3.2235 \(\int \frac{(a+b x)^{5/2} (A+B x)}{(d+e x)^{19/2}} \, dx\)
Optimal. Leaf size=309 \[ \frac{256 b^4 (a+b x)^{7/2} (-17 a B e+10 A b e+7 b B d)}{765765 e (d+e x)^{7/2} (b d-a e)^6}+\frac{128 b^3 (a+b x)^{7/2} (-17 a B e+10 A b e+7 b B d)}{109395 e (d+e x)^{9/2} (b d-a e)^5}+\frac{32 b^2 (a+b x)^{7/2} (-17 a B e+10 A b e+7 b B d)}{12155 e (d+e x)^{11/2} (b d-a e)^4}+\frac{16 b (a+b x)^{7/2} (-17 a B e+10 A b e+7 b B d)}{3315 e (d+e x)^{13/2} (b d-a e)^3}+\frac{2 (a+b x)^{7/2} (-17 a B e+10 A b e+7 b B d)}{255 e (d+e x)^{15/2} (b d-a e)^2}-\frac{2 (a+b x)^{7/2} (B d-A e)}{17 e (d+e x)^{17/2} (b d-a e)} \]
[Out]
(-2*(B*d - A*e)*(a + b*x)^(7/2))/(17*e*(b*d - a*e)*(d + e*x)^(17/2)) + (2*(7*b*B*d + 10*A*b*e - 17*a*B*e)*(a +
b*x)^(7/2))/(255*e*(b*d - a*e)^2*(d + e*x)^(15/2)) + (16*b*(7*b*B*d + 10*A*b*e - 17*a*B*e)*(a + b*x)^(7/2))/(
3315*e*(b*d - a*e)^3*(d + e*x)^(13/2)) + (32*b^2*(7*b*B*d + 10*A*b*e - 17*a*B*e)*(a + b*x)^(7/2))/(12155*e*(b*
d - a*e)^4*(d + e*x)^(11/2)) + (128*b^3*(7*b*B*d + 10*A*b*e - 17*a*B*e)*(a + b*x)^(7/2))/(109395*e*(b*d - a*e)
^5*(d + e*x)^(9/2)) + (256*b^4*(7*b*B*d + 10*A*b*e - 17*a*B*e)*(a + b*x)^(7/2))/(765765*e*(b*d - a*e)^6*(d + e
*x)^(7/2))
________________________________________________________________________________________
Rubi [A] time = 0.199307, antiderivative size = 309, 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} \[ \frac{256 b^4 (a+b x)^{7/2} (-17 a B e+10 A b e+7 b B d)}{765765 e (d+e x)^{7/2} (b d-a e)^6}+\frac{128 b^3 (a+b x)^{7/2} (-17 a B e+10 A b e+7 b B d)}{109395 e (d+e x)^{9/2} (b d-a e)^5}+\frac{32 b^2 (a+b x)^{7/2} (-17 a B e+10 A b e+7 b B d)}{12155 e (d+e x)^{11/2} (b d-a e)^4}+\frac{16 b (a+b x)^{7/2} (-17 a B e+10 A b e+7 b B d)}{3315 e (d+e x)^{13/2} (b d-a e)^3}+\frac{2 (a+b x)^{7/2} (-17 a B e+10 A b e+7 b B d)}{255 e (d+e x)^{15/2} (b d-a e)^2}-\frac{2 (a+b x)^{7/2} (B d-A e)}{17 e (d+e x)^{17/2} (b d-a e)} \]
Antiderivative was successfully verified.
