3.1172 \(\int (A+B x) (d+e x)^2 \left (b x+c x^2\right )^{3/2} \, dx\)

Optimal. Leaf size=345 \[ \frac{\left (b x+c x^2\right )^{5/2} \left (10 c e x (14 A c e-9 b B e+4 B c d)+14 A c e (24 c d-7 b e)+B \left (63 b^2 e^2-196 b c d e+48 c^2 d^2\right )\right )}{840 c^3}-\frac{b^2 (b+2 c x) \sqrt{b x+c x^2} \left (14 b^2 c e (A e+2 B d)-24 b c^2 d (2 A e+B d)+48 A c^3 d^2-9 b^3 B e^2\right )}{1024 c^5}+\frac{(b+2 c x) \left (b x+c x^2\right )^{3/2} \left (14 b^2 c e (A e+2 B d)-24 b c^2 d (2 A e+B d)+48 A c^3 d^2-9 b^3 B e^2\right )}{384 c^4}+\frac{b^4 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right ) \left (14 b^2 c e (A e+2 B d)-24 b c^2 d (2 A e+B d)+48 A c^3 d^2-9 b^3 B e^2\right )}{1024 c^{11/2}}+\frac{B \left (b x+c x^2\right )^{5/2} (d+e x)^2}{7 c} \]

[Out]

-(b^2*(48*A*c^3*d^2 - 9*b^3*B*e^2 + 14*b^2*c*e*(2*B*d + A*e) - 24*b*c^2*d*(B*d +
2*A*e))*(b + 2*c*x)*Sqrt[b*x + c*x^2])/(1024*c^5) + ((48*A*c^3*d^2 - 9*b^3*B*e^
2 + 14*b^2*c*e*(2*B*d + A*e) - 24*b*c^2*d*(B*d + 2*A*e))*(b + 2*c*x)*(b*x + c*x^
2)^(3/2))/(384*c^4) + (B*(d + e*x)^2*(b*x + c*x^2)^(5/2))/(7*c) + ((14*A*c*e*(24
*c*d - 7*b*e) + B*(48*c^2*d^2 - 196*b*c*d*e + 63*b^2*e^2) + 10*c*e*(4*B*c*d - 9*
b*B*e + 14*A*c*e)*x)*(b*x + c*x^2)^(5/2))/(840*c^3) + (b^4*(48*A*c^3*d^2 - 9*b^3
*B*e^2 + 14*b^2*c*e*(2*B*d + A*e) - 24*b*c^2*d*(B*d + 2*A*e))*ArcTanh[(Sqrt[c]*x
)/Sqrt[b*x + c*x^2]])/(1024*c^(11/2))

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Rubi [A]  time = 0.778669, antiderivative size = 345, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ \frac{\left (b x+c x^2\right )^{5/2} \left (10 c e x (14 A c e-9 b B e+4 B c d)+14 A c e (24 c d-7 b e)+B \left (63 b^2 e^2-196 b c d e+48 c^2 d^2\right )\right )}{840 c^3}-\frac{b^2 (b+2 c x) \sqrt{b x+c x^2} \left (14 b^2 c e (A e+2 B d)-24 b c^2 d (2 A e+B d)+48 A c^3 d^2-9 b^3 B e^2\right )}{1024 c^5}+\frac{(b+2 c x) \left (b x+c x^2\right )^{3/2} \left (14 b^2 c e (A e+2 B d)-24 b c^2 d (2 A e+B d)+48 A c^3 d^2-9 b^3 B e^2\right )}{384 c^4}+\frac{b^4 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right ) \left (14 b^2 c e (A e+2 B d)-24 b c^2 d (2 A e+B d)+48 A c^3 d^2-9 b^3 B e^2\right )}{1024 c^{11/2}}+\frac{B \left (b x+c x^2\right )^{5/2} (d+e x)^2}{7 c} \]

Antiderivative was successfully verified.

