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

Optimal. Leaf size=324 \[ -\frac{b \sqrt{1-d^2 x^2} \left (45 a^2 d^4+60 a c d^2+10 b^2 d^2+24 c^2\right )}{15 d^6}-\frac{x \sqrt{1-d^2 x^2} \left (24 a^2 c d^4+24 a b^2 d^4+18 a c^2 d^2+18 b^2 c d^2+5 c^3\right )}{16 d^6}+\frac{\sin ^{-1}(d x) \left (16 a^3 d^6+24 a^2 c d^4+24 a b^2 d^4+18 a c^2 d^2+18 b^2 c d^2+5 c^3\right )}{16 d^7}-\frac{b x^2 \sqrt{1-d^2 x^2} \left (30 a c d^2+5 b^2 d^2+12 c^2\right )}{15 d^4}-\frac{c x^3 \sqrt{1-d^2 x^2} \left (18 a c d^2+18 b^2 d^2+5 c^2\right )}{24 d^4}-\frac{3 b c^2 x^4 \sqrt{1-d^2 x^2}}{5 d^2}-\frac{c^3 x^5 \sqrt{1-d^2 x^2}}{6 d^2} \]

[Out]

-(b*(24*c^2 + 10*b^2*d^2 + 60*a*c*d^2 + 45*a^2*d^4)*Sqrt[1 - d^2*x^2])/(15*d^6)
- ((5*c^3 + 18*b^2*c*d^2 + 18*a*c^2*d^2 + 24*a*b^2*d^4 + 24*a^2*c*d^4)*x*Sqrt[1
- d^2*x^2])/(16*d^6) - (b*(12*c^2 + 5*b^2*d^2 + 30*a*c*d^2)*x^2*Sqrt[1 - d^2*x^2
])/(15*d^4) - (c*(5*c^2 + 18*b^2*d^2 + 18*a*c*d^2)*x^3*Sqrt[1 - d^2*x^2])/(24*d^
4) - (3*b*c^2*x^4*Sqrt[1 - d^2*x^2])/(5*d^2) - (c^3*x^5*Sqrt[1 - d^2*x^2])/(6*d^
2) + ((5*c^3 + 18*b^2*c*d^2 + 18*a*c^2*d^2 + 24*a*b^2*d^4 + 24*a^2*c*d^4 + 16*a^
3*d^6)*ArcSin[d*x])/(16*d^7)

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Rubi [A]  time = 1.4199, antiderivative size = 324, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{b \sqrt{1-d^2 x^2} \left (45 a^2 d^4+60 a c d^2+10 b^2 d^2+24 c^2\right )}{15 d^6}-\frac{x \sqrt{1-d^2 x^2} \left (24 a^2 c d^4+24 a b^2 d^4+18 a c^2 d^2+18 b^2 c d^2+5 c^3\right )}{16 d^6}+\frac{\sin ^{-1}(d x) \left (16 a^3 d^6+24 a^2 c d^4+24 a b^2 d^4+18 a c^2 d^2+18 b^2 c d^2+5 c^3\right )}{16 d^7}-\frac{b x^2 \sqrt{1-d^2 x^2} \left (30 a c d^2+5 b^2 d^2+12 c^2\right )}{15 d^4}-\frac{c x^3 \sqrt{1-d^2 x^2} \left (18 a c d^2+18 b^2 d^2+5 c^2\right )}{24 d^4}-\frac{3 b c^2 x^4 \sqrt{1-d^2 x^2}}{5 d^2}-\frac{c^3 x^5 \sqrt{1-d^2 x^2}}{6 d^2} \]

Antiderivative was successfully verified.

