3.19 \(\int \frac {x^7 (d+e x)}{(d^2-e^2 x^2)^{7/2}} \, dx\)

Optimal. Leaf size=161 \[ \frac {x^6 (d+e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {(32 d+35 e x) \sqrt {d^2-e^2 x^2}}{10 e^8}-\frac {7 d^2 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e^8}+\frac {x^2 (24 d+35 e x)}{15 e^6 \sqrt {d^2-e^2 x^2}}-\frac {x^4 (6 d+7 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}} \]

[Out]

1/5*x^6*(e*x+d)/e^2/(-e^2*x^2+d^2)^(5/2)-1/15*x^4*(7*e*x+6*d)/e^4/(-e^2*x^2+d^2)^(3/2)-7/2*d^2*arctan(e*x/(-e^
2*x^2+d^2)^(1/2))/e^8+1/15*x^2*(35*e*x+24*d)/e^6/(-e^2*x^2+d^2)^(1/2)+1/10*(35*e*x+32*d)*(-e^2*x^2+d^2)^(1/2)/
e^8

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Rubi [A]  time = 0.14, antiderivative size = 161, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {819, 780, 217, 203} \[ \frac {x^6 (d+e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {x^4 (6 d+7 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {x^2 (24 d+35 e x)}{15 e^6 \sqrt {d^2-e^2 x^2}}+\frac {(32 d+35 e x) \sqrt {d^2-e^2 x^2}}{10 e^8}-\frac {7 d^2 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e^8} \]

Antiderivative was successfully verified.

[In]

Int[(x^7*(d + e*x))/(d^2 - e^2*x^2)^(7/2),x]

[Out]

(x^6*(d + e*x))/(5*e^2*(d^2 - e^2*x^2)^(5/2)) - (x^4*(6*d + 7*e*x))/(15*e^4*(d^2 - e^2*x^2)^(3/2)) + (x^2*(24*
d + 35*e*x))/(15*e^6*Sqrt[d^2 - e^2*x^2]) + ((32*d + 35*e*x)*Sqrt[d^2 - e^2*x^2])/(10*e^8) - (7*d^2*ArcTan[(e*
x)/Sqrt[d^2 - e^2*x^2]])/(2*e^8)

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 780

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(((e*f + d*g)*(2*p
+ 3) + 2*e*g*(p + 1)*x)*(a + c*x^2)^(p + 1))/(2*c*(p + 1)*(2*p + 3)), x] - Dist[(a*e*g - c*d*f*(2*p + 3))/(c*
(2*p + 3)), Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, p}, x] &&  !LeQ[p, -1]

Rule 819

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x)^(
m - 1)*(a + c*x^2)^(p + 1)*(a*(e*f + d*g) - (c*d*f - a*e*g)*x))/(2*a*c*(p + 1)), x] - Dist[1/(2*a*c*(p + 1)),
Int[(d + e*x)^(m - 2)*(a + c*x^2)^(p + 1)*Simp[a*e*(e*f*(m - 1) + d*g*m) - c*d^2*f*(2*p + 3) + e*(a*e*g*m - c*
d*f*(m + 2*p + 2))*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && GtQ
[m, 1] && (EqQ[d, 0] || (EqQ[m, 2] && EqQ[p, -3] && RationalQ[a, c, d, e, f, g]) ||  !ILtQ[m + 2*p + 3, 0])

