3.2222 \(\int \frac{(a+b x)^{3/2} (A+B x)}{(d+e x)^{17/2}} \, dx\)

Optimal. Leaf size=304 \[ \frac{256 b^4 (a+b x)^{5/2} (-3 a B e+2 A b e+b B d)}{45045 e (d+e x)^{5/2} (b d-a e)^6}+\frac{128 b^3 (a+b x)^{5/2} (-3 a B e+2 A b e+b B d)}{9009 e (d+e x)^{7/2} (b d-a e)^5}+\frac{32 b^2 (a+b x)^{5/2} (-3 a B e+2 A b e+b B d)}{1287 e (d+e x)^{9/2} (b d-a e)^4}+\frac{16 b (a+b x)^{5/2} (-3 a B e+2 A b e+b B d)}{429 e (d+e x)^{11/2} (b d-a e)^3}+\frac{2 (a+b x)^{5/2} (-3 a B e+2 A b e+b B d)}{39 e (d+e x)^{13/2} (b d-a e)^2}-\frac{2 (a+b x)^{5/2} (B d-A e)}{15 e (d+e x)^{15/2} (b d-a e)} \]

[Out]

(-2*(B*d - A*e)*(a + b*x)^(5/2))/(15*e*(b*d - a*e)*(d + e*x)^(15/2)) + (2*(b*B*d + 2*A*b*e - 3*a*B*e)*(a + b*x
)^(5/2))/(39*e*(b*d - a*e)^2*(d + e*x)^(13/2)) + (16*b*(b*B*d + 2*A*b*e - 3*a*B*e)*(a + b*x)^(5/2))/(429*e*(b*
d - a*e)^3*(d + e*x)^(11/2)) + (32*b^2*(b*B*d + 2*A*b*e - 3*a*B*e)*(a + b*x)^(5/2))/(1287*e*(b*d - a*e)^4*(d +
e*x)^(9/2)) + (128*b^3*(b*B*d + 2*A*b*e - 3*a*B*e)*(a + b*x)^(5/2))/(9009*e*(b*d - a*e)^5*(d + e*x)^(7/2)) +
(256*b^4*(b*B*d + 2*A*b*e - 3*a*B*e)*(a + b*x)^(5/2))/(45045*e*(b*d - a*e)^6*(d + e*x)^(5/2))

________________________________________________________________________________________

Rubi [A]  time = 0.186685, antiderivative size = 304, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {78, 45, 37} \[ \frac{256 b^4 (a+b x)^{5/2} (-3 a B e+2 A b e+b B d)}{45045 e (d+e x)^{5/2} (b d-a e)^6}+\frac{128 b^3 (a+b x)^{5/2} (-3 a B e+2 A b e+b B d)}{9009 e (d+e x)^{7/2} (b d-a e)^5}+\frac{32 b^2 (a+b x)^{5/2} (-3 a B e+2 A b e+b B d)}{1287 e (d+e x)^{9/2} (b d-a e)^4}+\frac{16 b (a+b x)^{5/2} (-3 a B e+2 A b e+b B d)}{429 e (d+e x)^{11/2} (b d-a e)^3}+\frac{2 (a+b x)^{5/2} (-3 a B e+2 A b e+b B d)}{39 e (d+e x)^{13/2} (b d-a e)^2}-\frac{2 (a+b x)^{5/2} (B d-A e)}{15 e (d+e x)^{15/2} (b d-a e)} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^(3/2)*(A + B*x))/(d + e*x)^(17/2),x]

[Out]

(-2*(B*d - A*e)*(a + b*x)^(5/2))/(15*e*(b*d - a*e)*(d + e*x)^(15/2)) + (2*(b*B*d + 2*A*b*e - 3*a*B*e)*(a + b*x
)^(5/2))/(39*e*(b*d - a*e)^2*(d + e*x)^(13/2)) + (16*b*(b*B*d + 2*A*b*e - 3*a*B*e)*(a + b*x)^(5/2))/(429*e*(b*
d - a*e)^3*(d + e*x)^(11/2)) + (32*b^2*(b*B*d + 2*A*b*e - 3*a*B*e)*(a + b*x)^(5/2))/(1287*e*(b*d - a*e)^4*(d +
e*x)^(9/2)) + (128*b^3*(b*B*d + 2*A*b*e - 3*a*B*e)*(a + b*x)^(5/2))/(9009*e*(b*d - a*e)^5*(d + e*x)^(7/2)) +
(256*b^4*(b*B*d + 2*A*b*e - 3*a*B*e)*(a + b*x)^(5/2))/(45045*e*(b*d - a*e)^6*(d + e*x)^(5/2))