[In]
Int[((a + b*x)^(5/2)*(A + B*x))/(d + e*x)^(19/2),x]
[Out]
(-2*(B*d - A*e)*(a + b*x)^(7/2))/(17*e*(b*d - a*e)*(d + e*x)^(17/2)) + (2*(7*b*B*d + 10*A*b*e - 17*a*B*e)*(a +
b*x)^(7/2))/(255*e*(b*d - a*e)^2*(d + e*x)^(15/2)) + (16*b*(7*b*B*d + 10*A*b*e - 17*a*B*e)*(a + b*x)^(7/2))/(
3315*e*(b*d - a*e)^3*(d + e*x)^(13/2)) + (32*b^2*(7*b*B*d + 10*A*b*e - 17*a*B*e)*(a + b*x)^(7/2))/(12155*e*(b*
d - a*e)^4*(d + e*x)^(11/2)) + (128*b^3*(7*b*B*d + 10*A*b*e - 17*a*B*e)*(a + b*x)^(7/2))/(109395*e*(b*d - a*e)
^5*(d + e*x)^(9/2)) + (256*b^4*(7*b*B*d + 10*A*b*e - 17*a*B*e)*(a + b*x)^(7/2))/(765765*e*(b*d - a*e)^6*(d + e
*x)^(7/2))
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)^{5/2} (A+B x)}{(d+e x)^{19/2}} \, dx &=-\frac{2 (B d-A e) (a+b x)^{7/2}}{17 e (b d-a e) (d+e x)^{17/2}}+\frac{(7 b B d+10 A b e-17 a B e) \int \frac{(a+b x)^{5/2}}{(d+e x)^{17/2}} \, dx}{17 e (b d-a e)}\\ &=-\frac{2 (B d-A e) (a+b x)^{7/2}}{17 e (b d-a e) (d+e x)^{17/2}}+\frac{2 (7 b B d+10 A b e-17 a B e) (a+b x)^{7/2}}{255 e (b d-a e)^2 (d+e x)^{15/2}}+\frac{(8 b (7 b B d+10 A b e-17 a B e)) \int \frac{(a+b x)^{5/2}}{(d+e x)^{15/2}} \, dx}{255 e (b d-a e)^2}\\ &=-\frac{2 (B d-A e) (a+b x)^{7/2}}{17 e (b d-a e) (d+e x)^{17/2}}+\frac{2 (7 b B d+10 A b e-17 a B e) (a+b x)^{7/2}}{255 e (b d-a e)^2 (d+e x)^{15/2}}+\frac{16 b (7 b B d+10 A b e-17 a B e) (a+b x)^{7/2}}{3315 e (b d-a e)^3 (d+e x)^{13/2}}+\frac{\left (16 b^2 (7 b B d+10 A b e-17 a B e)\right ) \int \frac{(a+b x)^{5/2}}{(d+e x)^{13/2}} \, dx}{1105 e (b d-a e)^3}\\ &=-\frac{2 (B d-A e) (a+b x)^{7/2}}{17 e (b d-a e) (d+e x)^{17/2}}+\frac{2 (7 b B d+10 A b e-17 a B e) (a+b x)^{7/2}}{255 e (b d-a e)^2 (d+e x)^{15/2}}+\frac{16 b (7 b B d+10 A b e-17 a B e) (a+b x)^{7/2}}{3315 e (b d-a e)^3 (d+e x)^{13/2}}+\frac{32 b^2 (7 b B d+10 A b e-17 a B e) (a+b x)^{7/2}}{12155 e (b d-a e)^4 (d+e x)^{11/2}}+\frac{\left (64 b^3 (7 b B d+10 A b e-17 a B e)\right ) \int \frac{(a+b x)^{5/2}}{(d+e x)^{11/2}} \, dx}{12155 e (b d-a e)^4}\\ &=-\frac{2 (B d-A e) (a+b x)^{7/2}}{17 e (b d-a e) (d+e x)^{17/2}}+\frac{2 (7 b B d+10 A b e-17 a B e) (a+b x)^{7/2}}{255 e (b d-a e)^2 (d+e x)^{15/2}}+\frac{16 b (7 b B d+10 A b e-17 a B e) (a+b x)^{7/2}}{3315 e (b d-a e)^3 (d+e x)^{13/2}}+\frac{32 b^2 (7 b B d+10 A b e-17 a B e) (a+b