3.2540 \(\int (d+e x)^m \left (a+b x+c x^2\right )^2 \, dx\)

Optimal. Leaf size=178 \[ \frac{(d+e x)^{m+3} \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{e^5 (m+3)}+\frac{(d+e x)^{m+1} \left (a e^2-b d e+c d^2\right )^2}{e^5 (m+1)}-\frac{2 (2 c d-b e) (d+e x)^{m+2} \left (a e^2-b d e+c d^2\right )}{e^5 (m+2)}-\frac{2 c (2 c d-b e) (d+e x)^{m+4}}{e^5 (m+4)}+\frac{c^2 (d+e x)^{m+5}}{e^5 (m+5)} \]

[Out]

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

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Rubi [A]  time = 0.311752, antiderivative size = 178, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05 \[ \frac{(d+e x)^{m+3} \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{e^5 (m+3)}+\frac{(d+e x)^{m+1} \left (a e^2-b d e+c d^2\right )^2}{e^5 (m+1)}-\frac{2 (2 c d-b e) (d+e x)^{m+2} \left (a e^2-b d e+c d^2\right )}{e^5 (m+2)}-\frac{2 c (2 c d-b e) (d+e x)^{m+4}}{e^5 (m+4)}+\frac{c^2 (d+e x)^{m+5}}{e^5 (m+5)} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^m*(a + b*x + c*x^2)^2,x]

[Out]

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

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Rubi in Sympy [A]  time = 55.677, size = 165, normalized size = 0.93 \[ \frac{c^{2} \left (d + e x\right )^{m + 5}}{e^{5} \left (m + 5\right )} + \frac{2 c \left (d + e x\right )^{m + 4} \left (b e - 2 c d\right )}{e^{5} \left (m + 4\right )} + \frac{\left (d + e x\right )^{m + 1} \left (a e^{2} - b d e + c d^{2}\right )^{2}}{e^{5} \left (m + 1\right )} + \frac{2 \left (d + e x\right )^{m + 2} \left (b e - 2 c d\right ) \left (a e^{2} - b d e + c d^{2}\right )}{e^{5} \left (m + 2\right )} + \frac{\left (d + e x\right )^{m + 3} \left (2 a c e^{2} + b^{2} e^{2} - 6 b c d e + 6 c^{2} d^{2}\right )}{e^{5} \left (m + 3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**m*(c*x**2+b*x+a)**2,x)

[Out]

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

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Mathematica [A]  time = 0.411266, size = 302, normalized size = 1.7 \[ \frac{(d+e x)^{m+1} \left (e^2 \left (m^2+9 m+20\right ) \left (a^2 e^2 \left (m^2+5 m+6\right )+2 a b e (m+3) (e (m+1) x-d)+b^2 \left (2 d^2-2 d e (m+1) x+e^2 \left (m^2+3 m+2\right ) x^2\right )\right )+2 c e (m+5) \left (a e (m+4) \left (2 d^2-2 d e (m+1) x+e^2 \left (m^2+3 m+2\right ) x^2\right )+b \left (-6 d^3+6 d^2 e (m+1) x-3 d e^2 \left (m^2+3 m+2\right ) x^2+e^3 \left (m^3+6 m^2+11 m+6\right ) x^3\right )\right )+c^2 \left (24 d^4-24 d^3 e (m+1) x+12 d^2 e^2 \left (m^2+3 m+2\right ) x^2-4 d e^3 \left (m^3+6 m^2+11 m+6\right ) x^3+e^4 \left (m^4+10 m^3+35 m^2+50 m+24\right ) x^4\right )\right )}{e^5 (m+1) (m+2) (m+3) (m+4) (m+5)} \]

Antiderivative was successfully verified.

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

[Out]

((d + e*x)^(1 + m)*(c^2*(24*d^4 - 24*d^3*e*(1 + m)*x + 12*d^2*e^2*(2 + 3*m + m^2
)*x^2 - 4*d*e^3*(6 + 11*m + 6*m^2 + m^3)*x^3 + e^4*(24 + 50*m + 35*m^2 + 10*m^3
+ m^4)*x^4) + e^2*(20 + 9*m + m^2)*(a^2*e^2*(6 + 5*m + m^2) + 2*a*b*e*(3 + m)*(-
d + e*(1 + m)*x) + b^2*(2*d^2 - 2*d*e*(1 + m)*x + e^2*(2 + 3*m + m^2)*x^2)) + 2*
c*e*(5 + m)*(a*e*(4 + m)*(2*d^2 - 2*d*e*(1 + m)*x + e^2*(2 + 3*m + m^2)*x^2) + b
*(-6*d^3 + 6*d^2*e*(1 + m)*x - 3*d*e^2*(2 + 3*m + m^2)*x^2 + e^3*(6 + 11*m + 6*m
^2 + m^3)*x^3))))/(e^5*(1 + m)*(2 + m)*(3 + m)*(4 + m)*(5 + m))

