3.2485 \(\int \frac{A+B x}{(d+e x)^2 \left (a+b x+c x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=746 \[ \frac{e \sqrt{a+b x+c x^2} \left (-8 b c \left (B \left (5 a^2 e^4+3 a c d^2 e^2+2 c^2 d^4\right )+2 A c d e \left (9 a e^2+4 c d^2\right )\right )-16 c^2 \left (a B d e \left (2 c d^2-13 a e^2\right )-A \left (-8 a^2 e^4+9 a c d^2 e^2+2 c^2 d^4\right )\right )-2 b^3 e^2 \left (-3 a B e^2-10 A c d e+9 B c d^2\right )+4 b^2 c e \left (25 a A e^3-14 a B d e^2+3 A c d^2 e+10 B c d^3\right )+3 b^4 e^3 (3 B d-5 A e)\right )}{3 \left (b^2-4 a c\right )^2 (d+e x) \left (a e^2-b d e+c d^2\right )^3}+\frac{2 \left (c x \left ((2 c d-b e) \left (-2 b \left (-a B e^2+A c d e+2 B c d^2\right )+8 c \left (2 a A e^2-a B d e+A c d^2\right )+b^2 e (3 B d-5 A e)\right )+6 c e (b d-2 a e) (-2 a B e+A b e-2 A c d+b B d)\right )+\left (2 a c e+b^2 (-e)+b c d\right ) \left (-2 b \left (-a B e^2+A c d e+2 B c d^2\right )+8 c \left (2 a A e^2-a B d e+A c d^2\right )+b^2 e (3 B d-5 A e)\right )+6 a c e (2 c d-b e) (-2 a B e+A b e-2 A c d+b B d)\right )}{3 \left (b^2-4 a c\right )^2 (d+e x) \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )^2}+\frac{2 \left (-A \left (2 a c e+b^2 (-e)+b c d\right )+c x (-2 a B e+A b e-2 A c d+b B d)+a B (2 c d-b e)\right )}{3 \left (b^2-4 a c\right ) (d+e x) \left (a+b x+c x^2\right )^{3/2} \left (a e^2-b d e+c d^2\right )}+\frac{e^3 \left (5 A e (2 c d-b e)-B \left (8 c d^2-e (2 a e+3 b d)\right )\right ) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{2 \left (a e^2-b d e+c d^2\right )^{7/2}} \]

[Out]

(2*(a*B*(2*c*d - b*e) - A*(b*c*d - b^2*e + 2*a*c*e) + c*(b*B*d - 2*A*c*d + A*b*e
- 2*a*B*e)*x))/(3*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*(d + e*x)*(a + b*x + c*
x^2)^(3/2)) + (2*(6*a*c*e*(2*c*d - b*e)*(b*B*d - 2*A*c*d + A*b*e - 2*a*B*e) + (b
*c*d - b^2*e + 2*a*c*e)*(b^2*e*(3*B*d - 5*A*e) + 8*c*(A*c*d^2 - a*B*d*e + 2*a*A*
e^2) - 2*b*(2*B*c*d^2 + A*c*d*e - a*B*e^2)) + c*(6*c*e*(b*d - 2*a*e)*(b*B*d - 2*
A*c*d + A*b*e - 2*a*B*e) + (2*c*d - b*e)*(b^2*e*(3*B*d - 5*A*e) + 8*c*(A*c*d^2 -
a*B*d*e + 2*a*A*e^2) - 2*b*(2*B*c*d^2 + A*c*d*e - a*B*e^2)))*x))/(3*(b^2 - 4*a*
c)^2*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)*Sqrt[a + b*x + c*x^2]) + (e*(3*b^4*e^3*
(3*B*d - 5*A*e) - 2*b^3*e^2*(9*B*c*d^2 - 10*A*c*d*e - 3*a*B*e^2) + 4*b^2*c*e*(10
*B*c*d^3 + 3*A*c*d^2*e - 14*a*B*d*e^2 + 25*a*A*e^3) - 16*c^2*(a*B*d*e*(2*c*d^2 -
13*a*e^2) - A*(2*c^2*d^4 + 9*a*c*d^2*e^2 - 8*a^2*e^4)) - 8*b*c*(2*A*c*d*e*(4*c*
d^2 + 9*a*e^2) + B*(2*c^2*d^4 + 3*a*c*d^2*e^2 + 5*a^2*e^4)))*Sqrt[a + b*x + c*x^
2])/(3*(b^2 - 4*a*c)^2*(c*d^2 - b*d*e + a*e^2)^3*(d + e*x)) + (e^3*(5*A*e*(2*c*d
- b*e) - B*(8*c*d^2 - e*(3*b*d + 2*a*e)))*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])])/(2*(c*d^2 - b*d*e + a
*e^2)^(7/2))

