\(\int \frac {(d+e x)^{5/2}}{(a+b x+c x^2)^3} \, dx\) [2303]

Optimal result

Integrand size = 22, antiderivative size = 577 \[ \int \frac {(d+e x)^{5/2}}{\left (a+b x+c x^2\right )^3} \, dx=-\frac {(d+e x)^{3/2} (b d-2 a e+(2 c d-b e) x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac {3 \sqrt {d+e x} \left (3 b^2 d e+4 a c d e-4 b \left (c d^2+a e^2\right )-\left (8 c^2 d^2+b^2 e^2-4 c e (2 b d-a e)\right ) x\right )}{4 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac {3 \left (32 c^3 d^3-b^2 \left (b-\sqrt {b^2-4 a c}\right ) e^3-8 c^2 d e \left (6 b d-\sqrt {b^2-4 a c} d-3 a e\right )+2 c e^2 \left (9 b^2 d-4 b \sqrt {b^2-4 a c} d-6 a b e+2 a \sqrt {b^2-4 a c} e\right )\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{4 \sqrt {2} \sqrt {c} \left (b^2-4 a c\right )^{5/2} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}+\frac {3 \left (32 c^3 d^3-b^2 \left (b+\sqrt {b^2-4 a c}\right ) e^3-8 c^2 d e \left (6 b d+\sqrt {b^2-4 a c} d-3 a e\right )+2 c e^2 \left (9 b^2 d+4 b \sqrt {b^2-4 a c} d-6 a b e-2 a \sqrt {b^2-4 a c} e\right )\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{4 \sqrt {2} \sqrt {c} \left (b^2-4 a c\right )^{5/2} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \]

[Out]

-1/2*(e*x+d)^(3/2)*(b*d-2*a*e+(-b*e+2*c*d)*x)/(-4*a*c+b^2)/(c*x^2+b*x+a)^2-3/4*(3*b^2*d*e+4*a*c*d*e-4*b*(a*e^2
+c*d^2)-(8*c^2*d^2+b^2*e^2-4*c*e*(-a*e+2*b*d))*x)*(e*x+d)^(1/2)/(-4*a*c+b^2)^2/(c*x^2+b*x+a)-3/8*arctanh(2^(1/
2)*c^(1/2)*(e*x+d)^(1/2)/(2*c*d-e*(b-(-4*a*c+b^2)^(1/2)))^(1/2))*(32*c^3*d^3-b^2*e^3*(b-(-4*a*c+b^2)^(1/2))-8*
c^2*d*e*(6*b*d-3*a*e-d*(-4*a*c+b^2)^(1/2))+2*c*e^2*(9*b^2*d-6*a*b*e-4*b*d*(-4*a*c+b^2)^(1/2)+2*a*e*(-4*a*c+b^2
)^(1/2)))/(-4*a*c+b^2)^(5/2)*2^(1/2)/c^(1/2)/(2*c*d-e*(b-(-4*a*c+b^2)^(1/2)))^(1/2)+3/8*arctanh(2^(1/2)*c^(1/2
)*(e*x+d)^(1/2)/(2*c*d-e*(b+(-4*a*c+b^2)^(1/2)))^(1/2))*(32*c^3*d^3-b^2*e^3*(b+(-4*a*c+b^2)^(1/2))-8*c^2*d*e*(
6*b*d-3*a*e+d*(-4*a*c+b^2)^(1/2))+2*c*e^2*(9*b^2*d-6*a*b*e+4*b*d*(-4*a*c+b^2)^(1/2)-2*a*e*(-4*a*c+b^2)^(1/2)))
/(-4*a*c+b^2)^(5/2)*2^(1/2)/c^(1/2)/(2*c*d-e*(b+(-4*a*c+b^2)^(1/2)))^(1/2)

Rubi [A] (verified)

Time = 3.19 (sec) , antiderivative size = 577, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {752, 834, 840, 1180, 214} \[ \int \frac {(d+e x)^{5/2}}{\left (a+b x+c x^2\right )^3} \, dx=-\frac {3 \left (-8 c^2 d e \left (-d \sqrt {b^2-4 a c}-3 a e+6 b d\right )+2 c e^2 \left (-4 b d \sqrt {b^2-4 a c}+2 a e \sqrt {b^2-4 a c}-6 a b e+9 b^2 d\right )-b^2 e^3 \left (b-\sqrt {b^2-4 a c}\right )+32 c^3 d^3\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{4 \sqrt {2} \sqrt {c} \left (b^2-4 a c\right )^{5/2} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}+\frac {3 \left (-8 c^2 d e \left (d \sqrt {b^2-4 a c}-3 a e+6 b d\right )+2 c e^2 \left (4 b d \sqrt {b^2-4 a c}-2 a e \sqrt {b^2-4 a c}-6 a b e+9 b^2 d\right )-b^2 e^3 \left (\sqrt {b^2-4 a c}+b\right )+32 c^3 d^3\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{4 \sqrt {2} \sqrt {c} \left (b^2-4 a c\right )^{5/2} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}-\frac {3 \sqrt {d+e x} \left (-x \left (-4 c e (2 b d-a e)+b^2 e^2+8 c^2 d^2\right )-4 b \left (a e^2+c d^2\right )+4 a c d e+3 b^2 d e\right )}{4 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac {(d+e x)^{3/2} (-2 a e+x (2 c d-b e)+b d)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2} \]

[In]

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

[Out]

-1/2*((d + e*x)^(3/2)*(b*d - 2*a*e + (2*c*d - b*e)*x))/((b^2 - 4*a*c)*(a + b*x + c*x^2)^2) - (3*Sqrt[d + e*x]*
(3*b^2*d*e + 4*a*c*d*e - 4*b*(c*d^2 + a*e^2) - (8*c^2*d^2 + b^2*e^2 - 4*c*e*(2*b*d - a*e))*x))/(4*(b^2 - 4*a*c
)^2*(a + b*x + c*x^2)) - (3*(32*c^3*d^3 - b^2*(b - Sqrt[b^2 - 4*a*c])*e^3 - 8*c^2*d*e*(6*b*d - Sqrt[b^2 - 4*a*
c]*d - 3*a*e) + 2*c*e^2*(9*b^2*d - 4*b*Sqrt[b^2 - 4*a*c]*d - 6*a*b*e + 2*a*Sqrt[b^2 - 4*a*c]*e))*ArcTanh[(Sqrt
[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]])/(4*Sqrt[2]*Sqrt[c]*(b^2 - 4*a*c)^(5/2)*Sq
rt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]) + (3*(32*c^3*d^3 - b^2*(b + Sqrt[b^2 - 4*a*c])*e^3 - 8*c^2*d*e*(6*b*d +
Sqrt[b^2 - 4*a*c]*d - 3*a*e) + 2*c*e^2*(9*b^2*d + 4*b*Sqrt[b^2 - 4*a*c]*d - 6*a*b*e - 2*a*Sqrt[b^2 - 4*a*c]*e
))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/(4*Sqrt[2]*Sqrt[c]*(b^2 -
4*a*c)^(5/2)*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e])

Rule 214

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

Rule 752

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m - 1)*(d
*b - 2*a*e + (2*c*d - b*e)*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] + Dist[1/((p + 1)*(b^2 -
4*a*c)), Int[(d + e*x)^(m - 2)*Simp[e*(2*a*e*(m - 1) + b*d*(2*p - m + 4)) - 2*c*d^2*(2*p + 3) + e*(b*e - 2*d*
c)*(m + 2*p + 2)*x, x]*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] &
& NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && LtQ[p, -1] && GtQ[m, 1] && IntQuadraticQ[a, b, c, d,
e, m, p, x]

Rule 834

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*((f*b - 2*a*g + (2*c*f - b*g)*x)/((p + 1)*(b^2 - 4*a*c))), x] + Dist[1/
((p + 1)*(b^2 - 4*a*c)), Int[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1)*Simp[g*(2*a*e*m + b*d*(2*p + 3)) - f*
(b*e*m + 2*c*d*(2*p + 3)) - e*(2*c*f - b*g)*(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] &&
NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 0] && (IntegerQ[m] || IntegerQ[p]
|| IntegersQ[2*m, 2*p])

Rule 840

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2,
Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /
; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]

Rule 1180

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

Rubi steps \begin{align*} \text {integral}& = -\frac {(d+e x)^{3/2} (b d-2 a e+(2 c d-b e) x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac {\int \frac {\sqrt {d+e x} \left (\frac {3}{2} \left (4 c d^2-3 b d e+2 a e^2\right )+\frac {3}{2} e (2 c d-b e) x\right )}{\left (a+b x+c x^2\right )^2} \, dx}{2 \left (b^2-4 a c\right )} \\ & = -\frac {(d+e x)^{3/2} (b d-2 a e+(2 c d-b e) x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac {3 \sqrt {d+e x} \left (3 b^2 d e+4 a c d e-4 b \left (c d^2+a e^2\right )-\left (8 c^2 d^2+b^2 e^2-4 c e (2 b d-a e)\right ) x\right )}{4 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}+\frac {\int \frac {\frac {3}{4} \left (16 c^2 d^3+b e^2 (5 b d-4 a e)-4 c d e (5 b d-3 a e)\right )+\frac {3}{4} e \left (8 c^2 d^2+b^2 e^2-4 c e (2 b d-a e)\right ) x}{\sqrt {d+e x} \left (a+b x+c x^2\right )} \, dx}{2 \left (b^2-4 a c\right )^2} \\ & = -\frac {(d+e x)^{3/2} (b d-2 a e+(2 c d-b e) x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac {3 \sqrt {d+e x} \left (3 b^2 d e+4 a c d e-4 b \left (c d^2+a e^2\right )-\left (8 c^2 d^2+b^2 e^2-4 c e (2 b d-a e)\right ) x\right )}{4 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}+\frac {\text {Subst}\left (\int \frac {\frac {3}{4} e \left (16 c^2 d^3+b e^2 (5 b d-4 a e)-4 c d e (5 b d-3 a e)\right )-\frac {3}{4} d e \left (8 c^2 d^2+b^2 e^2-4 c e (2 b d-a e)\right )+\frac {3}{4} e \left (8 c^2 d^2+b^2 e^2-4 c e (2 b d-a e)\right ) x^2}{c d^2-b d e+a e^2+(-2 c d+b e) x^2+c x^4} \, dx,x,\sqrt {d+e x}\right )}{\left (b^2-4 a c\right )^2} \\ & = -\frac {(d+e x)^{3/2} (b d-2 a e+(2 c d-b e) x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac {3 \sqrt {d+e x} \left (3 b^2 d e+4 a c d e-4 b \left (c d^2+a e^2\right )-\left (8 c^2 d^2+b^2 e^2-4 c e (2 b d-a e)\right ) x\right )}{4 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac {\left (3 \left (32 c^3 d^3-b^2 \left (b+\sqrt {b^2-4 a c}\right ) e^3-8 c^2 d e \left (6 b d+\sqrt {b^2-4 a c} d-3 a e\right )+2 c e^2 \left (9 b^2 d+4 b \sqrt {b^2-4 a c} d-6 a b e-2 a \sqrt {b^2-4 a c} e\right )\right )\right ) \text {Subst}\left (\int \frac {1}{\frac {1}{2} \sqrt {b^2-4 a c} e+\frac {1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt {d+e x}\right )}{8 \left (b^2-4 a c\right )^{5/2}}+\frac {\left (3 \left (32 c^3 d^3-b^2 \left (b-\sqrt {b^2-4 a c}\right ) e^3-8 c^2 d e \left (6 b d-\sqrt {b^2-4 a c} d-3 a e\right )+2 c e^2 \left (9 b^2 d-4 b \sqrt {b^2-4 a c} d-6 a b e+2 a \sqrt {b^2-4 a c} e\right )\right )\right ) \text {Subst}\left (\int \frac {1}{-\frac {1}{2} \sqrt {b^2-4 a c} e+\frac {1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt {d+e x}\right )}{8 \left (b^2-4 a c\right )^{5/2}} \\ & = -\frac {(d+e x)^{3/2} (b d-2 a e+(2 c d-b e) x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac {3 \sqrt {d+e x} \left (3 b^2 d e+4 a c d e-4 b \left (c d^2+a e^2\right )-\left (8 c^2 d^2+b^2 e^2-4 c e (2 b d-a e)\right ) x\right )}{4 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac {3 \left (32 c^3 d^3-b^2 \left (b-\sqrt {b^2-4 a c}\right ) e^3-8 c^2 d e \left (6 b d-\sqrt {b^2-4 a c} d-3 a e\right )+2 c e^2 \left (9 b^2 d-4 b \sqrt {b^2-4 a c} d-6 a b e+2 a \sqrt {b^2-4 a c} e\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{4 \sqrt {2} \sqrt {c} \left (b^2-4 a c\right )^{5/2} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}+\frac {3 \left (32 c^3 d^3-b^2 \left (b+\sqrt {b^2-4 a c}\right ) e^3-8 c^2 d e \left (6 b d+\sqrt {b^2-4 a c} d-3 a e\right )+2 c e^2 \left (9 b^2 d+4 b \sqrt {b^2-4 a c} d-6 a b e-2 a \sqrt {b^2-4 a c} e\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{4 \sqrt {2} \sqrt {c} \left (b^2-4 a c\right )^{5/2} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \\ \end{align*}

