3.20.51 \(\int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^7} \, dx\)

Optimal. Leaf size=360 \[ -\frac {32 c^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-11 b e g+14 c d g+8 c e f)}{3465 e^2 (d+e x)^3 (2 c d-b e)^5}-\frac {16 c^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-11 b e g+14 c d g+8 c e f)}{1155 e^2 (d+e x)^4 (2 c d-b e)^4}-\frac {4 c \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-11 b e g+14 c d g+8 c e f)}{231 e^2 (d+e x)^5 (2 c d-b e)^3}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-11 b e g+14 c d g+8 c e f)}{99 e^2 (d+e x)^6 (2 c d-b e)^2}-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{11 e^2 (d+e x)^7 (2 c d-b e)} \]

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Rubi [A]  time = 0.58, antiderivative size = 360, normalized size of antiderivative = 1.00, number of steps used = 5, 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 {32 c^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-11 b e g+14 c d g+8 c e f)}{3465 e^2 (d+e x)^3 (2 c d-b e)^5}-\frac {16 c^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-11 b e g+14 c d g+8 c e f)}{1155 e^2 (d+e x)^4 (2 c d-b e)^4}-\frac {4 c \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-11 b e g+14 c d g+8 c e f)}{231 e^2 (d+e x)^5 (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} (-11 b e g+14 c d g+8 c e f)}{99 e^2 (d+e x)^6 (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}}{11 e^2 (d+e x)^7 (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)^7,x]

[Out]

(-2*(e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(11*e^2*(2*c*d - b*e)*(d + e*x)^7) - (2*(8*c*e*f
+ 14*c*d*g - 11*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(99*e^2*(2*c*d - b*e)^2*(d + e*x)^6) - (4*
c*(8*c*e*f + 14*c*d*g - 11*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(231*e^2*(2*c*d - b*e)^3*(d + e
*x)^5) - (16*c^2*(8*c*e*f + 14*c*d*g - 11*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(1155*e^2*(2*c*d
- b*e)^4*(d + e*x)^4) - (32*c^3*(8*c*e*f + 14*c*d*g - 11*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/
(3465*e^2*(2*c*d - b*e)^5*(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)^7} \, 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}}{11 e^2 (2 c d-b e) (d+e x)^7}+\frac {(8 c e f+14 c d g-11 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}{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 )^{3/2}}{11 e^2 (2 c d-b e) (d+e x)^7}-\frac {2 (8 c e f+14 c d g-11 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{99 e^2 (2 c d-b e)^2 (d+e x)^6}+\frac {(2 c (8 c e f+14 c d g-11 b e g)) \int \frac {\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^5} \, dx}{33 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}}{11 e^2 (2 c d-b e) (d+e x)^7}-\frac {2 (8 c e f+14 c d g-11 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{99 e^2 (2 c d-b e)^2 (d+e x)^6}-\frac {4 c (8 c e f+14 c d g-11 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{231 e^2 (2 c d-b e)^3 (d+e x)^5}+\frac {\left (8 c^2 (8 c e f+14 c d g-11 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}{231 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}}{11 e^2 (2 c d-b e) (d+e x)^7}-\frac {2 (8 c e f+14 c d g-11 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{99 e^2 (2 c d-b e)^2 (d+e x)^6}-\frac {4 c (8 c e f+14 c d g-11 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{231 e^2 (2 c d-b e)^3 (d+e x)^5}-\frac {16 c^2 (8 c e f+14 c d g-11 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{1155 e^2 (2 c d-b e)^4 (d+e x)^4}+\frac {\left (16 c^3 (8 c e f+14 c d g-11 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}{1155 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}}{11 e^2 (2 c d-b e) (d+e x)^7}-\frac {2 (8 c e f+14 c d g-11 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{99 e^2 (2 c d-b e)^2 (d+e x)^6}-\frac {4 c (8 c e f+14 c d g-11 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{231 e^2 (2 c d-b e)^3 (d+e x)^5}-\frac {16 c^2 (8 c e f+14 c d g-11 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{1155 e^2 (2 c d-b e)^4 (d+e x)^4}-\frac {32 c^3 (8 c e f+14 c d g-11 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3465 e^2 (2 c d-b e)^5 (d+e x)^3}\\ \end {align*}

