∫ (f+gx)√ _____________ cd2−bde−be2x−ce2x2 _______________________________________________

3.20.52 $\int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^8} \, dx$

Optimal. Leaf size=439 \[ -\frac {256 c^4 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-13 b e g+16 c d g+10 c e f)}{45045 e^2 (d+e x)^3 (2 c d-b e)^6}+\frac {128 c^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (13 b e g-2 c (8 d g+5 e f))}{15015 e^2 (d+e x)^4 (2 c d-b e)^5}-\frac {32 c^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-13 b e g+16 c d g+10 c e f)}{3003 e^2 (d+e x)^5 (2 c d-b e)^4}+\frac {16 c \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (13 b e g-2 c (8 d g+5 e f))}{1287 e^2 (d+e x)^6 (2 c d-b e)^3}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-13 b e g+16 c d g+10 c e f)}{143 e^2 (d+e x)^7 (2 c d-b e)^2}-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{13 e^2 (d+e x)^8 (2 c d-b e)} \]

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Rubi [A] time = 0.72, antiderivative size = 439, normalized size of antiderivative = 1.00, number of steps used = 6, 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 {256 c^4 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-13 b e g+16 c d g+10 c e f)}{45045 e^2 (d+e x)^3 (2 c d-b e)^6}+\frac {128 c^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (13 b e g-2 c (8 d g+5 e f))}{15015 e^2 (d+e x)^4 (2 c d-b e)^5}-\frac {32 c^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-13 b e g+16 c d g+10 c e f)}{3003 e^2 (d+e x)^5 (2 c d-b e)^4}+\frac {16 c \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (13 b e g-2 c (8 d g+5 e f))}{1287 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 )^{3/2} (-13 b e g+16 c d g+10 c e f)}{143 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 )^{3/2}}{13 e^2 (d+e x)^8 (2 c d-b e)} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

Int[((f + g*x)*Sqrt[c*d^2 - b*d*e - b*e^2*x - c*e^2*x^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)^(3/2))/(13*e^2*(2*c*d - b*e)*(d + e*x)^8) - (2*(10*c*e*f

+ 16*c*d*g - 13*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(143*e^2*(2*c*d - b*e)^2*(d + e*x)^7) + (

16*c*(13*b*e*g - 2*c*(5*e*f + 8*d*g))*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(1287*e^2*(2*c*d - b*e)^3*(

d + e*x)^6) - (32*c^2*(10*c*e*f + 16*c*d*g - 13*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(3003*e^2*

(2*c*d - b*e)^4*(d + e*x)^5) + (128*c^3*(13*b*e*g - 2*c*(5*e*f + 8*d*g))*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)

^(3/2))/(15015*e^2*(2*c*d - b*e)^5*(d + e*x)^4) - (256*c^4*(10*c*e*f + 16*c*d*g - 13*b*e*g)*(d*(c*d - b*e) - b

*e^2*x - c*e^2*x^2)^(3/2))/(45045*e^2*(2*c*d - b*e)^6*(d + e*x)^3)

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) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^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 )^{3/2}}{13 e^2 (2 c d-b e) (d+e x)^8}+\frac {(10 c e f+16 c d g-13 b e g) \int \frac {\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^7} \, dx}{13 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 )^{3/2}}{13 e^2 (2 c d-b e) (d+e x)^8}-\frac {2 (10 c e f+16 c d g-13 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{143 e^2 (2 c d-b e)^2 (d+e x)^7}+\frac {(8 c (10 c e f+16 c d g-13 b e g)) \int \frac {\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^6} \, dx}{143 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 )^{3/2}}{13 e^2 (2 c d-b e) (d+e x)^8}-\frac {2 (10 c e f+16 c d g-13 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{143 e^2 (2 c d-b e)^2 (d+e x)^7}-\frac {16 c (10 c e f+16 c d g-13 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{1287 e^2 (2 c d-b e)^3 (d+e x)^6}+\frac {\left (16 c^2 (10 c e f+16 c d g-13 b e g)\right ) \int \frac {\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^5} \, dx}{429 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 )^{3/2}}{13 e^2 (2 c d-b e) (d+e x)^8}-\frac {2 (10 c e f+16 c d g-13 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{143 e^2 (2 c d-b e)^2 (d+e x)^7}-\frac {16 c (10 c e f+16 c d g-13 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{1287 e^2 (2 c d-b e)^3 (d+e x)^6}-\frac {32 c^2 (10 c e f+16 c d g-13 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3003 e^2 (2 c d-b e)^4 (d+e x)^5}+\frac {\left (64 c^3 (10 c e f+16 c d g-13 b e g)\right ) \int \frac {\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^4} \, dx}{3003 e (2 c d-b e)^4}\\ &=-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{13 e^2 (2 c d-b e) (d+e x)^8}-\frac {2 (10 c e f+16 c d g-13 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{143 e^2 (2 c d-b e)^2 (d+e x)^7}-\frac {16 c (10 c e f+16 c d g-13 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{1287 e^2 (2 c d-b e)^3 (d+e x)^6}-\frac {32 c^2 (10 c e f+16 c d g-13 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3003 e^2 (2 c d-b e)^4 (d+e x)^5}-\frac {128 c^3 (10 c e f+16 c d g-13 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{15015 e^2 (2 c d-b e)^5 (d+e x)^4}+\frac {\left (128 c^4 (10 c e f+16 c d g-13 b e g)\right ) \int \frac {\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^3} \, dx}{15015 e (2 c d-b e)^5}\\ &=-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{13 e^2 (2 c d-b e) (d+e x)^8}-\frac {2 (10 c e f+16 c d g-13 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{143 e^2 (2 c d-b e)^2 (d+e x)^7}-\frac {16 c (10 c e f+16 c d g-13 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{1287 e^2 (2 c d-b e)^3 (d+e x)^6}-\frac {32 c^2 (10 c e f+16 c d g-13 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3003 e^2 (2 c d-b e)^4 (d+e x)^5}-\frac {128 c^3 (10 c e f+16 c d g-13 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{15015 e^2 (2 c d-b e)^5 (d+e x)^4}-\frac {256 c^4 (10 c e f+16 c d g-13 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{45045 e^2 (2 c d-b e)^6 (d+e x)^3}\\ \end {align*}

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Mathematica [A] time = 0.45, size = 176, normalized size = 0.40 \begin {gather*} -\frac {2 ((d+e x) (c (d-e x)-b e))^{3/2} \left (2 (d+e x) \left (8 c (d+e x) \left (2 c (d+e x) \left (4 c (d+e x) (-3 b e+8 c d+2 c e x)+15 (b e-2 c d)^2\right )+35 (2 c d-b e)^3\right )+315 (b e-2 c d)^4\right ) \left (c e (8 d g+5 e f)-\frac {13}{2} b e^2 g\right )-3465 e (b e-2 c d)^5 (e f-d g)\right )}{45045 e^3 (d+e x)^8 (b e-2 c d)^6} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-2*((d + e*x)*(-(b*e) + c*(d - e*x)))^(3/2)*(-3465*e*(-2*c*d + b*e)^5*(e*f - d*g) + 2*((-13*b*e^2*g)/2 + c*e*

(5*e*f + 8*d*g))*(d + e*x)*(315*(-2*c*d + b*e)^4 + 8*c*(d + e*x)*(35*(2*c*d - b*e)^3 + 2*c*(d + e*x)*(15*(-2*c

*d + b*e)^2 + 4*c*(d + e*x)*(8*c*d - 3*b*e + 2*c*e*x))))))/(45045*e^3*(-2*c*d + b*e)^6*(d + e*x)^8)

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IntegrateAlgebraic [F] time = 180.04, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}

Veriﬁcation is not applicable to the result.

