3.1440 \(\int \frac{(A+B x) (a+c x^2)^2}{(d+e x)^{9/2}} \, dx\)

Optimal. Leaf size=214 \[ -\frac{4 c \left (a B e^2-2 A c d e+5 B c d^2\right )}{e^6 \sqrt{d+e x}}+\frac{4 c \left (-a A e^3+3 a B d e^2-3 A c d^2 e+5 B c d^3\right )}{3 e^6 (d+e x)^{3/2}}-\frac{2 \left (a e^2+c d^2\right ) \left (a B e^2-4 A c d e+5 B c d^2\right )}{5 e^6 (d+e x)^{5/2}}+\frac{2 \left (a e^2+c d^2\right )^2 (B d-A e)}{7 e^6 (d+e x)^{7/2}}-\frac{2 c^2 \sqrt{d+e x} (5 B d-A e)}{e^6}+\frac{2 B c^2 (d+e x)^{3/2}}{3 e^6} \]

[Out]

(2*(B*d - A*e)*(c*d^2 + a*e^2)^2)/(7*e^6*(d + e*x)^(7/2)) - (2*(c*d^2 + a*e^2)*(5*B*c*d^2 - 4*A*c*d*e + a*B*e^
2))/(5*e^6*(d + e*x)^(5/2)) + (4*c*(5*B*c*d^3 - 3*A*c*d^2*e + 3*a*B*d*e^2 - a*A*e^3))/(3*e^6*(d + e*x)^(3/2))
- (4*c*(5*B*c*d^2 - 2*A*c*d*e + a*B*e^2))/(e^6*Sqrt[d + e*x]) - (2*c^2*(5*B*d - A*e)*Sqrt[d + e*x])/e^6 + (2*B
*c^2*(d + e*x)^(3/2))/(3*e^6)

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Rubi [A]  time = 0.0968576, antiderivative size = 214, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {772} \[ -\frac{4 c \left (a B e^2-2 A c d e+5 B c d^2\right )}{e^6 \sqrt{d+e x}}+\frac{4 c \left (-a A e^3+3 a B d e^2-3 A c d^2 e+5 B c d^3\right )}{3 e^6 (d+e x)^{3/2}}-\frac{2 \left (a e^2+c d^2\right ) \left (a B e^2-4 A c d e+5 B c d^2\right )}{5 e^6 (d+e x)^{5/2}}+\frac{2 \left (a e^2+c d^2\right )^2 (B d-A e)}{7 e^6 (d+e x)^{7/2}}-\frac{2 c^2 \sqrt{d+e x} (5 B d-A e)}{e^6}+\frac{2 B c^2 (d+e x)^{3/2}}{3 e^6} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a + c*x^2)^2)/(d + e*x)^(9/2),x]

[Out]

(2*(B*d - A*e)*(c*d^2 + a*e^2)^2)/(7*e^6*(d + e*x)^(7/2)) - (2*(c*d^2 + a*e^2)*(5*B*c*d^2 - 4*A*c*d*e + a*B*e^
2))/(5*e^6*(d + e*x)^(5/2)) + (4*c*(5*B*c*d^3 - 3*A*c*d^2*e + 3*a*B*d*e^2 - a*A*e^3))/(3*e^6*(d + e*x)^(3/2))
- (4*c*(5*B*c*d^2 - 2*A*c*d*e + a*B*e^2))/(e^6*Sqrt[d + e*x]) - (2*c^2*(5*B*d - A*e)*Sqrt[d + e*x])/e^6 + (2*B
*c^2*(d + e*x)^(3/2))/(3*e^6)

Rule 772

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegr
and[(d + e*x)^m*(f + g*x)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && IGtQ[p, 0]

Rubi steps

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

Mathematica [A]  time = 0.160092, size = 214, normalized size = 1. \[ -\frac{2 \left (A e \left (15 a^2 e^4+2 a c e^2 \left (8 d^2+28 d e x+35 e^2 x^2\right )-3 c^2 \left (560 d^2 e^2 x^2+448 d^3 e x+128 d^4+280 d e^3 x^3+35 e^4 x^4\right )\right )+B \left (3 a^2 e^4 (2 d+7 e x)+6 a c e^2 \left (56 d^2 e x+16 d^3+70 d e^2 x^2+35 e^3 x^3\right )+5 c^2 \left (1120 d^3 e^2 x^2+560 d^2 e^3 x^3+896 d^4 e x+256 d^5+70 d e^4 x^4-7 e^5 x^5\right )\right )\right )}{105 e^6 (d+e x)^{7/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a + c*x^2)^2)/(d + e*x)^(9/2),x]

