3.2254 \(\int \frac{(A+B x) (d+e x)^{5/2}}{(a+b x)^{5/2}} \, dx\)

Optimal. Leaf size=257 \[ -\frac{2 (d+e x)^{5/2} (-7 a B e+4 A b e+3 b B d)}{3 b^2 \sqrt{a+b x} (b d-a e)}+\frac{5 e \sqrt{a+b x} (d+e x)^{3/2} (-7 a B e+4 A b e+3 b B d)}{6 b^3 (b d-a e)}+\frac{5 e \sqrt{a+b x} \sqrt{d+e x} (-7 a B e+4 A b e+3 b B d)}{4 b^4}+\frac{5 \sqrt{e} (b d-a e) (-7 a B e+4 A b e+3 b B d) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{4 b^{9/2}}-\frac{2 (d+e x)^{7/2} (A b-a B)}{3 b (a+b x)^{3/2} (b d-a e)} \]

[Out]

(5*e*(3*b*B*d + 4*A*b*e - 7*a*B*e)*Sqrt[a + b*x]*Sqrt[d + e*x])/(4*b^4) + (5*e*(3*b*B*d + 4*A*b*e - 7*a*B*e)*S
qrt[a + b*x]*(d + e*x)^(3/2))/(6*b^3*(b*d - a*e)) - (2*(3*b*B*d + 4*A*b*e - 7*a*B*e)*(d + e*x)^(5/2))/(3*b^2*(
b*d - a*e)*Sqrt[a + b*x]) - (2*(A*b - a*B)*(d + e*x)^(7/2))/(3*b*(b*d - a*e)*(a + b*x)^(3/2)) + (5*Sqrt[e]*(b*
d - a*e)*(3*b*B*d + 4*A*b*e - 7*a*B*e)*ArcTanh[(Sqrt[e]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[d + e*x])])/(4*b^(9/2))

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Rubi [A]  time = 0.197994, antiderivative size = 257, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {78, 47, 50, 63, 217, 206} \[ -\frac{2 (d+e x)^{5/2} (-7 a B e+4 A b e+3 b B d)}{3 b^2 \sqrt{a+b x} (b d-a e)}+\frac{5 e \sqrt{a+b x} (d+e x)^{3/2} (-7 a B e+4 A b e+3 b B d)}{6 b^3 (b d-a e)}+\frac{5 e \sqrt{a+b x} \sqrt{d+e x} (-7 a B e+4 A b e+3 b B d)}{4 b^4}+\frac{5 \sqrt{e} (b d-a e) (-7 a B e+4 A b e+3 b B d) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{4 b^{9/2}}-\frac{2 (d+e x)^{7/2} (A b-a B)}{3 b (a+b x)^{3/2} (b d-a e)} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(d + e*x)^(5/2))/(a + b*x)^(5/2),x]

[Out]

(5*e*(3*b*B*d + 4*A*b*e - 7*a*B*e)*Sqrt[a + b*x]*Sqrt[d + e*x])/(4*b^4) + (5*e*(3*b*B*d + 4*A*b*e - 7*a*B*e)*S
qrt[a + b*x]*(d + e*x)^(3/2))/(6*b^3*(b*d - a*e)) - (2*(3*b*B*d + 4*A*b*e - 7*a*B*e)*(d + e*x)^(5/2))/(3*b^2*(
b*d - a*e)*Sqrt[a + b*x]) - (2*(A*b - a*B)*(d + e*x)^(7/2))/(3*b*(b*d - a*e)*(a + b*x)^(3/2)) + (5*Sqrt[e]*(b*
d - a*e)*(3*b*B*d + 4*A*b*e - 7*a*B*e)*ArcTanh[(Sqrt[e]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[d + e*x])])/(4*b^(9/2))

