3.810 \(\int \frac {(d^2-e^2 x^2)^{7/2}}{(d+e x)^8} \, dx\)
Optimal. Leaf size=143 \[ -\frac {2 \left (d^2-e^2 x^2\right )^{7/2}}{7 e (d+e x)^7}+\frac {2 \left (d^2-e^2 x^2\right )^{5/2}}{5 e (d+e x)^5}-\frac {2 \left (d^2-e^2 x^2\right )^{3/2}}{3 e (d+e x)^3}+\frac {2 \sqrt {d^2-e^2 x^2}}{e (d+e x)}+\frac {\tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e} \]
[Out]
-2/3*(-e^2*x^2+d^2)^(3/2)/e/(e*x+d)^3+2/5*(-e^2*x^2+d^2)^(5/2)/e/(e*x+d)^5-2/7*(-e^2*x^2+d^2)^(7/2)/e/(e*x+d)^
7+arctan(e*x/(-e^2*x^2+d^2)^(1/2))/e+2*(-e^2*x^2+d^2)^(1/2)/e/(e*x+d)
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Rubi [A] time = 0.04, antiderivative size = 143, normalized size of antiderivative = 1.00,
number of steps used = 6, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used =
{663, 217, 203} \[ -\frac {2 \left (d^2-e^2 x^2\right )^{7/2}}{7 e (d+e x)^7}+\frac {2 \left (d^2-e^2 x^2\right )^{5/2}}{5 e (d+e x)^5}-\frac {2 \left (d^2-e^2 x^2\right )^{3/2}}{3 e (d+e x)^3}+\frac {2 \sqrt {d^2-e^2 x^2}}{e (d+e x)}+\frac {\tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e} \]
Antiderivative was successfully verified.
[In]
Int[(d^2 - e^2*x^2)^(7/2)/(d + e*x)^8,x]
[Out]
(2*Sqrt[d^2 - e^2*x^2])/(e*(d + e*x)) - (2*(d^2 - e^2*x^2)^(3/2))/(3*e*(d + e*x)^3) + (2*(d^2 - e^2*x^2)^(5/2)
)/(5*e*(d + e*x)^5) - (2*(d^2 - e^2*x^2)^(7/2))/(7*e*(d + e*x)^7) + ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]]/e
Rule 203
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])
Rule 217
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] && !GtQ[a, 0]
Rule 663
Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m + 1)*(a + c*x^2)^p)/(
e*(m + p + 1)), x] - Dist[(c*p)/(e^2*(m + p + 1)), Int[(d + e*x)^(m + 2)*(a + c*x^2)^(p - 1), x], x] /; FreeQ[
{a, c, d, e}, x] && EqQ[c*d^2 + a*e^2, 0] && GtQ[p, 0] && (LtQ[m, -2] || EqQ[m + 2*p + 1, 0]) && NeQ[m + p + 1
, 0] && IntegerQ[2*p]
Rubi steps
\begin {align*} \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^8} \, dx &=-\frac {2 \left (d^2-e^2 x^2\right )^{7/2}}{7 e (d+e x)^7}-\int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^6} \, dx\\ &=\frac {2 \left (d^2-e^2 x^2\right )^{5/2}}{5 e (d+e x)^5}-\frac {2 \left (d^2-e^2 x^2\right )^{7/2}}{7 e (d+e x)^7}+\int \frac {\left (d^2-e^2 x^2\right )^{3/2}}{(d+e x)^4} \, dx\\ &=-\frac {2 \left (d^2-e^2 x^2\right )^{3/2}}{3 e (d+e x)^3}+\frac {2 \left (d^2-e^2 x^2\right )^{5/2}}{5 e (d+e x)^5}-\frac {2 \left (d^2-e^2 x^2\right )^{7/2}}{7 e (d+e x)^7}-\int \frac {\sqrt {d^2-e^2 x^2}}{(d+e x)^2} \, dx\\ &=\frac {2 \sqrt {d^2-e^2 x^2}}{e (d+e x)}-\frac {2 \left (d^2-e^2 x^2\right )^{3/2}}{3 e (d+e x)^3}+\frac {2 \left (d^2-e^2 x^2\right )^{5/2}}{5 e (d+e x)^5}-\frac {2 \left (d^2-e^2 x^2\right )^{7/2}}{7 e (d+e x)^7}+\int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx\\ &=\frac {2 \sqrt {d^2-e^2 x^2}}{e (d+e x)}-\frac {2 \left (d^2-e^2 x^2\right )^{3/2}}{3 e (d+e x)^3}+\frac {2 \left (d^2-e^2 x^2\right )^{5/2}}{5 e (d+e x)^5}-\frac {2 \left (d^2-e^2 x^2\right )^{7/2}}{7 e (d+e x)^7}+\operatorname {Subst}\left (\int \frac {1}{1+e^2 x^2} \, dx,x,\frac {x}{\sqrt {d^2-e^2 x^2}}\right )\\ &=\frac {2 \sqrt {d^2-e^2 x^2}}{e (d+e x)}-\frac {2 \left (d^2-e^2 x^2\right )^{3/2}}{3 e (d+e x)^3}+\frac {2 \left (d^2-e^2 x^2\right )^{5/2}}{5 e (d+e x)^5}-\frac {2 \left (d^2-e^2 x^2\right )^{7/2}}{7 e (d+e x)^7}+\frac {\tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 85, normalized size = 0.59 \[ \frac {\tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e}+\frac {8 \sqrt {d^2-e^2 x^2} \left (19 d^3+76 d^2 e x+71 d e^2 x^2+44 e^3 x^3\right )}{105 e (d+e x)^4} \]
Antiderivative was successfully verified.
