3.2261 \(\int \frac{A+B x}{(a+b x)^{5/2} (d+e x)^{9/2}} \, dx\)
Optimal. Leaf size=298 \[ -\frac{256 b^2 e \sqrt{a+b x} (7 a B e-10 A b e+3 b B d)}{105 \sqrt{d+e x} (b d-a e)^6}-\frac{128 b e \sqrt{a+b x} (7 a B e-10 A b e+3 b B d)}{105 (d+e x)^{3/2} (b d-a e)^5}-\frac{32 e \sqrt{a+b x} (7 a B e-10 A b e+3 b B d)}{35 (d+e x)^{5/2} (b d-a e)^4}-\frac{16 e \sqrt{a+b x} (7 a B e-10 A b e+3 b B d)}{21 b (d+e x)^{7/2} (b d-a e)^3}-\frac{2 (7 a B e-10 A b e+3 b B d)}{3 b \sqrt{a+b x} (d+e x)^{7/2} (b d-a e)^2}-\frac{2 (A b-a B)}{3 b (a+b x)^{3/2} (d+e x)^{7/2} (b d-a e)} \]
[Out]
(-2*(A*b - a*B))/(3*b*(b*d - a*e)*(a + b*x)^(3/2)*(d + e*x)^(7/2)) - (2*(3*b*B*d - 10*A*b*e + 7*a*B*e))/(3*b*(
b*d - a*e)^2*Sqrt[a + b*x]*(d + e*x)^(7/2)) - (16*e*(3*b*B*d - 10*A*b*e + 7*a*B*e)*Sqrt[a + b*x])/(21*b*(b*d -
a*e)^3*(d + e*x)^(7/2)) - (32*e*(3*b*B*d - 10*A*b*e + 7*a*B*e)*Sqrt[a + b*x])/(35*(b*d - a*e)^4*(d + e*x)^(5/
2)) - (128*b*e*(3*b*B*d - 10*A*b*e + 7*a*B*e)*Sqrt[a + b*x])/(105*(b*d - a*e)^5*(d + e*x)^(3/2)) - (256*b^2*e*
(3*b*B*d - 10*A*b*e + 7*a*B*e)*Sqrt[a + b*x])/(105*(b*d - a*e)^6*Sqrt[d + e*x])
________________________________________________________________________________________
Rubi [A] time = 0.207278, antiderivative size = 298, 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} \[ -\frac{256 b^2 e \sqrt{a+b x} (7 a B e-10 A b e+3 b B d)}{105 \sqrt{d+e x} (b d-a e)^6}-\frac{128 b e \sqrt{a+b x} (7 a B e-10 A b e+3 b B d)}{105 (d+e x)^{3/2} (b d-a e)^5}-\frac{32 e \sqrt{a+b x} (7 a B e-10 A b e+3 b B d)}{35 (d+e x)^{5/2} (b d-a e)^4}-\frac{16 e \sqrt{a+b x} (7 a B e-10 A b e+3 b B d)}{21 b (d+e x)^{7/2} (b d-a e)^3}-\frac{2 (7 a B e-10 A b e+3 b B d)}{3 b \sqrt{a+b x} (d+e x)^{7/2} (b d-a e)^2}-\frac{2 (A b-a B)}{3 b (a+b x)^{3/2} (d+e x)^{7/2} (b d-a e)} \]
Antiderivative was successfully verified.
