∫ (f+gx)(cd2−bde−be2x−ce2x2)5∕2 __________________________________________________

3.2205 \(\int \frac {(f+g x) (c d^2-b d e-b e^2 x-c e^2 x^2)^{5/2}}{(d+e x)^8} \, dx\)

Optimal. Leaf size=138 \[ \frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (9 b e g-2 c (8 d g+e f))}{63 e^2 (d+e x)^7 (2 c d-b e)^2}-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{9 e^2 (d+e x)^8 (2 c d-b e)} \]

[Out]

-2/9*(-d*g+e*f)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(7/2)/e^2/(-b*e+2*c*d)/(e*x+d)^8+2/63*(9*b*e*g-2*c*(8*d*g+e*f

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

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Rubi [A] time = 0.22, antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {792, 650} \[ \frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (9 b e g-2 c (8 d g+e f))}{63 e^2 (d+e x)^7 (2 c d-b e)^2}-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{9 e^2 (d+e x)^8 (2 c d-b e)} \]

Antiderivative was successfully veriﬁed.

[In]

Int[((f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2))/(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)^(7/2))/(9*e^2*(2*c*d - b*e)*(d + e*x)^8) + (2*(9*b*e*g -

2*c*(e*f + 8*d*g))*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(63*e^2*(2*c*d - b*e)^2*(d + e*x)^7)

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 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 )^{5/2}}{(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 )^{7/2}}{9 e^2 (2 c d-b e) (d+e x)^8}-\frac {\left (2 \left (\frac {7}{2} e \left (-2 c e^2 f+b e^2 g\right )-8 \left (-c e^3 f+\left (-c d e^2+b e^3\right ) g\right )\right )\right ) \int \frac {\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^7} \, dx}{9 e \left (-2 c d e^2+b e^3\right )}\\ &=-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{9 e^2 (2 c d-b e) (d+e x)^8}+\frac {2 (9 b e g-2 c (e f+8 d g)) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{63 e^2 (2 c d-b e)^2 (d+e x)^7}\\ \end {align*}

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Mathematica [A] time = 0.10, size = 104, normalized size = 0.75 \[ \frac {2 (b e-c d+c e x)^3 \sqrt {(d+e x) (c (d-e x)-b e)} \left (2 c \left (d^2 g+8 d e (f+g x)+e^2 f x\right )-b e (2 d g+7 e f+9 e g x)\right )}{63 e^2 (d+e x)^5 (b e-2 c d)^2} \]

Antiderivative was successfully veriﬁed.

[In]

Integrate[((f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2))/(d + e*x)^8,x]

[Out]

(2*(-(c*d) + b*e + c*e*x)^3*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))]*(-(b*e*(7*e*f + 2*d*g + 9*e*g*x)) + 2*c*(d^

2*g + e^2*f*x + 8*d*e*(f + g*x))))/(63*e^2*(-2*c*d + b*e)^2*(d + e*x)^5)

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

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)^(5/2)/(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 \[ \text {Timed out} \]

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)^(5/2)/(e*x+d)^8,x, algorithm="giac")

[Out]

Timed out

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maple [A] time = 0.06, size = 128, normalized size = 0.93 \[ -\frac {2 \left (c e x +b e -c d \right ) \left (9 b \,e^{2} g x -16 c d e g x -2 c \,e^{2} f x +2 b d e g +7 b \,e^{2} f -2 c \,d^{2} g -16 c d e f \right ) \left (-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}\right )^{\frac {5}{2}}}{63 \left (e x +d \right )^{7} \left (b^{2} e^{2}-4 b c d e +4 c^{2} d^{2}\right ) e^{2}} \]

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)^(5/2)/(e*x+d)^8,x)

[Out]

-2/63*(c*e*x+b*e-c*d)*(9*b*e^2*g*x-16*c*d*e*g*x-2*c*e^2*f*x+2*b*d*e*g+7*b*e^2*f-2*c*d^2*g-16*c*d*e*f)*(-c*e^2*

x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/(e*x+d)^7/e^2/(b^2*e^2-4*b*c*d*e+4*c^2*d^2)

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

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)^(5/2)/(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 = 26.34, size = 12294, normalized size = 89.09 \[ \text {result too large to display} \]

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)^(5/2))/(d + e*x)^8,x)

[Out]

(((d*((336*b^2*c^5*e^3*f - 2016*c^7*d^3*g + 384*b^3*c^4*e^3*g + 800*c^7*d^2*e*f - 1024*b*c^6*d*e^2*f + 3424*b*

c^6*d^2*e*g - 1968*b^2*c^5*d*e^2*g)/(945*e*(b*e - 2*c*d)^5) - (d*((16*c^5*(21*b^2*e^2*g + 50*c^2*d^2*g + 10*b*

c*e^2*f - 14*c^2*d*e*f - 64*b*c*d*e*g))/(945*(b*e - 2*c*d)^5) - (d*((32*c^6*e*(5*b*e*g - 7*c*d*g + c*e*f))/(94

5*(b*e - 2*c*d)^5) - (32*c^7*d*e*g)/(945*(b*e - 2*c*d)^5)))/e))/e))/e - (132*b^3*c^4*e^3*f + 126*b^4*c^3*e^3*g

- 1008*b*c^6*d^3*g + 400*b*c^6*d^2*e*f - 456*b^2*c^5*d*e^2*f + 1512*b^2*c^5*d^2*e*g - 756*b^3*c^4*d*e^2*g)/(9