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

[Out]

-(b*(24*c^2 + 10*b^2*d^2 + 60*a*c*d^2 + 45*a^2*d^4)*Sqrt[1 - d^2*x^2])/(15*d^6)
- ((5*c^3 + 18*b^2*c*d^2 + 18*a*c^2*d^2 + 24*a*b^2*d^4 + 24*a^2*c*d^4)*x*Sqrt[1
- d^2*x^2])/(16*d^6) - (b*(12*c^2 + 5*b^2*d^2 + 30*a*c*d^2)*x^2*Sqrt[1 - d^2*x^2
])/(15*d^4) - (c*(5*c^2 + 18*b^2*d^2 + 18*a*c*d^2)*x^3*Sqrt[1 - d^2*x^2])/(24*d^
4) - (3*b*c^2*x^4*Sqrt[1 - d^2*x^2])/(5*d^2) - (c^3*x^5*Sqrt[1 - d^2*x^2])/(6*d^
2) + ((5*c^3 + 18*b^2*c*d^2 + 18*a*c^2*d^2 + 24*a*b^2*d^4 + 24*a^2*c*d^4 + 16*a^
3*d^6)*ArcSin[d*x])/(16*d^7)

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Rubi in Sympy [A]  time = 169.637, size = 309, normalized size = 0.95 \[ - \frac{\left (8 b + 5 c x\right ) \sqrt{- d^{2} x^{2} + 1} \left (a + b x + c x^{2}\right )^{2}}{30 d^{2}} + \frac{\left (16 a^{3} d^{6} + 24 a^{2} c d^{4} + 24 a b^{2} d^{4} + 18 a c^{2} d^{2} + 18 b^{2} c d^{2} + 5 c^{3}\right ) \operatorname{asin}{\left (d x \right )}}{16 d^{7}} - \frac{\left (3 b \left (86 a c d^{2} + 2 b^{2} d^{2} + 71 c^{2}\right ) + 3 c x \left (6 b^{2} d^{2} + 25 c \left (2 a d^{2} + c\right )\right )\right ) \sqrt{- d^{2} x^{2} + 1} \left (a + b x + c x^{2}\right )}{360 c d^{4}} - \frac{\left (6 b \left (242 a^{2} c d^{4} - 2 a b^{2} d^{4} + 409 a c^{2} d^{2} + 80 b^{2} c d^{2} + 192 c^{3}\right ) + x \left (660 a^{2} c^{2} d^{4} + 144 a b^{2} c d^{4} + 660 a c^{3} d^{2} - 12 b^{4} d^{4} + 384 b^{2} c^{2} d^{2} + 225 c^{4}\right )\right ) \sqrt{- d^{2} x^{2} + 1}}{720 c d^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+b*x+a)**3/(-d*x+1)**(1/2)/(d*x+1)**(1/2),x)

[Out]

-(8*b + 5*c*x)*sqrt(-d**2*x**2 + 1)*(a + b*x + c*x**2)**2/(30*d**2) + (16*a**3*d
**6 + 24*a**2*c*d**4 + 24*a*b**2*d**4 + 18*a*c**2*d**2 + 18*b**2*c*d**2 + 5*c**3
)*asin(d*x)/(16*d**7) - (3*b*(86*a*c*d**2 + 2*b**2*d**2 + 71*c**2) + 3*c*x*(6*b*
*2*d**2 + 25*c*(2*a*d**2 + c)))*sqrt(-d**2*x**2 + 1)*(a + b*x + c*x**2)/(360*c*d
**4) - (6*b*(242*a**2*c*d**4 - 2*a*b**2*d**4 + 409*a*c**2*d**2 + 80*b**2*c*d**2
+ 192*c**3) + x*(660*a**2*c**2*d**4 + 144*a*b**2*c*d**4 + 660*a*c**3*d**2 - 12*b
**4*d**4 + 384*b**2*c**2*d**2 + 225*c**4))*sqrt(-d**2*x**2 + 1)/(720*c*d**6)

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Mathematica [A]  time = 0.364603, size = 229, normalized size = 0.71 \[ \frac{15 \sin ^{-1}(d x) \left (16 a^3 d^6+24 a^2 c d^4+24 a b^2 d^4+18 a c^2 d^2+18 b^2 c d^2+5 c^3\right )-d \sqrt{1-d^2 x^2} \left (48 b \left (15 a^2 d^4+10 a c d^2 \left (d^2 x^2+2\right )+c^2 \left (3 d^4 x^4+4 d^2 x^2+8\right )\right )+5 c x \left (72 a^2 d^4+18 a c d^2 \left (2 d^2 x^2+3\right )+c^2 \left (8 d^4 x^4+10 d^2 x^2+15\right )\right )+90 b^2 d^2 x \left (4 a d^2+c \left (2 d^2 x^2+3\right )\right )+80 b^3 d^2 \left (d^2 x^2+2\right )\right )}{240 d^7} \]

Antiderivative was successfully verified.

