3.2236 \(\int \frac{(A+B x) (d+e x)^{5/2}}{\sqrt{a+b x}} \, dx\)

Optimal. Leaf size=246 \[ -\frac{5 (b d-a e)^3 (7 a B e-8 A b e+b B d) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{64 b^{9/2} e^{3/2}}-\frac{\sqrt{a+b x} (d+e x)^{5/2} (7 a B e-8 A b e+b B d)}{24 b^2 e}-\frac{5 \sqrt{a+b x} (d+e x)^{3/2} (b d-a e) (7 a B e-8 A b e+b B d)}{96 b^3 e}-\frac{5 \sqrt{a+b x} \sqrt{d+e x} (b d-a e)^2 (7 a B e-8 A b e+b B d)}{64 b^4 e}+\frac{B \sqrt{a+b x} (d+e x)^{7/2}}{4 b e} \]

[Out]

(-5*(b*d - a*e)^2*(b*B*d - 8*A*b*e + 7*a*B*e)*Sqrt[a + b*x]*Sqrt[d + e*x])/(64*b^4*e) - (5*(b*d - a*e)*(b*B*d
- 8*A*b*e + 7*a*B*e)*Sqrt[a + b*x]*(d + e*x)^(3/2))/(96*b^3*e) - ((b*B*d - 8*A*b*e + 7*a*B*e)*Sqrt[a + b*x]*(d
+ e*x)^(5/2))/(24*b^2*e) + (B*Sqrt[a + b*x]*(d + e*x)^(7/2))/(4*b*e) - (5*(b*d - a*e)^3*(b*B*d - 8*A*b*e + 7*
a*B*e)*ArcTanh[(Sqrt[e]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[d + e*x])])/(64*b^(9/2)*e^(3/2))

________________________________________________________________________________________

Rubi [A]  time = 0.206779, antiderivative size = 246, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {80, 50, 63, 217, 206} \[ -\frac{5 (b d-a e)^3 (7 a B e-8 A b e+b B d) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{64 b^{9/2} e^{3/2}}-\frac{\sqrt{a+b x} (d+e x)^{5/2} (7 a B e-8 A b e+b B d)}{24 b^2 e}-\frac{5 \sqrt{a+b x} (d+e x)^{3/2} (b d-a e) (7 a B e-8 A b e+b B d)}{96 b^3 e}-\frac{5 \sqrt{a+b x} \sqrt{d+e x} (b d-a e)^2 (7 a B e-8 A b e+b B d)}{64 b^4 e}+\frac{B \sqrt{a+b x} (d+e x)^{7/2}}{4 b e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-5*(b*d - a*e)^2*(b*B*d - 8*A*b*e + 7*a*B*e)*Sqrt[a + b*x]*Sqrt[d + e*x])/(64*b^4*e) - (5*(b*d - a*e)*(b*B*d
- 8*A*b*e + 7*a*B*e)*Sqrt[a + b*x]*(d + e*x)^(3/2))/(96*b^3*e) - ((b*B*d - 8*A*b*e + 7*a*B*e)*Sqrt[a + b*x]*(d
+ e*x)^(5/2))/(24*b^2*e) + (B*Sqrt[a + b*x]*(d + e*x)^(7/2))/(4*b*e) - (5*(b*d - a*e)^3*(b*B*d - 8*A*b*e + 7*
a*B*e)*ArcTanh[(Sqrt[e]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[d + e*x])])/(64*b^(9/2)*e^(3/2))