[In]
Integrate[(d^2 - e^2*x^2)^(7/2)/(d + e*x)^8,x]
[Out]
(8*Sqrt[d^2 - e^2*x^2]*(19*d^3 + 76*d^2*e*x + 71*d*e^2*x^2 + 44*e^3*x^3))/(105*e*(d + e*x)^4) + ArcTan[(e*x)/S
qrt[d^2 - e^2*x^2]]/e
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fricas [A] time = 0.98, size = 200, normalized size = 1.40 \[ \frac {2 \, {\left (76 \, e^{4} x^{4} + 304 \, d e^{3} x^{3} + 456 \, d^{2} e^{2} x^{2} + 304 \, d^{3} e x + 76 \, d^{4} - 105 \, {\left (e^{4} x^{4} + 4 \, d e^{3} x^{3} + 6 \, d^{2} e^{2} x^{2} + 4 \, d^{3} e x + d^{4}\right )} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + 4 \, {\left (44 \, e^{3} x^{3} + 71 \, d e^{2} x^{2} + 76 \, d^{2} e x + 19 \, d^{3}\right )} \sqrt {-e^{2} x^{2} + d^{2}}\right )}}{105 \, {\left (e^{5} x^{4} + 4 \, d e^{4} x^{3} + 6 \, d^{2} e^{3} x^{2} + 4 \, d^{3} e^{2} x + d^{4} e\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((-e^2*x^2+d^2)^(7/2)/(e*x+d)^8,x, algorithm="fricas")
[Out]
2/105*(76*e^4*x^4 + 304*d*e^3*x^3 + 456*d^2*e^2*x^2 + 304*d^3*e*x + 76*d^4 - 105*(e^4*x^4 + 4*d*e^3*x^3 + 6*d^
2*e^2*x^2 + 4*d^3*e*x + d^4)*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) + 4*(44*e^3*x^3 + 71*d*e^2*x^2 + 76*d^2
*e*x + 19*d^3)*sqrt(-e^2*x^2 + d^2))/(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)
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((-e^2*x^2+d^2)^(7/2)/(e*x+d)^8,x, algorithm="giac")
[Out]
Exception raised: NotImplementedError >> Unable to parse Giac output: (53760*(-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)^2+80640*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(
2))^9*exp(1)^34*exp(2)^3+67200*(-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)
^4+33600*(-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)^5+10080*(-1/2*(-2*d*e
xp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^12*exp(1)^28*exp(2)^6+1680*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*ex
p(2))*exp(1))/x/exp(2))^13*exp(1)^26*exp(2)^7+57344*(-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)^2+200704*(-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)^3
+301056*(-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)^4+250880*(-1/2*(-2*d*ex
p(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^10*exp(1)^30*exp(2)^5+125440*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*e
xp(2))*exp(1))/x/exp(2))^11*exp(1)^28*exp(2)^6+40320*(-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)^7+6720*(-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)^
8+53760*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^6*exp(1)^36*exp(2)^2+197632*(-1/2*(-2*d*ex
p(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^7*exp(1)^34*exp(2)^3+328832*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*ex
p(2))*exp(1))/x/exp(2))^8*exp(1)^32*exp(2)^4+325248*(-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)^5+182560*(-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)^
6+76160*(-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)^7+30240*(-1/2*(-2*d*ex
p(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^12*exp(1)^24*exp(2)^8+5040*(-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)^9+200704*(-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)^3+430080*(-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)^4
+365568*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^8*exp(1)^30*exp(2)^5-78848*(-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)^6-380800*(-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)^7-250880*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2)
)^11*exp(1)^24*exp(2)^8-80640*(-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)^
9-13440*(-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)^10+80640*(-1/2*(-2*d*e
xp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^5*exp(1)^34*exp(2)^3+328832*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*e
xp(2))*exp(1))/x/exp(2))^6*exp(1)^32*exp(2)^4+1691136*(-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)^5+3568320*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^8*exp(1)^28*exp(2
)^6+3390800*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^9*exp(1)^26*exp(2)^7+1748320*(-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)^8+452480*(-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)^9+10080*(-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)^10-11760*(-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)^11+301056*(-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)^4-64512*(-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)^5+1017856*(-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)^6+3729152*(-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)^7+5040448*(-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)^8+3664640*(-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)^9+1388800*