45*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) - (((d*((696*b^2*c^5*e^3*f - 601

6*c^7*d^3*g + 1044*b^3*c^4*e^3*g + 1920*c^7*d^2*e*f - 2304*b*c^6*d*e^2*f + 9984*b*c^6*d^2*e*g - 5568*b^2*c^5*d

*e^2*g)/(945*e*(b*e - 2*c*d)^5) - (d*((8*c^5*(29*b^2*e^2*g + 80*c^2*d^2*g + 10*b*c*e^2*f - 16*c^2*d*e*f - 96*b

*c*d*e*g))/(315*(b*e - 2*c*d)^5) - (d*((16*c^6*e*(15*b*e*g - 24*c*d*g + 2*c*e*f))/(945*(b*e - 2*c*d)^5) - (32*

c^7*d*e*g)/(945*(b*e - 2*c*d)^5)))/e))/e))/e - (292*b^3*c^4*e^3*f + 376*b^4*c^3*e^3*g - 3008*b*c^6*d^3*g + 960

*b*c^6*d^2*e*f - 1056*b^2*c^5*d*e^2*f + 4512*b^2*c^5*d^2*e*g - 2256*b^3*c^4*d*e^2*g)/(945*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) - (((d*((840*b^2*c^5*e^3*f - 7616*c^7*d^3*g + 1308*b^3*

c^4*e^3*g + 2368*c^7*d^2*e*f - 2816*b*c^6*d*e^2*f + 12608*b*c^6*d^2*e*g - 7008*b^2*c^5*d*e^2*g)/(945*e*(b*e -

2*c*d)^5) - (d*((8*c^5*(105*b^2*e^2*g + 296*c^2*d^2*g + 34*b*c*e^2*f - 56*c^2*d*e*f - 352*b*c*d*e*g))/(945*(b*

e - 2*c*d)^5) - (d*((16*c^6*e*(17*b*e*g - 28*c*d*g + 2*c*e*f))/(945*(b*e - 2*c*d)^5) - (32*c^7*d*e*g)/(945*(b*

e - 2*c*d)^5)))/e))/e))/e - (356*b^3*c^4*e^3*f + 476*b^4*c^3*e^3*g - 3808*b*c^6*d^3*g + 1184*b*c^6*d^2*e*f - 1

296*b^2*c^5*d*e^2*f + 5712*b^2*c^5*d^2*e*g - 2856*b^3*c^4*d*e^2*g)/(945*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) - (((d*((984*b^2*c^5*e^3*f - 9216*c^7*d^3*g + 1572*b^3*c^4*e^3*g + 2816*c

^7*d^2*e*f - 3328*b*c^6*d*e^2*f + 15232*b*c^6*d^2*e*g - 8448*b^2*c^5*d*e^2*g)/(945*e*(b*e - 2*c*d)^5) - (d*((8

*c^5*(123*b^2*e^2*g + 352*c^2*d^2*g + 38*b*c*e^2*f - 64*c^2*d*e*f - 416*b*c*d*e*g))/(945*(b*e - 2*c*d)^5) - (d

*((16*c^6*e*(19*b*e*g - 32*c*d*g + 2*c*e*f))/(945*(b*e - 2*c*d)^5) - (32*c^7*d*e*g)/(945*(b*e - 2*c*d)^5)))/e)

)/e))/e - (420*b^3*c^4*e^3*f + 576*b^4*c^3*e^3*g - 4608*b*c^6*d^3*g + 1408*b*c^6*d^2*e*f - 1536*b^2*c^5*d*e^2*

f + 6912*b^2*c^5*d^2*e*g - 3456*b^3*c^4*d*e^2*g)/(945*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) - (((d*((d*((d*((4*c^4*e^2*(15*b*e*g - 24*c*d*g + 2*c*e*f))/(63*(5*b*e^2 - 10*c*d*e)*(b*e -

2*c*d)^2) - (8*c^5*d*e^2*g)/(63*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d)^2)))/e - (4*c^3*e*(37*b^2*e^2*g + 94*c^2*d

^2*g + 15*b*c*e^2*f - 24*c^2*d*e*f - 118*b*c*d*e*g))/(63*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d)^2)))/e + (148*b^2*

c^3*e^4*f + 164*b^3*c^2*e^4*g + 376*c^5*d^2*e^2*f - 832*c^5*d^3*e*g - 472*b*c^4*d*e^3*f + 1436*b*c^4*d^2*e^2*g

- 836*b^2*c^3*d*e^3*g)/(63*e*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d)^2)))/e - (640*c^5*d^4*g + 108*b^3*c^2*e^4*f +

68*b^4*c*e^4*g - 384*c^5*d^3*e*f - 1504*b*c^4*d^3*e*g + 764*b*c^4*d^2*e^2*f - 500*b^2*c^3*d*e^3*f - 488*b^3*c

^2*d*e^3*g + 1296*b^2*c^3*d^2*e^2*g)/(63*e*(5*b*e^2 - 10*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)^3 + (((d*((1480*b^2*c^5*e^3*f - 19456*c^7*d^3*g + 3088*b^3*c^4*e^3*g + 4608*c^7*d^2

