3.7.29 \(\int (d+e x)^m (f+g x)^2 (a+b x+c x^2) \, dx\)
Optimal. Leaf size=220 \[ \frac {(d+e x)^{m+3} \left (e g (a e g-3 b d g+2 b e f)+c \left (6 d^2 g^2-6 d e f g+e^2 f^2\right )\right )}{e^5 (m+3)}+\frac {(e f-d g)^2 (d+e x)^{m+1} \left (a e^2-b d e+c d^2\right )}{e^5 (m+1)}-\frac {(e f-d g) (d+e x)^{m+2} (2 c d (e f-2 d g)-e (2 a e g-3 b d g+b e f))}{e^5 (m+2)}+\frac {g (d+e x)^{m+4} (b e g-4 c d g+2 c e f)}{e^5 (m+4)}+\frac {c g^2 (d+e x)^{m+5}}{e^5 (m+5)} \]
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Rubi [A] time = 0.22, antiderivative size = 220, normalized size of antiderivative = 1.00,
number of steps used = 2, number of rules used = 1, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used =
{947} \begin {gather*} \frac {(d+e x)^{m+3} \left (e g (a e g-3 b d g+2 b e f)+c \left (6 d^2 g^2-6 d e f g+e^2 f^2\right )\right )}{e^5 (m+3)}+\frac {(e f-d g)^2 (d+e x)^{m+1} \left (a e^2-b d e+c d^2\right )}{e^5 (m+1)}-\frac {(e f-d g) (d+e x)^{m+2} (2 c d (e f-2 d g)-e (2 a e g-3 b d g+b e f))}{e^5 (m+2)}+\frac {g (d+e x)^{m+4} (b e g-4 c d g+2 c e f)}{e^5 (m+4)}+\frac {c g^2 (d+e x)^{m+5}}{e^5 (m+5)} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(d + e*x)^m*(f + g*x)^2*(a + b*x + c*x^2),x]
[Out]
((c*d^2 - b*d*e + a*e^2)*(e*f - d*g)^2*(d + e*x)^(1 + m))/(e^5*(1 + m)) - ((e*f - d*g)*(2*c*d*(e*f - 2*d*g) -
e*(b*e*f - 3*b*d*g + 2*a*e*g))*(d + e*x)^(2 + m))/(e^5*(2 + m)) + ((e*g*(2*b*e*f - 3*b*d*g + a*e*g) + c*(e^2*f
^2 - 6*d*e*f*g + 6*d^2*g^2))*(d + e*x)^(3 + m))/(e^5*(3 + m)) + (g*(2*c*e*f - 4*c*d*g + b*e*g)*(d + e*x)^(4 +
m))/(e^5*(4 + m)) + (c*g^2*(d + e*x)^(5 + m))/(e^5*(5 + m))
Rule 947
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] &
& NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && (IGtQ[m, 0] || (E
qQ[m, -2] && EqQ[p, 1] && EqQ[2*c*d - b*e, 0]))
Rubi steps
\begin {align*} \int (d+e x)^m (f+g x)^2 \left (a+b x+c x^2\right ) \, dx &=\int \left (\frac {\left (c d^2-b d e+a e^2\right ) (e f-d g)^2 (d+e x)^m}{e^4}+\frac {(e f-d g) (-2 c d (e f-2 d g)+e (b e f-3 b d g+2 a e g)) (d+e x)^{1+m}}{e^4}+\frac {\left (e g (2 b e f-3 b d g+a e g)+c \left (e^2 f^2-6 d e f g+6 d^2 g^2\right )\right ) (d+e x)^{2+m}}{e^4}+\frac {g (2 c e f-4 c d g+b e g) (d+e x)^{3+m}}{e^4}+\frac {c g^2 (d+e x)^{4+m}}{e^4}\right ) \, dx\\ &=\frac {\left (c d^2-b d e+a e^2\right ) (e f-d g)^2 (d+e x)^{1+m}}{e^5 (1+m)}-\frac {(e f-d g) (2 c d (e f-2 d g)-e (b e f-3 b d g+2 a e g)) (d+e x)^{2+m}}{e^5 (2+m)}+\frac {\left (e g (2 b e f-3 b d g+a e g)+c \left (e^2 f^2-6 d e f g+6 d^2 g^2\right )\right ) (d+e x)^{3+m}}{e^5 (3+m)}+\frac {g (2 c e f-4 c d g+b e g) (d+e x)^{4+m}}{e^5 (4+m)}+\frac {c g^2 (d+e x)^{5+m}}{e^5 (5+m)}\\ \end {align*}
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Mathematica [A] time = 0.27, size = 198, normalized size = 0.90 \begin {gather*} \frac {(d+e x)^{m+1} \left (\frac {(d+e x)^2 \left (e g (a e g-3 b d g+2 b e f)+c \left (6 d^2 g^2-6 d e f g+e^2 f^2\right )\right )}{m+3}+\frac {(e f-d g)^2 \left (e (a e-b d)+c d^2\right )}{m+1}+\frac {(d+e x) (e f-d g) (e (2 a e g-3 b d g+b e f)+2 c d (2 d g-e f))}{m+2}+\frac {g (d+e x)^3 (b e g-4 c d g+2 c e f)}{m+4}+\frac {c g^2 (d+e x)^4}{m+5}\right )}{e^5} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(d + e*x)^m*(f + g*x)^2*(a + b*x + c*x^2),x]
[Out]
((d + e*x)^(1 + m)*(((c*d^2 + e*(-(b*d) + a*e))*(e*f - d*g)^2)/(1 + m) + ((e*f - d*g)*(2*c*d*(-(e*f) + 2*d*g)
+ e*(b*e*f - 3*b*d*g + 2*a*e*g))*(d + e*x))/(2 + m) + ((e*g*(2*b*e*f - 3*b*d*g + a*e*g) + c*(e^2*f^2 - 6*d*e*f
*g + 6*d^2*g^2))*(d + e*x)^2)/(3 + m) + (g*(2*c*e*f - 4*c*d*g + b*e*g)*(d + e*x)^3)/(4 + m) + (c*g^2*(d + e*x)
^4)/(5 + m)))/e^5
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IntegrateAlgebraic [F] time = 0.11, size = 0, normalized size = 0.00 \begin {gather*} \int (d+e x)^m (f+g x)^2 \left (a+b x+c x^2\right ) \, dx \end {gather*}
Verification is not applicable to the result.
