3.606 \(\int \frac{\sqrt{d+e x} \sqrt{f+g x}}{a+c x^2} \, dx\)

Optimal. Leaf size=342 \[ \frac{\left (-\sqrt{-a} \sqrt{c} (d g+e f)-a e g+c d f\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x} \sqrt{\sqrt{c} f-\sqrt{-a} g}}{\sqrt{f+g x} \sqrt{\sqrt{c} d-\sqrt{-a} e}}\right )}{\sqrt{-a} c \sqrt{\sqrt{c} d-\sqrt{-a} e} \sqrt{\sqrt{c} f-\sqrt{-a} g}}-\frac{\left (\sqrt{-a} \sqrt{c} (d g+e f)-a e g+c d f\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x} \sqrt{\sqrt{-a} g+\sqrt{c} f}}{\sqrt{f+g x} \sqrt{\sqrt{-a} e+\sqrt{c} d}}\right )}{\sqrt{-a} c \sqrt{\sqrt{-a} e+\sqrt{c} d} \sqrt{\sqrt{-a} g+\sqrt{c} f}}+\frac{2 \sqrt{e} \sqrt{g} \tanh ^{-1}\left (\frac{\sqrt{g} \sqrt{d+e x}}{\sqrt{e} \sqrt{f+g x}}\right )}{c} \]

[Out]

(2*Sqrt[e]*Sqrt[g]*ArcTanh[(Sqrt[g]*Sqrt[d + e*x])/(Sqrt[e]*Sqrt[f + g*x])])/c + ((c*d*f - a*e*g - Sqrt[-a]*Sq
rt[c]*(e*f + d*g))*ArcTanh[(Sqrt[Sqrt[c]*f - Sqrt[-a]*g]*Sqrt[d + e*x])/(Sqrt[Sqrt[c]*d - Sqrt[-a]*e]*Sqrt[f +
g*x])])/(Sqrt[-a]*c*Sqrt[Sqrt[c]*d - Sqrt[-a]*e]*Sqrt[Sqrt[c]*f - Sqrt[-a]*g]) - ((c*d*f - a*e*g + Sqrt[-a]*S
qrt[c]*(e*f + d*g))*ArcTanh[(Sqrt[Sqrt[c]*f + Sqrt[-a]*g]*Sqrt[d + e*x])/(Sqrt[Sqrt[c]*d + Sqrt[-a]*e]*Sqrt[f
+ g*x])])/(Sqrt[-a]*c*Sqrt[Sqrt[c]*d + Sqrt[-a]*e]*Sqrt[Sqrt[c]*f + Sqrt[-a]*g])

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Rubi [A]  time = 2.08574, antiderivative size = 342, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {906, 63, 217, 206, 6725, 93, 208} \[ \frac{\left (-\sqrt{-a} \sqrt{c} (d g+e f)-a e g+c d f\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x} \sqrt{\sqrt{c} f-\sqrt{-a} g}}{\sqrt{f+g x} \sqrt{\sqrt{c} d-\sqrt{-a} e}}\right )}{\sqrt{-a} c \sqrt{\sqrt{c} d-\sqrt{-a} e} \sqrt{\sqrt{c} f-\sqrt{-a} g}}-\frac{\left (\sqrt{-a} \sqrt{c} (d g+e f)-a e g+c d f\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x} \sqrt{\sqrt{-a} g+\sqrt{c} f}}{\sqrt{f+g x} \sqrt{\sqrt{-a} e+\sqrt{c} d}}\right )}{\sqrt{-a} c \sqrt{\sqrt{-a} e+\sqrt{c} d} \sqrt{\sqrt{-a} g+\sqrt{c} f}}+\frac{2 \sqrt{e} \sqrt{g} \tanh ^{-1}\left (\frac{\sqrt{g} \sqrt{d+e x}}{\sqrt{e} \sqrt{f+g x}}\right )}{c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[e]*Sqrt[g]*ArcTanh[(Sqrt[g]*Sqrt[d + e*x])/(Sqrt[e]*Sqrt[f + g*x])])/c + ((c*d*f - a*e*g - Sqrt[-a]*Sq
rt[c]*(e*f + d*g))*ArcTanh[(Sqrt[Sqrt[c]*f - Sqrt[-a]*g]*Sqrt[d + e*x])/(Sqrt[Sqrt[c]*d - Sqrt[-a]*e]*Sqrt[f +
g*x])])/(Sqrt[-a]*c*Sqrt[Sqrt[c]*d - Sqrt[-a]*e]*Sqrt[Sqrt[c]*f - Sqrt[-a]*g]) - ((c*d*f - a*e*g + Sqrt[-a]*S
qrt[c]*(e*f + d*g))*ArcTanh[(Sqrt[Sqrt[c]*f + Sqrt[-a]*g]*Sqrt[d + e*x])/(Sqrt[Sqrt[c]*d + Sqrt[-a]*e]*Sqrt[f
+ g*x])])/(Sqrt[-a]*c*Sqrt[Sqrt[c]*d + Sqrt[-a]*e]*Sqrt[Sqrt[c]*f + Sqrt[-a]*g])

