3.1028 \(\int \frac{A+B x}{x^{3/2} \left (a+b x+c x^2\right )^3} \, dx\)

Optimal. Leaf size=664 \[ -\frac{-A \left (36 a^2 c^2-35 a b^2 c+5 b^4\right )+c x \left (a B \left (b^2-28 a c\right )-A \left (5 b^3-32 a b c\right )\right )+a b B \left (b^2-16 a c\right )}{4 a^2 \sqrt{x} \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}+\frac{3 \left (a b B \left (b^2-8 a c\right )-A \left (60 a^2 c^2-37 a b^2 c+5 b^4\right )\right )}{4 a^3 \sqrt{x} \left (b^2-4 a c\right )^2}+\frac{3 \sqrt{c} \left (a B \left (56 a^2 c^2-10 a b^2 c-8 a b c \sqrt{b^2-4 a c}+b^3 \sqrt{b^2-4 a c}+b^4\right )-A \left (60 a^2 c^2 \sqrt{b^2-4 a c}+124 a^2 b c^2-47 a b^3 c-37 a b^2 c \sqrt{b^2-4 a c}+5 b^4 \sqrt{b^2-4 a c}+5 b^5\right )\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{4 \sqrt{2} a^3 \left (b^2-4 a c\right )^{5/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{3 \sqrt{c} \left (a B \left (56 a^2 c^2-10 a b^2 c+8 a b c \sqrt{b^2-4 a c}-b^3 \sqrt{b^2-4 a c}+b^4\right )-A \left (-60 a^2 c^2 \sqrt{b^2-4 a c}+124 a^2 b c^2-47 a b^3 c+37 a b^2 c \sqrt{b^2-4 a c}-5 b^4 \sqrt{b^2-4 a c}+5 b^5\right )\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{4 \sqrt{2} a^3 \left (b^2-4 a c\right )^{5/2} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{c x (A b-2 a B)-2 a A c-a b B+A b^2}{2 a \sqrt{x} \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2} \]

[Out]

(3*(a*b*B*(b^2 - 8*a*c) - A*(5*b^4 - 37*a*b^2*c + 60*a^2*c^2)))/(4*a^3*(b^2 - 4*
a*c)^2*Sqrt[x]) + (A*b^2 - a*b*B - 2*a*A*c + (A*b - 2*a*B)*c*x)/(2*a*(b^2 - 4*a*
c)*Sqrt[x]*(a + b*x + c*x^2)^2) - (a*b*B*(b^2 - 16*a*c) - A*(5*b^4 - 35*a*b^2*c
+ 36*a^2*c^2) + c*(a*B*(b^2 - 28*a*c) - A*(5*b^3 - 32*a*b*c))*x)/(4*a^2*(b^2 - 4
*a*c)^2*Sqrt[x]*(a + b*x + c*x^2)) + (3*Sqrt[c]*(a*B*(b^4 - 10*a*b^2*c + 56*a^2*
c^2 + b^3*Sqrt[b^2 - 4*a*c] - 8*a*b*c*Sqrt[b^2 - 4*a*c]) - A*(5*b^5 - 47*a*b^3*c
+ 124*a^2*b*c^2 + 5*b^4*Sqrt[b^2 - 4*a*c] - 37*a*b^2*c*Sqrt[b^2 - 4*a*c] + 60*a
^2*c^2*Sqrt[b^2 - 4*a*c]))*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b - Sqrt[b^2 -
4*a*c]]])/(4*Sqrt[2]*a^3*(b^2 - 4*a*c)^(5/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) - (3*S
qrt[c]*(a*B*(b^4 - 10*a*b^2*c + 56*a^2*c^2 - b^3*Sqrt[b^2 - 4*a*c] + 8*a*b*c*Sqr
t[b^2 - 4*a*c]) - A*(5*b^5 - 47*a*b^3*c + 124*a^2*b*c^2 - 5*b^4*Sqrt[b^2 - 4*a*c
] + 37*a*b^2*c*Sqrt[b^2 - 4*a*c] - 60*a^2*c^2*Sqrt[b^2 - 4*a*c]))*ArcTan[(Sqrt[2
]*Sqrt[c]*Sqrt[x])/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(4*Sqrt[2]*a^3*(b^2 - 4*a*c)^(5
/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]])

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Rubi [A]  time = 4.12137, antiderivative size = 664, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217 \[ -\frac{-A \left (36 a^2 c^2-35 a b^2 c+5 b^4\right )+c x \left (a B \left (b^2-28 a c\right )-A \left (5 b^3-32 a b c\right )\right )+a b B \left (b^2-16 a c\right )}{4 a^2 \sqrt{x} \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}+\frac{3 \left (a b B \left (b^2-8 a c\right )-A \left (60 a^2 c^2-37 a b^2 c+5 b^4\right )\right )}{4 a^3 \sqrt{x} \left (b^2-4 a c\right )^2}+\frac{3 \sqrt{c} \left (a B \left (56 a^2 c^2-10 a b^2 c-8 a b c \sqrt{b^2-4 a c}+b^3 \sqrt{b^2-4 a c}+b^4\right )-A \left (60 a^2 c^2 \sqrt{b^2-4 a c}+124 a^2 b c^2-47 a b^3 c-37 a b^2 c \sqrt{b^2-4 a c}+5 b^4 \sqrt{b^2-4 a c}+5 b^5\right )\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{4 \sqrt{2} a^3 \left (b^2-4 a c\right )^{5/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{3 \sqrt{c} \left (a B \left (56 a^2 c^2-10 a b^2 c+8 a b c \sqrt{b^2-4 a c}-b^3 \sqrt{b^2-4 a c}+b^4\right )-A \left (-60 a^2 c^2 \sqrt{b^2-4 a c}+124 a^2 b c^2-47 a b^3 c+37 a b^2 c \sqrt{b^2-4 a c}-5 b^4 \sqrt{b^2-4 a c}+5 b^5\right )\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{4 \sqrt{2} a^3 \left (b^2-4 a c\right )^{5/2} \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{-A \left (b^2-2 a c\right )-c x (A b-2 a B)+a b B}{2 a \sqrt{x} \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

(3*(a*b*B*(b^2 - 8*a*c) - A*(5*b^4 - 37*a*b^2*c + 60*a^2*c^2)))/(4*a^3*(b^2 - 4*
a*c)^2*Sqrt[x]) - (a*b*B - A*(b^2 - 2*a*c) - (A*b - 2*a*B)*c*x)/(2*a*(b^2 - 4*a*
c)*Sqrt[x]*(a + b*x + c*x^2)^2) - (a*b*B*(b^2 - 16*a*c) - A*(5*b^4 - 35*a*b^2*c
+ 36*a^2*c^2) + c*(a*B*(b^2 - 28*a*c) - A*(5*b^3 - 32*a*b*c))*x)/(4*a^2*(b^2 - 4
*a*c)^2*Sqrt[x]*(a + b*x + c*x^2)) + (3*Sqrt[c]*(a*B*(b^4 - 10*a*b^2*c + 56*a^2*
c^2 + b^3*Sqrt[b^2 - 4*a*c] - 8*a*b*c*Sqrt[b^2 - 4*a*c]) - A*(5*b^5 - 47*a*b^3*c
+ 124*a^2*b*c^2 + 5*b^4*Sqrt[b^2 - 4*a*c] - 37*a*b^2*c*Sqrt[b^2 - 4*a*c] + 60*a
^2*c^2*Sqrt[b^2 - 4*a*c]))*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b - Sqrt[b^2 -
4*a*c]]])/(4*Sqrt[2]*a^3*(b^2 - 4*a*c)^(5/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) - (3*S
qrt[c]*(a*B*(b^4 - 10*a*b^2*c + 56*a^2*c^2 - b^3*Sqrt[b^2 - 4*a*c] + 8*a*b*c*Sqr
t[b^2 - 4*a*c]) - A*(5*b^5 - 47*a*b^3*c + 124*a^2*b*c^2 - 5*b^4*Sqrt[b^2 - 4*a*c
] + 37*a*b^2*c*Sqrt[b^2 - 4*a*c] - 60*a^2*c^2*Sqrt[b^2 - 4*a*c]))*ArcTan[(Sqrt[2
]*Sqrt[c]*Sqrt[x])/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(4*Sqrt[2]*a^3*(b^2 - 4*a*c)^(5
/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]])

