3.1207 \(\int (b d+2 c d x)^4 \left (a+b x+c x^2\right )^{5/2} \, dx\)

Optimal. Leaf size=249 \[ -\frac{3 d^4 \left (b^2-4 a c\right )^5 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{16384 c^{7/2}}-\frac{3 d^4 \left (b^2-4 a c\right )^4 (b+2 c x) \sqrt{a+b x+c x^2}}{8192 c^3}-\frac{d^4 \left (b^2-4 a c\right )^3 (b+2 c x)^3 \sqrt{a+b x+c x^2}}{4096 c^3}+\frac{d^4 \left (b^2-4 a c\right )^2 (b+2 c x)^5 \sqrt{a+b x+c x^2}}{1024 c^3}-\frac{d^4 \left (b^2-4 a c\right ) (b+2 c x)^5 \left (a+b x+c x^2\right )^{3/2}}{128 c^2}+\frac{d^4 (b+2 c x)^5 \left (a+b x+c x^2\right )^{5/2}}{20 c} \]

[Out]

(-3*(b^2 - 4*a*c)^4*d^4*(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(8192*c^3) - ((b^2 -
4*a*c)^3*d^4*(b + 2*c*x)^3*Sqrt[a + b*x + c*x^2])/(4096*c^3) + ((b^2 - 4*a*c)^2*
d^4*(b + 2*c*x)^5*Sqrt[a + b*x + c*x^2])/(1024*c^3) - ((b^2 - 4*a*c)*d^4*(b + 2*
c*x)^5*(a + b*x + c*x^2)^(3/2))/(128*c^2) + (d^4*(b + 2*c*x)^5*(a + b*x + c*x^2)
^(5/2))/(20*c) - (3*(b^2 - 4*a*c)^5*d^4*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a +
b*x + c*x^2])])/(16384*c^(7/2))

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Rubi [A]  time = 0.473208, antiderivative size = 249, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ -\frac{3 d^4 \left (b^2-4 a c\right )^5 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{16384 c^{7/2}}-\frac{3 d^4 \left (b^2-4 a c\right )^4 (b+2 c x) \sqrt{a+b x+c x^2}}{8192 c^3}-\frac{d^4 \left (b^2-4 a c\right )^3 (b+2 c x)^3 \sqrt{a+b x+c x^2}}{4096 c^3}+\frac{d^4 \left (b^2-4 a c\right )^2 (b+2 c x)^5 \sqrt{a+b x+c x^2}}{1024 c^3}-\frac{d^4 \left (b^2-4 a c\right ) (b+2 c x)^5 \left (a+b x+c x^2\right )^{3/2}}{128 c^2}+\frac{d^4 (b+2 c x)^5 \left (a+b x+c x^2\right )^{5/2}}{20 c} \]

Antiderivative was successfully verified.

[In]  Int[(b*d + 2*c*d*x)^4*(a + b*x + c*x^2)^(5/2),x]

[Out]

(-3*(b^2 - 4*a*c)^4*d^4*(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(8192*c^3) - ((b^2 -
4*a*c)^3*d^4*(b + 2*c*x)^3*Sqrt[a + b*x + c*x^2])/(4096*c^3) + ((b^2 - 4*a*c)^2*
d^4*(b + 2*c*x)^5*Sqrt[a + b*x + c*x^2])/(1024*c^3) - ((b^2 - 4*a*c)*d^4*(b + 2*
c*x)^5*(a + b*x + c*x^2)^(3/2))/(128*c^2) + (d^4*(b + 2*c*x)^5*(a + b*x + c*x^2)
^(5/2))/(20*c) - (3*(b^2 - 4*a*c)^5*d^4*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a +
b*x + c*x^2])])/(16384*c^(7/2))

