3.17.21 \(\int \frac {b+2 c x}{\sqrt {d+e x} (a+b x+c x^2)^2} \, dx\) [1621]

Optimal. Leaf size=364 \[ -\frac {\sqrt {d+e x} \left (\left (b^2-4 a c\right ) (c d-b e)-c \left (b^2-4 a c\right ) e x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )}+\frac {\sqrt {c} e \left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {2} \sqrt {b^2-4 a c} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e} \left (c d^2-b d e+a e^2\right )}-\frac {\sqrt {c} e \left (2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {2} \sqrt {b^2-4 a c} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} \left (c d^2-b d e+a e^2\right )} \]

[Out]

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

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Rubi [A]
time = 0.51, antiderivative size = 364, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {836, 840, 1180, 214} \begin {gather*} -\frac {\sqrt {d+e x} \left (\left (b^2-4 a c\right ) (c d-b e)-c e x \left (b^2-4 a c\right )\right )}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )}+\frac {\sqrt {c} e \left (2 c d-e \left (\sqrt {b^2-4 a c}+b\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 )}{\sqrt {2} \sqrt {b^2-4 a c} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )} \left (a e^2-b d e+c d^2\right )}-\frac {\sqrt {c} e \left (2 c d-e \left (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 (\sqrt {b^2-4 a c}+b\right )}}\right )}{\sqrt {2} \sqrt {b^2-4 a c} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )} \left (a e^2-b d e+c d^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 836

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[(d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2*a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x)
*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))), x] + Dist[1/((p + 1)*(b^2 - 4*a*
c)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2
*(p + m + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d*m + b*e*m) - b*d*(3*c*d -
b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, b,
c, d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && (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*} \int \frac {b+2 c x}{\sqrt {d+e x} \left (a+b x+c x^2\right )^2} \, dx &=-\frac {\sqrt {d+e x} \left (\left (b^2-4 a c\right ) (c d-b e)-c \left (b^2-4 a c\right ) e x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )}-\frac {\int \frac {\frac {1}{2} \left (b^2-4 a c\right ) e (c d-b e)-\frac {1}{2} c \left (b^2-4 a c\right ) e^2 x}{\sqrt {d+e x} \left (a+b x+c x^2\right )} \, dx}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )}\\ &=-\frac {\sqrt {d+e x} \left (\left (b^2-4 a c\right ) (c d-b e)-c \left (b^2-4 a c\right ) e x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )}-\frac {2 \text {Subst}\left (\int \frac {\frac {1}{2} c \left (b^2-4 a c\right ) d e^2+\frac {1}{2} \left (b^2-4 a c\right ) e^2 (c d-b e)-\frac {1}{2} c \left (b^2-4 a c\right ) e^2 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 ) \left (c d^2-b d e+a e^2\right )}\\ &=-\frac {\sqrt {d+e x} \left (\left (b^2-4 a c\right ) (c d-b e)-c \left (b^2-4 a c\right ) e x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )}+\frac {\left (c e \left (2 c d-\left (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 )}{2 \sqrt {b^2-4 a c} \left (c d^2-b d e+a e^2\right )}-\frac {\left (c e \left (2 c d-\left (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 )}{2 \sqrt {b^2-4 a c} \left (c d^2-b d e+a e^2\right )}\\ &=-\frac {\sqrt {d+e x} \left (\left (b^2-4 a c\right ) (c d-b e)-c \left (b^2-4 a c\right ) e x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )}+\frac {\sqrt {c} e \left (2 c d-\left (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} \sqrt {b^2-4 a c} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e} \left (c d^2-b d e+a e^2\right )}-\frac {\sqrt {c} e \left (2 c d-\left (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} \sqrt {b^2-4 a c} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} \left (c d^2-b d e+a e^2\right )}\\ \end {align*}

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Mathematica [A]
time = 1.51, size = 295, normalized size = 0.81 \begin {gather*} \frac {\frac {2 \sqrt {d+e x} (-c d+b e+c e x)}{a+x (b+c x)}+\frac {\sqrt {2} \sqrt {c} e \left (-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e\right ) \tan ^{-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 {b^2-4 a c} \sqrt {-2 c d+\left (b-\sqrt {b^2-4 a c}\right ) e}}+\frac {\sqrt {2} \sqrt {c} e \left (2 c d+\left (-b+\sqrt {b^2-4 a c}\right ) e\right ) \tan ^{-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 {b^2-4 a c} \sqrt {-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e}}}{2 \left (c d^2+e (-b d+a e)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 1.09, size = 433, normalized size = 1.19

