3.18 \(\int (e x)^m (A+B x^2) (c+d x^2)^3 \, dx\)
Optimal. Leaf size=121 \[ \frac {c^2 (e x)^{m+3} (3 A d+B c)}{e^3 (m+3)}+\frac {d^2 (e x)^{m+7} (A d+3 B c)}{e^7 (m+7)}+\frac {3 c d (e x)^{m+5} (A d+B c)}{e^5 (m+5)}+\frac {A c^3 (e x)^{m+1}}{e (m+1)}+\frac {B d^3 (e x)^{m+9}}{e^9 (m+9)} \]
[Out]
A*c^3*(e*x)^(1+m)/e/(1+m)+c^2*(3*A*d+B*c)*(e*x)^(3+m)/e^3/(3+m)+3*c*d*(A*d+B*c)*(e*x)^(5+m)/e^5/(5+m)+d^2*(A*d
+3*B*c)*(e*x)^(7+m)/e^7/(7+m)+B*d^3*(e*x)^(9+m)/e^9/(9+m)
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Rubi [A] time = 0.08, antiderivative size = 121, normalized size of antiderivative = 1.00,
number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used =
{448} \[ \frac {c^2 (e x)^{m+3} (3 A d+B c)}{e^3 (m+3)}+\frac {d^2 (e x)^{m+7} (A d+3 B c)}{e^7 (m+7)}+\frac {3 c d (e x)^{m+5} (A d+B c)}{e^5 (m+5)}+\frac {A c^3 (e x)^{m+1}}{e (m+1)}+\frac {B d^3 (e x)^{m+9}}{e^9 (m+9)} \]
Antiderivative was successfully verified.
[In]
Int[(e*x)^m*(A + B*x^2)*(c + d*x^2)^3,x]
[Out]
(A*c^3*(e*x)^(1 + m))/(e*(1 + m)) + (c^2*(B*c + 3*A*d)*(e*x)^(3 + m))/(e^3*(3 + m)) + (3*c*d*(B*c + A*d)*(e*x)
^(5 + m))/(e^5*(5 + m)) + (d^2*(3*B*c + A*d)*(e*x)^(7 + m))/(e^7*(7 + m)) + (B*d^3*(e*x)^(9 + m))/(e^9*(9 + m)
)
Rule 448
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[ExpandI
ntegrand[(e*x)^m*(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] &
& IGtQ[p, 0] && IGtQ[q, 0]
Rubi steps
\begin {align*} \int (e x)^m \left (A+B x^2\right ) \left (c+d x^2\right )^3 \, dx &=\int \left (A c^3 (e x)^m+\frac {c^2 (B c+3 A d) (e x)^{2+m}}{e^2}+\frac {3 c d (B c+A d) (e x)^{4+m}}{e^4}+\frac {d^2 (3 B c+A d) (e x)^{6+m}}{e^6}+\frac {B d^3 (e x)^{8+m}}{e^8}\right ) \, dx\\ &=\frac {A c^3 (e x)^{1+m}}{e (1+m)}+\frac {c^2 (B c+3 A d) (e x)^{3+m}}{e^3 (3+m)}+\frac {3 c d (B c+A d) (e x)^{5+m}}{e^5 (5+m)}+\frac {d^2 (3 B c+A d) (e x)^{7+m}}{e^7 (7+m)}+\frac {B d^3 (e x)^{9+m}}{e^9 (9+m)}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 90, normalized size = 0.74 \[ x (e x)^m \left (\frac {c^2 x^2 (3 A d+B c)}{m+3}+\frac {d^2 x^6 (A d+3 B c)}{m+7}+\frac {3 c d x^4 (A d+B c)}{m+5}+\frac {A c^3}{m+1}+\frac {B d^3 x^8}{m+9}\right ) \]
Antiderivative was successfully verified.
