3.2551 \(\int (d+e x)^m (a+b x+c x^2)^2 \, dx\)

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 = 0.113784, 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, Rules used = {698} \[ \frac{(d+e x)^{m+3} \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{e^5 (m+3)}+\frac{(d+e x)^{m+1} \left (a e^2-b d e+c d^2\right )^2}{e^5 (m+1)}-\frac{2 (2 c d-b e) (d+e x)^{m+2} \left (a e^2-b d e+c d^2\right )}{e^5 (m+2)}-\frac{2 c (2 c d-b e) (d+e x)^{m+4}}{e^5 (m+4)}+\frac{c^2 (d+e x)^{m+5}}{e^5 (m+5)} \]

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))

Rule 698

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*
e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

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

Mathematica [A]  time = 0.399677, size = 178, normalized size = 1. \[ \frac{(d+e x)^{m+1} \left (\frac{2 (d+e x) \left (\frac{6 (2 c d-b e) \left (e (a e-b d)+c d^2\right )}{m+2}-\frac{(d+e x) \left (4 c e (a e (m+4)-3 b d)-b^2 e^2 (m+1)+12 c^2 d^2\right )}{m+3}\right )}{e^4 (m+4) (m+5)}-\frac{2 (d+e x) (a+x (b+c x)) (b e (m+7)-6 c d+2 c e (m+4) x)}{e^2 (m+4) (m+5)}+(a+x (b+c x))^2\right )}{e (m+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.052, size = 822, normalized size = 4.6 \begin{align*}{\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\,{c}^{2}{x}^{4}{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\,{c}^{2}{d}^{3}ex+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 ) }} \end{align*}

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+156*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+120)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.75326, size = 1897, normalized size = 10.66 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^m*(c*x^2+b*x+a)^2,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 + 40*(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 = 11.1325, size = 9853, normalized size = 55.35 \begin{align*} \text{result too large to display} \end{align*}

