3.1546 \(\int (b+2 c x) (d+e x)^4 \sqrt{a+b x+c x^2} \, dx\)

Optimal. Leaf size=431 \[ \frac{\left (a+b x+c x^2\right )^{3/2} \left (8 c^2 e^2 \left (32 a^2 e^2+231 a b d e+87 b^2 d^2\right )+6 c e x (2 c d-b e) \left (-4 c e (19 a e+2 b d)+21 b^2 e^2+8 c^2 d^2\right )-14 b^2 c e^3 (34 a e+35 b d)-16 c^3 d^2 e (144 a e+13 b d)+105 b^4 e^4+128 c^4 d^4\right )}{1680 c^4}+\frac{(d+e x)^2 \left (a+b x+c x^2\right )^{3/2} \left (-4 c e (2 a e+b d)+3 b^2 e^2+4 c^2 d^2\right )}{35 c^2}-\frac{e \left (b^2-4 a c\right )^2 (2 c d-b e) \left (-4 c e (a e+2 b d)+3 b^2 e^2+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{256 c^{11/2}}+\frac{e \left (b^2-4 a c\right ) (b+2 c x) \sqrt{a+b x+c x^2} (2 c d-b e) \left (-4 c e (a e+2 b d)+3 b^2 e^2+8 c^2 d^2\right )}{128 c^5}+\frac{2}{7} (d+e x)^4 \left (a+b x+c x^2\right )^{3/2}+\frac{2 (d+e x)^3 \left (a+b x+c x^2\right )^{3/2} (2 c d-b e)}{21 c} \]

[Out]

((b^2 - 4*a*c)*e*(2*c*d - b*e)*(8*c^2*d^2 + 3*b^2*e^2 - 4*c*e*(2*b*d + a*e))*(b
+ 2*c*x)*Sqrt[a + b*x + c*x^2])/(128*c^5) + ((4*c^2*d^2 + 3*b^2*e^2 - 4*c*e*(b*d
+ 2*a*e))*(d + e*x)^2*(a + b*x + c*x^2)^(3/2))/(35*c^2) + (2*(2*c*d - b*e)*(d +
e*x)^3*(a + b*x + c*x^2)^(3/2))/(21*c) + (2*(d + e*x)^4*(a + b*x + c*x^2)^(3/2)
)/7 + ((128*c^4*d^4 + 105*b^4*e^4 - 14*b^2*c*e^3*(35*b*d + 34*a*e) - 16*c^3*d^2*
e*(13*b*d + 144*a*e) + 8*c^2*e^2*(87*b^2*d^2 + 231*a*b*d*e + 32*a^2*e^2) + 6*c*e
*(2*c*d - b*e)*(8*c^2*d^2 + 21*b^2*e^2 - 4*c*e*(2*b*d + 19*a*e))*x)*(a + b*x + c
*x^2)^(3/2))/(1680*c^4) - ((b^2 - 4*a*c)^2*e*(2*c*d - b*e)*(8*c^2*d^2 + 3*b^2*e^
2 - 4*c*e*(2*b*d + a*e))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])
/(256*c^(11/2))

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Rubi [A]  time = 1.76405, antiderivative size = 431, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ \frac{\left (a+b x+c x^2\right )^{3/2} \left (8 c^2 e^2 \left (32 a^2 e^2+231 a b d e+87 b^2 d^2\right )+6 c e x (2 c d-b e) \left (-4 c e (19 a e+2 b d)+21 b^2 e^2+8 c^2 d^2\right )-14 b^2 c e^3 (34 a e+35 b d)-16 c^3 d^2 e (144 a e+13 b d)+105 b^4 e^4+128 c^4 d^4\right )}{1680 c^4}+\frac{(d+e x)^2 \left (a+b x+c x^2\right )^{3/2} \left (-4 c e (2 a e+b d)+3 b^2 e^2+4 c^2 d^2\right )}{35 c^2}-\frac{e \left (b^2-4 a c\right )^2 (2 c d-b e) \left (-4 c e (a e+2 b d)+3 b^2 e^2+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{256 c^{11/2}}+\frac{e \left (b^2-4 a c\right ) (b+2 c x) \sqrt{a+b x+c x^2} (2 c d-b e) \left (-4 c e (a e+2 b d)+3 b^2 e^2+8 c^2 d^2\right )}{128 c^5}+\frac{2}{7} (d+e x)^4 \left (a+b x+c x^2\right )^{3/2}+\frac{2 (d+e x)^3 \left (a+b x+c x^2\right )^{3/2} (2 c d-b e)}{21 c} \]

