9*g^4)*log(abs(-sqrt(e^2*f + (e*x + d)*e*g - d*e*g)*sqrt(c*d*g) + sqrt(
-c*d*e^2*f*g + a*e^3*g^2 + (e^2*f + (e*x + d)*e*g - d*e*g)*c*d*g)))/(sqrt(c*d*g)*c^3*d^3*g^2))*d*abs(g)/g - (s
qrt(-c*d*e^2*f*g + a*e^3*g^2 + (e^2*f + (e*x + d)*e*g - d*e*g)*c*d*g)*sqrt(e^2*f + (e*x + d)*e*g - d*e*g)*(2*(
e^2*f + (e*x + d)*e*g - d*e*g)*(4*(e^2*f + (e*x + d)*e*g - d*e*g)*(6*(e^2*f + (e*x + d)*e*g - d*e*g)*(8*(e^2*f
+ (e*x + d)*e*g - d*e*g)/(e^4*g^4) - (41*c^8*d^8*e^8*f*g^19 - 40*c^8*d^9*e^7*g^20 - a*c^7*d^7*e^9*g^20)/(c^8*
d^8*e^10*g^23)) + (513*c^8*d^8*e^10*f^2*g^19 - 1000*c^8*d^9*e^9*f*g^20 - 26*a*c^7*d^7*e^11*f*g^20 + 480*c^8*d^
10*e^8*g^21 + 40*a*c^7*d^8*e^10*g^21 - 7*a^2*c^6*d^6*e^12*g^21)/(c^8*d^8*e^10*g^23)) - 5*(447*c^8*d^8*e^12*f^3
*g^19 - 1304*c^8*d^9*e^11*f^2*g^20 - 37*a*c^7*d^7*e^13*f^2*g^20 + 1248*c^8*d^10*e^10*f*g^21 + 112*a*c^7*d^8*e^
12*f*g^21 - 19*a^2*c^6*d^6*e^14*f*g^21 - 384*c^8*d^11*e^9*g^22 - 96*a*c^7*d^9*e^11*g^22 + 40*a^2*c^6*d^7*e^13*
g^22 - 7*a^3*c^5*d^5*e^15*g^22)/(c^8*d^8*e^10*g^23)) + 15*(193*c^8*d^8*e^14*f^4*g^19 - 744*c^8*d^9*e^13*f^3*g^
20 - 28*a*c^7*d^7*e^15*f^3*g^20 + 1056*c^8*d^10*e^12*f^2*g^21 + 120*a*c^7*d^8*e^14*f^2*g^21 - 18*a^2*c^6*d^6*e
^16*f^2*g^21 - 640*c^8*d^11*e^11*f*g^22 - 192*a*c^7*d^9*e^13*f*g^22 + 72*a^2*c^6*d^7*e^15*f*g^22 - 12*a^3*c^5*
d^5*e^17*f*g^22 + 128*c^8*d^12*e^10*g^23 + 128*a*c^7*d^10*e^12*g^23 - 96*a^2*c^6*d^8*e^14*g^23 + 40*a^3*c^5*d^
6*e^16*g^23 - 7*a^4*c^4*d^4*e^18*g^23)/(c^8*d^8*e^10*g^23)) + 15*(63*c^5*d^5*e^6*f^5 - 280*c^5*d^6*e^5*f^4*g -
35*a*c^4*d^4*e^7*f^4*g + 480*c^5*d^7*e^4*f^3*g^2 + 160*a*c^4*d^5*e^6*f^3*g^2 - 10*a^2*c^3*d^3*e^8*f^3*g^2 - 3
84*c^5*d^8*e^3*f^2*g^3 - 288*a*c^4*d^6*e^5*f^2*g^3 + 48*a^2*c^3*d^4*e^7*f^2*g^3 - 6*a^3*c^2*d^2*e^9*f^2*g^3 +
128*c^5*d^9*e^2*f*g^4 + 256*a*c^4*d^7*e^4*f*g^4 - 96*a^2*c^3*d^5*e^6*f*g^4 + 32*a^3*c^2*d^3*e^8*f*g^4 - 5*a^4*
c*d*e^10*f*g^4 - 128*a*c^4*d^8*e^3*g^5 + 128*a^2*c^3*d^6*e^5*g^5 - 96*a^3*c^2*d^4*e^7*g^5 + 40*a^4*c*d^2*e^9*g
^5 - 7*a^5*e^11*g^5)*log(abs(-sqrt(e^2*f + (e*x + d)*e*g - d*e*g)*sqrt(c*d*g) + sqrt(-c*d*e^2*f*g + a*e^3*g^2
+ (e^2*f + (e*x + d)*e*g - d*e*g)*c*d*g)))/(sqrt(c*d*g)*c^4*d^4*g^3))*abs(g)/g)/(e^3*g) - (105*c^5*d^5*e^6*f^5
*g*abs(g)*log(abs(-sqrt(e^2*f - d*e*g)*sqrt(c*d*g) + sqrt(-c*d^2*e*g^2 + a*e^3*g^2))) - 75*a*c^4*d^4*e^7*f^4*g