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Rubi [A]  time = 4.26574, antiderivative size = 744, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148 \[ \frac{e \sqrt{a+b x+c x^2} \left (-8 b c \left (B \left (5 a^2 e^4+3 a c d^2 e^2+2 c^2 d^4\right )+2 A c d e \left (9 a e^2+4 c d^2\right )\right )-16 c^2 \left (a B d e \left (2 c d^2-13 a e^2\right )-A \left (-8 a^2 e^4+9 a c d^2 e^2+2 c^2 d^4\right )\right )-2 b^3 e^2 \left (-3 a B e^2-10 A c d e+9 B c d^2\right )+4 b^2 c e \left (25 a A e^3-14 a B d e^2+3 A c d^2 e+10 B c d^3\right )+3 b^4 e^3 (3 B d-5 A e)\right )}{3 \left (b^2-4 a c\right )^2 (d+e x) \left (a e^2-b d e+c d^2\right )^3}+\frac{2 \left (c x \left ((2 c d-b e) \left (-2 b \left (-a B e^2+A c d e+2 B c d^2\right )+8 c \left (2 a A e^2-a B d e+A c d^2\right )+b^2 e (3 B d-5 A e)\right )+6 c e (b d-2 a e) (-2 a B e+A b e-2 A c d+b B d)\right )+\left (2 a c e+b^2 (-e)+b c d\right ) \left (-2 b \left (-a B e^2+A c d e+2 B c d^2\right )+8 c \left (2 a A e^2-a B d e+A c d^2\right )+b^2 e (3 B d-5 A e)\right )+6 a c e (2 c d-b e) (-2 a B e+A b e-2 A c d+b B d)\right )}{3 \left (b^2-4 a c\right )^2 (d+e x) \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )^2}+\frac{2 \left (-A \left (2 a c e+b^2 (-e)+b c d\right )+c x (-2 a B e+A b e-2 A c d+b B d)+a B (2 c d-b e)\right )}{3 \left (b^2-4 a c\right ) (d+e x) \left (a+b x+c x^2\right )^{3/2} \left (a e^2-b d e+c d^2\right )}-\frac{e^3 \left (-B e (2 a e+3 b d)-5 A e (2 c d-b e)+8 B c d^2\right ) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{2 \left (a e^2-b d e+c d^2\right )^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(a*B*(2*c*d - b*e) - A*(b*c*d - b^2*e + 2*a*c*e) + c*(b*B*d - 2*A*c*d + A*b*e
- 2*a*B*e)*x))/(3*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*(d + e*x)*(a + b*x + c*
x^2)^(3/2)) + (2*(6*a*c*e*(2*c*d - b*e)*(b*B*d - 2*A*c*d + A*b*e - 2*a*B*e) + (b
*c*d - b^2*e + 2*a*c*e)*(b^2*e*(3*B*d - 5*A*e) + 8*c*(A*c*d^2 - a*B*d*e + 2*a*A*
e^2) - 2*b*(2*B*c*d^2 + A*c*d*e - a*B*e^2)) + c*(6*c*e*(b*d - 2*a*e)*(b*B*d - 2*
A*c*d + A*b*e - 2*a*B*e) + (2*c*d - b*e)*(b^2*e*(3*B*d - 5*A*e) + 8*c*(A*c*d^2 -
a*B*d*e + 2*a*A*e^2) - 2*b*(2*B*c*d^2 + A*c*d*e - a*B*e^2)))*x))/(3*(b^2 - 4*a*
c)^2*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)*Sqrt[a + b*x + c*x^2]) + (e*(3*b^4*e^3*
(3*B*d - 5*A*e) - 2*b^3*e^2*(9*B*c*d^2 - 10*A*c*d*e - 3*a*B*e^2) + 4*b^2*c*e*(10
*B*c*d^3 + 3*A*c*d^2*e - 14*a*B*d*e^2 + 25*a*A*e^3) - 16*c^2*(a*B*d*e*(2*c*d^2 -
13*a*e^2) - A*(2*c^2*d^4 + 9*a*c*d^2*e^2 - 8*a^2*e^4)) - 8*b*c*(2*A*c*d*e*(4*c*
d^2 + 9*a*e^2) + B*(2*c^2*d^4 + 3*a*c*d^2*e^2 + 5*a^2*e^4)))*Sqrt[a + b*x + c*x^
2])/(3*(b^2 - 4*a*c)^2*(c*d^2 - b*d*e + a*e^2)^3*(d + e*x)) - (e^3*(8*B*c*d^2 -
B*e*(3*b*d + 2*a*e) - 5*A*e*(2*c*d - b*e))*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])])/(2*(c*d^2 - b*d*e + a
*e^2)^(7/2))

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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((B*x+A)/(e*x+d)**2/(c*x**2+b*x+a)**(5/2),x)

[Out]

Timed out