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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)/x**(3/2)/(c*x**2+b*x+a)**3,x)

[Out]

Timed out

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Mathematica [A]  time = 6.53799, size = 769, normalized size = 1.16 \[ -\frac{2 A}{a^3 \sqrt{x}}+\frac{2 a^2 B c \sqrt{x}-3 a A b c \sqrt{x}-2 a A c^2 x^{3/2}-a b^2 B \sqrt{x}-a b B c x^{3/2}+A b^3 \sqrt{x}+A b^2 c x^{3/2}}{2 a^2 \left (4 a c-b^2\right ) \left (a+b x+c x^2\right )^2}-\frac{3 \sqrt{c} \left (-56 a^3 B c^2+60 a^2 A c^2 \sqrt{b^2-4 a c}+124 a^2 A b c^2+10 a^2 b^2 B c+8 a^2 b B c \sqrt{b^2-4 a c}-47 a A b^3 c-37 a A b^2 c \sqrt{b^2-4 a c}+5 A b^4 \sqrt{b^2-4 a c}-a b^4 B-a b^3 B \sqrt{b^2-4 a c}+5 A b^5\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{4 \sqrt{2} a^3 \left (b^2-4 a c\right )^{5/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{3 \sqrt{c} \left (56 a^3 B c^2+60 a^2 A c^2 \sqrt{b^2-4 a c}-124 a^2 A b c^2-10 a^2 b^2 B c+8 a^2 b B c \sqrt{b^2-4 a c}+47 a A b^3 c-37 a A b^2 c \sqrt{b^2-4 a c}+5 A b^4 \sqrt{b^2-4 a c}+a b^4 B-a b^3 B \sqrt{b^2-4 a c}-5 A b^5\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{4 \sqrt{2} a^3 \left (b^2-4 a c\right )^{5/2} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{28 a^3 B c^2 \sqrt{x}-84 a^2 A b c^2 \sqrt{x}-52 a^2 A c^3 x^{3/2}-25 a^2 b^2 B c \sqrt{x}-24 a^2 b B c^2 x^{3/2}+52 a A b^3 c \sqrt{x}+47 a A b^2 c^2 x^{3/2}+3 a b^4 B \sqrt{x}+3 a b^3 B c x^{3/2}-7 A b^5 \sqrt{x}-7 A b^4 c x^{3/2}}{4 a^3 \left (4 a c-b^2\right )^2 \left (a+b x+c x^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

(-2*A)/(a^3*Sqrt[x]) + (A*b^3*Sqrt[x] - a*b^2*B*Sqrt[x] - 3*a*A*b*c*Sqrt[x] + 2*
a^2*B*c*Sqrt[x] + A*b^2*c*x^(3/2) - a*b*B*c*x^(3/2) - 2*a*A*c^2*x^(3/2))/(2*a^2*
(-b^2 + 4*a*c)*(a + b*x + c*x^2)^2) + (-7*A*b^5*Sqrt[x] + 3*a*b^4*B*Sqrt[x] + 52
*a*A*b^3*c*Sqrt[x] - 25*a^2*b^2*B*c*Sqrt[x] - 84*a^2*A*b*c^2*Sqrt[x] + 28*a^3*B*
c^2*Sqrt[x] - 7*A*b^4*c*x^(3/2) + 3*a*b^3*B*c*x^(3/2) + 47*a*A*b^2*c^2*x^(3/2) -
24*a^2*b*B*c^2*x^(3/2) - 52*a^2*A*c^3*x^(3/2))/(4*a^3*(-b^2 + 4*a*c)^2*(a + b*x
+ c*x^2)) - (3*Sqrt[c]*(5*A*b^5 - a*b^4*B - 47*a*A*b^3*c + 10*a^2*b^2*B*c + 124
*a^2*A*b*c^2 - 56*a^3*B*c^2 + 5*A*b^4*Sqrt[b^2 - 4*a*c] - a*b^3*B*Sqrt[b^2 - 4*a
*c] - 37*a*A*b^2*c*Sqrt[b^2 - 4*a*c] + 8*a^2*b*B*c*Sqrt[b^2 - 4*a*c] + 60*a^2*A*
c^2*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b - Sqrt[b^2 - 4*a*
c]]])/(4*Sqrt[2]*a^3*(b^2 - 4*a*c)^(5/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) - (3*Sqrt[
c]*(-5*A*b^5 + a*b^4*B + 47*a*A*b^3*c - 10*a^2*b^2*B*c - 124*a^2*A*b*c^2 + 56*a^
3*B*c^2 + 5*A*b^4*Sqrt[b^2 - 4*a*c] - a*b^3*B*Sqrt[b^2 - 4*a*c] - 37*a*A*b^2*c*S
qrt[b^2 - 4*a*c] + 8*a^2*b*B*c*Sqrt[b^2 - 4*a*c] + 60*a^2*A*c^2*Sqrt[b^2 - 4*a*c
])*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(4*Sqrt[2]*a^3
*(b^2 - 4*a*c)^(5/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]])

_______________________________________________________________________________________

Maple [B]  time = 0.131, size = 10168, normalized size = 15.3 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: RuntimeError

