3.80 \(\int \frac{1}{\sqrt{a+b x^2} (c+d x^2)^3} \, dx\)

Optimal. Leaf size=163 \[ \frac{\left (3 a^2 d^2-8 a b c d+8 b^2 c^2\right ) \tanh ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{c} \sqrt{a+b x^2}}\right )}{8 c^{5/2} (b c-a d)^{5/2}}-\frac{3 d x \sqrt{a+b x^2} (2 b c-a d)}{8 c^2 \left (c+d x^2\right ) (b c-a d)^2}-\frac{d x \sqrt{a+b x^2}}{4 c \left (c+d x^2\right )^2 (b c-a d)} \]

[Out]

-(d*x*Sqrt[a + b*x^2])/(4*c*(b*c - a*d)*(c + d*x^2)^2) - (3*d*(2*b*c - a*d)*x*Sqrt[a + b*x^2])/(8*c^2*(b*c - a
*d)^2*(c + d*x^2)) + ((8*b^2*c^2 - 8*a*b*c*d + 3*a^2*d^2)*ArcTanh[(Sqrt[b*c - a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2]
)])/(8*c^(5/2)*(b*c - a*d)^(5/2))

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Rubi [A]  time = 0.119151, antiderivative size = 163, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {414, 527, 12, 377, 208} \[ \frac{\left (3 a^2 d^2-8 a b c d+8 b^2 c^2\right ) \tanh ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{c} \sqrt{a+b x^2}}\right )}{8 c^{5/2} (b c-a d)^{5/2}}-\frac{3 d x \sqrt{a+b x^2} (2 b c-a d)}{8 c^2 \left (c+d x^2\right ) (b c-a d)^2}-\frac{d x \sqrt{a+b x^2}}{4 c \left (c+d x^2\right )^2 (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[a + b*x^2]*(c + d*x^2)^3),x]

[Out]

-(d*x*Sqrt[a + b*x^2])/(4*c*(b*c - a*d)*(c + d*x^2)^2) - (3*d*(2*b*c - a*d)*x*Sqrt[a + b*x^2])/(8*c^2*(b*c - a
*d)^2*(c + d*x^2)) + ((8*b^2*c^2 - 8*a*b*c*d + 3*a^2*d^2)*ArcTanh[(Sqrt[b*c - a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2]
)])/(8*c^(5/2)*(b*c - a*d)^(5/2))

Rule 414

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> -Simp[(b*x*(a + b*x^n)^(p + 1)*(
c + d*x^n)^(q + 1))/(a*n*(p + 1)*(b*c - a*d)), x] + Dist[1/(a*n*(p + 1)*(b*c - a*d)), Int[(a + b*x^n)^(p + 1)*
(c + d*x^n)^q*Simp[b*c + n*(p + 1)*(b*c - a*d) + d*b*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d,
n, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] &&  !( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomial
Q[a, b, c, d, n, p, q, x]

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]]