method	result	size
derivativedivides	\(32 e^{2} c^{2} \left (\frac {\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \left (\frac {\sqrt {e x +d}}{2 \left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) \left (2 c \left (e x +d \right )+b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right )}+\frac {\sqrt {2}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{4 \left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{4 e^{2} \left (4 a c -b^{2}\right )}-\frac {\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \left (\frac {\sqrt {e x +d}}{2 \left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) \left (-2 c \left (e x +d \right )-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right )}+\frac {\sqrt {2}\, \arctanh \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{4 \left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{4 e^{2} \left (4 a c -b^{2}\right )}\right )\)	\(433\)
default	\(32 e^{2} c^{2} \left (\frac {\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \left (\frac {\sqrt {e x +d}}{2 \left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) \left (2 c \left (e x +d \right )+b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right )}+\frac {\sqrt {2}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{4 \left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{4 e^{2} \left (4 a c -b^{2}\right )}-\frac {\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \left (\frac {\sqrt {e x +d}}{2 \left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) \left (-2 c \left (e x +d \right )-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right )}+\frac {\sqrt {2}\, \arctanh \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{4 \left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{4 e^{2} \left (4 a c -b^{2}\right )}\right )\)	\(433\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*c*x+b)/(c*x^2+b*x+a)^2/(e*x+d)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

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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)/(c*x^2+b*x+a)^2/(e*x+d)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 11145 vs. \(2 (331) = 662\).
time = 2.38, size = 11145, normalized size = 30.62 \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)/(c*x^2+b*x+a)^2/(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