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Maple [B]  time = 0.015, size = 822, normalized size = 4.6 \[{\frac{ \left ( ex+d \right ) ^{1+m} \left ({c}^{2}{e}^{4}{m}^{4}{x}^{4}+2\,bc{e}^{4}{m}^{4}{x}^{3}+10\,{c}^{2}{e}^{4}{m}^{3}{x}^{4}+2\,ac{e}^{4}{m}^{4}{x}^{2}+{b}^{2}{e}^{4}{m}^{4}{x}^{2}+22\,bc{e}^{4}{m}^{3}{x}^{3}-4\,{c}^{2}d{e}^{3}{m}^{3}{x}^{3}+35\,{c}^{2}{e}^{4}{m}^{2}{x}^{4}+2\,ab{e}^{4}{m}^{4}x+24\,ac{e}^{4}{m}^{3}{x}^{2}+12\,{b}^{2}{e}^{4}{m}^{3}{x}^{2}-6\,bcd{e}^{3}{m}^{3}{x}^{2}+82\,bc{e}^{4}{m}^{2}{x}^{3}-24\,{c}^{2}d{e}^{3}{m}^{2}{x}^{3}+50\,{c}^{2}{e}^{4}m{x}^{4}+{a}^{2}{e}^{4}{m}^{4}+26\,ab{e}^{4}{m}^{3}x-4\,acd{e}^{3}{m}^{3}x+98\,ac{e}^{4}{m}^{2}{x}^{2}-2\,{b}^{2}d{e}^{3}{m}^{3}x+49\,{b}^{2}{e}^{4}{m}^{2}{x}^{2}-48\,bcd{e}^{3}{m}^{2}{x}^{2}+122\,bc{e}^{4}m{x}^{3}+12\,{c}^{2}{d}^{2}{e}^{2}{m}^{2}{x}^{2}-44\,{c}^{2}d{e}^{3}m{x}^{3}+24\,{x}^{4}{c}^{2}{e}^{4}+14\,{a}^{2}{e}^{4}{m}^{3}-2\,abd{e}^{3}{m}^{3}+118\,ab{e}^{4}{m}^{2}x-40\,acd{e}^{3}{m}^{2}x+156\,ac{e}^{4}m{x}^{2}-20\,{b}^{2}d{e}^{3}{m}^{2}x+78\,{b}^{2}{e}^{4}m{x}^{2}+12\,bc{d}^{2}{e}^{2}{m}^{2}x-102\,bcd{e}^{3}m{x}^{2}+60\,{x}^{3}bc{e}^{4}+36\,{c}^{2}{d}^{2}{e}^{2}m{x}^{2}-24\,{x}^{3}{c}^{2}d{e}^{3}+71\,{a}^{2}{e}^{4}{m}^{2}-24\,abd{e}^{3}{m}^{2}+214\,ab{e}^{4}mx+4\,ac{d}^{2}{e}^{2}{m}^{2}-116\,acd{e}^{3}mx+80\,{x}^{2}ac{e}^{4}+2\,{b}^{2}{d}^{2}{e}^{2}{m}^{2}-58\,{b}^{2}d{e}^{3}mx+40\,{x}^{2}{b}^{2}{e}^{4}+72\,bc{d}^{2}{e}^{2}mx-60\,{x}^{2}bcd{e}^{3}-24\,{c}^{2}{d}^{3}emx+24\,{x}^{2}{c}^{2}{d}^{2}{e}^{2}+154\,{a}^{2}{e}^{4}m-94\,abd{e}^{3}m+120\,xab{e}^{4}+36\,ac{d}^{2}{e}^{2}m-80\,xacd{e}^{3}+18\,{b}^{2}{d}^{2}{e}^{2}m-40\,x{b}^{2}d{e}^{3}-12\,bc{d}^{3}em+60\,xbc{d}^{2}{e}^{2}-24\,x{c}^{2}{d}^{3}e+120\,{a}^{2}{e}^{4}-120\,abd{e}^{3}+80\,ac{d}^{2}{e}^{2}+40\,{b}^{2}{d}^{2}{e}^{2}-60\,{d}^{3}bce+24\,{c}^{2}{d}^{4} \right ) }{{e}^{5} \left ({m}^{5}+15\,{m}^{4}+85\,{m}^{3}+225\,{m}^{2}+274\,m+120 \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^m*(c*x^2+b*x+a)^2,x)