Rule 906

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

Rule 63

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

Rule 217

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

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 6725

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rule 93

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 208

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

Rubi steps

\begin{align*} \int \frac{\sqrt{d+e x} \sqrt{f+g x}}{a+c x^2} \, dx &=\frac{\int \frac{c d f-a e g+c (e f+d g) x}{\sqrt{d+e x} \sqrt{f+g x} \left (a+c x^2\right )} \, dx}{c}+\frac{(e g) \int \frac{1}{\sqrt{d+e x} \sqrt{f+g x}} \, dx}{c}\\ &=\frac{\int \left (\frac{-a \sqrt{c} (e f+d g)+\sqrt{-a} (c d f-a e g)}{2 a \left (\sqrt{-a}-\sqrt{c} x\right ) \sqrt{d+e x} \sqrt{f+g x}}+\frac{a \sqrt{c} (e f+d g)+\sqrt{-a} (c d f-a e g)}{2 a \left (\sqrt{-a}+\sqrt{c} x\right ) \sqrt{d+e x} \sqrt{f+g x}}\right ) \, dx}{c}+\frac{(2 g) \operatorname{Subst}\left (\int \frac{1}{\sqrt{f-\frac{d g}{e}+\frac{g x^2}{e}}} \, dx,x,\sqrt{d+e x}\right )}{c}\\ &=\frac{(2 g) \operatorname{Subst}\left (\int \frac{1}{1-\frac{g x^2}{e}} \, dx,x,\frac{\sqrt{d+e x}}{\sqrt{f+g x}}\right )}{c}-\frac{\left (c d f-a e g-\sqrt{-a} \sqrt{c} (e f+d g)\right ) \int \frac{1}{\left (\sqrt{-a}+\sqrt{c} x\right ) \sqrt{d+e x} \sqrt{f+g x}} \, dx}{2 \sqrt{-a} c}-\frac{\left (c d f-a e g+\sqrt{-a} \sqrt{c} (e f+d g)\right ) \int \frac{1}{\left (\sqrt{-a}-\sqrt{c} x\right ) \sqrt{d+e x} \sqrt{f+g x}} \, dx}{2 \sqrt{-a} c}\\ &=\frac{2 \sqrt{e} \sqrt{g} \tanh ^{-1}\left (\frac{\sqrt{g} \sqrt{d+e x}}{\sqrt{e} \sqrt{f+g x}}\right )}{c}-\frac{\left (c d f-a e g-\sqrt{-a} \sqrt{c} (e f+d g)\right ) \operatorname{Subst}\left (\int \frac{1}{-\sqrt{c} d+\sqrt{-a} e-\left (-\sqrt{c} f+\sqrt{-a} g\right ) x^2} \, dx,x,\frac{\sqrt{d+e x}}{\sqrt{f+g x}}\right )}{\sqrt{-a} c}-\frac{\left (c d f-a e g+\sqrt{-a} \sqrt{c} (e f+d g)\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{c} d+\sqrt{-a} e-\left (\sqrt{c} f+\sqrt{-a} g\right ) x^2} \, dx,x,\frac{\sqrt{d+e x}}{\sqrt{f+g x}}\right )}{\sqrt{-a} c}\\ &=\frac{2 \sqrt{e} \sqrt{g} \tanh ^{-1}\left (\frac{\sqrt{g} \sqrt{d+e x}}{\sqrt{e} \sqrt{f+g x}}\right )}{c}+\frac{\left (c d f-a e g-\sqrt{-a} \sqrt{c} (e f+d g)\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{c} f-\sqrt{-a} g} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{-a} e} \sqrt{f+g x}}\right )}{\sqrt{-a} c \sqrt{\sqrt{c} d-\sqrt{-a} e} \sqrt{\sqrt{c} f-\sqrt{-a} g}}-\frac{\left (c d f-a e g+\sqrt{-a} \sqrt{c} (e f+d g)\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{c} f+\sqrt{-a} g} \sqrt{d+e x}}{\sqrt{\sqrt{c} d+\sqrt{-a} e} \sqrt{f+g x}}\right )}{\sqrt{-a} c \sqrt{\sqrt{c} d+\sqrt{-a} e} \sqrt{\sqrt{c} f+\sqrt{-a} g}}\\ \end{align*}

