3.1973 \(\int \frac {1}{(d+e x)^3 (a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}} \, dx\)
Optimal. Leaf size=312 \[ \frac {1024 c^4 d^4 e \left (a e^2+c d^2+2 c d e x\right )}{63 \left (c d^2-a e^2\right )^7 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {128 c^3 d^3 \left (a e^2+c d^2+2 c d e x\right )}{63 \left (c d^2-a e^2\right )^5 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac {16 c^2 d^2}{21 (d+e x) \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac {8 c d}{21 (d+e x)^2 \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac {2}{9 (d+e x)^3 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \]
[Out]
2/9/(-a*e^2+c*d^2)/(e*x+d)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)+8/21*c*d/(-a*e^2+c*d^2)^2/(e*x+d)^2/(a*d*
e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)+16/21*c^2*d^2/(-a*e^2+c*d^2)^3/(e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3
/2)-128/63*c^3*d^3*(2*c*d*e*x+a*e^2+c*d^2)/(-a*e^2+c*d^2)^5/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)+1024/63*c^
4*d^4*e*(2*c*d*e*x+a*e^2+c*d^2)/(-a*e^2+c*d^2)^7/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)
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Rubi [A] time = 0.15, antiderivative size = 312, normalized size of antiderivative = 1.00,
number of steps used = 5, number of rules used = 3, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.081, Rules used =
{658, 614, 613} \[ \frac {1024 c^4 d^4 e \left (a e^2+c d^2+2 c d e x\right )}{63 \left (c d^2-a e^2\right )^7 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {128 c^3 d^3 \left (a e^2+c d^2+2 c d e x\right )}{63 \left (c d^2-a e^2\right )^5 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac {16 c^2 d^2}{21 (d+e x) \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac {8 c d}{21 (d+e x)^2 \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac {2}{9 (d+e x)^3 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In]
Int[1/((d + e*x)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)),x]
[Out]
2/(9*(c*d^2 - a*e^2)*(d + e*x)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) + (8*c*d)/(21*(c*d^2 - a*e^2)^
2*(d + e*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) + (16*c^2*d^2)/(21*(c*d^2 - a*e^2)^3*(d + e*x)*(a
*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) - (128*c^3*d^3*(c*d^2 + a*e^2 + 2*c*d*e*x))/(63*(c*d^2 - a*e^2)^5
*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) + (1024*c^4*d^4*e*(c*d^2 + a*e^2 + 2*c*d*e*x))/(63*(c*d^2 - a*
e^2)^7*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])
Rule 613
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[(-2*(b + 2*c*x))/((b^2 - 4*a*c)*Sqrt[a + b*x
+ c*x^2]), x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]
Rule 614
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b + 2*c*x)*(a + b*x + c*x^2)^(p + 1))/((p +
1)*(b^2 - 4*a*c)), x] - Dist[(2*c*(2*p + 3))/((p + 1)*(b^2 - 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1), x], x] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[p, -3/2] && IntegerQ[4*p]
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]
Rubi steps
\begin {align*} \int \frac {1}{(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx &=\frac {2}{9 \left (c d^2-a e^2\right ) (d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {(4 c d) \int \frac {1}{(d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx}{3 \left (c d^2-a e^2\right )}\\ &=\frac {2}{9 \left (c d^2-a e^2\right ) (d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {8 c d}{21 \left (c d^2-a e^2\right )^2 (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {\left (40 c^2 d^2\right ) \int \frac {1}{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx}{21 \left (c d^2-a e^2\right )^2}\\ &=\frac {2}{9 \left (c d^2-a e^2\right ) (d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {8 c d}{21 \left (c d^2-a e^2\right )^2 (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {16 c^2 d^2}{21 \left (c d^2-a e^2\right )^3 (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {\left (64 c^3 d^3\right ) \int \frac {1}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx}{21 \left (c d^2-a e^2\right )^3}\\ &=\frac {2}{9 \left (c d^2-a e^2\right ) (d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {8 c d}{21 \left (c d^2-a e^2\right )^2 (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {16 c^2 d^2}{21 \left (c d^2-a e^2\right )^3 (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {128 c^3 d^3 \left (c d^2+a e^2+2 c d e x\right )}{63 \left (c d^2-a e^2\right )^5 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {\left (512 c^4 d^4 e\right ) \int \frac {1}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx}{63 \left (c d^2-a e^2\right )^5}\\ &=\frac {2}{9 \left (c d^2-a e^2\right ) (d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {8 c d}{21 \left (c d^2-a e^2\right )^2 (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {16 c^2 d^2}{21 \left (c d^2-a e^2\right )^3 (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {128 c^3 d^3 \left (c d^2+a e^2+2 c d e x\right )}{63 \left (c d^2-a e^2\right )^5 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {1024 c^4 d^4 e \left (c d^2+a e^2+2 c d e x\right )}{63 \left (c d^2-a e^2\right )^7 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 336, normalized size = 1.08 \[ \frac {2 \left (7 a^6 e^{12}-6 a^5 c d e^{10} (9 d+2 e x)+3 a^4 c^2 d^2 e^8 \left (63 d^2+36 d e x+8 e^2 x^2\right )-4 a^3 c^3 d^3 e^6 \left (105 d^3+126 d^2 e x+72 d e^2 x^2+16 e^3 x^3\right )+3 a^2 c^4 d^4 e^4 \left (315 d^4+840 d^3 e x+1008 d^2 e^2 x^2+576 d e^3 x^3+128 e^4 x^4\right )+6 a c^5 d^5 e^2 \left (63 d^5+630 d^4 e x+1680 d^3 e^2 x^2+2016 d^2 e^3 x^3+1152 d e^4 x^4+256 e^5 x^5\right )+c^6 d^6 \left (-21 d^6+252 d^5 e x+2520 d^4 e^2 x^2+6720 d^3 e^3 x^3+8064 d^2 e^4 x^4+4608 d e^5 x^5+1024 e^6 x^6\right )\right )}{63 (d+e x)^3 \left (c d^2-a e^2\right )^7 ((d+e x) (a e+c d x))^{3/2}} \]
Antiderivative was successfully verified.