[Out]

(e*x+d)^(1+m)*(c^2*e^4*m^4*x^4+2*b*c*e^4*m^4*x^3+10*c^2*e^4*m^3*x^4+2*a*c*e^4*m^
4*x^2+b^2*e^4*m^4*x^2+22*b*c*e^4*m^3*x^3-4*c^2*d*e^3*m^3*x^3+35*c^2*e^4*m^2*x^4+
2*a*b*e^4*m^4*x+24*a*c*e^4*m^3*x^2+12*b^2*e^4*m^3*x^2-6*b*c*d*e^3*m^3*x^2+82*b*c
*e^4*m^2*x^3-24*c^2*d*e^3*m^2*x^3+50*c^2*e^4*m*x^4+a^2*e^4*m^4+26*a*b*e^4*m^3*x-
4*a*c*d*e^3*m^3*x+98*a*c*e^4*m^2*x^2-2*b^2*d*e^3*m^3*x+49*b^2*e^4*m^2*x^2-48*b*c
*d*e^3*m^2*x^2+122*b*c*e^4*m*x^3+12*c^2*d^2*e^2*m^2*x^2-44*c^2*d*e^3*m*x^3+24*c^
2*e^4*x^4+14*a^2*e^4*m^3-2*a*b*d*e^3*m^3+118*a*b*e^4*m^2*x-40*a*c*d*e^3*m^2*x+15
6*a*c*e^4*m*x^2-20*b^2*d*e^3*m^2*x+78*b^2*e^4*m*x^2+12*b*c*d^2*e^2*m^2*x-102*b*c
*d*e^3*m*x^2+60*b*c*e^4*x^3+36*c^2*d^2*e^2*m*x^2-24*c^2*d*e^3*x^3+71*a^2*e^4*m^2
-24*a*b*d*e^3*m^2+214*a*b*e^4*m*x+4*a*c*d^2*e^2*m^2-116*a*c*d*e^3*m*x+80*a*c*e^4
*x^2+2*b^2*d^2*e^2*m^2-58*b^2*d*e^3*m*x+40*b^2*e^4*x^2+72*b*c*d^2*e^2*m*x-60*b*c
*d*e^3*x^2-24*c^2*d^3*e*m*x+24*c^2*d^2*e^2*x^2+154*a^2*e^4*m-94*a*b*d*e^3*m+120*
a*b*e^4*x+36*a*c*d^2*e^2*m-80*a*c*d*e^3*x+18*b^2*d^2*e^2*m-40*b^2*d*e^3*x-12*b*c
*d^3*e*m+60*b*c*d^2*e^2*x-24*c^2*d^3*e*x+120*a^2*e^4-120*a*b*d*e^3+80*a*c*d^2*e^
2+40*b^2*d^2*e^2-60*b*c*d^3*e+24*c^2*d^4)/e^5/(m^5+15*m^4+85*m^3+225*m^2+274*m+1
20)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^2*(e*x + d)^m,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.276822, size = 1173, normalized size = 6.59 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^2*(e*x + d)^m,x, algorithm="fricas")

[Out]