Antiderivative was successfully verified.

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

[Out]

((b^2 - 4*a*c)*e*(2*c*d - b*e)*(8*c^2*d^2 + 3*b^2*e^2 - 4*c*e*(2*b*d + a*e))*(b
+ 2*c*x)*Sqrt[a + b*x + c*x^2])/(128*c^5) + ((4*c^2*d^2 + 3*b^2*e^2 - 4*c*e*(b*d
+ 2*a*e))*(d + e*x)^2*(a + b*x + c*x^2)^(3/2))/(35*c^2) + (2*(2*c*d - b*e)*(d +
e*x)^3*(a + b*x + c*x^2)^(3/2))/(21*c) + (2*(d + e*x)^4*(a + b*x + c*x^2)^(3/2)
)/7 + ((128*c^4*d^4 + 105*b^4*e^4 - 14*b^2*c*e^3*(35*b*d + 34*a*e) - 16*c^3*d^2*
e*(13*b*d + 144*a*e) + 8*c^2*e^2*(87*b^2*d^2 + 231*a*b*d*e + 32*a^2*e^2) + 6*c*e
*(2*c*d - b*e)*(8*c^2*d^2 + 21*b^2*e^2 - 4*c*e*(2*b*d + 19*a*e))*x)*(a + b*x + c
*x^2)^(3/2))/(1680*c^4) - ((b^2 - 4*a*c)^2*e*(2*c*d - b*e)*(8*c^2*d^2 + 3*b^2*e^
2 - 4*c*e*(2*b*d + a*e))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])
/(256*c^(11/2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2*c*x+b)*(e*x+d)**4*(c*x**2+b*x+a)**(1/2),x)

[Out]