[Out]

(-2*(A*e*(15*a^2*e^4 + 2*a*c*e^2*(8*d^2 + 28*d*e*x + 35*e^2*x^2) - 3*c^2*(128*d^4 + 448*d^3*e*x + 560*d^2*e^2*
x^2 + 280*d*e^3*x^3 + 35*e^4*x^4)) + B*(3*a^2*e^4*(2*d + 7*e*x) + 6*a*c*e^2*(16*d^3 + 56*d^2*e*x + 70*d*e^2*x^
2 + 35*e^3*x^3) + 5*c^2*(256*d^5 + 896*d^4*e*x + 1120*d^3*e^2*x^2 + 560*d^2*e^3*x^3 + 70*d*e^4*x^4 - 7*e^5*x^5
))))/(105*e^6*(d + e*x)^(7/2))

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Maple [A]  time = 0.007, size = 259, normalized size = 1.2 \begin{align*} -{\frac{-70\,B{c}^{2}{x}^{5}{e}^{5}-210\,A{c}^{2}{e}^{5}{x}^{4}+700\,B{c}^{2}d{e}^{4}{x}^{4}-1680\,A{c}^{2}d{e}^{4}{x}^{3}+420\,Bac{e}^{5}{x}^{3}+5600\,B{c}^{2}{d}^{2}{e}^{3}{x}^{3}+140\,Aac{e}^{5}{x}^{2}-3360\,A{c}^{2}{d}^{2}{e}^{3}{x}^{2}+840\,Bacd{e}^{4}{x}^{2}+11200\,B{c}^{2}{d}^{3}{e}^{2}{x}^{2}+112\,Aacd{e}^{4}x-2688\,A{c}^{2}{d}^{3}{e}^{2}x+42\,B{a}^{2}{e}^{5}x+672\,Bac{d}^{2}{e}^{3}x+8960\,B{c}^{2}{d}^{4}ex+30\,A{a}^{2}{e}^{5}+32\,A{d}^{2}ac{e}^{3}-768\,A{d}^{4}{c}^{2}e+12\,B{a}^{2}d{e}^{4}+192\,aBc{d}^{3}{e}^{2}+2560\,B{c}^{2}{d}^{5}}{105\,{e}^{6}} \left ( ex+d \right ) ^{-{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+a)^2/(e*x+d)^(9/2),x)

[Out]

-2/105/(e*x+d)^(7/2)*(-35*B*c^2*e^5*x^5-105*A*c^2*e^5*x^4+350*B*c^2*d*e^4*x^4-840*A*c^2*d*e^4*x^3+210*B*a*c*e^
5*x^3+2800*B*c^2*d^2*e^3*x^3+70*A*a*c*e^5*x^2-1680*A*c^2*d^2*e^3*x^2+420*B*a*c*d*e^4*x^2+5600*B*c^2*d^3*e^2*x^
2+56*A*a*c*d*e^4*x-1344*A*c^2*d^3*e^2*x+21*B*a^2*e^5*x+336*B*a*c*d^2*e^3*x+4480*B*c^2*d^4*e*x+15*A*a^2*e^5+16*
A*a*c*d^2*e^3-384*A*c^2*d^4*e+6*B*a^2*d*e^4+96*B*a*c*d^3*e^2+1280*B*c^2*d^5)/e^6

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Maxima [A]  time = 1.02172, size = 344, normalized size = 1.61 \begin{align*} \frac{2 \,{\left (\frac{35 \,{\left ({\left (e x + d\right )}^{\frac{3}{2}} B c^{2} - 3 \,{\left (5 \, B c^{2} d - A c^{2} e\right )} \sqrt{e x + d}\right )}}{e^{5}} + \frac{15 \, B c^{2} d^{5} - 15 \, A c^{2} d^{4} e + 30 \, B a c d^{3} e^{2} - 30 \, A a c d^{2} e^{3} + 15 \, B a^{2} d e^{4} - 15 \, A a^{2} e^{5} - 210 \,{\left (5 \, B c^{2} d^{2} - 2 \, A c^{2} d e + B a c e^{2}\right )}{\left (e x + d\right )}^{3} + 70 \,{\left (5 \, B c^{2} d^{3} - 3 \, A c^{2} d^{2} e + 3 \, B a c d e^{2} - A a c e^{3}\right )}{\left (e x + d\right )}^{2} - 21 \,{\left (5 \, B c^{2} d^{4} - 4 \, A c^{2} d^{3} e + 6 \, B a c d^{2} e^{2} - 4 \, A a c d e^{3} + B a^{2} e^{4}\right )}{\left (e x + d\right )}}{{\left (e x + d\right )}^{\frac{7}{2}} e^{5}}\right )}}{105 \, e} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^2/(e*x+d)^(9/2),x, algorithm="maxima")

