\(\int (d+e x)^m (a+c x^2)^3 \, dx\) [721]

Optimal result

Integrand size = 17, antiderivative size = 223 \[ \int (d+e x)^m \left (a+c x^2\right )^3 \, dx=\frac {\left (c d^2+a e^2\right )^3 (d+e x)^{1+m}}{e^7 (1+m)}-\frac {6 c d \left (c d^2+a e^2\right )^2 (d+e x)^{2+m}}{e^7 (2+m)}+\frac {3 c \left (c d^2+a e^2\right ) \left (5 c d^2+a e^2\right ) (d+e x)^{3+m}}{e^7 (3+m)}-\frac {4 c^2 d \left (5 c d^2+3 a e^2\right ) (d+e x)^{4+m}}{e^7 (4+m)}+\frac {3 c^2 \left (5 c d^2+a e^2\right ) (d+e x)^{5+m}}{e^7 (5+m)}-\frac {6 c^3 d (d+e x)^{6+m}}{e^7 (6+m)}+\frac {c^3 (d+e x)^{7+m}}{e^7 (7+m)} \]

[Out]

(a*e^2+c*d^2)^3*(e*x+d)^(1+m)/e^7/(1+m)-6*c*d*(a*e^2+c*d^2)^2*(e*x+d)^(2+m)/e^7/(2+m)+3*c*(a*e^2+c*d^2)*(a*e^2
+5*c*d^2)*(e*x+d)^(3+m)/e^7/(3+m)-4*c^2*d*(3*a*e^2+5*c*d^2)*(e*x+d)^(4+m)/e^7/(4+m)+3*c^2*(a*e^2+5*c*d^2)*(e*x
+d)^(5+m)/e^7/(5+m)-6*c^3*d*(e*x+d)^(6+m)/e^7/(6+m)+c^3*(e*x+d)^(7+m)/e^7/(7+m)

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {711} \[ \int (d+e x)^m \left (a+c x^2\right )^3 \, dx=-\frac {4 c^2 d \left (3 a e^2+5 c d^2\right ) (d+e x)^{m+4}}{e^7 (m+4)}+\frac {3 c^2 \left (a e^2+5 c d^2\right ) (d+e x)^{m+5}}{e^7 (m+5)}+\frac {\left (a e^2+c d^2\right )^3 (d+e x)^{m+1}}{e^7 (m+1)}-\frac {6 c d \left (a e^2+c d^2\right )^2 (d+e x)^{m+2}}{e^7 (m+2)}+\frac {3 c \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right ) (d+e x)^{m+3}}{e^7 (m+3)}-\frac {6 c^3 d (d+e x)^{m+6}}{e^7 (m+6)}+\frac {c^3 (d+e x)^{m+7}}{e^7 (m+7)} \]

[In]

Int[(d + e*x)^m*(a + c*x^2)^3,x]

[Out]

((c*d^2 + a*e^2)^3*(d + e*x)^(1 + m))/(e^7*(1 + m)) - (6*c*d*(c*d^2 + a*e^2)^2*(d + e*x)^(2 + m))/(e^7*(2 + m)
) + (3*c*(c*d^2 + a*e^2)*(5*c*d^2 + a*e^2)*(d + e*x)^(3 + m))/(e^7*(3 + m)) - (4*c^2*d*(5*c*d^2 + 3*a*e^2)*(d
+ e*x)^(4 + m))/(e^7*(4 + m)) + (3*c^2*(5*c*d^2 + a*e^2)*(d + e*x)^(5 + m))/(e^7*(5 + m)) - (6*c^3*d*(d + e*x)
^(6 + m))/(e^7*(6 + m)) + (c^3*(d + e*x)^(7 + m))/(e^7*(7 + m))

Rule 711

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + c*
x^2)^p, x], x] /; FreeQ[{a, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[p, 0]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\left (c d^2+a e^2\right )^3 (d+e x)^m}{e^6}-\frac {6 c d \left (c d^2+a e^2\right )^2 (d+e x)^{1+m}}{e^6}+\frac {3 c \left (c d^2+a e^2\right ) \left (5 c d^2+a e^2\right ) (d+e x)^{2+m}}{e^6}-\frac {4 c^2 d \left (5 c d^2+3 a e^2\right ) (d+e x)^{3+m}}{e^6}+\frac {3 c^2 \left (5 c d^2+a e^2\right ) (d+e x)^{4+m}}{e^6}-\frac {6 c^3 d (d+e x)^{5+m}}{e^6}+\frac {c^3 (d+e x)^{6+m}}{e^6}\right ) \, dx \\ & = \frac {\left (c d^2+a e^2\right )^3 (d+e x)^{1+m}}{e^7 (1+m)}-\frac {6 c d \left (c d^2+a e^2\right )^2 (d+e x)^{2+m}}{e^7 (2+m)}+\frac {3 c \left (c d^2+a e^2\right ) \left (5 c d^2+a e^2\right ) (d+e x)^{3+m}}{e^7 (3+m)}-\frac {4 c^2 d \left (5 c d^2+3 a e^2\right ) (d+e x)^{4+m}}{e^7 (4+m)}+\frac {3 c^2 \left (5 c d^2+a e^2\right ) (d+e x)^{5+m}}{e^7 (5+m)}-\frac {6 c^3 d (d+e x)^{6+m}}{e^7 (6+m)}+\frac {c^3 (d+e x)^{7+m}}{e^7 (7+m)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.78 (sec) , antiderivative size = 379, normalized size of antiderivative = 1.70 \[ \int (d+e x)^m \left (a+c x^2\right )^3 \, dx=\frac {(d+e x)^{1+m} \left (\left (a+c x^2\right )^3+\frac {6 \left (\left (c d^2+a e^2\right ) (6+m) \left (e^4 (1+m) (2+m) (3+m) (4+m) \left (a+c x^2\right )^2+4 \left (c d^2+a e^2\right ) (4+m) \left (a e^2 \left (6+5 m+m^2\right )+c \left (2 d^2-2 d e (1+m) x+e^2 \left (2+3 m+m^2\right ) x^2\right )\right )-4 c d (1+m) (d+e x) \left (a e^2 \left (12+7 m+m^2\right )+c \left (2 d^2-2 d e (2+m) x+e^2 \left (6+5 m+m^2\right ) x^2\right )\right )\right )-c d (1+m) (d+e x) \left (e^4 (2+m) (3+m) (4+m) (5+m) \left (a+c x^2\right )^2+4 \left (c d^2+a e^2\right ) (5+m) \left (a e^2 \left (12+7 m+m^2\right )+c \left (2 d^2-2 d e (2+m) x+e^2 \left (6+5 m+m^2\right ) x^2\right )\right )-4 c d (2+m) (d+e x) \left (a e^2 \left (20+9 m+m^2\right )+c \left (2 d^2-2 d e (3+m) x+e^2 \left (12+7 m+m^2\right ) x^2\right )\right )\right )\right )}{e^6 (1+m) (2+m) (3+m) (4+m) (5+m) (6+m)}\right )}{e (7+m)} \]

[In]

Integrate[(d + e*x)^m*(a + c*x^2)^3,x]

[Out]

((d + e*x)^(1 + m)*((a + c*x^2)^3 + (6*((c*d^2 + a*e^2)*(6 + m)*(e^4*(1 + m)*(2 + m)*(3 + m)*(4 + m)*(a + c*x^
2)^2 + 4*(c*d^2 + a*e^2)*(4 + m)*(a*e^2*(6 + 5*m + m^2) + c*(2*d^2 - 2*d*e*(1 + m)*x + e^2*(2 + 3*m + m^2)*x^2
)) - 4*c*d*(1 + m)*(d + e*x)*(a*e^2*(12 + 7*m + m^2) + c*(2*d^2 - 2*d*e*(2 + m)*x + e^2*(6 + 5*m + m^2)*x^2)))
- c*d*(1 + m)*(d + e*x)*(e^4*(2 + m)*(3 + m)*(4 + m)*(5 + m)*(a + c*x^2)^2 + 4*(c*d^2 + a*e^2)*(5 + m)*(a*e^2
*(12 + 7*m + m^2) + c*(2*d^2 - 2*d*e*(2 + m)*x + e^2*(6 + 5*m + m^2)*x^2)) - 4*c*d*(2 + m)*(d + e*x)*(a*e^2*(2
0 + 9*m + m^2) + c*(2*d^2 - 2*d*e*(3 + m)*x + e^2*(12 + 7*m + m^2)*x^2)))))/(e^6*(1 + m)*(2 + m)*(3 + m)*(4 +
m)*(5 + m)*(6 + m))))/(e*(7 + m))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(953\) vs. \(2(223)=446\).