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

[Out]

(-(d*Sqrt[1 - d^2*x^2]*(80*b^3*d^2*(2 + d^2*x^2) + 90*b^2*d^2*x*(4*a*d^2 + c*(3
+ 2*d^2*x^2)) + 48*b*(15*a^2*d^4 + 10*a*c*d^2*(2 + d^2*x^2) + c^2*(8 + 4*d^2*x^2
+ 3*d^4*x^4)) + 5*c*x*(72*a^2*d^4 + 18*a*c*d^2*(3 + 2*d^2*x^2) + c^2*(15 + 10*d
^2*x^2 + 8*d^4*x^4)))) + 15*(5*c^3 + 18*b^2*c*d^2 + 18*a*c^2*d^2 + 24*a*b^2*d^4
+ 24*a^2*c*d^4 + 16*a^3*d^6)*ArcSin[d*x])/(240*d^7)

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Maple [C]  time = 0.084, size = 602, normalized size = 1.9 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x+a)^3/(-d*x+1)^(1/2)/(d*x+1)^(1/2),x)

[Out]

-1/240*(-d*x+1)^(1/2)*(d*x+1)^(1/2)*(40*csgn(d)*x^5*c^3*d^5*(-d^2*x^2+1)^(1/2)+1
44*csgn(d)*x^4*b*c^2*d^5*(-d^2*x^2+1)^(1/2)+180*csgn(d)*x^3*a*c^2*d^5*(-d^2*x^2+
1)^(1/2)+180*csgn(d)*x^3*b^2*c*d^5*(-d^2*x^2+1)^(1/2)+480*csgn(d)*x^2*a*b*c*d^5*
(-d^2*x^2+1)^(1/2)+80*csgn(d)*x^2*b^3*d^5*(-d^2*x^2+1)^(1/2)+50*c^3*x^3*(-d^2*x^
2+1)^(1/2)*d^3*csgn(d)+360*x*(-d^2*x^2+1)^(1/2)*a^2*c*d^5*csgn(d)+360*x*(-d^2*x^
2+1)^(1/2)*a*b^2*d^5*csgn(d)+192*csgn(d)*d^3*(-d^2*x^2+1)^(1/2)*x^2*b*c^2+720*(-
d^2*x^2+1)^(1/2)*a^2*b*d^5*csgn(d)-240*arctan(csgn(d)*d*x/(-d^2*x^2+1)^(1/2))*a^
3*d^6+270*x*(-d^2*x^2+1)^(1/2)*a*c^2*d^3*csgn(d)+270*x*(-d^2*x^2+1)^(1/2)*b^2*c*
d^3*csgn(d)+960*(-d^2*x^2+1)^(1/2)*a*b*c*d^3*csgn(d)+160*(-d^2*x^2+1)^(1/2)*b^3*
d^3*csgn(d)-360*arctan(csgn(d)*d*x/(-d^2*x^2+1)^(1/2))*a^2*c*d^4-360*arctan(csgn
(d)*d*x/(-d^2*x^2+1)^(1/2))*a*b^2*d^4+75*x*(-d^2*x^2+1)^(1/2)*c^3*csgn(d)*d+384*
(-d^2*x^2+1)^(1/2)*b*c^2*csgn(d)*d-270*arctan(csgn(d)*d*x/(-d^2*x^2+1)^(1/2))*a*
c^2*d^2-270*arctan(csgn(d)*d*x/(-d^2*x^2+1)^(1/2))*b^2*c*d^2-75*arctan(csgn(d)*d
*x/(-d^2*x^2+1)^(1/2))*c^3)*csgn(d)/(-d^2*x^2+1)^(1/2)/d^7