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 47

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

Rule 50

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

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{(A+B x) (d+e x)^{5/2}}{(a+b x)^{5/2}} \, dx &=-\frac{2 (A b-a B) (d+e x)^{7/2}}{3 b (b d-a e) (a+b x)^{3/2}}+\frac{(3 b B d+4 A b e-7 a B e) \int \frac{(d+e x)^{5/2}}{(a+b x)^{3/2}} \, dx}{3 b (b d-a e)}\\ &=-\frac{2 (3 b B d+4 A b e-7 a B e) (d+e x)^{5/2}}{3 b^2 (b d-a e) \sqrt{a+b x}}-\frac{2 (A b-a B) (d+e x)^{7/2}}{3 b (b d-a e) (a+b x)^{3/2}}+\frac{(5 e (3 b B d+4 A b e-7 a B e)) \int \frac{(d+e x)^{3/2}}{\sqrt{a+b x}} \, dx}{3 b^2 (b d-a e)}\\ &=\frac{5 e (3 b B d+4 A b e-7 a B e) \sqrt{a+b x} (d+e x)^{3/2}}{6 b^3 (b d-a e)}-\frac{2 (3 b B d+4 A b e-7 a B e) (d+e x)^{5/2}}{3 b^2 (b d-a e) \sqrt{a+b x}}-\frac{2 (A b-a B) (d+e x)^{7/2}}{3 b (b d-a e) (a+b x)^{3/2}}+\frac{(5 e (3 b B d+4 A b e-7 a B e)) \int \frac{\sqrt{d+e x}}{\sqrt{a+b x}} \, dx}{4 b^3}\\ &=\frac{5 e (3 b B d+4 A b e-7 a B e) \sqrt{a+b x} \sqrt{d+e x}}{4 b^4}+\frac{5 e (3 b B d+4 A b e-7 a B e) \sqrt{a+b x} (d+e x)^{3/2}}{6 b^3 (b d-a e)}-\frac{2 (3 b B d+4 A b e-7 a B e) (d+e x)^{5/2}}{3 b^2 (b d-a e) \sqrt{a+b x}}-\frac{2 (A b-a B) (d+e x)^{7/2}}{3 b (b d-a e) (a+b x)^{3/2}}+\frac{(5 e (b d-a e) (3 b B d+4 A b e-7 a B e)) \int \frac{1}{\sqrt{a+b x} \sqrt{d+e x}} \, dx}{8 b^4}\\ &=\frac{5 e (3 b B d+4 A b e-7 a B e) \sqrt{a+b x} \sqrt{d+e x}}{4 b^4}+\frac{5 e (3 b B d+4 A b e-7 a B e) \sqrt{a+b x} (d+e x)^{3/2}}{6 b^3 (b d-a e)}-\frac{2 (3 b B d+4 A b e-7 a B e) (d+e x)^{5/2}}{3 b^2 (b d-a e) \sqrt{a+b x}}-\frac{2 (A b-a B) (d+e x)^{7/2}}{3 b (b d-a e) (a+b x)^{3/2}}+\frac{(5 e (b d-a e) (3 b B d+4 A b e-7 a B e)) \operatorname{Subst}\left (\int \frac{1}{\sqrt{d-\frac{a e}{b}+\frac{e x^2}{b}}} \, dx,x,\sqrt{a+b x}\right )}{4 b^5}\\ &=\frac{5 e (3 b B d+4 A b e-7 a B e) \sqrt{a+b x} \sqrt{d+e x}}{4 b^4}+\frac{5 e (3 b B d+4 A b e-7 a B e) \sqrt{a+b x} (d+e x)^{3/2}}{6 b^3 (b d-a e)}-\frac{2 (3 b B d+4 A b e-7 a B e) (d+e x)^{5/2}}{3 b^2 (b d-a e) \sqrt{a+b x}}-\frac{2 (A b-a B) (d+e x)^{7/2}}{3 b (b d-a e) (a+b x)^{3/2}}+\frac{(5 e (b d-a e) (3 b B d+4 A b e-7 a B e)) \operatorname{Subst}\left (\int \frac{1}{1-\frac{e x^2}{b}} \, dx,x,\frac{\sqrt{a+b x}}{\sqrt{d+e x}}\right )}{4 b^5}\\ &=\frac{5 e (3 b B d+4 A b e-7 a B e) \sqrt{a+b x} \sqrt{d+e x}}{4 b^4}+\frac{5 e (3 b B d+4 A b e-7 a B e) \sqrt{a+b x} (d+e x)^{3/2}}{6 b^3 (b d-a e)}-\frac{2 (3 b B d+4 A b e-7 a B e) (d+e x)^{5/2}}{3 b^2 (b d-a e) \sqrt{a+b x}}-\frac{2 (A b-a B) (d+e x)^{7/2}}{3 b (b d-a e) (a+b x)^{3/2}}+\frac{5 \sqrt{e} (b d-a e) (3 b B d+4 A b e-7 a B e) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{4 b^{9/2}}\\ \end{align*}