[In]

IntegrateAlgebraic[((f + g*x)*Sqrt[c*d^2 - b*d*e - b*e^2*x - c*e^2*x^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)^(1/2)/(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)^(1/2)/(e*x+d)^8,x, algorithm="giac")

[Out]

Timed out

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maple [A] time = 0.07, size = 782, normalized size = 1.78 \begin {gather*} -\frac {2 \left (c e x +b e -c d \right ) \left (1664 b \,c^{4} e^{6} g \,x^{5}-2048 c^{5} d \,e^{5} g \,x^{5}-1280 c^{5} e^{6} f \,x^{5}-2496 b^{2} c^{3} e^{6} g \,x^{4}+16384 b \,c^{4} d \,e^{5} g \,x^{4}+1920 b \,c^{4} e^{6} f \,x^{4}-16384 c^{5} d^{2} e^{4} g \,x^{4}-10240 c^{5} d \,e^{5} f \,x^{4}+3120 b^{3} c^{2} e^{6} g \,x^{3}-26304 b^{2} c^{3} d \,e^{5} g \,x^{3}-2400 b^{2} c^{3} e^{6} f \,x^{3}+76736 b \,c^{4} d^{2} e^{4} g \,x^{3}+17280 b \,c^{4} d \,e^{5} f \,x^{3}-60416 c^{5} d^{3} e^{3} g \,x^{3}-37760 c^{5} d^{2} e^{4} f \,x^{3}-3640 b^{4} c \,e^{6} g \,x^{2}+35680 b^{3} c^{2} d \,e^{5} g \,x^{2}+2800 b^{3} c^{2} e^{6} f \,x^{2}-134496 b^{2} c^{3} d^{2} e^{4} g \,x^{2}-24000 b^{2} c^{3} d \,e^{5} f \,x^{2}+231424 b \,c^{4} d^{3} e^{3} g \,x^{2}+73920 b \,c^{4} d^{2} e^{4} f \,x^{2}-139264 c^{5} d^{4} e^{2} g \,x^{2}-87040 c^{5} d^{3} e^{3} f \,x^{2}+4095 b^{5} e^{6} g x -45080 b^{4} c d \,e^{5} g x -3150 b^{4} c \,e^{6} f x +200600 b^{3} c^{2} d^{2} e^{4} g x +30800 b^{3} c^{2} d \,e^{5} f x -452064 b^{2} c^{3} d^{3} e^{3} g x -116400 b^{2} c^{3} d^{2} e^{4} f x +516656 b \,c^{4} d^{4} e^{2} g x +204480 b \,c^{4} d^{3} e^{3} f x -233216 c^{5} d^{5} e g x -145760 c^{5} d^{4} e^{2} f x +630 b^{5} d \,e^{5} g +3465 b^{5} e^{6} f -6790 b^{4} c \,d^{2} e^{4} g -37800 b^{4} c d \,e^{5} f +29440 b^{3} c^{2} d^{3} e^{3} g +166600 b^{3} c^{2} d^{2} e^{4} f -64176 b^{2} c^{3} d^{4} e^{2} g -372000 b^{2} c^{3} d^{3} e^{3} f +70048 b \,c^{4} d^{5} e g +423120 b \,c^{4} d^{4} e^{2} f -29152 c^{5} d^{6} g -198400 c^{5} d^{5} e f \right ) \sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}}{45045 \left (e x +d \right )^{7} \left (b^{6} e^{6}-12 b^{5} c d \,e^{5}+60 b^{4} c^{2} d^{2} e^{4}-160 b^{3} c^{3} d^{3} e^{3}+240 b^{2} c^{4} d^{4} e^{2}-192 b \,c^{5} d^{5} e +64 c^{6} d^{6}\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)^(1/2)/(e*x+d)^8,x)

[Out]

-2/45045*(c*e*x+b*e-c*d)*(1664*b*c^4*e^6*g*x^5-2048*c^5*d*e^5*g*x^5-1280*c^5*e^6*f*x^5-2496*b^2*c^3*e^6*g*x^4+

16384*b*c^4*d*e^5*g*x^4+1920*b*c^4*e^6*f*x^4-16384*c^5*d^2*e^4*g*x^4-10240*c^5*d*e^5*f*x^4+3120*b^3*c^2*e^6*g*

x^3-26304*b^2*c^3*d*e^5*g*x^3-2400*b^2*c^3*e^6*f*x^3+76736*b*c^4*d^2*e^4*g*x^3+17280*b*c^4*d*e^5*f*x^3-60416*c

^5*d^3*e^3*g*x^3-37760*c^5*d^2*e^4*f*x^3-3640*b^4*c*e^6*g*x^2+35680*b^3*c^2*d*e^5*g*x^2+2800*b^3*c^2*e^6*f*x^2

-134496*b^2*c^3*d^2*e^4*g*x^2-24000*b^2*c^3*d*e^5*f*x^2+231424*b*c^4*d^3*e^3*g*x^2+73920*b*c^4*d^2*e^4*f*x^2-1

39264*c^5*d^4*e^2*g*x^2-87040*c^5*d^3*e^3*f*x^2+4095*b^5*e^6*g*x-45080*b^4*c*d*e^5*g*x-3150*b^4*c*e^6*f*x+2006

00*b^3*c^2*d^2*e^4*g*x+30800*b^3*c^2*d*e^5*f*x-452064*b^2*c^3*d^3*e^3*g*x-116400*b^2*c^3*d^2*e^4*f*x+516656*b*

c^4*d^4*e^2*g*x+204480*b*c^4*d^3*e^3*f*x-233216*c^5*d^5*e*g*x-145760*c^5*d^4*e^2*f*x+630*b^5*d*e^5*g+3465*b^5*

e^6*f-6790*b^4*c*d^2*e^4*g-37800*b^4*c*d*e^5*f+29440*b^3*c^2*d^3*e^3*g+166600*b^3*c^2*d^2*e^4*f-64176*b^2*c^3*

d^4*e^2*g-372000*b^2*c^3*d^3*e^3*f+70048*b*c^4*d^5*e*g+423120*b*c^4*d^4*e^2*f-29152*c^5*d^6*g-198400*c^5*d^5*e

*f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)/(e*x+d)^7/e^2/(b^6*e^6-12*b^5*c*d*e^5+60*b^4*c^2*d^2*e^4-160*b^3*c^

3*d^3*e^3+240*b^2*c^4*d^4*e^2-192*b*c^5*d^5*e+64*c^6*d^6)

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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)^(1/2)/(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 = 55.43, size = 19572, normalized size = 44.58

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)^(1/2))/(d + e*x)^8,x)

[Out]

(((d*((8*c^2*(14*b*e*g - 27*c*d*g + c*e*f))/(143*(9*b*e^2 - 18*c*d*e)*(b*e - 2*c*d)^2) - (8*c^3*d*g)/(143*(9*b

*e^2 - 18*c*d*e)*(b*e - 2*c*d)^2)))/e - (8*c*(b*e - c*d)*(13*b*e*g - 26*c*d*g + c*e*f))/(143*e*(9*b*e^2 - 18*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)^5 - (((2*f*(b*e - c*d))/(13*b*e

^2 - 26*c*d*e) - (d*((2*b*e*g - 2*c*d*g + 2*c*e*f)/(13*b*e^2 - 26*c*d*e) - (2*c*d*g)/(13*b*e^2 - 26*c*d*e)))/e

)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x)^7 - (((d*((4*c^2*(15*b*e*g - 28*c*d*g + 2*c*e*f))/(14

3*(9*b*e^2 - 18*c*d*e)*(b*e - 2*c*d)^2) - (8*c^3*d*g)/(143*(9*b*e^2 - 18*c*d*e)*(b*e - 2*c*d)^2)))/e - (144*c^

3*d^2*g - 48*c^3*d*e*f + 28*b*c^2*e^2*f + 52*b^2*c*e^2*g - 176*b*c^2*d*e*g)/(143*e*(9*b*e^2 - 18*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)^5 - (((d*((4*c^2*e*f - 8*c^2*d*g + 6*b*c*e*

g)/(13*(11*b*e^2 - 22*c*d*e)*(b*e - 2*c*d)) - (4*c^2*d*g)/(13*(11*b*e^2 - 22*c*d*e)*(b*e - 2*c*d))))/e - (2*b*

(b*e*g - 2*c*d*g + c*e*f))/(13*(11*b*e^2 - 22*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)^6 - (((d*((16*c^4*e*f - 64*c^4*d*g + 40*b*c^3*e*g)/(1287*(7*b*e^2 - 14*c*d*e)*(b*e - 2*c*d)^3) -

(16*c^4*d*g)/(1287*(7*b*e^2 - 14*c*d*e)*(b*e - 2*c*d)^3)))/e - (8*b*c^2*(2*b*e*g - 4*c*d*g + c*e*f))/(1287*(7

*b*e^2 - 14*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)^4 + (((d*((16*c^4*

e*f - 240*c^4*d*g + 128*b*c^3*e*g)/(1287*(7*b*e^2 - 14*c*d*e)*(b*e - 2*c*d)^3) - (16*c^4*d*g)/(1287*(7*b*e^2 -

14*c*d*e)*(b*e - 2*c*d)^3)))/e - (4*b*c^2*(15*b*e*g - 30*c*d*g + 2*c*e*f))/(1287*(7*b*e^2 - 14*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)^4 + (((d*((16*c^4*e*f - 272*c^4*d*g + 144*b*

c^3*e*g)/(1287*(7*b*e^2 - 14*c*d*e)*(b*e - 2*c*d)^3) - (16*c^4*d*g)/(1287*(7*b*e^2 - 14*c*d*e)*(b*e - 2*c*d)^3

