3.312 \(\int \frac{x^2}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx\)

Optimal. Leaf size=200 \[ -\frac{\sqrt{d} \left (-a^2 d^2+10 a b c d+15 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{8 c^{3/2} (b c-a d)^4}+\frac{b^{3/2} (5 a d+b c) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 \sqrt{a} (b c-a d)^4}-\frac{d x (a d+11 b c)}{8 c \left (c+d x^2\right ) (b c-a d)^3}-\frac{3 d x}{4 \left (c+d x^2\right )^2 (b c-a d)^2}-\frac{x}{2 \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)} \]

[Out]

(-3*d*x)/(4*(b*c - a*d)^2*(c + d*x^2)^2) - x/(2*(b*c - a*d)*(a + b*x^2)*(c + d*x
^2)^2) - (d*(11*b*c + a*d)*x)/(8*c*(b*c - a*d)^3*(c + d*x^2)) + (b^(3/2)*(b*c +
5*a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*Sqrt[a]*(b*c - a*d)^4) - (Sqrt[d]*(15*b^2
*c^2 + 10*a*b*c*d - a^2*d^2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(8*c^(3/2)*(b*c - a*d)
^4)

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Rubi [A]  time = 0.603195, antiderivative size = 200, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -\frac{\sqrt{d} \left (-a^2 d^2+10 a b c d+15 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{8 c^{3/2} (b c-a d)^4}+\frac{b^{3/2} (5 a d+b c) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 \sqrt{a} (b c-a d)^4}-\frac{d x (a d+11 b c)}{8 c \left (c+d x^2\right ) (b c-a d)^3}-\frac{3 d x}{4 \left (c+d x^2\right )^2 (b c-a d)^2}-\frac{x}{2 \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

(-3*d*x)/(4*(b*c - a*d)^2*(c + d*x^2)^2) - x/(2*(b*c - a*d)*(a + b*x^2)*(c + d*x
^2)^2) - (d*(11*b*c + a*d)*x)/(8*c*(b*c - a*d)^3*(c + d*x^2)) + (b^(3/2)*(b*c +
5*a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*Sqrt[a]*(b*c - a*d)^4) - (Sqrt[d]*(15*b^2
*c^2 + 10*a*b*c*d - a^2*d^2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(8*c^(3/2)*(b*c - a*d)
^4)

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Rubi in Sympy [A]  time = 117.255, size = 178, normalized size = 0.89 \[ - \frac{3 d x}{4 \left (c + d x^{2}\right )^{2} \left (a d - b c\right )^{2}} + \frac{x}{2 \left (a + b x^{2}\right ) \left (c + d x^{2}\right )^{2} \left (a d - b c\right )} + \frac{d x \left (a d + 11 b c\right )}{8 c \left (c + d x^{2}\right ) \left (a d - b c\right )^{3}} + \frac{\sqrt{d} \left (a^{2} d^{2} - 10 a b c d - 15 b^{2} c^{2}\right ) \operatorname{atan}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{8 c^{\frac{3}{2}} \left (a d - b c\right )^{4}} + \frac{b^{\frac{3}{2}} \left (5 a d + b c\right ) \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2 \sqrt{a} \left (a d - b c\right )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2/(b*x**2+a)**2/(d*x**2+c)**3,x)

[Out]

-3*d*x/(4*(c + d*x**2)**2*(a*d - b*c)**2) + x/(2*(a + b*x**2)*(c + d*x**2)**2*(a
*d - b*c)) + d*x*(a*d + 11*b*c)/(8*c*(c + d*x**2)*(a*d - b*c)**3) + sqrt(d)*(a**
2*d**2 - 10*a*b*c*d - 15*b**2*c**2)*atan(sqrt(d)*x/sqrt(c))/(8*c**(3/2)*(a*d - b
*c)**4) + b**(3/2)*(5*a*d + b*c)*atan(sqrt(b)*x/sqrt(a))/(2*sqrt(a)*(a*d - b*c)*
*4)

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Mathematica [A]  time = 0.706733, size = 171, normalized size = 0.86 \[ \frac{\frac{\sqrt{d} \left (a^2 d^2-10 a b c d-15 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{c^{3/2}}+\frac{4 b^{3/2} (5 a d+b c) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a}}-\frac{4 b^2 x (b c-a d)}{a+b x^2}+\frac{d x (a d-b c) (a d+7 b c)}{c \left (c+d x^2\right )}-\frac{2 d x (b c-a d)^2}{\left (c+d x^2\right )^2}}{8 (b c-a d)^4} \]

Antiderivative was successfully verified.

