3.1742 \(\int \frac{(A+B x) (d+e x)^{7/2}}{a+b x} \, dx\)

Optimal. Leaf size=198 \[ \frac{2 (d+e x)^{7/2} (A b-a B)}{7 b^2}+\frac{2 (d+e x)^{5/2} (A b-a B) (b d-a e)}{5 b^3}+\frac{2 (d+e x)^{3/2} (A b-a B) (b d-a e)^2}{3 b^4}+\frac{2 \sqrt{d+e x} (A b-a B) (b d-a e)^3}{b^5}-\frac{2 (A b-a B) (b d-a e)^{7/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{11/2}}+\frac{2 B (d+e x)^{9/2}}{9 b e} \]

[Out]

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

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Rubi [A]  time = 0.214543, antiderivative size = 198, 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 = {80, 50, 63, 208} \[ \frac{2 (d+e x)^{7/2} (A b-a B)}{7 b^2}+\frac{2 (d+e x)^{5/2} (A b-a B) (b d-a e)}{5 b^3}+\frac{2 (d+e x)^{3/2} (A b-a B) (b d-a e)^2}{3 b^4}+\frac{2 \sqrt{d+e x} (A b-a B) (b d-a e)^3}{b^5}-\frac{2 (A b-a B) (b d-a e)^{7/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{11/2}}+\frac{2 B (d+e x)^{9/2}}{9 b e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.457044, size = 165, normalized size = 0.83 \[ \frac{2 \left (\frac{3 e (A b-a B) \left (7 (b d-a e) \left (5 (b d-a e) \left (\sqrt{b} \sqrt{d+e x} (-3 a e+4 b d+b e x)-3 (b d-a e)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )\right )+3 b^{5/2} (d+e x)^{5/2}\right )+15 b^{7/2} (d+e x)^{7/2}\right )}{35 b^{9/2}}+B (d+e x)^{9/2}\right )}{9 b e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(B*(d + e*x)^(9/2) + (3*(A*b - a*B)*e*(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]]))))/(35*b^(9/2))))/(9*b*e)

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Maple [B]  time = 0.014, size = 820, normalized size = 4.1 \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),x)

[Out]

2/7/b*A*(e*x+d)^(7/2)+2/((a*e-b*d)*b)^(1/2)*arctan(b*(e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2))*A*d^4+2/b*A*d^3*(e*x+d
)^(1/2)-2/7/b^2*B*(e*x+d)^(7/2)*a+2/5/b*A*(e*x+d)^(5/2)*d+2/3/b*A*(e*x+d)^(3/2)*d^2-2/3*e^2/b^4*B*(e*x+d)^(3/2
)*a^3-2/5/b^2*B*(e*x+d)^(5/2)*a*d-2/3/b^2*B*(e*x+d)^(3/2)*a*d^2-2/5*e/b^2*A*(e*x+d)^(5/2)*a+2*e^3/b^5*B*a^4*(e
*x+d)^(1/2)+2/5*e/b^3*B*(e*x+d)^(5/2)*a^2+2/3*e^2/b^3*A*(e*x+d)^(3/2)*a^2-2/b^2*B*a*d^3*(e*x+d)^(1/2)-2*e^3/b^
4*A*a^3*(e*x+d)^(1/2)+2/9*B*(e*x+d)^(9/2)/b/e+8*e/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^2*d^3-8*e^3/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^3*d+12*e^2/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^2*d^2-8*e/b/((a*e-b*d)*b)^(1/2)*arctan(b*
(e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2))*A*a*d^3+8*e^3/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^4*d-12*e^2/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^3*d^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*a*d^4-4/3*e/b^2*A*(e*x+d)^(3/2)*a*d+4/3*e/b^3*B*(e
*x+d)^(3/2)*a^2*d+6*e^2/b^3*A*a^2*d*(e*x+d)^(1/2)-6*e/b^2*A*a*d^2*(e*x+d)^(1/2)-6*e^2/b^4*B*a^3*d*(e*x+d)^(1/2
)+6*e/b^3*B*a^2*d^2*(e*x+d)^(1/2)+2*e^4/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^4-2*e^4/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^5

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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),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.45726, size = 1800, normalized size = 9.09 \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),x, algorithm="fricas")

[Out]