)))/e - (4*b*c^2*(17*b*e*g - 34*c*d*g + 2*c*e*f))/(1287*(7*b*e^2 - 14*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)^4 - (((d*((16*c^4*e*f - 448*c^4*d*g + 232*b*c^3*e*g)/(1287*(7*b*e^2 -

14*c*d*e)*(b*e - 2*c*d)^3) - (16*c^4*d*g)/(1287*(7*b*e^2 - 14*c*d*e)*(b*e - 2*c*d)^3)))/e - (8*b*c^2*(14*b*e*g

- 28*c*d*g + c*e*f))/(1287*(7*b*e^2 - 14*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)^4 + (((d*((32*c^5*e*f - 160*c^5*d*g + 96*b*c^4*e*g)/(9009*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d)^4) -

(32*c^5*d*g)/(9009*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d)^4)))/e - (8*b*c^3*(5*b*e*g - 10*c*d*g + 2*c*e*f))/(9009*

(5*b*e^2 - 10*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)^3 - (((d*((32*c^

5*e*f - 448*c^5*d*g + 240*b*c^4*e*g)/(9009*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d)^4) - (32*c^5*d*g)/(9009*(5*b*e^2

- 10*c*d*e)*(b*e - 2*c*d)^4)))/e - (16*b*c^3*(7*b*e*g - 14*c*d*g + c*e*f))/(9009*(5*b*e^2 - 10*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)^3 - (((d*((32*c^5*e*f - 512*c^5*d*g + 272*b*

c^4*e*g)/(9009*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d)^4) - (32*c^5*d*g)/(9009*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d)^4

)))/e - (16*b*c^3*(8*b*e*g - 16*c*d*g + c*e*f))/(9009*(5*b*e^2 - 10*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)^3 - (((d*((32*c^5*e*f - 576*c^5*d*g + 304*b*c^4*e*g)/(9009*(5*b*e^2 - 10

*c*d*e)*(b*e - 2*c*d)^4) - (32*c^5*d*g)/(9009*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d)^4)))/e - (16*b*c^3*(9*b*e*g -

18*c*d*g + c*e*f))/(9009*(5*b*e^2 - 10*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)^3 + (((d*((32*c^5*e*f - 800*c^5*d*g + 416*b*c^4*e*g)/(9009*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d)^4) - (

32*c^5*d*g)/(9009*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d)^4)))/e - (8*b*c^3*(25*b*e*g - 50*c*d*g + 2*c*e*f))/(9009*

(5*b*e^2 - 10*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)^3 + (((d*((32*c^

5*e*f - 864*c^5*d*g + 448*b*c^4*e*g)/(9009*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d)^4) - (32*c^5*d*g)/(9009*(5*b*e^2

- 10*c*d*e)*(b*e - 2*c*d)^4)))/e - (8*b*c^3*(27*b*e*g - 54*c*d*g + 2*c*e*f))/(9009*(5*b*e^2 - 10*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)^3 + (((d*((32*c^5*e*f - 928*c^5*d*g + 480*

b*c^4*e*g)/(9009*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d)^4) - (32*c^5*d*g)/(9009*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d)

^4)))/e - (8*b*c^3*(29*b*e*g - 58*c*d*g + 2*c*e*f))/(9009*(5*b*e^2 - 10*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)^3 - (((d*((32*c^5*e*f - 1216*c^5*d*g + 624*b*c^4*e*g)/(9009*(5*b*e^2

- 10*c*d*e)*(b*e - 2*c*d)^4) - (32*c^5*d*g)/(9009*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d)^4)))/e - (16*b*c^3*(19*b

*e*g - 38*c*d*g + c*e*f))/(9009*(5*b*e^2 - 10*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)^3 - (((d*((64*c^6*e*f - 384*c^6*d*g + 224*b*c^5*e*g)/(45045*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^

5) - (64*c^6*d*g)/(45045*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^5)))/e - (32*b*c^4*(3*b*e*g - 6*c*d*g + c*e*f))/(45

045*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^5))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x)^2 - (((d*((64

*c^6*e*f - 1408*c^6*d*g + 736*b*c^5*e*g)/(45045*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^5) - (64*c^6*d*g)/(45045*(3*

b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^5)))/e - (32*b*c^4*(11*b*e*g - 22*c*d*g + c*e*f))/(45045*(3*b*e^2 - 6*c*d*e)*(b

*e - 2*c*d)^5))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x)^2 - (((d*((64*c^6*e*f - 1536*c^6*d*g +

800*b*c^5*e*g)/(45045*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^5) - (64*c^6*d*g)/(45045*(3*b*e^2 - 6*c*d*e)*(b*e - 2*

c*d)^5)))/e - (32*b*c^4*(12*b*e*g - 24*c*d*g + c*e*f))/(45045*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^5))*(c*d^2 - c

*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x)^2 - (2*((d*((64*c^6*e*f - 1664*c^6*d*g + 864*b*c^5*e*g)/(45045*(3

*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^5) - (64*c^6*d*g)/(45045*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^5)))/e - (32*b*c^4*

(13*b*e*g - 26*c*d*g + c*e*f))/(45045*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^5))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2

*x)^(1/2))/(d + e*x)^2 - (((d*((64*c^6*e*f - 1792*c^6*d*g + 928*b*c^5*e*g)/(45045*(3*b*e^2 - 6*c*d*e)*(b*e - 2

*c*d)^5) - (64*c^6*d*g)/(45045*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^5)))/e - (32*b*c^4*(14*b*e*g - 28*c*d*g + c*e

*f))/(45045*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^5))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x)^2 - (

((d*((64*c^6*e*f - 1920*c^6*d*g + 992*b*c^5*e*g)/(45045*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^5) - (64*c^6*d*g)/(4

5045*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^5)))/e - (32*b*c^4*(15*b*e*g - 30*c*d*g + c*e*f))/(45045*(3*b*e^2 - 6*c

*d*e)*(b*e - 2*c*d)^5))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x)^2 - (((d*((64*c^6*e*f - 2944*c^

6*d*g + 1504*b*c^5*e*g)/(45045*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^5) - (64*c^6*d*g)/(45045*(3*b*e^2 - 6*c*d*e)*

(b*e - 2*c*d)^5)))/e - (32*b*c^4*(23*b*e*g - 46*c*d*g + c*e*f))/(45045*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^5))*(

c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x)^2 - (((d*((64*c^6*(13*b*e*g - 24*c*d*g + 2*c*e*f))/(1351

35*e*(b*e - 2*c*d)^7) - (128*c^7*d*g)/(135135*e*(b*e - 2*c*d)^7)))/e - (64*b*c^5*(6*b*e*g - 12*c*d*g + c*e*f))

/(135135*e*(b*e - 2*c*d)^7))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x) - (((d*((64*c^6*(15*b*e*g

- 28*c*d*g + 2*c*e*f))/(135135*e*(b*e - 2*c*d)^7) - (128*c^7*d*g)/(135135*e*(b*e - 2*c*d)^7)))/e - (64*b*c^5*(

7*b*e*g - 14*c*d*g + c*e*f))/(135135*e*(b*e - 2*c*d)^7))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x

) + (((d*((128*c^6*(4*b*e*g - 7*c*d*g + c*e*f))/(135135*e*(b*e - 2*c*d)^7) - (128*c^7*d*g)/(135135*e*(b*e - 2*

c*d)^7)))/e - (32*b*c^5*(7*b*e*g - 14*c*d*g + 2*c*e*f))/(135135*e*(b*e - 2*c*d)^7))*(c*d^2 - c*e^2*x^2 - b*d*e

- b*e^2*x)^(1/2))/(d + e*x) - (((d*((64*c^6*(17*b*e*g - 32*c*d*g + 2*c*e*f))/(135135*e*(b*e - 2*c*d)^7) - (12

8*c^7*d*g)/(135135*e*(b*e - 2*c*d)^7)))/e - (64*b*c^5*(8*b*e*g - 16*c*d*g + c*e*f))/(135135*e*(b*e - 2*c*d)^7)

)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x) - (((d*((64*c^6*(19*b*e*g - 36*c*d*g + 2*c*e*f))/(135

135*e*(b*e - 2*c*d)^7) - (128*c^7*d*g)/(135135*e*(b*e - 2*c*d)^7)))/e - (64*b*c^5*(9*b*e*g - 18*c*d*g + c*e*f)

)/(135135*e*(b*e - 2*c*d)^7))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x) - (((d*((64*c^6*(21*b*e*g