[In]
Int[(A + B*x)/((a + b*x)^(5/2)*(d + e*x)^(9/2)),x]
[Out]
(-2*(A*b - a*B))/(3*b*(b*d - a*e)*(a + b*x)^(3/2)*(d + e*x)^(7/2)) - (2*(3*b*B*d - 10*A*b*e + 7*a*B*e))/(3*b*(
b*d - a*e)^2*Sqrt[a + b*x]*(d + e*x)^(7/2)) - (16*e*(3*b*B*d - 10*A*b*e + 7*a*B*e)*Sqrt[a + b*x])/(21*b*(b*d -
a*e)^3*(d + e*x)^(7/2)) - (32*e*(3*b*B*d - 10*A*b*e + 7*a*B*e)*Sqrt[a + b*x])/(35*(b*d - a*e)^4*(d + e*x)^(5/
2)) - (128*b*e*(3*b*B*d - 10*A*b*e + 7*a*B*e)*Sqrt[a + b*x])/(105*(b*d - a*e)^5*(d + e*x)^(3/2)) - (256*b^2*e*
(3*b*B*d - 10*A*b*e + 7*a*B*e)*Sqrt[a + b*x])/(105*(b*d - a*e)^6*Sqrt[d + e*x])
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}{(a+b x)^{5/2} (d+e x)^{9/2}} \, dx &=-\frac{2 (A b-a B)}{3 b (b d-a e) (a+b x)^{3/2} (d+e x)^{7/2}}+\frac{(3 b B d-10 A b e+7 a B e) \int \frac{1}{(a+b x)^{3/2} (d+e x)^{9/2}} \, dx}{3 b (b d-a e)}\\ &=-\frac{2 (A b-a B)}{3 b (b d-a e) (a+b x)^{3/2} (d+e x)^{7/2}}-\frac{2 (3 b B d-10 A b e+7 a B e)}{3 b (b d-a e)^2 \sqrt{a+b x} (d+e x)^{7/2}}-\frac{(8 e (3 b B d-10 A b e+7 a B e)) \int \frac{1}{\sqrt{a+b x} (d+e x)^{9/2}} \, dx}{3 b (b d-a e)^2}\\ &=-\frac{2 (A b-a B)}{3 b (b d-a e) (a+b x)^{3/2} (d+e x)^{7/2}}-\frac{2 (3 b B d-10 A b e+7 a B e)}{3 b (b d-a e)^2 \sqrt{a+b x} (d+e x)^{7/2}}-\frac{16 e (3 b B d-10 A b e+7 a B e) \sqrt{a+b x}}{21 b (b d-a e)^3 (d+e x)^{7/2}}-\frac{(16 e (3 b B d-10 A b e+7 a B e)) \int \frac{1}{\sqrt{a+b x} (d+e x)^{7/2}} \, dx}{7 (b d-a e)^3}\\ &=-\frac{2 (A b-a B)}{3 b (b d-a e) (a+b x)^{3/2} (d+e x)^{7/2}}-\frac{2 (3 b B d-10 A b e+7 a B e)}{3 b (b d-a e)^2 \sqrt{a+b x} (d+e x)^{7/2}}-\frac{16 e (3 b B d-10 A b e+7 a B e) \sqrt{a+b x}}{21 b (b d-a e)^3 (d+e x)^{7/2}}-\frac{32 e (3 b B d-10 A b e+7 a B e) \sqrt{a+b x}}{35 (b d-a e)^4 (d+e x)^{5/2}}-\frac{(64 b e (3 b B d-10 A b e+7 a B e)) \int \frac{1}{\sqrt{a+b x} (d+e x)^{5/2}} \, dx}{35 (b d-a e)^4}\\ &=-\frac{2 (A b-a B)}{3 b (b d-a e) (a+b x)^{3/2} (d+e x)^{7/2}}-\frac{2 (3 b B d-10 A b e+7 a B e)}{3 b (b d-a e)^2 \sqrt{a+b x} (d+e x)^{7/2}}-\frac{16 e (3 b B d-10 A b e+7 a B e) \sqrt{a+b x}}{21 b (b d-a e)^3 (d+e x)^{7/2}}-\frac{32 e (3 b B d-10 A b e+7 a B e) \sqrt{a+b x}}{35 (b d-a e)^4 (d+e x)^{5/2}}-\frac{128 b