3.2210 \(\int \frac{\sqrt{a+b x} (A+B x)}{(d+e x)^{15/2}} \, dx\)

Optimal. Leaf size=309 \[ \frac{256 b^4 (a+b x)^{3/2} (-13 a B e+10 A b e+3 b B d)}{45045 e (d+e x)^{3/2} (b d-a e)^6}+\frac{128 b^3 (a+b x)^{3/2} (-13 a B e+10 A b e+3 b B d)}{15015 e (d+e x)^{5/2} (b d-a e)^5}+\frac{32 b^2 (a+b x)^{3/2} (-13 a B e+10 A b e+3 b B d)}{3003 e (d+e x)^{7/2} (b d-a e)^4}+\frac{16 b (a+b x)^{3/2} (-13 a B e+10 A b e+3 b B d)}{1287 e (d+e x)^{9/2} (b d-a e)^3}+\frac{2 (a+b x)^{3/2} (-13 a B e+10 A b e+3 b B d)}{143 e (d+e x)^{11/2} (b d-a e)^2}-\frac{2 (a+b x)^{3/2} (B d-A e)}{13 e (d+e x)^{13/2} (b d-a e)} \]

[Out]

(-2*(B*d - A*e)*(a + b*x)^(3/2))/(13*e*(b*d - a*e)*(d + e*x)^(13/2)) + (2*(3*b*B*d + 10*A*b*e - 13*a*B*e)*(a +
b*x)^(3/2))/(143*e*(b*d - a*e)^2*(d + e*x)^(11/2)) + (16*b*(3*b*B*d + 10*A*b*e - 13*a*B*e)*(a + b*x)^(3/2))/(
1287*e*(b*d - a*e)^3*(d + e*x)^(9/2)) + (32*b^2*(3*b*B*d + 10*A*b*e - 13*a*B*e)*(a + b*x)^(3/2))/(3003*e*(b*d
- a*e)^4*(d + e*x)^(7/2)) + (128*b^3*(3*b*B*d + 10*A*b*e - 13*a*B*e)*(a + b*x)^(3/2))/(15015*e*(b*d - a*e)^5*(
d + e*x)^(5/2)) + (256*b^4*(3*b*B*d + 10*A*b*e - 13*a*B*e)*(a + b*x)^(3/2))/(45045*e*(b*d - a*e)^6*(d + e*x)^(
3/2))

________________________________________________________________________________________

Rubi [A]  time = 0.209779, 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} \[ \frac{256 b^4 (a+b x)^{3/2} (-13 a B e+10 A b e+3 b B d)}{45045 e (d+e x)^{3/2} (b d-a e)^6}+\frac{128 b^3 (a+b x)^{3/2} (-13 a B e+10 A b e+3 b B d)}{15015 e (d+e x)^{5/2} (b d-a e)^5}+\frac{32 b^2 (a+b x)^{3/2} (-13 a B e+10 A b e+3 b B d)}{3003 e (d+e x)^{7/2} (b d-a e)^4}+\frac{16 b (a+b x)^{3/2} (-13 a B e+10 A b e+3 b B d)}{1287 e (d+e x)^{9/2} (b d-a e)^3}+\frac{2 (a+b x)^{3/2} (-13 a B e+10 A b e+3 b B d)}{143 e (d+e x)^{11/2} (b d-a e)^2}-\frac{2 (a+b x)^{3/2} (B d-A e)}{13 e (d+e x)^{13/2} (b d-a e)} \]

Antiderivative was successfully verified.

[In]

Int[(Sqrt[a + b*x]*(A + B*x))/(d + e*x)^(15/2),x]

[Out]

(-2*(B*d - A*e)*(a + b*x)^(3/2))/(13*e*(b*d - a*e)*(d + e*x)^(13/2)) + (2*(3*b*B*d + 10*A*b*e - 13*a*B*e)*(a +
b*x)^(3/2))/(143*e*(b*d - a*e)^2*(d + e*x)^(11/2)) + (16*b*(3*b*B*d + 10*A*b*e - 13*a*B*e)*(a + b*x)^(3/2))/(
1287*e*(b*d - a*e)^3*(d + e*x)^(9/2)) + (32*b^2*(3*b*B*d + 10*A*b*e - 13*a*B*e)*(a + b*x)^(3/2))/(3003*e*(b*d
- a*e)^4*(d + e*x)^(7/2)) + (128*b^3*(3*b*B*d + 10*A*b*e - 13*a*B*e)*(a + b*x)^(3/2))/(15015*e*(b*d - a*e)^5*(
d + e*x)^(5/2)) + (256*b^4*(3*b*B*d + 10*A*b*e - 13*a*B*e)*(a + b*x)^(3/2))/(45045*e*(b*d - a*e)^6*(d + e*x)^(
3/2))