Time = 2.25 (sec) , antiderivative size = 954, normalized size of antiderivative = 4.28

method	result	size
norman	\(\frac {c^{3} x^{7} {\mathrm e}^{m \ln \left (e x +d \right )}}{7+m}+\frac {d \left (a^{3} e^{6} m^{6}+27 a^{3} e^{6} m^{5}+295 a^{3} e^{6} m^{4}+6 a^{2} c \,d^{2} e^{4} m^{4}+1665 a^{3} e^{6} m^{3}+132 a^{2} c \,d^{2} e^{4} m^{3}+5104 a^{3} e^{6} m^{2}+1074 a^{2} c \,d^{2} e^{4} m^{2}+72 a \,c^{2} d^{4} e^{2} m^{2}+8028 a^{3} e^{6} m +3828 a^{2} c \,d^{2} e^{4} m +936 a \,c^{2} d^{4} e^{2} m +5040 e^{6} a^{3}+5040 d^{2} e^{4} a^{2} c +3024 d^{4} e^{2} c^{2} a +720 c^{3} d^{6}\right ) {\mathrm e}^{m \ln \left (e x +d \right )}}{e^{7} \left (m^{7}+28 m^{6}+322 m^{5}+1960 m^{4}+6769 m^{3}+13132 m^{2}+13068 m +5040\right )}+\frac {\left (a^{3} e^{6} m^{6}+27 a^{3} e^{6} m^{5}-6 a^{2} c \,d^{2} e^{4} m^{5}+295 a^{3} e^{6} m^{4}-132 a^{2} c \,d^{2} e^{4} m^{4}+1665 a^{3} e^{6} m^{3}-1074 a^{2} c \,d^{2} e^{4} m^{3}-72 a \,c^{2} d^{4} e^{2} m^{3}+5104 a^{3} e^{6} m^{2}-3828 a^{2} c \,d^{2} e^{4} m^{2}-936 a \,c^{2} d^{4} e^{2} m^{2}+8028 a^{3} e^{6} m -5040 a^{2} c \,d^{2} e^{4} m -3024 a \,c^{2} d^{4} e^{2} m -720 c^{3} d^{6} m +5040 e^{6} a^{3}\right ) x \,{\mathrm e}^{m \ln \left (e x +d \right )}}{e^{6} \left (m^{7}+28 m^{6}+322 m^{5}+1960 m^{4}+6769 m^{3}+13132 m^{2}+13068 m +5040\right )}+\frac {c^{3} d m \,x^{6} {\mathrm e}^{m \ln \left (e x +d \right )}}{e \left (m^{2}+13 m +42\right )}+\frac {3 \left (a \,e^{2} m^{2}+13 a \,e^{2} m -2 c \,d^{2} m +42 e^{2} a \right ) c^{2} x^{5} {\mathrm e}^{m \ln \left (e x +d \right )}}{e^{2} \left (m^{3}+18 m^{2}+107 m +210\right )}+\frac {3 \left (a^{2} e^{4} m^{4}+22 a^{2} e^{4} m^{3}-4 a c \,d^{2} e^{2} m^{3}+179 a^{2} e^{4} m^{2}-52 a c \,d^{2} e^{2} m^{2}+638 a^{2} e^{4} m -168 a c \,d^{2} e^{2} m -40 c^{2} d^{4} m +840 a^{2} e^{4}\right ) c \,x^{3} {\mathrm e}^{m \ln \left (e x +d \right )}}{e^{4} \left (m^{5}+25 m^{4}+245 m^{3}+1175 m^{2}+2754 m +2520\right )}+\frac {3 \left (a \,e^{2} m^{2}+13 a \,e^{2} m +42 e^{2} a +10 c \,d^{2}\right ) c^{2} d m \,x^{4} {\mathrm e}^{m \ln \left (e x +d \right )}}{e^{3} \left (m^{4}+22 m^{3}+179 m^{2}+638 m +840\right )}+\frac {3 \left (a^{2} e^{4} m^{4}+22 a^{2} e^{4} m^{3}+179 a^{2} e^{4} m^{2}+12 a c \,d^{2} e^{2} m^{2}+638 a^{2} e^{4} m +156 a c \,d^{2} e^{2} m +840 a^{2} e^{4}+504 a c \,d^{2} e^{2}+120 c^{2} d^{4}\right ) c d m \,x^{2} {\mathrm e}^{m \ln \left (e x +d \right )}}{e^{5} \left (m^{6}+27 m^{5}+295 m^{4}+1665 m^{3}+5104 m^{2}+8028 m +5040\right )}\)	\(954\)
gosper	\(\text {Expression too large to display}\)	\(1140\)
risch	\(\text {Expression too large to display}\)	\(1412\)
parallelrisch	\(\text {Expression too large to display}\)	\(2063\)

[In]

int((e*x+d)^m*(c*x^2+a)^3,x,method=_RETURNVERBOSE)

[Out]

c^3/(7+m)*x^7*exp(m*ln(e*x+d))+d*(a^3*e^6*m^6+27*a^3*e^6*m^5+295*a^3*e^6*m^4+6*a^2*c*d^2*e^4*m^4+1665*a^3*e^6*
m^3+132*a^2*c*d^2*e^4*m^3+5104*a^3*e^6*m^2+1074*a^2*c*d^2*e^4*m^2+72*a*c^2*d^4*e^2*m^2+8028*a^3*e^6*m+3828*a^2
*c*d^2*e^4*m+936*a*c^2*d^4*e^2*m+5040*a^3*e^6+5040*a^2*c*d^2*e^4+3024*a*c^2*d^4*e^2+720*c^3*d^6)/e^7/(m^7+28*m
^6+322*m^5+1960*m^4+6769*m^3+13132*m^2+13068*m+5040)*exp(m*ln(e*x+d))+(a^3*e^6*m^6+27*a^3*e^6*m^5-6*a^2*c*d^2*
e^4*m^5+295*a^3*e^6*m^4-132*a^2*c*d^2*e^4*m^4+1665*a^3*e^6*m^3-1074*a^2*c*d^2*e^4*m^3-72*a*c^2*d^4*e^2*m^3+510
4*a^3*e^6*m^2-3828*a^2*c*d^2*e^4*m^2-936*a*c^2*d^4*e^2*m^2+8028*a^3*e^6*m-5040*a^2*c*d^2*e^4*m-3024*a*c^2*d^4*
e^2*m-720*c^3*d^6*m+5040*a^3*e^6)/e^6/(m^7+28*m^6+322*m^5+1960*m^4+6769*m^3+13132*m^2+13068*m+5040)*x*exp(m*ln
(e*x+d))+c^3*d*m/e/(m^2+13*m+42)*x^6*exp(m*ln(e*x+d))+3*(a*e^2*m^2+13*a*e^2*m-2*c*d^2*m+42*a*e^2)/e^2*c^2/(m^3
+18*m^2+107*m+210)*x^5*exp(m*ln(e*x+d))+3*(a^2*e^4*m^4+22*a^2*e^4*m^3-4*a*c*d^2*e^2*m^3+179*a^2*e^4*m^2-52*a*c
*d^2*e^2*m^2+638*a^2*e^4*m-168*a*c*d^2*e^2*m-40*c^2*d^4*m+840*a^2*e^4)*c/e^4/(m^5+25*m^4+245*m^3+1175*m^2+2754
*m+2520)*x^3*exp(m*ln(e*x+d))+3*(a*e^2*m^2+13*a*e^2*m+42*a*e^2+10*c*d^2)*c^2*d/e^3*m/(m^4+22*m^3+179*m^2+638*m
+840)*x^4*exp(m*ln(e*x+d))+3*(a^2*e^4*m^4+22*a^2*e^4*m^3+179*a^2*e^4*m^2+12*a*c*d^2*e^2*m^2+638*a^2*e^4*m+156*
a*c*d^2*e^2*m+840*a^2*e^4+504*a*c*d^2*e^2+120*c^2*d^4)*c*d/e^5*m/(m^6+27*m^5+295*m^4+1665*m^3+5104*m^2+8028*m+
5040)*x^2*exp(m*ln(e*x+d))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1250 vs. \(2 (223) = 446\).

Time = 0.34 (sec) , antiderivative size = 1250, normalized size of antiderivative = 5.61 \[ \int (d+e x)^m \left (a+c x^2\right )^3 \, dx=\text {Too large to display} \]

[In]

integrate((e*x+d)^m*(c*x^2+a)^3,x, algorithm="fricas")

[Out]

(a^3*d*e^6*m^6 + 27*a^3*d*e^6*m^5 + 720*c^3*d^7 + 3024*a*c^2*d^5*e^2 + 5040*a^2*c*d^3*e^4 + 5040*a^3*d*e^6 + (
c^3*e^7*m^6 + 21*c^3*e^7*m^5 + 175*c^3*e^7*m^4 + 735*c^3*e^7*m^3 + 1624*c^3*e^7*m^2 + 1764*c^3*e^7*m + 720*c^3
*e^7)*x^7 + (c^3*d*e^6*m^6 + 15*c^3*d*e^6*m^5 + 85*c^3*d*e^6*m^4 + 225*c^3*d*e^6*m^3 + 274*c^3*d*e^6*m^2 + 120
*c^3*d*e^6*m)*x^6 + 3*(a*c^2*e^7*m^6 + 1008*a*c^2*e^7 - (2*c^3*d^2*e^5 - 23*a*c^2*e^7)*m^5 - (20*c^3*d^2*e^5 -
207*a*c^2*e^7)*m^4 - 5*(14*c^3*d^2*e^5 - 185*a*c^2*e^7)*m^3 - 4*(25*c^3*d^2*e^5 - 536*a*c^2*e^7)*m^2 - 12*(4*
c^3*d^2*e^5 - 201*a*c^2*e^7)*m)*x^5 + (6*a^2*c*d^3*e^4 + 295*a^3*d*e^6)*m^4 + 3*(a*c^2*d*e^6*m^6 + 19*a*c^2*d*
e^6*m^5 + (10*c^3*d^3*e^4 + 131*a*c^2*d*e^6)*m^4 + (60*c^3*d^3*e^4 + 401*a*c^2*d*e^6)*m^3 + 10*(11*c^3*d^3*e^4
+ 54*a*c^2*d*e^6)*m^2 + 12*(5*c^3*d^3*e^4 + 21*a*c^2*d*e^6)*m)*x^4 + 3*(44*a^2*c*d^3*e^4 + 555*a^3*d*e^6)*m^3
+ 3*(a^2*c*e^7*m^6 + 1680*a^2*c*e^7 - (4*a*c^2*d^2*e^5 - 25*a^2*c*e^7)*m^5 - (64*a*c^2*d^2*e^5 - 247*a^2*c*e^
7)*m^4 - (40*c^3*d^4*e^3 + 332*a*c^2*d^2*e^5 - 1219*a^2*c*e^7)*m^3 - 8*(15*c^3*d^4*e^3 + 76*a*c^2*d^2*e^5 - 38
9*a^2*c*e^7)*m^2 - 4*(20*c^3*d^4*e^3 + 84*a*c^2*d^2*e^5 - 949*a^2*c*e^7)*m)*x^3 + 2*(36*a*c^2*d^5*e^2 + 537*a^
2*c*d^3*e^4 + 2552*a^3*d*e^6)*m^2 + 3*(a^2*c*d*e^6*m^6 + 23*a^2*c*d*e^6*m^5 + 3*(4*a*c^2*d^3*e^4 + 67*a^2*c*d*
e^6)*m^4 + (168*a*c^2*d^3*e^4 + 817*a^2*c*d*e^6)*m^3 + 2*(60*c^3*d^5*e^2 + 330*a*c^2*d^3*e^4 + 739*a^2*c*d*e^6
)*m^2 + 24*(5*c^3*d^5*e^2 + 21*a*c^2*d^3*e^4 + 35*a^2*c*d*e^6)*m)*x^2 + 12*(78*a*c^2*d^5*e^2 + 319*a^2*c*d^3*e
^4 + 669*a^3*d*e^6)*m + (a^3*e^7*m^6 + 5040*a^3*e^7 - 3*(2*a^2*c*d^2*e^5 - 9*a^3*e^7)*m^5 - (132*a^2*c*d^2*e^5
- 295*a^3*e^7)*m^4 - 3*(24*a*c^2*d^4*e^3 + 358*a^2*c*d^2*e^5 - 555*a^3*e^7)*m^3 - 4*(234*a*c^2*d^4*e^3 + 957*
a^2*c*d^2*e^5 - 1276*a^3*e^7)*m^2 - 36*(20*c^3*d^6*e + 84*a*c^2*d^4*e^3 + 140*a^2*c*d^2*e^5 - 223*a^3*e^7)*m)*
x)*(e*x + d)^m/(e^7*m^7 + 28*e^7*m^6 + 322*e^7*m^5 + 1960*e^7*m^4 + 6769*e^7*m^3 + 13132*e^7*m^2 + 13068*e^7*m
+ 5040*e^7)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 15990 vs. \(2 (207) = 414\).