[In]
Integrate[1/((d + e*x)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)),x]
[Out]
(2*(7*a^6*e^12 - 6*a^5*c*d*e^10*(9*d + 2*e*x) + 3*a^4*c^2*d^2*e^8*(63*d^2 + 36*d*e*x + 8*e^2*x^2) - 4*a^3*c^3*
d^3*e^6*(105*d^3 + 126*d^2*e*x + 72*d*e^2*x^2 + 16*e^3*x^3) + 3*a^2*c^4*d^4*e^4*(315*d^4 + 840*d^3*e*x + 1008*
d^2*e^2*x^2 + 576*d*e^3*x^3 + 128*e^4*x^4) + 6*a*c^5*d^5*e^2*(63*d^5 + 630*d^4*e*x + 1680*d^3*e^2*x^2 + 2016*d
^2*e^3*x^3 + 1152*d*e^4*x^4 + 256*e^5*x^5) + c^6*d^6*(-21*d^6 + 252*d^5*e*x + 2520*d^4*e^2*x^2 + 6720*d^3*e^3*
x^3 + 8064*d^2*e^4*x^4 + 4608*d*e^5*x^5 + 1024*e^6*x^6)))/(63*(c*d^2 - a*e^2)^7*(d + e*x)^3*((a*e + c*d*x)*(d
+ e*x))^(3/2))
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(e*x+d)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algorithm="fricas")
[Out]
Timed out
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(e*x+d)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algorithm="giac")
[Out]
Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Eval
uation time: 0.57Unable to transpose Error: Bad Argument Value
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maple [A] time = 0.06, size = 536, normalized size = 1.72 \[ -\frac {2 \left (c d x +a e \right ) \left (1024 c^{6} d^{6} e^{6} x^{6}+1536 a \,c^{5} d^{5} e^{7} x^{5}+4608 c^{6} d^{7} e^{5} x^{5}+384 a^{2} c^{4} d^{4} e^{8} x^{4}+6912 a \,c^{5} d^{6} e^{6} x^{4}+8064 c^{6} d^{8} e^{4} x^{4}-64 a^{3} c^{3} d^{3} e^{9} x^{3}+1728 a^{2} c^{4} d^{5} e^{7} x^{3}+12096 a \,c^{5} d^{7} e^{5} x^{3}+6720 c^{6} d^{9} e^{3} x^{3}+24 a^{4} c^{2} d^{2} e^{10} x^{2}-288 a^{3} c^{3} d^{4} e^{8} x^{2}+3024 a^{2} c^{4} d^{6} e^{6} x^{2}+10080 a \,c^{5} d^{8} e^{4} x^{2}+2520 c^{6} d^{10} e^{2} x^{2}-12 a^{5} c d \,e^{11} x +108 a^{4} c^{2} d^{3} e^{9} x -504 a^{3} c^{3} d^{5} e^{7} x +2520 a^{2} c^{4} d^{7} e^{5} x +3780 a \,c^{5} d^{9} e^{3} x +252 c^{6} d^{11} e x +7 a^{6} e^{12}-54 a^{5} c \,d^{2} e^{10}+189 a^{4} c^{2} d^{4} e^{8}-420 a^{3} c^{3} d^{6} e^{6}+945 a^{2} c^{4} d^{8} e^{4}+378 a \,c^{5} d^{10} e^{2}-21 c^{6} d^{12}\right )}{63 \left (e x +d \right )^{2} \left (a^{7} e^{14}-7 a^{6} c \,d^{2} e^{12}+21 a^{5} c^{2} d^{4} e^{10}-35 a^{4} c^{3} d^{6} e^{8}+35 a^{3} c^{4} d^{8} e^{6}-21 a^{2} c^{5} d^{10} e^{4}+7 a \,c^{6} d^{12} e^{2}-c^{7} d^{14}\right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(e*x+d)^3/(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(5/2),x)
[Out]
-2/63*(c*d*x+a*e)*(1024*c^6*d^6*e^6*x^6+1536*a*c^5*d^5*e^7*x^5+4608*c^6*d^7*e^5*x^5+384*a^2*c^4*d^4*e^8*x^4+69
12*a*c^5*d^6*e^6*x^4+8064*c^6*d^8*e^4*x^4-64*a^3*c^3*d^3*e^9*x^3+1728*a^2*c^4*d^5*e^7*x^3+12096*a*c^5*d^7*e^5*
x^3+6720*c^6*d^9*e^3*x^3+24*a^4*c^2*d^2*e^10*x^2-288*a^3*c^3*d^4*e^8*x^2+3024*a^2*c^4*d^6*e^6*x^2+10080*a*c^5*
d^8*e^4*x^2+2520*c^6*d^10*e^2*x^2-12*a^5*c*d*e^11*x+108*a^4*c^2*d^3*e^9*x-504*a^3*c^3*d^5*e^7*x+2520*a^2*c^4*d
^7*e^5*x+3780*a*c^5*d^9*e^3*x+252*c^6*d^11*e*x+7*a^6*e^12-54*a^5*c*d^2*e^10+189*a^4*c^2*d^4*e^8-420*a^3*c^3*d^
6*e^6+945*a^2*c^4*d^8*e^4+378*a*c^5*d^10*e^2-21*c^6*d^12)/(e*x+d)^2/(a^7*e^14-7*a^6*c*d^2*e^12+21*a^5*c^2*d^4*