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

[Out]

-(b^2*(48*A*c^3*d^2 - 9*b^3*B*e^2 + 14*b^2*c*e*(2*B*d + A*e) - 24*b*c^2*d*(B*d +
2*A*e))*(b + 2*c*x)*Sqrt[b*x + c*x^2])/(1024*c^5) + ((48*A*c^3*d^2 - 9*b^3*B*e^
2 + 14*b^2*c*e*(2*B*d + A*e) - 24*b*c^2*d*(B*d + 2*A*e))*(b + 2*c*x)*(b*x + c*x^
2)^(3/2))/(384*c^4) + (B*(d + e*x)^2*(b*x + c*x^2)^(5/2))/(7*c) + ((14*A*c*e*(24
*c*d - 7*b*e) + B*(48*c^2*d^2 - 196*b*c*d*e + 63*b^2*e^2) + 10*c*e*(4*B*c*d - 9*
b*B*e + 14*A*c*e)*x)*(b*x + c*x^2)^(5/2))/(840*c^3) + (b^4*(48*A*c^3*d^2 - 9*b^3
*B*e^2 + 14*b^2*c*e*(2*B*d + A*e) - 24*b*c^2*d*(B*d + 2*A*e))*ArcTanh[(Sqrt[c]*x
)/Sqrt[b*x + c*x^2]])/(1024*c^(11/2))

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Rubi in Sympy [A]  time = 81.0191, size = 408, normalized size = 1.18 \[ \frac{B \left (d + e x\right )^{2} \left (b x + c x^{2}\right )^{\frac{5}{2}}}{7 c} + \frac{b^{4} \left (14 A b^{2} c e^{2} - 48 A b c^{2} d e + 48 A c^{3} d^{2} - 9 B b^{3} e^{2} + 28 B b^{2} c d e - 24 B b c^{2} d^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b x + c x^{2}}} \right )}}{1024 c^{\frac{11}{2}}} - \frac{b^{2} \left (b + 2 c x\right ) \sqrt{b x + c x^{2}} \left (14 A b^{2} c e^{2} - 48 A b c^{2} d e + 48 A c^{3} d^{2} - 9 B b^{3} e^{2} + 28 B b^{2} c d e - 24 B b c^{2} d^{2}\right )}{1024 c^{5}} + \frac{\left (b x + c x^{2}\right )^{\frac{5}{2}} \left (- \frac{49 A b c e^{2}}{2} + 84 A c^{2} d e + \frac{63 B b^{2} e^{2}}{4} - 49 B b c d e + 12 B c^{2} d^{2} + \frac{5 c e x \left (14 A c e - 9 B b e + 4 B c d\right )}{2}\right )}{210 c^{3}} + \frac{\left (b + 2 c x\right ) \left (b x + c x^{2}\right )^{\frac{3}{2}} \left (14 A b^{2} c e^{2} - 48 A b c^{2} d e + 48 A c^{3} d^{2} - 9 B b^{3} e^{2} + 28 B b^{2} c d e - 24 B b c^{2} d^{2}\right )}{384 c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*(d + e*x)**2*(b*x + c*x**2)**(5/2)/(7*c) + b**4*(14*A*b**2*c*e**2 - 48*A*b*c**
2*d*e + 48*A*c**3*d**2 - 9*B*b**3*e**2 + 28*B*b**2*c*d*e - 24*B*b*c**2*d**2)*ata
nh(sqrt(c)*x/sqrt(b*x + c*x**2))/(1024*c**(11/2)) - b**2*(b + 2*c*x)*sqrt(b*x +
c*x**2)*(14*A*b**2*c*e**2 - 48*A*b*c**2*d*e + 48*A*c**3*d**2 - 9*B*b**3*e**2 + 2
8*B*b**2*c*d*e - 24*B*b*c**2*d**2)/(1024*c**5) + (b*x + c*x**2)**(5/2)*(-49*A*b*
c*e**2/2 + 84*A*c**2*d*e + 63*B*b**2*e**2/4 - 49*B*b*c*d*e + 12*B*c**2*d**2 + 5*
c*e*x*(14*A*c*e - 9*B*b*e + 4*B*c*d)/2)/(210*c**3) + (b + 2*c*x)*(b*x + c*x**2)*
*(3/2)*(14*A*b**2*c*e**2 - 48*A*b*c**2*d*e + 48*A*c**3*d**2 - 9*B*b**3*e**2 + 28
*B*b**2*c*d*e - 24*B*b*c**2*d**2)/(384*c**4)