Time = 4.03 (sec) , antiderivative size = 15990, normalized size of antiderivative = 71.70 \[ \int (d+e x)^m \left (a+c x^2\right )^3 \, dx=\text {Too large to display} \]

[In]

integrate((e*x+d)**m*(c*x**2+a)**3,x)

[Out]

Piecewise((d**m*(a**3*x + a**2*c*x**3 + 3*a*c**2*x**5/5 + c**3*x**7/7), Eq(e, 0)), (-10*a**3*e**6/(60*d**6*e**
7 + 360*d**5*e**8*x + 900*d**4*e**9*x**2 + 1200*d**3*e**10*x**3 + 900*d**2*e**11*x**4 + 360*d*e**12*x**5 + 60*
e**13*x**6) - 3*a**2*c*d**2*e**4/(60*d**6*e**7 + 360*d**5*e**8*x + 900*d**4*e**9*x**2 + 1200*d**3*e**10*x**3 +
900*d**2*e**11*x**4 + 360*d*e**12*x**5 + 60*e**13*x**6) - 18*a**2*c*d*e**5*x/(60*d**6*e**7 + 360*d**5*e**8*x
+ 900*d**4*e**9*x**2 + 1200*d**3*e**10*x**3 + 900*d**2*e**11*x**4 + 360*d*e**12*x**5 + 60*e**13*x**6) - 45*a**
2*c*e**6*x**2/(60*d**6*e**7 + 360*d**5*e**8*x + 900*d**4*e**9*x**2 + 1200*d**3*e**10*x**3 + 900*d**2*e**11*x**
4 + 360*d*e**12*x**5 + 60*e**13*x**6) - 6*a*c**2*d**4*e**2/(60*d**6*e**7 + 360*d**5*e**8*x + 900*d**4*e**9*x**
2 + 1200*d**3*e**10*x**3 + 900*d**2*e**11*x**4 + 360*d*e**12*x**5 + 60*e**13*x**6) - 36*a*c**2*d**3*e**3*x/(60
*d**6*e**7 + 360*d**5*e**8*x + 900*d**4*e**9*x**2 + 1200*d**3*e**10*x**3 + 900*d**2*e**11*x**4 + 360*d*e**12*x
**5 + 60*e**13*x**6) - 90*a*c**2*d**2*e**4*x**2/(60*d**6*e**7 + 360*d**5*e**8*x + 900*d**4*e**9*x**2 + 1200*d*
*3*e**10*x**3 + 900*d**2*e**11*x**4 + 360*d*e**12*x**5 + 60*e**13*x**6) - 120*a*c**2*d*e**5*x**3/(60*d**6*e**7
+ 360*d**5*e**8*x + 900*d**4*e**9*x**2 + 1200*d**3*e**10*x**3 + 900*d**2*e**11*x**4 + 360*d*e**12*x**5 + 60*e
**13*x**6) - 90*a*c**2*e**6*x**4/(60*d**6*e**7 + 360*d**5*e**8*x + 900*d**4*e**9*x**2 + 1200*d**3*e**10*x**3 +
900*d**2*e**11*x**4 + 360*d*e**12*x**5 + 60*e**13*x**6) + 60*c**3*d**6*log(d/e + x)/(60*d**6*e**7 + 360*d**5*
e**8*x + 900*d**4*e**9*x**2 + 1200*d**3*e**10*x**3 + 900*d**2*e**11*x**4 + 360*d*e**12*x**5 + 60*e**13*x**6) +
147*c**3*d**6/(60*d**6*e**7 + 360*d**5*e**8*x + 900*d**4*e**9*x**2 + 1200*d**3*e**10*x**3 + 900*d**2*e**11*x*
*4 + 360*d*e**12*x**5 + 60*e**13*x**6) + 360*c**3*d**5*e*x*log(d/e + x)/(60*d**6*e**7 + 360*d**5*e**8*x + 900*
d**4*e**9*x**2 + 1200*d**3*e**10*x**3 + 900*d**2*e**11*x**4 + 360*d*e**12*x**5 + 60*e**13*x**6) + 822*c**3*d**
5*e*x/(60*d**6*e**7 + 360*d**5*e**8*x + 900*d**4*e**9*x**2 + 1200*d**3*e**10*x**3 + 900*d**2*e**11*x**4 + 360*
d*e**12*x**5 + 60*e**13*x**6) + 900*c**3*d**4*e**2*x**2*log(d/e + x)/(60*d**6*e**7 + 360*d**5*e**8*x + 900*d**
4*e**9*x**2 + 1200*d**3*e**10*x**3 + 900*d**2*e**11*x**4 + 360*d*e**12*x**5 + 60*e**13*x**6) + 1875*c**3*d**4*
e**2*x**2/(60*d**6*e**7 + 360*d**5*e**8*x + 900*d**4*e**9*x**2 + 1200*d**3*e**10*x**3 + 900*d**2*e**11*x**4 +
360*d*e**12*x**5 + 60*e**13*x**6) + 1200*c**3*d**3*e**3*x**3*log(d/e + x)/(60*d**6*e**7 + 360*d**5*e**8*x + 90
0*d**4*e**9*x**2 + 1200*d**3*e**10*x**3 + 900*d**2*e**11*x**4 + 360*d*e**12*x**5 + 60*e**13*x**6) + 2200*c**3*
d**3*e**3*x**3/(60*d**6*e**7 + 360*d**5*e**8*x + 900*d**4*e**9*x**2 + 1200*d**3*e**10*x**3 + 900*d**2*e**11*x*
*4 + 360*d*e**12*x**5 + 60*e**13*x**6) + 900*c**3*d**2*e**4*x**4*log(d/e + x)/(60*d**6*e**7 + 360*d**5*e**8*x
+ 900*d**4*e**9*x**2 + 1200*d**3*e**10*x**3 + 900*d**2*e**11*x**4 + 360*d*e**12*x**5 + 60*e**13*x**6) + 1350*c
**3*d**2*e**4*x**4/(60*d**6*e**7 + 360*d**5*e**8*x + 900*d**4*e**9*x**2 + 1200*d**3*e**10*x**3 + 900*d**2*e**1
1*x**4 + 360*d*e**12*x**5 + 60*e**13*x**6) + 360*c**3*d*e**5*x**5*log(d/e + x)/(60*d**6*e**7 + 360*d**5*e**8*x
+ 900*d**4*e**9*x**2 + 1200*d**3*e**10*x**3 + 900*d**2*e**11*x**4 + 360*d*e**12*x**5 + 60*e**13*x**6) + 360*c
**3*d*e**5*x**5/(60*d**6*e**7 + 360*d**5*e**8*x + 900*d**4*e**9*x**2 + 1200*d**3*e**10*x**3 + 900*d**2*e**11*x
**4 + 360*d*e**12*x**5 + 60*e**13*x**6) + 60*c**3*e**6*x**6*log(d/e + x)/(60*d**6*e**7 + 360*d**5*e**8*x + 900
*d**4*e**9*x**2 + 1200*d**3*e**10*x**3 + 900*d**2*e**11*x**4 + 360*d*e**12*x**5 + 60*e**13*x**6), Eq(m, -7)),
(-2*a**3*e**6/(10*d**5*e**7 + 50*d**4*e**8*x + 100*d**3*e**9*x**2 + 100*d**2*e**10*x**3 + 50*d*e**11*x**4 + 10
*e**12*x**5) - a**2*c*d**2*e**4/(10*d**5*e**7 + 50*d**4*e**8*x + 100*d**3*e**9*x**2 + 100*d**2*e**10*x**3 + 50
*d*e**11*x**4 + 10*e**12*x**5) - 5*a**2*c*d*e**5*x/(10*d**5*e**7 + 50*d**4*e**8*x + 100*d**3*e**9*x**2 + 100*d
**2*e**10*x**3 + 50*d*e**11*x**4 + 10*e**12*x**5) - 10*a**2*c*e**6*x**2/(10*d**5*e**7 + 50*d**4*e**8*x + 100*d
**3*e**9*x**2 + 100*d**2*e**10*x**3 + 50*d*e**11*x**4 + 10*e**12*x**5) - 6*a*c**2*d**4*e**2/(10*d**5*e**7 + 50
*d**4*e**8*x + 100*d**3*e**9*x**2 + 100*d**2*e**10*x**3 + 50*d*e**11*x**4 + 10*e**12*x**5) - 30*a*c**2*d**3*e*
*3*x/(10*d**5*e**7 + 50*d**4*e**8*x + 100*d**3*e**9*x**2 + 100*d**2*e**10*x**3 + 50*d*e**11*x**4 + 10*e**12*x*
*5) - 60*a*c**2*d**2*e**4*x**2/(10*d**5*e**7 + 50*d**4*e**8*x + 100*d**3*e**9*x**2 + 100*d**2*e**10*x**3 + 50*
d*e**11*x**4 + 10*e**12*x**5) - 60*a*c**2*d*e**5*x**3/(10*d**5*e**7 + 50*d**4*e**8*x + 100*d**3*e**9*x**2 + 10
0*d**2*e**10*x**3 + 50*d*e**11*x**4 + 10*e**12*x**5) - 30*a*c**2*e**6*x**4/(10*d**5*e**7 + 50*d**4*e**8*x + 10
0*d**3*e**9*x**2 + 100*d**2*e**10*x**3 + 50*d*e**11*x**4 + 10*e**12*x**5) - 60*c**3*d**6*log(d/e + x)/(10*d**5
*e**7 + 50*d**4*e**8*x + 100*d**3*e**9*x**2 + 100*d**2*e**10*x**3 + 50*d*e**11*x**4 + 10*e**12*x**5) - 137*c**
3*d**6/(10*d**5*e**7 + 50*d**4*e**8*x + 100*d**3*e**9*x**2 + 100*d**2*e**10*x**3 + 50*d*e**11*x**4 + 10*e**12*
x**5) - 300*c**3*d**5*e*x*log(d/e + x)/(10*d**5*e**7 + 50*d**4*e**8*x + 100*d**3*e**9*x**2 + 100*d**2*e**10*x*
*3 + 50*d*e**11*x**4 + 10*e**12*x**5) - 625*c**3*d**5*e*x/(10*d**5*e**7 + 50*d**4*e**8*x + 100*d**3*e**9*x**2
+ 100*d**2*e**10*x**3 + 50*d*e**11*x**4 + 10*e**12*x**5) - 600*c**3*d**4*e**2*x**2*log(d/e + x)/(10*d**5*e**7
+ 50*d**4*e**8*x + 100*d**3*e**9*x**2 + 100*d**2*e**10*x**3 + 50*d*e**11*x**4 + 10*e**12*x**5) - 1100*c**3*d**
4*e**2*x**2/(10*d**5*e**7 + 50*d**4*e**8*x + 100*d**3*e**9*x**2 + 100*d**2*e**10*x**3 + 50*d*e**11*x**4 + 10*e
**12*x**5) - 600*c**3*d**3*e**3*x**3*log(d/e + x)/(10*d**5*e**7 + 50*d**4*e**8*x + 100*d**3*e**9*x**2 + 100*d*
*2*e**10*x**3 + 50*d*e**11*x**4 + 10*e**12*x**5) - 900*c**3*d**3*e**3*x**3/(10*d**5*e**7 + 50*d**4*e**8*x + 10
0*d**3*e**9*x**2 + 100*d**2*e**10*x**3 + 50*d*e**11*x**4 + 10*e**12*x**5) - 300*c**3*d**2*e**4*x**4*log(d/e +
x)/(10*d**5*e**7 + 50*d**4*e**8*x + 100*d**3*e**9*x**2 + 100*d**2*e**10*x**3 + 50*d*e**11*x**4 + 10*e**12*x**5
) - 300*c**3*d**2*e**4*x**4/(10*d**5*e**7 + 50*d**4*e**8*x + 100*d**3*e**9*x**2 + 100*d**2*e**10*x**3 + 50*d*e
**11*x**4 + 10*e**12*x**5) - 60*c**3*d*e**5*x**5*log(d/e + x)/(10*d**5*e**7 + 50*d**4*e**8*x + 100*d**3*e**9*x
**2 + 100*d**2*e**10*x**3 + 50*d*e**11*x**4 + 10*e**12*x**5) + 10*c**3*e**6*x**6/(10*d**5*e**7 + 50*d**4*e**8*
x + 100*d**3*e**9*x**2 + 100*d**2*e**10*x**3 + 50*d*e**11*x**4 + 10*e**12*x**5), Eq(m, -6)), (-a**3*e**6/(4*d*
*4*e**7 + 16*d**3*e**8*x + 24*d**2*e**9*x**2 + 16*d*e**10*x**3 + 4*e**11*x**4) - a**2*c*d**2*e**4/(4*d**4*e**7
+ 16*d**3*e**8*x + 24*d**2*e**9*x**2 + 16*d*e**10*x**3 + 4*e**11*x**4) - 4*a**2*c*d*e**5*x/(4*d**4*e**7 + 16*
d**3*e**8*x + 24*d**2*e**9*x**2 + 16*d*e**10*x**3 + 4*e**11*x**4) - 6*a**2*c*e**6*x**2/(4*d**4*e**7 + 16*d**3*
e**8*x + 24*d**2*e**9*x**2 + 16*d*e**10*x**3 + 4*e**11*x**4) + 12*a*c**2*d**4*e**2*log(d/e + x)/(4*d**4*e**7 +
16*d**3*e**8*x + 24*d**2*e**9*x**2 + 16*d*e**10*x**3 + 4*e**11*x**4) + 25*a*c**2*d**4*e**2/(4*d**4*e**7 + 16*
d**3*e**8*x + 24*d**2*e**9*x**2 + 16*d*e**10*x**3 + 4*e**11*x**4) + 48*a*c**2*d**3*e**3*x*log(d/e + x)/(4*d**4
*e**7 + 16*d**3*e**8*x + 24*d**2*e**9*x**2 + 16*d*e**10*x**3 + 4*e**11*x**4) + 88*a*c**2*d**3*e**3*x/(4*d**4*e
**7 + 16*d**3*e**8*x + 24*d**2*e**9*x**2 + 16*d*e**10*x**3 + 4*e**11*x**4) + 72*a*c**2*d**2*e**4*x**2*log(d/e
+ x)/(4*d**4*e**7 + 16*d**3*e**8*x + 24*d**2*e**9*x**2 + 16*d*e**10*x**3 + 4*e**11*x**4) + 108*a*c**2*d**2*e**
4*x**2/(4*d**4*e**7 + 16*d**3*e**8*x + 24*d**2*e**9*x**2 + 16*d*e**10*x**3 + 4*e**11*x**4) + 48*a*c**2*d*e**5*
x**3*log(d/e + x)/(4*d**4*e**7 + 16*d**3*e**8*x + 24*d**2*e**9*x**2 + 16*d*e**10*x**3 + 4*e**11*x**4) + 48*a*c
**2*d*e**5*x**3/(4*d**4*e**7 + 16*d**3*e**8*x + 24*d**2*e**9*x**2 + 16*d*e**10*x**3 + 4*e**11*x**4) + 12*a*c**
2*e**6*x**4*log(d/e + x)/(4*d**4*e**7 + 16*d**3*e**8*x + 24*d**2*e**9*x**2 + 16*d*e**10*x**3 + 4*e**11*x**4) +
60*c**3*d**6*log(d/e + x)/(4*d**4*e**7 + 16*d**3*e**8*x + 24*d**2*e**9*x**2 + 16*d*e**10*x**3 + 4*e**11*x**4)
+ 125*c**3*d**6/(4*d**4*e**7 + 16*d**3*e**8*x + 24*d**2*e**9*x**2 + 16*d*e**10*x**3 + 4*e**11*x**4) + 240*c**
3*d**5*e*x*log(d/e + x)/(4*d**4*e**7 + 16*d**3*e**8*x + 24*d**2*e**9*x**2 + 16*d*e**10*x**3 + 4*e**11*x**4) +
440*c**3*d**5*e*x/(4*d**4*e**7 + 16*d**3*e**8*x + 24*d**2*e**9*x**2 + 16*d*e**10*x**3 + 4*e**11*x**4) + 360*c*
*3*d**4*e**2*x**2*log(d/e + x)/(4*d**4*e**7 + 16*d**3*e**8*x + 24*d**2*e**9*x**2 + 16*d*e**10*x**3 + 4*e**11*x
