3.23.5 \(\int \frac {(f+g x) (c d^2-b d e-b e^2 x-c e^2 x^2)^{5/2}}{(d+e x)^8} \, dx\) [2205]
Optimal. Leaf size=138 \[ -\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{9 e^2 (2 c d-b e) (d+e x)^8}+\frac {2 (9 b e g-2 c (e f+8 d g)) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{63 e^2 (2 c d-b e)^2 (d+e x)^7} \]
[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 = 0.13, 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 = {806, 664}
\begin {gather*} \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)} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[((f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(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 664
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 806
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 = 0.22, size = 104, normalized size = 0.75 \begin {gather*} \frac {2 (-c d+b e+c e x)^3 \sqrt {(d+e x) (-b e+c (d-e x))} \left (-b e (7 e f+2 d g+9 e g x)+2 c \left (d^2 g+e^2 f x+8 d e (f+g x)\right )\right )}{63 e^2 (-2 c d+b e)^2 (d+e x)^5} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[((f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(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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Maple [A]
time = 0.05, size = 212, normalized size = 1.54
| | |
method |
result |
size |
| | |
gosper |
\(-\frac {2 \left (c e x +b e -c d \right ) \left (9 b \,e^{2} g x -16 c d e g x -2 c \,e^{2} f x +2 b d e g +7 b \,e^{2} f -2 c \,d^{2} g -16 c d e f \right ) \left (-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}\right )^{\frac {5}{2}}}{63 \left (e x +d \right )^{7} e^{2} \left (b^{2} e^{2}-4 b c d e +4 c^{2} d^{2}\right )}\) |
\(128\) |
default |
\(-\frac {2 g \left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {7}{2}}}{7 e^{8} \left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )^{7}}+\frac {\left (-d g +e f \right ) \left (-\frac {2 \left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {7}{2}}}{9 \left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )^{8}}-\frac {4 c \,e^{2} \left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {7}{2}}}{63 \left (-b \,e^{2}+2 c d e \right )^{2} \left (x +\frac {d}{e}\right )^{7}}\right )}{e^{9}}\) |
\(212\) |
trager |
\(-\frac {2 \left (9 b \,c^{3} e^{5} g \,x^{4}-16 c^{4} d \,e^{4} g \,x^{4}-2 c^{4} e^{5} f \,x^{4}+27 b^{2} c^{2} e^{5} g \,x^{3}-73 b \,c^{3} d \,e^{4} g \,x^{3}+b \,c^{3} e^{5} f \,x^{3}+46 c^{4} d^{2} e^{3} g \,x^{3}-10 c^{4} d \,e^{4} f \,x^{3}+27 b^{3} c \,e^{5} g \,x^{2}-96 b^{2} c^{2} d \,e^{4} g \,x^{2}+15 b^{2} c^{2} e^{5} f \,x^{2}+111 b \,c^{3} d^{2} e^{3} g \,x^{2}-57 b \,c^{3} d \,e^{4} f \,x^{2}-42 c^{4} d^{3} e^{2} g \,x^{2}+42 c^{4} d^{2} e^{3} f \,x^{2}+9 b^{4} e^{5} g x -37 b^{3} c d \,e^{4} g x +19 b^{3} c \,e^{5} f x +57 b^{2} c^{2} d^{2} e^{3} g x -84 b^{2} c^{2} d \,e^{4} f x -39 b \,c^{3} d^{3} e^{2} g x +111 b \,c^{3} d^{2} e^{3} f x +10 c^{4} d^{4} e g x -46 c^{4} d^{3} e^{2} f x +2 b^{4} d \,e^{4} g +7 b^{4} e^{5} f -8 b^{3} c \,d^{2} e^{3} g -37 b^{3} c d \,e^{4} f +12 b^{2} c^{2} d^{3} e^{2} g +69 b^{2} c^{2} d^{2} e^{3} f -8 b \,c^{3} d^{4} e g -55 b \,c^{3} d^{3} e^{2} f +2 c^{4} d^{5} g +16 c^{4} d^{4} e f \right ) \sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}}{63 \left (b^{2} e^{2}-4 b c d e +4 c^{2} d^{2}\right ) e^{2} \left (e x +d \right )^{5}}\) |
\(509\) |
| | |
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|
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,method=_RETURNVERBOSE)
[Out]
-2/7*g/e^8/(-b*e^2+2*c*d*e)/(x+d/e)^7*(-c*e^2*(x+d/e)^2+(-b*e^2+2*c*d*e)*(x+d/e))^(7/2)+(-d*g+e*f)/e^9*(-2/9/(
-b*e^2+2*c*d*e)/(x+d/e)^8*(-c*e^2*(x+d/e)^2+(-b*e^2+2*c*d*e)*(x+d/e))^(7/2)-4/63*c*e^2/(-b*e^2+2*c*d*e)^2/(x+d
/e)^7*(-c*e^2*(x+d/e)^2+(-b*e^2+2*c*d*e)*(x+d/e))^(7/2))
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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)^(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(2*c*d-%e*b>0)', see `assume?`
for more det
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Fricas [F(-1)] Timed out
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)^(5/2)/(e*x+d)^8,x, algorithm="fricas")
[Out]
Timed out
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Sympy [F(-1)] Timed out
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)**(5/2)/(e*x+d)**8,x)
[Out]
Timed out
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 3739 vs.
