3.20.63 \(\int \frac {(f+g x) (c d^2-b d e-b e^2 x-c e^2 x^2)^{3/2}}{(d+e x)^8} \, dx\)

Optimal. Leaf size=285 \[ -\frac {16 c^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-11 b e g+16 c d g+6 c e f)}{3465 e^2 (d+e x)^5 (2 c d-b e)^4}-\frac {8 c \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-11 b e g+16 c d g+6 c e f)}{693 e^2 (d+e x)^6 (2 c d-b e)^3}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-11 b e g+16 c d g+6 c e f)}{99 e^2 (d+e x)^7 (2 c d-b e)^2}-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{11 e^2 (d+e x)^8 (2 c d-b e)} \]

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Rubi [A]  time = 0.45, antiderivative size = 285, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.068, Rules used = {792, 658, 650} \begin {gather*} -\frac {16 c^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-11 b e g+16 c d g+6 c e f)}{3465 e^2 (d+e x)^5 (2 c d-b e)^4}-\frac {8 c \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-11 b e g+16 c d g+6 c e f)}{693 e^2 (d+e x)^6 (2 c d-b e)^3}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-11 b e g+16 c d g+6 c e f)}{99 e^2 (d+e x)^7 (2 c d-b e)^2}-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{11 e^2 (d+e x)^8 (2 c d-b e)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(3/2))/(d + e*x)^8,x]

[Out]

(-2*(e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(11*e^2*(2*c*d - b*e)*(d + e*x)^8) - (2*(6*c*e*f
+ 16*c*d*g - 11*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(99*e^2*(2*c*d - b*e)^2*(d + e*x)^7) - (8*
c*(6*c*e*f + 16*c*d*g - 11*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(693*e^2*(2*c*d - b*e)^3*(d + e
*x)^6) - (16*c^2*(6*c*e*f + 16*c*d*g - 11*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(3465*e^2*(2*c*d
- b*e)^4*(d + e*x)^5)

Rule 650

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^m*(a +
b*x + c*x^2)^(p + 1))/((p + 1)*(2*c*d - b*e)), x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] &&
EqQ[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] && EqQ[m + 2*p + 2, 0]

Rule 658

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[(e*(d + e*x)^m*(a +
b*x + c*x^2)^(p + 1))/((m + p + 1)*(2*c*d - b*e)), x] + Dist[(c*Simplify[m + 2*p + 2])/((m + p + 1)*(2*c*d -
b*e)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c
, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] && ILtQ[Simplify[m + 2*p + 2], 0]

Rule 792

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[((d*g - e*f)*(d + e*x)^m*(a + b*x + c*x^2)^(p + 1))/((2*c*d - b*e)*(m + p + 1)), x] + Dist[(m*(g*(c*d - b*e)
+ c*e*f) + e*(p + 1)*(2*c*f - b*g))/(e*(2*c*d - b*e)*(m + p + 1)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p,
x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && ((L
tQ[m, -1] &&  !IGtQ[m + p + 1, 0]) || (LtQ[m, 0] && LtQ[p, -1]) || EqQ[m + 2*p + 2, 0]) && NeQ[m + p + 1, 0]

Rubi steps

\begin {align*} \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^8} \, dx &=-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{11 e^2 (2 c d-b e) (d+e x)^8}+\frac {(6 c e f+16 c d g-11 b e g) \int \frac {\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^7} \, dx}{11 e (2 c d-b e)}\\ &=-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{11 e^2 (2 c d-b e) (d+e x)^8}-\frac {2 (6 c e f+16 c d g-11 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{99 e^2 (2 c d-b e)^2 (d+e x)^7}+\frac {(4 c (6 c e f+16 c d g-11 b e g)) \int \frac {\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^6} \, dx}{99 e (2 c d-b e)^2}\\ &=-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{11 e^2 (2 c d-b e) (d+e x)^8}-\frac {2 (6 c e f+16 c d g-11 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{99 e^2 (2 c d-b e)^2 (d+e x)^7}-\frac {8 c (6 c e f+16 c d g-11 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{693 e^2 (2 c d-b e)^3 (d+e x)^6}+\frac {\left (8 c^2 (6 c e f+16 c d g-11 b e g)\right ) \int \frac {\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^5} \, dx}{693 e (2 c d-b e)^3}\\ &=-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{11 e^2 (2 c d-b e) (d+e x)^8}-\frac {2 (6 c e f+16 c d g-11 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{99 e^2 (2 c d-b e)^2 (d+e x)^7}-\frac {8 c (6 c e f+16 c d g-11 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{693 e^2 (2 c d-b e)^3 (d+e x)^6}-\frac {16 c^2 (6 c e f+16 c d g-11 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{3465 e^2 (2 c d-b e)^4 (d+e x)^5}\\ \end {align*}

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Mathematica [A]  time = 0.18, size = 249, normalized size = 0.87 \begin {gather*} -\frac {2 (b e-c d+c e x)^2 \sqrt {(d+e x) (c (d-e x)-b e)} \left (-35 b^3 e^3 (2 d g+9 e f+11 e g x)+10 b^2 c e^2 \left (43 d^2 g+d e (210 f+254 g x)+e^2 x (21 f+22 g x)\right )-4 b c^2 e \left (212 d^3 g+d^2 e (1185 f+1391 g x)+2 d e^2 x (135 f+128 g x)+2 e^3 x^2 (15 f+11 g x)\right )+8 c^3 \left (61 d^4 g+8 d^3 e (57 f+61 g x)+d^2 e^2 x (183 f+128 g x)+16 d e^3 x^2 (3 f+g x)+6 e^4 f x^3\right )\right )}{3465 e^2 (d+e x)^6 (b e-2 c d)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(3/2))/(d + e*x)^8,x]

[Out]

(-2*(-(c*d) + b*e + c*e*x)^2*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))]*(-35*b^3*e^3*(9*e*f + 2*d*g + 11*e*g*x) +
8*c^3*(61*d^4*g + 6*e^4*f*x^3 + 16*d*e^3*x^2*(3*f + g*x) + 8*d^3*e*(57*f + 61*g*x) + d^2*e^2*x*(183*f + 128*g*
x)) + 10*b^2*c*e^2*(43*d^2*g + e^2*x*(21*f + 22*g*x) + d*e*(210*f + 254*g*x)) - 4*b*c^2*e*(212*d^3*g + 2*e^3*x
^2*(15*f + 11*g*x) + 2*d*e^2*x*(135*f + 128*g*x) + d^2*e*(1185*f + 1391*g*x))))/(3465*e^2*(-2*c*d + b*e)^4*(d
+ e*x)^6)

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IntegrateAlgebraic [F]  time = 180.17, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(3/2))/(d + e*x)^8,x]

[Out]

$Aborted

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)/(e*x+d)^8,x, algorithm="fricas")

[Out]

Timed out

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)/(e*x+d)^8,x, algorithm="giac")

[Out]

Timed out

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maple [A]  time = 0.05, size = 382, normalized size = 1.34 \begin {gather*} -\frac {2 \left (c e x +b e -c d \right ) \left (88 b \,c^{2} e^{4} g \,x^{3}-128 c^{3} d \,e^{3} g \,x^{3}-48 c^{3} e^{4} f \,x^{3}-220 b^{2} c \,e^{4} g \,x^{2}+1024 b \,c^{2} d \,e^{3} g \,x^{2}+120 b \,c^{2} e^{4} f \,x^{2}-1024 c^{3} d^{2} e^{2} g \,x^{2}-384 c^{3} d \,e^{3} f \,x^{2}+385 b^{3} e^{4} g x -2540 b^{2} c d \,e^{3} g x -210 b^{2} c \,e^{4} f x +5564 b \,c^{2} d^{2} e^{2} g x +1080 b \,c^{2} d \,e^{3} f x -3904 c^{3} d^{3} e g x -1464 c^{3} d^{2} e^{2} f x +70 b^{3} d \,e^{3} g +315 b^{3} e^{4} f -430 b^{2} c \,d^{2} e^{2} g -2100 b^{2} c d \,e^{3} f +848 b \,c^{2} d^{3} e g +4740 b \,c^{2} d^{2} e^{2} f -488 c^{3} d^{4} g -3648 c^{3} d^{3} e f \right ) \left (-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}\right )^{\frac {3}{2}}}{3465 \left (e x +d \right )^{7} \left (b^{4} e^{4}-8 b^{3} c d \,e^{3}+24 b^{2} c^{2} d^{2} e^{2}-32 b \,c^{3} d^{3} e +16 c^{4} d^{4}\right ) e^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)/(e*x+d)^8,x)