e^10-35*a^4*c^3*d^6*e^8+35*a^3*c^4*d^8*e^6-21*a^2*c^5*d^10*e^4+7*a*c^6*d^12*e^2-c^7*d^14)/(c*d*e*x^2+a*e^2*x+c
*d^2*x+a*d*e)^(5/2)
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(e*x+d)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),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(a*e^2-c*d^2>0)', see `assume?`
for more details)Is a*e^2-c*d^2 zero or nonzero?
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mupad [B] time = 6.19, size = 10949, normalized size = 35.09 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/((d + e*x)^3*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2)),x)
[Out]
(((d*((d*((64*c^7*d^8*e^4)/(315*(a*e^2 - c*d^2)^10) - (8*c^6*d^6*e^4*(49*a*e^2 - 25*c*d^2))/(315*(a*e^2 - c*d^
2)^10)))/e + (8*c^5*d^5*e^3*(23*a^2*e^4 - 51*c^2*d^4 + 52*a*c*d^2*e^2))/(315*(a*e^2 - c*d^2)^10)))/e + (2*c^4*
d^4*e^2*(439*a^3*e^6 - 367*c^3*d^6 + 1305*a*c^2*d^4*e^2 - 1409*a^2*c*d^2*e^4))/(315*(a*e^2 - c*d^2)^10))*(x*(a
*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x) - (((d*((32*c^4*d^5*e^4)/(63*(a*e^2 - c*d^2)^6*(5*a*e^3 -
5*c*d^2*e)) - (4*c^3*d^3*e^4*(29*a*e^2 - 13*c*d^2))/(63*(a*e^2 - c*d^2)^6*(5*a*e^3 - 5*c*d^2*e))))/e + (2*c^2*
d^2*e^3*(25*a^2*e^4 - 17*c^2*d^4 + 8*a*c*d^2*e^2))/(63*(a*e^2 - c*d^2)^6*(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^3*d^4*e^4)/(9*(a*e^2 - c*d^2)^3*(7*a^3*e^7 - 7*c^3
*d^6*e + 21*a*c^2*d^4*e^3 - 21*a^2*c*d^2*e^5)) - (2*c^2*d^2*e^4*(25*a*e^2 - 9*c*d^2))/(9*(a*e^2 - c*d^2)^3*(7*
a^3*e^7 - 7*c^3*d^6*e + 21*a*c^2*d^4*e^3 - 21*a^2*c*d^2*e^5))))/e + (e^3*(18*c^3*d^5 - 54*a*c^2*d^3*e^2 + 52*a
^2*c*d*e^4))/(9*(a*e^2 - c*d^2)^3*(7*a^3*e^7 - 7*c^3*d^6*e + 21*a*c^2*d^4*e^3 - 21*a^2*c*d^2*e^5)))*(x*(a*e^2
+ c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x)^4 + (((d*((d*((d*((32*c^6*d^7*e^6)/(63*(a*e^2 - c*d^2)^6*(5*a^3
*e^7 - 5*c^3*d^6*e + 15*a*c^2*d^4*e^3 - 15*a^2*c*d^2*e^5)) - (4*c^5*d^5*e^6*(45*a*e^2 - 13*c*d^2))/(63*(a*e^2
- c*d^2)^6*(5*a^3*e^7 - 5*c^3*d^6*e + 15*a*c^2*d^4*e^3 - 15*a^2*c*d^2*e^5))))/e + (2*c^4*d^4*e^5*(a^2*e^4 - 17
3*c^2*d^4 + 268*a*c*d^2*e^2))/(63*(a*e^2 - c*d^2)^6*(5*a^3*e^7 - 5*c^3*d^6*e + 15*a*c^2*d^4*e^3 - 15*a^2*c*d^2
*e^5))))/e + (4*c^3*d^3*e^4*(165*a^3*e^6 - 63*c^3*d^6 + 362*a*c^2*d^4*e^2 - 496*a^2*c*d^2*e^4))/(63*(a*e^2 - c
*d^2)^6*(5*a^3*e^7 - 5*c^3*d^6*e + 15*a*c^2*d^4*e^3 - 15*a^2*c*d^2*e^5))))/e - (e^3*(126*c^6*d^10 - 756*a*c^5*
d^8*e^2 + 1858*a^2*c^4*d^6*e^4 - 1900*a^3*c^3*d^4*e^6 + 640*a^4*c^2*d^2*e^8))/(63*(a*e^2 - c*d^2)^6*(5*a^3*e^7
- 5*c^3*d^6*e + 15*a*c^2*d^4*e^3 - 15*a^2*c*d^2*e^5)))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*
x)^3 + (((32*c^4*d^5*e^3)/(63*(a*e^2 - c*d^2)^6*(3*a*e^3 - 3*c*d^2*e)) - (4*c^3*d^3*e^3*(19*a*e^2 - 11*c*d^2))
/(63*(a*e^2 - c*d^2)^6*(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*((d*((64*c^7*d^8*e^6)/(315*(a*e^2 - c*d^2)^9*(3*a*e^3 - 3*c*d^2*e)) - (8*c^6*d^6*e^6*(49*a*e^2 - 17*c*d
^2))/(315*(a*e^2 - c*d^2)^9*(3*a*e^3 - 3*c*d^2*e))))/e + (8*c^5*d^5*e^5*(23*a^2*e^4 - 76*c^2*d^4 + 101*a*c*d^2
*e^2))/(315*(a*e^2 - c*d^2)^9*(3*a*e^3 - 3*c*d^2*e))))/e + (2*c^4*d^4*e^4*(659*a^3*e^6 - 383*c^3*d^6 + 1757*a*