_______________________________________________________________________________________

Fricas [A]  time = 68.8703, size = 16899, normalized size = 25.45 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8*(16*A*a^2*b^4 - 128*A*a^3*b^2*c + 256*A*a^4*c^2 + 6*(60*A*a^2*c^4 + (8*B*a^
2*b - 37*A*a*b^2)*c^3 - (B*a*b^3 - 5*A*b^4)*c^2)*x^4 - 2*(28*(B*a^3 - 14*A*a^2*b
)*c^3 - (49*B*a^2*b^2 - 227*A*a*b^3)*c^2 + 6*(B*a*b^4 - 5*A*b^5)*c)*x^3 - 3*sqrt
(1/2)*(a^5*b^4 - 8*a^6*b^2*c + 16*a^7*c^2 + (a^3*b^4*c^2 - 8*a^4*b^2*c^3 + 16*a^
5*c^4)*x^4 + 2*(a^3*b^5*c - 8*a^4*b^3*c^2 + 16*a^5*b*c^3)*x^3 + (a^3*b^6 - 6*a^4
*b^4*c + 32*a^6*c^3)*x^2 + 2*(a^4*b^5 - 8*a^5*b^3*c + 16*a^6*b*c^2)*x)*sqrt(x)*s
qrt(-(B^2*a^2*b^9 - 10*A*B*a*b^10 + 25*A^2*b^11 + 1680*(4*A*B*a^6 - 11*A^2*a^5*b
)*c^5 + 840*(2*B^2*a^6*b - 16*A*B*a^5*b^2 + 33*A^2*a^4*b^3)*c^4 - 105*(8*B^2*a^5
*b^3 - 68*A*B*a^4*b^4 + 143*A^2*a^3*b^5)*c^3 + 3*(63*B^2*a^4*b^5 - 574*A*B*a^3*b
^6 + 1298*A^2*a^2*b^7)*c^2 - 3*(7*B^2*a^3*b^7 - 68*A*B*a^2*b^8 + 165*A^2*a*b^9)*
c + (a^7*b^10 - 20*a^8*b^8*c + 160*a^9*b^6*c^2 - 640*a^10*b^4*c^3 + 1280*a^11*b^
2*c^4 - 1024*a^12*c^5)*sqrt((B^4*a^4*b^8 - 20*A*B^3*a^3*b^9 + 150*A^2*B^2*a^2*b^
10 - 500*A^3*B*a*b^11 + 625*A^4*b^12 + 50625*A^4*a^6*c^6 - 450*(49*A^2*B^2*a^7 -
382*A^3*B*a^6*b + 694*A^4*a^5*b^2)*c^5 + (2401*B^4*a^8 - 37436*A*B^3*a^7*b + 21
8886*A^2*B^2*a^6*b^2 - 577016*A^3*B*a^5*b^3 + 591886*A^4*a^4*b^4)*c^4 - 2*(539*B
^4*a^7*b^2 - 9298*A*B^3*a^6*b^3 + 59592*A^2*B^2*a^5*b^4 - 168016*A^3*B*a^4*b^5 +
175655*A^4*a^3*b^6)*c^3 + 3*(73*B^4*a^6*b^4 - 1344*A*B^3*a^5*b^5 + 9228*A^2*B^2
*a^4*b^6 - 27980*A^3*B*a^3*b^7 + 31575*A^4*a^2*b^8)*c^2 - 2*(11*B^4*a^5*b^6 - 21
4*A*B^3*a^4*b^7 + 1560*A^2*B^2*a^3*b^8 - 5050*A^3*B*a^2*b^9 + 6125*A^4*a*b^10)*c
)/(a^14*b^10 - 20*a^15*b^8*c + 160*a^16*b^6*c^2 - 640*a^17*b^4*c^3 + 1280*a^18*b
^2*c^4 - 1024*a^19*c^5)))/(a^7*b^10 - 20*a^8*b^8*c + 160*a^9*b^6*c^2 - 640*a^10*
b^4*c^3 + 1280*a^11*b^2*c^4 - 1024*a^12*c^5))*log(27/2*sqrt(1/2)*(B^3*a^3*b^14 -
15*A*B^2*a^2*b^15 + 75*A^2*B*a*b^16 - 125*A^3*b^17 + 57600*(7*A^2*B*a^9 - 23*A^
3*a^8*b)*c^8 - 64*(1372*B^3*a^10 - 15204*A*B^2*a^9*b + 61326*A^2*B*a^8*b^2 - 888
23*A^3*a^7*b^3)*c^7 + 16*(7112*B^3*a^9*b^2 - 83292*A*B^2*a^8*b^3 + 330300*A^2*B*
a^7*b^4 - 446671*A^3*a^6*b^5)*c^6 - 4*(15920*B^3*a^8*b^4 - 197004*A*B^2*a^7*b^5
+ 811446*A^2*B*a^6*b^6 - 1115785*A^3*a^5*b^7)*c^5 + 3*(6696*B^3*a^7*b^6 - 87308*
A*B^2*a^6*b^7 + 377471*A^2*B*a^5*b^8 - 541178*A^3*a^4*b^9)*c^4 - 3*(1295*B^3*a^6
*b^8 - 17704*A*B^2*a^5*b^9 + 80329*A^2*B*a^4*b^10 - 120911*A^3*a^3*b^11)*c^3 + (
464*B^3*a^5*b^10 - 6609*A*B^2*a^4*b^11 + 31317*A^2*B*a^3*b^12 - 49360*A^3*a^2*b^
13)*c^2 - (32*B^3*a^4*b^12 - 471*A*B^2*a^3*b^13 + 2310*A^2*B*a^2*b^14 - 3775*A^3
*a*b^15)*c - (B*a^8*b^15 - 5*A*a^7*b^16 - 122880*A*a^15*c^8 - 4096*(11*B*a^15*b
- 79*A*a^14*b^2)*c^7 + 1536*(44*B*a^14*b^3 - 223*A*a^13*b^4)*c^6 - 256*(169*B*a^
13*b^5 - 770*A*a^12*b^6)*c^5 + 480*(32*B*a^12*b^7 - 143*A*a^11*b^8)*c^4 - 80*(41
*B*a^11*b^9 - 187*A*a^10*b^10)*c^3 + 2*(212*B*a^10*b^11 - 1003*A*a^9*b^12)*c^2 -
(31*B*a^9*b^13 - 152*A*a^8*b^14)*c)*sqrt((B^4*a^4*b^8 - 20*A*B^3*a^3*b^9 + 150*
A^2*B^2*a^2*b^10 - 500*A^3*B*a*b^11 + 625*A^4*b^12 + 50625*A^4*a^6*c^6 - 450*(49
*A^2*B^2*a^7 - 382*A^3*B*a^6*b + 694*A^4*a^5*b^2)*c^5 + (2401*B^4*a^8 - 37436*A*
B^3*a^7*b + 218886*A^2*B^2*a^6*b^2 - 577016*A^3*B*a^5*b^3 + 591886*A^4*a^4*b^4)*
c^4 - 2*(539*B^4*a^7*b^2 - 9298*A*B^3*a^6*b^3 + 59592*A^2*B^2*a^5*b^4 - 168016*A
^3*B*a^4*b^5 + 175655*A^4*a^3*b^6)*c^3 + 3*(73*B^4*a^6*b^4 - 1344*A*B^3*a^5*b^5
+ 9228*A^2*B^2*a^4*b^6 - 27980*A^3*B*a^3*b^7 + 31575*A^4*a^2*b^8)*c^2 - 2*(11*B^
4*a^5*b^6 - 214*A*B^3*a^4*b^7 + 1560*A^2*B^2*a^3*b^8 - 5050*A^3*B*a^2*b^9 + 6125