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 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{1}{\sqrt{a+b x^2} \left (c+d x^2\right )^3} \, dx &=-\frac{d x \sqrt{a+b x^2}}{4 c (b c-a d) \left (c+d x^2\right )^2}+\frac{\int \frac{4 b c-3 a d-2 b d x^2}{\sqrt{a+b x^2} \left (c+d x^2\right )^2} \, dx}{4 c (b c-a d)}\\ &=-\frac{d x \sqrt{a+b x^2}}{4 c (b c-a d) \left (c+d x^2\right )^2}-\frac{3 d (2 b c-a d) x \sqrt{a+b x^2}}{8 c^2 (b c-a d)^2 \left (c+d x^2\right )}+\frac{\int \frac{8 b^2 c^2-8 a b c d+3 a^2 d^2}{\sqrt{a+b x^2} \left (c+d x^2\right )} \, dx}{8 c^2 (b c-a d)^2}\\ &=-\frac{d x \sqrt{a+b x^2}}{4 c (b c-a d) \left (c+d x^2\right )^2}-\frac{3 d (2 b c-a d) x \sqrt{a+b x^2}}{8 c^2 (b c-a d)^2 \left (c+d x^2\right )}+\frac{\left (8 b^2 c^2-8 a b c d+3 a^2 d^2\right ) \int \frac{1}{\sqrt{a+b x^2} \left (c+d x^2\right )} \, dx}{8 c^2 (b c-a d)^2}\\ &=-\frac{d x \sqrt{a+b x^2}}{4 c (b c-a d) \left (c+d x^2\right )^2}-\frac{3 d (2 b c-a d) x \sqrt{a+b x^2}}{8 c^2 (b c-a d)^2 \left (c+d x^2\right )}+\frac{\left (8 b^2 c^2-8 a b c d+3 a^2 d^2\right ) \operatorname{Subst}\left (\int \frac{1}{c-(b c-a d) x^2} \, dx,x,\frac{x}{\sqrt{a+b x^2}}\right )}{8 c^2 (b c-a d)^2}\\ &=-\frac{d x \sqrt{a+b x^2}}{4 c (b c-a d) \left (c+d x^2\right )^2}-\frac{3 d (2 b c-a d) x \sqrt{a+b x^2}}{8 c^2 (b c-a d)^2 \left (c+d x^2\right )}+\frac{\left (8 b^2 c^2-8 a b c d+3 a^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{b c-a d} x}{\sqrt{c} \sqrt{a+b x^2}}\right )}{8 c^{5/2} (b c-a d)^{5/2}}\\ \end{align*}

Mathematica [A]  time = 0.622278, size = 192, normalized size = 1.18 \[ \frac{x \left (\frac{\left (c+d x^2\right )^2 \left (3 a^2 d^2-8 a b c d+8 b^2 c^2\right ) \tanh ^{-1}\left (\sqrt{\frac{x^2 (b c-a d)}{c \left (a+b x^2\right )}}\right )}{\sqrt{\frac{x^2 (b c-a d)}{c \left (a+b x^2\right )}}}-c d \left (a^2 (-d) \left (5 c+3 d x^2\right )+a b \left (8 c^2+c d x^2-3 d^2 x^4\right )+2 b^2 c x^2 \left (4 c+3 d x^2\right )\right )\right )}{8 c^3 \sqrt{a+b x^2} \left (c+d x^2\right )^2 (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[a + b*x^2]*(c + d*x^2)^3),x]

[Out]

(x*(-(c*d*(2*b^2*c*x^2*(4*c + 3*d*x^2) - a^2*d*(5*c + 3*d*x^2) + a*b*(8*c^2 + c*d*x^2 - 3*d^2*x^4))) + ((8*b^2
*c^2 - 8*a*b*c*d + 3*a^2*d^2)*(c + d*x^2)^2*ArcTanh[Sqrt[((b*c - a*d)*x^2)/(c*(a + b*x^2))]])/Sqrt[((b*c - a*d
)*x^2)/(c*(a + b*x^2))]))/(8*c^3*(b*c - a*d)^2*Sqrt[a + b*x^2]*(c + d*x^2)^2)

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Maple [B]  time = 0.02, size = 1815, normalized size = 11.1 \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)^(1/2)/(d*x^2+c)^3,x)

[Out]

