3.1732 \(\int (d+e x)^m (a^2+2 a b x+b^2 x^2)^2 \, dx\)

Optimal. Leaf size=142 \[ \frac{6 b^2 (b d-a e)^2 (d+e x)^{m+3}}{e^5 (m+3)}-\frac{4 b^3 (b d-a e) (d+e x)^{m+4}}{e^5 (m+4)}+\frac{(b d-a e)^4 (d+e x)^{m+1}}{e^5 (m+1)}-\frac{4 b (b d-a e)^3 (d+e x)^{m+2}}{e^5 (m+2)}+\frac{b^4 (d+e x)^{m+5}}{e^5 (m+5)} \]

[Out]

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

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Rubi [A]  time = 0.0673952, antiderivative size = 142, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {27, 43} \[ \frac{6 b^2 (b d-a e)^2 (d+e x)^{m+3}}{e^5 (m+3)}-\frac{4 b^3 (b d-a e) (d+e x)^{m+4}}{e^5 (m+4)}+\frac{(b d-a e)^4 (d+e x)^{m+1}}{e^5 (m+1)}-\frac{4 b (b d-a e)^3 (d+e x)^{m+2}}{e^5 (m+2)}+\frac{b^4 (d+e x)^{m+5}}{e^5 (m+5)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 0.104427, size = 121, normalized size = 0.85 \[ \frac{(d+e x)^{m+1} \left (\frac{6 b^2 (d+e x)^2 (b d-a e)^2}{m+3}-\frac{4 b^3 (d+e x)^3 (b d-a e)}{m+4}-\frac{4 b (d+e x) (b d-a e)^3}{m+2}+\frac{(b d-a e)^4}{m+1}+\frac{b^4 (d+e x)^4}{m+5}\right )}{e^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.051, size = 768, normalized size = 5.4 \begin{align*}{\frac{ \left ( ex+d \right ) ^{1+m} \left ({b}^{4}{e}^{4}{m}^{4}{x}^{4}+4\,a{b}^{3}{e}^{4}{m}^{4}{x}^{3}+10\,{b}^{4}{e}^{4}{m}^{3}{x}^{4}+6\,{a}^{2}{b}^{2}{e}^{4}{m}^{4}{x}^{2}+44\,a{b}^{3}{e}^{4}{m}^{3}{x}^{3}-4\,{b}^{4}d{e}^{3}{m}^{3}{x}^{3}+35\,{b}^{4}{e}^{4}{m}^{2}{x}^{4}+4\,{a}^{3}b{e}^{4}{m}^{4}x+72\,{a}^{2}{b}^{2}{e}^{4}{m}^{3}{x}^{2}-12\,a{b}^{3}d{e}^{3}{m}^{3}{x}^{2}+164\,a{b}^{3}{e}^{4}{m}^{2}{x}^{3}-24\,{b}^{4}d{e}^{3}{m}^{2}{x}^{3}+50\,{b}^{4}{e}^{4}m{x}^{4}+{a}^{4}{e}^{4}{m}^{4}+52\,{a}^{3}b{e}^{4}{m}^{3}x-12\,{a}^{2}{b}^{2}d{e}^{3}{m}^{3}x+294\,{a}^{2}{b}^{2}{e}^{4}{m}^{2}{x}^{2}-96\,a{b}^{3}d{e}^{3}{m}^{2}{x}^{2}+244\,a{b}^{3}{e}^{4}m{x}^{3}+12\,{b}^{4}{d}^{2}{e}^{2}{m}^{2}{x}^{2}-44\,{b}^{4}d{e}^{3}m{x}^{3}+24\,{b}^{4}{x}^{4}{e}^{4}+14\,{a}^{4}{e}^{4}{m}^{3}-4\,{a}^{3}bd{e}^{3}{m}^{3}+236\,{a}^{3}b{e}^{4}{m}^{2}x-120\,{a}^{2}{b}^{2}d{e}^{3}{m}^{2}x+468\,{a}^{2}{b}^{2}{e}^{4}m{x}^{2}+24\,a{b}^{3}{d}^{2}{e}^{2}{m}^{2}x-204\,a{b}^{3}d{e}^{3}m{x}^{2}+120\,a{b}^{3}{e}^{4}{x}^{3}+36\,{b}^{4}{d}^{2}{e}^{2}m{x}^{2}-24\,{b}^{4}d{e}^{3}{x}^{3}+71\,{a}^{4}{e}^{4}{m}^{2}-48\,{a}^{3}bd{e}^{3}{m}^{2}+428\,{a}^{3}b{e}^{4}mx+12\,{a}^{2}{b}^{2}{d}^{2}{e}^{2}{m}^{2}-348\,{a}^{2}{b}^{2}d{e}^{3}mx+240\,{a}^{2}{b}^{2}{e}^{4}{x}^{2}+144\,a{b}^{3}{d}^{2}{e}^{2}mx-120\,a{b}^{3}d{e}^{3}{x}^{2}-24\,{b}^{4}{d}^{3}emx+24\,{b}^{4}{d}^{2}{e}^{2}{x}^{2}+154\,{a}^{4}{e}^{4}m-188\,{a}^{3}bd{e}^{3}m+240\,{a}^{3}b{e}^{4}x+108\,{a}^{2}{b}^{2}{d}^{2}{e}^{2}m-240\,{a}^{2}{b}^{2}d{e}^{3}x-24\,a{b}^{3}{d}^{3}em+120\,a{b}^{3}{d}^{2}{e}^{2}x-24\,{b}^{4}{d}^{3}ex+120\,{a}^{4}{e}^{4}-240\,{a}^{3}bd{e}^{3}+240\,{d}^{2}{e}^{2}{a}^{2}{b}^{2}-120\,a{b}^{3}{d}^{3}e+24\,{b}^{4}{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*(b^2*x^2+2*a*b*x+a^2)^2,x)