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Maxima [A]  time = 0.794152, size = 554, normalized size = 1.71 \[ -\frac{\sqrt{-d^{2} x^{2} + 1} c^{3} x^{5}}{6 \, d^{2}} - \frac{3 \, \sqrt{-d^{2} x^{2} + 1} b c^{2} x^{4}}{5 \, d^{2}} + \frac{a^{3} \arcsin \left (\frac{d^{2} x}{\sqrt{d^{2}}}\right )}{\sqrt{d^{2}}} - \frac{5 \, \sqrt{-d^{2} x^{2} + 1} c^{3} x^{3}}{24 \, d^{4}} - \frac{3 \, \sqrt{-d^{2} x^{2} + 1}{\left (b^{2} c + a c^{2}\right )} x^{3}}{4 \, d^{2}} - \frac{3 \, \sqrt{-d^{2} x^{2} + 1} a^{2} b}{d^{2}} - \frac{4 \, \sqrt{-d^{2} x^{2} + 1} b c^{2} x^{2}}{5 \, d^{4}} - \frac{\sqrt{-d^{2} x^{2} + 1}{\left (b^{3} + 6 \, a b c\right )} x^{2}}{3 \, d^{2}} - \frac{3 \, \sqrt{-d^{2} x^{2} + 1}{\left (a b^{2} + a^{2} c\right )} x}{2 \, d^{2}} + \frac{3 \,{\left (a b^{2} + a^{2} c\right )} \arcsin \left (\frac{d^{2} x}{\sqrt{d^{2}}}\right )}{2 \, \sqrt{d^{2}} d^{2}} - \frac{5 \, \sqrt{-d^{2} x^{2} + 1} c^{3} x}{16 \, d^{6}} - \frac{9 \, \sqrt{-d^{2} x^{2} + 1}{\left (b^{2} c + a c^{2}\right )} x}{8 \, d^{4}} - \frac{8 \, \sqrt{-d^{2} x^{2} + 1} b c^{2}}{5 \, d^{6}} - \frac{2 \, \sqrt{-d^{2} x^{2} + 1}{\left (b^{3} + 6 \, a b c\right )}}{3 \, d^{4}} + \frac{5 \, c^{3} \arcsin \left (\frac{d^{2} x}{\sqrt{d^{2}}}\right )}{16 \, \sqrt{d^{2}} d^{6}} + \frac{9 \,{\left (b^{2} c + a c^{2}\right )} \arcsin \left (\frac{d^{2} x}{\sqrt{d^{2}}}\right )}{8 \, \sqrt{d^{2}} d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6*sqrt(-d^2*x^2 + 1)*c^3*x^5/d^2 - 3/5*sqrt(-d^2*x^2 + 1)*b*c^2*x^4/d^2 + a^3
*arcsin(d^2*x/sqrt(d^2))/sqrt(d^2) - 5/24*sqrt(-d^2*x^2 + 1)*c^3*x^3/d^4 - 3/4*s
qrt(-d^2*x^2 + 1)*(b^2*c + a*c^2)*x^3/d^2 - 3*sqrt(-d^2*x^2 + 1)*a^2*b/d^2 - 4/5
*sqrt(-d^2*x^2 + 1)*b*c^2*x^2/d^4 - 1/3*sqrt(-d^2*x^2 + 1)*(b^3 + 6*a*b*c)*x^2/d
^2 - 3/2*sqrt(-d^2*x^2 + 1)*(a*b^2 + a^2*c)*x/d^2 + 3/2*(a*b^2 + a^2*c)*arcsin(d
^2*x/sqrt(d^2))/(sqrt(d^2)*d^2) - 5/16*sqrt(-d^2*x^2 + 1)*c^3*x/d^6 - 9/8*sqrt(-
d^2*x^2 + 1)*(b^2*c + a*c^2)*x/d^4 - 8/5*sqrt(-d^2*x^2 + 1)*b*c^2/d^6 - 2/3*sqrt
(-d^2*x^2 + 1)*(b^3 + 6*a*b*c)/d^4 + 5/16*c^3*arcsin(d^2*x/sqrt(d^2))/(sqrt(d^2)
*d^6) + 9/8*(b^2*c + a*c^2)*arcsin(d^2*x/sqrt(d^2))/(sqrt(d^2)*d^4)

