3.7.53 \(\int \frac {(d^2-e^2 x^2)^{7/2}}{(d+e x)^{10}} \, dx\)
Optimal. Leaf size=67 \[ -\frac {\left (d^2-e^2 x^2\right )^{9/2}}{99 d^2 e (d+e x)^9}-\frac {\left (d^2-e^2 x^2\right )^{9/2}}{11 d e (d+e x)^{10}} \]
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Rubi [A] time = 0.02, antiderivative size = 67, normalized size of antiderivative = 1.00,
number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used =
{659, 651} \begin {gather*} -\frac {\left (d^2-e^2 x^2\right )^{9/2}}{99 d^2 e (d+e x)^9}-\frac {\left (d^2-e^2 x^2\right )^{9/2}}{11 d e (d+e x)^{10}} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(d^2 - e^2*x^2)^(7/2)/(d + e*x)^10,x]
[Out]
-(d^2 - e^2*x^2)^(9/2)/(11*d*e*(d + e*x)^10) - (d^2 - e^2*x^2)^(9/2)/(99*d^2*e*(d + e*x)^9)
Rule 651
Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^m*(a + c*x^2)^(p + 1))
/(2*c*d*(p + 1)), x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] && !IntegerQ[p] && EqQ[m + 2*p
+ 2, 0]
Rule 659
Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[(e*(d + e*x)^m*(a + c*x^2)^(p + 1)
)/(2*c*d*(m + p + 1)), x] + Dist[Simplify[m + 2*p + 2]/(2*d*(m + p + 1)), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p,
x], x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] && !IntegerQ[p] && ILtQ[Simplify[m + 2*p + 2
], 0]
Rubi steps
\begin {align*} \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^{10}} \, dx &=-\frac {\left (d^2-e^2 x^2\right )^{9/2}}{11 d e (d+e x)^{10}}+\frac {\int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^9} \, dx}{11 d}\\ &=-\frac {\left (d^2-e^2 x^2\right )^{9/2}}{11 d e (d+e x)^{10}}-\frac {\left (d^2-e^2 x^2\right )^{9/2}}{99 d^2 e (d+e x)^9}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 48, normalized size = 0.72 \begin {gather*} -\frac {(d-e x)^4 (10 d+e x) \sqrt {d^2-e^2 x^2}}{99 d^2 e (d+e x)^6} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(d^2 - e^2*x^2)^(7/2)/(d + e*x)^10,x]
[Out]
-1/99*((d - e*x)^4*(10*d + e*x)*Sqrt[d^2 - e^2*x^2])/(d^2*e*(d + e*x)^6)
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IntegrateAlgebraic [A] time = 0.75, size = 85, normalized size = 1.27 \begin {gather*} \frac {\sqrt {d^2-e^2 x^2} \left (-10 d^5+39 d^4 e x-56 d^3 e^2 x^2+34 d^2 e^3 x^3-6 d e^4 x^4-e^5 x^5\right )}{99 d^2 e (d+e x)^6} \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[(d^2 - e^2*x^2)^(7/2)/(d + e*x)^10,x]
[Out]
(Sqrt[d^2 - e^2*x^2]*(-10*d^5 + 39*d^4*e*x - 56*d^3*e^2*x^2 + 34*d^2*e^3*x^3 - 6*d*e^4*x^4 - e^5*x^5))/(99*d^2
*e*(d + e*x)^6)
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fricas [B] time = 0.50, size = 202, normalized size = 3.01 \begin {gather*} -\frac {10 \, e^{6} x^{6} + 60 \, d e^{5} x^{5} + 150 \, d^{2} e^{4} x^{4} + 200 \, d^{3} e^{3} x^{3} + 150 \, d^{4} e^{2} x^{2} + 60 \, d^{5} e x + 10 \, d^{6} + {\left (e^{5} x^{5} + 6 \, d e^{4} x^{4} - 34 \, d^{2} e^{3} x^{3} + 56 \, d^{3} e^{2} x^{2} - 39 \, d^{4} e x + 10 \, d^{5}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{99 \, {\left (d^{2} e^{7} x^{6} + 6 \, d^{3} e^{6} x^{5} + 15 \, d^{4} e^{5} x^{4} + 20 \, d^{5} e^{4} x^{3} + 15 \, d^{6} e^{3} x^{2} + 6 \, d^{7} e^{2} x + d^{8} e\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((-e^2*x^2+d^2)^(7/2)/(e*x+d)^10,x, algorithm="fricas")
[Out]
-1/99*(10*e^6*x^6 + 60*d*e^5*x^5 + 150*d^2*e^4*x^4 + 200*d^3*e^3*x^3 + 150*d^4*e^2*x^2 + 60*d^5*e*x + 10*d^6 +
(e^5*x^5 + 6*d*e^4*x^4 - 34*d^2*e^3*x^3 + 56*d^3*e^2*x^2 - 39*d^4*e*x + 10*d^5)*sqrt(-e^2*x^2 + d^2))/(d^2*e^
7*x^6 + 6*d^3*e^6*x^5 + 15*d^4*e^5*x^4 + 20*d^5*e^4*x^3 + 15*d^6*e^3*x^2 + 6*d^7*e^2*x + d^8*e)
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((-e^2*x^2+d^2)^(7/2)/(e*x+d)^10,x, algorithm="giac")
[Out]
Exception raised: NotImplementedError >> Unable to parse Giac output: (-5160960*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-
x^2*exp(2))*exp(1))/x/exp(2))^10*exp(1)^42*exp(2)^2-10321920*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))
/x/exp(2))^11*exp(1)^40*exp(2)^3-12042240*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^12*exp(1
)^38*exp(2)^4-9031680*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^13*exp(1)^36*exp(2)^5-451584
0*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^14*exp(1)^34*exp(2)^6-1505280*(-1/2*(-2*d*exp(1)
-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^15*exp(1)^32*exp(2)^7-322560*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2
))*exp(1))/x/exp(2))^16*exp(1)^30*exp(2)^8-40320*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^1
7*exp(1)^28*exp(2)^9-4423680*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^9*exp(1)^42*exp(2)^2-
19906560*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^10*exp(1)^40*exp(2)^3-39813120*(-1/2*(-2*
d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^11*exp(1)^38*exp(2)^4-46448640*(-1/2*(-2*d*exp(1)-2*sqrt(d^2