[Out]

(e*x+d)^(1+m)*(b^4*e^4*m^4*x^4+4*a*b^3*e^4*m^4*x^3+10*b^4*e^4*m^3*x^4+6*a^2*b^2*e^4*m^4*x^2+44*a*b^3*e^4*m^3*x
^3-4*b^4*d*e^3*m^3*x^3+35*b^4*e^4*m^2*x^4+4*a^3*b*e^4*m^4*x+72*a^2*b^2*e^4*m^3*x^2-12*a*b^3*d*e^3*m^3*x^2+164*
a*b^3*e^4*m^2*x^3-24*b^4*d*e^3*m^2*x^3+50*b^4*e^4*m*x^4+a^4*e^4*m^4+52*a^3*b*e^4*m^3*x-12*a^2*b^2*d*e^3*m^3*x+
294*a^2*b^2*e^4*m^2*x^2-96*a*b^3*d*e^3*m^2*x^2+244*a*b^3*e^4*m*x^3+12*b^4*d^2*e^2*m^2*x^2-44*b^4*d*e^3*m*x^3+2
4*b^4*e^4*x^4+14*a^4*e^4*m^3-4*a^3*b*d*e^3*m^3+236*a^3*b*e^4*m^2*x-120*a^2*b^2*d*e^3*m^2*x+468*a^2*b^2*e^4*m*x
^2+24*a*b^3*d^2*e^2*m^2*x-204*a*b^3*d*e^3*m*x^2+120*a*b^3*e^4*x^3+36*b^4*d^2*e^2*m*x^2-24*b^4*d*e^3*x^3+71*a^4
*e^4*m^2-48*a^3*b*d*e^3*m^2+428*a^3*b*e^4*m*x+12*a^2*b^2*d^2*e^2*m^2-348*a^2*b^2*d*e^3*m*x+240*a^2*b^2*e^4*x^2
+144*a*b^3*d^2*e^2*m*x-120*a*b^3*d*e^3*x^2-24*b^4*d^3*e*m*x+24*b^4*d^2*e^2*x^2+154*a^4*e^4*m-188*a^3*b*d*e^3*m
+240*a^3*b*e^4*x+108*a^2*b^2*d^2*e^2*m-240*a^2*b^2*d*e^3*x-24*a*b^3*d^3*e*m+120*a*b^3*d^2*e^2*x-24*b^4*d^3*e*x
+120*a^4*e^4-240*a^3*b*d*e^3+240*a^2*b^2*d^2*e^2-120*a*b^3*d^3*e+24*b^4*d^4)/e^5/(m^5+15*m^4+85*m^3+225*m^2+27
4*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*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.7684, size = 1890, normalized size = 13.31 \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*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="fricas")