e (3 b B d-10 A b e+7 a B e) \sqrt{a+b x}}{105 (b d-a e)^5 (d+e x)^{3/2}}-\frac{\left (128 b^2 e (3 b B d-10 A b e+7 a B e)\right ) \int \frac{1}{\sqrt{a+b x} (d+e x)^{3/2}} \, dx}{105 (b d-a e)^5}\\ &=-\frac{2 (A b-a B)}{3 b (b d-a e) (a+b x)^{3/2} (d+e x)^{7/2}}-\frac{2 (3 b B d-10 A b e+7 a B e)}{3 b (b d-a e)^2 \sqrt{a+b x} (d+e x)^{7/2}}-\frac{16 e (3 b B d-10 A b e+7 a B e) \sqrt{a+b x}}{21 b (b d-a e)^3 (d+e x)^{7/2}}-\frac{32 e (3 b B d-10 A b e+7 a B e) \sqrt{a+b x}}{35 (b d-a e)^4 (d+e x)^{5/2}}-\frac{128 b e (3 b B d-10 A b e+7 a B e) \sqrt{a+b x}}{105 (b d-a e)^5 (d+e x)^{3/2}}-\frac{256 b^2 e (3 b B d-10 A b e+7 a B e) \sqrt{a+b x}}{105 (b d-a e)^6 \sqrt{d+e x}}\\ \end{align*}
Mathematica [A] time = 0.216613, size = 160, normalized size = 0.54 \[ \frac{2 \left (2 (a+b x) \left (8 e (a+b x) \left (2 b (d+e x) \left (4 b (d+e x) (-a e+3 b d+2 b e x)+3 (b d-a e)^2\right )+5 (b d-a e)^3\right )+35 (b d-a e)^4\right ) \left (-\frac{7 a B e}{2}+5 A b e-\frac{3}{2} b B d\right )-35 (A b-a B) (b d-a e)^5\right )}{105 b (a+b x)^{3/2} (d+e x)^{7/2} (b d-a e)^6} \]
Antiderivative was successfully verified.
[In]
Integrate[(A + B*x)/((a + b*x)^(5/2)*(d + e*x)^(9/2)),x]
[Out]
(2*(-35*(A*b - a*B)*(b*d - a*e)^5 + 2*((-3*b*B*d)/2 + 5*A*b*e - (7*a*B*e)/2)*(a + b*x)*(35*(b*d - a*e)^4 + 8*e
*(a + b*x)*(5*(b*d - a*e)^3 + 2*b*(d + e*x)*(3*(b*d - a*e)^2 + 4*b*(d + e*x)*(3*b*d - a*e + 2*b*e*x))))))/(105
*b*(b*d - a*e)^6*(a + b*x)^(3/2)*(d + e*x)^(7/2))
________________________________________________________________________________________
Maple [B] time = 0.011, size = 722, normalized size = 2.4 \begin{align*} -{\frac{-2560\,A{b}^{5}{e}^{5}{x}^{5}+1792\,Ba{b}^{4}{e}^{5}{x}^{5}+768\,B{b}^{5}d{e}^{4}{x}^{5}-3840\,Aa{b}^{4}{e}^{5}{x}^{4}-8960\,A{b}^{5}d{e}^{4}{x}^{4}+2688\,B{a}^{2}{b}^{3}{e}^{5}{x}^{4}+7424\,Ba{b}^{4}d{e}^{4}{x}^{4}+2688\,B{b}^{5}{d}^{2}{e}^{3}{x}^{4}-960\,A{a}^{2}{b}^{3}{e}^{5}{x}^{3}-13440\,Aa{b}^{4}d{e}^{4}{x}^{3}-11200\,A{b}^{5}{d}^{2}{e}^{3}{x}^{3}+672\,B{a}^{3}{b}^{2}{e}^{5}{x}^{3}+9696\,B{a}^{2}{b}^{3}d{e}^{4}{x}^{3}+11872\,Ba{b}^{4}{d}^{2}{e}^{3}{x}^{3}+3360\,B{b}^{5}{d}^{3}{e}^{2}{x}^{3}+160\,A{a}^{3}{b}^{2}{e}^{5}{x}^