Mathematica [A]  time = 1.34225, size = 381, normalized size = 1.11 \[ \frac{-\frac{\left (a \sqrt{c} (d g+e f)-\sqrt{-a} c d f+\sqrt{-a} a e g\right ) \tan ^{-1}\left (\frac{\sqrt{d+e x} \sqrt{-\sqrt{-a} g-\sqrt{c} f}}{\sqrt{f+g x} \sqrt{\sqrt{-a} e+\sqrt{c} d}}\right )}{a \sqrt{\sqrt{-a} e+\sqrt{c} d} \sqrt{-\sqrt{-a} g-\sqrt{c} f}}+\frac{\left (a \sqrt{c} (d g+e f)+\sqrt{-a} c d f+(-a)^{3/2} e g\right ) \tan ^{-1}\left (\frac{\sqrt{d+e x} \sqrt{\sqrt{c} f-\sqrt{-a} g}}{\sqrt{f+g x} \sqrt{\sqrt{-a} e-\sqrt{c} d}}\right )}{a \sqrt{\sqrt{-a} e-\sqrt{c} d} \sqrt{\sqrt{c} f-\sqrt{-a} g}}+\frac{2 \sqrt{g} \sqrt{e f-d g} \sqrt{\frac{e (f+g x)}{e f-d g}} \sinh ^{-1}\left (\frac{\sqrt{g} \sqrt{d+e x}}{\sqrt{e f-d g}}\right )}{\sqrt{f+g x}}}{c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((2*Sqrt[g]*Sqrt[e*f - d*g]*Sqrt[(e*(f + g*x))/(e*f - d*g)]*ArcSinh[(Sqrt[g]*Sqrt[d + e*x])/Sqrt[e*f - d*g]])/
Sqrt[f + g*x] - ((-(Sqrt[-a]*c*d*f) + Sqrt[-a]*a*e*g + a*Sqrt[c]*(e*f + d*g))*ArcTan[(Sqrt[-(Sqrt[c]*f) - Sqrt
[-a]*g]*Sqrt[d + e*x])/(Sqrt[Sqrt[c]*d + Sqrt[-a]*e]*Sqrt[f + g*x])])/(a*Sqrt[Sqrt[c]*d + Sqrt[-a]*e]*Sqrt[-(S
qrt[c]*f) - Sqrt[-a]*g]) + ((Sqrt[-a]*c*d*f + (-a)^(3/2)*e*g + a*Sqrt[c]*(e*f + d*g))*ArcTan[(Sqrt[Sqrt[c]*f -
Sqrt[-a]*g]*Sqrt[d + e*x])/(Sqrt[-(Sqrt[c]*d) + Sqrt[-a]*e]*Sqrt[f + g*x])])/(a*Sqrt[-(Sqrt[c]*d) + Sqrt[-a]*
e]*Sqrt[Sqrt[c]*f - Sqrt[-a]*g]))/c