-x^2*exp(2))*exp(1))/x/exp(2))^12*exp(1)^36*exp(2)^5-34836480*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1)
)/x/exp(2))^13*exp(1)^34*exp(2)^6-17418240*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^14*exp(
1)^32*exp(2)^7-5806080*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^15*exp(1)^30*exp(2)^8-12902
40*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^16*exp(1)^28*exp(2)^9-161280*(-1/2*(-2*d*exp(1)
-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^17*exp(1)^26*exp(2)^10-5160960*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp
(2))*exp(1))/x/exp(2))^8*exp(1)^42*exp(2)^2-21254144*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2
))^9*exp(1)^40*exp(2)^3-38873088*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^10*exp(1)^38*exp(
2)^4-41619456*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^11*exp(1)^36*exp(2)^5-28987392*(-1/2
*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^12*exp(1)^34*exp(2)^6-13837824*(-1/2*(-2*d*exp(1)-2*sqr
t(d^2-x^2*exp(2))*exp(1))/x/exp(2))^13*exp(1)^32*exp(2)^7-4273920*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*ex
p(1))/x/exp(2))^14*exp(1)^30*exp(2)^8-994560*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^15*ex
p(1)^28*exp(2)^9-322560*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^16*exp(1)^26*exp(2)^10-403
20*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^17*exp(1)^24*exp(2)^11-19906560*(-1/2*(-2*d*exp
(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^8*exp(1)^40*exp(2)^3-61390848*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*e
xp(2))*exp(1))/x/exp(2))^9*exp(1)^38*exp(2)^4-57286656*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp
(2))^10*exp(1)^36*exp(2)^5+24772608*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^11*exp(1)^34*e
xp(2)^6+102660096*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^12*exp(1)^32*exp(2)^7+107476992*
(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^13*exp(1)^30*exp(2)^8+63544320*(-1/2*(-2*d*exp(1)-
2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^14*exp(1)^28*exp(2)^9+22901760*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(
2))*exp(1))/x/exp(2))^15*exp(1)^26*exp(2)^10+5160960*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2
))^16*exp(1)^24*exp(2)^11+645120*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^17*exp(1)^22*exp(
2)^12-10321920*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^7*exp(1)^40*exp(2)^3-38873088*(-1/2
*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^8*exp(1)^38*exp(2)^4-46614528*(-1/2*(-2*d*exp(1)-2*sqrt
(d^2-x^2*exp(2))*exp(1))/x/exp(2))^9*exp(1)^36*exp(2)^5+24302592*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp
(1))/x/exp(2))^10*exp(1)^34*exp(2)^6+135567360*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^11*
exp(1)^32*exp(2)^7+184488192*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^12*exp(1)^30*exp(2)^8
+145845504*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^13*exp(1)^28*exp(2)^9+73140480*(-1/2*(-
2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^14*exp(1)^26*exp(2)^10+24810240*(-1/2*(-2*d*exp(1)-2*sqrt(
d^2-x^2*exp(2))*exp(1))/x/exp(2))^15*exp(1)^24*exp(2)^11+6128640*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp
(1))/x/exp(2))^16*exp(1)^22*exp(2)^12+766080*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^17*ex
p(1)^20*exp(2)^13-39813120*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^7*exp(1)^38*exp(2)^4-57
286656*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^8*exp(1)^36*exp(2)^5+92516352*(-1/2*(-2*d*e
xp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^9*exp(1)^34*exp(2)^6+285235200*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^
2*exp(2))*exp(1))/x/exp(2))^10*exp(1)^32*exp(2)^7+262490112*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/
x/exp(2))^11*exp(1)^30*exp(2)^8+82295808*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^12*exp(1)
^28*exp(2)^9-49480704*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^13*exp(1)^26*exp(2)^10-65157
120*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^14*exp(1)^24*exp(2)^11-28600320*(-1/2*(-2*d*ex
p(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^15*exp(1)^22*exp(2)^12-6451200*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2
*exp(2))*exp(1))/x/exp(2))^16*exp(1)^20*exp(2)^13-806400*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/e
xp(2))^17*exp(1)^18*exp(2)^14-12042240*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^6*exp(1)^38
*exp(2)^4-41619456*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^7*exp(1)^36*exp(2)^5+13980672*(
-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^8*exp(1)^34*exp(2)^6+106127872*(-1/2*(-2*d*exp(1)-2