Timed out

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Mathematica [A]  time = 0.990497, size = 547, normalized size = 1.27 \[ \frac{\sqrt{a+x (b+c x)} \left (2048 a^3 c^3 e^4-16 a^2 c^2 e^2 \left (343 b^2 e^2-2 b c e (567 d+73 e x)+4 c^2 \left (336 d^2+105 d e x+16 e^2 x^2\right )\right )+8 a c \left (315 b^4 e^4-14 b^3 c e^3 (95 d+13 e x)+4 b^2 c^2 e^2 \left (525 d^2+189 d e x+31 e^2 x^2\right )-8 b c^3 e \left (175 d^3+147 d^2 e x+63 d e^2 x^2+11 e^3 x^3\right )+16 c^4 \left (70 d^4+105 d^3 e x+84 d^2 e^2 x^2+35 d e^3 x^3+6 e^4 x^4\right )\right )-315 b^6 e^4+210 b^5 c e^3 (7 d+e x)-28 b^4 c^2 e^2 \left (90 d^2+35 d e x+6 e^2 x^2\right )+16 b^3 c^3 e \left (105 d^3+105 d^2 e x+49 d e^2 x^2+9 e^3 x^3\right )-32 b^2 c^4 e x \left (35 d^3+42 d^2 e x+21 d e^2 x^2+4 e^3 x^3\right )+128 b c^5 x \left (70 d^4+175 d^3 e x+189 d^2 e^2 x^2+98 d e^3 x^3+20 e^4 x^4\right )+256 c^6 x^2 \left (35 d^4+105 d^3 e x+126 d^2 e^2 x^2+70 d e^3 x^3+15 e^4 x^4\right )\right )}{13440 c^5}+\frac{e \left (b^2-4 a c\right )^2 (b e-2 c d) \left (-4 c e (a e+2 b d)+3 b^2 e^2+8 c^2 d^2\right ) \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )}{256 c^{11/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + x*(b + c*x)]*(-315*b^6*e^4 + 2048*a^3*c^3*e^4 + 210*b^5*c*e^3*(7*d + e
*x) - 28*b^4*c^2*e^2*(90*d^2 + 35*d*e*x + 6*e^2*x^2) - 32*b^2*c^4*e*x*(35*d^3 +
42*d^2*e*x + 21*d*e^2*x^2 + 4*e^3*x^3) + 16*b^3*c^3*e*(105*d^3 + 105*d^2*e*x + 4
9*d*e^2*x^2 + 9*e^3*x^3) + 256*c^6*x^2*(35*d^4 + 105*d^3*e*x + 126*d^2*e^2*x^2 +
70*d*e^3*x^3 + 15*e^4*x^4) + 128*b*c^5*x*(70*d^4 + 175*d^3*e*x + 189*d^2*e^2*x^
2 + 98*d*e^3*x^3 + 20*e^4*x^4) - 16*a^2*c^2*e^2*(343*b^2*e^2 - 2*b*c*e*(567*d +
73*e*x) + 4*c^2*(336*d^2 + 105*d*e*x + 16*e^2*x^2)) + 8*a*c*(315*b^4*e^4 - 14*b^
3*c*e^3*(95*d + 13*e*x) + 4*b^2*c^2*e^2*(525*d^2 + 189*d*e*x + 31*e^2*x^2) - 8*b
*c^3*e*(175*d^3 + 147*d^2*e*x + 63*d*e^2*x^2 + 11*e^3*x^3) + 16*c^4*(70*d^4 + 10
5*d^3*e*x + 84*d^2*e^2*x^2 + 35*d*e^3*x^3 + 6*e^4*x^4))))/(13440*c^5) + ((b^2 -
4*a*c)^2*e*(-2*c*d + b*e)*(8*c^2*d^2 + 3*b^2*e^2 - 4*c*e*(2*b*d + a*e))*Log[b +
2*c*x + 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]])/(256*c^(11/2))

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Maple [B]  time = 0.025, size = 1537, normalized size = 3.6 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2*c*x+b)*(e*x+d)^4*(c*x^2+b*x+a)^(1/2),x)

[Out]