Mathematica [A] (verified)

Time = 13.14 (sec) , antiderivative size = 506, normalized size of antiderivative = 0.88 \[ \int \frac {(d+e x)^{5/2}}{\left (a+b x+c x^2\right )^3} \, dx=\frac {\sqrt {d+e x} \left (b^3 \left (-2 d^2-9 d e x+5 e^2 x^2\right )+b^2 \left (a e (-5 d+19 e x)+c x \left (8 d^2-37 d e x+3 e^2 x^2\right )\right )+4 c \left (6 c^2 d^2 x^3-a^2 e (7 d+e x)+a c x \left (10 d^2+d e x+3 e^2 x^2\right )\right )+4 b \left (3 a^2 e^2+3 c^2 d x^2 (3 d-2 e x)+a c \left (5 d^2-9 d e x+4 e^2 x^2\right )\right )\right )}{4 \left (b^2-4 a c\right )^2 (a+x (b+c x))^2}-\frac {3 \sqrt {4 c d+2 \left (-b+\sqrt {b^2-4 a c}\right ) e} \left (16 c^2 d^2+b \left (3 b+2 \sqrt {b^2-4 a c}\right ) e^2+4 c e \left (-4 b d-\sqrt {b^2-4 a c} d+a e\right )\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-b e+\sqrt {b^2-4 a c} e}}\right )}{8 \sqrt {c} \left (b^2-4 a c\right )^{5/2}}+\frac {3 \sqrt {4 c d-2 \left (b+\sqrt {b^2-4 a c}\right ) e} \left (16 c^2 d^2+b \left (3 b-2 \sqrt {b^2-4 a c}\right ) e^2+4 c e \left (-4 b d+\sqrt {b^2-4 a c} d+a e\right )\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{8 \sqrt {c} \left (b^2-4 a c\right )^{5/2}} \]

[In]

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

[Out]

(Sqrt[d + e*x]*(b^3*(-2*d^2 - 9*d*e*x + 5*e^2*x^2) + b^2*(a*e*(-5*d + 19*e*x) + c*x*(8*d^2 - 37*d*e*x + 3*e^2*
x^2)) + 4*c*(6*c^2*d^2*x^3 - a^2*e*(7*d + e*x) + a*c*x*(10*d^2 + d*e*x + 3*e^2*x^2)) + 4*b*(3*a^2*e^2 + 3*c^2*
d*x^2*(3*d - 2*e*x) + a*c*(5*d^2 - 9*d*e*x + 4*e^2*x^2))))/(4*(b^2 - 4*a*c)^2*(a + x*(b + c*x))^2) - (3*Sqrt[4
*c*d + 2*(-b + Sqrt[b^2 - 4*a*c])*e]*(16*c^2*d^2 + b*(3*b + 2*Sqrt[b^2 - 4*a*c])*e^2 + 4*c*e*(-4*b*d - Sqrt[b^
2 - 4*a*c]*d + a*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e]])/(8*Sqrt
[c]*(b^2 - 4*a*c)^(5/2)) + (3*Sqrt[4*c*d - 2*(b + Sqrt[b^2 - 4*a*c])*e]*(16*c^2*d^2 + b*(3*b - 2*Sqrt[b^2 - 4*
a*c])*e^2 + 4*c*e*(-4*b*d + Sqrt[b^2 - 4*a*c]*d + a*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (
b + Sqrt[b^2 - 4*a*c])*e]])/(8*Sqrt[c]*(b^2 - 4*a*c)^(5/2))

Maple [A] (verified)

Time = 1.10 (sec) , antiderivative size = 639, normalized size of antiderivative = 1.11

method	result	size
pseudoelliptic	\(\frac {-\frac {3 \sqrt {\left (b e -2 c d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}\right ) c}\, \sqrt {2}\, \left (c \,x^{2}+b x +a \right )^{2} e \left (\left (2 c^{2} d^{2}+\left (a \,e^{2}-2 b d e \right ) c +\frac {b^{2} e^{2}}{4}\right ) \sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}-3 \left (\frac {4 c^{2} d^{2}}{3}+e \left (a e -\frac {4 b d}{3}\right ) c +\frac {b^{2} e^{2}}{12}\right ) \left (b e -2 c d \right )\right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}\right ) c}}\right )}{32}+\frac {3 \sqrt {\left (-b e +2 c d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}\right ) c}\, \left (\frac {\sqrt {2}\, \left (c \,x^{2}+b x +a \right )^{2} e \left (\left (2 c^{2} d^{2}+\left (a \,e^{2}-2 b d e \right ) c +\frac {b^{2} e^{2}}{4}\right ) \sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}+3 \left (\frac {4 c^{2} d^{2}}{3}+e \left (a e -\frac {4 b d}{3}\right ) c +\frac {b^{2} e^{2}}{12}\right ) \left (b e -2 c d \right )\right ) \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}\right ) c}}\right )}{2}+\sqrt {\left (b e -2 c d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}\right ) c}\, \left (2 c^{3} d^{2} x^{3}+\frac {10 x \left (\frac {3 a \,e^{2} x^{2}}{10}+\frac {d x \left (-6 b x +a \right ) e}{10}+d^{2} \left (\frac {9 b x}{10}+a \right )\right ) c^{2}}{3}+\frac {\left (-\left (-\frac {3}{4} b^{2} x^{2}-4 a b x +a^{2}\right ) x \,e^{2}-7 \left (\frac {37}{28} b^{2} x^{2}+\frac {9}{7} a b x +a^{2}\right ) d e +5 \left (\frac {2 b x}{5}+a \right ) b \,d^{2}\right ) c}{3}+\left (\left (\frac {5 b x}{4}+a \right ) e +\frac {b d}{4}\right ) \left (\left (\frac {b x}{3}+a \right ) e -\frac {2 b d}{3}\right ) b \right ) \sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}\, \sqrt {e x +d}\right )}{16}}{\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}\, \sqrt {\left (-b e +2 c d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}\right ) c}\, \sqrt {\left (b e -2 c d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}\right ) c}\, \left (a c -\frac {b^{2}}{4}\right )^{2} \left (c \,x^{2}+b x +a \right )^{2}}\)	\(639\)
derivativedivides	\(2 e^{5} \left (\frac {\frac {3 c \left (4 a c \,e^{2}+b^{2} e^{2}-8 b c d e +8 c^{2} d^{2}\right ) \left (e x +d \right )^{\frac {7}{2}}}{8 e^{4} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {\left (16 a b c \,e^{3}-32 a \,c^{2} d \,e^{2}+5 b^{3} e^{3}-46 b^{2} c d \,e^{2}+108 b \,c^{2} d^{2} e -72 c^{3} d^{3}\right ) \left (e x +d \right )^{\frac {5}{2}}}{8 e^{4} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {\left (4 c \,a^{2} e^{4}-19 b^{2} a \,e^{4}+68 a b c d \,e^{3}-68 a \,c^{2} d^{2} e^{2}+19 b^{3} d \,e^{3}-91 b^{2} c \,d^{2} e^{2}+144 b \,c^{2} d^{3} e -72 c^{3} d^{4}\right ) \left (e x +d \right )^{\frac {3}{2}}}{8 e^{4} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {3 \left (a^{2} b \,e^{5}-2 a^{2} c d \,e^{4}-2 a \,b^{2} d \,e^{4}+6 a b c \,d^{2} e^{3}-4 a \,c^{2} d^{3} e^{2}+b^{3} d^{2} e^{3}-4 b^{2} c \,d^{3} e^{2}+5 b \,c^{2} d^{4} e -2 c^{3} d^{5}\right ) \sqrt {e x +d}}{2 e^{4} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}}{\left (c \left (e x +d \right )^{2}+b e \left (e x +d \right )-2 c d \left (e x +d \right )+a \,e^{2}-b d e +c \,d^{2}\right )^{2}}+\frac {3 c \left (\frac {\left (12 a b c \,e^{3}-24 a \,c^{2} d \,e^{2}+b^{3} e^{3}-18 b^{2} c d \,e^{2}+48 b \,c^{2} d^{2} e -32 c^{3} d^{3}+4 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a c \,e^{2}+\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} e^{2}-8 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b c d e +8 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c^{2} d^{2}\right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}-\frac {\left (-12 a b c \,e^{3}+24 a \,c^{2} d \,e^{2}-b^{3} e^{3}+18 b^{2} c d \,e^{2}-48 b \,c^{2} d^{2} e +32 c^{3} d^{3}+4 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a c \,e^{2}+\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} e^{2}-8 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b c d e +8 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c^{2} d^{2}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{2 e^{4} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}\right )\)	\(964\)
default	\(2 e^{5} \left (\frac {\frac {3 c \left (4 a c \,e^{2}+b^{2} e^{2}-8 b c d e +8 c^{2} d^{2}\right ) \left (e x +d \right )^{\frac {7}{2}}}{8 e^{4} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {\left (16 a b c \,e^{3}-32 a \,c^{2} d \,e^{2}+5 b^{3} e^{3}-46 b^{2} c d \,e^{2}+108 b \,c^{2} d^{2} e -72 c^{3} d^{3}\right ) \left (e x +d \right )^{\frac {5}{2}}}{8 e^{4} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {\left (4 c \,a^{2} e^{4}-19 b^{2} a \,e^{4}+68 a b c d \,e^{3}-68 a \,c^{2} d^{2} e^{2}+19 b^{3} d \,e^{3}-91 b^{2} c \,d^{2} e^{2}+144 b \,c^{2} d^{3} e -72 c^{3} d^{4}\right ) \left (e x +d \right )^{\frac {3}{2}}}{8 e^{4} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {3 \left (a^{2} b \,e^{5}-2 a^{2} c d \,e^{4}-2 a \,b^{2} d \,e^{4}+6 a b c \,d^{2} e^{3}-4 a \,c^{2} d^{3} e^{2}+b^{3} d^{2} e^{3}-4 b^{2} c \,d^{3} e^{2}+5 b \,c^{2} d^{4} e -2 c^{3} d^{5}\right ) \sqrt {e x +d}}{2 e^{4} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}}{\left (c \left (e x +d \right )^{2}+b e \left (e x +d \right )-2 c d \left (e x +d \right )+a \,e^{2}-b d e +c \,d^{2}\right )^{2}}+\frac {3 c \left (\frac {\left (12 a b c \,e^{3}-24 a \,c^{2} d \,e^{2}+b^{3} e^{3}-18 b^{2} c d \,e^{2}+48 b \,c^{2} d^{2} e -32 c^{3} d^{3}+4 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a c \,e^{2}+\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} e^{2}-8 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b c d e +8 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c^{2} d^{2}\right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}-\frac {\left (-12 a b c \,e^{3}+24 a \,c^{2} d \,e^{2}-b^{3} e^{3}+18 b^{2} c d \,e^{2}-48 b \,c^{2} d^{2} e +32 c^{3} d^{3}+4 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a c \,e^{2}+\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} e^{2}-8 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b c d e +8 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c^{2} d^{2}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{2 e^{4} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}\right )\)	\(964\)