3/16/c^2/(a*d-b*c)/(x-(-c*d)^(1/2)/d)*((x-(-c*d)^(1/2)/d)^2*b+2*b*(-c*d)^(1/2)/d*(x-(-c*d)^(1/2)/d)+(a*d-b*c)/
d)^(1/2)-3/16/c^2/d*b*(-c*d)^(1/2)/(a*d-b*c)/((a*d-b*c)/d)^(1/2)*ln((2*(a*d-b*c)/d+2*b*(-c*d)^(1/2)/d*(x-(-c*d
)^(1/2)/d)+2*((a*d-b*c)/d)^(1/2)*((x-(-c*d)^(1/2)/d)^2*b+2*b*(-c*d)^(1/2)/d*(x-(-c*d)^(1/2)/d)+(a*d-b*c)/d)^(1
/2))/(x-(-c*d)^(1/2)/d))-1/16/(-c*d)^(1/2)/c/(a*d-b*c)/(x+(-c*d)^(1/2)/d)^2*((x+(-c*d)^(1/2)/d)^2*b-2*b*(-c*d)
^(1/2)/d*(x+(-c*d)^(1/2)/d)+(a*d-b*c)/d)^(1/2)-3/16/c*b/(a*d-b*c)^2/(x+(-c*d)^(1/2)/d)*((x+(-c*d)^(1/2)/d)^2*b
-2*b*(-c*d)^(1/2)/d*(x+(-c*d)^(1/2)/d)+(a*d-b*c)/d)^(1/2)+3/16/(-c*d)^(1/2)*b^2/(a*d-b*c)^2/((a*d-b*c)/d)^(1/2
)*ln((2*(a*d-b*c)/d-2*b*(-c*d)^(1/2)/d*(x+(-c*d)^(1/2)/d)+2*((a*d-b*c)/d)^(1/2)*((x+(-c*d)^(1/2)/d)^2*b-2*b*(-
c*d)^(1/2)/d*(x+(-c*d)^(1/2)/d)+(a*d-b*c)/d)^(1/2))/(x+(-c*d)^(1/2)/d))+1/16/(-c*d)^(1/2)/c*b/(a*d-b*c)/((a*d-
b*c)/d)^(1/2)*ln((2*(a*d-b*c)/d-2*b*(-c*d)^(1/2)/d*(x+(-c*d)^(1/2)/d)+2*((a*d-b*c)/d)^(1/2)*((x+(-c*d)^(1/2)/d
)^2*b-2*b*(-c*d)^(1/2)/d*(x+(-c*d)^(1/2)/d)+(a*d-b*c)/d)^(1/2))/(x+(-c*d)^(1/2)/d))+3/16/c^2/(a*d-b*c)/(x+(-c*
d)^(1/2)/d)*((x+(-c*d)^(1/2)/d)^2*b-2*b*(-c*d)^(1/2)/d*(x+(-c*d)^(1/2)/d)+(a*d-b*c)/d)^(1/2)+3/16/c^2/d*b*(-c*
d)^(1/2)/(a*d-b*c)/((a*d-b*c)/d)^(1/2)*ln((2*(a*d-b*c)/d-2*b*(-c*d)^(1/2)/d*(x+(-c*d)^(1/2)/d)+2*((a*d-b*c)/d)
^(1/2)*((x+(-c*d)^(1/2)/d)^2*b-2*b*(-c*d)^(1/2)/d*(x+(-c*d)^(1/2)/d)+(a*d-b*c)/d)^(1/2))/(x+(-c*d)^(1/2)/d))-3
/16/(-c*d)^(1/2)/c^2/((a*d-b*c)/d)^(1/2)*ln((2*(a*d-b*c)/d+2*b*(-c*d)^(1/2)/d*(x-(-c*d)^(1/2)/d)+2*((a*d-b*c)/
d)^(1/2)*((x-(-c*d)^(1/2)/d)^2*b+2*b*(-c*d)^(1/2)/d*(x-(-c*d)^(1/2)/d)+(a*d-b*c)/d)^(1/2))/(x-(-c*d)^(1/2)/d))
+3/16/(-c*d)^(1/2)/c^2/((a*d-b*c)/d)^(1/2)*ln((2*(a*d-b*c)/d-2*b*(-c*d)^(1/2)/d*(x+(-c*d)^(1/2)/d)+2*((a*d-b*c
)/d)^(1/2)*((x+(-c*d)^(1/2)/d)^2*b-2*b*(-c*d)^(1/2)/d*(x+(-c*d)^(1/2)/d)+(a*d-b*c)/d)^(1/2))/(x+(-c*d)^(1/2)/d
))+1/16/(-c*d)^(1/2)/c/(a*d-b*c)/(x-(-c*d)^(1/2)/d)^2*((x-(-c*d)^(1/2)/d)^2*b+2*b*(-c*d)^(1/2)/d*(x-(-c*d)^(1/
2)/d)+(a*d-b*c)/d)^(1/2)-3/16/c*b/(a*d-b*c)^2/(x-(-c*d)^(1/2)/d)*((x-(-c*d)^(1/2)/d)^2*b+2*b*(-c*d)^(1/2)/d*(x
-(-c*d)^(1/2)/d)+(a*d-b*c)/d)^(1/2)-3/16/(-c*d)^(1/2)*b^2/(a*d-b*c)^2/((a*d-b*c)/d)^(1/2)*ln((2*(a*d-b*c)/d+2*
b*(-c*d)^(1/2)/d*(x-(-c*d)^(1/2)/d)+2*((a*d-b*c)/d)^(1/2)*((x-(-c*d)^(1/2)/d)^2*b+2*b*(-c*d)^(1/2)/d*(x-(-c*d)
^(1/2)/d)+(a*d-b*c)/d)^(1/2))/(x-(-c*d)^(1/2)/d))-1/16/(-c*d)^(1/2)/c*b/(a*d-b*c)/((a*d-b*c)/d)^(1/2)*ln((2*(a
*d-b*c)/d+2*b*(-c*d)^(1/2)/d*(x-(-c*d)^(1/2)/d)+2*((a*d-b*c)/d)^(1/2)*((x-(-c*d)^(1/2)/d)^2*b+2*b*(-c*d)^(1/2)
/d*(x-(-c*d)^(1/2)/d)+(a*d-b*c)/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}{\sqrt{b x^{2} + a}{\left (d x^{2} + c\right )}^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^2+a)^(1/2)/(d*x^2+c)^3,x, algorithm="maxima")