*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^9*exp(1)^32*exp(2)^7-73218816*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2)
)*exp(1))/x/exp(2))^10*exp(1)^30*exp(2)^8-467659008*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2)
)^11*exp(1)^28*exp(2)^9-658683648*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^12*exp(1)^26*exp
(2)^10-520402176*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^13*exp(1)^24*exp(2)^11-256757760*
(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^14*exp(1)^22*exp(2)^12-82467840*(-1/2*(-2*d*exp(1)
-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^15*exp(1)^20*exp(2)^13-17942400*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*ex
p(2))*exp(1))/x/exp(2))^16*exp(1)^18*exp(2)^14-2016000*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp
(2))^17*exp(1)^16*exp(2)^15-46448640*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^6*exp(1)^36*e
xp(2)^5+24772608*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^7*exp(1)^34*exp(2)^6+274913280*(-
1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^8*exp(1)^32*exp(2)^7+167503872*(-1/2*(-2*d*exp(1)-2*
sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^9*exp(1)^30*exp(2)^8-489618432*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2)
)*exp(1))/x/exp(2))^10*exp(1)^28*exp(2)^9-993881088*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2)
)^11*exp(1)^26*exp(2)^10-890609664*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^12*exp(1)^24*ex
p(2)^11-452325888*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^13*exp(1)^22*exp(2)^12-128620800
*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^14*exp(1)^20*exp(2)^13-22982400*(-1/2*(-2*d*exp(1
)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^15*exp(1)^18*exp(2)^14-3427200*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*ex
p(2))*exp(1))/x/exp(2))^16*exp(1)^16*exp(2)^15-201600*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(
2))^17*exp(1)^14*exp(2)^16-9031680*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^5*exp(1)^36*exp
(2)^5-28987392*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^6*exp(1)^34*exp(2)^6+114923520*(-1/
2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^7*exp(1)^32*exp(2)^7-349330176*(-1/2*(-2*d*exp(1)-2*sq
rt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^8*exp(1)^30*exp(2)^8-2772095488*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))
*exp(1))/x/exp(2))^9*exp(1)^28*exp(2)^9-6118071552*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))
^10*exp(1)^26*exp(2)^10-7415112960*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^11*exp(1)^24*ex
p(2)^11-5663196672*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^12*exp(1)^22*exp(2)^12-28074090
24*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^13*exp(1)^20*exp(2)^13-898430400*(-1/2*(-2*d*ex
p(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^14*exp(1)^18*exp(2)^14-174350400*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x
^2*exp(2))*exp(1))/x/exp(2))^15*exp(1)^16*exp(2)^15-15472800*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))
/x/exp(2))^16*exp(1)^14*exp(2)^16+50400*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^17*exp(1)^
12*exp(2)^17-34836480*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^5*exp(1)^34*exp(2)^6+1061007
36*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^6*exp(1)^32*exp(2)^7+231524352*(-1/2*(-2*d*exp(
1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^7*exp(1)^30*exp(2)^8-719926272*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*e
xp(2))*exp(1))/x/exp(2))^8*exp(1)^28*exp(2)^9-2781735168*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/e
xp(2))^9*exp(1)^26*exp(2)^10-4337603712*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^10*exp(1)^
24*exp(2)^11-4188978432*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^11*exp(1)^22*exp(2)^12-276
8465280*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^12*exp(1)^20*exp(2)^13-1311478560*(-1/2*(-
2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^13*exp(1)^18*exp(2)^14-443091600*(-1/2*(-2*d*exp(1)-2*sqrt
(d^2-x^2*exp(2))*exp(1))/x/exp(2))^14*exp(1)^16*exp(2)^15-92022000*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*e
xp(1))/x/exp(2))^15*exp(1)^14*exp(2)^16-8328600*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^16
*exp(1)^12*exp(2)^17-63000*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^17*exp(1)^10*exp(2)^18-
4515840*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^4*exp(1)^34*exp(2)^6-13837824*(-1/2*(-2*d*
exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^5*exp(1)^32*exp(2)^7+167715072*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x
^2*exp(2))*exp(1))/x/exp(2))^6*exp(1)^30*exp(2)^8-1434048768*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))