[In]
IntegrateAlgebraic[(d + e*x)^m*(f + g*x)^2*(a + b*x + c*x^2),x]
[Out]
Defer[IntegrateAlgebraic][(d + e*x)^m*(f + g*x)^2*(a + b*x + c*x^2), x]
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fricas [B] time = 0.66, size = 1381, normalized size = 6.28
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x+d)^m*(g*x+f)^2*(c*x^2+b*x+a),x, algorithm="fricas")
[Out]
(a*d*e^4*f^2*m^4 + (c*e^5*g^2*m^4 + 10*c*e^5*g^2*m^3 + 35*c*e^5*g^2*m^2 + 50*c*e^5*g^2*m + 24*c*e^5*g^2)*x^5 +
(60*c*e^5*f*g + 30*b*e^5*g^2 + (2*c*e^5*f*g + (c*d*e^4 + b*e^5)*g^2)*m^4 + (22*c*e^5*f*g + (6*c*d*e^4 + 11*b*
e^5)*g^2)*m^3 + (82*c*e^5*f*g + (11*c*d*e^4 + 41*b*e^5)*g^2)*m^2 + (122*c*e^5*f*g + (6*c*d*e^4 + 61*b*e^5)*g^2
)*m)*x^4 - (2*a*d^2*e^3*f*g + (b*d^2*e^3 - 14*a*d*e^4)*f^2)*m^3 + (40*c*e^5*f^2 + 80*b*e^5*f*g + 40*a*e^5*g^2
+ (c*e^5*f^2 + 2*(c*d*e^4 + b*e^5)*f*g + (b*d*e^4 + a*e^5)*g^2)*m^4 + 4*(3*c*e^5*f^2 + 2*(2*c*d*e^4 + 3*b*e^5)
*f*g - (c*d^2*e^3 - 2*b*d*e^4 - 3*a*e^5)*g^2)*m^3 + (49*c*e^5*f^2 + 2*(17*c*d*e^4 + 49*b*e^5)*f*g - (12*c*d^2*
e^3 - 17*b*d*e^4 - 49*a*e^5)*g^2)*m^2 + 2*(39*c*e^5*f^2 + 2*(5*c*d*e^4 + 39*b*e^5)*f*g - (4*c*d^2*e^3 - 5*b*d*
e^4 - 39*a*e^5)*g^2)*m)*x^3 + 20*(2*c*d^3*e^2 - 3*b*d^2*e^3 + 6*a*d*e^4)*f^2 - 20*(3*c*d^4*e - 4*b*d^3*e^2 + 6
*a*d^2*e^3)*f*g + 2*(12*c*d^5 - 15*b*d^4*e + 20*a*d^3*e^2)*g^2 + (2*a*d^3*e^2*g^2 + (2*c*d^3*e^2 - 12*b*d^2*e^
3 + 71*a*d*e^4)*f^2 + 4*(b*d^3*e^2 - 6*a*d^2*e^3)*f*g)*m^2 + (60*b*e^5*f^2 + 120*a*e^5*f*g + (a*d*e^4*g^2 + (c
*d*e^4 + b*e^5)*f^2 + 2*(b*d*e^4 + a*e^5)*f*g)*m^4 + ((10*c*d*e^4 + 13*b*e^5)*f^2 - 2*(3*c*d^2*e^3 - 10*b*d*e^
4 - 13*a*e^5)*f*g - (3*b*d^2*e^3 - 10*a*d*e^4)*g^2)*m^3 + ((29*c*d*e^4 + 59*b*e^5)*f^2 - 2*(18*c*d^2*e^3 - 29*
b*d*e^4 - 59*a*e^5)*f*g + (12*c*d^3*e^2 - 18*b*d^2*e^3 + 29*a*d*e^4)*g^2)*m^2 + ((20*c*d*e^4 + 107*b*e^5)*f^2
- 2*(15*c*d^2*e^3 - 20*b*d*e^4 - 107*a*e^5)*f*g + (12*c*d^3*e^2 - 15*b*d^2*e^3 + 20*a*d*e^4)*g^2)*m)*x^2 + ((1
8*c*d^3*e^2 - 47*b*d^2*e^3 + 154*a*d*e^4)*f^2 - 2*(6*c*d^4*e - 18*b*d^3*e^2 + 47*a*d^2*e^3)*f*g - 6*(b*d^4*e -
3*a*d^3*e^2)*g^2)*m + (120*a*e^5*f^2 + (2*a*d*e^4*f*g + (b*d*e^4 + a*e^5)*f^2)*m^4 - 2*(a*d^2*e^3*g^2 + (c*d^
2*e^3 - 6*b*d*e^4 - 7*a*e^5)*f^2 + 2*(b*d^2*e^3 - 6*a*d*e^4)*f*g)*m^3 - ((18*c*d^2*e^3 - 47*b*d*e^4 - 71*a*e^5
)*f^2 - 2*(6*c*d^3*e^2 - 18*b*d^2*e^3 + 47*a*d*e^4)*f*g - 6*(b*d^3*e^2 - 3*a*d^2*e^3)*g^2)*m^2 - 2*((20*c*d^2*
e^3 - 30*b*d*e^4 - 77*a*e^5)*f^2 - 10*(3*c*d^3*e^2 - 4*b*d^2*e^3 + 6*a*d*e^4)*f*g + (12*c*d^4*e - 15*b*d^3*e^2
+ 20*a*d^2*e^3)*g^2)*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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giac [B] time = 0.27, size = 2740, normalized size = 12.45
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x+d)^m*(g*x+f)^2*(c*x^2+b*x+a),x, algorithm="giac")
[Out]
((x*e + d)^m*c*g^2*m^4*x^5*e^5 + (x*e + d)^m*c*d*g^2*m^4*x^4*e^4 + 2*(x*e + d)^m*c*f*g*m^4*x^4*e^5 + (x*e + d)
^m*b*g^2*m^4*x^4*e^5 + 10*(x*e + d)^m*c*g^2*m^3*x^5*e^5 + 2*(x*e + d)^m*c*d*f*g*m^4*x^3*e^4 + (x*e + d)^m*b*d*
g^2*m^4*x^3*e^4 + 6*(x*e + d)^m*c*d*g^2*m^3*x^4*e^4 - 4*(x*e + d)^m*c*d^2*g^2*m^3*x^3*e^3 + (x*e + d)^m*c*f^2*
m^4*x^3*e^5 + 2*(x*e + d)^m*b*f*g*m^4*x^3*e^5 + (x*e + d)^m*a*g^2*m^4*x^3*e^5 + 22*(x*e + d)^m*c*f*g*m^3*x^4*e
^5 + 11*(x*e + d)^m*b*g^2*m^3*x^4*e^5 + 35*(x*e + d)^m*c*g^2*m^2*x^5*e^5 + (x*e + d)^m*c*d*f^2*m^4*x^2*e^4 + 2
*(x*e + d)^m*b*d*f*g*m^4*x^2*e^4 + (x*e + d)^m*a*d*g^2*m^4*x^2*e^4 + 16*(x*e + d)^m*c*d*f*g*m^3*x^3*e^4 + 8*(x
*e + d)^m*b*d*g^2*m^3*x^3*e^4 + 11*(x*e + d)^m*c*d*g^2*m^2*x^4*e^4 - 6*(x*e + d)^m*c*d^2*f*g*m^3*x^2*e^3 - 3*(
x*e + d)^m*b*d^2*g^2*m^3*x^2*e^3 - 12*(x*e + d)^m*c*d^2*g^2*m^2*x^3*e^3 + 12*(x*e + d)^m*c*d^3*g^2*m^2*x^2*e^2
+ (x*e + d)^m*b*f^2*m^4*x^2*e^5 + 2*(x*e + d)^m*a*f*g*m^4*x^2*e^5 + 12*(x*e + d)^m*c*f^2*m^3*x^3*e^5 + 24*(x*
e + d)^m*b*f*g*m^3*x^3*e^5 + 12*(x*e + d)^m*a*g^2*m^3*x^3*e^5 + 82*(x*e + d)^m*c*f*g*m^2*x^4*e^5 + 41*(x*e + d
)^m*b*g^2*m^2*x^4*e^5 + 50*(x*e + d)^m*c*g^2*m*x^5*e^5 + (x*e + d)^m*b*d*f^2*m^4*x*e^4 + 2*(x*e + d)^m*a*d*f*g
*m^4*x*e^4 + 10*(x*e + d)^m*c*d*f^2*m^3*x^2*e^4 + 20*(x*e + d)^m*b*d*f*g*m^3*x^2*e^4 + 10*(x*e + d)^m*a*d*g^2*
m^3*x^2*e^4 + 34*(x*e + d)^m*c*d*f*g*m^2*x^3*e^4 + 17*(x*e + d)^m*b*d*g^2*m^2*x^3*e^4 + 6*(x*e + d)^m*c*d*g^2*
m*x^4*e^4 - 2*(x*e + d)^m*c*d^2*f^2*m^3*x*e^3 - 4*(x*e + d)^m*b*d^2*f*g*m^3*x*e^3 - 2*(x*e + d)^m*a*d^2*g^2*m^
3*x*e^3 - 36*(x*e + d)^m*c*d^2*f*g*m^2*x^2*e^3 - 18*(x*e + d)^m*b*d^2*g^2*m^2*x^2*e^3 - 8*(x*e + d)^m*c*d^2*g^
2*m*x^3*e^3 + 12*(x*e + d)^m*c*d^3*f*g*m^2*x*e^2 + 6*(x*e + d)^m*b*d^3*g^2*m^2*x*e^2 + 12*(x*e + d)^m*c*d^3*g^
2*m*x^2*e^2 - 24*(x*e + d)^m*c*d^4*g^2*m*x*e + (x*e + d)^m*a*f^2*m^4*x*e^5 + 13*(x*e + d)^m*b*f^2*m^3*x^2*e^5
+ 26*(x*e + d)^m*a*f*g*m^3*x^2*e^5 + 49*(x*e + d)^m*c*f^2*m^2*x^3*e^5 + 98*(x*e + d)^m*b*f*g*m^2*x^3*e^5 + 49*
(x*e + d)^m*a*g^2*m^2*x^3*e^5 + 122*(x*e + d)^m*c*f*g*m*x^4*e^5 + 61*(x*e + d)^m*b*g^2*m*x^4*e^5 + 24*(x*e + d
)^m*c*g^2*x^5*e^5 + (x*e + d)^m*a*d*f^2*m^4*e^4 + 12*(x*e + d)^m*b*d*f^2*m^3*x*e^4 + 24*(x*e + d)^m*a*d*f*g*m^
3*x*e^4 + 29*(x*e + d)^m*c*d*f^2*m^2*x^2*e^4 + 58*(x*e + d)^m*b*d*f*g*m^2*x^2*e^4 + 29*(x*e + d)^m*a*d*g^2*m^2
*x^2*e^4 + 20*(x*e + d)^m*c*d*f*g*m*x^3*e^4 + 10*(x*e + d)^m*b*d*g^2*m*x^3*e^4 - (x*e + d)^m*b*d^2*f^2*m^3*e^3
- 2*(x*e + d)^m*a*d^2*f*g*m^3*e^3 - 18*(x*e + d)^m*c*d^2*f^2*m^2*x*e^3 - 36*(x*e + d)^m*b*d^2*f*g*m^2*x*e^3 -
18*(x*e + d)^m*a*d^2*g^2*m^2*x*e^3 - 30*(x*e + d)^m*c*d^2*f*g*m*x^2*e^3 - 15*(x*e + d)^m*b*d^2*g^2*m*x^2*e^3
+ 2*(x*e + d)^m*c*d^3*f^2*m^2*e^2 + 4*(x*e + d)^m*b*d^3*f*g*m^2*e^2 + 2*(x*e + d)^m*a*d^3*g^2*m^2*e^2 + 60*(x*
e + d)^m*c*d^3*f*g*m*x*e^2 + 30*(x*e + d)^m*b*d^3*g^2*m*x*e^2 - 12*(x*e + d)^m*c*d^4*f*g*m*e - 6*(x*e + d)^m*b
*d^4*g^2*m*e + 24*(x*e + d)^m*c*d^5*g^2 + 14*(x*e + d)^m*a*f^2*m^3*x*e^5 + 59*(x*e + d)^m*b*f^2*m^2*x^2*e^5 +
118*(x*e + d)^m*a*f*g*m^2*x^2*e^5 + 78*(x*e + d)^m*c*f^2*m*x^3*e^5 + 156*(x*e + d)^m*b*f*g*m*x^3*e^5 + 78*(x*e