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

[Out]

((-4*b^2*(b*c - a*d)*x)/(a + b*x^2) - (2*d*(b*c - a*d)^2*x)/(c + d*x^2)^2 + (d*(
-(b*c) + a*d)*(7*b*c + a*d)*x)/(c*(c + d*x^2)) + (4*b^(3/2)*(b*c + 5*a*d)*ArcTan
[(Sqrt[b]*x)/Sqrt[a]])/Sqrt[a] + (Sqrt[d]*(-15*b^2*c^2 - 10*a*b*c*d + a^2*d^2)*A
rcTan[(Sqrt[d]*x)/Sqrt[c]])/c^(3/2))/(8*(b*c - a*d)^4)

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Maple [B]  time = 0.022, size = 391, normalized size = 2. \[{\frac{{d}^{4}{x}^{3}{a}^{2}}{8\, \left ( ad-bc \right ) ^{4} \left ( d{x}^{2}+c \right ) ^{2}c}}+{\frac{3\,{d}^{3}{x}^{3}ab}{4\, \left ( ad-bc \right ) ^{4} \left ( d{x}^{2}+c \right ) ^{2}}}-{\frac{7\,{d}^{2}{x}^{3}{b}^{2}c}{8\, \left ( ad-bc \right ) ^{4} \left ( d{x}^{2}+c \right ) ^{2}}}+{\frac{5\,abc{d}^{2}x}{4\, \left ( ad-bc \right ) ^{4} \left ( d{x}^{2}+c \right ) ^{2}}}-{\frac{9\,{b}^{2}{c}^{2}dx}{8\, \left ( ad-bc \right ) ^{4} \left ( d{x}^{2}+c \right ) ^{2}}}-{\frac{{a}^{2}{d}^{3}x}{8\, \left ( ad-bc \right ) ^{4} \left ( d{x}^{2}+c \right ) ^{2}}}+{\frac{{a}^{2}{d}^{3}}{8\, \left ( ad-bc \right ) ^{4}c}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}-{\frac{5\,ab{d}^{2}}{4\, \left ( ad-bc \right ) ^{4}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}-{\frac{15\,{b}^{2}cd}{8\, \left ( ad-bc \right ) ^{4}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}+{\frac{{b}^{2}xad}{2\, \left ( ad-bc \right ) ^{4} \left ( b{x}^{2}+a \right ) }}-{\frac{{b}^{3}xc}{2\, \left ( ad-bc \right ) ^{4} \left ( b{x}^{2}+a \right ) }}+{\frac{5\,a{b}^{2}d}{2\, \left ( ad-bc \right ) ^{4}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{{b}^{3}c}{2\, \left ( ad-bc \right ) ^{4}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(b*x^2+a)^2/(d*x^2+c)^3,x)

[Out]

1/8*d^4/(a*d-b*c)^4/(d*x^2+c)^2/c*x^3*a^2+3/4*d^3/(a*d-b*c)^4/(d*x^2+c)^2*x^3*a*
b-7/8*d^2/(a*d-b*c)^4/(d*x^2+c)^2*x^3*b^2*c+5/4*d^2/(a*d-b*c)^4/(d*x^2+c)^2*a*b*
c*x-9/8*d/(a*d-b*c)^4/(d*x^2+c)^2*b^2*c^2*x-1/8*d^3/(a*d-b*c)^4/(d*x^2+c)^2*a^2*
x+1/8*d^3/(a*d-b*c)^4/c/(c*d)^(1/2)*arctan(x*d/(c*d)^(1/2))*a^2-5/4*d^2/(a*d-b*c
)^4/(c*d)^(1/2)*arctan(x*d/(c*d)^(1/2))*a*b-15/8*d/(a*d-b*c)^4*c/(c*d)^(1/2)*arc
tan(x*d/(c*d)^(1/2))*b^2+1/2*b^2/(a*d-b*c)^4*x/(b*x^2+a)*a*d-1/2*b^3/(a*d-b*c)^4
*x/(b*x^2+a)*c+5/2*b^2/(a*d-b*c)^4/(a*b)^(1/2)*arctan(x*b/(a*b)^(1/2))*a*d+1/2*b
^3/(a*d-b*c)^4/(a*b)^(1/2)*arctan(x*b/(a*b)^(1/2))*c