[In]
Integrate[(e*x)^m*(A + B*x^2)*(c + d*x^2)^3,x]
[Out]
x*(e*x)^m*((A*c^3)/(1 + m) + (c^2*(B*c + 3*A*d)*x^2)/(3 + m) + (3*c*d*(B*c + A*d)*x^4)/(5 + m) + (d^2*(3*B*c +
A*d)*x^6)/(7 + m) + (B*d^3*x^8)/(9 + m))
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fricas [B] time = 0.87, size = 381, normalized size = 3.15 \[ \frac {{\left ({\left (B d^{3} m^{4} + 16 \, B d^{3} m^{3} + 86 \, B d^{3} m^{2} + 176 \, B d^{3} m + 105 \, B d^{3}\right )} x^{9} + {\left ({\left (3 \, B c d^{2} + A d^{3}\right )} m^{4} + 405 \, B c d^{2} + 135 \, A d^{3} + 18 \, {\left (3 \, B c d^{2} + A d^{3}\right )} m^{3} + 104 \, {\left (3 \, B c d^{2} + A d^{3}\right )} m^{2} + 222 \, {\left (3 \, B c d^{2} + A d^{3}\right )} m\right )} x^{7} + 3 \, {\left ({\left (B c^{2} d + A c d^{2}\right )} m^{4} + 189 \, B c^{2} d + 189 \, A c d^{2} + 20 \, {\left (B c^{2} d + A c d^{2}\right )} m^{3} + 130 \, {\left (B c^{2} d + A c d^{2}\right )} m^{2} + 300 \, {\left (B c^{2} d + A c d^{2}\right )} m\right )} x^{5} + {\left ({\left (B c^{3} + 3 \, A c^{2} d\right )} m^{4} + 315 \, B c^{3} + 945 \, A c^{2} d + 22 \, {\left (B c^{3} + 3 \, A c^{2} d\right )} m^{3} + 164 \, {\left (B c^{3} + 3 \, A c^{2} d\right )} m^{2} + 458 \, {\left (B c^{3} + 3 \, A c^{2} d\right )} m\right )} x^{3} + {\left (A c^{3} m^{4} + 24 \, A c^{3} m^{3} + 206 \, A c^{3} m^{2} + 744 \, A c^{3} m + 945 \, A c^{3}\right )} x\right )} \left (e x\right )^{m}}{m^{5} + 25 \, m^{4} + 230 \, m^{3} + 950 \, m^{2} + 1689 \, m + 945} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x)^m*(B*x^2+A)*(d*x^2+c)^3,x, algorithm="fricas")
[Out]
((B*d^3*m^4 + 16*B*d^3*m^3 + 86*B*d^3*m^2 + 176*B*d^3*m + 105*B*d^3)*x^9 + ((3*B*c*d^2 + A*d^3)*m^4 + 405*B*c*
d^2 + 135*A*d^3 + 18*(3*B*c*d^2 + A*d^3)*m^3 + 104*(3*B*c*d^2 + A*d^3)*m^2 + 222*(3*B*c*d^2 + A*d^3)*m)*x^7 +
3*((B*c^2*d + A*c*d^2)*m^4 + 189*B*c^2*d + 189*A*c*d^2 + 20*(B*c^2*d + A*c*d^2)*m^3 + 130*(B*c^2*d + A*c*d^2)*
m^2 + 300*(B*c^2*d + A*c*d^2)*m)*x^5 + ((B*c^3 + 3*A*c^2*d)*m^4 + 315*B*c^3 + 945*A*c^2*d + 22*(B*c^3 + 3*A*c^
2*d)*m^3 + 164*(B*c^3 + 3*A*c^2*d)*m^2 + 458*(B*c^3 + 3*A*c^2*d)*m)*x^3 + (A*c^3*m^4 + 24*A*c^3*m^3 + 206*A*c^
3*m^2 + 744*A*c^3*m + 945*A*c^3)*x)*(e*x)^m/(m^5 + 25*m^4 + 230*m^3 + 950*m^2 + 1689*m + 945)
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giac [B] time = 0.51, size = 673, normalized size = 5.56 \[ \frac {B d^{3} m^{4} x^{9} x^{m} e^{m} + 16 \, B d^{3} m^{3} x^{9} x^{m} e^{m} + 3 \, B c d^{2} m^{4} x^{7} x^{m} e^{m} + A d^{3} m^{4} x^{7} x^{m} e^{m} + 86 \, B d^{3} m^{2} x^{9} x^{m} e^{m} + 54 \, B c d^{2} m^{3} x^{7} x^{m} e^{m} + 18 \, A d^{3} m^{3} x^{7} x^{m} e^{m} + 176 \, B d^{3} m x^{9} x^{m} e^{m} + 3 \, B c^{2} d m^{4} x^{5} x^{m} e^{m} + 3 \, A c d^{2} m^{4} x^{5} x^{m} e^{m} + 312 \, B c d^{2} m^{2} x^{7} x^{m} e^{m} + 104 \, A d^{3} m^{2} x^{7} x^{m} e^{m} + 105 \, B d^{3} x^{9} x^{m} e^{m} + 60 \, B c^{2} d m^{3} x^{5} x^{m} e^{m} + 60 \, A c d^{2} m^{3} x^{5} x^{m} e^{m} + 666 \, B c d^{2} m x^{7} x^{m} e^{m} + 222 \, A d^{3} m x^{7} x^{m} e^{m} + B c^{3} m^{4} x^{3} x^{m} e^{m} + 3 \, A c^{2} d m^{4} x^{3} x^{m} e^{m} + 390 \, B c^{2} d m^{2} x^{5} x^{m} e^{m} + 390 \, A c d^{2} m^{2} x^{5} x^{m} e^{m} + 405 \, B c d^{2} x^{7} x^{m} e^{m} + 135 \, A d^{3} x^{7} x^{m} e^{m} + 22 \, B c^{3} m^{3} x^{3} x^{m} e^{m} + 66 \, A c^{2} d m^{3} x^{3} x^{m} e^{m} + 900 \, B c^{2} d m x^{5} x^{m} e^{m} + 900 \, A c d^{2} m x^{5} x^{m} e^{m} + A c^{3} m^{4} x x^{m} e^{m} + 164 \, B c^{3} m^{2} x^{3} x^{m} e^{m} + 492 \, A c^{2} d m^{2} x^{3} x^{m} e^{m} + 567 \, B c^{2} d x^{5} x^{m} e^{m} + 567 \, A c d^{2} x^{5} x^{m} e^{m} + 24 \, A c^{3} m^{3} x x^{m} e^{m} + 458 \, B c^{3} m x^{3} x^{m} e^{m} + 1374 \, A c^{2} d m x^{3} x^{m} e^{m} + 206 \, A c^{3} m^{2} x x^{m} e^{m} + 315 \, B c^{3} x^{3} x^{m} e^{m} + 945 \, A c^{2} d x^{3} x^{m} e^{m} + 744 \, A c^{3} m x x^{m} e^{m} + 945 \, A c^{3} x x^{m} e^{m}}{m^{5} + 25 \, m^{4} + 230 \, m^{3} + 950 \, m^{2} + 1689 \, m + 945} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x)^m*(B*x^2+A)*(d*x^2+c)^3,x, algorithm="giac")
[Out]
(B*d^3*m^4*x^9*x^m*e^m + 16*B*d^3*m^3*x^9*x^m*e^m + 3*B*c*d^2*m^4*x^7*x^m*e^m + A*d^3*m^4*x^7*x^m*e^m + 86*B*d
^3*m^2*x^9*x^m*e^m + 54*B*c*d^2*m^3*x^7*x^m*e^m + 18*A*d^3*m^3*x^7*x^m*e^m + 176*B*d^3*m*x^9*x^m*e^m + 3*B*c^2
*d*m^4*x^5*x^m*e^m + 3*A*c*d^2*m^4*x^5*x^m*e^m + 312*B*c*d^2*m^2*x^7*x^m*e^m + 104*A*d^3*m^2*x^7*x^m*e^m + 105
*B*d^3*x^9*x^m*e^m + 60*B*c^2*d*m^3*x^5*x^m*e^m + 60*A*c*d^2*m^3*x^5*x^m*e^m + 666*B*c*d^2*m*x^7*x^m*e^m + 222
*A*d^3*m*x^7*x^m*e^m + B*c^3*m^4*x^3*x^m*e^m + 3*A*c^2*d*m^4*x^3*x^m*e^m + 390*B*c^2*d*m^2*x^5*x^m*e^m + 390*A
*c*d^2*m^2*x^5*x^m*e^m + 405*B*c*d^2*x^7*x^m*e^m + 135*A*d^3*x^7*x^m*e^m + 22*B*c^3*m^3*x^3*x^m*e^m + 66*A*c^2
*d*m^3*x^3*x^m*e^m + 900*B*c^2*d*m*x^5*x^m*e^m + 900*A*c*d^2*m*x^5*x^m*e^m + A*c^3*m^4*x*x^m*e^m + 164*B*c^3*m
^2*x^3*x^m*e^m + 492*A*c^2*d*m^2*x^3*x^m*e^m + 567*B*c^2*d*x^5*x^m*e^m + 567*A*c*d^2*x^5*x^m*e^m + 24*A*c^3*m^
3*x*x^m*e^m + 458*B*c^3*m*x^3*x^m*e^m + 1374*A*c^2*d*m*x^3*x^m*e^m + 206*A*c^3*m^2*x*x^m*e^m + 315*B*c^3*x^3*x
^m*e^m + 945*A*c^2*d*x^3*x^m*e^m + 744*A*c^3*m*x*x^m*e^m + 945*A*c^3*x*x^m*e^m)/(m^5 + 25*m^4 + 230*m^3 + 950*
m^2 + 1689*m + 945)