[In]

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

[Out]

3/16/(-4*(a*c-1/4*b^2)*e^2)^(1/2)*(-1/2*((b*e-2*c*d+(-4*(a*c-1/4*b^2)*e^2)^(1/2))*c)^(1/2)*2^(1/2)*(c*x^2+b*x+
a)^2*e*((2*c^2*d^2+(a*e^2-2*b*d*e)*c+1/4*b^2*e^2)*(-4*(a*c-1/4*b^2)*e^2)^(1/2)-3*(4/3*c^2*d^2+e*(a*e-4/3*b*d)*
c+1/12*b^2*e^2)*(b*e-2*c*d))*arctanh(c*(e*x+d)^(1/2)*2^(1/2)/((-b*e+2*c*d+(-4*(a*c-1/4*b^2)*e^2)^(1/2))*c)^(1/
2))+((-b*e+2*c*d+(-4*(a*c-1/4*b^2)*e^2)^(1/2))*c)^(1/2)*(1/2*2^(1/2)*(c*x^2+b*x+a)^2*e*((2*c^2*d^2+(a*e^2-2*b*
d*e)*c+1/4*b^2*e^2)*(-4*(a*c-1/4*b^2)*e^2)^(1/2)+3*(4/3*c^2*d^2+e*(a*e-4/3*b*d)*c+1/12*b^2*e^2)*(b*e-2*c*d))*a
rctan(c*(e*x+d)^(1/2)*2^(1/2)/((b*e-2*c*d+(-4*(a*c-1/4*b^2)*e^2)^(1/2))*c)^(1/2))+((b*e-2*c*d+(-4*(a*c-1/4*b^2
)*e^2)^(1/2))*c)^(1/2)*(2*c^3*d^2*x^3+10/3*x*(3/10*a*e^2*x^2+1/10*d*x*(-6*b*x+a)*e+d^2*(9/10*b*x+a))*c^2+1/3*(
-(-3/4*b^2*x^2-4*a*b*x+a^2)*x*e^2-7*(37/28*b^2*x^2+9/7*a*b*x+a^2)*d*e+5*(2/5*b*x+a)*b*d^2)*c+((5/4*b*x+a)*e+1/
4*b*d)*((1/3*b*x+a)*e-2/3*b*d)*b)*(-4*(a*c-1/4*b^2)*e^2)^(1/2)*(e*x+d)^(1/2)))/((-b*e+2*c*d+(-4*(a*c-1/4*b^2)*
e^2)^(1/2))*c)^(1/2)/((b*e-2*c*d+(-4*(a*c-1/4*b^2)*e^2)^(1/2))*c)^(1/2)/(a*c-1/4*b^2)^2/(c*x^2+b*x+a)^2

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 5124 vs. \(2 (509) = 1018\).

Time = 0.56 (sec) , antiderivative size = 5124, normalized size of antiderivative = 8.88 \[ \int \frac {(d+e x)^{5/2}}{\left (a+b x+c x^2\right )^3} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Too large to include

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2512 vs. \(2 (509) = 1018\).

Time = 1.26 (sec) , antiderivative size = 2512, normalized size of antiderivative = 4.35 \[ \int \frac {(d+e x)^{5/2}}{\left (a+b x+c x^2\right )^3} \, dx=\text {Too large to display} \]

[In]

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

[Out]

3/32*((b^4*e - 8*a*b^2*c*e + 16*a^2*c^2*e)^2*(8*c^2*d^2*e - 8*b*c*d*e^2 + (b^2 + 4*a*c)*e^3)*sqrt(-4*c^2*d + 2
*(b*c - sqrt(b^2 - 4*a*c)*c)*e) + 8*(2*(b^2*c^3 - 4*a*c^4)*sqrt(b^2 - 4*a*c)*d^3*e - 3*(b^3*c^2 - 4*a*b*c^3)*s
qrt(b^2 - 4*a*c)*d^2*e^2 + (b^4*c - 2*a*b^2*c^2 - 8*a^2*c^3)*sqrt(b^2 - 4*a*c)*d*e^3 - (a*b^3*c - 4*a^2*b*c^2)
*sqrt(b^2 - 4*a*c)*e^4)*sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e)*abs(b^4*e - 8*a*b^2*c*e + 16*a^2*c^2*
e) - (64*(b^6*c^4 - 12*a*b^4*c^5 + 48*a^2*b^2*c^6 - 64*a^3*c^7)*d^4*e - 128*(b^7*c^3 - 12*a*b^5*c^4 + 48*a^2*b
^3*c^5 - 64*a^3*b*c^6)*d^3*e^2 + 12*(7*b^8*c^2 - 80*a*b^6*c^3 + 288*a^2*b^4*c^4 - 256*a^3*b^2*c^5 - 256*a^4*c^
6)*d^2*e^3 - 4*(5*b^9*c - 48*a*b^7*c^2 + 96*a^2*b^5*c^3 + 256*a^3*b^3*c^4 - 768*a^4*b*c^5)*d*e^4 + (b^10 - 96*
a^2*b^6*c^2 + 512*a^3*b^4*c^3 - 768*a^4*b^2*c^4)*e^5)*sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e))*arctan
(2*sqrt(1/2)*sqrt(e*x + d)/sqrt(-(2*b^4*c*d - 16*a*b^2*c^2*d + 32*a^2*c^3*d - b^5*e + 8*a*b^3*c*e - 16*a^2*b*c
^2*e + sqrt((2*b^4*c*d - 16*a*b^2*c^2*d + 32*a^2*c^3*d - b^5*e + 8*a*b^3*c*e - 16*a^2*b*c^2*e)^2 - 4*(b^4*c*d^
2 - 8*a*b^2*c^2*d^2 + 16*a^2*c^3*d^2 - b^5*d*e + 8*a*b^3*c*d*e - 16*a^2*b*c^2*d*e + a*b^4*e^2 - 8*a^2*b^2*c*e^
2 + 16*a^3*c^2*e^2)*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)))/(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)))/(((b^6*c^2 - 12*
a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*sqrt(b^2 - 4*a*c)*d^2 - (b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a
^3*b*c^4)*sqrt(b^2 - 4*a*c)*d*e + (a*b^6*c - 12*a^2*b^4*c^2 + 48*a^3*b^2*c^3 - 64*a^4*c^4)*sqrt(b^2 - 4*a*c)*e
^2)*abs(b^4*e - 8*a*b^2*c*e + 16*a^2*c^2*e)*abs(c)) - 3/32*((b^4*e - 8*a*b^2*c*e + 16*a^2*c^2*e)^2*(8*c^2*d^2*
e - 8*b*c*d*e^2 + (b^2 + 4*a*c)*e^3)*sqrt(-4*c^2*d + 2*(b*c + sqrt(b^2 - 4*a*c)*c)*e) - 8*(2*(b^2*c^3 - 4*a*c^
4)*sqrt(b^2 - 4*a*c)*d^3*e - 3*(b^3*c^2 - 4*a*b*c^3)*sqrt(b^2 - 4*a*c)*d^2*e^2 + (b^4*c - 2*a*b^2*c^2 - 8*a^2*
c^3)*sqrt(b^2 - 4*a*c)*d*e^3 - (a*b^3*c - 4*a^2*b*c^2)*sqrt(b^2 - 4*a*c)*e^4)*sqrt(-4*c^2*d + 2*(b*c + sqrt(b^
2 - 4*a*c)*c)*e)*abs(b^4*e - 8*a*b^2*c*e + 16*a^2*c^2*e) - (64*(b^6*c^4 - 12*a*b^4*c^5 + 48*a^2*b^2*c^6 - 64*a
^3*c^7)*d^4*e - 128*(b^7*c^3 - 12*a*b^5*c^4 + 48*a^2*b^3*c^5 - 64*a^3*b*c^6)*d^3*e^2 + 12*(7*b^8*c^2 - 80*a*b^
6*c^3 + 288*a^2*b^4*c^4 - 256*a^3*b^2*c^5 - 256*a^4*c^6)*d^2*e^3 - 4*(5*b^9*c - 48*a*b^7*c^2 + 96*a^2*b^5*c^3
+ 256*a^3*b^3*c^4 - 768*a^4*b*c^5)*d*e^4 + (b^10 - 96*a^2*b^6*c^2 + 512*a^3*b^4*c^3 - 768*a^4*b^2*c^4)*e^5)*sq
rt(-4*c^2*d + 2*(b*c + sqrt(b^2 - 4*a*c)*c)*e))*arctan(2*sqrt(1/2)*sqrt(e*x + d)/sqrt(-(2*b^4*c*d - 16*a*b^2*c
^2*d + 32*a^2*c^3*d - b^5*e + 8*a*b^3*c*e - 16*a^2*b*c^2*e - sqrt((2*b^4*c*d - 16*a*b^2*c^2*d + 32*a^2*c^3*d -
b^5*e + 8*a*b^3*c*e - 16*a^2*b*c^2*e)^2 - 4*(b^4*c*d^2 - 8*a*b^2*c^2*d^2 + 16*a^2*c^3*d^2 - b^5*d*e + 8*a*b^3
*c*d*e - 16*a^2*b*c^2*d*e + a*b^4*e^2 - 8*a^2*b^2*c*e^2 + 16*a^3*c^2*e^2)*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)))
/(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)))/(((b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*sqrt(b^2 - 4*a*
c)*d^2 - (b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*sqrt(b^2 - 4*a*c)*d*e + (a*b^6*c - 12*a^2*b^4*
c^2 + 48*a^3*b^2*c^3 - 64*a^4*c^4)*sqrt(b^2 - 4*a*c)*e^2)*abs(b^4*e - 8*a*b^2*c*e + 16*a^2*c^2*e)*abs(c)) + 1/
4*(24*(e*x + d)^(7/2)*c^3*d^2*e - 72*(e*x + d)^(5/2)*c^3*d^3*e + 72*(e*x + d)^(3/2)*c^3*d^4*e - 24*sqrt(e*x +
d)*c^3*d^5*e - 24*(e*x + d)^(7/2)*b*c^2*d*e^2 + 108*(e*x + d)^(5/2)*b*c^2*d^2*e^2 - 144*(e*x + d)^(3/2)*b*c^2*
d^3*e^2 + 60*sqrt(e*x + d)*b*c^2*d^4*e^2 + 3*(e*x + d)^(7/2)*b^2*c*e^3 + 12*(e*x + d)^(7/2)*a*c^2*e^3 - 46*(e*
x + d)^(5/2)*b^2*c*d*e^3 - 32*(e*x + d)^(5/2)*a*c^2*d*e^3 + 91*(e*x + d)^(3/2)*b^2*c*d^2*e^3 + 68*(e*x + d)^(3
/2)*a*c^2*d^2*e^3 - 48*sqrt(e*x + d)*b^2*c*d^3*e^3 - 48*sqrt(e*x + d)*a*c^2*d^3*e^3 + 5*(e*x + d)^(5/2)*b^3*e^
4 + 16*(e*x + d)^(5/2)*a*b*c*e^4 - 19*(e*x + d)^(3/2)*b^3*d*e^4 - 68*(e*x + d)^(3/2)*a*b*c*d*e^4 + 12*sqrt(e*x
+ d)*b^3*d^2*e^4 + 72*sqrt(e*x + d)*a*b*c*d^2*e^4 + 19*(e*x + d)^(3/2)*a*b^2*e^5 - 4*(e*x + d)^(3/2)*a^2*c*e^
5 - 24*sqrt(e*x + d)*a*b^2*d*e^5 - 24*sqrt(e*x + d)*a^2*c*d*e^5 + 12*sqrt(e*x + d)*a^2*b*e^6)/((b^4 - 8*a*b^2*
c + 16*a^2*c^2)*((e*x + d)^2*c - 2*(e*x + d)*c*d + c*d^2 + (e*x + d)*b*e - b*d*e + a*e^2)^2)

