3.1323 \(\int \frac{(A+B x) (a+c x^2)^3}{(d+e x)^5} \, dx\)

Optimal. Leaf size=314 \[ \frac{c \left (4 A c d e \left (3 a e^2+5 c d^2\right )-B \left (3 a^2 e^4+30 a c d^2 e^2+35 c^2 d^4\right )\right )}{e^8 (d+e x)}-\frac{c^2 x \left (5 A c d e-3 B \left (a e^2+5 c d^2\right )\right )}{e^7}-\frac{c^2 \log (d+e x) \left (-3 a A e^3+15 a B d e^2-15 A c d^2 e+35 B c d^3\right )}{e^8}+\frac{3 c \left (a e^2+c d^2\right ) \left (-a A e^3+3 a B d e^2-5 A c d^2 e+7 B c d^3\right )}{2 e^8 (d+e x)^2}-\frac{\left (a e^2+c d^2\right )^2 \left (a B e^2-6 A c d e+7 B c d^2\right )}{3 e^8 (d+e x)^3}+\frac{\left (a e^2+c d^2\right )^3 (B d-A e)}{4 e^8 (d+e x)^4}-\frac{c^3 x^2 (5 B d-A e)}{2 e^6}+\frac{B c^3 x^3}{3 e^5} \]

[Out]

-((c^2*(5*A*c*d*e - 3*B*(5*c*d^2 + a*e^2))*x)/e^7) - (c^3*(5*B*d - A*e)*x^2)/(2*e^6) + (B*c^3*x^3)/(3*e^5) + (
(B*d - A*e)*(c*d^2 + a*e^2)^3)/(4*e^8*(d + e*x)^4) - ((c*d^2 + a*e^2)^2*(7*B*c*d^2 - 6*A*c*d*e + a*B*e^2))/(3*
e^8*(d + e*x)^3) + (3*c*(c*d^2 + a*e^2)*(7*B*c*d^3 - 5*A*c*d^2*e + 3*a*B*d*e^2 - a*A*e^3))/(2*e^8*(d + e*x)^2)
+ (c*(4*A*c*d*e*(5*c*d^2 + 3*a*e^2) - B*(35*c^2*d^4 + 30*a*c*d^2*e^2 + 3*a^2*e^4)))/(e^8*(d + e*x)) - (c^2*(3
5*B*c*d^3 - 15*A*c*d^2*e + 15*a*B*d*e^2 - 3*a*A*e^3)*Log[d + e*x])/e^8

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Rubi [A]  time = 0.381926, antiderivative size = 314, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {772} \[ \frac{c \left (4 A c d e \left (3 a e^2+5 c d^2\right )-B \left (3 a^2 e^4+30 a c d^2 e^2+35 c^2 d^4\right )\right )}{e^8 (d+e x)}-\frac{c^2 x \left (5 A c d e-3 B \left (a e^2+5 c d^2\right )\right )}{e^7}-\frac{c^2 \log (d+e x) \left (-3 a A e^3+15 a B d e^2-15 A c d^2 e+35 B c d^3\right )}{e^8}+\frac{3 c \left (a e^2+c d^2\right ) \left (-a A e^3+3 a B d e^2-5 A c d^2 e+7 B c d^3\right )}{2 e^8 (d+e x)^2}-\frac{\left (a e^2+c d^2\right )^2 \left (a B e^2-6 A c d e+7 B c d^2\right )}{3 e^8 (d+e x)^3}+\frac{\left (a e^2+c d^2\right )^3 (B d-A e)}{4 e^8 (d+e x)^4}-\frac{c^3 x^2 (5 B d-A e)}{2 e^6}+\frac{B c^3 x^3}{3 e^5} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a + c*x^2)^3)/(d + e*x)^5,x]

[Out]

-((c^2*(5*A*c*d*e - 3*B*(5*c*d^2 + a*e^2))*x)/e^7) - (c^3*(5*B*d - A*e)*x^2)/(2*e^6) + (B*c^3*x^3)/(3*e^5) + (
(B*d - A*e)*(c*d^2 + a*e^2)^3)/(4*e^8*(d + e*x)^4) - ((c*d^2 + a*e^2)^2*(7*B*c*d^2 - 6*A*c*d*e + a*B*e^2))/(3*
e^8*(d + e*x)^3) + (3*c*(c*d^2 + a*e^2)*(7*B*c*d^3 - 5*A*c*d^2*e + 3*a*B*d*e^2 - a*A*e^3))/(2*e^8*(d + e*x)^2)
+ (c*(4*A*c*d*e*(5*c*d^2 + 3*a*e^2) - B*(35*c^2*d^4 + 30*a*c*d^2*e^2 + 3*a^2*e^4)))/(e^8*(d + e*x)) - (c^2*(3
5*B*c*d^3 - 15*A*c*d^2*e + 15*a*B*d*e^2 - 3*a*A*e^3)*Log[d + e*x])/e^8

