3.1750 \(\int \frac{(A+B x) (d+e x)^{7/2}}{(a+b x)^2} \, dx\)

Optimal. Leaf size=256 \[ \frac{(d+e x)^{7/2} (-9 a B e+7 A b e+2 b B d)}{7 b^2 (b d-a e)}+\frac{(d+e x)^{5/2} (-9 a B e+7 A b e+2 b B d)}{5 b^3}+\frac{(d+e x)^{3/2} (b d-a e) (-9 a B e+7 A b e+2 b B d)}{3 b^4}+\frac{\sqrt{d+e x} (b d-a e)^2 (-9 a B e+7 A b e+2 b B d)}{b^5}-\frac{(b d-a e)^{5/2} (-9 a B e+7 A b e+2 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{11/2}}-\frac{(d+e x)^{9/2} (A b-a B)}{b (a+b x) (b d-a e)} \]

[Out]

((b*d - a*e)^2*(2*b*B*d + 7*A*b*e - 9*a*B*e)*Sqrt[d + e*x])/b^5 + ((b*d - a*e)*(2*b*B*d + 7*A*b*e - 9*a*B*e)*(
d + e*x)^(3/2))/(3*b^4) + ((2*b*B*d + 7*A*b*e - 9*a*B*e)*(d + e*x)^(5/2))/(5*b^3) + ((2*b*B*d + 7*A*b*e - 9*a*
B*e)*(d + e*x)^(7/2))/(7*b^2*(b*d - a*e)) - ((A*b - a*B)*(d + e*x)^(9/2))/(b*(b*d - a*e)*(a + b*x)) - ((b*d -
a*e)^(5/2)*(2*b*B*d + 7*A*b*e - 9*a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/b^(11/2)

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Rubi [A]  time = 0.233234, antiderivative size = 256, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {78, 50, 63, 208} \[ \frac{(d+e x)^{7/2} (-9 a B e+7 A b e+2 b B d)}{7 b^2 (b d-a e)}+\frac{(d+e x)^{5/2} (-9 a B e+7 A b e+2 b B d)}{5 b^3}+\frac{(d+e x)^{3/2} (b d-a e) (-9 a B e+7 A b e+2 b B d)}{3 b^4}+\frac{\sqrt{d+e x} (b d-a e)^2 (-9 a B e+7 A b e+2 b B d)}{b^5}-\frac{(b d-a e)^{5/2} (-9 a B e+7 A b e+2 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{11/2}}-\frac{(d+e x)^{9/2} (A b-a B)}{b (a+b x) (b d-a e)} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(d + e*x)^(7/2))/(a + b*x)^2,x]

[Out]

((b*d - a*e)^2*(2*b*B*d + 7*A*b*e - 9*a*B*e)*Sqrt[d + e*x])/b^5 + ((b*d - a*e)*(2*b*B*d + 7*A*b*e - 9*a*B*e)*(
d + e*x)^(3/2))/(3*b^4) + ((2*b*B*d + 7*A*b*e - 9*a*B*e)*(d + e*x)^(5/2))/(5*b^3) + ((2*b*B*d + 7*A*b*e - 9*a*
B*e)*(d + e*x)^(7/2))/(7*b^2*(b*d - a*e)) - ((A*b - a*B)*(d + e*x)^(9/2))/(b*(b*d - a*e)*(a + b*x)) - ((b*d -
a*e)^(5/2)*(2*b*B*d + 7*A*b*e - 9*a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/b^(11/2)

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 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 208

