3.20.45 \(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}}{(d+e x)^{10}} \, dx\) [1945]

Optimal. Leaf size=231 \[ \frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{13 \left (c d^2-a e^2\right ) (d+e x)^{10}}+\frac {12 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{143 \left (c d^2-a e^2\right )^2 (d+e x)^9}+\frac {16 c^2 d^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{429 \left (c d^2-a e^2\right )^3 (d+e x)^8}+\frac {32 c^3 d^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{3003 \left (c d^2-a e^2\right )^4 (d+e x)^7} \]

[Out]

2/13*(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)^10+12/143*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)^9+16/429*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)^8+32/3003*c^3*d^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/(-a*e^2+c*d^2)^4/(e*x+d)^7

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Rubi [A]
time = 0.09, 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 )^{7/2}}{3003 (d+e x)^7 \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 )^{7/2}}{429 (d+e x)^8 \left (c d^2-a e^2\right )^3}+\frac {12 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{143 (d+e x)^9 \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}}{13 (d+e x)^{10} \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)^10,x]

[Out]

(2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(13*(c*d^2 - a*e^2)*(d + e*x)^10) + (12*c*d*(a*d*e + (c*d^2
+ a*e^2)*x + c*d*e*x^2)^(7/2))/(143*(c*d^2 - a*e^2)^2*(d + e*x)^9) + (16*c^2*d^2*(a*d*e + (c*d^2 + a*e^2)*x +
c*d*e*x^2)^(7/2))/(429*(c*d^2 - a*e^2)^3*(d + e*x)^8) + (32*c^3*d^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7
/2))/(3003*(c*d^2 - a*e^2)^4*(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)^{10}} \, dx &=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{13 \left (c d^2-a e^2\right ) (d+e x)^{10}}+\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 )^{5/2}}{(d+e x)^9} \, dx}{13 \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}}{13 \left (c d^2-a e^2\right ) (d+e x)^{10}}+\frac {12 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{143 \left (c d^2-a e^2\right )^2 (d+e x)^9}+\frac {\left (24 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)^8} \, dx}{143 \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}}{13 \left (c d^2-a e^2\right ) (d+e x)^{10}}+\frac {12 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{143 \left (c d^2-a e^2\right )^2 (d+e x)^9}+\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}}{429 \left (c d^2-a e^2\right )^3 (d+e x)^8}+\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 )^{5/2}}{(d+e x)^7} \, dx}{429 \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 )^{7/2}}{13 \left (c d^2-a e^2\right ) (d+e x)^{10}}+\frac {12 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{143 \left (c d^2-a e^2\right )^2 (d+e x)^9}+\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}}{429 \left (c d^2-a e^2\right )^3 (d+e x)^8}+\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 )^{7/2}}{3003 \left (c d^2-a e^2\right )^4 (d+e x)^7}\\ \end {align*}

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Mathematica [A]
time = 0.18, size = 132, normalized size = 0.57 \begin {gather*} \frac {2 (a e+c d x)^4 ((a e+c d x) (d+e x))^{5/2} \left (-231 e^3+\frac {819 c d e^2 (d+e x)}{a e+c d x}-\frac {1001 c^2 d^2 e (d+e x)^2}{(a e+c d x)^2}+\frac {429 c^3 d^3 (d+e x)^3}{(a e+c d x)^3}\right )}{3003 \left (c d^2-a e^2\right )^4 (d+e x)^9} \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)^10,x]

[Out]

(2*(a*e + c*d*x)^4*((a*e + c*d*x)*(d + e*x))^(5/2)*(-231*e^3 + (819*c*d*e^2*(d + e*x))/(a*e + c*d*x) - (1001*c
^2*d^2*e*(d + e*x)^2)/(a*e + c*d*x)^2 + (429*c^3*d^3*(d + e*x)^3)/(a*e + c*d*x)^3))/(3003*(c*d^2 - a*e^2)^4*(d
+ e*x)^9)