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Fricas [A]  time = 0.289888, size = 1593, normalized size = 4.92 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/240*(240*c^3*d^11*x^11 + 864*b*c^2*d^11*x^10 + 11520*a^2*b*d^7*x^2 - 20*(61*c^
3*d^9 - 54*(b^2*c + a*c^2)*d^11)*x^9 - 480*(9*b*c^2*d^9 - (b^3 + 6*a*b*c)*d^11)*
x^8 + 30*(72*(a*b^2 + a^2*c)*d^11 + 37*c^3*d^7 - 174*(b^2*c + a*c^2)*d^9)*x^7 +
80*(45*a^2*b*d^11 + 48*b*c^2*d^7 - 28*(b^3 + 6*a*b*c)*d^9)*x^6 - 30*(456*(a*b^2
+ a^2*c)*d^9 + 31*c^3*d^5 - 42*(b^2*c + a*c^2)*d^7)*x^5 - 960*(15*a^2*b*d^9 - 2*
(b^3 + 6*a*b*c)*d^7)*x^4 + 640*(36*(a*b^2 + a^2*c)*d^7 + 5*c^3*d^3 + 18*(b^2*c +
a*c^2)*d^5)*x^3 - (40*c^3*d^11*x^11 + 144*b*c^2*d^11*x^10 + 11520*a^2*b*d^7*x^2
- 10*(67*c^3*d^9 - 18*(b^2*c + a*c^2)*d^11)*x^9 - 80*(30*b*c^2*d^9 - (b^3 + 6*a
*b*c)*d^11)*x^8 + 15*(24*(a*b^2 + a^2*c)*d^11 + 73*c^3*d^7 - 198*(b^2*c + a*c^2)
*d^9)*x^7 + 80*(9*a^2*b*d^11 + 48*b*c^2*d^7 - 16*(b^3 + 6*a*b*c)*d^9)*x^6 - 10*(
648*(a*b^2 + a^2*c)*d^9 + 23*c^3*d^5 - 378*(b^2*c + a*c^2)*d^7)*x^5 - 960*(9*a^2
*b*d^9 - 2*(b^3 + 6*a*b*c)*d^7)*x^4 + 80*(216*(a*b^2 + a^2*c)*d^7 + 25*c^3*d^3 +
90*(b^2*c + a*c^2)*d^5)*x^3 - 480*(24*(a*b^2 + a^2*c)*d^5 + 5*c^3*d + 18*(b^2*c
+ a*c^2)*d^3)*x)*sqrt(d*x + 1)*sqrt(-d*x + 1) - 480*(24*(a*b^2 + a^2*c)*d^5 + 5
*c^3*d + 18*(b^2*c + a*c^2)*d^3)*x + 30*(512*a^3*d^6 - (16*a^3*d^12 + 24*(a*b^2
+ a^2*c)*d^10 + 5*c^3*d^6 + 18*(b^2*c + a*c^2)*d^8)*x^6 + 768*(a*b^2 + a^2*c)*d^
4 + 18*(16*a^3*d^10 + 24*(a*b^2 + a^2*c)*d^8 + 5*c^3*d^4 + 18*(b^2*c + a*c^2)*d^
6)*x^4 + 160*c^3 + 576*(b^2*c + a*c^2)*d^2 - 48*(16*a^3*d^8 + 24*(a*b^2 + a^2*c)
*d^6 + 5*c^3*d^2 + 18*(b^2*c + a*c^2)*d^4)*x^2 - 2*(256*a^3*d^6 + 384*(a*b^2 + a
^2*c)*d^4 + 3*(16*a^3*d^10 + 24*(a*b^2 + a^2*c)*d^8 + 5*c^3*d^4 + 18*(b^2*c + a*
c^2)*d^6)*x^4 + 80*c^3 + 288*(b^2*c + a*c^2)*d^2 - 16*(16*a^3*d^8 + 24*(a*b^2 +
a^2*c)*d^6 + 5*c^3*d^2 + 18*(b^2*c + a*c^2)*d^4)*x^2)*sqrt(d*x + 1)*sqrt(-d*x +
1))*arctan((sqrt(d*x + 1)*sqrt(-d*x + 1) - 1)/(d*x)))/(d^13*x^6 - 18*d^11*x^4 +
48*d^9*x^2 - 32*d^7 + 2*(3*d^11*x^4 - 16*d^9*x^2 + 16*d^7)*sqrt(d*x + 1)*sqrt(-d
*x + 1))

