3.20.30 \(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}}{(d+e x)^8} \, dx\) [1930]

Optimal. Leaf size=231 \[ \frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{11 \left (c d^2-a e^2\right ) (d+e x)^8}+\frac {4 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{33 \left (c d^2-a e^2\right )^2 (d+e x)^7}+\frac {16 c^2 d^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{231 \left (c d^2-a e^2\right )^3 (d+e x)^6}+\frac {32 c^3 d^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{1155 \left (c d^2-a e^2\right )^4 (d+e x)^5} \]

[Out]

2/11*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(-a*e^2+c*d^2)/(e*x+d)^8+4/33*c*d*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^
2)^(5/2)/(-a*e^2+c*d^2)^2/(e*x+d)^7+16/231*c^2*d^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(-a*e^2+c*d^2)^3/(e
*x+d)^6+32/1155*c^3*d^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(-a*e^2+c*d^2)^4/(e*x+d)^5

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Rubi [A]
time = 0.08, antiderivative size = 231, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, Rules used = {672, 664} \begin {gather*} \frac {32 c^3 d^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{1155 (d+e x)^5 \left (c d^2-a e^2\right )^4}+\frac {16 c^2 d^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{231 (d+e x)^6 \left (c d^2-a e^2\right )^3}+\frac {4 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{33 (d+e x)^7 \left (c d^2-a e^2\right )^2}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{11 (d+e x)^8 \left (c d^2-a e^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(d + e*x)^8,x]

[Out]

(2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(11*(c*d^2 - a*e^2)*(d + e*x)^8) + (4*c*d*(a*d*e + (c*d^2 +
a*e^2)*x + c*d*e*x^2)^(5/2))/(33*(c*d^2 - a*e^2)^2*(d + e*x)^7) + (16*c^2*d^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d
*e*x^2)^(5/2))/(231*(c*d^2 - a*e^2)^3*(d + e*x)^6) + (32*c^3*d^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)
)/(1155*(c*d^2 - a*e^2)^4*(d + e*x)^5)

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 672

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]

Rubi steps

\begin {align*} \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^8} \, dx &=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{11 \left (c d^2-a e^2\right ) (d+e x)^8}+\frac {(6 c d) \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^7} \, dx}{11 \left (c d^2-a e^2\right )}\\ &=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{11 \left (c d^2-a e^2\right ) (d+e x)^8}+\frac {4 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{33 \left (c d^2-a e^2\right )^2 (d+e x)^7}+\frac {\left (8 c^2 d^2\right ) \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^6} \, dx}{33 \left (c d^2-a e^2\right )^2}\\ &=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{11 \left (c d^2-a e^2\right ) (d+e x)^8}+\frac {4 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{33 \left (c d^2-a e^2\right )^2 (d+e x)^7}+\frac {16 c^2 d^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{231 \left (c d^2-a e^2\right )^3 (d+e x)^6}+\frac {\left (16 c^3 d^3\right ) \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^5} \, dx}{231 \left (c d^2-a e^2\right )^3}\\ &=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{11 \left (c d^2-a e^2\right ) (d+e x)^8}+\frac {4 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{33 \left (c d^2-a e^2\right )^2 (d+e x)^7}+\frac {16 c^2 d^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{231 \left (c d^2-a e^2\right )^3 (d+e x)^6}+\frac {32 c^3 d^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{1155 \left (c d^2-a e^2\right )^4 (d+e x)^5}\\ \end {align*}

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Mathematica [A]
time = 0.17, size = 132, normalized size = 0.57 \begin {gather*} \frac {2 (a e+c d x)^4 ((a e+c d x) (d+e x))^{3/2} \left (-105 e^3+\frac {385 c d e^2 (d+e x)}{a e+c d x}-\frac {495 c^2 d^2 e (d+e x)^2}{(a e+c d x)^2}+\frac {231 c^3 d^3 (d+e x)^3}{(a e+c d x)^3}\right )}{1155 \left (c d^2-a e^2\right )^4 (d+e x)^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(d + e*x)^8,x]

[Out]

(2*(a*e + c*d*x)^4*((a*e + c*d*x)*(d + e*x))^(3/2)*(-105*e^3 + (385*c*d*e^2*(d + e*x))/(a*e + c*d*x) - (495*c^
2*d^2*e*(d + e*x)^2)/(a*e + c*d*x)^2 + (231*c^3*d^3*(d + e*x)^3)/(a*e + c*d*x)^3))/(1155*(c*d^2 - a*e^2)^4*(d
+ e*x)^7)

