3.85 \(\int \frac{\sqrt{a+c x^2} (d+e x+f x^2)}{(g+h x)^4} \, dx\)

Optimal. Leaf size=314 \[ -\frac{\sqrt{a+c x^2} \left (h x \left (2 a^2 f h^4+a c g h^2 (6 f g-e h)+c^2 \left (3 f g^4-d g^2 h^2\right )\right )+a^2 e h^5+a c g h^2 \left (d h^2+3 f g^2\right )+2 c^2 f g^5\right )}{2 h^3 (g+h x)^2 \left (a h^2+c g^2\right )^2}+\frac{c \tanh ^{-1}\left (\frac{a h-c g x}{\sqrt{a+c x^2} \sqrt{a h^2+c g^2}}\right ) \left (a^2 h^4 (4 f g-e h)+a c g h^2 \left (5 f g^2-d h^2\right )+2 c^2 f g^5\right )}{2 h^4 \left (a h^2+c g^2\right )^{5/2}}-\frac{\left (a+c x^2\right )^{3/2} \left (d h^2-e g h+f g^2\right )}{3 h (g+h x)^3 \left (a h^2+c g^2\right )}+\frac{\sqrt{c} f \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{h^4} \]

[Out]

-((2*c^2*f*g^5 + a^2*e*h^5 + a*c*g*h^2*(3*f*g^2 + d*h^2) + h*(2*a^2*f*h^4 + a*c*g*h^2*(6*f*g - e*h) + c^2*(3*f
*g^4 - d*g^2*h^2))*x)*Sqrt[a + c*x^2])/(2*h^3*(c*g^2 + a*h^2)^2*(g + h*x)^2) - ((f*g^2 - e*g*h + d*h^2)*(a + c
*x^2)^(3/2))/(3*h*(c*g^2 + a*h^2)*(g + h*x)^3) + (Sqrt[c]*f*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/h^4 + (c*(2*
c^2*f*g^5 + a^2*h^4*(4*f*g - e*h) + a*c*g*h^2*(5*f*g^2 - d*h^2))*ArcTanh[(a*h - c*g*x)/(Sqrt[c*g^2 + a*h^2]*Sq
rt[a + c*x^2])])/(2*h^4*(c*g^2 + a*h^2)^(5/2))

________________________________________________________________________________________

Rubi [A]  time = 0.505451, antiderivative size = 314, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207, Rules used = {1651, 811, 844, 217, 206, 725} \[ -\frac{\sqrt{a+c x^2} \left (h x \left (2 a^2 f h^4+a c g h^2 (6 f g-e h)+c^2 \left (3 f g^4-d g^2 h^2\right )\right )+a^2 e h^5+a c g h^2 \left (d h^2+3 f g^2\right )+2 c^2 f g^5\right )}{2 h^3 (g+h x)^2 \left (a h^2+c g^2\right )^2}+\frac{c \tanh ^{-1}\left (\frac{a h-c g x}{\sqrt{a+c x^2} \sqrt{a h^2+c g^2}}\right ) \left (a^2 h^4 (4 f g-e h)+a c g h^2 \left (5 f g^2-d h^2\right )+2 c^2 f g^5\right )}{2 h^4 \left (a h^2+c g^2\right )^{5/2}}-\frac{\left (a+c x^2\right )^{3/2} \left (d h^2-e g h+f g^2\right )}{3 h (g+h x)^3 \left (a h^2+c g^2\right )}+\frac{\sqrt{c} f \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{h^4} \]

Antiderivative was successfully verified.