*e*f - 5216*b*c^6*d*e^2*f + 31488*b*c^6*d^2*e*g - 17048*b^2*c^5*d*e^2*g)/(945*e*(b*e - 2*c*d)^5) - (d*((8*c^5*

(185*b^2*e^2*g + 576*c^2*d^2*g + 44*b*c*e^2*f - 76*c^2*d*e*f - 652*b*c*d*e*g))/(945*(b*e - 2*c*d)^5) - (d*((32

*c^6*e*(11*b*e*g - 19*c*d*g + c*e*f))/(945*(b*e - 2*c*d)^5) - (32*c^7*d*e*g)/(945*(b*e - 2*c*d)^5)))/e))/e))/e

- (8*b*c^3*(152*b^3*e^3*g - 1216*c^3*d^3*g + 82*b^2*c*e^3*f + 288*c^3*d^2*e*f - 307*b*c^2*d*e^2*f + 1824*b*c^

2*d^2*e*g - 912*b^2*c*d*e^2*g))/(945*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

) + (((d*((1704*b^2*c^5*e^3*f - 23296*c^7*d^3*g + 3672*b^3*c^4*e^3*g + 5376*c^7*d^2*e*f - 6048*b*c^6*d*e^2*f +

37632*b*c^6*d^2*e*g - 20328*b^2*c^5*d*e^2*g)/(945*e*(b*e - 2*c*d)^5) - (d*((8*c^5*(71*b^2*e^2*g + 224*c^2*d^2

*g + 16*b*c*e^2*f - 28*c^2*d*e*f - 252*b*c*d*e*g))/(315*(b*e - 2*c*d)^5) - (d*((32*c^6*e*(12*b*e*g - 21*c*d*g

+ c*e*f))/(945*(b*e - 2*c*d)^5) - (32*c^7*d*e*g)/(945*(b*e - 2*c*d)^5)))/e))/e))/e - (8*b*c^3*(182*b^3*e^3*g -

1456*c^3*d^3*g + 95*b^2*c*e^3*f + 336*c^3*d^2*e*f - 357*b*c^2*d*e^2*f + 2184*b*c^2*d^2*e*g - 1092*b^2*c*d*e^2

*g))/(945*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) - (((d*((2992*b^2*c^5*e^3

*f - 60928*c^7*d^3*g + 8992*b^3*c^4*e^3*g + 10080*c^7*d^2*e*f - 10976*b*c^6*d*e^2*f + 96432*b*c^6*d^2*e*g - 50

960*b^2*c^5*d*e^2*g)/(945*e*(b*e - 2*c*d)^5) - (d*((16*c^5*(187*b^2*e^2*g + 630*c^2*d^2*g + 31*b*c*e^2*f - 56*

c^2*d*e*f - 686*b*c*d*e*g))/(945*(b*e - 2*c*d)^5) - (d*((16*c^6*e*(31*b*e*g - 56*c*d*g + 2*c*e*f))/(945*(b*e -

2*c*d)^5) - (32*c^7*d*e*g)/(945*(b*e - 2*c*d)^5)))/e))/e))/e - (16*b*c^3*(238*b^3*e^3*g - 1904*c^3*d^3*g + 86

*b^2*c*e^3*f + 315*c^3*d^2*e*f - 329*b*c^2*d*e^2*f + 2856*b*c^2*d^2*e*g - 1428*b^2*c*d*e^2*g))/(945*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) + (((d*((356*b^2*c^4*e^3*f - 2624*c^6*d^3*g +

476*b^3*c^3*e^3*g + 960*c^6*d^2*e*f - 1168*b*c^5*d*e^2*f + 4416*b*c^5*d^2*e*g - 2500*b^2*c^4*d*e^2*g)/(315*(3

*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^3) + (d*((d*((16*c^5*e^2*(8*b*e*g - 13*c*d*g + c*e*f))/(315*(3*b*e^2 - 6*c*d*e

)*(b*e - 2*c*d)^3) - (16*c^6*d*e^2*g)/(315*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^3)))/e - (356*b^2*c^4*e^3*g + 128

*b*c^5*e^3*f - 208*c^6*d*e^2*f + 960*c^6*d^2*e*g - 1168*b*c^5*d*e^2*g)/(315*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^

3)))/e))/e - (148*b^3*c^3*e^3*f + 164*b^4*c^2*e^3*g - 1312*b*c^5*d^3*g + 480*b*c^5*d^2*e*f - 532*b^2*c^4*d*e^2

*f + 1968*b^2*c^4*d^2*e*g - 984*b^3*c^3*d*e^2*g)/(315*(3*b*e^2 - 6*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)^2 - (((d*((552*b^2*c^5*e^4*f + 780*b^3*c^4*e^4*g + 1472*c^7*d^2*e^2*f - 4

416*c^7*d^3*e*g - 1792*b*c^6*d*e^3*f + 7360*b*c^6*d^2*e^2*g - 4128*b^2*c^5*d*e^3*g)/(945*e^2*(b*e - 2*c*d)^5)

- (d*((8*c^5*(69*b^2*e^2*g + 184*c^2*d^2*g + 26*b*c*e^2*f - 40*c^2*d*e*f - 224*b*c*d*e*g))/(945*(b*e - 2*c*d)^

5) - (d*((16*c^6*e*(13*b*e*g - 20*c*d*g + 2*c*e*f))/(945*(b*e - 2*c*d)^5) - (32*c^7*d*e*g)/(945*(b*e - 2*c*d)^