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> -Simp[((b*e - a*f
)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1)
+ c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f,
n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ
[p, n]))))

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

\begin{align*} \int \frac{(a+b x)^{3/2} (A+B x)}{(d+e x)^{17/2}} \, dx &=-\frac{2 (B d-A e) (a+b x)^{5/2}}{15 e (b d-a e) (d+e x)^{15/2}}+\frac{(b B d+2 A b e-3 a B e) \int \frac{(a+b x)^{3/2}}{(d+e x)^{15/2}} \, dx}{3 e (b d-a e)}\\ &=-\frac{2 (B d-A e) (a+b x)^{5/2}}{15 e (b d-a e) (d+e x)^{15/2}}+\frac{2 (b B d+2 A b e-3 a B e) (a+b x)^{5/2}}{39 e (b d-a e)^2 (d+e x)^{13/2}}+\frac{(8 b (b B d+2 A b e-3 a B e)) \int \frac{(a+b x)^{3/2}}{(d+e x)^{13/2}} \, dx}{39 e (b d-a e)^2}\\ &=-\frac{2 (B d-A e) (a+b x)^{5/2}}{15 e (b d-a e) (d+e x)^{15/2}}+\frac{2 (b B d+2 A b e-3 a B e) (a+b x)^{5/2}}{39 e (b d-a e)^2 (d+e x)^{13/2}}+\frac{16 b (b B d+2 A b e-3 a B e) (a+b x)^{5/2}}{429 e (b d-a e)^3 (d+e x)^{11/2}}+\frac{\left (16 b^2 (b B d+2 A b e-3 a B e)\right ) \int \frac{(a+b x)^{3/2}}{(d+e x)^{11/2}} \, dx}{143 e (b d-a e)^3}\\ &=-\frac{2 (B d-A e) (a+b x)^{5/2}}{15 e (b d-a e) (d+e x)^{15/2}}+\frac{2 (b B d+2 A b e-3 a B e) (a+b x)^{5/2}}{39 e (b d-a e)^2 (d+e x)^{13/2}}+\frac{16 b (b B d+2 A b e-3 a B e) (a+b x)^{5/2}}{429 e (b d-a e)^3 (d+e x)^{11/2}}+\frac{32 b^2 (b B d+2 A b e-3 a B e) (a+b x)^{5/2}}{1287 e (b d-a e)^4 (d+e x)^{9/2}}+\frac{\left (64 b^3 (b B d+2 A b e-3 a B e)\right ) \int \frac{(a+b x)^{3/2}}{(d+e x)^{9/2}} \, dx}{1287 e (b d-a e)^4}\\ &=-\frac{2 (B d-A e) (a+b x)^{5/2}}{15 e (b d-a e) (d+e x)^{15/2}}+\frac{2 (b B d+2 A b e-3 a B e) (a+b x)^{5/2}}{39 e (b d-a e)^2 (d+e x)^{13/2}}+\frac{16 b (b B d+2 A b e-3 a B e) (a+b x)^{5/2}}{429 e (b d-a e)^3 (d+e x)^{11/2}}+\frac{32 b^2 (b B d+2 A b e-3 a B e) (a+b x)^{5/2}}{1287 e (b d-a e)^4 (d+e x)^{9/2}}+\frac{128 b^3 (b B d+2 A b e-3 a B e) (a+b x)^{5/2}}{9009 e (b d-a e)^5 (d+e x)^{7/2}}+\frac{\left (128 b^4 (b B d+2 A b e-3 a B e)\right ) \int \frac{(a+b x)^{3/2}}{(d+e x)^{7/2}} \, dx}{9009 e (b d-a e)^5}\\ &=-\frac{2 (B d-A e) (a+b x)^{5/2}}{15 e (b d-a e) (d+e x)^{15/2}}+\frac{2 (b B d+2 A b e-3 a B e) (a+b x)^{5/2}}{39 e (b d-a e)^2 (d+e x)^{13/2}}+\frac{16 b (b B d+2 A b e-3 a B e) (a+b x)^{5/2}}{429 e (b d-a e)^3 (d+e x)^{11/2}}+\frac{32 b^2 (b B d+2 A b e-3 a B e) (a+b x)^{5/2}}{1287 e (b d-a e)^4 (d+e x)^{9/2}}+\frac{128 b^3 (b B d+2 A b e-3 a B e) (a+b x)^{5/2}}{9009 e (b d-a e)^5 (d+e x)^{7/2}}+\frac{256 b^4 (b B d+2 A b e-3 a B e) (a+b x)^{5/2}}{45045 e (b d-a e)^6 (d+e x)^{5/2}}\\ \end{align*}