-1/2*(sqrt(1/2)*(c^2*d^2*x^2 + b*c*d^2*x + a*c*d^2 + (a*c*x^2 + a*b*x + a^2)*e^2 - (b*c*d*x^2 + b^2*d*x + a*b*
d)*e)*sqrt((2*c^3*d^3*e^2 - 3*b*c^2*d^2*e^3 + 3*(b^2*c - 2*a*c^2)*d*e^4 - (b^3 - 3*a*b*c)*e^5 + ((b^2*c^3 - 4*
a*c^4)*d^6 - 3*(b^3*c^2 - 4*a*b*c^3)*d^5*e + 3*(b^4*c - 3*a*b^2*c^2 - 4*a^2*c^3)*d^4*e^2 - (b^5 + 2*a*b^3*c -
24*a^2*b*c^2)*d^3*e^3 + 3*(a*b^4 - 3*a^2*b^2*c - 4*a^3*c^2)*d^2*e^4 - 3*(a^2*b^3 - 4*a^3*b*c)*d*e^5 + (a^3*b^2
- 4*a^4*c)*e^6)*sqrt((9*c^4*d^4*e^6 - 18*b*c^3*d^3*e^7 + 3*(5*b^2*c^2 - 2*a*c^3)*d^2*e^8 - 6*(b^3*c - a*b*c^2
)*d*e^9 + (b^4 - 2*a*b^2*c + a^2*c^2)*e^10)/((b^2*c^6 - 4*a*c^7)*d^12 - 6*(b^3*c^5 - 4*a*b*c^6)*d^11*e + 3*(5*
b^4*c^4 - 18*a*b^2*c^5 - 8*a^2*c^6)*d^10*e^2 - 10*(2*b^5*c^3 - 5*a*b^3*c^4 - 12*a^2*b*c^5)*d^9*e^3 + 15*(b^6*c
^2 - 15*a^2*b^2*c^4 - 4*a^3*c^5)*d^8*e^4 - 6*(b^7*c + 6*a*b^5*c^2 - 30*a^2*b^3*c^3 - 40*a^3*b*c^4)*d^7*e^5 + (
b^8 + 26*a*b^6*c - 30*a^2*b^4*c^2 - 340*a^3*b^2*c^3 - 80*a^4*c^4)*d^6*e^6 - 6*(a*b^7 + 6*a^2*b^5*c - 30*a^3*b^
3*c^2 - 40*a^4*b*c^3)*d^5*e^7 + 15*(a^2*b^6 - 15*a^4*b^2*c^2 - 4*a^5*c^3)*d^4*e^8 - 10*(2*a^3*b^5 - 5*a^4*b^3*
c - 12*a^5*b*c^2)*d^3*e^9 + 3*(5*a^4*b^4 - 18*a^5*b^2*c - 8*a^6*c^2)*d^2*e^10 - 6*(a^5*b^3 - 4*a^6*b*c)*d*e^11
+ (a^6*b^2 - 4*a^7*c)*e^12)))/((b^2*c^3 - 4*a*c^4)*d^6 - 3*(b^3*c^2 - 4*a*b*c^3)*d^5*e + 3*(b^4*c - 3*a*b^2*c
^2 - 4*a^2*c^3)*d^4*e^2 - (b^5 + 2*a*b^3*c - 24*a^2*b*c^2)*d^3*e^3 + 3*(a*b^4 - 3*a^2*b^2*c - 4*a^3*c^2)*d^2*e
^4 - 3*(a^2*b^3 - 4*a^3*b*c)*d*e^5 + (a^3*b^2 - 4*a^4*c)*e^6))*log(sqrt(1/2)*(6*(b^2*c^3 - 4*a*c^4)*d^3*e^4 -
9*(b^3*c^2 - 4*a*b*c^3)*d^2*e^5 + (5*b^4*c - 22*a*b^2*c^2 + 8*a^2*c^3)*d*e^6 - (b^5 - 5*a*b^3*c + 4*a^2*b*c^2)
*e^7 - (2*(b^2*c^5 - 4*a*c^6)*d^8 - 8*(b^3*c^4 - 4*a*b*c^5)*d^7*e + (13*b^4*c^3 - 48*a*b^2*c^4 - 16*a^2*c^5)*d
^6*e^2 - (11*b^5*c^2 - 32*a*b^3*c^3 - 48*a^2*b*c^4)*d^5*e^3 + 5*(b^6*c - a*b^4*c^2 - 12*a^2*b^2*c^3)*d^4*e^4 -
(b^7 + 6*a*b^5*c - 40*a^2*b^3*c^2)*d^3*e^5 + (3*a*b^6 - 9*a^2*b^4*c - 16*a^3*b^2*c^2 + 16*a^4*c^3)*d^2*e^6 -
(3*a^2*b^5 - 16*a^3*b^3*c + 16*a^4*b*c^2)*d*e^7 + (a^3*b^4 - 6*a^4*b^2*c + 8*a^5*c^2)*e^8)*sqrt((9*c^4*d^4*e^6
- 18*b*c^3*d^3*e^7 + 3*(5*b^2*c^2 - 2*a*c^3)*d^2*e^8 - 6*(b^3*c - a*b*c^2)*d*e^9 + (b^4 - 2*a*b^2*c + a^2*c^2
)*e^10)/((b^2*c^6 - 4*a*c^7)*d^12 - 6*(b^3*c^5 - 4*a*b*c^6)*d^11*e + 3*(5*b^4*c^4 - 18*a*b^2*c^5 - 8*a^2*c^6)*
d^10*e^2 - 10*(2*b^5*c^3 - 5*a*b^3*c^4 - 12*a^2*b*c^5)*d^9*e^3 + 15*(b^6*c^2 - 15*a^2*b^2*c^4 - 4*a^3*c^5)*d^8
*e^4 - 6*(b^7*c + 6*a*b^5*c^2 - 30*a^2*b^3*c^3 - 40*a^3*b*c^4)*d^7*e^5 + (b^8 + 26*a*b^6*c - 30*a^2*b^4*c^2 -