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Maple [B]  time = 0.378, size = 1569, normalized size = 4.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(1/2)*(g*x+f)^(1/2)/(c*x^2+a),x)

[Out]

1/2*(e*x+d)^(1/2)*(g*x+f)^(1/2)*(2*ln(1/2*(2*e*g*x+2*(e*g*x^2+d*g*x+e*f*x+d*f)^(1/2)*(e*g)^(1/2)+d*g+e*f)/(e*g
)^(1/2))*(((-a*c)^(1/2)*d*g+(-a*c)^(1/2)*e*f-a*e*g+c*d*f)/c)^(1/2)*(-((-a*c)^(1/2)*d*g+(-a*c)^(1/2)*e*f+a*e*g-
c*d*f)/c)^(1/2)*(-a*c)^(1/2)*e*g-(e*g)^(1/2)*(((-a*c)^(1/2)*d*g+(-a*c)^(1/2)*e*f-a*e*g+c*d*f)/c)^(1/2)*ln((-2*
(-a*c)^(1/2)*x*e*g+x*c*d*g+x*c*e*f+2*(-((-a*c)^(1/2)*d*g+(-a*c)^(1/2)*e*f+a*e*g-c*d*f)/c)^(1/2)*(e*g*x^2+d*g*x
+e*f*x+d*f)^(1/2)*c-(-a*c)^(1/2)*d*g-(-a*c)^(1/2)*e*f+2*c*d*f)/(c*x+(-a*c)^(1/2)))*(-a*c)^(1/2)*d*g-(e*g)^(1/2
)*(((-a*c)^(1/2)*d*g+(-a*c)^(1/2)*e*f-a*e*g+c*d*f)/c)^(1/2)*ln((-2*(-a*c)^(1/2)*x*e*g+x*c*d*g+x*c*e*f+2*(-((-a
*c)^(1/2)*d*g+(-a*c)^(1/2)*e*f+a*e*g-c*d*f)/c)^(1/2)*(e*g*x^2+d*g*x+e*f*x+d*f)^(1/2)*c-(-a*c)^(1/2)*d*g-(-a*c)
^(1/2)*e*f+2*c*d*f)/(c*x+(-a*c)^(1/2)))*(-a*c)^(1/2)*e*f-(e*g)^(1/2)*(((-a*c)^(1/2)*d*g+(-a*c)^(1/2)*e*f-a*e*g
+c*d*f)/c)^(1/2)*ln((-2*(-a*c)^(1/2)*x*e*g+x*c*d*g+x*c*e*f+2*(-((-a*c)^(1/2)*d*g+(-a*c)^(1/2)*e*f+a*e*g-c*d*f)
/c)^(1/2)*(e*g*x^2+d*g*x+e*f*x+d*f)^(1/2)*c-(-a*c)^(1/2)*d*g-(-a*c)^(1/2)*e*f+2*c*d*f)/(c*x+(-a*c)^(1/2)))*a*e
*g+(e*g)^(1/2)*(((-a*c)^(1/2)*d*g+(-a*c)^(1/2)*e*f-a*e*g+c*d*f)/c)^(1/2)*ln((-2*(-a*c)^(1/2)*x*e*g+x*c*d*g+x*c
*e*f+2*(-((-a*c)^(1/2)*d*g+(-a*c)^(1/2)*e*f+a*e*g-c*d*f)/c)^(1/2)*(e*g*x^2+d*g*x+e*f*x+d*f)^(1/2)*c-(-a*c)^(1/
2)*d*g-(-a*c)^(1/2)*e*f+2*c*d*f)/(c*x+(-a*c)^(1/2)))*c*d*f-(e*g)^(1/2)*ln((2*(-a*c)^(1/2)*x*e*g+x*c*d*g+x*c*e*