3/2*b/c*a*(c*x^2+b*x+a)^(1/2)*x*d^2*e^2-b^2/c^2*a*(c*x^2+b*x+a)^(1/2)*x*d*e^3-2/
5*b/c*x^2*(c*x^2+b*x+a)^(3/2)*d*e^3+19/70*a/c^2*x*(c*x^2+b*x+a)^(3/2)*b*e^4-3/5*
b/c*x*(c*x^2+b*x+a)^(3/2)*d^2*e^2-3/40*b^3/c^3*x*(c*x^2+b*x+a)^(3/2)*e^4-7/24*b^
3/c^3*(c*x^2+b*x+a)^(3/2)*d*e^3+7/64*b^5/c^4*(c*x^2+b*x+a)^(1/2)*d*e^3+1/8*b^3/c
^2*(c*x^2+b*x+a)^(1/2)*d^3*e-a*(c*x^2+b*x+a)^(1/2)*x*d^3*e-a^2/c^(1/2)*ln((1/2*b
+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*d^3*e+1/8*b^4/c^4*a*(c*x^2+b*x+a)^(1/2)*e^4+5
/16*b^3/c^(7/2)*a^2*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*e^4+2/7*e^4*x^4*
(c*x^2+b*x+a)^(3/2)-1/3*b/c*(c*x^2+b*x+a)^(3/2)*d^3*e-17/60*b^2/c^3*a*(c*x^2+b*x
+a)^(3/2)*e^4-1/8*a^2/c^3*(c*x^2+b*x+a)^(1/2)*b^2*e^4-1/4*a^3/c^(5/2)*ln((1/2*b+
c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*b*e^4-3/64*b^5/c^4*(c*x^2+b*x+a)^(1/2)*x*e^4+1
/2*a^3/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*d*e^3-1/16*b^4/c^(5/2
)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*d^3*e+1/2*b^2/c^2*(c*x^2+b*x+a)^(3
/2)*d^2*e^2-3/16*b^4/c^3*(c*x^2+b*x+a)^(1/2)*d^2*e^2-8/35/c*e^4*a*x^2*(c*x^2+b*x
+a)^(3/2)+3/32*b^5/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*d^2*e^2-7
/128*b^6/c^(9/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*d*e^3-8/5*a/c*(c*x^
2+b*x+a)^(3/2)*d^2*e^2-7/64*b^5/c^(9/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/
2))*a*e^4-2/21*x^3*(c*x^2+b*x+a)^(3/2)/c*b*e^4+15/32*b^4/c^(7/2)*ln((1/2*b+c*x)/
c^(1/2)+(c*x^2+b*x+a)^(1/2))*a*d*e^3+1/4*b^2/c*(c*x^2+b*x+a)^(1/2)*x*d^3*e-9/8*b
^2/c^(5/2)*a^2*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*d*e^3+1/2*b^2/c^(3/2)
*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a*d^3*e+11/10*b/c^2*a*(c*x^2+b*x+a)
^(3/2)*d*e^3-1/2*b^3/c^3*a*(c*x^2+b*x+a)^(1/2)*d*e^3-1/4*a^2/c^2*(c*x^2+b*x+a)^(
1/2)*x*b*e^4+1/2*a^2/c*(c*x^2+b*x+a)^(1/2)*x*d*e^3+1/4*a^2/c^2*(c*x^2+b*x+a)^(1/
2)*b*d*e^3+3/2*b/c^(3/2)*a^2*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*d^2*e^2
-1/2*a/c*(c*x^2+b*x+a)^(1/2)*b*d^3*e+7/20*b^2/c^2*x*(c*x^2+b*x+a)^(3/2)*d*e^3-3/
8*b^3/c^2*(c*x^2+b*x+a)^(1/2)*x*d^2*e^2+3/4*b^2/c^2*a*(c*x^2+b*x+a)^(1/2)*d^2*e^
2-a/c*x*(c*x^2+b*x+a)^(3/2)*d*e^3-3/4*b^3/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+
b*x+a)^(1/2))*a*d^2*e^2+7/32*b^4/c^3*(c*x^2+b*x+a)^(1/2)*x*d*e^3+1/4*b^3/c^3*a*(
c*x^2+b*x+a)^(1/2)*x*e^4+2/3*(c*x^2+b*x+a)^(3/2)*d^4+3/35*b^2/c^2*x^2*(c*x^2+b*x
+a)^(3/2)*e^4+3/256*b^7/c^(11/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*e^4
+12/5*x^2*(c*x^2+b*x+a)^(3/2)*d^2*e^2+4/3*x^3*(c*x^2+b*x+a)^(3/2)*d*e^3+1/16*b^4
/c^4*(c*x^2+b*x+a)^(3/2)*e^4-3/128*b^6/c^5*(c*x^2+b*x+a)^(1/2)*e^4+2*x*(c*x^2+b*
x+a)^(3/2)*d^3*e+16/105/c^2*e^4*a^2*(c*x^2+b*x+a)^(3/2)