5)))/e))/e))/e - (4032*c^7*d^4*g + 428*b^3*c^4*e^4*f + 428*b^4*c^3*e^4*g - 1600*c^7*d^3*e*f - 9472*b*c^6*d^3*e

*g + 3136*b*c^6*d^2*e^2*f - 2016*b^2*c^5*d*e^3*f - 3072*b^3*c^4*d*e^3*g + 8160*b^2*c^5*d^2*e^2*g)/(945*e^2*(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) + (((d*((1032*b^2*c^5*e^4*f + 1920*b^3*c

^4*e^4*g + 3072*c^7*d^2*e^2*f - 11776*c^7*d^3*e*g - 3552*b*c^6*d*e^3*f + 19200*b*c^6*d^2*e^2*g - 10488*b^2*c^5

*d*e^3*g)/(945*e^2*(b*e - 2*c*d)^5) - (d*((8*c^5*(43*b^2*e^2*g + 128*c^2*d^2*g + 12*b*c*e^2*f - 20*c^2*d*e*f -

148*b*c*d*e*g))/(315*(b*e - 2*c*d)^5) - (d*((32*c^6*e*(9*b*e*g - 15*c*d*g + c*e*f))/(945*(b*e - 2*c*d)^5) - (

32*c^7*d*e*g)/(945*(b*e - 2*c*d)^5)))/e))/e))/e - (8384*c^7*d^4*g + 712*b^3*c^4*e^4*f + 1128*b^4*c^3*e^4*g - 2

112*c^7*d^3*e*f - 21600*b*c^6*d^3*e*g + 4704*b*c^6*d^2*e^2*f - 3240*b^2*c^5*d*e^3*f - 7816*b^3*c^4*d*e^3*g + 1

9824*b^2*c^5*d^2*e^2*g)/(945*e^2*(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) + ((

(d*((1224*b^2*c^5*e^4*f + 2376*b^3*c^4*e^4*g + 3712*c^7*d^2*e^2*f - 14720*c^7*d^3*e*g - 4256*b*c^6*d*e^3*f + 2

3936*b*c^6*d^2*e^2*g - 13032*b^2*c^5*d*e^3*g)/(945*e^2*(b*e - 2*c*d)^5) - (d*((8*c^5*(153*b^2*e^2*g + 464*c^2*

d^2*g + 40*b*c*e^2*f - 68*c^2*d*e*f - 532*b*c*d*e*g))/(945*(b*e - 2*c*d)^5) - (d*((32*c^6*e*(10*b*e*g - 17*c*d

*g + c*e*f))/(945*(b*e - 2*c*d)^5) - (32*c^7*d*e*g)/(945*(b*e - 2*c*d)^5)))/e))/e))/e - (15232*c^7*d^4*g + 112

8*b^3*c^4*e^4*f + 1576*b^4*c^3*e^4*g - 4736*c^7*d^3*e*f - 35456*b*c^6*d^3*e*g + 8960*b*c^6*d^2*e^2*f - 5544*b^

2*c^5*d*e^3*f - 11360*b^3*c^4*d*e^3*g + 30336*b^2*c^5*d^2*e^2*g)/(945*e^2*(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) + (((d*((1416*b^2*c^5*e^4*f + 2832*b^3*c^4*e^4*g + 4352*c^7*d^2*e^2*f - 1

7664*c^7*d^3*e*g - 4960*b*c^6*d*e^3*f + 28672*b*c^6*d^2*e^2*g - 15576*b^2*c^5*d*e^3*g)/(945*e^2*(b*e - 2*c*d)^

5) - (d*((8*c^5*(177*b^2*e^2*g + 544*c^2*d^2*g + 44*b*c*e^2*f - 76*c^2*d*e*f - 620*b*c*d*e*g))/(945*(b*e - 2*c

*d)^5) - (d*((32*c^6*e*(11*b*e*g - 19*c*d*g + c*e*f))/(945*(b*e - 2*c*d)^5) - (32*c^7*d*e*g)/(945*(b*e - 2*c*d

)^5)))/e))/e))/e - (18432*c^7*d^4*g + 1328*b^3*c^4*e^4*f + 1904*b^4*c^3*e^4*g - 5632*c^7*d^3*e*f - 42880*b*c^6

*d^3*e*g + 10624*b*c^6*d^2*e^2*f - 6552*b^2*c^5*d*e^3*f - 13728*b^3*c^4*d*e^3*g + 36672*b^2*c^5*d^2*e^2*g)/(94

5*e^2*(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) - (((d*((1984*b^2*c^5*e^4*f + 5

056*b^3*c^4*e^4*g + 6432*c^7*d^2*e^2*f - 33280*c^7*d^3*e*g - 7136*b*c^6*d*e^3*f + 53136*b*c^6*d^2*e^2*g - 2835

2*b^2*c^5*d*e^3*g)/(945*e^2*(b*e - 2*c*d)^5) - (d*((16*c^5*(124*b^2*e^2*g + 402*c^2*d^2*g + 25*b*c*e^2*f - 44*

c^2*d*e*f - 446*b*c*d*e*g))/(945*(b*e - 2*c*d)^5) - (d*((16*c^6*e*(25*b*e*g - 44*c*d*g + 2*c*e*f))/(945*(b*e -