*A^4*a*b^10)*c)/(a^14*b^10 - 20*a^15*b^8*c + 160*a^16*b^6*c^2 - 640*a^17*b^4*c^3
+ 1280*a^18*b^2*c^4 - 1024*a^19*c^5)))*sqrt(-(B^2*a^2*b^9 - 10*A*B*a*b^10 + 25*
A^2*b^11 + 1680*(4*A*B*a^6 - 11*A^2*a^5*b)*c^5 + 840*(2*B^2*a^6*b - 16*A*B*a^5*b
^2 + 33*A^2*a^4*b^3)*c^4 - 105*(8*B^2*a^5*b^3 - 68*A*B*a^4*b^4 + 143*A^2*a^3*b^5
)*c^3 + 3*(63*B^2*a^4*b^5 - 574*A*B*a^3*b^6 + 1298*A^2*a^2*b^7)*c^2 - 3*(7*B^2*a
^3*b^7 - 68*A*B*a^2*b^8 + 165*A^2*a*b^9)*c + (a^7*b^10 - 20*a^8*b^8*c + 160*a^9*
b^6*c^2 - 640*a^10*b^4*c^3 + 1280*a^11*b^2*c^4 - 1024*a^12*c^5)*sqrt((B^4*a^4*b^
8 - 20*A*B^3*a^3*b^9 + 150*A^2*B^2*a^2*b^10 - 500*A^3*B*a*b^11 + 625*A^4*b^12 +
50625*A^4*a^6*c^6 - 450*(49*A^2*B^2*a^7 - 382*A^3*B*a^6*b + 694*A^4*a^5*b^2)*c^5
+ (2401*B^4*a^8 - 37436*A*B^3*a^7*b + 218886*A^2*B^2*a^6*b^2 - 577016*A^3*B*a^5
*b^3 + 591886*A^4*a^4*b^4)*c^4 - 2*(539*B^4*a^7*b^2 - 9298*A*B^3*a^6*b^3 + 59592
*A^2*B^2*a^5*b^4 - 168016*A^3*B*a^4*b^5 + 175655*A^4*a^3*b^6)*c^3 + 3*(73*B^4*a^
6*b^4 - 1344*A*B^3*a^5*b^5 + 9228*A^2*B^2*a^4*b^6 - 27980*A^3*B*a^3*b^7 + 31575*
A^4*a^2*b^8)*c^2 - 2*(11*B^4*a^5*b^6 - 214*A*B^3*a^4*b^7 + 1560*A^2*B^2*a^3*b^8
- 5050*A^3*B*a^2*b^9 + 6125*A^4*a*b^10)*c)/(a^14*b^10 - 20*a^15*b^8*c + 160*a^16
*b^6*c^2 - 640*a^17*b^4*c^3 + 1280*a^18*b^2*c^4 - 1024*a^19*c^5)))/(a^7*b^10 - 2
0*a^8*b^8*c + 160*a^9*b^6*c^2 - 640*a^10*b^4*c^3 + 1280*a^11*b^2*c^4 - 1024*a^12
*c^5)) + 27*(810000*A^4*a^5*c^9 + 405000*(2*A^3*B*a^5*b - 7*A^4*a^4*b^2)*c^8 - (
38416*B^4*a^7 - 422576*A*B^3*a^6*b + 1439376*A^2*B^2*a^5*b^2 - 1018856*A^3*B*a^4
*b^3 - 1957349*A^4*a^3*b^4)*c^7 + (19208*B^4*a^6*b^2 - 239896*A*B^3*a^5*b^3 + 95
5704*A^2*B^2*a^4*b^4 - 1067347*A^3*B*a^3*b^5 - 571030*A^4*a^2*b^6)*c^6 - (4189*B
^4*a^5*b^4 - 56807*A*B^3*a^4*b^5 + 251349*A^2*B^2*a^3*b^6 - 344630*A^3*B*a^2*b^7
- 77825*A^4*a*b^8)*c^5 + 3*(149*B^4*a^4*b^6 - 2161*A*B^3*a^3*b^7 + 10380*A^2*B^
2*a^2*b^8 - 16225*A^3*B*a*b^9 - 1375*A^4*b^10)*c^4 - 21*(B^4*a^3*b^8 - 15*A*B^3*
a^2*b^9 + 75*A^2*B^2*a*b^10 - 125*A^3*B*b^11)*c^3)*sqrt(x)) + 3*sqrt(1/2)*(a^5*b
^4 - 8*a^6*b^2*c + 16*a^7*c^2 + (a^3*b^4*c^2 - 8*a^4*b^2*c^3 + 16*a^5*c^4)*x^4 +
2*(a^3*b^5*c - 8*a^4*b^3*c^2 + 16*a^5*b*c^3)*x^3 + (a^3*b^6 - 6*a^4*b^4*c + 32*
a^6*c^3)*x^2 + 2*(a^4*b^5 - 8*a^5*b^3*c + 16*a^6*b*c^2)*x)*sqrt(x)*sqrt(-(B^2*a^
2*b^9 - 10*A*B*a*b^10 + 25*A^2*b^11 + 1680*(4*A*B*a^6 - 11*A^2*a^5*b)*c^5 + 840*
(2*B^2*a^6*b - 16*A*B*a^5*b^2 + 33*A^2*a^4*b^3)*c^4 - 105*(8*B^2*a^5*b^3 - 68*A*
B*a^4*b^4 + 143*A^2*a^3*b^5)*c^3 + 3*(63*B^2*a^4*b^5 - 574*A*B*a^3*b^6 + 1298*A^
2*a^2*b^7)*c^2 - 3*(7*B^2*a^3*b^7 - 68*A*B*a^2*b^8 + 165*A^2*a*b^9)*c + (a^7*b^1
0 - 20*a^8*b^8*c + 160*a^9*b^6*c^2 - 640*a^10*b^4*c^3 + 1280*a^11*b^2*c^4 - 1024
*a^12*c^5)*sqrt((B^4*a^4*b^8 - 20*A*B^3*a^3*b^9 + 150*A^2*B^2*a^2*b^10 - 500*A^3
*B*a*b^11 + 625*A^4*b^12 + 50625*A^4*a^6*c^6 - 450*(49*A^2*B^2*a^7 - 382*A^3*B*a
^6*b + 694*A^4*a^5*b^2)*c^5 + (2401*B^4*a^8 - 37436*A*B^3*a^7*b + 218886*A^2*B^2
*a^6*b^2 - 577016*A^3*B*a^5*b^3 + 591886*A^4*a^4*b^4)*c^4 - 2*(539*B^4*a^7*b^2 -
9298*A*B^3*a^6*b^3 + 59592*A^2*B^2*a^5*b^4 - 168016*A^3*B*a^4*b^5 + 175655*A^4*
a^3*b^6)*c^3 + 3*(73*B^4*a^6*b^4 - 1344*A*B^3*a^5*b^5 + 9228*A^2*B^2*a^4*b^6 - 2
7980*A^3*B*a^3*b^7 + 31575*A^4*a^2*b^8)*c^2 - 2*(11*B^4*a^5*b^6 - 214*A*B^3*a^4*
b^7 + 1560*A^2*B^2*a^3*b^8 - 5050*A^3*B*a^2*b^9 + 6125*A^4*a*b^10)*c)/(a^14*b^10
- 20*a^15*b^8*c + 160*a^16*b^6*c^2 - 640*a^17*b^4*c^3 + 1280*a^18*b^2*c^4 - 102
4*a^19*c^5)))/(a^7*b^10 - 20*a^8*b^8*c + 160*a^9*b^6*c^2 - 640*a^10*b^4*c^3 + 12
80*a^11*b^2*c^4 - 1024*a^12*c^5))*log(-27/2*sqrt(1/2)*(B^3*a^3*b^14 - 15*A*B^2*a
^2*b^15 + 75*A^2*B*a*b^16 - 125*A^3*b^17 + 57600*(7*A^2*B*a^9 - 23*A^3*a^8*b)*c^
8 - 64*(1372*B^3*a^10 - 15204*A*B^2*a^9*b + 61326*A^2*B*a^8*b^2 - 88823*A^3*a^7*