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maple [B] time = 0.01, size = 475, normalized size = 3.93 \[ \frac {\left (B \,d^{3} m^{4} x^{8}+16 B \,d^{3} m^{3} x^{8}+A \,d^{3} m^{4} x^{6}+3 B c \,d^{2} m^{4} x^{6}+86 B \,d^{3} m^{2} x^{8}+18 A \,d^{3} m^{3} x^{6}+54 B c \,d^{2} m^{3} x^{6}+176 B \,d^{3} m \,x^{8}+3 A c \,d^{2} m^{4} x^{4}+104 A \,d^{3} m^{2} x^{6}+3 B \,c^{2} d \,m^{4} x^{4}+312 B c \,d^{2} m^{2} x^{6}+105 B \,d^{3} x^{8}+60 A c \,d^{2} m^{3} x^{4}+222 A \,d^{3} m \,x^{6}+60 B \,c^{2} d \,m^{3} x^{4}+666 B c \,d^{2} m \,x^{6}+3 A \,c^{2} d \,m^{4} x^{2}+390 A c \,d^{2} m^{2} x^{4}+135 A \,d^{3} x^{6}+B \,c^{3} m^{4} x^{2}+390 B \,c^{2} d \,m^{2} x^{4}+405 B c \,d^{2} x^{6}+66 A \,c^{2} d \,m^{3} x^{2}+900 A c \,d^{2} m \,x^{4}+22 B \,c^{3} m^{3} x^{2}+900 B \,c^{2} d m \,x^{4}+A \,c^{3} m^{4}+492 A \,c^{2} d \,m^{2} x^{2}+567 A c \,d^{2} x^{4}+164 B \,c^{3} m^{2} x^{2}+567 B \,c^{2} d \,x^{4}+24 A \,c^{3} m^{3}+1374 A \,c^{2} d m \,x^{2}+458 B \,c^{3} m \,x^{2}+206 A \,c^{3} m^{2}+945 A \,c^{2} d \,x^{2}+315 B \,c^{3} x^{2}+744 A \,c^{3} m +945 A \,c^{3}\right ) x \left (e x \right )^{m}}{\left (m +9\right ) \left (m +7\right ) \left (m +5\right ) \left (m +3\right ) \left (m +1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((e*x)^m*(B*x^2+A)*(d*x^2+c)^3,x)
[Out]
x*(B*d^3*m^4*x^8+16*B*d^3*m^3*x^8+A*d^3*m^4*x^6+3*B*c*d^2*m^4*x^6+86*B*d^3*m^2*x^8+18*A*d^3*m^3*x^6+54*B*c*d^2
*m^3*x^6+176*B*d^3*m*x^8+3*A*c*d^2*m^4*x^4+104*A*d^3*m^2*x^6+3*B*c^2*d*m^4*x^4+312*B*c*d^2*m^2*x^6+105*B*d^3*x
^8+60*A*c*d^2*m^3*x^4+222*A*d^3*m*x^6+60*B*c^2*d*m^3*x^4+666*B*c*d^2*m*x^6+3*A*c^2*d*m^4*x^2+390*A*c*d^2*m^2*x
^4+135*A*d^3*x^6+B*c^3*m^4*x^2+390*B*c^2*d*m^2*x^4+405*B*c*d^2*x^6+66*A*c^2*d*m^3*x^2+900*A*c*d^2*m*x^4+22*B*c
^3*m^3*x^2+900*B*c^2*d*m*x^4+A*c^3*m^4+492*A*c^2*d*m^2*x^2+567*A*c*d^2*x^4+164*B*c^3*m^2*x^2+567*B*c^2*d*x^4+2
4*A*c^3*m^3+1374*A*c^2*d*m*x^2+458*B*c^3*m*x^2+206*A*c^3*m^2+945*A*c^2*d*x^2+315*B*c^3*x^2+744*A*c^3*m+945*A*c
^3)*(e*x)^m/(m+9)/(m+7)/(m+5)/(m+3)/(m+1)
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maxima [A] time = 1.64, size = 162, normalized size = 1.34 \[ \frac {B d^{3} e^{m} x^{9} x^{m}}{m + 9} + \frac {3 \, B c d^{2} e^{m} x^{7} x^{m}}{m + 7} + \frac {A d^{3} e^{m} x^{7} x^{m}}{m + 7} + \frac {3 \, B c^{2} d e^{m} x^{5} x^{m}}{m + 5} + \frac {3 \, A c d^{2} e^{m} x^{5} x^{m}}{m + 5} + \frac {B c^{3} e^{m} x^{3} x^{m}}{m + 3} + \frac {3 \, A c^{2} d e^{m} x^{3} x^{m}}{m + 3} + \frac {\left (e x\right )^{m + 1} A c^{3}}{e {\left (m + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x)^m*(B*x^2+A)*(d*x^2+c)^3,x, algorithm="maxima")
[Out]
B*d^3*e^m*x^9*x^m/(m + 9) + 3*B*c*d^2*e^m*x^7*x^m/(m + 7) + A*d^3*e^m*x^7*x^m/(m + 7) + 3*B*c^2*d*e^m*x^5*x^m/
(m + 5) + 3*A*c*d^2*e^m*x^5*x^m/(m + 5) + B*c^3*e^m*x^3*x^m/(m + 3) + 3*A*c^2*d*e^m*x^3*x^m/(m + 3) + (e*x)^(m
+ 1)*A*c^3/(e*(m + 1))
