### 3.2141 $$\int \frac{(a+b x+c x^2)^3}{(d+e x)^6} \, dx$$

Optimal. Leaf size=256 $-\frac{3 c \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^7 (d+e x)}+\frac{(2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{2 e^7 (d+e x)^2}-\frac{\left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^7 (d+e x)^3}+\frac{3 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{4 e^7 (d+e x)^4}-\frac{\left (a e^2-b d e+c d^2\right )^3}{5 e^7 (d+e x)^5}-\frac{3 c^2 (2 c d-b e) \log (d+e x)}{e^7}+\frac{c^3 x}{e^6}$

[Out]

(c^3*x)/e^6 - (c*d^2 - b*d*e + a*e^2)^3/(5*e^7*(d + e*x)^5) + (3*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^2)/(4*e
^7*(d + e*x)^4) - ((c*d^2 - b*d*e + a*e^2)*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e)))/(e^7*(d + e*x)^3) + ((2*
c*d - b*e)*(10*c^2*d^2 + b^2*e^2 - 2*c*e*(5*b*d - 3*a*e)))/(2*e^7*(d + e*x)^2) - (3*c*(5*c^2*d^2 + b^2*e^2 - c
*e*(5*b*d - a*e)))/(e^7*(d + e*x)) - (3*c^2*(2*c*d - b*e)*Log[d + e*x])/e^7

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Rubi [A]  time = 0.244684, antiderivative size = 256, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.05, Rules used = {698} $-\frac{3 c \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^7 (d+e x)}+\frac{(2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{2 e^7 (d+e x)^2}-\frac{\left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^7 (d+e x)^3}+\frac{3 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{4 e^7 (d+e x)^4}-\frac{\left (a e^2-b d e+c d^2\right )^3}{5 e^7 (d+e x)^5}-\frac{3 c^2 (2 c d-b e) \log (d+e x)}{e^7}+\frac{c^3 x}{e^6}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a + b*x + c*x^2)^3/(d + e*x)^6,x]

[Out]

(c^3*x)/e^6 - (c*d^2 - b*d*e + a*e^2)^3/(5*e^7*(d + e*x)^5) + (3*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^2)/(4*e
^7*(d + e*x)^4) - ((c*d^2 - b*d*e + a*e^2)*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e)))/(e^7*(d + e*x)^3) + ((2*
c*d - b*e)*(10*c^2*d^2 + b^2*e^2 - 2*c*e*(5*b*d - 3*a*e)))/(2*e^7*(d + e*x)^2) - (3*c*(5*c^2*d^2 + b^2*e^2 - c
*e*(5*b*d - a*e)))/(e^7*(d + e*x)) - (3*c^2*(2*c*d - b*e)*Log[d + e*x])/e^7

Rule 698

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*
e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin{align*} \int \frac{\left (a+b x+c x^2\right )^3}{(d+e x)^6} \, dx &=\int \left (\frac{c^3}{e^6}+\frac{\left (c d^2-b d e+a e^2\right )^3}{e^6 (d+e x)^6}+\frac{3 (-2 c d+b e) \left (c d^2-b d e+a e^2\right )^2}{e^6 (d+e x)^5}+\frac{3 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2-5 b c d e+b^2 e^2+a c e^2\right )}{e^6 (d+e x)^4}+\frac{(2 c d-b e) \left (-10 c^2 d^2-b^2 e^2+2 c e (5 b d-3 a e)\right )}{e^6 (d+e x)^3}+\frac{3 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )}{e^6 (d+e x)^2}-\frac{3 c^2 (2 c d-b e)}{e^6 (d+e x)}\right ) \, dx\\ &=\frac{c^3 x}{e^6}-\frac{\left (c d^2-b d e+a e^2\right )^3}{5 e^7 (d+e x)^5}+\frac{3 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^2}{4 e^7 (d+e x)^4}-\frac{\left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )}{e^7 (d+e x)^3}+\frac{(2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right )}{2 e^7 (d+e x)^2}-\frac{3 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )}{e^7 (d+e x)}-\frac{3 c^2 (2 c d-b e) \log (d+e x)}{e^7}\\ \end{align*}