(a^2*d*e^4*m^4 + 24*c^2*d^5 - 60*b*c*d^4*e - 120*a*b*d^2*e^3 + 120*a^2*d*e^4 + 4
0*(b^2 + 2*a*c)*d^3*e^2 + (c^2*e^5*m^4 + 10*c^2*e^5*m^3 + 35*c^2*e^5*m^2 + 50*c^
2*e^5*m + 24*c^2*e^5)*x^5 + (60*b*c*e^5 + (c^2*d*e^4 + 2*b*c*e^5)*m^4 + 2*(3*c^2
*d*e^4 + 11*b*c*e^5)*m^3 + (11*c^2*d*e^4 + 82*b*c*e^5)*m^2 + 2*(3*c^2*d*e^4 + 61
*b*c*e^5)*m)*x^4 - 2*(a*b*d^2*e^3 - 7*a^2*d*e^4)*m^3 + (40*(b^2 + 2*a*c)*e^5 + (
2*b*c*d*e^4 + (b^2 + 2*a*c)*e^5)*m^4 - 4*(c^2*d^2*e^3 - 4*b*c*d*e^4 - 3*(b^2 + 2
*a*c)*e^5)*m^3 - (12*c^2*d^2*e^3 - 34*b*c*d*e^4 - 49*(b^2 + 2*a*c)*e^5)*m^2 - 2*
(4*c^2*d^2*e^3 - 10*b*c*d*e^4 - 39*(b^2 + 2*a*c)*e^5)*m)*x^3 - (24*a*b*d^2*e^3 -
71*a^2*d*e^4 - 2*(b^2 + 2*a*c)*d^3*e^2)*m^2 + (120*a*b*e^5 + (2*a*b*e^5 + (b^2
+ 2*a*c)*d*e^4)*m^4 - 2*(3*b*c*d^2*e^3 - 13*a*b*e^5 - 5*(b^2 + 2*a*c)*d*e^4)*m^3
+ (12*c^2*d^3*e^2 - 36*b*c*d^2*e^3 + 118*a*b*e^5 + 29*(b^2 + 2*a*c)*d*e^4)*m^2
+ 2*(6*c^2*d^3*e^2 - 15*b*c*d^2*e^3 + 107*a*b*e^5 + 10*(b^2 + 2*a*c)*d*e^4)*m)*x
^2 - 2*(6*b*c*d^4*e + 47*a*b*d^2*e^3 - 77*a^2*d*e^4 - 9*(b^2 + 2*a*c)*d^3*e^2)*m
+ (120*a^2*e^5 + (2*a*b*d*e^4 + a^2*e^5)*m^4 + 2*(12*a*b*d*e^4 + 7*a^2*e^5 - (b
^2 + 2*a*c)*d^2*e^3)*m^3 + (12*b*c*d^3*e^2 + 94*a*b*d*e^4 + 71*a^2*e^5 - 18*(b^2
+ 2*a*c)*d^2*e^3)*m^2 - 2*(12*c^2*d^4*e - 30*b*c*d^3*e^2 - 60*a*b*d*e^4 - 77*a^
2*e^5 + 20*(b^2 + 2*a*c)*d^2*e^3)*m)*x)*(e*x + d)^m/(e^5*m^5 + 15*e^5*m^4 + 85*e
^5*m^3 + 225*e^5*m^2 + 274*e^5*m + 120*e^5)

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Sympy [A]  time = 27.4454, size = 10178, normalized size = 57.18 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**m*(c*x**2+b*x+a)**2,x)

[Out]