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Mathematica [A]  time = 0.26, size = 335, normalized size = 0.93 \begin {gather*} \frac {2 ((d+e x) (c (d-e x)-b e))^{3/2} \left (35 b^4 e^4 (2 d g+9 e f+11 e g x)-10 b^3 c e^3 \left (61 d^2 g+d e (280 f+346 g x)+e^2 x (28 f+33 g x)\right )+12 b^2 c^2 e^2 \left (167 d^3 g+d^2 e (790 f+986 g x)+d e^2 x (180 f+211 g x)+2 e^3 x^2 (10 f+11 g x)\right )-8 b c^3 e \left (365 d^4 g+2 d^3 e (912 f+1141 g x)+3 d^2 e^2 x (244 f+277 g x)+4 d e^3 x^2 (48 f+49 g x)+2 e^4 x^3 (12 f+11 g x)\right )+16 c^4 \left (91 d^5 g+d^4 e (547 f+637 g x)+7 d^3 e^2 x (52 f+45 g x)+2 d^2 e^3 x^2 (90 f+49 g x)+14 d e^4 x^3 (4 f+g x)+8 e^5 f x^4\right )\right )}{3465 e^2 (d+e x)^7 (b e-2 c d)^5} \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)^7,x]

[Out]

(2*((d + e*x)*(-(b*e) + c*(d - e*x)))^(3/2)*(35*b^4*e^4*(9*e*f + 2*d*g + 11*e*g*x) - 10*b^3*c*e^3*(61*d^2*g +
e^2*x*(28*f + 33*g*x) + d*e*(280*f + 346*g*x)) + 16*c^4*(91*d^5*g + 8*e^5*f*x^4 + 14*d*e^4*x^3*(4*f + g*x) + 7
*d^3*e^2*x*(52*f + 45*g*x) + 2*d^2*e^3*x^2*(90*f + 49*g*x) + d^4*e*(547*f + 637*g*x)) + 12*b^2*c^2*e^2*(167*d^
3*g + 2*e^3*x^2*(10*f + 11*g*x) + d*e^2*x*(180*f + 211*g*x) + d^2*e*(790*f + 986*g*x)) - 8*b*c^3*e*(365*d^4*g
+ 2*e^4*x^3*(12*f + 11*g*x) + 4*d*e^3*x^2*(48*f + 49*g*x) + 3*d^2*e^2*x*(244*f + 277*g*x) + 2*d^3*e*(912*f + 1
141*g*x))))/(3465*e^2*(-2*c*d + b*e)^5*(d + e*x)^7)

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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)^7,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)^7,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)^7,x, algorithm="giac")

[Out]

Timed out

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maple [A]  time = 0.06, size = 564, normalized size = 1.57 \begin {gather*} -\frac {2 \left (c e x +b e -c d \right ) \left (-176 b \,c^{3} e^{5} g \,x^{4}+224 c^{4} d \,e^{4} g \,x^{4}+128 c^{4} e^{5} f \,x^{4}+264 b^{2} c^{2} e^{5} g \,x^{3}-1568 b \,c^{3} d \,e^{4} g \,x^{3}-192 b \,c^{3} e^{5} f \,x^{3}+1568 c^{4} d^{2} e^{3} g \,x^{3}+896 c^{4} d \,e^{4} f \,x^{3}-330 b^{3} c \,e^{5} g \,x^{2}+2532 b^{2} c^{2} d \,e^{4} g \,x^{2}+240 b^{2} c^{2} e^{5} f \,x^{2}-6648 b \,c^{3} d^{2} e^{3} g \,x^{2}-1536 b \,c^{3} d \,e^{4} f \,x^{2}+5040 c^{4} d^{3} e^{2} g \,x^{2}+2880 c^{4} d^{2} e^{3} f \,x^{2}+385 b^{4} e^{5} g x -3460 b^{3} c d \,e^{4} g x -280 b^{3} c \,e^{5} f x +11832 b^{2} c^{2} d^{2} e^{3} g x +2160 b^{2} c^{2} d \,e^{4} f x -18256 b \,c^{3} d^{3} e^{2} g x -5856 b \,c^{3} d^{2} e^{3} f x +10192 c^{4} d^{4} e g x +5824 c^{4} d^{3} e^{2} f x +70 b^{4} d \,e^{4} g +315 b^{4} e^{5} f -610 b^{3} c \,d^{2} e^{3} g -2800 b^{3} c d \,e^{4} f +2004 b^{2} c^{2} d^{3} e^{2} g +9480 b^{2} c^{2} d^{2} e^{3} f -2920 b \,c^{3} d^{4} e g -14592 b \,c^{3} d^{3} e^{2} f +1456 c^{4} d^{5} g +8752 c^{4} d^{4} e f \right ) \sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}}{3465 \left (e x +d \right )^{6} \left (b^{5} e^{5}-10 b^{4} c d \,e^{4}+40 b^{3} c^{2} d^{2} e^{3}-80 b^{2} c^{3} d^{3} e^{2}+80 b \,c^{4} d^{4} e -32 c^{5} d^{5}\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)^7,x)

