\(\int \frac {(d+e x)^{3/2}}{(a+b x+c x^2)^2} \, dx\) [2297]

Optimal result

Integrand size = 22, antiderivative size = 363 \[ \int \frac {(d+e x)^{3/2}}{\left (a+b x+c x^2\right )^2} \, dx=-\frac {\sqrt {d+e x} (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {\left (8 c^2 d^2+b \left (b-\sqrt {b^2-4 a c}\right ) e^2-2 c e \left (4 b d-\sqrt {b^2-4 a c} d-2 a e\right )\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {2} \sqrt {c} \left (b^2-4 a c\right )^{3/2} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}-\frac {\left (8 c^2 d^2+b \left (b+\sqrt {b^2-4 a c}\right ) e^2-2 c e \left (4 b d+\sqrt {b^2-4 a c} d-2 a e\right )\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {2} \sqrt {c} \left (b^2-4 a c\right )^{3/2} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \]

[Out]

-(b*d-2*a*e+(-b*e+2*c*d)*x)*(e*x+d)^(1/2)/(-4*a*c+b^2)/(c*x^2+b*x+a)+1/2*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))*(8*c^2*d^2+b*e^2*(b-(-4*a*c+b^2)^(1/2))-2*c*e*(4*b*d-2*a*e-d*(-4*a*c+
b^2)^(1/2)))/(-4*a*c+b^2)^(3/2)*2^(1/2)/c^(1/2)/(2*c*d-e*(b-(-4*a*c+b^2)^(1/2)))^(1/2)-1/2*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))*(8*c^2*d^2+b*e^2*(b+(-4*a*c+b^2)^(1/2))-2*c*e*(4*b*
d-2*a*e+d*(-4*a*c+b^2)^(1/2)))/(-4*a*c+b^2)^(3/2)*2^(1/2)/c^(1/2)/(2*c*d-e*(b+(-4*a*c+b^2)^(1/2)))^(1/2)

Rubi [A] (verified)

Time = 0.81 (sec) , antiderivative size = 363, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {752, 840, 1180, 214} \[ \int \frac {(d+e x)^{3/2}}{\left (a+b x+c x^2\right )^2} \, dx=\frac {\left (-2 c e \left (-d \sqrt {b^2-4 a c}-2 a e+4 b d\right )+b e^2 \left (b-\sqrt {b^2-4 a c}\right )+8 c^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{\sqrt {2} \sqrt {c} \left (b^2-4 a c\right )^{3/2} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}-\frac {\left (-2 c e \left (d \sqrt {b^2-4 a c}-2 a e+4 b d\right )+b e^2 \left (\sqrt {b^2-4 a c}+b\right )+8 c^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{\sqrt {2} \sqrt {c} \left (b^2-4 a c\right )^{3/2} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}-\frac {\sqrt {d+e x} (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )} \]

[In]

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

[Out]

-((Sqrt[d + e*x]*(b*d - 2*a*e + (2*c*d - b*e)*x))/((b^2 - 4*a*c)*(a + b*x + c*x^2))) + ((8*c^2*d^2 + b*(b - Sq
rt[b^2 - 4*a*c])*e^2 - 2*c*e*(4*b*d - Sqrt[b^2 - 4*a*c]*d - 2*a*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sq
rt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]])/(Sqrt[2]*Sqrt[c]*(b^2 - 4*a*c)^(3/2)*Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*
c])*e]) - ((8*c^2*d^2 + b*(b + Sqrt[b^2 - 4*a*c])*e^2 - 2*c*e*(4*b*d + Sqrt[b^2 - 4*a*c]*d - 2*a*e))*ArcTanh[(
Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/(Sqrt[2]*Sqrt[c]*(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 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 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 {\sqrt {d+e x} (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {\int \frac {\frac {1}{2} \left (4 c d^2-3 b d e+2 a e^2\right )+\frac {1}{2} e (2 c d-b e) x}{\sqrt {d+e x} \left (a+b x+c x^2\right )} \, dx}{-b^2+4 a c} \\ & = -\frac {\sqrt {d+e x} (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {2 \text {Subst}\left (\int \frac {-\frac {1}{2} d e (2 c d-b e)+\frac {1}{2} e \left (4 c d^2-3 b d e+2 a e^2\right )+\frac {1}{2} e (2 c d-b 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 )}{b^2-4 a c} \\ & = -\frac {\sqrt {d+e x} (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {\left (8 c^2 d^2+b \left (b-\sqrt {b^2-4 a c}\right ) e^2-2 c e \left (4 b d-\sqrt {b^2-4 a c} d-2 a 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 )}{2 \left (b^2-4 a c\right )^{3/2}}+\frac {\left (8 c^2 d^2+b \left (b+\sqrt {b^2-4 a c}\right ) e^2-2 c e \left (4 b d+\sqrt {b^2-4 a c} d-2 a 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 )}{2 \left (b^2-4 a c\right )^{3/2}} \\ & = -\frac {\sqrt {d+e x} (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {\left (8 c^2 d^2+b \left (b-\sqrt {b^2-4 a c}\right ) e^2-2 c e \left (4 b d-\sqrt {b^2-4 a c} d-2 a 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 )}{\sqrt {2} \sqrt {c} \left (b^2-4 a c\right )^{3/2} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}-\frac {\left (8 c^2 d^2+b \left (b+\sqrt {b^2-4 a c}\right ) e^2-2 c e \left (4 b d+\sqrt {b^2-4 a c} d-2 a 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 )}{\sqrt {2} \sqrt {c} \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 [C] (verified)

Result contains complex when optimal does not.