f+2*(e*g*x^2+d*g*x+e*f*x+d*f)^(1/2)*(((-a*c)^(1/2)*d*g+(-a*c)^(1/2)*e*f-a*e*g+c*d*f)/c)^(1/2)*c+(-a*c)^(1/2)*d
*g+(-a*c)^(1/2)*e*f+2*c*d*f)/(c*x-(-a*c)^(1/2)))*(-((-a*c)^(1/2)*d*g+(-a*c)^(1/2)*e*f+a*e*g-c*d*f)/c)^(1/2)*(-
a*c)^(1/2)*d*g-(e*g)^(1/2)*ln((2*(-a*c)^(1/2)*x*e*g+x*c*d*g+x*c*e*f+2*(e*g*x^2+d*g*x+e*f*x+d*f)^(1/2)*(((-a*c)
^(1/2)*d*g+(-a*c)^(1/2)*e*f-a*e*g+c*d*f)/c)^(1/2)*c+(-a*c)^(1/2)*d*g+(-a*c)^(1/2)*e*f+2*c*d*f)/(c*x-(-a*c)^(1/
2)))*(-((-a*c)^(1/2)*d*g+(-a*c)^(1/2)*e*f+a*e*g-c*d*f)/c)^(1/2)*(-a*c)^(1/2)*e*f+(e*g)^(1/2)*ln((2*(-a*c)^(1/2
)*x*e*g+x*c*d*g+x*c*e*f+2*(e*g*x^2+d*g*x+e*f*x+d*f)^(1/2)*(((-a*c)^(1/2)*d*g+(-a*c)^(1/2)*e*f-a*e*g+c*d*f)/c)^
(1/2)*c+(-a*c)^(1/2)*d*g+(-a*c)^(1/2)*e*f+2*c*d*f)/(c*x-(-a*c)^(1/2)))*(-((-a*c)^(1/2)*d*g+(-a*c)^(1/2)*e*f+a*
e*g-c*d*f)/c)^(1/2)*a*e*g-(e*g)^(1/2)*ln((2*(-a*c)^(1/2)*x*e*g+x*c*d*g+x*c*e*f+2*(e*g*x^2+d*g*x+e*f*x+d*f)^(1/
2)*(((-a*c)^(1/2)*d*g+(-a*c)^(1/2)*e*f-a*e*g+c*d*f)/c)^(1/2)*c+(-a*c)^(1/2)*d*g+(-a*c)^(1/2)*e*f+2*c*d*f)/(c*x
-(-a*c)^(1/2)))*(-((-a*c)^(1/2)*d*g+(-a*c)^(1/2)*e*f+a*e*g-c*d*f)/c)^(1/2)*c*d*f)/(e*g*x^2+d*g*x+e*f*x+d*f)^(1
/2)/(-a*c)^(1/2)/c/(e*g)^(1/2)/(((-a*c)^(1/2)*d*g+(-a*c)^(1/2)*e*f-a*e*g+c*d*f)/c)^(1/2)/(-((-a*c)^(1/2)*d*g+(
-a*c)^(1/2)*e*f+a*e*g-c*d*f)/c)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{e x + d} \sqrt{g x + f}}{c x^{2} + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(1/2)*(g*x+f)^(1/2)/(c*x^2+a),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{d + e x} \sqrt{f + g x}}{a + c x^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**(1/2)*(g*x+f)**(1/2)/(c*x**2+a),x)

[Out]

Integral(sqrt(d + e*x)*sqrt(f + g*x)/(a + c*x**2), x)