- 40*c*d*g + 2*c*e*f))/(135135*e*(b*e - 2*c*d)^7) - (128*c^7*d*g)/(135135*e*(b*e - 2*c*d)^7)))/e - (64*b*c^5*

(10*b*e*g - 20*c*d*g + c*e*f))/(135135*e*(b*e - 2*c*d)^7))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e

*x) + (((d*((128*c^6*(10*b*e*g - 19*c*d*g + c*e*f))/(135135*e*(b*e - 2*c*d)^7) - (128*c^7*d*g)/(135135*e*(b*e

- 2*c*d)^7)))/e - (32*b*c^5*(19*b*e*g - 38*c*d*g + 2*c*e*f))/(135135*e*(b*e - 2*c*d)^7))*(c*d^2 - c*e^2*x^2 -

b*d*e - b*e^2*x)^(1/2))/(d + e*x) - (((d*((64*c^6*(29*b*e*g - 56*c*d*g + 2*c*e*f))/(135135*e*(b*e - 2*c*d)^7)

- (128*c^7*d*g)/(135135*e*(b*e - 2*c*d)^7)))/e - (64*b*c^5*(14*b*e*g - 28*c*d*g + c*e*f))/(135135*e*(b*e - 2*c

*d)^7))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x) + (((d*((128*c^6*(11*b*e*g - 21*c*d*g + c*e*f))

/(135135*e*(b*e - 2*c*d)^7) - (128*c^7*d*g)/(135135*e*(b*e - 2*c*d)^7)))/e - (32*b*c^5*(21*b*e*g - 42*c*d*g +

2*c*e*f))/(135135*e*(b*e - 2*c*d)^7))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x) - (((d*((64*c^6*(

31*b*e*g - 60*c*d*g + 2*c*e*f))/(135135*e*(b*e - 2*c*d)^7) - (128*c^7*d*g)/(135135*e*(b*e - 2*c*d)^7)))/e - (6

4*b*c^5*(15*b*e*g - 30*c*d*g + c*e*f))/(135135*e*(b*e - 2*c*d)^7))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2)

)/(d + e*x) + (2*((d*((128*c^6*(12*b*e*g - 23*c*d*g + c*e*f))/(135135*e*(b*e - 2*c*d)^7) - (128*c^7*d*g)/(1351

35*e*(b*e - 2*c*d)^7)))/e - (32*b*c^5*(23*b*e*g - 46*c*d*g + 2*c*e*f))/(135135*e*(b*e - 2*c*d)^7))*(c*d^2 - c*

e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x) - (2*((d*((64*c^6*(33*b*e*g - 64*c*d*g + 2*c*e*f))/(135135*e*(b*e

- 2*c*d)^7) - (128*c^7*d*g)/(135135*e*(b*e - 2*c*d)^7)))/e - (64*b*c^5*(16*b*e*g - 32*c*d*g + c*e*f))/(135135*

e*(b*e - 2*c*d)^7))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x) + (2*((d*((128*c^6*(13*b*e*g - 25*c

*d*g + c*e*f))/(135135*e*(b*e - 2*c*d)^7) - (128*c^7*d*g)/(135135*e*(b*e - 2*c*d)^7)))/e - (32*b*c^5*(25*b*e*g

- 50*c*d*g + 2*c*e*f))/(135135*e*(b*e - 2*c*d)^7))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x) + (

2*((d*((128*c^6*(14*b*e*g - 27*c*d*g + c*e*f))/(135135*e*(b*e - 2*c*d)^7) - (128*c^7*d*g)/(135135*e*(b*e - 2*c

*d)^7)))/e - (32*b*c^5*(27*b*e*g - 54*c*d*g + 2*c*e*f))/(135135*e*(b*e - 2*c*d)^7))*(c*d^2 - c*e^2*x^2 - b*d*e

- b*e^2*x)^(1/2))/(d + e*x) - (2*((d*((64*c^6*(35*b*e*g - 68*c*d*g + 2*c*e*f))/(135135*e*(b*e - 2*c*d)^7) - (

128*c^7*d*g)/(135135*e*(b*e - 2*c*d)^7)))/e - (64*b*c^5*(17*b*e*g - 34*c*d*g + c*e*f))/(135135*e*(b*e - 2*c*d)

^7))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x) + (((d*((128*c^6*(15*b*e*g - 29*c*d*g + c*e*f))/(1

35135*e*(b*e - 2*c*d)^7) - (128*c^7*d*g)/(135135*e*(b*e - 2*c*d)^7)))/e - (32*b*c^5*(29*b*e*g - 58*c*d*g + 2*c

*e*f))/(135135*e*(b*e - 2*c*d)^7))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x) - (2*((d*((64*c^6*(3

7*b*e*g - 72*c*d*g + 2*c*e*f))/(135135*e*(b*e - 2*c*d)^7) - (128*c^7*d*g)/(135135*e*(b*e - 2*c*d)^7)))/e - (64

*b*c^5*(18*b*e*g - 36*c*d*g + c*e*f))/(135135*e*(b*e - 2*c*d)^7))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))

/(d + e*x) + (((d*((128*c^6*(16*b*e*g - 31*c*d*g + c*e*f))/(135135*e*(b*e - 2*c*d)^7) - (128*c^7*d*g)/(135135*

e*(b*e - 2*c*d)^7)))/e - (32*b*c^5*(31*b*e*g - 62*c*d*g + 2*c*e*f))/(135135*e*(b*e - 2*c*d)^7))*(c*d^2 - c*e^2

*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x) - (((d*((64*c^6*(39*b*e*g - 76*c*d*g + 2*c*e*f))/(135135*e*(b*e - 2*c

*d)^7) - (128*c^7*d*g)/(135135*e*(b*e - 2*c*d)^7)))/e - (64*b*c^5*(19*b*e*g - 38*c*d*g + c*e*f))/(135135*e*(b*

e - 2*c*d)^7))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x) - (((d*((64*c^6*(41*b*e*g - 80*c*d*g + 2

*c*e*f))/(135135*e*(b*e - 2*c*d)^7) - (128*c^7*d*g)/(135135*e*(b*e - 2*c*d)^7)))/e - (64*b*c^5*(20*b*e*g - 40*

c*d*g + c*e*f))/(135135*e*(b*e - 2*c*d)^7))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x) + (((d*((12

8*c^6*(20*b*e*g - 39*c*d*g + c*e*f))/(135135*e*(b*e - 2*c*d)^7) - (128*c^7*d*g)/(135135*e*(b*e - 2*c*d)^7)))/e

- (32*b*c^5*(39*b*e*g - 78*c*d*g + 2*c*e*f))/(135135*e*(b*e - 2*c*d)^7))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x

)^(1/2))/(d + e*x) + (((d*((128*c^6*(21*b*e*g - 41*c*d*g + c*e*f))/(135135*e*(b*e - 2*c*d)^7) - (128*c^7*d*g)/

(135135*e*(b*e - 2*c*d)^7)))/e - (32*b*c^5*(41*b*e*g - 82*c*d*g + 2*c*e*f))/(135135*e*(b*e - 2*c*d)^7))*(c*d^2

- c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x) + (((d*((128*c^6*(22*b*e*g - 43*c*d*g + c*e*f))/(135135*e*(b*

e - 2*c*d)^7) - (128*c^7*d*g)/(135135*e*(b*e - 2*c*d)^7)))/e - (32*b*c^5*(43*b*e*g - 86*c*d*g + 2*c*e*f))/(135

135*e*(b*e - 2*c*d)^7))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x) + (((d*((128*c^6*(23*b*e*g - 45

*c*d*g + c*e*f))/(135135*e*(b*e - 2*c*d)^7) - (128*c^7*d*g)/(135135*e*(b*e - 2*c*d)^7)))/e - (32*b*c^5*(45*b*e

*g - 90*c*d*g + 2*c*e*f))/(135135*e*(b*e - 2*c*d)^7))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x) -

(((d*((64*c^6*(53*b*e*g - 104*c*d*g + 2*c*e*f))/(135135*e*(b*e - 2*c*d)^7) - (128*c^7*d*g)/(135135*e*(b*e - 2

*c*d)^7)))/e - (64*b*c^5*(26*b*e*g - 52*c*d*g + c*e*f))/(135135*e*(b*e - 2*c*d)^7))*(c*d^2 - c*e^2*x^2 - b*d*e

- b*e^2*x)^(1/2))/(d + e*x) + (((d*((128*c^6*(24*b*e*g - 47*c*d*g + c*e*f))/(135135*e*(b*e - 2*c*d)^7) - (128