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 )^3}{(d+e x)^5} \, dx &=\int \left (-\frac{c^2 \left (-15 B c d^2+5 A c d e-3 a B e^2\right )}{e^7}+\frac{c^3 (-5 B d+A e) x}{e^6}+\frac{B c^3 x^2}{e^5}+\frac{(-B d+A e) \left (c d^2+a e^2\right )^3}{e^7 (d+e x)^5}+\frac{\left (c d^2+a e^2\right )^2 \left (7 B c d^2-6 A c d e+a B e^2\right )}{e^7 (d+e x)^4}+\frac{3 c \left (c d^2+a e^2\right ) \left (-7 B c d^3+5 A c d^2 e-3 a B d e^2+a A e^3\right )}{e^7 (d+e x)^3}-\frac{c \left (-35 B c^2 d^4+20 A c^2 d^3 e-30 a B c d^2 e^2+12 a A c d e^3-3 a^2 B e^4\right )}{e^7 (d+e x)^2}+\frac{c^2 \left (-35 B c d^3+15 A c d^2 e-15 a B d e^2+3 a A e^3\right )}{e^7 (d+e x)}\right ) \, dx\\ &=-\frac{c^2 \left (5 A c d e-3 B \left (5 c d^2+a e^2\right )\right ) x}{e^7}-\frac{c^3 (5 B d-A e) x^2}{2 e^6}+\frac{B c^3 x^3}{3 e^5}+\frac{(B d-A e) \left (c d^2+a e^2\right )^3}{4 e^8 (d+e x)^4}-\frac{\left (c d^2+a e^2\right )^2 \left (7 B c d^2-6 A c d e+a B e^2\right )}{3 e^8 (d+e x)^3}+\frac{3 c \left (c d^2+a e^2\right ) \left (7 B c d^3-5 A c d^2 e+3 a B d e^2-a A e^3\right )}{2 e^8 (d+e x)^2}+\frac{c \left (4 A c d e \left (5 c d^2+3 a e^2\right )-B \left (35 c^2 d^4+30 a c d^2 e^2+3 a^2 e^4\right )\right )}{e^8 (d+e x)}-\frac{c^2 \left (35 B c d^3-15 A c d^2 e+15 a B d e^2-3 a A e^3\right ) \log (d+e x)}{e^8}\\ \end{align*}

Mathematica [A]  time = 0.223573, size = 405, normalized size = 1.29 \[ \frac{3 A e \left (-a^2 c e^4 \left (d^2+4 d e x+6 e^2 x^2\right )-a^3 e^6+a c^2 d e^2 \left (88 d^2 e x+25 d^3+108 d e^2 x^2+48 e^3 x^3\right )+c^3 \left (132 d^4 e^2 x^2-32 d^3 e^3 x^3-68 d^2 e^4 x^4+168 d^5 e x+57 d^6-12 d e^5 x^5+2 e^6 x^6\right )\right )-B \left (9 a^2 c e^4 \left (4 d^2 e x+d^3+6 d e^2 x^2+4 e^3 x^3\right )+a^3 e^6 (d+4 e x)+3 a c^2 e^2 \left (252 d^3 e^2 x^2+48 d^2 e^3 x^3+248 d^4 e x+77 d^5-48 d e^4 x^4-12 e^5 x^5\right )+c^3 \left (444 d^5 e^2 x^2-544 d^4 e^3 x^3-556 d^3 e^4 x^4-84 d^2 e^5 x^5+856 d^6 e x+319 d^7+14 d e^6 x^6-4 e^7 x^7\right )\right )+12 c^2 (d+e x)^4 \log (d+e x) \left (3 A e \left (a e^2+5 c d^2\right )-5 B \left (3 a d e^2+7 c d^3\right )\right )}{12 e^8 (d+e x)^4} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a + c*x^2)^3)/(d + e*x)^5,x]