Time = 3.39 (sec) , antiderivative size = 410, normalized size of antiderivative = 1.13 \[ \int \frac {(d+e x)^{3/2}}{\left (a+b x+c x^2\right )^2} \, dx=\frac {\frac {2 \sqrt {d+e x} (-b d+2 a e-2 c d x+b e x)}{a+x (b+c x)}+\frac {\sqrt {2} \left (8 i c^2 d^2+b \left (i b+\sqrt {-b^2+4 a c}\right ) e^2+2 i c e \left (-4 b d+i \sqrt {-b^2+4 a c} d+2 a e\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e-i \sqrt {-b^2+4 a c} e}}\right )}{\sqrt {c} \sqrt {-b^2+4 a c} \sqrt {-2 c d+\left (b-i \sqrt {-b^2+4 a c}\right ) e}}+\frac {\sqrt {2} \left (-8 i c^2 d^2+b \left (-i b+\sqrt {-b^2+4 a c}\right ) e^2-2 c e \left (-4 i b d+\sqrt {-b^2+4 a c} d+2 i a e\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e+i \sqrt {-b^2+4 a c} e}}\right )}{\sqrt {c} \sqrt {-b^2+4 a c} \sqrt {-2 c d+\left (b+i \sqrt {-b^2+4 a c}\right ) e}}}{2 \left (b^2-4 a c\right )} \]

[In]

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

[Out]

((2*Sqrt[d + e*x]*(-(b*d) + 2*a*e - 2*c*d*x + b*e*x))/(a + x*(b + c*x)) + (Sqrt[2]*((8*I)*c^2*d^2 + b*(I*b + S
qrt[-b^2 + 4*a*c])*e^2 + (2*I)*c*e*(-4*b*d + I*Sqrt[-b^2 + 4*a*c]*d + 2*a*e))*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[d +
e*x])/Sqrt[-2*c*d + b*e - I*Sqrt[-b^2 + 4*a*c]*e]])/(Sqrt[c]*Sqrt[-b^2 + 4*a*c]*Sqrt[-2*c*d + (b - I*Sqrt[-b^
2 + 4*a*c])*e]) + (Sqrt[2]*((-8*I)*c^2*d^2 + b*((-I)*b + Sqrt[-b^2 + 4*a*c])*e^2 - 2*c*e*((-4*I)*b*d + Sqrt[-b
^2 + 4*a*c]*d + (2*I)*a*e))*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[-2*c*d + b*e + I*Sqrt[-b^2 + 4*a*c]*e]
])/(Sqrt[c]*Sqrt[-b^2 + 4*a*c]*Sqrt[-2*c*d + (b + I*Sqrt[-b^2 + 4*a*c])*e]))/(2*(b^2 - 4*a*c))

Maple [A] (verified)

Time = 0.54 (sec) , antiderivative size = 454, normalized size of antiderivative = 1.25