+ d)^m*a*g^2*m*x^3*e^5 + 60*(x*e + d)^m*c*f*g*x^4*e^5 + 30*(x*e + d)^m*b*g^2*x^4*e^5 + 14*(x*e + d)^m*a*d*f^2
*m^3*e^4 + 47*(x*e + d)^m*b*d*f^2*m^2*x*e^4 + 94*(x*e + d)^m*a*d*f*g*m^2*x*e^4 + 20*(x*e + d)^m*c*d*f^2*m*x^2*
e^4 + 40*(x*e + d)^m*b*d*f*g*m*x^2*e^4 + 20*(x*e + d)^m*a*d*g^2*m*x^2*e^4 - 12*(x*e + d)^m*b*d^2*f^2*m^2*e^3 -
24*(x*e + d)^m*a*d^2*f*g*m^2*e^3 - 40*(x*e + d)^m*c*d^2*f^2*m*x*e^3 - 80*(x*e + d)^m*b*d^2*f*g*m*x*e^3 - 40*(
x*e + d)^m*a*d^2*g^2*m*x*e^3 + 18*(x*e + d)^m*c*d^3*f^2*m*e^2 + 36*(x*e + d)^m*b*d^3*f*g*m*e^2 + 18*(x*e + d)^
m*a*d^3*g^2*m*e^2 - 60*(x*e + d)^m*c*d^4*f*g*e - 30*(x*e + d)^m*b*d^4*g^2*e + 71*(x*e + d)^m*a*f^2*m^2*x*e^5 +
107*(x*e + d)^m*b*f^2*m*x^2*e^5 + 214*(x*e + d)^m*a*f*g*m*x^2*e^5 + 40*(x*e + d)^m*c*f^2*x^3*e^5 + 80*(x*e +
d)^m*b*f*g*x^3*e^5 + 40*(x*e + d)^m*a*g^2*x^3*e^5 + 71*(x*e + d)^m*a*d*f^2*m^2*e^4 + 60*(x*e + d)^m*b*d*f^2*m*
x*e^4 + 120*(x*e + d)^m*a*d*f*g*m*x*e^4 - 47*(x*e + d)^m*b*d^2*f^2*m*e^3 - 94*(x*e + d)^m*a*d^2*f*g*m*e^3 + 40
*(x*e + d)^m*c*d^3*f^2*e^2 + 80*(x*e + d)^m*b*d^3*f*g*e^2 + 40*(x*e + d)^m*a*d^3*g^2*e^2 + 154*(x*e + d)^m*a*f
^2*m*x*e^5 + 60*(x*e + d)^m*b*f^2*x^2*e^5 + 120*(x*e + d)^m*a*f*g*x^2*e^5 + 154*(x*e + d)^m*a*d*f^2*m*e^4 - 60
*(x*e + d)^m*b*d^2*f^2*e^3 - 120*(x*e + d)^m*a*d^2*f*g*e^3 + 120*(x*e + d)^m*a*f^2*x*e^5 + 120*(x*e + d)^m*a*d
*f^2*e^4)/(m^5*e^5 + 15*m^4*e^5 + 85*m^3*e^5 + 225*m^2*e^5 + 274*m*e^5 + 120*e^5)
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maple [B] time = 0.02, size = 1347, normalized size = 6.12 \begin {gather*} \frac {\left (c \,e^{4} g^{2} m^{4} x^{4}+b \,e^{4} g^{2} m^{4} x^{3}+2 c \,e^{4} f g \,m^{4} x^{3}+10 c \,e^{4} g^{2} m^{3} x^{4}+a \,e^{4} g^{2} m^{4} x^{2}+2 b \,e^{4} f g \,m^{4} x^{2}+11 b \,e^{4} g^{2} m^{3} x^{3}-4 c d \,e^{3} g^{2} m^{3} x^{3}+c \,e^{4} f^{2} m^{4} x^{2}+22 c \,e^{4} f g \,m^{3} x^{3}+35 c \,e^{4} g^{2} m^{2} x^{4}+2 a \,e^{4} f g \,m^{4} x +12 a \,e^{4} g^{2} m^{3} x^{2}-3 b d \,e^{3} g^{2} m^{3} x^{2}+b \,e^{4} f^{2} m^{4} x +24 b \,e^{4} f g \,m^{3} x^{2}+41 b \,e^{4} g^{2} m^{2} x^{3}-6 c d \,e^{3} f g \,m^{3} x^{2}-24 c d \,e^{3} g^{2} m^{2} x^{3}+12 c \,e^{4} f^{2} m^{3} x^{2}+82 c \,e^{4} f g \,m^{2} x^{3}+50 c \,e^{4} g^{2} m \,x^{4}-2 a d \,e^{3} g^{2} m^{3} x +a \,e^{4} f^{2} m^{4}+26 a \,e^{4} f g \,m^{3} x +49 a \,e^{4} g^{2} m^{2} x^{2}-4 b d \,e^{3} f g \,m^{3} x -24 b d \,e^{3} g^{2} m^{2} x^{2}+13 b \,e^{4} f^{2} m^{3} x +98 b \,e^{4} f g \,m^{2} x^{2}+61 b \,e^{4} g^{2} m \,x^{3}+12 c \,d^{2} e^{2} g^{2} m^{2} x^{2}-2 c d \,e^{3} f^{2} m^{3} x -48 c d \,e^{3} f g \,m^{2} x^{2}-44 c d \,e^{3} g^{2} m \,x^{3}+49 c \,e^{4} f^{2} m^{2} x^{2}+122 c \,e^{4} f g m \,x^{3}+24 c \,g^{2} x^{4} e^{4}-2 a d \,e^{3} f g \,m^{3}-20 a d \,e^{3} g^{2} m^{2} x +14 a \,e^{4} f^{2} m^{3}+118 a \,e^{4} f g \,m^{2} x +78 a \,e^{4} g^{2} m \,x^{2}+6 b \,d^{2} e^{2} g^{2} m^{2} x -b d \,e^{3} f^{2} m^{3}-40 b d \,e^{3} f g \,m^{2} x -51 b d \,e^{3} g^{2} m \,x^{2}+59 b \,e^{4} f^{2} m^{2} x +156 b \,e^{4} f g m \,x^{2}+30 b \,e^{4} g^{2} x^{3}+12 c \,d^{2} e^{2} f g \,m^{2} x +36 c \,d^{2} e^{2} g^{2} m \,x^{2}-20 c d \,e^{3} f^{2} m^{2} x -102 c d \,e^{3} f g m \,x^{2}-24 c d \,e^{3} g^{2} x^{3}+78 c \,e^{4} f^{2} m \,x^{2}+60 c \,e^{4} f g \,x^{3}+2 a \,d^{2} e^{2} g^{2} m^{2}-24 a d \,e^{3} f g \,m^{2}-58 a d \,e^{3} g^{2} m x +71 a \,e^{4} f^{2} m^{2}+214 a \,e^{4} f g m x +40 a \,e^{4} g^{2} x^{2}+4 b \,d^{2} e^{2} f g \,m^{2}+36 b \,d^{2} e^{2} g^{2} m x -12 b d \,e^{3} f^{2} m^{2}-116 b d \,e^{3} f g m x -30 b d \,e^{3} g^{2} x^{2}+107 b \,e^{4} f^{2} m x +80 b \,e^{4} f g \,x^{2}-24 c \,d^{3} e \,g^{2} m x +2 c \,d^{2} e^{2} f^{2} m^{2}+72 c \,d^{2} e^{2} f g m x +24 c \,d^{2} e^{2} g^{2} x^{2}-58 c d \,e^{3} f^{2} m x -60 c d \,e^{3} f g \,x^{2}+40 c \,e^{4} f^{2} x^{2}+18 a \,d^{2} e^{2} g^{2} m -94 a d \,e^{3} f g m -40 a d \,e^{3} g^{2} x +154 a \,e^{4} f^{2} m +120 a \,e^{4} f g x -6 b \,d^{3} e \,g^{2} m +36 b \,d^{2} e^{2} f g m +30 b \,d^{2} e^{2} g^{2} x -47 b d \,e^{3} f^{2} m -80 b d \,e^{3} f g x +60 b \,e^{4} f^{2} x -12 c \,d^{3} e f g m -24 c \,d^{3} e \,g^{2} x +18 c \,d^{2} e^{2} f^{2} m +60 c \,d^{2} e^{2} f g x -40 c d \,e^{3} f^{2} x +40 a \,d^{2} e^{2} g^{2}-120 a d \,e^{3} f g +120 a \,e^{4} f^{2}-30 b \,d^{3} e \,g^{2}+80 b \,d^{2} e^{2} f g -60 b d \,e^{3} f^{2}+24 c \,d^{4} g^{2}-60 c \,d^{3} e f g +40 c \,d^{2} e^{2} f^{2}\right ) \left (e x +d \right )^{m +1}}{\left (m^{5}+15 m^{4}+85 m^{3}+225 m^{2}+274 m +120\right ) e^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((e*x+d)^m*(g*x+f)^2*(c*x^2+b*x+a),x)
[Out]
(e*x+d)^(1+m)*(c*e^4*g^2*m^4*x^4+b*e^4*g^2*m^4*x^3+2*c*e^4*f*g*m^4*x^3+10*c*e^4*g^2*m^3*x^4+a*e^4*g^2*m^4*x^2+
2*b*e^4*f*g*m^4*x^2+11*b*e^4*g^2*m^3*x^3-4*c*d*e^3*g^2*m^3*x^3+c*e^4*f^2*m^4*x^2+22*c*e^4*f*g*m^3*x^3+35*c*e^4
*g^2*m^2*x^4+2*a*e^4*f*g*m^4*x+12*a*e^4*g^2*m^3*x^2-3*b*d*e^3*g^2*m^3*x^2+b*e^4*f^2*m^4*x+24*b*e^4*f*g*m^3*x^2
+41*b*e^4*g^2*m^2*x^3-6*c*d*e^3*f*g*m^3*x^2-24*c*d*e^3*g^2*m^2*x^3+12*c*e^4*f^2*m^3*x^2+82*c*e^4*f*g*m^2*x^3+5
0*c*e^4*g^2*m*x^4-2*a*d*e^3*g^2*m^3*x+a*e^4*f^2*m^4+26*a*e^4*f*g*m^3*x+49*a*e^4*g^2*m^2*x^2-4*b*d*e^3*f*g*m^3*
x-24*b*d*e^3*g^2*m^2*x^2+13*b*e^4*f^2*m^3*x+98*b*e^4*f*g*m^2*x^2+61*b*e^4*g^2*m*x^3+12*c*d^2*e^2*g^2*m^2*x^2-2
*c*d*e^3*f^2*m^3*x-48*c*d*e^3*f*g*m^2*x^2-44*c*d*e^3*g^2*m*x^3+49*c*e^4*f^2*m^2*x^2+122*c*e^4*f*g*m*x^3+24*c*e
^4*g^2*x^4-2*a*d*e^3*f*g*m^3-20*a*d*e^3*g^2*m^2*x+14*a*e^4*f^2*m^3+118*a*e^4*f*g*m^2*x+78*a*e^4*g^2*m*x^2+6*b*
d^2*e^2*g^2*m^2*x-b*d*e^3*f^2*m^3-40*b*d*e^3*f*g*m^2*x-51*b*d*e^3*g^2*m*x^2+59*b*e^4*f^2*m^2*x+156*b*e^4*f*g*m
*x^2+30*b*e^4*g^2*x^3+12*c*d^2*e^2*f*g*m^2*x+36*c*d^2*e^2*g^2*m*x^2-20*c*d*e^3*f^2*m^2*x-102*c*d*e^3*f*g*m*x^2
-24*c*d*e^3*g^2*x^3+78*c*e^4*f^2*m*x^2+60*c*e^4*f*g*x^3+2*a*d^2*e^2*g^2*m^2-24*a*d*e^3*f*g*m^2-58*a*d*e^3*g^2*
m*x+71*a*e^4*f^2*m^2+214*a*e^4*f*g*m*x+40*a*e^4*g^2*x^2+4*b*d^2*e^2*f*g*m^2+36*b*d^2*e^2*g^2*m*x-12*b*d*e^3*f^