(-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)^10+255360*(-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)^11+20160*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))
*exp(1))/x/exp(2))^13*exp(1)^16*exp(2)^12+67200*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^4*
exp(1)^32*exp(2)^4+325248*(-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)^5+534
2400*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^6*exp(1)^28*exp(2)^6+16528384*(-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)^7+23906624*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*e
xp(2))*exp(1))/x/exp(2))^8*exp(1)^24*exp(2)^8+21140336*(-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)^9+11792480*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^10*exp(1)^20*ex
p(2)^10+4043200*(-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)^11+813120*(-1/
2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^12*exp(1)^16*exp(2)^12+75600*(-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)^13+250880*(-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)^5-723968*(-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)^6+7008512*(-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)^7+2043955
2*(-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)^8+23982336*(-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)^9+15710464*(-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)^10+6202560*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2)
)^10*exp(1)^18*exp(2)^11+1646400*(-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)^12+312480*(-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)^13+30240*(-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)^14+33600*(-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)^5+236320*(-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)^6+8928080*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^5*exp(1)^26*
exp(2)^7+36357104*(-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)^8+59975216*(-
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)^9+53901064*(-1/2*(-2*d*exp(1)-2*s
qrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^8*exp(1)^20*exp(2)^10+28133476*(-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)^11+8689870*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^1
0*exp(1)^16*exp(2)^12+1610630*(-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)^
13+172095*(-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)^14+8715*(-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)^15+125440*(-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)^6-748160*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/ex
p(2))^4*exp(1)^26*exp(2)^7+13888448*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^5*exp(1)^24*ex
p(2)^8+33914496*(-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)^9+34642048*(-1/
2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^7*exp(1)^20*exp(2)^10+19550720*(-1/2*(-2*d*exp(1)-2*sq
rt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^8*exp(1)^18*exp(2)^11+6273120*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*e
xp(1))/x/exp(2))^9*exp(1)^16*exp(2)^12+982800*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^10*e
xp(1)^14*exp(2)^13-85680*(-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)^14-73
080*(-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)^15-9240*(-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)^16+10080*(-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)^6+103040*(-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)^7+8188880*(-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)^8+44
412256*(-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)^9+70938784*(-1/2*(-2*d*e
xp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^6*exp(1)^20*exp(2)^10+53067952*(-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)^11+20015030*(-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)^12+3643430*(-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)^13+350560*(-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)^14+77000*(-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)^15+23730*(-1/2*(-2*d*exp(1)-2*sqr
t(d^2-x^2*exp(2))*exp(1))/x/exp(2))^12*exp(1)^8*exp(2)^16+2730*(-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)^17+34944*(-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)^7-426496*(-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)^8+13351296*(