[Out]

(3*A*e*(-(a^3*e^6) - a^2*c*e^4*(d^2 + 4*d*e*x + 6*e^2*x^2) + a*c^2*d*e^2*(25*d^3 + 88*d^2*e*x + 108*d*e^2*x^2
+ 48*e^3*x^3) + c^3*(57*d^6 + 168*d^5*e*x + 132*d^4*e^2*x^2 - 32*d^3*e^3*x^3 - 68*d^2*e^4*x^4 - 12*d*e^5*x^5 +
2*e^6*x^6)) - B*(a^3*e^6*(d + 4*e*x) + 9*a^2*c*e^4*(d^3 + 4*d^2*e*x + 6*d*e^2*x^2 + 4*e^3*x^3) + 3*a*c^2*e^2*
(77*d^5 + 248*d^4*e*x + 252*d^3*e^2*x^2 + 48*d^2*e^3*x^3 - 48*d*e^4*x^4 - 12*e^5*x^5) + c^3*(319*d^7 + 856*d^6
*e*x + 444*d^5*e^2*x^2 - 544*d^4*e^3*x^3 - 556*d^3*e^4*x^4 - 84*d^2*e^5*x^5 + 14*d*e^6*x^6 - 4*e^7*x^7)) + 12*
c^2*(3*A*e*(5*c*d^2 + a*e^2) - 5*B*(7*c*d^3 + 3*a*d*e^2))*(d + e*x)^4*Log[d + e*x])/(12*e^8*(d + e*x)^4)

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Maple [B]  time = 0.013, size = 632, normalized size = 2. \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)*(c*x^2+a)^3/(e*x+d)^5,x)

[Out]

1/3*B*c^3*x^3/e^5-3/e^4/(e*x+d)^3*B*a^2*c*d^2+12*c^2/e^5/(e*x+d)*A*d*a-30*c^2/e^6/(e*x+d)*B*a*d^2-9*c^2/e^5/(e
*x+d)^2*A*d^2*a+9/2*c/e^4/(e*x+d)^2*B*a^2*d+15*c^2/e^6/(e*x+d)^2*a*B*d^3+2/e^3/(e*x+d)^3*A*a^2*c*d+4/e^5/(e*x+
d)^3*A*a*c^2*d^3-5/e^6/(e*x+d)^3*B*a*c^2*d^4-15*c^2/e^6*ln(e*x+d)*a*B*d-3/4/e^3/(e*x+d)^4*A*d^2*a^2*c-3/4/e^5/
(e*x+d)^4*A*d^4*a*c^2+3/4/e^4/(e*x+d)^4*B*d^3*a^2*c+3/4/e^6/(e*x+d)^4*B*a*c^2*d^5-1/4/e/(e*x+d)^4*A*a^3+1/2*c^
3/e^5*A*x^2-1/3/e^2/(e*x+d)^3*B*a^3+3*c^2/e^5*a*B*x+15*c^3/e^7*B*d^2*x-5/2*c^3/e^6*B*x^2*d-5*c^3/e^6*A*d*x+1/4
/e^8/(e*x+d)^4*B*c^3*d^7+20*c^3/e^7/(e*x+d)*A*d^3-3*c/e^4/(e*x+d)*B*a^2-35*c^3/e^8/(e*x+d)*B*d^4-3/2*c/e^3/(e*
x+d)^2*A*a^2-15/2*c^3/e^7/(e*x+d)^2*A*d^4+21/2*c^3/e^8/(e*x+d)^2*B*d^5+2/e^7/(e*x+d)^3*A*c^3*d^5-7/3/e^8/(e*x+
d)^3*B*c^3*d^6-35*c^3/e^8*ln(e*x+d)*B*d^3-1/4/e^7/(e*x+d)^4*A*c^3*d^6+1/4/e^2/(e*x+d)^4*B*d*a^3+3*c^2/e^5*ln(e
*x+d)*a*A+15*c^3/e^7*ln(e*x+d)*A*d^2

