3.2152 \(\int (a+b x) (d+e x)^m (a^2+2 a b x+b^2 x^2)^{3/2} \, dx\)
Optimal. Leaf size=277 \[ \frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4 (d+e x)^{m+1}}{e^5 (m+1) (a+b x)}-\frac{4 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3 (d+e x)^{m+2}}{e^5 (m+2) (a+b x)}+\frac{6 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2 (d+e x)^{m+3}}{e^5 (m+3) (a+b x)}-\frac{4 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e) (d+e x)^{m+4}}{e^5 (m+4) (a+b x)}+\frac{b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{m+5}}{e^5 (m+5) (a+b x)} \]
[Out]
((b*d - a*e)^4*(d + e*x)^(1 + m)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^5*(1 + m)*(a + b*x)) - (4*b*(b*d - a*e)^3*(
d + e*x)^(2 + m)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^5*(2 + m)*(a + b*x)) + (6*b^2*(b*d - a*e)^2*(d + e*x)^(3 +
m)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^5*(3 + m)*(a + b*x)) - (4*b^3*(b*d - a*e)*(d + e*x)^(4 + m)*Sqrt[a^2 + 2*
a*b*x + b^2*x^2])/(e^5*(4 + m)*(a + b*x)) + (b^4*(d + e*x)^(5 + m)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^5*(5 + m)
*(a + b*x))
________________________________________________________________________________________
Rubi [A] time = 0.149806, antiderivative size = 277, normalized size of antiderivative = 1.,
number of steps used = 4, number of rules used = 3, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used =
{770, 21, 43} \[ \frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4 (d+e x)^{m+1}}{e^5 (m+1) (a+b x)}-\frac{4 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3 (d+e x)^{m+2}}{e^5 (m+2) (a+b x)}+\frac{6 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2 (d+e x)^{m+3}}{e^5 (m+3) (a+b x)}-\frac{4 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e) (d+e x)^{m+4}}{e^5 (m+4) (a+b x)}+\frac{b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{m+5}}{e^5 (m+5) (a+b x)} \]
Antiderivative was successfully verified.
[In]
Int[(a + b*x)*(d + e*x)^m*(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]
[Out]
((b*d - a*e)^4*(d + e*x)^(1 + m)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^5*(1 + m)*(a + b*x)) - (4*b*(b*d - a*e)^3*(
d + e*x)^(2 + m)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^5*(2 + m)*(a + b*x)) + (6*b^2*(b*d - a*e)^2*(d + e*x)^(3 +
m)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^5*(3 + m)*(a + b*x)) - (4*b^3*(b*d - a*e)*(d + e*x)^(4 + m)*Sqrt[a^2 + 2*
a*b*x + b^2*x^2])/(e^5*(4 + m)*(a + b*x)) + (b^4*(d + e*x)^(5 + m)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^5*(5 + m)
*(a + b*x))
Rule 770
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dis
t[(a + b*x + c*x^2)^FracPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(f + g*x)*(b/2 + c
*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && EqQ[b^2 - 4*a*c, 0]
Rule 21
Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])
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 (a+b x) (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int (a+b x) \left (a b+b^2 x\right )^3 (d+e x)^m \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac{\left (b \sqrt{a^2+2 a b x+b^2 x^2}\right ) \int (a+b x)^4 (d+e x)^m \, dx}{a b+b^2 x}\\ &=\frac{\left (b \sqrt{a^2+2 a b x+b^2 x^2}\right ) \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}{a b+b^2 x}\\ &=\frac{(b d-a e)^4 (d+e x)^{1+m} \sqrt{a^2+2 a b x+b^2 x^2}}{e^5 (1+m) (a+b x)}-\frac{4 b (b d-a e)^3 (d+e x)^{2+m} \sqrt{a^2+2 a b x+b^2 x^2}}{e^5 (2+m) (a+b x)}+\frac{6 b^2 (b d-a e)^2 (d+e x)^{3+m} \sqrt{a^2+2 a b x+b^2 x^2}}{e^5 (3+m) (a+b x)}-\frac{4 b^3 (b d-a e) (d+e x)^{4+m} \sqrt{a^2+2 a b x+b^2 x^2}}{e^5 (4+m) (a+b x)}+\frac{b^4 (d+e x)^{5+m} \sqrt{a^2+2 a b x+b^2 x^2}}{e^5 (5+m) (a+b x)}\\ \end{align*}
Mathematica [A] time = 0.165291, size = 139, normalized size = 0.5 \[ \frac{\sqrt{(a+b x)^2} (d+e x)^{m+1} \left (\frac{6 b^2 (d+e x)^2 (b d-a e)^2}{m+3}-\frac{4 b^3 (d+e x)^3 (b d-a e)}{m+4}-\frac{4 b (d+e x) (b d-a e)^3}{m+2}+\frac{(b d-a e)^4}{m+1}+\frac{b^4 (d+e x)^4}{m+5}\right )}{e^5 (a+b x)} \]
Antiderivative was successfully verified.