*c^7*d*g)/(135135*e*(b*e - 2*c*d)^7)))/e - (32*b*c^5*(47*b*e*g - 94*c*d*g + 2*c*e*f))/(135135*e*(b*e - 2*c*d)^

7))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x) + (((d*((4*c*(8*b*e*g - 15*c*d*g + c*e*f))/(13*(11*

b*e^2 - 22*c*d*e)*(b*e - 2*c*d)) - (4*c^2*d*g)/(13*(11*b*e^2 - 22*c*d*e)*(b*e - 2*c*d))))/e - (4*(b*e - c*d)*(

7*b*e*g - 14*c*d*g + c*e*f))/(13*e*(11*b*e^2 - 22*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)^6 + (((d*((64*c^5*(7*b*e*g - 13*c*d*g + c*e*f))/(45045*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^5)

- (64*c^6*d*g)/(45045*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^5)))/e - (16*b*c^4*(13*b*e*g - 26*c*d*g + 2*c*e*f))/(4

5045*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^5))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x)^2 + (((d*((6

4*c^5*(8*b*e*g - 15*c*d*g + c*e*f))/(45045*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^5) - (64*c^6*d*g)/(45045*(3*b*e^2

- 6*c*d*e)*(b*e - 2*c*d)^5)))/e - (16*b*c^4*(15*b*e*g - 30*c*d*g + 2*c*e*f))/(45045*(3*b*e^2 - 6*c*d*e)*(b*e

- 2*c*d)^5))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x)^2 + (((d*((64*c^5*(9*b*e*g - 17*c*d*g + c*

e*f))/(45045*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^5) - (64*c^6*d*g)/(45045*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^5)))

/e - (16*b*c^4*(17*b*e*g - 34*c*d*g + 2*c*e*f))/(45045*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^5))*(c*d^2 - c*e^2*x^

2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x)^2 + (((d*((64*c^5*(10*b*e*g - 19*c*d*g + c*e*f))/(45045*(3*b*e^2 - 6*c*d

*e)*(b*e - 2*c*d)^5) - (64*c^6*d*g)/(45045*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^5)))/e - (16*b*c^4*(19*b*e*g - 38

*c*d*g + 2*c*e*f))/(45045*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^5))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(

d + e*x)^2 + (((d*((64*c^5*(17*b*e*g - 33*c*d*g + c*e*f))/(45045*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^5) - (64*c^

6*d*g)/(45045*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^5)))/e - (16*b*c^4*(33*b*e*g - 66*c*d*g + 2*c*e*f))/(45045*(3*

b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^5))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x)^2 + (((d*((64*c^5*(1

8*b*e*g - 35*c*d*g + c*e*f))/(45045*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^5) - (64*c^6*d*g)/(45045*(3*b*e^2 - 6*c*

d*e)*(b*e - 2*c*d)^5)))/e - (16*b*c^4*(35*b*e*g - 70*c*d*g + 2*c*e*f))/(45045*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d

)^5))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x)^2 + (((d*((64*c^5*(19*b*e*g - 37*c*d*g + c*e*f))/

(45045*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^5) - (64*c^6*d*g)/(45045*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^5)))/e - (

16*b*c^4*(37*b*e*g - 74*c*d*g + 2*c*e*f))/(45045*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^5))*(c*d^2 - c*e^2*x^2 - b*

d*e - b*e^2*x)^(1/2))/(d + e*x)^2 + (((d*((64*c^5*(20*b*e*g - 39*c*d*g + c*e*f))/(45045*(3*b*e^2 - 6*c*d*e)*(b

*e - 2*c*d)^5) - (64*c^6*d*g)/(45045*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^5)))/e - (16*b*c^4*(39*b*e*g - 78*c*d*g

+ 2*c*e*f))/(45045*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^5))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*

x)^2 + (((d*((16*c^3*(7*b*e*g - 13*c*d*g + c*e*f))/(1287*(7*b*e^2 - 14*c*d*e)*(b*e - 2*c*d)^3) - (16*c^4*d*g)/

(1287*(7*b*e^2 - 14*c*d*e)*(b*e - 2*c*d)^3)))/e - (320*c^4*d^2*g + 112*b^2*c^2*e^2*g - 80*c^4*d*e*f + 48*b*c^3

*e^2*f - 384*b*c^3*d*e*g)/(1287*e*(7*b*e^2 - 14*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)^4 - (((d*((8*c^3*(25*b*e*g - 48*c*d*g + 2*c*e*f))/(1287*(7*b*e^2 - 14*c*d*e)*(b*e - 2*c*d)^3

) - (16*c^4*d*g)/(1287*(7*b*e^2 - 14*c*d*e)*(b*e - 2*c*d)^3)))/e - (48*c^4*d^2*g + 184*b^2*c^2*e^2*g + 304*c^4

*d*e*f - 144*b*c^3*e^2*f - 392*b*c^3*d*e*g)/(1287*e*(7*b*e^2 - 14*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)^4 - (((d*((8*c^3*(27*b*e*g - 52*c*d*g + 2*c*e*f))/(1287*(7*b*e^2 - 14*c*d*

e)*(b*e - 2*c*d)^3) - (16*c^4*d*g)/(1287*(7*b*e^2 - 14*c*d*e)*(b*e - 2*c*d)^3)))/e - (1360*c^4*d^2*g + 424*b^2

*c^2*e^2*g - 80*c^4*d*e*f + 48*b*c^3*e^2*f - 1528*b*c^3*d*e*g)/(1287*e*(7*b*e^2 - 14*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)^4 + (((d*((32*c^4*(11*b*e*g - 21*c*d*g + c*e*f))/(9009*

(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d)^4) - (32*c^5*d*g)/(9009*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d)^4)))/e + (128*c^

5*d^2*g - 224*b^2*c^3*e^2*g - 352*c^5*d*e*f + 160*b*c^4*e^2*f + 384*b*c^4*d*e*g)/(9009*e*(5*b*e^2 - 10*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)^3 - (((d*((16*c^4*(13*b*e*g - 24*c*d*

g + 2*c*e*f))/(9009*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d)^4) - (32*c^5*d*g)/(9009*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c

*d)^4)))/e - (640*c^5*d^2*g + 224*b^2*c^3*e^2*g - 128*c^5*d*e*f + 80*b*c^4*e^2*f - 768*b*c^4*d*e*g)/(9009*e*(5

*b*e^2 - 10*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)^3 + (((d*((32*c^4*

(12*b*e*g - 23*c*d*g + c*e*f))/(9009*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d)^4) - (32*c^5*d*g)/(9009*(5*b*e^2 - 10*

c*d*e)*(b*e - 2*c*d)^4)))/e - (2048*c^5*d^2*g + 664*b^2*c^3*e^2*g - 128*c^5*d*e*f + 80*b*c^4*e^2*f - 2352*b*c^

4*d*e*g)/(9009*e*(5*b*e^2 - 10*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)

^3 + (((d*((32*c^4*(13*b*e*g - 25*c*d*g + c*e*f))/(9009*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d)^4) - (32*c^5*d*g)/(

9009*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d)^4)))/e - (2304*c^5*d^2*g + 744*b^2*c^3*e^2*g - 128*c^5*d*e*f + 80*b*c^

4*e^2*f - 2640*b*c^4*d*e*g)/(9009*e*(5*b*e^2 - 10*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)^3 - (((d*((16*c^4*(33*b*e*g - 64*c*d*g + 2*c*e*f))/(9009*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d

)^4) - (32*c^5*d*g)/(9009*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d)^4)))/e - (2912*c^5*d^2*g + 496*b^2*c^3*e^2*g - 19

52*c^5*d*e*f + 992*b*c^4*e^2*f - 2448*b*c^4*d*e*g)/(9009*e*(5*b*e^2 - 10*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)^3 - (((d*((16*c^4*(37*b*e*g - 72*c*d*g + 2*c*e*f))/(9009*(5*b*e^2 -

10*c*d*e)*(b*e - 2*c*d)^4) - (32*c^5*d*g)/(9009*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d)^4)))/e - (3712*c^5*d^2*g +

1184*b^2*c^3*e^2*g - 128*c^5*d*e*f + 80*b*c^4*e^2*f - 4224*b*c^4*d*e*g)/(9009*e*(5*b*e^2 - 10*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)^3 - (((d*((16*c^4*(35*b*e*g - 68*c*d*g + 2*c*

e*f))/(9009*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d)^4) - (32*c^5*d*g)/(9009*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d)^4)))

/e + (4704*c^5*d^2*g + 816*b^2*c^3*e^2*g - 352*c^5*d*e*f + 160*b*c^4*e^2*f - 3984*b*c^4*d*e*g)/(9009*e*(5*b*e^