**4) + 540*c**3*d**4*e**2*x**2/(4*d**4*e**7 + 16*d**3*e**8*x + 24*d**2*e**9*x**2 + 16*d*e**10*x**3 + 4*e**11*x
**4) + 240*c**3*d**3*e**3*x**3*log(d/e + x)/(4*d**4*e**7 + 16*d**3*e**8*x + 24*d**2*e**9*x**2 + 16*d*e**10*x**
3 + 4*e**11*x**4) + 240*c**3*d**3*e**3*x**3/(4*d**4*e**7 + 16*d**3*e**8*x + 24*d**2*e**9*x**2 + 16*d*e**10*x**
3 + 4*e**11*x**4) + 60*c**3*d**2*e**4*x**4*log(d/e + x)/(4*d**4*e**7 + 16*d**3*e**8*x + 24*d**2*e**9*x**2 + 16
*d*e**10*x**3 + 4*e**11*x**4) - 12*c**3*d*e**5*x**5/(4*d**4*e**7 + 16*d**3*e**8*x + 24*d**2*e**9*x**2 + 16*d*e
**10*x**3 + 4*e**11*x**4) + 2*c**3*e**6*x**6/(4*d**4*e**7 + 16*d**3*e**8*x + 24*d**2*e**9*x**2 + 16*d*e**10*x*
*3 + 4*e**11*x**4), Eq(m, -5)), (-a**3*e**6/(3*d**3*e**7 + 9*d**2*e**8*x + 9*d*e**9*x**2 + 3*e**10*x**3) - 3*a
**2*c*d**2*e**4/(3*d**3*e**7 + 9*d**2*e**8*x + 9*d*e**9*x**2 + 3*e**10*x**3) - 9*a**2*c*d*e**5*x/(3*d**3*e**7
+ 9*d**2*e**8*x + 9*d*e**9*x**2 + 3*e**10*x**3) - 9*a**2*c*e**6*x**2/(3*d**3*e**7 + 9*d**2*e**8*x + 9*d*e**9*x
**2 + 3*e**10*x**3) - 36*a*c**2*d**4*e**2*log(d/e + x)/(3*d**3*e**7 + 9*d**2*e**8*x + 9*d*e**9*x**2 + 3*e**10*
x**3) - 66*a*c**2*d**4*e**2/(3*d**3*e**7 + 9*d**2*e**8*x + 9*d*e**9*x**2 + 3*e**10*x**3) - 108*a*c**2*d**3*e**
3*x*log(d/e + x)/(3*d**3*e**7 + 9*d**2*e**8*x + 9*d*e**9*x**2 + 3*e**10*x**3) - 162*a*c**2*d**3*e**3*x/(3*d**3
*e**7 + 9*d**2*e**8*x + 9*d*e**9*x**2 + 3*e**10*x**3) - 108*a*c**2*d**2*e**4*x**2*log(d/e + x)/(3*d**3*e**7 +
9*d**2*e**8*x + 9*d*e**9*x**2 + 3*e**10*x**3) - 108*a*c**2*d**2*e**4*x**2/(3*d**3*e**7 + 9*d**2*e**8*x + 9*d*e
**9*x**2 + 3*e**10*x**3) - 36*a*c**2*d*e**5*x**3*log(d/e + x)/(3*d**3*e**7 + 9*d**2*e**8*x + 9*d*e**9*x**2 + 3
*e**10*x**3) + 9*a*c**2*e**6*x**4/(3*d**3*e**7 + 9*d**2*e**8*x + 9*d*e**9*x**2 + 3*e**10*x**3) - 60*c**3*d**6*
log(d/e + x)/(3*d**3*e**7 + 9*d**2*e**8*x + 9*d*e**9*x**2 + 3*e**10*x**3) - 110*c**3*d**6/(3*d**3*e**7 + 9*d**
2*e**8*x + 9*d*e**9*x**2 + 3*e**10*x**3) - 180*c**3*d**5*e*x*log(d/e + x)/(3*d**3*e**7 + 9*d**2*e**8*x + 9*d*e
**9*x**2 + 3*e**10*x**3) - 270*c**3*d**5*e*x/(3*d**3*e**7 + 9*d**2*e**8*x + 9*d*e**9*x**2 + 3*e**10*x**3) - 18
0*c**3*d**4*e**2*x**2*log(d/e + x)/(3*d**3*e**7 + 9*d**2*e**8*x + 9*d*e**9*x**2 + 3*e**10*x**3) - 180*c**3*d**
4*e**2*x**2/(3*d**3*e**7 + 9*d**2*e**8*x + 9*d*e**9*x**2 + 3*e**10*x**3) - 60*c**3*d**3*e**3*x**3*log(d/e + x)
/(3*d**3*e**7 + 9*d**2*e**8*x + 9*d*e**9*x**2 + 3*e**10*x**3) + 15*c**3*d**2*e**4*x**4/(3*d**3*e**7 + 9*d**2*e
**8*x + 9*d*e**9*x**2 + 3*e**10*x**3) - 3*c**3*d*e**5*x**5/(3*d**3*e**7 + 9*d**2*e**8*x + 9*d*e**9*x**2 + 3*e*
*10*x**3) + c**3*e**6*x**6/(3*d**3*e**7 + 9*d**2*e**8*x + 9*d*e**9*x**2 + 3*e**10*x**3), Eq(m, -4)), (-2*a**3*
e**6/(4*d**2*e**7 + 8*d*e**8*x + 4*e**9*x**2) + 12*a**2*c*d**2*e**4*log(d/e + x)/(4*d**2*e**7 + 8*d*e**8*x + 4
*e**9*x**2) + 18*a**2*c*d**2*e**4/(4*d**2*e**7 + 8*d*e**8*x + 4*e**9*x**2) + 24*a**2*c*d*e**5*x*log(d/e + x)/(
4*d**2*e**7 + 8*d*e**8*x + 4*e**9*x**2) + 24*a**2*c*d*e**5*x/(4*d**2*e**7 + 8*d*e**8*x + 4*e**9*x**2) + 12*a**
2*c*e**6*x**2*log(d/e + x)/(4*d**2*e**7 + 8*d*e**8*x + 4*e**9*x**2) + 72*a*c**2*d**4*e**2*log(d/e + x)/(4*d**2
*e**7 + 8*d*e**8*x + 4*e**9*x**2) + 108*a*c**2*d**4*e**2/(4*d**2*e**7 + 8*d*e**8*x + 4*e**9*x**2) + 144*a*c**2
*d**3*e**3*x*log(d/e + x)/(4*d**2*e**7 + 8*d*e**8*x + 4*e**9*x**2) + 144*a*c**2*d**3*e**3*x/(4*d**2*e**7 + 8*d
*e**8*x + 4*e**9*x**2) + 72*a*c**2*d**2*e**4*x**2*log(d/e + x)/(4*d**2*e**7 + 8*d*e**8*x + 4*e**9*x**2) - 24*a
*c**2*d*e**5*x**3/(4*d**2*e**7 + 8*d*e**8*x + 4*e**9*x**2) + 6*a*c**2*e**6*x**4/(4*d**2*e**7 + 8*d*e**8*x + 4*
e**9*x**2) + 60*c**3*d**6*log(d/e + x)/(4*d**2*e**7 + 8*d*e**8*x + 4*e**9*x**2) + 90*c**3*d**6/(4*d**2*e**7 +
8*d*e**8*x + 4*e**9*x**2) + 120*c**3*d**5*e*x*log(d/e + x)/(4*d**2*e**7 + 8*d*e**8*x + 4*e**9*x**2) + 120*c**3
*d**5*e*x/(4*d**2*e**7 + 8*d*e**8*x + 4*e**9*x**2) + 60*c**3*d**4*e**2*x**2*log(d/e + x)/(4*d**2*e**7 + 8*d*e*
*8*x + 4*e**9*x**2) - 20*c**3*d**3*e**3*x**3/(4*d**2*e**7 + 8*d*e**8*x + 4*e**9*x**2) + 5*c**3*d**2*e**4*x**4/
(4*d**2*e**7 + 8*d*e**8*x + 4*e**9*x**2) - 2*c**3*d*e**5*x**5/(4*d**2*e**7 + 8*d*e**8*x + 4*e**9*x**2) + c**3*
e**6*x**6/(4*d**2*e**7 + 8*d*e**8*x + 4*e**9*x**2), Eq(m, -3)), (-10*a**3*e**6/(10*d*e**7 + 10*e**8*x) - 60*a*
*2*c*d**2*e**4*log(d/e + x)/(10*d*e**7 + 10*e**8*x) - 60*a**2*c*d**2*e**4/(10*d*e**7 + 10*e**8*x) - 60*a**2*c*
d*e**5*x*log(d/e + x)/(10*d*e**7 + 10*e**8*x) + 30*a**2*c*e**6*x**2/(10*d*e**7 + 10*e**8*x) - 120*a*c**2*d**4*
e**2*log(d/e + x)/(10*d*e**7 + 10*e**8*x) - 120*a*c**2*d**4*e**2/(10*d*e**7 + 10*e**8*x) - 120*a*c**2*d**3*e**
3*x*log(d/e + x)/(10*d*e**7 + 10*e**8*x) + 60*a*c**2*d**2*e**4*x**2/(10*d*e**7 + 10*e**8*x) - 20*a*c**2*d*e**5
*x**3/(10*d*e**7 + 10*e**8*x) + 10*a*c**2*e**6*x**4/(10*d*e**7 + 10*e**8*x) - 60*c**3*d**6*log(d/e + x)/(10*d*
e**7 + 10*e**8*x) - 60*c**3*d**6/(10*d*e**7 + 10*e**8*x) - 60*c**3*d**5*e*x*log(d/e + x)/(10*d*e**7 + 10*e**8*
x) + 30*c**3*d**4*e**2*x**2/(10*d*e**7 + 10*e**8*x) - 10*c**3*d**3*e**3*x**3/(10*d*e**7 + 10*e**8*x) + 5*c**3*
d**2*e**4*x**4/(10*d*e**7 + 10*e**8*x) - 3*c**3*d*e**5*x**5/(10*d*e**7 + 10*e**8*x) + 2*c**3*e**6*x**6/(10*d*e
**7 + 10*e**8*x), Eq(m, -2)), (a**3*log(d/e + x)/e + 3*a**2*c*d**2*log(d/e + x)/e**3 - 3*a**2*c*d*x/e**2 + 3*a
**2*c*x**2/(2*e) + 3*a*c**2*d**4*log(d/e + x)/e**5 - 3*a*c**2*d**3*x/e**4 + 3*a*c**2*d**2*x**2/(2*e**3) - a*c*
*2*d*x**3/e**2 + 3*a*c**2*x**4/(4*e) + c**3*d**6*log(d/e + x)/e**7 - c**3*d**5*x/e**6 + c**3*d**4*x**2/(2*e**5
) - c**3*d**3*x**3/(3*e**4) + c**3*d**2*x**4/(4*e**3) - c**3*d*x**5/(5*e**2) + c**3*x**6/(6*e), Eq(m, -1)), (a
**3*d*e**6*m**6*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 131
32*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 27*a**3*d*e**6*m**5*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e*
*7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 295*a**3*d*e**6*m**4
*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 +
13068*e**7*m + 5040*e**7) + 1665*a**3*d*e**6*m**3*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 196
0*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 5104*a**3*d*e**6*m**2*(d + e*x)**
m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m
+ 5040*e**7) + 8028*a**3*d*e**6*m*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6
769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 5040*a**3*d*e**6*(d + e*x)**m/(e**7*m**7 + 28*e*
*7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + a**3
*e**7*m**6*x*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*
e**7*m**2 + 13068*e**7*m + 5040*e**7) + 27*a**3*e**7*m**5*x*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*
m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 295*a**3*e**7*m**4*x*(d
+ e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 130
68*e**7*m + 5040*e**7) + 1665*a**3*e**7*m**3*x*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e
**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 5104*a**3*e**7*m**2*x*(d + e*x)**m/(
e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m +
5040*e**7) + 8028*a**3*e**7*m*x*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769
*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 5040*a**3*e**7*x*(d + e*x)**m/(e**7*m**7 + 28*e**7*
m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 6*a**2*
c*d**3*e**4*m**4*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13
132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 132*a**2*c*d**3*e**4*m**3*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 +
322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 1074*a**2*c*d
**3*e**4*m**2*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132
*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 3828*a**2*c*d**3*e**4*m*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*
e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 5040*a**2*c*d**3*e
**4*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2
+ 13068*e**7*m + 5040*e**7) - 6*a**2*c*d**2*e**5*m**5*x*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**
5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) - 132*a**2*c*d**2*e**5*m**4*
x*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 +
13068*e**7*m + 5040*e**7) - 1074*a**2*c*d**2*e**5*m**3*x*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m*
*5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) - 3828*a**2*c*d**2*e**5*m**
2*x*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2
+ 13068*e**7*m + 5040*e**7) - 5040*a**2*c*d**2*e**5*m*x*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**
5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 3*a**2*c*d*e**6*m**6*x**2*
(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 1
3068*e**7*m + 5040*e**7) + 69*a**2*c*d*e**6*m**5*x**2*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 +