[Out]

-2/3465*(c*e*x+b*e-c*d)*(-176*b*c^3*e^5*g*x^4+224*c^4*d*e^4*g*x^4+128*c^4*e^5*f*x^4+264*b^2*c^2*e^5*g*x^3-1568
*b*c^3*d*e^4*g*x^3-192*b*c^3*e^5*f*x^3+1568*c^4*d^2*e^3*g*x^3+896*c^4*d*e^4*f*x^3-330*b^3*c*e^5*g*x^2+2532*b^2
*c^2*d*e^4*g*x^2+240*b^2*c^2*e^5*f*x^2-6648*b*c^3*d^2*e^3*g*x^2-1536*b*c^3*d*e^4*f*x^2+5040*c^4*d^3*e^2*g*x^2+
2880*c^4*d^2*e^3*f*x^2+385*b^4*e^5*g*x-3460*b^3*c*d*e^4*g*x-280*b^3*c*e^5*f*x+11832*b^2*c^2*d^2*e^3*g*x+2160*b
^2*c^2*d*e^4*f*x-18256*b*c^3*d^3*e^2*g*x-5856*b*c^3*d^2*e^3*f*x+10192*c^4*d^4*e*g*x+5824*c^4*d^3*e^2*f*x+70*b^
4*d*e^4*g+315*b^4*e^5*f-610*b^3*c*d^2*e^3*g-2800*b^3*c*d*e^4*f+2004*b^2*c^2*d^3*e^2*g+9480*b^2*c^2*d^2*e^3*f-2
920*b*c^3*d^4*e*g-14592*b*c^3*d^3*e^2*f+1456*c^4*d^5*g+8752*c^4*d^4*e*f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2
)/(e*x+d)^6/e^2/(b^5*e^5-10*b^4*c*d*e^4+40*b^3*c^2*d^2*e^3-80*b^2*c^3*d^3*e^2+80*b*c^4*d^4*e-32*c^5*d^5)

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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)^7,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 = 28.09, size = 10084, normalized size = 28.01

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)^7,x)

[Out]