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Rubi in Sympy [A]  time = 91.5409, size = 240, normalized size = 0.96 \[ \frac{d^{4} \left (b + 2 c x\right )^{5} \left (a + b x + c x^{2}\right )^{\frac{5}{2}}}{20 c} - \frac{d^{4} \left (b + 2 c x\right )^{5} \left (- 4 a c + b^{2}\right ) \left (a + b x + c x^{2}\right )^{\frac{3}{2}}}{128 c^{2}} + \frac{d^{4} \left (b + 2 c x\right )^{5} \left (- 4 a c + b^{2}\right )^{2} \sqrt{a + b x + c x^{2}}}{1024 c^{3}} - \frac{d^{4} \left (b + 2 c x\right )^{3} \left (- 4 a c + b^{2}\right )^{3} \sqrt{a + b x + c x^{2}}}{4096 c^{3}} - \frac{3 d^{4} \left (b + 2 c x\right ) \left (- 4 a c + b^{2}\right )^{4} \sqrt{a + b x + c x^{2}}}{8192 c^{3}} - \frac{3 d^{4} \left (- 4 a c + b^{2}\right )^{5} \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{16384 c^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2*c*d*x+b*d)**4*(c*x**2+b*x+a)**(5/2),x)

[Out]

d**4*(b + 2*c*x)**5*(a + b*x + c*x**2)**(5/2)/(20*c) - d**4*(b + 2*c*x)**5*(-4*a
*c + b**2)*(a + b*x + c*x**2)**(3/2)/(128*c**2) + d**4*(b + 2*c*x)**5*(-4*a*c +
b**2)**2*sqrt(a + b*x + c*x**2)/(1024*c**3) - d**4*(b + 2*c*x)**3*(-4*a*c + b**2
)**3*sqrt(a + b*x + c*x**2)/(4096*c**3) - 3*d**4*(b + 2*c*x)*(-4*a*c + b**2)**4*
sqrt(a + b*x + c*x**2)/(8192*c**3) - 3*d**4*(-4*a*c + b**2)**5*atanh((b + 2*c*x)
/(2*sqrt(c)*sqrt(a + b*x + c*x**2)))/(16384*c**(7/2))

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Mathematica [A]  time = 0.537218, size = 312, normalized size = 1.25 \[ \frac{d^4 \left (2 \sqrt{c} (b+2 c x) \sqrt{a+x (b+c x)} \left (32 b^4 c^2 \left (64 a^2+1047 a c x^2+2084 c^2 x^4\right )+128 b^3 c^3 x \left (233 a^2+1184 a c x^2+1288 c^2 x^4\right )+128 b^2 c^3 \left (35 a^3+729 a^2 c x^2+2272 a c^2 x^4+1624 c^3 x^6\right )+512 b c^4 x \left (5 a^3+248 a^2 c x^2+504 a c^2 x^4+256 c^3 x^6\right )+256 c^4 \left (-15 a^4+10 a^3 c x^2+248 a^2 c^2 x^4+336 a c^3 x^6+128 c^4 x^8\right )+8 b^6 c \left (11 c x^2-35 a\right )+32 b^5 c^2 x \left (23 a+360 c x^2\right )+15 b^8-40 b^7 c x\right )-15 \left (b^2-4 a c\right )^5 \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )\right )}{81920 c^{7/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(b*d + 2*c*d*x)^4*(a + b*x + c*x^2)^(5/2),x]

[Out]

(d^4*(2*Sqrt[c]*(b + 2*c*x)*Sqrt[a + x*(b + c*x)]*(15*b^8 - 40*b^7*c*x + 8*b^6*c
*(-35*a + 11*c*x^2) + 32*b^5*c^2*x*(23*a + 360*c*x^2) + 128*b^3*c^3*x*(233*a^2 +
1184*a*c*x^2 + 1288*c^2*x^4) + 32*b^4*c^2*(64*a^2 + 1047*a*c*x^2 + 2084*c^2*x^4
) + 512*b*c^4*x*(5*a^3 + 248*a^2*c*x^2 + 504*a*c^2*x^4 + 256*c^3*x^6) + 128*b^2*
c^3*(35*a^3 + 729*a^2*c*x^2 + 2272*a*c^2*x^4 + 1624*c^3*x^6) + 256*c^4*(-15*a^4
+ 10*a^3*c*x^2 + 248*a^2*c^2*x^4 + 336*a*c^3*x^6 + 128*c^4*x^8)) - 15*(b^2 - 4*a
*c)^5*Log[b + 2*c*x + 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]]))/(81920*c^(7/2))