1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 603*a**2*c*d*e**6*m**4*x**2*(
d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13
068*e**7*m + 5040*e**7) + 2451*a**2*c*d*e**6*m**3*x**2*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5
+ 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 4434*a**2*c*d*e**6*m**2*x**2
*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 +
13068*e**7*m + 5040*e**7) + 2520*a**2*c*d*e**6*m*x**2*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 +
1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 3*a**2*c*e**7*m**6*x**3*(d +
e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*
e**7*m + 5040*e**7) + 75*a**2*c*e**7*m**5*x**3*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e
**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 741*a**2*c*e**7*m**4*x**3*(d + e*x)*
*m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*
m + 5040*e**7) + 3657*a**2*c*e**7*m**3*x**3*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7
*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 9336*a**2*c*e**7*m**2*x**3*(d + e*x)**m
/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m
+ 5040*e**7) + 11388*a**2*c*e**7*m*x**3*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**
4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 5040*a**2*c*e**7*x**3*(d + e*x)**m/(e**7*m*
*7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e*
*7) + 72*a*c**2*d**5*e**2*m**2*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*
e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 936*a*c**2*d**5*e**2*m*(d + e*x)**m/(e**7*m**7 + 28*
e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 30
24*a*c**2*d**5*e**2*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 +
13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) - 72*a*c**2*d**4*e**3*m**3*x*(d + e*x)**m/(e**7*m**7 + 28*e**7*m*
*6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) - 936*a*c**
2*d**4*e**3*m**2*x*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 +
13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) - 3024*a*c**2*d**4*e**3*m*x*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6
+ 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 36*a*c**2*d
**3*e**4*m**4*x**2*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 +
13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 504*a*c**2*d**3*e**4*m**3*x**2*(d + e*x)**m/(e**7*m**7 + 28*e**7
*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 1980*a
*c**2*d**3*e**4*m**2*x**2*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*
m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 1512*a*c**2*d**3*e**4*m*x**2*(d + e*x)**m/(e**7*m**7 + 28
*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) - 1
2*a*c**2*d**2*e**5*m**5*x**3*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e*
*7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) - 192*a*c**2*d**2*e**5*m**4*x**3*(d + e*x)**m/(e**7*m**7
+ 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7
) - 996*a*c**2*d**2*e**5*m**3*x**3*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6
769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) - 1824*a*c**2*d**2*e**5*m**2*x**3*(d + e*x)**m/(e*
*7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 50
40*e**7) - 1008*a*c**2*d**2*e**5*m*x**3*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**
4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 3*a*c**2*d*e**6*m**6*x**4*(d + e*x)**m/(e**
7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 504
0*e**7) + 57*a*c**2*d*e**6*m**5*x**4*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 +
6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 393*a*c**2*d*e**6*m**4*x**4*(d + e*x)**m/(e**7
*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040
*e**7) + 1203*a*c**2*d*e**6*m**3*x**4*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4
+ 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 1620*a*c**2*d*e**6*m**2*x**4*(d + e*x)**m/(e*
*7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 50
40*e**7) + 756*a*c**2*d*e**6*m*x**4*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 +
6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 3*a*c**2*e**7*m**6*x**5*(d + e*x)**m/(e**7*m**7
+ 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7
) + 69*a*c**2*e**7*m**5*x**5*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e*
*7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 621*a*c**2*e**7*m**4*x**5*(d + e*x)**m/(e**7*m**7 + 28
*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 2
775*a*c**2*e**7*m**3*x**5*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*
m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 6432*a*c**2*e**7*m**2*x**5*(d + e*x)**m/(e**7*m**7 + 28*e
**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 723
6*a*c**2*e**7*m*x**5*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3
+ 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 3024*a*c**2*e**7*x**5*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 +
322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 720*c**3*d**7
*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 +
13068*e**7*m + 5040*e**7) - 720*c**3*d**6*e*m*x*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*
e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 360*c**3*d**5*e**2*m**2*x**2*(d + e
*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e
**7*m + 5040*e**7) + 360*c**3*d**5*e**2*m*x**2*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e
**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) - 120*c**3*d**4*e**3*m**3*x**3*(d + e*
x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e*
*7*m + 5040*e**7) - 360*c**3*d**4*e**3*m**2*x**3*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960
*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) - 240*c**3*d**4*e**3*m*x**3*(d + e*x
)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**
7*m + 5040*e**7) + 30*c**3*d**3*e**4*m**4*x**4*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e
**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 180*c**3*d**3*e**4*m**3*x**4*(d + e*
x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e*
*7*m + 5040*e**7) + 330*c**3*d**3*e**4*m**2*x**4*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960
*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 180*c**3*d**3*e**4*m*x**4*(d + e*x
)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**
7*m + 5040*e**7) - 6*c**3*d**2*e**5*m**5*x**5*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e*
*7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) - 60*c**3*d**2*e**5*m**4*x**5*(d + e*x)
**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7
*m + 5040*e**7) - 210*c**3*d**2*e**5*m**3*x**5*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e
**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) - 300*c**3*d**2*e**5*m**2*x**5*(d + e*
x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e*
*7*m + 5040*e**7) - 144*c**3*d**2*e**5*m*x**5*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e*
*7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + c**3*d*e**6*m**6*x**6*(d + e*x)**m/(e
**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5
040*e**7) + 15*c**3*d*e**6*m**5*x**6*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 +
6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 85*c**3*d*e**6*m**4*x**6*(d + e*x)**m/(e**7*m*
*7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e*
*7) + 225*c**3*d*e**6*m**3*x**6*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769
*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 274*c**3*d*e**6*m**2*x**6*(d + e*x)**m/(e**7*m**7 +
28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7)
+ 120*c**3*d*e**6*m*x**6*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m
**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + c**3*e**7*m**6*x**7*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6
+ 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 21*c**3*e**
7*m**5*x**7*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e
**7*m**2 + 13068*e**7*m + 5040*e**7) + 175*c**3*e**7*m**4*x**7*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e*
*7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 735*c**3*e**7*m**3*x
**7*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2
+ 13068*e**7*m + 5040*e**7) + 1624*c**3*e**7*m**2*x**7*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5
+ 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 1764*c**3*e**7*m*x**7*(d +
e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*
e**7*m + 5040*e**7) + 720*c**3*e**7*x**7*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m*
*4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7), True))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 472 vs. \(2 (223) = 446\).