(((d*((8*c^2*(12*b*e*g - 23*c*d*g + c*e*f))/(99*(7*b*e^2 - 14*c*d*e)*(b*e - 2*c*d)^2) - (8*c^3*d*g)/(99*(7*b*e
^2 - 14*c*d*e)*(b*e - 2*c*d)^2)))/e - (8*c*(b*e - c*d)*(11*b*e*g - 22*c*d*g + c*e*f))/(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*((64*c^6*e*f - 1152*c^6*d
*g + 608*b*c^5*e*g)/(10395*e*(b*e - 2*c*d)^6) - (64*c^6*d*g)/(10395*e*(b*e - 2*c*d)^6)))/e - (32*b*c^4*(9*b*e*
g - 18*c*d*g + c*e*f))/(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*((64*c^6*e*f - 1280*c^6*d*g + 672*b*c^5*e*g)/(10395*e*(b*e - 2*c*d)^6) - (64*c^6*d*g)/(10395*e*(b*e - 2*c*d)
^6)))/e - (32*b*c^4*(10*b*e*g - 20*c*d*g + c*e*f))/(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) - (2*((d*((64*c^6*e*f - 1408*c^6*d*g + 736*b*c^5*e*g)/(10395*e*(b*e - 2*c*d)^6) - (64*c
^6*d*g)/(10395*e*(b*e - 2*c*d)^6)))/e - (32*b*c^4*(11*b*e*g - 22*c*d*g + c*e*f))/(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*((64*c^6*e*f - 1536*c^6*d*g + 800*b*c^5*e*g)/(1039
5*e*(b*e - 2*c*d)^6) - (64*c^6*d*g)/(10395*e*(b*e - 2*c*d)^6)))/e - (32*b*c^4*(12*b*e*g - 24*c*d*g + c*e*f))/(
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*((64*c^6*e*f - 1664*c^
6*d*g + 864*b*c^5*e*g)/(10395*e*(b*e - 2*c*d)^6) - (64*c^6*d*g)/(10395*e*(b*e - 2*c*d)^6)))/e - (32*b*c^4*(13*
b*e*g - 26*c*d*g + c*e*f))/(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*((64*c^6*e*f - 2432*c^6*d*g + 1248*b*c^5*e*g)/(10395*e*(b*e - 2*c*d)^6) - (64*c^6*d*g)/(10395*e*(b*e - 2
*c*d)^6)))/e - (32*b*c^4*(19*b*e*g - 38*c*d*g + c*e*f))/(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*((4*c^2*e*f - 8*c^2*d*g + 6*b*c*e*g)/(11*(9*b*e^2 - 18*c*d*e)*(b*e - 2*c*d)
) - (4*c^2*d*g)/(11*(9*b*e^2 - 18*c*d*e)*(b*e - 2*c*d))))/e - (2*b*(b*e*g - 2*c*d*g + c*e*f))/(11*(9*b*e^2 - 1
8*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*f*(b*e - c*d))/(11*b*
e^2 - 22*c*d*e) - (d*((2*b*e*g - 2*c*d*g + 2*c*e*f)/(11*b*e^2 - 22*c*d*e) - (2*c*d*g)/(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*((4*c^2*(13*b*e*g - 24*c*d*g + 2*c*e*f))/(9
9*(7*b*e^2 - 14*c*d*e)*(b*e - 2*c*d)^2) - (8*c^3*d*g)/(99*(7*b*e^2 - 14*c*d*e)*(b*e - 2*c*d)^2)))/e - (120*c^3
*d^2*g - 40*c^3*d*e*f + 24*b*c^2*e^2*f + 44*b^2*c*e^2*g - 148*b*c^2*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*((64*c^6*e*f - 384*c^6*d*g + 224*b*c
^5*e*g)/(10395*e*(b*e - 2*c*d)^6) - (64*c^6*d*g)/(10395*e*(b*e - 2*c*d)^6)))/e - (32*b*c^4*(3*b*e*g - 6*c*d*g
+ c*e*f))/(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*((16*c^4*e*
f - 64*c^4*d*g + 40*b*c^3*e*g)/(693*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d)^3) - (16*c^4*d*g)/(693*(5*b*e^2 - 10*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))/(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*((16*c^4*e*f - 208*c^4*d*g + 112*b*c^3*e*g)/(6
93*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d)^3) - (16*c^4*d*g)/(693*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d)^3)))/e - (4*b*
c^2*(13*b*e*g - 26*c*d*g + 2*c*e*f))/(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*((16*c^4*e*f - 240*c^4*d*g + 128*b*c^3*e*g)/(693*(5*b*e^2 - 10*c*d*e)*(b*e
- 2*c*d)^3) - (16*c^4*d*g)/(693*(5*b*e^2 - 10*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))/(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*((16*c^4*e*f - 384*c^4*d*g + 200*b*c^3*e*g)/(693*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d)^3) - (16*c^4*d*g)/(6