method	result	size
pseudoelliptic	\(-\frac {\sqrt {2}\, \left (c \,x^{2}+b x +a \right ) e \sqrt {\left (b e -2 c d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}\right ) c}\, \left (\left (-\frac {b e}{4}+\frac {c d}{2}\right ) \sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}+2 c^{2} d^{2}+\left (a \,e^{2}-2 b d e \right ) c +\frac {b^{2} e^{2}}{4}\right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}\right ) c}}\right )+\sqrt {\left (-b e +2 c d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}\right ) c}\, \left (\sqrt {2}\, \left (c \,x^{2}+b x +a \right ) \left (\left (\frac {b e}{4}-\frac {c d}{2}\right ) \sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}+2 c^{2} d^{2}+\left (a \,e^{2}-2 b d e \right ) c +\frac {b^{2} e^{2}}{4}\right ) e \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}\right ) c}}\right )+\left (-c d x +\left (\frac {b x}{2}+a \right ) e -\frac {b d}{2}\right ) \sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}\, \sqrt {\left (b e -2 c d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}\right ) c}\, \sqrt {e x +d}\right )}{2 \sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}\, \sqrt {\left (-b e +2 c d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}\right ) c}\, \sqrt {\left (b e -2 c d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}\right ) c}\, \left (a c -\frac {b^{2}}{4}\right ) \left (c \,x^{2}+b x +a \right )}\)	\(454\)
derivativedivides	\(2 e^{3} \left (\frac {-\frac {\left (b e -2 c d \right ) \left (e x +d \right )^{\frac {3}{2}}}{2 e^{2} \left (4 a c -b^{2}\right )}-\frac {\left (a \,e^{2}-b d e +c \,d^{2}\right ) \sqrt {e x +d}}{e^{2} \left (4 a c -b^{2}\right )}}{c \left (e x +d \right )^{2}+b e \left (e x +d \right )-2 c d \left (e x +d \right )+a \,e^{2}-b d e +c \,d^{2}}+\frac {2 c \left (\frac {\left (-4 a c \,e^{2}-b^{2} e^{2}+8 b c d e -8 c^{2} d^{2}-\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b e +2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c d \right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}-\frac {\left (4 a c \,e^{2}+b^{2} e^{2}-8 b c d e +8 c^{2} d^{2}-\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b e +2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c d \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{e^{2} \left (4 a c -b^{2}\right )}\right )\)	\(481\)
default	\(2 e^{3} \left (\frac {-\frac {\left (b e -2 c d \right ) \left (e x +d \right )^{\frac {3}{2}}}{2 e^{2} \left (4 a c -b^{2}\right )}-\frac {\left (a \,e^{2}-b d e +c \,d^{2}\right ) \sqrt {e x +d}}{e^{2} \left (4 a c -b^{2}\right )}}{c \left (e x +d \right )^{2}+b e \left (e x +d \right )-2 c d \left (e x +d \right )+a \,e^{2}-b d e +c \,d^{2}}+\frac {2 c \left (\frac {\left (-4 a c \,e^{2}-b^{2} e^{2}+8 b c d e -8 c^{2} d^{2}-\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b e +2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c d \right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}-\frac {\left (4 a c \,e^{2}+b^{2} e^{2}-8 b c d e +8 c^{2} d^{2}-\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b e +2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c d \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{e^{2} \left (4 a c -b^{2}\right )}\right )\)	\(481\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2396 vs. \(2 (314) = 628\).

Time = 0.46 (sec) , antiderivative size = 2396, normalized size of antiderivative = 6.60 \[ \int \frac {(d+e x)^{3/2}}{\left (a+b x+c x^2\right )^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/2*(sqrt(1/2)*(a*b^2 - 4*a^2*c + (b^2*c - 4*a*c^2)*x^2 + (b^3 - 4*a*b*c)*x)*sqrt((32*c^3*d^3 - 48*b*c^2*d^2*e
+ 6*(3*b^2*c + 4*a*c^2)*d*e^2 - (b^3 + 12*a*b*c)*e^3 + (b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4)*s
qrt(e^6/(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)))/(b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a
^3*c^4))*log(sqrt(1/2)*((b^4 - 8*a*b^2*c + 16*a^2*c^2)*e^4 + 2*sqrt(e^6/(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c
^4 - 64*a^3*c^5))*(2*(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d - (b^7*c - 12*a*b^5*c^2 + 48*a^2
*b^3*c^3 - 64*a^3*b*c^4)*e))*sqrt((32*c^3*d^3 - 48*b*c^2*d^2*e + 6*(3*b^2*c + 4*a*c^2)*d*e^2 - (b^3 + 12*a*b*c
)*e^3 + (b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4)*sqrt(e^6/(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4
- 64*a^3*c^5)))/(b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4)) + (16*c^2*d^2*e^3 - 16*b*c*d*e^4 + (3*b
^2 + 4*a*c)*e^5)*sqrt(e*x + d)) - sqrt(1/2)*(a*b^2 - 4*a^2*c + (b^2*c - 4*a*c^2)*x^2 + (b^3 - 4*a*b*c)*x)*sqrt
((32*c^3*d^3 - 48*b*c^2*d^2*e + 6*(3*b^2*c + 4*a*c^2)*d*e^2 - (b^3 + 12*a*b*c)*e^3 + (b^6*c - 12*a*b^4*c^2 + 4
8*a^2*b^2*c^3 - 64*a^3*c^4)*sqrt(e^6/(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)))/(b^6*c - 12*a*b^
4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4))*log(-sqrt(1/2)*((b^4 - 8*a*b^2*c + 16*a^2*c^2)*e^4 + 2*sqrt(e^6/(b^6*c^2
- 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5))*(2*(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d -
(b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*e))*sqrt((32*c^3*d^3 - 48*b*c^2*d^2*e + 6*(3*b^2*c + 4*
a*c^2)*d*e^2 - (b^3 + 12*a*b*c)*e^3 + (b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4)*sqrt(e^6/(b^6*c^2 -
12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)))/(b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4)) + (16*c^2
*d^2*e^3 - 16*b*c*d*e^4 + (3*b^2 + 4*a*c)*e^5)*sqrt(e*x + d)) + sqrt(1/2)*(a*b^2 - 4*a^2*c + (b^2*c - 4*a*c^2)
*x^2 + (b^3 - 4*a*b*c)*x)*sqrt((32*c^3*d^3 - 48*b*c^2*d^2*e + 6*(3*b^2*c + 4*a*c^2)*d*e^2 - (b^3 + 12*a*b*c)*e
^3 - (b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4)*sqrt(e^6/(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 -
64*a^3*c^5)))/(b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4))*log(sqrt(1/2)*((b^4 - 8*a*b^2*c + 16*a^2*c
^2)*e^4 - 2*sqrt(e^6/(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5))*(2*(b^6*c^2 - 12*a*b^4*c^3 + 48*a
^2*b^2*c^4 - 64*a^3*c^5)*d - (b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*e))*sqrt((32*c^3*d^3 - 48*
b*c^2*d^2*e + 6*(3*b^2*c + 4*a*c^2)*d*e^2 - (b^3 + 12*a*b*c)*e^3 - (b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64
*a^3*c^4)*sqrt(e^6/(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)))/(b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2
*c^3 - 64*a^3*c^4)) + (16*c^2*d^2*e^3 - 16*b*c*d*e^4 + (3*b^2 + 4*a*c)*e^5)*sqrt(e*x + d)) - sqrt(1/2)*(a*b^2
- 4*a^2*c + (b^2*c - 4*a*c^2)*x^2 + (b^3 - 4*a*b*c)*x)*sqrt((32*c^3*d^3 - 48*b*c^2*d^2*e + 6*(3*b^2*c + 4*a*c^
2)*d*e^2 - (b^3 + 12*a*b*c)*e^3 - (b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4)*sqrt(e^6/(b^6*c^2 - 12*
a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)))/(b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4))*log(-sqrt(1/2
)*((b^4 - 8*a*b^2*c + 16*a^2*c^2)*e^4 - 2*sqrt(e^6/(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5))*(2*
(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d - (b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c
^4)*e))*sqrt((32*c^3*d^3 - 48*b*c^2*d^2*e + 6*(3*b^2*c + 4*a*c^2)*d*e^2 - (b^3 + 12*a*b*c)*e^3 - (b^6*c - 12*a
*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4)*sqrt(e^6/(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)))/(b^6
*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4)) + (16*c^2*d^2*e^3 - 16*b*c*d*e^4 + (3*b^2 + 4*a*c)*e^5)*sqrt
(e*x + d)) - 2*(b*d - 2*a*e + (2*c*d - b*e)*x)*sqrt(e*x + d))/(a*b^2 - 4*a^2*c + (b^2*c - 4*a*c^2)*x^2 + (b^3
- 4*a*b*c)*x)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1193 vs. \(2 (314) = 628\).