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Maxima [A]  time = 1.08517, size = 657, normalized size = 2.09 \begin{align*} -\frac{319 \, B c^{3} d^{7} - 171 \, A c^{3} d^{6} e + 231 \, B a c^{2} d^{5} e^{2} - 75 \, A a c^{2} d^{4} e^{3} + 9 \, B a^{2} c d^{3} e^{4} + 3 \, A a^{2} c d^{2} e^{5} + B a^{3} d e^{6} + 3 \, A a^{3} e^{7} + 12 \,{\left (35 \, B c^{3} d^{4} e^{3} - 20 \, A c^{3} d^{3} e^{4} + 30 \, B a c^{2} d^{2} e^{5} - 12 \, A a c^{2} d e^{6} + 3 \, B a^{2} c e^{7}\right )} x^{3} + 18 \,{\left (63 \, B c^{3} d^{5} e^{2} - 35 \, A c^{3} d^{4} e^{3} + 50 \, B a c^{2} d^{3} e^{4} - 18 \, A a c^{2} d^{2} e^{5} + 3 \, B a^{2} c d e^{6} + A a^{2} c e^{7}\right )} x^{2} + 4 \,{\left (259 \, B c^{3} d^{6} e - 141 \, A c^{3} d^{5} e^{2} + 195 \, B a c^{2} d^{4} e^{3} - 66 \, A a c^{2} d^{3} e^{4} + 9 \, B a^{2} c d^{2} e^{5} + 3 \, A a^{2} c d e^{6} + B a^{3} e^{7}\right )} x}{12 \,{\left (e^{12} x^{4} + 4 \, d e^{11} x^{3} + 6 \, d^{2} e^{10} x^{2} + 4 \, d^{3} e^{9} x + d^{4} e^{8}\right )}} + \frac{2 \, B c^{3} e^{2} x^{3} - 3 \,{\left (5 \, B c^{3} d e - A c^{3} e^{2}\right )} x^{2} + 6 \,{\left (15 \, B c^{3} d^{2} - 5 \, A c^{3} d e + 3 \, B a c^{2} e^{2}\right )} x}{6 \, e^{7}} - \frac{{\left (35 \, B c^{3} d^{3} - 15 \, A c^{3} d^{2} e + 15 \, B a c^{2} d e^{2} - 3 \, A a c^{2} e^{3}\right )} \log \left (e x + d\right )}{e^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^3/(e*x+d)^5,x, algorithm="maxima")

[Out]

-1/12*(319*B*c^3*d^7 - 171*A*c^3*d^6*e + 231*B*a*c^2*d^5*e^2 - 75*A*a*c^2*d^4*e^3 + 9*B*a^2*c*d^3*e^4 + 3*A*a^
2*c*d^2*e^5 + B*a^3*d*e^6 + 3*A*a^3*e^7 + 12*(35*B*c^3*d^4*e^3 - 20*A*c^3*d^3*e^4 + 30*B*a*c^2*d^2*e^5 - 12*A*
a*c^2*d*e^6 + 3*B*a^2*c*e^7)*x^3 + 18*(63*B*c^3*d^5*e^2 - 35*A*c^3*d^4*e^3 + 50*B*a*c^2*d^3*e^4 - 18*A*a*c^2*d
^2*e^5 + 3*B*a^2*c*d*e^6 + A*a^2*c*e^7)*x^2 + 4*(259*B*c^3*d^6*e - 141*A*c^3*d^5*e^2 + 195*B*a*c^2*d^4*e^3 - 6
6*A*a*c^2*d^3*e^4 + 9*B*a^2*c*d^2*e^5 + 3*A*a^2*c*d*e^6 + B*a^3*e^7)*x)/(e^12*x^4 + 4*d*e^11*x^3 + 6*d^2*e^10*
x^2 + 4*d^3*e^9*x + d^4*e^8) + 1/6*(2*B*c^3*e^2*x^3 - 3*(5*B*c^3*d*e - A*c^3*e^2)*x^2 + 6*(15*B*c^3*d^2 - 5*A*
c^3*d*e + 3*B*a*c^2*e^2)*x)/e^7 - (35*B*c^3*d^3 - 15*A*c^3*d^2*e + 15*B*a*c^2*d*e^2 - 3*A*a*c^2*e^3)*log(e*x +
d)/e^8

