3.992 \(\int e^{\tanh ^{-1}(a x)} x^m (1-a^2 x^2)^{5/2} \, dx\)

Optimal. Leaf size=82 \[ \frac {a^5 x^{m+6}}{m+6}+\frac {a^4 x^{m+5}}{m+5}-\frac {2 a^3 x^{m+4}}{m+4}-\frac {2 a^2 x^{m+3}}{m+3}+\frac {a x^{m+2}}{m+2}+\frac {x^{m+1}}{m+1} \]

[Out]

x^(1+m)/(1+m)+a*x^(2+m)/(2+m)-2*a^2*x^(3+m)/(3+m)-2*a^3*x^(4+m)/(4+m)+a^4*x^(5+m)/(5+m)+a^5*x^(6+m)/(6+m)

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Rubi [A]  time = 0.12, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {6150, 88} \[ -\frac {2 a^2 x^{m+3}}{m+3}-\frac {2 a^3 x^{m+4}}{m+4}+\frac {a^4 x^{m+5}}{m+5}+\frac {a^5 x^{m+6}}{m+6}+\frac {a x^{m+2}}{m+2}+\frac {x^{m+1}}{m+1} \]

Antiderivative was successfully verified.

[In]

Int[E^ArcTanh[a*x]*x^m*(1 - a^2*x^2)^(5/2),x]

[Out]

x^(1 + m)/(1 + m) + (a*x^(2 + m))/(2 + m) - (2*a^2*x^(3 + m))/(3 + m) - (2*a^3*x^(4 + m))/(4 + m) + (a^4*x^(5
+ m))/(5 + m) + (a^5*x^(6 + m))/(6 + m)

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rule 6150

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[c^p, Int[x^m*(1 -
a*x)^(p - n/2)*(1 + a*x)^(p + n/2), x], x] /; FreeQ[{a, c, d, m, n, p}, x] && EqQ[a^2*c + d, 0] && (IntegerQ[p
] || GtQ[c, 0])

Rubi steps

\begin {align*} \int e^{\tanh ^{-1}(a x)} x^m \left (1-a^2 x^2\right )^{5/2} \, dx &=\int x^m (1-a x)^2 (1+a x)^3 \, dx\\ &=\int \left (x^m+a x^{1+m}-2 a^2 x^{2+m}-2 a^3 x^{3+m}+a^4 x^{4+m}+a^5 x^{5+m}\right ) \, dx\\ &=\frac {x^{1+m}}{1+m}+\frac {a x^{2+m}}{2+m}-\frac {2 a^2 x^{3+m}}{3+m}-\frac {2 a^3 x^{4+m}}{4+m}+\frac {a^4 x^{5+m}}{5+m}+\frac {a^5 x^{6+m}}{6+m}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 70, normalized size = 0.85 \[ x^{m+1} \left (\frac {a^5 x^5}{m+6}+\frac {a^4 x^4}{m+5}-\frac {2 a^3 x^3}{m+4}-\frac {2 a^2 x^2}{m+3}+\frac {a x}{m+2}+\frac {1}{m+1}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[E^ArcTanh[a*x]*x^m*(1 - a^2*x^2)^(5/2),x]

[Out]

x^(1 + m)*((1 + m)^(-1) + (a*x)/(2 + m) - (2*a^2*x^2)/(3 + m) - (2*a^3*x^3)/(4 + m) + (a^4*x^4)/(5 + m) + (a^5
*x^5)/(6 + m))