Time = 0.54 (sec) , antiderivative size = 1193, normalized size of antiderivative = 3.29 \[ \int \frac {(d+e x)^{3/2}}{\left (a+b x+c x^2\right )^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-(2*(e*x + d)^(3/2)*c*d*e - 2*sqrt(e*x + d)*c*d^2*e - (e*x + d)^(3/2)*b*e^2 + 2*sqrt(e*x + d)*b*d*e^2 - 2*sqrt
(e*x + d)*a*e^3)/(((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)*(b^2 - 4*a*c)) - 1
/8*(sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e)*(b^2*e - 4*a*c*e)^2*(2*c*d*e - b*e^2) + 4*(sqrt(b^2 - 4*a
*c)*c^2*d^2*e - sqrt(b^2 - 4*a*c)*b*c*d*e^2 + sqrt(b^2 - 4*a*c)*a*c*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) - (16*(b^2*c^3 - 4*a*c^4)*d^3*e - 24*(b^3*c^2 - 4*a*b*c^3)*d^2*e^2 + 2*(5*b^4
*c - 16*a*b^2*c^2 - 16*a^2*c^3)*d*e^3 - (b^5 - 16*a^2*b*c^2)*e^4)*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^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^2 - 4*a*c^3)*sqrt(b^2 - 4*a*c)*d^2 - (b^3*c - 4*a*b*c^2)*sqr
t(b^2 - 4*a*c)*d*e + (a*b^2*c - 4*a^2*c^2)*sqrt(b^2 - 4*a*c)*e^2)*abs(b^2*e - 4*a*c*e)*abs(c)) + 1/8*(sqrt(-4*
c^2*d + 2*(b*c + sqrt(b^2 - 4*a*c)*c)*e)*(b^2*e - 4*a*c*e)^2*(2*c*d*e - b*e^2) - 4*(sqrt(b^2 - 4*a*c)*c^2*d^2*
e - sqrt(b^2 - 4*a*c)*b*c*d*e^2 + sqrt(b^2 - 4*a*c)*a*c*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) - (16*(b^2*c^3 - 4*a*c^4)*d^3*e - 24*(b^3*c^2 - 4*a*b*c^3)*d^2*e^2 + 2*(5*b^4*c - 16*a*b^
2*c^2 - 16*a^2*c^3)*d*e^3 - (b^5 - 16*a^2*b*c^2)*e^4)*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^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^2 - 4*a*c^3)*sqrt(b^2 - 4*a*c)*d^2 - (b^3*c - 4*a*b*c^2)*sqrt(b^2 - 4*a*
c)*d*e + (a*b^2*c - 4*a^2*c^2)*sqrt(b^2 - 4*a*c)*e^2)*abs(b^2*e - 4*a*c*e)*abs(c))