^2*abs(g)*log(abs(-sqrt(e^2*f - d*e*g)*sqrt(c*d*g) + sqrt(-c*d^2*e*g^2 + a*e^3*g^2))) - 30*a^2*c^3*d^3*e^8*f^3
*g^3*abs(g)*log(abs(-sqrt(e^2*f - d*e*g)*sqrt(c*d*g) + sqrt(-c*d^2*e*g^2 + a*e^3*g^2))) - 30*a^3*c^2*d^2*e^9*f
^2*g^4*abs(g)*log(abs(-sqrt(e^2*f - d*e*g)*sqrt(c*d*g) + sqrt(-c*d^2*e*g^2 + a*e^3*g^2))) - 75*a^4*c*d*e^10*f*
g^5*abs(g)*log(abs(-sqrt(e^2*f - d*e*g)*sqrt(c*d*g) + sqrt(-c*d^2*e*g^2 + a*e^3*g^2))) + 105*a^5*e^11*g^6*abs(
g)*log(abs(-sqrt(e^2*f - d*e*g)*sqrt(c*d*g) + sqrt(-c*d^2*e*g^2 + a*e^3*g^2))) + 105*sqrt(-c*d^2*e*g^2 + a*e^3
*g^2)*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*c^4*d^4*e^4*f^4*abs(g) + 70*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(e^2*f -
d*e*g)*sqrt(c*d*g)*c^4*d^5*e^3*f^3*g*abs(g) - 40*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(e^2*f - d*e*g)*sqrt(c*d*g
)*a*c^3*d^3*e^5*f^3*g*abs(g) + 56*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*c^4*d^6*e^2*f
^2*g^2*abs(g) - 22*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*a*c^3*d^4*e^4*f^2*g^2*abs(g)
- 34*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*a^2*c^2*d^2*e^6*f^2*g^2*abs(g) + 48*sqrt(
-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*c^4*d^7*e*f*g^3*abs(g) - 16*sqrt(-c*d^2*e*g^2 + a*e^
3*g^2)*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*a*c^3*d^5*e^3*f*g^3*abs(g) - 22*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(e^2
*f - d*e*g)*sqrt(c*d*g)*a^2*c^2*d^3*e^5*f*g^3*abs(g) - 40*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(e^2*f - d*e*g)*s
qrt(c*d*g)*a^3*c*d*e^7*f*g^3*abs(g) - 384*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*c^4*d
^8*g^4*abs(g) + 48*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*a*c^3*d^6*e^2*g^4*abs(g) + 5
6*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*a^2*c^2*d^4*e^4*g^4*abs(g) + 70*sqrt(-c*d^2*e
*g^2 + a*e^3*g^2)*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*a^3*c*d^2*e^6*g^4*abs(g) + 105*sqrt(-c*d^2*e*g^2 + a*e^3*g^2
)*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*a^4*e^8*g^4*abs(g))/(sqrt(c*d*g)*c^4*d^4*e^3*g^6))*abs(e)^2/e^4 - 160*a*c*d*
f*((24*((c*d*e^2*f*g - a*e^3*g^2)*log(abs(-sqrt(e^2*f + (e*x + d)*e*g - d*e*g)*sqrt(c*d*g) + sqrt(-c*d*e^2*f*g
+ a*e^3*g^2 + (e^2*f + (e*x + d)*e*g - d*e*g)*c*d*g)))/sqrt(c*d*g) + sqrt(-c*d*e^2*f*g + a*e^3*g^2 + (e^2*f +