Rubi steps

\begin {align*} \int \frac {x^7 (d+e x)}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx &=\frac {x^6 (d+e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\int \frac {x^5 \left (6 d^3+7 d^2 e x\right )}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 d^2 e^2}\\ &=\frac {x^6 (d+e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {x^4 (6 d+7 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {\int \frac {x^3 \left (24 d^5+35 d^4 e x\right )}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{15 d^4 e^4}\\ &=\frac {x^6 (d+e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {x^4 (6 d+7 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {x^2 (24 d+35 e x)}{15 e^6 \sqrt {d^2-e^2 x^2}}-\frac {\int \frac {x \left (48 d^7+105 d^6 e x\right )}{\sqrt {d^2-e^2 x^2}} \, dx}{15 d^6 e^6}\\ &=\frac {x^6 (d+e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {x^4 (6 d+7 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {x^2 (24 d+35 e x)}{15 e^6 \sqrt {d^2-e^2 x^2}}+\frac {(32 d+35 e x) \sqrt {d^2-e^2 x^2}}{10 e^8}-\frac {\left (7 d^2\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx}{2 e^7}\\ &=\frac {x^6 (d+e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {x^4 (6 d+7 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {x^2 (24 d+35 e x)}{15 e^6 \sqrt {d^2-e^2 x^2}}+\frac {(32 d+35 e x) \sqrt {d^2-e^2 x^2}}{10 e^8}-\frac {\left (7 d^2\right ) \operatorname {Subst}\left (\int \frac {1}{1+e^2 x^2} \, dx,x,\frac {x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e^7}\\ &=\frac {x^6 (d+e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {x^4 (6 d+7 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {x^2 (24 d+35 e x)}{15 e^6 \sqrt {d^2-e^2 x^2}}+\frac {(32 d+35 e x) \sqrt {d^2-e^2 x^2}}{10 e^8}-\frac {7 d^2 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e^8}\\ \end {align*}

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Mathematica [A]  time = 0.10, size = 155, normalized size = 0.96 \[ \frac {96 d^6+9 d^5 e x-249 d^4 e^2 x^2+4 d^3 e^3 x^3+176 d^2 e^4 x^4-105 d^2 (d-e x)^2 (d+e x) \sqrt {d^2-e^2 x^2} \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-15 d e^5 x^5-15 e^6 x^6}{30 e^8 (d-e x)^2 (d+e x) \sqrt {d^2-e^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^7*(d + e*x))/(d^2 - e^2*x^2)^(7/2),x]

[Out]

(96*d^6 + 9*d^5*e*x - 249*d^4*e^2*x^2 + 4*d^3*e^3*x^3 + 176*d^2*e^4*x^4 - 15*d*e^5*x^5 - 15*e^6*x^6 - 105*d^2*
(d - e*x)^2*(d + e*x)*Sqrt[d^2 - e^2*x^2]*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(30*e^8*(d - e*x)^2*(d + e*x)*Sqr
t[d^2 - e^2*x^2])

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fricas [A]  time = 0.98, size = 278, normalized size = 1.73 \[ \frac {96 \, d^{2} e^{5} x^{5} - 96 \, d^{3} e^{4} x^{4} - 192 \, d^{4} e^{3} x^{3} + 192 \, d^{5} e^{2} x^{2} + 96 \, d^{6} e x - 96 \, d^{7} + 210 \, {\left (d^{2} e^{5} x^{5} - d^{3} e^{4} x^{4} - 2 \, d^{4} e^{3} x^{3} + 2 \, d^{5} e^{2} x^{2} + d^{6} e x - d^{7}\right )} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + {\left (15 \, e^{6} x^{6} + 15 \, d e^{5} x^{5} - 176 \, d^{2} e^{4} x^{4} - 4 \, d^{3} e^{3} x^{3} + 249 \, d^{4} e^{2} x^{2} - 9 \, d^{5} e x - 96 \, d^{6}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{30 \, {\left (e^{13} x^{5} - d e^{12} x^{4} - 2 \, d^{2} e^{11} x^{3} + 2 \, d^{3} e^{10} x^{2} + d^{4} e^{9} x - d^{5} e^{8}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(e*x+d)/(-e^2*x^2+d^2)^(7/2),x, algorithm="fricas")

[Out]

1/30*(96*d^2*e^5*x^5 - 96*d^3*e^4*x^4 - 192*d^4*e^3*x^3 + 192*d^5*e^2*x^2 + 96*d^6*e*x - 96*d^7 + 210*(d^2*e^5
*x^5 - d^3*e^4*x^4 - 2*d^4*e^3*x^3 + 2*d^5*e^2*x^2 + d^6*e*x - d^7)*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x))
+ (15*e^6*x^6 + 15*d*e^5*x^5 - 176*d^2*e^4*x^4 - 4*d^3*e^3*x^3 + 249*d^4*e^2*x^2 - 9*d^5*e*x - 96*d^6)*sqrt(-e
^2*x^2 + d^2))/(e^13*x^5 - d*e^12*x^4 - 2*d^2*e^11*x^3 + 2*d^3*e^10*x^2 + d^4*e^9*x - d^5*e^8)