{2}-3360\,A{a}^{2}{b}^{3}d{e}^{4}{x}^{2}-16800\,Aa{b}^{4}{d}^{2}{e}^{3}{x}^{2}-5600\,A{b}^{5}{d}^{3}{e}^{2}{x}^{2}-112\,B{a}^{4}b{e}^{5}{x}^{2}+2304\,B{a}^{3}{b}^{2}d{e}^{4}{x}^{2}+12768\,B{a}^{2}{b}^{3}{d}^{2}{e}^{3}{x}^{2}+8960\,Ba{b}^{4}{d}^{3}{e}^{2}{x}^{2}+1680\,B{b}^{5}{d}^{4}e{x}^{2}-60\,A{a}^{4}b{e}^{5}x+560\,A{a}^{3}{b}^{2}d{e}^{4}x-4200\,A{a}^{2}{b}^{3}{d}^{2}{e}^{3}x-8400\,Aa{b}^{4}{d}^{3}{e}^{2}x-700\,A{b}^{5}{d}^{4}ex+42\,B{a}^{5}{e}^{5}x-374\,B{a}^{4}bd{e}^{4}x+2772\,B{a}^{3}{b}^{2}{d}^{2}{e}^{3}x+7140\,B{a}^{2}{b}^{3}{d}^{3}{e}^{2}x+3010\,Ba{b}^{4}{d}^{4}ex+210\,B{b}^{5}{d}^{5}x+30\,A{a}^{5}{e}^{5}-210\,A{a}^{4}bd{e}^{4}+700\,A{a}^{3}{b}^{2}{d}^{2}{e}^{3}-2100\,A{a}^{2}{b}^{3}{d}^{3}{e}^{2}-1050\,Aa{b}^{4}{d}^{4}e+70\,A{b}^{5}{d}^{5}+12\,B{a}^{5}d{e}^{4}-112\,B{a}^{4}b{d}^{2}{e}^{3}+840\,B{a}^{3}{b}^{2}{d}^{3}{e}^{2}+1680\,B{a}^{2}{b}^{3}{d}^{4}e+140\,Ba{b}^{4}{d}^{5}}{105\,{a}^{6}{e}^{6}-630\,{a}^{5}bd{e}^{5}+1575\,{a}^{4}{b}^{2}{d}^{2}{e}^{4}-2100\,{a}^{3}{b}^{3}{d}^{3}{e}^{3}+1575\,{a}^{2}{b}^{4}{d}^{4}{e}^{2}-630\,a{b}^{5}{d}^{5}e+105\,{b}^{6}{d}^{6}} \left ( bx+a \right ) ^{-{\frac{3}{2}}} \left ( ex+d \right ) ^{-{\frac{7}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((B*x+A)/(b*x+a)^(5/2)/(e*x+d)^(9/2),x)
[Out]
-2/105*(-1280*A*b^5*e^5*x^5+896*B*a*b^4*e^5*x^5+384*B*b^5*d*e^4*x^5-1920*A*a*b^4*e^5*x^4-4480*A*b^5*d*e^4*x^4+
1344*B*a^2*b^3*e^5*x^4+3712*B*a*b^4*d*e^4*x^4+1344*B*b^5*d^2*e^3*x^4-480*A*a^2*b^3*e^5*x^3-6720*A*a*b^4*d*e^4*
x^3-5600*A*b^5*d^2*e^3*x^3+336*B*a^3*b^2*e^5*x^3+4848*B*a^2*b^3*d*e^4*x^3+5936*B*a*b^4*d^2*e^3*x^3+1680*B*b^5*
d^3*e^2*x^3+80*A*a^3*b^2*e^5*x^2-1680*A*a^2*b^3*d*e^4*x^2-8400*A*a*b^4*d^2*e^3*x^2-2800*A*b^5*d^3*e^2*x^2-56*B
*a^4*b*e^5*x^2+1152*B*a^3*b^2*d*e^4*x^2+6384*B*a^2*b^3*d^2*e^3*x^2+4480*B*a*b^4*d^3*e^2*x^2+840*B*b^5*d^4*e*x^
2-30*A*a^4*b*e^5*x+280*A*a^3*b^2*d*e^4*x-2100*A*a^2*b^3*d^2*e^3*x-4200*A*a*b^4*d^3*e^2*x-350*A*b^5*d^4*e*x+21*
B*a^5*e^5*x-187*B*a^4*b*d*e^4*x+1386*B*a^3*b^2*d^2*e^3*x+3570*B*a^2*b^3*d^3*e^2*x+1505*B*a*b^4*d^4*e*x+105*B*b
^5*d^5*x+15*A*a^5*e^5-105*A*a^4*b*d*e^4+350*A*a^3*b^2*d^2*e^3-1050*A*a^2*b^3*d^3*e^2-525*A*a*b^4*d^4*e+35*A*b^