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{\sqrt{a+b x} (A+B x)}{(d+e x)^{15/2}} \, dx &=-\frac{2 (B d-A e) (a+b x)^{3/2}}{13 e (b d-a e) (d+e x)^{13/2}}+\frac{(3 b B d+10 A b e-13 a B e) \int \frac{\sqrt{a+b x}}{(d+e x)^{13/2}} \, dx}{13 e (b d-a e)}\\ &=-\frac{2 (B d-A e) (a+b x)^{3/2}}{13 e (b d-a e) (d+e x)^{13/2}}+\frac{2 (3 b B d+10 A b e-13 a B e) (a+b x)^{3/2}}{143 e (b d-a e)^2 (d+e x)^{11/2}}+\frac{(8 b (3 b B d+10 A b e-13 a B e)) \int \frac{\sqrt{a+b x}}{(d+e x)^{11/2}} \, dx}{143 e (b d-a e)^2}\\ &=-\frac{2 (B d-A e) (a+b x)^{3/2}}{13 e (b d-a e) (d+e x)^{13/2}}+\frac{2 (3 b B d+10 A b e-13 a B e) (a+b x)^{3/2}}{143 e (b d-a e)^2 (d+e x)^{11/2}}+\frac{16 b (3 b B d+10 A b e-13 a B e) (a+b x)^{3/2}}{1287 e (b d-a e)^3 (d+e x)^{9/2}}+\frac{\left (16 b^2 (3 b B d+10 A b e-13 a B e)\right ) \int \frac{\sqrt{a+b x}}{(d+e x)^{9/2}} \, dx}{429 e (b d-a e)^3}\\ &=-\frac{2 (B d-A e) (a+b x)^{3/2}}{13 e (b d-a e) (d+e x)^{13/2}}+\frac{2 (3 b B d+10 A b e-13 a B e) (a+b x)^{3/2}}{143 e (b d-a e)^2 (d+e x)^{11/2}}+\frac{16 b (3 b B d+10 A b e-13 a B e) (a+b x)^{3/2}}{1287 e (b d-a e)^3 (d+e x)^{9/2}}+\frac{32 b^2 (3 b B d+10 A b e-13 a B e) (a+b x)^{3/2}}{3003 e (b d-a e)^4 (d+e x)^{7/2}}+\frac{\left (64 b^3 (3 b B d+10 A b e-13 a B e)\right ) \int \frac{\sqrt{a+b x}}{(d+e x)^{7/2}} \, dx}{3003 e (b d-a e)^4}\\ &=-\frac{2 (B d-A e) (a+b x)^{3/2}}{13 e (b d-a e) (d+e x)^{13/2}}+\frac{2 (3 b B d+10 A b e-13 a B e) (a+b x)^{3/2}}{143 e (b d-a e)^2 (d+e x)^{11/2}}+\frac{16 b (3 b B d+10 A b e-13 a B e) (a+b x)^{3/2}}{1287 e (b d-a e)^3 (d+e x)^{9/2}}+\frac{32 b^2 (3 b B d+10 A b e-13 a B e) (a+b x)^{3/2}}{3003 e (b d-a e)^4 (d+e x)^{7/2}}+\frac{128 b^3 (3 b B d+10 A b e-13 a B e) (a+b x)^{3/2}}{15015 e (b d-a e)^5 (d+e x)^{5/2}}+\frac{\left (128 b^4 (3 b B d+10 A b e-13 a B e)\right ) \int \frac{\sqrt{a+b x}}{(d+e x)^{5/2}} \, dx}{15015 e (b d-a e)^5}\\ &=-\frac{2 (B d-A e) (a+b x)^{3/2}}{13 e (b d-a e) (d+e x)^{13/2}}+\frac{2 (3 b B d+10 A b e-13 a B e) (a+b x)^{3/2}}{143 e (b d-a e)^2 (d+e x)^{11/2}}+\frac{16 b (3 b B d+10 A b e-13 a B e) (a+b x)^{3/2}}{1287 e (b d-a e)^3 (d+e x)^{9/2}}+\frac{32 b^2 (3 b B d+10 A b e-13 a B e) (a+b x)^{3/2}}{3003 e (b d-a e)^4 (d+e x)^{7/2}}+\frac{128 b^3 (3 b B d+10 A b e-13 a B e) (a+b x)^{3/2}}{15015 e (b d-a e)^5 (d+e x)^{5/2}}+\frac{256 b^4 (3 b B d+10 A b e-13 a B e) (a+b x)^{3/2}}{45045 e (b d-a e)^6 (d+e x)^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.300559, size = 160, normalized size = 0.52 \[ \frac{2 (a+b x)^{3/2} \left (3465 (B d-A e)-\frac{2 (d+e x) \left (8 b (d+e x) \left (2 b (d+e x) \left (4 b (d+e x) (-3 a e+5 b d+2 b e x)+15 (b d-a e)^2\right )+35 (b d-a e)^3\right )+315 (b d-a e)^4\right ) \left (-\frac{13 a B e}{2}+5 A b e+\frac{3 b B d}{2}\right )}{(b d-a e)^5}\right )}{45045 e (d+e x)^{13/2} (a e-b d)} \]