-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)^9+28594944*(-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)^10+23569280*(-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)^11+7454720*(-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)^12-1730960*(-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)^1
3-2131920*(-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)^14-567840*(-1/2*(-2*d
*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^10*exp(1)^10*exp(2)^15+16800*(-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)^16+30240*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/ex
p(2))^12*exp(1)^6*exp(2)^17+3360*(-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
)^18+19152*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^2*exp(1)^24*exp(2)^8+4442032*(-1/2*(-2*
d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^3*exp(1)^22*exp(2)^9+33861408*(-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)^10+44991576*(-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)^11+22397900*(-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)^12+4208540*(-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)^13+135971
5*(-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)^14+1428875*(-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)^15+647640*(-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)^16+141960*(-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)^17+15960*(-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)^18+
840*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^13*exp(2)^20-138880*(-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)^9+7448448*(-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)^10+14516992*(-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)^11+6608000*(-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)^12-4051
040*(-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)^13-4037600*(-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)^14+292040*(-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)^15+1425480*(-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)^16+618240*(-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)^
17+107520*(-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)^18+240*exp(1)^24*exp(
2)^8+8400*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^12*exp(2)^20+1447936*(-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)^10+16596720*(-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)^11+14407820*(-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)^12+2231740*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^5*exp(1)^14*exp(2
)^13+1661800*(-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)^14+4086600*(-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)^15+2797410*(-1/2*(-2*d*exp(1)-2*sqrt(d
^2-x^2*exp(2))*exp(1))/x/exp(2))^8*exp(1)^8*exp(2)^16+969570*(-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)^17+204960*(-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)^18+896*exp(1)^22*exp(2)^9+26880*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^11*exp(2)^
20+2444736*(-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)^11+4663232*(-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)^12-1816192*(-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)^13-4457600*(-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)^14+1097600*(-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)^15+3920000*(-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)^16+2158240*
(-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)^17+487200*(-1/2*(-2*d*exp(1)-2*s
qrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^9*exp(1)^4*exp(2)^18+568*exp(1)^20*exp(2)^10+50400*(-1/2*(-2*d*exp(1)-2*
sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^10*exp(2)^20+5063814*(-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)^12+1122814*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^3*exp(1)^1
4*exp(2)^13-465899*(-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)^14+3391745*(
-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)^15+4116000*(-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)^16+2156000*(-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)^17+665560*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^8*ex
p(1)^4*exp(2)^18-2688*exp(1)^18*exp(2)^11+121800*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^9
*exp(2)^20+991312*(-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)^13-2286704*(-
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)^14-574056*(-1/2*(-2*d*exp(1)-2*sq
rt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^4*exp(1)^10*exp(2)^15+3513720*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*e
xp(1))/x/exp(2))^5*exp(1)^8*exp(2)^16+3055360*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^6*ex
