3.20.44 \(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}}{(d+e x)^9} \, dx\) [1944]

Optimal. Leaf size=171 \[ \frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{11 \left (c d^2-a e^2\right ) (d+e x)^9}+\frac {8 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{99 \left (c d^2-a e^2\right )^2 (d+e x)^8}+\frac {16 c^2 d^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{693 \left (c d^2-a e^2\right )^3 (d+e x)^7} \]

[Out]

2/11*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/(-a*e^2+c*d^2)/(e*x+d)^9+8/99*c*d*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^
2)^(7/2)/(-a*e^2+c*d^2)^2/(e*x+d)^8+16/693*c^2*d^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/(-a*e^2+c*d^2)^3/(e
*x+d)^7

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Rubi [A]
time = 0.06, antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps used = 3, 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 {16 c^2 d^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{693 (d+e x)^7 \left (c d^2-a e^2\right )^3}+\frac {8 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{99 (d+e x)^8 \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 )^{7/2}}{11 (d+e x)^9 \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)^(5/2)/(d + e*x)^9,x]

[Out]

(2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(11*(c*d^2 - a*e^2)*(d + e*x)^9) + (8*c*d*(a*d*e + (c*d^2 +
a*e^2)*x + c*d*e*x^2)^(7/2))/(99*(c*d^2 - a*e^2)^2*(d + e*x)^8) + (16*c^2*d^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d
*e*x^2)^(7/2))/(693*(c*d^2 - a*e^2)^3*(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 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 )^{5/2}}{(d+e x)^9} \, dx &=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{11 \left (c d^2-a e^2\right ) (d+e x)^9}+\frac {(4 c d) \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^8} \, 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 )^{7/2}}{11 \left (c d^2-a e^2\right ) (d+e x)^9}+\frac {8 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{99 \left (c d^2-a e^2\right )^2 (d+e x)^8}+\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 )^{5/2}}{(d+e x)^7} \, dx}{99 \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 )^{7/2}}{11 \left (c d^2-a e^2\right ) (d+e x)^9}+\frac {8 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{99 \left (c d^2-a e^2\right )^2 (d+e x)^8}+\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 )^{7/2}}{693 \left (c d^2-a e^2\right )^3 (d+e x)^7}\\ \end {align*}

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Mathematica [A]
time = 0.14, size = 104, normalized size = 0.61 \begin {gather*} \frac {2 (a e+c d x)^3 ((a e+c d x) (d+e x))^{5/2} \left (63 e^2-\frac {154 c d e (d+e x)}{a e+c d x}+\frac {99 c^2 d^2 (d+e x)^2}{(a e+c d x)^2}\right )}{693 \left (c d^2-a e^2\right )^3 (d+e x)^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(d + e*x)^9,x]

[Out]

(2*(a*e + c*d*x)^3*((a*e + c*d*x)*(d + e*x))^(5/2)*(63*e^2 - (154*c*d*e*(d + e*x))/(a*e + c*d*x) + (99*c^2*d^2
*(d + e*x)^2)/(a*e + c*d*x)^2))/(693*(c*d^2 - a*e^2)^3*(d + e*x)^8)

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Maple [A]
time = 0.72, size = 212, normalized size = 1.24

method	result	size
gosper	\(-\frac {2 \left (c d x +a e \right ) \left (8 e^{2} x^{2} c^{2} d^{2}-28 a c d \,e^{3} x +44 c^{2} d^{3} e x +63 a^{2} e^{4}-154 a c \,d^{2} e^{2}+99 c^{2} d^{4}\right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {5}{2}}}{693 \left (e x +d \right )^{8} \left (e^{6} a^{3}-3 e^{4} d^{2} a^{2} c +3 d^{4} e^{2} c^{2} a -d^{6} c^{3}\right )}\)	\(146\)
default	\(\frac {-\frac {2 \left (c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {7}{2}}}{11 \left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{9}}-\frac {4 c d e \left (-\frac {2 \left (c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {7}{2}}}{9 \left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{8}}+\frac {4 c d e \left (c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {7}{2}}}{63 \left (e^{2} a -c \,d^{2}\right )^{2} \left (x +\frac {d}{e}\right )^{7}}\right )}{11 \left (e^{2} a -c \,d^{2}\right )}}{e^{9}}\)	\(212\)
trager	\(-\frac {2 \left (8 c^{5} d^{5} e^{2} x^{5}-4 a \,c^{4} d^{4} e^{3} x^{4}+44 c^{5} d^{6} e \,x^{4}+3 a^{2} c^{3} d^{3} e^{4} x^{3}-22 a \,c^{4} d^{5} e^{2} x^{3}+99 c^{5} d^{7} x^{3}+113 a^{3} c^{2} d^{2} e^{5} x^{2}-330 a^{2} c^{3} d^{4} e^{3} x^{2}+297 a \,c^{4} d^{6} e \,x^{2}+161 a^{4} c d \,e^{6} x -418 a^{3} c^{2} d^{3} e^{4} x +297 a^{2} c^{3} d^{5} e^{2} x +63 a^{5} e^{7}-154 a^{4} c \,d^{2} e^{5}+99 a^{3} c^{2} d^{4} e^{3}\right ) \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}}{693 \left (e^{6} a^{3}-3 e^{4} d^{2} a^{2} c +3 d^{4} e^{2} c^{2} a -d^{6} c^{3}\right ) \left (e x +d \right )^{6}}\)	\(285\)