Mathematica [A]  time = 0.255258, size = 396, normalized size = 1.55 $-\frac{2 c e^2 \left (a^2 e^2 \left (d^2+5 d e x+10 e^2 x^2\right )+3 a b e \left (5 d^2 e x+d^3+10 d e^2 x^2+10 e^3 x^3\right )+6 b^2 \left (10 d^2 e^2 x^2+5 d^3 e x+d^4+10 d e^3 x^3+5 e^4 x^4\right )\right )+e^3 \left (3 a^2 b e^2 (d+5 e x)+4 a^3 e^3+2 a b^2 e \left (d^2+5 d e x+10 e^2 x^2\right )+b^3 \left (5 d^2 e x+d^3+10 d e^2 x^2+10 e^3 x^3\right )\right )+c^2 e \left (12 a e \left (10 d^2 e^2 x^2+5 d^3 e x+d^4+10 d e^3 x^3+5 e^4 x^4\right )-b d \left (1100 d^2 e^2 x^2+625 d^3 e x+137 d^4+900 d e^3 x^3+300 e^4 x^4\right )\right )+60 c^2 (d+e x)^5 (2 c d-b e) \log (d+e x)+2 c^3 \left (600 d^4 e^2 x^2+400 d^3 e^3 x^3+50 d^2 e^4 x^4+375 d^5 e x+87 d^6-50 d e^5 x^5-10 e^6 x^6\right )}{20 e^7 (d+e x)^5}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(a + b*x + c*x^2)^3/(d + e*x)^6,x]

[Out]

-(2*c^3*(87*d^6 + 375*d^5*e*x + 600*d^4*e^2*x^2 + 400*d^3*e^3*x^3 + 50*d^2*e^4*x^4 - 50*d*e^5*x^5 - 10*e^6*x^6
) + e^3*(4*a^3*e^3 + 3*a^2*b*e^2*(d + 5*e*x) + 2*a*b^2*e*(d^2 + 5*d*e*x + 10*e^2*x^2) + b^3*(d^3 + 5*d^2*e*x +
10*d*e^2*x^2 + 10*e^3*x^3)) + 2*c*e^2*(a^2*e^2*(d^2 + 5*d*e*x + 10*e^2*x^2) + 3*a*b*e*(d^3 + 5*d^2*e*x + 10*d
*e^2*x^2 + 10*e^3*x^3) + 6*b^2*(d^4 + 5*d^3*e*x + 10*d^2*e^2*x^2 + 10*d*e^3*x^3 + 5*e^4*x^4)) + c^2*e*(12*a*e*
(d^4 + 5*d^3*e*x + 10*d^2*e^2*x^2 + 10*d*e^3*x^3 + 5*e^4*x^4) - b*d*(137*d^4 + 625*d^3*e*x + 1100*d^2*e^2*x^2
+ 900*d*e^3*x^3 + 300*e^4*x^4)) + 60*c^2*(2*c*d - b*e)*(d + e*x)^5*Log[d + e*x])/(20*e^7*(d + e*x)^5)

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Maple [B]  time = 0.051, size = 688, normalized size = 2.7 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+b*x+a)^3/(e*x+d)^6,x)

[Out]