Antiderivative was successfully verified.

[In]

Integrate[(Sqrt[a + b*x]*(A + B*x))/(d + e*x)^(15/2),x]

[Out]

(2*(a + b*x)^(3/2)*(3465*(B*d - A*e) - (2*((3*b*B*d)/2 + 5*A*b*e - (13*a*B*e)/2)*(d + e*x)*(315*(b*d - a*e)^4
+ 8*b*(d + e*x)*(35*(b*d - a*e)^3 + 2*b*(d + e*x)*(15*(b*d - a*e)^2 + 4*b*(d + e*x)*(5*b*d - 3*a*e + 2*b*e*x))
)))/(b*d - a*e)^5))/(45045*e*(-(b*d) + a*e)*(d + e*x)^(13/2))

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Maple [B]  time = 0.012, size = 722, normalized size = 2.3 \begin{align*} -{\frac{-2560\,A{b}^{5}{e}^{5}{x}^{5}+3328\,Ba{b}^{4}{e}^{5}{x}^{5}-768\,B{b}^{5}d{e}^{4}{x}^{5}+3840\,Aa{b}^{4}{e}^{5}{x}^{4}-16640\,A{b}^{5}d{e}^{4}{x}^{4}-4992\,B{a}^{2}{b}^{3}{e}^{5}{x}^{4}+22784\,Ba{b}^{4}d{e}^{4}{x}^{4}-4992\,B{b}^{5}{d}^{2}{e}^{3}{x}^{4}-4800\,A{a}^{2}{b}^{3}{e}^{5}{x}^{3}+24960\,Aa{b}^{4}d{e}^{4}{x}^{3}-45760\,A{b}^{5}{d}^{2}{e}^{3}{x}^{3}+6240\,B{a}^{3}{b}^{2}{e}^{5}{x}^{3}-33888\,B{a}^{2}{b}^{3}d{e}^{4}{x}^{3}+66976\,Ba{b}^{4}{d}^{2}{e}^{3}{x}^{3}-13728\,B{b}^{5}{d}^{3}{e}^{2}{x}^{3}+5600\,A{a}^{3}{b}^{2}{e}^{5}{x}^{2}-31200\,A{a}^{2}{b}^{3}d{e}^{4}{x}^{2}+68640\,Aa{b}^{4}{d}^{2}{e}^{3}{x}^{2}-68640\,A{b}^{5}{d}^{3}{e}^{2}{x}^{2}-7280\,B{a}^{4}b{e}^{5}{x}^{2}+42240\,B{a}^{3}{b}^{2}d{e}^{4}{x}^{2}-98592\,B{a}^{2}{b}^{3}{d}^{2}{e}^{3}{x}^{2}+109824\,Ba{b}^{4}{d}^{3}{e}^{2}{x}^{2}-20592\,B{b}^{5}{d}^{4}e{x}^{2}-6300\,A{a}^{4}b{e}^{5}x+36400\,A{a}^{3}{b}^{2}d{e}^{4}x-85800\,A{a}^{2}{b}^{3}{d}^{2}{e}^{3}x+102960\,Aa{b}^{4}{d}^{3}{e}^{2}x-60060\,A{b}^{5}{d}^{4}ex+8190\,B{a}^{5}{e}^{5}x-49210\,B{a}^{4}bd{e}^{4}x+122460\,B{a}^{3}{b}^{2}{d}^{2}{e}^{3}x-159588\,B{a}^{2}{b}^{3}{d}^{3}{e}^{2}x+108966\,Ba{b}^{4}{d}^{4}ex-18018\,B{b}^{5}{d}^{5}x+6930\,A{a}^{5}{e}^{5}-40950\,A{a}^{4}bd{e}^{4}+100100\,A{a}^{3}{b}^{2}{d}^{2}{e}^{3}-128700\,A{a}^{2}{b}^{3}{d}^{3}{e}^{2}+90090\,Aa{b}^{4}{d}^{4}e-30030\,A{b}^{5}{d}^{5}+1260\,B{a}^{5}d{e}^{4}-7280\,B{a}^{4}b{d}^{2}{e}^{3}+17160\,B{a}^{3}{b}^{2}{d}^{3}{e}^{2}-20592\,B{a}^{2}{b}^{3}{d}^{4}e+12012\,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{3}{2}}} \left ( ex+d \right ) ^{-{\frac{13}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(b*x+a)^(1/2)/(e*x+d)^(15/2),x)