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*d**2*e**4/(12*d**6*e**5 + 48*d**5*e**6*x + 72*d**4*e**7*x**2 + 48*d**3*e**8*x**3 + 12*d**2*e**9*x**4) - 2*a
*b*d**3*e**3/(12*d**6*e**5 + 48*d**5*e**6*x + 72*d**4*e**7*x**2 + 48*d**3*e**8*x**3 + 12*d**2*e**9*x**4) - 8*a
*b*d**2*e**4*x/(12*d**6*e**5 + 48*d**5*e**6*x + 72*d**4*e**7*x**2 + 48*d**3*e**8*x**3 + 12*d**2*e**9*x**4) + 8
*a*c*d*e**5*x**3/(12*d**6*e**5 + 48*d**5*e**6*x + 72*d**4*e**7*x**2 + 48*d**3*e**8*x**3 + 12*d**2*e**9*x**4) +
2*a*c*e**6*x**4/(12*d**6*e**5 + 48*d**5*e**6*x + 72*d**4*e**7*x**2 + 48*d**3*e**8*x**3 + 12*d**2*e**9*x**4) +
4*b**2*d*e**5*x**3/(12*d**6*e**5 + 48*d**5*e**6*x + 72*d**4*e**7*x**2 + 48*d**3*e**8*x**3 + 12*d**2*e**9*x**4
) + b**2*e**6*x**4/(12*d**6*e**5 + 48*d**5*e**6*x + 72*d**4*e**7*x**2 + 48*d**3*e**8*x**3 + 12*d**2*e**9*x**4)
+ 6*b*c*d*e**5*x**4/(12*d**6*e**5 + 48*d**5*e**6*x + 72*d**4*e**7*x**2 + 48*d**3*e**8*x**3 + 12*d**2*e**9*x**
4) + 12*c**2*d**6*log(d/e + x)/(12*d**6*e**5 + 48*d**5*e**6*x + 72*d**4*e**7*x**2 + 48*d**3*e**8*x**3 + 12*d**
2*e**9*x**4) + 7*c**2*d**6/(12*d**6*e**5 + 48*d**5*e**6*x + 72*d**4*e**7*x**2 + 48*d**3*e**8*x**3 + 12*d**2*e*
*9*x**4) + 48*c**2*d**5*e*x*log(d/e + x)/(12*d**6*e**5 + 48*d**5*e**6*x + 72*d**4*e**7*x**2 + 48*d**3*e**8*x**
3 + 12*d**2*e**9*x**4) + 16*c**2*d**5*e*x/(12*d**6*e**5 + 48*d**5*e**6*x + 72*d**4*e**7*x**2 + 48*d**3*e**8*x*
*3 + 12*d**2*e**9*x**4) + 72*c**2*d**4*e**2*x**2*log(d/e + x)/(12*d**6*e**5 + 48*d**5*e**6*x + 72*d**4*e**7*x*
*2 + 48*d**3*e**8*x**3 + 12*d**2*e**9*x**4) + 48*c**2*d**3*e**3*x**3*log(d/e + x)/(12*d**6*e**5 + 48*d**5*e**6
*x + 72*d**4*e**7*x**2 + 48*d**3*e**8*x**3 + 12*d**2*e**9*x**4) - 24*c**2*d**3*e**3*x**3/(12*d**6*e**5 + 48*d*
*5*e**6*x + 72*d**4*e**7*x**2 + 48*d**3*e**8*x**3 + 12*d**2*e**9*x**4) + 12*c**2*d**2*e**4*x**4*log(d/e + x)/(
12*d**6*e**5 + 48*d**5*e**6*x + 72*d**4*e**7*x**2 + 48*d**3*e**8*x**3 + 12*d**2*e**9*x**4) - 18*c**2*d**2*e**4
*x**4/(12*d**6*e**5 + 48*d**5*e**6*x + 72*d**4*e**7*x**2 + 48*d**3*e**8*x**3 + 12*d**2*e**9*x**4), Eq(m, -5)),
(-a**2*d*e**4/(3*d**4*e**5 + 9*d**3*e**6*x + 9*d**2*e**7*x**2 + 3*d*e**8*x**3) - a*b*d**2*e**3/(3*d**4*e**5 +
9*d**3*e**6*x + 9*d**2*e**7*x**2 + 3*d*e**8*x**3) - 3*a*b*d*e**4*x/(3*d**4*e**5 + 9*d**3*e**6*x + 9*d**2*e**7
*x**2 + 3*d*e**8*x**3) + 2*a*c*e**5*x**3/(3*d**4*e**5 + 9*d**3*e**6*x + 9*d**2*e**7*x**2 + 3*d*e**8*x**3) + b*
*2*e**5*x**3/(3*d**4*e**5 + 9*d**3*e**6*x + 9*d**2*e**7*x**2 + 3*d*e**8*x**3) + 6*b*c*d**4*e*log(d/e + x)/(3*d
**4*e**5 + 9*d**3*e**6*x + 9*d**2*e**7*x**2 + 3*d*e**8*x**3) + 5*b*c*d**4*e/(3*d**4*e**5 + 9*d**3*e**6*x + 9*d
**2*e**7*x**2 + 3*d*e**8*x**3) + 18*b*c*d**3*e**2*x*log(d/e + x)/(3*d**4*e**5 + 9*d**3*e**6*x + 9*d**2*e**7*x*
*2 + 3*d*e**8*x**3) + 9*b*c*d**3*e**2*x/(3*d**4*e**5 + 9*d**3*e**6*x + 9*d**2*e**7*x**2 + 3*d*e**8*x**3) + 18*
b*c*d**2*e**3*x**2*log(d/e + x)/(3*d**4*e**5 + 9*d**3*e**6*x + 9*d**2*e**7*x**2 + 3*d*e**8*x**3) + 6*b*c*d*e**
4*x**3*log(d/e + x)/(3*d**4*e**5 + 9*d**3*e**6*x + 9*d**2*e**7*x**2 + 3*d*e**8*x**3) - 6*b*c*d*e**4*x**3/(3*d*
*4*e**5 + 9*d**3*e**6*x + 9*d**2*e**7*x**2 + 3*d*e**8*x**3) - 12*c**2*d**5*log(d/e + x)/(3*d**4*e**5 + 9*d**3*
e**6*x + 9*d**2*e**7*x**2 + 3*d*e**8*x**3) - 10*c**2*d**5/(3*d**4*e**5 + 9*d**3*e**6*x + 9*d**2*e**7*x**2 + 3*
d*e**8*x**3) - 36*c**2*d**4*e*x*log(d/e + x)/(3*d**4*e**5 + 9*d**3*e**6*x + 9*d**2*e**7*x**2 + 3*d*e**8*x**3)