b^3)*c^7 + 16*(7112*B^3*a^9*b^2 - 83292*A*B^2*a^8*b^3 + 330300*A^2*B*a^7*b^4 - 4
46671*A^3*a^6*b^5)*c^6 - 4*(15920*B^3*a^8*b^4 - 197004*A*B^2*a^7*b^5 + 811446*A^
2*B*a^6*b^6 - 1115785*A^3*a^5*b^7)*c^5 + 3*(6696*B^3*a^7*b^6 - 87308*A*B^2*a^6*b
^7 + 377471*A^2*B*a^5*b^8 - 541178*A^3*a^4*b^9)*c^4 - 3*(1295*B^3*a^6*b^8 - 1770
4*A*B^2*a^5*b^9 + 80329*A^2*B*a^4*b^10 - 120911*A^3*a^3*b^11)*c^3 + (464*B^3*a^5
*b^10 - 6609*A*B^2*a^4*b^11 + 31317*A^2*B*a^3*b^12 - 49360*A^3*a^2*b^13)*c^2 - (
32*B^3*a^4*b^12 - 471*A*B^2*a^3*b^13 + 2310*A^2*B*a^2*b^14 - 3775*A^3*a*b^15)*c
- (B*a^8*b^15 - 5*A*a^7*b^16 - 122880*A*a^15*c^8 - 4096*(11*B*a^15*b - 79*A*a^14
*b^2)*c^7 + 1536*(44*B*a^14*b^3 - 223*A*a^13*b^4)*c^6 - 256*(169*B*a^13*b^5 - 77
0*A*a^12*b^6)*c^5 + 480*(32*B*a^12*b^7 - 143*A*a^11*b^8)*c^4 - 80*(41*B*a^11*b^9
- 187*A*a^10*b^10)*c^3 + 2*(212*B*a^10*b^11 - 1003*A*a^9*b^12)*c^2 - (31*B*a^9*
b^13 - 152*A*a^8*b^14)*c)*sqrt((B^4*a^4*b^8 - 20*A*B^3*a^3*b^9 + 150*A^2*B^2*a^2
*b^10 - 500*A^3*B*a*b^11 + 625*A^4*b^12 + 50625*A^4*a^6*c^6 - 450*(49*A^2*B^2*a^
7 - 382*A^3*B*a^6*b + 694*A^4*a^5*b^2)*c^5 + (2401*B^4*a^8 - 37436*A*B^3*a^7*b +
218886*A^2*B^2*a^6*b^2 - 577016*A^3*B*a^5*b^3 + 591886*A^4*a^4*b^4)*c^4 - 2*(53
9*B^4*a^7*b^2 - 9298*A*B^3*a^6*b^3 + 59592*A^2*B^2*a^5*b^4 - 168016*A^3*B*a^4*b^
5 + 175655*A^4*a^3*b^6)*c^3 + 3*(73*B^4*a^6*b^4 - 1344*A*B^3*a^5*b^5 + 9228*A^2*
B^2*a^4*b^6 - 27980*A^3*B*a^3*b^7 + 31575*A^4*a^2*b^8)*c^2 - 2*(11*B^4*a^5*b^6 -
214*A*B^3*a^4*b^7 + 1560*A^2*B^2*a^3*b^8 - 5050*A^3*B*a^2*b^9 + 6125*A^4*a*b^10
)*c)/(a^14*b^10 - 20*a^15*b^8*c + 160*a^16*b^6*c^2 - 640*a^17*b^4*c^3 + 1280*a^1
8*b^2*c^4 - 1024*a^19*c^5)))*sqrt(-(B^2*a^2*b^9 - 10*A*B*a*b^10 + 25*A^2*b^11 +
1680*(4*A*B*a^6 - 11*A^2*a^5*b)*c^5 + 840*(2*B^2*a^6*b - 16*A*B*a^5*b^2 + 33*A^2
*a^4*b^3)*c^4 - 105*(8*B^2*a^5*b^3 - 68*A*B*a^4*b^4 + 143*A^2*a^3*b^5)*c^3 + 3*(
63*B^2*a^4*b^5 - 574*A*B*a^3*b^6 + 1298*A^2*a^2*b^7)*c^2 - 3*(7*B^2*a^3*b^7 - 68
*A*B*a^2*b^8 + 165*A^2*a*b^9)*c + (a^7*b^10 - 20*a^8*b^8*c + 160*a^9*b^6*c^2 - 6
40*a^10*b^4*c^3 + 1280*a^11*b^2*c^4 - 1024*a^12*c^5)*sqrt((B^4*a^4*b^8 - 20*A*B^
3*a^3*b^9 + 150*A^2*B^2*a^2*b^10 - 500*A^3*B*a*b^11 + 625*A^4*b^12 + 50625*A^4*a
^6*c^6 - 450*(49*A^2*B^2*a^7 - 382*A^3*B*a^6*b + 694*A^4*a^5*b^2)*c^5 + (2401*B^
4*a^8 - 37436*A*B^3*a^7*b + 218886*A^2*B^2*a^6*b^2 - 577016*A^3*B*a^5*b^3 + 5918
86*A^4*a^4*b^4)*c^4 - 2*(539*B^4*a^7*b^2 - 9298*A*B^3*a^6*b^3 + 59592*A^2*B^2*a^
5*b^4 - 168016*A^3*B*a^4*b^5 + 175655*A^4*a^3*b^6)*c^3 + 3*(73*B^4*a^6*b^4 - 134
4*A*B^3*a^5*b^5 + 9228*A^2*B^2*a^4*b^6 - 27980*A^3*B*a^3*b^7 + 31575*A^4*a^2*b^8
)*c^2 - 2*(11*B^4*a^5*b^6 - 214*A*B^3*a^4*b^7 + 1560*A^2*B^2*a^3*b^8 - 5050*A^3*
B*a^2*b^9 + 6125*A^4*a*b^10)*c)/(a^14*b^10 - 20*a^15*b^8*c + 160*a^16*b^6*c^2 -
640*a^17*b^4*c^3 + 1280*a^18*b^2*c^4 - 1024*a^19*c^5)))/(a^7*b^10 - 20*a^8*b^8*c
+ 160*a^9*b^6*c^2 - 640*a^10*b^4*c^3 + 1280*a^11*b^2*c^4 - 1024*a^12*c^5)) + 27
*(810000*A^4*a^5*c^9 + 405000*(2*A^3*B*a^5*b - 7*A^4*a^4*b^2)*c^8 - (38416*B^4*a
^7 - 422576*A*B^3*a^6*b + 1439376*A^2*B^2*a^5*b^2 - 1018856*A^3*B*a^4*b^3 - 1957
349*A^4*a^3*b^4)*c^7 + (19208*B^4*a^6*b^2 - 239896*A*B^3*a^5*b^3 + 955704*A^2*B^
2*a^4*b^4 - 1067347*A^3*B*a^3*b^5 - 571030*A^4*a^2*b^6)*c^6 - (4189*B^4*a^5*b^4
- 56807*A*B^3*a^4*b^5 + 251349*A^2*B^2*a^3*b^6 - 344630*A^3*B*a^2*b^7 - 77825*A^
4*a*b^8)*c^5 + 3*(149*B^4*a^4*b^6 - 2161*A*B^3*a^3*b^7 + 10380*A^2*B^2*a^2*b^8 -
16225*A^3*B*a*b^9 - 1375*A^4*b^10)*c^4 - 21*(B^4*a^3*b^8 - 15*A*B^3*a^2*b^9 + 7
5*A^2*B^2*a*b^10 - 125*A^3*B*b^11)*c^3)*sqrt(x)) - 3*sqrt(1/2)*(a^5*b^4 - 8*a^6*
b^2*c + 16*a^7*c^2 + (a^3*b^4*c^2 - 8*a^4*b^2*c^3 + 16*a^5*c^4)*x^4 + 2*(a^3*b^5
*c - 8*a^4*b^3*c^2 + 16*a^5*b*c^3)*x^3 + (a^3*b^6 - 6*a^4*b^4*c + 32*a^6*c^3)*x^