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.31835, size = 518, normalized size = 1.6 \[ -\frac{{\left (720 \, a^{2} b d^{41} - 360 \, a b^{2} d^{40} - 360 \, a^{2} c d^{40} + 240 \, b^{3} d^{39} + 1440 \, a b c d^{39} - 450 \, b^{2} c d^{38} - 450 \, a c^{2} d^{38} + 720 \, b c^{2} d^{37} - 165 \, c^{3} d^{36} +{\left (360 \, a b^{2} d^{40} + 360 \, a^{2} c d^{40} - 160 \, b^{3} d^{39} - 960 \, a b c d^{39} + 810 \, b^{2} c d^{38} + 810 \, a c^{2} d^{38} - 960 \, b c^{2} d^{37} + 425 \, c^{3} d^{36} + 2 \,{\left (40 \, b^{3} d^{39} + 240 \, a b c d^{39} - 270 \, b^{2} c d^{38} - 270 \, a c^{2} d^{38} + 528 \, b c^{2} d^{37} - 275 \, c^{3} d^{36} +{\left (90 \, b^{2} c d^{38} + 90 \, a c^{2} d^{38} - 288 \, b c^{2} d^{37} + 225 \, c^{3} d^{36} + 4 \,{\left (5 \,{\left (d x + 1\right )} c^{3} d^{36} + 18 \, b c^{2} d^{37} - 25 \, c^{3} d^{36}\right )}{\left (d x + 1\right )}\right )}{\left (d x + 1\right )}\right )}{\left (d x + 1\right )}\right )}{\left (d x + 1\right )}\right )} \sqrt{d x + 1} \sqrt{-d x + 1} - 30 \,{\left (16 \, a^{3} d^{42} + 24 \, a b^{2} d^{40} + 24 \, a^{2} c d^{40} + 18 \, b^{2} c d^{38} + 18 \, a c^{2} d^{38} + 5 \, c^{3} d^{36}\right )} \arcsin \left (\frac{1}{2} \, \sqrt{2} \sqrt{d x + 1}\right )}{21626880 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/21626880*((720*a^2*b*d^41 - 360*a*b^2*d^40 - 360*a^2*c*d^40 + 240*b^3*d^39 +
1440*a*b*c*d^39 - 450*b^2*c*d^38 - 450*a*c^2*d^38 + 720*b*c^2*d^37 - 165*c^3*d^3
6 + (360*a*b^2*d^40 + 360*a^2*c*d^40 - 160*b^3*d^39 - 960*a*b*c*d^39 + 810*b^2*c
*d^38 + 810*a*c^2*d^38 - 960*b*c^2*d^37 + 425*c^3*d^36 + 2*(40*b^3*d^39 + 240*a*
b*c*d^39 - 270*b^2*c*d^38 - 270*a*c^2*d^38 + 528*b*c^2*d^37 - 275*c^3*d^36 + (90
*b^2*c*d^38 + 90*a*c^2*d^38 - 288*b*c^2*d^37 + 225*c^3*d^36 + 4*(5*(d*x + 1)*c^3
*d^36 + 18*b*c^2*d^37 - 25*c^3*d^36)*(d*x + 1))*(d*x + 1))*(d*x + 1))*(d*x + 1))
*sqrt(d*x + 1)*sqrt(-d*x + 1) - 30*(16*a^3*d^42 + 24*a*b^2*d^40 + 24*a^2*c*d^40
+ 18*b^2*c*d^38 + 18*a*c^2*d^38 + 5*c^3*d^36)*arcsin(1/2*sqrt(2)*sqrt(d*x + 1)))
/d