2*m^2-116*b*d*e^3*f*g*m*x-30*b*d*e^3*g^2*x^2+107*b*e^4*f^2*m*x+80*b*e^4*f*g*x^2-24*c*d^3*e*g^2*m*x+2*c*d^2*e^2
*f^2*m^2+72*c*d^2*e^2*f*g*m*x+24*c*d^2*e^2*g^2*x^2-58*c*d*e^3*f^2*m*x-60*c*d*e^3*f*g*x^2+40*c*e^4*f^2*x^2+18*a
*d^2*e^2*g^2*m-94*a*d*e^3*f*g*m-40*a*d*e^3*g^2*x+154*a*e^4*f^2*m+120*a*e^4*f*g*x-6*b*d^3*e*g^2*m+36*b*d^2*e^2*
f*g*m+30*b*d^2*e^2*g^2*x-47*b*d*e^3*f^2*m-80*b*d*e^3*f*g*x+60*b*e^4*f^2*x-12*c*d^3*e*f*g*m-24*c*d^3*e*g^2*x+18
*c*d^2*e^2*f^2*m+60*c*d^2*e^2*f*g*x-40*c*d*e^3*f^2*x+40*a*d^2*e^2*g^2-120*a*d*e^3*f*g+120*a*e^4*f^2-30*b*d^3*e
*g^2+80*b*d^2*e^2*f*g-60*b*d*e^3*f^2+24*c*d^4*g^2-60*c*d^3*e*f*g+40*c*d^2*e^2*f^2)/e^5/(m^5+15*m^4+85*m^3+225*
m^2+274*m+120)
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maxima [B] time = 0.58, size = 684, normalized size = 3.11 \begin {gather*} \frac {{\left (e^{2} {\left (m + 1\right )} x^{2} + d e m x - d^{2}\right )} {\left (e x + d\right )}^{m} b f^{2}}{{\left (m^{2} + 3 \, m + 2\right )} e^{2}} + \frac {2 \, {\left (e^{2} {\left (m + 1\right )} x^{2} + d e m x - d^{2}\right )} {\left (e x + d\right )}^{m} a f g}{{\left (m^{2} + 3 \, m + 2\right )} e^{2}} + \frac {{\left (e x + d\right )}^{m + 1} a f^{2}}{e {\left (m + 1\right )}} + \frac {{\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} c f^{2}}{{\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{3}} + \frac {2 \, {\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} b f g}{{\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{3}} + \frac {{\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 g^{2}}{{\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{3}} + \frac {2 \, {\left ({\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{4} x^{4} + {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d e^{3} x^{3} - 3 \, {\left (m^{2} + m\right )} d^{2} e^{2} x^{2} + 6 \, d^{3} e m x - 6 \, d^{4}\right )} {\left (e x + d\right )}^{m} c f g}{{\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} e^{4}} + \frac {{\left ({\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{4} x^{4} + {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d e^{3} x^{3} - 3 \, {\left (m^{2} + m\right )} d^{2} e^{2} x^{2} + 6 \, d^{3} e m x - 6 \, d^{4}\right )} {\left (e x + d\right )}^{m} b g^{2}}{{\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} e^{4}} + \frac {{\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} c g^{2}}{{\left (m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120\right )} e^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x+d)^m*(g*x+f)^2*(c*x^2+b*x+a),x, algorithm="maxima")
[Out]
(e^2*(m + 1)*x^2 + d*e*m*x - d^2)*(e*x + d)^m*b*f^2/((m^2 + 3*m + 2)*e^2) + 2*(e^2*(m + 1)*x^2 + d*e*m*x - d^2
)*(e*x + d)^m*a*f*g/((m^2 + 3*m + 2)*e^2) + (e*x + d)^(m + 1)*a*f^2/(e*(m + 1)) + ((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*c*f^2/((m^3 + 6*m^2 + 11*m + 6)*e^3) + 2*((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*b*f*g/((m^3 + 6*m^2 + 11*m + 6)*e^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*g^2/((m^3 + 6*m^2 + 11*m + 6)*
e^3) + 2*((m^3 + 6*m^2 + 11*m + 6)*e^4*x^4 + (m^3 + 3*m^2 + 2*m)*d*e^3*x^3 - 3*(m^2 + m)*d^2*e^2*x^2 + 6*d^3*e
*m*x - 6*d^4)*(e*x + d)^m*c*f*g/((m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*e^4) + ((m^3 + 6*m^2 + 11*m + 6)*e^4*x^4
+ (m^3 + 3*m^2 + 2*m)*d*e^3*x^3 - 3*(m^2 + m)*d^2*e^2*x^2 + 6*d^3*e*m*x - 6*d^4)*(e*x + d)^m*b*g^2/((m^4 + 10*
m^3 + 35*m^2 + 50*m + 24)*e^4) + ((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*c
*g^2/((m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m + 120)*e^5)
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mupad [B] time = 3.94, size = 1354, normalized size = 6.15 \begin {gather*} \frac {{\left (d+e\,x\right )}^m\,\left (24\,c\,d^5\,g^2-12\,c\,d^4\,e\,f\,g\,m-60\,c\,d^4\,e\,f\,g-6\,b\,d^4\,e\,g^2\,m-30\,b\,d^4\,e\,g^2+2\,c\,d^3\,e^2\,f^2\,m^2+18\,c\,d^3\,e^2\,f^2\,m+40\,c\,d^3\,e^2\,f^2+4\,b\,d^3\,e^2\,f\,g\,m^2+36\,b\,d^3\,e^2\,f\,g\,m+80\,b\,d^3\,e^2\,f\,g+2\,a\,d^3\,e^2\,g^2\,m^2+18\,a\,d^3\,e^2\,g^2\,m+40\,a\,d^3\,e^2\,g^2-b\,d^2\,e^3\,f^2\,m^3-12\,b\,d^2\,e^3\,f^2\,m^2-47\,b\,d^2\,e^3\,f^2\,m-60\,b\,d^2\,e^3\,f^2-2\,a\,d^2\,e^3\,f\,g\,m^3-24\,a\,d^2\,e^3\,f\,g\,m^2-94\,a\,d^2\,e^3\,f\,g\,m-120\,a\,d^2\,e^3\,f\,g+a\,d\,e^4\,f^2\,m^4+14\,a\,d\,e^4\,f^2\,m^3+71\,a\,d\,e^4\,f^2\,m^2+154\,a\,d\,e^4\,f^2\,m+120\,a\,d\,e^4\,f^2\right )}{e^5\,\left (m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120\right )}+\frac {x\,{\left (d+e\,x\right )}^m\,\left (-24\,c\,d^4\,e\,g^2\,m+12\,c\,d^3\,e^2\,f\,g\,m^2+60\,c\,d^3\,e^2\,f\,g\,m+6\,b\,d^3\,e^2\,g^2\,m^2+30\,b\,d^3\,e^2\,g^2\,m-2\,c\,d^2\,e^3\,f^2\,m^3-18\,c\,d^2\,e^3\,f^2\,m^2-40\,c\,d^2\,e^3\,f^2\,m-4\,b\,d^2\,e^3\,f\,g\,m^3-36\,b\,d^2\,e^3\,f\,g\,m^2-80\,b\,d^2\,e^3\,f\,g\,m-2\,a\,d^2\,e^3\,g^2\,m^3-18\,a\,d^2\,e^3\,g^2\,m^2-40\,a\,d^2\,e^3\,g^2\,m+b\,d\,e^4\,f^2\,m^4+12\,b\,d\,e^4\,f^2\,m^3+47\,b\,d\,e^4\,f^2\,m^2+60\,b\,d\,e^4\,f^2\,m+2\,a\,d\,e^4\,f\,g\,m^4+24\,a\,d\,e^4\,f\,g\,m^3+94\,a\,d\,e^4\,f\,g\,m^2+120\,a\,d\,e^4\,f\,g\,m+a\,e^5\,f^2\,m^4+14\,a\,e^5\,f^2\,m^3+71\,a\,e^5\,f^2\,m^2+154\,a\,e^5\,f^2\,m+120\,a\,e^5\,f^2\right )}{e^5\,\left (m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120\right )}+\frac {c\,g^2\,x^5\,{\left (d+e\,x\right )}^m\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )}{m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120}+\frac {x^2\,\left (m+1\right )\,{\left (d+e\,x\right )}^m\,\left (12\,c\,d^3\,g^2\,m-6\,c\,d^2\,e\,f\,g\,m^2-30\,c\,d^2\,e\,f\,g\,m-3\,b\,d^2\,e\,g^2\,m^2-15\,b\,d^2\,e\,g^2\,m+c\,d\,e^2\,f^2\,m^3+9\,c\,d\,e^2\,f^2\,m^2+20\,c\,d\,e^2\,f^2\,m+2\,b\,d\,e^2\,f\,g\,m^3+18\,b\,d\,e^2\,f\,g\,m^2+40\,b\,d\,e^2\,f\,g\,m+a\,d\,e^2\,g^2\,m^3+9\,a\,d\,e^2\,g^2\,m^2+20\,a\,d\,e^2\,g^2\,m+b\,e^3\,f^2\,m^3+12\,b\,e^3\,f^2\,m^2+47\,b\,e^3\,f^2\,m+60\,b\,e^3\,f^2+2\,a\,e^3\,f\,g\,m^3+24\,a\,e^3\,f\,g\,m^2+94\,a\,e^3\,f\,g\,m+120\,a\,e^3\,f\,g\right )}{e^3\,\left (m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120\right )}+\frac {x^3\,{\left (d+e\,x\right )}^m\,\left (m^2+3\,m+2\right )\,\left (-4\,c\,d^2\,g^2\,m+2\,c\,d\,e\,f\,g\,m^2+10\,c\,d\,e\,f\,g\,m+b\,d\,e\,g^2\,m^2+5\,b\,d\,e\,g^2\,m+c\,e^2\,f^2\,m^2+9\,c\,e^2\,f^2\,m+20\,c\,e^2\,f^2+2\,b\,e^2\,f\,g\,m^2+18\,b\,e^2\,f\,g\,m+40\,b\,e^2\,f\,g+a\,e^2\,g^2\,m^2+9\,a\,e^2\,g^2\,m+20\,a\,e^2\,g^2\right )}{e^2\,\left (m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120\right )}+\frac {g\,x^4\,{\left (d+e\,x\right )}^m\,\left (m^3+6\,m^2+11\,m+6\right )\,\left (5\,b\,e\,g+10\,c\,e\,f+b\,e\,g\,m+c\,d\,g\,m+2\,c\,e\,f\,m\right )}{e\,\left (m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((f + g*x)^2*(d + e*x)^m*(a + b*x + c*x^2),x)