2 - 10*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)^3 - (((d*((32*c^5*(19*b

*e*g - 36*c*d*g + 2*c*e*f))/(45045*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^5) - (64*c^6*d*g)/(45045*(3*b*e^2 - 6*c*d

*e)*(b*e - 2*c*d)^5)))/e + (64*c^6*d^2*g - 352*b^2*c^4*e^2*g - 320*c^6*d*e*f + 128*b*c^5*e^2*f + 672*b*c^5*d*e

*g)/(45045*e*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^5))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x)^2 +

(((d*((64*c^5*(6*b*e*g - 11*c*d*g + c*e*f))/(45045*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^5) - (64*c^6*d*g)/(45045*

(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^5)))/e - (1152*c^6*d^2*g + 416*b^2*c^4*e^2*g - 192*c^6*d*e*f + 128*b*c^5*e^2

*f - 1408*b*c^5*d*e*g)/(45045*e*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^5))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1

/2))/(d + e*x)^2 + (((d*((64*c^5*(15*b*e*g - 29*c*d*g + c*e*f))/(45045*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^5) -

(64*c^6*d*g)/(45045*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^5)))/e + (3584*c^6*d^2*g + 352*b^2*c^4*e^2*g - 320*c^6*d

*e*f + 128*b*c^5*e^2*f - 2496*b*c^5*d*e*g)/(45045*e*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^5))*(c*d^2 - c*e^2*x^2 -

b*d*e - b*e^2*x)^(1/2))/(d + e*x)^2 - (((d*((32*c^5*(21*b*e*g - 40*c*d*g + 2*c*e*f))/(45045*(3*b*e^2 - 6*c*d*

e)*(b*e - 2*c*d)^5) - (64*c^6*d*g)/(45045*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^5)))/e - (2880*c^6*d^2*g + 992*b^2

*c^4*e^2*g - 192*c^6*d*e*f + 128*b*c^5*e^2*f - 3424*b*c^5*d*e*g)/(45045*e*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^5)

)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x)^2 + (((d*((64*c^5*(16*b*e*g - 31*c*d*g + c*e*f))/(450

45*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^5) - (64*c^6*d*g)/(45045*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^5)))/e + (4224

*c^6*d^2*g + 480*b^2*c^4*e^2*g - 320*c^6*d*e*f + 128*b*c^5*e^2*f - 3072*b*c^5*d*e*g)/(45045*e*(3*b*e^2 - 6*c*d

*e)*(b*e - 2*c*d)^5))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x)^2 - (((d*((32*c^5*(23*b*e*g - 44*

c*d*g + 2*c*e*f))/(45045*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^5) - (64*c^6*d*g)/(45045*(3*b*e^2 - 6*c*d*e)*(b*e -

2*c*d)^5)))/e - (3264*c^6*d^2*g + 1120*b^2*c^4*e^2*g - 192*c^6*d*e*f + 128*b*c^5*e^2*f - 3872*b*c^5*d*e*g)/(4

5045*e*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^5))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x)^2 - (((d*(

(32*c^5*(25*b*e*g - 48*c*d*g + 2*c*e*f))/(45045*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^5) - (64*c^6*d*g)/(45045*(3*

b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^5)))/e - (3648*c^6*d^2*g + 1248*b^2*c^4*e^2*g - 192*c^6*d*e*f + 128*b*c^5*e^2*f

- 4320*b*c^5*d*e*g)/(45045*e*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^5))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2

))/(d + e*x)^2 + (((d*((64*c^5*(16*b*e*g - 31*c*d*g + c*e*f))/(45045*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^5) - (6

4*c^6*d*g)/(45045*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^5)))/e - (4992*c^6*d^2*g + 1696*b^2*c^4*e^2*g - 192*c^6*d*

e*f + 128*b*c^5*e^2*f - 5888*b*c^5*d*e*g)/(45045*e*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^5))*(c*d^2 - c*e^2*x^2 -

b*d*e - b*e^2*x)^(1/2))/(d + e*x)^2 + (((d*((64*c^5*(14*b*e*g - 27*c*d*g + c*e*f))/(45045*(3*b*e^2 - 6*c*d*e)*

(b*e - 2*c*d)^5) - (64*c^6*d*g)/(45045*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^5)))/e - (4736*c^6*d^2*g + 1216*b^2*c

^4*e^2*g - 1600*c^6*d*e*f + 832*b*c^5*e^2*f - 4800*b*c^5*d*e*g)/(45045*e*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^5))

*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x)^2 + (((d*((64*c^5*(17*b*e*g - 33*c*d*g + c*e*f))/(4504

5*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^5) - (64*c^6*d*g)/(45045*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^5)))/e - (5376*

c^6*d^2*g + 1824*b^2*c^4*e^2*g - 192*c^6*d*e*f + 128*b*c^5*e^2*f - 6336*b*c^5*d*e*g)/(45045*e*(3*b*e^2 - 6*c*d

*e)*(b*e - 2*c*d)^5))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x)^2 + (((d*((64*c^5*(18*b*e*g - 35*

c*d*g + c*e*f))/(45045*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^5) - (64*c^6*d*g)/(45045*(3*b*e^2 - 6*c*d*e)*(b*e - 2

*c*d)^5)))/e - (5760*c^6*d^2*g + 1952*b^2*c^4*e^2*g - 192*c^6*d*e*f + 128*b*c^5*e^2*f - 6784*b*c^5*d*e*g)/(450

45*e*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^5))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x)^2 - (((d*((3

2*c^5*(43*b*e*g - 84*c*d*g + 2*c*e*f))/(45045*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^5) - (64*c^6*d*g)/(45045*(3*b*

e^2 - 6*c*d*e)*(b*e - 2*c*d)^5)))/e + (7744*c^6*d^2*g + 1184*b^2*c^4*e^2*g - 320*c^6*d*e*f + 128*b*c^5*e^2*f -

6240*b*c^5*d*e*g)/(45045*e*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^5))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))

/(d + e*x)^2 - (((d*((32*c^5*(39*b*e*g - 76*c*d*g + 2*c*e*f))/(45045*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^5) - (6

4*c^6*d*g)/(45045*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^5)))/e - (1184*b^2*c^4*e^2*g - 5312*c^6*d^2*g + 7616*c^6*d

*e*f - 3776*b*c^5*e^2*f + 288*b*c^5*d*e*g)/(45045*e*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^5))*(c*d^2 - c*e^2*x^2 -

b*d*e - b*e^2*x)^(1/2))/(d + e*x)^2 - (((d*((32*c^5*(45*b*e*g - 88*c*d*g + 2*c*e*f))/(45045*(3*b*e^2 - 6*c*d*

e)*(b*e - 2*c*d)^5) - (64*c^6*d*g)/(45045*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^5)))/e - (7488*c^6*d^2*g + 2528*b^

2*c^4*e^2*g - 192*c^6*d*e*f + 128*b*c^5*e^2*f - 8800*b*c^5*d*e*g)/(45045*e*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^5

))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x)^2 - (((d*((32*c^5*(41*b*e*g - 80*c*d*g + 2*c*e*f))/(

45045*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^5) - (64*c^6*d*g)/(45045*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^5)))/e - (2

5536*c^6*d^2*g + 6624*b^2*c^4*e^2*g - 1600*c^6*d*e*f + 832*b*c^5*e^2*f - 26016*b*c^5*d*e*g)/(45045*e*(3*b*e^2

- 6*c*d*e)*(b*e - 2*c*d)^5))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x)^2 + (((d*((16*c^3*(19*b*e*

g - 37*c*d*g + c*e*f))/(1287*(7*b*e^2 - 14*c*d*e)*(b*e - 2*c*d)^3) - (16*c^4*d*g)/(1287*(7*b*e^2 - 14*c*d*e)*(

b*e - 2*c*d)^3)))/e - (16*c^2*(b*e - c*d)*(18*b*e*g - 36*c*d*g + c*e*f))/(1287*e*(7*b*e^2 - 14*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)^4 + (((d*((32*c^4*(23*b*e*g - 45*c*d*g + c*e*

f))/(9009*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d)^4) - (32*c^5*d*g)/(9009*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d)^4)))/e

- (32*c^3*(b*e - c*d)*(22*b*e*g - 44*c*d*g + c*e*f))/(9009*e*(5*b*e^2 - 10*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)^3 + (((d*((64*c^5*(26*b*e*g - 51*c*d*g + c*e*f))/(45045*(3*b*e^2

- 6*c*d*e)*(b*e - 2*c*d)^5) - (64*c^6*d*g)/(45045*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^5)))/e - (64*c^4*(b*e - c