/x/exp(2))^7*exp(1)^28*exp(2)^9-8371395072*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^8*exp(1
)^26*exp(2)^10-18690300768*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^9*exp(1)^24*exp(2)^11-2
4180183216*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^10*exp(1)^22*exp(2)^12-20602312416*(-1/
2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^11*exp(1)^20*exp(2)^13-12350484720*(-1/2*(-2*d*exp(1)-
2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^12*exp(1)^18*exp(2)^14-5411610540*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*e
xp(2))*exp(1))/x/exp(2))^13*exp(1)^16*exp(2)^15-1725731910*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x
/exp(2))^14*exp(1)^14*exp(2)^16-380418570*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^15*exp(1
)^12*exp(2)^17-52583265*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^16*exp(1)^10*exp(2)^18-349
8705*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^17*exp(1)^8*exp(2)^19-17418240*(-1/2*(-2*d*ex
p(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^4*exp(1)^32*exp(2)^7+110057472*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2
*exp(2))*exp(1))/x/exp(2))^5*exp(1)^30*exp(2)^8+22729728*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/e
xp(2))^6*exp(1)^28*exp(2)^9-1740284928*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^7*exp(1)^26
*exp(2)^10-5660099712*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^8*exp(1)^24*exp(2)^11-960914
9568*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^9*exp(1)^22*exp(2)^12-10539829440*(-1/2*(-2*d
*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^10*exp(1)^20*exp(2)^13-8006019840*(-1/2*(-2*d*exp(1)-2*sqrt(d
^2-x^2*exp(2))*exp(1))/x/exp(2))^11*exp(1)^18*exp(2)^14-4362464400*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*e
xp(1))/x/exp(2))^12*exp(1)^16*exp(2)^15-1715338800*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))
^13*exp(1)^14*exp(2)^16-486606960*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^14*exp(1)^12*exp
(2)^17-103973520*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^15*exp(1)^10*exp(2)^18-16299360*(
-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^16*exp(1)^8*exp(2)^19-1300320*(-1/2*(-2*d*exp(1)-2*
sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^17*exp(1)^6*exp(2)^20-1505280*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))
*exp(1))/x/exp(2))^3*exp(1)^32*exp(2)^7-5048064*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^4*
exp(1)^30*exp(2)^8+134620416*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^5*exp(1)^28*exp(2)^9-
2167651584*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^6*exp(1)^26*exp(2)^10-13084814592*(-1/2
*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^7*exp(1)^24*exp(2)^11-31282922160*(-1/2*(-2*d*exp(1)-2*
sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^8*exp(1)^22*exp(2)^12-43233005424*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp
(2))*exp(1))/x/exp(2))^9*exp(1)^20*exp(2)^13-39512670120*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/e
xp(2))^10*exp(1)^18*exp(2)^14-25426376640*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^11*exp(1
)^16*exp(2)^15-11737785990*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^12*exp(1)^14*exp(2)^16-
3842785170*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^13*exp(1)^12*exp(2)^17-878252130*(-1/2*
(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^14*exp(1)^10*exp(2)^18-139609470*(-1/2*(-2*d*exp(1)-2*sq
rt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^15*exp(1)^8*exp(2)^19-14928480*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*
exp(1))/x/exp(2))^16*exp(1)^6*exp(2)^20-856800*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^17*
exp(1)^4*exp(2)^21-5806080*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^3*exp(1)^30*exp(2)^8+62
641152*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^4*exp(1)^28*exp(2)^9-92574720*(-1/2*(-2*d*e
xp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^5*exp(1)^26*exp(2)^10-1991463936*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-
x^2*exp(2))*exp(1))/x/exp(2))^6*exp(1)^24*exp(2)^11-7160207616*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1
))/x/exp(2))^7*exp(1)^22*exp(2)^12-13521993408*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^8*e
xp(1)^20*exp(2)^13-15845830560*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^9*exp(1)^18*exp(2)^
14-12488832720*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^10*exp(1)^16*exp(2)^15-6913917360*(
-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^11*exp(1)^14*exp(2)^16-2731271760*(-1/2*(-2*d*exp(1
)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^12*exp(1)^12*exp(2)^17-771256080*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*