[Out]
((d + e*x)^m*(24*c*d^5*g^2 + 40*a*d^3*e^2*g^2 - 60*b*d^2*e^3*f^2 + 40*c*d^3*e^2*f^2 + 120*a*d*e^4*f^2 - 30*b*d
^4*e*g^2 - 120*a*d^2*e^3*f*g + 80*b*d^3*e^2*f*g + 154*a*d*e^4*f^2*m - 6*b*d^4*e*g^2*m + 71*a*d*e^4*f^2*m^2 + 1
4*a*d*e^4*f^2*m^3 + a*d*e^4*f^2*m^4 + 18*a*d^3*e^2*g^2*m - 47*b*d^2*e^3*f^2*m + 18*c*d^3*e^2*f^2*m - 60*c*d^4*
e*f*g + 2*a*d^3*e^2*g^2*m^2 - 12*b*d^2*e^3*f^2*m^2 - b*d^2*e^3*f^2*m^3 + 2*c*d^3*e^2*f^2*m^2 - 12*c*d^4*e*f*g*
m - 94*a*d^2*e^3*f*g*m + 36*b*d^3*e^2*f*g*m - 24*a*d^2*e^3*f*g*m^2 - 2*a*d^2*e^3*f*g*m^3 + 4*b*d^3*e^2*f*g*m^2
))/(e^5*(274*m + 225*m^2 + 85*m^3 + 15*m^4 + m^5 + 120)) + (x*(d + e*x)^m*(120*a*e^5*f^2 + 71*a*e^5*f^2*m^2 +
14*a*e^5*f^2*m^3 + a*e^5*f^2*m^4 + 154*a*e^5*f^2*m + 60*b*d*e^4*f^2*m - 24*c*d^4*e*g^2*m - 40*a*d^2*e^3*g^2*m
+ 47*b*d*e^4*f^2*m^2 + 12*b*d*e^4*f^2*m^3 + b*d*e^4*f^2*m^4 + 30*b*d^3*e^2*g^2*m - 40*c*d^2*e^3*f^2*m - 18*a*d
^2*e^3*g^2*m^2 - 2*a*d^2*e^3*g^2*m^3 + 6*b*d^3*e^2*g^2*m^2 - 18*c*d^2*e^3*f^2*m^2 - 2*c*d^2*e^3*f^2*m^3 + 120*
a*d*e^4*f*g*m + 94*a*d*e^4*f*g*m^2 + 24*a*d*e^4*f*g*m^3 + 2*a*d*e^4*f*g*m^4 - 80*b*d^2*e^3*f*g*m + 60*c*d^3*e^
2*f*g*m - 36*b*d^2*e^3*f*g*m^2 - 4*b*d^2*e^3*f*g*m^3 + 12*c*d^3*e^2*f*g*m^2))/(e^5*(274*m + 225*m^2 + 85*m^3 +
15*m^4 + m^5 + 120)) + (c*g^2*x^5*(d + e*x)^m*(50*m + 35*m^2 + 10*m^3 + m^4 + 24))/(274*m + 225*m^2 + 85*m^3
+ 15*m^4 + m^5 + 120) + (x^2*(m + 1)*(d + e*x)^m*(60*b*e^3*f^2 + 12*b*e^3*f^2*m^2 + b*e^3*f^2*m^3 + 120*a*e^3*
f*g + 47*b*e^3*f^2*m + 12*c*d^3*g^2*m + 20*a*d*e^2*g^2*m - 15*b*d^2*e*g^2*m + 20*c*d*e^2*f^2*m + 24*a*e^3*f*g*
m^2 + 2*a*e^3*f*g*m^3 + 9*a*d*e^2*g^2*m^2 + a*d*e^2*g^2*m^3 - 3*b*d^2*e*g^2*m^2 + 9*c*d*e^2*f^2*m^2 + c*d*e^2*
f^2*m^3 + 94*a*e^3*f*g*m + 40*b*d*e^2*f*g*m - 30*c*d^2*e*f*g*m + 18*b*d*e^2*f*g*m^2 + 2*b*d*e^2*f*g*m^3 - 6*c*
d^2*e*f*g*m^2))/(e^3*(274*m + 225*m^2 + 85*m^3 + 15*m^4 + m^5 + 120)) + (x^3*(d + e*x)^m*(3*m + m^2 + 2)*(20*a
*e^2*g^2 + 20*c*e^2*f^2 + a*e^2*g^2*m^2 + c*e^2*f^2*m^2 + 40*b*e^2*f*g + 9*a*e^2*g^2*m - 4*c*d^2*g^2*m + 9*c*e
^2*f^2*m + b*d*e*g^2*m^2 + 2*b*e^2*f*g*m^2 + 5*b*d*e*g^2*m + 18*b*e^2*f*g*m + 2*c*d*e*f*g*m^2 + 10*c*d*e*f*g*m
))/(e^2*(274*m + 225*m^2 + 85*m^3 + 15*m^4 + m^5 + 120)) + (g*x^4*(d + e*x)^m*(11*m + 6*m^2 + m^3 + 6)*(5*b*e*
g + 10*c*e*f + b*e*g*m + c*d*g*m + 2*c*e*f*m))/(e*(274*m + 225*m^2 + 85*m^3 + 15*m^4 + m^5 + 120))
________________________________________________________________________________________
sympy [A] time = 14.70, size = 15757, normalized size = 71.62
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x+d)**m*(g*x+f)**2*(c*x**2+b*x+a),x)
[Out]
Piecewise((d**m*(a*f**2*x + a*f*g*x**2 + a*g**2*x**3/3 + b*f**2*x**2/2 + 2*b*f*g*x**3/3 + b*g**2*x**4/4 + c*f*
*2*x**3/3 + c*f*g*x**4/2 + c*g**2*x**5/5), Eq(e, 0)), (-a*d**2*e**2*g**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) - 2*a*d*e**3*f*g/(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*a*d*e**3*g**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) - 3*a*e**4*f**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*e**4*f*g*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 + 1
2*e**9*x**4) - 6*a*e**4*g**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) - 3*b*d**3*e*g**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)
- 2*b*d**2*e**2*f*g/(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*b
*d**2*e**2*g**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) - b*d*e*
*3*f**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*b*d*e**3*f*g*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) - 18*b*d*e**3*g**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) - 4*b*e**4*f**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) - 12*b*e**4*f*g*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) - 12*b*e**4*g**2*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*d**4*g**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) + 25*c*d**4*g**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*c*d**3*e*f*g/(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*d**3*e*g**2*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*d**3*e*g**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) - c*d**2*e**2*f**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*c*d**2*e**2*f*g*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*d**2*e**2*g**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*d**2*e**2*g**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) - 4*c*d*e**3*f**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*c*d*e**3*f*g*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*d*e**3*g**2*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*d*e**3*g**2*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) - 6*c*e**4*f**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) - 24*c*e**4*f*g*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*e**4*g**2*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)), (-2*a*d**2*e**2*g**2/(6*d**3*e**5 + 1
8*d**2*e**6*x + 18*d*e**7*x**2 + 6*e**8*x**3) - 2*a*d*e**3*f*g/(6*d**3*e**5 + 18*d**2*e**6*x + 18*d*e**7*x**2