Mupad [B] (verification not implemented)

Time = 27.22 (sec) , antiderivative size = 13637, normalized size of antiderivative = 23.63 \[ \int \frac {(d+e x)^{5/2}}{\left (a+b x+c x^2\right )^3} \, dx=\text {Too large to display} \]

[In]

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

[Out]

log((3*2^(1/2)*((3*2^(1/2)*((12*c^2*e^3*(b*e - 2*c*d)*(a*e^2 + c*d^2 - b*d*e))/(4*a*c - b^2) + (3*2^(1/2)*c^2*
e^2*(4*a*c - b^2)*(b*e - 2*c*d)*(d + e*x)^(1/2)*(-(b^15*e^5 + e^5*(-(4*a*c - b^2)^15)^(1/2) + 524288*a^5*c^10*
d^5 - 512*b^10*c^5*d^5 + 10240*a*b^8*c^6*d^5 - 81920*a^7*b*c^7*e^5 + 163840*a^7*c^8*d*e^4 + 1280*b^11*c^4*d^4*
e - 81920*a^2*b^6*c^7*d^5 + 327680*a^3*b^4*c^8*d^5 - 655360*a^4*b^2*c^9*d^5 - 560*a^2*b^11*c^2*e^5 + 4160*a^3*
b^9*c^3*e^5 - 11520*a^4*b^7*c^4*e^5 - 1024*a^5*b^5*c^5*e^5 + 61440*a^6*b^3*c^6*e^5 + 655360*a^6*c^9*d^3*e^2 -
1120*b^12*c^3*d^3*e^2 + 400*b^13*c^2*d^2*e^3 + 20*a*b^13*c*e^5 - 50*b^14*c*d*e^4 - 166400*a^2*b^8*c^5*d^3*e^2
+ 44800*a^2*b^9*c^4*d^2*e^3 + 614400*a^3*b^6*c^6*d^3*e^2 - 102400*a^3*b^7*c^5*d^2*e^3 - 1024000*a^4*b^4*c^7*d^
3*e^2 - 102400*a^4*b^5*c^6*d^2*e^3 + 327680*a^5*b^2*c^8*d^3*e^2 + 819200*a^5*b^3*c^7*d^2*e^3 - 25600*a*b^9*c^5
*d^4*e + 600*a*b^12*c^2*d*e^4 - 1310720*a^5*b*c^9*d^4*e + 21760*a*b^10*c^4*d^3*e^2 - 7040*a*b^11*c^3*d^2*e^3 +
204800*a^2*b^7*c^6*d^4*e - 160*a^2*b^10*c^3*d*e^4 - 819200*a^3*b^5*c^7*d^4*e - 28800*a^3*b^8*c^4*d*e^4 + 1638
400*a^4*b^3*c^8*d^4*e + 166400*a^4*b^6*c^5*d*e^4 - 358400*a^5*b^4*c^6*d*e^4 - 983040*a^6*b*c^8*d^2*e^3 + 20480
0*a^6*b^2*c^7*d*e^4)/(c*(4*a*c - b^2)^10))^(1/2))/2)*(-(b^15*e^5 + e^5*(-(4*a*c - b^2)^15)^(1/2) + 524288*a^5*
c^10*d^5 - 512*b^10*c^5*d^5 + 10240*a*b^8*c^6*d^5 - 81920*a^7*b*c^7*e^5 + 163840*a^7*c^8*d*e^4 + 1280*b^11*c^4
*d^4*e - 81920*a^2*b^6*c^7*d^5 + 327680*a^3*b^4*c^8*d^5 - 655360*a^4*b^2*c^9*d^5 - 560*a^2*b^11*c^2*e^5 + 4160
*a^3*b^9*c^3*e^5 - 11520*a^4*b^7*c^4*e^5 - 1024*a^5*b^5*c^5*e^5 + 61440*a^6*b^3*c^6*e^5 + 655360*a^6*c^9*d^3*e
^2 - 1120*b^12*c^3*d^3*e^2 + 400*b^13*c^2*d^2*e^3 + 20*a*b^13*c*e^5 - 50*b^14*c*d*e^4 - 166400*a^2*b^8*c^5*d^3
*e^2 + 44800*a^2*b^9*c^4*d^2*e^3 + 614400*a^3*b^6*c^6*d^3*e^2 - 102400*a^3*b^7*c^5*d^2*e^3 - 1024000*a^4*b^4*c
^7*d^3*e^2 - 102400*a^4*b^5*c^6*d^2*e^3 + 327680*a^5*b^2*c^8*d^3*e^2 + 819200*a^5*b^3*c^7*d^2*e^3 - 25600*a*b^
9*c^5*d^4*e + 600*a*b^12*c^2*d*e^4 - 1310720*a^5*b*c^9*d^4*e + 21760*a*b^10*c^4*d^3*e^2 - 7040*a*b^11*c^3*d^2*
e^3 + 204800*a^2*b^7*c^6*d^4*e - 160*a^2*b^10*c^3*d*e^4 - 819200*a^3*b^5*c^7*d^4*e - 28800*a^3*b^8*c^4*d*e^4 +
1638400*a^4*b^3*c^8*d^4*e + 166400*a^4*b^6*c^5*d*e^4 - 358400*a^5*b^4*c^6*d*e^4 - 983040*a^6*b*c^8*d^2*e^3 +
204800*a^6*b^2*c^7*d*e^4)/(c*(4*a*c - b^2)^10))^(1/2))/16 - (9*c*e^2*(d + e*x)^(1/2)*(b^6*e^6 + 512*c^6*d^6 -
32*a^3*c^3*e^6 + 640*a*c^5*d^4*e^2 + 64*a^2*b^2*c^2*e^6 + 160*a^2*c^4*d^2*e^4 + 1760*b^2*c^4*d^4*e^2 - 960*b^3
*c^3*d^3*e^3 + 250*b^4*c^2*d^2*e^4 + 14*a*b^4*c*e^6 - 1536*b*c^5*d^5*e - 26*b^5*c*d*e^5 - 1280*a*b*c^4*d^3*e^3
- 240*a*b^3*c^2*d*e^5 - 160*a^2*b*c^3*d*e^5 + 880*a*b^2*c^3*d^2*e^4))/(8*(4*a*c - b^2)^4))*(-(b^15*e^5 + e^5*
(-(4*a*c - b^2)^15)^(1/2) + 524288*a^5*c^10*d^5 - 512*b^10*c^5*d^5 + 10240*a*b^8*c^6*d^5 - 81920*a^7*b*c^7*e^5
+ 163840*a^7*c^8*d*e^4 + 1280*b^11*c^4*d^4*e - 81920*a^2*b^6*c^7*d^5 + 327680*a^3*b^4*c^8*d^5 - 655360*a^4*b^
2*c^9*d^5 - 560*a^2*b^11*c^2*e^5 + 4160*a^3*b^9*c^3*e^5 - 11520*a^4*b^7*c^4*e^5 - 1024*a^5*b^5*c^5*e^5 + 61440
*a^6*b^3*c^6*e^5 + 655360*a^6*c^9*d^3*e^2 - 1120*b^12*c^3*d^3*e^2 + 400*b^13*c^2*d^2*e^3 + 20*a*b^13*c*e^5 - 5
0*b^14*c*d*e^4 - 166400*a^2*b^8*c^5*d^3*e^2 + 44800*a^2*b^9*c^4*d^2*e^3 + 614400*a^3*b^6*c^6*d^3*e^2 - 102400*
a^3*b^7*c^5*d^2*e^3 - 1024000*a^4*b^4*c^7*d^3*e^2 - 102400*a^4*b^5*c^6*d^2*e^3 + 327680*a^5*b^2*c^8*d^3*e^2 +
819200*a^5*b^3*c^7*d^2*e^3 - 25600*a*b^9*c^5*d^4*e + 600*a*b^12*c^2*d*e^4 - 1310720*a^5*b*c^9*d^4*e + 21760*a*
b^10*c^4*d^3*e^2 - 7040*a*b^11*c^3*d^2*e^3 + 204800*a^2*b^7*c^6*d^4*e - 160*a^2*b^10*c^3*d*e^4 - 819200*a^3*b^
5*c^7*d^4*e - 28800*a^3*b^8*c^4*d*e^4 + 1638400*a^4*b^3*c^8*d^4*e + 166400*a^4*b^6*c^5*d*e^4 - 358400*a^5*b^4*
c^6*d*e^4 - 983040*a^6*b*c^8*d^2*e^3 + 204800*a^6*b^2*c^7*d*e^4)/(c*(4*a*c - b^2)^10))^(1/2))/16 - (3*(576*a^4
*c^4*e^11 + 18432*c^8*d^8*e^3 + 41472*a*c^7*d^6*e^5 - 73728*b*c^7*d^7*e^4 + 540*a^2*b^4*c^2*e^11 + 1584*a^3*b^
2*c^3*e^11 + 31104*a^2*c^6*d^4*e^7 + 8640*a^3*c^5*d^2*e^9 + 118656*b^2*c^6*d^6*e^5 - 97920*b^3*c^5*d^5*e^6 + 4
3704*b^4*c^4*d^4*e^7 - 10224*b^5*c^3*d^3*e^8 + 1125*b^6*c^2*d^2*e^9 + 45*a*b^6*c*e^11 - 45*b^7*c*d*e^10 + 4017
6*a^2*b^2*c^4*d^2*e^9 - 124416*a*b*c^6*d^5*e^6 - 1620*a*b^5*c^2*d*e^10 - 8640*a^3*b*c^4*d*e^10 + 139968*a*b^2*
c^5*d^4*e^7 - 72576*a*b^3*c^4*d^3*e^8 + 17172*a*b^4*c^3*d^2*e^9 - 62208*a^2*b*c^5*d^3*e^8 - 9072*a^2*b^3*c^3*d
*e^10))/(64*(4*a*c - b^2)^6))*(-(9*(b^15*e^5 + e^5*(-(4*a*c - b^2)^15)^(1/2) + 524288*a^5*c^10*d^5 - 512*b^10*