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

Rubi steps

\begin{align*} \int \frac{(A+B x) (d+e x)^{7/2}}{(a+b x)^2} \, dx &=-\frac{(A b-a B) (d+e x)^{9/2}}{b (b d-a e) (a+b x)}+\frac{(2 b B d+7 A b e-9 a B e) \int \frac{(d+e x)^{7/2}}{a+b x} \, dx}{2 b (b d-a e)}\\ &=\frac{(2 b B d+7 A b e-9 a B e) (d+e x)^{7/2}}{7 b^2 (b d-a e)}-\frac{(A b-a B) (d+e x)^{9/2}}{b (b d-a e) (a+b x)}+\frac{(2 b B d+7 A b e-9 a B e) \int \frac{(d+e x)^{5/2}}{a+b x} \, dx}{2 b^2}\\ &=\frac{(2 b B d+7 A b e-9 a B e) (d+e x)^{5/2}}{5 b^3}+\frac{(2 b B d+7 A b e-9 a B e) (d+e x)^{7/2}}{7 b^2 (b d-a e)}-\frac{(A b-a B) (d+e x)^{9/2}}{b (b d-a e) (a+b x)}+\frac{((b d-a e) (2 b B d+7 A b e-9 a B e)) \int \frac{(d+e x)^{3/2}}{a+b x} \, dx}{2 b^3}\\ &=\frac{(b d-a e) (2 b B d+7 A b e-9 a B e) (d+e x)^{3/2}}{3 b^4}+\frac{(2 b B d+7 A b e-9 a B e) (d+e x)^{5/2}}{5 b^3}+\frac{(2 b B d+7 A b e-9 a B e) (d+e x)^{7/2}}{7 b^2 (b d-a e)}-\frac{(A b-a B) (d+e x)^{9/2}}{b (b d-a e) (a+b x)}+\frac{\left ((b d-a e)^2 (2 b B d+7 A b e-9 a B e)\right ) \int \frac{\sqrt{d+e x}}{a+b x} \, dx}{2 b^4}\\ &=\frac{(b d-a e)^2 (2 b B d+7 A b e-9 a B e) \sqrt{d+e x}}{b^5}+\frac{(b d-a e) (2 b B d+7 A b e-9 a B e) (d+e x)^{3/2}}{3 b^4}+\frac{(2 b B d+7 A b e-9 a B e) (d+e x)^{5/2}}{5 b^3}+\frac{(2 b B d+7 A b e-9 a B e) (d+e x)^{7/2}}{7 b^2 (b d-a e)}-\frac{(A b-a B) (d+e x)^{9/2}}{b (b d-a e) (a+b x)}+\frac{\left ((b d-a e)^3 (2 b B d+7 A b e-9 a B e)\right ) \int \frac{1}{(a+b x) \sqrt{d+e x}} \, dx}{2 b^5}\\ &=\frac{(b d-a e)^2 (2 b B d+7 A b e-9 a B e) \sqrt{d+e x}}{b^5}+\frac{(b d-a e) (2 b B d+7 A b e-9 a B e) (d+e x)^{3/2}}{3 b^4}+\frac{(2 b B d+7 A b e-9 a B e) (d+e x)^{5/2}}{5 b^3}+\frac{(2 b B d+7 A b e-9 a B e) (d+e x)^{7/2}}{7 b^2 (b d-a e)}-\frac{(A b-a B) (d+e x)^{9/2}}{b (b d-a e) (a+b x)}+\frac{\left ((b d-a e)^3 (2 b B d+7 A b e-9 a B e)\right ) \operatorname{Subst}\left (\int \frac{1}{a-\frac{b d}{e}+\frac{b x^2}{e}} \, dx,x,\sqrt{d+e x}\right )}{b^5 e}\\ &=\frac{(b d-a e)^2 (2 b B d+7 A b e-9 a B e) \sqrt{d+e x}}{b^5}+\frac{(b d-a e) (2 b B d+7 A b e-9 a B e) (d+e x)^{3/2}}{3 b^4}+\frac{(2 b B d+7 A b e-9 a B e) (d+e x)^{5/2}}{5 b^3}+\frac{(2 b B d+7 A b e-9 a B e) (d+e x)^{7/2}}{7 b^2 (b d-a e)}-\frac{(A b-a B) (d+e x)^{9/2}}{b (b d-a e) (a+b x)}-\frac{(b d-a e)^{5/2} (2 b B d+7 A b e-9 a B e) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{11/2}}\\ \end{align*}

Mathematica [A]  time = 0.541665, size = 193, normalized size = 0.75 \[ \frac{\frac{2 \left (-\frac{9 a B e}{2}+\frac{7 A b e}{2}+b B d\right ) \left (7 (b d-a e) \left (5 (b d-a e) \left (\sqrt{b} \sqrt{d+e x} (-3 a e+4 b d+b e x)-3 (b d-a e)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )\right )+3 b^{5/2} (d+e x)^{5/2}\right )+15 b^{7/2} (d+e x)^{7/2}\right )}{105 b^{9/2}}+\frac{(d+e x)^{9/2} (a B-A b)}{a+b x}}{b (b d-a e)} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(d + e*x)^(7/2))/(a + b*x)^2,x]