Mathematica [C]  time = 0.155771, size = 136, normalized size = 0.53 \[ \frac{2 \sqrt{d+e x} \left (b^3 (d+e x)^3 (a B-A b)-\frac{(a+b x) (b d-a e)^2 (-7 a B e+4 A b e+3 b B d) \, _2F_1\left (-\frac{5}{2},-\frac{1}{2};\frac{1}{2};\frac{e (a+b x)}{a e-b d}\right )}{\sqrt{\frac{b (d+e x)}{b d-a e}}}\right )}{3 b^4 (a+b x)^{3/2} (b d-a e)} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(d + e*x)^(5/2))/(a + b*x)^(5/2),x]

[Out]

(2*Sqrt[d + e*x]*(b^3*(-(A*b) + a*B)*(d + e*x)^3 - ((b*d - a*e)^2*(3*b*B*d + 4*A*b*e - 7*a*B*e)*(a + b*x)*Hype
rgeometric2F1[-5/2, -1/2, 1/2, (e*(a + b*x))/(-(b*d) + a*e)])/Sqrt[(b*(d + e*x))/(b*d - a*e)]))/(3*b^4*(b*d -
a*e)*(a + b*x)^(3/2))

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Maple [B]  time = 0.023, size = 1250, normalized size = 4.9 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(e*x+d)^(5/2)/(b*x+a)^(5/2),x)

[Out]

-1/24*(e*x+d)^(1/2)*(-316*B*x*a*b^2*d*e*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)+280*B*x*a^2*b*e^2*(b*e)^(1/2)*((b*
x+a)*(e*x+d))^(1/2)+80*A*a*b^2*d*e*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)-230*B*a^2*b*d*e*(b*e)^(1/2)*((b*x+a)*(e
*x+d))^(1/2)-120*A*ln(1/2*(2*b*x*e+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*x*a*b^3*d*e^2+2
10*B*a^3*e^2*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)-105*B*ln(1/2*(2*b*x*e+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a
*e+b*d)/(b*e)^(1/2))*a^4*e^3+60*A*ln(1/2*(2*b*x*e+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*
x^2*a*b^3*e^3-60*A*ln(1/2*(2*b*x*e+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*x^2*b^4*d*e^2-1
05*B*ln(1/2*(2*b*x*e+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*x^2*a^2*b^2*e^3-45*B*ln(1/2*(
2*b*x*e+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*x^2*b^4*d^2*e+60*A*ln(1/2*(2*b*x*e+2*((b*x
+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a^3*b*e^3+16*A*b^3*d^2*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2
)-45*B*ln(1/2*(2*b*x*e+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a^2*b^2*d^2*e+150*B*ln(1/2*
(2*b*x*e+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*x^2*a*b^3*d*e^2-90*B*ln(1/2*(2*b*x*e+2*((
b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*x*a*b^3*d^2*e-12*B*x^3*b^3*e^2*(b*e)^(1/2)*((b*x+a)*(e
*x+d))^(1/2)-24*A*x^2*b^3*e^2*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)+48*B*x*b^3*d^2*(b*e)^(1/2)*((b*x+a)*(e*x+d))
^(1/2)-120*A*a^2*b*e^2*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)+32*B*a*b^2*d^2*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)+
120*A*ln(1/2*(2*b*x*e+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*x*a^2*b^2*e^3-210*B*ln(1/2*(
2*b*x*e+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*x*a^3*b*e^3-60*A*ln(1/2*(2*b*x*e+2*((b*x+a
)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a^2*b^2*d*e^2+150*B*ln(1/2*(2*b*x*e+2*((b*x+a)*(e*x+d))^(1/
2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a^3*b*d*e^2+42*B*x^2*a*b^2*e^2*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)-54*B*x
^2*b^3*d*e*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)-160*A*x*a*b^2*e^2*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)+112*A*x*b
^3*d*e*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)+300*B*ln(1/2*(2*b*x*e+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d
)/(b*e)^(1/2))*x*a^2*b^2*d*e^2)/((b*x+a)*(e*x+d))^(1/2)/(b*e)^(1/2)/(b*x+a)^(3/2)/b^4