2 + 2*(a^4*b^5 - 8*a^5*b^3*c + 16*a^6*b*c^2)*x)*sqrt(x)*sqrt(-(B^2*a^2*b^9 - 10*
A*B*a*b^10 + 25*A^2*b^11 + 1680*(4*A*B*a^6 - 11*A^2*a^5*b)*c^5 + 840*(2*B^2*a^6*
b - 16*A*B*a^5*b^2 + 33*A^2*a^4*b^3)*c^4 - 105*(8*B^2*a^5*b^3 - 68*A*B*a^4*b^4 +
143*A^2*a^3*b^5)*c^3 + 3*(63*B^2*a^4*b^5 - 574*A*B*a^3*b^6 + 1298*A^2*a^2*b^7)*
c^2 - 3*(7*B^2*a^3*b^7 - 68*A*B*a^2*b^8 + 165*A^2*a*b^9)*c - (a^7*b^10 - 20*a^8*
b^8*c + 160*a^9*b^6*c^2 - 640*a^10*b^4*c^3 + 1280*a^11*b^2*c^4 - 1024*a^12*c^5)*
sqrt((B^4*a^4*b^8 - 20*A*B^3*a^3*b^9 + 150*A^2*B^2*a^2*b^10 - 500*A^3*B*a*b^11 +
625*A^4*b^12 + 50625*A^4*a^6*c^6 - 450*(49*A^2*B^2*a^7 - 382*A^3*B*a^6*b + 694*
A^4*a^5*b^2)*c^5 + (2401*B^4*a^8 - 37436*A*B^3*a^7*b + 218886*A^2*B^2*a^6*b^2 -
577016*A^3*B*a^5*b^3 + 591886*A^4*a^4*b^4)*c^4 - 2*(539*B^4*a^7*b^2 - 9298*A*B^3
*a^6*b^3 + 59592*A^2*B^2*a^5*b^4 - 168016*A^3*B*a^4*b^5 + 175655*A^4*a^3*b^6)*c^
3 + 3*(73*B^4*a^6*b^4 - 1344*A*B^3*a^5*b^5 + 9228*A^2*B^2*a^4*b^6 - 27980*A^3*B*
a^3*b^7 + 31575*A^4*a^2*b^8)*c^2 - 2*(11*B^4*a^5*b^6 - 214*A*B^3*a^4*b^7 + 1560*
A^2*B^2*a^3*b^8 - 5050*A^3*B*a^2*b^9 + 6125*A^4*a*b^10)*c)/(a^14*b^10 - 20*a^15*
b^8*c + 160*a^16*b^6*c^2 - 640*a^17*b^4*c^3 + 1280*a^18*b^2*c^4 - 1024*a^19*c^5)
))/(a^7*b^10 - 20*a^8*b^8*c + 160*a^9*b^6*c^2 - 640*a^10*b^4*c^3 + 1280*a^11*b^2
*c^4 - 1024*a^12*c^5))*log(27/2*sqrt(1/2)*(B^3*a^3*b^14 - 15*A*B^2*a^2*b^15 + 75
*A^2*B*a*b^16 - 125*A^3*b^17 + 57600*(7*A^2*B*a^9 - 23*A^3*a^8*b)*c^8 - 64*(1372
*B^3*a^10 - 15204*A*B^2*a^9*b + 61326*A^2*B*a^8*b^2 - 88823*A^3*a^7*b^3)*c^7 + 1
6*(7112*B^3*a^9*b^2 - 83292*A*B^2*a^8*b^3 + 330300*A^2*B*a^7*b^4 - 446671*A^3*a^
6*b^5)*c^6 - 4*(15920*B^3*a^8*b^4 - 197004*A*B^2*a^7*b^5 + 811446*A^2*B*a^6*b^6
- 1115785*A^3*a^5*b^7)*c^5 + 3*(6696*B^3*a^7*b^6 - 87308*A*B^2*a^6*b^7 + 377471*
A^2*B*a^5*b^8 - 541178*A^3*a^4*b^9)*c^4 - 3*(1295*B^3*a^6*b^8 - 17704*A*B^2*a^5*
b^9 + 80329*A^2*B*a^4*b^10 - 120911*A^3*a^3*b^11)*c^3 + (464*B^3*a^5*b^10 - 6609
*A*B^2*a^4*b^11 + 31317*A^2*B*a^3*b^12 - 49360*A^3*a^2*b^13)*c^2 - (32*B^3*a^4*b
^12 - 471*A*B^2*a^3*b^13 + 2310*A^2*B*a^2*b^14 - 3775*A^3*a*b^15)*c + (B*a^8*b^1
5 - 5*A*a^7*b^16 - 122880*A*a^15*c^8 - 4096*(11*B*a^15*b - 79*A*a^14*b^2)*c^7 +
1536*(44*B*a^14*b^3 - 223*A*a^13*b^4)*c^6 - 256*(169*B*a^13*b^5 - 770*A*a^12*b^6
)*c^5 + 480*(32*B*a^12*b^7 - 143*A*a^11*b^8)*c^4 - 80*(41*B*a^11*b^9 - 187*A*a^1
0*b^10)*c^3 + 2*(212*B*a^10*b^11 - 1003*A*a^9*b^12)*c^2 - (31*B*a^9*b^13 - 152*A
*a^8*b^14)*c)*sqrt((B^4*a^4*b^8 - 20*A*B^3*a^3*b^9 + 150*A^2*B^2*a^2*b^10 - 500*
A^3*B*a*b^11 + 625*A^4*b^12 + 50625*A^4*a^6*c^6 - 450*(49*A^2*B^2*a^7 - 382*A^3*
B*a^6*b + 694*A^4*a^5*b^2)*c^5 + (2401*B^4*a^8 - 37436*A*B^3*a^7*b + 218886*A^2*
B^2*a^6*b^2 - 577016*A^3*B*a^5*b^3 + 591886*A^4*a^4*b^4)*c^4 - 2*(539*B^4*a^7*b^
2 - 9298*A*B^3*a^6*b^3 + 59592*A^2*B^2*a^5*b^4 - 168016*A^3*B*a^4*b^5 + 175655*A
^4*a^3*b^6)*c^3 + 3*(73*B^4*a^6*b^4 - 1344*A*B^3*a^5*b^5 + 9228*A^2*B^2*a^4*b^6
- 27980*A^3*B*a^3*b^7 + 31575*A^4*a^2*b^8)*c^2 - 2*(11*B^4*a^5*b^6 - 214*A*B^3*a
^4*b^7 + 1560*A^2*B^2*a^3*b^8 - 5050*A^3*B*a^2*b^9 + 6125*A^4*a*b^10)*c)/(a^14*b
^10 - 20*a^15*b^8*c + 160*a^16*b^6*c^2 - 640*a^17*b^4*c^3 + 1280*a^18*b^2*c^4 -
1024*a^19*c^5)))*sqrt(-(B^2*a^2*b^9 - 10*A*B*a*b^10 + 25*A^2*b^11 + 1680*(4*A*B*
a^6 - 11*A^2*a^5*b)*c^5 + 840*(2*B^2*a^6*b - 16*A*B*a^5*b^2 + 33*A^2*a^4*b^3)*c^
4 - 105*(8*B^2*a^5*b^3 - 68*A*B*a^4*b^4 + 143*A^2*a^3*b^5)*c^3 + 3*(63*B^2*a^4*b
^5 - 574*A*B*a^3*b^6 + 1298*A^2*a^2*b^7)*c^2 - 3*(7*B^2*a^3*b^7 - 68*A*B*a^2*b^8
+ 165*A^2*a*b^9)*c - (a^7*b^10 - 20*a^8*b^8*c + 160*a^9*b^6*c^2 - 640*a^10*b^4*
c^3 + 1280*a^11*b^2*c^4 - 1024*a^12*c^5)*sqrt((B^4*a^4*b^8 - 20*A*B^3*a^3*b^9 +
150*A^2*B^2*a^2*b^10 - 500*A^3*B*a*b^11 + 625*A^4*b^12 + 50625*A^4*a^6*c^6 - 450