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Maple [B]  time = 0.03, size = 920, normalized size = 3.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2*c*d*x+b*d)^4*(c*x^2+b*x+a)^(5/2),x)

[Out]

11/40*d^4*b^3*(c*x^2+b*x+a)^(7/2)+3/16*d^4*c^(3/2)*a^5*ln((1/2*b+c*x)/c^(1/2)+(c
*x^2+b*x+a)^(1/2))+8/5*d^4*c^3*x^3*(c*x^2+b*x+a)^(7/2)+1/160*d^4*b^4*(c*x^2+b*x+
a)^(5/2)*x+1/320*d^4*b^5/c*(c*x^2+b*x+a)^(5/2)-1/1024*d^4*b^7/c^2*(c*x^2+b*x+a)^
(3/2)+3/8192*d^4*b^9/c^3*(c*x^2+b*x+a)^(1/2)-3/16384*d^4*b^10/c^(7/2)*ln((1/2*b+
c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-1/40*d^4*b^3*a*(c*x^2+b*x+a)^(5/2)-3/64*d^4*b^
3*a^2*(c*x^2+b*x+a)^(3/2)-3/32*d^4*b^3*a^3*(c*x^2+b*x+a)^(1/2)+3/4096*d^4*b^8/c^
2*(c*x^2+b*x+a)^(1/2)*x+9/256*d^4*b^5/c*(c*x^2+b*x+a)^(1/2)*a^2-3/512*d^4*b^7/c^
2*(c*x^2+b*x+a)^(1/2)*a+27/20*d^4*c*b^2*x*(c*x^2+b*x+a)^(7/2)+1/10*d^4*c^2*a^2*(
c*x^2+b*x+a)^(5/2)*x+1/20*d^4*c*a^2*(c*x^2+b*x+a)^(5/2)*b+1/8*d^4*c^2*a^3*(c*x^2
+b*x+a)^(3/2)*x+1/16*d^4*c*a^3*(c*x^2+b*x+a)^(3/2)*b+3/32*d^4*c*a^4*(c*x^2+b*x+a
)^(1/2)*b-3/5*d^4*c^2*a*x*(c*x^2+b*x+a)^(7/2)+15/128*d^4*b^4/c^(1/2)*ln((1/2*b+c
*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a^3-15/512*d^4*b^6/c^(3/2)*ln((1/2*b+c*x)/c^(1/
2)+(c*x^2+b*x+a)^(1/2))*a^2+15/4096*d^4*b^8/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^
2+b*x+a)^(1/2))*a-3/10*d^4*c*b*a*(c*x^2+b*x+a)^(7/2)+12/5*d^4*c^2*b*x^2*(c*x^2+b
*x+a)^(7/2)-15/64*d^4*c^(1/2)*b^2*a^4*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2)
)+3/128*d^4*b^4*(c*x^2+b*x+a)^(3/2)*x*a-1/512*d^4*b^6/c*(c*x^2+b*x+a)^(3/2)*x+3/
256*d^4*b^5/c*(c*x^2+b*x+a)^(3/2)*a+9/128*d^4*b^4*(c*x^2+b*x+a)^(1/2)*x*a^2-1/20
*d^4*c*b^2*a*(c*x^2+b*x+a)^(5/2)*x-3/32*d^4*c*b^2*a^2*(c*x^2+b*x+a)^(3/2)*x-3/16
*d^4*c*b^2*a^3*(c*x^2+b*x+a)^(1/2)*x-3/256*d^4*b^6/c*(c*x^2+b*x+a)^(1/2)*x*a+3/1
6*d^4*c^2*a^4*(c*x^2+b*x+a)^(1/2)*x

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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((2*c*d*x + b*d)^4*(c*x^2 + b*x + a)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.326857, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*d*x + b*d)^4*(c*x^2 + b*x + a)^(5/2),x, algorithm="fricas")

[Out]