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Mathematica [A]  time = 7.59717, size = 1078, normalized size = 1.45 \[ \frac{\left (-8 B c d^2+3 b B e d+10 A c e d-5 A b e^2+2 a B e^2\right ) \left (c x^2+b x+a\right )^{5/2} \log (d+e x) e^3}{2 \left (c d^2-b e d+a e^2\right )^{7/2} (a+x (b+c x))^{5/2}}-\frac{\left (-8 B c d^2+3 b B e d+10 A c e d-5 A b e^2+2 a B e^2\right ) \left (c x^2+b x+a\right )^{5/2} \log \left (-b d-2 c x d+2 a e+b e x+2 \sqrt{c d^2-b e d+a e^2} \sqrt{c x^2+b x+a}\right ) e^3}{2 \left (c d^2-b e d+a e^2\right )^{7/2} (a+x (b+c x))^{5/2}}+\frac{\left (c x^2+b x+a\right )^3 \left (-\frac{(A e-B d) e^4}{\left (c d^2-b e d+a e^2\right )^3 (d+e x)}+\frac{2 \left (-6 A e^4 b^5+3 B d e^3 b^5+3 a B e^4 b^4+11 A c d e^3 b^4-9 B c d^2 e^2 b^4-6 A c e^4 x b^4+3 B c d e^3 x b^4+43 a A c e^4 b^3-20 a B c d e^3 b^3+3 A c^2 d^2 e^2 b^3+10 B c^2 d^3 e b^3+3 a B c e^4 x b^3+10 A c^2 d e^3 x b^3-9 B c^2 d^2 e^2 x b^3-4 B c^3 d^4 b^2-22 a^2 B c e^4 b^2-84 a A c^2 d e^3 b^2+30 a B c^2 d^2 e^2 b^2-16 A c^3 d^3 e b^2+38 a A c^2 e^4 x b^2-16 a B c^2 d e^3 x b^2+6 A c^3 d^2 e^2 x b^2+20 B c^3 d^3 e x b^2+8 A c^4 d^4 b-68 a^2 A c^2 e^4 b+64 a^2 B c^2 d e^3 b+36 a A c^3 d^2 e^2 b-8 a B c^3 d^3 e b-8 B c^4 d^4 x b-20 a^2 B c^2 e^4 x b-72 a A c^3 d e^3 x b-12 a B c^3 d^2 e^2 x b-32 A c^4 d^3 e x b+24 a^3 B c^2 e^4+96 a^2 A c^3 d e^3-72 a^2 B c^3 d^2 e^2+16 A c^5 d^4 x-40 a^2 A c^3 e^4 x+80 a^2 B c^3 d e^3 x+72 a A c^4 d^2 e^2 x-16 a B c^4 d^3 e x\right )}{3 \left (4 a c-b^2\right )^2 \left (c d^2-b e d+a e^2\right )^3 \left (c x^2+b x+a\right )}+\frac{2 \left (A e^2 b^3-a B e^2 b^2-2 A c d e b^2+A c e^2 x b^2+A c^2 d^2 b-3 a A c e^2 b+2 a B c d e b-B c^2 d^2 x b-a B c e^2 x b-2 A c^2 d e x b-2 a B c^2 d^2+2 a^2 B c e^2+4 a A c^2 d e+2 A c^3 d^2 x-2 a A c^2 e^2 x+4 a B c^2 d e x\right )}{3 \left (4 a c-b^2\right ) \left (c d^2-b e d+a e^2\right )^2 \left (c x^2+b x+a\right )^2}\right )}{(a+x (b+c x))^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

((a + b*x + c*x^2)^3*(-((e^4*(-(B*d) + A*e))/((c*d^2 - b*d*e + a*e^2)^3*(d + e*x
))) + (2*(A*b*c^2*d^2 - 2*a*B*c^2*d^2 - 2*A*b^2*c*d*e + 2*a*b*B*c*d*e + 4*a*A*c^
2*d*e + A*b^3*e^2 - a*b^2*B*e^2 - 3*a*A*b*c*e^2 + 2*a^2*B*c*e^2 - b*B*c^2*d^2*x
+ 2*A*c^3*d^2*x - 2*A*b*c^2*d*e*x + 4*a*B*c^2*d*e*x + A*b^2*c*e^2*x - a*b*B*c*e^
2*x - 2*a*A*c^2*e^2*x))/(3*(-b^2 + 4*a*c)*(c*d^2 - b*d*e + a*e^2)^2*(a + b*x + c
*x^2)^2) + (2*(-4*b^2*B*c^3*d^4 + 8*A*b*c^4*d^4 + 10*b^3*B*c^2*d^3*e - 16*A*b^2*
c^3*d^3*e - 8*a*b*B*c^3*d^3*e - 9*b^4*B*c*d^2*e^2 + 3*A*b^3*c^2*d^2*e^2 + 30*a*b
^2*B*c^2*d^2*e^2 + 36*a*A*b*c^3*d^2*e^2 - 72*a^2*B*c^3*d^2*e^2 + 3*b^5*B*d*e^3 +
11*A*b^4*c*d*e^3 - 20*a*b^3*B*c*d*e^3 - 84*a*A*b^2*c^2*d*e^3 + 64*a^2*b*B*c^2*d
*e^3 + 96*a^2*A*c^3*d*e^3 - 6*A*b^5*e^4 + 3*a*b^4*B*e^4 + 43*a*A*b^3*c*e^4 - 22*
a^2*b^2*B*c*e^4 - 68*a^2*A*b*c^2*e^4 + 24*a^3*B*c^2*e^4 - 8*b*B*c^4*d^4*x + 16*A
*c^5*d^4*x + 20*b^2*B*c^3*d^3*e*x - 32*A*b*c^4*d^3*e*x - 16*a*B*c^4*d^3*e*x - 9*
b^3*B*c^2*d^2*e^2*x + 6*A*b^2*c^3*d^2*e^2*x - 12*a*b*B*c^3*d^2*e^2*x + 72*a*A*c^
4*d^2*e^2*x + 3*b^4*B*c*d*e^3*x + 10*A*b^3*c^2*d*e^3*x - 16*a*b^2*B*c^2*d*e^3*x
- 72*a*A*b*c^3*d*e^3*x + 80*a^2*B*c^3*d*e^3*x - 6*A*b^4*c*e^4*x + 3*a*b^3*B*c*e^
4*x + 38*a*A*b^2*c^2*e^4*x - 20*a^2*b*B*c^2*e^4*x - 40*a^2*A*c^3*e^4*x))/(3*(-b^
2 + 4*a*c)^2*(c*d^2 - b*d*e + a*e^2)^3*(a + b*x + c*x^2))))/(a + x*(b + c*x))^(5
/2) + (e^3*(-8*B*c*d^2 + 3*b*B*d*e + 10*A*c*d*e - 5*A*b*e^2 + 2*a*B*e^2)*(a + b*
x + c*x^2)^(5/2)*Log[d + e*x])/(2*(c*d^2 - b*d*e + a*e^2)^(7/2)*(a + x*(b + c*x)