93*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d)^3)))/e - (8*b*c^2*(12*b*e*g - 24*c*d*g + c*e*f))/(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*((32*c^5*e*f - 160*c^5*d*
g + 96*b*c^4*e*g)/(3465*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^4) - (32*c^5*d*g)/(3465*(3*b*e^2 - 6*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))/(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*((32*c^5*e*f - 384*c^5*d*g + 208*b*c^4*e*g)/(3465*(3*b*e
^2 - 6*c*d*e)*(b*e - 2*c*d)^4) - (32*c^5*d*g)/(3465*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^4)))/e - (16*b*c^3*(6*b*
e*g - 12*c*d*g + c*e*f))/(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*((32*c^5*e*f - 448*c^5*d*g + 240*b*c^4*e*g)/(3465*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^4)
- (32*c^5*d*g)/(3465*(3*b*e^2 - 6*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))/(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*((32*c^5
*e*f - 512*c^5*d*g + 272*b*c^4*e*g)/(3465*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^4) - (32*c^5*d*g)/(3465*(3*b*e^2 -
6*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))/(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*((32*c^5*e*f - 672*c^5*d*g + 352*b*c^4*
e*g)/(3465*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^4) - (32*c^5*d*g)/(3465*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^4)))/e
- (8*b*c^3*(21*b*e*g - 42*c*d*g + 2*c*e*f))/(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*((32*c^5*e*f - 736*c^5*d*g + 384*b*c^4*e*g)/(3465*(3*b*e^2 - 6*c*d*e
)*(b*e - 2*c*d)^4) - (32*c^5*d*g)/(3465*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^4)))/e - (8*b*c^3*(23*b*e*g - 46*c*d
*g + 2*c*e*f))/(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*((32*c^5*e*f - 800*c^5*d*g + 416*b*c^4*e*g)/(3465*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^4) - (32*c^5*
d*g)/(3465*(3*b*e^2 - 6*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))/(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*((32*c^5*e*f - 1
024*c^5*d*g + 528*b*c^4*e*g)/(3465*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^4) - (32*c^5*d*g)/(3465*(3*b*e^2 - 6*c*d*
e)*(b*e - 2*c*d)^4)))/e - (16*b*c^3*(16*b*e*g - 32*c*d*g + c*e*f))/(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*((64*c^5*(6*b*e*g - 11*c*d*g + c*e*f))/(10395
*e*(b*e - 2*c*d)^6) - (64*c^6*d*g)/(10395*e*(b*e - 2*c*d)^6)))/e - (16*b*c^4*(11*b*e*g - 22*c*d*g + 2*c*e*f))/
(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*((64*c^5*(7*b*e*g - 1
3*c*d*g + c*e*f))/(10395*e*(b*e - 2*c*d)^6) - (64*c^6*d*g)/(10395*e*(b*e - 2*c*d)^6)))/e - (16*b*c^4*(13*b*e*g
- 26*c*d*g + 2*c*e*f))/(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*((64*c^5*(8*b*e*g - 15*c*d*g + c*e*f))/(10395*e*(b*e - 2*c*d)^6) - (64*c^6*d*g)/(10395*e*(b*e - 2*c*d)^6)))
/e - (16*b*c^4*(15*b*e*g - 30*c*d*g + 2*c*e*f))/(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*((64*c^5*(9*b*e*g - 17*c*d*g + c*e*f))/(10395*e*(b*e - 2*c*d)^6) - (64*c^6*d*g)/(10
395*e*(b*e - 2*c*d)^6)))/e - (16*b*c^4*(17*b*e*g - 34*c*d*g + 2*c*e*f))/(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*((64*c^5*(14*b*e*g - 27*c*d*g + c*e*f))/(10395*e*(b*e - 2*c