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

method	result	size
gosper	\(-\frac {2 \left (c d x +a e \right ) \left (-16 c^{3} d^{3} e^{3} x^{3}+56 a \,c^{2} d^{2} e^{4} x^{2}-104 c^{3} d^{4} e^{2} x^{2}-126 a^{2} c d \,e^{5} x +364 a \,c^{2} d^{3} e^{3} x -286 c^{3} d^{5} e x +231 e^{6} a^{3}-819 e^{4} d^{2} a^{2} c +1001 d^{4} e^{2} c^{2} a -429 d^{6} c^{3}\right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {5}{2}}}{3003 \left (e x +d \right )^{9} \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right )}\)	\(217\)
default	\(\frac {-\frac {2 \left (c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {7}{2}}}{13 \left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{10}}-\frac {6 c d e \left (-\frac {2 \left (c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {7}{2}}}{11 \left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{9}}-\frac {4 c d e \left (-\frac {2 \left (c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {7}{2}}}{9 \left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{8}}+\frac {4 c d e \left (c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {7}{2}}}{63 \left (e^{2} a -c \,d^{2}\right )^{2} \left (x +\frac {d}{e}\right )^{7}}\right )}{11 \left (e^{2} a -c \,d^{2}\right )}\right )}{13 \left (e^{2} a -c \,d^{2}\right )}}{e^{10}}\)	\(293\)
trager	\(-\frac {2 \left (-16 c^{6} d^{6} e^{3} x^{6}+8 a \,c^{5} d^{5} e^{4} x^{5}-104 c^{6} d^{7} e^{2} x^{5}-6 a^{2} c^{4} e^{5} d^{4} x^{4}+52 a \,c^{5} d^{6} e^{3} x^{4}-286 c^{6} d^{8} e \,x^{4}+5 a^{3} c^{3} d^{3} e^{6} x^{3}-39 a^{2} c^{4} d^{5} e^{4} x^{3}+143 a \,c^{5} d^{7} e^{2} x^{3}-429 c^{6} d^{9} x^{3}+371 a^{4} c^{2} d^{2} e^{7} x^{2}-1469 a^{3} c^{3} d^{4} e^{5} x^{2}+2145 a^{2} c^{4} d^{6} e^{3} x^{2}-1287 a \,c^{5} d^{8} e \,x^{2}+567 a^{5} c d \,e^{8} x -2093 a^{4} c^{2} d^{3} e^{6} x +2717 a^{3} c^{3} d^{5} e^{4} x -1287 a^{2} c^{4} d^{7} e^{2} x +231 a^{6} e^{9}-819 a^{5} c \,d^{2} e^{7}+1001 a^{4} c^{2} d^{4} e^{5}-429 a^{3} c^{3} d^{6} e^{3}\right ) \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}}{3003 \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right ) \left (e x +d \right )^{7}}\)	\(407\)

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)^10,x,method=_RETURNVERBOSE)

[Out]

1/e^10*(-2/13/(a*e^2-c*d^2)/(x+d/e)^10*(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(7/2)-6/13*c*d*e/(a*e^2-c*d^2)*
(-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)^10,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. 834 vs. \(2 (219) = 438\).
time = 165.13, size = 834, normalized size = 3.61 \begin {gather*} \frac {2 \, {\left (429 \, c^{6} d^{9} x^{3} - 567 \, a^{5} c d x e^{8} - 231 \, a^{6} e^{9} - 7 \, {\left (53 \, a^{4} c^{2} d^{2} x^{2} - 117 \, a^{5} c d^{2}\right )} e^{7} - {\left (5 \, a^{3} c^{3} d^{3} x^{3} - 2093 \, a^{4} c^{2} d^{3} x\right )} e^{6} + {\left (6 \, a^{2} c^{4} d^{4} x^{4} + 1469 \, a^{3} c^{3} d^{4} x^{2} - 1001 \, a^{4} c^{2} d^{4}\right )} e^{5} - {\left (8 \, a c^{5} d^{5} x^{5} - 39 \, a^{2} c^{4} d^{5} x^{3} + 2717 \, a^{3} c^{3} d^{5} x\right )} e^{4} + {\left (16 \, c^{6} d^{6} x^{6} - 52 \, a c^{5} d^{6} x^{4} - 2145 \, a^{2} c^{4} d^{6} x^{2} + 429 \, a^{3} c^{3} d^{6}\right )} e^{3} + 13 \, {\left (8 \, c^{6} d^{7} x^{5} - 11 \, a c^{5} d^{7} x^{3} + 99 \, a^{2} c^{4} d^{7} x\right )} e^{2} + 143 \, {\left (2 \, c^{6} d^{8} x^{4} + 9 \, a c^{5} d^{8} x^{2}\right )} e\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e}}{3003 \, {\left (7 \, c^{4} d^{14} x e + c^{4} d^{15} + a^{4} x^{7} e^{15} + 7 \, a^{4} d x^{6} e^{14} - {\left (4 \, a^{3} c d^{2} x^{7} - 21 \, a^{4} d^{2} x^{5}\right )} e^{13} - 7 \, {\left (4 \, a^{3} c d^{3} x^{6} - 5 \, a^{4} d^{3} x^{4}\right )} e^{12} + {\left (6 \, a^{2} c^{2} d^{4} x^{7} - 84 \, a^{3} c d^{4} x^{5} + 35 \, a^{4} d^{4} x^{3}\right )} e^{11} + 7 \, {\left (6 \, a^{2} c^{2} d^{5} x^{6} - 20 \, a^{3} c d^{5} x^{4} + 3 \, a^{4} d^{5} x^{2}\right )} e^{10} - {\left (4 \, a c^{3} d^{6} x^{7} - 126 \, a^{2} c^{2} d^{6} x^{5} + 140 \, a^{3} c d^{6} x^{3} - 7 \, a^{4} d^{6} x\right )} e^{9} - {\left (28 \, a c^{3} d^{7} x^{6} - 210 \, a^{2} c^{2} d^{7} x^{4} + 84 \, a^{3} c d^{7} x^{2} - a^{4} d^{7}\right )} e^{8} + {\left (c^{4} d^{8} x^{7} - 84 \, a c^{3} d^{8} x^{5} + 210 \, a^{2} c^{2} d^{8} x^{3} - 28 \, a^{3} c d^{8} x\right )} e^{7} + {\left (7 \, c^{4} d^{9} x^{6} - 140 \, a c^{3} d^{9} x^{4} + 126 \, a^{2} c^{2} d^{9} x^{2} - 4 \, a^{3} c d^{9}\right )} e^{6} + 7 \, {\left (3 \, c^{4} d^{10} x^{5} - 20 \, a c^{3} d^{10} x^{3} + 6 \, a^{2} c^{2} d^{10} x\right )} e^{5} + {\left (35 \, c^{4} d^{11} x^{4} - 84 \, a c^{3} d^{11} x^{2} + 6 \, a^{2} c^{2} d^{11}\right )} e^{4} + 7 \, {\left (5 \, c^{4} d^{12} x^{3} - 4 \, a c^{3} d^{12} x\right )} e^{3} + {\left (21 \, c^{4} d^{13} x^{2} - 4 \, a c^{3} d^{13}\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)^10,x, algorithm="fricas")