[-1/315*(315*((B*a*b^3 - A*b^4)*d^3*e - 3*(B*a^2*b^2 - A*a*b^3)*d^2*e^2 + 3*(B*a^3*b - A*a^2*b^2)*d*e^3 - (B*a
^4 - A*a^3*b)*e^4)*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*(35*B*b^4*e^4*x^4 + 35*B*b^4*d^4 - 528*(B*a*b^3 - A*b^4)*d^3*e + 1218*(B*a^2*b^2 - A*a*b^3)*d^2*e^2
- 1050*(B*a^3*b - A*a^2*b^2)*d*e^3 + 315*(B*a^4 - A*a^3*b)*e^4 + 5*(28*B*b^4*d*e^3 - 9*(B*a*b^3 - A*b^4)*e^4)*
x^3 + 3*(70*B*b^4*d^2*e^2 - 66*(B*a*b^3 - A*b^4)*d*e^3 + 21*(B*a^2*b^2 - A*a*b^3)*e^4)*x^2 + (140*B*b^4*d^3*e
- 366*(B*a*b^3 - A*b^4)*d^2*e^2 + 336*(B*a^2*b^2 - A*a*b^3)*d*e^3 - 105*(B*a^3*b - A*a^2*b^2)*e^4)*x)*sqrt(e*x
+ d))/(b^5*e), 2/315*(315*((B*a*b^3 - A*b^4)*d^3*e - 3*(B*a^2*b^2 - A*a*b^3)*d^2*e^2 + 3*(B*a^3*b - A*a^2*b^2
)*d*e^3 - (B*a^4 - A*a^3*b)*e^4)*sqrt(-(b*d - a*e)/b)*arctan(-sqrt(e*x + d)*b*sqrt(-(b*d - a*e)/b)/(b*d - a*e)
) + (35*B*b^4*e^4*x^4 + 35*B*b^4*d^4 - 528*(B*a*b^3 - A*b^4)*d^3*e + 1218*(B*a^2*b^2 - A*a*b^3)*d^2*e^2 - 1050
*(B*a^3*b - A*a^2*b^2)*d*e^3 + 315*(B*a^4 - A*a^3*b)*e^4 + 5*(28*B*b^4*d*e^3 - 9*(B*a*b^3 - A*b^4)*e^4)*x^3 +
3*(70*B*b^4*d^2*e^2 - 66*(B*a*b^3 - A*b^4)*d*e^3 + 21*(B*a^2*b^2 - A*a*b^3)*e^4)*x^2 + (140*B*b^4*d^3*e - 366*
(B*a*b^3 - A*b^4)*d^2*e^2 + 336*(B*a^2*b^2 - A*a*b^3)*d*e^3 - 105*(B*a^3*b - A*a^2*b^2)*e^4)*x)*sqrt(e*x + d))
/(b^5*e)]

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Sympy [A]  time = 86.3754, size = 337, normalized size = 1.7 \begin{align*} \frac{2 B \left (d + e x\right )^{\frac{9}{2}}}{9 b e} + \frac{\left (d + e x\right )^{\frac{7}{2}} \left (2 A b - 2 B a\right )}{7 b^{2}} + \frac{\left (d + e x\right )^{\frac{5}{2}} \left (- 2 A a b e + 2 A b^{2} d + 2 B a^{2} e - 2 B a b d\right )}{5 b^{3}} + \frac{\left (d + e x\right )^{\frac{3}{2}} \left (2 A a^{2} b e^{2} - 4 A a b^{2} d e + 2 A b^{3} d^{2} - 2 B a^{3} e^{2} + 4 B a^{2} b d e - 2 B a b^{2} d^{2}\right )}{3 b^{4}} + \frac{\sqrt{d + e x} \left (- 2 A a^{3} b e^{3} + 6 A a^{2} b^{2} d e^{2} - 6 A a b^{3} d^{2} e + 2 A b^{4} d^{3} + 2 B a^{4} e^{3} - 6 B a^{3} b d e^{2} + 6 B a^{2} b^{2} d^{2} e - 2 B a b^{3} d^{3}\right )}{b^{5}} - \frac{2 \left (- A b + B a\right ) \left (a e - b d\right )^{4} \operatorname{atan}{\left (\frac{\sqrt{d + e x}}{\sqrt{\frac{a e - b d}{b}}} \right )}}{b^{6} \sqrt{\frac{a e - b d}{b}}} \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),x)

[Out]

2*B*(d + e*x)**(9/2)/(9*b*e) + (d + e*x)**(7/2)*(2*A*b - 2*B*a)/(7*b**2) + (d + e*x)**(5/2)*(-2*A*a*b*e + 2*A*
b**2*d + 2*B*a**2*e - 2*B*a*b*d)/(5*b**3) + (d + e*x)**(3/2)*(2*A*a**2*b*e**2 - 4*A*a*b**2*d*e + 2*A*b**3*d**2
- 2*B*a**3*e**2 + 4*B*a**2*b*d*e - 2*B*a*b**2*d**2)/(3*b**4) + sqrt(d + e*x)*(-2*A*a**3*b*e**3 + 6*A*a**2*b**
2*d*e**2 - 6*A*a*b**3*d**2*e + 2*A*b**4*d**3 + 2*B*a**4*e**3 - 6*B*a**3*b*d*e**2 + 6*B*a**2*b**2*d**2*e - 2*B*
a*b**3*d**3)/b**5 - 2*(-A*b + B*a)*(a*e - b*d)**4*atan(sqrt(d + e*x)/sqrt((a*e - b*d)/b))/(b**6*sqrt((a*e - b*
d)/b))