*d)*(25*b*e*g - 50*c*d*g + c*e*f))/(45045*e*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^5))*(c*d^2 - c*e^2*x^2 - b*d*e -

b*e^2*x)^(1/2))/(d + e*x)^2 + (((1792*c^7*d^2*g + 704*b^2*c^5*e^2*g - 256*c^7*d*e*f + 192*b*c^6*e^2*f - 2304*

b*c^6*d*e*g)/(135135*e^2*(b*e - 2*c*d)^7) - (d*((64*c^6*(11*b*e*g - 20*c*d*g + 2*c*e*f))/(135135*e*(b*e - 2*c*

d)^7) - (128*c^7*d*g)/(135135*e*(b*e - 2*c*d)^7)))/e)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x) +

(((2304*c^7*d^2*g - 1152*b^2*c^5*e^2*g - 2944*c^7*d*e*f + 1408*b*c^6*e^2*f + 1152*b*c^6*d*e*g)/(135135*e^2*(b

*e - 2*c*d)^7) + (d*((128*c^6*(16*b*e*g - 31*c*d*g + c*e*f))/(135135*e*(b*e - 2*c*d)^7) - (128*c^7*d*g)/(13513

5*e*(b*e - 2*c*d)^7)))/e)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x) - (((3584*c^7*d^2*g + 1376*b^

2*c^5*e^2*g - 256*c^7*d*e*f + 192*b*c^6*e^2*f - 4544*b*c^6*d*e*g)/(135135*e^2*(b*e - 2*c*d)^7) - (d*((128*c^6*

(9*b*e*g - 17*c*d*g + c*e*f))/(135135*e*(b*e - 2*c*d)^7) - (128*c^7*d*g)/(135135*e*(b*e - 2*c*d)^7)))/e)*(c*d^

2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x) - (((4096*c^7*d^2*g + 1568*b^2*c^5*e^2*g - 256*c^7*d*e*f + 1

92*b*c^6*e^2*f - 5184*b*c^6*d*e*g)/(135135*e^2*(b*e - 2*c*d)^7) - (d*((128*c^6*(10*b*e*g - 19*c*d*g + c*e*f))/

(135135*e*(b*e - 2*c*d)^7) - (128*c^7*d*g)/(135135*e*(b*e - 2*c*d)^7)))/e)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*

x)^(1/2))/(d + e*x) - (((4608*c^7*d^2*g + 1760*b^2*c^5*e^2*g - 256*c^7*d*e*f + 192*b*c^6*e^2*f - 5824*b*c^6*d*

e*g)/(135135*e^2*(b*e - 2*c*d)^7) - (d*((128*c^6*(11*b*e*g - 21*c*d*g + c*e*f))/(135135*e*(b*e - 2*c*d)^7) - (

128*c^7*d*g)/(135135*e*(b*e - 2*c*d)^7)))/e)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x) - (((5120*

c^7*d^2*g + 1952*b^2*c^5*e^2*g - 256*c^7*d*e*f + 192*b*c^6*e^2*f - 6464*b*c^6*d*e*g)/(135135*e^2*(b*e - 2*c*d)

^7) - (d*((128*c^6*(12*b*e*g - 23*c*d*g + c*e*f))/(135135*e*(b*e - 2*c*d)^7) - (128*c^7*d*g)/(135135*e*(b*e -

2*c*d)^7)))/e)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x) + (((4992*c^7*d^2*g + 1728*b^2*c^5*e^2*g

- 896*c^7*d*e*f + 512*b*c^6*e^2*f - 5952*b*c^6*d*e*g)/(135135*e^2*(b*e - 2*c*d)^7) - (d*((64*c^6*(23*b*e*g -

44*c*d*g + 2*c*e*f))/(135135*e*(b*e - 2*c*d)^7) - (128*c^7*d*g)/(135135*e*(b*e - 2*c*d)^7)))/e)*(c*d^2 - c*e^2

*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x) + (((5888*c^7*d^2*g + 2240*b^2*c^5*e^2*g - 256*c^7*d*e*f + 192*b*c^6*

e^2*f - 7424*b*c^6*d*e*g)/(135135*e^2*(b*e - 2*c*d)^7) - (d*((64*c^6*(27*b*e*g - 52*c*d*g + 2*c*e*f))/(135135*

e*(b*e - 2*c*d)^7) - (128*c^7*d*g)/(135135*e*(b*e - 2*c*d)^7)))/e)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2)

)/(d + e*x) + (((6400*c^7*d^2*g + 2432*b^2*c^5*e^2*g - 256*c^7*d*e*f + 192*b*c^6*e^2*f - 8064*b*c^6*d*e*g)/(13

5135*e^2*(b*e - 2*c*d)^7) - (d*((64*c^6*(29*b*e*g - 56*c*d*g + 2*c*e*f))/(135135*e*(b*e - 2*c*d)^7) - (128*c^7

*d*g)/(135135*e*(b*e - 2*c*d)^7)))/e)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x) + (2*((6912*c^7*d

^2*g + 2624*b^2*c^5*e^2*g - 256*c^7*d*e*f + 192*b*c^6*e^2*f - 8704*b*c^6*d*e*g)/(135135*e^2*(b*e - 2*c*d)^7) -

(d*((64*c^6*(31*b*e*g - 60*c*d*g + 2*c*e*f))/(135135*e*(b*e - 2*c*d)^7) - (128*c^7*d*g)/(135135*e*(b*e - 2*c*

d)^7)))/e)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x) + (((7424*c^7*d^2*g + 2816*b^2*c^5*e^2*g - 2

56*c^7*d*e*f + 192*b*c^6*e^2*f - 9344*b*c^6*d*e*g)/(135135*e^2*(b*e - 2*c*d)^7) - (d*((64*c^6*(33*b*e*g - 64*c

*d*g + 2*c*e*f))/(135135*e*(b*e - 2*c*d)^7) - (128*c^7*d*g)/(135135*e*(b*e - 2*c*d)^7)))/e)*(c*d^2 - c*e^2*x^2

- b*d*e - b*e^2*x)^(1/2))/(d + e*x) + (((7936*c^7*d^2*g + 3008*b^2*c^5*e^2*g - 256*c^7*d*e*f + 192*b*c^6*e^2*

f - 9984*b*c^6*d*e*g)/(135135*e^2*(b*e - 2*c*d)^7) - (d*((64*c^6*(35*b*e*g - 68*c*d*g + 2*c*e*f))/(135135*e*(b

*e - 2*c*d)^7) - (128*c^7*d*g)/(135135*e*(b*e - 2*c*d)^7)))/e)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d

+ e*x) - (((8704*c^7*d^2*g + 3296*b^2*c^5*e^2*g - 256*c^7*d*e*f + 192*b*c^6*e^2*f - 10944*b*c^6*d*e*g)/(13513

5*e^2*(b*e - 2*c*d)^7) - (d*((128*c^6*(19*b*e*g - 37*c*d*g + c*e*f))/(135135*e*(b*e - 2*c*d)^7) - (128*c^7*d*g

)/(135135*e*(b*e - 2*c*d)^7)))/e)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x) - (((9216*c^7*d^2*g +

3488*b^2*c^5*e^2*g - 256*c^7*d*e*f + 192*b*c^6*e^2*f - 11584*b*c^6*d*e*g)/(135135*e^2*(b*e - 2*c*d)^7) - (d*(

(128*c^6*(20*b*e*g - 39*c*d*g + c*e*f))/(135135*e*(b*e - 2*c*d)^7) - (128*c^7*d*g)/(135135*e*(b*e - 2*c*d)^7))

)/e)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x) - (((9728*c^7*d^2*g + 3680*b^2*c^5*e^2*g - 256*c^7

*d*e*f + 192*b*c^6*e^2*f - 12224*b*c^6*d*e*g)/(135135*e^2*(b*e - 2*c*d)^7) - (d*((128*c^6*(21*b*e*g - 41*c*d*g

+ c*e*f))/(135135*e*(b*e - 2*c*d)^7) - (128*c^7*d*g)/(135135*e*(b*e - 2*c*d)^7)))/e)*(c*d^2 - c*e^2*x^2 - b*d

*e - b*e^2*x)^(1/2))/(d + e*x) - (((10240*c^7*d^2*g + 3872*b^2*c^5*e^2*g - 256*c^7*d*e*f + 192*b*c^6*e^2*f - 1

2864*b*c^6*d*e*g)/(135135*e^2*(b*e - 2*c*d)^7) - (d*((128*c^6*(22*b*e*g - 43*c*d*g + c*e*f))/(135135*e*(b*e -