[In]
Integrate[(a + b*x)*(d + e*x)^m*(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]
[Out]
(Sqrt[(a + b*x)^2]*(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
*(a + b*x))
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Maple [B] time = 0.01, size = 784, normalized size = 2.8 \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\,{x}^{4}{b}^{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\,{x}^{3}a{b}^{3}{e}^{4}+36\,{b}^{4}{d}^{2}{e}^{2}m{x}^{2}-24\,{x}^{3}{b}^{4}d{e}^{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\,{x}^{2}{a}^{2}{b}^{2}{e}^{4}+144\,a{b}^{3}{d}^{2}{e}^{2}mx-120\,{x}^{2}a{b}^{3}d{e}^{3}-24\,{b}^{4}{d}^{3}emx+24\,{x}^{2}{b}^{4}{d}^{2}{e}^{2}+154\,{a}^{4}{e}^{4}m-188\,{a}^{3}bd{e}^{3}m+240\,x{a}^{3}b{e}^{4}+108\,{a}^{2}{b}^{2}{d}^{2}{e}^{2}m-240\,x{a}^{2}{b}^{2}d{e}^{3}-24\,a{b}^{3}{d}^{3}em+120\,xa{b}^{3}{d}^{2}{e}^{2}-24\,x{b}^{4}{d}^{3}e+120\,{a}^{4}{e}^{4}-240\,d{e}^{3}{a}^{3}b+240\,{a}^{2}{b}^{2}{d}^{2}{e}^{2}-120\,a{b}^{3}{d}^{3}e+24\,{b}^{4}{d}^{4} \right ) }{ \left ( bx+a \right ) ^{3}{e}^{5} \left ({m}^{5}+15\,{m}^{4}+85\,{m}^{3}+225\,{m}^{2}+274\,m+120 \right ) } \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((b*x+a)*(e*x+d)^m*(b^2*x^2+2*a*b*x+a^2)^(3/2),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)*((b*x+a)^2)^(3/2)/(b*x+a)^3/e^5/(
m^5+15*m^4+85*m^3+225*m^2+274*m+120)
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Maxima [B] time = 1.10999, size = 1021, normalized size = 3.69 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)*(e*x+d)^m*(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm="maxima")
[Out]
((m^3 + 6*m^2 + 11*m + 6)*b^3*e^4*x^4 - 3*(m^2 + 7*m + 12)*a^2*b*d^2*e^2 + (m^3 + 9*m^2 + 26*m + 24)*a^3*d*e^3
+ 6*a*b^2*d^3*e*(m + 4) - 6*b^3*d^4 + ((m^3 + 3*m^2 + 2*m)*b^3*d*e^3 + 3*(m^3 + 7*m^2 + 14*m + 8)*a*b^2*e^4)*
x^3 - 3*((m^2 + m)*b^3*d^2*e^2 - (m^3 + 5*m^2 + 4*m)*a*b^2*d*e^3 - (m^3 + 8*m^2 + 19*m + 12)*a^2*b*e^4)*x^2 -
(6*(m^2 + 4*m)*a*b^2*d^2*e^2 - 3*(m^3 + 7*m^2 + 12*m)*a^2*b*d*e^3 - (m^3 + 9*m^2 + 26*m + 24)*a^3*e^4 - 6*b^3*
d^3*e*m)*x)*(e*x + d)^m*a/((m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*e^4) + ((m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*b^3
*e^5*x^5 + 6*(m^2 + 9*m + 20)*a^2*b*d^3*e^2 - (m^3 + 12*m^2 + 47*m + 60)*a^3*d^2*e^3 - 18*a*b^2*d^4*e*(m + 5)