[Out]

2/105*(35*((e*x + d)^(3/2)*B*c^2 - 3*(5*B*c^2*d - A*c^2*e)*sqrt(e*x + d))/e^5 + (15*B*c^2*d^5 - 15*A*c^2*d^4*e
+ 30*B*a*c*d^3*e^2 - 30*A*a*c*d^2*e^3 + 15*B*a^2*d*e^4 - 15*A*a^2*e^5 - 210*(5*B*c^2*d^2 - 2*A*c^2*d*e + B*a*
c*e^2)*(e*x + d)^3 + 70*(5*B*c^2*d^3 - 3*A*c^2*d^2*e + 3*B*a*c*d*e^2 - A*a*c*e^3)*(e*x + d)^2 - 21*(5*B*c^2*d^
4 - 4*A*c^2*d^3*e + 6*B*a*c*d^2*e^2 - 4*A*a*c*d*e^3 + B*a^2*e^4)*(e*x + d))/((e*x + d)^(7/2)*e^5))/e

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Fricas [A]  time = 1.48935, size = 643, normalized size = 3. \begin{align*} \frac{2 \,{\left (35 \, B c^{2} e^{5} x^{5} - 1280 \, B c^{2} d^{5} + 384 \, A c^{2} d^{4} e - 96 \, B a c d^{3} e^{2} - 16 \, A a c d^{2} e^{3} - 6 \, B a^{2} d e^{4} - 15 \, A a^{2} e^{5} - 35 \,{\left (10 \, B c^{2} d e^{4} - 3 \, A c^{2} e^{5}\right )} x^{4} - 70 \,{\left (40 \, B c^{2} d^{2} e^{3} - 12 \, A c^{2} d e^{4} + 3 \, B a c e^{5}\right )} x^{3} - 70 \,{\left (80 \, B c^{2} d^{3} e^{2} - 24 \, A c^{2} d^{2} e^{3} + 6 \, B a c d e^{4} + A a c e^{5}\right )} x^{2} - 7 \,{\left (640 \, B c^{2} d^{4} e - 192 \, A c^{2} d^{3} e^{2} + 48 \, B a c d^{2} e^{3} + 8 \, A a c d e^{4} + 3 \, B a^{2} e^{5}\right )} x\right )} \sqrt{e x + d}}{105 \,{\left (e^{10} x^{4} + 4 \, d e^{9} x^{3} + 6 \, d^{2} e^{8} x^{2} + 4 \, d^{3} e^{7} x + d^{4} e^{6}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^2/(e*x+d)^(9/2),x, algorithm="fricas")

[Out]

2/105*(35*B*c^2*e^5*x^5 - 1280*B*c^2*d^5 + 384*A*c^2*d^4*e - 96*B*a*c*d^3*e^2 - 16*A*a*c*d^2*e^3 - 6*B*a^2*d*e
^4 - 15*A*a^2*e^5 - 35*(10*B*c^2*d*e^4 - 3*A*c^2*e^5)*x^4 - 70*(40*B*c^2*d^2*e^3 - 12*A*c^2*d*e^4 + 3*B*a*c*e^
5)*x^3 - 70*(80*B*c^2*d^3*e^2 - 24*A*c^2*d^2*e^3 + 6*B*a*c*d*e^4 + A*a*c*e^5)*x^2 - 7*(640*B*c^2*d^4*e - 192*A
*c^2*d^3*e^2 + 48*B*a*c*d^2*e^3 + 8*A*a*c*d*e^4 + 3*B*a^2*e^5)*x)*sqrt(e*x + d)/(e^10*x^4 + 4*d*e^9*x^3 + 6*d^
2*e^8*x^2 + 4*d^3*e^7*x + d^4*e^6)

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Sympy [A]  time = 8.81553, size = 1855, normalized size = 8.67 \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)*(c*x**2+a)**2/(e*x+d)**(9/2),x)

[Out]