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.96483, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/16*(2*(11*b^3*c^2*d^2 - 10*a*b^2*c*d^3 - a^2*b*d^4)*x^5 + 2*(17*b^3*c^3*d -
11*a*b^2*c^2*d^2 - 5*a^2*b*c*d^3 - a^3*d^4)*x^3 - 4*(a*b^2*c^4 + 5*a^2*b*c^3*d +
(b^3*c^2*d^2 + 5*a*b^2*c*d^3)*x^6 + (2*b^3*c^3*d + 11*a*b^2*c^2*d^2 + 5*a^2*b*c
*d^3)*x^4 + (b^3*c^4 + 7*a*b^2*c^3*d + 10*a^2*b*c^2*d^2)*x^2)*sqrt(-b/a)*log((b*
x^2 + 2*a*x*sqrt(-b/a) - a)/(b*x^2 + a)) + (15*a*b^2*c^4 + 10*a^2*b*c^3*d - a^3*
c^2*d^2 + (15*b^3*c^2*d^2 + 10*a*b^2*c*d^3 - a^2*b*d^4)*x^6 + (30*b^3*c^3*d + 35
*a*b^2*c^2*d^2 + 8*a^2*b*c*d^3 - a^3*d^4)*x^4 + (15*b^3*c^4 + 40*a*b^2*c^3*d + 1
9*a^2*b*c^2*d^2 - 2*a^3*c*d^3)*x^2)*sqrt(-d/c)*log((d*x^2 + 2*c*x*sqrt(-d/c) - c
)/(d*x^2 + c)) + 2*(4*b^3*c^4 + 5*a*b^2*c^3*d - 10*a^2*b*c^2*d^2 + a^3*c*d^3)*x)
/(a*b^4*c^7 - 4*a^2*b^3*c^6*d + 6*a^3*b^2*c^5*d^2 - 4*a^4*b*c^4*d^3 + a^5*c^3*d^
4 + (b^5*c^5*d^2 - 4*a*b^4*c^4*d^3 + 6*a^2*b^3*c^3*d^4 - 4*a^3*b^2*c^2*d^5 + a^4
*b*c*d^6)*x^6 + (2*b^5*c^6*d - 7*a*b^4*c^5*d^2 + 8*a^2*b^3*c^4*d^3 - 2*a^3*b^2*c
^3*d^4 - 2*a^4*b*c^2*d^5 + a^5*c*d^6)*x^4 + (b^5*c^7 - 2*a*b^4*c^6*d - 2*a^2*b^3
*c^5*d^2 + 8*a^3*b^2*c^4*d^3 - 7*a^4*b*c^3*d^4 + 2*a^5*c^2*d^5)*x^2), -1/8*((11*
b^3*c^2*d^2 - 10*a*b^2*c*d^3 - a^2*b*d^4)*x^5 + (17*b^3*c^3*d - 11*a*b^2*c^2*d^2
- 5*a^2*b*c*d^3 - a^3*d^4)*x^3 + (15*a*b^2*c^4 + 10*a^2*b*c^3*d - a^3*c^2*d^2 +
(15*b^3*c^2*d^2 + 10*a*b^2*c*d^3 - a^2*b*d^4)*x^6 + (30*b^3*c^3*d + 35*a*b^2*c^
2*d^2 + 8*a^2*b*c*d^3 - a^3*d^4)*x^4 + (15*b^3*c^4 + 40*a*b^2*c^3*d + 19*a^2*b*c
^2*d^2 - 2*a^3*c*d^3)*x^2)*sqrt(d/c)*arctan(d*x/(c*sqrt(d/c))) - 2*(a*b^2*c^4 +
5*a^2*b*c^3*d + (b^3*c^2*d^2 + 5*a*b^2*c*d^3)*x^6 + (2*b^3*c^3*d + 11*a*b^2*c^2*
d^2 + 5*a^2*b*c*d^3)*x^4 + (b^3*c^4 + 7*a*b^2*c^3*d + 10*a^2*b*c^2*d^2)*x^2)*sqr
t(-b/a)*log((b*x^2 + 2*a*x*sqrt(-b/a) - a)/(b*x^2 + a)) + (4*b^3*c^4 + 5*a*b^2*c
^3*d - 10*a^2*b*c^2*d^2 + a^3*c*d^3)*x)/(a*b^4*c^7 - 4*a^2*b^3*c^6*d + 6*a^3*b^2