Piecewise((d**m*(a**2*x + a*b*x**2 + 2*a*c*x**3/3 + b**2*x**3/3 + b*c*x**4/2 + c
**2*x**5/5), Eq(e, 0)), (-3*a**2*e**4/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e
**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) - 2*a*b*d*e**3/(12*d**4*e**5 + 48*d**3
*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) - 8*a*b*e**4*x/(12*
d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4)
- 2*a*c*d**2*e**2/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8
*x**3 + 12*e**9*x**4) - 8*a*c*d*e**3*x/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*
e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) - 12*a*c*e**4*x**2/(12*d**4*e**5 + 48
*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) - b**2*d**2*e*
*2/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9
*x**4) - 4*b**2*d*e**3*x/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48
*d*e**8*x**3 + 12*e**9*x**4) - 6*b**2*e**4*x**2/(12*d**4*e**5 + 48*d**3*e**6*x +
72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) - 6*b*c*d**3*e/(12*d**4*e**5
+ 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) - 24*b*c*
d**2*e**2*x/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3
+ 12*e**9*x**4) - 36*b*c*d*e**3*x**2/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e*
*7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) - 24*b*c*e**4*x**3/(12*d**4*e**5 + 48*d
**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) + 12*c**2*d**4*l
og(d/e + x)/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3
+ 12*e**9*x**4) + 25*c**2*d**4/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**
2 + 48*d*e**8*x**3 + 12*e**9*x**4) + 48*c**2*d**3*e*x*log(d/e + x)/(12*d**4*e**5
+ 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) + 88*c**2
*d**3*e*x/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 +
12*e**9*x**4) + 72*c**2*d**2*e**2*x**2*log(d/e + x)/(12*d**4*e**5 + 48*d**3*e**6
*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) + 108*c**2*d**2*e**2*x**
2/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*
x**4) + 48*c**2*d*e**3*x**3*log(d/e + x)/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**
2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) + 48*c**2*d*e**3*x**3/(12*d**4*e**5
+ 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) + 12*c**2
*e**4*x**4*log(d/e + x)/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*
d*e**8*x**3 + 12*e**9*x**4), Eq(m, -5)), (-a**2*d**2*e**4/(3*d**5*e**5 + 9*d**4*
e**6*x + 9*d**3*e**7*x**2 + 3*d**2*e**8*x**3) + 3*a*b*d*e**5*x**2/(3*d**5*e**5 +
9*d**4*e**6*x + 9*d**3*e**7*x**2 + 3*d**2*e**8*x**3) + a*b*e**6*x**3/(3*d**5*e*
*5 + 9*d**4*e**6*x + 9*d**3*e**7*x**2 + 3*d**2*e**8*x**3) + 2*a*c*d*e**5*x**3/(3
*d**5*e**5 + 9*d**4*e**6*x + 9*d**3*e**7*x**2 + 3*d**2*e**8*x**3) + b**2*d*e**5*
x**3/(3*d**5*e**5 + 9*d**4*e**6*x + 9*d**3*e**7*x**2 + 3*d**2*e**8*x**3) + 6*b*c
*d**5*e*log(d/e + x)/(3*d**5*e**5 + 9*d**4*e**6*x + 9*d**3*e**7*x**2 + 3*d**2*e*
*8*x**3) + 2*b*c*d**5*e/(3*d**5*e**5 + 9*d**4*e**6*x + 9*d**3*e**7*x**2 + 3*d**2
*e**8*x**3) + 18*b*c*d**4*e**2*x*log(d/e + x)/(3*d**5*e**5 + 9*d**4*e**6*x + 9*d
**3*e**7*x**2 + 3*d**2*e**8*x**3) + 18*b*c*d**3*e**3*x**2*log(d/e + x)/(3*d**5*e
**5 + 9*d**4*e**6*x + 9*d**3*e**7*x**2 + 3*d**2*e**8*x**3) - 9*b*c*d**3*e**3*x**
2/(3*d**5*e**5 + 9*d**4*e**6*x + 9*d**3*e**7*x**2 + 3*d**2*e**8*x**3) + 6*b*c*d*
*2*e**4*x**3*log(d/e + x)/(3*d**5*e**5 + 9*d**4*e**6*x + 9*d**3*e**7*x**2 + 3*d*
*2*e**8*x**3) - 9*b*c*d**2*e**4*x**3/(3*d**5*e**5 + 9*d**4*e**6*x + 9*d**3*e**7*
x**2 + 3*d**2*e**8*x**3) - 12*c**2*d**6*log(d/e + x)/(3*d**5*e**5 + 9*d**4*e**6*
x + 9*d**3*e**7*x**2 + 3*d**2*e**8*x**3) - 4*c**2*d**6/(3*d**5*e**5 + 9*d**4*e**
6*x + 9*d**3*e**7*x**2 + 3*d**2*e**8*x**3) - 36*c**2*d**5*e*x*log(d/e + x)/(3*d*
*5*e**5 + 9*d**4*e**6*x + 9*d**3*e**7*x**2 + 3*d**2*e**8*x**3) - 36*c**2*d**4*e*
*2*x**2*log(d/e + x)/(3*d**5*e**5 + 9*d**4*e**6*x + 9*d**3*e**7*x**2 + 3*d**2*e*
*8*x**3) + 18*c**2*d**4*e**2*x**2/(3*d**5*e**5 + 9*d**4*e**6*x + 9*d**3*e**7*x**
2 + 3*d**2*e**8*x**3) - 12*c**2*d**3*e**3*x**3*log(d/e + x)/(3*d**5*e**5 + 9*d**
4*e**6*x + 9*d**3*e**7*x**2 + 3*d**2*e**8*x**3) + 18*c**2*d**3*e**3*x**3/(3*d**5
*e**5 + 9*d**4*e**6*x + 9*d**3*e**7*x**2 + 3*d**2*e**8*x**3) + 3*c**2*d**2*e**4*
x**4/(3*d**5*e**5 + 9*d**4*e**6*x + 9*d**3*e**7*x**2 + 3*d**2*e**8*x**3), Eq(m,