Mupad [B] (verification not implemented)

Time = 6.25 (sec) , antiderivative size = 5326, normalized size of antiderivative = 14.67 \[ \int \frac {(d+e x)^{3/2}}{\left (a+b x+c x^2\right )^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

log((c*e^3*(b*e - 2*c*d)*(16*c^3*d^4 + 3*a*b^2*e^4 + 4*a^2*c*e^4 - 3*b^3*d*e^3 + 20*a*c^2*d^2*e^2 + 19*b^2*c*d
^2*e^2 - 32*b*c^2*d^3*e - 20*a*b*c*d*e^3))/(4*a*c - b^2)^3 - (2^(1/2)*((2^(1/2)*(8*c^2*e^3*(a*e^2 + c*d^2 - b*
d*e) - 2*2^(1/2)*c^2*e^2*(4*a*c - b^2)*(b*e - 2*c*d)*(d + e*x)^(1/2)*(-(b^9*e^3 + e^3*(-(4*a*c - b^2)^9)^(1/2)
+ 2048*a^3*c^6*d^3 - 32*b^6*c^3*d^3 + 384*a*b^4*c^4*d^3 - 768*a^4*b*c^4*e^3 + 1536*a^4*c^5*d*e^2 + 48*b^7*c^2
*d^2*e - 1536*a^2*b^2*c^5*d^3 - 96*a^2*b^5*c^2*e^3 + 512*a^3*b^3*c^3*e^3 - 18*b^8*c*d*e^2 - 576*a*b^5*c^3*d^2*
e + 192*a*b^6*c^2*d*e^2 - 3072*a^3*b*c^5*d^2*e + 2304*a^2*b^3*c^4*d^2*e - 576*a^2*b^4*c^3*d*e^2)/(c*(4*a*c - b
^2)^6))^(1/2))*(-(b^9*e^3 + e^3*(-(4*a*c - b^2)^9)^(1/2) + 2048*a^3*c^6*d^3 - 32*b^6*c^3*d^3 + 384*a*b^4*c^4*d
^3 - 768*a^4*b*c^4*e^3 + 1536*a^4*c^5*d*e^2 + 48*b^7*c^2*d^2*e - 1536*a^2*b^2*c^5*d^3 - 96*a^2*b^5*c^2*e^3 + 5
12*a^3*b^3*c^3*e^3 - 18*b^8*c*d*e^2 - 576*a*b^5*c^3*d^2*e + 192*a*b^6*c^2*d*e^2 - 3072*a^3*b*c^5*d^2*e + 2304*
a^2*b^3*c^4*d^2*e - 576*a^2*b^4*c^3*d*e^2)/(c*(4*a*c - b^2)^6))^(1/2))/4 + (2*c*e^2*(d + e*x)^(1/2)*(b^4*e^4 +
32*c^4*d^4 + 8*a^2*c^2*e^4 + 24*a*c^3*d^2*e^2 + 42*b^2*c^2*d^2*e^2 + 2*a*b^2*c*e^4 - 64*b*c^3*d^3*e - 10*b^3*
c*d*e^3 - 24*a*b*c^2*d*e^3))/(4*a*c - b^2)^2)*(-(b^9*e^3 + e^3*(-(4*a*c - b^2)^9)^(1/2) + 2048*a^3*c^6*d^3 - 3
2*b^6*c^3*d^3 + 384*a*b^4*c^4*d^3 - 768*a^4*b*c^4*e^3 + 1536*a^4*c^5*d*e^2 + 48*b^7*c^2*d^2*e - 1536*a^2*b^2*c
^5*d^3 - 96*a^2*b^5*c^2*e^3 + 512*a^3*b^3*c^3*e^3 - 18*b^8*c*d*e^2 - 576*a*b^5*c^3*d^2*e + 192*a*b^6*c^2*d*e^2
- 3072*a^3*b*c^5*d^2*e + 2304*a^2*b^3*c^4*d^2*e - 576*a^2*b^4*c^3*d*e^2)/(c*(4*a*c - b^2)^6))^(1/2))/4)*(-(b^
9*e^3 + e^3*(-(4*a*c - b^2)^9)^(1/2) + 2048*a^3*c^6*d^3 - 32*b^6*c^3*d^3 + 384*a*b^4*c^4*d^3 - 768*a^4*b*c^4*e
^3 + 1536*a^4*c^5*d*e^2 + 48*b^7*c^2*d^2*e - 1536*a^2*b^2*c^5*d^3 - 96*a^2*b^5*c^2*e^3 + 512*a^3*b^3*c^3*e^3 -
18*b^8*c*d*e^2 - 576*a*b^5*c^3*d^2*e + 192*a*b^6*c^2*d*e^2 - 3072*a^3*b*c^5*d^2*e + 2304*a^2*b^3*c^4*d^2*e -