[-1/163840*(15*(b^10 - 20*a*b^8*c + 160*a^2*b^6*c^2 - 640*a^3*b^4*c^3 + 1280*a^4
*b^2*c^4 - 1024*a^5*c^5)*d^4*log(-4*(2*c^2*x + b*c)*sqrt(c*x^2 + b*x + a) - (8*c
^2*x^2 + 8*b*c*x + b^2 + 4*a*c)*sqrt(c)) - 4*(65536*c^9*d^4*x^9 + 294912*b*c^8*d
^4*x^8 + 6144*(89*b^2*c^7 + 28*a*c^8)*d^4*x^7 + 21504*(25*b^3*c^6 + 28*a*b*c^7)*
d^4*x^6 + 256*(1165*b^4*c^5 + 3280*a*b^2*c^6 + 496*a^2*c^7)*d^4*x^5 + 128*(701*b
^5*c^4 + 4640*a*b^3*c^5 + 2480*a^2*b*c^6)*d^4*x^4 + 16*(731*b^6*c^3 + 13660*a*b^
4*c^4 + 19600*a^2*b^2*c^5 + 320*a^3*c^6)*d^4*x^3 + 8*(b^7*c^2 + 4372*a*b^5*c^3 +
19120*a^2*b^3*c^4 + 960*a^3*b*c^5)*d^4*x^2 - 2*(5*b^8*c - 88*a*b^6*c^2 - 16960*
a^2*b^4*c^3 - 5760*a^3*b^2*c^4 + 3840*a^4*c^5)*d^4*x + (15*b^9 - 280*a*b^7*c + 2
048*a^2*b^5*c^2 + 4480*a^3*b^3*c^3 - 3840*a^4*b*c^4)*d^4)*sqrt(c*x^2 + b*x + a)*
sqrt(c))/c^(7/2), -1/81920*(15*(b^10 - 20*a*b^8*c + 160*a^2*b^6*c^2 - 640*a^3*b^
4*c^3 + 1280*a^4*b^2*c^4 - 1024*a^5*c^5)*d^4*arctan(1/2*(2*c*x + b)*sqrt(-c)/(sq
rt(c*x^2 + b*x + a)*c)) - 2*(65536*c^9*d^4*x^9 + 294912*b*c^8*d^4*x^8 + 6144*(89
*b^2*c^7 + 28*a*c^8)*d^4*x^7 + 21504*(25*b^3*c^6 + 28*a*b*c^7)*d^4*x^6 + 256*(11
65*b^4*c^5 + 3280*a*b^2*c^6 + 496*a^2*c^7)*d^4*x^5 + 128*(701*b^5*c^4 + 4640*a*b
^3*c^5 + 2480*a^2*b*c^6)*d^4*x^4 + 16*(731*b^6*c^3 + 13660*a*b^4*c^4 + 19600*a^2
*b^2*c^5 + 320*a^3*c^6)*d^4*x^3 + 8*(b^7*c^2 + 4372*a*b^5*c^3 + 19120*a^2*b^3*c^
4 + 960*a^3*b*c^5)*d^4*x^2 - 2*(5*b^8*c - 88*a*b^6*c^2 - 16960*a^2*b^4*c^3 - 576
0*a^3*b^2*c^4 + 3840*a^4*c^5)*d^4*x + (15*b^9 - 280*a*b^7*c + 2048*a^2*b^5*c^2 +
4480*a^3*b^3*c^3 - 3840*a^4*b*c^4)*d^4)*sqrt(c*x^2 + b*x + a)*sqrt(-c))/(sqrt(-
c)*c^3)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ d^{4} \left (\int a^{2} b^{4} \sqrt{a + b x + c x^{2}}\, dx + \int b^{6} x^{2} \sqrt{a + b x + c x^{2}}\, dx + \int 16 c^{6} x^{8} \sqrt{a + b x + c x^{2}}\, dx + \int 2 a b^{5} x \sqrt{a + b x + c x^{2}}\, dx + \int 32 a c^{5} x^{6} \sqrt{a + b x + c x^{2}}\, dx + \int 16 a^{2} c^{4} x^{4} \sqrt{a + b x + c x^{2}}\, dx + \int 64 b c^{5} x^{7} \sqrt{a + b x + c x^{2}}\, dx + \int 104 b^{2} c^{4} x^{6} \sqrt{a + b x + c x^{2}}\, dx + \int 88 b^{3} c^{3} x^{5} \sqrt{a + b x + c x^{2}}\, dx + \int 41 b^{4} c^{2} x^{4} \sqrt{a + b x + c x^{2}}\, dx + \int 10 b^{5} c x^{3} \sqrt{a + b x + c x^{2}}\, dx + \int 96 a b c^{4} x^{5} \sqrt{a + b x + c x^{2}}\, dx + \int 112 a b^{2} c^{3} x^{4} \sqrt{a + b x + c x^{2}}\, dx + \int 64 a b^{3} c^{2} x^{3} \sqrt{a + b x + c x^{2}}\, dx + \int 18 a b^{4} c x^{2} \sqrt{a + b x + c x^{2}}\, dx + \int 32 a^{2} b c^{3} x^{3} \sqrt{a + b x + c x^{2}}\, dx + \int 24 a^{2} b^{2} c^{2} x^{2} \sqrt{a + b x + c x^{2}}\, dx + \int 8 a^{2} b^{3} c x \sqrt{a + b x + c x^{2}}\, dx\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*d*x+b*d)**4*(c*x**2+b*x+a)**(5/2),x)