2*c*d)^5) - (32*c^7*d*e*g)/(945*(b*e - 2*c*d)^5)))/e))/e))/e - (28288*c^7*d^4*g + 1712*b^3*c^4*e^4*f + 3440*b

^4*c^3*e^4*g - 6528*c^7*d^3*e*f - 69952*b*c^6*d^3*e*g + 13008*b*c^6*d^2*e^2*f - 8288*b^2*c^5*d*e^3*f - 24176*b

^3*c^4*d*e^3*g + 62496*b^2*c^5*d^2*e^2*g)/(945*e^2*(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) - (((d*((2256*b^2*c^5*e^4*f + 5952*b^3*c^4*e^4*g + 7392*c^7*d^2*e^2*f - 39424*c^7*d^3*e*g - 8160

*b*c^6*d*e^3*f + 62832*b*c^6*d^2*e^2*g - 33456*b^2*c^5*d*e^3*g)/(945*e^2*(b*e - 2*c*d)^5) - (d*((16*c^5*(47*b^

2*e^2*g + 154*c^2*d^2*g + 9*b*c*e^2*f - 16*c^2*d*e*f - 170*b*c*d*e*g))/(315*(b*e - 2*c*d)^5) - (d*((16*c^6*e*(

27*b*e*g - 48*c*d*g + 2*c*e*f))/(945*(b*e - 2*c*d)^5) - (32*c^7*d*e*g)/(945*(b*e - 2*c*d)^5)))/e))/e))/e - (27

392*c^7*d^4*g + 1504*b^3*c^4*e^4*f + 3936*b^4*c^3*e^4*g - 3840*c^7*d^3*e*f - 72576*b*c^6*d^3*e*g + 9456*b*c^6*

d^2*e^2*f - 6768*b^2*c^5*d*e^3*f - 27040*b^3*c^4*d*e^3*g + 67776*b^2*c^5*d^2*e^2*g)/(945*e^2*(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) - (((d*((2592*b^2*c^5*e^4*f + 7200*b^3*c^4*e^4*g + 860

8*c^7*d^2*e^2*f - 48128*c^7*d^3*e*g - 9440*b*c^6*d*e^3*f + 76496*b*c^6*d^2*e^2*g - 40608*b^2*c^5*d*e^3*g)/(945

*e^2*(b*e - 2*c*d)^5) - (d*((16*c^5*(162*b^2*e^2*g + 538*c^2*d^2*g + 29*b*c*e^2*f - 52*c^2*d*e*f - 590*b*c*d*e

*g))/(945*(b*e - 2*c*d)^5) - (d*((16*c^6*e*(29*b*e*g - 52*c*d*g + 2*c*e*f))/(945*(b*e - 2*c*d)^5) - (32*c^7*d*

e*g)/(945*(b*e - 2*c*d)^5)))/e))/e))/e - (57856*c^7*d^4*g + 2784*b^3*c^4*e^4*f + 5824*b^4*c^3*e^4*g - 12800*c^

7*d^3*e*f - 133376*b*c^6*d^3*e*g + 23504*b*c^6*d^2*e^2*f - 14112*b^2*c^5*d*e^3*f - 42176*b^3*c^4*d*e^3*g + 113

280*b^2*c^5*d^2*e^2*g)/(945*e^2*(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*b^4*e^3*g - 16*b*c^3*d^3*g + 6*b^3*c*e^3*f + 14*b*c^3*d^2*e*f - 12*b^3*c*d*e^2*g - 18*b^2*c^2*d*e^2*f + 24*b

^2*c^2*d^2*e*g)/(9*(7*b*e^2 - 14*c*d*e)*(b*e - 2*c*d)) - (d*((18*b^2*c^2*e^3*f - 32*c^4*d^3*g + 10*b^3*c*e^3*g

+ 28*c^4*d^2*e*f - 44*b*c^3*d*e^2*f + 62*b*c^3*d^2*e*g - 42*b^2*c^2*d*e^2*g)/(9*(7*b*e^2 - 14*c*d*e)*(b*e - 2

*c*d)) + (d*((d*((2*c^3*e^2*(7*b*e*g - 8*c*d*g + 2*c*e*f))/(9*(7*b*e^2 - 14*c*d*e)*(b*e - 2*c*d)) - (4*c^4*d*e

^2*g)/(9*(7*b*e^2 - 14*c*d*e)*(b*e - 2*c*d))))/e - (18*b^2*c^2*e^3*g + 14*b*c^3*e^3*f - 16*c^4*d*e^2*f + 28*c^

4*d^2*e*g - 44*b*c^3*d*e^2*g)/(9*(7*b*e^2 - 14*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)^4 - (((d*((896*b^2*c^4*e^3*f - 11264*c^6*d^3*g + 1808*b^3*c^3*e^3*g + 2832*c^6*d^2*e*f

- 3184*b*c^5*d*e^2*f + 18312*b*c^5*d^2*e*g - 9952*b^2*c^4*d*e^2*g)/(315*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^3) +

(d*((d*((8*c^5*e^2*(25*b*e*g - 44*c*d*g + 2*c*e*f))/(315*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^3) - (16*c^6*d*e^2

*g)/(315*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^3)))/e - (896*b^2*c^4*e^3*g + 200*b*c^5*e^3*f - 352*c^6*d*e^2*f + 2

832*c^6*d^2*e*g - 3184*b*c^5*d*e^2*g)/(315*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^3)))/e))/e - (8*b*c^2*(88*b^3*e^3

