3.17.25 \(\int \frac {(b+2 c x) (d+e x)^{3/2}}{(a+b x+c x^2)^3} \, dx\) [1625]

Optimal. Leaf size=322 \[ -\frac {(d+e x)^{3/2}}{2 \left (a+b x+c x^2\right )^2}-\frac {3 e (b+2 c x) \sqrt {d+e x}}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {3 \sqrt {c} e \left (4 c d-\left (2 b-\sqrt {b^2-4 a c}\right ) e\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{2 \sqrt {2} \left (b^2-4 a c\right )^{3/2} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}-\frac {3 \sqrt {c} e \left (4 c d-\left (2 b+\sqrt {b^2-4 a c}\right ) e\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{2 \sqrt {2} \left (b^2-4 a c\right )^{3/2} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \]

[Out]

-1/2*(e*x+d)^(3/2)/(c*x^2+b*x+a)^2-3/4*e*(2*c*x+b)*(e*x+d)^(1/2)/(-4*a*c+b^2)/(c*x^2+b*x+a)+3/4*e*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))*c^(1/2)*(4*c*d-e*(2*b-(-4*a*c+b^2)^(1/2)))/(
-4*a*c+b^2)^(3/2)*2^(1/2)/(2*c*d-e*(b-(-4*a*c+b^2)^(1/2)))^(1/2)-3/4*e*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))*c^(1/2)*(4*c*d-e*(2*b+(-4*a*c+b^2)^(1/2)))/(-4*a*c+b^2)^(3/2)*2^(1/2)/(
2*c*d-e*(b+(-4*a*c+b^2)^(1/2)))^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.53, antiderivative size = 322, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {782, 750, 840, 1180, 214} \begin {gather*} -\frac {3 e (b+2 c x) \sqrt {d+e x}}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {3 \sqrt {c} e \left (4 c d-e \left (2 b-\sqrt {b^2-4 a c}\right )\right ) \tanh ^{-1}\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 )}{2 \sqrt {2} \left (b^2-4 a c\right )^{3/2} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}-\frac {3 \sqrt {c} e \left (4 c d-e \left (\sqrt {b^2-4 a c}+2 b\right )\right ) \tanh ^{-1}\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 )}{2 \sqrt {2} \left (b^2-4 a c\right )^{3/2} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}-\frac {(d+e x)^{3/2}}{2 \left (a+b x+c x^2\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*(d + e*x)^(3/2)/(a + b*x + c*x^2)^2 - (3*e*(b + 2*c*x)*Sqrt[d + e*x])/(4*(b^2 - 4*a*c)*(a + b*x + c*x^2))
+ (3*Sqrt[c]*e*(4*c*d - (2*b - 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]])/(2*Sqrt[2]*(b^2 - 4*a*c)^(3/2)*Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]) - (3*Sqrt[c]
*e*(4*c*d - (2*b + 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]])/(2*Sqrt[2]*(b^2 - 4*a*c)^(3/2)*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 750

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^m*(b + 2*c
*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
- 1)*(b*e*m + 2*c*d*(2*p + 3) + 2*c*e*(m + 2*p + 3)*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
, 0] && (LtQ[m, 1] || (ILtQ[m + 2*p + 3, 0] && NeQ[m, 2])) && IntQuadraticQ[a, b, c, d, e, m, p, x]