x)^{7/2}}{12155 e (b d-a e)^4 (d+e x)^{11/2}}+\frac{128 b^3 (7 b B d+10 A b e-17 a B e) (a+b x)^{7/2}}{109395 e (b d-a e)^5 (d+e x)^{9/2}}+\frac{\left (128 b^4 (7 b B d+10 A b e-17 a B e)\right ) \int \frac{(a+b x)^{5/2}}{(d+e x)^{9/2}} \, dx}{109395 e (b d-a e)^5}\\ &=-\frac{2 (B d-A e) (a+b x)^{7/2}}{17 e (b d-a e) (d+e x)^{17/2}}+\frac{2 (7 b B d+10 A b e-17 a B e) (a+b x)^{7/2}}{255 e (b d-a e)^2 (d+e x)^{15/2}}+\frac{16 b (7 b B d+10 A b e-17 a B e) (a+b x)^{7/2}}{3315 e (b d-a e)^3 (d+e x)^{13/2}}+\frac{32 b^2 (7 b B d+10 A b e-17 a B e) (a+b x)^{7/2}}{12155 e (b d-a e)^4 (d+e x)^{11/2}}+\frac{128 b^3 (7 b B d+10 A b e-17 a B e) (a+b x)^{7/2}}{109395 e (b d-a e)^5 (d+e x)^{9/2}}+\frac{256 b^4 (7 b B d+10 A b e-17 a B e) (a+b x)^{7/2}}{765765 e (b d-a e)^6 (d+e x)^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.368241, size = 160, normalized size = 0.52 \[ \frac{2 (a+b x)^{7/2} \left (45045 (B d-A e)-\frac{2 (d+e x) \left (8 b (d+e x) \left (2 b (d+e x) \left (4 b (d+e x) (-7 a e+9 b d+2 b e x)+63 (b d-a e)^2\right )+231 (b d-a e)^3\right )+3003 (b d-a e)^4\right ) \left (-\frac{17 a B e}{2}+5 A b e+\frac{7 b B d}{2}\right )}{(b d-a e)^5}\right )}{765765 e (d+e x)^{17/2} (a e-b d)} \]
Antiderivative was successfully verified.
[In]
Integrate[((a + b*x)^(5/2)*(A + B*x))/(d + e*x)^(19/2),x]
[Out]
(2*(a + b*x)^(7/2)*(45045*(B*d - A*e) - (2*((7*b*B*d)/2 + 5*A*b*e - (17*a*B*e)/2)*(d + e*x)*(3003*(b*d - a*e)^
4 + 8*b*(d + e*x)*(231*(b*d - a*e)^3 + 2*b*(d + e*x)*(63*(b*d - a*e)^2 + 4*b*(d + e*x)*(9*b*d - 7*a*e + 2*b*e*
x)))))/(b*d - a*e)^5))/(765765*e*(-(b*d) + a*e)*(d + e*x)^(17/2))
________________________________________________________________________________________
Maple [B] time = 0.012, size = 722, normalized size = 2.3 \begin{align*} -{\frac{-2560\,A{b}^{5}{e}^{5}{x}^{5}+4352\,Ba{b}^{4}{e}^{5}{x}^{5}-1792\,B{b}^{5}d{e}^{4}{x}^{5}+8960\,Aa{b}^{4}{e}^{5}{x}^{4}-21760\,A{b}^{5}d{e}^{4}{x}^{4}-15232\,B{a}^{2}{b}^{3}{e}^{5}{x}^{4}+43264\,Ba{b}^{4}d{e}^{4}{x}^{4}-15232\,B{b}^{5}{d}^{2}{e}^{3}{x}^{4}-20160\,A{a}^{2}{b}^{3}{e}^{5}{x}^{3}+76160\,Aa{b}^{4}d{e}^{4}{x}^{3}-81600\,A{b}^{5}{d}^{2}{e}^{3}{x}^{3}+34272\,B{a}^{3}{b}^{2}{e}^{5}{x}^{3}-143584\,B{a}^{2}{b}^{3}d{e}^{4}{x}^{3}+192032\,Ba{b}^{4}{d}^{2}{e}^{3}{x}^{3}-57120\,B