Time = 0.20 (sec) , antiderivative size = 472, normalized size of antiderivative = 2.12 \[ \int (d+e x)^m \left (a+c x^2\right )^3 \, dx=\frac {{\left (e x + d\right )}^{m + 1} a^{3}}{e {\left (m + 1\right )}} + \frac {3 \, {\left ({\left (m^{2} + 3 \, m + 2\right )} e^{3} x^{3} + {\left (m^{2} + m\right )} d e^{2} x^{2} - 2 \, d^{2} e m x + 2 \, d^{3}\right )} {\left (e x + d\right )}^{m} a^{2} c}{{\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{3}} + \frac {3 \, {\left ({\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} e^{5} x^{5} + {\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} d e^{4} x^{4} - 4 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d^{2} e^{3} x^{3} + 12 \, {\left (m^{2} + m\right )} d^{3} e^{2} x^{2} - 24 \, d^{4} e m x + 24 \, d^{5}\right )} {\left (e x + d\right )}^{m} a c^{2}}{{\left (m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120\right )} e^{5}} + \frac {{\left ({\left (m^{6} + 21 \, m^{5} + 175 \, m^{4} + 735 \, m^{3} + 1624 \, m^{2} + 1764 \, m + 720\right )} e^{7} x^{7} + {\left (m^{6} + 15 \, m^{5} + 85 \, m^{4} + 225 \, m^{3} + 274 \, m^{2} + 120 \, m\right )} d e^{6} x^{6} - 6 \, {\left (m^{5} + 10 \, m^{4} + 35 \, m^{3} + 50 \, m^{2} + 24 \, m\right )} d^{2} e^{5} x^{5} + 30 \, {\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} d^{3} e^{4} x^{4} - 120 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d^{4} e^{3} x^{3} + 360 \, {\left (m^{2} + m\right )} d^{5} e^{2} x^{2} - 720 \, d^{6} e m x + 720 \, d^{7}\right )} {\left (e x + d\right )}^{m} c^{3}}{{\left (m^{7} + 28 \, m^{6} + 322 \, m^{5} + 1960 \, m^{4} + 6769 \, m^{3} + 13132 \, m^{2} + 13068 \, m + 5040\right )} e^{7}} \]