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Giac [B]  time = 36.9141, size = 4077, normalized size = 11.92 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(sqrt(g)*e^(3/2)*log((sqrt(x*e + d)*sqrt(g)*e^(1/2) - sqrt((x*e + d)*g*e - d*g*e + f*e^2))^2)/c - 2*(c^2*d^6*
g^(13/2)*e^(11/2) - 4*(sqrt(x*e + d)*sqrt(g)*e^(1/2) - sqrt((x*e + d)*g*e - d*g*e + f*e^2))^4*c^2*d^4*g^(9/2)*
e^(7/2) + 4*(sqrt(x*e + d)*sqrt(g)*e^(1/2) - sqrt((x*e + d)*g*e - d*g*e + f*e^2))^6*c^2*d^3*g^(7/2)*e^(5/2) -
(sqrt(x*e + d)*sqrt(g)*e^(1/2) - sqrt((x*e + d)*g*e - d*g*e + f*e^2))^8*c^2*d^2*g^(5/2)*e^(3/2) - 4*(sqrt(x*e
+ d)*sqrt(g)*e^(1/2) - sqrt((x*e + d)*g*e - d*g*e + f*e^2))^4*c^2*d^3*f*g^(7/2)*e^(9/2) + 4*(sqrt(x*e + d)*sqr
t(g)*e^(1/2) - sqrt((x*e + d)*g*e - d*g*e + f*e^2))^6*c^2*d^2*f*g^(5/2)*e^(7/2) - 9*c^2*d^4*f^2*g^(9/2)*e^(15/
2) - 2*a*c*d^4*g^(13/2)*e^(15/2) - 16*(sqrt(x*e + d)*sqrt(g)*e^(1/2) - sqrt((x*e + d)*g*e - d*g*e + f*e^2))^4*
c^2*d^2*f^2*g^(5/2)*e^(11/2) - 28*(sqrt(x*e + d)*sqrt(g)*e^(1/2) - sqrt((x*e + d)*g*e - d*g*e + f*e^2))^4*a*c*
d^2*g^(9/2)*e^(11/2) + 4*(sqrt(x*e + d)*sqrt(g)*e^(1/2) - sqrt((x*e + d)*g*e - d*g*e + f*e^2))^6*c^2*d*f^2*g^(
3/2)*e^(9/2) + 8*(sqrt(x*e + d)*sqrt(g)*e^(1/2) - sqrt((x*e + d)*g*e - d*g*e + f*e^2))^6*a*c*d*g^(7/2)*e^(9/2)
- (sqrt(x*e + d)*sqrt(g)*e^(1/2) - sqrt((x*e + d)*g*e - d*g*e + f*e^2))^8*c^2*f^2*sqrt(g)*e^(7/2) - 2*(sqrt(x
*e + d)*sqrt(g)*e^(1/2) - sqrt((x*e + d)*g*e - d*g*e + f*e^2))^8*a*c*g^(5/2)*e^(7/2) + 16*c^2*d^3*f^3*g^(7/2)*
e^(17/2) + 8*a*c*d^3*f*g^(11/2)*e^(17/2) - 4*(sqrt(x*e + d)*sqrt(g)*e^(1/2) - sqrt((x*e + d)*g*e - d*g*e + f*e
^2))^4*c^2*d*f^3*g^(3/2)*e^(13/2) - 8*(sqrt(x*e + d)*sqrt(g)*e^(1/2) - sqrt((x*e + d)*g*e - d*g*e + f*e^2))^4*
a*c*d*f*g^(7/2)*e^(13/2) + 4*(sqrt(x*e + d)*sqrt(g)*e^(1/2) - sqrt((x*e + d)*g*e - d*g*e + f*e^2))^6*c^2*f^3*s