*d)^6) - (64*c^6*d*g)/(10395*e*(b*e - 2*c*d)^6)))/e - (16*b*c^4*(27*b*e*g - 54*c*d*g + 2*c*e*f))/(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*((64*c^5*(15*b*e*g - 29*c*d*g + c*
e*f))/(10395*e*(b*e - 2*c*d)^6) - (64*c^6*d*g)/(10395*e*(b*e - 2*c*d)^6)))/e - (16*b*c^4*(29*b*e*g - 58*c*d*g
+ 2*c*e*f))/(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*((64*c^5*
(16*b*e*g - 31*c*d*g + c*e*f))/(10395*e*(b*e - 2*c*d)^6) - (64*c^6*d*g)/(10395*e*(b*e - 2*c*d)^6)))/e - (16*b*
c^4*(31*b*e*g - 62*c*d*g + 2*c*e*f))/(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*((64*c^5*(17*b*e*g - 33*c*d*g + c*e*f))/(10395*e*(b*e - 2*c*d)^6) - (64*c^6*d*g)/(10395*e*(b*e
- 2*c*d)^6)))/e - (16*b*c^4*(33*b*e*g - 66*c*d*g + 2*c*e*f))/(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*((4*c*(7*b*e*g - 13*c*d*g + c*e*f))/(11*(9*b*e^2 - 18*c*d*e)*(b*e - 2
*c*d)) - (4*c^2*d*g)/(11*(9*b*e^2 - 18*c*d*e)*(b*e - 2*c*d))))/e - (4*(b*e - c*d)*(6*b*e*g - 12*c*d*g + c*e*f)
)/(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 + (((d*(
(16*c^3*(6*b*e*g - 11*c*d*g + c*e*f))/(693*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d)^3) - (16*c^4*d*g)/(693*(5*b*e^2
- 10*c*d*e)*(b*e - 2*c*d)^3)))/e - (256*c^4*d^2*g + 92*b^2*c^2*e^2*g - 64*c^4*d*e*f + 40*b*c^3*e^2*f - 312*b*c
^3*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*((8*c^3*(21*b*e*g - 40*c*d*g + 2*c*e*f))/(693*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d)^3) - (16*c^4*d*g)/(
693*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d)^3)))/e - (112*c^4*d^2*g + 152*b^2*c^2*e^2*g + 176*c^4*d*e*f - 80*b*c^3*
e^2*f - 360*b*c^3*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*((8*c^3*(23*b*e*g - 44*c*d*g + 2*c*e*f))/(693*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d)^3) -
(16*c^4*d*g)/(693*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d)^3)))/e - (960*c^4*d^2*g + 312*b^2*c^2*e^2*g - 64*c^4*d*e
*f + 40*b*c^3*e^2*f - 1104*b*c^3*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*((32*c^4*(9*b*e*g - 17*c*d*g + c*e*f))/(3465*(3*b*e^2 - 6*c*d*e)*(b*e
- 2*c*d)^4) - (32*c^5*d*g)/(3465*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^4)))/e - (128*c^5*d^2*g + 208*b^2*c^3*e^2*g
+ 160*c^5*d*e*f - 64*b*c^4*e^2*f - 480*b*c^4*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 - (((d*((16*c^4*(11*b*e*g - 20*c*d*g + 2*c*e*f))/(3465*(3*b*e^2
- 6*c*d*e)*(b*e - 2*c*d)^4) - (32*c^5*d*g)/(3465*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^4)))/e - (480*c^5*d^2*g + 1
76*b^2*c^3*e^2*g - 96*c^5*d*e*f + 64*b*c^4*e^2*f - 592*b*c^4*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 - (((d*((16*c^4*(29*b*e*g - 56*c*d*g + 2*c*e*f))/
(3465*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^4) - (32*c^5*d*g)/(3465*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^4)))/e + (16
32*c^5*d^2*g + 144*b^2*c^3*e^2*g - 160*c^5*d*e*f + 64*b*c^4*e^2*f - 1104*b*c^4*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 + (((d*((32*c^4*(10*b*e*g - 19*
c*d*g + c*e*f))/(3465*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^4) - (32*c^5*d*g)/(3465*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c