\(2 (133) = 266\).
time = 1.04, size = 3739, normalized size = 27.09 \begin {gather*} \text {Too large to display} \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)^(5/2)/(e*x+d)^8,x, algorithm="giac")
[Out]
-2/63*(8*c^7*d^8*g + c^7*d^7*f*e - 28*b*c^6*d^7*g*e - 9*(sqrt(-c*e^2)*x - sqrt(-c*x^2*e^2 + c*d^2 - b*x*e^2 -
b*d*e))*sqrt(-c)*c^6*d^7*g - 9*(sqrt(-c*e^2)*x - sqrt(-c*x^2*e^2 + c*d^2 - b*x*e^2 - b*d*e))*sqrt(-c)*c^6*d^6*
f*e - 189*(sqrt(-c*e^2)*x - sqrt(-c*x^2*e^2 + c*d^2 - b*x*e^2 - b*d*e))*b*sqrt(-c)*c^5*d^6*g*e - 225*(sqrt(-c*
e^2)*x - sqrt(-c*x^2*e^2 + c*d^2 - b*x*e^2 - b*d*e))^2*c^6*d^6*g + b*c^6*d^6*f*e^2 + 96*b^2*c^5*d^6*g*e^2 + 27
*(sqrt(-c*e^2)*x - sqrt(-c*x^2*e^2 + c*d^2 - b*x*e^2 - b*d*e))^2*c^6*d^5*f*e + 378*(sqrt(-c*e^2)*x - sqrt(-c*x
^2*e^2 + c*d^2 - b*x*e^2 - b*d*e))^2*b*c^5*d^5*g*e + 189*(sqrt(-c*e^2)*x - sqrt(-c*x^2*e^2 + c*d^2 - b*x*e^2 -
b*d*e))^3*sqrt(-c)*c^5*d^5*g + 54*(sqrt(-c*e^2)*x - sqrt(-c*x^2*e^2 + c*d^2 - b*x*e^2 - b*d*e))*b*sqrt(-c)*c^
5*d^5*f*e^2 + 459*(sqrt(-c*e^2)*x - sqrt(-c*x^2*e^2 + c*d^2 - b*x*e^2 - b*d*e))*b^2*sqrt(-c)*c^4*d^5*g*e^2 + 1
89*(sqrt(-c*e^2)*x - sqrt(-c*x^2*e^2 + c*d^2 - b*x*e^2 - b*d*e))^3*sqrt(-c)*c^5*d^4*f*e + 819*(sqrt(-c*e^2)*x
- sqrt(-c*x^2*e^2 + c*d^2 - b*x*e^2 - b*d*e))^3*b*sqrt(-c)*c^4*d^4*g*e + 693*(sqrt(-c*e^2)*x - sqrt(-c*x^2*e^2
+ c*d^2 - b*x*e^2 - b*d*e))^4*c^5*d^4*g - 15*b^2*c^5*d^5*f*e^3 - 170*b^3*c^4*d^5*g*e^3 - 351*(sqrt(-c*e^2)*x
- sqrt(-c*x^2*e^2 + c*d^2 - b*x*e^2 - b*d*e))^2*b*c^5*d^4*f*e^2 - 999*(sqrt(-c*e^2)*x - sqrt(-c*x^2*e^2 + c*d^
2 - b*x*e^2 - b*d*e))^2*b^2*c^4*d^4*g*e^2 - 189*(sqrt(-c*e^2)*x - sqrt(-c*x^2*e^2 + c*d^2 - b*x*e^2 - b*d*e))^
4*c^5*d^3*f*e - 504*(sqrt(-c*e^2)*x - sqrt(-c*x^2*e^2 + c*d^2 - b*x*e^2 - b*d*e))^4*b*c^4*d^3*g*e - 315*(sqrt(
-c*e^2)*x - sqrt(-c*x^2*e^2 + c*d^2 - b*x*e^2 - b*d*e))^5*sqrt(-c)*c^4*d^3*g - 243*(sqrt(-c*e^2)*x - sqrt(-c*x