exp(2))*exp(1))/x/exp(2))^13*exp(1)^10*exp(2)^18-149526720*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x
/exp(2))^14*exp(1)^8*exp(2)^19-15731520*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^15*exp(1)^
6*exp(2)^20-80640*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^16*exp(1)^4*exp(2)^21+80640*(-1/
2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^17*exp(2)^23-322560*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*
exp(2))*exp(1))/x/exp(2))^2*exp(1)^30*exp(2)^8-1252608*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp
(2))^3*exp(1)^28*exp(2)^9+70463232*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^4*exp(1)^26*exp
(2)^10-1850655744*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^5*exp(1)^24*exp(2)^11-1311609331
2*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^6*exp(1)^22*exp(2)^12-33986028384*(-1/2*(-2*d*ex
p(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^7*exp(1)^20*exp(2)^13-49348504296*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-
x^2*exp(2))*exp(1))/x/exp(2))^8*exp(1)^18*exp(2)^14-46332269340*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(
1))/x/exp(2))^9*exp(1)^16*exp(2)^15-29715364350*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^10
*exp(1)^14*exp(2)^16-13222991250*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^11*exp(1)^12*exp(
2)^17-4102043190*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^12*exp(1)^10*exp(2)^18-894966030*
(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^13*exp(1)^8*exp(2)^19-134527680*(-1/2*(-2*d*exp(1)
-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^14*exp(1)^6*exp(2)^20-11914560*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp
(2))*exp(1))/x/exp(2))^15*exp(1)^4*exp(2)^21-362880*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2)
)^16*exp(2)^23-1198080*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^2*exp(1)^28*exp(2)^9+224778
24*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^3*exp(1)^26*exp(2)^10-73939968*(-1/2*(-2*d*exp(
1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^4*exp(1)^24*exp(2)^11-1465809408*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2
*exp(2))*exp(1))/x/exp(2))^5*exp(1)^22*exp(2)^12-6316291968*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/
x/exp(2))^6*exp(1)^20*exp(2)^13-12863354112*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^7*exp(
1)^18*exp(2)^14-15342085584*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^8*exp(1)^16*exp(2)^15-
11944018800*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^9*exp(1)^14*exp(2)^16-6353892720*(-1/2
*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^10*exp(1)^12*exp(2)^17-2318636880*(-1/2*(-2*d*exp(1)-2*
sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^11*exp(1)^10*exp(2)^18-557014080*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(
2))*exp(1))/x/exp(2))^12*exp(1)^8*exp(2)^19-74168640*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2
))^13*exp(1)^6*exp(2)^20-1774080*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^14*exp(1)^4*exp(2
)^21+483840*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^15*exp(2)^23-61056*(-1/2*(-2*d*exp(1)-
2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^2*exp(1)^26*exp(2)^10+23569152*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(
2))*exp(1))/x/exp(2))^3*exp(1)^24*exp(2)^11-1001340288*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp
(2))^4*exp(1)^22*exp(2)^12-9075386880*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^5*exp(1)^20*
exp(2)^13-25589855088*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^6*exp(1)^18*exp(2)^14-384749
52768*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^7*exp(1)^16*exp(2)^15-35817747966*(-1/2*(-2*
d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^8*exp(1)^14*exp(2)^16-21844478250*(-1/2*(-2*d*exp(1)-2*sqrt(
d^2-x^2*exp(2))*exp(1))/x/exp(2))^9*exp(1)^12*exp(2)^17-8981927010*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*e
xp(1))/x/exp(2))^10*exp(1)^10*exp(2)^18-2545790310*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))
^11*exp(1)^8*exp(2)^19-494598720*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^12*exp(1)^6*exp(2
)^20-58685760*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^13*exp(1)^4*exp(2)^21-2903040*(-1/2*
(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^14*exp(2)^23+4940928*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*e
xp(2))*exp(1))/x/exp(2))^2*exp(1)^24*exp(2)^11-31234560*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/ex
p(2))^3*exp(1)^22*exp(2)^12-740694528*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^4*exp(1)^20*
exp(2)^13-3980787552*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^5*exp(1)^18*exp(2)^14-8518561