*d)^4)))/e - (1344*c^5*d^2*g + 464*b^2*c^3*e^2*g - 96*c^5*d*e*f + 64*b*c^4*e^2*f - 1600*b*c^4*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 + (((d*((32*c^4*
(11*b*e*g - 21*c*d*g + c*e*f))/(3465*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^4) - (32*c^5*d*g)/(3465*(3*b*e^2 - 6*c*
d*e)*(b*e - 2*c*d)^4)))/e - (1536*c^5*d^2*g + 528*b^2*c^3*e^2*g - 96*c^5*d*e*f + 64*b*c^4*e^2*f - 1824*b*c^4*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 -
(((d*((16*c^4*(27*b*e*g - 52*c*d*g + 2*c*e*f))/(3465*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^4) - (32*c^5*d*g)/(346
5*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^4)))/e - (1568*c^5*d^2*g + 400*b^2*c^3*e^2*g - 800*c^5*d*e*f + 416*b*c^4*e
^2*f - 1584*b*c^4*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 - (((d*((16*c^4*(31*b*e*g - 60*c*d*g + 2*c*e*f))/(3465*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^4)
- (32*c^5*d*g)/(3465*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^4)))/e - (2400*c^5*d^2*g + 816*b^2*c^3*e^2*g - 96*c^5*d
*e*f + 64*b*c^4*e^2*f - 2832*b*c^4*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 + (((d*((16*c^3*(16*b*e*g - 31*c*d*g + c*e*f))/(693*(5*b*e^2 - 10*c*d*e)*(b
*e - 2*c*d)^3) - (16*c^4*d*g)/(693*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d)^3)))/e - (16*c^2*(b*e - c*d)*(15*b*e*g -
30*c*d*g + c*e*f))/(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*((32*c^4*(19*b*e*g - 37*c*d*g + c*e*f))/(3465*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^4) - (32*c
^5*d*g)/(3465*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^4)))/e - (32*c^3*(b*e - c*d)*(18*b*e*g - 36*c*d*g + c*e*f))/(3
465*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 - (((768*
c^6*d^2*g + 304*b^2*c^4*e^2*g - 128*c^6*d*e*f + 96*b*c^5*e^2*f - 992*b*c^5*d*e*g)/(10395*e^2*(b*e - 2*c*d)^6)
- (d*((64*c^5*(5*b*e*g - 9*c*d*g + c*e*f))/(10395*e*(b*e - 2*c*d)^6) - (64*c^6*d*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) + (((1664*c^6*d^2*g + 640*b^2*c^4*e^2*g - 128*c^6*
d*e*f + 96*b*c^5*e^2*f - 2112*b*c^5*d*e*g)/(10395*e^2*(b*e - 2*c*d)^6) - (d*((32*c^5*(17*b*e*g - 32*c*d*g + 2*
c*e*f))/(10395*e*(b*e - 2*c*d)^6) - (64*c^6*d*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) + (((1920*c^6*d^2*g + 736*b^2*c^4*e^2*g - 128*c^6*d*e*f + 96*b*c^5*e^2*f - 2432*b*c^5
*d*e*g)/(10395*e^2*(b*e - 2*c*d)^6) - (d*((32*c^5*(19*b*e*g - 36*c*d*g + 2*c*e*f))/(10395*e*(b*e - 2*c*d)^6) -
(64*c^6*d*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) + (((320*c
^6*d^2*g + 928*b^2*c^4*e^2*g + 1472*c^6*d*e*f - 704*b*c^5*e^2*f - 2016*b*c^5*d*e*g)/(10395*e^2*(b*e - 2*c*d)^6
) - (d*((32*c^5*(31*b*e*g - 60*c*d*g + 2*c*e*f))/(10395*e*(b*e - 2*c*d)^6) - (64*c^6*d*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) + (((2176*c^6*d^2*g + 832*b^2*c^4*e^2*g - 12
8*c^6*d*e*f + 96*b*c^5*e^2*f - 2752*b*c^5*d*e*g)/(10395*e^2*(b*e - 2*c*d)^6) - (d*((32*c^5*(21*b*e*g - 40*c*d*
g + 2*c*e*f))/(10395*e*(b*e - 2*c*d)^6) - (64*c^6*d*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) - (((2048*c^6*d^2*g + 736*b^2*c^4*e^2*g - 448*c^6*d*e*f + 256*b*c^5*e^2*f - 249
6*b*c^5*d*e*g)/(10395*e^2*(b*e - 2*c*d)^6) - (d*((64*c^5*(11*b*e*g - 21*c*d*g + c*e*f))/(10395*e*(b*e - 2*c*d)
^6) - (64*c^6*d*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) - (((
