3.880 \(\int \frac{(f+g x)^2}{(d+e x) \left (a+b x+c x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=240 \[ \frac{2 \left (-x \left (c (2 a g (2 e f-d g)-b f (2 d g+e f))+b g^2 (b d-a e)+2 c^2 d f^2\right )-b \left (a g (d g+2 e f)+c d f^2\right )+2 a \left (a e g^2-c f (e f-2 d g)\right )+b^2 e f^2\right )}{\left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}+\frac{(e f-d g)^2 \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{\left (a e^2-b d e+c d^2\right )^{3/2}} \]

[Out]

(2*(b^2*e*f^2 + 2*a*(a*e*g^2 - c*f*(e*f - 2*d*g)) - b*(c*d*f^2 + a*g*(2*e*f + d*
g)) - (2*c^2*d*f^2 + b*(b*d - a*e)*g^2 + c*(2*a*g*(2*e*f - d*g) - b*f*(e*f + 2*d
*g)))*x))/((b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*Sqrt[a + b*x + c*x^2]) + ((e*f
- d*g)^2*ArcTanh[(b*d - 2*a*e + (2*c*d - b*e)*x)/(2*Sqrt[c*d^2 - b*d*e + a*e^2]*
Sqrt[a + b*x + c*x^2])])/(c*d^2 - b*d*e + a*e^2)^(3/2)

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Rubi [A]  time = 0.655042, antiderivative size = 240, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.138 \[ \frac{2 \left (-x \left (b g^2 (b d-a e)+2 a c g (2 e f-d g)-b c f (2 d g+e f)+2 c^2 d f^2\right )-b \left (a g (d g+2 e f)+c d f^2\right )+2 a \left (a e g^2-c f (e f-2 d g)\right )+b^2 e f^2\right )}{\left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}+\frac{(e f-d g)^2 \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{\left (a e^2-b d e+c d^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(b^2*e*f^2 + 2*a*(a*e*g^2 - c*f*(e*f - 2*d*g)) - b*(c*d*f^2 + a*g*(2*e*f + d*
g)) - (2*c^2*d*f^2 + b*(b*d - a*e)*g^2 + 2*a*c*g*(2*e*f - d*g) - b*c*f*(e*f + 2*
d*g))*x))/((b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*Sqrt[a + b*x + c*x^2]) + ((e*f
- d*g)^2*ArcTanh[(b*d - 2*a*e + (2*c*d - b*e)*x)/(2*Sqrt[c*d^2 - b*d*e + a*e^2]*
Sqrt[a + b*x + c*x^2])])/(c*d^2 - b*d*e + a*e^2)^(3/2)

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Rubi in Sympy [A]  time = 103.567, size = 240, normalized size = 1. \[ - \frac{\left (d g - e f\right )^{2} \operatorname{atanh}{\left (\frac{2 a e - b d + x \left (b e - 2 c d\right )}{2 \sqrt{a + b x + c x^{2}} \sqrt{a e^{2} - b d e + c d^{2}}} \right )}}{\left (a e^{2} - b d e + c d^{2}\right )^{\frac{3}{2}}} + \frac{g^{2} \left (4 a + 2 b x\right )}{e \left (- 4 a c + b^{2}\right ) \sqrt{a + b x + c x^{2}}} + \frac{g \left (2 b + 4 c x\right ) \left (d g - 2 e f\right )}{e^{2} \left (- 4 a c + b^{2}\right ) \sqrt{a + b x + c x^{2}}} + \frac{2 \left (d g - e f\right )^{2} \left (- 2 a c e + b^{2} e - b c d + c x \left (b e - 2 c d\right )\right )}{e^{2} \left (- 4 a c + b^{2}\right ) \sqrt{a + b x + c x^{2}} \left (a e^{2} - b d e + c d^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(d*g - e*f)**2*atanh((2*a*e - b*d + x*(b*e - 2*c*d))/(2*sqrt(a + b*x + c*x**2)*