c^5*d^5 + 10240*a*b^8*c^6*d^5 - 81920*a^7*b*c^7*e^5 + 163840*a^7*c^8*d*e^4 + 1280*b^11*c^4*d^4*e - 81920*a^2*b
^6*c^7*d^5 + 327680*a^3*b^4*c^8*d^5 - 655360*a^4*b^2*c^9*d^5 - 560*a^2*b^11*c^2*e^5 + 4160*a^3*b^9*c^3*e^5 - 1
1520*a^4*b^7*c^4*e^5 - 1024*a^5*b^5*c^5*e^5 + 61440*a^6*b^3*c^6*e^5 + 655360*a^6*c^9*d^3*e^2 - 1120*b^12*c^3*d
^3*e^2 + 400*b^13*c^2*d^2*e^3 + 20*a*b^13*c*e^5 - 50*b^14*c*d*e^4 - 166400*a^2*b^8*c^5*d^3*e^2 + 44800*a^2*b^9
*c^4*d^2*e^3 + 614400*a^3*b^6*c^6*d^3*e^2 - 102400*a^3*b^7*c^5*d^2*e^3 - 1024000*a^4*b^4*c^7*d^3*e^2 - 102400*
a^4*b^5*c^6*d^2*e^3 + 327680*a^5*b^2*c^8*d^3*e^2 + 819200*a^5*b^3*c^7*d^2*e^3 - 25600*a*b^9*c^5*d^4*e + 600*a*
b^12*c^2*d*e^4 - 1310720*a^5*b*c^9*d^4*e + 21760*a*b^10*c^4*d^3*e^2 - 7040*a*b^11*c^3*d^2*e^3 + 204800*a^2*b^7
*c^6*d^4*e - 160*a^2*b^10*c^3*d*e^4 - 819200*a^3*b^5*c^7*d^4*e - 28800*a^3*b^8*c^4*d*e^4 + 1638400*a^4*b^3*c^8
*d^4*e + 166400*a^4*b^6*c^5*d*e^4 - 358400*a^5*b^4*c^6*d*e^4 - 983040*a^6*b*c^8*d^2*e^3 + 204800*a^6*b^2*c^7*d
*e^4))/(128*(b^20*c + 1048576*a^10*c^11 - 40*a*b^18*c^2 + 720*a^2*b^16*c^3 - 7680*a^3*b^14*c^4 + 53760*a^4*b^1
2*c^5 - 258048*a^5*b^10*c^6 + 860160*a^6*b^8*c^7 - 1966080*a^7*b^6*c^8 + 2949120*a^8*b^4*c^9 - 2621440*a^9*b^2
*c^10)))^(1/2) - log((((12*c^2*e^3*(b*e - 2*c*d)*(a*e^2 + c*d^2 - b*d*e))/(4*a*c - b^2) - 8*c^2*e^2*(4*a*c - b
^2)*(b*e - 2*c*d)*(d + e*x)^(1/2)*(-((9*b^15*e^5)/128 + (9*e^5*(-(4*a*c - b^2)^15)^(1/2))/128 + 36864*a^5*c^10
*d^5 - 36*b^10*c^5*d^5 + 720*a*b^8*c^6*d^5 - 5760*a^7*b*c^7*e^5 + 11520*a^7*c^8*d*e^4 + 90*b^11*c^4*d^4*e - 57
60*a^2*b^6*c^7*d^5 + 23040*a^3*b^4*c^8*d^5 - 46080*a^4*b^2*c^9*d^5 - (315*a^2*b^11*c^2*e^5)/8 + (585*a^3*b^9*c
^3*e^5)/2 - 810*a^4*b^7*c^4*e^5 - 72*a^5*b^5*c^5*e^5 + 4320*a^6*b^3*c^6*e^5 + 46080*a^6*c^9*d^3*e^2 - (315*b^1
2*c^3*d^3*e^2)/4 + (225*b^13*c^2*d^2*e^3)/8 + (45*a*b^13*c*e^5)/32 - (225*b^14*c*d*e^4)/64 - 11700*a^2*b^8*c^5
*d^3*e^2 + 3150*a^2*b^9*c^4*d^2*e^3 + 43200*a^3*b^6*c^6*d^3*e^2 - 7200*a^3*b^7*c^5*d^2*e^3 - 72000*a^4*b^4*c^7
*d^3*e^2 - 7200*a^4*b^5*c^6*d^2*e^3 + 23040*a^5*b^2*c^8*d^3*e^2 + 57600*a^5*b^3*c^7*d^2*e^3 - 1800*a*b^9*c^5*d
^4*e + (675*a*b^12*c^2*d*e^4)/16 - 92160*a^5*b*c^9*d^4*e + 1530*a*b^10*c^4*d^3*e^2 - 495*a*b^11*c^3*d^2*e^3 +
14400*a^2*b^7*c^6*d^4*e - (45*a^2*b^10*c^3*d*e^4)/4 - 57600*a^3*b^5*c^7*d^4*e - 2025*a^3*b^8*c^4*d*e^4 + 11520
0*a^4*b^3*c^8*d^4*e + 11700*a^4*b^6*c^5*d*e^4 - 25200*a^5*b^4*c^6*d*e^4 - 69120*a^6*b*c^8*d^2*e^3 + 14400*a^6*
b^2*c^7*d*e^4)/(c*(4*a*c - b^2)^10))^(1/2))*(-((9*b^15*e^5)/128 + (9*e^5*(-(4*a*c - b^2)^15)^(1/2))/128 + 3686
4*a^5*c^10*d^5 - 36*b^10*c^5*d^5 + 720*a*b^8*c^6*d^5 - 5760*a^7*b*c^7*e^5 + 11520*a^7*c^8*d*e^4 + 90*b^11*c^4*
d^4*e - 5760*a^2*b^6*c^7*d^5 + 23040*a^3*b^4*c^8*d^5 - 46080*a^4*b^2*c^9*d^5 - (315*a^2*b^11*c^2*e^5)/8 + (585
*a^3*b^9*c^3*e^5)/2 - 810*a^4*b^7*c^4*e^5 - 72*a^5*b^5*c^5*e^5 + 4320*a^6*b^3*c^6*e^5 + 46080*a^6*c^9*d^3*e^2
- (315*b^12*c^3*d^3*e^2)/4 + (225*b^13*c^2*d^2*e^3)/8 + (45*a*b^13*c*e^5)/32 - (225*b^14*c*d*e^4)/64 - 11700*a
^2*b^8*c^5*d^3*e^2 + 3150*a^2*b^9*c^4*d^2*e^3 + 43200*a^3*b^6*c^6*d^3*e^2 - 7200*a^3*b^7*c^5*d^2*e^3 - 72000*a
^4*b^4*c^7*d^3*e^2 - 7200*a^4*b^5*c^6*d^2*e^3 + 23040*a^5*b^2*c^8*d^3*e^2 + 57600*a^5*b^3*c^7*d^2*e^3 - 1800*a
*b^9*c^5*d^4*e + (675*a*b^12*c^2*d*e^4)/16 - 92160*a^5*b*c^9*d^4*e + 1530*a*b^10*c^4*d^3*e^2 - 495*a*b^11*c^3*
d^2*e^3 + 14400*a^2*b^7*c^6*d^4*e - (45*a^2*b^10*c^3*d*e^4)/4 - 57600*a^3*b^5*c^7*d^4*e - 2025*a^3*b^8*c^4*d*e
^4 + 115200*a^4*b^3*c^8*d^4*e + 11700*a^4*b^6*c^5*d*e^4 - 25200*a^5*b^4*c^6*d*e^4 - 69120*a^6*b*c^8*d^2*e^3 +
14400*a^6*b^2*c^7*d*e^4)/(c*(4*a*c - b^2)^10))^(1/2) + (9*c*e^2*(d + e*x)^(1/2)*(b^6*e^6 + 512*c^6*d^6 - 32*a^
3*c^3*e^6 + 640*a*c^5*d^4*e^2 + 64*a^2*b^2*c^2*e^6 + 160*a^2*c^4*d^2*e^4 + 1760*b^2*c^4*d^4*e^2 - 960*b^3*c^3*
d^3*e^3 + 250*b^4*c^2*d^2*e^4 + 14*a*b^4*c*e^6 - 1536*b*c^5*d^5*e - 26*b^5*c*d*e^5 - 1280*a*b*c^4*d^3*e^3 - 24
0*a*b^3*c^2*d*e^5 - 160*a^2*b*c^3*d*e^5 + 880*a*b^2*c^3*d^2*e^4))/(8*(4*a*c - b^2)^4))*(-((9*b^15*e^5)/128 + (
9*e^5*(-(4*a*c - b^2)^15)^(1/2))/128 + 36864*a^5*c^10*d^5 - 36*b^10*c^5*d^5 + 720*a*b^8*c^6*d^5 - 5760*a^7*b*c
^7*e^5 + 11520*a^7*c^8*d*e^4 + 90*b^11*c^4*d^4*e - 5760*a^2*b^6*c^7*d^5 + 23040*a^3*b^4*c^8*d^5 - 46080*a^4*b^
2*c^9*d^5 - (315*a^2*b^11*c^2*e^5)/8 + (585*a^3*b^9*c^3*e^5)/2 - 810*a^4*b^7*c^4*e^5 - 72*a^5*b^5*c^5*e^5 + 43
20*a^6*b^3*c^6*e^5 + 46080*a^6*c^9*d^3*e^2 - (315*b^12*c^3*d^3*e^2)/4 + (225*b^13*c^2*d^2*e^3)/8 + (45*a*b^13*
c*e^5)/32 - (225*b^14*c*d*e^4)/64 - 11700*a^2*b^8*c^5*d^3*e^2 + 3150*a^2*b^9*c^4*d^2*e^3 + 43200*a^3*b^6*c^6*d
^3*e^2 - 7200*a^3*b^7*c^5*d^2*e^3 - 72000*a^4*b^4*c^7*d^3*e^2 - 7200*a^4*b^5*c^6*d^2*e^3 + 23040*a^5*b^2*c^8*d
^3*e^2 + 57600*a^5*b^3*c^7*d^2*e^3 - 1800*a*b^9*c^5*d^4*e + (675*a*b^12*c^2*d*e^4)/16 - 92160*a^5*b*c^9*d^4*e
+ 1530*a*b^10*c^4*d^3*e^2 - 495*a*b^11*c^3*d^2*e^3 + 14400*a^2*b^7*c^6*d^4*e - (45*a^2*b^10*c^3*d*e^4)/4 - 576