)^(5/2)) - (e^3*(-8*B*c*d^2 + 3*b*B*d*e + 10*A*c*d*e - 5*A*b*e^2 + 2*a*B*e^2)*(a
+ b*x + c*x^2)^(5/2)*Log[-(b*d) + 2*a*e - 2*c*d*x + b*e*x + 2*Sqrt[c*d^2 - b*d*
e + a*e^2]*Sqrt[a + b*x + c*x^2]])/(2*(c*d^2 - b*d*e + a*e^2)^(7/2)*(a + x*(b +
c*x))^(5/2))

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Maple [B]  time = 0.037, size = 6675, normalized size = 9. \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

result too large to display

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 13.781, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/12*(4*(2*(4*(2*B*a^2 - 3*A*a*b)*c^4 + (2*B*a*b^2 + A*b^3)*c^3)*d^5 - 2*(16*A
*a^2*c^4 + 8*(B*a^2*b - 4*A*a*b^2)*c^3 + 3*(2*B*a*b^3 + A*b^4)*c^2)*d^4*e + 6*(4
*(6*B*a^3 - A*a^2*b)*c^3 - (6*B*a^2*b^2 + 7*A*a*b^3)*c^2 + (2*B*a*b^4 + A*b^5)*c
)*d^3*e^2 - 2*(2*B*a*b^5 + A*b^6 + 112*A*a^3*c^3 + 4*(16*B*a^3*b - 21*A*a^2*b^2)
*c^2 - 2*(8*B*a^2*b^3 - 3*A*a*b^4)*c)*d^2*e^3 - (11*B*a^2*b^4 - 14*A*a*b^5 + 16*
(7*B*a^4 - 10*A*a^3*b)*c^2 - 20*(4*B*a^3*b^2 - 5*A*a^2*b^3)*c)*d*e^4 + 3*(A*a^2*
b^4 - 8*A*a^3*b^2*c + 16*A*a^4*c^2)*e^5 + (16*(B*b*c^5 - 2*A*c^6)*d^4*e - 8*(5*B
*b^2*c^4 - 4*(B*a + 2*A*b)*c^5)*d^3*e^2 + 6*(3*B*b^3*c^3 - 24*A*a*c^5 + 2*(2*B*a
*b - A*b^2)*c^4)*d^2*e^3 - (9*B*b^4*c^2 + 16*(13*B*a^2 - 9*A*a*b)*c^4 - 4*(14*B*
a*b^2 - 5*A*b^3)*c^3)*d*e^4 + (128*A*a^2*c^4 + 20*(2*B*a^2*b - 5*A*a*b^2)*c^3 -
3*(2*B*a*b^3 - 5*A*b^4)*c^2)*e^5)*x^4 + 2*(8*(B*b*c^5 - 2*A*c^6)*d^5 - 8*(B*b^2*
c^4 - (2*B*a + A*b)*c^5)*d^4*e - 3*(7*B*b^3*c^3 + 24*A*a*c^5 - 2*(6*B*a*b + 7*A*
b^2)*c^4)*d^3*e^2 + (15*B*b^4*c^2 - 4*(2*B*a^2 + 9*A*a*b)*c^4 - (2*B*a*b^2 + 19*
A*b^3)*c^3)*d^2*e^3 - (9*B*b^5*c + 56*A*a^2*c^4 + 2*(86*B*a^2*b - 59*A*a*b^2)*c^
3 - 3*(19*B*a*b^3 - 5*A*b^4)*c^2)*d*e^4 - 3*(4*(2*B*a^3 - 13*A*a^2*b)*c^3 - 7*(2
*B*a^2*b^2 - 5*A*a*b^3)*c^2 + (2*B*a*b^4 - 5*A*b^5)*c)*e^5)*x^3 + 3*(8*(B*b^2*c^
4 - 2*A*b*c^5)*d^5 - 2*(9*B*b^3*c^3 + 8*A*a*c^5 - 2*(6*B*a*b + 7*A*b^2)*c^4)*d^4
*e + 2*(3*B*b^4*c^2 + 4*(6*B*a^2 - 5*A*a*b)*c^4 - (10*B*a*b^2 - A*b^3)*c^3)*d^3*
e^2 + 2*(B*b^5*c - 56*A*a^2*c^4 - 2*(6*B*a^2*b - 17*A*a*b^2)*c^3 + (3*B*a*b^3 -
7*A*b^4)*c^2)*d^2*e^3 - (3*B*b^6 - 14*B*a*b^4*c + 56*(2*B*a^3 - A*a^2*b)*c^3 + 2
*(2*B*a^2*b^2 + A*a*b^3)*c^2)*d*e^4 - (2*B*a*b^5 - 5*A*b^6 - 16*A*a^2*b^2*c^2 -
64*A*a^3*c^3 - 6*(2*B*a^2*b^3 - 5*A*a*b^4)*c)*e^5)*x^2 + 2*(3*(B*b^3*c^3 - 8*A*a
*c^5 + 2*(2*B*a*b - A*b^2)*c^4)*d^5 - (9*B*b^4*c^2 - 4*(2*B*a^2 + 9*A*a*b)*c^4 +
(10*B*a*b^2 - 13*A*b^3)*c^3)*d^4*e + (9*B*b^5*c - 88*A*a^2*c^4 + 2*(50*B*a^2*b
- 11*A*a*b^2)*c^3 - 3*(11*B*a*b^3 + A*b^4)*c^2)*d^3*e^2 - 3*(B*b^6 + 4*(2*B*a^3
+ 3*A*a^2*b)*c^3 + (22*B*a^2*b^2 - 19*A*a*b^3)*c^2 - (8*B*a*b^4 - 3*A*b^5)*c)*d^
2*e^3 - (8*B*a*b^5 - 5*A*b^6 + 64*A*a^3*c^3 + 4*(28*B*a^3*b - 27*A*a^2*b^2)*c^2
- 2*(29*B*a^2*b^3 - 21*A*a*b^4)*c)*d*e^4 - 2*(2*B*a^2*b^4 - 5*A*a*b^5 + 16*(B*a^
4 - 4*A*a^3*b)*c^2 - (14*B*a^3*b^2 - 37*A*a^2*b^3)*c)*e^5)*x)*sqrt(c*d^2 - b*d*e
+ a*e^2)*sqrt(c*x^2 + b*x + a) - 3*(8*(B*a^2*b^4*c - 8*B*a^3*b^2*c^2 + 16*B*a^4
*c^3)*d^3*e^3 - (3*B*a^2*b^5 + 160*A*a^4*c^3 + 16*(3*B*a^4*b - 5*A*a^3*b^2)*c^2
- 2*(12*B*a^3*b^3 - 5*A*a^2*b^4)*c)*d^2*e^4 - (2*B*a^3*b^4 - 5*A*a^2*b^5 + 16*(2