(e*x + d)*e*g - d*e*g)*c*d*g)*sqrt(e^2*f + (e*x + d)*e*g - d*e*g))*d*e*f*abs(g)/g^2 - 24*((c*d*e^2*f*g - a*e^
3*g^2)*log(abs(-sqrt(e^2*f + (e*x + d)*e*g - d*e*g)*sqrt(c*d*g) + sqrt(-c*d*e^2*f*g + a*e^3*g^2 + (e^2*f + (e*
x + d)*e*g - d*e*g)*c*d*g)))/sqrt(c*d*g) + sqrt(-c*d*e^2*f*g + a*e^3*g^2 + (e^2*f + (e*x + d)*e*g - d*e*g)*c*d
*g)*sqrt(e^2*f + (e*x + d)*e*g - d*e*g))*d^2*abs(g)/g - (sqrt(-c*d*e^2*f*g + a*e^3*g^2 + (e^2*f + (e*x + d)*e*
g - d*e*g)*c*d*g)*sqrt(e^2*f + (e*x + d)*e*g - d*e*g)*(2*(e^2*f + (e*x + d)*e*g - d*e*g)*(4*(e^2*f + (e*x + d)
*e*g - d*e*g)/(e^2*g^2) - (13*c^4*d^4*e^3*f*g^5 - 12*c^4*d^5*e^2*g^6 - a*c^3*d^3*e^4*g^6)/(c^4*d^4*e^3*g^7)) +
3*(11*c^4*d^4*e^5*f^2*g^5 - 20*c^4*d^5*e^4*f*g^6 - 2*a*c^3*d^3*e^6*f*g^6 + 8*c^4*d^6*e^3*g^7 + 4*a*c^3*d^4*e^
5*g^7 - a^2*c^2*d^2*e^7*g^7)/(c^4*d^4*e^3*g^7)) + 3*(5*c^3*d^3*e^4*f^3 - 12*c^3*d^4*e^3*f^2*g - 3*a*c^2*d^2*e^
5*f^2*g + 8*c^3*d^5*e^2*f*g^2 + 8*a*c^2*d^3*e^4*f*g^2 - a^2*c*d*e^6*f*g^2 - 8*a*c^2*d^4*e^3*g^3 + 4*a^2*c*d^2*
e^5*g^3 - a^3*e^7*g^3)*log(abs(-sqrt(e^2*f + (e*x + d)*e*g - d*e*g)*sqrt(c*d*g) + sqrt(-c*d*e^2*f*g + a*e^3*g^
2 + (e^2*f + (e*x + d)*e*g - d*e*g)*c*d*g)))/(sqrt(c*d*g)*c^2*d^2*g))*abs(g)/g - 6*(sqrt(-c*d*e^2*f*g + a*e^3*
g^2 + (e^2*f + (e*x + d)*e*g - d*e*g)*c*d*g)*(2*e^2*f + 2*(e*x + d)*e*g - 2*d*e*g - (5*c^2*d^2*e^2*f - 4*c^2*d
^3*e*g - a*c*d*e^3*g)/(c^2*d^2))*sqrt(e^2*f + (e*x + d)*e*g - d*e*g) - (3*c^2*d^2*e^4*f^2*g - 4*c^2*d^3*e^3*f*
g^2 - 2*a*c*d*e^5*f*g^2 + 4*a*c*d^2*e^4*g^3 - a^2*e^6*g^3)*log(abs(-sqrt(e^2*f + (e*x + d)*e*g - d*e*g)*sqrt(c
*d*g) + sqrt(-c*d*e^2*f*g + a*e^3*g^2 + (e^2*f + (e*x + d)*e*g - d*e*g)*c*d*g)))/(sqrt(c*d*g)*c*d))*f*abs(g)/g
^3 + 12*(sqrt(-c*d*e^2*f*g + a*e^3*g^2 + (e^2*f + (e*x + d)*e*g - d*e*g)*c*d*g)*(2*e^2*f + 2*(e*x + d)*e*g - 2
*d*e*g - (5*c^2*d^2*e^2*f - 4*c^2*d^3*e*g - a*c*d*e^3*g)/(c^2*d^2))*sqrt(e^2*f + (e*x + d)*e*g - d*e*g) - (3*c
^2*d^2*e^4*f^2*g - 4*c^2*d^3*e^3*f*g^2 - 2*a*c*d*e^5*f*g^2 + 4*a*c*d^2*e^4*g^3 - a^2*e^6*g^3)*log(abs(-sqrt(e^
2*f + (e*x + d)*e*g - d*e*g)*sqrt(c*d*g) + sqrt(-c*d*e^2*f*g + a*e^3*g^2 + (e^2*f + (e*x + d)*e*g - d*e*g)*c*d
*g)))/(sqrt(c*d*g)*c*d))*d*abs(g)/(e*g^2))/g - (3*c^3*d^3*e^4*f^3*g*abs(g)*log(abs(-sqrt(e^2*f - d*e*g)*sqrt(c