+ 6*e**8*x**3) - 6*a*d*e**3*g**2*x/(6*d**3*e**5 + 18*d**2*e**6*x + 18*d*e**7*x**2 + 6*e**8*x**3) - 2*a*e**4*f*
*2/(6*d**3*e**5 + 18*d**2*e**6*x + 18*d*e**7*x**2 + 6*e**8*x**3) - 6*a*e**4*f*g*x/(6*d**3*e**5 + 18*d**2*e**6*
x + 18*d*e**7*x**2 + 6*e**8*x**3) - 6*a*e**4*g**2*x**2/(6*d**3*e**5 + 18*d**2*e**6*x + 18*d*e**7*x**2 + 6*e**8
*x**3) + 6*b*d**3*e*g**2*log(d/e + x)/(6*d**3*e**5 + 18*d**2*e**6*x + 18*d*e**7*x**2 + 6*e**8*x**3) + 11*b*d**
3*e*g**2/(6*d**3*e**5 + 18*d**2*e**6*x + 18*d*e**7*x**2 + 6*e**8*x**3) - 4*b*d**2*e**2*f*g/(6*d**3*e**5 + 18*d
**2*e**6*x + 18*d*e**7*x**2 + 6*e**8*x**3) + 18*b*d**2*e**2*g**2*x*log(d/e + x)/(6*d**3*e**5 + 18*d**2*e**6*x
+ 18*d*e**7*x**2 + 6*e**8*x**3) + 27*b*d**2*e**2*g**2*x/(6*d**3*e**5 + 18*d**2*e**6*x + 18*d*e**7*x**2 + 6*e**
8*x**3) - b*d*e**3*f**2/(6*d**3*e**5 + 18*d**2*e**6*x + 18*d*e**7*x**2 + 6*e**8*x**3) - 12*b*d*e**3*f*g*x/(6*d
**3*e**5 + 18*d**2*e**6*x + 18*d*e**7*x**2 + 6*e**8*x**3) + 18*b*d*e**3*g**2*x**2*log(d/e + x)/(6*d**3*e**5 +
18*d**2*e**6*x + 18*d*e**7*x**2 + 6*e**8*x**3) + 18*b*d*e**3*g**2*x**2/(6*d**3*e**5 + 18*d**2*e**6*x + 18*d*e*
*7*x**2 + 6*e**8*x**3) - 3*b*e**4*f**2*x/(6*d**3*e**5 + 18*d**2*e**6*x + 18*d*e**7*x**2 + 6*e**8*x**3) - 12*b*
e**4*f*g*x**2/(6*d**3*e**5 + 18*d**2*e**6*x + 18*d*e**7*x**2 + 6*e**8*x**3) + 6*b*e**4*g**2*x**3*log(d/e + x)/
(6*d**3*e**5 + 18*d**2*e**6*x + 18*d*e**7*x**2 + 6*e**8*x**3) - 24*c*d**4*g**2*log(d/e + x)/(6*d**3*e**5 + 18*
d**2*e**6*x + 18*d*e**7*x**2 + 6*e**8*x**3) - 44*c*d**4*g**2/(6*d**3*e**5 + 18*d**2*e**6*x + 18*d*e**7*x**2 +
6*e**8*x**3) + 12*c*d**3*e*f*g*log(d/e + x)/(6*d**3*e**5 + 18*d**2*e**6*x + 18*d*e**7*x**2 + 6*e**8*x**3) + 22
*c*d**3*e*f*g/(6*d**3*e**5 + 18*d**2*e**6*x + 18*d*e**7*x**2 + 6*e**8*x**3) - 72*c*d**3*e*g**2*x*log(d/e + x)/
(6*d**3*e**5 + 18*d**2*e**6*x + 18*d*e**7*x**2 + 6*e**8*x**3) - 108*c*d**3*e*g**2*x/(6*d**3*e**5 + 18*d**2*e**
6*x + 18*d*e**7*x**2 + 6*e**8*x**3) - 2*c*d**2*e**2*f**2/(6*d**3*e**5 + 18*d**2*e**6*x + 18*d*e**7*x**2 + 6*e*
*8*x**3) + 36*c*d**2*e**2*f*g*x*log(d/e + x)/(6*d**3*e**5 + 18*d**2*e**6*x + 18*d*e**7*x**2 + 6*e**8*x**3) + 5
4*c*d**2*e**2*f*g*x/(6*d**3*e**5 + 18*d**2*e**6*x + 18*d*e**7*x**2 + 6*e**8*x**3) - 72*c*d**2*e**2*g**2*x**2*l
og(d/e + x)/(6*d**3*e**5 + 18*d**2*e**6*x + 18*d*e**7*x**2 + 6*e**8*x**3) - 72*c*d**2*e**2*g**2*x**2/(6*d**3*e
**5 + 18*d**2*e**6*x + 18*d*e**7*x**2 + 6*e**8*x**3) - 6*c*d*e**3*f**2*x/(6*d**3*e**5 + 18*d**2*e**6*x + 18*d*
e**7*x**2 + 6*e**8*x**3) + 36*c*d*e**3*f*g*x**2*log(d/e + x)/(6*d**3*e**5 + 18*d**2*e**6*x + 18*d*e**7*x**2 +
6*e**8*x**3) + 36*c*d*e**3*f*g*x**2/(6*d**3*e**5 + 18*d**2*e**6*x + 18*d*e**7*x**2 + 6*e**8*x**3) - 24*c*d*e**
3*g**2*x**3*log(d/e + x)/(6*d**3*e**5 + 18*d**2*e**6*x + 18*d*e**7*x**2 + 6*e**8*x**3) - 6*c*e**4*f**2*x**2/(6
*d**3*e**5 + 18*d**2*e**6*x + 18*d*e**7*x**2 + 6*e**8*x**3) + 12*c*e**4*f*g*x**3*log(d/e + x)/(6*d**3*e**5 + 1
8*d**2*e**6*x + 18*d*e**7*x**2 + 6*e**8*x**3) + 6*c*e**4*g**2*x**4/(6*d**3*e**5 + 18*d**2*e**6*x + 18*d*e**7*x
**2 + 6*e**8*x**3), Eq(m, -4)), (2*a*d**2*e**2*g**2*log(d/e + x)/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + 3*
a*d**2*e**2*g**2/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) - 2*a*d*e**3*f*g/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*
x**2) + 4*a*d*e**3*g**2*x*log(d/e + x)/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + 4*a*d*e**3*g**2*x/(2*d**2*e*
*5 + 4*d*e**6*x + 2*e**7*x**2) - a*e**4*f**2/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) - 4*a*e**4*f*g*x/(2*d**2
*e**5 + 4*d*e**6*x + 2*e**7*x**2) + 2*a*e**4*g**2*x**2*log(d/e + x)/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) -
6*b*d**3*e*g**2*log(d/e + x)/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) - 9*b*d**3*e*g**2/(2*d**2*e**5 + 4*d*e*
*6*x + 2*e**7*x**2) + 4*b*d**2*e**2*f*g*log(d/e + x)/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + 6*b*d**2*e**2*
f*g/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) - 12*b*d**2*e**2*g**2*x*log(d/e + x)/(2*d**2*e**5 + 4*d*e**6*x +
2*e**7*x**2) - 12*b*d**2*e**2*g**2*x/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) - b*d*e**3*f**2/(2*d**2*e**5 + 4
*d*e**6*x + 2*e**7*x**2) + 8*b*d*e**3*f*g*x*log(d/e + x)/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + 8*b*d*e**3
*f*g*x/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) - 6*b*d*e**3*g**2*x**2*log(d/e + x)/(2*d**2*e**5 + 4*d*e**6*x
+ 2*e**7*x**2) - 2*b*e**4*f**2*x/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + 4*b*e**4*f*g*x**2*log(d/e + x)/(2*
d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + 2*b*e**4*g**2*x**3/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + 12*c*d**
4*g**2*log(d/e + x)/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + 18*c*d**4*g**2/(2*d**2*e**5 + 4*d*e**6*x + 2*e*
*7*x**2) - 12*c*d**3*e*f*g*log(d/e + x)/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) - 18*c*d**3*e*f*g/(2*d**2*e**
5 + 4*d*e**6*x + 2*e**7*x**2) + 24*c*d**3*e*g**2*x*log(d/e + x)/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + 24*
c*d**3*e*g**2*x/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + 2*c*d**2*e**2*f**2*log(d/e + x)/(2*d**2*e**5 + 4*d*
e**6*x + 2*e**7*x**2) + 3*c*d**2*e**2*f**2/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) - 24*c*d**2*e**2*f*g*x*log
(d/e + x)/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) - 24*c*d**2*e**2*f*g*x/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x
**2) + 12*c*d**2*e**2*g**2*x**2*log(d/e + x)/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + 4*c*d*e**3*f**2*x*log(
d/e + x)/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + 4*c*d*e**3*f**2*x/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2)
- 12*c*d*e**3*f*g*x**2*log(d/e + x)/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) - 4*c*d*e**3*g**2*x**3/(2*d**2*e