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giac [A]  time = 0.28, size = 120, normalized size = 0.75 \[ -\frac {7}{2} \, d^{2} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-8\right )} \mathrm {sgn}\relax (d) - \frac {{\left (96 \, d^{7} e^{\left (-8\right )} + {\left (105 \, d^{6} e^{\left (-7\right )} - {\left (240 \, d^{5} e^{\left (-6\right )} + {\left (245 \, d^{4} e^{\left (-5\right )} - {\left (180 \, d^{3} e^{\left (-4\right )} + {\left (161 \, d^{2} e^{\left (-3\right )} - 15 \, {\left (x e^{\left (-1\right )} + 2 \, d e^{\left (-2\right )}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{30 \, {\left (x^{2} e^{2} - d^{2}\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(e*x+d)/(-e^2*x^2+d^2)^(7/2),x, algorithm="giac")

[Out]

-7/2*d^2*arcsin(x*e/d)*e^(-8)*sgn(d) - 1/30*(96*d^7*e^(-8) + (105*d^6*e^(-7) - (240*d^5*e^(-6) + (245*d^4*e^(-
5) - (180*d^3*e^(-4) + (161*d^2*e^(-3) - 15*(x*e^(-1) + 2*d*e^(-2))*x)*x)*x)*x)*x)*x)*sqrt(-x^2*e^2 + d^2)/(x^
2*e^2 - d^2)^3

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maple [A]  time = 0.07, size = 227, normalized size = 1.41 \[ -\frac {x^{7}}{2 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e}-\frac {d \,x^{6}}{\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e^{2}}+\frac {7 d^{2} x^{5}}{10 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e^{3}}+\frac {6 d^{3} x^{4}}{\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e^{4}}-\frac {8 d^{5} x^{2}}{\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e^{6}}-\frac {7 d^{2} x^{3}}{6 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} e^{5}}+\frac {16 d^{7}}{5 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e^{8}}+\frac {7 d^{2} x}{2 \sqrt {-e^{2} x^{2}+d^{2}}\, e^{7}}-\frac {7 d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}\, e^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7*(e*x+d)/(-e^2*x^2+d^2)^(7/2),x)

[Out]

-1/2*x^7/e/(-e^2*x^2+d^2)^(5/2)+7/10*d^2/e^3*x^5/(-e^2*x^2+d^2)^(5/2)-7/6*d^2/e^5*x^3/(-e^2*x^2+d^2)^(3/2)+7/2
*d^2/e^7*x/(-e^2*x^2+d^2)^(1/2)-7/2*d^2/e^7/(e^2)^(1/2)*arctan((e^2)^(1/2)/(-e^2*x^2+d^2)^(1/2)*x)-d*x^6/e^2/(
-e^2*x^2+d^2)^(5/2)+6*d^3/e^4*x^4/(-e^2*x^2+d^2)^(5/2)-8*d^5/e^6*x^2/(-e^2*x^2+d^2)^(5/2)+16/5*d^7/e^8/(-e^2*x
^2+d^2)^(5/2)

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maxima [B]  time = 1.03, size = 312, normalized size = 1.94 \[ -\frac {x^{7}}{2 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e} + \frac {7 \, d^{2} x {\left (\frac {15 \, x^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{2}} - \frac {20 \, d^{2} x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{4}} + \frac {8 \, d^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{6}}\right )}}{30 \, e} - \frac {d x^{6}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{2}} - \frac {7 \, d^{2} x {\left (\frac {3 \, x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{2}} - \frac {2 \, d^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{4}}\right )}}{6 \, e^{3}} + \frac {6 \, d^{3} x^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{4}} - \frac {8 \, d^{5} x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{6}} + \frac {16 \, d^{7}}{5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{8}} + \frac {14 \, d^{4} x}{15 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{7}} - \frac {49 \, d^{2} x}{30 \, \sqrt {-e^{2} x^{2} + d^{2}} e^{7}} - \frac {7 \, d^{2} \arcsin \left (\frac {e x}{d}\right )}{2 \, e^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(e*x+d)/(-e^2*x^2+d^2)^(7/2),x, algorithm="maxima")

[Out]