^2*e^2 + c*d^2 - b*x*e^2 - b*d*e))*b^2*sqrt(-c)*c^4*d^4*f*e^3 - 675*(sqrt(-c*e^2)*x - sqrt(-c*x^2*e^2 + c*d^2
- b*x*e^2 - b*d*e))*b^3*sqrt(-c)*c^3*d^4*g*e^3 - 756*(sqrt(-c*e^2)*x - sqrt(-c*x^2*e^2 + c*d^2 - b*x*e^2 - b*d
*e))^3*b*sqrt(-c)*c^4*d^3*f*e^2 - 1134*(sqrt(-c*e^2)*x - sqrt(-c*x^2*e^2 + c*d^2 - b*x*e^2 - b*d*e))^3*b^2*sqr
t(-c)*c^3*d^3*g*e^2 - 315*(sqrt(-c*e^2)*x - sqrt(-c*x^2*e^2 + c*d^2 - b*x*e^2 - b*d*e))^5*sqrt(-c)*c^4*d^2*f*e
- 819*(sqrt(-c*e^2)*x - sqrt(-c*x^2*e^2 + c*d^2 - b*x*e^2 - b*d*e))^5*b*sqrt(-c)*c^3*d^2*g*e - 483*(sqrt(-c*e
^2)*x - sqrt(-c*x^2*e^2 + c*d^2 - b*x*e^2 - b*d*e))^6*c^4*d^2*g + 53*b^3*c^4*d^4*f*e^4 + 184*b^4*c^3*d^4*g*e^4
+ 918*(sqrt(-c*e^2)*x - sqrt(-c*x^2*e^2 + c*d^2 - b*x*e^2 - b*d*e))^2*b^2*c^4*d^3*f*e^3 + 1080*(sqrt(-c*e^2)*
x - sqrt(-c*x^2*e^2 + c*d^2 - b*x*e^2 - b*d*e))^2*b^3*c^3*d^3*g*e^3 + 1071*(sqrt(-c*e^2)*x - sqrt(-c*x^2*e^2 +
c*d^2 - b*x*e^2 - b*d*e))^4*b*c^4*d^2*f*e^2 + 1134*(sqrt(-c*e^2)*x - sqrt(-c*x^2*e^2 + c*d^2 - b*x*e^2 - b*d*
e))^4*b^2*c^3*d^2*g*e^2 + 105*(sqrt(-c*e^2)*x - sqrt(-c*x^2*e^2 + c*d^2 - b*x*e^2 - b*d*e))^6*c^4*d*f*e + 210*
(sqrt(-c*e^2)*x - sqrt(-c*x^2*e^2 + c*d^2 - b*x*e^2 - b*d*e))^6*b*c^3*d*g*e + 63*(sqrt(-c*e^2)*x - sqrt(-c*x^2
*e^2 + c*d^2 - b*x*e^2 - b*d*e))^7*sqrt(-c)*c^3*d*g + 468*(sqrt(-c*e^2)*x - sqrt(-c*x^2*e^2 + c*d^2 - b*x*e^2
- b*d*e))*b^3*sqrt(-c)*c^3*d^3*f*e^4 + 549*(sqrt(-c*e^2)*x - sqrt(-c*x^2*e^2 + c*d^2 - b*x*e^2 - b*d*e))*b^4*s
qrt(-c)*c^2*d^3*g*e^4 + 1638*(sqrt(-c*e^2)*x - sqrt(-c*x^2*e^2 + c*d^2 - b*x*e^2 - b*d*e))^3*b^2*sqrt(-c)*c^3*
d^2*f*e^3 + 1050*(sqrt(-c*e^2)*x - sqrt(-c*x^2*e^2 + c*d^2 - b*x*e^2 - b*d*e))^3*b^3*sqrt(-c)*c^2*d^2*g*e^3 +
630*(sqrt(-c*e^2)*x - sqrt(-c*x^2*e^2 + c*d^2 - b*x*e^2 - b*d*e))^5*b*sqrt(-c)*c^3*d*f*e^2 + 567*(sqrt(-c*e^2)