qrt(g)*e^(11/2) + 8*(sqrt(x*e + d)*sqrt(g)*e^(1/2) - sqrt((x*e + d)*g*e - d*g*e + f*e^2))^6*a*c*f*g^(5/2)*e^(1
1/2) - 9*c^2*d^2*f^4*g^(5/2)*e^(19/2) - 12*a*c*d^2*f^2*g^(9/2)*e^(19/2) - 4*(sqrt(x*e + d)*sqrt(g)*e^(1/2) - s
qrt((x*e + d)*g*e - d*g*e + f*e^2))^4*c^2*f^4*sqrt(g)*e^(15/2) - 28*(sqrt(x*e + d)*sqrt(g)*e^(1/2) - sqrt((x*e
+ d)*g*e - d*g*e + f*e^2))^4*a*c*f^2*g^(5/2)*e^(15/2) - 32*(sqrt(x*e + d)*sqrt(g)*e^(1/2) - sqrt((x*e + d)*g*
e - d*g*e + f*e^2))^4*a^2*g^(9/2)*e^(15/2) + 8*a*c*d*f^3*g^(7/2)*e^(21/2) + c^2*f^6*sqrt(g)*e^(23/2) - 2*a*c*f
^4*g^(5/2)*e^(23/2))*log(abs(c^2*d^4*g^4*e^4 - 4*(sqrt(x*e + d)*sqrt(g)*e^(1/2) - sqrt((x*e + d)*g*e - d*g*e +
f*e^2))^2*c^2*d^3*g^3*e^3 + 6*(sqrt(x*e + d)*sqrt(g)*e^(1/2) - sqrt((x*e + d)*g*e - d*g*e + f*e^2))^4*c^2*d^2
*g^2*e^2 - 4*(sqrt(x*e + d)*sqrt(g)*e^(1/2) - sqrt((x*e + d)*g*e - d*g*e + f*e^2))^6*c^2*d*g*e + (sqrt(x*e + d
)*sqrt(g)*e^(1/2) - sqrt((x*e + d)*g*e - d*g*e + f*e^2))^8*c^2 - 4*c^2*d^3*f*g^3*e^5 + 4*(sqrt(x*e + d)*sqrt(g
)*e^(1/2) - sqrt((x*e + d)*g*e - d*g*e + f*e^2))^2*c^2*d^2*f*g^2*e^4 + 4*(sqrt(x*e + d)*sqrt(g)*e^(1/2) - sqrt
((x*e + d)*g*e - d*g*e + f*e^2))^4*c^2*d*f*g*e^3 - 4*(sqrt(x*e + d)*sqrt(g)*e^(1/2) - sqrt((x*e + d)*g*e - d*g
*e + f*e^2))^6*c^2*f*e^2 + 6*c^2*d^2*f^2*g^2*e^6 + 4*(sqrt(x*e + d)*sqrt(g)*e^(1/2) - sqrt((x*e + d)*g*e - d*g
*e + f*e^2))^2*c^2*d*f^2*g*e^5 + 6*(sqrt(x*e + d)*sqrt(g)*e^(1/2) - sqrt((x*e + d)*g*e - d*g*e + f*e^2))^4*c^2
*f^2*e^4 + 16*(sqrt(x*e + d)*sqrt(g)*e^(1/2) - sqrt((x*e + d)*g*e - d*g*e + f*e^2))^4*a*c*g^2*e^4 - 4*c^2*d*f^
3*g*e^7 - 4*(sqrt(x*e + d)*sqrt(g)*e^(1/2) - sqrt((x*e + d)*g*e - d*g*e + f*e^2))^2*c^2*f^3*e^6 + c^2*f^4*e^8)
)/(c^3*d^6*g^6*e^4 - 6*(sqrt(x*e + d)*sqrt(g)*e^(1/2) - sqrt((x*e + d)*g*e - d*g*e + f*e^2))^4*c^3*d^4*g^4*e^2
+ 8*(sqrt(x*e + d)*sqrt(g)*e^(1/2) - sqrt((x*e + d)*g*e - d*g*e + f*e^2))^6*c^3*d^3*g^3*e - 3*(sqrt(x*e + d)*