Rule 80

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

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}}{\sqrt{a+b x}} \, dx &=\frac{B \sqrt{a+b x} (d+e x)^{7/2}}{4 b e}+\frac{\left (4 A b e-B \left (\frac{b d}{2}+\frac{7 a e}{2}\right )\right ) \int \frac{(d+e x)^{5/2}}{\sqrt{a+b x}} \, dx}{4 b e}\\ &=-\frac{(b B d-8 A b e+7 a B e) \sqrt{a+b x} (d+e x)^{5/2}}{24 b^2 e}+\frac{B \sqrt{a+b x} (d+e x)^{7/2}}{4 b e}-\frac{(5 (b d-a e) (b B d-8 A b e+7 a B e)) \int \frac{(d+e x)^{3/2}}{\sqrt{a+b x}} \, dx}{48 b^2 e}\\ &=-\frac{5 (b d-a e) (b B d-8 A b e+7 a B e) \sqrt{a+b x} (d+e x)^{3/2}}{96 b^3 e}-\frac{(b B d-8 A b e+7 a B e) \sqrt{a+b x} (d+e x)^{5/2}}{24 b^2 e}+\frac{B \sqrt{a+b x} (d+e x)^{7/2}}{4 b e}-\frac{\left (5 (b d-a e)^2 (b B d-8 A b e+7 a B e)\right ) \int \frac{\sqrt{d+e x}}{\sqrt{a+b x}} \, dx}{64 b^3 e}\\ &=-\frac{5 (b d-a e)^2 (b B d-8 A b e+7 a B e) \sqrt{a+b x} \sqrt{d+e x}}{64 b^4 e}-\frac{5 (b d-a e) (b B d-8 A b e+7 a B e) \sqrt{a+b x} (d+e x)^{3/2}}{96 b^3 e}-\frac{(b B d-8 A b e+7 a B e) \sqrt{a+b x} (d+e x)^{5/2}}{24 b^2 e}+\frac{B \sqrt{a+b x} (d+e x)^{7/2}}{4 b e}-\frac{\left (5 (b d-a e)^3 (b B d-8 A b e+7 a B e)\right ) \int \frac{1}{\sqrt{a+b x} \sqrt{d+e x}} \, dx}{128 b^4 e}\\ &=-\frac{5 (b d-a e)^2 (b B d-8 A b e+7 a B e) \sqrt{a+b x} \sqrt{d+e x}}{64 b^4 e}-\frac{5 (b d-a e) (b B d-8 A b e+7 a B e) \sqrt{a+b x} (d+e x)^{3/2}}{96 b^3 e}-\frac{(b B d-8 A b e+7 a B e) \sqrt{a+b x} (d+e x)^{5/2}}{24 b^2 e}+\frac{B \sqrt{a+b x} (d+e x)^{7/2}}{4 b e}-\frac{\left (5 (b d-a e)^3 (b B d-8 A b e+7 a B e)\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{d-\frac{a e}{b}+\frac{e x^2}{b}}} \, dx,x,\sqrt{a+b x}\right )}{64 b^5 e}\\ &=-\frac{5 (b d-a e)^2 (b B d-8 A b e+7 a B e) \sqrt{a+b x} \sqrt{d+e x}}{64 b^4 e}-\frac{5 (b d-a e) (b B d-8 A b e+7 a B e) \sqrt{a+b x} (d+e x)^{3/2}}{96 b^3 e}-\frac{(b B d-8 A b e+7 a B e) \sqrt{a+b x} (d+e x)^{5/2}}{24 b^2 e}+\frac{B \sqrt{a+b x} (d+e x)^{7/2}}{4 b e}-\frac{\left (5 (b d-a e)^3 (b B d-8 A b e+7 a B e)\right ) \operatorname{Subst}\left (\int \frac{1}{1-\frac{e x^2}{b}} \, dx,x,\frac{\sqrt{a+b x}}{\sqrt{d+e x}}\right )}{64 b^5 e}\\ &=-\frac{5 (b d-a e)^2 (b B d-8 A b e+7 a B e) \sqrt{a+b x} \sqrt{d+e x}}{64 b^4 e}-\frac{5 (b d-a e) (b B d-8 A b e+7 a B e) \sqrt{a+b x} (d+e x)^{3/2}}{96 b^3 e}-\frac{(b B d-8 A b e+7 a B e) \sqrt{a+b x} (d+e x)^{5/2}}{24 b^2 e}+\frac{B \sqrt{a+b x} (d+e x)^{7/2}}{4 b e}-\frac{5 (b d-a e)^3 (b B d-8 A b e+7 a B e) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{64 b^{9/2} e^{3/2}}\\ \end{align*}

Mathematica [A]  time = 1.46573, size = 210, normalized size = 0.85 \[ \frac{\sqrt{d+e x} \left (48 b^3 B \sqrt{e} \sqrt{a+b x} (d+e x)^3-\frac{(7 a B e-8 A b e+b B d) \left (\sqrt{e} \sqrt{a+b x} \sqrt{\frac{b (d+e x)}{b d-a e}} \left (15 a^2 e^2-10 a b e (4 d+e x)+b^2 \left (33 d^2+26 d e x+8 e^2 x^2\right )\right )+15 (b d-a e)^{5/2} \sinh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b d-a e}}\right )\right )}{\sqrt{\frac{b (d+e x)}{b d-a e}}}\right )}{192 b^4 e^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[d + e*x]*(48*b^3*B*Sqrt[e]*Sqrt[a + b*x]*(d + e*x)^3 - ((b*B*d - 8*A*b*e + 7*a*B*e)*(Sqrt[e]*Sqrt[a + b*
x]*Sqrt[(b*(d + e*x))/(b*d - a*e)]*(15*a^2*e^2 - 10*a*b*e*(4*d + e*x) + b^2*(33*d^2 + 26*d*e*x + 8*e^2*x^2)) +
15*(b*d - a*e)^(5/2)*ArcSinh[(Sqrt[e]*Sqrt[a + b*x])/Sqrt[b*d - a*e]]))/Sqrt[(b*(d + e*x))/(b*d - a*e)]))/(19
2*b^4*e^(3/2))