*x - sqrt(-c*x^2*e^2 + c*d^2 - b*x*e^2 - b*d*e))^5*b^2*sqrt(-c)*c^2*d*g*e^2 + 63*(sqrt(-c*e^2)*x - sqrt(-c*x^2
*e^2 + c*d^2 - b*x*e^2 - b*d*e))^7*sqrt(-c)*c^3*f*e + 189*(sqrt(-c*e^2)*x - sqrt(-c*x^2*e^2 + c*d^2 - b*x*e^2
- b*d*e))^7*b*sqrt(-c)*c^2*g*e + 63*(sqrt(-c*e^2)*x - sqrt(-c*x^2*e^2 + c*d^2 - b*x*e^2 - b*d*e))^8*c^3*g - 85
*b^4*c^3*d^3*f*e^5 - 120*b^5*c^2*d^3*g*e^5 - 1278*(sqrt(-c*e^2)*x - sqrt(-c*x^2*e^2 + c*d^2 - b*x*e^2 - b*d*e)
)^2*b^3*c^3*d^2*f*e^4 - 639*(sqrt(-c*e^2)*x - sqrt(-c*x^2*e^2 + c*d^2 - b*x*e^2 - b*d*e))^2*b^4*c^2*d^2*g*e^4
- 1323*(sqrt(-c*e^2)*x - sqrt(-c*x^2*e^2 + c*d^2 - b*x*e^2 - b*d*e))^4*b^2*c^3*d*f*e^3 - 630*(sqrt(-c*e^2)*x -
sqrt(-c*x^2*e^2 + c*d^2 - b*x*e^2 - b*d*e))^4*b^3*c^2*d*g*e^3 - 273*(sqrt(-c*e^2)*x - sqrt(-c*x^2*e^2 + c*d^2
- b*x*e^2 - b*d*e))^6*b*c^3*f*e^2 - 315*(sqrt(-c*e^2)*x - sqrt(-c*x^2*e^2 + c*d^2 - b*x*e^2 - b*d*e))^6*b^2*c
^2*g*e^2 - 495*(sqrt(-c*e^2)*x - sqrt(-c*x^2*e^2 + c*d^2 - b*x*e^2 - b*d*e))*b^4*sqrt(-c)*c^2*d^2*f*e^5 - 243*
(sqrt(-c*e^2)*x - sqrt(-c*x^2*e^2 + c*d^2 - b*x*e^2 - b*d*e))*b^5*sqrt(-c)*c*d^2*g*e^5 - 1428*(sqrt(-c*e^2)*x
- sqrt(-c*x^2*e^2 + c*d^2 - b*x*e^2 - b*d*e))^3*b^3*sqrt(-c)*c^2*d*f*e^4 - 399*(sqrt(-c*e^2)*x - sqrt(-c*x^2*e
^2 + c*d^2 - b*x*e^2 - b*d*e))^3*b^4*sqrt(-c)*c*d*g*e^4 - 567*(sqrt(-c*e^2)*x - sqrt(-c*x^2*e^2 + c*d^2 - b*x*
e^2 - b*d*e))^5*b^2*sqrt(-c)*c^2*f*e^3 - 315*(sqrt(-c*e^2)*x - sqrt(-c*x^2*e^2 + c*d^2 - b*x*e^2 - b*d*e))^5*b
^3*sqrt(-c)*c*g*e^3 + 75*b^5*c^2*d^2*f*e^6 + 44*b^6*c*d^2*g*e^6 + 855*(sqrt(-c*e^2)*x - sqrt(-c*x^2*e^2 + c*d^
2 - b*x*e^2 - b*d*e))^2*b^4*c^2*d*f*e^5 + 162*(...
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Mupad [B]
time = 26.34, size = 2500, normalized size = 18.12 \begin {gather*} \text {Too large to display} \end {gather*}
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)^...
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