[Out]

d**4*(Integral(a**2*b**4*sqrt(a + b*x + c*x**2), x) + Integral(b**6*x**2*sqrt(a
+ b*x + c*x**2), x) + Integral(16*c**6*x**8*sqrt(a + b*x + c*x**2), x) + Integra
l(2*a*b**5*x*sqrt(a + b*x + c*x**2), x) + Integral(32*a*c**5*x**6*sqrt(a + b*x +
c*x**2), x) + Integral(16*a**2*c**4*x**4*sqrt(a + b*x + c*x**2), x) + Integral(
64*b*c**5*x**7*sqrt(a + b*x + c*x**2), x) + Integral(104*b**2*c**4*x**6*sqrt(a +
b*x + c*x**2), x) + Integral(88*b**3*c**3*x**5*sqrt(a + b*x + c*x**2), x) + Int
egral(41*b**4*c**2*x**4*sqrt(a + b*x + c*x**2), x) + Integral(10*b**5*c*x**3*sqr
t(a + b*x + c*x**2), x) + Integral(96*a*b*c**4*x**5*sqrt(a + b*x + c*x**2), x) +
Integral(112*a*b**2*c**3*x**4*sqrt(a + b*x + c*x**2), x) + Integral(64*a*b**3*c
**2*x**3*sqrt(a + b*x + c*x**2), x) + Integral(18*a*b**4*c*x**2*sqrt(a + b*x + c
*x**2), x) + Integral(32*a**2*b*c**3*x**3*sqrt(a + b*x + c*x**2), x) + Integral(
24*a**2*b**2*c**2*x**2*sqrt(a + b*x + c*x**2), x) + Integral(8*a**2*b**3*c*x*sqr
t(a + b*x + c*x**2), x))