968*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^6*exp(1)^16*exp(2)^15-10104840144*(-1/2*(-2*d*
exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^7*exp(1)^14*exp(2)^16-7539110208*(-1/2*(-2*d*exp(1)-2*sqrt(d^2
-x^2*exp(2))*exp(1))/x/exp(2))^8*exp(1)^12*exp(2)^17-3649867200*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(
1))/x/exp(2))^9*exp(1)^10*exp(2)^18-1113981120*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^10*
exp(1)^8*exp(2)^19-188596800*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^11*exp(1)^6*exp(2)^20
-9408000*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^12*exp(1)^4*exp(2)^21-4480*exp(1)^26*exp(
2)^10+1128960*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^13*exp(2)^23+4290480*(-1/2*(-2*d*exp
(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^2*exp(1)^22*exp(2)^12-350790336*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2
*exp(2))*exp(1))/x/exp(2))^3*exp(1)^20*exp(2)^13-4393080288*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/
x/exp(2))^4*exp(1)^18*exp(2)^14-13783500612*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^5*exp(
1)^16*exp(2)^15-21224357490*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^6*exp(1)^14*exp(2)^16-
18956802942*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^7*exp(1)^12*exp(2)^17-10629509184*(-1/
2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^8*exp(1)^10*exp(2)^18-3939364800*(-1/2*(-2*d*exp(1)-2*
sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^9*exp(1)^8*exp(2)^19-980925120*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2)
)*exp(1))/x/exp(2))^10*exp(1)^6*exp(2)^20-150373440*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2)
)^11*exp(1)^4*exp(2)^21-17280*exp(1)^24*exp(2)^11-10160640*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x
/exp(2))^12*exp(2)^23-7927488*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^2*exp(1)^20*exp(2)^1
3-247617792*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^3*exp(1)^18*exp(2)^14-1797141168*(-1/2
*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^4*exp(1)^16*exp(2)^15-4025023632*(-1/2*(-2*d*exp(1)-2*s
qrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^5*exp(1)^14*exp(2)^16-4686493392*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2
))*exp(1))/x/exp(2))^6*exp(1)^12*exp(2)^17-3214073520*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(
2))^7*exp(1)^10*exp(2)^18-1304916480*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^8*exp(1)^8*ex
p(2)^19-287884800*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^9*exp(1)^6*exp(2)^20-23708160*(-
1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^10*exp(1)^4*exp(2)^21-2384*exp(1)^22*exp(2)^12+11289
60*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^11*exp(2)^23-76483368*(-1/2*(-2*d*exp(1)-2*sqrt
(d^2-x^2*exp(2))*exp(1))/x/exp(2))^2*exp(1)^18*exp(2)^14-1463585472*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*
exp(1))/x/exp(2))^3*exp(1)^16*exp(2)^15-5414919858*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))
^4*exp(1)^14*exp(2)^16-8382681846*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^5*exp(1)^12*exp(
2)^17-6952538166*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^6*exp(1)^10*exp(2)^18-3513338010*
(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^7*exp(1)^8*exp(2)^19-1148152320*(-1/2*(-2*d*exp(1)
-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^8*exp(1)^6*exp(2)^20-228614400*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp
(2))*exp(1))/x/exp(2))^9*exp(1)^4*exp(2)^21+71232*exp(1)^20*exp(2)^13-20321280*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x
^2*exp(2))*exp(1))/x/exp(2))^10*exp(2)^23-52278768*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))
^2*exp(1)^16*exp(2)^15-580447632*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^3*exp(1)^14*exp(2
)^16-1374869808*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^4*exp(1)^12*exp(2)^17-1534483440*(
-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^5*exp(1)^10*exp(2)^18-902045760*(-1/2*(-2*d*exp(1)-
2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^6*exp(1)^8*exp(2)^19-272018880*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(
2))*exp(1))/x/exp(2))^7*exp(1)^6*exp(2)^20-33868800*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2)
)^8*exp(1)^4*exp(2)^21+72504*exp(1)^18*exp(2)^14-321081930*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x
/exp(2))^2*exp(1)^14*exp(2)^16-1551059046*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^3*exp(1)
^12*exp(2)^17-2337264954*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^4*exp(1)^10*exp(2)^18-176
8815090*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^5*exp(1)^8*exp(2)^19-802858560*(-1/2*(-2*d