[Out]

-2/45045*(b*x+a)^(3/2)*(-1280*A*b^5*e^5*x^5+1664*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^4-8320
*A*b^5*d*e^4*x^4-2496*B*a^2*b^3*e^5*x^4+11392*B*a*b^4*d*e^4*x^4-2496*B*b^5*d^2*e^3*x^4-2400*A*a^2*b^3*e^5*x^3+
12480*A*a*b^4*d*e^4*x^3-22880*A*b^5*d^2*e^3*x^3+3120*B*a^3*b^2*e^5*x^3-16944*B*a^2*b^3*d*e^4*x^3+33488*B*a*b^4
*d^2*e^3*x^3-6864*B*b^5*d^3*e^2*x^3+2800*A*a^3*b^2*e^5*x^2-15600*A*a^2*b^3*d*e^4*x^2+34320*A*a*b^4*d^2*e^3*x^2
-34320*A*b^5*d^3*e^2*x^2-3640*B*a^4*b*e^5*x^2+21120*B*a^3*b^2*d*e^4*x^2-49296*B*a^2*b^3*d^2*e^3*x^2+54912*B*a*
b^4*d^3*e^2*x^2-10296*B*b^5*d^4*e*x^2-3150*A*a^4*b*e^5*x+18200*A*a^3*b^2*d*e^4*x-42900*A*a^2*b^3*d^2*e^3*x+514
80*A*a*b^4*d^3*e^2*x-30030*A*b^5*d^4*e*x+4095*B*a^5*e^5*x-24605*B*a^4*b*d*e^4*x+61230*B*a^3*b^2*d^2*e^3*x-7979
4*B*a^2*b^3*d^3*e^2*x+54483*B*a*b^4*d^4*e*x-9009*B*b^5*d^5*x+3465*A*a^5*e^5-20475*A*a^4*b*d*e^4+50050*A*a^3*b^
2*d^2*e^3-64350*A*a^2*b^3*d^3*e^2+45045*A*a*b^4*d^4*e-15015*A*b^5*d^5+630*B*a^5*d*e^4-3640*B*a^4*b*d^2*e^3+858
0*B*a^3*b^2*d^3*e^2-10296*B*a^2*b^3*d^4*e+6006*B*a*b^4*d^5)/(e*x+d)^(13/2)/(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)

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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((B*x+A)*(b*x+a)^(1/2)/(e*x+d)^(15/2),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)*(b*x+a)^(1/2)/(e*x+d)^(15/2),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)**(1/2)/(e*x+d)**(15/2),x)

[Out]

Timed out

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Giac [B]  time = 3.96019, size = 1755, normalized size = 5.68 \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)^(1/2)/(e*x+d)^(15/2),x, algorithm="giac")

[Out]