[In]

Int[(Sqrt[a + c*x^2]*(d + e*x + f*x^2))/(g + h*x)^4,x]

[Out]

-((2*c^2*f*g^5 + a^2*e*h^5 + a*c*g*h^2*(3*f*g^2 + d*h^2) + h*(2*a^2*f*h^4 + a*c*g*h^2*(6*f*g - e*h) + c^2*(3*f
*g^4 - d*g^2*h^2))*x)*Sqrt[a + c*x^2])/(2*h^3*(c*g^2 + a*h^2)^2*(g + h*x)^2) - ((f*g^2 - e*g*h + d*h^2)*(a + c
*x^2)^(3/2))/(3*h*(c*g^2 + a*h^2)*(g + h*x)^3) + (Sqrt[c]*f*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/h^4 + (c*(2*
c^2*f*g^5 + a^2*h^4*(4*f*g - e*h) + a*c*g*h^2*(5*f*g^2 - d*h^2))*ArcTanh[(a*h - c*g*x)/(Sqrt[c*g^2 + a*h^2]*Sq
rt[a + c*x^2])])/(2*h^4*(c*g^2 + a*h^2)^(5/2))

Rule 1651

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, d
+ e*x, x], R = PolynomialRemainder[Pq, d + e*x, x]}, Simp[(e*R*(d + e*x)^(m + 1)*(a + c*x^2)^(p + 1))/((m + 1
)*(c*d^2 + a*e^2)), x] + Dist[1/((m + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p*ExpandToSum[(m
+ 1)*(c*d^2 + a*e^2)*Q + c*d*R*(m + 1) - c*e*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, c, d, e, p}, x] && Po
lyQ[Pq, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[m, -1]

Rule 811

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

Rule 844

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

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 725

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> -Subst[Int[1/(c*d^2 + a*e^2 - x^2), x], x,
(a*e - c*d*x)/Sqrt[a + c*x^2]] /; FreeQ[{a, c, d, e}, x]