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.43291, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/53760*(4*(3840*c^6*e^4*x^6 + 8960*a*c^5*d^4 + 2560*(7*c^6*d*e^3 + b*c^5*e^4)*
x^5 + 560*(3*b^3*c^3 - 20*a*b*c^4)*d^3*e - 168*(15*b^4*c^2 - 100*a*b^2*c^3 + 128
*a^2*c^4)*d^2*e^2 + 14*(105*b^5*c - 760*a*b^3*c^2 + 1296*a^2*b*c^3)*d*e^3 - (315
*b^6 - 2520*a*b^4*c + 5488*a^2*b^2*c^2 - 2048*a^3*c^3)*e^4 + 128*(252*c^6*d^2*e^
2 + 98*b*c^5*d*e^3 - (b^2*c^4 - 6*a*c^5)*e^4)*x^4 + 16*(1680*c^6*d^3*e + 1512*b*
c^5*d^2*e^2 - 14*(3*b^2*c^4 - 20*a*c^5)*d*e^3 + (9*b^3*c^3 - 44*a*b*c^4)*e^4)*x^
3 + 8*(1120*c^6*d^4 + 2800*b*c^5*d^3*e - 168*(b^2*c^4 - 8*a*c^5)*d^2*e^2 + 14*(7
*b^3*c^3 - 36*a*b*c^4)*d*e^3 - (21*b^4*c^2 - 124*a*b^2*c^3 + 128*a^2*c^4)*e^4)*x
^2 + 2*(4480*b*c^5*d^4 - 560*(b^2*c^4 - 12*a*c^5)*d^3*e + 168*(5*b^3*c^3 - 28*a*
b*c^4)*d^2*e^2 - 14*(35*b^4*c^2 - 216*a*b^2*c^3 + 240*a^2*c^4)*d*e^3 + (105*b^5*
c - 728*a*b^3*c^2 + 1168*a^2*b*c^3)*e^4)*x)*sqrt(c*x^2 + b*x + a)*sqrt(c) + 105*
(16*(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*d^3*e - 24*(b^5*c^2 - 8*a*b^3*c^3 + 16*
a^2*b*c^4)*d^2*e^2 + 2*(7*b^6*c - 60*a*b^4*c^2 + 144*a^2*b^2*c^3 - 64*a^3*c^4)*d
*e^3 - (3*b^7 - 28*a*b^5*c + 80*a^2*b^3*c^2 - 64*a^3*b*c^3)*e^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)))/c^(
11/2), 1/26880*(2*(3840*c^6*e^4*x^6 + 8960*a*c^5*d^4 + 2560*(7*c^6*d*e^3 + b*c^5
*e^4)*x^5 + 560*(3*b^3*c^3 - 20*a*b*c^4)*d^3*e - 168*(15*b^4*c^2 - 100*a*b^2*c^3
+ 128*a^2*c^4)*d^2*e^2 + 14*(105*b^5*c - 760*a*b^3*c^2 + 1296*a^2*b*c^3)*d*e^3
- (315*b^6 - 2520*a*b^4*c + 5488*a^2*b^2*c^2 - 2048*a^3*c^3)*e^4 + 128*(252*c^6*
d^2*e^2 + 98*b*c^5*d*e^3 - (b^2*c^4 - 6*a*c^5)*e^4)*x^4 + 16*(1680*c^6*d^3*e + 1
512*b*c^5*d^2*e^2 - 14*(3*b^2*c^4 - 20*a*c^5)*d*e^3 + (9*b^3*c^3 - 44*a*b*c^4)*e
^4)*x^3 + 8*(1120*c^6*d^4 + 2800*b*c^5*d^3*e - 168*(b^2*c^4 - 8*a*c^5)*d^2*e^2 +
14*(7*b^3*c^3 - 36*a*b*c^4)*d*e^3 - (21*b^4*c^2 - 124*a*b^2*c^3 + 128*a^2*c^4)*
e^4)*x^2 + 2*(4480*b*c^5*d^4 - 560*(b^2*c^4 - 12*a*c^5)*d^3*e + 168*(5*b^3*c^3 -
28*a*b*c^4)*d^2*e^2 - 14*(35*b^4*c^2 - 216*a*b^2*c^3 + 240*a^2*c^4)*d*e^3 + (10
5*b^5*c - 728*a*b^3*c^2 + 1168*a^2*b*c^3)*e^4)*x)*sqrt(c*x^2 + b*x + a)*sqrt(-c)
- 105*(16*(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*d^3*e - 24*(b^5*c^2 - 8*a*b^3*c^
3 + 16*a^2*b*c^4)*d^2*e^2 + 2*(7*b^6*c - 60*a*b^4*c^2 + 144*a^2*b^2*c^3 - 64*a^3
*c^4)*d*e^3 - (3*b^7 - 28*a*b^5*c + 80*a^2*b^3*c^2 - 64*a^3*b*c^3)*e^4)*arctan(1
/2*(2*c*x + b)*sqrt(-c)/(sqrt(c*x^2 + b*x + a)*c)))/(sqrt(-c)*c^5)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \left (b + 2 c x\right ) \left (d + e x\right )^{4} \sqrt{a + b x + c x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*x+b)*(e*x+d)**4*(c*x**2+b*x+a)**(1/2),x)