+ 24*b^3*d^5 + ((m^4 + 6*m^3 + 11*m^2 + 6*m)*b^3*d*e^4 + 3*(m^4 + 11*m^3 + 41*m^2 + 61*m + 30)*a*b^2*e^5)*x^4
- (4*(m^3 + 3*m^2 + 2*m)*b^3*d^2*e^3 - 3*(m^4 + 8*m^3 + 17*m^2 + 10*m)*a*b^2*d*e^4 - 3*(m^4 + 12*m^3 + 49*m^2
+ 78*m + 40)*a^2*b*e^5)*x^3 + (12*(m^2 + m)*b^3*d^3*e^2 - 9*(m^3 + 6*m^2 + 5*m)*a*b^2*d^2*e^3 + 3*(m^4 + 10*m^
3 + 29*m^2 + 20*m)*a^2*b*d*e^4 + (m^4 + 13*m^3 + 59*m^2 + 107*m + 60)*a^3*e^5)*x^2 + (18*(m^2 + 5*m)*a*b^2*d^3
*e^2 - 6*(m^3 + 9*m^2 + 20*m)*a^2*b*d^2*e^3 + (m^4 + 12*m^3 + 47*m^2 + 60*m)*a^3*d*e^4 - 24*b^3*d^4*e*m)*x)*(e
*x + d)^m*b/((m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m + 120)*e^5)
________________________________________________________________________________________
Fricas [B] time = 1.18485, size = 1890, normalized size = 6.82 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)*(e*x+d)^m*(b^2*x^2+2*a*b*x+a^2)^(3/2),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 [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)*(e*x+d)**m*(b**2*x**2+2*a*b*x+a**2)**(3/2),x)
[Out]
Timed out
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Giac [B] time = 1.27909, size = 2631, normalized size = 9.5 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)*(e*x+d)^m*(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm="giac")
[Out]
((x*e + d)^m*b^4*m^4*x^5*e^5*sgn(b*x + a) + (x*e + d)^m*b^4*d*m^4*x^4*e^4*sgn(b*x + a) + 4*(x*e + d)^m*a*b^3*m
^4*x^4*e^5*sgn(b*x + a) + 10*(x*e + d)^m*b^4*m^3*x^5*e^5*sgn(b*x + a) + 4*(x*e + d)^m*a*b^3*d*m^4*x^3*e^4*sgn(
b*x + a) + 6*(x*e + d)^m*b^4*d*m^3*x^4*e^4*sgn(b*x + a) - 4*(x*e + d)^m*b^4*d^2*m^3*x^3*e^3*sgn(b*x + a) + 6*(
x*e + d)^m*a^2*b^2*m^4*x^3*e^5*sgn(b*x + a) + 44*(x*e + d)^m*a*b^3*m^3*x^4*e^5*sgn(b*x + a) + 35*(x*e + d)^m*b
^4*m^2*x^5*e^5*sgn(b*x + a) + 6*(x*e + d)^m*a^2*b^2*d*m^4*x^2*e^4*sgn(b*x + a) + 32*(x*e + d)^m*a*b^3*d*m^3*x^
3*e^4*sgn(b*x + a) + 11*(x*e + d)^m*b^4*d*m^2*x^4*e^4*sgn(b*x + a) - 12*(x*e + d)^m*a*b^3*d^2*m^3*x^2*e^3*sgn(
b*x + a) - 12*(x*e + d)^m*b^4*d^2*m^2*x^3*e^3*sgn(b*x + a) + 12*(x*e + d)^m*b^4*d^3*m^2*x^2*e^2*sgn(b*x + a) +
4*(x*e + d)^m*a^3*b*m^4*x^2*e^5*sgn(b*x + a) + 72*(x*e + d)^m*a^2*b^2*m^3*x^3*e^5*sgn(b*x + a) + 164*(x*e + d
)^m*a*b^3*m^2*x^4*e^5*sgn(b*x + a) + 50*(x*e + d)^m*b^4*m*x^5*e^5*sgn(b*x + a) + 4*(x*e + d)^m*a^3*b*d*m^4*x*e