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mupad [B] time = 1.13, size = 280, normalized size = 2.31 \[ {\left (e\,x\right )}^m\,\left (\frac {A\,c^3\,x\,\left (m^4+24\,m^3+206\,m^2+744\,m+945\right )}{m^5+25\,m^4+230\,m^3+950\,m^2+1689\,m+945}+\frac {B\,d^3\,x^9\,\left (m^4+16\,m^3+86\,m^2+176\,m+105\right )}{m^5+25\,m^4+230\,m^3+950\,m^2+1689\,m+945}+\frac {c^2\,x^3\,\left (3\,A\,d+B\,c\right )\,\left (m^4+22\,m^3+164\,m^2+458\,m+315\right )}{m^5+25\,m^4+230\,m^3+950\,m^2+1689\,m+945}+\frac {d^2\,x^7\,\left (A\,d+3\,B\,c\right )\,\left (m^4+18\,m^3+104\,m^2+222\,m+135\right )}{m^5+25\,m^4+230\,m^3+950\,m^2+1689\,m+945}+\frac {3\,c\,d\,x^5\,\left (A\,d+B\,c\right )\,\left (m^4+20\,m^3+130\,m^2+300\,m+189\right )}{m^5+25\,m^4+230\,m^3+950\,m^2+1689\,m+945}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((A + B*x^2)*(e*x)^m*(c + d*x^2)^3,x)
[Out]
(e*x)^m*((A*c^3*x*(744*m + 206*m^2 + 24*m^3 + m^4 + 945))/(1689*m + 950*m^2 + 230*m^3 + 25*m^4 + m^5 + 945) +
(B*d^3*x^9*(176*m + 86*m^2 + 16*m^3 + m^4 + 105))/(1689*m + 950*m^2 + 230*m^3 + 25*m^4 + m^5 + 945) + (c^2*x^3
*(3*A*d + B*c)*(458*m + 164*m^2 + 22*m^3 + m^4 + 315))/(1689*m + 950*m^2 + 230*m^3 + 25*m^4 + m^5 + 945) + (d^
2*x^7*(A*d + 3*B*c)*(222*m + 104*m^2 + 18*m^3 + m^4 + 135))/(1689*m + 950*m^2 + 230*m^3 + 25*m^4 + m^5 + 945)
+ (3*c*d*x^5*(A*d + B*c)*(300*m + 130*m^2 + 20*m^3 + m^4 + 189))/(1689*m + 950*m^2 + 230*m^3 + 25*m^4 + m^5 +
945))
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sympy [A] time = 8.51, size = 2220, normalized size = 18.35 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x)**m*(B*x**2+A)*(d*x**2+c)**3,x)
[Out]
Piecewise(((-A*c**3/(8*x**8) - A*c**2*d/(2*x**6) - 3*A*c*d**2/(4*x**4) - A*d**3/(2*x**2) - B*c**3/(6*x**6) - 3
*B*c**2*d/(4*x**4) - 3*B*c*d**2/(2*x**2) + B*d**3*log(x))/e**9, Eq(m, -9)), ((-A*c**3/(6*x**6) - 3*A*c**2*d/(4
*x**4) - 3*A*c*d**2/(2*x**2) + A*d**3*log(x) - B*c**3/(4*x**4) - 3*B*c**2*d/(2*x**2) + 3*B*c*d**2*log(x) + B*d
**3*x**2/2)/e**7, Eq(m, -7)), ((-A*c**3/(4*x**4) - 3*A*c**2*d/(2*x**2) + 3*A*c*d**2*log(x) + A*d**3*x**2/2 - B
*c**3/(2*x**2) + 3*B*c**2*d*log(x) + 3*B*c*d**2*x**2/2 + B*d**3*x**4/4)/e**5, Eq(m, -5)), ((-A*c**3/(2*x**2) +
3*A*c**2*d*log(x) + 3*A*c*d**2*x**2/2 + A*d**3*x**4/4 + B*c**3*log(x) + 3*B*c**2*d*x**2/2 + 3*B*c*d**2*x**4/4
+ B*d**3*x**6/6)/e**3, Eq(m, -3)), ((A*c**3*log(x) + 3*A*c**2*d*x**2/2 + 3*A*c*d**2*x**4/4 + A*d**3*x**6/6 +
B*c**3*x**2/2 + 3*B*c**2*d*x**4/4 + B*c*d**2*x**6/2 + B*d**3*x**8/8)/e, Eq(m, -1)), (A*c**3*e**m*m**4*x*x**m/(
m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 24*A*c**3*e**m*m**3*x*x**m/(m**5 + 25*m**4 + 230*m**3 +
950*m**2 + 1689*m + 945) + 206*A*c**3*e**m*m**2*x*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945)
+ 744*A*c**3*e**m*m*x*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 945*A*c**3*e**m*x*x**m/(m**
5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 3*A*c**2*d*e**m*m**4*x**3*x**m/(m**5 + 25*m**4 + 230*m**3
+ 950*m**2 + 1689*m + 945) + 66*A*c**2*d*e**m*m**3*x**3*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m +
945) + 492*A*c**2*d*e**m*m**2*x**3*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 1374*A*c**2*d*