*B*a^5 - 5*A*a^4*b)*c^2 - 8*(2*B*a^4*b^2 - 5*A*a^3*b^3)*c)*d*e^5 + (8*(B*b^4*c^3
- 8*B*a*b^2*c^4 + 16*B*a^2*c^5)*d^2*e^4 - (3*B*b^5*c^2 + 160*A*a^2*c^5 + 16*(3*
B*a^2*b - 5*A*a*b^2)*c^4 - 2*(12*B*a*b^3 - 5*A*b^4)*c^3)*d*e^5 - (16*(2*B*a^3 -
5*A*a^2*b)*c^4 - 8*(2*B*a^2*b^2 - 5*A*a*b^3)*c^3 + (2*B*a*b^4 - 5*A*b^5)*c^2)*e^
6)*x^5 + (8*(B*b^4*c^3 - 8*B*a*b^2*c^4 + 16*B*a^2*c^5)*d^3*e^3 + (13*B*b^5*c^2 -
160*A*a^2*c^5 + 16*(13*B*a^2*b + 5*A*a*b^2)*c^4 - 2*(52*B*a*b^3 + 5*A*b^4)*c^3)
*d^2*e^4 - (6*B*b^6*c + 16*(2*B*a^3 + 15*A*a^2*b)*c^4 + 40*(2*B*a^2*b^2 - 3*A*a*
b^3)*c^3 - (46*B*a*b^4 - 15*A*b^5)*c^2)*d*e^5 - 2*(16*(2*B*a^3*b - 5*A*a^2*b^2)*
c^3 - 8*(2*B*a^2*b^3 - 5*A*a*b^4)*c^2 + (2*B*a*b^5 - 5*A*b^6)*c)*e^6)*x^4 + (16*
(B*b^5*c^2 - 8*B*a*b^3*c^3 + 16*B*a^2*b*c^4)*d^3*e^3 + 2*(B*b^6*c - 10*A*b^5*c^2
+ 32*(4*B*a^3 - 5*A*a^2*b)*c^4 - 16*(3*B*a^2*b^2 - 5*A*a*b^3)*c^3)*d^2*e^4 - (3
*B*b^7 - 14*B*a*b^5*c + 320*A*a^3*c^4 + 160*(B*a^3*b - A*a^2*b^2)*c^3 - 4*(8*B*a
^2*b^3 - 5*A*a*b^4)*c^2)*d*e^5 - (2*B*a*b^6 - 5*A*b^7 + 32*(2*B*a^4 - 5*A*a^3*b)
*c^3 - 6*(2*B*a^2*b^4 - 5*A*a*b^5)*c)*e^6)*x^3 + (8*(B*b^6*c - 6*B*a*b^4*c^2 + 3
2*B*a^3*c^4)*d^3*e^3 - (3*B*b^7 - 160*B*a^3*b*c^3 + 320*A*a^3*c^4 + 4*(32*B*a^2*
b^3 - 15*A*a*b^4)*c^2 - 2*(17*B*a*b^5 - 5*A*b^6)*c)*d^2*e^4 - (8*B*a*b^6 - 5*A*b
^7 + 32*(2*B*a^4 + 5*A*a^3*b)*c^3 + 32*(3*B*a^3*b^2 - 5*A*a^2*b^3)*c^2 - 10*(6*B
*a^2*b^4 - 5*A*a*b^5)*c)*d*e^5 - 2*(2*B*a^2*b^5 - 5*A*a*b^6 + 16*(2*B*a^4*b - 5*
A*a^3*b^2)*c^2 - 8*(2*B*a^3*b^3 - 5*A*a^2*b^4)*c)*e^6)*x^2 + (16*(B*a*b^5*c - 8*
B*a^2*b^3*c^2 + 16*B*a^3*b*c^3)*d^3*e^3 - 2*(3*B*a*b^6 - 32*(2*B*a^4 - 5*A*a^3*b
)*c^3 + 80*(B*a^3*b^2 - A*a^2*b^3)*c^2 - 2*(14*B*a^2*b^4 - 5*A*a*b^5)*c)*d^2*e^4
- (7*B*a^2*b^5 - 10*A*a*b^6 + 160*A*a^4*c^3 + 16*(7*B*a^4*b - 15*A*a^3*b^2)*c^2
- 2*(28*B*a^3*b^3 - 45*A*a^2*b^4)*c)*d*e^5 - (2*B*a^3*b^4 - 5*A*a^2*b^5 + 16*(2
*B*a^5 - 5*A*a^4*b)*c^2 - 8*(2*B*a^4*b^2 - 5*A*a^3*b^3)*c)*e^6)*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^2*b^4*c^3 - 8*a^3*b^2*c^4 + 16*a^4*c^5)*d^7 - 3*(a^2*b
^5*c^2 - 8*a^3*b^3*c^3 + 16*a^4*b*c^4)*d^6*e + 3*(a^2*b^6*c - 7*a^3*b^4*c^2 + 8*
a^4*b^2*c^3 + 16*a^5*c^4)*d^5*e^2 - (a^2*b^7 - 2*a^3*b^5*c - 32*a^4*b^3*c^2 + 96
*a^5*b*c^3)*d^4*e^3 + 3*(a^3*b^6 - 7*a^4*b^4*c + 8*a^5*b^2*c^2 + 16*a^6*c^3)*d^3
*e^4 - 3*(a^4*b^5 - 8*a^5*b^3*c + 16*a^6*b*c^2)*d^2*e^5 + (a^5*b^4 - 8*a^6*b^2*c
+ 16*a^7*c^2)*d*e^6 + ((b^4*c^5 - 8*a*b^2*c^6 + 16*a^2*c^7)*d^6*e - 3*(b^5*c^4
- 8*a*b^3*c^5 + 16*a^2*b*c^6)*d^5*e^2 + 3*(b^6*c^3 - 7*a*b^4*c^4 + 8*a^2*b^2*c^5
+ 16*a^3*c^6)*d^4*e^3 - (b^7*c^2 - 2*a*b^5*c^3 - 32*a^2*b^3*c^4 + 96*a^3*b*c^5)
*d^3*e^4 + 3*(a*b^6*c^2 - 7*a^2*b^4*c^3 + 8*a^3*b^2*c^4 + 16*a^4*c^5)*d^2*e^5 -
3*(a^2*b^5*c^2 - 8*a^3*b^3*c^3 + 16*a^4*b*c^4)*d*e^6 + (a^3*b^4*c^2 - 8*a^4*b^2*
c^3 + 16*a^5*c^4)*e^7)*x^5 + ((b^4*c^5 - 8*a*b^2*c^6 + 16*a^2*c^7)*d^7 - (b^5*c^
4 - 8*a*b^3*c^5 + 16*a^2*b*c^6)*d^6*e - 3*(b^6*c^3 - 9*a*b^4*c^4 + 24*a^2*b^2*c^
5 - 16*a^3*c^6)*d^5*e^2 + 5*(b^7*c^2 - 8*a*b^5*c^3 + 16*a^2*b^3*c^4)*d^4*e^3 - (
2*b^8*c - 7*a*b^6*c^2 - 43*a^2*b^4*c^3 + 168*a^3*b^2*c^4 - 48*a^4*c^5)*d^3*e^4 +
3*(2*a*b^7*c - 15*a^2*b^5*c^2 + 24*a^3*b^3*c^3 + 16*a^4*b*c^4)*d^2*e^5 - (6*a^2