*g - 704*c^3*d^3*g + 50*b^2*c*e^3*f + 177*c^3*d^2*e*f - 188*b*c^2*d*e^2*f + 1056*b*c^2*d^2*e*g - 528*b^2*c*d*e

^2*g))/(315*(3*b*e^2 - 6*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)^2 + (

((d*((d*((d*((8*c^4*e^2*(12*b*e*g - 21*c*d*g + c*e*f))/(63*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d)^2) - (8*c^5*d*e^

2*g)/(63*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d)^2)))/e - (8*c^3*e*(50*b^2*e^2*g + 155*c^2*d^2*g + 12*b*c*e^2*f - 2

1*c^2*d*e*f - 176*b*c*d*e*g))/(63*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d)^2)))/e + (400*b^2*c^3*e^4*f + 704*b^3*c^2

*e^4*g + 1240*c^5*d^2*e^2*f - 4216*c^5*d^3*e*g - 1408*b*c^4*d*e^3*f + 6944*b*c^4*d^2*e^2*g - 3824*b^2*c^3*d*e^

3*g)/(63*e*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d)^2)))/e - (8*c*(b*e - c*d)*(49*b^3*e^3*g - 392*c^3*d^3*g + 39*b^2

*c*e^3*f + 135*c^3*d^2*e*f - 145*b*c^2*d*e^2*f + 588*b*c^2*d^2*e*g - 294*b^2*c*d*e^2*g))/(63*e*(5*b*e^2 - 10*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)^3 + (((d*((d*((d*((16*c^5*e^2*(

7*b*e*g - 11*c*d*g + c*e*f))/(315*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^3) - (16*c^6*d*e^2*g)/(315*(3*b*e^2 - 6*c*

d*e)*(b*e - 2*c*d)^3)))/e - (4*c^4*e*(73*b^2*e^2*g + 192*c^2*d^2*g + 28*b*c*e^2*f - 44*c^2*d*e*f - 236*b*c*d*e

*g))/(315*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^3)))/e + (292*b^2*c^4*e^4*f + 376*b^3*c^3*e^4*g + 768*c^6*d^2*e^2*

f - 2048*c^6*d^3*e*g - 944*b*c^5*d*e^3*f + 3456*b*c^5*d^2*e^2*g - 1964*b^2*c^4*d*e^3*g)/(315*e*(3*b*e^2 - 6*c*

d*e)*(b*e - 2*c*d)^3)))/e - (1824*c^6*d^4*g + 228*b^3*c^3*e^4*f + 188*b^4*c^2*e^4*g - 864*c^6*d^3*e*f - 4240*b

*c^5*d^3*e*g + 1680*b*c^5*d^2*e^2*f - 1076*b^2*c^4*d*e^3*f - 1356*b^3*c^3*d*e^3*g + 3624*b^2*c^4*d^2*e^2*g)/(3

15*e*(3*b*e^2 - 6*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)^2 - (((d*((d

*((d*((8*c^5*e^2*(21*b*e*g - 36*c*d*g + 2*c*e*f))/(315*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^3) - (16*c^6*d*e^2*g)

/(315*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^3)))/e - (8*c^4*e*(82*b^2*e^2*g + 250*c^2*d^2*g + 21*b*c*e^2*f - 36*c^

2*d*e*f - 286*b*c*d*e*g))/(315*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^3)))/e + (656*b^2*c^4*e^4*f + 1216*b^3*c^3*e^

4*g + 2000*c^6*d^2*e^2*f - 7424*c^6*d^3*e*g - 2288*b*c^5*d*e^3*f + 12136*b*c^5*d^2*e^2*g - 6640*b^2*c^4*d*e^3*

g)/(315*e*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^3)))/e - (5312*c^6*d^4*g + 456*b^3*c^3*e^4*f + 712*b^4*c^2*e^4*g -

1344*c^6*d^3*e*f - 13664*b*c^5*d^3*e*g + 3016*b*c^5*d^2*e^2*f - 2080*b^2*c^4*d*e^3*f - 4936*b^3*c^3*d*e^3*g +

12528*b^2*c^4*d^2*e^2*g)/(315*e*(3*b*e^2 - 6*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)^2 - (((d*((d*((d*((8*c^5*e^2*(23*b*e*g - 40*c*d*g + 2*c*e*f))/(315*(3*b*e^2 - 6*c*d*e)*(b*e -

2*c*d)^3) - (16*c^6*d*e^2*g)/(315*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^3)))/e - (8*c^4*e*(95*b^2*e^2*g + 294*c^2*

d^2*g + 23*b*c*e^2*f - 40*c^2*d*e*f - 334*b*c*d*e*g))/(315*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^3)))/e + (760*b^2

*c^4*e^4*f + 1456*b^3*c^3*e^4*g + 2352*c^6*d^2*e^2*f - 8960*c^6*d^3*e*g - 2672*b*c^5*d*e^3*f + 14616*b*c^5*d^2

*e^2*g - 7976*b^2*c^4*d*e^3*g)/(315*e*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^3)))/e - (9600*c^6*d^4*g + 768*b^3*c^3

*e^4*f + 944*b^4*c^2*e^4*g - 3456*c^6*d^3*e*f - 21952*b*c^5*d^3*e*g + 6360*b*c^5*d^2*e^2*f - 3848*b^2*c^4*d*e^