sqrt(a*e**2 - b*d*e + c*d**2)))/(a*e**2 - b*d*e + c*d**2)**(3/2) + g**2*(4*a + 2
*b*x)/(e*(-4*a*c + b**2)*sqrt(a + b*x + c*x**2)) + g*(2*b + 4*c*x)*(d*g - 2*e*f)
/(e**2*(-4*a*c + b**2)*sqrt(a + b*x + c*x**2)) + 2*(d*g - e*f)**2*(-2*a*c*e + b*
*2*e - b*c*d + c*x*(b*e - 2*c*d))/(e**2*(-4*a*c + b**2)*sqrt(a + b*x + c*x**2)*(
a*e**2 - b*d*e + c*d**2))

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Mathematica [A]  time = 0.987066, size = 265, normalized size = 1.1 \[ \frac{2 \left (-2 a^2 e g^2+a b g (d g+2 e f-e g x)-2 a c d g (2 f+g x)+2 a c e f (f+2 g x)+b^2 \left (d g^2 x-e f^2\right )+b c f (d (f-2 g x)-e f x)+2 c^2 d f^2 x\right )}{\left (b^2-4 a c\right ) \sqrt{a+x (b+c x)} \left (e (b d-a e)-c d^2\right )}+\frac{(e f-d g)^2 \log (d+e x)}{\left (e (a e-b d)+c d^2\right )^{3/2}}-\frac{(e f-d g)^2 \log \left (2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}+2 a e-b d+b e x-2 c d x\right )}{\left (e (a e-b d)+c d^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(-2*a^2*e*g^2 + 2*c^2*d*f^2*x - 2*a*c*d*g*(2*f + g*x) + 2*a*c*e*f*(f + 2*g*x)
+ a*b*g*(2*e*f + d*g - e*g*x) + b^2*(-(e*f^2) + d*g^2*x) + b*c*f*(-(e*f*x) + d*
(f - 2*g*x))))/((b^2 - 4*a*c)*(-(c*d^2) + e*(b*d - a*e))*Sqrt[a + x*(b + c*x)])
+ ((e*f - d*g)^2*Log[d + e*x])/(c*d^2 + e*(-(b*d) + a*e))^(3/2) - ((e*f - d*g)^2
*Log[-(b*d) + 2*a*e - 2*c*d*x + b*e*x + 2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a
+ x*(b + c*x)]])/(c*d^2 + e*(-(b*d) + a*e))^(3/2)

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Maple [B]  time = 0.021, size = 2123, normalized size = 8.9 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-g^2/e/c/(c*x^2+b*x+a)^(1/2)+e/(a*e^2-b*d*e+c*d^2)/((x+d/e)^2*c+(b*e-2*c*d)/e*(x
+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*f^2-2*g^2/e*b/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/
2)*x-g^2/e*b^2/c/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)-2*g^2/e^2*d/(4*a*c-b^2)/(c*x^2+
b*x+a)^(1/2)*b+4*g/e*f/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*b+2/(a*e^2-b*d*e+c*d^2)/(
(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(x+d/
e)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b
*d*e+c*d^2)/e^2)^(1/2))/(x+d/e))*d*f*g-e/(a*e^2-b*d*e+c*d^2)/(4*a*c-b^2)/((x+d/e