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Maple [A]
time = 0.71, size = 293, normalized size = 1.27

method	result	size
gosper	\(-\frac {2 \left (c d x +a e \right ) \left (-16 c^{3} d^{3} e^{3} x^{3}+40 a \,c^{2} d^{2} e^{4} x^{2}-88 c^{3} d^{4} e^{2} x^{2}-70 a^{2} c d \,e^{5} x +220 a \,c^{2} d^{3} e^{3} x -198 c^{3} d^{5} e x +105 e^{6} a^{3}-385 e^{4} d^{2} a^{2} c +495 d^{4} e^{2} c^{2} a -231 d^{6} c^{3}\right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {3}{2}}}{1155 \left (e x +d \right )^{7} \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right )}\)	\(217\)
default	\(\frac {-\frac {2 \left (c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{11 \left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{8}}-\frac {6 c d e \left (-\frac {2 \left (c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{9 \left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{7}}-\frac {4 c d e \left (-\frac {2 \left (c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{7 \left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{6}}+\frac {4 c d e \left (c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{35 \left (e^{2} a -c \,d^{2}\right )^{2} \left (x +\frac {d}{e}\right )^{5}}\right )}{9 \left (e^{2} a -c \,d^{2}\right )}\right )}{11 \left (e^{2} a -c \,d^{2}\right )}}{e^{8}}\)	\(293\)
trager	\(-\frac {2 \left (-16 c^{5} d^{5} e^{3} x^{5}+8 a \,c^{4} d^{4} e^{4} x^{4}-88 c^{5} d^{6} e^{2} x^{4}-6 a^{2} c^{3} d^{3} e^{5} x^{3}+44 a \,c^{4} d^{5} e^{3} x^{3}-198 c^{5} d^{7} e \,x^{3}+5 a^{3} c^{2} d^{2} e^{6} x^{2}-33 a^{2} c^{3} d^{4} e^{4} x^{2}+99 a \,c^{4} d^{6} e^{2} x^{2}-231 c^{5} d^{8} x^{2}+140 a^{4} c d \,e^{7} x -550 a^{3} c^{2} d^{3} e^{5} x +792 a^{2} c^{3} d^{5} e^{3} x -462 a \,c^{4} d^{7} e x +105 a^{5} e^{8}-385 a^{4} c \,d^{2} e^{6}+495 a^{3} c^{2} d^{4} e^{4}-231 a^{2} c^{3} d^{6} e^{2}\right ) \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}}{1155 \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right ) \left (e x +d \right )^{6}}\)	\(339\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d)^8,x,method=_RETURNVERBOSE)

[Out]

1/e^8*(-2/11/(a*e^2-c*d^2)/(x+d/e)^8*(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(5/2)-6/11*c*d*e/(a*e^2-c*d^2)*(-
2/9/(a*e^2-c*d^2)/(x+d/e)^7*(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(5/2)-4/9*c*d*e/(a*e^2-c*d^2)*(-2/7/(a*e^2
-c*d^2)/(x+d/e)^6*(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(5/2)+4/35*c*d*e/(a*e^2-c*d^2)^2/(x+d/e)^5*(c*d*e*(x
+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(5/2))))