{b}^{5}{d}^{3}{e}^{2}{x}^{3}+36960\,A{a}^{3}{b}^{2}{e}^{5}{x}^{2}-171360\,A{a}^{2}{b}^{3}d{e}^{4}{x}^{2}+285600\,Aa{b}^{4}{d}^{2}{e}^{3}{x}^{2}-176800\,A{b}^{5}{d}^{3}{e}^{2}{x}^{2}-62832\,B{a}^{4}b{e}^{5}{x}^{2}+317184\,B{a}^{3}{b}^{2}d{e}^{4}{x}^{2}-605472\,B{a}^{2}{b}^{3}{d}^{2}{e}^{3}{x}^{2}+500480\,Ba{b}^{4}{d}^{3}{e}^{2}{x}^{2}-123760\,B{b}^{5}{d}^{4}e{x}^{2}-60060\,A{a}^{4}b{e}^{5}x+314160\,A{a}^{3}{b}^{2}d{e}^{4}x-642600\,A{a}^{2}{b}^{3}{d}^{2}{e}^{3}x+618800\,Aa{b}^{4}{d}^{3}{e}^{2}x-243100\,A{b}^{5}{d}^{4}ex+102102\,B{a}^{5}{e}^{5}x-576114\,B{a}^{4}bd{e}^{4}x+1312332\,B{a}^{3}{b}^{2}{d}^{2}{e}^{3}x-1501780\,B{a}^{2}{b}^{3}{d}^{3}{e}^{2}x+846430\,Ba{b}^{4}{d}^{4}ex-170170\,B{b}^{5}{d}^{5}x+90090\,A{a}^{5}{e}^{5}-510510\,A{a}^{4}bd{e}^{4}+1178100\,A{a}^{3}{b}^{2}{d}^{2}{e}^{3}-1392300\,A{a}^{2}{b}^{3}{d}^{3}{e}^{2}+850850\,Aa{b}^{4}{d}^{4}e-218790\,A{b}^{5}{d}^{5}+12012\,B{a}^{5}d{e}^{4}-62832\,B{a}^{4}b{d}^{2}{e}^{3}+128520\,B{a}^{3}{b}^{2}{d}^{3}{e}^{2}-123760\,B{a}^{2}{b}^{3}{d}^{4}e+48620\,Ba{b}^{4}{d}^{5}}{765765\,{a}^{6}{e}^{6}-4594590\,{a}^{5}bd{e}^{5}+11486475\,{a}^{4}{b}^{2}{d}^{2}{e}^{4}-15315300\,{a}^{3}{b}^{3}{d}^{3}{e}^{3}+11486475\,{a}^{2}{b}^{4}{d}^{4}{e}^{2}-4594590\,a{b}^{5}{d}^{5}e+765765\,{b}^{6}{d}^{6}} \left ( bx+a \right ) ^{{\frac{7}{2}}} \left ( ex+d \right ) ^{-{\frac{17}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((b*x+a)^(5/2)*(B*x+A)/(e*x+d)^(19/2),x)
[Out]
-2/765765*(b*x+a)^(7/2)*(-1280*A*b^5*e^5*x^5+2176*B*a*b^4*e^5*x^5-896*B*b^5*d*e^4*x^5+4480*A*a*b^4*e^5*x^4-108
80*A*b^5*d*e^4*x^4-7616*B*a^2*b^3*e^5*x^4+21632*B*a*b^4*d*e^4*x^4-7616*B*b^5*d^2*e^3*x^4-10080*A*a^2*b^3*e^5*x
^3+38080*A*a*b^4*d*e^4*x^3-40800*A*b^5*d^2*e^3*x^3+17136*B*a^3*b^2*e^5*x^3-71792*B*a^2*b^3*d*e^4*x^3+96016*B*a
*b^4*d^2*e^3*x^3-28560*B*b^5*d^3*e^2*x^3+18480*A*a^3*b^2*e^5*x^2-85680*A*a^2*b^3*d*e^4*x^2+142800*A*a*b^4*d^2*
e^3*x^2-88400*A*b^5*d^3*e^2*x^2-31416*B*a^4*b*e^5*x^2+158592*B*a^3*b^2*d*e^4*x^2-302736*B*a^2*b^3*d^2*e^3*x^2+
250240*B*a*b^4*d^3*e^2*x^2-61880*B*b^5*d^4*e*x^2-30030*A*a^4*b*e^5*x+157080*A*a^3*b^2*d*e^4*x-321300*A*a^2*b^3
*d^2*e^3*x+309400*A*a*b^4*d^3*e^2*x-121550*A*b^5*d^4*e*x+51051*B*a^5*e^5*x-288057*B*a^4*b*d*e^4*x+656166*B*a^3
*b^2*d^2*e^3*x-750890*B*a^2*b^3*d^3*e^2*x+423215*B*a*b^4*d^4*e*x-85085*B*b^5*d^5*x+45045*A*a^5*e^5-255255*A*a^