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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)*(e*x+d)^(5/2)/(b*x+a)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 18.2358, size = 1922, normalized size = 7.48 \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)^(5/2)/(b*x+a)^(5/2),x, algorithm="fricas")

[Out]

[1/48*(15*(3*B*a^2*b^2*d^2 - 2*(5*B*a^3*b - 2*A*a^2*b^2)*d*e + (7*B*a^4 - 4*A*a^3*b)*e^2 + (3*B*b^4*d^2 - 2*(5
*B*a*b^3 - 2*A*b^4)*d*e + (7*B*a^2*b^2 - 4*A*a*b^3)*e^2)*x^2 + 2*(3*B*a*b^3*d^2 - 2*(5*B*a^2*b^2 - 2*A*a*b^3)*
d*e + (7*B*a^3*b - 4*A*a^2*b^2)*e^2)*x)*sqrt(e/b)*log(8*b^2*e^2*x^2 + b^2*d^2 + 6*a*b*d*e + a^2*e^2 + 4*(2*b^2
*e*x + b^2*d + a*b*e)*sqrt(b*x + a)*sqrt(e*x + d)*sqrt(e/b) + 8*(b^2*d*e + a*b*e^2)*x) + 4*(6*B*b^3*e^2*x^3 -
8*(2*B*a*b^2 + A*b^3)*d^2 + 5*(23*B*a^2*b - 8*A*a*b^2)*d*e - 15*(7*B*a^3 - 4*A*a^2*b)*e^2 + 3*(9*B*b^3*d*e - (
7*B*a*b^2 - 4*A*b^3)*e^2)*x^2 - 2*(12*B*b^3*d^2 - (79*B*a*b^2 - 28*A*b^3)*d*e + 10*(7*B*a^2*b - 4*A*a*b^2)*e^2
)*x)*sqrt(b*x + a)*sqrt(e*x + d))/(b^6*x^2 + 2*a*b^5*x + a^2*b^4), -1/24*(15*(3*B*a^2*b^2*d^2 - 2*(5*B*a^3*b -
2*A*a^2*b^2)*d*e + (7*B*a^4 - 4*A*a^3*b)*e^2 + (3*B*b^4*d^2 - 2*(5*B*a*b^3 - 2*A*b^4)*d*e + (7*B*a^2*b^2 - 4*
A*a*b^3)*e^2)*x^2 + 2*(3*B*a*b^3*d^2 - 2*(5*B*a^2*b^2 - 2*A*a*b^3)*d*e + (7*B*a^3*b - 4*A*a^2*b^2)*e^2)*x)*sqr
t(-e/b)*arctan(1/2*(2*b*e*x + b*d + a*e)*sqrt(b*x + a)*sqrt(e*x + d)*sqrt(-e/b)/(b*e^2*x^2 + a*d*e + (b*d*e +
a*e^2)*x)) - 2*(6*B*b^3*e^2*x^3 - 8*(2*B*a*b^2 + A*b^3)*d^2 + 5*(23*B*a^2*b - 8*A*a*b^2)*d*e - 15*(7*B*a^3 - 4
*A*a^2*b)*e^2 + 3*(9*B*b^3*d*e - (7*B*a*b^2 - 4*A*b^3)*e^2)*x^2 - 2*(12*B*b^3*d^2 - (79*B*a*b^2 - 28*A*b^3)*d*
e + 10*(7*B*a^2*b - 4*A*a*b^2)*e^2)*x)*sqrt(b*x + a)*sqrt(e*x + d))/(b^6*x^2 + 2*a*b^5*x + a^2*b^4)]