[Out]

integrate(1/(sqrt(b*x^2 + a)*(d*x^2 + c)^3), x)

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Fricas [B]  time = 4.63851, size = 1750, normalized size = 10.74 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^2+a)^(1/2)/(d*x^2+c)^3,x, algorithm="fricas")

[Out]

[1/32*((8*b^2*c^4 - 8*a*b*c^3*d + 3*a^2*c^2*d^2 + (8*b^2*c^2*d^2 - 8*a*b*c*d^3 + 3*a^2*d^4)*x^4 + 2*(8*b^2*c^3
*d - 8*a*b*c^2*d^2 + 3*a^2*c*d^3)*x^2)*sqrt(b*c^2 - a*c*d)*log(((8*b^2*c^2 - 8*a*b*c*d + a^2*d^2)*x^4 + a^2*c^
2 + 2*(4*a*b*c^2 - 3*a^2*c*d)*x^2 + 4*((2*b*c - a*d)*x^3 + a*c*x)*sqrt(b*c^2 - a*c*d)*sqrt(b*x^2 + a))/(d^2*x^
4 + 2*c*d*x^2 + c^2)) - 4*(3*(2*b^2*c^3*d^2 - 3*a*b*c^2*d^3 + a^2*c*d^4)*x^3 + (8*b^2*c^4*d - 13*a*b*c^3*d^2 +
5*a^2*c^2*d^3)*x)*sqrt(b*x^2 + a))/(b^3*c^8 - 3*a*b^2*c^7*d + 3*a^2*b*c^6*d^2 - a^3*c^5*d^3 + (b^3*c^6*d^2 -
3*a*b^2*c^5*d^3 + 3*a^2*b*c^4*d^4 - a^3*c^3*d^5)*x^4 + 2*(b^3*c^7*d - 3*a*b^2*c^6*d^2 + 3*a^2*b*c^5*d^3 - a^3*
c^4*d^4)*x^2), -1/16*((8*b^2*c^4 - 8*a*b*c^3*d + 3*a^2*c^2*d^2 + (8*b^2*c^2*d^2 - 8*a*b*c*d^3 + 3*a^2*d^4)*x^4
+ 2*(8*b^2*c^3*d - 8*a*b*c^2*d^2 + 3*a^2*c*d^3)*x^2)*sqrt(-b*c^2 + a*c*d)*arctan(1/2*sqrt(-b*c^2 + a*c*d)*((2
*b*c - a*d)*x^2 + a*c)*sqrt(b*x^2 + a)/((b^2*c^2 - a*b*c*d)*x^3 + (a*b*c^2 - a^2*c*d)*x)) + 2*(3*(2*b^2*c^3*d^
2 - 3*a*b*c^2*d^3 + a^2*c*d^4)*x^3 + (8*b^2*c^4*d - 13*a*b*c^3*d^2 + 5*a^2*c^2*d^3)*x)*sqrt(b*x^2 + a))/(b^3*c
^8 - 3*a*b^2*c^7*d + 3*a^2*b*c^6*d^2 - a^3*c^5*d^3 + (b^3*c^6*d^2 - 3*a*b^2*c^5*d^3 + 3*a^2*b*c^4*d^4 - a^3*c^
3*d^5)*x^4 + 2*(b^3*c^7*d - 3*a*b^2*c^6*d^2 + 3*a^2*b*c^5*d^3 - a^3*c^4*d^4)*x^2)]