[Out]

-2/3465*(c*e*x+b*e-c*d)*(88*b*c^2*e^4*g*x^3-128*c^3*d*e^3*g*x^3-48*c^3*e^4*f*x^3-220*b^2*c*e^4*g*x^2+1024*b*c^
2*d*e^3*g*x^2+120*b*c^2*e^4*f*x^2-1024*c^3*d^2*e^2*g*x^2-384*c^3*d*e^3*f*x^2+385*b^3*e^4*g*x-2540*b^2*c*d*e^3*
g*x-210*b^2*c*e^4*f*x+5564*b*c^2*d^2*e^2*g*x+1080*b*c^2*d*e^3*f*x-3904*c^3*d^3*e*g*x-1464*c^3*d^2*e^2*f*x+70*b
^3*d*e^3*g+315*b^3*e^4*f-430*b^2*c*d^2*e^2*g-2100*b^2*c*d*e^3*f+848*b*c^2*d^3*e*g+4740*b*c^2*d^2*e^2*f-488*c^3
*d^4*g-3648*c^3*d^3*e*f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)/(e*x+d)^7/e^2/(b^4*e^4-8*b^3*c*d*e^3+24*b^2*c^
2*d^2*e^2-32*b*c^3*d^3*e+16*c^4*d^4)

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(b*e-2*c*d>0)', see `assume?` f
or more details)Is b*e-2*c*d zero or nonzero?

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mupad [B]  time = 42.70, size = 16485, normalized size = 57.84

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((f + g*x)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(3/2))/(d + e*x)^8,x)

[Out]