*d*g) + sqrt(-c*d^2*e*g^2 + a*e^3*g^2))) - 3*a*c^2*d^2*e^5*f^2*g^2*abs(g)*log(abs(-sqrt(e^2*f - d*e*g)*sqrt(c*
d*g) + sqrt(-c*d^2*e*g^2 + a*e^3*g^2))) - 3*a^2*c*d*e^6*f*g^3*abs(g)*log(abs(-sqrt(e^2*f - d*e*g)*sqrt(c*d*g)
+ sqrt(-c*d^2*e*g^2 + a*e^3*g^2))) + 3*a^3*e^7*g^4*abs(g)*log(abs(-sqrt(e^2*f - d*e*g)*sqrt(c*d*g) + sqrt(-c*d
^2*e*g^2 + a*e^3*g^2))) + 3*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*c^2*d^2*e^2*f^2*abs
(g) + 2*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*c^2*d^3*e*f*g*abs(g) - 2*sqrt(-c*d^2*e*
g^2 + a*e^3*g^2)*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*a*c*d*e^3*f*g*abs(g) - 8*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(
e^2*f - d*e*g)*sqrt(c*d*g)*c^2*d^4*g^2*abs(g) + 2*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(e^2*f - d*e*g)*sqrt(c*d*
g)*a*c*d^2*e^2*g^2*abs(g) + 3*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*a^2*e^4*g^2*abs(g
))/(sqrt(c*d*g)*c^2*d^2*g^4))*abs(e)^2/e^4 - 80*a^2*g*((24*((c*d*e^2*f*g - a*e^3*g^2)*log(abs(-sqrt(e^2*f + (e
*x + d)*e*g - d*e*g)*sqrt(c*d*g) + sqrt(-c*d*e^2*f*g + a*e^3*g^2 + (e^2*f + (e*x + d)*e*g - d*e*g)*c*d*g)))/sq
rt(c*d*g) + sqrt(-c*d*e^2*f*g + a*e^3*g^2 + (e^2*f + (e*x + d)*e*g - d*e*g)*c*d*g)*sqrt(e^2*f + (e*x + d)*e*g
- d*e*g))*d*e*f*abs(g)/g^2 - 24*((c*d*e^2*f*g - a*e^3*g^2)*log(abs(-sqrt(e^2*f + (e*x + d)*e*g - d*e*g)*sqrt(c
*d*g) + sqrt(-c*d*e^2*f*g + a*e^3*g^2 + (e^2*f + (e*x + d)*e*g - d*e*g)*c*d*g)))/sqrt(c*d*g) + sqrt(-c*d*e^2*f
*g + a*e^3*g^2 + (e^2*f + (e*x + d)*e*g - d*e*g)*c*d*g)*sqrt(e^2*f + (e*x + d)*e*g - d*e*g))*d^2*abs(g)/g - (s
qrt(-c*d*e^2*f*g + a*e^3*g^2 + (e^2*f + (e*x + d)*e*g - d*e*g)*c*d*g)*sqrt(e^2*f + (e*x + d)*e*g - d*e*g)*(2*(
e^2*f + (e*x + d)*e*g - d*e*g)*(4*(e^2*f + (e*x + d)*e*g - d*e*g)/(e^2*g^2) - (13*c^4*d^4*e^3*f*g^5 - 12*c^4*d
^5*e^2*g^6 - a*c^3*d^3*e^4*g^6)/(c^4*d^4*e^3*g^7)) + 3*(11*c^4*d^4*e^5*f^2*g^5 - 20*c^4*d^5*e^4*f*g^6 - 2*a*c^
3*d^3*e^6*f*g^6 + 8*c^4*d^6*e^3*g^7 + 4*a*c^3*d^4*e^5*g^7 - a^2*c^2*d^2*e^7*g^7)/(c^4*d^4*e^3*g^7)) + 3*(5*c^3
*d^3*e^4*f^3 - 12*c^3*d^4*e^3*f^2*g - 3*a*c^2*d^2*e^5*f^2*g + 8*c^3*d^5*e^2*f*g^2 + 8*a*c^2*d^3*e^4*f*g^2 - a^
2*c*d*e^6*f*g^2 - 8*a*c^2*d^4*e^3*g^3 + 4*a^2*c*d^2*e^5*g^3 - a^3*e^7*g^3)*log(abs(-sqrt(e^2*f + (e*x + d)*e*g
- d*e*g)*sqrt(c*d*g) + sqrt(-c*d*e^2*f*g + a*e^3*g^2 + (e^2*f + (e*x + d)*e*g - d*e*g)*c*d*g)))/(sqrt(c*d*g)*