-1/33210777600*((8*(2*(4*(b*x + a)*(2*(3*B*b^14*d*abs(b)*e^10 - 13*B*a*b^13*abs(b)*e^11 + 10*A*b^14*abs(b)*e^1
1)*(b*x + a)/(b^28*d^7*e^14 - 7*a*b^27*d^6*e^15 + 21*a^2*b^26*d^5*e^16 - 35*a^3*b^25*d^4*e^17 + 35*a^4*b^24*d^
3*e^18 - 21*a^5*b^23*d^2*e^19 + 7*a^6*b^22*d*e^20 - a^7*b^21*e^21) + 13*(3*B*b^15*d^2*abs(b)*e^9 - 16*B*a*b^14
*d*abs(b)*e^10 + 10*A*b^15*d*abs(b)*e^10 + 13*B*a^2*b^13*abs(b)*e^11 - 10*A*a*b^14*abs(b)*e^11)/(b^28*d^7*e^14
- 7*a*b^27*d^6*e^15 + 21*a^2*b^26*d^5*e^16 - 35*a^3*b^25*d^4*e^17 + 35*a^4*b^24*d^3*e^18 - 21*a^5*b^23*d^2*e^
19 + 7*a^6*b^22*d*e^20 - a^7*b^21*e^21)) + 143*(3*B*b^16*d^3*abs(b)*e^8 - 19*B*a*b^15*d^2*abs(b)*e^9 + 10*A*b^
16*d^2*abs(b)*e^9 + 29*B*a^2*b^14*d*abs(b)*e^10 - 20*A*a*b^15*d*abs(b)*e^10 - 13*B*a^3*b^13*abs(b)*e^11 + 10*A
*a^2*b^14*abs(b)*e^11)/(b^28*d^7*e^14 - 7*a*b^27*d^6*e^15 + 21*a^2*b^26*d^5*e^16 - 35*a^3*b^25*d^4*e^17 + 35*a
^4*b^24*d^3*e^18 - 21*a^5*b^23*d^2*e^19 + 7*a^6*b^22*d*e^20 - a^7*b^21*e^21))*(b*x + a) + 429*(3*B*b^17*d^4*ab
s(b)*e^7 - 22*B*a*b^16*d^3*abs(b)*e^8 + 10*A*b^17*d^3*abs(b)*e^8 + 48*B*a^2*b^15*d^2*abs(b)*e^9 - 30*A*a*b^16*
d^2*abs(b)*e^9 - 42*B*a^3*b^14*d*abs(b)*e^10 + 30*A*a^2*b^15*d*abs(b)*e^10 + 13*B*a^4*b^13*abs(b)*e^11 - 10*A*
a^3*b^14*abs(b)*e^11)/(b^28*d^7*e^14 - 7*a*b^27*d^6*e^15 + 21*a^2*b^26*d^5*e^16 - 35*a^3*b^25*d^4*e^17 + 35*a^
4*b^24*d^3*e^18 - 21*a^5*b^23*d^2*e^19 + 7*a^6*b^22*d*e^20 - a^7*b^21*e^21))*(b*x + a) + 3003*(3*B*b^18*d^5*ab
s(b)*e^6 - 25*B*a*b^17*d^4*abs(b)*e^7 + 10*A*b^18*d^4*abs(b)*e^7 + 70*B*a^2*b^16*d^3*abs(b)*e^8 - 40*A*a*b^17*
d^3*abs(b)*e^8 - 90*B*a^3*b^15*d^2*abs(b)*e^9 + 60*A*a^2*b^16*d^2*abs(b)*e^9 + 55*B*a^4*b^14*d*abs(b)*e^10 - 4
0*A*a^3*b^15*d*abs(b)*e^10 - 13*B*a^5*b^13*abs(b)*e^11 + 10*A*a^4*b^14*abs(b)*e^11)/(b^28*d^7*e^14 - 7*a*b^27*
d^6*e^15 + 21*a^2*b^26*d^5*e^16 - 35*a^3*b^25*d^4*e^17 + 35*a^4*b^24*d^3*e^18 - 21*a^5*b^23*d^2*e^19 + 7*a^6*b
^22*d*e^20 - a^7*b^21*e^21))*(b*x + a) - 15015*(B*a*b^18*d^5*abs(b)*e^6 - A*b^19*d^5*abs(b)*e^6 - 5*B*a^2*b^17
*d^4*abs(b)*e^7 + 5*A*a*b^18*d^4*abs(b)*e^7 + 10*B*a^3*b^16*d^3*abs(b)*e^8 - 10*A*a^2*b^17*d^3*abs(b)*e^8 - 10
*B*a^4*b^15*d^2*abs(b)*e^9 + 10*A*a^3*b^16*d^2*abs(b)*e^9 + 5*B*a^5*b^14*d*abs(b)*e^10 - 5*A*a^4*b^15*d*abs(b)
*e^10 - B*a^6*b^13*abs(b)*e^11 + A*a^5*b^14*abs(b)*e^11)/(b^28*d^7*e^14 - 7*a*b^27*d^6*e^15 + 21*a^2*b^26*d^5*
e^16 - 35*a^3*b^25*d^4*e^17 + 35*a^4*b^24*d^3*e^18 - 21*a^5*b^23*d^2*e^19 + 7*a^6*b^22*d*e^20 - a^7*b^21*e^21)
)*(b*x + a)^(3/2)/(b^2*d + (b*x + a)*b*e - a*b*e)^(13/2)