(((d*((d*((64*c^6*(7*b*e*g - 12*c*d*g + c*e*f))/(10395*(b*e - 2*c*d)^6) - (64*c^7*d*g)/(10395*(b*e - 2*c*d)^6)
))/e - (3904*c^7*d^2*g + 1184*b^2*c^5*e^2*g - 768*c^7*d*e*f + 448*b*c^6*e^2*f - 4288*b*c^6*d*e*g)/(10395*e*(b*
e - 2*c*d)^6)))/e + (8*b*c^4*(61*b^2*e^2*g + 244*c^2*d^2*g + 26*b*c*e^2*f - 48*c^2*d*e*f - 244*b*c*d*e*g))/(10
395*e*(b*e - 2*c*d)^6))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x) - (((d*((d*((32*c^6*(9*b*e*g -
14*c*d*g + 2*c*e*f))/(10395*(b*e - 2*c*d)^6) - (64*c^7*d*g)/(10395*(b*e - 2*c*d)^6)))/e - (1664*c^7*d^2*g + 54
4*b^2*c^5*e^2*g - 448*c^7*d*e*f + 288*b*c^6*e^2*f - 1888*b*c^6*d*e*g)/(10395*e*(b*e - 2*c*d)^6)))/e + (16*b*c^
4*(13*b^2*e^2*g + 52*c^2*d^2*g + 8*b*c*e^2*f - 14*c^2*d*e*f - 52*b*c*d*e*g))/(10395*e*(b*e - 2*c*d)^6))*(c*d^2
- c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x) + (((d*((d*((64*c^6*(8*b*e*g - 14*c*d*g + c*e*f))/(10395*(b*e
- 2*c*d)^6) - (64*c^7*d*g)/(10395*(b*e - 2*c*d)^6)))/e - (4800*c^7*d^2*g + 1440*b^2*c^5*e^2*g - 896*c^7*d*e*f
+ 512*b*c^6*e^2*f - 5248*b*c^6*d*e*g)/(10395*e*(b*e - 2*c*d)^6)))/e + (8*b*c^4*(75*b^2*e^2*g + 300*c^2*d^2*g
+ 30*b*c*e^2*f - 56*c^2*d*e*f - 300*b*c*d*e*g))/(10395*e*(b*e - 2*c*d)^6))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*
x)^(1/2))/(d + e*x) + (((d*((d*((64*c^6*(9*b*e*g - 16*c*d*g + c*e*f))/(10395*(b*e - 2*c*d)^6) - (64*c^7*d*g)/(
10395*(b*e - 2*c*d)^6)))/e - (5696*c^7*d^2*g + 1696*b^2*c^5*e^2*g - 1024*c^7*d*e*f + 576*b*c^6*e^2*f - 6208*b*
c^6*d*e*g)/(10395*e*(b*e - 2*c*d)^6)))/e + (8*b*c^4*(89*b^2*e^2*g + 356*c^2*d^2*g + 34*b*c*e^2*f - 64*c^2*d*e*
f - 356*b*c*d*e*g))/(10395*e*(b*e - 2*c*d)^6))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x) + (((d*(
(d*((64*c^6*(10*b*e*g - 18*c*d*g + c*e*f))/(10395*(b*e - 2*c*d)^6) - (64*c^7*d*g)/(10395*(b*e - 2*c*d)^6)))/e
- (6592*c^7*d^2*g + 1952*b^2*c^5*e^2*g - 1152*c^7*d*e*f + 640*b*c^6*e^2*f - 7168*b*c^6*d*e*g)/(10395*e*(b*e -
2*c*d)^6)))/e + (8*b*c^4*(103*b^2*e^2*g + 412*c^2*d^2*g + 38*b*c*e^2*f - 72*c^2*d*e*f - 412*b*c*d*e*g))/(10395
*e*(b*e - 2*c*d)^6))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x) - (((d*((d*((32*c^6*(21*b*e*g - 38
*c*d*g + 2*c*e*f))/(10395*(b*e - 2*c*d)^6) - (64*c^7*d*g)/(10395*(b*e - 2*c*d)^6)))/e - (9280*c^7*d^2*g + 2640
*b^2*c^5*e^2*g - 1216*c^7*d*e*f + 672*b*c^6*e^2*f - 9888*b*c^6*d*e*g)/(10395*e*(b*e - 2*c*d)^6)))/e + (8*b*c^4
*(145*b^2*e^2*g + 580*c^2*d^2*g + 40*b*c*e^2*f - 76*c^2*d*e*f - 580*b*c*d*e*g))/(10395*e*(b*e - 2*c*d)^6))*(c*
d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x) - (((d*((d*((32*c^6*(23*b*e*g - 42*c*d*g + 2*c*e*f))/(1039
5*(b*e - 2*c*d)^6) - (64*c^7*d*g)/(10395*(b*e - 2*c*d)^6)))/e - (10816*c^7*d^2*g + 3056*b^2*c^5*e^2*g - 1344*c
^7*d*e*f + 736*b*c^6*e^2*f - 11488*b*c^6*d*e*g)/(10395*e*(b*e - 2*c*d)^6)))/e + (8*b*c^4*(169*b^2*e^2*g + 676*
c^2*d^2*g + 44*b*c*e^2*f - 84*c^2*d*e*f - 676*b*c*d*e*g))/(10395*e*(b*e - 2*c*d)^6))*(c*d^2 - c*e^2*x^2 - b*d*
e - b*e^2*x)^(1/2))/(d + e*x) - (((d*((d*((32*c^6*(25*b*e*g - 46*c*d*g + 2*c*e*f))/(10395*(b*e - 2*c*d)^6) - (
64*c^7*d*g)/(10395*(b*e - 2*c*d)^6)))/e - (12352*c^7*d^2*g + 3472*b^2*c^5*e^2*g - 1472*c^7*d*e*f + 800*b*c^6*e
^2*f - 13088*b*c^6*d*e*g)/(10395*e*(b*e - 2*c*d)^6)))/e + (8*b*c^4*(193*b^2*e^2*g + 772*c^2*d^2*g + 48*b*c*e^2
*f - 92*c^2*d*e*f - 772*b*c*d*e*g))/(10395*e*(b*e - 2*c*d)^6))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d
+ e*x) - (((d*((d*((32*c^6*(25*b*e*g - 46*c*d*g + 2*c*e*f))/(10395*(b*e - 2*c*d)^6) - (64*c^7*d*g)/(10395*(b*
e - 2*c*d)^6)))/e - (12864*c^7*d^2*g + 3600*b^2*c^5*e^2*g - 1472*c^7*d*e*f + 800*b*c^6*e^2*f - 13600*b*c^6*d*e
*g)/(10395*e*(b*e - 2*c*d)^6)))/e + (8*b*c^4*(201*b^2*e^2*g + 804*c^2*d^2*g + 48*b*c*e^2*f - 92*c^2*d*e*f - 80
4*b*c*d*e*g))/(10395*e*(b*e - 2*c*d)^6))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x) - (((d*((d*((3
2*c^6*(27*b*e*g - 50*c*d*g + 2*c*e*f))/(10395*(b*e - 2*c*d)^6) - (64*c^7*d*g)/(10395*(b*e - 2*c*d)^6)))/e - (1
4656*c^7*d^2*g + 4080*b^2*c^5*e^2*g - 1600*c^7*d*e*f + 864*b*c^6*e^2*f - 15456*b*c^6*d*e*g)/(10395*e*(b*e - 2*
c*d)^6)))/e + (8*b*c^4*(229*b^2*e^2*g + 916*c^2*d^2*g + 52*b*c*e^2*f - 100*c^2*d*e*f - 916*b*c*d*e*g))/(10395*
e*(b*e - 2*c*d)^6))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x) - (((d*((d*((32*c^6*(29*b*e*g - 54*
c*d*g + 2*c*e*f))/(10395*(b*e - 2*c*d)^6) - (64*c^7*d*g)/(10395*(b*e - 2*c*d)^6)))/e - (16960*c^7*d^2*g + 4688
*b^2*c^5*e^2*g - 1728*c^7*d*e*f + 928*b*c^6*e^2*f - 17824*b*c^6*d*e*g)/(10395*e*(b*e - 2*c*d)^6)))/e + (8*b*c^
4*(265*b^2*e^2*g + 1060*c^2*d^2*g + 56*b*c*e^2*f - 108*c^2*d*e*f - 1060*b*c*d*e*g))/(10395*e*(b*e - 2*c*d)^6))
*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x) + (((d*((d*((64*c^6*(15*b*e*g - 28*c*d*g + c*e*f))/(10
395*(b*e - 2*c*d)^6) - (64*c^7*d*g)/(10395*(b*e - 2*c*d)^6)))/e - (20224*c^7*d^2*g + 5520*b^2*c^5*e^2*g - 1792
*c^7*d*e*f + 960*b*c^6*e^2*f - 21120*b*c^6*d*e*g)/(10395*e*(b*e - 2*c*d)^6)))/e + (16*b*c^4*(158*b^2*e^2*g + 6
32*c^2*d^2*g + 29*b*c*e^2*f - 56*c^2*d*e*f - 632*b*c*d*e*g))/(10395*e*(b*e - 2*c*d)^6))*(c*d^2 - c*e^2*x^2 - b
*d*e - b*e^2*x)^(1/2))/(d + e*x) + (((d*((d*((64*c^6*(16*b*e*g - 30*c*d*g + c*e*f))/(10395*(b*e - 2*c*d)^6) -
(64*c^7*d*g)/(10395*(b*e - 2*c*d)^6)))/e - (22656*c^7*d^2*g + 6160*b^2*c^5*e^2*g - 1920*c^7*d*e*f + 1024*b*c^6
*e^2*f - 23616*b*c^6*d*e*g)/(10395*e*(b*e - 2*c*d)^6)))/e + (16*b*c^4*(177*b^2*e^2*g + 708*c^2*d^2*g + 31*b*c*