[Out]

(((-(A*b) + a*B)*(d + e*x)^(9/2))/(a + b*x) + (2*(b*B*d + (7*A*b*e)/2 - (9*a*B*e)/2)*(15*b^(7/2)*(d + e*x)^(7/
2) + 7*(b*d - a*e)*(3*b^(5/2)*(d + e*x)^(5/2) + 5*(b*d - a*e)*(Sqrt[b]*Sqrt[d + e*x]*(4*b*d - 3*a*e + b*e*x) -
3*(b*d - a*e)^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]]))))/(105*b^(9/2)))/(b*(b*d - a*e))

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Maple [B]  time = 0.021, size = 915, normalized size = 3.6 \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)^(7/2)/(b*x+a)^2,x)

[Out]

-29/b^4/((a*e-b*d)*b)^(1/2)*arctan(b*(e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2))*B*a^3*d*e^3+33/b^3/((a*e-b*d)*b)^(1/2)
*arctan(b*(e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2))*B*a^2*d^2*e^2-15/b^2/((a*e-b*d)*b)^(1/2)*arctan(b*(e*x+d)^(1/2)/(
(a*e-b*d)*b)^(1/2))*B*a*d^3*e-3/b^3*(e*x+d)^(1/2)/(b*e*x+a*e)*A*a^2*d*e^3+3/b^2*(e*x+d)^(1/2)/(b*e*x+a*e)*A*a*
d^2*e^2+3/b^4*(e*x+d)^(1/2)/(b*e*x+a*e)*B*a^3*d*e^3-3/b^3*(e*x+d)^(1/2)/(b*e*x+a*e)*B*a^2*d^2*e^2+1/b^2*(e*x+d
)^(1/2)/(b*e*x+a*e)*B*a*d^3*e+21/b^3/((a*e-b*d)*b)^(1/2)*arctan(b*(e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2))*A*a^2*d*e
^3-21/b^2/((a*e-b*d)*b)^(1/2)*arctan(b*(e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2))*A*a*d^2*e^2+1/b^4*(e*x+d)^(1/2)/(b*e
*x+a*e)*A*a^3*e^4-1/b*(e*x+d)^(1/2)/(b*e*x+a*e)*A*d^3*e-1/b^5*(e*x+d)^(1/2)/(b*e*x+a*e)*B*a^4*e^4-7/b^4/((a*e-
b*d)*b)^(1/2)*arctan(b*(e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2))*A*a^3*e^4+7/b/((a*e-b*d)*b)^(1/2)*arctan(b*(e*x+d)^(
1/2)/((a*e-b*d)*b)^(1/2))*A*d^3*e+9/b^5/((a*e-b*d)*b)^(1/2)*arctan(b*(e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2))*B*a^4*
e^4-8/3/b^3*B*(e*x+d)^(3/2)*a*d*e-12/b^3*A*a*d*e^2*(e*x+d)^(1/2)+18/b^4*B*a^2*d*e^2*(e*x+d)^(1/2)-12/b^3*B*a*d
^2*e*(e*x+d)^(1/2)+2/b^2*B*d^3*(e*x+d)^(1/2)+2/5/b^2*A*(e*x+d)^(5/2)*e+4/3/b^2*A*(e*x+d)^(3/2)*d*e+2/b^4*B*(e*
x+d)^(3/2)*a^2*e^2+2/7/b^2*B*(e*x+d)^(7/2)-4/5/b^3*B*(e*x+d)^(5/2)*a*e-4/3/b^3*A*(e*x+d)^(3/2)*a*e^2+2/b/((a*e
-b*d)*b)^(1/2)*arctan(b*(e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2))*B*d^4+6/b^4*A*a^2*e^3*(e*x+d)^(1/2)+6/b^2*A*d^2*e*(
e*x+d)^(1/2)-8/b^5*a^3*e^3*B*(e*x+d)^(1/2)+2/5/b^2*B*(e*x+d)^(5/2)*d+2/3/b^2*B*(e*x+d)^(3/2)*d^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)^(7/2)/(b*x+a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.45688, size = 2191, normalized size = 8.56 \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)^(7/2)/(b*x+a)^2,x, algorithm="fricas")

[Out]