[In]

integrate((e*x+d)^m*(c*x^2+a)^3,x, algorithm="maxima")

[Out]

(e*x + d)^(m + 1)*a^3/(e*(m + 1)) + 3*((m^2 + 3*m + 2)*e^3*x^3 + (m^2 + m)*d*e^2*x^2 - 2*d^2*e*m*x + 2*d^3)*(e
*x + d)^m*a^2*c/((m^3 + 6*m^2 + 11*m + 6)*e^3) + 3*((m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*e^5*x^5 + (m^4 + 6*m^3
+ 11*m^2 + 6*m)*d*e^4*x^4 - 4*(m^3 + 3*m^2 + 2*m)*d^2*e^3*x^3 + 12*(m^2 + m)*d^3*e^2*x^2 - 24*d^4*e*m*x + 24*
d^5)*(e*x + d)^m*a*c^2/((m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m + 120)*e^5) + ((m^6 + 21*m^5 + 175*m^4 + 735*
m^3 + 1624*m^2 + 1764*m + 720)*e^7*x^7 + (m^6 + 15*m^5 + 85*m^4 + 225*m^3 + 274*m^2 + 120*m)*d*e^6*x^6 - 6*(m^
5 + 10*m^4 + 35*m^3 + 50*m^2 + 24*m)*d^2*e^5*x^5 + 30*(m^4 + 6*m^3 + 11*m^2 + 6*m)*d^3*e^4*x^4 - 120*(m^3 + 3*
m^2 + 2*m)*d^4*e^3*x^3 + 360*(m^2 + m)*d^5*e^2*x^2 - 720*d^6*e*m*x + 720*d^7)*(e*x + d)^m*c^3/((m^7 + 28*m^6 +
322*m^5 + 1960*m^4 + 6769*m^3 + 13132*m^2 + 13068*m + 5040)*e^7)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2085 vs. \(2 (223) = 446\).

Time = 0.31 (sec) , antiderivative size = 2085, normalized size of antiderivative = 9.35 \[ \int (d+e x)^m \left (a+c x^2\right )^3 \, dx=\text {Too large to display} \]

[In]

integrate((e*x+d)^m*(c*x^2+a)^3,x, algorithm="giac")

[Out]

((e*x + d)^m*c^3*e^7*m^6*x^7 + (e*x + d)^m*c^3*d*e^6*m^6*x^6 + 21*(e*x + d)^m*c^3*e^7*m^5*x^7 + 3*(e*x + d)^m*
a*c^2*e^7*m^6*x^5 + 15*(e*x + d)^m*c^3*d*e^6*m^5*x^6 + 175*(e*x + d)^m*c^3*e^7*m^4*x^7 + 3*(e*x + d)^m*a*c^2*d
*e^6*m^6*x^4 - 6*(e*x + d)^m*c^3*d^2*e^5*m^5*x^5 + 69*(e*x + d)^m*a*c^2*e^7*m^5*x^5 + 85*(e*x + d)^m*c^3*d*e^6
*m^4*x^6 + 735*(e*x + d)^m*c^3*e^7*m^3*x^7 + 3*(e*x + d)^m*a^2*c*e^7*m^6*x^3 + 57*(e*x + d)^m*a*c^2*d*e^6*m^5*
x^4 - 60*(e*x + d)^m*c^3*d^2*e^5*m^4*x^5 + 621*(e*x + d)^m*a*c^2*e^7*m^4*x^5 + 225*(e*x + d)^m*c^3*d*e^6*m^3*x
^6 + 1624*(e*x + d)^m*c^3*e^7*m^2*x^7 + 3*(e*x + d)^m*a^2*c*d*e^6*m^6*x^2 - 12*(e*x + d)^m*a*c^2*d^2*e^5*m^5*x
^3 + 75*(e*x + d)^m*a^2*c*e^7*m^5*x^3 + 30*(e*x + d)^m*c^3*d^3*e^4*m^4*x^4 + 393*(e*x + d)^m*a*c^2*d*e^6*m^4*x
^4 - 210*(e*x + d)^m*c^3*d^2*e^5*m^3*x^5 + 2775*(e*x + d)^m*a*c^2*e^7*m^3*x^5 + 274*(e*x + d)^m*c^3*d*e^6*m^2*
x^6 + 1764*(e*x + d)^m*c^3*e^7*m*x^7 + (e*x + d)^m*a^3*e^7*m^6*x + 69*(e*x + d)^m*a^2*c*d*e^6*m^5*x^2 - 192*(e
*x + d)^m*a*c^2*d^2*e^5*m^4*x^3 + 741*(e*x + d)^m*a^2*c*e^7*m^4*x^3 + 180*(e*x + d)^m*c^3*d^3*e^4*m^3*x^4 + 12
03*(e*x + d)^m*a*c^2*d*e^6*m^3*x^4 - 300*(e*x + d)^m*c^3*d^2*e^5*m^2*x^5 + 6432*(e*x + d)^m*a*c^2*e^7*m^2*x^5
+ 120*(e*x + d)^m*c^3*d*e^6*m*x^6 + 720*(e*x + d)^m*c^3*e^7*x^7 + (e*x + d)^m*a^3*d*e^6*m^6 - 6*(e*x + d)^m*a^
2*c*d^2*e^5*m^5*x + 27*(e*x + d)^m*a^3*e^7*m^5*x + 36*(e*x + d)^m*a*c^2*d^3*e^4*m^4*x^2 + 603*(e*x + d)^m*a^2*
c*d*e^6*m^4*x^2 - 120*(e*x + d)^m*c^3*d^4*e^3*m^3*x^3 - 996*(e*x + d)^m*a*c^2*d^2*e^5*m^3*x^3 + 3657*(e*x + d)
^m*a^2*c*e^7*m^3*x^3 + 330*(e*x + d)^m*c^3*d^3*e^4*m^2*x^4 + 1620*(e*x + d)^m*a*c^2*d*e^6*m^2*x^4 - 144*(e*x +
d)^m*c^3*d^2*e^5*m*x^5 + 7236*(e*x + d)^m*a*c^2*e^7*m*x^5 + 27*(e*x + d)^m*a^3*d*e^6*m^5 - 132*(e*x + d)^m*a^
2*c*d^2*e^5*m^4*x + 295*(e*x + d)^m*a^3*e^7*m^4*x + 504*(e*x + d)^m*a*c^2*d^3*e^4*m^3*x^2 + 2451*(e*x + d)^m*a
^2*c*d*e^6*m^3*x^2 - 360*(e*x + d)^m*c^3*d^4*e^3*m^2*x^3 - 1824*(e*x + d)^m*a*c^2*d^2*e^5*m^2*x^3 + 9336*(e*x
+ d)^m*a^2*c*e^7*m^2*x^3 + 180*(e*x + d)^m*c^3*d^3*e^4*m*x^4 + 756*(e*x + d)^m*a*c^2*d*e^6*m*x^4 + 3024*(e*x +
d)^m*a*c^2*e^7*x^5 + 6*(e*x + d)^m*a^2*c*d^3*e^4*m^4 + 295*(e*x + d)^m*a^3*d*e^6*m^4 - 72*(e*x + d)^m*a*c^2*d
^4*e^3*m^3*x - 1074*(e*x + d)^m*a^2*c*d^2*e^5*m^3*x + 1665*(e*x + d)^m*a^3*e^7*m^3*x + 360*(e*x + d)^m*c^3*d^5
*e^2*m^2*x^2 + 1980*(e*x + d)^m*a*c^2*d^3*e^4*m^2*x^2 + 4434*(e*x + d)^m*a^2*c*d*e^6*m^2*x^2 - 240*(e*x + d)^m
*c^3*d^4*e^3*m*x^3 - 1008*(e*x + d)^m*a*c^2*d^2*e^5*m*x^3 + 11388*(e*x + d)^m*a^2*c*e^7*m*x^3 + 132*(e*x + d)^
m*a^2*c*d^3*e^4*m^3 + 1665*(e*x + d)^m*a^3*d*e^6*m^3 - 936*(e*x + d)^m*a*c^2*d^4*e^3*m^2*x - 3828*(e*x + d)^m*
a^2*c*d^2*e^5*m^2*x + 5104*(e*x + d)^m*a^3*e^7*m^2*x + 360*(e*x + d)^m*c^3*d^5*e^2*m*x^2 + 1512*(e*x + d)^m*a*
c^2*d^3*e^4*m*x^2 + 2520*(e*x + d)^m*a^2*c*d*e^6*m*x^2 + 5040*(e*x + d)^m*a^2*c*e^7*x^3 + 72*(e*x + d)^m*a*c^2
*d^5*e^2*m^2 + 1074*(e*x + d)^m*a^2*c*d^3*e^4*m^2 + 5104*(e*x + d)^m*a^3*d*e^6*m^2 - 720*(e*x + d)^m*c^3*d^6*e
*m*x - 3024*(e*x + d)^m*a*c^2*d^4*e^3*m*x - 5040*(e*x + d)^m*a^2*c*d^2*e^5*m*x + 8028*(e*x + d)^m*a^3*e^7*m*x
+ 936*(e*x + d)^m*a*c^2*d^5*e^2*m + 3828*(e*x + d)^m*a^2*c*d^3*e^4*m + 8028*(e*x + d)^m*a^3*d*e^6*m + 5040*(e*
x + d)^m*a^3*e^7*x + 720*(e*x + d)^m*c^3*d^7 + 3024*(e*x + d)^m*a*c^2*d^5*e^2 + 5040*(e*x + d)^m*a^2*c*d^3*e^4
+ 5040*(e*x + d)^m*a^3*d*e^6)/(e^7*m^7 + 28*e^7*m^6 + 322*e^7*m^5 + 1960*e^7*m^4 + 6769*e^7*m^3 + 13132*e^7*m
^2 + 13068*e^7*m + 5040*e^7)