3*f - 6864*b^3*c^3*d*e^3*g + 18528*b^2*c^4*d^2*e^2*g)/(315*e*(3*b*e^2 - 6*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)^2 + (((d*((3712*b^2*c^5*e^4*f + 13120*b^3*c^4*e^4*g + 12768*c^7*d^

2*e^2*f - 91168*c^7*d^3*e*g - 13760*b*c^6*d*e^3*f + 143136*b*c^6*d^2*e^2*g - 75008*b^2*c^5*d*e^3*g)/(945*e^2*(

b*e - 2*c*d)^5) - (d*((32*c^5*(116*b^2*e^2*g + 399*c^2*d^2*g + 17*b*c*e^2*f - 31*c^2*d*e*f - 430*b*c*d*e*g))/(

945*(b*e - 2*c*d)^5) - (d*((32*c^6*e*(17*b*e*g - 31*c*d*g + c*e*f))/(945*(b*e - 2*c*d)^5) - (32*c^7*d*e*g)/(94

5*(b*e - 2*c*d)^5)))/e))/e))/e - (32*c^3*(b*e - c*d)*(310*b^3*e^3*g - 2480*c^3*d^3*g + 100*b^2*c*e^3*f + 369*c

^3*d^2*e*f - 384*b*c^2*d*e^2*f + 3720*b*c^2*d^2*e*g - 1860*b^2*c*d*e^2*g))/(945*e^2*(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) + (((d*((d*((d*((4*c^3*e^2*(8*b*e*g - 13*c*d*g + c*e*f))/(9*(7*

b*e^2 - 14*c*d*e)*(b*e - 2*c*d)) - (4*c^4*d*e^2*g)/(9*(7*b*e^2 - 14*c*d*e)*(b*e - 2*c*d))))/e - (4*c^2*e*(18*b

^2*e^2*g + 43*c^2*d^2*g + 8*b*c*e^2*f - 13*c^2*d*e*f - 56*b*c*d*e*g))/(9*(7*b*e^2 - 14*c*d*e)*(b*e - 2*c*d))))

/e + (72*b^2*c^2*e^4*f + 172*c^4*d^2*e^2*f + 64*b^3*c*e^4*g - 284*c^4*d^3*e*g - 224*b*c^3*d*e^3*f + 512*b*c^3*

d^2*e^2*g - 312*b^2*c^2*d*e^3*g)/(9*e*(7*b*e^2 - 14*c*d*e)*(b*e - 2*c*d))))/e - (4*(b*e - c*d)*(5*b^3*e^3*g -

40*c^3*d^3*g + 11*b^2*c*e^3*f + 31*c^3*d^2*e*f - 37*b*c^2*d*e^2*f + 60*b*c^2*d^2*e*g - 30*b^2*c*d*e^2*g))/(9*e

*(7*b*e^2 - 14*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)^4 - (((d*((132*b^

2*c^4*e^3*f - 608*c^6*d^3*g + 126*b^3*c^3*e^3*g + 288*c^6*d^2*e*f - 384*b*c^5*d*e^2*f + 1056*b*c^5*d^2*e*g - 6

24*b^2*c^4*d*e^2*g)/(315*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^3) + (d*((d*((8*c^5*e^2*(9*b*e*g - 12*c*d*g + 2*c*e

*f))/(315*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^3) - (16*c^6*d*e^2*g)/(315*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^3)))/

e - (4*c^4*e*(11*b^2*e^2*g + 24*c^2*d^2*g + 6*b*c*e^2*f - 8*c^2*d*e*f - 32*b*c*d*e*g))/(105*(3*b*e^2 - 6*c*d*e

)*(b*e - 2*c*d)^3)))/e))/e - (50*b^3*c^3*e^3*f + 38*b^4*c^2*e^3*g - 304*b*c^5*d^3*g + 144*b*c^5*d^2*e*f - 168*

b^2*c^4*d*e^2*f + 456*b^2*c^4*d^2*e*g - 228*b^3*c^3*d*e^2*g)/(315*(3*b*e^2 - 6*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)^2 + (((d*((420*b^2*c^4*e^3*f - 3200*c^6*d^3*g + 576*b^3*c^3*e

^3*g + 1152*c^6*d^2*e*f - 1392*b*c^5*d*e^2*f + 5376*b*c^5*d^2*e*g - 3036*b^2*c^4*d*e^2*g)/(315*(3*b*e^2 - 6*c*

d*e)*(b*e - 2*c*d)^3) + (d*((d*((16*c^5*e^2*(9*b*e*g - 15*c*d*g + c*e*f))/(315*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*

d)^3) - (16*c^6*d*e^2*g)/(315*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^3)))/e - (4*c^4*e*(35*b^2*e^2*g + 96*c^2*d^2*g

+ 12*b*c*e^2*f - 20*c^2*d*e*f - 116*b*c*d*e*g))/(105*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^3)))/e))/e - (176*b^3*

c^3*e^3*f + 200*b^4*c^2*e^3*g - 1600*b*c^5*d^3*g + 576*b*c^5*d^2*e*f - 636*b^2*c^4*d*e^2*f + 2400*b^2*c^4*d^2*

e*g - 1200*b^3*c^3*d*e^2*g)/(315*(3*b*e^2 - 6*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)^2 - (((2*f*(b*e - c*d)^3)/(9*b*e^2 - 18*c*d*e) - (d*((2*(b*e - c*d)^2*(b*e*g - c*d*g + 3*c*e*f