00*a^3*b^5*c^7*d^4*e - 2025*a^3*b^8*c^4*d*e^4 + 115200*a^4*b^3*c^8*d^4*e + 11700*a^4*b^6*c^5*d*e^4 - 25200*a^5
*b^4*c^6*d*e^4 - 69120*a^6*b*c^8*d^2*e^3 + 14400*a^6*b^2*c^7*d*e^4)/(c*(4*a*c - b^2)^10))^(1/2) - (3*(576*a^4*
c^4*e^11 + 18432*c^8*d^8*e^3 + 41472*a*c^7*d^6*e^5 - 73728*b*c^7*d^7*e^4 + 540*a^2*b^4*c^2*e^11 + 1584*a^3*b^2
*c^3*e^11 + 31104*a^2*c^6*d^4*e^7 + 8640*a^3*c^5*d^2*e^9 + 118656*b^2*c^6*d^6*e^5 - 97920*b^3*c^5*d^5*e^6 + 43
704*b^4*c^4*d^4*e^7 - 10224*b^5*c^3*d^3*e^8 + 1125*b^6*c^2*d^2*e^9 + 45*a*b^6*c*e^11 - 45*b^7*c*d*e^10 + 40176
*a^2*b^2*c^4*d^2*e^9 - 124416*a*b*c^6*d^5*e^6 - 1620*a*b^5*c^2*d*e^10 - 8640*a^3*b*c^4*d*e^10 + 139968*a*b^2*c
^5*d^4*e^7 - 72576*a*b^3*c^4*d^3*e^8 + 17172*a*b^4*c^3*d^2*e^9 - 62208*a^2*b*c^5*d^3*e^8 - 9072*a^2*b^3*c^3*d*
e^10))/(64*(4*a*c - b^2)^6))*(-((9*b^15*e^5)/128 + (9*e^5*(-(4*a*c - b^2)^15)^(1/2))/128 + 36864*a^5*c^10*d^5
- 36*b^10*c^5*d^5 + 720*a*b^8*c^6*d^5 - 5760*a^7*b*c^7*e^5 + 11520*a^7*c^8*d*e^4 + 90*b^11*c^4*d^4*e - 5760*a^
2*b^6*c^7*d^5 + 23040*a^3*b^4*c^8*d^5 - 46080*a^4*b^2*c^9*d^5 - (315*a^2*b^11*c^2*e^5)/8 + (585*a^3*b^9*c^3*e^
5)/2 - 810*a^4*b^7*c^4*e^5 - 72*a^5*b^5*c^5*e^5 + 4320*a^6*b^3*c^6*e^5 + 46080*a^6*c^9*d^3*e^2 - (315*b^12*c^3
*d^3*e^2)/4 + (225*b^13*c^2*d^2*e^3)/8 + (45*a*b^13*c*e^5)/32 - (225*b^14*c*d*e^4)/64 - 11700*a^2*b^8*c^5*d^3*
e^2 + 3150*a^2*b^9*c^4*d^2*e^3 + 43200*a^3*b^6*c^6*d^3*e^2 - 7200*a^3*b^7*c^5*d^2*e^3 - 72000*a^4*b^4*c^7*d^3*
e^2 - 7200*a^4*b^5*c^6*d^2*e^3 + 23040*a^5*b^2*c^8*d^3*e^2 + 57600*a^5*b^3*c^7*d^2*e^3 - 1800*a*b^9*c^5*d^4*e
+ (675*a*b^12*c^2*d*e^4)/16 - 92160*a^5*b*c^9*d^4*e + 1530*a*b^10*c^4*d^3*e^2 - 495*a*b^11*c^3*d^2*e^3 + 14400
*a^2*b^7*c^6*d^4*e - (45*a^2*b^10*c^3*d*e^4)/4 - 57600*a^3*b^5*c^7*d^4*e - 2025*a^3*b^8*c^4*d*e^4 + 115200*a^4
*b^3*c^8*d^4*e + 11700*a^4*b^6*c^5*d*e^4 - 25200*a^5*b^4*c^6*d*e^4 - 69120*a^6*b*c^8*d^2*e^3 + 14400*a^6*b^2*c
^7*d*e^4)/(b^20*c + 1048576*a^10*c^11 - 40*a*b^18*c^2 + 720*a^2*b^16*c^3 - 7680*a^3*b^14*c^4 + 53760*a^4*b^12*
c^5 - 258048*a^5*b^10*c^6 + 860160*a^6*b^8*c^7 - 1966080*a^7*b^6*c^8 + 2949120*a^8*b^4*c^9 - 2621440*a^9*b^2*c
^10))^(1/2) - log((((12*c^2*e^3*(b*e - 2*c*d)*(a*e^2 + c*d^2 - b*d*e))/(4*a*c - b^2) - 8*c^2*e^2*(4*a*c - b^2)
*(b*e - 2*c*d)*(d + e*x)^(1/2)*(((9*e^5*(-(4*a*c - b^2)^15)^(1/2))/128 - (9*b^15*e^5)/128 - 36864*a^5*c^10*d^5
+ 36*b^10*c^5*d^5 - 720*a*b^8*c^6*d^5 + 5760*a^7*b*c^7*e^5 - 11520*a^7*c^8*d*e^4 - 90*b^11*c^4*d^4*e + 5760*a
^2*b^6*c^7*d^5 - 23040*a^3*b^4*c^8*d^5 + 46080*a^4*b^2*c^9*d^5 + (315*a^2*b^11*c^2*e^5)/8 - (585*a^3*b^9*c^3*e
^5)/2 + 810*a^4*b^7*c^4*e^5 + 72*a^5*b^5*c^5*e^5 - 4320*a^6*b^3*c^6*e^5 - 46080*a^6*c^9*d^3*e^2 + (315*b^12*c^
3*d^3*e^2)/4 - (225*b^13*c^2*d^2*e^3)/8 - (45*a*b^13*c*e^5)/32 + (225*b^14*c*d*e^4)/64 + 11700*a^2*b^8*c^5*d^3
*e^2 - 3150*a^2*b^9*c^4*d^2*e^3 - 43200*a^3*b^6*c^6*d^3*e^2 + 7200*a^3*b^7*c^5*d^2*e^3 + 72000*a^4*b^4*c^7*d^3
*e^2 + 7200*a^4*b^5*c^6*d^2*e^3 - 23040*a^5*b^2*c^8*d^3*e^2 - 57600*a^5*b^3*c^7*d^2*e^3 + 1800*a*b^9*c^5*d^4*e
- (675*a*b^12*c^2*d*e^4)/16 + 92160*a^5*b*c^9*d^4*e - 1530*a*b^10*c^4*d^3*e^2 + 495*a*b^11*c^3*d^2*e^3 - 1440
0*a^2*b^7*c^6*d^4*e + (45*a^2*b^10*c^3*d*e^4)/4 + 57600*a^3*b^5*c^7*d^4*e + 2025*a^3*b^8*c^4*d*e^4 - 115200*a^
4*b^3*c^8*d^4*e - 11700*a^4*b^6*c^5*d*e^4 + 25200*a^5*b^4*c^6*d*e^4 + 69120*a^6*b*c^8*d^2*e^3 - 14400*a^6*b^2*
c^7*d*e^4)/(c*(4*a*c - b^2)^10))^(1/2))*(((9*e^5*(-(4*a*c - b^2)^15)^(1/2))/128 - (9*b^15*e^5)/128 - 36864*a^5
*c^10*d^5 + 36*b^10*c^5*d^5 - 720*a*b^8*c^6*d^5 + 5760*a^7*b*c^7*e^5 - 11520*a^7*c^8*d*e^4 - 90*b^11*c^4*d^4*e
+ 5760*a^2*b^6*c^7*d^5 - 23040*a^3*b^4*c^8*d^5 + 46080*a^4*b^2*c^9*d^5 + (315*a^2*b^11*c^2*e^5)/8 - (585*a^3*
b^9*c^3*e^5)/2 + 810*a^4*b^7*c^4*e^5 + 72*a^5*b^5*c^5*e^5 - 4320*a^6*b^3*c^6*e^5 - 46080*a^6*c^9*d^3*e^2 + (31
5*b^12*c^3*d^3*e^2)/4 - (225*b^13*c^2*d^2*e^3)/8 - (45*a*b^13*c*e^5)/32 + (225*b^14*c*d*e^4)/64 + 11700*a^2*b^
8*c^5*d^3*e^2 - 3150*a^2*b^9*c^4*d^2*e^3 - 43200*a^3*b^6*c^6*d^3*e^2 + 7200*a^3*b^7*c^5*d^2*e^3 + 72000*a^4*b^
4*c^7*d^3*e^2 + 7200*a^4*b^5*c^6*d^2*e^3 - 23040*a^5*b^2*c^8*d^3*e^2 - 57600*a^5*b^3*c^7*d^2*e^3 + 1800*a*b^9*
c^5*d^4*e - (675*a*b^12*c^2*d*e^4)/16 + 92160*a^5*b*c^9*d^4*e - 1530*a*b^10*c^4*d^3*e^2 + 495*a*b^11*c^3*d^2*e
^3 - 14400*a^2*b^7*c^6*d^4*e + (45*a^2*b^10*c^3*d*e^4)/4 + 57600*a^3*b^5*c^7*d^4*e + 2025*a^3*b^8*c^4*d*e^4 -
115200*a^4*b^3*c^8*d^4*e - 11700*a^4*b^6*c^5*d*e^4 + 25200*a^5*b^4*c^6*d*e^4 + 69120*a^6*b*c^8*d^2*e^3 - 14400
*a^6*b^2*c^7*d*e^4)/(c*(4*a*c - b^2)^10))^(1/2) + (9*c*e^2*(d + e*x)^(1/2)*(b^6*e^6 + 512*c^6*d^6 - 32*a^3*c^3
*e^6 + 640*a*c^5*d^4*e^2 + 64*a^2*b^2*c^2*e^6 + 160*a^2*c^4*d^2*e^4 + 1760*b^2*c^4*d^4*e^2 - 960*b^3*c^3*d^3*e
^3 + 250*b^4*c^2*d^2*e^4 + 14*a*b^4*c*e^6 - 1536*b*c^5*d^5*e - 26*b^5*c*d*e^5 - 1280*a*b*c^4*d^3*e^3 - 240*a*b
^3*c^2*d*e^5 - 160*a^2*b*c^3*d*e^5 + 880*a*b^2*c^3*d^2*e^4))/(8*(4*a*c - b^2)^4))*(((9*e^5*(-(4*a*c - b^2)^15)