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Sympy [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(1/(b*x**2+a)**(1/2)/(d*x**2+c)**3,x)

[Out]

Exception raised: ValueError

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Giac [B]  time = 3.54621, size = 726, normalized size = 4.45 \begin{align*} -\frac{1}{8} \, b^{\frac{5}{2}}{\left (\frac{{\left (8 \, b^{2} c^{2} - 8 \, a b c d + 3 \, a^{2} d^{2}\right )} \arctan \left (\frac{{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} d + 2 \, b c - a d}{2 \, \sqrt{-b^{2} c^{2} + a b c d}}\right )}{{\left (b^{4} c^{4} - 2 \, a b^{3} c^{3} d + a^{2} b^{2} c^{2} d^{2}\right )} \sqrt{-b^{2} c^{2} + a b c d}} + \frac{2 \,{\left (8 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{6} b^{2} c^{2} d - 8 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{6} a b c d^{2} + 3 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{6} a^{2} d^{3} + 48 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} b^{3} c^{3} - 72 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} a b^{2} c^{2} d + 42 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} a^{2} b c d^{2} - 9 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} a^{3} d^{3} + 40 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} a^{2} b^{2} c^{2} d - 40 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} a^{3} b c d^{2} + 9 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} a^{4} d^{3} + 6 \, a^{4} b c d^{2} - 3 \, a^{5} d^{3}\right )}}{{\left (b^{4} c^{4} - 2 \, a b^{3} c^{3} d + a^{2} b^{2} c^{2} d^{2}\right )}{\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} d + 4 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} b c - 2 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} a d + a^{2} d\right )}^{2}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^2+a)^(1/2)/(d*x^2+c)^3,x, algorithm="giac")

[Out]

-1/8*b^(5/2)*((8*b^2*c^2 - 8*a*b*c*d + 3*a^2*d^2)*arctan(1/2*((sqrt(b)*x - sqrt(b*x^2 + a))^2*d + 2*b*c - a*d)
/sqrt(-b^2*c^2 + a*b*c*d))/((b^4*c^4 - 2*a*b^3*c^3*d + a^2*b^2*c^2*d^2)*sqrt(-b^2*c^2 + a*b*c*d)) + 2*(8*(sqrt
(b)*x - sqrt(b*x^2 + a))^6*b^2*c^2*d - 8*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a*b*c*d^2 + 3*(sqrt(b)*x - sqrt(b*x^2
+ a))^6*a^2*d^3 + 48*(sqrt(b)*x - sqrt(b*x^2 + a))^4*b^3*c^3 - 72*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a*b^2*c^2*d
+ 42*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^2*b*c*d^2 - 9*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^3*d^3 + 40*(sqrt(b)*x
- sqrt(b*x^2 + a))^2*a^2*b^2*c^2*d - 40*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^3*b*c*d^2 + 9*(sqrt(b)*x - sqrt(b*x^
2 + a))^2*a^4*d^3 + 6*a^4*b*c*d^2 - 3*a^5*d^3)/((b^4*c^4 - 2*a*b^3*c^3*d + a^2*b^2*c^2*d^2)*((sqrt(b)*x - sqrt
(b*x^2 + a))^4*d + 4*(sqrt(b)*x - sqrt(b*x^2 + a))^2*b*c - 2*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a*d + a^2*d)^2))