c^2*d^4*e^2 - 2161*a^2*c*d^2*e^4))/(315*(a*e^2 - c*d^2)^9*(3*a*e^3 - 3*c*d^2*e))))/e + (e*(514*c^7*d^11*e^2 -
1290*a*c^6*d^9*e^4 + 178*a^2*c^5*d^7*e^6 + 1322*a^3*c^4*d^5*e^8 - 660*a^4*c^3*d^3*e^10))/(315*(a*e^2 - c*d^2)^
9*(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 - ((x*((a*(((a*e^2 + c*d^
2)*((64*c^9*d^9*e^6*(a*e^2 + c*d^2))/(315*(a*e^2 - c*d^2)^9*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (16
*c^9*d^9*e^6*(57*a*e^2 - 25*c*d^2))/(315*(a*e^2 - c*d^2)^9*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/(c*d
*e) + (16*c^8*d^8*e^5*(24*a^2*e^4 - 17*c^2*d^4 + 9*a*c*d^2*e^2))/(105*(a*e^2 - c*d^2)^9*(c^3*d^5*e - 2*a*c^2*d
^3*e^3 + a^2*c*d*e^5)) - (128*a*c^9*d^10*e^7)/(315*(a*e^2 - c*d^2)^9*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^
5)) + (8*c^8*d^8*e^5*(a*e^2 + c*d^2)*(57*a*e^2 - 25*c*d^2))/(315*(a*e^2 - c*d^2)^9*(c^3*d^5*e - 2*a*c^2*d^3*e^
3 + a^2*c*d*e^5))))/c + ((a*e^2 + c*d^2)*((a*((64*c^9*d^9*e^6*(a*e^2 + c*d^2))/(315*(a*e^2 - c*d^2)^9*(c^3*d^5
*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (16*c^9*d^9*e^6*(57*a*e^2 - 25*c*d^2))/(315*(a*e^2 - c*d^2)^9*(c^3*d^5*
e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/c - ((a*e^2 + c*d^2)*(((a*e^2 + c*d^2)*((64*c^9*d^9*e^6*(a*e^2 + c*d^2))
/(315*(a*e^2 - c*d^2)^9*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (16*c^9*d^9*e^6*(57*a*e^2 - 25*c*d^2))/
(315*(a*e^2 - c*d^2)^9*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/(c*d*e) + (16*c^8*d^8*e^5*(24*a^2*e^4 -
17*c^2*d^4 + 9*a*c*d^2*e^2))/(105*(a*e^2 - c*d^2)^9*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (128*a*c^9*
d^10*e^7)/(315*(a*e^2 - c*d^2)^9*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) + (8*c^8*d^8*e^5*(a*e^2 + c*d^2)
*(57*a*e^2 - 25*c*d^2))/(315*(a*e^2 - c*d^2)^9*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/(c*d*e) - (4*c^7
*d^7*e^4*(487*a^3*e^6 - 507*c^3*d^6 + 1929*a*c^2*d^4*e^2 - 2037*a^2*c*d^2*e^4))/(315*(a*e^2 - c*d^2)^9*(c^3*d^
5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) + (8*c^7*d^7*e^4*(a*e^2 + c*d^2)*(24*a^2*e^4 - 17*c^2*d^4 + 9*a*c*d^2*e^
2))/(105*(a*e^2 - c*d^2)^9*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/(c*d*e) + (2*c^2*d^2*e^2*(15080*c^8*
d^12*e - 63362*a*c^7*d^10*e^3 + 100830*a^2*c^6*d^8*e^5 - 71294*a^3*c^5*d^6*e^7 + 18554*a^4*c^4*d^4*e^9))/(945*
(a*e^2 - c*d^2)^9*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) + (2*c^6*d^6*e^3*(a*e^2 + c*d^2)*(487*a^3*e^6 -
507*c^3*d^6 + 1929*a*c^2*d^4*e^2 - 2037*a^2*c*d^2*e^4))/(315*(a*e^2 - c*d^2)^9*(c^3*d^5*e - 2*a*c^2*d^3*e^3 +
a^2*c*d*e^5))) + (a*((a*((64*c^9*d^9*e^6*(a*e^2 + c*d^2))/(315*(a*e^2 - c*d^2)^9*(c^3*d^5*e - 2*a*c^2*d^3*e^3
+ a^2*c*d*e^5)) - (16*c^9*d^9*e^6*(57*a*e^2 - 25*c*d^2))/(315*(a*e^2 - c*d^2)^9*(c^3*d^5*e - 2*a*c^2*d^3*e^3
+ a^2*c*d*e^5))))/c - ((a*e^2 + c*d^2)*(((a*e^2 + c*d^2)*((64*c^9*d^9*e^6*(a*e^2 + c*d^2))/(315*(a*e^2 - c*d^2
)^9*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (16*c^9*d^9*e^6*(57*a*e^2 - 25*c*d^2))/(315*(a*e^2 - c*d^2)
^9*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/(c*d*e) + (16*c^8*d^8*e^5*(24*a^2*e^4 - 17*c^2*d^4 + 9*a*c*d