Rule 782

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

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*} \int \frac {(b+2 c x) (d+e x)^{3/2}}{\left (a+b x+c x^2\right )^3} \, dx &=-\frac {(d+e x)^{3/2}}{2 \left (a+b x+c x^2\right )^2}+\frac {1}{4} (3 e) \int \frac {\sqrt {d+e x}}{\left (a+b x+c x^2\right )^2} \, dx\\ &=-\frac {(d+e x)^{3/2}}{2 \left (a+b x+c x^2\right )^2}-\frac {3 e (b+2 c x) \sqrt {d+e x}}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {(3 e) \int \frac {-2 c d+\frac {b e}{2}-c e x}{\sqrt {d+e x} \left (a+b x+c x^2\right )} \, dx}{4 \left (b^2-4 a c\right )}\\ &=-\frac {(d+e x)^{3/2}}{2 \left (a+b x+c x^2\right )^2}-\frac {3 e (b+2 c x) \sqrt {d+e x}}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {(3 e) \text {Subst}\left (\int \frac {c d e+e \left (-2 c d+\frac {b e}{2}\right )-c e 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 )}{2 \left (b^2-4 a c\right )}\\ &=-\frac {(d+e x)^{3/2}}{2 \left (a+b x+c x^2\right )^2}-\frac {3 e (b+2 c x) \sqrt {d+e x}}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {\left (3 c e \left (4 c d-\left (2 b-\sqrt {b^2-4 a c}\right ) e\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 )}{4 \left (b^2-4 a c\right )^{3/2}}+\frac {\left (3 c e \left (4 c d-\left (2 b+\sqrt {b^2-4 a c}\right ) e\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 )}{4 \left (b^2-4 a c\right )^{3/2}}\\ &=-\frac {(d+e x)^{3/2}}{2 \left (a+b x+c x^2\right )^2}-\frac {3 e (b+2 c x) \sqrt {d+e x}}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {3 \sqrt {c} e \left (4 c d-\left (2 b-\sqrt {b^2-4 a c}\right ) e\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 )}{2 \sqrt {2} \left (b^2-4 a c\right )^{3/2} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}-\frac {3 \sqrt {c} e \left (4 c d-\left (2 b+\sqrt {b^2-4 a c}\right ) e\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 )}{2 \sqrt {2} \left (b^2-4 a c\right )^{3/2} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 15.05, size = 501, normalized size = 1.56 \begin {gather*} \frac {\frac {\left (b^2-4 a c\right ) (d+e x)^{5/2} (-c d+b e+c e x)}{(a+x (b+c x))^2}-\frac {e (d+e x)^{5/2} \left (b^3 e^2+b^2 c e (-4 d+e x)+b c \left (-a e^2+3 c d (d-2 e x)\right )+2 c^2 \left (3 c d^2 x+a e (2 d+e x)\right )\right )}{2 \left (c d^2+e (-b d+a e)\right ) (a+x (b+c x))}+\frac {1}{2} e \left (-3 e (-2 c d+b e) \sqrt {d+e x}+\frac {e \left (6 c^2 d^2+b^2 e^2+2 c e (-3 b d+a e)\right ) (d+e x)^{3/2}}{c d^2+e (-b d+a e)}+\frac {3 \sqrt {2} \sqrt {c} \left (c d^2+e (-b d+a e)\right ) \left (\frac {\left (4 c d+\left (-2 b+\sqrt {b^2-4 a c}\right ) e\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-b e+\sqrt {b^2-4 a c} e}}\right )}{\sqrt {2 c d+\left (-b+\sqrt {b^2-4 a c}\right ) e}}+\frac {\left (-4 c d+\left (2 b+\sqrt {b^2-4 a c}\right ) e\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 )}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {b^2-4 a c}}\right )}{2 \left (b^2-4 a c\right ) \left (c d^2+e (-b d+a e)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(((b^2 - 4*a*c)*(d + e*x)^(5/2)*(-(c*d) + b*e + c*e*x))/(a + x*(b + c*x))^2 - (e*(d + e*x)^(5/2)*(b^3*e^2 + b^
2*c*e*(-4*d + e*x) + b*c*(-(a*e^2) + 3*c*d*(d - 2*e*x)) + 2*c^2*(3*c*d^2*x + a*e*(2*d + e*x))))/(2*(c*d^2 + e*
(-(b*d) + a*e))*(a + x*(b + c*x))) + (e*(-3*e*(-2*c*d + b*e)*Sqrt[d + e*x] + (e*(6*c^2*d^2 + b^2*e^2 + 2*c*e*(
-3*b*d + a*e))*(d + e*x)^(3/2))/(c*d^2 + e*(-(b*d) + a*e)) + (3*Sqrt[2]*Sqrt[c]*(c*d^2 + e*(-(b*d) + a*e))*(((
4*c*d + (-2*b + Sqrt[b^2 - 4*a*c])*e)*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - b*e + Sqrt[b^2 - 4*
a*c]*e]])/Sqrt[2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e] + ((-4*c*d + (2*b + 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]])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]))/Sqr
t[b^2 - 4*a*c]))/2)/(2*(b^2 - 4*a*c)*(c*d^2 + e*(-(b*d) + a*e)))