5*d^5+6*B*a^5*d*e^4-56*B*a^4*b*d^2*e^3+420*B*a^3*b^2*d^3*e^2+840*B*a^2*b^3*d^4*e+70*B*a*b^4*d^5)/(b*x+a)^(3/2)
/(e*x+d)^(7/2)/(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)/(b*x+a)^(5/2)/(e*x+d)^(9/2),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)/(b*x+a)^(5/2)/(e*x+d)^(9/2),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)/(b*x+a)**(5/2)/(e*x+d)**(9/2),x)
[Out]
Timed out
________________________________________________________________________________________
Giac [B] time = 19.4697, size = 3706, normalized size = 12.44 \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)/(b*x+a)^(5/2)/(e*x+d)^(9/2),x, algorithm="giac")
[Out]
1/80640*(((b*x + a)*((279*B*b^22*d^13*abs(b)*e^7 - 2837*B*a*b^21*d^12*abs(b)*e^8 - 790*A*b^22*d^12*abs(b)*e^8
+ 12282*B*a^2*b^20*d^11*abs(b)*e^9 + 9480*A*a*b^21*d^11*abs(b)*e^9 - 27654*B*a^3*b^19*d^10*abs(b)*e^10 - 52140
*A*a^2*b^20*d^10*abs(b)*e^10 + 25685*B*a^4*b^18*d^9*abs(b)*e^11 + 173800*A*a^3*b^19*d^9*abs(b)*e^11 + 31977*B*
a^5*b^17*d^8*abs(b)*e^12 - 391050*A*a^4*b^18*d^8*abs(b)*e^12 - 146916*B*a^6*b^16*d^7*abs(b)*e^13 + 625680*A*a^
5*b^17*d^7*abs(b)*e^13 + 251196*B*a^7*b^15*d^6*abs(b)*e^14 - 729960*A*a^6*b^16*d^6*abs(b)*e^14 - 266607*B*a^8*
b^14*d^5*abs(b)*e^15 + 625680*A*a^7*b^15*d^5*abs(b)*e^15 + 191565*B*a^9*b^13*d^4*abs(b)*e^16 - 391050*A*a^8*b^
14*d^4*abs(b)*e^16 - 94006*B*a^10*b^12*d^3*abs(b)*e^17 + 173800*A*a^9*b^13*d^3*abs(b)*e^17 + 30378*B*a^11*b^11
*d^2*abs(b)*e^18 - 52140*A*a^10*b^12*d^2*abs(b)*e^18 - 5853*B*a^12*b^10*d*abs(b)*e^19 + 9480*A*a^11*b^11*d*abs
(b)*e^19 + 511*B*a^13*b^9*abs(b)*e^20 - 790*A*a^12*b^10*abs(b)*e^20)*(b*x + a)/(b^16*d^4*e^8 - 4*a*b^15*d^3*e^
9 + 6*a^2*b^14*d^2*e^10 - 4*a^3*b^13*d*e^11 + a^4*b^12*e^12) + 7*(132*B*b^23*d^14*abs(b)*e^6 - 1483*B*a*b^22*d
^13*abs(b)*e^7 - 365*A*b^23*d^13*abs(b)*e^7 + 7267*B*a^2*b^21*d^12*abs(b)*e^8 + 4745*A*a*b^22*d^12*abs(b)*e^8
- 19578*B*a^3*b^20*d^11*abs(b)*e^9 - 28470*A*a^2*b^21*d^11*abs(b)*e^9 + 27742*B*a^4*b^19*d^10*abs(b)*e^10 + 10
4390*A*a^3*b^20*d^10*abs(b)*e^10 - 3289*B*a^5*b^18*d^9*abs(b)*e^11 - 260975*A*a^4*b^19*d^9*abs(b)*e^11 - 73359
*B*a^6*b^17*d^8*abs(b)*e^12 + 469755*A*a^5*b^18*d^8*abs(b)*e^12 + 173316*B*a^7*b^16*d^7*abs(b)*e^13 - 626340*A