*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^6*exp(1)^6*exp(2)^20-215409600*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-
x^2*exp(2))*exp(1))/x/exp(2))^7*exp(1)^4*exp(2)^21-104304*exp(1)^16*exp(2)^15-25401600*(-1/2*(-2*d*exp(1)-2*sq
rt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^8*exp(2)^23-127459152*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x
/exp(2))^2*exp(1)^12*exp(2)^17-335468400*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^3*exp(1)^
10*exp(2)^18-344180160*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^4*exp(1)^8*exp(2)^19-156139
200*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^5*exp(1)^6*exp(2)^20-28976640*(-1/2*(-2*d*exp(
1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^6*exp(1)^4*exp(2)^21-660266*exp(1)^14*exp(2)^16-1128960*(-1/2*(-2*
d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^7*exp(2)^23-311122566*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(
2))*exp(1))/x/exp(2))^2*exp(1)^10*exp(2)^18-447288450*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(
2))^3*exp(1)^8*exp(2)^19-317016000*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^4*exp(1)^6*exp(
2)^20-124205760*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^5*exp(1)^4*exp(2)^21-380328*exp(1)
^12*exp(2)^17-20321280*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^6*exp(2)^23-57775680*(-1/2*
(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^2*exp(1)^8*exp(2)^19-50070720*(-1/2*(-2*d*exp(1)-2*sqrt(
d^2-x^2*exp(2))*exp(1))/x/exp(2))^3*exp(1)^6*exp(2)^20-14676480*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(
1))/x/exp(2))^4*exp(1)^4*exp(2)^21-2295919*exp(1)^10*exp(2)^18-1128960*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2
))*exp(1))/x/exp(2))^5*exp(2)^23-58766400*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^2*exp(1)
^6*exp(2)^20-40340160*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^3*exp(1)^4*exp(2)^21-919200*
exp(1)^8*exp(2)^19-10160640*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^4*exp(2)^23-4032000*(-
1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^2*exp(1)^4*exp(2)^21-2382240*exp(1)^6*exp(2)^20-4838
40*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^3*exp(2)^23-456960*exp(1)^4*exp(2)^21-2903040*(
-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^2*exp(2)^23-362880*exp(2)^23-1146880*(-1/2*(-2*d*ex
p(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^9*exp(1)^44*exp(2)+40320*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp
(1))*exp(2)^23/x/exp(2)+2837520*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))*exp(1)^4*exp(2)^21/x/exp(2)+346248
0*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))*exp(1)^6*exp(2)^20/x/exp(2)+39381615/2*(-2*d*exp(1)-2*sqrt(d^2-x
^2*exp(2))*exp(1))*exp(1)^8*exp(2)^19/x/exp(2)+8241300*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))*exp(1)^10*e
xp(2)^18/x/exp(2)+20688471*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))*exp(1)^12*exp(2)^17/x/exp(2)+3322152*(-
2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))*exp(1)^14*exp(2)^16/x/exp(2)+4934394*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp
(2))*exp(1))*exp(1)^16*exp(2)^15/x/exp(2)+535536*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))*exp(1)^18*exp(2)^
14/x/exp(2)-269496*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))*exp(1)^20*exp(2)^13/x/exp(2)-318528*(-2*d*exp(1
)-2*sqrt(d^2-x^2*exp(2))*exp(1))*exp(1)^22*exp(2)^12/x/exp(2)+1296*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))
*exp(1)^24*exp(2)^11/x/exp(2)+74880*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))*exp(1)^26*exp(2)^10/x/exp(2)+2
0160*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))*exp(1)^28*exp(2)^9/x/exp(2))/((-1/2*(-2*d*exp(1)-2*sqrt(d^2-x
^2*exp(2))*exp(1))/x/exp(2))^2*exp(2)-(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x+exp(2))^9/(-20160*d^2*exp(
1)^29-20160*d^2*exp(1)^25*exp(2)^2+322560*d^2*exp(1)^23*exp(2)^3+383040*d^2*exp(1)^21*exp(2)^4-403200*d^2*exp(
1)^19*exp(2)^5-907200*d^2*exp(1)^17*exp(2)^6+907200*d^2*exp(1)^13*exp(2)^8+403200*d^2*exp(1)^11*exp(2)^9-38304
0*d^2*exp(1)^9*exp(2)^10-322560*d^2*exp(1)^7*exp(2)^11+20160*d^2*exp(1)^5*exp(2)^12-80640*d^2*exp(1)^27*exp(2)
+100800*d^2*exp(1)*exp(2)^14)+1/2*(-640*exp(1)^16*exp(2)^7-640*exp(1)^14*exp(2)^8-5600*exp(1)^12*exp(2)^9-2760
*exp(1)^10*exp(2)^10-8675*exp(1)^8*exp(2)^11-2080*exp(1)^6*exp(2)^12-2848*exp(1)^4*exp(2)^13-384*exp(2)^15)*at
an((-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x+exp(2))/sqrt(-exp(1)^4+exp(2)^2))/sqrt(-exp(1)^4+exp(2)
^2)/(-32*d^2*exp(1)^29-32*d^2*exp(1)^25*exp(2)^2+512*d^2*exp(1)^23*exp(2)^3+608*d^2*exp(1)^21*exp(2)^4-640*d^2
*exp(1)^19*exp(2)^5-1440*d^2*exp(1)^17*exp(2)^6+1440*d^2*exp(1)^13*exp(2)^8+640*d^2*exp(1)^11*exp(2)^9-608*d^2