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fricas [B]  time = 0.67, size = 285, normalized size = 3.48 \[ \frac {{\left ({\left (a^{5} m^{5} + 15 \, a^{5} m^{4} + 85 \, a^{5} m^{3} + 225 \, a^{5} m^{2} + 274 \, a^{5} m + 120 \, a^{5}\right )} x^{6} + {\left (a^{4} m^{5} + 16 \, a^{4} m^{4} + 95 \, a^{4} m^{3} + 260 \, a^{4} m^{2} + 324 \, a^{4} m + 144 \, a^{4}\right )} x^{5} - 2 \, {\left (a^{3} m^{5} + 17 \, a^{3} m^{4} + 107 \, a^{3} m^{3} + 307 \, a^{3} m^{2} + 396 \, a^{3} m + 180 \, a^{3}\right )} x^{4} - 2 \, {\left (a^{2} m^{5} + 18 \, a^{2} m^{4} + 121 \, a^{2} m^{3} + 372 \, a^{2} m^{2} + 508 \, a^{2} m + 240 \, a^{2}\right )} x^{3} + {\left (a m^{5} + 19 \, a m^{4} + 137 \, a m^{3} + 461 \, a m^{2} + 702 \, a m + 360 \, a\right )} x^{2} + {\left (m^{5} + 20 \, m^{4} + 155 \, m^{3} + 580 \, m^{2} + 1044 \, m + 720\right )} x\right )} x^{m}}{m^{6} + 21 \, m^{5} + 175 \, m^{4} + 735 \, m^{3} + 1624 \, m^{2} + 1764 \, m + 720} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)*(-a^2*x^2+1)^2*x^m,x, algorithm="fricas")

[Out]

((a^5*m^5 + 15*a^5*m^4 + 85*a^5*m^3 + 225*a^5*m^2 + 274*a^5*m + 120*a^5)*x^6 + (a^4*m^5 + 16*a^4*m^4 + 95*a^4*
m^3 + 260*a^4*m^2 + 324*a^4*m + 144*a^4)*x^5 - 2*(a^3*m^5 + 17*a^3*m^4 + 107*a^3*m^3 + 307*a^3*m^2 + 396*a^3*m
+ 180*a^3)*x^4 - 2*(a^2*m^5 + 18*a^2*m^4 + 121*a^2*m^3 + 372*a^2*m^2 + 508*a^2*m + 240*a^2)*x^3 + (a*m^5 + 19
*a*m^4 + 137*a*m^3 + 461*a*m^2 + 702*a*m + 360*a)*x^2 + (m^5 + 20*m^4 + 155*m^3 + 580*m^2 + 1044*m + 720)*x)*x
^m/(m^6 + 21*m^5 + 175*m^4 + 735*m^3 + 1624*m^2 + 1764*m + 720)

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giac [B]  time = 0.22, size = 460, normalized size = 5.61 \[ \frac {a^{5} m^{5} x^{6} x^{m} + 15 \, a^{5} m^{4} x^{6} x^{m} + a^{4} m^{5} x^{5} x^{m} + 85 \, a^{5} m^{3} x^{6} x^{m} + 16 \, a^{4} m^{4} x^{5} x^{m} + 225 \, a^{5} m^{2} x^{6} x^{m} - 2 \, a^{3} m^{5} x^{4} x^{m} + 95 \, a^{4} m^{3} x^{5} x^{m} + 274 \, a^{5} m x^{6} x^{m} - 34 \, a^{3} m^{4} x^{4} x^{m} + 260 \, a^{4} m^{2} x^{5} x^{m} + 120 \, a^{5} x^{6} x^{m} - 2 \, a^{2} m^{5} x^{3} x^{m} - 214 \, a^{3} m^{3} x^{4} x^{m} + 324 \, a^{4} m x^{5} x^{m} - 36 \, a^{2} m^{4} x^{3} x^{m} - 614 \, a^{3} m^{2} x^{4} x^{m} + 144 \, a^{4} x^{5} x^{m} + a m^{5} x^{2} x^{m} - 242 \, a^{2} m^{3} x^{3} x^{m} - 792 \, a^{3} m x^{4} x^{m} + 19 \, a m^{4} x^{2} x^{m} - 744 \, a^{2} m^{2} x^{3} x^{m} - 360 \, a^{3} x^{4} x^{m} + m^{5} x x^{m} + 137 \, a m^{3} x^{2} x^{m} - 1016 \, a^{2} m x^{3} x^{m} + 20 \, m^{4} x x^{m} + 461 \, a m^{2} x^{2} x^{m} - 480 \, a^{2} x^{3} x^{m} + 155 \, m^{3} x x^{m} + 702 \, a m x^{2} x^{m} + 580 \, m^{2} x x^{m} + 360 \, a x^{2} x^{m} + 1044 \, m x x^{m} + 720 \, x x^{m}}{m^{6} + 21 \, m^{5} + 175 \, m^{4} + 735 \, m^{3} + 1624 \, m^{2} + 1764 \, m + 720} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)*(-a^2*x^2+1)^2*x^m,x, algorithm="giac")