)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b^2*f^2+2/(a*e^2-b*d*
e+c*d^2)/(4*a*c-b^2)/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)
^(1/2)*b^2*d*f*g+2/(a*e^2-b*d*e+c*d^2)/(4*a*c-b^2)/((x+d/e)^2*c+(b*e-2*c*d)/e*(x
+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b*c*d*f^2-4*g^2/e^2*d/(4*a*c-b^2)/(c*x^2+b*
x+a)^(1/2)*c*x+8*g/e*f/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*c*x+4/(a*e^2-b*d*e+c*d^2)
/(4*a*c-b^2)/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*x
*c^2*d*f^2-1/e/(a*e^2-b*d*e+c*d^2)/(4*a*c-b^2)/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e
)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b^2*d^2*g^2-e/(a*e^2-b*d*e+c*d^2)/((a*e^2-b*d*e
+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(x+d/e)+2*((a*e^2
-b*d*e+c*d^2)/e^2)^(1/2)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/
e^2)^(1/2))/(x+d/e))*f^2-2/(a*e^2-b*d*e+c*d^2)/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e
)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*d*f*g+1/e/(a*e^2-b*d*e+c*d^2)/((x+d/e)^2*c+(b*e
-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*d^2*g^2-1/e/(a*e^2-b*d*e+c*d^2)
/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(x+
d/e)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2
-b*d*e+c*d^2)/e^2)^(1/2))/(x+d/e))*d^2*g^2+4/e^2/(a*e^2-b*d*e+c*d^2)/(4*a*c-b^2)
/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*x*c^2*d^3*g^2
+2/e^2/(a*e^2-b*d*e+c*d^2)/(4*a*c-b^2)/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2
-b*d*e+c*d^2)/e^2)^(1/2)*b*c*d^3*g^2+4/(a*e^2-b*d*e+c*d^2)/(4*a*c-b^2)/((x+d/e)^
2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*x*b*c*d*f*g-2/e/(a*e^2-
b*d*e+c*d^2)/(4*a*c-b^2)/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/
e^2)^(1/2)*x*b*c*d^2*g^2-8/e/(a*e^2-b*d*e+c*d^2)/(4*a*c-b^2)/((x+d/e)^2*c+(b*e-2
*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*x*c^2*d^2*f*g-4/e/(a*e^2-b*d*e+c*
d^2)/(4*a*c-b^2)/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/
2)*b*c*d^2*f*g-2*e/(a*e^2-b*d*e+c*d^2)/(4*a*c-b^2)/((x+d/e)^2*c+(b*e-2*c*d)/e*(x
+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*x*b*c*f^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((g*x + f)^2/((c*x^2 + b*x + a)^(3/2)*(e*x + d)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.03328, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/2*(4*sqrt(c*d^2 - b*d*e + a*e^2)*((b*c*d - (b^2 - 2*a*c)*e)*f^2 - 2*(2*a*c*d
- a*b*e)*f*g + (a*b*d - 2*a^2*e)*g^2 + ((2*c^2*d - b*c*e)*f^2 - 2*(b*c*d - 2*a*