4*b*d*e^4+589050*A*a^3*b^2*d^2*e^3-696150*A*a^2*b^3*d^3*e^2+425425*A*a*b^4*d^4*e-109395*A*b^5*d^5+6006*B*a^5*d
*e^4-31416*B*a^4*b*d^2*e^3+64260*B*a^3*b^2*d^3*e^2-61880*B*a^2*b^3*d^4*e+24310*B*a*b^4*d^5)/(e*x+d)^(17/2)/(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)^(5/2)*(B*x+A)/(e*x+d)^(19/2),x, algorithm="maxima")
[Out]
Exception raised: ValueError
________________________________________________________________________________________
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)^(5/2)*(B*x+A)/(e*x+d)^(19/2),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)**(5/2)*(B*x+A)/(e*x+d)**(19/2),x)
[Out]
Timed out
________________________________________________________________________________________
Giac [B] time = 8.60805, size = 2479, normalized size = 8.02 \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)^(5/2)*(B*x+A)/(e*x+d)^(19/2),x, algorithm="giac")
[Out]
-1/200740700160*((8*(2*(4*(b*x + a)*(2*(7*B*b^20*d^3*abs(b)*e^12 - 31*B*a*b^19*d^2*abs(b)*e^13 + 10*A*b^20*d^2
*abs(b)*e^13 + 41*B*a^2*b^18*d*abs(b)*e^14 - 20*A*a*b^19*d*abs(b)*e^14 - 17*B*a^3*b^17*abs(b)*e^15 + 10*A*a^2*
b^18*abs(b)*e^15)*(b*x + a)/(b^36*d^9*e^18 - 9*a*b^35*d^8*e^19 + 36*a^2*b^34*d^7*e^20 - 84*a^3*b^33*d^6*e^21 +
126*a^4*b^32*d^5*e^22 - 126*a^5*b^31*d^4*e^23 + 84*a^6*b^30*d^3*e^24 - 36*a^7*b^29*d^2*e^25 + 9*a^8*b^28*d*e^
26 - a^9*b^27*e^27) + 17*(7*B*b^21*d^4*abs(b)*e^11 - 38*B*a*b^20*d^3*abs(b)*e^12 + 10*A*b^21*d^3*abs(b)*e^12 +
72*B*a^2*b^19*d^2*abs(b)*e^13 - 30*A*a*b^20*d^2*abs(b)*e^13 - 58*B*a^3*b^18*d*abs(b)*e^14 + 30*A*a^2*b^19*d*a
bs(b)*e^14 + 17*B*a^4*b^17*abs(b)*e^15 - 10*A*a^3*b^18*abs(b)*e^15)/(b^36*d^9*e^18 - 9*a*b^35*d^8*e^19 + 36*a^
2*b^34*d^7*e^20 - 84*a^3*b^33*d^6*e^21 + 126*a^4*b^32*d^5*e^22 - 126*a^5*b^31*d^4*e^23 + 84*a^6*b^30*d^3*e^24
- 36*a^7*b^29*d^2*e^25 + 9*a^8*b^28*d*e^26 - a^9*b^27*e^27)) + 255*(7*B*b^22*d^5*abs(b)*e^10 - 45*B*a*b^21*d^4
*abs(b)*e^11 + 10*A*b^22*d^4*abs(b)*e^11 + 110*B*a^2*b^20*d^3*abs(b)*e^12 - 40*A*a*b^21*d^3*abs(b)*e^12 - 130*
B*a^3*b^19*d^2*abs(b)*e^13 + 60*A*a^2*b^20*d^2*abs(b)*e^13 + 75*B*a^4*b^18*d*abs(b)*e^14 - 40*A*a^3*b^19*d*abs
(b)*e^14 - 17*B*a^5*b^17*abs(b)*e^15 + 10*A*a^4*b^18*abs(b)*e^15)/(b^36*d^9*e^18 - 9*a*b^35*d^8*e^19 + 36*a^2*