e^2*f - 60*c^2*d*e*f - 708*b*c*d*e*g))/(10395*e*(b*e - 2*c*d)^6))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))
/(d + e*x) + (((d*((d*((64*c^6*(17*b*e*g - 32*c*d*g + c*e*f))/(10395*(b*e - 2*c*d)^6) - (64*c^7*d*g)/(10395*(b
*e - 2*c*d)^6)))/e - (25600*c^7*d^2*g + 6928*b^2*c^5*e^2*g - 2048*c^7*d*e*f + 1088*b*c^6*e^2*f - 26624*b*c^6*d
*e*g)/(10395*e*(b*e - 2*c*d)^6)))/e + (16*b*c^4*(200*b^2*e^2*g + 800*c^2*d^2*g + 33*b*c*e^2*f - 64*c^2*d*e*f -
800*b*c*d*e*g))/(10395*e*(b*e - 2*c*d)^6))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x) + (((d*((d*
((64*c^6*(18*b*e*g - 34*c*d*g + c*e*f))/(10395*(b*e - 2*c*d)^6) - (64*c^7*d*g)/(10395*(b*e - 2*c*d)^6)))/e - (
29056*c^7*d^2*g + 7824*b^2*c^5*e^2*g - 2176*c^7*d*e*f + 1152*b*c^6*e^2*f - 30144*b*c^6*d*e*g)/(10395*e*(b*e -
2*c*d)^6)))/e + (16*b*c^4*(227*b^2*e^2*g + 908*c^2*d^2*g + 35*b*c*e^2*f - 68*c^2*d*e*f - 908*b*c*d*e*g))/(1039
5*e*(b*e - 2*c*d)^6))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x) - (((d*((d*((32*c^6*(41*b*e*g - 7
8*c*d*g + 2*c*e*f))/(10395*(b*e - 2*c*d)^6) - (64*c^7*d*g)/(10395*(b*e - 2*c*d)^6)))/e - (39936*c^7*d^2*g + 10
624*b^2*c^5*e^2*g - 2496*c^7*d*e*f + 1312*b*c^6*e^2*f - 41184*b*c^6*d*e*g)/(10395*e*(b*e - 2*c*d)^6)))/e + (32
*b*c^4*(156*b^2*e^2*g + 624*c^2*d^2*g + 20*b*c*e^2*f - 39*c^2*d*e*f - 624*b*c*d*e*g))/(10395*e*(b*e - 2*c*d)^6
))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x) + (((d*((d*((8*c^3*e*(13*b*e*g - 24*c*d*g + c*e*f))/
(99*(7*b*e^2 - 14*c*d*e)*(b*e - 2*c*d)^2) - (8*c^4*d*e*g)/(99*(7*b*e^2 - 14*c*d*e)*(b*e - 2*c*d)^2)))/e - (424
*b^2*c^2*e^3*g + 104*b*c^3*e^3*f - 192*c^4*d*e^2*f + 1496*c^4*d^2*e*g - 1592*b*c^3*d*e^2*g)/(99*e*(7*b*e^2 - 1
4*c*d*e)*(b*e - 2*c*d)^2)))/e + (8*c*(b*e - c*d)*(41*b^2*e^2*g + 164*c^2*d^2*g + 12*b*c*e^2*f - 23*c^2*d*e*f -
164*b*c*d*e*g))/(99*e*(7*b*e^2 - 14*c*d*e)*(b*e - 2*c*d)^2))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d
+ e*x)^4 + (((d*((d*((8*c^3*e*(3*b*e*g - 4*c*d*g + c*e*f))/(99*(7*b*e^2 - 14*c*d*e)*(b*e - 2*c*d)^2) - (8*c^4*
d*e*g)/(99*(7*b*e^2 - 14*c*d*e)*(b*e - 2*c*d)^2)))/e - (64*c^4*d^2*g + 26*b^2*c^2*e^2*g - 32*c^4*d*e*f + 24*b*
c^3*e^2*f - 80*b*c^3*d*e*g)/(99*(7*b*e^2 - 14*c*d*e)*(b*e - 2*c*d)^2)))/e + (2*b*c*(4*b^2*e^2*g + 16*c^2*d^2*g
+ 5*b*c*e^2*f - 8*c^2*d*e*f - 16*b*c*d*e*g))/(99*(7*b*e^2 - 14*c*d*e)*(b*e - 2*c*d)^2))*(c*d^2 - c*e^2*x^2 -
b*d*e - b*e^2*x)^(1/2))/(d + e*x)^4 - (((d*((d*((4*c^3*e*(17*b*e*g - 30*c*d*g + 2*c*e*f))/(99*(7*b*e^2 - 14*c*
d*e)*(b*e - 2*c*d)^2) - (8*c^4*d*e*g)/(99*(7*b*e^2 - 14*c*d*e)*(b*e - 2*c*d)^2)))/e - (416*c^4*d^2*g + 136*b^2
*c^2*e^2*g - 120*c^4*d*e*f + 68*b*c^3*e^2*f - 476*b*c^3*d*e*g)/(99*(7*b*e^2 - 14*c*d*e)*(b*e - 2*c*d)^2)))/e +
(4*b*c*(13*b^2*e^2*g + 52*c^2*d^2*g + 8*b*c*e^2*f - 15*c^2*d*e*f - 52*b*c*d*e*g))/(99*(7*b*e^2 - 14*c*d*e)*(b
*e - 2*c*d)^2))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x)^4 + (((d*((d*((16*c^4*e*(7*b*e*g - 12*c
*d*g + c*e*f))/(693*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d)^3) - (16*c^5*d*e*g)/(693*(5*b*e^2 - 10*c*d*e)*(b*e - 2*
c*d)^3)))/e - (244*b^2*c^3*e^3*g + 112*b*c^4*e^3*f - 192*c^5*d*e^2*f + 768*c^5*d^2*e*g - 864*b*c^4*d*e^2*g)/(6
93*e*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d)^3)))/e + (132*b^2*c^3*e^3*f - 832*c^5*d^3*g + 160*b^3*c^2*e^3*g + 320*
c^5*d^2*e*f - 416*b*c^4*d*e^2*f + 1472*b*c^4*d^2*e*g - 848*b^2*c^3*d*e^2*g)/(693*e*(5*b*e^2 - 10*c*d*e)*(b*e -
2*c*d)^3))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x)^3 - (((d*((d*((8*c^4*e*(23*b*e*g - 42*c*d*g
+ 2*c*e*f))/(693*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d)^3) - (16*c^5*d*e*g)/(693*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*
d)^3)))/e - (712*b^2*c^3*e^3*g + 184*b*c^4*e^3*f - 336*c^5*d*e^2*f + 2496*c^5*d^2*e*g - 2664*b*c^4*d*e^2*g)/(6
93*e*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d)^3)))/e - (1792*c^5*d^3*g - 72*b^2*c^3*e^3*f - 544*b^3*c^2*e^3*g + 64*c
^5*d^2*e*f + 104*b*c^4*d*e^2*f - 3968*b*c^4*d^2*e*g + 2624*b^2*c^3*d*e^2*g)/(693*e*(5*b*e^2 - 10*c*d*e)*(b*e -
2*c*d)^3))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x)^3 - (((d*((d*((16*c^5*e*(13*b*e*g - 22*c*d*
g + 2*c*e*f))/(3465*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^4) - (32*c^6*d*e*g)/(3465*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c
*d)^4)))/e - (488*b^2*c^4*e^3*g + 208*b*c^5*e^3*f - 352*c^6*d*e^2*f + 1568*c^6*d^2*e*g - 1744*b*c^5*d*e^2*g)/(
3465*e*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^4)))/e + (240*b^2*c^4*e^3*f - 1824*c^6*d^3*g + 352*b^3*c^3*e^3*g + 57
6*c^6*d^2*e*f - 752*b*c^5*d*e^2*f + 3232*b*c^5*d^2*e*g - 1864*b^2*c^4*d*e^2*g)/(3465*e*(3*b*e^2 - 6*c*d*e)*(b*
e - 2*c*d)^4))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x)^2 - (((d*((d*((8*c^4*e*(25*b*e*g - 46*c*
d*g + 2*c*e*f))/(693*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d)^3) - (16*c^5*d*e*g)/(693*(5*b*e^2 - 10*c*d*e)*(b*e - 2
*c*d)^3)))/e - (816*b^2*c^3*e^3*g + 200*b*c^4*e^3*f - 368*c^5*d*e^2*f + 2880*c^5*d^2*e*g - 3064*b*c^4*d*e^2*g)
/(693*e*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d)^3)))/e + (352*b^2*c^3*e^3*f - 4352*c^5*d^3*g + 776*b^3*c^2*e^3*g +
1024*c^5*d^2*e*f - 1208*b*c^4*d*e^2*f + 7456*b*c^4*d^2*e*g - 4192*b^2*c^3*d*e^2*g)/(693*e*(5*b*e^2 - 10*c*d*e)
*(b*e - 2*c*d)^3))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x)^3 + (((d*((d*((32*c^5*e*(10*b*e*g -