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Fricas [B]  time = 1.77124, size = 1602, normalized size = 5.1 \begin{align*} \frac{4 \, B c^{3} e^{7} x^{7} - 319 \, B c^{3} d^{7} + 171 \, A c^{3} d^{6} e - 231 \, B a c^{2} d^{5} e^{2} + 75 \, A a c^{2} d^{4} e^{3} - 9 \, B a^{2} c d^{3} e^{4} - 3 \, A a^{2} c d^{2} e^{5} - B a^{3} d e^{6} - 3 \, A a^{3} e^{7} - 2 \,{\left (7 \, B c^{3} d e^{6} - 3 \, A c^{3} e^{7}\right )} x^{6} + 12 \,{\left (7 \, B c^{3} d^{2} e^{5} - 3 \, A c^{3} d e^{6} + 3 \, B a c^{2} e^{7}\right )} x^{5} + 4 \,{\left (139 \, B c^{3} d^{3} e^{4} - 51 \, A c^{3} d^{2} e^{5} + 36 \, B a c^{2} d e^{6}\right )} x^{4} + 4 \,{\left (136 \, B c^{3} d^{4} e^{3} - 24 \, A c^{3} d^{3} e^{4} - 36 \, B a c^{2} d^{2} e^{5} + 36 \, A a c^{2} d e^{6} - 9 \, B a^{2} c e^{7}\right )} x^{3} - 6 \,{\left (74 \, B c^{3} d^{5} e^{2} - 66 \, A c^{3} d^{4} e^{3} + 126 \, B a c^{2} d^{3} e^{4} - 54 \, A a c^{2} d^{2} e^{5} + 9 \, B a^{2} c d e^{6} + 3 \, A a^{2} c e^{7}\right )} x^{2} - 4 \,{\left (214 \, B c^{3} d^{6} e - 126 \, A c^{3} d^{5} e^{2} + 186 \, B a c^{2} d^{4} e^{3} - 66 \, A a c^{2} d^{3} e^{4} + 9 \, B a^{2} c d^{2} e^{5} + 3 \, A a^{2} c d e^{6} + B a^{3} e^{7}\right )} x - 12 \,{\left (35 \, B c^{3} d^{7} - 15 \, A c^{3} d^{6} e + 15 \, B a c^{2} d^{5} e^{2} - 3 \, A a c^{2} d^{4} e^{3} +{\left (35 \, B c^{3} d^{3} e^{4} - 15 \, A c^{3} d^{2} e^{5} + 15 \, B a c^{2} d e^{6} - 3 \, A a c^{2} e^{7}\right )} x^{4} + 4 \,{\left (35 \, B c^{3} d^{4} e^{3} - 15 \, A c^{3} d^{3} e^{4} + 15 \, B a c^{2} d^{2} e^{5} - 3 \, A a c^{2} d e^{6}\right )} x^{3} + 6 \,{\left (35 \, B c^{3} d^{5} e^{2} - 15 \, A c^{3} d^{4} e^{3} + 15 \, B a c^{2} d^{3} e^{4} - 3 \, A a c^{2} d^{2} e^{5}\right )} x^{2} + 4 \,{\left (35 \, B c^{3} d^{6} e - 15 \, A c^{3} d^{5} e^{2} + 15 \, B a c^{2} d^{4} e^{3} - 3 \, A a c^{2} d^{3} e^{4}\right )} x\right )} \log \left (e x + d\right )}{12 \,{\left (e^{12} x^{4} + 4 \, d e^{11} x^{3} + 6 \, d^{2} e^{10} x^{2} + 4 \, d^{3} e^{9} x + d^{4} e^{8}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^3/(e*x+d)^5,x, algorithm="fricas")

[Out]