[Out]

(a^4*d*e^4*m^4 + 24*b^4*d^5 - 120*a*b^3*d^4*e + 240*a^2*b^2*d^3*e^2 - 240*a^3*b*d^2*e^3 + 120*a^4*d*e^4 + (b^4
*e^5*m^4 + 10*b^4*e^5*m^3 + 35*b^4*e^5*m^2 + 50*b^4*e^5*m + 24*b^4*e^5)*x^5 + (120*a*b^3*e^5 + (b^4*d*e^4 + 4*
a*b^3*e^5)*m^4 + 2*(3*b^4*d*e^4 + 22*a*b^3*e^5)*m^3 + (11*b^4*d*e^4 + 164*a*b^3*e^5)*m^2 + 2*(3*b^4*d*e^4 + 12
2*a*b^3*e^5)*m)*x^4 - 2*(2*a^3*b*d^2*e^3 - 7*a^4*d*e^4)*m^3 + 2*(120*a^2*b^2*e^5 + (2*a*b^3*d*e^4 + 3*a^2*b^2*
e^5)*m^4 - 2*(b^4*d^2*e^3 - 8*a*b^3*d*e^4 - 18*a^2*b^2*e^5)*m^3 - (6*b^4*d^2*e^3 - 34*a*b^3*d*e^4 - 147*a^2*b^
2*e^5)*m^2 - 2*(2*b^4*d^2*e^3 - 10*a*b^3*d*e^4 - 117*a^2*b^2*e^5)*m)*x^3 + (12*a^2*b^2*d^3*e^2 - 48*a^3*b*d^2*
e^3 + 71*a^4*d*e^4)*m^2 + 2*(120*a^3*b*e^5 + (3*a^2*b^2*d*e^4 + 2*a^3*b*e^5)*m^4 - 2*(3*a*b^3*d^2*e^3 - 15*a^2
*b^2*d*e^4 - 13*a^3*b*e^5)*m^3 + (6*b^4*d^3*e^2 - 36*a*b^3*d^2*e^3 + 87*a^2*b^2*d*e^4 + 118*a^3*b*e^5)*m^2 + 2
*(3*b^4*d^3*e^2 - 15*a*b^3*d^2*e^3 + 30*a^2*b^2*d*e^4 + 107*a^3*b*e^5)*m)*x^2 - 2*(12*a*b^3*d^4*e - 54*a^2*b^2
*d^3*e^2 + 94*a^3*b*d^2*e^3 - 77*a^4*d*e^4)*m + (120*a^4*e^5 + (4*a^3*b*d*e^4 + a^4*e^5)*m^4 - 2*(6*a^2*b^2*d^
2*e^3 - 24*a^3*b*d*e^4 - 7*a^4*e^5)*m^3 + (24*a*b^3*d^3*e^2 - 108*a^2*b^2*d^2*e^3 + 188*a^3*b*d*e^4 + 71*a^4*e
^5)*m^2 - 2*(12*b^4*d^4*e - 60*a*b^3*d^3*e^2 + 120*a^2*b^2*d^2*e^3 - 120*a^3*b*d*e^4 - 77*a^4*e^5)*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 = 9.75118, size = 8531, normalized size = 60.08 \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*(b**2*x**2+2*a*b*x+a**2)**2,x)

[Out]