c^2*d^2*g))*abs(g)/g - 6*(sqrt(-c*d*e^2*f*g + a*e^3*g^2 + (e^2*f + (e*x + d)*e*g - d*e*g)*c*d*g)*(2*e^2*f + 2*
(e*x + d)*e*g - 2*d*e*g - (5*c^2*d^2*e^2*f - 4*c^2*d^3*e*g - a*c*d*e^3*g)/(c^2*d^2))*sqrt(e^2*f + (e*x + d)*e*
g - d*e*g) - (3*c^2*d^2*e^4*f^2*g - 4*c^2*d^3*e^3*f*g^2 - 2*a*c*d*e^5*f*g^2 + 4*a*c*d^2*e^4*g^3 - a^2*e^6*g^3)
*log(abs(-sqrt(e^2*f + (e*x + d)*e*g - d*e*g)*sqrt(c*d*g) + sqrt(-c*d*e^2*f*g + a*e^3*g^2 + (e^2*f + (e*x + d)
*e*g - d*e*g)*c*d*g)))/(sqrt(c*d*g)*c*d))*f*abs(g)/g^3 + 12*(sqrt(-c*d*e^2*f*g + a*e^3*g^2 + (e^2*f + (e*x + d
)*e*g - d*e*g)*c*d*g)*(2*e^2*f + 2*(e*x + d)*e*g - 2*d*e*g - (5*c^2*d^2*e^2*f - 4*c^2*d^3*e*g - a*c*d*e^3*g)/(
c^2*d^2))*sqrt(e^2*f + (e*x + d)*e*g - d*e*g) - (3*c^2*d^2*e^4*f^2*g - 4*c^2*d^3*e^3*f*g^2 - 2*a*c*d*e^5*f*g^2
+ 4*a*c*d^2*e^4*g^3 - a^2*e^6*g^3)*log(abs(-sqrt(e^2*f + (e*x + d)*e*g - d*e*g)*sqrt(c*d*g) + sqrt(-c*d*e^2*f
*g + a*e^3*g^2 + (e^2*f + (e*x + d)*e*g - d*e*g)*c*d*g)))/(sqrt(c*d*g)*c*d))*d*abs(g)/(e*g^2))/g - (3*c^3*d^3*
e^4*f^3*g*abs(g)*log(abs(-sqrt(e^2*f - d*e*g)*sqrt(c*d*g) + sqrt(-c*d^2*e*g^2 + a*e^3*g^2))) - 3*a*c^2*d^2*e^5
*f^2*g^2*abs(g)*log(abs(-sqrt(e^2*f - d*e*g)*sqrt(c*d*g) + sqrt(-c*d^2*e*g^2 + a*e^3*g^2))) - 3*a^2*c*d*e^6*f*
g^3*abs(g)*log(abs(-sqrt(e^2*f - d*e*g)*sqrt(c*d*g) + sqrt(-c*d^2*e*g^2 + a*e^3*g^2))) + 3*a^3*e^7*g^4*abs(g)*
log(abs(-sqrt(e^2*f - d*e*g)*sqrt(c*d*g) + sqrt(-c*d^2*e*g^2 + a*e^3*g^2))) + 3*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)
*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*c^2*d^2*e^2*f^2*abs(g) + 2*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(e^2*f - d*e*g)
*sqrt(c*d*g)*c^2*d^3*e*f*g*abs(g) - 2*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*a*c*d*e^3
*f*g*abs(g) - 8*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*c^2*d^4*g^2*abs(g) + 2*sqrt(-c*
d^2*e*g^2 + a*e^3*g^2)*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*a*c*d^2*e^2*g^2*abs(g) + 3*sqrt(-c*d^2*e*g^2 + a*e^3*g^
2)*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*a^2*e^4*g^2*abs(g))/(sqrt(c*d*g)*c^2*d^2*g^4))*abs(e)^2/e^3)/e

Mupad [F(-1)]

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

[In]

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

[Out]

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