-1/2*x^7/((-e^2*x^2 + d^2)^(5/2)*e) + 7/30*d^2*x*(15*x^4/((-e^2*x^2 + d^2)^(5/2)*e^2) - 20*d^2*x^2/((-e^2*x^2
+ d^2)^(5/2)*e^4) + 8*d^4/((-e^2*x^2 + d^2)^(5/2)*e^6))/e - d*x^6/((-e^2*x^2 + d^2)^(5/2)*e^2) - 7/6*d^2*x*(3*
x^2/((-e^2*x^2 + d^2)^(3/2)*e^2) - 2*d^2/((-e^2*x^2 + d^2)^(3/2)*e^4))/e^3 + 6*d^3*x^4/((-e^2*x^2 + d^2)^(5/2)
*e^4) - 8*d^5*x^2/((-e^2*x^2 + d^2)^(5/2)*e^6) + 16/5*d^7/((-e^2*x^2 + d^2)^(5/2)*e^8) + 14/15*d^4*x/((-e^2*x^
2 + d^2)^(3/2)*e^7) - 49/30*d^2*x/(sqrt(-e^2*x^2 + d^2)*e^7) - 7/2*d^2*arcsin(e*x/d)/e^8

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^7\,\left (d+e\,x\right )}{{\left (d^2-e^2\,x^2\right )}^{7/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^7*(d + e*x))/(d^2 - e^2*x^2)^(7/2),x)

[Out]

int((x^7*(d + e*x))/(d^2 - e^2*x^2)^(7/2), x)

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sympy [B]  time = 66.42, size = 2004, normalized size = 12.45 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**7*(e*x+d)/(-e**2*x**2+d**2)**(7/2),x)

[Out]