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Mathematica [A]  time = 1.42005, size = 421, normalized size = 1.22 \[ \frac{(x (b+c x))^{3/2} \left (\frac{\sqrt{x} \left (-210 b^5 c e (7 A e+14 B d+3 B e x)+28 b^4 c^2 \left (5 A e (36 d+7 e x)+2 B \left (45 d^2+35 d e x+9 e^2 x^2\right )\right )-16 b^3 c^3 \left (7 A \left (45 d^2+30 d e x+7 e^2 x^2\right )+B x \left (105 d^2+98 d e x+27 e^2 x^2\right )\right )+96 b^2 c^4 x \left (7 A \left (5 d^2+4 d e x+e^2 x^2\right )+2 B x \left (7 d^2+7 d e x+2 e^2 x^2\right )\right )+128 b c^5 x^2 \left (7 A \left (45 d^2+66 d e x+26 e^2 x^2\right )+B x \left (231 d^2+364 d e x+150 e^2 x^2\right )\right )+256 c^6 x^3 \left (7 A \left (15 d^2+24 d e x+10 e^2 x^2\right )+4 B x \left (21 d^2+35 d e x+15 e^2 x^2\right )\right )+945 b^6 B e^2\right )}{105 c^5 (b+c x)}-\frac{b^4 \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right ) \left (-14 b^2 c e (A e+2 B d)+24 b c^2 d (2 A e+B d)-48 A c^3 d^2+9 b^3 B e^2\right )}{c^{11/2} (b+c x)^{3/2}}\right )}{1024 x^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

((x*(b + c*x))^(3/2)*((Sqrt[x]*(945*b^6*B*e^2 - 210*b^5*c*e*(14*B*d + 7*A*e + 3*
B*e*x) + 96*b^2*c^4*x*(7*A*(5*d^2 + 4*d*e*x + e^2*x^2) + 2*B*x*(7*d^2 + 7*d*e*x
+ 2*e^2*x^2)) + 28*b^4*c^2*(5*A*e*(36*d + 7*e*x) + 2*B*(45*d^2 + 35*d*e*x + 9*e^
2*x^2)) + 256*c^6*x^3*(7*A*(15*d^2 + 24*d*e*x + 10*e^2*x^2) + 4*B*x*(21*d^2 + 35
*d*e*x + 15*e^2*x^2)) - 16*b^3*c^3*(7*A*(45*d^2 + 30*d*e*x + 7*e^2*x^2) + B*x*(1
05*d^2 + 98*d*e*x + 27*e^2*x^2)) + 128*b*c^5*x^2*(7*A*(45*d^2 + 66*d*e*x + 26*e^
2*x^2) + B*x*(231*d^2 + 364*d*e*x + 150*e^2*x^2))))/(105*c^5*(b + c*x)) - (b^4*(
-48*A*c^3*d^2 + 9*b^3*B*e^2 - 14*b^2*c*e*(2*B*d + A*e) + 24*b*c^2*d*(B*d + 2*A*e
))*Log[c*Sqrt[x] + Sqrt[c]*Sqrt[b + c*x]])/(c^(11/2)*(b + c*x)^(3/2))))/(1024*x^
(3/2))