^2*e^2))/(105*(a*e^2 - c*d^2)^9*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (128*a*c^9*d^10*e^7)/(315*(a*e^
2 - c*d^2)^9*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) + (8*c^8*d^8*e^5*(a*e^2 + c*d^2)*(57*a*e^2 - 25*c*d^
2))/(315*(a*e^2 - c*d^2)^9*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/(c*d*e) - (4*c^7*d^7*e^4*(487*a^3*e^
6 - 507*c^3*d^6 + 1929*a*c^2*d^4*e^2 - 2037*a^2*c*d^2*e^4))/(315*(a*e^2 - c*d^2)^9*(c^3*d^5*e - 2*a*c^2*d^3*e^
3 + a^2*c*d*e^5)) + (8*c^7*d^7*e^4*(a*e^2 + c*d^2)*(24*a^2*e^4 - 17*c^2*d^4 + 9*a*c*d^2*e^2))/(105*(a*e^2 - c*
d^2)^9*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/c + (c*d*e*(a*e^2 + c*d^2)*(15080*c^8*d^12*e - 63362*a*c
^7*d^10*e^3 + 100830*a^2*c^6*d^8*e^5 - 71294*a^3*c^5*d^6*e^7 + 18554*a^4*c^4*d^4*e^9))/(945*(a*e^2 - c*d^2)^9*
(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/((a*e + c*d*x)*(d
+ e*x)) - (2*e^3*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/((d + e*x)^5*(9*a^3*e^7 - 9*c^3*d^6*e + 27*a*
c^2*d^4*e^3 - 27*a^2*c*d^2*e^5)) + ((x*((a*((a*(((a*e^2 + c*d^2)*((64*c^10*d^10*e^7*(a*e^2 + c*d^2))/(945*(a*e
^2 - c*d^2)^9*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (16*c^10*d^10*e^7*(65*a*e^2 - 17*c*d^2))/(945*(a*
e^2 - c*d^2)^9*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/(c*d*e) + (16*c^9*d^9*e^6*(129*a^2*e^4 - 76*c^2*
d^4 + 67*a*c*d^2*e^2))/(945*(a*e^2 - c*d^2)^9*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (128*a*c^10*d^11*
e^8)/(945*(a*e^2 - c*d^2)^9*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) + (8*c^9*d^9*e^6*(a*e^2 + c*d^2)*(65*
a*e^2 - 17*c*d^2))/(945*(a*e^2 - c*d^2)^9*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/c + ((a*e^2 + c*d^2)*
((a*((64*c^10*d^10*e^7*(a*e^2 + c*d^2))/(945*(a*e^2 - c*d^2)^9*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) -
(16*c^10*d^10*e^7*(65*a*e^2 - 17*c*d^2))/(945*(a*e^2 - c*d^2)^9*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))
/c - ((a*e^2 + c*d^2)*(((a*e^2 + c*d^2)*((64*c^10*d^10*e^7*(a*e^2 + c*d^2))/(945*(a*e^2 - c*d^2)^9*(c^3*d^5*e
- 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (16*c^10*d^10*e^7*(65*a*e^2 - 17*c*d^2))/(945*(a*e^2 - c*d^2)^9*(c^3*d^5*e
- 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/(c*d*e) + (16*c^9*d^9*e^6*(129*a^2*e^4 - 76*c^2*d^4 + 67*a*c*d^2*e^2))/(9
45*(a*e^2 - c*d^2)^9*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (128*a*c^10*d^11*e^8)/(945*(a*e^2 - c*d^2)
^9*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) + (8*c^9*d^9*e^6*(a*e^2 + c*d^2)*(65*a*e^2 - 17*c*d^2))/(945*(
a*e^2 - c*d^2)^9*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/(c*d*e) - (4*c^8*d^8*e^5*(199*a^3*e^6 - 303*c^
3*d^6 + 2125*a*c^2*d^4*e^2 - 2661*a^2*c*d^2*e^4))/(945*(a*e^2 - c*d^2)^9*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*
d*e^5)) + (8*c^8*d^8*e^5*(a*e^2 + c*d^2)*(129*a^2*e^4 - 76*c^2*d^4 + 67*a*c*d^2*e^2))/(945*(a*e^2 - c*d^2)^9*(
c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/(c*d*e) - (4*c^7*d^7*e^4*(174*a^4*e^8 + 1747*c^4*d^8 - 6079*a*c^
3*d^6*e^2 - 1293*a^3*c*d^2*e^6 + 5931*a^2*c^2*d^4*e^4))/(945*(a*e^2 - c*d^2)^9*(c^3*d^5*e - 2*a*c^2*d^3*e^3 +