*c^5*d^2 - 4*a^4*b*c^4*d^3 + a^5*c^3*d^4 + (b^5*c^5*d^2 - 4*a*b^4*c^4*d^3 + 6*a^
2*b^3*c^3*d^4 - 4*a^3*b^2*c^2*d^5 + a^4*b*c*d^6)*x^6 + (2*b^5*c^6*d - 7*a*b^4*c^
5*d^2 + 8*a^2*b^3*c^4*d^3 - 2*a^3*b^2*c^3*d^4 - 2*a^4*b*c^2*d^5 + a^5*c*d^6)*x^4
+ (b^5*c^7 - 2*a*b^4*c^6*d - 2*a^2*b^3*c^5*d^2 + 8*a^3*b^2*c^4*d^3 - 7*a^4*b*c^
3*d^4 + 2*a^5*c^2*d^5)*x^2), -1/16*(2*(11*b^3*c^2*d^2 - 10*a*b^2*c*d^3 - a^2*b*d
^4)*x^5 + 2*(17*b^3*c^3*d - 11*a*b^2*c^2*d^2 - 5*a^2*b*c*d^3 - a^3*d^4)*x^3 - 8*
(a*b^2*c^4 + 5*a^2*b*c^3*d + (b^3*c^2*d^2 + 5*a*b^2*c*d^3)*x^6 + (2*b^3*c^3*d +
11*a*b^2*c^2*d^2 + 5*a^2*b*c*d^3)*x^4 + (b^3*c^4 + 7*a*b^2*c^3*d + 10*a^2*b*c^2*
d^2)*x^2)*sqrt(b/a)*arctan(b*x/(a*sqrt(b/a))) + (15*a*b^2*c^4 + 10*a^2*b*c^3*d -
a^3*c^2*d^2 + (15*b^3*c^2*d^2 + 10*a*b^2*c*d^3 - a^2*b*d^4)*x^6 + (30*b^3*c^3*d
+ 35*a*b^2*c^2*d^2 + 8*a^2*b*c*d^3 - a^3*d^4)*x^4 + (15*b^3*c^4 + 40*a*b^2*c^3*
d + 19*a^2*b*c^2*d^2 - 2*a^3*c*d^3)*x^2)*sqrt(-d/c)*log((d*x^2 + 2*c*x*sqrt(-d/c
) - c)/(d*x^2 + c)) + 2*(4*b^3*c^4 + 5*a*b^2*c^3*d - 10*a^2*b*c^2*d^2 + a^3*c*d^
3)*x)/(a*b^4*c^7 - 4*a^2*b^3*c^6*d + 6*a^3*b^2*c^5*d^2 - 4*a^4*b*c^4*d^3 + a^5*c
^3*d^4 + (b^5*c^5*d^2 - 4*a*b^4*c^4*d^3 + 6*a^2*b^3*c^3*d^4 - 4*a^3*b^2*c^2*d^5
+ a^4*b*c*d^6)*x^6 + (2*b^5*c^6*d - 7*a*b^4*c^5*d^2 + 8*a^2*b^3*c^4*d^3 - 2*a^3*
b^2*c^3*d^4 - 2*a^4*b*c^2*d^5 + a^5*c*d^6)*x^4 + (b^5*c^7 - 2*a*b^4*c^6*d - 2*a^
2*b^3*c^5*d^2 + 8*a^3*b^2*c^4*d^3 - 7*a^4*b*c^3*d^4 + 2*a^5*c^2*d^5)*x^2), -1/8*
((11*b^3*c^2*d^2 - 10*a*b^2*c*d^3 - a^2*b*d^4)*x^5 + (17*b^3*c^3*d - 11*a*b^2*c^
2*d^2 - 5*a^2*b*c*d^3 - a^3*d^4)*x^3 - 4*(a*b^2*c^4 + 5*a^2*b*c^3*d + (b^3*c^2*d
^2 + 5*a*b^2*c*d^3)*x^6 + (2*b^3*c^3*d + 11*a*b^2*c^2*d^2 + 5*a^2*b*c*d^3)*x^4 +
(b^3*c^4 + 7*a*b^2*c^3*d + 10*a^2*b*c^2*d^2)*x^2)*sqrt(b/a)*arctan(b*x/(a*sqrt(
b/a))) + (15*a*b^2*c^4 + 10*a^2*b*c^3*d - a^3*c^2*d^2 + (15*b^3*c^2*d^2 + 10*a*b
^2*c*d^3 - a^2*b*d^4)*x^6 + (30*b^3*c^3*d + 35*a*b^2*c^2*d^2 + 8*a^2*b*c*d^3 - a
^3*d^4)*x^4 + (15*b^3*c^4 + 40*a*b^2*c^3*d + 19*a^2*b*c^2*d^2 - 2*a^3*c*d^3)*x^2