Piecewise((-30*A*a**2*e**5/(105*d**3*e**6*sqrt(d + e*x) + 315*d**2*e**7*x*sqrt(d + e*x) + 315*d*e**8*x**2*sqrt
(d + e*x) + 105*e**9*x**3*sqrt(d + e*x)) - 32*A*a*c*d**2*e**3/(105*d**3*e**6*sqrt(d + e*x) + 315*d**2*e**7*x*s
qrt(d + e*x) + 315*d*e**8*x**2*sqrt(d + e*x) + 105*e**9*x**3*sqrt(d + e*x)) - 112*A*a*c*d*e**4*x/(105*d**3*e**
6*sqrt(d + e*x) + 315*d**2*e**7*x*sqrt(d + e*x) + 315*d*e**8*x**2*sqrt(d + e*x) + 105*e**9*x**3*sqrt(d + e*x))
- 140*A*a*c*e**5*x**2/(105*d**3*e**6*sqrt(d + e*x) + 315*d**2*e**7*x*sqrt(d + e*x) + 315*d*e**8*x**2*sqrt(d +
e*x) + 105*e**9*x**3*sqrt(d + e*x)) + 768*A*c**2*d**4*e/(105*d**3*e**6*sqrt(d + e*x) + 315*d**2*e**7*x*sqrt(d
+ e*x) + 315*d*e**8*x**2*sqrt(d + e*x) + 105*e**9*x**3*sqrt(d + e*x)) + 2688*A*c**2*d**3*e**2*x/(105*d**3*e**
6*sqrt(d + e*x) + 315*d**2*e**7*x*sqrt(d + e*x) + 315*d*e**8*x**2*sqrt(d + e*x) + 105*e**9*x**3*sqrt(d + e*x))
+ 3360*A*c**2*d**2*e**3*x**2/(105*d**3*e**6*sqrt(d + e*x) + 315*d**2*e**7*x*sqrt(d + e*x) + 315*d*e**8*x**2*s
qrt(d + e*x) + 105*e**9*x**3*sqrt(d + e*x)) + 1680*A*c**2*d*e**4*x**3/(105*d**3*e**6*sqrt(d + e*x) + 315*d**2*
e**7*x*sqrt(d + e*x) + 315*d*e**8*x**2*sqrt(d + e*x) + 105*e**9*x**3*sqrt(d + e*x)) + 210*A*c**2*e**5*x**4/(10
5*d**3*e**6*sqrt(d + e*x) + 315*d**2*e**7*x*sqrt(d + e*x) + 315*d*e**8*x**2*sqrt(d + e*x) + 105*e**9*x**3*sqrt
(d + e*x)) - 12*B*a**2*d*e**4/(105*d**3*e**6*sqrt(d + e*x) + 315*d**2*e**7*x*sqrt(d + e*x) + 315*d*e**8*x**2*s
qrt(d + e*x) + 105*e**9*x**3*sqrt(d + e*x)) - 42*B*a**2*e**5*x/(105*d**3*e**6*sqrt(d + e*x) + 315*d**2*e**7*x*
sqrt(d + e*x) + 315*d*e**8*x**2*sqrt(d + e*x) + 105*e**9*x**3*sqrt(d + e*x)) - 192*B*a*c*d**3*e**2/(105*d**3*e
**6*sqrt(d + e*x) + 315*d**2*e**7*x*sqrt(d + e*x) + 315*d*e**8*x**2*sqrt(d + e*x) + 105*e**9*x**3*sqrt(d + e*x
)) - 672*B*a*c*d**2*e**3*x/(105*d**3*e**6*sqrt(d + e*x) + 315*d**2*e**7*x*sqrt(d + e*x) + 315*d*e**8*x**2*sqrt
(d + e*x) + 105*e**9*x**3*sqrt(d + e*x)) - 840*B*a*c*d*e**4*x**2/(105*d**3*e**6*sqrt(d + e*x) + 315*d**2*e**7*
x*sqrt(d + e*x) + 315*d*e**8*x**2*sqrt(d + e*x) + 105*e**9*x**3*sqrt(d + e*x)) - 420*B*a*c*e**5*x**3/(105*d**3
*e**6*sqrt(d + e*x) + 315*d**2*e**7*x*sqrt(d + e*x) + 315*d*e**8*x**2*sqrt(d + e*x) + 105*e**9*x**3*sqrt(d + e
*x)) - 2560*B*c**2*d**5/(105*d**3*e**6*sqrt(d + e*x) + 315*d**2*e**7*x*sqrt(d + e*x) + 315*d*e**8*x**2*sqrt(d
+ e*x) + 105*e**9*x**3*sqrt(d + e*x)) - 8960*B*c**2*d**4*e*x/(105*d**3*e**6*sqrt(d + e*x) + 315*d**2*e**7*x*sq
rt(d + e*x) + 315*d*e**8*x**2*sqrt(d + e*x) + 105*e**9*x**3*sqrt(d + e*x)) - 11200*B*c**2*d**3*e**2*x**2/(105*
d**3*e**6*sqrt(d + e*x) + 315*d**2*e**7*x*sqrt(d + e*x) + 315*d*e**8*x**2*sqrt(d + e*x) + 105*e**9*x**3*sqrt(d
+ e*x)) - 5600*B*c**2*d**2*e**3*x**3/(105*d**3*e**6*sqrt(d + e*x) + 315*d**2*e**7*x*sqrt(d + e*x) + 315*d*e**
8*x**2*sqrt(d + e*x) + 105*e**9*x**3*sqrt(d + e*x)) - 700*B*c**2*d*e**4*x**4/(105*d**3*e**6*sqrt(d + e*x) + 31
5*d**2*e**7*x*sqrt(d + e*x) + 315*d*e**8*x**2*sqrt(d + e*x) + 105*e**9*x**3*sqrt(d + e*x)) + 70*B*c**2*e**5*x*
*5/(105*d**3*e**6*sqrt(d + e*x) + 315*d**2*e**7*x*sqrt(d + e*x) + 315*d*e**8*x**2*sqrt(d + e*x) + 105*e**9*x**
3*sqrt(d + e*x)), Ne(e, 0)), ((A*a**2*x + 2*A*a*c*x**3/3 + A*c**2*x**5/5 + B*a**2*x**2/2 + B*a*c*x**4/2 + B*c*
*2*x**6/6)/d**(9/2), True))