________________________________________________________________________________________

Maple [A]
time = 0.93, size = 490, normalized size = 1.52

method	result	size
derivativedivides	\(2 e^{4} \left (\frac {\frac {3 c^{2} \left (e x +d \right )^{\frac {7}{2}}}{4 \left (4 a c -b^{2}\right ) e^{2}}+\frac {9 c \left (b e -2 c d \right ) \left (e x +d \right )^{\frac {5}{2}}}{8 \left (4 a c -b^{2}\right ) e^{2}}-\frac {\left (2 a c \,e^{2}-5 b^{2} e^{2}+18 b c d e -18 c^{2} d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}}{8 e^{2} \left (4 a c -b^{2}\right )}+\frac {3 \left (a b \,e^{3}-2 a d \,e^{2} c -b^{2} d \,e^{2}+3 d^{2} e b c -2 c^{2} d^{3}\right ) \sqrt {e x +d}}{8 e^{2} \left (4 a c -b^{2}\right )}}{\left (\left (e x +d \right )^{2} c +b e \left (e x +d \right )-2 c d \left (e x +d \right )+e^{2} a -b d e +c \,d^{2}\right )^{2}}+\frac {3 c \left (\frac {\left (2 b e -4 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{4 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}-\frac {\left (-2 b e +4 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) \sqrt {2}\, \arctanh \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{4 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{2 \left (4 a c -b^{2}\right ) e^{2}}\right )\)	\(490\)
default	\(2 e^{4} \left (\frac {\frac {3 c^{2} \left (e x +d \right )^{\frac {7}{2}}}{4 \left (4 a c -b^{2}\right ) e^{2}}+\frac {9 c \left (b e -2 c d \right ) \left (e x +d \right )^{\frac {5}{2}}}{8 \left (4 a c -b^{2}\right ) e^{2}}-\frac {\left (2 a c \,e^{2}-5 b^{2} e^{2}+18 b c d e -18 c^{2} d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}}{8 e^{2} \left (4 a c -b^{2}\right )}+\frac {3 \left (a b \,e^{3}-2 a d \,e^{2} c -b^{2} d \,e^{2}+3 d^{2} e b c -2 c^{2} d^{3}\right ) \sqrt {e x +d}}{8 e^{2} \left (4 a c -b^{2}\right )}}{\left (\left (e x +d \right )^{2} c +b e \left (e x +d \right )-2 c d \left (e x +d \right )+e^{2} a -b d e +c \,d^{2}\right )^{2}}+\frac {3 c \left (\frac {\left (2 b e -4 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{4 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}-\frac {\left (-2 b e +4 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) \sqrt {2}\, \arctanh \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{4 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{2 \left (4 a c -b^{2}\right ) e^{2}}\right )\)	\(490\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*e^4*((3/4*c^2/(4*a*c-b^2)/e^2*(e*x+d)^(7/2)+9/8*c*(b*e-2*c*d)/(4*a*c-b^2)/e^2*(e*x+d)^(5/2)-1/8*(2*a*c*e^2-5
*b^2*e^2+18*b*c*d*e-18*c^2*d^2)/e^2/(4*a*c-b^2)*(e*x+d)^(3/2)+3/8*(a*b*e^3-2*a*c*d*e^2-b^2*d*e^2+3*b*c*d^2*e-2
*c^2*d^3)/e^2/(4*a*c-b^2)*(e*x+d)^(1/2))/((e*x+d)^2*c+b*e*(e*x+d)-2*c*d*(e*x+d)+e^2*a-b*d*e+c*d^2)^2+3/2/(4*a*
c-b^2)/e^2*c*(1/4*(2*b*e-4*c*d+(-e^2*(4*a*c-b^2))^(1/2))/(-e^2*(4*a*c-b^2))^(1/2)*2^(1/2)/((b*e-2*c*d+(-e^2*(4
*a*c-b^2))^(1/2))*c)^(1/2)*arctan(c*(e*x+d)^(1/2)*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))-1/4*
(-2*b*e+4*c*d+(-e^2*(4*a*c-b^2))^(1/2))/(-e^2*(4*a*c-b^2))^(1/2)*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2)
)*c)^(1/2)*arctanh(c*(e*x+d)^(1/2)*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))))