576*a^2*b^4*c^3*d*e^2)/(8*(b^12*c + 4096*a^6*c^7 - 24*a*b^10*c^2 + 240*a^2*b^8*c^3 - 1280*a^3*b^6*c^4 + 3840*a
^4*b^4*c^5 - 6144*a^5*b^2*c^6)))^(1/2) + log((c*e^3*(b*e - 2*c*d)*(16*c^3*d^4 + 3*a*b^2*e^4 + 4*a^2*c*e^4 - 3*
b^3*d*e^3 + 20*a*c^2*d^2*e^2 + 19*b^2*c*d^2*e^2 - 32*b*c^2*d^3*e - 20*a*b*c*d*e^3))/(4*a*c - b^2)^3 - (2^(1/2)
*((2^(1/2)*(8*c^2*e^3*(a*e^2 + c*d^2 - b*d*e) - 2*2^(1/2)*c^2*e^2*(4*a*c - b^2)*(b*e - 2*c*d)*(d + e*x)^(1/2)*
((e^3*(-(4*a*c - b^2)^9)^(1/2) - b^9*e^3 - 2048*a^3*c^6*d^3 + 32*b^6*c^3*d^3 - 384*a*b^4*c^4*d^3 + 768*a^4*b*c
^4*e^3 - 1536*a^4*c^5*d*e^2 - 48*b^7*c^2*d^2*e + 1536*a^2*b^2*c^5*d^3 + 96*a^2*b^5*c^2*e^3 - 512*a^3*b^3*c^3*e
^3 + 18*b^8*c*d*e^2 + 576*a*b^5*c^3*d^2*e - 192*a*b^6*c^2*d*e^2 + 3072*a^3*b*c^5*d^2*e - 2304*a^2*b^3*c^4*d^2*
e + 576*a^2*b^4*c^3*d*e^2)/(c*(4*a*c - b^2)^6))^(1/2))*((e^3*(-(4*a*c - b^2)^9)^(1/2) - b^9*e^3 - 2048*a^3*c^6
*d^3 + 32*b^6*c^3*d^3 - 384*a*b^4*c^4*d^3 + 768*a^4*b*c^4*e^3 - 1536*a^4*c^5*d*e^2 - 48*b^7*c^2*d^2*e + 1536*a
^2*b^2*c^5*d^3 + 96*a^2*b^5*c^2*e^3 - 512*a^3*b^3*c^3*e^3 + 18*b^8*c*d*e^2 + 576*a*b^5*c^3*d^2*e - 192*a*b^6*c
^2*d*e^2 + 3072*a^3*b*c^5*d^2*e - 2304*a^2*b^3*c^4*d^2*e + 576*a^2*b^4*c^3*d*e^2)/(c*(4*a*c - b^2)^6))^(1/2))/
4 + (2*c*e^2*(d + e*x)^(1/2)*(b^4*e^4 + 32*c^4*d^4 + 8*a^2*c^2*e^4 + 24*a*c^3*d^2*e^2 + 42*b^2*c^2*d^2*e^2 + 2
*a*b^2*c*e^4 - 64*b*c^3*d^3*e - 10*b^3*c*d*e^3 - 24*a*b*c^2*d*e^3))/(4*a*c - b^2)^2)*((e^3*(-(4*a*c - b^2)^9)^
(1/2) - b^9*e^3 - 2048*a^3*c^6*d^3 + 32*b^6*c^3*d^3 - 384*a*b^4*c^4*d^3 + 768*a^4*b*c^4*e^3 - 1536*a^4*c^5*d*e
^2 - 48*b^7*c^2*d^2*e + 1536*a^2*b^2*c^5*d^3 + 96*a^2*b^5*c^2*e^3 - 512*a^3*b^3*c^3*e^3 + 18*b^8*c*d*e^2 + 576
*a*b^5*c^3*d^2*e - 192*a*b^6*c^2*d*e^2 + 3072*a^3*b*c^5*d^2*e - 2304*a^2*b^3*c^4*d^2*e + 576*a^2*b^4*c^3*d*e^2
)/(c*(4*a*c - b^2)^6))^(1/2))/4)*((e^3*(-(4*a*c - b^2)^9)^(1/2) - b^9*e^3 - 2048*a^3*c^6*d^3 + 32*b^6*c^3*d^3
- 384*a*b^4*c^4*d^3 + 768*a^4*b*c^4*e^3 - 1536*a^4*c^5*d*e^2 - 48*b^7*c^2*d^2*e + 1536*a^2*b^2*c^5*d^3 + 96*a^
2*b^5*c^2*e^3 - 512*a^3*b^3*c^3*e^3 + 18*b^8*c*d*e^2 + 576*a*b^5*c^3*d^2*e - 192*a*b^6*c^2*d*e^2 + 3072*a^3*b*
c^5*d^2*e - 2304*a^2*b^3*c^4*d^2*e + 576*a^2*b^4*c^3*d*e^2)/(8*(b^12*c + 4096*a^6*c^7 - 24*a*b^10*c^2 + 240*a^