- 18*c**2*d**4*e*x/(3*d**4*e**5 + 9*d**3*e**6*x + 9*d**2*e**7*x**2 + 3*d*e**8*x**3) - 36*c**2*d**3*e**2*x**2*l
og(d/e + x)/(3*d**4*e**5 + 9*d**3*e**6*x + 9*d**2*e**7*x**2 + 3*d*e**8*x**3) - 12*c**2*d**2*e**3*x**3*log(d/e
+ x)/(3*d**4*e**5 + 9*d**3*e**6*x + 9*d**2*e**7*x**2 + 3*d*e**8*x**3) + 12*c**2*d**2*e**3*x**3/(3*d**4*e**5 +
9*d**3*e**6*x + 9*d**2*e**7*x**2 + 3*d*e**8*x**3) + 3*c**2*d*e**4*x**4/(3*d**4*e**5 + 9*d**3*e**6*x + 9*d**2*e
**7*x**2 + 3*d*e**8*x**3), Eq(m, -4)), (-a**2*e**4/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) - 2*a*b*d*e**3/(2*
d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) - 4*a*b*e**4*x/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + 4*a*c*d**2*e**
2*log(d/e + x)/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + 6*a*c*d**2*e**2/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x
**2) + 8*a*c*d*e**3*x*log(d/e + x)/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + 8*a*c*d*e**3*x/(2*d**2*e**5 + 4*
d*e**6*x + 2*e**7*x**2) + 4*a*c*e**4*x**2*log(d/e + x)/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + 2*b**2*d**2*
e**2*log(d/e + x)/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + 3*b**2*d**2*e**2/(2*d**2*e**5 + 4*d*e**6*x + 2*e*
*7*x**2) + 4*b**2*d*e**3*x*log(d/e + x)/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + 4*b**2*d*e**3*x/(2*d**2*e**
5 + 4*d*e**6*x + 2*e**7*x**2) + 2*b**2*e**4*x**2*log(d/e + x)/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) - 12*b*
c*d**3*e*log(d/e + x)/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) - 18*b*c*d**3*e/(2*d**2*e**5 + 4*d*e**6*x + 2*e
**7*x**2) - 24*b*c*d**2*e**2*x*log(d/e + x)/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) - 24*b*c*d**2*e**2*x/(2*d
**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) - 12*b*c*d*e**3*x**2*log(d/e + x)/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2
) + 4*b*c*e**4*x**3/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + 12*c**2*d**4*log(d/e + x)/(2*d**2*e**5 + 4*d*e*
*6*x + 2*e**7*x**2) + 18*c**2*d**4/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + 24*c**2*d**3*e*x*log(d/e + x)/(2
*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + 24*c**2*d**3*e*x/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + 12*c**2*d
**2*e**2*x**2*log(d/e + x)/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) - 4*c**2*d*e**3*x**3/(2*d**2*e**5 + 4*d*e*
*6*x + 2*e**7*x**2) + c**2*e**4*x**4/(2*d**2*e**5 + 4*d*e**6*x + 2*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*log(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 + 274*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 + 2
25*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 + 225*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 + 120*e**5) + 9
4*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 + 2
25*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) + 120*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 + 1
5*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) + 2
0*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 + 12
0*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 + 8
5*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 + 225*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 + 225*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 + 85*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 + 274*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 + 274*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 + 274*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 + 8