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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((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d)^8,x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 710 vs. \(2 (219) = 438\).
time = 79.95, size = 710, normalized size = 3.07 \begin {gather*} \frac {2 \, {\left (231 \, c^{5} d^{8} x^{2} - 140 \, a^{4} c d x e^{7} - 105 \, a^{5} e^{8} - 5 \, {\left (a^{3} c^{2} d^{2} x^{2} - 77 \, a^{4} c d^{2}\right )} e^{6} + 2 \, {\left (3 \, a^{2} c^{3} d^{3} x^{3} + 275 \, a^{3} c^{2} d^{3} x\right )} e^{5} - {\left (8 \, a c^{4} d^{4} x^{4} - 33 \, a^{2} c^{3} d^{4} x^{2} + 495 \, a^{3} c^{2} d^{4}\right )} e^{4} + 4 \, {\left (4 \, c^{5} d^{5} x^{5} - 11 \, a c^{4} d^{5} x^{3} - 198 \, a^{2} c^{3} d^{5} x\right )} e^{3} + 11 \, {\left (8 \, c^{5} d^{6} x^{4} - 9 \, a c^{4} d^{6} x^{2} + 21 \, a^{2} c^{3} d^{6}\right )} e^{2} + 66 \, {\left (3 \, c^{5} d^{7} x^{3} + 7 \, a c^{4} d^{7} x\right )} e\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e}}{1155 \, {\left (6 \, c^{4} d^{13} x e + c^{4} d^{14} + a^{4} x^{6} e^{14} + 6 \, a^{4} d x^{5} e^{13} - {\left (4 \, a^{3} c d^{2} x^{6} - 15 \, a^{4} d^{2} x^{4}\right )} e^{12} - 4 \, {\left (6 \, a^{3} c d^{3} x^{5} - 5 \, a^{4} d^{3} x^{3}\right )} e^{11} + 3 \, {\left (2 \, a^{2} c^{2} d^{4} x^{6} - 20 \, a^{3} c d^{4} x^{4} + 5 \, a^{4} d^{4} x^{2}\right )} e^{10} + 2 \, {\left (18 \, a^{2} c^{2} d^{5} x^{5} - 40 \, a^{3} c d^{5} x^{3} + 3 \, a^{4} d^{5} x\right )} e^{9} - {\left (4 \, a c^{3} d^{6} x^{6} - 90 \, a^{2} c^{2} d^{6} x^{4} + 60 \, a^{3} c d^{6} x^{2} - a^{4} d^{6}\right )} e^{8} - 24 \, {\left (a c^{3} d^{7} x^{5} - 5 \, a^{2} c^{2} d^{7} x^{3} + a^{3} c d^{7} x\right )} e^{7} + {\left (c^{4} d^{8} x^{6} - 60 \, a c^{3} d^{8} x^{4} + 90 \, a^{2} c^{2} d^{8} x^{2} - 4 \, a^{3} c d^{8}\right )} e^{6} + 2 \, {\left (3 \, c^{4} d^{9} x^{5} - 40 \, a c^{3} d^{9} x^{3} + 18 \, a^{2} c^{2} d^{9} x\right )} e^{5} + 3 \, {\left (5 \, c^{4} d^{10} x^{4} - 20 \, a c^{3} d^{10} x^{2} + 2 \, a^{2} c^{2} d^{10}\right )} e^{4} + 4 \, {\left (5 \, c^{4} d^{11} x^{3} - 6 \, a c^{3} d^{11} x\right )} e^{3} + {\left (15 \, c^{4} d^{12} x^{2} - 4 \, a c^{3} d^{12}\right )} e^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d)^8,x, algorithm="fricas")

[Out]

2/1155*(231*c^5*d^8*x^2 - 140*a^4*c*d*x*e^7 - 105*a^5*e^8 - 5*(a^3*c^2*d^2*x^2 - 77*a^4*c*d^2)*e^6 + 2*(3*a^2*
c^3*d^3*x^3 + 275*a^3*c^2*d^3*x)*e^5 - (8*a*c^4*d^4*x^4 - 33*a^2*c^3*d^4*x^2 + 495*a^3*c^2*d^4)*e^4 + 4*(4*c^5
*d^5*x^5 - 11*a*c^4*d^5*x^3 - 198*a^2*c^3*d^5*x)*e^3 + 11*(8*c^5*d^6*x^4 - 9*a*c^4*d^6*x^2 + 21*a^2*c^3*d^6)*e
^2 + 66*(3*c^5*d^7*x^3 + 7*a*c^4*d^7*x)*e)*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)/(6*c^4*d^13*x*e + c^4*d
^14 + a^4*x^6*e^14 + 6*a^4*d*x^5*e^13 - (4*a^3*c*d^2*x^6 - 15*a^4*d^2*x^4)*e^12 - 4*(6*a^3*c*d^3*x^5 - 5*a^4*d
^3*x^3)*e^11 + 3*(2*a^2*c^2*d^4*x^6 - 20*a^3*c*d^4*x^4 + 5*a^4*d^4*x^2)*e^10 + 2*(18*a^2*c^2*d^5*x^5 - 40*a^3*
c*d^5*x^3 + 3*a^4*d^5*x)*e^9 - (4*a*c^3*d^6*x^6 - 90*a^2*c^2*d^6*x^4 + 60*a^3*c*d^6*x^2 - a^4*d^6)*e^8 - 24*(a
*c^3*d^7*x^5 - 5*a^2*c^2*d^7*x^3 + a^3*c*d^7*x)*e^7 + (c^4*d^8*x^6 - 60*a*c^3*d^8*x^4 + 90*a^2*c^2*d^8*x^2 - 4
*a^3*c*d^8)*e^6 + 2*(3*c^4*d^9*x^5 - 40*a*c^3*d^9*x^3 + 18*a^2*c^2*d^9*x)*e^5 + 3*(5*c^4*d^10*x^4 - 20*a*c^3*d
^10*x^2 + 2*a^2*c^2*d^10)*e^4 + 4*(5*c^4*d^11*x^3 - 6*a*c^3*d^11*x)*e^3 + (15*c^4*d^12*x^2 - 4*a*c^3*d^12)*e^2
)