sqrt(g)*e^(1/2) - sqrt((x*e + d)*g*e - d*g*e + f*e^2))^8*c^3*d^2*g^2 + 2*c^3*d^5*f*g^5*e^5 - 24*(sqrt(x*e + d)
*sqrt(g)*e^(1/2) - sqrt((x*e + d)*g*e - d*g*e + f*e^2))^4*c^3*d^3*f*g^3*e^3 + 24*(sqrt(x*e + d)*sqrt(g)*e^(1/2
) - sqrt((x*e + d)*g*e - d*g*e + f*e^2))^6*c^3*d^2*f*g^2*e^2 - 2*(sqrt(x*e + d)*sqrt(g)*e^(1/2) - sqrt((x*e +
d)*g*e - d*g*e + f*e^2))^8*c^3*d*f*g*e - 17*c^3*d^4*f^2*g^4*e^6 - 8*a*c^2*d^4*g^6*e^6 - 68*(sqrt(x*e + d)*sqrt
(g)*e^(1/2) - sqrt((x*e + d)*g*e - d*g*e + f*e^2))^4*c^3*d^2*f^2*g^2*e^4 - 96*(sqrt(x*e + d)*sqrt(g)*e^(1/2) -
sqrt((x*e + d)*g*e - d*g*e + f*e^2))^4*a*c^2*d^2*g^4*e^4 + 24*(sqrt(x*e + d)*sqrt(g)*e^(1/2) - sqrt((x*e + d)
*g*e - d*g*e + f*e^2))^6*c^3*d*f^2*g*e^3 + 32*(sqrt(x*e + d)*sqrt(g)*e^(1/2) - sqrt((x*e + d)*g*e - d*g*e + f*
e^2))^6*a*c^2*d*g^3*e^3 - 3*(sqrt(x*e + d)*sqrt(g)*e^(1/2) - sqrt((x*e + d)*g*e - d*g*e + f*e^2))^8*c^3*f^2*e^
2 - 8*(sqrt(x*e + d)*sqrt(g)*e^(1/2) - sqrt((x*e + d)*g*e - d*g*e + f*e^2))^8*a*c^2*g^2*e^2 + 28*c^3*d^3*f^3*g
^3*e^7 + 32*a*c^2*d^3*f*g^5*e^7 - 24*(sqrt(x*e + d)*sqrt(g)*e^(1/2) - sqrt((x*e + d)*g*e - d*g*e + f*e^2))^4*c
^3*d*f^3*g*e^5 - 64*(sqrt(x*e + d)*sqrt(g)*e^(1/2) - sqrt((x*e + d)*g*e - d*g*e + f*e^2))^4*a*c^2*d*f*g^3*e^5
+ 8*(sqrt(x*e + d)*sqrt(g)*e^(1/2) - sqrt((x*e + d)*g*e - d*g*e + f*e^2))^6*c^3*f^3*e^4 + 32*(sqrt(x*e + d)*sq
rt(g)*e^(1/2) - sqrt((x*e + d)*g*e - d*g*e + f*e^2))^6*a*c^2*f*g^2*e^4 - 17*c^3*d^2*f^4*g^2*e^8 - 48*a*c^2*d^2
*f^2*g^4*e^8 - 6*(sqrt(x*e + d)*sqrt(g)*e^(1/2) - sqrt((x*e + d)*g*e - d*g*e + f*e^2))^4*c^3*f^4*e^6 - 96*(sqr
t(x*e + d)*sqrt(g)*e^(1/2) - sqrt((x*e + d)*g*e - d*g*e + f*e^2))^4*a*c^2*f^2*g^2*e^6 - 128*(sqrt(x*e + d)*sqr
t(g)*e^(1/2) - sqrt((x*e + d)*g*e - d*g*e + f*e^2))^4*a^2*c*g^4*e^6 + 2*c^3*d*f^5*g*e^9 + 32*a*c^2*d*f^3*g^3*e
^9 + c^3*f^6*e^10 - 8*a*c^2*f^4*g^2*e^10))*e^(-1)