18*c*d*g + c*e*f))/(3465*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^4) - (32*c^6*d*e*g)/(3465*(3*b*e^2 - 6*c*d*e)*(b*e
- 2*c*d)^4)))/e - (1160*b^2*c^4*e^3*g + 320*b*c^5*e^3*f - 576*c^6*d*e^2*f + 4032*c^6*d^2*e*g - 4320*b*c^5*d*e^
2*g)/(3465*e*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^4)))/e - (2528*c^6*d^3*g - 144*b^2*c^4*e^3*f - 824*b^3*c^3*e^3*
g + 32*c^6*d^2*e*f + 256*b*c^5*d*e^2*f - 5824*b*c^5*d^2*e*g + 3928*b^2*c^4*d*e^2*g)/(3465*e*(3*b*e^2 - 6*c*d*e
)*(b*e - 2*c*d)^4))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x)^2 + (((d*((d*((32*c^5*e*(11*b*e*g -
20*c*d*g + c*e*f))/(3465*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^4) - (32*c^6*d*e*g)/(3465*(3*b*e^2 - 6*c*d*e)*(b*e
- 2*c*d)^4)))/e - (1352*b^2*c^4*e^3*g + 352*b*c^5*e^3*f - 640*c^6*d*e^2*f + 4736*c^6*d^2*e*g - 5056*b*c^5*d*e
^2*g)/(3465*e*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^4)))/e + (528*b^2*c^4*e^3*f - 7008*c^6*d^3*g + 1288*b^3*c^3*e^
3*g + 1440*c^6*d^2*e*f - 1760*b*c^5*d*e^2*f + 12160*b*c^5*d^2*e*g - 6904*b^2*c^4*d*e^2*g)/(3465*e*(3*b*e^2 - 6
*c*d*e)*(b*e - 2*c*d)^4))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x)^2 + (((d*((d*((32*c^5*e*(12*b
*e*g - 22*c*d*g + c*e*f))/(3465*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^4) - (32*c^6*d*e*g)/(3465*(3*b*e^2 - 6*c*d*e
)*(b*e - 2*c*d)^4)))/e - (1544*b^2*c^4*e^3*g + 384*b*c^5*e^3*f - 704*c^6*d*e^2*f + 5440*c^6*d^2*e*g - 5792*b*c
^5*d*e^2*g)/(3465*e*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^4)))/e + (592*b^2*c^4*e^3*f - 8160*c^6*d^3*g + 1496*b^3*
c^3*e^3*g + 1632*c^6*d^2*e*f - 1984*b*c^5*d*e^2*f + 14144*b*c^5*d^2*e*g - 8024*b^2*c^4*d*e^2*g)/(3465*e*(3*b*e
^2 - 6*c*d*e)*(b*e - 2*c*d)^4))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x)^2 + (((400*b^2*c^5*e^3*
f - 3328*c^7*d^3*g + 680*b^3*c^4*e^3*g + 896*c^7*d^2*e*f - 1216*b*c^6*d*e^2*f + 6048*b*c^6*d^2*e*g - 3552*b^2*
c^5*d*e^2*g)/(10395*e^2*(b*e - 2*c*d)^6) + (d*((d*((64*c^6*(6*b*e*g - 10*c*d*g + c*e*f))/(10395*(b*e - 2*c*d)^
6) - (64*c^7*d*g)/(10395*(b*e - 2*c*d)^6)))/e - (32*c^5*(29*b^2*e^2*g + 94*c^2*d^2*g + 12*b*c*e^2*f - 20*c^2*d
*e*f - 104*b*c*d*e*g))/(10395*e*(b*e - 2*c*d)^6)))/e)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x) -
(((352*b^2*c^5*e^3*f - 4160*c^7*d^3*g + 1248*b^3*c^4*e^3*g + 384*c^7*d^2*e*f - 864*b*c^6*d*e^2*f + 9152*b*c^6
*d^2*e*g - 6032*b^2*c^5*d*e^2*g)/(10395*e^2*(b*e - 2*c*d)^6) + (d*((d*((32*c^6*(17*b*e*g - 30*c*d*g + 2*c*e*f)
)/(10395*(b*e - 2*c*d)^6) - (64*c^7*d*g)/(10395*(b*e - 2*c*d)^6)))/e - (16*c^5*(113*b^2*e^2*g + 388*c^2*d^2*g
+ 34*b*c*e^2*f - 60*c^2*d*e*f - 418*b*c*d*e*g))/(10395*e*(b*e - 2*c*d)^6)))/e)*(c*d^2 - c*e^2*x^2 - b*d*e - b*
e^2*x)^(1/2))/(d + e*x) - (((736*b^2*c^5*e^3*f - 9600*c^7*d^3*g + 1912*b^3*c^4*e^3*g + 1792*c^7*d^2*e*f - 2336
*b*c^6*d*e^2*f + 17248*b*c^6*d^2*e*g - 10048*b^2*c^5*d*e^2*g)/(10395*e^2*(b*e - 2*c*d)^6) + (d*((d*((32*c^6*(1
9*b*e*g - 34*c*d*g + 2*c*e*f))/(10395*(b*e - 2*c*d)^6) - (64*c^7*d*g)/(10395*(b*e - 2*c*d)^6)))/e - (16*c^5*(1
35*b^2*e^2*g + 468*c^2*d^2*g + 38*b*c*e^2*f - 68*c^2*d*e*f - 502*b*c*d*e*g))/(10395*e*(b*e - 2*c*d)^6)))/e)*(c
*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x) - (((7616*c^7*d^3*g - 352*b^2*c^5*e^3*f - 2832*b^3*c^4*e^
3*g + 192*c^7*d^2*e*f + 576*b*c^6*d*e^2*f - 18944*b*c^6*d^2*e*g + 13232*b^2*c^5*d*e^2*g)/(10395*e^2*(b*e - 2*c
*d)^6) - (d*((d*((64*c^6*(13*b*e*g - 24*c*d*g + c*e*f))/(10395*(b*e - 2*c*d)^6) - (64*c^7*d*g)/(10395*(b*e - 2
*c*d)^6)))/e - (16*c^5*(257*b^2*e^2*g + 928*c^2*d^2*g + 52*b*c*e^2*f - 96*c^2*d*e*f - 976*b*c*d*e*g))/(10395*e
*(b*e - 2*c*d)^6)))/e)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x) - (((8384*c^7*d^3*g - 352*b^2*c^
5*e^3*f - 3184*b^3*c^4*e^3*g + 320*c^7*d^2*e*f + 512*b*c^6*d*e^2*f - 21120*b*c^6*d^2*e*g + 14832*b^2*c^5*d*e^2
*g)/(10395*e^2*(b*e - 2*c*d)^6) - (d*((d*((64*c^6*(14*b*e*g - 26*c*d*g + c*e*f))/(10395*(b*e - 2*c*d)^6) - (64
*c^7*d*g)/(10395*(b*e - 2*c*d)^6)))/e - (16*c^5*(289*b^2*e^2*g + 1048*c^2*d^2*g + 56*b*c*e^2*f - 104*c^2*d*e*f
- 1100*b*c*d*e*g))/(10395*e*(b*e - 2*c*d)^6)))/e)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x) - ((
(832*b^2*c^5*e^3*f - 11392*c^7*d^3*g + 2264*b^3*c^4*e^3*g + 2048*c^7*d^2*e*f - 2656*b*c^6*d*e^2*f + 20448*b*c^
6*d^2*e*g - 11904*b^2*c^5*d*e^2*g)/(10395*e^2*(b*e - 2*c*d)^6) + (d*((d*((32*c^6*(21*b*e*g - 38*c*d*g + 2*c*e*
f))/(10395*(b*e - 2*c*d)^6) - (64*c^7*d*g)/(10395*(b*e - 2*c*d)^6)))/e - (16*c^5*(157*b^2*e^2*g + 548*c^2*d^2*
g + 42*b*c*e^2*f - 76*c^2*d*e*f - 586*b*c*d*e*g))/(10395*e*(b*e - 2*c*d)^6)))/e)*(c*d^2 - c*e^2*x^2 - b*d*e -
b*e^2*x)^(1/2))/(d + e*x) + (((5504*c^7*d^3*g - 352*b^2*c^5*e^3*f - 4768*b^3*c^4*e^3*g + 896*c^7*d^2*e*f + 224
*b*c^6*d*e^2*f - 24576*b*c^6*d^2*e*g + 20448*b^2*c^5*d*e^2*g)/(10395*e^2*(b*e - 2*c*d)^6) - (d*((d*((32*c^6*(3
7*b*e*g - 70*c*d*g + 2*c*e*f))/(10395*(b*e - 2*c*d)^6) - (64*c^7*d*g)/(10395*(b*e - 2*c*d)^6)))/e - (32*c^5*(2
66*b^2*e^2*g + 992*c^2*d^2*g + 37*b*c*e^2*f - 70*c^2*d*e*f - 1027*b*c*d*e*g))/(10395*e*(b*e - 2*c*d)^6)))/e)*(
c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x) - (((928*b^2*c^5*e^3*f - 13184*c^7*d^3*g + 2616*b^3*c^4*
e^3*g + 2304*c^7*d^2*e*f - 2976*b*c^6*d*e^2*f + 23648*b*c^6*d^2*e*g - 13760*b^2*c^5*d*e^2*g)/(10395*e^2*(b*e -