340*a^3*b^2*c^3 - 80*a^4*c^4)*d^6*e^6 - 6*(a*b^7 + 6*a^2*b^5*c - 30*a^3*b^3*c^2 - 40*a^4*b*c^3)*d^5*e^7 + 15*(
a^2*b^6 - 15*a^4*b^2*c^2 - 4*a^5*c^3)*d^4*e^8 - 10*(2*a^3*b^5 - 5*a^4*b^3*c - 12*a^5*b*c^2)*d^3*e^9 + 3*(5*a^4
*b^4 - 18*a^5*b^2*c - 8*a^6*c^2)*d^2*e^10 - 6*(a^5*b^3 - 4*a^6*b*c)*d*e^11 + (a^6*b^2 - 4*a^7*c)*e^12)))*sqrt(
(2*c^3*d^3*e^2 - 3*b*c^2*d^2*e^3 + 3*(b^2*c - 2*a*c^2)*d*e^4 - (b^3 - 3*a*b*c)*e^5 + ((b^2*c^3 - 4*a*c^4)*d^6
- 3*(b^3*c^2 - 4*a*b*c^3)*d^5*e + 3*(b^4*c - 3*a*b^2*c^2 - 4*a^2*c^3)*d^4*e^2 - (b^5 + 2*a*b^3*c - 24*a^2*b*c^
2)*d^3*e^3 + 3*(a*b^4 - 3*a^2*b^2*c - 4*a^3*c^2)*d^2*e^4 - 3*(a^2*b^3 - 4*a^3*b*c)*d*e^5 + (a^3*b^2 - 4*a^4*c)
*e^6)*sqrt((9*c^4*d^4*e^6 - 18*b*c^3*d^3*e^7 + 3*(5*b^2*c^2 - 2*a*c^3)*d^2*e^8 - 6*(b^3*c - a*b*c^2)*d*e^9 + (
b^4 - 2*a*b^2*c + a^2*c^2)*e^10)/((b^2*c^6 - 4*a*c^7)*d^12 - 6*(b^3*c^5 - 4*a*b*c^6)*d^11*e + 3*(5*b^4*c^4 - 1
8*a*b^2*c^5 - 8*a^2*c^6)*d^10*e^2 - 10*(2*b^5*c^3 - 5*a*b^3*c^4 - 12*a^2*b*c^5)*d^9*e^3 + 15*(b^6*c^2 - 15*a^2
*b^2*c^4 - 4*a^3*c^5)*d^8*e^4 - 6*(b^7*c + 6*a*b^5*c^2 - 30*a^2*b^3*c^3 - 40*a^3*b*c^4)*d^7*e^5 + (b^8 + 26*a*
b^6*c - 30*a^2*b^4*c^2 - 340*a^3*b^2*c^3 - 80*a^4*c^4)*d^6*e^6 - 6*(a*b^7 + 6*a^2*b^5*c - 30*a^3*b^3*c^2 - 40*
a^4*b*c^3)*d^5*e^7 + 15*(a^2*b^6 - 15*a^4*b^2*c^2 - 4*a^5*c^3)*d^4*e^8 - 10*(2*a^3*b^5 - 5*a^4*b^3*c - 12*a^5*
b*c^2)*d^3*e^9 + 3*(5*a^4*b^4 - 18*a^5*b^2*c - 8*a^6*c^2)*d^2*e^10 - 6*(a^5*b^3 - 4*a^6*b*c)*d*e^11 + (a^6*b^2
- 4*a^7*c)*e^12)))/((b^2*c^3 - 4*a*c^4)*d^6 - 3*(b^3*c^2 - 4*a*b*c^3)*d^5*e + 3*(b^4*c - 3*a*b^2*c^2 - 4*a^2*
c^3)*d^4*e^2 - (b^5 + 2*a*b^3*c - 24*a^2*b*c^2)*d^3*e^3 + 3*(a*b^4 - 3*a^2*b^2*c - 4*a^3*c^2)*d^2*e^4 - 3*(a^2
*b^3 - 4*a^3*b*c)*d*e^5 + (a^3*b^2 - 4*a^4*c)*e^6)) - 2*(3*c^4*d^2*e^4 - 3*b*c^3*d*e^5 + (b^2*c^2 - a*c^3)*e^6
)*sqrt(x*e + d)) - sqrt(1/2)*(c^2*d^2*x^2 + b*c*d^2*x + a*c*d^2 + (a*c*x^2 + a*b*x + a^2)*e^2 - (b*c*d*x^2 + b
^2*d*x + a*b*d)*e)*sqrt((2*c^3*d^3*e^2 - 3*b*c^2*d^2*e^3 + 3*(b^2*c - 2*a*c^2)*d*e^4 - (b^3 - 3*a*b*c)*e^5 + (
(b^2*c^3 - 4*a*c^4)*d^6 - 3*(b^3*c^2 - 4*a*b*c^3)*d^5*e + 3*(b^4*c - 3*a*b^2*c^2 - 4*a^2*c^3)*d^4*e^2 - (b^5 +
2*a*b^3*c - 24*a^2*b*c^2)*d^3*e^3 + 3*(a*b^4 - 3*a^2*b^2*c - 4*a^3*c^2)*d^2*e^4 - 3*(a^2*b^3 - 4*a^3*b*c)*d*e
^5 + (a^3*b^2 - 4*a^4*c)*e^6)*sqrt((9*c^4*d^4*e^6 - 18*b*c^3*d^3*e^7 + 3*(5*b^2*c^2 - 2*a*c^3)*d^2*e^8 - 6*(b^
3*c - a*b*c^2)*d*e^9 + (b^4 - 2*a*b^2*c + a^2*c^2)*e^10)/((b^2*c^6 - 4*a*c^7)*d^12 - 6*(b^3*c^5 - 4*a*b*c^6)*d
^11*e + 3*(5*b^4*c^4 - 18*a*b^2*c^5 - 8*a^2*c^6...