**5 + 4*d*e**6*x + 2*e**7*x**2) + 2*c*e**4*f**2*x**2*log(d/e + x)/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + 4
*c*e**4*f*g*x**3/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + c*e**4*g**2*x**4/(2*d**2*e**5 + 4*d*e**6*x + 2*e**
7*x**2), Eq(m, -3)), (-12*a*d**2*e**2*g**2*log(d/e + x)/(6*d*e**5 + 6*e**6*x) - 12*a*d**2*e**2*g**2/(6*d*e**5
+ 6*e**6*x) + 12*a*d*e**3*f*g*log(d/e + x)/(6*d*e**5 + 6*e**6*x) + 12*a*d*e**3*f*g/(6*d*e**5 + 6*e**6*x) - 12*
a*d*e**3*g**2*x*log(d/e + x)/(6*d*e**5 + 6*e**6*x) - 6*a*e**4*f**2/(6*d*e**5 + 6*e**6*x) + 12*a*e**4*f*g*x*log
(d/e + x)/(6*d*e**5 + 6*e**6*x) + 6*a*e**4*g**2*x**2/(6*d*e**5 + 6*e**6*x) + 18*b*d**3*e*g**2*log(d/e + x)/(6*
d*e**5 + 6*e**6*x) + 18*b*d**3*e*g**2/(6*d*e**5 + 6*e**6*x) - 24*b*d**2*e**2*f*g*log(d/e + x)/(6*d*e**5 + 6*e*
*6*x) - 24*b*d**2*e**2*f*g/(6*d*e**5 + 6*e**6*x) + 18*b*d**2*e**2*g**2*x*log(d/e + x)/(6*d*e**5 + 6*e**6*x) +
6*b*d*e**3*f**2*log(d/e + x)/(6*d*e**5 + 6*e**6*x) + 6*b*d*e**3*f**2/(6*d*e**5 + 6*e**6*x) - 24*b*d*e**3*f*g*x
*log(d/e + x)/(6*d*e**5 + 6*e**6*x) - 9*b*d*e**3*g**2*x**2/(6*d*e**5 + 6*e**6*x) + 6*b*e**4*f**2*x*log(d/e + x
)/(6*d*e**5 + 6*e**6*x) + 12*b*e**4*f*g*x**2/(6*d*e**5 + 6*e**6*x) + 3*b*e**4*g**2*x**3/(6*d*e**5 + 6*e**6*x)
- 24*c*d**4*g**2*log(d/e + x)/(6*d*e**5 + 6*e**6*x) - 24*c*d**4*g**2/(6*d*e**5 + 6*e**6*x) + 36*c*d**3*e*f*g*l
og(d/e + x)/(6*d*e**5 + 6*e**6*x) + 36*c*d**3*e*f*g/(6*d*e**5 + 6*e**6*x) - 24*c*d**3*e*g**2*x*log(d/e + x)/(6
*d*e**5 + 6*e**6*x) - 12*c*d**2*e**2*f**2*log(d/e + x)/(6*d*e**5 + 6*e**6*x) - 12*c*d**2*e**2*f**2/(6*d*e**5 +
6*e**6*x) + 36*c*d**2*e**2*f*g*x*log(d/e + x)/(6*d*e**5 + 6*e**6*x) + 12*c*d**2*e**2*g**2*x**2/(6*d*e**5 + 6*
e**6*x) - 12*c*d*e**3*f**2*x*log(d/e + x)/(6*d*e**5 + 6*e**6*x) - 18*c*d*e**3*f*g*x**2/(6*d*e**5 + 6*e**6*x) -
4*c*d*e**3*g**2*x**3/(6*d*e**5 + 6*e**6*x) + 6*c*e**4*f**2*x**2/(6*d*e**5 + 6*e**6*x) + 6*c*e**4*f*g*x**3/(6*
d*e**5 + 6*e**6*x) + 2*c*e**4*g**2*x**4/(6*d*e**5 + 6*e**6*x), Eq(m, -2)), (a*d**2*g**2*log(d/e + x)/e**3 - 2*
a*d*f*g*log(d/e + x)/e**2 - a*d*g**2*x/e**2 + a*f**2*log(d/e + x)/e + 2*a*f*g*x/e + a*g**2*x**2/(2*e) - b*d**3
*g**2*log(d/e + x)/e**4 + 2*b*d**2*f*g*log(d/e + x)/e**3 + b*d**2*g**2*x/e**3 - b*d*f**2*log(d/e + x)/e**2 - 2
*b*d*f*g*x/e**2 - b*d*g**2*x**2/(2*e**2) + b*f**2*x/e + b*f*g*x**2/e + b*g**2*x**3/(3*e) + c*d**4*g**2*log(d/e
+ x)/e**5 - 2*c*d**3*f*g*log(d/e + x)/e**4 - c*d**3*g**2*x/e**4 + c*d**2*f**2*log(d/e + x)/e**3 + 2*c*d**2*f*
g*x/e**3 + c*d**2*g**2*x**2/(2*e**3) - c*d*f**2*x/e**2 - c*d*f*g*x**2/e**2 - c*d*g**2*x**3/(3*e**2) + c*f**2*x
**2/(2*e) + 2*c*f*g*x**3/(3*e) + c*g**2*x**4/(4*e), Eq(m, -1)), (2*a*d**3*e**2*g**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*a*d**3*e**2*g**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*a*d**3*e**2*g**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*d**2*e**3*f*g*
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*d**
2*e**3*f*g*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*d**2*e**3*f*g*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 + 1
20*e**5) - 120*a*d**2*e**3*f*g*(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*d**2*e**3*g**2*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) - 18*a*d**2*e**3*g**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) - 40*a*d**2*e**3*g**2*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) + a*d*e**4*f**2*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*d*e**4*f**2*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*d*e**4*f**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) + 154*a*d*e**4*f**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) + 120*a*d*e**4*f**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*d*e**4*f*g*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*d*e*
*4*f*g*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) +
94*a*d*e**4*f*g*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 + 1
20*e**5) + 120*a*d*e**4*f*g*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) + a*d*e**4*g**2*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*a*d*e**4*g**2*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*a*d*e**4*g**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) + 20*a*d*e**4*g**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) + a*e**5*f**2*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*e**5*f**2*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*e**5*f**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) + 154*a*e**5*f**2*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*e**5
*f**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) + 2*a*e
**5*f*g*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*e**5*f*g*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*e**5*f*g*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*e**5*f*g*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) + 120*a*e**5*f*g*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) + a*e**5*g**2*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*a*e**5*g**2*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) + 49*a*e**5*g**2*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*a*e**5*g**2*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*a*e**5*g**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) - 6*b*d**4*e*g**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) - 30*b*d**4*e*
g**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*b*d**3
*e**2*f*g*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*b*d**3*e**2*f*g*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 + 12
0*e**5) + 80*b*d**3*e**2*f*g*(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*b*d**3*e**2*g**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) + 30*b*d**3*e**2*g**2*m*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) - b*d**2*e**3*f**2*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) - 12*b*d**2*e**3*f**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) - 47*b*d**2*e**3*f**2*m*(d + e*x)**m/(e**5*m**5 + 1