[Out]

(a^5*m^5*x^6*x^m + 15*a^5*m^4*x^6*x^m + a^4*m^5*x^5*x^m + 85*a^5*m^3*x^6*x^m + 16*a^4*m^4*x^5*x^m + 225*a^5*m^
2*x^6*x^m - 2*a^3*m^5*x^4*x^m + 95*a^4*m^3*x^5*x^m + 274*a^5*m*x^6*x^m - 34*a^3*m^4*x^4*x^m + 260*a^4*m^2*x^5*
x^m + 120*a^5*x^6*x^m - 2*a^2*m^5*x^3*x^m - 214*a^3*m^3*x^4*x^m + 324*a^4*m*x^5*x^m - 36*a^2*m^4*x^3*x^m - 614
*a^3*m^2*x^4*x^m + 144*a^4*x^5*x^m + a*m^5*x^2*x^m - 242*a^2*m^3*x^3*x^m - 792*a^3*m*x^4*x^m + 19*a*m^4*x^2*x^
m - 744*a^2*m^2*x^3*x^m - 360*a^3*x^4*x^m + m^5*x*x^m + 137*a*m^3*x^2*x^m - 1016*a^2*m*x^3*x^m + 20*m^4*x*x^m
+ 461*a*m^2*x^2*x^m - 480*a^2*x^3*x^m + 155*m^3*x*x^m + 702*a*m*x^2*x^m + 580*m^2*x*x^m + 360*a*x^2*x^m + 1044
*m*x*x^m + 720*x*x^m)/(m^6 + 21*m^5 + 175*m^4 + 735*m^3 + 1624*m^2 + 1764*m + 720)

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maple [B]  time = 0.03, size = 338, normalized size = 4.12 \[ \frac {x^{1+m} \left (a^{5} m^{5} x^{5}+15 a^{5} m^{4} x^{5}+85 a^{5} m^{3} x^{5}+a^{4} m^{5} x^{4}+225 a^{5} m^{2} x^{5}+16 a^{4} m^{4} x^{4}+274 a^{5} m \,x^{5}+95 a^{4} m^{3} x^{4}-2 a^{3} m^{5} x^{3}+120 x^{5} a^{5}+260 a^{4} m^{2} x^{4}-34 a^{3} m^{4} x^{3}+324 a^{4} m \,x^{4}-214 a^{3} m^{3} x^{3}-2 a^{2} m^{5} x^{2}+144 x^{4} a^{4}-614 a^{3} m^{2} x^{3}-36 a^{2} m^{4} x^{2}-792 a^{3} m \,x^{3}-242 a^{2} m^{3} x^{2}+a \,m^{5} x -360 x^{3} a^{3}-744 a^{2} m^{2} x^{2}+19 a \,m^{4} x -1016 a^{2} m \,x^{2}+137 a \,m^{3} x +m^{5}-480 a^{2} x^{2}+461 a \,m^{2} x +20 m^{4}+702 a m x +155 m^{3}+360 a x +580 m^{2}+1044 m +720\right )}{\left (6+m \right ) \left (5+m \right ) \left (4+m \right ) \left (3+m \right ) \left (2+m \right ) \left (1+m \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x+1)*(-a^2*x^2+1)^2*x^m,x)