^(1/2))/128 - (9*b^15*e^5)/128 - 36864*a^5*c^10*d^5 + 36*b^10*c^5*d^5 - 720*a*b^8*c^6*d^5 + 5760*a^7*b*c^7*e^5
- 11520*a^7*c^8*d*e^4 - 90*b^11*c^4*d^4*e + 5760*a^2*b^6*c^7*d^5 - 23040*a^3*b^4*c^8*d^5 + 46080*a^4*b^2*c^9*
d^5 + (315*a^2*b^11*c^2*e^5)/8 - (585*a^3*b^9*c^3*e^5)/2 + 810*a^4*b^7*c^4*e^5 + 72*a^5*b^5*c^5*e^5 - 4320*a^6
*b^3*c^6*e^5 - 46080*a^6*c^9*d^3*e^2 + (315*b^12*c^3*d^3*e^2)/4 - (225*b^13*c^2*d^2*e^3)/8 - (45*a*b^13*c*e^5)
/32 + (225*b^14*c*d*e^4)/64 + 11700*a^2*b^8*c^5*d^3*e^2 - 3150*a^2*b^9*c^4*d^2*e^3 - 43200*a^3*b^6*c^6*d^3*e^2
+ 7200*a^3*b^7*c^5*d^2*e^3 + 72000*a^4*b^4*c^7*d^3*e^2 + 7200*a^4*b^5*c^6*d^2*e^3 - 23040*a^5*b^2*c^8*d^3*e^2
- 57600*a^5*b^3*c^7*d^2*e^3 + 1800*a*b^9*c^5*d^4*e - (675*a*b^12*c^2*d*e^4)/16 + 92160*a^5*b*c^9*d^4*e - 1530
*a*b^10*c^4*d^3*e^2 + 495*a*b^11*c^3*d^2*e^3 - 14400*a^2*b^7*c^6*d^4*e + (45*a^2*b^10*c^3*d*e^4)/4 + 57600*a^3
*b^5*c^7*d^4*e + 2025*a^3*b^8*c^4*d*e^4 - 115200*a^4*b^3*c^8*d^4*e - 11700*a^4*b^6*c^5*d*e^4 + 25200*a^5*b^4*c
^6*d*e^4 + 69120*a^6*b*c^8*d^2*e^3 - 14400*a^6*b^2*c^7*d*e^4)/(c*(4*a*c - b^2)^10))^(1/2) - (3*(576*a^4*c^4*e^
11 + 18432*c^8*d^8*e^3 + 41472*a*c^7*d^6*e^5 - 73728*b*c^7*d^7*e^4 + 540*a^2*b^4*c^2*e^11 + 1584*a^3*b^2*c^3*e
^11 + 31104*a^2*c^6*d^4*e^7 + 8640*a^3*c^5*d^2*e^9 + 118656*b^2*c^6*d^6*e^5 - 97920*b^3*c^5*d^5*e^6 + 43704*b^
4*c^4*d^4*e^7 - 10224*b^5*c^3*d^3*e^8 + 1125*b^6*c^2*d^2*e^9 + 45*a*b^6*c*e^11 - 45*b^7*c*d*e^10 + 40176*a^2*b
^2*c^4*d^2*e^9 - 124416*a*b*c^6*d^5*e^6 - 1620*a*b^5*c^2*d*e^10 - 8640*a^3*b*c^4*d*e^10 + 139968*a*b^2*c^5*d^4
*e^7 - 72576*a*b^3*c^4*d^3*e^8 + 17172*a*b^4*c^3*d^2*e^9 - 62208*a^2*b*c^5*d^3*e^8 - 9072*a^2*b^3*c^3*d*e^10))
/(64*(4*a*c - b^2)^6))*(((9*e^5*(-(4*a*c - b^2)^15)^(1/2))/128 - (9*b^15*e^5)/128 - 36864*a^5*c^10*d^5 + 36*b^
10*c^5*d^5 - 720*a*b^8*c^6*d^5 + 5760*a^7*b*c^7*e^5 - 11520*a^7*c^8*d*e^4 - 90*b^11*c^4*d^4*e + 5760*a^2*b^6*c
^7*d^5 - 23040*a^3*b^4*c^8*d^5 + 46080*a^4*b^2*c^9*d^5 + (315*a^2*b^11*c^2*e^5)/8 - (585*a^3*b^9*c^3*e^5)/2 +
810*a^4*b^7*c^4*e^5 + 72*a^5*b^5*c^5*e^5 - 4320*a^6*b^3*c^6*e^5 - 46080*a^6*c^9*d^3*e^2 + (315*b^12*c^3*d^3*e^
2)/4 - (225*b^13*c^2*d^2*e^3)/8 - (45*a*b^13*c*e^5)/32 + (225*b^14*c*d*e^4)/64 + 11700*a^2*b^8*c^5*d^3*e^2 - 3
150*a^2*b^9*c^4*d^2*e^3 - 43200*a^3*b^6*c^6*d^3*e^2 + 7200*a^3*b^7*c^5*d^2*e^3 + 72000*a^4*b^4*c^7*d^3*e^2 + 7
200*a^4*b^5*c^6*d^2*e^3 - 23040*a^5*b^2*c^8*d^3*e^2 - 57600*a^5*b^3*c^7*d^2*e^3 + 1800*a*b^9*c^5*d^4*e - (675*
a*b^12*c^2*d*e^4)/16 + 92160*a^5*b*c^9*d^4*e - 1530*a*b^10*c^4*d^3*e^2 + 495*a*b^11*c^3*d^2*e^3 - 14400*a^2*b^
7*c^6*d^4*e + (45*a^2*b^10*c^3*d*e^4)/4 + 57600*a^3*b^5*c^7*d^4*e + 2025*a^3*b^8*c^4*d*e^4 - 115200*a^4*b^3*c^
8*d^4*e - 11700*a^4*b^6*c^5*d*e^4 + 25200*a^5*b^4*c^6*d*e^4 + 69120*a^6*b*c^8*d^2*e^3 - 14400*a^6*b^2*c^7*d*e^
4)/(b^20*c + 1048576*a^10*c^11 - 40*a*b^18*c^2 + 720*a^2*b^16*c^3 - 7680*a^3*b^14*c^4 + 53760*a^4*b^12*c^5 - 2
58048*a^5*b^10*c^6 + 860160*a^6*b^8*c^7 - 1966080*a^7*b^6*c^8 + 2949120*a^8*b^4*c^9 - 2621440*a^9*b^2*c^10))^(
1/2) + log((3*2^(1/2)*((3*2^(1/2)*((12*c^2*e^3*(b*e - 2*c*d)*(a*e^2 + c*d^2 - b*d*e))/(4*a*c - b^2) + (3*2^(1/
2)*c^2*e^2*(4*a*c - b^2)*(b*e - 2*c*d)*(d + e*x)^(1/2)*((e^5*(-(4*a*c - b^2)^15)^(1/2) - b^15*e^5 - 524288*a^5
*c^10*d^5 + 512*b^10*c^5*d^5 - 10240*a*b^8*c^6*d^5 + 81920*a^7*b*c^7*e^5 - 163840*a^7*c^8*d*e^4 - 1280*b^11*c^
4*d^4*e + 81920*a^2*b^6*c^7*d^5 - 327680*a^3*b^4*c^8*d^5 + 655360*a^4*b^2*c^9*d^5 + 560*a^2*b^11*c^2*e^5 - 416
0*a^3*b^9*c^3*e^5 + 11520*a^4*b^7*c^4*e^5 + 1024*a^5*b^5*c^5*e^5 - 61440*a^6*b^3*c^6*e^5 - 655360*a^6*c^9*d^3*
e^2 + 1120*b^12*c^3*d^3*e^2 - 400*b^13*c^2*d^2*e^3 - 20*a*b^13*c*e^5 + 50*b^14*c*d*e^4 + 166400*a^2*b^8*c^5*d^
3*e^2 - 44800*a^2*b^9*c^4*d^2*e^3 - 614400*a^3*b^6*c^6*d^3*e^2 + 102400*a^3*b^7*c^5*d^2*e^3 + 1024000*a^4*b^4*
c^7*d^3*e^2 + 102400*a^4*b^5*c^6*d^2*e^3 - 327680*a^5*b^2*c^8*d^3*e^2 - 819200*a^5*b^3*c^7*d^2*e^3 + 25600*a*b
^9*c^5*d^4*e - 600*a*b^12*c^2*d*e^4 + 1310720*a^5*b*c^9*d^4*e - 21760*a*b^10*c^4*d^3*e^2 + 7040*a*b^11*c^3*d^2
*e^3 - 204800*a^2*b^7*c^6*d^4*e + 160*a^2*b^10*c^3*d*e^4 + 819200*a^3*b^5*c^7*d^4*e + 28800*a^3*b^8*c^4*d*e^4
- 1638400*a^4*b^3*c^8*d^4*e - 166400*a^4*b^6*c^5*d*e^4 + 358400*a^5*b^4*c^6*d*e^4 + 983040*a^6*b*c^8*d^2*e^3 -
204800*a^6*b^2*c^7*d*e^4)/(c*(4*a*c - b^2)^10))^(1/2))/2)*((e^5*(-(4*a*c - b^2)^15)^(1/2) - b^15*e^5 - 524288
*a^5*c^10*d^5 + 512*b^10*c^5*d^5 - 10240*a*b^8*c^6*d^5 + 81920*a^7*b*c^7*e^5 - 163840*a^7*c^8*d*e^4 - 1280*b^1
1*c^4*d^4*e + 81920*a^2*b^6*c^7*d^5 - 327680*a^3*b^4*c^8*d^5 + 655360*a^4*b^2*c^9*d^5 + 560*a^2*b^11*c^2*e^5 -
4160*a^3*b^9*c^3*e^5 + 11520*a^4*b^7*c^4*e^5 + 1024*a^5*b^5*c^5*e^5 - 61440*a^6*b^3*c^6*e^5 - 655360*a^6*c^9*
d^3*e^2 + 1120*b^12*c^3*d^3*e^2 - 400*b^13*c^2*d^2*e^3 - 20*a*b^13*c*e^5 + 50*b^14*c*d*e^4 + 166400*a^2*b^8*c^
5*d^3*e^2 - 44800*a^2*b^9*c^4*d^2*e^3 - 614400*a^3*b^6*c^6*d^3*e^2 + 102400*a^3*b^7*c^5*d^2*e^3 + 1024000*a^4*