a^2*c*d*e^5)) + (2*c^7*d^7*e^4*(a*e^2 + c*d^2)*(199*a^3*e^6 - 303*c^3*d^6 + 2125*a*c^2*d^4*e^2 - 2661*a^2*c*d^
2*e^4))/(945*(a*e^2 - c*d^2)^9*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/c + ((a*e^2 + c*d^2)*((a*((a*((6
4*c^10*d^10*e^7*(a*e^2 + c*d^2))/(945*(a*e^2 - c*d^2)^9*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (16*c^1
0*d^10*e^7*(65*a*e^2 - 17*c*d^2))/(945*(a*e^2 - c*d^2)^9*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/c - ((
a*e^2 + c*d^2)*(((a*e^2 + c*d^2)*((64*c^10*d^10*e^7*(a*e^2 + c*d^2))/(945*(a*e^2 - c*d^2)^9*(c^3*d^5*e - 2*a*c
^2*d^3*e^3 + a^2*c*d*e^5)) - (16*c^10*d^10*e^7*(65*a*e^2 - 17*c*d^2))/(945*(a*e^2 - c*d^2)^9*(c^3*d^5*e - 2*a*
c^2*d^3*e^3 + a^2*c*d*e^5))))/(c*d*e) + (16*c^9*d^9*e^6*(129*a^2*e^4 - 76*c^2*d^4 + 67*a*c*d^2*e^2))/(945*(a*e
^2 - c*d^2)^9*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (128*a*c^10*d^11*e^8)/(945*(a*e^2 - c*d^2)^9*(c^3
*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) + (8*c^9*d^9*e^6*(a*e^2 + c*d^2)*(65*a*e^2 - 17*c*d^2))/(945*(a*e^2 -
c*d^2)^9*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/(c*d*e) - (4*c^8*d^8*e^5*(199*a^3*e^6 - 303*c^3*d^6 +
2125*a*c^2*d^4*e^2 - 2661*a^2*c*d^2*e^4))/(945*(a*e^2 - c*d^2)^9*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))
+ (8*c^8*d^8*e^5*(a*e^2 + c*d^2)*(129*a^2*e^4 - 76*c^2*d^4 + 67*a*c*d^2*e^2))/(945*(a*e^2 - c*d^2)^9*(c^3*d^5
*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/c - ((a*e^2 + c*d^2)*((a*(((a*e^2 + c*d^2)*((64*c^10*d^10*e^7*(a*e^2 +
c*d^2))/(945*(a*e^2 - c*d^2)^9*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (16*c^10*d^10*e^7*(65*a*e^2 - 17
*c*d^2))/(945*(a*e^2 - c*d^2)^9*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/(c*d*e) + (16*c^9*d^9*e^6*(129*
a^2*e^4 - 76*c^2*d^4 + 67*a*c*d^2*e^2))/(945*(a*e^2 - c*d^2)^9*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) -
(128*a*c^10*d^11*e^8)/(945*(a*e^2 - c*d^2)^9*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) + (8*c^9*d^9*e^6*(a*
e^2 + c*d^2)*(65*a*e^2 - 17*c*d^2))/(945*(a*e^2 - c*d^2)^9*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/c +
((a*e^2 + c*d^2)*((a*((64*c^10*d^10*e^7*(a*e^2 + c*d^2))/(945*(a*e^2 - c*d^2)^9*(c^3*d^5*e - 2*a*c^2*d^3*e^3 +
a^2*c*d*e^5)) - (16*c^10*d^10*e^7*(65*a*e^2 - 17*c*d^2))/(945*(a*e^2 - c*d^2)^9*(c^3*d^5*e - 2*a*c^2*d^3*e^3
+ a^2*c*d*e^5))))/c - ((a*e^2 + c*d^2)*(((a*e^2 + c*d^2)*((64*c^10*d^10*e^7*(a*e^2 + c*d^2))/(945*(a*e^2 - c*d
^2)^9*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (16*c^10*d^10*e^7*(65*a*e^2 - 17*c*d^2))/(945*(a*e^2 - c*
d^2)^9*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/(c*d*e) + (16*c^9*d^9*e^6*(129*a^2*e^4 - 76*c^2*d^4 + 67
*a*c*d^2*e^2))/(945*(a*e^2 - c*d^2)^9*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (128*a*c^10*d^11*e^8)/(94
5*(a*e^2 - c*d^2)^9*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) + (8*c^9*d^9*e^6*(a*e^2 + c*d^2)*(65*a*e^2 -
17*c*d^2))/(945*(a*e^2 - c*d^2)^9*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/(c*d*e) - (4*c^8*d^8*e^5*(199
*a^3*e^6 - 303*c^3*d^6 + 2125*a*c^2*d^4*e^2 - 2661*a^2*c*d^2*e^4))/(945*(a*e^2 - c*d^2)^9*(c^3*d^5*e - 2*a*c^2
