3.63 \(\int \frac{1}{(a+b x^2)^2 (c+d x^2) \sqrt{e+f x^2}} \, dx\)

Optimal. Leaf size=203 \[ \frac{b \left (4 a^2 d f-2 a b c f-3 a b d e+b^2 c e\right ) \tan ^{-1}\left (\frac{x \sqrt{b e-a f}}{\sqrt{a} \sqrt{e+f x^2}}\right )}{2 a^{3/2} (b c-a d)^2 (b e-a f)^{3/2}}+\frac{b^2 x \sqrt{e+f x^2}}{2 a \left (a+b x^2\right ) (b c-a d) (b e-a f)}+\frac{d^2 \tan ^{-1}\left (\frac{x \sqrt{d e-c f}}{\sqrt{c} \sqrt{e+f x^2}}\right )}{\sqrt{c} (b c-a d)^2 \sqrt{d e-c f}} \]

[Out]

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

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Rubi [A]  time = 0.265115, antiderivative size = 203, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {546, 377, 205, 527, 12} \[ \frac{b \left (4 a^2 d f-2 a b c f-3 a b d e+b^2 c e\right ) \tan ^{-1}\left (\frac{x \sqrt{b e-a f}}{\sqrt{a} \sqrt{e+f x^2}}\right )}{2 a^{3/2} (b c-a d)^2 (b e-a f)^{3/2}}+\frac{b^2 x \sqrt{e+f x^2}}{2 a \left (a+b x^2\right ) (b c-a d) (b e-a f)}+\frac{d^2 \tan ^{-1}\left (\frac{x \sqrt{d e-c f}}{\sqrt{c} \sqrt{e+f x^2}}\right )}{\sqrt{c} (b c-a d)^2 \sqrt{d e-c f}} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b*x^2)^2*(c + d*x^2)*Sqrt[e + f*x^2]),x]

[Out]

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

Rule 546

Int[(((c_) + (d_.)*(x_)^2)^(q_)*((e_) + (f_.)*(x_)^2)^(r_))/((a_) + (b_.)*(x_)^2), x_Symbol] :> Dist[b^2/(b*c
- a*d)^2, Int[((c + d*x^2)^(q + 2)*(e + f*x^2)^r)/(a + b*x^2), x], x] - Dist[d/(b*c - a*d)^2, Int[(c + d*x^2)^
q*(e + f*x^2)^r*(2*b*c - a*d + b*d*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, r}, x] && LtQ[q, -1]

Rule 377

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]

Rule 205

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

Rule 527

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> -Simp[
((b*e - a*f)*x*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*n*(b*c - a*d)*(p + 1)), x] + Dist[1/(a*n*(b*c - a*d
)*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*f)
*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, q}, x] && LtQ[p, -1]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rubi steps

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

Mathematica [C]  time = 2.5579, size = 531, normalized size = 2.62 \[ \frac{\frac{b x \sqrt{e+f x^2} (b c-a d) \left (-30 f x^2 \sqrt{\frac{a x^2 \left (e+f x^2\right ) (b e-a f)}{e^2 \left (a+b x^2\right )^2}}-45 e \sqrt{\frac{a x^2 \left (e+f x^2\right ) (b e-a f)}{e^2 \left (a+b x^2\right )^2}}+16 f x^2 \sqrt{\frac{a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} \left (\frac{x^2 (b e-a f)}{e \left (a+b x^2\right )}\right )^{5/2} \, _2F_1\left (2,3;\frac{7}{2};\frac{(b e-a f) x^2}{e \left (b x^2+a\right )}\right )+16 e \sqrt{\frac{a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} \left (\frac{x^2 (b e-a f)}{e \left (a+b x^2\right )}\right )^{5/2} \, _2F_1\left (2,3;\frac{7}{2};\frac{(b e-a f) x^2}{e \left (b x^2+a\right )}\right )+30 f x^2 \sin ^{-1}\left (\sqrt{\frac{x^2 (b e-a f)}{e \left (a+b x^2\right )}}\right )+45 e \sin ^{-1}\left (\sqrt{\frac{x^2 (b e-a f)}{e \left (a+b x^2\right )}}\right )\right )}{e^2 \left (a+b x^2\right )^2 \left (\frac{x^2 (b e-a f)}{e \left (a+b x^2\right )}\right )^{3/2} \sqrt{\frac{a \left (e+f x^2\right )}{e \left (a+b x^2\right )}}}-\frac{30 b d \tan ^{-1}\left (\frac{x \sqrt{b e-a f}}{\sqrt{a} \sqrt{e+f x^2}}\right )}{\sqrt{a} \sqrt{b e-a f}}+\frac{30 d^2 \tan ^{-1}\left (\frac{x \sqrt{d e-c f}}{\sqrt{c} \sqrt{e+f x^2}}\right )}{\sqrt{c} \sqrt{d e-c f}}}{30 (b c-a d)^2} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/((a + b*x^2)^2*(c + d*x^2)*Sqrt[e + f*x^2]),x]

[Out]