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 6601 vs. \(2 (279) = 558\).
time = 1.63, size = 6601, normalized size = 20.50 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(3*sqrt(1/2)*((b^2*c^2 - 4*a*c^3)*x^4 + a^2*b^2 - 4*a^3*c + 2*(b^3*c - 4*a*b*c^2)*x^3 + (b^4 - 2*a*b^2*c -
8*a^2*c^2)*x^2 + 2*(a*b^3 - 4*a^2*b*c)*x)*sqrt((32*c^3*d^3*e^2 - 48*b*c^2*d^2*e^3 + 6*(3*b^2*c + 4*a*c^2)*d*e
^4 - (b^3 + 12*a*b*c)*e^5 + ((b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4)*d^2 - (b^7 - 12*a*b^5*c + 48
*a^2*b^3*c^2 - 64*a^3*b*c^3)*d*e + (a*b^6 - 12*a^2*b^4*c + 48*a^3*b^2*c^2 - 64*a^4*c^3)*e^2)*e^5/sqrt((b^6*c^2
- 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d^4 - 2*(b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*
d^3*e + (b^8 - 10*a*b^6*c + 24*a^2*b^4*c^2 + 32*a^3*b^2*c^3 - 128*a^4*c^4)*d^2*e^2 - 2*(a*b^7 - 12*a^2*b^5*c +
48*a^3*b^3*c^2 - 64*a^4*b*c^3)*d*e^3 + (a^2*b^6 - 12*a^3*b^4*c + 48*a^4*b^2*c^2 - 64*a^5*c^3)*e^4))/((b^6*c -
12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4)*d^2 - (b^7 - 12*a*b^5*c + 48*a^2*b^3*c^2 - 64*a^3*b*c^3)*d*e + (a
*b^6 - 12*a^2*b^4*c + 48*a^3*b^2*c^2 - 64*a^4*c^3)*e^2))*log(27/2*sqrt(1/2)*(2*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c
^3)*d*e^6 - (b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*e^7 - (8*(b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6)*d
^4 - 16*(b^7*c^2 - 12*a*b^5*c^3 + 48*a^2*b^3*c^4 - 64*a^3*b*c^5)*d^3*e + 3*(3*b^8*c - 32*a*b^6*c^2 + 96*a^2*b^
4*c^3 - 256*a^4*c^5)*d^2*e^2 - (b^9 - 96*a^2*b^5*c^2 + 512*a^3*b^3*c^3 - 768*a^4*b*c^4)*d*e^3 + (a*b^8 - 8*a^2
*b^6*c + 128*a^4*b^2*c^3 - 256*a^5*c^4)*e^4)*e^5/sqrt((b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d
^4 - 2*(b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*d^3*e + (b^8 - 10*a*b^6*c + 24*a^2*b^4*c^2 + 32*
a^3*b^2*c^3 - 128*a^4*c^4)*d^2*e^2 - 2*(a*b^7 - 12*a^2*b^5*c + 48*a^3*b^3*c^2 - 64*a^4*b*c^3)*d*e^3 + (a^2*b^6
- 12*a^3*b^4*c + 48*a^4*b^2*c^2 - 64*a^5*c^3)*e^4))*sqrt((32*c^3*d^3*e^2 - 48*b*c^2*d^2*e^3 + 6*(3*b^2*c + 4*
a*c^2)*d*e^4 - (b^3 + 12*a*b*c)*e^5 + ((b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4)*d^2 - (b^7 - 12*a*
b^5*c + 48*a^2*b^3*c^2 - 64*a^3*b*c^3)*d*e + (a*b^6 - 12*a^2*b^4*c + 48*a^3*b^2*c^2 - 64*a^4*c^3)*e^2)*e^5/sqr
t((b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d^4 - 2*(b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a
^3*b*c^4)*d^3*e + (b^8 - 10*a*b^6*c + 24*a^2*b^4*c^2 + 32*a^3*b^2*c^3 - 128*a^4*c^4)*d^2*e^2 - 2*(a*b^7 - 12*a
^2*b^5*c + 48*a^3*b^3*c^2 - 64*a^4*b*c^3)*d*e^3 + (a^2*b^6 - 12*a^3*b^4*c + 48*a^4*b^2*c^2 - 64*a^5*c^3)*e^4))
/((b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4)*d^2 - (b^7 - 12*a*b^5*c + 48*a^2*b^3*c^2 - 64*a^3*b*c^3
)*d*e + (a*b^6 - 12*a^2*b^4*c + 48*a^3*b^2*c^2 - 64*a^4*c^3)*e^2)) + 27*(16*c^3*d^2*e^6 - 16*b*c^2*d*e^7 + (3*
b^2*c + 4*a*c^2)*e^8)*sqrt(x*e + d)) - 3*sqrt(1/2)*((b^2*c^2 - 4*a*c^3)*x^4 + a^2*b^2 - 4*a^3*c + 2*(b^3*c - 4
*a*b*c^2)*x^3 + (b^4 - 2*a*b^2*c - 8*a^2*c^2)*x^2 + 2*(a*b^3 - 4*a^2*b*c)*x)*sqrt((32*c^3*d^3*e^2 - 48*b*c^2*d
^2*e^3 + 6*(3*b^2*c + 4*a*c^2)*d*e^4 - (b^3 + 12*a*b*c)*e^5 + ((b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3
*c^4)*d^2 - (b^7 - 12*a*b^5*c + 48*a^2*b^3*c^2 - 64*a^3*b*c^3)*d*e + (a*b^6 - 12*a^2*b^4*c + 48*a^3*b^2*c^2 -
64*a^4*c^3)*e^2)*e^5/sqrt((b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d^4 - 2*(b^7*c - 12*a*b^5*c^2
+ 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*d^3*e + (b^8 - 10*a*b^6*c + 24*a^2*b^4*c^2 + 32*a^3*b^2*c^3 - 128*a^4*c^4)*d
^2*e^2 - 2*(a*b^7 - 12*a^2*b^5*c + 48*a^3*b^3*c^2 - 64*a^4*b*c^3)*d*e^3 + (a^2*b^6 - 12*a^3*b^4*c + 48*a^4*b^2
*c^2 - 64*a^5*c^3)*e^4))/((b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4)*d^2 - (b^7 - 12*a*b^5*c + 48*a^
2*b^3*c^2 - 64*a^3*b*c^3)*d*e + (a*b^6 - 12*a^2*b^4*c + 48*a^3*b^2*c^2 - 64*a^4*c^3)*e^2))*log(-27/2*sqrt(1/2)
*(2*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*d*e^6 - (b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*e^7 - (8*(b^6*c^3 - 12*a*b^4*c
^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6)*d^4 - 16*(b^7*c^2 - 12*a*b^5*c^3 + 48*a^2*b^3*c^4 - 64*a^3*b*c^5)*d^3*e + 3*
(3*b^8*c - 32*a*b^6*c^2 + 96*a^2*b^4*c^3 - 256*a^4*c^5)*d^2*e^2 - (b^9 - 96*a^2*b^5*c^2 + 512*a^3*b^3*c^3 - 76
8*a^4*b*c^4)*d*e^3 + (a*b^8 - 8*a^2*b^6*c + 128*a^4*b^2*c^3 - 256*a^5*c^4)*e^4)*e^5/sqrt((b^6*c^2 - 12*a*b^4*c
^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d^4 - 2*(b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*d^3*e + (b^8
- 10*a*b^6*c + 24*a^2*b^4*c^2 + 32*a^3*b^2*c^3 - 128*a^4*c^4)*d^2*e^2 - 2*(a*b^7 - 12*a^2*b^5*c + 48*a^3*b^3*c
^2 - 64*a^4*b*c^3)*d*e^3 + (a^2*b^6 - 12*a^3*b^4*c + 48*a^4*b^2*c^2 - 64*a^5*c^3)*e^4))*sqrt((32*c^3*d^3*e^2 -
48*b*c^2*d^2*e^3 + 6*(3*b^2*c + 4*a*c^2)*d*e^4 - (b^3 + 12*a*b*c)*e^5 + ((b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c
^3 - 64*a^3*c^4)*d^2 - (b^7 - 12*a*b^5*c + 48*a^2*b^3*c^2 - 64*a^3*b*c^3)*d*e + (a*b^6 - 12*a^2*b^4*c + 48*a^3
*b^2*c^2 - 64*a^4*c^3)*e^2)*e^5/sqrt((b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d^4 - 2*(b^7*c - 1
2*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*d^3*e + (b^8 - 10*a*b^6*c + 24*a^2*b^4*c^2 + 32*a^3*b^2*c^3 - 128
*a^4*c^4)*d^2*e^2 - 2*(a*b^7 - 12*a^2*b^5*c + 48*a^3*b^3*c^2 - 64*a^4*b*c^3)*d*e^3 + (a^2*b^6 - 12*a^3*b^4*c +
48*a^4*b^2*c^2 - 64*a^5*c^3)*e^4))/((b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4)*d^2 - (b^7 - 12*a*b^
5*c + 48*a^2*b^3*c^2 - 64*a^3*b*c^3)*d*e + (a*b...