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)^(5/2)/(e*x+d)^9,x,method=_RETURNVERBOSE)

[Out]

1/e^9*(-2/11/(a*e^2-c*d^2)/(x+d/e)^9*(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(7/2)-4/11*c*d*e/(a*e^2-c*d^2)*(-
2/9/(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))^(7/2)+4/63*c*d*e/(a*e^2-c*d^2)^2/(x+d/e)^7
*(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(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((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^9,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. 567 vs. \(2 (162) = 324\).
time = 84.55, size = 567, normalized size = 3.32 \begin {gather*} \frac {2 \, {\left (99 \, c^{5} d^{7} x^{3} + 161 \, a^{4} c d x e^{6} + 63 \, a^{5} e^{7} + {\left (113 \, a^{3} c^{2} d^{2} x^{2} - 154 \, a^{4} c d^{2}\right )} e^{5} + {\left (3 \, a^{2} c^{3} d^{3} x^{3} - 418 \, a^{3} c^{2} d^{3} x\right )} e^{4} - {\left (4 \, a c^{4} d^{4} x^{4} + 330 \, a^{2} c^{3} d^{4} x^{2} - 99 \, a^{3} c^{2} d^{4}\right )} e^{3} + {\left (8 \, c^{5} d^{5} x^{5} - 22 \, a c^{4} d^{5} x^{3} + 297 \, a^{2} c^{3} d^{5} x\right )} e^{2} + 11 \, {\left (4 \, c^{5} d^{6} x^{4} + 27 \, a c^{4} d^{6} x^{2}\right )} e\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e}}{693 \, {\left (6 \, c^{3} d^{11} x e + c^{3} d^{12} - a^{3} x^{6} e^{12} - 6 \, a^{3} d x^{5} e^{11} + 3 \, {\left (a^{2} c d^{2} x^{6} - 5 \, a^{3} d^{2} x^{4}\right )} e^{10} + 2 \, {\left (9 \, a^{2} c d^{3} x^{5} - 10 \, a^{3} d^{3} x^{3}\right )} e^{9} - 3 \, {\left (a c^{2} d^{4} x^{6} - 15 \, a^{2} c d^{4} x^{4} + 5 \, a^{3} d^{4} x^{2}\right )} e^{8} - 6 \, {\left (3 \, a c^{2} d^{5} x^{5} - 10 \, a^{2} c d^{5} x^{3} + a^{3} d^{5} x\right )} e^{7} + {\left (c^{3} d^{6} x^{6} - 45 \, a c^{2} d^{6} x^{4} + 45 \, a^{2} c d^{6} x^{2} - a^{3} d^{6}\right )} e^{6} + 6 \, {\left (c^{3} d^{7} x^{5} - 10 \, a c^{2} d^{7} x^{3} + 3 \, a^{2} c d^{7} x\right )} e^{5} + 3 \, {\left (5 \, c^{3} d^{8} x^{4} - 15 \, a c^{2} d^{8} x^{2} + a^{2} c d^{8}\right )} e^{4} + 2 \, {\left (10 \, c^{3} d^{9} x^{3} - 9 \, a c^{2} d^{9} x\right )} e^{3} + 3 \, {\left (5 \, c^{3} d^{10} x^{2} - a c^{2} d^{10}\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)^(5/2)/(e*x+d)^9,x, algorithm="fricas")

[Out]