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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((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2)/(e*x+d)**8,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d)^8,x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Unable to divide, perhaps due to rounding error%%%{%%%{1,[0,0,6]%%%},[12]%%%}+%%%{%%{[%%%{-12,[0,1,5]%%%},0
]:[1,0,%%%{

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Mupad [B]
time = 5.15, size = 2657, normalized size = 11.50 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(3/2)/(d + e*x)^8,x)

[Out]

(((d*((4*c^3*d^4)/(11*(a*e^2 - c*d^2)*(9*a*e^3 - 9*c*d^2*e)) - (2*c^2*d^2*(5*a*e^2 - c*d^2))/(11*(a*e^2 - c*d^
2)*(9*a*e^3 - 9*c*d^2*e))))/e + (2*a*c^2*d^3*e^2 - 2*c^3*d^5 + 4*a^2*c*d*e^4)/(11*e*(a*e^2 - c*d^2)*(9*a*e^3 -
9*c*d^2*e)))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x)^5 - (((228*c^4*d^5 - 284*a*c^3*d^3*e^2)
/(693*e*(a*e^2 - c*d^2)^2*(5*a*e^3 - 5*c*d^2*e)) + (8*c^4*d^5)/(99*e*(a*e^2 - c*d^2)^2*(5*a*e^3 - 5*c*d^2*e)))
*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x)^3 - (((d*((16*c^5*d^6)/(693*(a*e^2 - c*d^2)^3*(5*a*e
^3 - 5*c*d^2*e)) - (16*c^4*d^4*(17*a*e^2 - 15*c*d^2))/(693*(a*e^2 - c*d^2)^3*(5*a*e^3 - 5*c*d^2*e))))/e + (16*
a*c^3*d^3*e*(16*a*e^2 - 15*c*d^2))/(693*(a*e^2 - c*d^2)^3*(5*a*e^3 - 5*c*d^2*e)))*(x*(a*e^2 + c*d^2) + a*d*e +
c*d*e*x^2)^(1/2))/(d + e*x)^3 - (((d*((32*c^6*d^7)/(3465*(a*e^2 - c*d^2)^4*(3*a*e^3 - 3*c*d^2*e)) - (64*c^5*d
^5*(10*a*e^2 - 9*c*d^2))/(3465*(a*e^2 - c*d^2)^4*(3*a*e^3 - 3*c*d^2*e))))/e + (32*a*c^4*d^4*e*(19*a*e^2 - 18*c
*d^2))/(3465*(a*e^2 - c*d^2)^4*(3*a*e^3 - 3*c*d^2*e)))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x
)^2 - (((22*c^3*d^4 - 58*a*c^2*d^2*e^2)/(99*e*(a*e^2 - c*d^2)*(7*a*e^3 - 7*c*d^2*e)) + (4*c^3*d^4)/(11*e*(a*e^
2 - c*d^2)*(7*a*e^3 - 7*c*d^2*e)))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x)^4 - (((32*c^6*d^7)
/(3465*e^2*(a*e^2 - c*d^2)^5) - (16*c^5*d^5*(71*a*e^2 - 65*c*d^2))/(10395*e^2*(a*e^2 - c*d^2)^5))*(x*(a*e^2 +
c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x) - (((d*((88*c^4*d^5 - 104*a*c^3*d^3*e^2)/(99*(a*e^2 - c*d^2)^2*(7
*a*e^3 - 7*c*d^2*e)) + (8*c^4*d^5)/(99*(a*e^2 - c*d^2)^2*(7*a*e^3 - 7*c*d^2*e))))/e + (8*a*c^2*d^2*e*(12*a*e^2
- 11*c*d^2))/(99*(a*e^2 - c*d^2)^2*(7*a*e^3 - 7*c*d^2*e)))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d
+ e*x)^4 + (((d*((64*c^7*d^8)/(10395*e*(a*e^2 - c*d^2)^6) - (32*c^6*d^6*(41*a*e^2 - 37*c*d^2))/(10395*e*(a*e^2