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Giac [B]  time = 2.0714, size = 753, normalized size = 3.8 \begin{align*} -\frac{2 \,{\left (B a b^{4} d^{4} - A b^{5} d^{4} - 4 \, B a^{2} b^{3} d^{3} e + 4 \, A a b^{4} d^{3} e + 6 \, B a^{3} b^{2} d^{2} e^{2} - 6 \, A a^{2} b^{3} d^{2} e^{2} - 4 \, B a^{4} b d e^{3} + 4 \, A a^{3} b^{2} d e^{3} + B a^{5} e^{4} - A a^{4} 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{2 \,{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} B b^{8} e^{8} - 45 \,{\left (x e + d\right )}^{\frac{7}{2}} B a b^{7} e^{9} + 45 \,{\left (x e + d\right )}^{\frac{7}{2}} A b^{8} e^{9} - 63 \,{\left (x e + d\right )}^{\frac{5}{2}} B a b^{7} d e^{9} + 63 \,{\left (x e + d\right )}^{\frac{5}{2}} A b^{8} d e^{9} - 105 \,{\left (x e + d\right )}^{\frac{3}{2}} B a b^{7} d^{2} e^{9} + 105 \,{\left (x e + d\right )}^{\frac{3}{2}} A b^{8} d^{2} e^{9} - 315 \, \sqrt{x e + d} B a b^{7} d^{3} e^{9} + 315 \, \sqrt{x e + d} A b^{8} d^{3} e^{9} + 63 \,{\left (x e + d\right )}^{\frac{5}{2}} B a^{2} b^{6} e^{10} - 63 \,{\left (x e + d\right )}^{\frac{5}{2}} A a b^{7} e^{10} + 210 \,{\left (x e + d\right )}^{\frac{3}{2}} B a^{2} b^{6} d e^{10} - 210 \,{\left (x e + d\right )}^{\frac{3}{2}} A a b^{7} d e^{10} + 945 \, \sqrt{x e + d} B a^{2} b^{6} d^{2} e^{10} - 945 \, \sqrt{x e + d} A a b^{7} d^{2} e^{10} - 105 \,{\left (x e + d\right )}^{\frac{3}{2}} B a^{3} b^{5} e^{11} + 105 \,{\left (x e + d\right )}^{\frac{3}{2}} A a^{2} b^{6} e^{11} - 945 \, \sqrt{x e + d} B a^{3} b^{5} d e^{11} + 945 \, \sqrt{x e + d} A a^{2} b^{6} d e^{11} + 315 \, \sqrt{x e + d} B a^{4} b^{4} e^{12} - 315 \, \sqrt{x e + d} A a^{3} b^{5} e^{12}\right )} e^{\left (-9\right )}}{315 \, b^{9}} \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),x, algorithm="giac")

[Out]

-2*(B*a*b^4*d^4 - A*b^5*d^4 - 4*B*a^2*b^3*d^3*e + 4*A*a*b^4*d^3*e + 6*B*a^3*b^2*d^2*e^2 - 6*A*a^2*b^3*d^2*e^2
- 4*B*a^4*b*d*e^3 + 4*A*a^3*b^2*d*e^3 + B*a^5*e^4 - A*a^4*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) + 2/315*(35*(x*e + d)^(9/2)*B*b^8*e^8 - 45*(x*e + d)^(7/2)*B*a*b^7*e^9 + 45*(x*e +
d)^(7/2)*A*b^8*e^9 - 63*(x*e + d)^(5/2)*B*a*b^7*d*e^9 + 63*(x*e + d)^(5/2)*A*b^8*d*e^9 - 105*(x*e + d)^(3/2)*B
*a*b^7*d^2*e^9 + 105*(x*e + d)^(3/2)*A*b^8*d^2*e^9 - 315*sqrt(x*e + d)*B*a*b^7*d^3*e^9 + 315*sqrt(x*e + d)*A*b
^8*d^3*e^9 + 63*(x*e + d)^(5/2)*B*a^2*b^6*e^10 - 63*(x*e + d)^(5/2)*A*a*b^7*e^10 + 210*(x*e + d)^(3/2)*B*a^2*b
^6*d*e^10 - 210*(x*e + d)^(3/2)*A*a*b^7*d*e^10 + 945*sqrt(x*e + d)*B*a^2*b^6*d^2*e^10 - 945*sqrt(x*e + d)*A*a*
b^7*d^2*e^10 - 105*(x*e + d)^(3/2)*B*a^3*b^5*e^11 + 105*(x*e + d)^(3/2)*A*a^2*b^6*e^11 - 945*sqrt(x*e + d)*B*a
^3*b^5*d*e^11 + 945*sqrt(x*e + d)*A*a^2*b^6*d*e^11 + 315*sqrt(x*e + d)*B*a^4*b^4*e^12 - 315*sqrt(x*e + d)*A*a^
3*b^5*e^12)*e^(-9)/b^9