-3*c/e^5/(e*x+d)*b^2-15*c^3/e^7/(e*x+d)*d^2-3/4/e^2/(e*x+d)^4*a^2*b-3/4/e^4/(e*x+d)^4*b^3*d^2+3/2/e^7/(e*x+d)^
4*c^3*d^5-1/e^3/(e*x+d)^3*a^2*c-1/e^3/(e*x+d)^3*a*b^2+1/e^4/(e*x+d)^3*b^3*d-5/e^7/(e*x+d)^3*c^3*d^4+10/e^7/(e*
x+d)^2*c^3*d^3+3*c^2/e^6*ln(e*x+d)*b-3*c^2/e^5/(e*x+d)*a+6/e^4/(e*x+d)^3*a*b*c*d+3/5/e^2/(e*x+d)^5*d*a^2*b-3/5
/e^3/(e*x+d)^5*a^2*c*d^2-9/2/e^4/(e*x+d)^4*a*b*c*d^2-3/5/e^3/(e*x+d)^5*d^2*a*b^2-3/5/e^5/(e*x+d)^5*a*c^2*d^4-3
/5/e^5/(e*x+d)^5*d^4*b^2*c+10/e^6/(e*x+d)^3*b*c^2*d^3-3/e^4/(e*x+d)^2*a*b*c+6/e^5/(e*x+d)^2*c^2*a*d+6/e^5/(e*x
+d)^2*b^2*c*d-15/e^6/(e*x+d)^2*b*c^2*d^2+3/2/e^3/(e*x+d)^4*a^2*c*d+3/2/e^3/(e*x+d)^4*a*b^2*d-6/e^5/(e*x+d)^3*a
*c^2*d^2-6/e^5/(e*x+d)^3*b^2*c*d^2+3/e^5/(e*x+d)^4*a*c^2*d^3+3/e^5/(e*x+d)^4*b^2*c*d^3-15/4/e^6/(e*x+d)^4*b*c^
2*d^4+1/5/e^4/(e*x+d)^5*d^3*b^3-1/5/e^7/(e*x+d)^5*c^3*d^6-1/5/e/(e*x+d)^5*a^3-1/2/e^4/(e*x+d)^2*b^3+3/5/e^6/(e
*x+d)^5*d^5*b*c^2+15*c^2/e^6/(e*x+d)*b*d+c^3*x/e^6+6/5/e^4/(e*x+d)^5*d^3*a*b*c-6*c^3*d*ln(e*x+d)/e^7

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Maxima [A]  time = 1.06975, size = 606, normalized size = 2.37 \begin{align*} -\frac{174 \, c^{3} d^{6} - 137 \, b c^{2} d^{5} e + 3 \, a^{2} b d e^{5} + 4 \, a^{3} e^{6} + 12 \,{\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} +{\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} + 2 \,{\left (a b^{2} + a^{2} c\right )} d^{2} e^{4} + 60 \,{\left (5 \, c^{3} d^{2} e^{4} - 5 \, b c^{2} d e^{5} +{\left (b^{2} c + a c^{2}\right )} e^{6}\right )} x^{4} + 10 \,{\left (100 \, c^{3} d^{3} e^{3} - 90 \, b c^{2} d^{2} e^{4} + 12 \,{\left (b^{2} c + a c^{2}\right )} d e^{5} +{\left (b^{3} + 6 \, a b c\right )} e^{6}\right )} x^{3} + 10 \,{\left (130 \, c^{3} d^{4} e^{2} - 110 \, b c^{2} d^{3} e^{3} + 12 \,{\left (b^{2} c + a c^{2}\right )} d^{2} e^{4} +{\left (b^{3} + 6 \, a b c\right )} d e^{5} + 2 \,{\left (a b^{2} + a^{2} c\right )} e^{6}\right )} x^{2} + 5 \,{\left (154 \, c^{3} d^{5} e - 125 \, b c^{2} d^{4} e^{2} + 3 \, a^{2} b e^{6} + 12 \,{\left (b^{2} c + a c^{2}\right )} d^{3} e^{3} +{\left (b^{3} + 6 \, a b c\right )} d^{2} e^{4} + 2 \,{\left (a b^{2} + a^{2} c\right )} d e^{5}\right )} x}{20 \,{\left (e^{12} x^{5} + 5 \, d e^{11} x^{4} + 10 \, d^{2} e^{10} x^{3} + 10 \, d^{3} e^{9} x^{2} + 5 \, d^{4} e^{8} x + d^{5} e^{7}\right )}} + \frac{c^{3} x}{e^{6}} - \frac{3 \,{\left (2 \, c^{3} d - b c^{2} e\right )} \log \left (e x + d\right )}{e^{7}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^3/(e*x+d)^6,x, algorithm="maxima")

[Out]