Mathematica [A]  time = 0.331472, size = 155, normalized size = 0.51 \[ \frac{2 (a+b x)^{5/2} \left (15015 (B d-A e)-\frac{5 (d+e x) \left (8 b (d+e x) \left (2 b (d+e x) \left (4 b (d+e x) (-5 a e+7 b d+2 b e x)+35 (b d-a e)^2\right )+105 (b d-a e)^3\right )+1155 (b d-a e)^4\right ) (-3 a B e+2 A b e+b B d)}{(b d-a e)^5}\right )}{225225 e (d+e x)^{15/2} (a e-b d)} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^(3/2)*(A + B*x))/(d + e*x)^(17/2),x]

[Out]

(2*(a + b*x)^(5/2)*(15015*(B*d - A*e) - (5*(b*B*d + 2*A*b*e - 3*a*B*e)*(d + e*x)*(1155*(b*d - a*e)^4 + 8*b*(d
+ e*x)*(105*(b*d - a*e)^3 + 2*b*(d + e*x)*(35*(b*d - a*e)^2 + 4*b*(d + e*x)*(7*b*d - 5*a*e + 2*b*e*x)))))/(b*d
- a*e)^5))/(225225*e*(-(b*d) + a*e)*(d + e*x)^(15/2))

________________________________________________________________________________________

Maple [B]  time = 0.01, size = 722, normalized size = 2.4 \begin{align*} -{\frac{-512\,A{b}^{5}{e}^{5}{x}^{5}+768\,Ba{b}^{4}{e}^{5}{x}^{5}-256\,B{b}^{5}d{e}^{4}{x}^{5}+1280\,Aa{b}^{4}{e}^{5}{x}^{4}-3840\,A{b}^{5}d{e}^{4}{x}^{4}-1920\,B{a}^{2}{b}^{3}{e}^{5}{x}^{4}+6400\,Ba{b}^{4}d{e}^{4}{x}^{4}-1920\,B{b}^{5}{d}^{2}{e}^{3}{x}^{4}-2240\,A{a}^{2}{b}^{3}{e}^{5}{x}^{3}+9600\,Aa{b}^{4}d{e}^{4}{x}^{3}-12480\,A{b}^{5}{d}^{2}{e}^{3}{x}^{3}+3360\,B{a}^{3}{b}^{2}{e}^{5}{x}^{3}-15520\,B{a}^{2}{b}^{3}d{e}^{4}{x}^{3}+23520\,Ba{b}^{4}{d}^{2}{e}^{3}{x}^{3}-6240\,B{b}^{5}{d}^{3}{e}^{2}{x}^{3}+3360\,A{a}^{3}{b}^{2}{e}^{5}{x}^{2}-16800\,A{a}^{2}{b}^{3}d{e}^{4}{x}^{2}+31200\,Aa{b}^{4}{d}^{2}{e}^{3}{x}^{2}-22880\,A{b}^{5}{d}^{3}{e}^{2}{x}^{2}-5040\,B{a}^{4}b{e}^{5}{x}^{2}+26880\,B{a}^{3}{b}^{2}d{e}^{4}{x}^{2}-55200\,B{a}^{2}{b}^{3}{d}^{2}{e}^{3}{x}^{2}+49920\,Ba{b}^{4}{d}^{3}{e}^{2}{x}^{2}-11440\,B{b}^{5}{d}^{4}e{x}^{2}-4620\,A{a}^{4}b{e}^{5}x+25200\,A{a}^{3}{b}^{2}d{e}^{4}x-54600\,A{a}^{2}{b}^{3}{d}^{2}{e}^{3}x+57200\,Aa{b}^{4}{d}^{3}{e}^{2}x-25740\,A{b}^{5}{d}^{4}ex+6930\,B{a}^{5}{e}^{5}x-40110\,B{a}^{4}bd{e}^{4}x+94500\,B{a}^{3}{b}