p(1)^6*exp(2)^17+940800*(-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)^18+17310
*exp(1)^16*exp(2)^12+126000*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^8*exp(2)^20-599144*(-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)^14+505904*(-1/2*(-2*d*exp(1)-2*sqr
t(d^2-x^2*exp(2))*exp(1))/x/exp(2))^3*exp(1)^10*exp(2)^15+2300886*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*ex
p(1))/x/exp(2))^4*exp(1)^8*exp(2)^16+2029230*(-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)^17+931840*(-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)^18+32368*
exp(1)^14*exp(2)^13+235200*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^7*exp(2)^20-872032*(-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)^15+829920*(-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)^16+1945440*(-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)^17+924000*(-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)^18+68521*exp(1)^12*exp(2)^14+168000*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^6*ex
p(2)^20+347592*(-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)^16+775320*(-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)^17+612360*(-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)^18+12824*exp(1)^10*exp(2)^15+231000*(-1/2*(-2*d*exp(1)-2*sqrt
(d^2-x^2*exp(2))*exp(1))/x/exp(2))^5*exp(2)^20+448000*(-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)^17+456960*(-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)^
18-10794*exp(1)^8*exp(2)^16+126000*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^4*exp(2)^20+162
400*(-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)^18-12320*exp(1)^6*exp(2)^17+
114240*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^3*exp(2)^20+5320*exp(1)^4*exp(2)^18+50400*(
-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^2*exp(2)^20+8400*exp(2)^20+15360*(-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)-11340*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))
*exp(2)^20/x/exp(2)-45360*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))*exp(1)^4*exp(2)^18/x/exp(2)-35875*(-2*d*
exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))*exp(1)^6*exp(2)^17/x/exp(2)+81620*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*ex
p(1))*exp(1)^8*exp(2)^16/x/exp(2)+159831/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))*exp(1)^10*exp(2)^15/x/e
xp(2)-74648*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))*exp(1)^12*exp(2)^14/x/exp(2)-441847*(-2*d*exp(1)-2*sqr
t(d^2-x^2*exp(2))*exp(1))*exp(1)^14*exp(2)^13/x/exp(2)-216496*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))*exp(
1)^16*exp(2)^12/x/exp(2)-127050*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))*exp(1)^18*exp(2)^11/x/exp(2)+12096
*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))*exp(1)^20*exp(2)^10/x/exp(2)-1456*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp
(2))*exp(1))*exp(1)^22*exp(2)^9/x/exp(2)-2912*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))*exp(1)^24*exp(2)^8/x
/exp(2)-840*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))*exp(1)^26*exp(2)^7/x/exp(2))/((-1/2*(-2*d*exp(1)-2*sqr
t(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))^7/(840*exp(
1)^27+2520*exp(1)^23*exp(2)^2-6720*exp(1)^21*exp(2)^3-11760*exp(1)^19*exp(2)^4+11760*exp(1)^15*exp(2)^6+6720*e
xp(1)^13*exp(2)^7-2520*exp(1)^11*exp(2)^8-3360*exp(1)^9*exp(2)^9-840*exp(1)^7*exp(2)^10+3360*exp(1)^25*exp(2))
+1/2*(-128*exp(1)^20*exp(2)^5-256*exp(1)^18*exp(2)^6-544*exp(1)^16*exp(2)^7-32*exp(1)^14*exp(2)^8+93*exp(1)^12
*exp(2)^9+24*exp(1)^10*exp(2)^10-266*exp(1)^8*exp(2)^11-160*exp(1)^6*exp(2)^12+40*exp(1)^4*exp(2)^13+80*exp(2)
^15)*atan((-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)/(-4*exp(1)^29-12*exp(1)^25*exp(2)^2+32*exp(1)^23*exp(2)^3+56*exp(1)^21*exp(2)^4-56*exp(1)^17*exp(2)
^6-32*exp(1)^15*exp(2)^7+12*exp(1)^13*exp(2)^8+16*exp(1)^11*exp(2)^9+4*exp(1)^9*exp(2)^10-16*exp(1)^27*exp(2))
+sign(d)*asin(x*exp(2)/d/exp(1))/exp(1)
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maple [B] time = 0.05, size = 496, normalized size = 3.47 \[ \frac {\arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}}\right )}{\sqrt {e^{2}}}+\frac {\sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}\, x}{d^{2}}+\frac {2 \left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {3}{2}} x}{3 d^{4}}+\frac {8 \left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {5}{2}} x}{15 d^{6}}+\frac {16 \left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {7}{2}}}{35 d^{7} e}-\frac {\left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {9}{2}}}{7 \left (x +\frac {d}{e}\right )^{8} d \,e^{9}}+\frac {\left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {9}{2}}}{35 \left (x +\frac {d}{e}\right )^{7} d^{2} e^{8}}-\frac {2 \left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {9}{2}}}{105 \left (x +\frac {d}{e}\right )^{6} d^{3} e^{7}}+\frac {2 \left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {9}{2}}}{35 \left (x +\frac {d}{e}\right )^{5} d^{4} e^{6}}+\frac {8 \left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {9}{2}}}{35 \left (x +\frac {d}{e}\right )^{4} d^{5} e^{5}}+\frac {8 \left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {9}{2}}}{21 \left (x +\frac {d}{e}\right )^{3} d^{6} e^{4}}+\frac {16 \left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {9}{2}}}{35 \left (x +\frac {d}{e}\right )^{2} d^{7} e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((-e^2*x^2+d^2)^(7/2)/(e*x+d)^8,x)
[Out]
-1/7/e^9/d/(x+d/e)^8*(2*(x+d/e)*d*e-(x+d/e)^2*e^2)^(9/2)+1/35/e^8/d^2/(x+d/e)^7*(2*(x+d/e)*d*e-(x+d/e)^2*e^2)^
(9/2)-2/105/e^7/d^3/(x+d/e)^6*(2*(x+d/e)*d*e-(x+d/e)^2*e^2)^(9/2)+2/35/e^6/d^4/(x+d/e)^5*(2*(x+d/e)*d*e-(x+d/e
)^2*e^2)^(9/2)+8/35/e^5/d^5/(x+d/e)^4*(2*(x+d/e)*d*e-(x+d/e)^2*e^2)^(9/2)+8/21/e^4/d^6/(x+d/e)^3*(2*(x+d/e)*d*
e-(x+d/e)^2*e^2)^(9/2)+16/35/e^3/d^7/(x+d/e)^2*(2*(x+d/e)*d*e-(x+d/e)^2*e^2)^(9/2)+16/35/e/d^7*(2*(x+d/e)*d*e-
(x+d/e)^2*e^2)^(7/2)+8/15/d^6*(2*(x+d/e)*d*e-(x+d/e)^2*e^2)^(5/2)*x+2/3/d^4*(2*(x+d/e)*d*e-(x+d/e)^2*e^2)^(3/2
)*x+1/d^2*(2*(x+d/e)*d*e-(x+d/e)^2*e^2)^(1/2)*x+1/(e^2)^(1/2)*arctan((e^2)^(1/2)/(2*(x+d/e)*d*e-(x+d/e)^2*e^2)
^(1/2)*x)
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maxima [B] time = 3.12, size = 623, normalized size = 4.36 \[ -\frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}}}{7 \, {\left (e^{8} x^{7} + 7 \, d e^{7} x^{6} + 21 \, d^{2} e^{6} x^{5} + 35 \, d^{3} e^{5} x^{4} + 35 \, d^{4} e^{4} x^{3} + 21 \, d^{5} e^{3} x^{2} + 7 \, d^{6} e^{2} x + d^{7} e\right )}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d}{e^{7} x^{6} + 6 \, d e^{6} x^{5} + 15 \, d^{2} e^{5} x^{4} + 20 \, d^{3} e^{4} x^{3} + 15 \, d^{4} e^{3} x^{2} + 6 \, d^{5} e^{2} x + d^{6} e} + \frac {5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{2}}{2 \, {\left (e^{6} x^{5} + 5 \, d e^{5} x^{4} + 10 \, d^{2} e^{4} x^{3} + 10 \, d^{3} e^{3} x^{2} + 5 \, d^{4} e^{2} x + d^{5} e\right )}} - \frac {15 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{3}}{7 \, {\left (e^{5} x^{4} + 4 \, d e^{4} x^{3} + 6 \, d^{2} e^{3} x^{2} + 4 \, d^{3} e^{2} x + d^{4} e\right )}} + \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}}{5 \, {\left (e^{6} x^{5} + 5 \, d e^{5} x^{4} + 10 \, d^{2} e^{4} x^{3} + 10 \, d^{3} e^{3} x^{2} + 5 \, d^{4} e^{2} x + d^{5} e\right )}} + \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d}{e^{5} x^{4} + 4 \, d e^{4} x^{3} + 6 \, d^{2} e^{3} x^{2} + 4 \, d^{3} e^{2} x + d^{4} e} - \frac {69 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{2}}{70 \, {\left (e^{4} x^{3} + 3 \, d e^{3} x^{2} + 3 \, d^{2} e^{2} x + d^{3} e\right )}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}}}{3 \, {\left (e^{4} x^{3} + 3 \, d e^{3} x^{2} + 3 \, d^{2} e^{2} x + d^{3} e\right )}} - \frac {34 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{105 \, {\left (e^{3} x^{2} + 2 \, d e^{2} x + d^{2} e\right )}} + \frac {\arcsin \left (\frac {e x}{d}\right )}{e} + \frac {281 \, \sqrt {-e^{2} x^{2} + d^{2}}}{105 \, {\left (e^{2} x + d e\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((-e^2*x^2+d^2)^(7/2)/(e*x+d)^8,x, algorithm="maxima")
[Out]
-1/7*(-e^2*x^2 + d^2)^(7/2)/(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) - (-e^2*x^2 + d^2)^(5/2)*d/(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) + 5/2*(-e^2*x^2 + d^2)^(3/2)*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) - 15/7*sqrt(-e^2*x^2 + d^2)*d^3/(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) + 1/5*(-e^2*x^2 + d^2)^(5/2)/(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) + (-e^2*x^2 + d^2)^(3/2)*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) - 69/70*sqrt(-e^2*x^2 + d^2)*d^2/(e^4*x^3 + 3*d*e^3*x^2 + 3*d^2*e^2*x + d^3*e) - 1/3*(-e^2*x^2 + d^
2)^(3/2)/(e^4*x^3 + 3*d*e^3*x^2 + 3*d^2*e^2*x + d^3*e) - 34/105*sqrt(-e^2*x^2 + d^2)*d/(e^3*x^2 + 2*d*e^2*x +
d^2*e) + arcsin(e*x/d)/e + 281/105*sqrt(-e^2*x^2 + d^2)/(e^2*x + d*e)
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (d^2-e^2\,x^2\right )}^{7/2}}{{\left (d+e\,x\right )}^8} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((d^2 - e^2*x^2)^(7/2)/(d + e*x)^8,x)
[Out]
int((d^2 - e^2*x^2)^(7/2)/(d + e*x)^8, x)
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sympy [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((-e**2*x**2+d**2)**(7/2)/(e*x+d)**8,x)
[Out]
Timed out
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