- c*d^2)^6)))/e + (32*c^5*d^5*(20*a^2*e^4 - 19*c^2*d^4 + a*c*d^2*e^2))/(10395*e^2*(a*e^2 - c*d^2)^6))*(x*(a*e
^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x) + (((d*((16*c^5*d^6)/(693*(a*e^2 - c*d^2)^3*(5*a*e^3 - 5*c*d
^2*e)) - (8*c^4*d^4*(27*a*e^2 - 23*c*d^2))/(693*(a*e^2 - c*d^2)^3*(5*a*e^3 - 5*c*d^2*e))))/e + (8*c^3*d^3*(13*
a^2*e^4 - 12*c^2*d^4 + a*c*d^2*e^2))/(693*e*(a*e^2 - c*d^2)^3*(5*a*e^3 - 5*c*d^2*e)))*(x*(a*e^2 + c*d^2) + a*d
*e + c*d*e*x^2)^(1/2))/(d + e*x)^3 + (((d*((32*c^6*d^7)/(3465*(a*e^2 - c*d^2)^4*(3*a*e^3 - 3*c*d^2*e)) - (16*c
^5*d^5*(35*a*e^2 - 31*c*d^2))/(3465*(a*e^2 - c*d^2)^4*(3*a*e^3 - 3*c*d^2*e))))/e + (16*c^4*d^4*(17*a^2*e^4 - 1
6*c^2*d^4 + a*c*d^2*e^2))/(3465*e*(a*e^2 - c*d^2)^4*(3*a*e^3 - 3*c*d^2*e)))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e
*x^2)^(1/2))/(d + e*x)^2 - (((2*a^2*e^3)/(11*a*e^3 - 11*c*d^2*e) + (d*((2*c^2*d^3)/(11*a*e^3 - 11*c*d^2*e) - (
4*a*c*d*e^2)/(11*a*e^3 - 11*c*d^2*e)))/e)*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x)^6 - (((16*c
^5*d^6)/(693*e*(a*e^2 - c*d^2)^3*(3*a*e^3 - 3*c*d^2*e)) - (8*c^4*d^4*(83*a*e^2 - 73*c*d^2))/(3465*e*(a*e^2 - c
*d^2)^3*(3*a*e^3 - 3*c*d^2*e)))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x)^2 - (((d*((64*c^7*d^8
)/(10395*e*(a*e^2 - c*d^2)^6) - (128*c^6*d^6*(11*a*e^2 - 10*c*d^2))/(10395*e*(a*e^2 - c*d^2)^6)))/e + (64*a*c^
5*d^5*(21*a*e^2 - 20*c*d^2))/(10395*(a*e^2 - c*d^2)^6))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*
x) - (((d*((24*c^3*d^4 - 32*a*c^2*d^2*e^2)/(11*(a*e^2 - c*d^2)*(9*a*e^3 - 9*c*d^2*e)) + (4*c^3*d^4)/(11*(a*e^2
- c*d^2)*(9*a*e^3 - 9*c*d^2*e))))/e + (4*a*c*d*e*(7*a*e^2 - 6*c*d^2))/(11*(a*e^2 - c*d^2)*(9*a*e^3 - 9*c*d^2*
e)))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x)^5 + (((d*((8*c^4*d^5)/(99*(a*e^2 - c*d^2)^2*(7*a
*e^3 - 7*c*d^2*e)) - (4*c^3*d^3*(17*a*e^2 - 13*c*d^2))/(99*(a*e^2 - c*d^2)^2*(7*a*e^3 - 7*c*d^2*e))))/e + (4*a
*c^3*d^4*e^2 - 28*c^4*d^6 + 32*a^2*c^2*d^2*e^4)/(99*e*(a*e^2 - c*d^2)^2*(7*a*e^3 - 7*c*d^2*e)))*(x*(a*e^2 + c*
d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x)^4 - (16*c^5*d^5*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(115
5*e^2*(a*e^2 - c*d^2)^4*(d + e*x)) - (472*c^4*d^4*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(3465*e*(a*e^
2 - c*d^2)^2*(3*a*e^3 - 3*c*d^2*e)*(d + e*x)^2) - (32*c^3*d^3*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(
231*e*(a*e^2 - c*d^2)*(5*a*e^3 - 5*c*d^2*e)*(d + e*x)^3)

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