2*c*d)^6) + (d*((d*((32*c^6*(23*b*e*g - 42*c*d*g + 2*c*e*f))/(10395*(b*e - 2*c*d)^6) - (64*c^7*d*g)/(10395*(b
*e - 2*c*d)^6)))/e - (16*c^5*(179*b^2*e^2*g + 628*c^2*d^2*g + 46*b*c*e^2*f - 84*c^2*d*e*f - 670*b*c*d*e*g))/(1
0395*e*(b*e - 2*c*d)^6)))/e)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x) + (((992*b^2*c^5*e^3*f - 1
3504*c^7*d^3*g + 2992*b^3*c^4*e^3*g + 2496*c^7*d^2*e*f - 3200*b*c^6*d*e^2*f + 25472*b*c^6*d^2*e*g - 15344*b^2*
c^5*d*e^2*g)/(10395*e^2*(b*e - 2*c*d)^6) + (d*((d*((64*c^6*(12*b*e*g - 22*c*d*g + c*e*f))/(10395*(b*e - 2*c*d)
^6) - (64*c^7*d*g)/(10395*(b*e - 2*c*d)^6)))/e - (16*c^5*(225*b^2*e^2*g + 808*c^2*d^2*g + 48*b*c*e^2*f - 88*c^
2*d*e*f - 852*b*c*d*e*g))/(10395*e*(b*e - 2*c*d)^6)))/e)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x
) - (((224*b^2*c^5*e^3*f - 20608*c^7*d^3*g + 5920*b^3*c^4*e^3*g - 1152*c^7*d^2*e*f + 160*b*c^6*d*e^2*f + 44288
*b*c^6*d^2*e*g - 28832*b^2*c^5*d*e^2*g)/(10395*e^2*(b*e - 2*c*d)^6) + (d*((d*((32*c^6*(33*b*e*g - 62*c*d*g + 2
*c*e*f))/(10395*(b*e - 2*c*d)^6) - (64*c^7*d*g)/(10395*(b*e - 2*c*d)^6)))/e - (32*c^5*(216*b^2*e^2*g + 800*c^2
*d^2*g + 33*b*c*e^2*f - 62*c^2*d*e*f - 831*b*c*d*e*g))/(10395*e*(b*e - 2*c*d)^6)))/e)*(c*d^2 - c*e^2*x^2 - b*d
*e - b*e^2*x)^(1/2))/(d + e*x) + (((1168*b^2*c^5*e^3*f - 25728*c^7*d^3*g + 5008*b^3*c^4*e^3*g + 2944*c^7*d^2*e
*f - 3776*b*c^6*d*e^2*f + 45760*b*c^6*d^2*e*g - 26464*b^2*c^5*d*e^2*g)/(10395*e^2*(b*e - 2*c*d)^6) + (d*((d*((
64*c^6*(14*b*e*g - 26*c*d*g + c*e*f))/(10395*(b*e - 2*c*d)^6) - (64*c^7*d*g)/(10395*(b*e - 2*c*d)^6)))/e - (16
*c^5*(297*b^2*e^2*g + 1080*c^2*d^2*g + 56*b*c*e^2*f - 104*c^2*d*e*f - 1132*b*c*d*e*g))/(10395*e*(b*e - 2*c*d)^
6)))/e)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x) + (((1264*b^2*c^5*e^3*f - 29312*c^7*d^3*g + 569
6*b^3*c^4*e^3*g + 3200*c^7*d^2*e*f - 4096*b*c^6*d*e^2*f + 52096*b*c^6*d^2*e*g - 30112*b^2*c^5*d*e^2*g)/(10395*
e^2*(b*e - 2*c*d)^6) + (d*((d*((64*c^6*(15*b*e*g - 28*c*d*g + c*e*f))/(10395*(b*e - 2*c*d)^6) - (64*c^7*d*g)/(
10395*(b*e - 2*c*d)^6)))/e - (16*c^5*(333*b^2*e^2*g + 1216*c^2*d^2*g + 60*b*c*e^2*f - 112*c^2*d*e*f - 1272*b*c
*d*e*g))/(10395*e*(b*e - 2*c*d)^6)))/e)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x) + (((1360*b^2*c
^5*e^3*f - 33920*c^7*d^3*g + 6576*b^3*c^4*e^3*g + 3456*c^7*d^2*e*f - 4416*b*c^6*d*e^2*f + 60224*b*c^6*d^2*e*g
- 34784*b^2*c^5*d*e^2*g)/(10395*e^2*(b*e - 2*c*d)^6) + (d*((d*((64*c^6*(16*b*e*g - 30*c*d*g + c*e*f))/(10395*(
b*e - 2*c*d)^6) - (64*c^7*d*g)/(10395*(b*e - 2*c*d)^6)))/e - (16*c^5*(377*b^2*e^2*g + 1384*c^2*d^2*g + 64*b*c*
e^2*f - 120*c^2*d*e*f - 1444*b*c*d*e*g))/(10395*e*(b*e - 2*c*d)^6)))/e)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^
(1/2))/(d + e*x) - (((2400*b^2*c^5*e^3*f - 40960*c^7*d^3*g + 7744*b^3*c^4*e^3*g + 7424*c^7*d^2*e*f - 8480*b*c^
6*d*e^2*f + 71936*b*c^6*d^2*e*g - 41216*b^2*c^5*d*e^2*g)/(10395*e^2*(b*e - 2*c*d)^6) + (d*((d*((32*c^6*(35*b*e
*g - 66*c*d*g + 2*c*e*f))/(10395*(b*e - 2*c*d)^6) - (64*c^7*d*g)/(10395*(b*e - 2*c*d)^6)))/e - (32*c^5*(239*b^
2*e^2*g + 888*c^2*d^2*g + 35*b*c*e^2*f - 66*c^2*d*e*f - 921*b*c*d*e*g))/(10395*e*(b*e - 2*c*d)^6)))/e)*(c*d^2
- c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x) - (((1696*b^2*c^5*e^3*f - 58112*c^7*d^3*g + 11168*b^3*c^4*e^3*
g + 4352*c^7*d^2*e*f - 5536*b*c^6*d*e^2*f + 102784*b*c^6*d^2*e*g - 59200*b^2*c^5*d*e^2*g)/(10395*e^2*(b*e - 2*
c*d)^6) + (d*((d*((32*c^6*(39*b*e*g - 74*c*d*g + 2*c*e*f))/(10395*(b*e - 2*c*d)^6) - (64*c^7*d*g)/(10395*(b*e
- 2*c*d)^6)))/e - (32*c^5*(297*b^2*e^2*g + 1112*c^2*d^2*g + 39*b*c*e^2*f - 74*c^2*d*e*f - 1149*b*c*d*e*g))/(10
395*e*(b*e - 2*c*d)^6)))/e)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x) + (((d*((d*((4*c^2*e*(8*b*e
*g - 14*c*d*g + c*e*f))/(11*(9*b*e^2 - 18*c*d*e)*(b*e - 2*c*d)) - (4*c^3*d*e*g)/(11*(9*b*e^2 - 18*c*d*e)*(b*e
- 2*c*d))))/e - (32*b*c^2*e^3*f + 52*b^2*c*e^3*g - 56*c^3*d*e^2*f + 148*c^3*d^2*e*g - 176*b*c^2*d*e^2*g)/(11*e
*(9*b*e^2 - 18*c*d*e)*(b*e - 2*c*d))))/e + (4*(b*e - c*d)*(6*b^2*e^2*g + 24*c^2*d^2*g + 7*b*c*e^2*f - 13*c^2*d
*e*f - 24*b*c*d*e*g))/(11*e*(9*b*e^2 - 18*c*d*e)*(b*e - 2*c*d)))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/
(d + e*x)^5 - (((2*b^3*e^2*g + 8*b*c^2*d^2*g + 4*b^2*c*e^2*f - 6*b*c^2*d*e*f - 8*b^2*c*d*e*g)/(11*(9*b*e^2 - 1
8*c*d*e)*(b*e - 2*c*d)) - (d*((16*c^3*d^2*g - 12*c^3*d*e*f + 10*b*c^2*e^2*f + 8*b^2*c*e^2*g - 22*b*c^2*d*e*g)/
(11*(9*b*e^2 - 18*c*d*e)*(b*e - 2*c*d)) - (d*((2*c^2*e*(5*b*e*g - 6*c*d*g + 2*c*e*f))/(11*(9*b*e^2 - 18*c*d*e)
*(b*e - 2*c*d)) - (4*c^3*d*e*g)/(11*(9*b*e^2 - 18*c*d*e)*(b*e - 2*c*d))))/e))/e)*(c*d^2 - c*e^2*x^2 - b*d*e -
b*e^2*x)^(1/2))/(d + e*x)^5 - (((d*((d*((8*c^4*e*(7*b*e*g - 10*c*d*g + 2*c*e*f))/(693*(5*b*e^2 - 10*c*d*e)*(b*
e - 2*c*d)^3) - (16*c^5*d*e*g)/(693*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d)^3)))/e - (208*c^5*d^2*g + 76*b^2*c^3*e^
2*g - 80*c^5*d*e*f + 56*b*c^4*e^2*f - 248*b*c^4*d*e*g)/(693*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d)^3)))/e + (2*b*c
^2*(13*b^2*e^2*g + 52*c^2*d^2*g + 12*b*c*e^2*f - 20*c^2*d*e*f - 52*b*c*d*e*g))/(693*(5*b*e^2 - 10*c*d*e)*(b*e
- 2*c*d)^3))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x)^3 + (((d*((d*((16*c^4*e*(8*b*e*g - 14*c*d*
g + c*e*f))/(693*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d)^3) - (16*c^5*d*e*g)/(693*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d