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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)/(c*x**2+b*x+a)**2/(e*x+d)**(1/2),x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1534 vs. \(2 (331) = 662\).
time = 2.55, size = 1534, normalized size = 4.21 \begin {gather*} \frac {{\left (x e + d\right )}^{\frac {3}{2}} c e^{2} - 2 \, \sqrt {x e + d} c d e^{2} + \sqrt {x e + d} b e^{3}}{{\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 )} {\left (c d^{2} - b d e + a e^{2}\right )}} + \frac {{\left ({\left (c d^{2} e - b d e^{2} + a e^{3}\right )}^{2} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c - \sqrt {b^{2} - 4 \, a c} c\right )} e} {\left (b^{2} - 4 \, a c\right )} e^{2} - 2 \, {\left (2 \, \sqrt {b^{2} - 4 \, a c} c^{2} d^{3} e^{2} - 3 \, \sqrt {b^{2} - 4 \, a c} b c d^{2} e^{3} - \sqrt {b^{2} - 4 \, a c} a b e^{5} + {\left (b^{2} + 2 \, a c\right )} \sqrt {b^{2} - 4 \, a c} d e^{4}\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c - \sqrt {b^{2} - 4 \, a c} c\right )} e} {\left | c d^{2} e - b d e^{2} + a e^{3} \right |} + {\left (4 \, c^{4} d^{6} e^{2} - 12 \, b c^{3} d^{5} e^{3} + {\left (13 \, b^{2} c^{2} + 8 \, a c^{3}\right )} d^{4} e^{4} + a^{2} b^{2} e^{8} - 2 \, {\left (3 \, b^{3} c + 8 \, a b c^{2}\right )} d^{3} e^{5} + {\left (b^{4} + 10 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} d^{2} e^{6} - 2 \, {\left (a b^{3} + 2 \, a^{2} b c\right )} d e^{7}\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 \, c^{2} d^{3} - 3 \, b c d^{2} e + b^{2} d e^{2} + 2 \, a c d e^{2} - a b e^{3} + \sqrt {{\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e + b^{2} d e^{2} + 2 \, a c d e^{2} - a b e^{3}\right )}^{2} - 4 \, {\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2} + 2 \, a c d^{2} e^{2} - 2 \, a b d e^{3} + a^{2} e^{4}\right )} {\left (c^{2} d^{2} - b c d e + a c e^{2}\right )}}}{c^{2} d^{2} - b c d e + a c e^{2}}}}\right )}{8 \, {\left (\sqrt {b^{2} - 4 \, a c} c^{3} d^{6} - 3 \, \sqrt {b^{2} - 4 \, a c} b c^{2} d^{5} e + 3 \, {\left (b^{2} c + a c^{2}\right )} \sqrt {b^{2} - 4 \, a c} d^{4} e^{2} - 3 \, \sqrt {b^{2} - 4 \, a c} a^{2} b d e^{5} - {\left (b^{3} + 6 \, a b c\right )} \sqrt {b^{2} - 4 \, a c} d^{3} e^{3} + \sqrt {b^{2} - 4 \, a c} a^{3} e^{6} + 3 \, {\left (a b^{2} + a^{2} c\right )} \sqrt {b^{2} - 4 \, a c} d^{2} e^{4}\right )} {\left | c d^{2} e - b d e^{2} + a e^{3} \right |} {\left | c \right |}} - \frac {{\left ({\left (c d^{2} e - b d e^{2} + a e^{3}\right )}^{2} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c + \sqrt {b^{2} - 4 \, a c} c\right )} e} {\left (b^{2} - 4 \, a c\right )} e^{2} + 2 \, {\left (2 \, \sqrt {b^{2} - 4 \, a c} c^{2} d^{3} e^{2} - 3 \, \sqrt {b^{2} - 4 \, a c} b c d^{2} e^{3} - \sqrt {b^{2} - 4 \, a c} a b e^{5} + {\left (b^{2} + 2 \, a c\right )} \sqrt {b^{2} - 4 \, a c} d e^{4}\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c + \sqrt {b^{2} - 4 \, a c} c\right )} e} {\left | c d^{2} e - b d e^{2} + a e^{3} \right |} + {\left (4 \, c^{4} d^{6} e^{2} - 12 \, b c^{3} d^{5} e^{3} + {\left (13 \, b^{2} c^{2} + 8 \, a c^{3}\right )} d^{4} e^{4} + a^{2} b^{2} e^{8} - 2 \, {\left (3 \, b^{3} c + 8 \, a b c^{2}\right )} d^{3} e^{5} + {\left (b^{4} + 10 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} d^{2} e^{6} - 2 \, {\left (a b^{3} + 2 \, a^{2} b c\right )} d e^{7}\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 \, c^{2} d^{3} - 3 \, b c d^{2} e + b^{2} d e^{2} + 2 \, a c d e^{2} - a b e^{3} - \sqrt {{\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e + b^{2} d e^{2} + 2 \, a c d e^{2} - a b e^{3}\right )}^{2} - 4 \, {\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2} + 2 \, a c d^{2} e^{2} - 2 \, a b d e^{3} + a^{2} e^{4}\right )} {\left (c^{2} d^{2} - b c d e + a c e^{2}\right )}}}{c^{2} d^{2} - b c d e + a c e^{2}}}}\right )}{8 \, {\left (\sqrt {b^{2} - 4 \, a c} c^{3} d^{6} - 3 \, \sqrt {b^{2} - 4 \, a c} b c^{2} d^{5} e + 3 \, {\left (b^{2} c + a c^{2}\right )} \sqrt {b^{2} - 4 \, a c} d^{4} e^{2} - 3 \, \sqrt {b^{2} - 4 \, a c} a^{2} b d e^{5} - {\left (b^{3} + 6 \, a b c\right )} \sqrt {b^{2} - 4 \, a c} d^{3} e^{3} + \sqrt {b^{2} - 4 \, a c} a^{3} e^{6} + 3 \, {\left (a b^{2} + a^{2} c\right )} \sqrt {b^{2} - 4 \, a c} d^{2} e^{4}\right )} {\left | c d^{2} e - b d e^{2} + a e^{3} \right |} {\left | c \right |}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)/(c*x^2+b*x+a)^2/(e*x+d)^(1/2),x, algorithm="giac")