*b^6*c - 49*a^3*b^4*c^2 + 104*a^4*b^2*c^3 - 16*a^5*c^4)*d*e^6 + 2*(a^3*b^5*c - 8
*a^4*b^3*c^2 + 16*a^5*b*c^3)*e^7)*x^4 + (2*(b^5*c^4 - 8*a*b^3*c^5 + 16*a^2*b*c^6
)*d^7 - (5*b^6*c^3 - 42*a*b^4*c^4 + 96*a^2*b^2*c^5 - 32*a^3*c^6)*d^6*e + 3*(b^7*
c^2 - 8*a*b^5*c^3 + 16*a^2*b^3*c^4)*d^5*e^2 + (b^8*c - 11*a*b^6*c^2 + 46*a^2*b^4
*c^3 - 96*a^3*b^2*c^4 + 96*a^4*c^5)*d^4*e^3 - (b^9 - 6*a*b^7*c + 6*a^2*b^5*c^2 -
16*a^3*b^3*c^3 + 96*a^4*b*c^4)*d^3*e^4 + 3*(a*b^8 - 7*a^2*b^6*c + 10*a^3*b^4*c^
2 + 32*a^5*c^4)*d^2*e^5 - (3*a^2*b^7 - 20*a^3*b^5*c + 16*a^4*b^3*c^2 + 64*a^5*b*
c^3)*d*e^6 + (a^3*b^6 - 6*a^4*b^4*c + 32*a^6*c^3)*e^7)*x^3 + ((b^6*c^3 - 6*a*b^4
*c^4 + 32*a^3*c^6)*d^7 - (3*b^7*c^2 - 20*a*b^5*c^3 + 16*a^2*b^3*c^4 + 64*a^3*b*c
^5)*d^6*e + 3*(b^8*c - 7*a*b^6*c^2 + 10*a^2*b^4*c^3 + 32*a^4*c^5)*d^5*e^2 - (b^9
- 6*a*b^7*c + 6*a^2*b^5*c^2 - 16*a^3*b^3*c^3 + 96*a^4*b*c^4)*d^4*e^3 + (a*b^8 -
11*a^2*b^6*c + 46*a^3*b^4*c^2 - 96*a^4*b^2*c^3 + 96*a^5*c^4)*d^3*e^4 + 3*(a^2*b
^7 - 8*a^3*b^5*c + 16*a^4*b^3*c^2)*d^2*e^5 - (5*a^3*b^6 - 42*a^4*b^4*c + 96*a^5*
b^2*c^2 - 32*a^6*c^3)*d*e^6 + 2*(a^4*b^5 - 8*a^5*b^3*c + 16*a^6*b*c^2)*e^7)*x^2
+ (2*(a*b^5*c^3 - 8*a^2*b^3*c^4 + 16*a^3*b*c^5)*d^7 - (6*a*b^6*c^2 - 49*a^2*b^4*
c^3 + 104*a^3*b^2*c^4 - 16*a^4*c^5)*d^6*e + 3*(2*a*b^7*c - 15*a^2*b^5*c^2 + 24*a
^3*b^3*c^3 + 16*a^4*b*c^4)*d^5*e^2 - (2*a*b^8 - 7*a^2*b^6*c - 43*a^3*b^4*c^2 + 1
68*a^4*b^2*c^3 - 48*a^5*c^4)*d^4*e^3 + 5*(a^2*b^7 - 8*a^3*b^5*c + 16*a^4*b^3*c^2
)*d^3*e^4 - 3*(a^3*b^6 - 9*a^4*b^4*c + 24*a^5*b^2*c^2 - 16*a^6*c^3)*d^2*e^5 - (a
^4*b^5 - 8*a^5*b^3*c + 16*a^6*b*c^2)*d*e^6 + (a^5*b^4 - 8*a^6*b^2*c + 16*a^7*c^2
)*e^7)*x)*sqrt(c*d^2 - b*d*e + a*e^2)), -1/6*(2*(2*(4*(2*B*a^2 - 3*A*a*b)*c^4 +
(2*B*a*b^2 + A*b^3)*c^3)*d^5 - 2*(16*A*a^2*c^4 + 8*(B*a^2*b - 4*A*a*b^2)*c^3 + 3
*(2*B*a*b^3 + A*b^4)*c^2)*d^4*e + 6*(4*(6*B*a^3 - A*a^2*b)*c^3 - (6*B*a^2*b^2 +
7*A*a*b^3)*c^2 + (2*B*a*b^4 + A*b^5)*c)*d^3*e^2 - 2*(2*B*a*b^5 + A*b^6 + 112*A*a
^3*c^3 + 4*(16*B*a^3*b - 21*A*a^2*b^2)*c^2 - 2*(8*B*a^2*b^3 - 3*A*a*b^4)*c)*d^2*
e^3 - (11*B*a^2*b^4 - 14*A*a*b^5 + 16*(7*B*a^4 - 10*A*a^3*b)*c^2 - 20*(4*B*a^3*b
^2 - 5*A*a^2*b^3)*c)*d*e^4 + 3*(A*a^2*b^4 - 8*A*a^3*b^2*c + 16*A*a^4*c^2)*e^5 +
(16*(B*b*c^5 - 2*A*c^6)*d^4*e - 8*(5*B*b^2*c^4 - 4*(B*a + 2*A*b)*c^5)*d^3*e^2 +
6*(3*B*b^3*c^3 - 24*A*a*c^5 + 2*(2*B*a*b - A*b^2)*c^4)*d^2*e^3 - (9*B*b^4*c^2 +
16*(13*B*a^2 - 9*A*a*b)*c^4 - 4*(14*B*a*b^2 - 5*A*b^3)*c^3)*d*e^4 + (128*A*a^2*c
^4 + 20*(2*B*a^2*b - 5*A*a*b^2)*c^3 - 3*(2*B*a*b^3 - 5*A*b^4)*c^2)*e^5)*x^4 + 2*
(8*(B*b*c^5 - 2*A*c^6)*d^5 - 8*(B*b^2*c^4 - (2*B*a + A*b)*c^5)*d^4*e - 3*(7*B*b^
3*c^3 + 24*A*a*c^5 - 2*(6*B*a*b + 7*A*b^2)*c^4)*d^3*e^2 + (15*B*b^4*c^2 - 4*(2*B
*a^2 + 9*A*a*b)*c^4 - (2*B*a*b^2 + 19*A*b^3)*c^3)*d^2*e^3 - (9*B*b^5*c + 56*A*a^
2*c^4 + 2*(86*B*a^2*b - 59*A*a*b^2)*c^3 - 3*(19*B*a*b^3 - 5*A*b^4)*c^2)*d*e^4 -
3*(4*(2*B*a^3 - 13*A*a^2*b)*c^3 - 7*(2*B*a^2*b^2 - 5*A*a*b^3)*c^2 + (2*B*a*b^4 -
5*A*b^5)*c)*e^5)*x^3 + 3*(8*(B*b^2*c^4 - 2*A*b*c^5)*d^5 - 2*(9*B*b^3*c^3 + 8*A*
a*c^5 - 2*(6*B*a*b + 7*A*b^2)*c^4)*d^4*e + 2*(3*B*b^4*c^2 + 4*(6*B*a^2 - 5*A*a*b