Mupad [B] (verification not implemented)

Time = 10.11 (sec) , antiderivative size = 1144, normalized size of antiderivative = 5.13 \[ \int (d+e x)^m \left (a+c x^2\right )^3 \, dx=\frac {{\left (d+e\,x\right )}^m\,\left (a^3\,d\,e^6\,m^6+27\,a^3\,d\,e^6\,m^5+295\,a^3\,d\,e^6\,m^4+1665\,a^3\,d\,e^6\,m^3+5104\,a^3\,d\,e^6\,m^2+8028\,a^3\,d\,e^6\,m+5040\,a^3\,d\,e^6+6\,a^2\,c\,d^3\,e^4\,m^4+132\,a^2\,c\,d^3\,e^4\,m^3+1074\,a^2\,c\,d^3\,e^4\,m^2+3828\,a^2\,c\,d^3\,e^4\,m+5040\,a^2\,c\,d^3\,e^4+72\,a\,c^2\,d^5\,e^2\,m^2+936\,a\,c^2\,d^5\,e^2\,m+3024\,a\,c^2\,d^5\,e^2+720\,c^3\,d^7\right )}{e^7\,\left (m^7+28\,m^6+322\,m^5+1960\,m^4+6769\,m^3+13132\,m^2+13068\,m+5040\right )}-\frac {x\,{\left (d+e\,x\right )}^m\,\left (-a^3\,e^7\,m^6-27\,a^3\,e^7\,m^5-295\,a^3\,e^7\,m^4-1665\,a^3\,e^7\,m^3-5104\,a^3\,e^7\,m^2-8028\,a^3\,e^7\,m-5040\,a^3\,e^7+6\,a^2\,c\,d^2\,e^5\,m^5+132\,a^2\,c\,d^2\,e^5\,m^4+1074\,a^2\,c\,d^2\,e^5\,m^3+3828\,a^2\,c\,d^2\,e^5\,m^2+5040\,a^2\,c\,d^2\,e^5\,m+72\,a\,c^2\,d^4\,e^3\,m^3+936\,a\,c^2\,d^4\,e^3\,m^2+3024\,a\,c^2\,d^4\,e^3\,m+720\,c^3\,d^6\,e\,m\right )}{e^7\,\left (m^7+28\,m^6+322\,m^5+1960\,m^4+6769\,m^3+13132\,m^2+13068\,m+5040\right )}+\frac {c^3\,x^7\,{\left (d+e\,x\right )}^m\,\left (m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720\right )}{m^7+28\,m^6+322\,m^5+1960\,m^4+6769\,m^3+13132\,m^2+13068\,m+5040}+\frac {3\,c^2\,x^5\,{\left (d+e\,x\right )}^m\,\left (-2\,c\,d^2\,m+a\,e^2\,m^2+13\,a\,e^2\,m+42\,a\,e^2\right )\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )}{e^2\,\left (m^7+28\,m^6+322\,m^5+1960\,m^4+6769\,m^3+13132\,m^2+13068\,m+5040\right )}+\frac {3\,c\,x^3\,{\left (d+e\,x\right )}^m\,\left (m^2+3\,m+2\right )\,\left (a^2\,e^4\,m^4+22\,a^2\,e^4\,m^3+179\,a^2\,e^4\,m^2+638\,a^2\,e^4\,m+840\,a^2\,e^4-4\,a\,c\,d^2\,e^2\,m^3-52\,a\,c\,d^2\,e^2\,m^2-168\,a\,c\,d^2\,e^2\,m-40\,c^2\,d^4\,m\right )}{e^4\,\left (m^7+28\,m^6+322\,m^5+1960\,m^4+6769\,m^3+13132\,m^2+13068\,m+5040\right )}+\frac {c^3\,d\,m\,x^6\,{\left (d+e\,x\right )}^m\,\left (m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120\right )}{e\,\left (m^7+28\,m^6+322\,m^5+1960\,m^4+6769\,m^3+13132\,m^2+13068\,m+5040\right )}+\frac {3\,c^2\,d\,m\,x^4\,{\left (d+e\,x\right )}^m\,\left (m^3+6\,m^2+11\,m+6\right )\,\left (10\,c\,d^2+a\,e^2\,m^2+13\,a\,e^2\,m+42\,a\,e^2\right )}{e^3\,\left (m^7+28\,m^6+322\,m^5+1960\,m^4+6769\,m^3+13132\,m^2+13068\,m+5040\right )}+\frac {3\,c\,d\,m\,x^2\,\left (m+1\right )\,{\left (d+e\,x\right )}^m\,\left (a^2\,e^4\,m^4+22\,a^2\,e^4\,m^3+179\,a^2\,e^4\,m^2+638\,a^2\,e^4\,m+840\,a^2\,e^4+12\,a\,c\,d^2\,e^2\,m^2+156\,a\,c\,d^2\,e^2\,m+504\,a\,c\,d^2\,e^2+120\,c^2\,d^4\right )}{e^5\,\left (m^7+28\,m^6+322\,m^5+1960\,m^4+6769\,m^3+13132\,m^2+13068\,m+5040\right )} \]

[In]

int((a + c*x^2)^3*(d + e*x)^m,x)

[Out]

((d + e*x)^m*(720*c^3*d^7 + 5040*a^3*d*e^6 + 3024*a*c^2*d^5*e^2 + 5040*a^2*c*d^3*e^4 + 5104*a^3*d*e^6*m^2 + 16
65*a^3*d*e^6*m^3 + 295*a^3*d*e^6*m^4 + 27*a^3*d*e^6*m^5 + a^3*d*e^6*m^6 + 8028*a^3*d*e^6*m + 936*a*c^2*d^5*e^2
*m + 3828*a^2*c*d^3*e^4*m + 72*a*c^2*d^5*e^2*m^2 + 1074*a^2*c*d^3*e^4*m^2 + 132*a^2*c*d^3*e^4*m^3 + 6*a^2*c*d^
3*e^4*m^4))/(e^7*(13068*m + 13132*m^2 + 6769*m^3 + 1960*m^4 + 322*m^5 + 28*m^6 + m^7 + 5040)) - (x*(d + e*x)^m
*(720*c^3*d^6*e*m - 8028*a^3*e^7*m - 5104*a^3*e^7*m^2 - 1665*a^3*e^7*m^3 - 295*a^3*e^7*m^4 - 27*a^3*e^7*m^5 -
a^3*e^7*m^6 - 5040*a^3*e^7 + 3024*a*c^2*d^4*e^3*m + 5040*a^2*c*d^2*e^5*m + 936*a*c^2*d^4*e^3*m^2 + 3828*a^2*c*
d^2*e^5*m^2 + 72*a*c^2*d^4*e^3*m^3 + 1074*a^2*c*d^2*e^5*m^3 + 132*a^2*c*d^2*e^5*m^4 + 6*a^2*c*d^2*e^5*m^5))/(e
^7*(13068*m + 13132*m^2 + 6769*m^3 + 1960*m^4 + 322*m^5 + 28*m^6 + m^7 + 5040)) + (c^3*x^7*(d + e*x)^m*(1764*m
+ 1624*m^2 + 735*m^3 + 175*m^4 + 21*m^5 + m^6 + 720))/(13068*m + 13132*m^2 + 6769*m^3 + 1960*m^4 + 322*m^5 +
28*m^6 + m^7 + 5040) + (3*c^2*x^5*(d + e*x)^m*(42*a*e^2 + a*e^2*m^2 + 13*a*e^2*m - 2*c*d^2*m)*(50*m + 35*m^2 +
10*m^3 + m^4 + 24))/(e^2*(13068*m + 13132*m^2 + 6769*m^3 + 1960*m^4 + 322*m^5 + 28*m^6 + m^7 + 5040)) + (3*c*
x^3*(d + e*x)^m*(3*m + m^2 + 2)*(840*a^2*e^4 + 638*a^2*e^4*m - 40*c^2*d^4*m + 179*a^2*e^4*m^2 + 22*a^2*e^4*m^3
+ a^2*e^4*m^4 - 168*a*c*d^2*e^2*m - 52*a*c*d^2*e^2*m^2 - 4*a*c*d^2*e^2*m^3))/(e^4*(13068*m + 13132*m^2 + 6769
*m^3 + 1960*m^4 + 322*m^5 + 28*m^6 + m^7 + 5040)) + (c^3*d*m*x^6*(d + e*x)^m*(274*m + 225*m^2 + 85*m^3 + 15*m^
4 + m^5 + 120))/(e*(13068*m + 13132*m^2 + 6769*m^3 + 1960*m^4 + 322*m^5 + 28*m^6 + m^7 + 5040)) + (3*c^2*d*m*x
^4*(d + e*x)^m*(11*m + 6*m^2 + m^3 + 6)*(42*a*e^2 + 10*c*d^2 + a*e^2*m^2 + 13*a*e^2*m))/(e^3*(13068*m + 13132*
m^2 + 6769*m^3 + 1960*m^4 + 322*m^5 + 28*m^6 + m^7 + 5040)) + (3*c*d*m*x^2*(m + 1)*(d + e*x)^m*(840*a^2*e^4 +
120*c^2*d^4 + 638*a^2*e^4*m + 179*a^2*e^4*m^2 + 22*a^2*e^4*m^3 + a^2*e^4*m^4 + 504*a*c*d^2*e^2 + 156*a*c*d^2*e
^2*m + 12*a*c*d^2*e^2*m^2))/(e^5*(13068*m + 13132*m^2 + 6769*m^3 + 1960*m^4 + 322*m^5 + 28*m^6 + m^7 + 5040))