b^4*c^7*d^3*e^2 + 102400*a^4*b^5*c^6*d^2*e^3 - 327680*a^5*b^2*c^8*d^3*e^2 - 819200*a^5*b^3*c^7*d^2*e^3 + 25600
*a*b^9*c^5*d^4*e - 600*a*b^12*c^2*d*e^4 + 1310720*a^5*b*c^9*d^4*e - 21760*a*b^10*c^4*d^3*e^2 + 7040*a*b^11*c^3
*d^2*e^3 - 204800*a^2*b^7*c^6*d^4*e + 160*a^2*b^10*c^3*d*e^4 + 819200*a^3*b^5*c^7*d^4*e + 28800*a^3*b^8*c^4*d*
e^4 - 1638400*a^4*b^3*c^8*d^4*e - 166400*a^4*b^6*c^5*d*e^4 + 358400*a^5*b^4*c^6*d*e^4 + 983040*a^6*b*c^8*d^2*e
^3 - 204800*a^6*b^2*c^7*d*e^4)/(c*(4*a*c - b^2)^10))^(1/2))/16 - (9*c*e^2*(d + e*x)^(1/2)*(b^6*e^6 + 512*c^6*d
^6 - 32*a^3*c^3*e^6 + 640*a*c^5*d^4*e^2 + 64*a^2*b^2*c^2*e^6 + 160*a^2*c^4*d^2*e^4 + 1760*b^2*c^4*d^4*e^2 - 96
0*b^3*c^3*d^3*e^3 + 250*b^4*c^2*d^2*e^4 + 14*a*b^4*c*e^6 - 1536*b*c^5*d^5*e - 26*b^5*c*d*e^5 - 1280*a*b*c^4*d^
3*e^3 - 240*a*b^3*c^2*d*e^5 - 160*a^2*b*c^3*d*e^5 + 880*a*b^2*c^3*d^2*e^4))/(8*(4*a*c - b^2)^4))*((e^5*(-(4*a*
c - b^2)^15)^(1/2) - b^15*e^5 - 524288*a^5*c^10*d^5 + 512*b^10*c^5*d^5 - 10240*a*b^8*c^6*d^5 + 81920*a^7*b*c^7
*e^5 - 163840*a^7*c^8*d*e^4 - 1280*b^11*c^4*d^4*e + 81920*a^2*b^6*c^7*d^5 - 327680*a^3*b^4*c^8*d^5 + 655360*a^
4*b^2*c^9*d^5 + 560*a^2*b^11*c^2*e^5 - 4160*a^3*b^9*c^3*e^5 + 11520*a^4*b^7*c^4*e^5 + 1024*a^5*b^5*c^5*e^5 - 6
1440*a^6*b^3*c^6*e^5 - 655360*a^6*c^9*d^3*e^2 + 1120*b^12*c^3*d^3*e^2 - 400*b^13*c^2*d^2*e^3 - 20*a*b^13*c*e^5
+ 50*b^14*c*d*e^4 + 166400*a^2*b^8*c^5*d^3*e^2 - 44800*a^2*b^9*c^4*d^2*e^3 - 614400*a^3*b^6*c^6*d^3*e^2 + 102
400*a^3*b^7*c^5*d^2*e^3 + 1024000*a^4*b^4*c^7*d^3*e^2 + 102400*a^4*b^5*c^6*d^2*e^3 - 327680*a^5*b^2*c^8*d^3*e^
2 - 819200*a^5*b^3*c^7*d^2*e^3 + 25600*a*b^9*c^5*d^4*e - 600*a*b^12*c^2*d*e^4 + 1310720*a^5*b*c^9*d^4*e - 2176
0*a*b^10*c^4*d^3*e^2 + 7040*a*b^11*c^3*d^2*e^3 - 204800*a^2*b^7*c^6*d^4*e + 160*a^2*b^10*c^3*d*e^4 + 819200*a^
3*b^5*c^7*d^4*e + 28800*a^3*b^8*c^4*d*e^4 - 1638400*a^4*b^3*c^8*d^4*e - 166400*a^4*b^6*c^5*d*e^4 + 358400*a^5*
b^4*c^6*d*e^4 + 983040*a^6*b*c^8*d^2*e^3 - 204800*a^6*b^2*c^7*d*e^4)/(c*(4*a*c - b^2)^10))^(1/2))/16 - (3*(576
*a^4*c^4*e^11 + 18432*c^8*d^8*e^3 + 41472*a*c^7*d^6*e^5 - 73728*b*c^7*d^7*e^4 + 540*a^2*b^4*c^2*e^11 + 1584*a^
3*b^2*c^3*e^11 + 31104*a^2*c^6*d^4*e^7 + 8640*a^3*c^5*d^2*e^9 + 118656*b^2*c^6*d^6*e^5 - 97920*b^3*c^5*d^5*e^6
+ 43704*b^4*c^4*d^4*e^7 - 10224*b^5*c^3*d^3*e^8 + 1125*b^6*c^2*d^2*e^9 + 45*a*b^6*c*e^11 - 45*b^7*c*d*e^10 +
40176*a^2*b^2*c^4*d^2*e^9 - 124416*a*b*c^6*d^5*e^6 - 1620*a*b^5*c^2*d*e^10 - 8640*a^3*b*c^4*d*e^10 + 139968*a*
b^2*c^5*d^4*e^7 - 72576*a*b^3*c^4*d^3*e^8 + 17172*a*b^4*c^3*d^2*e^9 - 62208*a^2*b*c^5*d^3*e^8 - 9072*a^2*b^3*c
^3*d*e^10))/(64*(4*a*c - b^2)^6))*((9*(e^5*(-(4*a*c - b^2)^15)^(1/2) - b^15*e^5 - 524288*a^5*c^10*d^5 + 512*b^
10*c^5*d^5 - 10240*a*b^8*c^6*d^5 + 81920*a^7*b*c^7*e^5 - 163840*a^7*c^8*d*e^4 - 1280*b^11*c^4*d^4*e + 81920*a^
2*b^6*c^7*d^5 - 327680*a^3*b^4*c^8*d^5 + 655360*a^4*b^2*c^9*d^5 + 560*a^2*b^11*c^2*e^5 - 4160*a^3*b^9*c^3*e^5
+ 11520*a^4*b^7*c^4*e^5 + 1024*a^5*b^5*c^5*e^5 - 61440*a^6*b^3*c^6*e^5 - 655360*a^6*c^9*d^3*e^2 + 1120*b^12*c^
3*d^3*e^2 - 400*b^13*c^2*d^2*e^3 - 20*a*b^13*c*e^5 + 50*b^14*c*d*e^4 + 166400*a^2*b^8*c^5*d^3*e^2 - 44800*a^2*
b^9*c^4*d^2*e^3 - 614400*a^3*b^6*c^6*d^3*e^2 + 102400*a^3*b^7*c^5*d^2*e^3 + 1024000*a^4*b^4*c^7*d^3*e^2 + 1024
00*a^4*b^5*c^6*d^2*e^3 - 327680*a^5*b^2*c^8*d^3*e^2 - 819200*a^5*b^3*c^7*d^2*e^3 + 25600*a*b^9*c^5*d^4*e - 600
*a*b^12*c^2*d*e^4 + 1310720*a^5*b*c^9*d^4*e - 21760*a*b^10*c^4*d^3*e^2 + 7040*a*b^11*c^3*d^2*e^3 - 204800*a^2*
b^7*c^6*d^4*e + 160*a^2*b^10*c^3*d*e^4 + 819200*a^3*b^5*c^7*d^4*e + 28800*a^3*b^8*c^4*d*e^4 - 1638400*a^4*b^3*
c^8*d^4*e - 166400*a^4*b^6*c^5*d*e^4 + 358400*a^5*b^4*c^6*d*e^4 + 983040*a^6*b*c^8*d^2*e^3 - 204800*a^6*b^2*c^
7*d*e^4))/(128*(b^20*c + 1048576*a^10*c^11 - 40*a*b^18*c^2 + 720*a^2*b^16*c^3 - 7680*a^3*b^14*c^4 + 53760*a^4*
b^12*c^5 - 258048*a^5*b^10*c^6 + 860160*a^6*b^8*c^7 - 1966080*a^7*b^6*c^8 + 2949120*a^8*b^4*c^9 - 2621440*a^9*
b^2*c^10)))^(1/2) + (((d + e*x)^(3/2)*(19*a*b^2*e^5 - 4*a^2*c*e^5 - 19*b^3*d*e^4 + 72*c^3*d^4*e + 68*a*c^2*d^2
*e^3 - 144*b*c^2*d^3*e^2 + 91*b^2*c*d^2*e^3 - 68*a*b*c*d*e^4))/(4*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) - (3*(d + e*
x)^(1/2)*(2*c^3*d^5*e - a^2*b*e^6 - b^3*d^2*e^4 + 4*a*c^2*d^3*e^3 - 5*b*c^2*d^4*e^2 + 4*b^2*c*d^3*e^3 + 2*a*b^
2*d*e^5 + 2*a^2*c*d*e^5 - 6*a*b*c*d^2*e^4))/(b^4 + 16*a^2*c^2 - 8*a*b^2*c) + ((b*e - 2*c*d)*(d + e*x)^(5/2)*(5
*b^2*e^3 + 36*c^2*d^2*e + 16*a*c*e^3 - 36*b*c*d*e^2))/(4*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + (3*c*e*(d + e*x)^(7
/2)*(b^2*e^2 + 8*c^2*d^2 + 4*a*c*e^2 - 8*b*c*d*e))/(4*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)))/(c^2*(d + e*x)^4 - (d +
e*x)*(4*c^2*d^3 + 2*b^2*d*e^2 - 2*a*b*e^3 + 4*a*c*d*e^2 - 6*b*c*d^2*e) - (4*c^2*d - 2*b*c*e)*(d + e*x)^3 + (d
+ e*x)^2*(b^2*e^2 + 6*c^2*d^2 + 2*a*c*e^2 - 6*b*c*d*e) + a^2*e^4 + c^2*d^4 + b^2*d^2*e^2 - 2*a*b*d*e^3 - 2*b*
c*d^3*e + 2*a*c*d^2*e^2)