[Out]

2/3003*(429*c^6*d^9*x^3 - 567*a^5*c*d*x*e^8 - 231*a^6*e^9 - 7*(53*a^4*c^2*d^2*x^2 - 117*a^5*c*d^2)*e^7 - (5*a^
3*c^3*d^3*x^3 - 2093*a^4*c^2*d^3*x)*e^6 + (6*a^2*c^4*d^4*x^4 + 1469*a^3*c^3*d^4*x^2 - 1001*a^4*c^2*d^4)*e^5 -
(8*a*c^5*d^5*x^5 - 39*a^2*c^4*d^5*x^3 + 2717*a^3*c^3*d^5*x)*e^4 + (16*c^6*d^6*x^6 - 52*a*c^5*d^6*x^4 - 2145*a^
2*c^4*d^6*x^2 + 429*a^3*c^3*d^6)*e^3 + 13*(8*c^6*d^7*x^5 - 11*a*c^5*d^7*x^3 + 99*a^2*c^4*d^7*x)*e^2 + 143*(2*c
^6*d^8*x^4 + 9*a*c^5*d^8*x^2)*e)*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)/(7*c^4*d^14*x*e + c^4*d^15 + a^4*
x^7*e^15 + 7*a^4*d*x^6*e^14 - (4*a^3*c*d^2*x^7 - 21*a^4*d^2*x^5)*e^13 - 7*(4*a^3*c*d^3*x^6 - 5*a^4*d^3*x^4)*e^
12 + (6*a^2*c^2*d^4*x^7 - 84*a^3*c*d^4*x^5 + 35*a^4*d^4*x^3)*e^11 + 7*(6*a^2*c^2*d^5*x^6 - 20*a^3*c*d^5*x^4 +
3*a^4*d^5*x^2)*e^10 - (4*a*c^3*d^6*x^7 - 126*a^2*c^2*d^6*x^5 + 140*a^3*c*d^6*x^3 - 7*a^4*d^6*x)*e^9 - (28*a*c^
3*d^7*x^6 - 210*a^2*c^2*d^7*x^4 + 84*a^3*c*d^7*x^2 - a^4*d^7)*e^8 + (c^4*d^8*x^7 - 84*a*c^3*d^8*x^5 + 210*a^2*
c^2*d^8*x^3 - 28*a^3*c*d^8*x)*e^7 + (7*c^4*d^9*x^6 - 140*a*c^3*d^9*x^4 + 126*a^2*c^2*d^9*x^2 - 4*a^3*c*d^9)*e^
6 + 7*(3*c^4*d^10*x^5 - 20*a*c^3*d^10*x^3 + 6*a^2*c^2*d^10*x)*e^5 + (35*c^4*d^11*x^4 - 84*a*c^3*d^11*x^2 + 6*a
^2*c^2*d^11)*e^4 + 7*(5*c^4*d^12*x^3 - 4*a*c^3*d^12*x)*e^3 + (21*c^4*d^13*x^2 - 4*a*c^3*d^13)*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)**10,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)^10,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,[0,0,7]%%%},[14]%%%}+%%%{%%{[%%%{-14,[0,1,6]%%%},0
]:[1,0,%%%{