Piecewise((d**m*(a**4*x + 2*a**3*b*x**2 + 2*a**2*b**2*x**3 + a*b**3*x**4 + b**4*x**5/5), Eq(e, 0)), (-3*a**4*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) - 4*a**3*
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) - 16*a
**3*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)
+ 24*a**2*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) + 6*a**2*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) + 12*a*b**3*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*b**4*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*b**4*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*b**4*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*b**4*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*b**4*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*b**4*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*b**4*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*b**4*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*b**4*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**4*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) - 2*a**3*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) - 6*a**3*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) + 6*a**2*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) + 12*a*b**3*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) + 10*a*b**3*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) + 36*a*b**3
*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) + 18*a*b**3*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) + 36*a*b**3*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) + 12*a*b**3*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) - 12*a*b**3*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*b**4*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*b**4*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*b**4*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*b**4*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*b**4*d**3*e**2*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) - 12*b**4*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*b**4*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*b**4*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**4*e**4/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) - 4*a**3*b*d*e**3/(2*d**2*e**5 + 4*d*e**6*
x + 2*e**7*x**2) - 8*a**3*b*e**4*x/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + 12*a**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) + 18*a**2*b**2*d**2*e**2/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2)
+ 24*a**2*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) + 24*a**2*b**2*d*e**3*x/(2*d**2
*e**5 + 4*d*e**6*x + 2*e**7*x**2) + 12*a**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) - 24*a*b**3*d**3*e*log(d/e + x)/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) - 36*a*b**3*d**3*e/(2*d**2*e**5 +
4*d*e**6*x + 2*e**7*x**2) - 48*a*b**3*d**2*e**2*x*log(d/e + x)/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) - 48*a
*b**3*d**2*e**2*x/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) - 24*a*b**3*d*e**3*x**2*log(d/e + x)/(2*d**2*e**5 +
4*d*e**6*x + 2*e**7*x**2) + 8*a*b**3*e**4*x**3/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + 12*b**4*d**4*log(d/
e + x)/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + 18*b**4*d**4/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + 24*b
**4*d**3*e*x*log(d/e + x)/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + 24*b**4*d**3*e*x/(2*d**2*e**5 + 4*d*e**6*
x + 2*e**7*x**2) + 12*b**4*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*b**4*d*e**
3*x**3/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + b**4*e**4*x**4/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2), Eq(
m, -3)), (-3*a**4*e**4/(3*d*e**5 + 3*e**6*x) + 12*a**3*b*d*e**3*log(d/e + x)/(3*d*e**5 + 3*e**6*x) + 12*a**3*b
*d*e**3/(3*d*e**5 + 3*e**6*x) + 12*a**3*b*e**4*x*log(d/e + x)/(3*d*e**5 + 3*e**6*x) - 36*a**2*b**2*d**2*e**2*l
og(d/e + x)/(3*d*e**5 + 3*e**6*x) - 36*a**2*b**2*d**2*e**2/(3*d*e**5 + 3*e**6*x) - 36*a**2*b**2*d*e**3*x*log(d
/e + x)/(3*d*e**5 + 3*e**6*x) + 18*a**2*b**2*e**4*x**2/(3*d*e**5 + 3*e**6*x) + 36*a*b**3*d**3*e*log(d/e + x)/(
3*d*e**5 + 3*e**6*x) + 36*a*b**3*d**3*e/(3*d*e**5 + 3*e**6*x) + 36*a*b**3*d**2*e**2*x*log(d/e + x)/(3*d*e**5 +
3*e**6*x) - 18*a*b**3*d*e**3*x**2/(3*d*e**5 + 3*e**6*x) + 6*a*b**3*e**4*x**3/(3*d*e**5 + 3*e**6*x) - 12*b**4*
d**4*log(d/e + x)/(3*d*e**5 + 3*e**6*x) - 12*b**4*d**4/(3*d*e**5 + 3*e**6*x) - 12*b**4*d**3*e*x*log(d/e + x)/(
3*d*e**5 + 3*e**6*x) + 6*b**4*d**2*e**2*x**2/(3*d*e**5 + 3*e**6*x) - 2*b**4*d*e**3*x**3/(3*d*e**5 + 3*e**6*x)
+ b**4*e**4*x**4/(3*d*e**5 + 3*e**6*x), Eq(m, -2)), (a**4*log(d/e + x)/e - 4*a**3*b*d*log(d/e + x)/e**2 + 4*a*