e**m*m*x**3*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 945*A*c**2*d*e**m*x**3*x**m/(m**5 + 2
5*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 3*A*c*d**2*e**m*m**4*x**5*x**m/(m**5 + 25*m**4 + 230*m**3 + 950
*m**2 + 1689*m + 945) + 60*A*c*d**2*e**m*m**3*x**5*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945)
+ 390*A*c*d**2*e**m*m**2*x**5*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 900*A*c*d**2*e**m*m
*x**5*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 567*A*c*d**2*e**m*x**5*x**m/(m**5 + 25*m**4
+ 230*m**3 + 950*m**2 + 1689*m + 945) + A*d**3*e**m*m**4*x**7*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 16
89*m + 945) + 18*A*d**3*e**m*m**3*x**7*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 104*A*d**3
*e**m*m**2*x**7*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 222*A*d**3*e**m*m*x**7*x**m/(m**5
+ 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 135*A*d**3*e**m*x**7*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*
m**2 + 1689*m + 945) + B*c**3*e**m*m**4*x**3*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 22*B
*c**3*e**m*m**3*x**3*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 164*B*c**3*e**m*m**2*x**3*x*
*m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 458*B*c**3*e**m*m*x**3*x**m/(m**5 + 25*m**4 + 230*m
**3 + 950*m**2 + 1689*m + 945) + 315*B*c**3*e**m*x**3*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 94
5) + 3*B*c**2*d*e**m*m**4*x**5*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 60*B*c**2*d*e**m*m
**3*x**5*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 390*B*c**2*d*e**m*m**2*x**5*x**m/(m**5 +
25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 900*B*c**2*d*e**m*m*x**5*x**m/(m**5 + 25*m**4 + 230*m**3 + 95
0*m**2 + 1689*m + 945) + 567*B*c**2*d*e**m*x**5*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 3
*B*c*d**2*e**m*m**4*x**7*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 54*B*c*d**2*e**m*m**3*x*
*7*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 312*B*c*d**2*e**m*m**2*x**7*x**m/(m**5 + 25*m*
*4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 666*B*c*d**2*e**m*m*x**7*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2
+ 1689*m + 945) + 405*B*c*d**2*e**m*x**7*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + B*d**3*
e**m*m**4*x**9*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 16*B*d**3*e**m*m**3*x**9*x**m/(m**
5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 86*B*d**3*e**m*m**2*x**9*x**m/(m**5 + 25*m**4 + 230*m**3 +
950*m**2 + 1689*m + 945) + 176*B*d**3*e**m*m*x**9*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945)
+ 105*B*d**3*e**m*x**9*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945), True))
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