)*c^4 - (10*B*a*b^2 - A*b^3)*c^3)*d^3*e^2 + 2*(B*b^5*c - 56*A*a^2*c^4 - 2*(6*B*a
^2*b - 17*A*a*b^2)*c^3 + (3*B*a*b^3 - 7*A*b^4)*c^2)*d^2*e^3 - (3*B*b^6 - 14*B*a*
b^4*c + 56*(2*B*a^3 - A*a^2*b)*c^3 + 2*(2*B*a^2*b^2 + A*a*b^3)*c^2)*d*e^4 - (2*B
*a*b^5 - 5*A*b^6 - 16*A*a^2*b^2*c^2 - 64*A*a^3*c^3 - 6*(2*B*a^2*b^3 - 5*A*a*b^4)
*c)*e^5)*x^2 + 2*(3*(B*b^3*c^3 - 8*A*a*c^5 + 2*(2*B*a*b - A*b^2)*c^4)*d^5 - (9*B
*b^4*c^2 - 4*(2*B*a^2 + 9*A*a*b)*c^4 + (10*B*a*b^2 - 13*A*b^3)*c^3)*d^4*e + (9*B
*b^5*c - 88*A*a^2*c^4 + 2*(50*B*a^2*b - 11*A*a*b^2)*c^3 - 3*(11*B*a*b^3 + A*b^4)
*c^2)*d^3*e^2 - 3*(B*b^6 + 4*(2*B*a^3 + 3*A*a^2*b)*c^3 + (22*B*a^2*b^2 - 19*A*a*
b^3)*c^2 - (8*B*a*b^4 - 3*A*b^5)*c)*d^2*e^3 - (8*B*a*b^5 - 5*A*b^6 + 64*A*a^3*c^
3 + 4*(28*B*a^3*b - 27*A*a^2*b^2)*c^2 - 2*(29*B*a^2*b^3 - 21*A*a*b^4)*c)*d*e^4 -
2*(2*B*a^2*b^4 - 5*A*a*b^5 + 16*(B*a^4 - 4*A*a^3*b)*c^2 - (14*B*a^3*b^2 - 37*A*
a^2*b^3)*c)*e^5)*x)*sqrt(-c*d^2 + b*d*e - a*e^2)*sqrt(c*x^2 + b*x + a) - 3*(8*(B
*a^2*b^4*c - 8*B*a^3*b^2*c^2 + 16*B*a^4*c^3)*d^3*e^3 - (3*B*a^2*b^5 + 160*A*a^4*
c^3 + 16*(3*B*a^4*b - 5*A*a^3*b^2)*c^2 - 2*(12*B*a^3*b^3 - 5*A*a^2*b^4)*c)*d^2*e
^4 - (2*B*a^3*b^4 - 5*A*a^2*b^5 + 16*(2*B*a^5 - 5*A*a^4*b)*c^2 - 8*(2*B*a^4*b^2
- 5*A*a^3*b^3)*c)*d*e^5 + (8*(B*b^4*c^3 - 8*B*a*b^2*c^4 + 16*B*a^2*c^5)*d^2*e^4
- (3*B*b^5*c^2 + 160*A*a^2*c^5 + 16*(3*B*a^2*b - 5*A*a*b^2)*c^4 - 2*(12*B*a*b^3
- 5*A*b^4)*c^3)*d*e^5 - (16*(2*B*a^3 - 5*A*a^2*b)*c^4 - 8*(2*B*a^2*b^2 - 5*A*a*b
^3)*c^3 + (2*B*a*b^4 - 5*A*b^5)*c^2)*e^6)*x^5 + (8*(B*b^4*c^3 - 8*B*a*b^2*c^4 +
16*B*a^2*c^5)*d^3*e^3 + (13*B*b^5*c^2 - 160*A*a^2*c^5 + 16*(13*B*a^2*b + 5*A*a*b
^2)*c^4 - 2*(52*B*a*b^3 + 5*A*b^4)*c^3)*d^2*e^4 - (6*B*b^6*c + 16*(2*B*a^3 + 15*
A*a^2*b)*c^4 + 40*(2*B*a^2*b^2 - 3*A*a*b^3)*c^3 - (46*B*a*b^4 - 15*A*b^5)*c^2)*d
*e^5 - 2*(16*(2*B*a^3*b - 5*A*a^2*b^2)*c^3 - 8*(2*B*a^2*b^3 - 5*A*a*b^4)*c^2 + (
2*B*a*b^5 - 5*A*b^6)*c)*e^6)*x^4 + (16*(B*b^5*c^2 - 8*B*a*b^3*c^3 + 16*B*a^2*b*c
^4)*d^3*e^3 + 2*(B*b^6*c - 10*A*b^5*c^2 + 32*(4*B*a^3 - 5*A*a^2*b)*c^4 - 16*(3*B
*a^2*b^2 - 5*A*a*b^3)*c^3)*d^2*e^4 - (3*B*b^7 - 14*B*a*b^5*c + 320*A*a^3*c^4 + 1
60*(B*a^3*b - A*a^2*b^2)*c^3 - 4*(8*B*a^2*b^3 - 5*A*a*b^4)*c^2)*d*e^5 - (2*B*a*b
^6 - 5*A*b^7 + 32*(2*B*a^4 - 5*A*a^3*b)*c^3 - 6*(2*B*a^2*b^4 - 5*A*a*b^5)*c)*e^6
)*x^3 + (8*(B*b^6*c - 6*B*a*b^4*c^2 + 32*B*a^3*c^4)*d^3*e^3 - (3*B*b^7 - 160*B*a
^3*b*c^3 + 320*A*a^3*c^4 + 4*(32*B*a^2*b^3 - 15*A*a*b^4)*c^2 - 2*(17*B*a*b^5 - 5
*A*b^6)*c)*d^2*e^4 - (8*B*a*b^6 - 5*A*b^7 + 32*(2*B*a^4 + 5*A*a^3*b)*c^3 + 32*(3
*B*a^3*b^2 - 5*A*a^2*b^3)*c^2 - 10*(6*B*a^2*b^4 - 5*A*a*b^5)*c)*d*e^5 - 2*(2*B*a
^2*b^5 - 5*A*a*b^6 + 16*(2*B*a^4*b - 5*A*a^3*b^2)*c^2 - 8*(2*B*a^3*b^3 - 5*A*a^2
*b^4)*c)*e^6)*x^2 + (16*(B*a*b^5*c - 8*B*a^2*b^3*c^2 + 16*B*a^3*b*c^3)*d^3*e^3 -
2*(3*B*a*b^6 - 32*(2*B*a^4 - 5*A*a^3*b)*c^3 + 80*(B*a^3*b^2 - A*a^2*b^3)*c^2 -
2*(14*B*a^2*b^4 - 5*A*a*b^5)*c)*d^2*e^4 - (7*B*a^2*b^5 - 10*A*a*b^6 + 160*A*a^4*
c^3 + 16*(7*B*a^4*b - 15*A*a^3*b^2)*c^2 - 2*(28*B*a^3*b^3 - 45*A*a^2*b^4)*c)*d*e
^5 - (2*B*a^3*b^4 - 5*A*a^2*b^5 + 16*(2*B*a^5 - 5*A*a^4*b)*c^2 - 8*(2*B*a^4*b^2
- 5*A*a^3*b^3)*c)*e^6)*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^2*b^4*