*d^3*e^3 + a^2*c*d*e^5)) + (8*c^8*d^8*e^5*(a*e^2 + c*d^2)*(129*a^2*e^4 - 76*c^2*d^4 + 67*a*c*d^2*e^2))/(945*(a
*e^2 - c*d^2)^9*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/(c*d*e) - (4*c^7*d^7*e^4*(174*a^4*e^8 + 1747*c^
4*d^8 - 6079*a*c^3*d^6*e^2 - 1293*a^3*c*d^2*e^6 + 5931*a^2*c^2*d^4*e^4))/(945*(a*e^2 - c*d^2)^9*(c^3*d^5*e - 2
*a*c^2*d^3*e^3 + a^2*c*d*e^5)) + (2*c^7*d^7*e^4*(a*e^2 + c*d^2)*(199*a^3*e^6 - 303*c^3*d^6 + 2125*a*c^2*d^4*e^
2 - 2661*a^2*c*d^2*e^4))/(945*(a*e^2 - c*d^2)^9*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/(c*d*e) + (2*c^
2*d^2*e^2*(1890*c^9*d^14*e - 16438*a*c^8*d^12*e^3 + 45034*a^2*c^7*d^10*e^5 - 52942*a^3*c^6*d^8*e^7 + 27764*a^4
*c^5*d^6*e^9 - 5692*a^5*c^4*d^4*e^11))/(945*(a*e^2 - c*d^2)^9*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (
2*c^6*d^6*e^3*(a*e^2 + c*d^2)*(174*a^4*e^8 + 1747*c^4*d^8 - 6079*a*c^3*d^6*e^2 - 1293*a^3*c*d^2*e^6 + 5931*a^2
*c^2*d^4*e^4))/(945*(a*e^2 - c*d^2)^9*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/(c*d*e) + (2*c^2*d^2*e^2*
(630*c^9*d^15 - 5670*a*c^8*d^13*e^2 + 22394*a^2*c^7*d^11*e^4 - 44870*a^3*c^6*d^9*e^6 + 46888*a^4*c^5*d^7*e^8 -
24308*a^5*c^4*d^5*e^10 + 5000*a^6*c^3*d^3*e^12))/(945*(a*e^2 - c*d^2)^9*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*
d*e^5)) - (c*d*e*(a*e^2 + c*d^2)*(1890*c^9*d^14*e - 16438*a*c^8*d^12*e^3 + 45034*a^2*c^7*d^10*e^5 - 52942*a^3*
c^6*d^8*e^7 + 27764*a^4*c^5*d^6*e^9 - 5692*a^5*c^4*d^4*e^11))/(945*(a*e^2 - c*d^2)^9*(c^3*d^5*e - 2*a*c^2*d^3*
e^3 + a^2*c*d*e^5))) + (a*((a*((a*((64*c^10*d^10*e^7*(a*e^2 + c*d^2))/(945*(a*e^2 - c*d^2)^9*(c^3*d^5*e - 2*a*
c^2*d^3*e^3 + a^2*c*d*e^5)) - (16*c^10*d^10*e^7*(65*a*e^2 - 17*c*d^2))/(945*(a*e^2 - c*d^2)^9*(c^3*d^5*e - 2*a
*c^2*d^3*e^3 + a^2*c*d*e^5))))/c - ((a*e^2 + c*d^2)*(((a*e^2 + c*d^2)*((64*c^10*d^10*e^7*(a*e^2 + c*d^2))/(945
*(a*e^2 - c*d^2)^9*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (16*c^10*d^10*e^7*(65*a*e^2 - 17*c*d^2))/(94
5*(a*e^2 - c*d^2)^9*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/(c*d*e) + (16*c^9*d^9*e^6*(129*a^2*e^4 - 76
*c^2*d^4 + 67*a*c*d^2*e^2))/(945*(a*e^2 - c*d^2)^9*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (128*a*c^10*
d^11*e^8)/(945*(a*e^2 - c*d^2)^9*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) + (8*c^9*d^9*e^6*(a*e^2 + c*d^2)
*(65*a*e^2 - 17*c*d^2))/(945*(a*e^2 - c*d^2)^9*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/(c*d*e) - (4*c^8
*d^8*e^5*(199*a^3*e^6 - 303*c^3*d^6 + 2125*a*c^2*d^4*e^2 - 2661*a^2*c*d^2*e^4))/(945*(a*e^2 - c*d^2)^9*(c^3*d^
5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) + (8*c^8*d^8*e^5*(a*e^2 + c*d^2)*(129*a^2*e^4 - 76*c^2*d^4 + 67*a*c*d^2*
e^2))/(945*(a*e^2 - c*d^2)^9*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/c - ((a*e^2 + c*d^2)*((a*(((a*e^2
+ c*d^2)*((64*c^10*d^10*e^7*(a*e^2 + c*d^2))/(945*(a*e^2 - c*d^2)^9*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5
)) - (16*c^10*d^10*e^7*(65*a*e^2 - 17*c*d^2))/(945*(a*e^2 - c*d^2)^9*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^
5))))/(c*d*e) + (16*c^9*d^9*e^6*(129*a^2*e^4 - 76*c^2*d^4 + 67*a*c*d^2*e^2))/(945*(a*e^2 - c*d^2)^9*(c^3*d^5*e