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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)**(5/2)/(b*x+a)**(5/2),x)

[Out]

Timed out

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Giac [B]  time = 3.9677, size = 1737, normalized size = 6.76 \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)^(5/2)/(b*x+a)^(5/2),x, algorithm="giac")

[Out]

1/4*sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*sqrt(b*x + a)*(2*(b*x + a)*B*abs(b)*e^2/b^6 + (9*B*b^12*d*abs(b)*e^3 -
13*B*a*b^11*abs(b)*e^4 + 4*A*b^12*abs(b)*e^4)*e^(-2)/b^17) - 5/8*(3*B*b^(5/2)*d^2*abs(b)*e^(1/2) - 10*B*a*b^(
3/2)*d*abs(b)*e^(3/2) + 4*A*b^(5/2)*d*abs(b)*e^(3/2) + 7*B*a^2*sqrt(b)*abs(b)*e^(5/2) - 4*A*a*b^(3/2)*abs(b)*e
^(5/2))*log((sqrt(b*x + a)*sqrt(b)*e^(1/2) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e))^2)/b^6 - 4/3*(3*B*b^(15/2)*d
^5*abs(b)*e^(1/2) - 22*B*a*b^(13/2)*d^4*abs(b)*e^(3/2) + 7*A*b^(15/2)*d^4*abs(b)*e^(3/2) - 6*(sqrt(b*x + a)*sq
rt(b)*e^(1/2) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e))^2*B*b^(11/2)*d^4*abs(b)*e^(1/2) + 58*B*a^2*b^(11/2)*d^3*a
bs(b)*e^(5/2) - 28*A*a*b^(13/2)*d^3*abs(b)*e^(5/2) + 36*(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^(9/2)*d^3*abs(b)*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))^2*A*b^(11/2)*d^3*abs(b)*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^(7/2)*d^3*abs(b)*e^(1/2) - 72*B*a^3*b^(9/2)*d^2*abs(b)*e^(7/2) + 42*A*a^2*b^(11/2)*d^2*abs(b)*
e^(7/2) - 72*(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^(7/2)*d^2*abs(b)*
e^(5/2) + 36*(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^(9/2)*d^2*abs(b)*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))^4*B*a*b^(5/2)*d^2*abs(b)*e^(3
/2) + 9*(sqrt(b*x + a)*sqrt(b)*e^(1/2) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e))^4*A*b^(7/2)*d^2*abs(b)*e^(3/2) +
43*B*a^4*b^(7/2)*d*abs(b)*e^(9/2) - 28*A*a^3*b^(9/2)*d*abs(b)*e^(9/2) + 60*(sqrt(b*x + a)*sqrt(b)*e^(1/2) - s
qrt(b^2*d + (b*x + a)*b*e - a*b*e))^2*B*a^3*b^(5/2)*d*abs(b)*e^(7/2) - 36*(sqrt(b*x + a)*sqrt(b)*e^(1/2) - sqr
t(b^2*d + (b*x + a)*b*e - a*b*e))^2*A*a^2*b^(7/2)*d*abs(b)*e^(7/2) + 27*(sqrt(b*x + a)*sqrt(b)*e^(1/2) - sqrt(
b^2*d + (b*x + a)*b*e - a*b*e))^4*B*a^2*b^(3/2)*d*abs(b)*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))^4*A*a*b^(5/2)*d*abs(b)*e^(5/2) - 10*B*a^5*b^(5/2)*abs(b)*e^(11/2) + 7*A*a^4*b^(7
/2)*abs(b)*e^(11/2) - 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^4*b^(3/2)
*abs(b)*e^(9/2) + 12*(sqrt(b*x + a)*sqrt(b)*e^(1/2) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e))^2*A*a^3*b^(5/2)*abs
(b)*e^(9/2) - 12*(sqrt(b*x + a)*sqrt(b)*e^(1/2) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e))^4*B*a^3*sqrt(b)*abs(b)*
e^(7/2) + 9*(sqrt(b*x + a)*sqrt(b)*e^(1/2) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e))^4*A*a^2*b^(3/2)*abs(b)*e^(7/
2))/((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*b^5)