-4)), (-a**2*d*e**4/(2*d**3*e**5 + 4*d**2*e**6*x + 2*d*e**7*x**2) + 2*a*b*e**5*x
**2/(2*d**3*e**5 + 4*d**2*e**6*x + 2*d*e**7*x**2) + 4*a*c*d**3*e**2*log(d/e + x)
/(2*d**3*e**5 + 4*d**2*e**6*x + 2*d*e**7*x**2) + 2*a*c*d**3*e**2/(2*d**3*e**5 +
4*d**2*e**6*x + 2*d*e**7*x**2) + 8*a*c*d**2*e**3*x*log(d/e + x)/(2*d**3*e**5 + 4
*d**2*e**6*x + 2*d*e**7*x**2) + 4*a*c*d*e**4*x**2*log(d/e + x)/(2*d**3*e**5 + 4*
d**2*e**6*x + 2*d*e**7*x**2) - 4*a*c*d*e**4*x**2/(2*d**3*e**5 + 4*d**2*e**6*x +
2*d*e**7*x**2) + 2*b**2*d**3*e**2*log(d/e + x)/(2*d**3*e**5 + 4*d**2*e**6*x + 2*
d*e**7*x**2) + b**2*d**3*e**2/(2*d**3*e**5 + 4*d**2*e**6*x + 2*d*e**7*x**2) + 4*
b**2*d**2*e**3*x*log(d/e + x)/(2*d**3*e**5 + 4*d**2*e**6*x + 2*d*e**7*x**2) + 2*
b**2*d*e**4*x**2*log(d/e + x)/(2*d**3*e**5 + 4*d**2*e**6*x + 2*d*e**7*x**2) - 2*
b**2*d*e**4*x**2/(2*d**3*e**5 + 4*d**2*e**6*x + 2*d*e**7*x**2) - 12*b*c*d**4*e*l
og(d/e + x)/(2*d**3*e**5 + 4*d**2*e**6*x + 2*d*e**7*x**2) - 6*b*c*d**4*e/(2*d**3
*e**5 + 4*d**2*e**6*x + 2*d*e**7*x**2) - 24*b*c*d**3*e**2*x*log(d/e + x)/(2*d**3
*e**5 + 4*d**2*e**6*x + 2*d*e**7*x**2) - 12*b*c*d**2*e**3*x**2*log(d/e + x)/(2*d
**3*e**5 + 4*d**2*e**6*x + 2*d*e**7*x**2) + 12*b*c*d**2*e**3*x**2/(2*d**3*e**5 +
4*d**2*e**6*x + 2*d*e**7*x**2) + 4*b*c*d*e**4*x**3/(2*d**3*e**5 + 4*d**2*e**6*x
+ 2*d*e**7*x**2) + 12*c**2*d**5*log(d/e + x)/(2*d**3*e**5 + 4*d**2*e**6*x + 2*d
*e**7*x**2) + 6*c**2*d**5/(2*d**3*e**5 + 4*d**2*e**6*x + 2*d*e**7*x**2) + 24*c**
2*d**4*e*x*log(d/e + x)/(2*d**3*e**5 + 4*d**2*e**6*x + 2*d*e**7*x**2) + 12*c**2*
d**3*e**2*x**2*log(d/e + x)/(2*d**3*e**5 + 4*d**2*e**6*x + 2*d*e**7*x**2) - 12*c
**2*d**3*e**2*x**2/(2*d**3*e**5 + 4*d**2*e**6*x + 2*d*e**7*x**2) - 4*c**2*d**2*e
**3*x**3/(2*d**3*e**5 + 4*d**2*e**6*x + 2*d*e**7*x**2) + c**2*d*e**4*x**4/(2*d**
3*e**5 + 4*d**2*e**6*x + 2*d*e**7*x**2), Eq(m, -3)), (-3*a**2*e**4/(3*d*e**5 + 3
*e**6*x) + 6*a*b*d*e**3*log(d/e + x)/(3*d*e**5 + 3*e**6*x) + 6*a*b*d*e**3/(3*d*e
**5 + 3*e**6*x) + 6*a*b*e**4*x*log(d/e + x)/(3*d*e**5 + 3*e**6*x) - 12*a*c*d**2*
e**2*log(d/e + x)/(3*d*e**5 + 3*e**6*x) - 12*a*c*d**2*e**2/(3*d*e**5 + 3*e**6*x)
- 12*a*c*d*e**3*x*log(d/e + x)/(3*d*e**5 + 3*e**6*x) + 6*a*c*e**4*x**2/(3*d*e**
5 + 3*e**6*x) - 6*b**2*d**2*e**2*log(d/e + x)/(3*d*e**5 + 3*e**6*x) - 6*b**2*d**
2*e**2/(3*d*e**5 + 3*e**6*x) - 6*b**2*d*e**3*x*log(d/e + x)/(3*d*e**5 + 3*e**6*x
) + 3*b**2*e**4*x**2/(3*d*e**5 + 3*e**6*x) + 18*b*c*d**3*e*log(d/e + x)/(3*d*e**
5 + 3*e**6*x) + 18*b*c*d**3*e/(3*d*e**5 + 3*e**6*x) + 18*b*c*d**2*e**2*x*log(d/e
+ x)/(3*d*e**5 + 3*e**6*x) - 9*b*c*d*e**3*x**2/(3*d*e**5 + 3*e**6*x) + 3*b*c*e*
*4*x**3/(3*d*e**5 + 3*e**6*x) - 12*c**2*d**4*log(d/e + x)/(3*d*e**5 + 3*e**6*x)
- 12*c**2*d**4/(3*d*e**5 + 3*e**6*x) - 12*c**2*d**3*e*x*log(d/e + x)/(3*d*e**5 +
3*e**6*x) + 6*c**2*d**2*e**2*x**2/(3*d*e**5 + 3*e**6*x) - 2*c**2*d*e**3*x**3/(3
*d*e**5 + 3*e**6*x) + c**2*e**4*x**4/(3*d*e**5 + 3*e**6*x), Eq(m, -2)), (a**2*lo
g(d/e + x)/e - 2*a*b*d*log(d/e + x)/e**2 + 2*a*b*x/e + 2*a*c*d**2*log(d/e + x)/e
**3 - 2*a*c*d*x/e**2 + a*c*x**2/e + b**2*d**2*log(d/e + x)/e**3 - b**2*d*x/e**2
+ b**2*x**2/(2*e) - 2*b*c*d**3*log(d/e + x)/e**4 + 2*b*c*d**2*x/e**3 - b*c*d*x**
2/e**2 + 2*b*c*x**3/(3*e) + c**2*d**4*log(d/e + x)/e**5 - c**2*d**3*x/e**4 + c**
2*d**2*x**2/(2*e**3) - c**2*d*x**3/(3*e**2) + c**2*x**4/(4*e), Eq(m, -1)), (a**2
*d*e**4*m**4*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m*
*2 + 274*e**5*m + 120*e**5) + 14*a**2*d*e**4*m**3*(d + e*x)**m/(e**5*m**5 + 15*e
**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 71*a**2*d*e**
4*m**2*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 2
74*e**5*m + 120*e**5) + 154*a**2*d*e**4*m*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4
+ 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 120*a**2*d*e**4*(d +
e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m +
120*e**5) + a**2*e**5*m**4*x*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m*
*3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 14*a**2*e**5*m**3*x*(d + e*x)**m/(
e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5)
+ 71*a**2*e**5*m**2*x*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 2
25*e**5*m**2 + 274*e**5*m + 120*e**5) + 154*a**2*e**5*m*x*(d + e*x)**m/(e**5*m**
5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 120*a