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Maple [B]  time = 0.016, size = 949, normalized size = 2.8 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5*(c*x^2+b*x)^(5/2)/c*B*d^2+1/4*A*d^2*(c*x^2+b*x)^(3/2)*x-9/2048*B*e^2*b^7/c^(
11/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^(1/2))-3/64*B*e^2*b^3/c^3*(c*x^2+b*x)^(
3/2)*x-1/8*b/c*(c*x^2+b*x)^(3/2)*x*B*d^2-1/8*b^2/c^2*(c*x^2+b*x)^(3/2)*A*d*e+3/6
4*b^3/c^2*(c*x^2+b*x)^(1/2)*x*B*d^2+3/64*b^4/c^3*(c*x^2+b*x)^(1/2)*A*d*e+9/1024*
B*e^2*b^6/c^5*(c*x^2+b*x)^(1/2)+7/48*b^2/c^2*(c*x^2+b*x)^(3/2)*x*B*d*e-7/128*b^4
/c^3*(c*x^2+b*x)^(1/2)*x*B*d*e-1/4*b/c*(c*x^2+b*x)^(3/2)*x*A*d*e+3/32*b^3/c^2*(c
*x^2+b*x)^(1/2)*x*A*d*e-3/256*b^5/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^(1/
2))*B*d^2+2/5*(c*x^2+b*x)^(5/2)/c*A*d*e-1/16*b^2/c^2*(c*x^2+b*x)^(3/2)*B*d^2+3/1
28*b^4/c^3*(c*x^2+b*x)^(1/2)*B*d^2-3/128*B*e^2*b^4/c^4*(c*x^2+b*x)^(3/2)-3/64*A*
d^2*b^3/c^2*(c*x^2+b*x)^(1/2)+1/8*A*d^2/c*(c*x^2+b*x)^(3/2)*b+3/128*A*d^2*b^4/c^
(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^(1/2))+1/6*x*(c*x^2+b*x)^(5/2)/c*A*e^2-
7/60*b/c^2*(c*x^2+b*x)^(5/2)*A*e^2-3/128*b^5/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x
^2+b*x)^(1/2))*A*d*e+7/192*b^3/c^3*(c*x^2+b*x)^(3/2)*A*e^2-7/512*b^5/c^4*(c*x^2+
b*x)^(1/2)*A*e^2+7/1024*b^6/c^(9/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^(1/2))*A*
e^2+1/7*B*e^2*x^2*(c*x^2+b*x)^(5/2)/c+3/40*B*e^2*b^2/c^3*(c*x^2+b*x)^(5/2)-3/28*
B*e^2*b/c^2*x*(c*x^2+b*x)^(5/2)-3/32*A*d^2*b^2/c*(c*x^2+b*x)^(1/2)*x+1/3*x*(c*x^
2+b*x)^(5/2)/c*B*d*e-7/30*b/c^2*(c*x^2+b*x)^(5/2)*B*d*e+7/96*b^2/c^2*(c*x^2+b*x)
^(3/2)*x*A*e^2+7/96*b^3/c^3*(c*x^2+b*x)^(3/2)*B*d*e-7/256*b^4/c^3*(c*x^2+b*x)^(1
/2)*x*A*e^2-7/256*b^5/c^4*(c*x^2+b*x)^(1/2)*B*d*e+7/512*b^6/c^(9/2)*ln((1/2*b+c*
x)/c^(1/2)+(c*x^2+b*x)^(1/2))*B*d*e+9/512*B*e^2*b^5/c^4*(c*x^2+b*x)^(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((c*x^2 + b*x)^(3/2)*(B*x + A)*(e*x + d)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.312583, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/215040*(2*(15360*B*c^6*e^2*x^6 + 1280*(28*B*c^6*d*e + (15*B*b*c^5 + 14*A*c^6)