*a^6*b^17*d^7*abs(b)*e^13 - 229944*B*a^8*b^15*d^6*abs(b)*e^14 + 626340*A*a^7*b^16*d^6*abs(b)*e^14 + 205491*B*a
^9*b^14*d^5*abs(b)*e^15 - 469755*A*a^8*b^15*d^5*abs(b)*e^15 - 128843*B*a^10*b^13*d^4*abs(b)*e^16 + 260975*A*a^
9*b^14*d^4*abs(b)*e^16 + 56342*B*a^11*b^12*d^3*abs(b)*e^17 - 104390*A*a^10*b^13*d^3*abs(b)*e^17 - 16458*B*a^12
*b^11*d^2*abs(b)*e^18 + 28470*A*a^11*b^12*d^2*abs(b)*e^18 + 2897*B*a^13*b^10*d*abs(b)*e^19 - 4745*A*a^12*b^11*
d*abs(b)*e^19 - 233*B*a^14*b^9*abs(b)*e^20 + 365*A*a^13*b^10*abs(b)*e^20)/(b^16*d^4*e^8 - 4*a*b^15*d^3*e^9 + 6
*a^2*b^14*d^2*e^10 - 4*a^3*b^13*d*e^11 + a^4*b^12*e^12)) + 350*(3*B*b^24*d^15*abs(b)*e^5 - 37*B*a*b^23*d^14*ab
s(b)*e^6 - 8*A*b^24*d^14*abs(b)*e^6 + 203*B*a^2*b^22*d^13*abs(b)*e^7 + 112*A*a*b^23*d^13*abs(b)*e^7 - 637*B*a^
3*b^21*d^12*abs(b)*e^8 - 728*A*a^2*b^22*d^12*abs(b)*e^8 + 1183*B*a^4*b^20*d^11*abs(b)*e^9 + 2912*A*a^3*b^21*d^
11*abs(b)*e^9 - 1001*B*a^5*b^19*d^10*abs(b)*e^10 - 8008*A*a^4*b^20*d^10*abs(b)*e^10 - 1001*B*a^6*b^18*d^9*abs(
b)*e^11 + 16016*A*a^5*b^19*d^9*abs(b)*e^11 + 4719*B*a^7*b^17*d^8*abs(b)*e^12 - 24024*A*a^6*b^18*d^8*abs(b)*e^1
2 - 8151*B*a^8*b^16*d^7*abs(b)*e^13 + 27456*A*a^7*b^17*d^7*abs(b)*e^13 + 9009*B*a^9*b^15*d^6*abs(b)*e^14 - 240
24*A*a^8*b^16*d^6*abs(b)*e^14 - 7007*B*a^10*b^14*d^5*abs(b)*e^15 + 16016*A*a^9*b^15*d^5*abs(b)*e^15 + 3913*B*a
^11*b^13*d^4*abs(b)*e^16 - 8008*A*a^10*b^14*d^4*abs(b)*e^16 - 1547*B*a^12*b^12*d^3*abs(b)*e^17 + 2912*A*a^11*b
^13*d^3*abs(b)*e^17 + 413*B*a^13*b^11*d^2*abs(b)*e^18 - 728*A*a^12*b^12*d^2*abs(b)*e^18 - 67*B*a^14*b^10*d*abs
(b)*e^19 + 112*A*a^13*b^11*d*abs(b)*e^19 + 5*B*a^15*b^9*abs(b)*e^20 - 8*A*a^14*b^10*abs(b)*e^20)/(b^16*d^4*e^8
- 4*a*b^15*d^3*e^9 + 6*a^2*b^14*d^2*e^10 - 4*a^3*b^13*d*e^11 + a^4*b^12*e^12))*(b*x + a) + 210*(2*B*b^25*d^16
*abs(b)*e^4 - 27*B*a*b^24*d^15*abs(b)*e^5 - 5*A*b^25*d^15*abs(b)*e^5 + 165*B*a^2*b^23*d^14*abs(b)*e^6 + 75*A*a
*b^24*d^14*abs(b)*e^6 - 595*B*a^3*b^22*d^13*abs(b)*e^7 - 525*A*a^2*b^23*d^13*abs(b)*e^7 + 1365*B*a^4*b^21*d^12
*abs(b)*e^8 + 2275*A*a^3*b^22*d^12*abs(b)*e^8 - 1911*B*a^5*b^20*d^11*abs(b)*e^9 - 6825*A*a^4*b^21*d^11*abs(b)*