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Maple [B]  time = 0.024, size = 968, normalized size = 3.9 \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)^(1/2),x)

[Out]

-1/384*(e*x+d)^(1/2)*(b*x+a)^(1/2)*(360*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*b^3*d^2*e^2-236*B*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)*x*b^3*d^2*e+112*B*x^2*a*b^2*e^3*(b*e)^(1/2)*((
b*x+a)*(e*x+d))^(1/2)-272*B*x^2*b^3*d*e^2*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)-416*A*(b*e)^(1/2)*((b*x+a)*(e*x+
d))^(1/2)*x*b^3*d*e^2-530*B*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)*a^2*b*d*e^2+382*B*(b*e)^(1/2)*((b*x+a)*(e*x+d)
)^(1/2)*a*b^2*d^2*e-140*B*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)*x*a^2*b*e^3+160*A*(b*e)^(1/2)*((b*x+a)*(e*x+d))^
(1/2)*x*a*b^2*e^3+640*A*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)*a*b^2*d*e^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))*a^3*b*e^4-30*B*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)*b^3*d^3-120*A*l
n(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))*b^4*d^3*e+210*B*(b*e)^(1/2)*((b*x+a
)*(e*x+d))^(1/2)*a^3*e^3+344*B*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)*x*a*b^2*d*e^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^4+15*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))*b^4*d^4-360*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^3-96*B*x^3*b^3*e^3*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)-128*A*x^2*b^3*e^3*(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))*a^3*b
*d*e^3-270*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^2+60*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*b^3*d^3*e-240*A*(b*e)^(1/2)*((b*
x+a)*(e*x+d))^(1/2)*a^2*b*e^3-528*A*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)*b^3*d^2*e)/b^4/e/((b*x+a)*(e*x+d))^(1/
2)/(b*e)^(1/2)

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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)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.17159, size = 1709, normalized size = 6.95 \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)^(1/2),x, algorithm="fricas")

[Out]

[-1/768*(15*(B*b^4*d^4 + 4*(B*a*b^3 - 2*A*b^4)*d^3*e - 6*(3*B*a^2*b^2 - 4*A*a*b^3)*d^2*e^2 + 4*(5*B*a^3*b - 6*
A*a^2*b^2)*d*e^3 - (7*B*a^4 - 8*A*a^3*b)*e^4)*sqrt(b*e)*log(8*b^2*e^2*x^2 + b^2*d^2 + 6*a*b*d*e + a^2*e^2 + 4*
(2*b*e*x + b*d + a*e)*sqrt(b*e)*sqrt(b*x + a)*sqrt(e*x + d) + 8*(b^2*d*e + a*b*e^2)*x) - 4*(48*B*b^4*e^4*x^3 +
15*B*b^4*d^3*e - (191*B*a*b^3 - 264*A*b^4)*d^2*e^2 + 5*(53*B*a^2*b^2 - 64*A*a*b^3)*d*e^3 - 15*(7*B*a^3*b - 8*
A*a^2*b^2)*e^4 + 8*(17*B*b^4*d*e^3 - (7*B*a*b^3 - 8*A*b^4)*e^4)*x^2 + 2*(59*B*b^4*d^2*e^2 - 2*(43*B*a*b^3 - 52
*A*b^4)*d*e^3 + 5*(7*B*a^2*b^2 - 8*A*a*b^3)*e^4)*x)*sqrt(b*x + a)*sqrt(e*x + d))/(b^5*e^2), 1/384*(15*(B*b^4*d
^4 + 4*(B*a*b^3 - 2*A*b^4)*d^3*e - 6*(3*B*a^2*b^2 - 4*A*a*b^3)*d^2*e^2 + 4*(5*B*a^3*b - 6*A*a^2*b^2)*d*e^3 - (
7*B*a^4 - 8*A*a^3*b)*e^4)*sqrt(-b*e)*arctan(1/2*(2*b*e*x + b*d + a*e)*sqrt(-b*e)*sqrt(b*x + a)*sqrt(e*x + d)/(
b^2*e^2*x^2 + a*b*d*e + (b^2*d*e + a*b*e^2)*x)) + 2*(48*B*b^4*e^4*x^3 + 15*B*b^4*d^3*e - (191*B*a*b^3 - 264*A*
b^4)*d^2*e^2 + 5*(53*B*a^2*b^2 - 64*A*a*b^3)*d*e^3 - 15*(7*B*a^3*b - 8*A*a^2*b^2)*e^4 + 8*(17*B*b^4*d*e^3 - (7
*B*a*b^3 - 8*A*b^4)*e^4)*x^2 + 2*(59*B*b^4*d^2*e^2 - 2*(43*B*a*b^3 - 52*A*b^4)*d*e^3 + 5*(7*B*a^2*b^2 - 8*A*a*
b^3)*e^4)*x)*sqrt(b*x + a)*sqrt(e*x + d))/(b^5*e^2)]