*(49*A^2*B^2*a^7 - 382*A^3*B*a^6*b + 694*A^4*a^5*b^2)*c^5 + (2401*B^4*a^8 - 3743
6*A*B^3*a^7*b + 218886*A^2*B^2*a^6*b^2 - 577016*A^3*B*a^5*b^3 + 591886*A^4*a^4*b
^4)*c^4 - 2*(539*B^4*a^7*b^2 - 9298*A*B^3*a^6*b^3 + 59592*A^2*B^2*a^5*b^4 - 1680
16*A^3*B*a^4*b^5 + 175655*A^4*a^3*b^6)*c^3 + 3*(73*B^4*a^6*b^4 - 1344*A*B^3*a^5*
b^5 + 9228*A^2*B^2*a^4*b^6 - 27980*A^3*B*a^3*b^7 + 31575*A^4*a^2*b^8)*c^2 - 2*(1
1*B^4*a^5*b^6 - 214*A*B^3*a^4*b^7 + 1560*A^2*B^2*a^3*b^8 - 5050*A^3*B*a^2*b^9 +
6125*A^4*a*b^10)*c)/(a^14*b^10 - 20*a^15*b^8*c + 160*a^16*b^6*c^2 - 640*a^17*b^4
*c^3 + 1280*a^18*b^2*c^4 - 1024*a^19*c^5)))/(a^7*b^10 - 20*a^8*b^8*c + 160*a^9*b
^6*c^2 - 640*a^10*b^4*c^3 + 1280*a^11*b^2*c^4 - 1024*a^12*c^5)) + 27*(810000*A^4
*a^5*c^9 + 405000*(2*A^3*B*a^5*b - 7*A^4*a^4*b^2)*c^8 - (38416*B^4*a^7 - 422576*
A*B^3*a^6*b + 1439376*A^2*B^2*a^5*b^2 - 1018856*A^3*B*a^4*b^3 - 1957349*A^4*a^3*
b^4)*c^7 + (19208*B^4*a^6*b^2 - 239896*A*B^3*a^5*b^3 + 955704*A^2*B^2*a^4*b^4 -
1067347*A^3*B*a^3*b^5 - 571030*A^4*a^2*b^6)*c^6 - (4189*B^4*a^5*b^4 - 56807*A*B^
3*a^4*b^5 + 251349*A^2*B^2*a^3*b^6 - 344630*A^3*B*a^2*b^7 - 77825*A^4*a*b^8)*c^5
+ 3*(149*B^4*a^4*b^6 - 2161*A*B^3*a^3*b^7 + 10380*A^2*B^2*a^2*b^8 - 16225*A^3*B
*a*b^9 - 1375*A^4*b^10)*c^4 - 21*(B^4*a^3*b^8 - 15*A*B^3*a^2*b^9 + 75*A^2*B^2*a*
b^10 - 125*A^3*B*b^11)*c^3)*sqrt(x)) + 3*sqrt(1/2)*(a^5*b^4 - 8*a^6*b^2*c + 16*a
^7*c^2 + (a^3*b^4*c^2 - 8*a^4*b^2*c^3 + 16*a^5*c^4)*x^4 + 2*(a^3*b^5*c - 8*a^4*b
^3*c^2 + 16*a^5*b*c^3)*x^3 + (a^3*b^6 - 6*a^4*b^4*c + 32*a^6*c^3)*x^2 + 2*(a^4*b
^5 - 8*a^5*b^3*c + 16*a^6*b*c^2)*x)*sqrt(x)*sqrt(-(B^2*a^2*b^9 - 10*A*B*a*b^10 +
25*A^2*b^11 + 1680*(4*A*B*a^6 - 11*A^2*a^5*b)*c^5 + 840*(2*B^2*a^6*b - 16*A*B*a
^5*b^2 + 33*A^2*a^4*b^3)*c^4 - 105*(8*B^2*a^5*b^3 - 68*A*B*a^4*b^4 + 143*A^2*a^3
*b^5)*c^3 + 3*(63*B^2*a^4*b^5 - 574*A*B*a^3*b^6 + 1298*A^2*a^2*b^7)*c^2 - 3*(7*B
^2*a^3*b^7 - 68*A*B*a^2*b^8 + 165*A^2*a*b^9)*c - (a^7*b^10 - 20*a^8*b^8*c + 160*
a^9*b^6*c^2 - 640*a^10*b^4*c^3 + 1280*a^11*b^2*c^4 - 1024*a^12*c^5)*sqrt((B^4*a^
4*b^8 - 20*A*B^3*a^3*b^9 + 150*A^2*B^2*a^2*b^10 - 500*A^3*B*a*b^11 + 625*A^4*b^1
2 + 50625*A^4*a^6*c^6 - 450*(49*A^2*B^2*a^7 - 382*A^3*B*a^6*b + 694*A^4*a^5*b^2)
*c^5 + (2401*B^4*a^8 - 37436*A*B^3*a^7*b + 218886*A^2*B^2*a^6*b^2 - 577016*A^3*B
*a^5*b^3 + 591886*A^4*a^4*b^4)*c^4 - 2*(539*B^4*a^7*b^2 - 9298*A*B^3*a^6*b^3 + 5
9592*A^2*B^2*a^5*b^4 - 168016*A^3*B*a^4*b^5 + 175655*A^4*a^3*b^6)*c^3 + 3*(73*B^
4*a^6*b^4 - 1344*A*B^3*a^5*b^5 + 9228*A^2*B^2*a^4*b^6 - 27980*A^3*B*a^3*b^7 + 31
575*A^4*a^2*b^8)*c^2 - 2*(11*B^4*a^5*b^6 - 214*A*B^3*a^4*b^7 + 1560*A^2*B^2*a^3*
b^8 - 5050*A^3*B*a^2*b^9 + 6125*A^4*a*b^10)*c)/(a^14*b^10 - 20*a^15*b^8*c + 160*
a^16*b^6*c^2 - 640*a^17*b^4*c^3 + 1280*a^18*b^2*c^4 - 1024*a^19*c^5)))/(a^7*b^10
- 20*a^8*b^8*c + 160*a^9*b^6*c^2 - 640*a^10*b^4*c^3 + 1280*a^11*b^2*c^4 - 1024*
a^12*c^5))*log(-27/2*sqrt(1/2)*(B^3*a^3*b^14 - 15*A*B^2*a^2*b^15 + 75*A^2*B*a*b^
16 - 125*A^3*b^17 + 57600*(7*A^2*B*a^9 - 23*A^3*a^8*b)*c^8 - 64*(1372*B^3*a^10 -
15204*A*B^2*a^9*b + 61326*A^2*B*a^8*b^2 - 88823*A^3*a^7*b^3)*c^7 + 16*(7112*B^3
*a^9*b^2 - 83292*A*B^2*a^8*b^3 + 330300*A^2*B*a^7*b^4 - 446671*A^3*a^6*b^5)*c^6
- 4*(15920*B^3*a^8*b^4 - 197004*A*B^2*a^7*b^5 + 811446*A^2*B*a^6*b^6 - 1115785*A
^3*a^5*b^7)*c^5 + 3*(6696*B^3*a^7*b^6 - 87308*A*B^2*a^6*b^7 + 377471*A^2*B*a^5*b
^8 - 541178*A^3*a^4*b^9)*c^4 - 3*(1295*B^3*a^6*b^8 - 17704*A*B^2*a^5*b^9 + 80329
*A^2*B*a^4*b^10 - 120911*A^3*a^3*b^11)*c^3 + (464*B^3*a^5*b^10 - 6609*A*B^2*a^4*
b^11 + 31317*A^2*B*a^3*b^12 - 49360*A^3*a^2*b^13)*c^2 - (32*B^3*a^4*b^12 - 471*A
*B^2*a^3*b^13 + 2310*A^2*B*a^2*b^14 - 3775*A^3*a*b^15)*c + (B*a^8*b^15 - 5*A*a^7