((-30*b*d*ArcTan[(Sqrt[b*e - a*f]*x)/(Sqrt[a]*Sqrt[e + f*x^2])])/(Sqrt[a]*Sqrt[b*e - a*f]) + (30*d^2*ArcTan[(S
qrt[d*e - c*f]*x)/(Sqrt[c]*Sqrt[e + f*x^2])])/(Sqrt[c]*Sqrt[d*e - c*f]) + (b*(b*c - a*d)*x*Sqrt[e + f*x^2]*(-4
5*e*Sqrt[(a*(b*e - a*f)*x^2*(e + f*x^2))/(e^2*(a + b*x^2)^2)] - 30*f*x^2*Sqrt[(a*(b*e - a*f)*x^2*(e + f*x^2))/
(e^2*(a + b*x^2)^2)] + 45*e*ArcSin[Sqrt[((b*e - a*f)*x^2)/(e*(a + b*x^2))]] + 30*f*x^2*ArcSin[Sqrt[((b*e - a*f
)*x^2)/(e*(a + b*x^2))]] + 16*e*(((b*e - a*f)*x^2)/(e*(a + b*x^2)))^(5/2)*Sqrt[(a*(e + f*x^2))/(e*(a + b*x^2))
]*Hypergeometric2F1[2, 3, 7/2, ((b*e - a*f)*x^2)/(e*(a + b*x^2))] + 16*f*x^2*(((b*e - a*f)*x^2)/(e*(a + b*x^2)
))^(5/2)*Sqrt[(a*(e + f*x^2))/(e*(a + b*x^2))]*Hypergeometric2F1[2, 3, 7/2, ((b*e - a*f)*x^2)/(e*(a + b*x^2))]
))/(e^2*(((b*e - a*f)*x^2)/(e*(a + b*x^2)))^(3/2)*(a + b*x^2)^2*Sqrt[(a*(e + f*x^2))/(e*(a + b*x^2))]))/(30*(b
*c - a*d)^2)

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Maple [B]  time = 0.056, size = 1865, normalized size = 9.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{2} + a\right )}^{2}{\left (d x^{2} + c\right )} \sqrt{f x^{2} + e}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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

[Out]

Timed out

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Giac [B]  time = 16.9323, size = 647, normalized size = 3.19 \begin{align*} -\frac{1}{2} \,{\left (\frac{2 \, d^{2} \arctan \left (\frac{{\left (\sqrt{f} x - \sqrt{f x^{2} + e}\right )}^{2} d + 2 \, c f - d e}{2 \, \sqrt{-c^{2} f^{2} + c d f e}}\right )}{{\left (b^{2} c^{2} f^{2} - 2 \, a b c d f^{2} + a^{2} d^{2} f^{2}\right )} \sqrt{-c^{2} f^{2} + c d f e}} + \frac{{\left (2 \, a b^{2} c f - 4 \, a^{2} b d f - b^{3} c e + 3 \, a b^{2} d e\right )} \arctan \left (\frac{{\left (\sqrt{f} x - \sqrt{f x^{2} + e}\right )}^{2} b + 2 \, a f - b e}{2 \, \sqrt{-a^{2} f^{2} + a b f e}}\right )}{{\left (a^{2} b^{2} c^{2} f^{3} - 2 \, a^{3} b c d f^{3} + a^{4} d^{2} f^{3} - a b^{3} c^{2} f^{2} e + 2 \, a^{2} b^{2} c d f^{2} e - a^{3} b d^{2} f^{2} e\right )} \sqrt{-a^{2} f^{2} + a b f e}} + \frac{2 \,{\left (2 \,{\left (\sqrt{f} x - \sqrt{f x^{2} + e}\right )}^{2} a b f -{\left (\sqrt{f} x - \sqrt{f x^{2} + e}\right )}^{2} b^{2} e + b^{2} e^{2}\right )}}{{\left (a^{2} b c f^{3} - a^{3} d f^{3} - a b^{2} c f^{2} e + a^{2} b d f^{2} e\right )}{\left ({\left (\sqrt{f} x - \sqrt{f x^{2} + e}\right )}^{4} b + 4 \,{\left (\sqrt{f} x - \sqrt{f x^{2} + e}\right )}^{2} a f - 2 \,{\left (\sqrt{f} x - \sqrt{f x^{2} + e}\right )}^{2} b e + b e^{2}\right )}}\right )} f^{\frac{5}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(2*d^2*arctan(1/2*((sqrt(f)*x - sqrt(f*x^2 + e))^2*d + 2*c*f - d*e)/sqrt(-c^2*f^2 + c*d*f*e))/((b^2*c^2*f
^2 - 2*a*b*c*d*f^2 + a^2*d^2*f^2)*sqrt(-c^2*f^2 + c*d*f*e)) + (2*a*b^2*c*f - 4*a^2*b*d*f - b^3*c*e + 3*a*b^2*d
*e)*arctan(1/2*((sqrt(f)*x - sqrt(f*x^2 + e))^2*b + 2*a*f - b*e)/sqrt(-a^2*f^2 + a*b*f*e))/((a^2*b^2*c^2*f^3 -
2*a^3*b*c*d*f^3 + a^4*d^2*f^3 - a*b^3*c^2*f^2*e + 2*a^2*b^2*c*d*f^2*e - a^3*b*d^2*f^2*e)*sqrt(-a^2*f^2 + a*b*
f*e)) + 2*(2*(sqrt(f)*x - sqrt(f*x^2 + e))^2*a*b*f - (sqrt(f)*x - sqrt(f*x^2 + e))^2*b^2*e + b^2*e^2)/((a^2*b*
c*f^3 - a^3*d*f^3 - a*b^2*c*f^2*e + a^2*b*d*f^2*e)*((sqrt(f)*x - sqrt(f*x^2 + e))^4*b + 4*(sqrt(f)*x - sqrt(f*
x^2 + e))^2*a*f - 2*(sqrt(f)*x - sqrt(f*x^2 + e))^2*b*e + b*e^2)))*f^(5/2)