2*c*d)^7) - (128*c^7*d*g)/(135135*e*(b*e - 2*c*d)^7)))/e)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*

x) + (((12032*c^7*d^2*g + 4544*b^2*c^5*e^2*g - 256*c^7*d*e*f + 192*b*c^6*e^2*f - 15104*b*c^6*d*e*g)/(135135*e^

2*(b*e - 2*c*d)^7) - (d*((64*c^6*(51*b*e*g - 100*c*d*g + 2*c*e*f))/(135135*e*(b*e - 2*c*d)^7) - (128*c^7*d*g)/

(135135*e*(b*e - 2*c*d)^7)))/e)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x) - (((14848*c^7*d^2*g +

4544*b^2*c^5*e^2*g - 896*c^7*d*e*f + 512*b*c^6*e^2*f - 16512*b*c^6*d*e*g)/(135135*e^2*(b*e - 2*c*d)^7) - (d*((

128*c^6*(17*b*e*g - 33*c*d*g + c*e*f))/(135135*e*(b*e - 2*c*d)^7) - (128*c^7*d*g)/(135135*e*(b*e - 2*c*d)^7)))

/e)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x) - (((16640*c^7*d^2*g + 5056*b^2*c^5*e^2*g - 896*c^7

*d*e*f + 512*b*c^6*e^2*f - 18432*b*c^6*d*e*g)/(135135*e^2*(b*e - 2*c*d)^7) - (d*((128*c^6*(18*b*e*g - 35*c*d*g

+ c*e*f))/(135135*e*(b*e - 2*c*d)^7) - (128*c^7*d*g)/(135135*e*(b*e - 2*c*d)^7)))/e)*(c*d^2 - c*e^2*x^2 - b*d

*e - b*e^2*x)^(1/2))/(d + e*x) + (((20608*c^7*d^2*g + 2624*b^2*c^5*e^2*g - 15488*c^7*d*e*f + 7808*b*c^6*e^2*f

- 15552*b*c^6*d*e*g)/(135135*e^2*(b*e - 2*c*d)^7) - (d*((64*c^6*(43*b*e*g - 84*c*d*g + 2*c*e*f))/(135135*e*(b*

e - 2*c*d)^7) - (128*c^7*d*g)/(135135*e*(b*e - 2*c*d)^7)))/e)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d

+ e*x) + (((26496*c^7*d^2*g + 7872*b^2*c^5*e^2*g - 896*c^7*d*e*f + 512*b*c^6*e^2*f - 28992*b*c^6*d*e*g)/(13513

5*e^2*(b*e - 2*c*d)^7) - (d*((64*c^6*(47*b*e*g - 92*c*d*g + 2*c*e*f))/(135135*e*(b*e - 2*c*d)^7) - (128*c^7*d*

g)/(135135*e*(b*e - 2*c*d)^7)))/e)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x) - (((40576*c^7*d^2*g

+ 8000*b^2*c^5*e^2*g - 2944*c^7*d*e*f + 1408*b*c^6*e^2*f - 36288*b*c^6*d*e*g)/(135135*e^2*(b*e - 2*c*d)^7) +

(d*((64*c^6*(45*b*e*g - 88*c*d*g + 2*c*e*f))/(135135*e*(b*e - 2*c*d)^7) - (128*c^7*d*g)/(135135*e*(b*e - 2*c*d

)^7)))/e)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x) + (((d*((128*c^6*(28*b*e*g - 55*c*d*g + c*e*f

))/(135135*e*(b*e - 2*c*d)^7) - (128*c^7*d*g)/(135135*e*(b*e - 2*c*d)^7)))/e - (128*c^5*(b*e - c*d)*(27*b*e*g

- 54*c*d*g + c*e*f))/(135135*e^2*(b*e - 2*c*d)^7))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x) + ((

(d*((8*c^3*e*f - 24*c^3*d*g + 16*b*c^2*e*g)/(143*(9*b*e^2 - 18*c*d*e)*(b*e - 2*c*d)^2) - (8*c^3*d*g)/(143*(9*b

*e^2 - 18*c*d*e)*(b*e - 2*c*d)^2)))/e - (2*b*c*(3*b*e*g - 6*c*d*g + 2*c*e*f))/(143*(9*b*e^2 - 18*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)^5 - (((d*((8*c^3*e*f - 128*c^3*d*g + 68*b*c

^2*e*g)/(143*(9*b*e^2 - 18*c*d*e)*(b*e - 2*c*d)^2) - (8*c^3*d*g)/(143*(9*b*e^2 - 18*c*d*e)*(b*e - 2*c*d)^2)))/

e - (4*b*c*(8*b*e*g - 16*c*d*g + c*e*f))/(143*(9*b*e^2 - 18*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)^5 - (((d*((64*c^6*(25*b*e*g - 48*c*d*g + 2*c*e*f))/(135135*e*(b*e - 2*c*d)^7) -

(128*c^7*d*g)/(135135*e*(b*e - 2*c*d)^7)))/e + (384*c^7*d^2*g - 704*b^2*c^5*e^2*g - 128*c^7*d*e*f + 1216*b*c^6

*d*e*g)/(135135*e^2*(b*e - 2*c*d)^7))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x) - (((d*((64*c^6*(

27*b*e*g - 52*c*d*g + 2*c*e*f))/(135135*e*(b*e - 2*c*d)^7) - (128*c^7*d*g)/(135135*e*(b*e - 2*c*d)^7)))/e + (6

40*c^7*d^2*g - 704*b^2*c^5*e^2*g - 128*c^7*d*e*f + 1088*b*c^6*d*e*g)/(135135*e^2*(b*e - 2*c*d)^7))*(c*d^2 - c*

e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x) - (((d*((64*c^6*(29*b*e*g - 56*c*d*g + 2*c*e*f))/(135135*e*(b*e -

2*c*d)^7) - (128*c^7*d*g)/(135135*e*(b*e - 2*c*d)^7)))/e + (896*c^7*d^2*g - 704*b^2*c^5*e^2*g - 128*c^7*d*e*f

+ 960*b*c^6*d*e*g)/(135135*e^2*(b*e - 2*c*d)^7))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x) + (((d

*((128*c^6*(18*b*e*g - 35*c*d*g + c*e*f))/(135135*e*(b*e - 2*c*d)^7) - (128*c^7*d*g)/(135135*e*(b*e - 2*c*d)^7

)))/e + (1792*c^7*d^2*g - 704*b^2*c^5*e^2*g - 128*c^7*d*e*f + 512*b*c^6*d*e*g)/(135135*e^2*(b*e - 2*c*d)^7))*(

c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x) + (((d*((128*c^6*(19*b*e*g - 37*c*d*g + c*e*f))/(135135*

e*(b*e - 2*c*d)^7) - (128*c^7*d*g)/(135135*e*(b*e - 2*c*d)^7)))/e + (2048*c^7*d^2*g - 704*b^2*c^5*e^2*g - 128*

c^7*d*e*f + 384*b*c^6*d*e*g)/(135135*e^2*(b*e - 2*c*d)^7))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e

*x) + (((d*((128*c^6*(8*b*e*g - 15*c*d*g + c*e*f))/(135135*e*(b*e - 2*c*d)^7) - (128*c^7*d*g)/(135135*e*(b*e -

2*c*d)^7)))/e - (768*c^7*d^2*g + 704*b^2*c^5*e^2*g + 128*c^7*d*e*f - 1792*b*c^6*d*e*g)/(135135*e^2*(b*e - 2*c

*d)^7))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x) + (((d*((128*c^6*(20*b*e*g - 39*c*d*g + c*e*f))

/(135135*e*(b*e - 2*c*d)^7) - (128*c^7*d*g)/(135135*e*(b*e - 2*c*d)^7)))/e + (2304*c^7*d^2*g - 704*b^2*c^5*e^2

*g - 128*c^7*d*e*f + 256*b*c^6*d*e*g)/(135135*e^2*(b*e - 2*c*d)^7))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2

))/(d + e*x) - (((d*((64*c^6*(49*b*e*g - 96*c*d*g + 2*c*e*f))/(135135*e*(b*e - 2*c*d)^7) - (128*c^7*d*g)/(1351

35*e*(b*e - 2*c*d)^7)))/e - (704*b^2*c^5*e^2*g - 3456*c^7*d^2*g + 128*c^7*d*e*f + 320*b*c^6*d*e*g)/(135135*e^2

*(b*e - 2*c*d)^7))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x)

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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {- \left (d + e x\right ) \left (b e - c d + c e x\right )} \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)**(1/2)/(e*x+d)**8,x)

[Out]

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

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