2*b^8*c^3 - 1280*a^3*b^6*c^4 + 3840*a^4*b^4*c^5 - 6144*a^5*b^2*c^6)))^(1/2) - ((2*(d + e*x)^(1/2)*(a*e^3 - b*d
*e^2 + c*d^2*e))/(4*a*c - b^2) + (e*(b*e - 2*c*d)*(d + e*x)^(3/2))/(4*a*c - b^2))/((b*e - 2*c*d)*(d + e*x) + c
*(d + e*x)^2 + a*e^2 + c*d^2 - b*d*e) - log((c*e^3*(b*e - 2*c*d)*(16*c^3*d^4 + 3*a*b^2*e^4 + 4*a^2*c*e^4 - 3*b
^3*d*e^3 + 20*a*c^2*d^2*e^2 + 19*b^2*c*d^2*e^2 - 32*b*c^2*d^3*e - 20*a*b*c*d*e^3))/(4*a*c - b^2)^3 - ((8*c^2*e
^3*(a*e^2 + c*d^2 - b*d*e) + 8*c^2*e^2*(4*a*c - b^2)*(b*e - 2*c*d)*(d + e*x)^(1/2)*(((e^3*(-(4*a*c - b^2)^9)^(
1/2))/8 - (b^9*e^3)/8 - 256*a^3*c^6*d^3 + 4*b^6*c^3*d^3 - 48*a*b^4*c^4*d^3 + 96*a^4*b*c^4*e^3 - 192*a^4*c^5*d*
e^2 - 6*b^7*c^2*d^2*e + 192*a^2*b^2*c^5*d^3 + 12*a^2*b^5*c^2*e^3 - 64*a^3*b^3*c^3*e^3 + (9*b^8*c*d*e^2)/4 + 72
*a*b^5*c^3*d^2*e - 24*a*b^6*c^2*d*e^2 + 384*a^3*b*c^5*d^2*e - 288*a^2*b^3*c^4*d^2*e + 72*a^2*b^4*c^3*d*e^2)/(c
*(4*a*c - b^2)^6))^(1/2))*(((e^3*(-(4*a*c - b^2)^9)^(1/2))/8 - (b^9*e^3)/8 - 256*a^3*c^6*d^3 + 4*b^6*c^3*d^3 -
48*a*b^4*c^4*d^3 + 96*a^4*b*c^4*e^3 - 192*a^4*c^5*d*e^2 - 6*b^7*c^2*d^2*e + 192*a^2*b^2*c^5*d^3 + 12*a^2*b^5*
c^2*e^3 - 64*a^3*b^3*c^3*e^3 + (9*b^8*c*d*e^2)/4 + 72*a*b^5*c^3*d^2*e - 24*a*b^6*c^2*d*e^2 + 384*a^3*b*c^5*d^2
*e - 288*a^2*b^3*c^4*d^2*e + 72*a^2*b^4*c^3*d*e^2)/(c*(4*a*c - b^2)^6))^(1/2) - (2*c*e^2*(d + e*x)^(1/2)*(b^4*
e^4 + 32*c^4*d^4 + 8*a^2*c^2*e^4 + 24*a*c^3*d^2*e^2 + 42*b^2*c^2*d^2*e^2 + 2*a*b^2*c*e^4 - 64*b*c^3*d^3*e - 10
*b^3*c*d*e^3 - 24*a*b*c^2*d*e^3))/(4*a*c - b^2)^2)*(((e^3*(-(4*a*c - b^2)^9)^(1/2))/8 - (b^9*e^3)/8 - 256*a^3*
c^6*d^3 + 4*b^6*c^3*d^3 - 48*a*b^4*c^4*d^3 + 96*a^4*b*c^4*e^3 - 192*a^4*c^5*d*e^2 - 6*b^7*c^2*d^2*e + 192*a^2*
b^2*c^5*d^3 + 12*a^2*b^5*c^2*e^3 - 64*a^3*b^3*c^3*e^3 + (9*b^8*c*d*e^2)/4 + 72*a*b^5*c^3*d^2*e - 24*a*b^6*c^2*
d*e^2 + 384*a^3*b*c^5*d^2*e - 288*a^2*b^3*c^4*d^2*e + 72*a^2*b^4*c^3*d*e^2)/(c*(4*a*c - b^2)^6))^(1/2))*(((e^3
*(-(4*a*c - b^2)^9)^(1/2))/8 - (b^9*e^3)/8 - 256*a^3*c^6*d^3 + 4*b^6*c^3*d^3 - 48*a*b^4*c^4*d^3 + 96*a^4*b*c^4
*e^3 - 192*a^4*c^5*d*e^2 - 6*b^7*c^2*d^2*e + 192*a^2*b^2*c^5*d^3 + 12*a^2*b^5*c^2*e^3 - 64*a^3*b^3*c^3*e^3 + (
9*b^8*c*d*e^2)/4 + 72*a*b^5*c^3*d^2*e - 24*a*b^6*c^2*d*e^2 + 384*a^3*b*c^5*d^2*e - 288*a^2*b^3*c^4*d^2*e + 72*