^{2}{d}^{2}{e}^{3}x-113100\,B{a}^{2}{b}^{3}{d}^{3}{e}^{2}x+67210\,Ba{b}^{4}{d}^{4}ex-12870\,B{b}^{5}{d}^{5}x+6006\,A{a}^{5}{e}^{5}-34650\,A{a}^{4}bd{e}^{4}+81900\,A{a}^{3}{b}^{2}{d}^{2}{e}^{3}-100100\,A{a}^{2}{b}^{3}{d}^{3}{e}^{2}+64350\,Aa{b}^{4}{d}^{4}e-18018\,A{b}^{5}{d}^{5}+924\,B{a}^{5}d{e}^{4}-5040\,B{a}^{4}b{d}^{2}{e}^{3}+10920\,B{a}^{3}{b}^{2}{d}^{3}{e}^{2}-11440\,B{a}^{2}{b}^{3}{d}^{4}e+5148\,Ba{b}^{4}{d}^{5}}{45045\,{a}^{6}{e}^{6}-270270\,{a}^{5}bd{e}^{5}+675675\,{a}^{4}{b}^{2}{d}^{2}{e}^{4}-900900\,{a}^{3}{b}^{3}{d}^{3}{e}^{3}+675675\,{a}^{2}{b}^{4}{d}^{4}{e}^{2}-270270\,a{b}^{5}{d}^{5}e+45045\,{b}^{6}{d}^{6}} \left ( bx+a \right ) ^{{\frac{5}{2}}} \left ( ex+d \right ) ^{-{\frac{15}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^(3/2)*(B*x+A)/(e*x+d)^(17/2),x)

[Out]

-2/45045*(b*x+a)^(5/2)*(-256*A*b^5*e^5*x^5+384*B*a*b^4*e^5*x^5-128*B*b^5*d*e^4*x^5+640*A*a*b^4*e^5*x^4-1920*A*
b^5*d*e^4*x^4-960*B*a^2*b^3*e^5*x^4+3200*B*a*b^4*d*e^4*x^4-960*B*b^5*d^2*e^3*x^4-1120*A*a^2*b^3*e^5*x^3+4800*A
*a*b^4*d*e^4*x^3-6240*A*b^5*d^2*e^3*x^3+1680*B*a^3*b^2*e^5*x^3-7760*B*a^2*b^3*d*e^4*x^3+11760*B*a*b^4*d^2*e^3*
x^3-3120*B*b^5*d^3*e^2*x^3+1680*A*a^3*b^2*e^5*x^2-8400*A*a^2*b^3*d*e^4*x^2+15600*A*a*b^4*d^2*e^3*x^2-11440*A*b
^5*d^3*e^2*x^2-2520*B*a^4*b*e^5*x^2+13440*B*a^3*b^2*d*e^4*x^2-27600*B*a^2*b^3*d^2*e^3*x^2+24960*B*a*b^4*d^3*e^
2*x^2-5720*B*b^5*d^4*e*x^2-2310*A*a^4*b*e^5*x+12600*A*a^3*b^2*d*e^4*x-27300*A*a^2*b^3*d^2*e^3*x+28600*A*a*b^4*
d^3*e^2*x-12870*A*b^5*d^4*e*x+3465*B*a^5*e^5*x-20055*B*a^4*b*d*e^4*x+47250*B*a^3*b^2*d^2*e^3*x-56550*B*a^2*b^3
*d^3*e^2*x+33605*B*a*b^4*d^4*e*x-6435*B*b^5*d^5*x+3003*A*a^5*e^5-17325*A*a^4*b*d*e^4+40950*A*a^3*b^2*d^2*e^3-5
0050*A*a^2*b^3*d^3*e^2+32175*A*a*b^4*d^4*e-9009*A*b^5*d^5+462*B*a^5*d*e^4-2520*B*a^4*b*d^2*e^3+5460*B*a^3*b^2*
d^3*e^2-5720*B*a^2*b^3*d^4*e+2574*B*a*b^4*d^5)/(e*x+d)^(15/2)/(a^6*e^6-6*a^5*b*d*e^5+15*a^4*b^2*d^2*e^4-20*a^3
*b^3*d^3*e^3+15*a^2*b^4*d^4*e^2-6*a*b^5*d^5*e+b^6*d^6)