[Out]

((x*e + d)^(3/2)*c*e^2 - 2*sqrt(x*e + d)*c*d*e^2 + sqrt(x*e + d)*b*e^3)/(((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)*(c*d^2 - b*d*e + a*e^2)) + 1/8*((c*d^2*e - b*d*e^2 + a*e^3)^2*sqrt(-4*c^2
*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e)*(b^2 - 4*a*c)*e^2 - 2*(2*sqrt(b^2 - 4*a*c)*c^2*d^3*e^2 - 3*sqrt(b^2 - 4*
a*c)*b*c*d^2*e^3 - sqrt(b^2 - 4*a*c)*a*b*e^5 + (b^2 + 2*a*c)*sqrt(b^2 - 4*a*c)*d*e^4)*sqrt(-4*c^2*d + 2*(b*c -
sqrt(b^2 - 4*a*c)*c)*e)*abs(c*d^2*e - b*d*e^2 + a*e^3) + (4*c^4*d^6*e^2 - 12*b*c^3*d^5*e^3 + (13*b^2*c^2 + 8*
a*c^3)*d^4*e^4 + a^2*b^2*e^8 - 2*(3*b^3*c + 8*a*b*c^2)*d^3*e^5 + (b^4 + 10*a*b^2*c + 4*a^2*c^2)*d^2*e^6 - 2*(a
*b^3 + 2*a^2*b*c)*d*e^7)*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)/sq
rt(-(2*c^2*d^3 - 3*b*c*d^2*e + b^2*d*e^2 + 2*a*c*d*e^2 - a*b*e^3 + sqrt((2*c^2*d^3 - 3*b*c*d^2*e + b^2*d*e^2 +
2*a*c*d*e^2 - a*b*e^3)^2 - 4*(c^2*d^4 - 2*b*c*d^3*e + b^2*d^2*e^2 + 2*a*c*d^2*e^2 - 2*a*b*d*e^3 + a^2*e^4)*(c
^2*d^2 - b*c*d*e + a*c*e^2)))/(c^2*d^2 - b*c*d*e + a*c*e^2)))/((sqrt(b^2 - 4*a*c)*c^3*d^6 - 3*sqrt(b^2 - 4*a*c
)*b*c^2*d^5*e + 3*(b^2*c + a*c^2)*sqrt(b^2 - 4*a*c)*d^4*e^2 - 3*sqrt(b^2 - 4*a*c)*a^2*b*d*e^5 - (b^3 + 6*a*b*c
)*sqrt(b^2 - 4*a*c)*d^3*e^3 + sqrt(b^2 - 4*a*c)*a^3*e^6 + 3*(a*b^2 + a^2*c)*sqrt(b^2 - 4*a*c)*d^2*e^4)*abs(c*d
^2*e - b*d*e^2 + a*e^3)*abs(c)) - 1/8*((c*d^2*e - b*d*e^2 + a*e^3)^2*sqrt(-4*c^2*d + 2*(b*c + sqrt(b^2 - 4*a*c
)*c)*e)*(b^2 - 4*a*c)*e^2 + 2*(2*sqrt(b^2 - 4*a*c)*c^2*d^3*e^2 - 3*sqrt(b^2 - 4*a*c)*b*c*d^2*e^3 - sqrt(b^2 -
4*a*c)*a*b*e^5 + (b^2 + 2*a*c)*sqrt(b^2 - 4*a*c)*d*e^4)*sqrt(-4*c^2*d + 2*(b*c + sqrt(b^2 - 4*a*c)*c)*e)*abs(c
*d^2*e - b*d*e^2 + a*e^3) + (4*c^4*d^6*e^2 - 12*b*c^3*d^5*e^3 + (13*b^2*c^2 + 8*a*c^3)*d^4*e^4 + a^2*b^2*e^8 -
2*(3*b^3*c + 8*a*b*c^2)*d^3*e^5 + (b^4 + 10*a*b^2*c + 4*a^2*c^2)*d^2*e^6 - 2*(a*b^3 + 2*a^2*b*c)*d*e^7)*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*c^2*d^3 - 3*b*c*d^2*e +
b^2*d*e^2 + 2*a*c*d*e^2 - a*b*e^3 - sqrt((2*c^2*d^3 - 3*b*c*d^2*e + b^2*d*e^2 + 2*a*c*d*e^2 - a*b*e^3)^2 - 4*
(c^2*d^4 - 2*b*c*d^3*e + b^2*d^2*e^2 + 2*a*c*d^2*e^2 - 2*a*b*d*e^3 + a^2*e^4)*(c^2*d^2 - b*c*d*e + a*c*e^2)))/
(c^2*d^2 - b*c*d*e + a*c*e^2)))/((sqrt(b^2 - 4*a*c)*c^3*d^6 - 3*sqrt(b^2 - 4*a*c)*b*c^2*d^5*e + 3*(b^2*c + a*c
^2)*sqrt(b^2 - 4*a*c)*d^4*e^2 - 3*sqrt(b^2 - 4*a*c)*a^2*b*d*e^5 - (b^3 + 6*a*b*c)*sqrt(b^2 - 4*a*c)*d^3*e^3 +
sqrt(b^2 - 4*a*c)*a^3*e^6 + 3*(a*b^2 + a^2*c)*sqrt(b^2 - 4*a*c)*d^2*e^4)*abs(c*d^2*e - b*d*e^2 + a*e^3)*abs(c)
)

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Mupad [B]
time = 5.32, size = 2500, normalized size = 6.87 \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)^(1/2)*(a + b*x + c*x^2)^2),x)

[Out]