[Out]

Integral((b + 2*c*x)*(d + e*x)**4*sqrt(a + b*x + c*x**2), x)

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GIAC/XCAS [A]  time = 0.288297, size = 1025, normalized size = 2.38 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/13440*sqrt(c*x^2 + b*x + a)*(2*(4*(2*(8*(10*(3*c*x*e^4 + 2*(7*c^7*d*e^3 + b*c^
6*e^4)/c^6)*x + (252*c^7*d^2*e^2 + 98*b*c^6*d*e^3 - b^2*c^5*e^4 + 6*a*c^6*e^4)/c
^6)*x + (1680*c^7*d^3*e + 1512*b*c^6*d^2*e^2 - 42*b^2*c^5*d*e^3 + 280*a*c^6*d*e^
3 + 9*b^3*c^4*e^4 - 44*a*b*c^5*e^4)/c^6)*x + (1120*c^7*d^4 + 2800*b*c^6*d^3*e -
168*b^2*c^5*d^2*e^2 + 1344*a*c^6*d^2*e^2 + 98*b^3*c^4*d*e^3 - 504*a*b*c^5*d*e^3
- 21*b^4*c^3*e^4 + 124*a*b^2*c^4*e^4 - 128*a^2*c^5*e^4)/c^6)*x + (4480*b*c^6*d^4
- 560*b^2*c^5*d^3*e + 6720*a*c^6*d^3*e + 840*b^3*c^4*d^2*e^2 - 4704*a*b*c^5*d^2
*e^2 - 490*b^4*c^3*d*e^3 + 3024*a*b^2*c^4*d*e^3 - 3360*a^2*c^5*d*e^3 + 105*b^5*c
^2*e^4 - 728*a*b^3*c^3*e^4 + 1168*a^2*b*c^4*e^4)/c^6)*x + (8960*a*c^6*d^4 + 1680
*b^3*c^4*d^3*e - 11200*a*b*c^5*d^3*e - 2520*b^4*c^3*d^2*e^2 + 16800*a*b^2*c^4*d^
2*e^2 - 21504*a^2*c^5*d^2*e^2 + 1470*b^5*c^2*d*e^3 - 10640*a*b^3*c^3*d*e^3 + 181
44*a^2*b*c^4*d*e^3 - 315*b^6*c*e^4 + 2520*a*b^4*c^2*e^4 - 5488*a^2*b^2*c^3*e^4 +
2048*a^3*c^4*e^4)/c^6) + 1/256*(16*b^4*c^3*d^3*e - 128*a*b^2*c^4*d^3*e + 256*a^
2*c^5*d^3*e - 24*b^5*c^2*d^2*e^2 + 192*a*b^3*c^3*d^2*e^2 - 384*a^2*b*c^4*d^2*e^2
+ 14*b^6*c*d*e^3 - 120*a*b^4*c^2*d*e^3 + 288*a^2*b^2*c^3*d*e^3 - 128*a^3*c^4*d*
e^3 - 3*b^7*e^4 + 28*a*b^5*c*e^4 - 80*a^2*b^3*c^2*e^4 + 64*a^3*b*c^3*e^4)*ln(abs
(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) - b))/c^(11/2)