________________________________________________________________________________________

Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1189 vs. \(2 (279) = 558\).
time = 3.81, size = 1189, normalized size = 3.69 \begin {gather*} -\frac {3 \, {\left (\sqrt {-4 \, c^{2} d + 2 \, {\left (b c - \sqrt {b^{2} - 4 \, a c} c\right )} e} {\left (b^{2} e - 4 \, a c e\right )}^{2} e^{2} + {\left (2 \, \sqrt {b^{2} - 4 \, a c} c d e^{2} - \sqrt {b^{2} - 4 \, a c} b e^{3}\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c - \sqrt {b^{2} - 4 \, a c} c\right )} e} {\left | b^{2} e - 4 \, a c e \right |} - 2 \, {\left (4 \, {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d^{2} e^{2} - 4 \, {\left (b^{3} c - 4 \, a b c^{2}\right )} d e^{3} + {\left (b^{4} - 4 \, a b^{2} c\right )} e^{4}\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c - \sqrt {b^{2} - 4 \, a c} c\right )} e}\right )} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {x e + d}}{\sqrt {-\frac {2 \, b^{2} c d - 8 \, a c^{2} d - b^{3} e + 4 \, a b c e + \sqrt {{\left (2 \, b^{2} c d - 8 \, a c^{2} d - b^{3} e + 4 \, a b c e\right )}^{2} - 4 \, {\left (b^{2} c d^{2} - 4 \, a c^{2} d^{2} - b^{3} d e + 4 \, a b c d e + a b^{2} e^{2} - 4 \, a^{2} c e^{2}\right )} {\left (b^{2} c - 4 \, a c^{2}\right )}}}{b^{2} c - 4 \, a c^{2}}}}\right )}{16 \, {\left ({\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt {b^{2} - 4 \, a c} d^{2} - {\left (b^{3} - 4 \, a b c\right )} \sqrt {b^{2} - 4 \, a c} d e + {\left (a b^{2} - 4 \, a^{2} c\right )} \sqrt {b^{2} - 4 \, a c} e^{2}\right )} {\left | b^{2} e - 4 \, a c e \right |} {\left | c \right |}} + \frac {3 \, {\left (\sqrt {-4 \, c^{2} d + 2 \, {\left (b c + \sqrt {b^{2} - 4 \, a c} c\right )} e} {\left (b^{2} e - 4 \, a c e\right )}^{2} e^{2} - {\left (2 \, \sqrt {b^{2} - 4 \, a c} c d e^{2} - \sqrt {b^{2} - 4 \, a c} b e^{3}\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c + \sqrt {b^{2} - 4 \, a c} c\right )} e} {\left | b^{2} e - 4 \, a c e \right |} - 2 \, {\left (4 \, {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d^{2} e^{2} - 4 \, {\left (b^{3} c - 4 \, a b c^{2}\right )} d e^{3} + {\left (b^{4} - 4 \, a b^{2} c\right )} e^{4}\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c + \sqrt {b^{2} - 4 \, a c} c\right )} e}\right )} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {x e + d}}{\sqrt {-\frac {2 \, b^{2} c d - 8 \, a c^{2} d - b^{3} e + 4 \, a b c e - \sqrt {{\left (2 \, b^{2} c d - 8 \, a c^{2} d - b^{3} e + 4 \, a b c e\right )}^{2} - 4 \, {\left (b^{2} c d^{2} - 4 \, a c^{2} d^{2} - b^{3} d e + 4 \, a b c d e + a b^{2} e^{2} - 4 \, a^{2} c e^{2}\right )} {\left (b^{2} c - 4 \, a c^{2}\right )}}}{b^{2} c - 4 \, a c^{2}}}}\right )}{16 \, {\left ({\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt {b^{2} - 4 \, a c} d^{2} - {\left (b^{3} - 4 \, a b c\right )} \sqrt {b^{2} - 4 \, a c} d e + {\left (a b^{2} - 4 \, a^{2} c\right )} \sqrt {b^{2} - 4 \, a c} e^{2}\right )} {\left | b^{2} e - 4 \, a c e \right |} {\left | c \right |}} - \frac {6 \, {\left (x e + d\right )}^{\frac {7}{2}} c^{2} e^{2} - 18 \, {\left (x e + d\right )}^{\frac {5}{2}} c^{2} d e^{2} + 18 \, {\left (x e + d\right )}^{\frac {3}{2}} c^{2} d^{2} e^{2} - 6 \, \sqrt {x e + d} c^{2} d^{3} e^{2} + 9 \, {\left (x e + d\right )}^{\frac {5}{2}} b c e^{3} - 18 \, {\left (x e + d\right )}^{\frac {3}{2}} b c d e^{3} + 9 \, \sqrt {x e + d} b c d^{2} e^{3} + 5 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{2} e^{4} - 2 \, {\left (x e + d\right )}^{\frac {3}{2}} a c e^{4} - 3 \, \sqrt {x e + d} b^{2} d e^{4} - 6 \, \sqrt {x e + d} a c d e^{4} + 3 \, \sqrt {x e + d} a b e^{5}}{4 \, {\left ({\left (x e + d\right )}^{2} c - 2 \, {\left (x e + d\right )} c d + c d^{2} + {\left (x e + d\right )} b e - b d e + a e^{2}\right )}^{2} {\left (b^{2} - 4 \, a c\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/16*(sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e)*(b^2*e - 4*a*c*e)^2*e^2 + (2*sqrt(b^2 - 4*a*c)*c*d*e^2
- sqrt(b^2 - 4*a*c)*b*e^3)*sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e)*abs(b^2*e - 4*a*c*e) - 2*(4*(b^2*
c^2 - 4*a*c^3)*d^2*e^2 - 4*(b^3*c - 4*a*b*c^2)*d*e^3 + (b^4 - 4*a*b^2*c)*e^4)*sqrt(-4*c^2*d + 2*(b*c - sqrt(b^
2 - 4*a*c)*c)*e))*arctan(2*sqrt(1/2)*sqrt(x*e + d)/sqrt(-(2*b^2*c*d - 8*a*c^2*d - b^3*e + 4*a*b*c*e + sqrt((2*
b^2*c*d - 8*a*c^2*d - b^3*e + 4*a*b*c*e)^2 - 4*(b^2*c*d^2 - 4*a*c^2*d^2 - b^3*d*e + 4*a*b*c*d*e + a*b^2*e^2 -
4*a^2*c*e^2)*(b^2*c - 4*a*c^2)))/(b^2*c - 4*a*c^2)))/(((b^2*c - 4*a*c^2)*sqrt(b^2 - 4*a*c)*d^2 - (b^3 - 4*a*b*
c)*sqrt(b^2 - 4*a*c)*d*e + (a*b^2 - 4*a^2*c)*sqrt(b^2 - 4*a*c)*e^2)*abs(b^2*e - 4*a*c*e)*abs(c)) + 3/16*(sqrt(
-4*c^2*d + 2*(b*c + sqrt(b^2 - 4*a*c)*c)*e)*(b^2*e - 4*a*c*e)^2*e^2 - (2*sqrt(b^2 - 4*a*c)*c*d*e^2 - sqrt(b^2
- 4*a*c)*b*e^3)*sqrt(-4*c^2*d + 2*(b*c + sqrt(b^2 - 4*a*c)*c)*e)*abs(b^2*e - 4*a*c*e) - 2*(4*(b^2*c^2 - 4*a*c^
3)*d^2*e^2 - 4*(b^3*c - 4*a*b*c^2)*d*e^3 + (b^4 - 4*a*b^2*c)*e^4)*sqrt(-4*c^2*d + 2*(b*c + sqrt(b^2 - 4*a*c)*c
)*e))*arctan(2*sqrt(1/2)*sqrt(x*e + d)/sqrt(-(2*b^2*c*d - 8*a*c^2*d - b^3*e + 4*a*b*c*e - sqrt((2*b^2*c*d - 8*
a*c^2*d - b^3*e + 4*a*b*c*e)^2 - 4*(b^2*c*d^2 - 4*a*c^2*d^2 - b^3*d*e + 4*a*b*c*d*e + a*b^2*e^2 - 4*a^2*c*e^2)
*(b^2*c - 4*a*c^2)))/(b^2*c - 4*a*c^2)))/(((b^2*c - 4*a*c^2)*sqrt(b^2 - 4*a*c)*d^2 - (b^3 - 4*a*b*c)*sqrt(b^2
- 4*a*c)*d*e + (a*b^2 - 4*a^2*c)*sqrt(b^2 - 4*a*c)*e^2)*abs(b^2*e - 4*a*c*e)*abs(c)) - 1/4*(6*(x*e + d)^(7/2)*
c^2*e^2 - 18*(x*e + d)^(5/2)*c^2*d*e^2 + 18*(x*e + d)^(3/2)*c^2*d^2*e^2 - 6*sqrt(x*e + d)*c^2*d^3*e^2 + 9*(x*e
+ d)^(5/2)*b*c*e^3 - 18*(x*e + d)^(3/2)*b*c*d*e^3 + 9*sqrt(x*e + d)*b*c*d^2*e^3 + 5*(x*e + d)^(3/2)*b^2*e^4 -
2*(x*e + d)^(3/2)*a*c*e^4 - 3*sqrt(x*e + d)*b^2*d*e^4 - 6*sqrt(x*e + d)*a*c*d*e^4 + 3*sqrt(x*e + d)*a*b*e^5)/
(((x*e + d)^2*c - 2*(x*e + d)*c*d + c*d^2 + (x*e + d)*b*e - b*d*e + a*e^2)^2*(b^2 - 4*a*c))