[Out]

x^(1+m)*(a^5*m^5*x^5+15*a^5*m^4*x^5+85*a^5*m^3*x^5+a^4*m^5*x^4+225*a^5*m^2*x^5+16*a^4*m^4*x^4+274*a^5*m*x^5+95
*a^4*m^3*x^4-2*a^3*m^5*x^3+120*a^5*x^5+260*a^4*m^2*x^4-34*a^3*m^4*x^3+324*a^4*m*x^4-214*a^3*m^3*x^3-2*a^2*m^5*
x^2+144*a^4*x^4-614*a^3*m^2*x^3-36*a^2*m^4*x^2-792*a^3*m*x^3-242*a^2*m^3*x^2+a*m^5*x-360*a^3*x^3-744*a^2*m^2*x
^2+19*a*m^4*x-1016*a^2*m*x^2+137*a*m^3*x+m^5-480*a^2*x^2+461*a*m^2*x+20*m^4+702*a*m*x+155*m^3+360*a*x+580*m^2+
1044*m+720)/(6+m)/(5+m)/(4+m)/(3+m)/(2+m)/(1+m)

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maxima [A]  time = 0.32, size = 82, normalized size = 1.00 \[ \frac {a^{5} x^{m + 6}}{m + 6} + \frac {a^{4} x^{m + 5}}{m + 5} - \frac {2 \, a^{3} x^{m + 4}}{m + 4} - \frac {2 \, a^{2} x^{m + 3}}{m + 3} + \frac {a x^{m + 2}}{m + 2} + \frac {x^{m + 1}}{m + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)*(-a^2*x^2+1)^2*x^m,x, algorithm="maxima")

[Out]

a^5*x^(m + 6)/(m + 6) + a^4*x^(m + 5)/(m + 5) - 2*a^3*x^(m + 4)/(m + 4) - 2*a^2*x^(m + 3)/(m + 3) + a*x^(m + 2
)/(m + 2) + x^(m + 1)/(m + 1)

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mupad [B]  time = 1.16, size = 374, normalized size = 4.56 \[ \frac {x\,x^m\,\left (m^5+20\,m^4+155\,m^3+580\,m^2+1044\,m+720\right )}{m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720}+\frac {a\,x^m\,x^2\,\left (m^5+19\,m^4+137\,m^3+461\,m^2+702\,m+360\right )}{m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720}+\frac {a^5\,x^m\,x^6\,\left (m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120\right )}{m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720}+\frac {a^4\,x^m\,x^5\,\left (m^5+16\,m^4+95\,m^3+260\,m^2+324\,m+144\right )}{m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720}-\frac {2\,a^3\,x^m\,x^4\,\left (m^5+17\,m^4+107\,m^3+307\,m^2+396\,m+180\right )}{m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720}-\frac {2\,a^2\,x^m\,x^3\,\left (m^5+18\,m^4+121\,m^3+372\,m^2+508\,m+240\right )}{m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m*(a^2*x^2 - 1)^2*(a*x + 1),x)

[Out]

(x*x^m*(1044*m + 580*m^2 + 155*m^3 + 20*m^4 + m^5 + 720))/(1764*m + 1624*m^2 + 735*m^3 + 175*m^4 + 21*m^5 + m^
6 + 720) + (a*x^m*x^2*(702*m + 461*m^2 + 137*m^3 + 19*m^4 + m^5 + 360))/(1764*m + 1624*m^2 + 735*m^3 + 175*m^4
+ 21*m^5 + m^6 + 720) + (a^5*x^m*x^6*(274*m + 225*m^2 + 85*m^3 + 15*m^4 + m^5 + 120))/(1764*m + 1624*m^2 + 73
5*m^3 + 175*m^4 + 21*m^5 + m^6 + 720) + (a^4*x^m*x^5*(324*m + 260*m^2 + 95*m^3 + 16*m^4 + m^5 + 144))/(1764*m
+ 1624*m^2 + 735*m^3 + 175*m^4 + 21*m^5 + m^6 + 720) - (2*a^3*x^m*x^4*(396*m + 307*m^2 + 107*m^3 + 17*m^4 + m^
5 + 180))/(1764*m + 1624*m^2 + 735*m^3 + 175*m^4 + 21*m^5 + m^6 + 720) - (2*a^2*x^m*x^3*(508*m + 372*m^2 + 121
*m^3 + 18*m^4 + m^5 + 240))/(1764*m + 1624*m^2 + 735*m^3 + 175*m^4 + 21*m^5 + m^6 + 720)