________________________________________________________________________________________

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((b*x+a)^(3/2)*(B*x+A)/(e*x+d)^(17/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [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)^(3/2)*(B*x+A)/(e*x+d)^(17/2),x, algorithm="fricas")

[Out]

Timed out

________________________________________________________________________________________

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)**(3/2)*(B*x+A)/(e*x+d)**(17/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 5.43721, size = 2099, normalized size = 6.9 \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)^(3/2)*(B*x+A)/(e*x+d)^(17/2),x, algorithm="giac")

[Out]

-1/2952069120*((8*(2*(4*(b*x + a)*(2*(B*b^17*d^2*abs(b)*e^11 - 4*B*a*b^16*d*abs(b)*e^12 + 2*A*b^17*d*abs(b)*e^
12 + 3*B*a^2*b^15*abs(b)*e^13 - 2*A*a*b^16*abs(b)*e^13)*(b*x + a)/(b^32*d^8*e^16 - 8*a*b^31*d^7*e^17 + 28*a^2*
b^30*d^6*e^18 - 56*a^3*b^29*d^5*e^19 + 70*a^4*b^28*d^4*e^20 - 56*a^5*b^27*d^3*e^21 + 28*a^6*b^26*d^2*e^22 - 8*
a^7*b^25*d*e^23 + a^8*b^24*e^24) + 15*(B*b^18*d^3*abs(b)*e^10 - 5*B*a*b^17*d^2*abs(b)*e^11 + 2*A*b^18*d^2*abs(
b)*e^11 + 7*B*a^2*b^16*d*abs(b)*e^12 - 4*A*a*b^17*d*abs(b)*e^12 - 3*B*a^3*b^15*abs(b)*e^13 + 2*A*a^2*b^16*abs(
b)*e^13)/(b^32*d^8*e^16 - 8*a*b^31*d^7*e^17 + 28*a^2*b^30*d^6*e^18 - 56*a^3*b^29*d^5*e^19 + 70*a^4*b^28*d^4*e^
20 - 56*a^5*b^27*d^3*e^21 + 28*a^6*b^26*d^2*e^22 - 8*a^7*b^25*d*e^23 + a^8*b^24*e^24)) + 195*(B*b^19*d^4*abs(b
)*e^9 - 6*B*a*b^18*d^3*abs(b)*e^10 + 2*A*b^19*d^3*abs(b)*e^10 + 12*B*a^2*b^17*d^2*abs(b)*e^11 - 6*A*a*b^18*d^2
*abs(b)*e^11 - 10*B*a^3*b^16*d*abs(b)*e^12 + 6*A*a^2*b^17*d*abs(b)*e^12 + 3*B*a^4*b^15*abs(b)*e^13 - 2*A*a^3*b
^16*abs(b)*e^13)/(b^32*d^8*e^16 - 8*a*b^31*d^7*e^17 + 28*a^2*b^30*d^6*e^18 - 56*a^3*b^29*d^5*e^19 + 70*a^4*b^2