Rubi steps

\begin{align*} \int \frac{\sqrt{a+c x^2} \left (d+e x+f x^2\right )}{(g+h x)^4} \, dx &=-\frac{\left (f g^2-e g h+d h^2\right ) \left (a+c x^2\right )^{3/2}}{3 h \left (c g^2+a h^2\right ) (g+h x)^3}-\frac{\int \frac{\left (-3 (c d g-a f g+a e h)-3 f \left (\frac{c g^2}{h}+a h\right ) x\right ) \sqrt{a+c x^2}}{(g+h x)^3} \, dx}{3 \left (c g^2+a h^2\right )}\\ &=-\frac{\left (2 c^2 f g^5+a^2 e h^5+a c g h^2 \left (3 f g^2+d h^2\right )+h \left (2 a^2 f h^4+a c g h^2 (6 f g-e h)+c^2 \left (3 f g^4-d g^2 h^2\right )\right ) x\right ) \sqrt{a+c x^2}}{2 h^3 \left (c g^2+a h^2\right )^2 (g+h x)^2}-\frac{\left (f g^2-e g h+d h^2\right ) \left (a+c x^2\right )^{3/2}}{3 h \left (c g^2+a h^2\right ) (g+h x)^3}+\frac{\int \frac{-6 a c \left (a h^2 (2 f g-e h)+c \left (f g^3-d g h^2\right )\right )+\frac{12 c f \left (c g^2+a h^2\right )^2 x}{h}}{(g+h x) \sqrt{a+c x^2}} \, dx}{12 h^2 \left (c g^2+a h^2\right )^2}\\ &=-\frac{\left (2 c^2 f g^5+a^2 e h^5+a c g h^2 \left (3 f g^2+d h^2\right )+h \left (2 a^2 f h^4+a c g h^2 (6 f g-e h)+c^2 \left (3 f g^4-d g^2 h^2\right )\right ) x\right ) \sqrt{a+c x^2}}{2 h^3 \left (c g^2+a h^2\right )^2 (g+h x)^2}-\frac{\left (f g^2-e g h+d h^2\right ) \left (a+c x^2\right )^{3/2}}{3 h \left (c g^2+a h^2\right ) (g+h x)^3}+\frac{(c f) \int \frac{1}{\sqrt{a+c x^2}} \, dx}{h^4}-\frac{\left (c \left (2 c^2 f g^5+a^2 h^4 (4 f g-e h)+a c g h^2 \left (5 f g^2-d h^2\right )\right )\right ) \int \frac{1}{(g+h x) \sqrt{a+c x^2}} \, dx}{2 h^4 \left (c g^2+a h^2\right )^2}\\ &=-\frac{\left (2 c^2 f g^5+a^2 e h^5+a c g h^2 \left (3 f g^2+d h^2\right )+h \left (2 a^2 f h^4+a c g h^2 (6 f g-e h)+c^2 \left (3 f g^4-d g^2 h^2\right )\right ) x\right ) \sqrt{a+c x^2}}{2 h^3 \left (c g^2+a h^2\right )^2 (g+h x)^2}-\frac{\left (f g^2-e g h+d h^2\right ) \left (a+c x^2\right )^{3/2}}{3 h \left (c g^2+a h^2\right ) (g+h x)^3}+\frac{(c f) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{a+c x^2}}\right )}{h^4}+\frac{\left (c \left (2 c^2 f g^5+a^2 h^4 (4 f g-e h)+a c g h^2 \left (5 f g^2-d h^2\right )\right )\right ) \operatorname{Subst}\left (\int \frac{1}{c g^2+a h^2-x^2} \, dx,x,\frac{a h-c g x}{\sqrt{a+c x^2}}\right )}{2 h^4 \left (c g^2+a h^2\right )^2}\\ &=-\frac{\left (2 c^2 f g^5+a^2 e h^5+a c g h^2 \left (3 f g^2+d h^2\right )+h \left (2 a^2 f h^4+a c g h^2 (6 f g-e h)+c^2 \left (3 f g^4-d g^2 h^2\right )\right ) x\right ) \sqrt{a+c x^2}}{2 h^3 \left (c g^2+a h^2\right )^2 (g+h x)^2}-\frac{\left (f g^2-e g h+d h^2\right ) \left (a+c x^2\right )^{3/2}}{3 h \left (c g^2+a h^2\right ) (g+h x)^3}+\frac{\sqrt{c} f \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{h^4}+\frac{c \left (2 c^2 f g^5+a^2 h^4 (4 f g-e h)+a c g h^2 \left (5 f g^2-d h^2\right )\right ) \tanh ^{-1}\left (\frac{a h-c g x}{\sqrt{c g^2+a h^2} \sqrt{a+c x^2}}\right )}{2 h^4 \left (c g^2+a h^2\right )^{5/2}}\\ \end{align*}

Mathematica [A]  time = 0.984721, size = 382, normalized size = 1.22 \[ \frac{\frac{h \sqrt{a+c x^2} \left (-\frac{(g+h x)^2 \left (6 a^2 f h^4+a c h^2 \left (h (2 d h-5 e g)+20 f g^2\right )+c^2 \left (11 f g^4-g^2 h (d h+2 e g)\right )\right )}{\left (a h^2+c g^2\right )^2}+\frac{(g+h x) \left (-3 a h^2 (e h-2 f g)+c g h (d h-4 e g)+7 c f g^3\right )}{a h^2+c g^2}-2 \left (h (d h-e g)+f g^2\right )\right )}{(g+h x)^3}+\frac{3 c \log \left (\sqrt{a+c x^2} \sqrt{a h^2+c g^2}+a h-c g x\right ) \left (a^2 h^4 (4 f g-e h)+a c g h^2 \left (5 f g^2-d h^2\right )+2 c^2 f g^5\right )}{\left (a h^2+c g^2\right )^{5/2}}-\frac{3 c \log (g+h x) \left (a^2 h^4 (4 f g-e h)+a c g h^2 \left (5 f g^2-d h^2\right )+2 c^2 f g^5\right )}{\left (a h^2+c g^2\right )^{5/2}}+6 \sqrt{c} f \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right )}{6 h^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(Sqrt[a + c*x^2]*(d + e*x + f*x^2))/(g + h*x)^4,x]