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sympy [A]  time = 1.38, size = 1760, normalized size = 21.46 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)*(-a**2*x**2+1)**2*x**m,x)

[Out]

Piecewise((a**5*log(x) - a**4/x + a**3/x**2 + 2*a**2/(3*x**3) - a/(4*x**4) - 1/(5*x**5), Eq(m, -6)), (a**5*x +
a**4*log(x) + 2*a**3/x + a**2/x**2 - a/(3*x**3) - 1/(4*x**4), Eq(m, -5)), (a**5*x**2/2 + a**4*x - 2*a**3*log(
x) + 2*a**2/x - a/(2*x**2) - 1/(3*x**3), Eq(m, -4)), (a**5*x**3/3 + a**4*x**2/2 - 2*a**3*x - 2*a**2*log(x) - a
/x - 1/(2*x**2), Eq(m, -3)), (a**5*x**4/4 + a**4*x**3/3 - a**3*x**2 - 2*a**2*x + a*log(x) - 1/x, Eq(m, -2)), (
a**5*x**5/5 + a**4*x**4/4 - 2*a**3*x**3/3 - a**2*x**2 + a*x + log(x), Eq(m, -1)), (a**5*m**5*x**6*x**m/(m**6 +
21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 15*a**5*m**4*x**6*x**m/(m**6 + 21*m**5 + 175*m**4
+ 735*m**3 + 1624*m**2 + 1764*m + 720) + 85*a**5*m**3*x**6*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*
m**2 + 1764*m + 720) + 225*a**5*m**2*x**6*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 72
0) + 274*a**5*m*x**6*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 120*a**5*x**6*x*
*m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + a**4*m**5*x**5*x**m/(m**6 + 21*m**5 + 1
75*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 16*a**4*m**4*x**5*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3
+ 1624*m**2 + 1764*m + 720) + 95*a**4*m**3*x**5*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*
m + 720) + 260*a**4*m**2*x**5*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 324*a**
4*m*x**5*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 144*a**4*x**5*x**m/(m**6 + 2
1*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) - 2*a**3*m**5*x**4*x**m/(m**6 + 21*m**5 + 175*m**4 +
735*m**3 + 1624*m**2 + 1764*m + 720) - 34*a**3*m**4*x**4*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**
2 + 1764*m + 720) - 214*a**3*m**3*x**4*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720)
- 614*a**3*m**2*x**4*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) - 792*a**3*m*x**4*
x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) - 360*a**3*x**4*x**m/(m**6 + 21*m**5 +
175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) - 2*a**2*m**5*x**3*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3
+ 1624*m**2 + 1764*m + 720) - 36*a**2*m**4*x**3*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*
m + 720) - 242*a**2*m**3*x**3*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) - 744*a**
2*m**2*x**3*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) - 1016*a**2*m*x**3*x**m/(m*
*6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) - 480*a**2*x**3*x**m/(m**6 + 21*m**5 + 175*m**4
+ 735*m**3 + 1624*m**2 + 1764*m + 720) + a*m**5*x**2*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 +
1764*m + 720) + 19*a*m**4*x**2*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 137*a
*m**3*x**2*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 461*a*m**2*x**2*x**m/(m**6
+ 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 702*a*m*x**2*x**m/(m**6 + 21*m**5 + 175*m**4 +
735*m**3 + 1624*m**2 + 1764*m + 720) + 360*a*x**2*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 176
4*m + 720) + m**5*x*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 20*m**4*x*x**m/(m
**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 155*m**3*x*x**m/(m**6 + 21*m**5 + 175*m**4 +
735*m**3 + 1624*m**2 + 1764*m + 720) + 580*m**2*x*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 17
64*m + 720) + 1044*m*x*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 720*x*x**m/(m*
*6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720), True))

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