*3*b*x/e + 6*a**2*b**2*d**2*log(d/e + x)/e**3 - 6*a**2*b**2*d*x/e**2 + 3*a**2*b**2*x**2/e - 4*a*b**3*d**3*log(
d/e + x)/e**4 + 4*a*b**3*d**2*x/e**3 - 2*a*b**3*d*x**2/e**2 + 4*a*b**3*x**3/(3*e) + b**4*d**4*log(d/e + x)/e**
5 - b**4*d**3*x/e**4 + b**4*d**2*x**2/(2*e**3) - b**4*d*x**3/(3*e**2) + b**4*x**4/(4*e), Eq(m, -1)), (a**4*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
**4*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**4*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 + 12
0*e**5) + 154*a**4*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**4*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**4*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**4*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**4*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**4*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**4*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) - 4*a**3*b*d**2*e**3*m**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) - 48*a**3*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) - 188*a**3*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) - 240*a**3*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) + 4*a**3*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) + 48*a**3*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) + 188*
a**3*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) + 240*a**3*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) + 4*a**3*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) + 52*a**3*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 + 2
25*e**5*m**2 + 274*e**5*m + 120*e**5) + 236*a**3*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) + 428*a**3*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) + 240*a**3*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) + 12*a**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) + 108*a**2*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) + 240*a**2*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) -
12*a**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) - 108*a**2*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) - 240*a**2*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) + 6*a**2*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) + 60*a**2*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) + 174*a**2*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) + 120*a**2*
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) + 6*a**2*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) + 72*a**2*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) + 294*a**2*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) + 468*a**2*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) + 240*a**2*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) - 24*a*b**3*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) - 120*a*b**3*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) + 24*a*b**3*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) + 120*a*b**3
*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)
- 12*a*b**3*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) - 72*a*b**3*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) - 60*a*b**3*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) + 4*a*b**3*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) + 32*a*b**3*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) + 68*a*b**3*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) + 40*a*b**3*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) + 4*a*b**3
*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)
+ 44*a*b**3*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) + 164*a*b**3*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) + 244*a*b**3*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) + 120*a*b**3*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*b**4*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*b**4*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*b**4*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*b**4*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*b**4*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*b**4*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*
b**4*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) + b**4*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*b**4*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*b**4*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*b**4*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) + b**4*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*b**4*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*b**4*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*b**4*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*b**4*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))