e^9 + 1001*B*a^6*b^19*d^10*abs(b)*e^10 + 15015*A*a^5*b^20*d^10*abs(b)*e^10 + 2145*B*a^7*b^18*d^9*abs(b)*e^11 -
25025*A*a^6*b^19*d^9*abs(b)*e^11 - 6435*B*a^8*b^17*d^8*abs(b)*e^12 + 32175*A*a^7*b^18*d^8*abs(b)*e^12 + 9295*
B*a^9*b^16*d^7*abs(b)*e^13 - 32175*A*a^8*b^17*d^7*abs(b)*e^13 - 9009*B*a^10*b^15*d^6*abs(b)*e^14 + 25025*A*a^9
*b^16*d^6*abs(b)*e^14 + 6279*B*a^11*b^14*d^5*abs(b)*e^15 - 15015*A*a^10*b^15*d^5*abs(b)*e^15 - 3185*B*a^12*b^1
3*d^4*abs(b)*e^16 + 6825*A*a^11*b^14*d^4*abs(b)*e^16 + 1155*B*a^13*b^12*d^3*abs(b)*e^17 - 2275*A*a^12*b^13*d^3
*abs(b)*e^17 - 285*B*a^14*b^11*d^2*abs(b)*e^18 + 525*A*a^13*b^12*d^2*abs(b)*e^18 + 43*B*a^15*b^10*d*abs(b)*e^1
9 - 75*A*a^14*b^11*d*abs(b)*e^19 - 3*B*a^16*b^9*abs(b)*e^20 + 5*A*a^15*b^10*abs(b)*e^20)/(b^16*d^4*e^8 - 4*a*b
^15*d^3*e^9 + 6*a^2*b^14*d^2*e^10 - 4*a^3*b^13*d*e^11 + a^4*b^12*e^12))*sqrt(b*x + a)/(b^2*d + (b*x + a)*b*e -
a*b*e)^(7/2) - 4/3*(3*B*b^(19/2)*d^3*e^(1/2) + 5*B*a*b^(17/2)*d^2*e^(3/2) - 14*A*b^(19/2)*d^2*e^(3/2) - 6*(sq
rt(b*x + a)*sqrt(b)*e^(1/2) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e))^2*B*b^(15/2)*d^2*e^(1/2) - 19*B*a^2*b^(15/2
)*d*e^(5/2) + 28*A*a*b^(17/2)*d*e^(5/2) - 18*(sqrt(b*x + a)*sqrt(b)*e^(1/2) - sqrt(b^2*d + (b*x + a)*b*e - a*b
*e))^2*B*a*b^(13/2)*d*e^(3/2) + 30*(sqrt(b*x + a)*sqrt(b)*e^(1/2) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e))^2*A*b
^(15/2)*d*e^(3/2) + 3*(sqrt(b*x + a)*sqrt(b)*e^(1/2) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e))^4*B*b^(11/2)*d*e^(
1/2) + 11*B*a^3*b^(13/2)*e^(7/2) - 14*A*a^2*b^(15/2)*e^(7/2) + 24*(sqrt(b*x + a)*sqrt(b)*e^(1/2) - sqrt(b^2*d
+ (b*x + a)*b*e - a*b*e))^2*B*a^2*b^(11/2)*e^(5/2) - 30*(sqrt(b*x + a)*sqrt(b)*e^(1/2) - sqrt(b^2*d + (b*x + a
)*b*e - a*b*e))^2*A*a*b^(13/2)*e^(5/2) + 9*(sqrt(b*x + a)*sqrt(b)*e^(1/2) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e
))^4*B*a*b^(9/2)*e^(3/2) - 12*(sqrt(b*x + a)*sqrt(b)*e^(1/2) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e))^4*A*b^(11/
2)*e^(3/2))/((b^5*d^5*abs(b) - 5*a*b^4*d^4*abs(b)*e + 10*a^2*b^3*d^3*abs(b)*e^2 - 10*a^3*b^2*d^2*abs(b)*e^3 +
5*a^4*b*d*abs(b)*e^4 - a^5*abs(b)*e^5)*(b^2*d - a*b*e - (sqrt(b*x + a)*sqrt(b)*e^(1/2) - sqrt(b^2*d + (b*x + a
)*b*e - a*b*e))^2)^3)