)^3)))/e - (928*c^5*d^2*g + 292*b^2*c^3*e^2*g - 224*c^5*d*e*f + 128*b*c^4*e^2*f - 1040*b*c^4*d*e*g)/(693*(5*b*
e^2 - 10*c*d*e)*(b*e - 2*c*d)^3)))/e + (4*b*c^2*(29*b^2*e^2*g + 116*c^2*d^2*g + 15*b*c*e^2*f - 28*c^2*d*e*f -
116*b*c*d*e*g))/(693*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d)^3))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d +
e*x)^3 + (((d*((d*((16*c^4*e*(9*b*e*g - 16*c*d*g + c*e*f))/(693*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d)^3) - (16*c^
5*d*e*g)/(693*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d)^3)))/e - (1088*c^5*d^2*g + 340*b^2*c^3*e^2*g - 256*c^5*d*e*f
+ 144*b*c^4*e^2*f - 1216*b*c^4*d*e*g)/(693*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d)^3)))/e + (4*b*c^2*(34*b^2*e^2*g
+ 136*c^2*d^2*g + 17*b*c*e^2*f - 32*c^2*d*e*f - 136*b*c*d*e*g))/(693*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d)^3))*(c
*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x)^3 - (((d*((d*((8*c^4*e*(27*b*e*g - 50*c*d*g + 2*c*e*f))/(
693*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d)^3) - (16*c^5*d*e*g)/(693*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d)^3)))/e - (3
392*c^5*d^2*g + 952*b^2*c^3*e^2*g - 400*c^5*d*e*f + 216*b*c^4*e^2*f - 3592*b*c^4*d*e*g)/(693*(5*b*e^2 - 10*c*d
*e)*(b*e - 2*c*d)^3)))/e + (8*b*c^2*(53*b^2*e^2*g + 212*c^2*d^2*g + 13*b*c*e^2*f - 25*c^2*d*e*f - 212*b*c*d*e*
g))/(693*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d)^3))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x)^3 + (((
d*((d*((32*c^5*e*(4*b*e*g - 6*c*d*g + c*e*f))/(3465*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^4) - (32*c^6*d*e*g)/(346
5*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^4)))/e - (608*c^6*d^2*g + 208*b^2*c^4*e^2*g - 192*c^6*d*e*f + 128*b*c^5*e^
2*f - 704*b*c^5*d*e*g)/(3465*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^4)))/e + (4*b*c^3*(19*b^2*e^2*g + 76*c^2*d^2*g
+ 14*b*c*e^2*f - 24*c^2*d*e*f - 76*b*c*d*e*g))/(3465*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^4))*(c*d^2 - c*e^2*x^2
- b*d*e - b*e^2*x)^(1/2))/(d + e*x)^2 - (((d*((d*((16*c^5*e*(15*b*e*g - 26*c*d*g + 2*c*e*f))/(3465*(3*b*e^2 -
6*c*d*e)*(b*e - 2*c*d)^4) - (32*c^6*d*e*g)/(3465*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^4)))/e - (1952*c^6*d^2*g +
600*b^2*c^4*e^2*g - 416*c^6*d*e*f + 240*b*c^5*e^2*f - 2160*b*c^5*d*e*g)/(3465*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d
)^4)))/e + (4*b*c^3*(61*b^2*e^2*g + 244*c^2*d^2*g + 28*b*c*e^2*f - 52*c^2*d*e*f - 244*b*c*d*e*g))/(3465*(3*b*e
^2 - 6*c*d*e)*(b*e - 2*c*d)^4))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x)^2 - (((d*((d*((16*c^5*e
*(17*b*e*g - 30*c*d*g + 2*c*e*f))/(3465*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^4) - (32*c^6*d*e*g)/(3465*(3*b*e^2 -
6*c*d*e)*(b*e - 2*c*d)^4)))/e - (2336*c^6*d^2*g + 712*b^2*c^4*e^2*g - 480*c^6*d*e*f + 272*b*c^5*e^2*f - 2576*
b*c^5*d*e*g)/(3465*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^4)))/e + (4*b*c^3*(73*b^2*e^2*g + 292*c^2*d^2*g + 32*b*c*
e^2*f - 60*c^2*d*e*f - 292*b*c*d*e*g))/(3465*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^4))*(c*d^2 - c*e^2*x^2 - b*d*e
- b*e^2*x)^(1/2))/(d + e*x)^2 - (((d*((d*((16*c^5*e*(19*b*e*g - 34*c*d*g + 2*c*e*f))/(3465*(3*b*e^2 - 6*c*d*e)
*(b*e - 2*c*d)^4) - (32*c^6*d*e*g)/(3465*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^4)))/e - (2720*c^6*d^2*g + 824*b^2*
c^4*e^2*g - 544*c^6*d*e*f + 304*b*c^5*e^2*f - 2992*b*c^5*d*e*g)/(3465*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^4)))/e
+ (4*b*c^3*(85*b^2*e^2*g + 340*c^2*d^2*g + 36*b*c*e^2*f - 68*c^2*d*e*f - 340*b*c*d*e*g))/(3465*(3*b*e^2 - 6*c
*d*e)*(b*e - 2*c*d)^4))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x)^2 + (((d*((d*((32*c^5*e*(12*b*e
*g - 22*c*d*g + c*e*f))/(3465*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^4) - (32*c^6*d*e*g)/(3465*(3*b*e^2 - 6*c*d*e)*
(b*e - 2*c*d)^4)))/e - (5696*c^6*d^2*g + 1608*b^2*c^4*e^2*g - 704*c^6*d*e*f + 384*b*c^5*e^2*f - 6048*b*c^5*d*e
*g)/(3465*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^4)))/e + (8*b*c^3*(89*b^2*e^2*g + 356*c^2*d^2*g + 23*b*c*e^2*f - 4
4*c^2*d*e*f - 356*b*c*d*e*g))/(3465*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^4))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x
)^(1/2))/(d + e*x)^2 + (((d*((d*((32*c^5*e*(13*b*e*g - 24*c*d*g + c*e*f))/(3465*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c
*d)^4) - (32*c^6*d*e*g)/(3465*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^4)))/e - (6528*c^6*d^2*g + 1832*b^2*c^4*e^2*g
- 768*c^6*d*e*f + 416*b*c^5*e^2*f - 6912*b*c^5*d*e*g)/(3465*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^4)))/e + (8*b*c^
3*(102*b^2*e^2*g + 408*c^2*d^2*g + 25*b*c*e^2*f - 48*c^2*d*e*f - 408*b*c*d*e*g))/(3465*(3*b*e^2 - 6*c*d*e)*(b*
e - 2*c*d)^4))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x)^2 + (((d*((d*((32*c^5*e*(14*b*e*g - 26*c
*d*g + c*e*f))/(3465*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^4) - (32*c^6*d*e*g)/(3465*(3*b*e^2 - 6*c*d*e)*(b*e - 2*
c*d)^4)))/e - (7616*c^6*d^2*g + 2120*b^2*c^4*e^2*g - 832*c^6*d*e*f + 448*b*c^5*e^2*f - 8032*b*c^5*d*e*g)/(3465
*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^4)))/e + (8*b*c^3*(119*b^2*e^2*g + 476*c^2*d^2*g + 27*b*c*e^2*f - 52*c^2*d*
e*f - 476*b*c*d*e*g))/(3465*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^4))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))