^4*sgn(b*x + a) + 60*(x*e + d)^m*a^2*b^2*d*m^3*x^2*e^4*sgn(b*x + a) + 68*(x*e + d)^m*a*b^3*d*m^2*x^3*e^4*sgn(b
*x + a) + 6*(x*e + d)^m*b^4*d*m*x^4*e^4*sgn(b*x + a) - 12*(x*e + d)^m*a^2*b^2*d^2*m^3*x*e^3*sgn(b*x + a) - 72*
(x*e + d)^m*a*b^3*d^2*m^2*x^2*e^3*sgn(b*x + a) - 8*(x*e + d)^m*b^4*d^2*m*x^3*e^3*sgn(b*x + a) + 24*(x*e + d)^m
*a*b^3*d^3*m^2*x*e^2*sgn(b*x + a) + 12*(x*e + d)^m*b^4*d^3*m*x^2*e^2*sgn(b*x + a) - 24*(x*e + d)^m*b^4*d^4*m*x
*e*sgn(b*x + a) + (x*e + d)^m*a^4*m^4*x*e^5*sgn(b*x + a) + 52*(x*e + d)^m*a^3*b*m^3*x^2*e^5*sgn(b*x + a) + 294
*(x*e + d)^m*a^2*b^2*m^2*x^3*e^5*sgn(b*x + a) + 244*(x*e + d)^m*a*b^3*m*x^4*e^5*sgn(b*x + a) + 24*(x*e + d)^m*
b^4*x^5*e^5*sgn(b*x + a) + (x*e + d)^m*a^4*d*m^4*e^4*sgn(b*x + a) + 48*(x*e + d)^m*a^3*b*d*m^3*x*e^4*sgn(b*x +
a) + 174*(x*e + d)^m*a^2*b^2*d*m^2*x^2*e^4*sgn(b*x + a) + 40*(x*e + d)^m*a*b^3*d*m*x^3*e^4*sgn(b*x + a) - 4*(
x*e + d)^m*a^3*b*d^2*m^3*e^3*sgn(b*x + a) - 108*(x*e + d)^m*a^2*b^2*d^2*m^2*x*e^3*sgn(b*x + a) - 60*(x*e + d)^
m*a*b^3*d^2*m*x^2*e^3*sgn(b*x + a) + 12*(x*e + d)^m*a^2*b^2*d^3*m^2*e^2*sgn(b*x + a) + 120*(x*e + d)^m*a*b^3*d
^3*m*x*e^2*sgn(b*x + a) - 24*(x*e + d)^m*a*b^3*d^4*m*e*sgn(b*x + a) + 24*(x*e + d)^m*b^4*d^5*sgn(b*x + a) + 14
*(x*e + d)^m*a^4*m^3*x*e^5*sgn(b*x + a) + 236*(x*e + d)^m*a^3*b*m^2*x^2*e^5*sgn(b*x + a) + 468*(x*e + d)^m*a^2
*b^2*m*x^3*e^5*sgn(b*x + a) + 120*(x*e + d)^m*a*b^3*x^4*e^5*sgn(b*x + a) + 14*(x*e + d)^m*a^4*d*m^3*e^4*sgn(b*
x + a) + 188*(x*e + d)^m*a^3*b*d*m^2*x*e^4*sgn(b*x + a) + 120*(x*e + d)^m*a^2*b^2*d*m*x^2*e^4*sgn(b*x + a) - 4
8*(x*e + d)^m*a^3*b*d^2*m^2*e^3*sgn(b*x + a) - 240*(x*e + d)^m*a^2*b^2*d^2*m*x*e^3*sgn(b*x + a) + 108*(x*e + d
)^m*a^2*b^2*d^3*m*e^2*sgn(b*x + a) - 120*(x*e + d)^m*a*b^3*d^4*e*sgn(b*x + a) + 71*(x*e + d)^m*a^4*m^2*x*e^5*s
gn(b*x + a) + 428*(x*e + d)^m*a^3*b*m*x^2*e^5*sgn(b*x + a) + 240*(x*e + d)^m*a^2*b^2*x^3*e^5*sgn(b*x + a) + 71
*(x*e + d)^m*a^4*d*m^2*e^4*sgn(b*x + a) + 240*(x*e + d)^m*a^3*b*d*m*x*e^4*sgn(b*x + a) - 188*(x*e + d)^m*a^3*b
*d^2*m*e^3*sgn(b*x + a) + 240*(x*e + d)^m*a^2*b^2*d^3*e^2*sgn(b*x + a) + 154*(x*e + d)^m*a^4*m*x*e^5*sgn(b*x +
a) + 240*(x*e + d)^m*a^3*b*x^2*e^5*sgn(b*x + a) + 154*(x*e + d)^m*a^4*d*m*e^4*sgn(b*x + a) - 240*(x*e + d)^m*
a^3*b*d^2*e^3*sgn(b*x + a) + 120*(x*e + d)^m*a^4*x*e^5*sgn(b*x + a) + 120*(x*e + d)^m*a^4*d*e^4*sgn(b*x + a))/
(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)