**2*e**5*x*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2
+ 274*e**5*m + 120*e**5) - 2*a*b*d**2*e**3*m**3*(d + e*x)**m/(e**5*m**5 + 15*e*
*5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) - 24*a*b*d**2*e*
*3*m**2*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 +
274*e**5*m + 120*e**5) - 94*a*b*d**2*e**3*m*(d + e*x)**m/(e**5*m**5 + 15*e**5*m*
*4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) - 120*a*b*d**2*e**3*(
d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*
m + 120*e**5) + 2*a*b*d*e**4*m**4*x*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*
e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 24*a*b*d*e**4*m**3*x*(d + e
*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 1
20*e**5) + 94*a*b*d*e**4*m**2*x*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5
*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 120*a*b*d*e**4*m*x*(d + e*x)**m
/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**
5) + 2*a*b*e**5*m**4*x**2*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3
+ 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 26*a*b*e**5*m**3*x**2*(d + e*x)**m/(e
**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5)
+ 118*a*b*e**5*m**2*x**2*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 +
225*e**5*m**2 + 274*e**5*m + 120*e**5) + 214*a*b*e**5*m*x**2*(d + e*x)**m/(e**5
*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 1
20*a*b*e**5*x**2*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**
5*m**2 + 274*e**5*m + 120*e**5) + 4*a*c*d**3*e**2*m**2*(d + e*x)**m/(e**5*m**5 +
15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 36*a*c*d
**3*e**2*m*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2
+ 274*e**5*m + 120*e**5) + 80*a*c*d**3*e**2*(d + e*x)**m/(e**5*m**5 + 15*e**5*m
**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) - 4*a*c*d**2*e**3*m*
*3*x*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274
*e**5*m + 120*e**5) - 36*a*c*d**2*e**3*m**2*x*(d + e*x)**m/(e**5*m**5 + 15*e**5*
m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) - 80*a*c*d**2*e**3*
m*x*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*
e**5*m + 120*e**5) + 2*a*c*d*e**4*m**4*x**2*(d + e*x)**m/(e**5*m**5 + 15*e**5*m*
*4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 20*a*c*d*e**4*m**3*
x**2*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274
*e**5*m + 120*e**5) + 58*a*c*d*e**4*m**2*x**2*(d + e*x)**m/(e**5*m**5 + 15*e**5*
m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 40*a*c*d*e**4*m*x
**2*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*
e**5*m + 120*e**5) + 2*a*c*e**5*m**4*x**3*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4
+ 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 24*a*c*e**5*m**3*x**3
*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**
5*m + 120*e**5) + 98*a*c*e**5*m**2*x**3*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 +
85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 156*a*c*e**5*m*x**3*(d
+ e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m
+ 120*e**5) + 80*a*c*e**5*x**3*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*
m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 2*b**2*d**3*e**2*m**2*(d + e*x)*
*m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e
**5) + 18*b**2*d**3*e**2*m*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3
+ 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 40*b**2*d**3*e**2*(d + e*x)**m/(e**5
*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) - 2
*b**2*d**2*e**3*m**3*x*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 2
25*e**5*m**2 + 274*e**5*m + 120*e**5) - 18*b**2*d**2*e**3*m**2*x*(d + e*x)**m/(e
**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5)
- 40*b**2*d**2*e**3*m*x*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 +
225*e**5*m**2 + 274*e**5*m + 120*e**5) + b**2*d*e**4*m**4*x**2*(d + e*x)**m/(e**
5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) +
10*b**2*d*e**4*m**3*x**2*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 +
225*e**5*m**2 + 274*e**5*m + 120*e**5) + 29*b**2*d*e**4*m**2*x**2*(d + e*x)**m/
(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5
) + 20*b**2*d*e**4*m*x**2*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3
+ 225*e**5*m**2 + 274*e**5*m + 120*e**5) + b**2*e**5*m**4*x**3*(d + e*x)**m/(e**