c*e)*f*g - (a*b*e - (b^2 - 2*a*c)*d)*g^2)*x)*sqrt(c*x^2 + b*x + a) - ((a*b^2 - 4
*a^2*c)*e^2*f^2 - 2*(a*b^2 - 4*a^2*c)*d*e*f*g + (a*b^2 - 4*a^2*c)*d^2*g^2 + ((b^
2*c - 4*a*c^2)*e^2*f^2 - 2*(b^2*c - 4*a*c^2)*d*e*f*g + (b^2*c - 4*a*c^2)*d^2*g^2
)*x^2 + ((b^3 - 4*a*b*c)*e^2*f^2 - 2*(b^3 - 4*a*b*c)*d*e*f*g + (b^3 - 4*a*b*c)*d
^2*g^2)*x)*log(((8*a*b*d*e - 8*a^2*e^2 - (b^2 + 4*a*c)*d^2 - (8*c^2*d^2 - 8*b*c*
d*e + (b^2 + 4*a*c)*e^2)*x^2 - 2*(4*b*c*d^2 + 4*a*b*e^2 - (3*b^2 + 4*a*c)*d*e)*x
)*sqrt(c*d^2 - b*d*e + a*e^2) - 4*(b*c*d^3 + 3*a*b*d*e^2 - 2*a^2*e^3 - (b^2 + 2*
a*c)*d^2*e + (2*c^2*d^3 - 3*b*c*d^2*e - a*b*e^3 + (b^2 + 2*a*c)*d*e^2)*x)*sqrt(c
*x^2 + b*x + a))/(e^2*x^2 + 2*d*e*x + d^2)))/(((a*b^2*c - 4*a^2*c^2)*d^2 - (a*b^
3 - 4*a^2*b*c)*d*e + (a^2*b^2 - 4*a^3*c)*e^2 + ((b^2*c^2 - 4*a*c^3)*d^2 - (b^3*c
- 4*a*b*c^2)*d*e + (a*b^2*c - 4*a^2*c^2)*e^2)*x^2 + ((b^3*c - 4*a*b*c^2)*d^2 -
(b^4 - 4*a*b^2*c)*d*e + (a*b^3 - 4*a^2*b*c)*e^2)*x)*sqrt(c*d^2 - b*d*e + a*e^2))
, -(2*sqrt(-c*d^2 + b*d*e - a*e^2)*((b*c*d - (b^2 - 2*a*c)*e)*f^2 - 2*(2*a*c*d -
a*b*e)*f*g + (a*b*d - 2*a^2*e)*g^2 + ((2*c^2*d - b*c*e)*f^2 - 2*(b*c*d - 2*a*c*
e)*f*g - (a*b*e - (b^2 - 2*a*c)*d)*g^2)*x)*sqrt(c*x^2 + b*x + a) + ((a*b^2 - 4*a
^2*c)*e^2*f^2 - 2*(a*b^2 - 4*a^2*c)*d*e*f*g + (a*b^2 - 4*a^2*c)*d^2*g^2 + ((b^2*
c - 4*a*c^2)*e^2*f^2 - 2*(b^2*c - 4*a*c^2)*d*e*f*g + (b^2*c - 4*a*c^2)*d^2*g^2)*
x^2 + ((b^3 - 4*a*b*c)*e^2*f^2 - 2*(b^3 - 4*a*b*c)*d*e*f*g + (b^3 - 4*a*b*c)*d^2
*g^2)*x)*arctan(-1/2*sqrt(-c*d^2 + b*d*e - a*e^2)*(b*d - 2*a*e + (2*c*d - b*e)*x
)/((c*d^2 - b*d*e + a*e^2)*sqrt(c*x^2 + b*x + a))))/(((a*b^2*c - 4*a^2*c^2)*d^2
- (a*b^3 - 4*a^2*b*c)*d*e + (a^2*b^2 - 4*a^3*c)*e^2 + ((b^2*c^2 - 4*a*c^3)*d^2 -
(b^3*c - 4*a*b*c^2)*d*e + (a*b^2*c - 4*a^2*c^2)*e^2)*x^2 + ((b^3*c - 4*a*b*c^2)
*d^2 - (b^4 - 4*a*b^2*c)*d*e + (a*b^3 - 4*a^2*b*c)*e^2)*x)*sqrt(-c*d^2 + b*d*e -
a*e^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((g*x+f)**2/(e*x+d)/(c*x**2+b*x+a)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.282558, size = 1022, normalized size = 4.26 \[ -\frac{2 \,{\left (\frac{{\left (2 \, c^{3} d^{3} f^{2} - 2 \, b c^{2} d^{3} f g + b^{2} c d^{3} g^{2} - 2 \, a c^{2} d^{3} g^{2} - 3 \, b c^{2} d^{2} f^{2} e + 2 \, b^{2} c d^{2} f g e + 4 \, a c^{2} d^{2} f g e - b^{3} d^{2} g^{2} e + a b c d^{2} g^{2} e + b^{2} c d f^{2} e^{2} + 2 \, a c^{2} d f^{2} e^{2} - 6 \, a b c d f g e^{2} + 2 \, a b^{2} d g^{2} e^{2} - 2 \, a^{2} c d g^{2} e^{2} - a b c f^{2} e^{3} + 4 \, a^{2} c f g e^{3} - a^{2} b g^{2} e^{3}\right )} x}{b^{2} c^{2} d^{4} - 4 \, a c^{3} d^{4} - 2 \, b^{3} c d^{3} e + 8 \, a b c^{2} d^{3} e + b^{4} d^{2} e^{2} - 2 \, a b^{2} c d^{2} e^{2} - 8 \, a^{2} c^{2} d^{2} e^{2} - 2 \, a b^{3} d e^{3} + 8 \, a^{2} b c d e^{3} + a^{2} b^{2} e^{4} - 4 \, a^{3} c e^{4}} + \frac{b c^{2} d^{3} f^{2} - 4 \, a c^{2} d^{3} f g + a b c d^{3} g^{2} - 2 \, b^{2} c d^{2} f^{2} e + 2 \, a c^{2} d^{2} f^{2} e + 6 \, a b c d^{2} f g e - a b^{2} d^{2} g^{2} e - 2 \, a^{2} c d^{2} g^{2} e + b^{3} d f^{2} e^{2} - a b c d f^{2} e^{2} - 2 \, a b^{2} d f g e^{2} - 4 \, a^{2} c d f g e^{2} + 3 \, a^{2} b d g^{2} e^{2} - a b^{2} f^{2} e^{3} + 2 \, a^{2} c f^{2} e^{3} + 2 \, a^{2} b f g e^{3} - 2 \, a^{3} g^{2} e^{3}}{b^{2} c^{2} d^{4} - 4 \, a c^{3} d^{4} - 2 \, b^{3} c d^{3} e + 8 \, a b c^{2} d^{3} e + b^{4} d^{2} e^{2} - 2 \, a b^{2} c d^{2} e^{2} - 8 \, a^{2} c^{2} d^{2} e^{2} - 2 \, a b^{3} d e^{3} + 8 \, a^{2} b c d e^{3} + a^{2} b^{2} e^{4} - 4 \, a^{3} c e^{4}}\right )}}{\sqrt{c x^{2} + b x + a}} + \frac{2 \,{\left (d^{2} g^{2} - 2 \, d f g e + f^{2} e^{2}\right )} \arctan \left (-\frac{{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} e + \sqrt{c} d}{\sqrt{-c d^{2} + b d e - a e^{2}}}\right )}{{\left (c d^{2} - b d e + a e^{2}\right )} \sqrt{-c d^{2} + b d e - a e^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*((2*c^3*d^3*f^2 - 2*b*c^2*d^3*f*g + b^2*c*d^3*g^2 - 2*a*c^2*d^3*g^2 - 3*b*c^2
*d^2*f^2*e + 2*b^2*c*d^2*f*g*e + 4*a*c^2*d^2*f*g*e - b^3*d^2*g^2*e + a*b*c*d^2*g
^2*e + b^2*c*d*f^2*e^2 + 2*a*c^2*d*f^2*e^2 - 6*a*b*c*d*f*g*e^2 + 2*a*b^2*d*g^2*e
^2 - 2*a^2*c*d*g^2*e^2 - a*b*c*f^2*e^3 + 4*a^2*c*f*g*e^3 - a^2*b*g^2*e^3)*x/(b^2
*c^2*d^4 - 4*a*c^3*d^4 - 2*b^3*c*d^3*e + 8*a*b*c^2*d^3*e + b^4*d^2*e^2 - 2*a*b^2
*c*d^2*e^2 - 8*a^2*c^2*d^2*e^2 - 2*a*b^3*d*e^3 + 8*a^2*b*c*d*e^3 + a^2*b^2*e^4 -
4*a^3*c*e^4) + (b*c^2*d^3*f^2 - 4*a*c^2*d^3*f*g + a*b*c*d^3*g^2 - 2*b^2*c*d^2*f
^2*e + 2*a*c^2*d^2*f^2*e + 6*a*b*c*d^2*f*g*e - a*b^2*d^2*g^2*e - 2*a^2*c*d^2*g^2
*e + b^3*d*f^2*e^2 - a*b*c*d*f^2*e^2 - 2*a*b^2*d*f*g*e^2 - 4*a^2*c*d*f*g*e^2 + 3
*a^2*b*d*g^2*e^2 - a*b^2*f^2*e^3 + 2*a^2*c*f^2*e^3 + 2*a^2*b*f*g*e^3 - 2*a^3*g^2
*e^3)/(b^2*c^2*d^4 - 4*a*c^3*d^4 - 2*b^3*c*d^3*e + 8*a*b*c^2*d^3*e + b^4*d^2*e^2
- 2*a*b^2*c*d^2*e^2 - 8*a^2*c^2*d^2*e^2 - 2*a*b^3*d*e^3 + 8*a^2*b*c*d*e^3 + a^2
*b^2*e^4 - 4*a^3*c*e^4))/sqrt(c*x^2 + b*x + a) + 2*(d^2*g^2 - 2*d*f*g*e + f^2*e^
2)*arctan(-((sqrt(c)*x - sqrt(c*x^2 + b*x + a))*e + sqrt(c)*d)/sqrt(-c*d^2 + b*d
*e - a*e^2))/((c*d^2 - b*d*e + a*e^2)*sqrt(-c*d^2 + b*d*e - a*e^2))