________________________________________________________________________________________

Mupad [B]
time = 7.15, size = 2500, normalized size = 7.76 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(((d + e*x)^(3/2)*(5*b^2*e^4 + 18*c^2*d^2*e^2 - 2*a*c*e^4 - 18*b*c*d*e^3))/(4*(4*a*c - b^2)) - (3*(d + e*x)^(1
/2)*(b^2*d*e^4 + 2*c^2*d^3*e^2 - a*b*e^5 + 2*a*c*d*e^4 - 3*b*c*d^2*e^3))/(4*(4*a*c - b^2)) + (3*c^2*e^2*(d + e
*x)^(7/2))/(2*(4*a*c - b^2)) + (9*c*e^2*(b*e - 2*c*d)*(d + e*x)^(5/2))/(4*(4*a*c - b^2)))/(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) - atan(((((3*(32*b^7*c^2*e^5 - 384*a*b^5*c^3*e^5 - 2048*a^3*b*c^5*e^5 + 4096*a^3*c
^6*d*e^4 - 64*b^6*c^3*d*e^4 + 1536*a^2*b^3*c^4*e^5 + 768*a*b^4*c^4*d*e^4 - 3072*a^2*b^2*c^5*d*e^4))/(32*(b^6 -
64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c)) - ((d + e*x)^(1/2)*((9*(e^5*(-(4*a*c - b^2)^9)^(1/2) - b^9*e^5 + 7
68*a^4*b*c^4*e^5 - 1536*a^4*c^5*d*e^4 + 96*a^2*b^5*c^2*e^5 - 512*a^3*b^3*c^3*e^5 - 2048*a^3*c^6*d^3*e^2 + 32*b
^6*c^3*d^3*e^2 - 48*b^7*c^2*d^2*e^3 + 18*b^8*c*d*e^4 + 1536*a^2*b^2*c^5*d^3*e^2 - 2304*a^2*b^3*c^4*d^2*e^3 - 1
92*a*b^6*c^2*d*e^4 - 384*a*b^4*c^4*d^3*e^2 + 576*a*b^5*c^3*d^2*e^3 + 576*a^2*b^4*c^3*d*e^4 + 3072*a^3*b*c^5*d^
2*e^3))/(128*(a*b^12*e^2 + b^12*c*d^2 + 4096*a^6*c^7*d^2 + 4096*a^7*c^6*e^2 - b^13*d*e - 24*a*b^10*c^2*d^2 - 2
4*a^2*b^10*c*e^2 + 240*a^2*b^8*c^3*d^2 - 1280*a^3*b^6*c^4*d^2 + 3840*a^4*b^4*c^5*d^2 - 6144*a^5*b^2*c^6*d^2 +
240*a^3*b^8*c^2*e^2 - 1280*a^4*b^6*c^3*e^2 + 3840*a^5*b^4*c^4*e^2 - 6144*a^6*b^2*c^5*e^2 - 4096*a^6*b*c^6*d*e
- 240*a^2*b^9*c^2*d*e + 1280*a^3*b^7*c^3*d*e - 3840*a^4*b^5*c^4*d*e + 6144*a^5*b^3*c^5*d*e + 24*a*b^11*c*d*e))
)^(1/2)*(32*b^7*c^2*e^3 - 384*a*b^5*c^3*e^3 - 2048*a^3*b*c^5*e^3 + 4096*a^3*c^6*d*e^2 - 64*b^6*c^3*d*e^2 + 153
6*a^2*b^3*c^4*e^3 + 768*a*b^4*c^4*d*e^2 - 3072*a^2*b^2*c^5*d*e^2))/(4*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)))*((9*(e^
5*(-(4*a*c - b^2)^9)^(1/2) - b^9*e^5 + 768*a^4*b*c^4*e^5 - 1536*a^4*c^5*d*e^4 + 96*a^2*b^5*c^2*e^5 - 512*a^3*b
^3*c^3*e^5 - 2048*a^3*c^6*d^3*e^2 + 32*b^6*c^3*d^3*e^2 - 48*b^7*c^2*d^2*e^3 + 18*b^8*c*d*e^4 + 1536*a^2*b^2*c^
5*d^3*e^2 - 2304*a^2*b^3*c^4*d^2*e^3 - 192*a*b^6*c^2*d*e^4 - 384*a*b^4*c^4*d^3*e^2 + 576*a*b^5*c^3*d^2*e^3 + 5
76*a^2*b^4*c^3*d*e^4 + 3072*a^3*b*c^5*d^2*e^3))/(128*(a*b^12*e^2 + b^12*c*d^2 + 4096*a^6*c^7*d^2 + 4096*a^7*c^
6*e^2 - b^13*d*e - 24*a*b^10*c^2*d^2 - 24*a^2*b^10*c*e^2 + 240*a^2*b^8*c^3*d^2 - 1280*a^3*b^6*c^4*d^2 + 3840*a
^4*b^4*c^5*d^2 - 6144*a^5*b^2*c^6*d^2 + 240*a^3*b^8*c^2*e^2 - 1280*a^4*b^6*c^3*e^2 + 3840*a^5*b^4*c^4*e^2 - 61