5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) +
12*b**2*e**5*m**3*x**3*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 2
25*e**5*m**2 + 274*e**5*m + 120*e**5) + 49*b**2*e**5*m**2*x**3*(d + e*x)**m/(e**
5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) +
78*b**2*e**5*m*x**3*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*
e**5*m**2 + 274*e**5*m + 120*e**5) + 40*b**2*e**5*x**3*(d + e*x)**m/(e**5*m**5 +
15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) - 12*b*c*d
**4*e*m*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 +
274*e**5*m + 120*e**5) - 60*b*c*d**4*e*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 +
85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 12*b*c*d**3*e**2*m**2*x*
(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5
*m + 120*e**5) + 60*b*c*d**3*e**2*m*x*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 8
5*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) - 6*b*c*d**2*e**3*m**3*x**2
*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**
5*m + 120*e**5) - 36*b*c*d**2*e**3*m**2*x**2*(d + e*x)**m/(e**5*m**5 + 15*e**5*m
**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) - 30*b*c*d**2*e**3*m
*x**2*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 27
4*e**5*m + 120*e**5) + 2*b*c*d*e**4*m**4*x**3*(d + e*x)**m/(e**5*m**5 + 15*e**5*
m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 16*b*c*d*e**4*m**
3*x**3*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 2
74*e**5*m + 120*e**5) + 34*b*c*d*e**4*m**2*x**3*(d + e*x)**m/(e**5*m**5 + 15*e**
5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 20*b*c*d*e**4*m
*x**3*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 27
4*e**5*m + 120*e**5) + 2*b*c*e**5*m**4*x**4*(d + e*x)**m/(e**5*m**5 + 15*e**5*m*
*4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 22*b*c*e**5*m**3*x*
*4*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e
**5*m + 120*e**5) + 82*b*c*e**5*m**2*x**4*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4
+ 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 122*b*c*e**5*m*x**4*(
d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*
m + 120*e**5) + 60*b*c*e**5*x**4*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**
5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 24*c**2*d**5*(d + e*x)**m/(e**
5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) -
24*c**2*d**4*e*m*x*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e
**5*m**2 + 274*e**5*m + 120*e**5) + 12*c**2*d**3*e**2*m**2*x**2*(d + e*x)**m/(e*
*5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) +
12*c**2*d**3*e**2*m*x**2*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3
+ 225*e**5*m**2 + 274*e**5*m + 120*e**5) - 4*c**2*d**2*e**3*m**3*x**3*(d + e*x)*
*m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e
**5) - 12*c**2*d**2*e**3*m**2*x**3*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e
**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) - 8*c**2*d**2*e**3*m*x**3*(d +
e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m +
120*e**5) + c**2*d*e**4*m**4*x**4*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e
**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 6*c**2*d*e**4*m**3*x**4*(d +
e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m +
120*e**5) + 11*c**2*d*e**4*m**2*x**4*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 8
5*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 6*c**2*d*e**4*m*x**4*(d +
e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m +
120*e**5) + c**2*e**5*m**4*x**5*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**
5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 10*c**2*e**5*m**3*x**5*(d + e*
x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 12
0*e**5) + 35*c**2*e**5*m**2*x**5*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**
5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 50*c**2*e**5*m*x**5*(d + e*x)*
*m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e
**5) + 24*c**2*e**5*x**5*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 +
225*e**5*m**2 + 274*e**5*m + 120*e**5), True))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.221878, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^2*(e*x + d)^m,x, algorithm="giac")

[Out]

Done