*e^2)*x^5 + 128*(168*B*c^6*d^2 + 28*(13*B*b*c^5 + 12*A*c^6)*d*e + (3*B*b^2*c^4 +
182*A*b*c^5)*e^2)*x^4 + 48*(56*(11*B*b*c^5 + 10*A*c^6)*d^2 + 28*(B*b^2*c^4 + 44
*A*b*c^5)*d*e - (9*B*b^3*c^3 - 14*A*b^2*c^4)*e^2)*x^3 + 2520*(B*b^4*c^2 - 2*A*b^
3*c^3)*d^2 - 420*(7*B*b^5*c - 12*A*b^4*c^2)*d*e + 105*(9*B*b^6 - 14*A*b^5*c)*e^2
+ 56*(24*(B*b^2*c^4 + 30*A*b*c^5)*d^2 - 4*(7*B*b^3*c^3 - 12*A*b^2*c^4)*d*e + (9
*B*b^4*c^2 - 14*A*b^3*c^3)*e^2)*x^2 - 70*(24*(B*b^3*c^3 - 2*A*b^2*c^4)*d^2 - 4*(
7*B*b^4*c^2 - 12*A*b^3*c^3)*d*e + (9*B*b^5*c - 14*A*b^4*c^2)*e^2)*x)*sqrt(c*x^2
+ b*x)*sqrt(c) - 105*(24*(B*b^5*c^2 - 2*A*b^4*c^3)*d^2 - 4*(7*B*b^6*c - 12*A*b^5
*c^2)*d*e + (9*B*b^7 - 14*A*b^6*c)*e^2)*log((2*c*x + b)*sqrt(c) + 2*sqrt(c*x^2 +
b*x)*c))/c^(11/2), 1/107520*((15360*B*c^6*e^2*x^6 + 1280*(28*B*c^6*d*e + (15*B*
b*c^5 + 14*A*c^6)*e^2)*x^5 + 128*(168*B*c^6*d^2 + 28*(13*B*b*c^5 + 12*A*c^6)*d*e
+ (3*B*b^2*c^4 + 182*A*b*c^5)*e^2)*x^4 + 48*(56*(11*B*b*c^5 + 10*A*c^6)*d^2 + 2
8*(B*b^2*c^4 + 44*A*b*c^5)*d*e - (9*B*b^3*c^3 - 14*A*b^2*c^4)*e^2)*x^3 + 2520*(B
*b^4*c^2 - 2*A*b^3*c^3)*d^2 - 420*(7*B*b^5*c - 12*A*b^4*c^2)*d*e + 105*(9*B*b^6
- 14*A*b^5*c)*e^2 + 56*(24*(B*b^2*c^4 + 30*A*b*c^5)*d^2 - 4*(7*B*b^3*c^3 - 12*A*
b^2*c^4)*d*e + (9*B*b^4*c^2 - 14*A*b^3*c^3)*e^2)*x^2 - 70*(24*(B*b^3*c^3 - 2*A*b
^2*c^4)*d^2 - 4*(7*B*b^4*c^2 - 12*A*b^3*c^3)*d*e + (9*B*b^5*c - 14*A*b^4*c^2)*e^
2)*x)*sqrt(c*x^2 + b*x)*sqrt(-c) - 105*(24*(B*b^5*c^2 - 2*A*b^4*c^3)*d^2 - 4*(7*
B*b^6*c - 12*A*b^5*c^2)*d*e + (9*B*b^7 - 14*A*b^6*c)*e^2)*arctan(sqrt(c*x^2 + b*
x)*sqrt(-c)/(c*x)))/(sqrt(-c)*c^5)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \left (x \left (b + c x\right )\right )^{\frac{3}{2}} \left (A + B x\right ) \left (d + e x\right )^{2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((x*(b + c*x))**(3/2)*(A + B*x)*(d + e*x)**2, x)