2/693*(99*c^5*d^7*x^3 + 161*a^4*c*d*x*e^6 + 63*a^5*e^7 + (113*a^3*c^2*d^2*x^2 - 154*a^4*c*d^2)*e^5 + (3*a^2*c^
3*d^3*x^3 - 418*a^3*c^2*d^3*x)*e^4 - (4*a*c^4*d^4*x^4 + 330*a^2*c^3*d^4*x^2 - 99*a^3*c^2*d^4)*e^3 + (8*c^5*d^5
*x^5 - 22*a*c^4*d^5*x^3 + 297*a^2*c^3*d^5*x)*e^2 + 11*(4*c^5*d^6*x^4 + 27*a*c^4*d^6*x^2)*e)*sqrt(c*d^2*x + a*x
*e^2 + (c*d*x^2 + a*d)*e)/(6*c^3*d^11*x*e + c^3*d^12 - a^3*x^6*e^12 - 6*a^3*d*x^5*e^11 + 3*(a^2*c*d^2*x^6 - 5*
a^3*d^2*x^4)*e^10 + 2*(9*a^2*c*d^3*x^5 - 10*a^3*d^3*x^3)*e^9 - 3*(a*c^2*d^4*x^6 - 15*a^2*c*d^4*x^4 + 5*a^3*d^4
*x^2)*e^8 - 6*(3*a*c^2*d^5*x^5 - 10*a^2*c*d^5*x^3 + a^3*d^5*x)*e^7 + (c^3*d^6*x^6 - 45*a*c^2*d^6*x^4 + 45*a^2*
c*d^6*x^2 - a^3*d^6)*e^6 + 6*(c^3*d^7*x^5 - 10*a*c^2*d^7*x^3 + 3*a^2*c*d^7*x)*e^5 + 3*(5*c^3*d^8*x^4 - 15*a*c^
2*d^8*x^2 + a^2*c*d^8)*e^4 + 2*(10*c^3*d^9*x^3 - 9*a*c^2*d^9*x)*e^3 + 3*(5*c^3*d^10*x^2 - a*c^2*d^10)*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)**(5/2)/(e*x+d)**9,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)^(5/2)/(e*x+d)^9,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 = 7.21, size = 2500, normalized size = 14.62 \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)^(5/2)/(d + e*x)^9,x)

[Out]