[Out]

((h*Sqrt[a + c*x^2]*(-2*(f*g^2 + h*(-(e*g) + d*h)) + ((7*c*f*g^3 + c*g*h*(-4*e*g + d*h) - 3*a*h^2*(-2*f*g + e*
h))*(g + h*x))/(c*g^2 + a*h^2) - ((6*a^2*f*h^4 + c^2*(11*f*g^4 - g^2*h*(2*e*g + d*h)) + a*c*h^2*(20*f*g^2 + h*
(-5*e*g + 2*d*h)))*(g + h*x)^2)/(c*g^2 + a*h^2)^2))/(g + h*x)^3 - (3*c*(2*c^2*f*g^5 + a^2*h^4*(4*f*g - e*h) +
a*c*g*h^2*(5*f*g^2 - d*h^2))*Log[g + h*x])/(c*g^2 + a*h^2)^(5/2) + 6*Sqrt[c]*f*Log[c*x + Sqrt[c]*Sqrt[a + c*x^
2]] + (3*c*(2*c^2*f*g^5 + a^2*h^4*(4*f*g - e*h) + a*c*g*h^2*(5*f*g^2 - d*h^2))*Log[a*h - c*g*x + Sqrt[c*g^2 +
a*h^2]*Sqrt[a + c*x^2]])/(c*g^2 + a*h^2)^(5/2))/(6*h^4)

________________________________________________________________________________________

Maple [B]  time = 0.244, size = 5565, normalized size = 17.7 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x^2+e*x+d)*(c*x^2+a)^(1/2)/(h*x+g)^4,x)

[Out]

result too large to display

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^2+e*x+d)*(c*x^2+a)^(1/2)/(h*x+g)^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

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((f*x^2+e*x+d)*(c*x^2+a)^(1/2)/(h*x+g)^4,x, algorithm="fricas")

[Out]

Timed out

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{a + c x^{2}} \left (d + e x + f x^{2}\right )}{\left (g + h x\right )^{4}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x**2+e*x+d)*(c*x**2+a)**(1/2)/(h*x+g)**4,x)

[Out]

Integral(sqrt(a + c*x**2)*(d + e*x + f*x**2)/(g + h*x)**4, x)

________________________________________________________________________________________

Giac [B]  time = 1.47842, size = 2321, normalized size = 7.39 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^2+e*x+d)*(c*x^2+a)^(1/2)/(h*x+g)^4,x, algorithm="giac")

[Out]