)*sqrt(d/c)*arctan(d*x/(c*sqrt(d/c))) + (4*b^3*c^4 + 5*a*b^2*c^3*d - 10*a^2*b*c^
2*d^2 + a^3*c*d^3)*x)/(a*b^4*c^7 - 4*a^2*b^3*c^6*d + 6*a^3*b^2*c^5*d^2 - 4*a^4*b
*c^4*d^3 + a^5*c^3*d^4 + (b^5*c^5*d^2 - 4*a*b^4*c^4*d^3 + 6*a^2*b^3*c^3*d^4 - 4*
a^3*b^2*c^2*d^5 + a^4*b*c*d^6)*x^6 + (2*b^5*c^6*d - 7*a*b^4*c^5*d^2 + 8*a^2*b^3*
c^4*d^3 - 2*a^3*b^2*c^3*d^4 - 2*a^4*b*c^2*d^5 + a^5*c*d^6)*x^4 + (b^5*c^7 - 2*a*
b^4*c^6*d - 2*a^2*b^3*c^5*d^2 + 8*a^3*b^2*c^4*d^3 - 7*a^4*b*c^3*d^4 + 2*a^5*c^2*
d^5)*x^2)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2/(b*x**2+a)**2/(d*x**2+c)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.25718, size = 428, normalized size = 2.14 \[ -\frac{b^{2} x}{2 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )}{\left (b x^{2} + a\right )}} + \frac{{\left (b^{3} c + 5 \, a b^{2} d\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{2 \,{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \sqrt{a b}} - \frac{{\left (15 \, b^{2} c^{2} d + 10 \, a b c d^{2} - a^{2} d^{3}\right )} \arctan \left (\frac{d x}{\sqrt{c d}}\right )}{8 \,{\left (b^{4} c^{5} - 4 \, a b^{3} c^{4} d + 6 \, a^{2} b^{2} c^{3} d^{2} - 4 \, a^{3} b c^{2} d^{3} + a^{4} c d^{4}\right )} \sqrt{c d}} - \frac{7 \, b c d^{2} x^{3} + a d^{3} x^{3} + 9 \, b c^{2} d x - a c d^{2} x}{8 \,{\left (b^{3} c^{4} - 3 \, a b^{2} c^{3} d + 3 \, a^{2} b c^{2} d^{2} - a^{3} c d^{3}\right )}{\left (d x^{2} + c\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*b^2*x/((b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*(b*x^2 + a)) + 1
/2*(b^3*c + 5*a*b^2*d)*arctan(b*x/sqrt(a*b))/((b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b
^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*sqrt(a*b)) - 1/8*(15*b^2*c^2*d + 10*a*b*c*
d^2 - a^2*d^3)*arctan(d*x/sqrt(c*d))/((b^4*c^5 - 4*a*b^3*c^4*d + 6*a^2*b^2*c^3*d
^2 - 4*a^3*b*c^2*d^3 + a^4*c*d^4)*sqrt(c*d)) - 1/8*(7*b*c*d^2*x^3 + a*d^3*x^3 +
9*b*c^2*d*x - a*c*d^2*x)/((b^3*c^4 - 3*a*b^2*c^3*d + 3*a^2*b*c^2*d^2 - a^3*c*d^3
)*(d*x^2 + c)^2)