44*a^6*b^2*c^5*e^2 - 4096*a^6*b*c^6*d*e - 240*a^2*b^9*c^2*d*e + 1280*a^3*b^7*c^3*d*e - 3840*a^4*b^5*c^4*d*e +
6144*a^5*b^3*c^5*d*e + 24*a*b^11*c*d*e)))^(1/2) + ((d + e*x)^(1/2)*(36*a*c^4*e^6 - 45*b^2*c^3*e^6 - 144*c^5*d^
2*e^4 + 144*b*c^4*d*e^5))/(4*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)))*((9*(e^5*(-(4*a*c - b^2)^9)^(1/2) - b^9*e^5 + 76
8*a^4*b*c^4*e^5 - 1536*a^4*c^5*d*e^4 + 96*a^2*b^5*c^2*e^5 - 512*a^3*b^3*c^3*e^5 - 2048*a^3*c^6*d^3*e^2 + 32*b^
6*c^3*d^3*e^2 - 48*b^7*c^2*d^2*e^3 + 18*b^8*c*d*e^4 + 1536*a^2*b^2*c^5*d^3*e^2 - 2304*a^2*b^3*c^4*d^2*e^3 - 19
2*a*b^6*c^2*d*e^4 - 384*a*b^4*c^4*d^3*e^2 + 576*a*b^5*c^3*d^2*e^3 + 576*a^2*b^4*c^3*d*e^4 + 3072*a^3*b*c^5*d^2
*e^3))/(128*(a*b^12*e^2 + b^12*c*d^2 + 4096*a^6*c^7*d^2 + 4096*a^7*c^6*e^2 - b^13*d*e - 24*a*b^10*c^2*d^2 - 24
*a^2*b^10*c*e^2 + 240*a^2*b^8*c^3*d^2 - 1280*a^3*b^6*c^4*d^2 + 3840*a^4*b^4*c^5*d^2 - 6144*a^5*b^2*c^6*d^2 + 2
40*a^3*b^8*c^2*e^2 - 1280*a^4*b^6*c^3*e^2 + 3840*a^5*b^4*c^4*e^2 - 6144*a^6*b^2*c^5*e^2 - 4096*a^6*b*c^6*d*e -
240*a^2*b^9*c^2*d*e + 1280*a^3*b^7*c^3*d*e - 3840*a^4*b^5*c^4*d*e + 6144*a^5*b^3*c^5*d*e + 24*a*b^11*c*d*e)))
^(1/2)*1i - (((3*(32*b^7*c^2*e^5 - 384*a*b^5*c^3*e^5 - 2048*a^3*b*c^5*e^5 + 4096*a^3*c^6*d*e^4 - 64*b^6*c^3*d*
e^4 + 1536*a^2*b^3*c^4*e^5 + 768*a*b^4*c^4*d*e^4 - 3072*a^2*b^2*c^5*d*e^4))/(32*(b^6 - 64*a^3*c^3 + 48*a^2*b^2
*c^2 - 12*a*b^4*c)) + ((d + e*x)^(1/2)*((9*(e^5*(-(4*a*c - b^2)^9)^(1/2) - b^9*e^5 + 768*a^4*b*c^4*e^5 - 1536*
a^4*c^5*d*e^4 + 96*a^2*b^5*c^2*e^5 - 512*a^3*b^3*c^3*e^5 - 2048*a^3*c^6*d^3*e^2 + 32*b^6*c^3*d^3*e^2 - 48*b^7*
c^2*d^2*e^3 + 18*b^8*c*d*e^4 + 1536*a^2*b^2*c^5*d^3*e^2 - 2304*a^2*b^3*c^4*d^2*e^3 - 192*a*b^6*c^2*d*e^4 - 384
*a*b^4*c^4*d^3*e^2 + 576*a*b^5*c^3*d^2*e^3 + 576*a^2*b^4*c^3*d*e^4 + 3072*a^3*b*c^5*d^2*e^3))/(128*(a*b^12*e^2
+ b^12*c*d^2 + 4096*a^6*c^7*d^2 + 4096*a^7*c^6*e^2 - b^13*d*e - 24*a*b^10*c^2*d^2 - 24*a^2*b^10*c*e^2 + 240*a
^2*b^8*c^3*d^2 - 1280*a^3*b^6*c^4*d^2 + 3840*a^4*b^4*c^5*d^2 - 6144*a^5*b^2*c^6*d^2 + 240*a^3*b^8*c^2*e^2 - 12
80*a^4*b^6*c^3*e^2 + 3840*a^5*b^4*c^4*e^2 - 6144*a^6*b^2*c^5*e^2 - 4096*a^6*b*c^6*d*e - 240*a^2*b^9*c^2*d*e +
1280*a^3*b^7*c^3*d*e - 3840*a^4*b^5*c^4*d*e + 6144*a^5*b^3*c^5*d*e + 24*a*b^11*c*d*e)))^(1/2)*(32*b^7*c^2*e^3
- 384*a*b^5*c^3*e^3 - 2048*a^3*b*c^5*e^3 + 4096*a^3*c^6*d*e^2 - 64*b^6*c^3*d*e^2 + 1536*a^2*b^3*c^4*e^3 + 768*
a*b^4*c^4*d*e^2 - 3072*a^2*b^2*c^5*d*e^2))/(4*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)))*((9*(e^5*(-(4*a*c - b^2)^9)^(1/
2) - b^9*e^5 + 768*a^4*b*c^4*e^5 - 1536*a^4*c^5...

________________________________________________________________________________________