d*Piecewise((16*d**6/(5*d**4*e**8*sqrt(d**2 - e**2*x**2) - 10*d**2*e**10*x**2*sqrt(d**2 - e**2*x**2) + 5*e**12
*x**4*sqrt(d**2 - e**2*x**2)) - 40*d**4*e**2*x**2/(5*d**4*e**8*sqrt(d**2 - e**2*x**2) - 10*d**2*e**10*x**2*sqr
t(d**2 - e**2*x**2) + 5*e**12*x**4*sqrt(d**2 - e**2*x**2)) + 30*d**2*e**4*x**4/(5*d**4*e**8*sqrt(d**2 - e**2*x
**2) - 10*d**2*e**10*x**2*sqrt(d**2 - e**2*x**2) + 5*e**12*x**4*sqrt(d**2 - e**2*x**2)) - 5*e**6*x**6/(5*d**4*
e**8*sqrt(d**2 - e**2*x**2) - 10*d**2*e**10*x**2*sqrt(d**2 - e**2*x**2) + 5*e**12*x**4*sqrt(d**2 - e**2*x**2))
, Ne(e, 0)), (x**8/(8*(d**2)**(7/2)), True)) + e*Piecewise((-210*I*d**7*sqrt(-1 + e**2*x**2/d**2)*acosh(e*x/d)
/(-60*d**5*e**9*sqrt(-1 + e**2*x**2/d**2) + 120*d**3*e**11*x**2*sqrt(-1 + e**2*x**2/d**2) - 60*d*e**13*x**4*sq
rt(-1 + e**2*x**2/d**2)) + 105*pi*d**7*sqrt(-1 + e**2*x**2/d**2)/(-60*d**5*e**9*sqrt(-1 + e**2*x**2/d**2) + 12
0*d**3*e**11*x**2*sqrt(-1 + e**2*x**2/d**2) - 60*d*e**13*x**4*sqrt(-1 + e**2*x**2/d**2)) + 210*I*d**6*e*x/(-60
*d**5*e**9*sqrt(-1 + e**2*x**2/d**2) + 120*d**3*e**11*x**2*sqrt(-1 + e**2*x**2/d**2) - 60*d*e**13*x**4*sqrt(-1
+ e**2*x**2/d**2)) + 420*I*d**5*e**2*x**2*sqrt(-1 + e**2*x**2/d**2)*acosh(e*x/d)/(-60*d**5*e**9*sqrt(-1 + e**
2*x**2/d**2) + 120*d**3*e**11*x**2*sqrt(-1 + e**2*x**2/d**2) - 60*d*e**13*x**4*sqrt(-1 + e**2*x**2/d**2)) - 21
0*pi*d**5*e**2*x**2*sqrt(-1 + e**2*x**2/d**2)/(-60*d**5*e**9*sqrt(-1 + e**2*x**2/d**2) + 120*d**3*e**11*x**2*s
qrt(-1 + e**2*x**2/d**2) - 60*d*e**13*x**4*sqrt(-1 + e**2*x**2/d**2)) - 490*I*d**4*e**3*x**3/(-60*d**5*e**9*sq
rt(-1 + e**2*x**2/d**2) + 120*d**3*e**11*x**2*sqrt(-1 + e**2*x**2/d**2) - 60*d*e**13*x**4*sqrt(-1 + e**2*x**2/
d**2)) - 210*I*d**3*e**4*x**4*sqrt(-1 + e**2*x**2/d**2)*acosh(e*x/d)/(-60*d**5*e**9*sqrt(-1 + e**2*x**2/d**2)
+ 120*d**3*e**11*x**2*sqrt(-1 + e**2*x**2/d**2) - 60*d*e**13*x**4*sqrt(-1 + e**2*x**2/d**2)) + 105*pi*d**3*e**
4*x**4*sqrt(-1 + e**2*x**2/d**2)/(-60*d**5*e**9*sqrt(-1 + e**2*x**2/d**2) + 120*d**3*e**11*x**2*sqrt(-1 + e**2
*x**2/d**2) - 60*d*e**13*x**4*sqrt(-1 + e**2*x**2/d**2)) + 322*I*d**2*e**5*x**5/(-60*d**5*e**9*sqrt(-1 + e**2*
x**2/d**2) + 120*d**3*e**11*x**2*sqrt(-1 + e**2*x**2/d**2) - 60*d*e**13*x**4*sqrt(-1 + e**2*x**2/d**2)) - 30*I
*e**7*x**7/(-60*d**5*e**9*sqrt(-1 + e**2*x**2/d**2) + 120*d**3*e**11*x**2*sqrt(-1 + e**2*x**2/d**2) - 60*d*e**
13*x**4*sqrt(-1 + e**2*x**2/d**2)), Abs(e**2*x**2/d**2) > 1), (-105*d**7*sqrt(1 - e**2*x**2/d**2)*asin(e*x/d)/
(30*d**5*e**9*sqrt(1 - e**2*x**2/d**2) - 60*d**3*e**11*x**2*sqrt(1 - e**2*x**2/d**2) + 30*d*e**13*x**4*sqrt(1
- e**2*x**2/d**2)) + 105*d**6*e*x/(30*d**5*e**9*sqrt(1 - e**2*x**2/d**2) - 60*d**3*e**11*x**2*sqrt(1 - e**2*x*
*2/d**2) + 30*d*e**13*x**4*sqrt(1 - e**2*x**2/d**2)) + 210*d**5*e**2*x**2*sqrt(1 - e**2*x**2/d**2)*asin(e*x/d)
/(30*d**5*e**9*sqrt(1 - e**2*x**2/d**2) - 60*d**3*e**11*x**2*sqrt(1 - e**2*x**2/d**2) + 30*d*e**13*x**4*sqrt(1
- e**2*x**2/d**2)) - 245*d**4*e**3*x**3/(30*d**5*e**9*sqrt(1 - e**2*x**2/d**2) - 60*d**3*e**11*x**2*sqrt(1 -
e**2*x**2/d**2) + 30*d*e**13*x**4*sqrt(1 - e**2*x**2/d**2)) - 105*d**3*e**4*x**4*sqrt(1 - e**2*x**2/d**2)*asin
(e*x/d)/(30*d**5*e**9*sqrt(1 - e**2*x**2/d**2) - 60*d**3*e**11*x**2*sqrt(1 - e**2*x**2/d**2) + 30*d*e**13*x**4
*sqrt(1 - e**2*x**2/d**2)) + 161*d**2*e**5*x**5/(30*d**5*e**9*sqrt(1 - e**2*x**2/d**2) - 60*d**3*e**11*x**2*sq
rt(1 - e**2*x**2/d**2) + 30*d*e**13*x**4*sqrt(1 - e**2*x**2/d**2)) - 15*e**7*x**7/(30*d**5*e**9*sqrt(1 - e**2*
x**2/d**2) - 60*d**3*e**11*x**2*sqrt(1 - e**2*x**2/d**2) + 30*d*e**13*x**4*sqrt(1 - e**2*x**2/d**2)), True))

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