[1/210*(105*(2*B*a*b^3*d^3 - (13*B*a^2*b^2 - 7*A*a*b^3)*d^2*e + 2*(10*B*a^3*b - 7*A*a^2*b^2)*d*e^2 - (9*B*a^4
- 7*A*a^3*b)*e^3 + (2*B*b^4*d^3 - (13*B*a*b^3 - 7*A*b^4)*d^2*e + 2*(10*B*a^2*b^2 - 7*A*a*b^3)*d*e^2 - (9*B*a^3
*b - 7*A*a^2*b^2)*e^3)*x)*sqrt((b*d - a*e)/b)*log((b*e*x + 2*b*d - a*e - 2*sqrt(e*x + d)*b*sqrt((b*d - a*e)/b)
)/(b*x + a)) + 2*(30*B*b^4*e^3*x^4 + (457*B*a*b^3 - 105*A*b^4)*d^3 - 7*(277*B*a^2*b^2 - 161*A*a*b^3)*d^2*e + 3
5*(69*B*a^3*b - 49*A*a^2*b^2)*d*e^2 - 105*(9*B*a^4 - 7*A*a^3*b)*e^3 + 6*(22*B*b^4*d*e^2 - (9*B*a*b^3 - 7*A*b^4
)*e^3)*x^3 + 2*(122*B*b^4*d^2*e - 2*(79*B*a*b^3 - 56*A*b^4)*d*e^2 + 7*(9*B*a^2*b^2 - 7*A*a*b^3)*e^3)*x^2 + 2*(
176*B*b^4*d^3 - 2*(345*B*a*b^3 - 203*A*b^4)*d^2*e + 14*(59*B*a^2*b^2 - 42*A*a*b^3)*d*e^2 - 35*(9*B*a^3*b - 7*A
*a^2*b^2)*e^3)*x)*sqrt(e*x + d))/(b^6*x + a*b^5), -1/105*(105*(2*B*a*b^3*d^3 - (13*B*a^2*b^2 - 7*A*a*b^3)*d^2*
e + 2*(10*B*a^3*b - 7*A*a^2*b^2)*d*e^2 - (9*B*a^4 - 7*A*a^3*b)*e^3 + (2*B*b^4*d^3 - (13*B*a*b^3 - 7*A*b^4)*d^2
*e + 2*(10*B*a^2*b^2 - 7*A*a*b^3)*d*e^2 - (9*B*a^3*b - 7*A*a^2*b^2)*e^3)*x)*sqrt(-(b*d - a*e)/b)*arctan(-sqrt(
e*x + d)*b*sqrt(-(b*d - a*e)/b)/(b*d - a*e)) - (30*B*b^4*e^3*x^4 + (457*B*a*b^3 - 105*A*b^4)*d^3 - 7*(277*B*a^
2*b^2 - 161*A*a*b^3)*d^2*e + 35*(69*B*a^3*b - 49*A*a^2*b^2)*d*e^2 - 105*(9*B*a^4 - 7*A*a^3*b)*e^3 + 6*(22*B*b^
4*d*e^2 - (9*B*a*b^3 - 7*A*b^4)*e^3)*x^3 + 2*(122*B*b^4*d^2*e - 2*(79*B*a*b^3 - 56*A*b^4)*d*e^2 + 7*(9*B*a^2*b
^2 - 7*A*a*b^3)*e^3)*x^2 + 2*(176*B*b^4*d^3 - 2*(345*B*a*b^3 - 203*A*b^4)*d^2*e + 14*(59*B*a^2*b^2 - 42*A*a*b^
3)*d*e^2 - 35*(9*B*a^3*b - 7*A*a^2*b^2)*e^3)*x)*sqrt(e*x + d))/(b^6*x + a*b^5)]

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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)**(7/2)/(b*x+a)**2,x)

[Out]