- 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (128*a*c^10*d^11*e^8)/(945*(a*e^2 - c*d^2)^9*(c^3*d^5*e - 2*a*c^2*d^3*e^3
+ a^2*c*d*e^5)) + (8*c^9*d^9*e^6*(a*e^2 + c*d^2)*(65*a*e^2 - 17*c*d^2))/(945*(a*e^2 - c*d^2)^9*(c^3*d^5*e - 2
*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/c + ((a*e^2 + c*d^2)*((a*((64*c^10*d^10*e^7*(a*e^2 + c*d^2))/(945*(a*e^2 - c*
d^2)^9*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (16*c^10*d^10*e^7*(65*a*e^2 - 17*c*d^2))/(945*(a*e^2 - c
*d^2)^9*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/c - ((a*e^2 + c*d^2)*(((a*e^2 + c*d^2)*((64*c^10*d^10*e
^7*(a*e^2 + c*d^2))/(945*(a*e^2 - c*d^2)^9*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (16*c^10*d^10*e^7*(6
5*a*e^2 - 17*c*d^2))/(945*(a*e^2 - c*d^2)^9*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/(c*d*e) + (16*c^9*d
^9*e^6*(129*a^2*e^4 - 76*c^2*d^4 + 67*a*c*d^2*e^2))/(945*(a*e^2 - c*d^2)^9*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*
c*d*e^5)) - (128*a*c^10*d^11*e^8)/(945*(a*e^2 - c*d^2)^9*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) + (8*c^9
*d^9*e^6*(a*e^2 + c*d^2)*(65*a*e^2 - 17*c*d^2))/(945*(a*e^2 - c*d^2)^9*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*
e^5))))/(c*d*e) - (4*c^8*d^8*e^5*(199*a^3*e^6 - 303*c^3*d^6 + 2125*a*c^2*d^4*e^2 - 2661*a^2*c*d^2*e^4))/(945*(
a*e^2 - c*d^2)^9*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) + (8*c^8*d^8*e^5*(a*e^2 + c*d^2)*(129*a^2*e^4 -
76*c^2*d^4 + 67*a*c*d^2*e^2))/(945*(a*e^2 - c*d^2)^9*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/(c*d*e) -
(4*c^7*d^7*e^4*(174*a^4*e^8 + 1747*c^4*d^8 - 6079*a*c^3*d^6*e^2 - 1293*a^3*c*d^2*e^6 + 5931*a^2*c^2*d^4*e^4))/
(945*(a*e^2 - c*d^2)^9*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) + (2*c^7*d^7*e^4*(a*e^2 + c*d^2)*(199*a^3*
e^6 - 303*c^3*d^6 + 2125*a*c^2*d^4*e^2 - 2661*a^2*c*d^2*e^4))/(945*(a*e^2 - c*d^2)^9*(c^3*d^5*e - 2*a*c^2*d^3*
e^3 + a^2*c*d*e^5))))/(c*d*e) + (2*c^2*d^2*e^2*(1890*c^9*d^14*e - 16438*a*c^8*d^12*e^3 + 45034*a^2*c^7*d^10*e^
5 - 52942*a^3*c^6*d^8*e^7 + 27764*a^4*c^5*d^6*e^9 - 5692*a^5*c^4*d^4*e^11))/(945*(a*e^2 - c*d^2)^9*(c^3*d^5*e
- 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (2*c^6*d^6*e^3*(a*e^2 + c*d^2)*(174*a^4*e^8 + 1747*c^4*d^8 - 6079*a*c^3*d^
6*e^2 - 1293*a^3*c*d^2*e^6 + 5931*a^2*c^2*d^4*e^4))/(945*(a*e^2 - c*d^2)^9*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*
c*d*e^5))))/c + (c*d*e*(a*e^2 + c*d^2)*(630*c^9*d^15 - 5670*a*c^8*d^13*e^2 + 22394*a^2*c^7*d^11*e^4 - 44870*a^
3*c^6*d^9*e^6 + 46888*a^4*c^5*d^7*e^8 - 24308*a^5*c^4*d^5*e^10 + 5000*a^6*c^3*d^3*e^12))/(945*(a*e^2 - c*d^2)^
9*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/((a*e + c*d*x)^
2*(d + e*x)^2) + (8*c^4*d^4*e^2*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(27*(a*e^2 - c*d^2)^7*(d + e*x)
)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {5}{2}} \left (d + e x\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(e*x+d)**3/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2),x)
[Out]
Integral(1/(((d + e*x)*(a*e + c*d*x))**(5/2)*(d + e*x)**3), x)
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