*b^16 - 122880*A*a^15*c^8 - 4096*(11*B*a^15*b - 79*A*a^14*b^2)*c^7 + 1536*(44*B*
a^14*b^3 - 223*A*a^13*b^4)*c^6 - 256*(169*B*a^13*b^5 - 770*A*a^12*b^6)*c^5 + 480
*(32*B*a^12*b^7 - 143*A*a^11*b^8)*c^4 - 80*(41*B*a^11*b^9 - 187*A*a^10*b^10)*c^3
+ 2*(212*B*a^10*b^11 - 1003*A*a^9*b^12)*c^2 - (31*B*a^9*b^13 - 152*A*a^8*b^14)*
c)*sqrt((B^4*a^4*b^8 - 20*A*B^3*a^3*b^9 + 150*A^2*B^2*a^2*b^10 - 500*A^3*B*a*b^1
1 + 625*A^4*b^12 + 50625*A^4*a^6*c^6 - 450*(49*A^2*B^2*a^7 - 382*A^3*B*a^6*b + 6
94*A^4*a^5*b^2)*c^5 + (2401*B^4*a^8 - 37436*A*B^3*a^7*b + 218886*A^2*B^2*a^6*b^2
- 577016*A^3*B*a^5*b^3 + 591886*A^4*a^4*b^4)*c^4 - 2*(539*B^4*a^7*b^2 - 9298*A*
B^3*a^6*b^3 + 59592*A^2*B^2*a^5*b^4 - 168016*A^3*B*a^4*b^5 + 175655*A^4*a^3*b^6)
*c^3 + 3*(73*B^4*a^6*b^4 - 1344*A*B^3*a^5*b^5 + 9228*A^2*B^2*a^4*b^6 - 27980*A^3
*B*a^3*b^7 + 31575*A^4*a^2*b^8)*c^2 - 2*(11*B^4*a^5*b^6 - 214*A*B^3*a^4*b^7 + 15
60*A^2*B^2*a^3*b^8 - 5050*A^3*B*a^2*b^9 + 6125*A^4*a*b^10)*c)/(a^14*b^10 - 20*a^
15*b^8*c + 160*a^16*b^6*c^2 - 640*a^17*b^4*c^3 + 1280*a^18*b^2*c^4 - 1024*a^19*c
^5)))*sqrt(-(B^2*a^2*b^9 - 10*A*B*a*b^10 + 25*A^2*b^11 + 1680*(4*A*B*a^6 - 11*A^
2*a^5*b)*c^5 + 840*(2*B^2*a^6*b - 16*A*B*a^5*b^2 + 33*A^2*a^4*b^3)*c^4 - 105*(8*
B^2*a^5*b^3 - 68*A*B*a^4*b^4 + 143*A^2*a^3*b^5)*c^3 + 3*(63*B^2*a^4*b^5 - 574*A*
B*a^3*b^6 + 1298*A^2*a^2*b^7)*c^2 - 3*(7*B^2*a^3*b^7 - 68*A*B*a^2*b^8 + 165*A^2*
a*b^9)*c - (a^7*b^10 - 20*a^8*b^8*c + 160*a^9*b^6*c^2 - 640*a^10*b^4*c^3 + 1280*
a^11*b^2*c^4 - 1024*a^12*c^5)*sqrt((B^4*a^4*b^8 - 20*A*B^3*a^3*b^9 + 150*A^2*B^2
*a^2*b^10 - 500*A^3*B*a*b^11 + 625*A^4*b^12 + 50625*A^4*a^6*c^6 - 450*(49*A^2*B^
2*a^7 - 382*A^3*B*a^6*b + 694*A^4*a^5*b^2)*c^5 + (2401*B^4*a^8 - 37436*A*B^3*a^7
*b + 218886*A^2*B^2*a^6*b^2 - 577016*A^3*B*a^5*b^3 + 591886*A^4*a^4*b^4)*c^4 - 2
*(539*B^4*a^7*b^2 - 9298*A*B^3*a^6*b^3 + 59592*A^2*B^2*a^5*b^4 - 168016*A^3*B*a^
4*b^5 + 175655*A^4*a^3*b^6)*c^3 + 3*(73*B^4*a^6*b^4 - 1344*A*B^3*a^5*b^5 + 9228*
A^2*B^2*a^4*b^6 - 27980*A^3*B*a^3*b^7 + 31575*A^4*a^2*b^8)*c^2 - 2*(11*B^4*a^5*b
^6 - 214*A*B^3*a^4*b^7 + 1560*A^2*B^2*a^3*b^8 - 5050*A^3*B*a^2*b^9 + 6125*A^4*a*
b^10)*c)/(a^14*b^10 - 20*a^15*b^8*c + 160*a^16*b^6*c^2 - 640*a^17*b^4*c^3 + 1280
*a^18*b^2*c^4 - 1024*a^19*c^5)))/(a^7*b^10 - 20*a^8*b^8*c + 160*a^9*b^6*c^2 - 64
0*a^10*b^4*c^3 + 1280*a^11*b^2*c^4 - 1024*a^12*c^5)) + 27*(810000*A^4*a^5*c^9 +
405000*(2*A^3*B*a^5*b - 7*A^4*a^4*b^2)*c^8 - (38416*B^4*a^7 - 422576*A*B^3*a^6*b
+ 1439376*A^2*B^2*a^5*b^2 - 1018856*A^3*B*a^4*b^3 - 1957349*A^4*a^3*b^4)*c^7 +
(19208*B^4*a^6*b^2 - 239896*A*B^3*a^5*b^3 + 955704*A^2*B^2*a^4*b^4 - 1067347*A^3
*B*a^3*b^5 - 571030*A^4*a^2*b^6)*c^6 - (4189*B^4*a^5*b^4 - 56807*A*B^3*a^4*b^5 +
251349*A^2*B^2*a^3*b^6 - 344630*A^3*B*a^2*b^7 - 77825*A^4*a*b^8)*c^5 + 3*(149*B
^4*a^4*b^6 - 2161*A*B^3*a^3*b^7 + 10380*A^2*B^2*a^2*b^8 - 16225*A^3*B*a*b^9 - 13
75*A^4*b^10)*c^4 - 21*(B^4*a^3*b^8 - 15*A*B^3*a^2*b^9 + 75*A^2*B^2*a*b^10 - 125*
A^3*B*b^11)*c^3)*sqrt(x)) - 2*(3*B*a*b^5 - 15*A*b^6 - 324*A*a^3*c^3 - (4*B*a^3*b
+ 25*A*a^2*b^2)*c^2 - (20*B*a^2*b^3 - 91*A*a*b^4)*c)*x^2 - 2*(5*B*a^2*b^4 - 25*
A*a*b^5 + 4*(11*B*a^4 - 91*A*a^3*b)*c^2 - (37*B*a^3*b^2 - 194*A*a^2*b^3)*c)*x)/(
(a^5*b^4 - 8*a^6*b^2*c + 16*a^7*c^2 + (a^3*b^4*c^2 - 8*a^4*b^2*c^3 + 16*a^5*c^4)
*x^4 + 2*(a^3*b^5*c - 8*a^4*b^3*c^2 + 16*a^5*b*c^3)*x^3 + (a^3*b^6 - 6*a^4*b^4*c
+ 32*a^6*c^3)*x^2 + 2*(a^4*b^5 - 8*a^5*b^3*c + 16*a^6*b*c^2)*x)*sqrt(x))

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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)/x**(3/2)/(c*x**2+b*x+a)**3,x)

[Out]

Timed out

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GIAC/XCAS [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)/((c*x^2 + b*x + a)^3*x^(3/2)),x, algorithm="giac")

[Out]

Timed out