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Giac [B]  time = 1.1804, size = 2064, normalized size = 14.54 \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*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="giac")

[Out]

((x*e + d)^m*b^4*m^4*x^5*e^5 + (x*e + d)^m*b^4*d*m^4*x^4*e^4 + 4*(x*e + d)^m*a*b^3*m^4*x^4*e^5 + 10*(x*e + d)^
m*b^4*m^3*x^5*e^5 + 4*(x*e + d)^m*a*b^3*d*m^4*x^3*e^4 + 6*(x*e + d)^m*b^4*d*m^3*x^4*e^4 - 4*(x*e + d)^m*b^4*d^
2*m^3*x^3*e^3 + 6*(x*e + d)^m*a^2*b^2*m^4*x^3*e^5 + 44*(x*e + d)^m*a*b^3*m^3*x^4*e^5 + 35*(x*e + d)^m*b^4*m^2*
x^5*e^5 + 6*(x*e + d)^m*a^2*b^2*d*m^4*x^2*e^4 + 32*(x*e + d)^m*a*b^3*d*m^3*x^3*e^4 + 11*(x*e + d)^m*b^4*d*m^2*
x^4*e^4 - 12*(x*e + d)^m*a*b^3*d^2*m^3*x^2*e^3 - 12*(x*e + d)^m*b^4*d^2*m^2*x^3*e^3 + 12*(x*e + d)^m*b^4*d^3*m
^2*x^2*e^2 + 4*(x*e + d)^m*a^3*b*m^4*x^2*e^5 + 72*(x*e + d)^m*a^2*b^2*m^3*x^3*e^5 + 164*(x*e + d)^m*a*b^3*m^2*
x^4*e^5 + 50*(x*e + d)^m*b^4*m*x^5*e^5 + 4*(x*e + d)^m*a^3*b*d*m^4*x*e^4 + 60*(x*e + d)^m*a^2*b^2*d*m^3*x^2*e^
4 + 68*(x*e + d)^m*a*b^3*d*m^2*x^3*e^4 + 6*(x*e + d)^m*b^4*d*m*x^4*e^4 - 12*(x*e + d)^m*a^2*b^2*d^2*m^3*x*e^3
- 72*(x*e + d)^m*a*b^3*d^2*m^2*x^2*e^3 - 8*(x*e + d)^m*b^4*d^2*m*x^3*e^3 + 24*(x*e + d)^m*a*b^3*d^3*m^2*x*e^2
+ 12*(x*e + d)^m*b^4*d^3*m*x^2*e^2 - 24*(x*e + d)^m*b^4*d^4*m*x*e + (x*e + d)^m*a^4*m^4*x*e^5 + 52*(x*e + d)^m
*a^3*b*m^3*x^2*e^5 + 294*(x*e + d)^m*a^2*b^2*m^2*x^3*e^5 + 244*(x*e + d)^m*a*b^3*m*x^4*e^5 + 24*(x*e + d)^m*b^
4*x^5*e^5 + (x*e + d)^m*a^4*d*m^4*e^4 + 48*(x*e + d)^m*a^3*b*d*m^3*x*e^4 + 174*(x*e + d)^m*a^2*b^2*d*m^2*x^2*e
^4 + 40*(x*e + d)^m*a*b^3*d*m*x^3*e^4 - 4*(x*e + d)^m*a^3*b*d^2*m^3*e^3 - 108*(x*e + d)^m*a^2*b^2*d^2*m^2*x*e^
3 - 60*(x*e + d)^m*a*b^3*d^2*m*x^2*e^3 + 12*(x*e + d)^m*a^2*b^2*d^3*m^2*e^2 + 120*(x*e + d)^m*a*b^3*d^3*m*x*e^
2 - 24*(x*e + d)^m*a*b^3*d^4*m*e + 24*(x*e + d)^m*b^4*d^5 + 14*(x*e + d)^m*a^4*m^3*x*e^5 + 236*(x*e + d)^m*a^3
*b*m^2*x^2*e^5 + 468*(x*e + d)^m*a^2*b^2*m*x^3*e^5 + 120*(x*e + d)^m*a*b^3*x^4*e^5 + 14*(x*e + d)^m*a^4*d*m^3*
e^4 + 188*(x*e + d)^m*a^3*b*d*m^2*x*e^4 + 120*(x*e + d)^m*a^2*b^2*d*m*x^2*e^4 - 48*(x*e + d)^m*a^3*b*d^2*m^2*e
^3 - 240*(x*e + d)^m*a^2*b^2*d^2*m*x*e^3 + 108*(x*e + d)^m*a^2*b^2*d^3*m*e^2 - 120*(x*e + d)^m*a*b^3*d^4*e + 7
1*(x*e + d)^m*a^4*m^2*x*e^5 + 428*(x*e + d)^m*a^3*b*m*x^2*e^5 + 240*(x*e + d)^m*a^2*b^2*x^3*e^5 + 71*(x*e + d)
^m*a^4*d*m^2*e^4 + 240*(x*e + d)^m*a^3*b*d*m*x*e^4 - 188*(x*e + d)^m*a^3*b*d^2*m*e^3 + 240*(x*e + d)^m*a^2*b^2
*d^3*e^2 + 154*(x*e + d)^m*a^4*m*x*e^5 + 240*(x*e + d)^m*a^3*b*x^2*e^5 + 154*(x*e + d)^m*a^4*d*m*e^4 - 240*(x*
e + d)^m*a^3*b*d^2*e^3 + 120*(x*e + d)^m*a^4*x*e^5 + 120*(x*e + d)^m*a^4*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)