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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)**(1/2),x)

[Out]

Timed out

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Giac [B]  time = 2.70194, size = 1438, normalized size = 5.85 \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)^(1/2),x, algorithm="giac")

[Out]

-1/192*(192*((b^2*d - a*b*e)*e^(-1/2)*log(abs(-sqrt(b*x + a)*sqrt(b)*e^(1/2) + sqrt(b^2*d + (b*x + a)*b*e - a*
b*e)))/sqrt(b) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*sqrt(b*x + a))*A*d^2*abs(b)/b^2 - 16*(sqrt(b^2*d + (b*x +
a)*b*e - a*b*e)*sqrt(b*x + a)*(2*(b*x + a)*(4*(b*x + a)/b^2 + (b^6*d*e^3 - 13*a*b^5*e^4)*e^(-4)/b^7) - 3*(b^7
*d^2*e^2 + 2*a*b^6*d*e^3 - 11*a^2*b^5*e^4)*e^(-4)/b^7) - 3*(b^3*d^3 + a*b^2*d^2*e + 3*a^2*b*d*e^2 - 5*a^3*e^3)
*e^(-5/2)*log(abs(-sqrt(b*x + a)*sqrt(b)*e^(1/2) + sqrt(b^2*d + (b*x + a)*b*e - a*b*e)))/b^(3/2))*B*d*abs(b)*e
/b^2 - 4*(sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*sqrt(b*x + a)*(2*(b*x + a)*e^(-2)/b^4 + (b*d*e - 5*a*e^2)*e^(-4)
/b^4) + (b^2*d^2 + 2*a*b*d*e - 3*a^2*e^2)*e^(-7/2)*log(abs(-sqrt(b*x + a)*sqrt(b)*e^(1/2) + sqrt(b^2*d + (b*x
+ a)*b*e - a*b*e)))/b^(7/2))*B*d^2*abs(b)/b^3 - 8*(sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*sqrt(b*x + a)*(2*(b*x +
a)*(4*(b*x + a)/b^2 + (b^6*d*e^3 - 13*a*b^5*e^4)*e^(-4)/b^7) - 3*(b^7*d^2*e^2 + 2*a*b^6*d*e^3 - 11*a^2*b^5*e^
4)*e^(-4)/b^7) - 3*(b^3*d^3 + a*b^2*d^2*e + 3*a^2*b*d*e^2 - 5*a^3*e^3)*e^(-5/2)*log(abs(-sqrt(b*x + a)*sqrt(b)
*e^(1/2) + sqrt(b^2*d + (b*x + a)*b*e - a*b*e)))/b^(3/2))*A*abs(b)*e^2/b^2 - (sqrt(b^2*d + (b*x + a)*b*e - a*b
*e)*(2*(b*x + a)*(4*(b*x + a)*(6*(b*x + a)/b^3 + (b^12*d*e^5 - 25*a*b^11*e^6)*e^(-6)/b^14) - (5*b^13*d^2*e^4 +
14*a*b^12*d*e^5 - 163*a^2*b^11*e^6)*e^(-6)/b^14) + 3*(5*b^14*d^3*e^3 + 9*a*b^13*d^2*e^4 + 15*a^2*b^12*d*e^5 -
93*a^3*b^11*e^6)*e^(-6)/b^14)*sqrt(b*x + a) + 3*(5*b^4*d^4 + 4*a*b^3*d^3*e + 6*a^2*b^2*d^2*e^2 + 20*a^3*b*d*e
^3 - 35*a^4*e^4)*e^(-7/2)*log(abs(-sqrt(b*x + a)*sqrt(b)*e^(1/2) + sqrt(b^2*d + (b*x + a)*b*e - a*b*e)))/b^(5/
2))*B*abs(b)*e^2/b^2 - 8*(sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*sqrt(b*x + a)*(2*(b*x + a)*e^(-2)/b^4 + (b*d*e -
5*a*e^2)*e^(-4)/b^4) + (b^2*d^2 + 2*a*b*d*e - 3*a^2*e^2)*e^(-7/2)*log(abs(-sqrt(b*x + a)*sqrt(b)*e^(1/2) + sq
rt(b^2*d + (b*x + a)*b*e - a*b*e)))/b^(7/2))*A*d*abs(b)*e/b^3)/b