a^2*b^4*c^3*d*e^2)/(b^12*c + 4096*a^6*c^7 - 24*a*b^10*c^2 + 240*a^2*b^8*c^3 - 1280*a^3*b^6*c^4 + 3840*a^4*b^4*
c^5 - 6144*a^5*b^2*c^6))^(1/2) - log((c*e^3*(b*e - 2*c*d)*(16*c^3*d^4 + 3*a*b^2*e^4 + 4*a^2*c*e^4 - 3*b^3*d*e^
3 + 20*a*c^2*d^2*e^2 + 19*b^2*c*d^2*e^2 - 32*b*c^2*d^3*e - 20*a*b*c*d*e^3))/(4*a*c - b^2)^3 - ((8*c^2*e^3*(a*e
^2 + c*d^2 - b*d*e) + 8*c^2*e^2*(4*a*c - b^2)*(b*e - 2*c*d)*(d + e*x)^(1/2)*(-((b^9*e^3)/8 + (e^3*(-(4*a*c - b
^2)^9)^(1/2))/8 + 256*a^3*c^6*d^3 - 4*b^6*c^3*d^3 + 48*a*b^4*c^4*d^3 - 96*a^4*b*c^4*e^3 + 192*a^4*c^5*d*e^2 +
6*b^7*c^2*d^2*e - 192*a^2*b^2*c^5*d^3 - 12*a^2*b^5*c^2*e^3 + 64*a^3*b^3*c^3*e^3 - (9*b^8*c*d*e^2)/4 - 72*a*b^5
*c^3*d^2*e + 24*a*b^6*c^2*d*e^2 - 384*a^3*b*c^5*d^2*e + 288*a^2*b^3*c^4*d^2*e - 72*a^2*b^4*c^3*d*e^2)/(c*(4*a*
c - b^2)^6))^(1/2))*(-((b^9*e^3)/8 + (e^3*(-(4*a*c - b^2)^9)^(1/2))/8 + 256*a^3*c^6*d^3 - 4*b^6*c^3*d^3 + 48*a
*b^4*c^4*d^3 - 96*a^4*b*c^4*e^3 + 192*a^4*c^5*d*e^2 + 6*b^7*c^2*d^2*e - 192*a^2*b^2*c^5*d^3 - 12*a^2*b^5*c^2*e
^3 + 64*a^3*b^3*c^3*e^3 - (9*b^8*c*d*e^2)/4 - 72*a*b^5*c^3*d^2*e + 24*a*b^6*c^2*d*e^2 - 384*a^3*b*c^5*d^2*e +
288*a^2*b^3*c^4*d^2*e - 72*a^2*b^4*c^3*d*e^2)/(c*(4*a*c - b^2)^6))^(1/2) - (2*c*e^2*(d + e*x)^(1/2)*(b^4*e^4 +
32*c^4*d^4 + 8*a^2*c^2*e^4 + 24*a*c^3*d^2*e^2 + 42*b^2*c^2*d^2*e^2 + 2*a*b^2*c*e^4 - 64*b*c^3*d^3*e - 10*b^3*
c*d*e^3 - 24*a*b*c^2*d*e^3))/(4*a*c - b^2)^2)*(-((b^9*e^3)/8 + (e^3*(-(4*a*c - b^2)^9)^(1/2))/8 + 256*a^3*c^6*
d^3 - 4*b^6*c^3*d^3 + 48*a*b^4*c^4*d^3 - 96*a^4*b*c^4*e^3 + 192*a^4*c^5*d*e^2 + 6*b^7*c^2*d^2*e - 192*a^2*b^2*
c^5*d^3 - 12*a^2*b^5*c^2*e^3 + 64*a^3*b^3*c^3*e^3 - (9*b^8*c*d*e^2)/4 - 72*a*b^5*c^3*d^2*e + 24*a*b^6*c^2*d*e^
2 - 384*a^3*b*c^5*d^2*e + 288*a^2*b^3*c^4*d^2*e - 72*a^2*b^4*c^3*d*e^2)/(c*(4*a*c - b^2)^6))^(1/2))*(-((b^9*e^
3)/8 + (e^3*(-(4*a*c - b^2)^9)^(1/2))/8 + 256*a^3*c^6*d^3 - 4*b^6*c^3*d^3 + 48*a*b^4*c^4*d^3 - 96*a^4*b*c^4*e^
3 + 192*a^4*c^5*d*e^2 + 6*b^7*c^2*d^2*e - 192*a^2*b^2*c^5*d^3 - 12*a^2*b^5*c^2*e^3 + 64*a^3*b^3*c^3*e^3 - (9*b
^8*c*d*e^2)/4 - 72*a*b^5*c^3*d^2*e + 24*a*b^6*c^2*d*e^2 - 384*a^3*b*c^5*d^2*e + 288*a^2*b^3*c^4*d^2*e - 72*a^2
*b^4*c^3*d*e^2)/(b^12*c + 4096*a^6*c^7 - 24*a*b^10*c^2 + 240*a^2*b^8*c^3 - 1280*a^3*b^6*c^4 + 3840*a^4*b^4*c^5
- 6144*a^5*b^2*c^6))^(1/2)