*exp(1)^9*exp(2)^10-512*d^2*exp(1)^7*exp(2)^11+32*d^2*exp(1)^5*exp(2)^12-128*d^2*exp(1)^27*exp(2)+160*d^2*exp(
1)*exp(2)^14)
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maple [A] time = 0.04, size = 43, normalized size = 0.64 \begin {gather*} -\frac {\left (-e x +d \right ) \left (e x +10 d \right ) \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{99 \left (e x +d \right )^{9} d^{2} e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((-e^2*x^2+d^2)^(7/2)/(e*x+d)^10,x)
[Out]
-1/99*(-e*x+d)*(e*x+10*d)*(-e^2*x^2+d^2)^(7/2)/(e*x+d)^9/d^2/e
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maxima [B] time = 1.52, size = 659, normalized size = 9.84 \begin {gather*} -\frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}}}{2 \, {\left (e^{10} x^{9} + 9 \, d e^{9} x^{8} + 36 \, d^{2} e^{8} x^{7} + 84 \, d^{3} e^{7} x^{6} + 126 \, d^{4} e^{6} x^{5} + 126 \, d^{5} e^{5} x^{4} + 84 \, d^{6} e^{4} x^{3} + 36 \, d^{7} e^{3} x^{2} + 9 \, d^{8} e^{2} x + d^{9} e\right )}} + \frac {7 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d}{6 \, {\left (e^{9} x^{8} + 8 \, d e^{8} x^{7} + 28 \, d^{2} e^{7} x^{6} + 56 \, d^{3} e^{6} x^{5} + 70 \, d^{4} e^{5} x^{4} + 56 \, d^{5} e^{4} x^{3} + 28 \, d^{6} e^{3} x^{2} + 8 \, d^{7} e^{2} x + d^{8} e\right )}} - \frac {35 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{2}}{24 \, {\left (e^{8} x^{7} + 7 \, d e^{7} x^{6} + 21 \, d^{2} e^{6} x^{5} + 35 \, d^{3} e^{5} x^{4} + 35 \, d^{4} e^{4} x^{3} + 21 \, d^{5} e^{3} x^{2} + 7 \, d^{6} e^{2} x + d^{7} e\right )}} + \frac {35 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{3}}{44 \, {\left (e^{7} x^{6} + 6 \, d e^{6} x^{5} + 15 \, d^{2} e^{5} x^{4} + 20 \, d^{3} e^{4} x^{3} + 15 \, d^{4} e^{3} x^{2} + 6 \, d^{5} e^{2} x + d^{6} e\right )}} - \frac {35 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{2}}{792 \, {\left (e^{6} x^{5} + 5 \, d e^{5} x^{4} + 10 \, d^{2} e^{4} x^{3} + 10 \, d^{3} e^{3} x^{2} + 5 \, d^{4} e^{2} x + d^{5} e\right )}} - \frac {5 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{198 \, {\left (e^{5} x^{4} + 4 \, d e^{4} x^{3} + 6 \, d^{2} e^{3} x^{2} + 4 \, d^{3} e^{2} x + d^{4} e\right )}} - \frac {\sqrt {-e^{2} x^{2} + d^{2}}}{66 \, {\left (e^{4} x^{3} + 3 \, d e^{3} x^{2} + 3 \, d^{2} e^{2} x + d^{3} e\right )}} - \frac {\sqrt {-e^{2} x^{2} + d^{2}}}{99 \, {\left (d e^{3} x^{2} + 2 \, d^{2} e^{2} x + d^{3} e\right )}} - \frac {\sqrt {-e^{2} x^{2} + d^{2}}}{99 \, {\left (d^{2} e^{2} x + d^{3} e\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((-e^2*x^2+d^2)^(7/2)/(e*x+d)^10,x, algorithm="maxima")
[Out]
-1/2*(-e^2*x^2 + d^2)^(7/2)/(e^10*x^9 + 9*d*e^9*x^8 + 36*d^2*e^8*x^7 + 84*d^3*e^7*x^6 + 126*d^4*e^6*x^5 + 126*
d^5*e^5*x^4 + 84*d^6*e^4*x^3 + 36*d^7*e^3*x^2 + 9*d^8*e^2*x + d^9*e) + 7/6*(-e^2*x^2 + d^2)^(5/2)*d/(e^9*x^8 +
8*d*e^8*x^7 + 28*d^2*e^7*x^6 + 56*d^3*e^6*x^5 + 70*d^4*e^5*x^4 + 56*d^5*e^4*x^3 + 28*d^6*e^3*x^2 + 8*d^7*e^2*
x + d^8*e) - 35/24*(-e^2*x^2 + d^2)^(3/2)*d^2/(e^8*x^7 + 7*d*e^7*x^6 + 21*d^2*e^6*x^5 + 35*d^3*e^5*x^4 + 35*d^
4*e^4*x^3 + 21*d^5*e^3*x^2 + 7*d^6*e^2*x + d^7*e) + 35/44*sqrt(-e^2*x^2 + d^2)*d^3/(e^7*x^6 + 6*d*e^6*x^5 + 15
*d^2*e^5*x^4 + 20*d^3*e^4*x^3 + 15*d^4*e^3*x^2 + 6*d^5*e^2*x + d^6*e) - 35/792*sqrt(-e^2*x^2 + d^2)*d^2/(e^6*x
^5 + 5*d*e^5*x^4 + 10*d^2*e^4*x^3 + 10*d^3*e^3*x^2 + 5*d^4*e^2*x + d^5*e) - 5/198*sqrt(-e^2*x^2 + d^2)*d/(e^5*
x^4 + 4*d*e^4*x^3 + 6*d^2*e^3*x^2 + 4*d^3*e^2*x + d^4*e) - 1/66*sqrt(-e^2*x^2 + d^2)/(e^4*x^3 + 3*d*e^3*x^2 +
3*d^2*e^2*x + d^3*e) - 1/99*sqrt(-e^2*x^2 + d^2)/(d*e^3*x^2 + 2*d^2*e^2*x + d^3*e) - 1/99*sqrt(-e^2*x^2 + d^2)
/(d^2*e^2*x + d^3*e)
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mupad [B] time = 2.12, size = 170, normalized size = 2.54 \begin {gather*} \frac {16\,\sqrt {d^2-e^2\,x^2}}{33\,e\,{\left (d+e\,x\right )}^3}-\frac {184\,d\,\sqrt {d^2-e^2\,x^2}}{99\,e\,{\left (d+e\,x\right )}^4}-\frac {\sqrt {d^2-e^2\,x^2}}{99\,d\,e\,{\left (d+e\,x\right )}^2}-\frac {\sqrt {d^2-e^2\,x^2}}{99\,d^2\,e\,\left (d+e\,x\right )}+\frac {272\,d^2\,\sqrt {d^2-e^2\,x^2}}{99\,e\,{\left (d+e\,x\right )}^5}-\frac {16\,d^3\,\sqrt {d^2-e^2\,x^2}}{11\,e\,{\left (d+e\,x\right )}^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((d^2 - e^2*x^2)^(7/2)/(d + e*x)^10,x)
[Out]
(16*(d^2 - e^2*x^2)^(1/2))/(33*e*(d + e*x)^3) - (184*d*(d^2 - e^2*x^2)^(1/2))/(99*e*(d + e*x)^4) - (d^2 - e^2*
x^2)^(1/2)/(99*d*e*(d + e*x)^2) - (d^2 - e^2*x^2)^(1/2)/(99*d^2*e*(d + e*x)) + (272*d^2*(d^2 - e^2*x^2)^(1/2))
/(99*e*(d + e*x)^5) - (16*d^3*(d^2 - e^2*x^2)^(1/2))/(11*e*(d + e*x)^6)
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sympy [F(-1)] 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((-e**2*x**2+d**2)**(7/2)/(e*x+d)**10,x)
[Out]
Timed out
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