5*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) - 60*b*d**2*e**3*f**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*b*d**2*e**3*f*g*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*b*d**2*e**3*f*g*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*b*d**
2*e**3*f*g*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)
- 3*b*d**2*e**3*g**2*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) - 18*b*d**2*e**3*g**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) - 15*b*d**2*e**3*g**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) + b*d*e**4*f**2*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) + 12*b*d*e**4*f**2*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) + 47*b*d*e**4*f**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*d*e**4*f**2*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*b*d*e**4*f*g*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*b*d*e**4
*f*g*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*b*d*e**4*f*g*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*b*d*e**4*f*g*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*d*e**4*g**2*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) + 8*b*d*e**4*g**2*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) + 17*b*d*e**4*g**2*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) + 10*b*d*e**4*g**2*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) + b*e**5*f**2*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) + 13*b*e**5*f**2*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) + 59*b*e**5*f**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) + 107
*b*e**5*f**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) + 60*b*e**5*f**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) + 2*b*e**5*f*g*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 + 2
74*e**5*m + 120*e**5) + 24*b*e**5*f*g*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*b*e**5*f*g*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*b*e**5*f*g*m*x**3*(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) + 80*b*e**5*f*g*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) + b*e**5*g**2*m**4*x**4*(d + e*x)**m/(e**5*m**5 + 1
5*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 11*b*e**5*g**2*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) + 41*b*e**5*g**2*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) + 61*b*e**5*g**2*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) + 30*b*e*
*5*g**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) +
24*c*d**5*g**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*d**4*e*f*g*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*c*d**4*e*f*g*(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) - 24*c*d**4*e*g**2*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*c*d**3*e**2*f**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*c*d**3*e**2*f**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*c*d**3*e**2*f**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*d**3*e**2*f*g*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*c*d**3*e**2*f*g*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*d**3*e**2*g**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*d**3*e**2*g**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) - 2*c*d**2*e*
*3*f**2*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)
- 18*c*d**2*e**3*f**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) - 40*c*d**2*e**3*f**2*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) - 6*c*d**2*e**3*f*g*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*c*d**2*e**3*f*g*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*c*d**2*e**3*f*g*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*d**2*e**3*g**2*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*d**2*e**3*g**2*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*d*
*2*e**3*g**2*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*d*e**4*f**2*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*c*d*e**4*f**2*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*c*d*e**4*f**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) + 20*c*d*e**4*f**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) + 2*c*d*e**4*f*g*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*c*d*e**4*f*g*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) + 34*c*d*e**4*f*g*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*c*d*e**4*f*g
*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*d*e
**4*g**2*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*d*e**4*g**2*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*d*e**4*g**2*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) + 6*c*d*e**4*g**2*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*e**5*f**2*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*c*e**5*f**2*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) + 49*c*e**5*f**2*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*c*e**5*f**2*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*c*e**5*f**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) + 2*c*e**5*f*g*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*c*e**
5*f*g*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*c*e**5*f*g*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*c*e**5*f*g*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 + 27
4*e**5*m + 120*e**5) + 60*c*e**5*f*g*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*e**5*g**2*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*e**5*g**2*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 + 120*e**5) + 35*c*e**5*g**2*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*e**5*g**2*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*e**5*g**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), True))
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