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Mupad [B]
time = 9.57, size = 2500, normalized size = 10.82 \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)^10,x)

[Out]

(((d*((8*c^5*d^6)/(143*e*(a*e^2 - c*d^2)^2*(7*a*e^3 - 7*c*d^2*e)) - (4*c^4*d^4*(21*a*e^2 - 17*c*d^2))/(143*e*(
a*e^2 - c*d^2)^2*(7*a*e^3 - 7*c*d^2*e))))/e + (4*c^3*d^3*(110*a^2*e^4 + 53*c^2*d^4 - 157*a*c*d^2*e^2))/(429*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 - (((2*
a^3*e^4)/(13*a*e^3 - 13*c*d^2*e) - (d*((d*((2*c^3*d^4)/(13*a*e^3 - 13*c*d^2*e) - (6*a*c^2*d^2*e^2)/(13*a*e^3 -
13*c*d^2*e)))/e + (6*a^2*c*d*e^3)/(13*a*e^3 - 13*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)^7 + (((d*((16*c^6*d^7)/(1287*e*(a*e^2 - c*d^2)^3*(5*a*e^3 - 5*c*d^2*e)) - (8*c^5*d^5*(33*a*e^2 - 29
*c*d^2))/(1287*e*(a*e^2 - c*d^2)^3*(5*a*e^3 - 5*c*d^2*e))))/e + (8*c^4*d^4*(112*a^2*e^4 + 81*c^2*d^4 - 191*a*c
*d^2*e^2))/(1287*e^2*(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^7*d^8)/(9009*e*(a*e^2 - c*d^2)^4*(3*a*e^3 - 3*c*d^2*e)) - (16*c^6*d^6*(43*a*e^2 - 39
*c*d^2))/(9009*e*(a*e^2 - c*d^2)^4*(3*a*e^3 - 3*c*d^2*e))))/e + (16*c^5*d^5*(1089*a^2*e^4 + 884*c^2*d^4 - 1963
*a*c*d^2*e^2))/(45045*e^2*(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 + (((38*c^4*d^5 + 94*a*c^3*d^3*e^2)/(429*e^2*(a*e^2 - c*d^2)*(7*a*e^3 - 7*c*d^2*e)) - (4*c^4*
d^5)/(13*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 + (((348*c^5*d^6 - 292*a*c^4*d^4*e^2)/(1001*e^2*(a*e^2 - c*d^2)^2*(5*a*e^3 - 5*c*d^2*e)) - (8*c^5*d^6)/(143*
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 + (((
d*((d*((128*c^9*d^10)/(135135*e*(a*e^2 - c*d^2)^7) - (128*c^8*d^8*(10*a*e^2 - 9*c*d^2))/(45045*e*(a*e^2 - c*d^
2)^7)))/e + (128*c^7*d^7*(379*a^2*e^4 + 322*c^2*d^4 - 698*a*c*d^2*e^2))/(135135*e^2*(a*e^2 - c*d^2)^7)))/e - (
128*a*c^6*d^6*(350*a^2*e^4 + 322*c^2*d^4 - 671*a*c*d^2*e^2))/(135135*e*(a*e^2 - c*d^2)^7))*(x*(a*e^2 + c*d^2)
+ a*d*e + c*d*e*x^2)^(1/2))/(d + e*x) - (((d*((d*((128*c^9*d^10)/(135135*e*(a*e^2 - c*d^2)^7) - (64*c^8*d^8*(1
9*a*e^2 - 17*c*d^2))/(45045*e*(a*e^2 - c*d^2)^7)))/e + (128*c^7*d^7*(337*a^2*e^4 + 283*c^2*d^4 - 617*a*c*d^2*e
^2))/(135135*e^2*(a*e^2 - c*d^2)^7)))/e - (64*c^6*d^6*(323*a^3*e^6 + 296*c^3*d^6 - 322*a*c^2*d^4*e^2 - 295*a^2
*c*d^2*e^4))/(135135*e^3*(a*e^2 - c*d^2)^7))*(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)/(13*(a*e^2 - c*d^2)*(11*a*e^3 - 11*c*d^2*e)) - (4*c^3*d^3*(10*a*e^2 - 7*c*d^2))/(13*(a*e^2 - c