c^3 - 8*a^3*b^2*c^4 + 16*a^4*c^5)*d^7 - 3*(a^2*b^5*c^2 - 8*a^3*b^3*c^3 + 16*a^4*
b*c^4)*d^6*e + 3*(a^2*b^6*c - 7*a^3*b^4*c^2 + 8*a^4*b^2*c^3 + 16*a^5*c^4)*d^5*e^
2 - (a^2*b^7 - 2*a^3*b^5*c - 32*a^4*b^3*c^2 + 96*a^5*b*c^3)*d^4*e^3 + 3*(a^3*b^6
- 7*a^4*b^4*c + 8*a^5*b^2*c^2 + 16*a^6*c^3)*d^3*e^4 - 3*(a^4*b^5 - 8*a^5*b^3*c
+ 16*a^6*b*c^2)*d^2*e^5 + (a^5*b^4 - 8*a^6*b^2*c + 16*a^7*c^2)*d*e^6 + ((b^4*c^5
- 8*a*b^2*c^6 + 16*a^2*c^7)*d^6*e - 3*(b^5*c^4 - 8*a*b^3*c^5 + 16*a^2*b*c^6)*d^
5*e^2 + 3*(b^6*c^3 - 7*a*b^4*c^4 + 8*a^2*b^2*c^5 + 16*a^3*c^6)*d^4*e^3 - (b^7*c^
2 - 2*a*b^5*c^3 - 32*a^2*b^3*c^4 + 96*a^3*b*c^5)*d^3*e^4 + 3*(a*b^6*c^2 - 7*a^2*
b^4*c^3 + 8*a^3*b^2*c^4 + 16*a^4*c^5)*d^2*e^5 - 3*(a^2*b^5*c^2 - 8*a^3*b^3*c^3 +
16*a^4*b*c^4)*d*e^6 + (a^3*b^4*c^2 - 8*a^4*b^2*c^3 + 16*a^5*c^4)*e^7)*x^5 + ((b
^4*c^5 - 8*a*b^2*c^6 + 16*a^2*c^7)*d^7 - (b^5*c^4 - 8*a*b^3*c^5 + 16*a^2*b*c^6)*
d^6*e - 3*(b^6*c^3 - 9*a*b^4*c^4 + 24*a^2*b^2*c^5 - 16*a^3*c^6)*d^5*e^2 + 5*(b^7
*c^2 - 8*a*b^5*c^3 + 16*a^2*b^3*c^4)*d^4*e^3 - (2*b^8*c - 7*a*b^6*c^2 - 43*a^2*b
^4*c^3 + 168*a^3*b^2*c^4 - 48*a^4*c^5)*d^3*e^4 + 3*(2*a*b^7*c - 15*a^2*b^5*c^2 +
24*a^3*b^3*c^3 + 16*a^4*b*c^4)*d^2*e^5 - (6*a^2*b^6*c - 49*a^3*b^4*c^2 + 104*a^
4*b^2*c^3 - 16*a^5*c^4)*d*e^6 + 2*(a^3*b^5*c - 8*a^4*b^3*c^2 + 16*a^5*b*c^3)*e^7
)*x^4 + (2*(b^5*c^4 - 8*a*b^3*c^5 + 16*a^2*b*c^6)*d^7 - (5*b^6*c^3 - 42*a*b^4*c^
4 + 96*a^2*b^2*c^5 - 32*a^3*c^6)*d^6*e + 3*(b^7*c^2 - 8*a*b^5*c^3 + 16*a^2*b^3*c
^4)*d^5*e^2 + (b^8*c - 11*a*b^6*c^2 + 46*a^2*b^4*c^3 - 96*a^3*b^2*c^4 + 96*a^4*c
^5)*d^4*e^3 - (b^9 - 6*a*b^7*c + 6*a^2*b^5*c^2 - 16*a^3*b^3*c^3 + 96*a^4*b*c^4)*
d^3*e^4 + 3*(a*b^8 - 7*a^2*b^6*c + 10*a^3*b^4*c^2 + 32*a^5*c^4)*d^2*e^5 - (3*a^2
*b^7 - 20*a^3*b^5*c + 16*a^4*b^3*c^2 + 64*a^5*b*c^3)*d*e^6 + (a^3*b^6 - 6*a^4*b^
4*c + 32*a^6*c^3)*e^7)*x^3 + ((b^6*c^3 - 6*a*b^4*c^4 + 32*a^3*c^6)*d^7 - (3*b^7*
c^2 - 20*a*b^5*c^3 + 16*a^2*b^3*c^4 + 64*a^3*b*c^5)*d^6*e + 3*(b^8*c - 7*a*b^6*c
^2 + 10*a^2*b^4*c^3 + 32*a^4*c^5)*d^5*e^2 - (b^9 - 6*a*b^7*c + 6*a^2*b^5*c^2 - 1
6*a^3*b^3*c^3 + 96*a^4*b*c^4)*d^4*e^3 + (a*b^8 - 11*a^2*b^6*c + 46*a^3*b^4*c^2 -
96*a^4*b^2*c^3 + 96*a^5*c^4)*d^3*e^4 + 3*(a^2*b^7 - 8*a^3*b^5*c + 16*a^4*b^3*c^
2)*d^2*e^5 - (5*a^3*b^6 - 42*a^4*b^4*c + 96*a^5*b^2*c^2 - 32*a^6*c^3)*d*e^6 + 2*
(a^4*b^5 - 8*a^5*b^3*c + 16*a^6*b*c^2)*e^7)*x^2 + (2*(a*b^5*c^3 - 8*a^2*b^3*c^4
+ 16*a^3*b*c^5)*d^7 - (6*a*b^6*c^2 - 49*a^2*b^4*c^3 + 104*a^3*b^2*c^4 - 16*a^4*c
^5)*d^6*e + 3*(2*a*b^7*c - 15*a^2*b^5*c^2 + 24*a^3*b^3*c^3 + 16*a^4*b*c^4)*d^5*e
^2 - (2*a*b^8 - 7*a^2*b^6*c - 43*a^3*b^4*c^2 + 168*a^4*b^2*c^3 - 48*a^5*c^4)*d^4
*e^3 + 5*(a^2*b^7 - 8*a^3*b^5*c + 16*a^4*b^3*c^2)*d^3*e^4 - 3*(a^3*b^6 - 9*a^4*b
^4*c + 24*a^5*b^2*c^2 - 16*a^6*c^3)*d^2*e^5 - (a^4*b^5 - 8*a^5*b^3*c + 16*a^6*b*
c^2)*d*e^6 + (a^5*b^4 - 8*a^6*b^2*c + 16*a^7*c^2)*e^7)*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((B*x+A)/(e*x+d)**2/(c*x**2+b*x+a)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{B x + A}{{\left (c x^{2} + b x + a\right )}^{\frac{5}{2}}{\left (e x + d\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((B*x + A)/((c*x^2 + b*x + a)^(5/2)*(e*x + d)^2), x)