Timed out

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Giac [B]  time = 1.96398, size = 815, normalized size = 3.18 \begin{align*} \frac{{\left (2 \, B b^{4} d^{4} - 15 \, B a b^{3} d^{3} e + 7 \, A b^{4} d^{3} e + 33 \, B a^{2} b^{2} d^{2} e^{2} - 21 \, A a b^{3} d^{2} e^{2} - 29 \, B a^{3} b d e^{3} + 21 \, A a^{2} b^{2} d e^{3} + 9 \, B a^{4} e^{4} - 7 \, A a^{3} b e^{4}\right )} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right )}{\sqrt{-b^{2} d + a b e} b^{5}} + \frac{\sqrt{x e + d} B a b^{3} d^{3} e - \sqrt{x e + d} A b^{4} d^{3} e - 3 \, \sqrt{x e + d} B a^{2} b^{2} d^{2} e^{2} + 3 \, \sqrt{x e + d} A a b^{3} d^{2} e^{2} + 3 \, \sqrt{x e + d} B a^{3} b d e^{3} - 3 \, \sqrt{x e + d} A a^{2} b^{2} d e^{3} - \sqrt{x e + d} B a^{4} e^{4} + \sqrt{x e + d} A a^{3} b e^{4}}{{\left ({\left (x e + d\right )} b - b d + a e\right )} b^{5}} + \frac{2 \,{\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} B b^{12} + 21 \,{\left (x e + d\right )}^{\frac{5}{2}} B b^{12} d + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} B b^{12} d^{2} + 105 \, \sqrt{x e + d} B b^{12} d^{3} - 42 \,{\left (x e + d\right )}^{\frac{5}{2}} B a b^{11} e + 21 \,{\left (x e + d\right )}^{\frac{5}{2}} A b^{12} e - 140 \,{\left (x e + d\right )}^{\frac{3}{2}} B a b^{11} d e + 70 \,{\left (x e + d\right )}^{\frac{3}{2}} A b^{12} d e - 630 \, \sqrt{x e + d} B a b^{11} d^{2} e + 315 \, \sqrt{x e + d} A b^{12} d^{2} e + 105 \,{\left (x e + d\right )}^{\frac{3}{2}} B a^{2} b^{10} e^{2} - 70 \,{\left (x e + d\right )}^{\frac{3}{2}} A a b^{11} e^{2} + 945 \, \sqrt{x e + d} B a^{2} b^{10} d e^{2} - 630 \, \sqrt{x e + d} A a b^{11} d e^{2} - 420 \, \sqrt{x e + d} B a^{3} b^{9} e^{3} + 315 \, \sqrt{x e + d} A a^{2} b^{10} e^{3}\right )}}{105 \, b^{14}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^(7/2)/(b*x+a)^2,x, algorithm="giac")

[Out]

(2*B*b^4*d^4 - 15*B*a*b^3*d^3*e + 7*A*b^4*d^3*e + 33*B*a^2*b^2*d^2*e^2 - 21*A*a*b^3*d^2*e^2 - 29*B*a^3*b*d*e^3
+ 21*A*a^2*b^2*d*e^3 + 9*B*a^4*e^4 - 7*A*a^3*b*e^4)*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))/(sqrt(-b^2*d
+ a*b*e)*b^5) + (sqrt(x*e + d)*B*a*b^3*d^3*e - sqrt(x*e + d)*A*b^4*d^3*e - 3*sqrt(x*e + d)*B*a^2*b^2*d^2*e^2
+ 3*sqrt(x*e + d)*A*a*b^3*d^2*e^2 + 3*sqrt(x*e + d)*B*a^3*b*d*e^3 - 3*sqrt(x*e + d)*A*a^2*b^2*d*e^3 - sqrt(x*e
+ d)*B*a^4*e^4 + sqrt(x*e + d)*A*a^3*b*e^4)/(((x*e + d)*b - b*d + a*e)*b^5) + 2/105*(15*(x*e + d)^(7/2)*B*b^1
2 + 21*(x*e + d)^(5/2)*B*b^12*d + 35*(x*e + d)^(3/2)*B*b^12*d^2 + 105*sqrt(x*e + d)*B*b^12*d^3 - 42*(x*e + d)^
(5/2)*B*a*b^11*e + 21*(x*e + d)^(5/2)*A*b^12*e - 140*(x*e + d)^(3/2)*B*a*b^11*d*e + 70*(x*e + d)^(3/2)*A*b^12*
d*e - 630*sqrt(x*e + d)*B*a*b^11*d^2*e + 315*sqrt(x*e + d)*A*b^12*d^2*e + 105*(x*e + d)^(3/2)*B*a^2*b^10*e^2 -
70*(x*e + d)^(3/2)*A*a*b^11*e^2 + 945*sqrt(x*e + d)*B*a^2*b^10*d*e^2 - 630*sqrt(x*e + d)*A*a*b^11*d*e^2 - 420
*sqrt(x*e + d)*B*a^3*b^9*e^3 + 315*sqrt(x*e + d)*A*a^2*b^10*e^3)/b^14