8*d^4*e^20 - 56*a^5*b^27*d^3*e^21 + 28*a^6*b^26*d^2*e^22 - 8*a^7*b^25*d*e^23 + a^8*b^24*e^24))*(b*x + a) + 715
*(B*b^20*d^5*abs(b)*e^8 - 7*B*a*b^19*d^4*abs(b)*e^9 + 2*A*b^20*d^4*abs(b)*e^9 + 18*B*a^2*b^18*d^3*abs(b)*e^10
- 8*A*a*b^19*d^3*abs(b)*e^10 - 22*B*a^3*b^17*d^2*abs(b)*e^11 + 12*A*a^2*b^18*d^2*abs(b)*e^11 + 13*B*a^4*b^16*d
*abs(b)*e^12 - 8*A*a^3*b^17*d*abs(b)*e^12 - 3*B*a^5*b^15*abs(b)*e^13 + 2*A*a^4*b^16*abs(b)*e^13)/(b^32*d^8*e^1
6 - 8*a*b^31*d^7*e^17 + 28*a^2*b^30*d^6*e^18 - 56*a^3*b^29*d^5*e^19 + 70*a^4*b^28*d^4*e^20 - 56*a^5*b^27*d^3*e
^21 + 28*a^6*b^26*d^2*e^22 - 8*a^7*b^25*d*e^23 + a^8*b^24*e^24))*(b*x + a) + 6435*(B*b^21*d^6*abs(b)*e^7 - 8*B
*a*b^20*d^5*abs(b)*e^8 + 2*A*b^21*d^5*abs(b)*e^8 + 25*B*a^2*b^19*d^4*abs(b)*e^9 - 10*A*a*b^20*d^4*abs(b)*e^9 -
40*B*a^3*b^18*d^3*abs(b)*e^10 + 20*A*a^2*b^19*d^3*abs(b)*e^10 + 35*B*a^4*b^17*d^2*abs(b)*e^11 - 20*A*a^3*b^18
*d^2*abs(b)*e^11 - 16*B*a^5*b^16*d*abs(b)*e^12 + 10*A*a^4*b^17*d*abs(b)*e^12 + 3*B*a^6*b^15*abs(b)*e^13 - 2*A*
a^5*b^16*abs(b)*e^13)/(b^32*d^8*e^16 - 8*a*b^31*d^7*e^17 + 28*a^2*b^30*d^6*e^18 - 56*a^3*b^29*d^5*e^19 + 70*a^
4*b^28*d^4*e^20 - 56*a^5*b^27*d^3*e^21 + 28*a^6*b^26*d^2*e^22 - 8*a^7*b^25*d*e^23 + a^8*b^24*e^24))*(b*x + a)
- 9009*(B*a*b^21*d^6*abs(b)*e^7 - A*b^22*d^6*abs(b)*e^7 - 6*B*a^2*b^20*d^5*abs(b)*e^8 + 6*A*a*b^21*d^5*abs(b)*
e^8 + 15*B*a^3*b^19*d^4*abs(b)*e^9 - 15*A*a^2*b^20*d^4*abs(b)*e^9 - 20*B*a^4*b^18*d^3*abs(b)*e^10 + 20*A*a^3*b
^19*d^3*abs(b)*e^10 + 15*B*a^5*b^17*d^2*abs(b)*e^11 - 15*A*a^4*b^18*d^2*abs(b)*e^11 - 6*B*a^6*b^16*d*abs(b)*e^
12 + 6*A*a^5*b^17*d*abs(b)*e^12 + B*a^7*b^15*abs(b)*e^13 - A*a^6*b^16*abs(b)*e^13)/(b^32*d^8*e^16 - 8*a*b^31*d
^7*e^17 + 28*a^2*b^30*d^6*e^18 - 56*a^3*b^29*d^5*e^19 + 70*a^4*b^28*d^4*e^20 - 56*a^5*b^27*d^3*e^21 + 28*a^6*b
^26*d^2*e^22 - 8*a^7*b^25*d*e^23 + a^8*b^24*e^24))*(b*x + a)^(5/2)/(b^2*d + (b*x + a)*b*e - a*b*e)^(15/2)