*d^2)*(11*a*e^3 - 11*c*d^2*e))))/e + (28*c^4*d^6 - 112*a*c^3*d^4*e^2 + 96*a^2*c^2*d^2*e^4)/(13*e*(a*e^2 - c*d^
2)*(11*a*e^3 - 11*c*d^2*e))))/e - (4*a*c*d*(15*a^2*e^4 + 7*c^2*d^4 - 21*a*c*d^2*e^2))/(13*(a*e^2 - c*d^2)*(11*
a*e^3 - 11*c*d^2*e)))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x)^6 - (((32*c^7*d^8)/(9009*e^3*(a
*e^2 - c*d^2)^5) + (16*c^6*d^6*(511*a*e^2 - 521*c*d^2))/(45045*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*((4*c^4*d^5)/(13*e*(a*e^2 - c*d^2)*(9*a*e^3 - 9*c*d^2*e)) - (2*c^3*d^
3*(7*a*e^2 - 3*c*d^2))/(13*e*(a*e^2 - c*d^2)*(9*a*e^3 - 9*c*d^2*e))))/e + (28*c^4*d^6 - 122*a*c^3*d^4*e^2 + 13
8*a^2*c^2*d^2*e^4)/(143*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)/(143*(a*e^2 - c*d^2)^2*(9*a*e^3 - 9*c*d^2*e)) - (8*c^4*d^4*(16*a*e^2
- 13*c*d^2))/(143*(a*e^2 - c*d^2)^2*(9*a*e^3 - 9*c*d^2*e))))/e + (8*c^3*d^3*(84*a^2*e^4 + 55*c^2*d^4 - 136*a*c
*d^2*e^2))/(143*e*(a*e^2 - c*d^2)^2*(9*a*e^3 - 9*c*d^2*e))))/e - (8*a*c^2*d^2*(69*a^2*e^4 + 55*c^2*d^4 - 123*a
*c*d^2*e^2))/(143*(a*e^2 - c*d^2)^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*((16*c^6*d^7)/(1287*(a*e^2 - c*d^2)^3*(7*a*e^3 - 7*c*d^2*e)) - (16*c^5*d^5*(7*a*e^2 - 6*c*
d^2))/(429*(a*e^2 - c*d^2)^3*(7*a*e^3 - 7*c*d^2*e))))/e + (16*c^4*d^4*(164*a^2*e^4 + 125*c^2*d^4 - 286*a*c*d^2
*e^2))/(1287*e*(a*e^2 - c*d^2)^3*(7*a*e^3 - 7*c*d^2*e))))/e - (16*a*c^3*d^3*(144*a^2*e^4 + 125*c^2*d^4 - 268*a
*c*d^2*e^2))/(1287*(a*e^2 - c*d^2)^3*(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*((32*c^7*d^8)/(9009*(a*e^2 - c*d^2)^4*(5*a*e^3 - 5*c*d^2*e)) - (32*c^6*d^6*(25*a*e^2 - 22
*c*d^2))/(9009*(a*e^2 - c*d^2)^4*(5*a*e^3 - 5*c*d^2*e))))/e + (32*c^5*d^5*(248*a^2*e^4 + 201*c^2*d^4 - 446*a*c
*d^2*e^2))/(9009*e*(a*e^2 - c*d^2)^4*(5*a*e^3 - 5*c*d^2*e))))/e - (32*a*c^4*d^4*(224*a^2*e^4 + 201*c^2*d^4 - 4
24*a*c*d^2*e^2))/(9009*(a*e^2 - c*d^2)^4*(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*((64*c^8*d^9)/(45045*(a*e^2 - c*d^2)^5*(3*a*e^3 - 3*c*d^2*e)) - (64*c^7*d^7*(28*a*e^2
- 25*c*d^2))/(45045*(a*e^2 - c*d^2)^5*(3*a*e^3 - 3*c*d^2*e))))/e + (64*c^6*d^6*(323*a^2*e^4 + 270*c^2*d^4 - 5
90*a*c*d^2*e^2))/(45045*e*(a*e^2 - c*d^2)^5*(3*a*e^3 - 3*c*d^2*e))))/e - (64*a*c^5*d^5*(296*a^2*e^4 + 270*c^2*
d^4 - 565*a*c*d^2*e^2))/(45045*(a*e^2 - c*d^2)^...

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