-(2*c^3*f*g^5 + 5*a*c^2*f*g^3*h^2 - a*c^2*d*g*h^4 + 4*a^2*c*f*g*h^4 - a^2*c*h^5*e)*arctan(-((sqrt(c)*x - sqrt(
c*x^2 + a))*h + sqrt(c)*g)/sqrt(-c*g^2 - a*h^2))/((c^2*g^4*h^4 + 2*a*c*g^2*h^6 + a^2*h^8)*sqrt(-c*g^2 - a*h^2)
) - sqrt(c)*f*log(abs(-sqrt(c)*x + sqrt(c*x^2 + a)))/h^4 - 1/3*(18*(sqrt(c)*x - sqrt(c*x^2 + a))^5*c^3*f*g^5*h
^2 + 33*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a*c^2*f*g^3*h^4 + 3*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a*c^2*d*g*h^6 + 12
*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a^2*c*f*g*h^6 - 6*(sqrt(c)*x - sqrt(c*x^2 + a))^5*c^3*g^4*h^3*e - 12*(sqrt(c)
*x - sqrt(c*x^2 + a))^5*a*c^2*g^2*h^5*e - 3*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a^2*c*h^7*e + 54*(sqrt(c)*x - sqrt
(c*x^2 + a))^4*c^(7/2)*f*g^6*h - 6*(sqrt(c)*x - sqrt(c*x^2 + a))^4*c^(7/2)*d*g^4*h^3 + 87*(sqrt(c)*x - sqrt(c*
x^2 + a))^4*a*c^(5/2)*f*g^4*h^3 + 3*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a*c^(5/2)*d*g^2*h^5 + 12*(sqrt(c)*x - sqrt
(c*x^2 + a))^4*a^2*c^(3/2)*f*g^2*h^5 - 6*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^2*c^(3/2)*d*h^7 - 6*(sqrt(c)*x - sq
rt(c*x^2 + a))^4*a^3*sqrt(c)*f*h^7 - 12*(sqrt(c)*x - sqrt(c*x^2 + a))^4*c^(7/2)*g^5*h^2*e - 24*(sqrt(c)*x - sq
rt(c*x^2 + a))^4*a*c^(5/2)*g^3*h^4*e + 3*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^2*c^(3/2)*g*h^6*e + 44*(sqrt(c)*x -
sqrt(c*x^2 + a))^3*c^4*f*g^7 - 4*(sqrt(c)*x - sqrt(c*x^2 + a))^3*c^4*d*g^5*h^2 + 14*(sqrt(c)*x - sqrt(c*x^2 +
a))^3*a*c^3*f*g^5*h^2 + 14*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a*c^3*d*g^3*h^4 - 96*(sqrt(c)*x - sqrt(c*x^2 + a))
^3*a^2*c^2*f*g^3*h^4 - 12*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^2*c^2*d*g*h^6 - 36*(sqrt(c)*x - sqrt(c*x^2 + a))^3
*a^3*c*f*g*h^6 - 8*(sqrt(c)*x - sqrt(c*x^2 + a))^3*c^4*g^6*h*e - 8*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a*c^3*g^4*h
^3*e + 30*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^2*c^2*g^2*h^5*e - 78*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a*c^(7/2)*f*g
^6*h + 6*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a*c^(7/2)*d*g^4*h^3 - 120*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^2*c^(5/2)
*f*g^4*h^3 - 24*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^2*c^(5/2)*d*g^2*h^5 + 12*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^4
*sqrt(c)*f*h^7 + 12*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a*c^(7/2)*g^5*h^2*e + 30*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a
^2*c^(5/2)*g^3*h^4*e - 12*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^3*c^(3/2)*g*h^6*e + 48*(sqrt(c)*x - sqrt(c*x^2 + a
))*a^2*c^3*f*g^5*h^2 - 6*(sqrt(c)*x - sqrt(c*x^2 + a))*a^2*c^3*d*g^3*h^4 + 87*(sqrt(c)*x - sqrt(c*x^2 + a))*a^
3*c^2*f*g^3*h^4 + 9*(sqrt(c)*x - sqrt(c*x^2 + a))*a^3*c^2*d*g*h^6 + 24*(sqrt(c)*x - sqrt(c*x^2 + a))*a^4*c*f*g
*h^6 - 6*(sqrt(c)*x - sqrt(c*x^2 + a))*a^2*c^3*g^4*h^3*e - 18*(sqrt(c)*x - sqrt(c*x^2 + a))*a^3*c^2*g^2*h^5*e
+ 3*(sqrt(c)*x - sqrt(c*x^2 + a))*a^4*c*h^7*e - 11*a^3*c^(5/2)*f*g^4*h^3 + a^3*c^(5/2)*d*g^2*h^5 - 20*a^4*c^(3
/2)*f*g^2*h^5 - 2*a^4*c^(3/2)*d*h^7 - 6*a^5*sqrt(c)*f*h^7 + 2*a^3*c^(5/2)*g^3*h^4*e + 5*a^4*c^(3/2)*g*h^6*e)/(
(c^2*g^4*h^4 + 2*a*c*g^2*h^6 + a^2*h^8)*((sqrt(c)*x - sqrt(c*x^2 + a))^2*h + 2*(sqrt(c)*x - sqrt(c*x^2 + a))*s
qrt(c)*g - a*h)^3)