((c*e^2*(d + e*x)^(3/2))/(a*e^2 + c*d^2 - b*d*e) + (e^2*(b*e - 2*c*d)*(d + e*x)^(1/2))/(a*e^2 + c*d^2 - b*d*e)
)/((b*e - 2*c*d)*(d + e*x) + c*(d + e*x)^2 + a*e^2 + c*d^2 - b*d*e) - atan(((((4*a*b^3*c^2*e^7 - 16*a^2*b*c^3*
e^7 + 32*a*c^5*d^3*e^4 + 32*a^2*c^4*d*e^6 - 4*b^4*c^2*d*e^6 - 8*b^2*c^4*d^3*e^4 + 12*b^3*c^3*d^2*e^5 - 48*a*b*
c^4*d^2*e^5 + 8*a*b^2*c^3*d*e^6)/(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)
- (2*(d + e*x)^(1/2)*(-(b^5*e^5 - b^2*e^5*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*e^5 + 8*a*c^4*d^3*e^2 - 24*
a^2*c^3*d*e^4 - 2*b^2*c^3*d^3*e^2 + 3*b^3*c^2*d^2*e^3 - 3*c^2*d^2*e^3*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^3*c*e^5
+ a*c*e^5*(-(4*a*c - b^2)^3)^(1/2) - 3*b^4*c*d*e^4 + 3*b*c*d*e^4*(-(4*a*c - b^2)^3)^(1/2) - 12*a*b*c^3*d^2*e^
3 + 18*a*b^2*c^2*d*e^4)/(8*(16*a^2*c^5*d^6 + a^3*b^4*e^6 + 16*a^5*c^2*e^6 + b^4*c^3*d^6 - b^7*d^3*e^3 - 8*a*b^
2*c^4*d^6 - 8*a^4*b^2*c*e^6 + 3*a*b^6*d^2*e^4 - 3*a^2*b^5*d*e^5 - 3*b^5*c^2*d^5*e + 3*b^6*c*d^4*e^2 + 48*a^3*c
^4*d^4*e^2 + 48*a^4*c^3*d^2*e^4 + 24*a^2*b^2*c^3*d^4*e^2 + 32*a^2*b^3*c^2*d^3*e^3 + 24*a^3*b^2*c^2*d^2*e^4 + 2
4*a*b^3*c^3*d^5*e + 2*a*b^5*c*d^3*e^3 - 48*a^2*b*c^4*d^5*e + 24*a^3*b^3*c*d*e^5 - 48*a^4*b*c^2*d*e^5 - 21*a*b^
4*c^2*d^4*e^2 - 21*a^2*b^4*c*d^2*e^4 - 96*a^3*b*c^3*d^3*e^3)))^(1/2)*(32*a*c^6*d^5*e^2 - 16*a^3*b*c^3*e^7 + 32
*a^3*c^4*d*e^6 + 4*a^2*b^3*c^2*e^7 + 64*a^2*c^5*d^3*e^4 - 8*b^2*c^5*d^5*e^2 + 20*b^3*c^4*d^4*e^3 - 16*b^4*c^3*
d^3*e^4 + 4*b^5*c^2*d^2*e^5 - 80*a*b*c^5*d^4*e^3 - 8*a*b^4*c^2*d*e^6 + 48*a*b^2*c^4*d^3*e^4 + 8*a*b^3*c^3*d^2*
e^5 - 96*a^2*b*c^4*d^2*e^5 + 24*a^2*b^2*c^3*d*e^6))/(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))*(-(b^5*e^5 - b^2*e^5*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*e^5 + 8*a*c^4*d^3*e^2 - 24*a
^2*c^3*d*e^4 - 2*b^2*c^3*d^3*e^2 + 3*b^3*c^2*d^2*e^3 - 3*c^2*d^2*e^3*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^3*c*e^5
+ a*c*e^5*(-(4*a*c - b^2)^3)^(1/2) - 3*b^4*c*d*e^4 + 3*b*c*d*e^4*(-(4*a*c - b^2)^3)^(1/2) - 12*a*b*c^3*d^2*e^3
+ 18*a*b^2*c^2*d*e^4)/(8*(16*a^2*c^5*d^6 + a^3*b^4*e^6 + 16*a^5*c^2*e^6 + b^4*c^3*d^6 - b^7*d^3*e^3 - 8*a*b^2
*c^4*d^6 - 8*a^4*b^2*c*e^6 + 3*a*b^6*d^2*e^4 - 3*a^2*b^5*d*e^5 - 3*b^5*c^2*d^5*e + 3*b^6*c*d^4*e^2 + 48*a^3*c^
4*d^4*e^2 + 48*a^4*c^3*d^2*e^4 + 24*a^2*b^2*c^3*d^4*e^2 + 32*a^2*b^3*c^2*d^3*e^3 + 24*a^3*b^2*c^2*d^2*e^4 + 24
*a*b^3*c^3*d^5*e + 2*a*b^5*c*d^3*e^3 - 48*a^2*b*c^4*d^5*e + 24*a^3*b^3*c*d*e^5 - 48*a^4*b*c^2*d*e^5 - 21*a*b^4
*c^2*d^4*e^2 - 21*a^2*b^4*c*d^2*e^4 - 96*a^3*b*c^3*d^3*e^3)))^(1/2) + (2*(d + e*x)^(1/2)*(2*a*c^4*e^6 - b^2*c^