5*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 + 1
5*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 + 85*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 + 1
20*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 + 120*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 [B]  time = 1.25513, size = 2318, normalized size = 13.02 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((x*e + d)^m*c^2*m^4*x^5*e^5 + (x*e + d)^m*c^2*d*m^4*x^4*e^4 + 2*(x*e + d)^m*b*c*m^4*x^4*e^5 + 10*(x*e + d)^m*
c^2*m^3*x^5*e^5 + 2*(x*e + d)^m*b*c*d*m^4*x^3*e^4 + 6*(x*e + d)^m*c^2*d*m^3*x^4*e^4 - 4*(x*e + d)^m*c^2*d^2*m^
3*x^3*e^3 + (x*e + d)^m*b^2*m^4*x^3*e^5 + 2*(x*e + d)^m*a*c*m^4*x^3*e^5 + 22*(x*e + d)^m*b*c*m^3*x^4*e^5 + 35*
(x*e + d)^m*c^2*m^2*x^5*e^5 + (x*e + d)^m*b^2*d*m^4*x^2*e^4 + 2*(x*e + d)^m*a*c*d*m^4*x^2*e^4 + 16*(x*e + d)^m
*b*c*d*m^3*x^3*e^4 + 11*(x*e + d)^m*c^2*d*m^2*x^4*e^4 - 6*(x*e + d)^m*b*c*d^2*m^3*x^2*e^3 - 12*(x*e + d)^m*c^2
*d^2*m^2*x^3*e^3 + 12*(x*e + d)^m*c^2*d^3*m^2*x^2*e^2 + 2*(x*e + d)^m*a*b*m^4*x^2*e^5 + 12*(x*e + d)^m*b^2*m^3
*x^3*e^5 + 24*(x*e + d)^m*a*c*m^3*x^3*e^5 + 82*(x*e + d)^m*b*c*m^2*x^4*e^5 + 50*(x*e + d)^m*c^2*m*x^5*e^5 + 2*
(x*e + d)^m*a*b*d*m^4*x*e^4 + 10*(x*e + d)^m*b^2*d*m^3*x^2*e^4 + 20*(x*e + d)^m*a*c*d*m^3*x^2*e^4 + 34*(x*e +
d)^m*b*c*d*m^2*x^3*e^4 + 6*(x*e + d)^m*c^2*d*m*x^4*e^4 - 2*(x*e + d)^m*b^2*d^2*m^3*x*e^3 - 4*(x*e + d)^m*a*c*d
^2*m^3*x*e^3 - 36*(x*e + d)^m*b*c*d^2*m^2*x^2*e^3 - 8*(x*e + d)^m*c^2*d^2*m*x^3*e^3 + 12*(x*e + d)^m*b*c*d^3*m
^2*x*e^2 + 12*(x*e + d)^m*c^2*d^3*m*x^2*e^2 - 24*(x*e + d)^m*c^2*d^4*m*x*e + (x*e + d)^m*a^2*m^4*x*e^5 + 26*(x
*e + d)^m*a*b*m^3*x^2*e^5 + 49*(x*e + d)^m*b^2*m^2*x^3*e^5 + 98*(x*e + d)^m*a*c*m^2*x^3*e^5 + 122*(x*e + d)^m*
b*c*m*x^4*e^5 + 24*(x*e + d)^m*c^2*x^5*e^5 + (x*e + d)^m*a^2*d*m^4*e^4 + 24*(x*e + d)^m*a*b*d*m^3*x*e^4 + 29*(
x*e + d)^m*b^2*d*m^2*x^2*e^4 + 58*(x*e + d)^m*a*c*d*m^2*x^2*e^4 + 20*(x*e + d)^m*b*c*d*m*x^3*e^4 - 2*(x*e + d)
^m*a*b*d^2*m^3*e^3 - 18*(x*e + d)^m*b^2*d^2*m^2*x*e^3 - 36*(x*e + d)^m*a*c*d^2*m^2*x*e^3 - 30*(x*e + d)^m*b*c*
d^2*m*x^2*e^3 + 2*(x*e + d)^m*b^2*d^3*m^2*e^2 + 4*(x*e + d)^m*a*c*d^3*m^2*e^2 + 60*(x*e + d)^m*b*c*d^3*m*x*e^2
- 12*(x*e + d)^m*b*c*d^4*m*e + 24*(x*e + d)^m*c^2*d^5 + 14*(x*e + d)^m*a^2*m^3*x*e^5 + 118*(x*e + d)^m*a*b*m^
2*x^2*e^5 + 78*(x*e + d)^m*b^2*m*x^3*e^5 + 156*(x*e + d)^m*a*c*m*x^3*e^5 + 60*(x*e + d)^m*b*c*x^4*e^5 + 14*(x*
e + d)^m*a^2*d*m^3*e^4 + 94*(x*e + d)^m*a*b*d*m^2*x*e^4 + 20*(x*e + d)^m*b^2*d*m*x^2*e^4 + 40*(x*e + d)^m*a*c*
d*m*x^2*e^4 - 24*(x*e + d)^m*a*b*d^2*m^2*e^3 - 40*(x*e + d)^m*b^2*d^2*m*x*e^3 - 80*(x*e + d)^m*a*c*d^2*m*x*e^3
+ 18*(x*e + d)^m*b^2*d^3*m*e^2 + 36*(x*e + d)^m*a*c*d^3*m*e^2 - 60*(x*e + d)^m*b*c*d^4*e + 71*(x*e + d)^m*a^2
*m^2*x*e^5 + 214*(x*e + d)^m*a*b*m*x^2*e^5 + 40*(x*e + d)^m*b^2*x^3*e^5 + 80*(x*e + d)^m*a*c*x^3*e^5 + 71*(x*e
+ d)^m*a^2*d*m^2*e^4 + 120*(x*e + d)^m*a*b*d*m*x*e^4 - 94*(x*e + d)^m*a*b*d^2*m*e^3 + 40*(x*e + d)^m*b^2*d^3*
e^2 + 80*(x*e + d)^m*a*c*d^3*e^2 + 154*(x*e + d)^m*a^2*m*x*e^5 + 120*(x*e + d)^m*a*b*x^2*e^5 + 154*(x*e + d)^m
*a^2*d*m*e^4 - 120*(x*e + d)^m*a*b*d^2*e^3 + 120*(x*e + d)^m*a^2*x*e^5 + 120*(x*e + d)^m*a^2*d*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)