1/12*(4*B*c^3*e^7*x^7 - 319*B*c^3*d^7 + 171*A*c^3*d^6*e - 231*B*a*c^2*d^5*e^2 + 75*A*a*c^2*d^4*e^3 - 9*B*a^2*c
*d^3*e^4 - 3*A*a^2*c*d^2*e^5 - B*a^3*d*e^6 - 3*A*a^3*e^7 - 2*(7*B*c^3*d*e^6 - 3*A*c^3*e^7)*x^6 + 12*(7*B*c^3*d
^2*e^5 - 3*A*c^3*d*e^6 + 3*B*a*c^2*e^7)*x^5 + 4*(139*B*c^3*d^3*e^4 - 51*A*c^3*d^2*e^5 + 36*B*a*c^2*d*e^6)*x^4
+ 4*(136*B*c^3*d^4*e^3 - 24*A*c^3*d^3*e^4 - 36*B*a*c^2*d^2*e^5 + 36*A*a*c^2*d*e^6 - 9*B*a^2*c*e^7)*x^3 - 6*(74
*B*c^3*d^5*e^2 - 66*A*c^3*d^4*e^3 + 126*B*a*c^2*d^3*e^4 - 54*A*a*c^2*d^2*e^5 + 9*B*a^2*c*d*e^6 + 3*A*a^2*c*e^7
)*x^2 - 4*(214*B*c^3*d^6*e - 126*A*c^3*d^5*e^2 + 186*B*a*c^2*d^4*e^3 - 66*A*a*c^2*d^3*e^4 + 9*B*a^2*c*d^2*e^5
+ 3*A*a^2*c*d*e^6 + B*a^3*e^7)*x - 12*(35*B*c^3*d^7 - 15*A*c^3*d^6*e + 15*B*a*c^2*d^5*e^2 - 3*A*a*c^2*d^4*e^3
+ (35*B*c^3*d^3*e^4 - 15*A*c^3*d^2*e^5 + 15*B*a*c^2*d*e^6 - 3*A*a*c^2*e^7)*x^4 + 4*(35*B*c^3*d^4*e^3 - 15*A*c^
3*d^3*e^4 + 15*B*a*c^2*d^2*e^5 - 3*A*a*c^2*d*e^6)*x^3 + 6*(35*B*c^3*d^5*e^2 - 15*A*c^3*d^4*e^3 + 15*B*a*c^2*d^
3*e^4 - 3*A*a*c^2*d^2*e^5)*x^2 + 4*(35*B*c^3*d^6*e - 15*A*c^3*d^5*e^2 + 15*B*a*c^2*d^4*e^3 - 3*A*a*c^2*d^3*e^4
)*x)*log(e*x + d))/(e^12*x^4 + 4*d*e^11*x^3 + 6*d^2*e^10*x^2 + 4*d^3*e^9*x + d^4*e^8)

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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)*(c*x**2+a)**3/(e*x+d)**5,x)

[Out]

Timed out

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Giac [B]  time = 1.20216, size = 873, normalized size = 2.78 \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)^3/(e*x+d)^5,x, algorithm="giac")

[Out]

1/6*(2*B*c^3 - 3*(7*B*c^3*d*e - A*c^3*e^2)*e^(-1)/(x*e + d) + 18*(7*B*c^3*d^2*e^2 - 2*A*c^3*d*e^3 + B*a*c^2*e^
4)*e^(-2)/(x*e + d)^2)*(x*e + d)^3*e^(-8) + (35*B*c^3*d^3 - 15*A*c^3*d^2*e + 15*B*a*c^2*d*e^2 - 3*A*a*c^2*e^3)
*e^(-8)*log(abs(x*e + d)*e^(-1)/(x*e + d)^2) - 1/12*(420*B*c^3*d^4*e^36/(x*e + d) - 126*B*c^3*d^5*e^36/(x*e +
d)^2 + 28*B*c^3*d^6*e^36/(x*e + d)^3 - 3*B*c^3*d^7*e^36/(x*e + d)^4 - 240*A*c^3*d^3*e^37/(x*e + d) + 90*A*c^3*
d^4*e^37/(x*e + d)^2 - 24*A*c^3*d^5*e^37/(x*e + d)^3 + 3*A*c^3*d^6*e^37/(x*e + d)^4 + 360*B*a*c^2*d^2*e^38/(x*
e + d) - 180*B*a*c^2*d^3*e^38/(x*e + d)^2 + 60*B*a*c^2*d^4*e^38/(x*e + d)^3 - 9*B*a*c^2*d^5*e^38/(x*e + d)^4 -
144*A*a*c^2*d*e^39/(x*e + d) + 108*A*a*c^2*d^2*e^39/(x*e + d)^2 - 48*A*a*c^2*d^3*e^39/(x*e + d)^3 + 9*A*a*c^2
*d^4*e^39/(x*e + d)^4 + 36*B*a^2*c*e^40/(x*e + d) - 54*B*a^2*c*d*e^40/(x*e + d)^2 + 36*B*a^2*c*d^2*e^40/(x*e +
d)^3 - 9*B*a^2*c*d^3*e^40/(x*e + d)^4 + 18*A*a^2*c*e^41/(x*e + d)^2 - 24*A*a^2*c*d*e^41/(x*e + d)^3 + 9*A*a^2
*c*d^2*e^41/(x*e + d)^4 + 4*B*a^3*e^42/(x*e + d)^3 - 3*B*a^3*d*e^42/(x*e + d)^4 + 3*A*a^3*e^43/(x*e + d)^4)*e^
(-44)