/(d + e*x)^2 - (((d*((d*((16*c^5*e*(35*b*e*g - 66*c*d*g + 2*c*e*f))/(3465*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^4)
- (32*c^6*d*e*g)/(3465*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^4)))/e - (13440*c^6*d^2*g + 3632*b^2*c^4*e^2*g - 105
6*c^6*d*e*f + 560*b*c^5*e^2*f - 13968*b*c^5*d*e*g)/(3465*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^4)))/e + (16*b*c^3*
(105*b^2*e^2*g + 420*c^2*d^2*g + 17*b*c*e^2*f - 33*c^2*d*e*f - 420*b*c*d*e*g))/(3465*(3*b*e^2 - 6*c*d*e)*(b*e
- 2*c*d)^4))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x)^2 - (((d*((d*((16*c^5*e*(31*b*e*g - 58*c*d
*g + 2*c*e*f))/(3465*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^4) - (32*c^6*d*e*g)/(3465*(3*b*e^2 - 6*c*d*e)*(b*e - 2*
c*d)^4)))/e - (16*c^4*(177*b^2*e^2*g + 648*c^2*d^2*g + 31*b*c*e^2*f - 58*c^2*d*e*f - 677*b*c*d*e*g))/(3465*(3*
b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^4)))/e - (2176*c^6*d^3*g + 208*b^2*c^4*e^3*f - 1792*b^3*c^3*e^3*g + 1792*c^6*d^
2*e*f - 1328*b*c^5*d*e^2*f - 9344*b*c^5*d^2*e*g + 7712*b^2*c^4*d*e^2*g)/(3465*e*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c
*d)^4))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x)^2 - (((d*((d*((16*c^5*e*(29*b*e*g - 54*c*d*g +
2*c*e*f))/(3465*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^4) - (32*c^6*d*e*g)/(3465*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^
4)))/e - (16*c^4*(158*b^2*e^2*g + 576*c^2*d^2*g + 29*b*c*e^2*f - 54*c^2*d*e*f - 603*b*c*d*e*g))/(3465*(3*b*e^2
- 6*c*d*e)*(b*e - 2*c*d)^4)))/e + (816*b^2*c^4*e^3*f - 9920*c^6*d^3*g + 2096*b^3*c^3*e^3*g + 2368*c^6*d^2*e*f
- 2800*b*c^5*d*e^2*f + 18304*b*c^5*d^2*e*g - 10864*b^2*c^4*d*e^2*g)/(3465*e*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)
^4))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x)^2 - (((d*((d*((16*c^5*e*(33*b*e*g - 62*c*d*g + 2*c
*e*f))/(3465*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^4) - (32*c^6*d*e*g)/(3465*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^4))
)/e - (16*c^4*(200*b^2*e^2*g + 736*c^2*d^2*g + 33*b*c*e^2*f - 62*c^2*d*e*f - 767*b*c*d*e*g))/(3465*(3*b*e^2 -
6*c*d*e)*(b*e - 2*c*d)^4)))/e + (880*b^2*c^4*e^3*f - 22848*c^6*d^3*g + 4016*b^3*c^3*e^3*g + 2496*c^6*d^2*e*f -
2992*b*c^5*d*e^2*f + 38912*b*c^5*d^2*e*g - 21776*b^2*c^4*d*e^2*g)/(3465*e*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^4
))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x)^2 + (((d*((d*((64*c^6*(22*b*e*g - 42*c*d*g + c*e*f))
/(10395*(b*e - 2*c*d)^6) - (64*c^7*d*g)/(10395*(b*e - 2*c*d)^6)))/e - (64*c^5*(196*b^2*e^2*g + 741*c^2*d^2*g +
22*b*c*e^2*f - 42*c^2*d*e*f - 762*b*c*d*e*g))/(10395*e*(b*e - 2*c*d)^6)))/e + (64*c^4*(b*e - c*d)*(175*b^2*e^
2*g + 700*c^2*d^2*g + 21*b*c*e^2*f - 41*c^2*d*e*f - 700*b*c*d*e*g))/(10395*e^2*(b*e - 2*c*d)^6))*(c*d^2 - c*e^
2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x) + (((2*f*(b*e - c*d)^2)/(11*b*e^2 - 22*c*d*e) + (d*((d*((2*c*e*(2*b*
e*g - 2*c*d*g + c*e*f))/(11*b*e^2 - 22*c*d*e) - (2*c^2*d*e*g)/(11*b*e^2 - 22*c*d*e)))/e - (2*(b*e - c*d)*(b*e*
g - c*d*g + 2*c*e*f))/(11*b*e^2 - 22*c*d*e)))/e)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x)^6 - ((
(d*((d*((4*c^3*e*(15*b*e*g - 26*c*d*g + 2*c*e*f))/(99*(7*b*e^2 - 14*c*d*e)*(b*e - 2*c*d)^2) - (8*c^4*d*e*g)/(9
9*(7*b*e^2 - 14*c*d*e)*(b*e - 2*c*d)^2)))/e - (116*b^2*c^2*e^3*g + 60*b*c^3*e^3*f - 104*c^4*d*e^2*f + 352*c^4*
d^2*e*g - 404*b*c^3*d*e^2*g)/(99*e*(7*b*e^2 - 14*c*d*e)*(b*e - 2*c*d)^2)))/e + (68*b^2*c^2*e^3*f - 320*c^4*d^3
*g + 64*b^3*c*e^3*g + 160*c^4*d^2*e*f - 212*b*c^3*d*e^2*f + 576*b*c^3*d^2*e*g - 336*b^2*c^2*d*e^2*g)/(99*e*(7*
b*e^2 - 14*c*d*e)*(b*e - 2*c*d)^2))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x)^4 + (((d*((d*((16*c
^4*e*(17*b*e*g - 32*c*d*g + c*e*f))/(693*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d)^3) - (16*c^5*d*e*g)/(693*(5*b*e^2
- 10*c*d*e)*(b*e - 2*c*d)^3)))/e - (16*c^3*(105*b^2*e^2*g + 387*c^2*d^2*g + 17*b*c*e^2*f - 32*c^2*d*e*f - 403*
b*c*d*e*g))/(693*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d)^3)))/e + (16*c^2*(b*e - c*d)*(89*b^2*e^2*g + 356*c^2*d^2*g
+ 16*b*c*e^2*f - 31*c^2*d*e*f - 356*b*c*d*e*g))/(693*e*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d)^3))*(c*d^2 - c*e^2*
x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x)^3 + (((d*((d*((32*c^5*e*(20*b*e*g - 38*c*d*g + c*e*f))/(3465*(3*b*e^2
- 6*c*d*e)*(b*e - 2*c*d)^4) - (32*c^6*d*e*g)/(3465*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^4)))/e - (32*c^4*(156*b^2
*e^2*g + 585*c^2*d^2*g + 20*b*c*e^2*f - 38*c^2*d*e*f - 604*b*c*d*e*g))/(3465*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)
^4)))/e + (32*c^3*(b*e - c*d)*(137*b^2*e^2*g + 548*c^2*d^2*g + 19*b*c*e^2*f - 37*c^2*d*e*f - 548*b*c*d*e*g))/(
3465*e*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^4))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x)^2

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (- \left (d + e x\right ) \left (b e - c d + c e x\right )\right )^{\frac {3}{2}} \left (f + g x\right )}{\left (d + e x\right )^{8}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)*(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(3/2)/(e*x+d)**8,x)

[Out]

Integral((-(d + e*x)*(b*e - c*d + c*e*x))**(3/2)*(f + g*x)/(d + e*x)**8, x)

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