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GIAC/XCAS [A]  time = 0.28508, size = 699, normalized size = 2.03 \[ \frac{1}{107520} \, \sqrt{c x^{2} + b x}{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \,{\left (10 \,{\left (12 \, B c x e^{2} + \frac{28 \, B c^{7} d e + 15 \, B b c^{6} e^{2} + 14 \, A c^{7} e^{2}}{c^{6}}\right )} x + \frac{168 \, B c^{7} d^{2} + 364 \, B b c^{6} d e + 336 \, A c^{7} d e + 3 \, B b^{2} c^{5} e^{2} + 182 \, A b c^{6} e^{2}}{c^{6}}\right )} x + \frac{3 \,{\left (616 \, B b c^{6} d^{2} + 560 \, A c^{7} d^{2} + 28 \, B b^{2} c^{5} d e + 1232 \, A b c^{6} d e - 9 \, B b^{3} c^{4} e^{2} + 14 \, A b^{2} c^{5} e^{2}\right )}}{c^{6}}\right )} x + \frac{7 \,{\left (24 \, B b^{2} c^{5} d^{2} + 720 \, A b c^{6} d^{2} - 28 \, B b^{3} c^{4} d e + 48 \, A b^{2} c^{5} d e + 9 \, B b^{4} c^{3} e^{2} - 14 \, A b^{3} c^{4} e^{2}\right )}}{c^{6}}\right )} x - \frac{35 \,{\left (24 \, B b^{3} c^{4} d^{2} - 48 \, A b^{2} c^{5} d^{2} - 28 \, B b^{4} c^{3} d e + 48 \, A b^{3} c^{4} d e + 9 \, B b^{5} c^{2} e^{2} - 14 \, A b^{4} c^{3} e^{2}\right )}}{c^{6}}\right )} x + \frac{105 \,{\left (24 \, B b^{4} c^{3} d^{2} - 48 \, A b^{3} c^{4} d^{2} - 28 \, B b^{5} c^{2} d e + 48 \, A b^{4} c^{3} d e + 9 \, B b^{6} c e^{2} - 14 \, A b^{5} c^{2} e^{2}\right )}}{c^{6}}\right )} + \frac{{\left (24 \, B b^{5} c^{2} d^{2} - 48 \, A b^{4} c^{3} d^{2} - 28 \, B b^{6} c d e + 48 \, A b^{5} c^{2} d e + 9 \, B b^{7} e^{2} - 14 \, A b^{6} c e^{2}\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{2048 \, c^{\frac{11}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/107520*sqrt(c*x^2 + b*x)*(2*(4*(2*(8*(10*(12*B*c*x*e^2 + (28*B*c^7*d*e + 15*B*
b*c^6*e^2 + 14*A*c^7*e^2)/c^6)*x + (168*B*c^7*d^2 + 364*B*b*c^6*d*e + 336*A*c^7*
d*e + 3*B*b^2*c^5*e^2 + 182*A*b*c^6*e^2)/c^6)*x + 3*(616*B*b*c^6*d^2 + 560*A*c^7
*d^2 + 28*B*b^2*c^5*d*e + 1232*A*b*c^6*d*e - 9*B*b^3*c^4*e^2 + 14*A*b^2*c^5*e^2)
/c^6)*x + 7*(24*B*b^2*c^5*d^2 + 720*A*b*c^6*d^2 - 28*B*b^3*c^4*d*e + 48*A*b^2*c^
5*d*e + 9*B*b^4*c^3*e^2 - 14*A*b^3*c^4*e^2)/c^6)*x - 35*(24*B*b^3*c^4*d^2 - 48*A
*b^2*c^5*d^2 - 28*B*b^4*c^3*d*e + 48*A*b^3*c^4*d*e + 9*B*b^5*c^2*e^2 - 14*A*b^4*
c^3*e^2)/c^6)*x + 105*(24*B*b^4*c^3*d^2 - 48*A*b^3*c^4*d^2 - 28*B*b^5*c^2*d*e +
48*A*b^4*c^3*d*e + 9*B*b^6*c*e^2 - 14*A*b^5*c^2*e^2)/c^6) + 1/2048*(24*B*b^5*c^2
*d^2 - 48*A*b^4*c^3*d^2 - 28*B*b^6*c*d*e + 48*A*b^5*c^2*d*e + 9*B*b^7*e^2 - 14*A
*b^6*c*e^2)*ln(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x))*sqrt(c) - b))/c^(11/2)