3*e^6 - 2*c^5*d^2*e^4 + 2*b*c^4*d*e^5))/(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))*(-(b^5*e^5 - b^2*e^5*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*e^5 + 8*a*c^4*d^3*e^2 - 24*a^2*c^3*d*e^4
- 2*b^2*c^3*d^3*e^2 + 3*b^3*c^2*d^2*e^3 - 3*c^2*d^2*e^3*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^3*c*e^5 + a*c*e^5*(-
(4*a*c - b^2)^3)^(1/2) - 3*b^4*c*d*e^4 + 3*b*c*d*e^4*(-(4*a*c - b^2)^3)^(1/2) - 12*a*b*c^3*d^2*e^3 + 18*a*b^2*
c^2*d*e^4)/(8*(16*a^2*c^5*d^6 + a^3*b^4*e^6 + 16*a^5*c^2*e^6 + b^4*c^3*d^6 - b^7*d^3*e^3 - 8*a*b^2*c^4*d^6 - 8
*a^4*b^2*c*e^6 + 3*a*b^6*d^2*e^4 - 3*a^2*b^5*d*e^5 - 3*b^5*c^2*d^5*e + 3*b^6*c*d^4*e^2 + 48*a^3*c^4*d^4*e^2 +
48*a^4*c^3*d^2*e^4 + 24*a^2*b^2*c^3*d^4*e^2 + 32*a^2*b^3*c^2*d^3*e^3 + 24*a^3*b^2*c^2*d^2*e^4 + 24*a*b^3*c^3*d
^5*e + 2*a*b^5*c*d^3*e^3 - 48*a^2*b*c^4*d^5*e + 24*a^3*b^3*c*d*e^5 - 48*a^4*b*c^2*d*e^5 - 21*a*b^4*c^2*d^4*e^2
- 21*a^2*b^4*c*d^2*e^4 - 96*a^3*b*c^3*d^3*e^3)))^(1/2)*1i - (((4*a*b^3*c^2*e^7 - 16*a^2*b*c^3*e^7 + 32*a*c^5*
d^3*e^4 + 32*a^2*c^4*d*e^6 - 4*b^4*c^2*d*e^6 - 8*b^2*c^4*d^3*e^4 + 12*b^3*c^3*d^2*e^5 - 48*a*b*c^4*d^2*e^5 + 8
*a*b^2*c^3*d*e^6)/(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) + (2*(d + e*x)
^(1/2)*(-(b^5*e^5 - b^2*e^5*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*e^5 + 8*a*c^4*d^3*e^2 - 24*a^2*c^3*d*e^4 -
2*b^2*c^3*d^3*e^2 + 3*b^3*c^2*d^2*e^3 - 3*c^2*d^2*e^3*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^3*c*e^5 + a*c*e^5*(-(4
*a*c - b^2)^3)^(1/2) - 3*b^4*c*d*e^4 + 3*b*c*d*e^4*(-(4*a*c - b^2)^3)^(1/2) - 12*a*b*c^3*d^2*e^3 + 18*a*b^2*c^
2*d*e^4)/(8*(16*a^2*c^5*d^6 + a^3*b^4*e^6 + 16*a^5*c^2*e^6 + b^4*c^3*d^6 - b^7*d^3*e^3 - 8*a*b^2*c^4*d^6 - 8*a
^4*b^2*c*e^6 + 3*a*b^6*d^2*e^4 - 3*a^2*b^5*d*e^5 - 3*b^5*c^2*d^5*e + 3*b^6*c*d^4*e^2 + 48*a^3*c^4*d^4*e^2 + 48
*a^4*c^3*d^2*e^4 + 24*a^2*b^2*c^3*d^4*e^2 + 32*a^2*b^3*c^2*d^3*e^3 + 24*a^3*b^2*c^2*d^2*e^4 + 24*a*b^3*c^3*d^5
*e + 2*a*b^5*c*d^3*e^3 - 48*a^2*b*c^4*d^5*e + 24*a^3*b^3*c*d*e^5 - 48*a^4*b*c^2*d*e^5 - 21*a*b^4*c^2*d^4*e^2 -
21*a^2*b^4*c*d^2*e^4 - 96*a^3*b*c^3*d^3*e^3)))^(1/2)*(32*a*c^6*d^5*e^2 - 16*a^3*b*c^3*e^7 + 32*a^3*c^4*d*e^6
+ 4*a^2*b^3*c^2*e^7 + 64*a^2*c^5*d^3*e^4 - 8*b^2*c^5*d^5*e^2 + 20*b^3*c^4*d^4*e^3 - 16*b^4*c^3*d^3*e^4 + 4*b^5
*c^2*d^2*e^5 - 80*a*b*c^5*d^4*e^3 - 8*a*b^4*c^2*d*e^6 + 48*a*b^2*c^4*d^3*e^4 + 8*a*b^3*c^3*d^2*e^5 - 96*a^2*b*
c^4*d^2*e^5 + 24*a^2*b^2*c^3*d*e^6))/(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))*(-(b^5*e^5 - b^2*e^5*(-(4*a*c - b^2)^3)^(...

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