2816*c^6*d^2*g + 1072*b^2*c^4*e^2*g - 128*c^6*d*e*f + 96*b*c^5*e^2*f - 3552*b*c^5*d*e*g)/(10395*e^2*(b*e - 2*c
*d)^6) - (d*((64*c^5*(13*b*e*g - 25*c*d*g + c*e*f))/(10395*e*(b*e - 2*c*d)^6) - (64*c^6*d*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) - (((3072*c^6*d^2*g + 1168*b^2*c^4*e^2*g
- 128*c^6*d*e*f + 96*b*c^5*e^2*f - 3872*b*c^5*d*e*g)/(10395*e^2*(b*e - 2*c*d)^6) - (d*((64*c^5*(14*b*e*g - 27*
c*d*g + c*e*f))/(10395*e*(b*e - 2*c*d)^6) - (64*c^6*d*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) - (((3328*c^6*d^2*g + 1264*b^2*c^4*e^2*g - 128*c^6*d*e*f + 96*b*c^5*e^2*f - 4
192*b*c^5*d*e*g)/(10395*e^2*(b*e - 2*c*d)^6) - (d*((64*c^5*(15*b*e*g - 29*c*d*g + c*e*f))/(10395*e*(b*e - 2*c*
d)^6) - (64*c^6*d*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) + (
((4224*c^6*d^2*g + 1600*b^2*c^4*e^2*g - 128*c^6*d*e*f + 96*b*c^5*e^2*f - 5312*b*c^5*d*e*g)/(10395*e^2*(b*e - 2
*c*d)^6) - (d*((32*c^5*(37*b*e*g - 72*c*d*g + 2*c*e*f))/(10395*e*(b*e - 2*c*d)^6) - (64*c^6*d*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) + (((6976*c^6*d^2*g + 2144*b^2*c^4*e^
2*g - 448*c^6*d*e*f + 256*b*c^5*e^2*f - 7776*b*c^5*d*e*g)/(10395*e^2*(b*e - 2*c*d)^6) - (d*((32*c^5*(33*b*e*g
- 64*c*d*g + 2*c*e*f))/(10395*e*(b*e - 2*c*d)^6) - (64*c^6*d*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) + (((d*((64*c^5*(21*b*e*g - 41*c*d*g + c*e*f))/(10395*e*(b*e - 2*c*d)^
6) - (64*c^6*d*g)/(10395*e*(b*e - 2*c*d)^6)))/e - (64*c^4*(b*e - c*d)*(20*b*e*g - 40*c*d*g + c*e*f))/(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) + (((d*((8*c^3*e*f - 24*c^3*d*g + 1
6*b*c^2*e*g)/(99*(7*b*e^2 - 14*c*d*e)*(b*e - 2*c*d)^2) - (8*c^3*d*g)/(99*(7*b*e^2 - 14*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))/(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*((8*c^3*e*f - 112*c^3*d*g + 60*b*c^2*e*g)/(99*(7*b*e^2 - 14*c*d*e)*(
b*e - 2*c*d)^2) - (8*c^3*d*g)/(99*(7*b*e^2 - 14*c*d*e)*(b*e - 2*c*d)^2)))/e - (4*b*c*(7*b*e*g - 14*c*d*g + c*e
*f))/(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*((64*c^5*(12*b*e*g - 23*c*d*g + c*e*f))/(10395*e*(b*e - 2*c*d)^6) - (64*c^6*d*g)/(10395*e*(b*e - 2*c*d)^6)))
/e + (128*c^6*d^2*g - 352*b^2*c^4*e^2*g - 64*c^6*d*e*f + 640*b*c^5*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) + (((d*((64*c^5*(13*b*e*g - 25*c*d*g + c*e*f))/(10395*e*(b*e -
2*c*d)^6) - (64*c^6*d*g)/(10395*e*(b*e - 2*c*d)^6)))/e + (256*c^6*d^2*g - 352*b^2*c^4*e^2*g - 64*c^6*d*e*f +
576*b*c^5*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) - (((d*((
32*c^5*(35*b*e*g - 68*c*d*g + 2*c*e*f))/(10395*e*(b*e - 2*c*d)^6) - (64*c^6*d*g)/(10395*e*(b*e - 2*c*d)^6)))/e
+ (832*c^6*d^2*g - 352*b^2*c^4*e^2*g - 64*c^6*d*e*f + 288*b*c^5*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) - (((d*((32*c^5*(15*b*e*g - 28*c*d*g + 2*c*e*f))/(10395*e*(b*e -
2*c*d)^6) - (64*c^6*d*g)/(10395*e*(b*e - 2*c*d)^6)))/e - (448*c^6*d^2*g + 352*b^2*c^4*e^2*g + 64*c^6*d*e*f -
928*b*c^5*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)

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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 )^{7}}\, 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)**7,x)

[Out]

Integral(sqrt(-(d + e*x)*(b*e - c*d + c*e*x))*(f + g*x)/(d + e*x)**7, x)

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