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GIAC/XCAS [A]  time = 0.237341, size = 738, normalized size = 2.96 \[ \frac{1}{40960} \, \sqrt{c x^{2} + b x + a}{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \,{\left (2 \,{\left (4 \,{\left (2 \,{\left (16 \,{\left (2 \, c^{6} d^{4} x + 9 \, b c^{5} d^{4}\right )} x + \frac{3 \,{\left (89 \, b^{2} c^{13} d^{4} + 28 \, a c^{14} d^{4}\right )}}{c^{9}}\right )} x + \frac{21 \,{\left (25 \, b^{3} c^{12} d^{4} + 28 \, a b c^{13} d^{4}\right )}}{c^{9}}\right )} x + \frac{1165 \, b^{4} c^{11} d^{4} + 3280 \, a b^{2} c^{12} d^{4} + 496 \, a^{2} c^{13} d^{4}}{c^{9}}\right )} x + \frac{701 \, b^{5} c^{10} d^{4} + 4640 \, a b^{3} c^{11} d^{4} + 2480 \, a^{2} b c^{12} d^{4}}{c^{9}}\right )} x + \frac{731 \, b^{6} c^{9} d^{4} + 13660 \, a b^{4} c^{10} d^{4} + 19600 \, a^{2} b^{2} c^{11} d^{4} + 320 \, a^{3} c^{12} d^{4}}{c^{9}}\right )} x + \frac{b^{7} c^{8} d^{4} + 4372 \, a b^{5} c^{9} d^{4} + 19120 \, a^{2} b^{3} c^{10} d^{4} + 960 \, a^{3} b c^{11} d^{4}}{c^{9}}\right )} x - \frac{5 \, b^{8} c^{7} d^{4} - 88 \, a b^{6} c^{8} d^{4} - 16960 \, a^{2} b^{4} c^{9} d^{4} - 5760 \, a^{3} b^{2} c^{10} d^{4} + 3840 \, a^{4} c^{11} d^{4}}{c^{9}}\right )} x + \frac{15 \, b^{9} c^{6} d^{4} - 280 \, a b^{7} c^{7} d^{4} + 2048 \, a^{2} b^{5} c^{8} d^{4} + 4480 \, a^{3} b^{3} c^{9} d^{4} - 3840 \, a^{4} b c^{10} d^{4}}{c^{9}}\right )} + \frac{3 \,{\left (b^{10} d^{4} - 20 \, a b^{8} c d^{4} + 160 \, a^{2} b^{6} c^{2} d^{4} - 640 \, a^{3} b^{4} c^{3} d^{4} + 1280 \, a^{4} b^{2} c^{4} d^{4} - 1024 \, a^{5} c^{5} d^{4}\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{16384 \, c^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*d*x + b*d)^4*(c*x^2 + b*x + a)^(5/2),x, algorithm="giac")

[Out]

1/40960*sqrt(c*x^2 + b*x + a)*(2*(4*(2*(8*(2*(4*(2*(16*(2*c^6*d^4*x + 9*b*c^5*d^
4)*x + 3*(89*b^2*c^13*d^4 + 28*a*c^14*d^4)/c^9)*x + 21*(25*b^3*c^12*d^4 + 28*a*b
*c^13*d^4)/c^9)*x + (1165*b^4*c^11*d^4 + 3280*a*b^2*c^12*d^4 + 496*a^2*c^13*d^4)
/c^9)*x + (701*b^5*c^10*d^4 + 4640*a*b^3*c^11*d^4 + 2480*a^2*b*c^12*d^4)/c^9)*x
+ (731*b^6*c^9*d^4 + 13660*a*b^4*c^10*d^4 + 19600*a^2*b^2*c^11*d^4 + 320*a^3*c^1
2*d^4)/c^9)*x + (b^7*c^8*d^4 + 4372*a*b^5*c^9*d^4 + 19120*a^2*b^3*c^10*d^4 + 960
*a^3*b*c^11*d^4)/c^9)*x - (5*b^8*c^7*d^4 - 88*a*b^6*c^8*d^4 - 16960*a^2*b^4*c^9*
d^4 - 5760*a^3*b^2*c^10*d^4 + 3840*a^4*c^11*d^4)/c^9)*x + (15*b^9*c^6*d^4 - 280*
a*b^7*c^7*d^4 + 2048*a^2*b^5*c^8*d^4 + 4480*a^3*b^3*c^9*d^4 - 3840*a^4*b*c^10*d^
4)/c^9) + 3/16384*(b^10*d^4 - 20*a*b^8*c*d^4 + 160*a^2*b^6*c^2*d^4 - 640*a^3*b^4
*c^3*d^4 + 1280*a^4*b^2*c^4*d^4 - 1024*a^5*c^5*d^4)*ln(abs(-2*(sqrt(c)*x - sqrt(
c*x^2 + b*x + a))*sqrt(c) - b))/c^(7/2)