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

g)/(9*b*e^2 - 18*c*d*e)))/e - (6*c*e*(b*e - c*d)*(b*e*g - c*d*g + c*e*f))/(9*b*e^2 - 18*c*d*e)))/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*((d*((16*c^5*e^2*(15*b*e*g - 27*c*d*g + c*e*f))/

(315*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^3) - (16*c^6*d*e^2*g)/(315*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^3)))/e - (

16*c^4*e*(86*b^2*e^2*g + 287*c^2*d^2*g + 15*b*c*e^2*f - 27*c^2*d*e*f - 314*b*c*d*e*g))/(315*(3*b*e^2 - 6*c*d*e

)*(b*e - 2*c*d)^3)))/e + (1376*b^2*c^4*e^4*f + 3808*b^3*c^3*e^4*g + 4592*c^6*d^2*e^2*f - 25424*c^6*d^3*e*g - 5

024*b*c^5*d*e^3*f + 40432*b*c^5*d^2*e^2*g - 21472*b^2*c^4*d*e^3*g)/(315*e*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^3)

))/e - (16*c^2*(b*e - c*d)*(166*b^3*e^3*g - 1328*c^3*d^3*g + 72*b^2*c*e^3*f + 261*c^3*d^2*e*f - 274*b*c^2*d*e^

2*f + 1992*b*c^2*d^2*e*g - 996*b^2*c*d*e^2*g))/(315*e*(3*b*e^2 - 6*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)^2 + (((d*((1992*b^2*c^5*e^3*f - 28928*c^7*d^3*g + 4512*b^3*c^4*e^3*g + 64

00*c^7*d^2*e*f - 7136*b*c^6*d*e^2*f + 46592*b*c^6*d^2*e*g - 25080*b^2*c^5*d*e^2*g)/(945*e*(b*e - 2*c*d)^5) - (

d*((1992*b^2*c^5*e^3*g + 416*b*c^6*e^3*f - 736*c^7*d*e^2*f + 6400*c^7*d^2*e*g - 7136*b*c^6*d*e^2*g)/(945*e*(b*

e - 2*c*d)^5) - (d*((32*c^6*e*(13*b*e*g - 23*c*d*g + c*e*f))/(945*(b*e - 2*c*d)^5) - (32*c^7*d*e*g)/(945*(b*e

- 2*c*d)^5)))/e))/e))/e - (8*b*c^3*(226*b^3*e^3*g - 1808*c^3*d^3*g + 112*b^2*c*e^3*f + 400*c^3*d^2*e*f - 423*b

*c^2*d*e^2*f + 2712*b*c^2*d^2*e*g - 1356*b^2*c*d*e^2*g))/(945*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) + (((d*((50*b^2*c^3*e^3*f - 160*c^5*d^3*g + 38*b^3*c^2*e^3*g + 96*c^5*d^2*e*f - 136

*b*c^4*d*e^2*f + 288*b*c^4*d^2*e*g - 178*b^2*c^3*d*e^2*g)/(63*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d)^2) + (d*((d*(

(8*c^4*e^2*(4*b*e*g - 5*c*d*g + c*e*f))/(63*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d)^2) - (8*c^5*d*e^2*g)/(63*(5*b*e

^2 - 10*c*d*e)*(b*e - 2*c*d)^2)))/e - (50*b^2*c^3*e^3*g + 32*b*c^4*e^3*f - 40*c^5*d*e^2*f + 96*c^5*d^2*e*g - 1

36*b*c^4*d*e^2*g)/(63*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d)^2)))/e))/e - (18*b^3*c^2*e^3*f - 80*b*c^4*d^3*g + 10*

b^4*c*e^3*g + 48*b*c^4*d^2*e*f - 58*b^2*c^3*d*e^2*f + 120*b^2*c^3*d^2*e*g - 60*b^3*c^2*d*e^2*g)/(63*(5*b*e^2 -

10*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)^3 - (((d*((176*b^2*c^3*e^3

*f - 1024*c^5*d^3*g + 200*b^3*c^2*e^3*g + 456*c^5*d^2*e*f - 568*b*c^4*d*e^2*f + 1764*b*c^4*d^2*e*g - 1024*b^2*

c^3*d*e^2*g)/(63*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d)^2) + (d*((d*((4*c^4*e^2*(17*b*e*g - 28*c*d*g + 2*c*e*f))/(

63*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d)^2) - (8*c^5*d*e^2*g)/(63*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d)^2)))/e - (17

6*b^2*c^3*e^3*g + 68*b*c^4*e^3*f - 112*c^5*d*e^2*f + 456*c^5*d^2*e*g - 568*b*c^4*d*e^2*g)/(63*(5*b*e^2 - 10*c*

d*e)*(b*e - 2*c*d)^2)))/e))/e - (72*b^3*c^2*e^3*f - 512*b*c^4*d^3*g + 64*b^4*c*e^3*g + 228*b*c^4*d^2*e*f - 256

*b^2*c^3*d*e^2*f + 768*b^2*c^3*d^2*e*g - 384*b^3*c^2*d*e^2*g)/(63*(5*b*e^2 - 10*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)^3

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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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)**(5/2)/(e*x+d)**8,x)

[Out]

Timed out

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