(((d*((8*c^5*d^6)/(99*e*(a*e^2 - c*d^2)^2*(5*a*e^3 - 5*c*d^2*e)) - (4*c^4*d^4*(19*a*e^2 - 15*c*d^2))/(99*e*(a*
e^2 - c*d^2)^2*(5*a*e^3 - 5*c*d^2*e))))/e + (4*c^3*d^3*(33*a^2*e^4 + 16*c^2*d^4 - 47*a*c*d^2*e^2))/(99*e^2*(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 - (((2*a^3*e
^4)/(11*a*e^3 - 11*c*d^2*e) - (d*((d*((2*c^3*d^4)/(11*a*e^3 - 11*c*d^2*e) - (6*a*c^2*d^2*e^2)/(11*a*e^3 - 11*c
*d^2*e)))/e + (6*a^2*c*d*e^3)/(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 + (((d*((16*c^6*d^7)/(693*e*(a*e^2 - c*d^2)^3*(3*a*e^3 - 3*c*d^2*e)) - (8*c^5*d^5*(29*a*e^2 - 25*c*d^2
))/(693*e*(a*e^2 - c*d^2)^3*(3*a*e^3 - 3*c*d^2*e))))/e + (8*c^4*d^4*(466*a^2*e^4 + 331*c^2*d^4 - 787*a*c*d^2*e
^2))/(3465*e^2*(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 + (((6*c^4*d^5 + 22*a*c^3*d^3*e^2)/(77*e^2*(a*e^2 - c*d^2)*(5*a*e^3 - 5*c*d^2*e)) - (4*c^4*d^5)/(11*e^2*
(a*e^2 - c*d^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 + (((188*c^
5*d^6 - 148*a*c^4*d^4*e^2)/(495*e^2*(a*e^2 - c*d^2)^2*(3*a*e^3 - 3*c*d^2*e)) - (8*c^5*d^6)/(99*e^2*(a*e^2 - c*
d^2)^2*(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*((d*((64*c^8*
d^9)/(10395*e*(a*e^2 - c*d^2)^6) - (64*c^7*d^7*(23*a*e^2 - 20*c*d^2))/(10395*e*(a*e^2 - c*d^2)^6)))/e + (64*c^
6*d^6*(218*a^2*e^4 + 175*c^2*d^4 - 390*a*c*d^2*e^2))/(10395*e^2*(a*e^2 - c*d^2)^6)))/e - (64*a*c^5*d^5*(196*a^
2*e^4 + 175*c^2*d^4 - 370*a*c*d^2*e^2))/(10395*e*(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) - (((16*c^6*d^7)/(693*e^3*(a*e^2 - c*d^2)^4) + (8*c^5*d^5*(497*a*e^2 - 527*c*d^2))/(10395*e^3*
(a*e^2 - c*d^2)^4))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x) + (((d*((4*c^4*d^5)/(11*e*(a*e^2
- c*d^2)*(7*a*e^3 - 7*c*d^2*e)) - (2*c^3*d^3*(7*a*e^2 - 3*c*d^2))/(11*e*(a*e^2 - c*d^2)*(7*a*e^3 - 7*c*d^2*e))
))/e + (8*c^4*d^6 - 34*a*c^3*d^4*e^2 + 38*a^2*c^2*d^2*e^4)/(33*e^2*(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 + (((4992*c^8*d^11 - 5600*a*c^7*d^9*e^2 - 4960*a^2*c^6
*d^7*e^4 + 5632*a^3*c^5*d^5*e^6)/(10395*e^3*(a*e^2 - c*d^2)^6) - (d*((d*((64*c^8*d^9)/(10395*e*(a*e^2 - c*d^2)
^6) - (32*c^7*d^7*(43*a*e^2 - 37*c*d^2))/(10395*e*(a*e^2 - c*d^2)^6)))/e + (32*c^6*d^6*(373*a^2*e^4 + 293*c^2*
d^4 - 660*a*c*d^2*e^2))/(10395*e^2*(a*e^2 - c*d^2)^6)))/e)*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d +
e*x) + (((d*((d*((8*c^5*d^6)/(99*(a*e^2 - c*d^2)^2*(7*a*e^3 - 7*c*d^2*e)) - (8*c^4*d^4*(14*a*e^2 - 11*c*d^2))
/(99*(a*e^2 - c*d^2)^2*(7*a*e^3 - 7*c*d^2*e))))/e + (8*c^3*d^3*(66*a^2*e^4 + 41*c^2*d^4 - 104*a*c*d^2*e^2))/(9
9*e*(a*e^2 - c*d^2)^2*(7*a*e^3 - 7*c*d^2*e))))/e - (8*a*c^2*d^2*(53*a^2*e^4 + 41*c^2*d^4 - 93*a*c*d^2*e^2))/(9
9*(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*
((d*((16*c^6*d^7)/(693*(a*e^2 - c*d^2)^3*(5*a*e^3 - 5*c*d^2*e)) - (16*c^5*d^5*(6*a*e^2 - 5*c*d^2))/(231*(a*e^2
- c*d^2)^3*(5*a*e^3 - 5*c*d^2*e))))/e + (16*c^4*d^4*(122*a^2*e^4 + 89*c^2*d^4 - 208*a*c*d^2*e^2))/(693*e*(a*e
^2 - c*d^2)^3*(5*a*e^3 - 5*c*d^2*e))))/e - (16*a*c^3*d^3*(105*a^2*e^4 + 89*c^2*d^4 - 193*a*c*d^2*e^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*((d*
((32*c^7*d^8)/(3465*(a*e^2 - c*d^2)^4*(3*a*e^3 - 3*c*d^2*e)) - (32*c^6*d^6*(7*a*e^2 - 6*c*d^2))/(1155*(a*e^2 -
c*d^2)^4*(3*a*e^3 - 3*c*d^2*e))))/e + (32*c^5*d^5*(176*a^2*e^4 + 137*c^2*d^4 - 310*a*c*d^2*e^2))/(3465*e*(a*e
^2 - c*d^2)^4*(3*a*e^3 - 3*c*d^2*e))))/e - (32*a*c^4*d^4*(156*a^2*e^4 + 137*c^2*d^4 - 292*a*c*d^2*e^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 + (((d*((
32*c^7*d^8)/(3465*e^2*(a*e^2 - c*d^2)^5) - (16*c^6*d^6*(37*a*e^2 - 33*c*d^2))/(3465*e^2*(a*e^2 - c*d^2)^5)))/e
+ (16*c^5*d^5*(542*a^2*e^4 + 437*c^2*d^4 - 973*a*c*d^2*e^2))/(10395*e^3*(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*((d*((4*c^4*d^5)/(11*(a*e^2 - c*d^2)*(9*a*e^3 - 9*c*d^2*e)) - (
2*c^3*d^3*(7*a*e^2 - c*d^2))/(11*(a*e^2 - c*d^2)*(9*a*e^3 - 9*c*d^2*e))))/e + (2*c^2*d^2*(9*a^2*e^4 + c^2*d^4
- 4*a*c*d^2*e^2))/(11*e*(a*e^2 - c*d^2)*(9*a*e^3 - 9*c*d^2*e))))/e - (2*c^4*d^7 - 4*a*c^3*d^5*e^2 + 6*a^3*c*d*
e^6)/(11*e^2*(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*((d*((8*c^5*d^6)/(99*(a*e^2 - c*d^2)^2*(7*a*e^3 - 7*c*d^2*e)) - (4*c^4*d^4*(19*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*c^3*d^3*(51*a^2*e^4 + 19*c^2*d^4 - 64*a*c*d^2*e^2))/(99*e*(
a*e^2 - c*d^2)^2*(7*a*e^3 - 7*c*d^2*e))))/e - (52*c^5*d^8 - 80*a*c^4*d^6*e^2 - 48*a^2*c^3*d^4*e^4 + 84*a^3*c^2
*d^2*e^6)/(99*e^2*(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*((d*((16*c^6*d^7)/(693*(a*e^2 -...

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