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Giac [A]  time = 1.19981, size = 427, normalized size = 2. \begin{align*} \frac{2}{3} \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} B c^{2} e^{12} - 15 \, \sqrt{x e + d} B c^{2} d e^{12} + 3 \, \sqrt{x e + d} A c^{2} e^{13}\right )} e^{\left (-18\right )} - \frac{2 \,{\left (1050 \,{\left (x e + d\right )}^{3} B c^{2} d^{2} - 350 \,{\left (x e + d\right )}^{2} B c^{2} d^{3} + 105 \,{\left (x e + d\right )} B c^{2} d^{4} - 15 \, B c^{2} d^{5} - 420 \,{\left (x e + d\right )}^{3} A c^{2} d e + 210 \,{\left (x e + d\right )}^{2} A c^{2} d^{2} e - 84 \,{\left (x e + d\right )} A c^{2} d^{3} e + 15 \, A c^{2} d^{4} e + 210 \,{\left (x e + d\right )}^{3} B a c e^{2} - 210 \,{\left (x e + d\right )}^{2} B a c d e^{2} + 126 \,{\left (x e + d\right )} B a c d^{2} e^{2} - 30 \, B a c d^{3} e^{2} + 70 \,{\left (x e + d\right )}^{2} A a c e^{3} - 84 \,{\left (x e + d\right )} A a c d e^{3} + 30 \, A a c d^{2} e^{3} + 21 \,{\left (x e + d\right )} B a^{2} e^{4} - 15 \, B a^{2} d e^{4} + 15 \, A a^{2} e^{5}\right )} e^{\left (-6\right )}}{105 \,{\left (x e + d\right )}^{\frac{7}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^2/(e*x+d)^(9/2),x, algorithm="giac")

[Out]

2/3*((x*e + d)^(3/2)*B*c^2*e^12 - 15*sqrt(x*e + d)*B*c^2*d*e^12 + 3*sqrt(x*e + d)*A*c^2*e^13)*e^(-18) - 2/105*
(1050*(x*e + d)^3*B*c^2*d^2 - 350*(x*e + d)^2*B*c^2*d^3 + 105*(x*e + d)*B*c^2*d^4 - 15*B*c^2*d^5 - 420*(x*e +
d)^3*A*c^2*d*e + 210*(x*e + d)^2*A*c^2*d^2*e - 84*(x*e + d)*A*c^2*d^3*e + 15*A*c^2*d^4*e + 210*(x*e + d)^3*B*a
*c*e^2 - 210*(x*e + d)^2*B*a*c*d*e^2 + 126*(x*e + d)*B*a*c*d^2*e^2 - 30*B*a*c*d^3*e^2 + 70*(x*e + d)^2*A*a*c*e
^3 - 84*(x*e + d)*A*a*c*d*e^3 + 30*A*a*c*d^2*e^3 + 21*(x*e + d)*B*a^2*e^4 - 15*B*a^2*d*e^4 + 15*A*a^2*e^5)*e^(
-6)/(x*e + d)^(7/2)