b^34*d^7*e^20 - 84*a^3*b^33*d^6*e^21 + 126*a^4*b^32*d^5*e^22 - 126*a^5*b^31*d^4*e^23 + 84*a^6*b^30*d^3*e^24 -
36*a^7*b^29*d^2*e^25 + 9*a^8*b^28*d*e^26 - a^9*b^27*e^27))*(b*x + a) + 1105*(7*B*b^23*d^6*abs(b)*e^9 - 52*B*a*
b^22*d^5*abs(b)*e^10 + 10*A*b^23*d^5*abs(b)*e^10 + 155*B*a^2*b^21*d^4*abs(b)*e^11 - 50*A*a*b^22*d^4*abs(b)*e^1
1 - 240*B*a^3*b^20*d^3*abs(b)*e^12 + 100*A*a^2*b^21*d^3*abs(b)*e^12 + 205*B*a^4*b^19*d^2*abs(b)*e^13 - 100*A*a
^3*b^20*d^2*abs(b)*e^13 - 92*B*a^5*b^18*d*abs(b)*e^14 + 50*A*a^4*b^19*d*abs(b)*e^14 + 17*B*a^6*b^17*abs(b)*e^1
5 - 10*A*a^5*b^18*abs(b)*e^15)/(b^36*d^9*e^18 - 9*a*b^35*d^8*e^19 + 36*a^2*b^34*d^7*e^20 - 84*a^3*b^33*d^6*e^2
1 + 126*a^4*b^32*d^5*e^22 - 126*a^5*b^31*d^4*e^23 + 84*a^6*b^30*d^3*e^24 - 36*a^7*b^29*d^2*e^25 + 9*a^8*b^28*d
*e^26 - a^9*b^27*e^27))*(b*x + a) + 12155*(7*B*b^24*d^7*abs(b)*e^8 - 59*B*a*b^23*d^6*abs(b)*e^9 + 10*A*b^24*d^
6*abs(b)*e^9 + 207*B*a^2*b^22*d^5*abs(b)*e^10 - 60*A*a*b^23*d^5*abs(b)*e^10 - 395*B*a^3*b^21*d^4*abs(b)*e^11 +
150*A*a^2*b^22*d^4*abs(b)*e^11 + 445*B*a^4*b^20*d^3*abs(b)*e^12 - 200*A*a^3*b^21*d^3*abs(b)*e^12 - 297*B*a^5*
b^19*d^2*abs(b)*e^13 + 150*A*a^4*b^20*d^2*abs(b)*e^13 + 109*B*a^6*b^18*d*abs(b)*e^14 - 60*A*a^5*b^19*d*abs(b)*
e^14 - 17*B*a^7*b^17*abs(b)*e^15 + 10*A*a^6*b^18*abs(b)*e^15)/(b^36*d^9*e^18 - 9*a*b^35*d^8*e^19 + 36*a^2*b^34
*d^7*e^20 - 84*a^3*b^33*d^6*e^21 + 126*a^4*b^32*d^5*e^22 - 126*a^5*b^31*d^4*e^23 + 84*a^6*b^30*d^3*e^24 - 36*a
^7*b^29*d^2*e^25 + 9*a^8*b^28*d*e^26 - a^9*b^27*e^27))*(b*x + a) - 109395*(B*a*b^24*d^7*abs(b)*e^8 - A*b^25*d^
7*abs(b)*e^8 - 7*B*a^2*b^23*d^6*abs(b)*e^9 + 7*A*a*b^24*d^6*abs(b)*e^9 + 21*B*a^3*b^22*d^5*abs(b)*e^10 - 21*A*
a^2*b^23*d^5*abs(b)*e^10 - 35*B*a^4*b^21*d^4*abs(b)*e^11 + 35*A*a^3*b^22*d^4*abs(b)*e^11 + 35*B*a^5*b^20*d^3*a
bs(b)*e^12 - 35*A*a^4*b^21*d^3*abs(b)*e^12 - 21*B*a^6*b^19*d^2*abs(b)*e^13 + 21*A*a^5*b^20*d^2*abs(b)*e^13 + 7
*B*a^7*b^18*d*abs(b)*e^14 - 7*A*a^6*b^19*d*abs(b)*e^14 - B*a^8*b^17*abs(b)*e^15 + A*a^7*b^18*abs(b)*e^15)/(b^3
6*d^9*e^18 - 9*a*b^35*d^8*e^19 + 36*a^2*b^34*d^7*e^20 - 84*a^3*b^33*d^6*e^21 + 126*a^4*b^32*d^5*e^22 - 126*a^5
*b^31*d^4*e^23 + 84*a^6*b^30*d^3*e^24 - 36*a^7*b^29*d^2*e^25 + 9*a^8*b^28*d*e^26 - a^9*b^27*e^27))*(b*x + a)^(
7/2)/(b^2*d + (b*x + a)*b*e - a*b*e)^(17/2)