-1/20*(174*c^3*d^6 - 137*b*c^2*d^5*e + 3*a^2*b*d*e^5 + 4*a^3*e^6 + 12*(b^2*c + a*c^2)*d^4*e^2 + (b^3 + 6*a*b*c
)*d^3*e^3 + 2*(a*b^2 + a^2*c)*d^2*e^4 + 60*(5*c^3*d^2*e^4 - 5*b*c^2*d*e^5 + (b^2*c + a*c^2)*e^6)*x^4 + 10*(100
*c^3*d^3*e^3 - 90*b*c^2*d^2*e^4 + 12*(b^2*c + a*c^2)*d*e^5 + (b^3 + 6*a*b*c)*e^6)*x^3 + 10*(130*c^3*d^4*e^2 -
110*b*c^2*d^3*e^3 + 12*(b^2*c + a*c^2)*d^2*e^4 + (b^3 + 6*a*b*c)*d*e^5 + 2*(a*b^2 + a^2*c)*e^6)*x^2 + 5*(154*c
^3*d^5*e - 125*b*c^2*d^4*e^2 + 3*a^2*b*e^6 + 12*(b^2*c + a*c^2)*d^3*e^3 + (b^3 + 6*a*b*c)*d^2*e^4 + 2*(a*b^2 +
a^2*c)*d*e^5)*x)/(e^12*x^5 + 5*d*e^11*x^4 + 10*d^2*e^10*x^3 + 10*d^3*e^9*x^2 + 5*d^4*e^8*x + d^5*e^7) + c^3*x
/e^6 - 3*(2*c^3*d - b*c^2*e)*log(e*x + d)/e^7

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Fricas [B]  time = 2.09305, size = 1247, normalized size = 4.87 \begin{align*} \frac{20 \, c^{3} e^{6} x^{6} + 100 \, c^{3} d e^{5} x^{5} - 174 \, c^{3} d^{6} + 137 \, b c^{2} d^{5} e - 3 \, a^{2} b d e^{5} - 4 \, a^{3} e^{6} - 12 \,{\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} -{\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} - 2 \,{\left (a b^{2} + a^{2} c\right )} d^{2} e^{4} - 20 \,{\left (5 \, c^{3} d^{2} e^{4} - 15 \, b c^{2} d e^{5} + 3 \,{\left (b^{2} c + a c^{2}\right )} e^{6}\right )} x^{4} - 10 \,{\left (80 \, c^{3} d^{3} e^{3} - 90 \, b c^{2} d^{2} e^{4} + 12 \,{\left (b^{2} c + a c^{2}\right )} d e^{5} +{\left (b^{3} + 6 \, a b c\right )} e^{6}\right )} x^{3} - 10 \,{\left (120 \, c^{3} d^{4} e^{2} - 110 \, b c^{2} d^{3} e^{3} + 12 \,{\left (b^{2} c + a c^{2}\right )} d^{2} e^{4} +{\left (b^{3} + 6 \, a b c\right )} d e^{5} + 2 \,{\left (a b^{2} + a^{2} c\right )} e^{6}\right )} x^{2} - 5 \,{\left (150 \, c^{3} d^{5} e - 125 \, b c^{2} d^{4} e^{2} + 3 \, a^{2} b e^{6} + 12 \,{\left (b^{2} c + a c^{2}\right )} d^{3} e^{3} +{\left (b^{3} + 6 \, a b c\right )} d^{2} e^{4} + 2 \,{\left (a b^{2} + a^{2} c\right )} d e^{5}\right )} x - 60 \,{\left (2 \, c^{3} d^{6} - b c^{2} d^{5} e +{\left (2 \, c^{3} d e^{5} - b c^{2} e^{6}\right )} x^{5} + 5 \,{\left (2 \, c^{3} d^{2} e^{4} - b c^{2} d e^{5}\right )} x^{4} + 10 \,{\left (2 \, c^{3} d^{3} e^{3} - b c^{2} d^{2} e^{4}\right )} x^{3} + 10 \,{\left (2 \, c^{3} d^{4} e^{2} - b c^{2} d^{3} e^{3}\right )} x^{2} + 5 \,{\left (2 \, c^{3} d^{5} e - b c^{2} d^{4} e^{2}\right )} x\right )} \log \left (e x + d\right )}{20 \,{\left (e^{12} x^{5} + 5 \, d e^{11} x^{4} + 10 \, d^{2} e^{10} x^{3} + 10 \, d^{3} e^{9} x^{2} + 5 \, d^{4} e^{8} x + d^{5} e^{7}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^3/(e*x+d)^6,x, algorithm="fricas")

[Out]

1/20*(20*c^3*e^6*x^6 + 100*c^3*d*e^5*x^5 - 174*c^3*d^6 + 137*b*c^2*d^5*e - 3*a^2*b*d*e^5 - 4*a^3*e^6 - 12*(b^2
*c + a*c^2)*d^4*e^2 - (b^3 + 6*a*b*c)*d^3*e^3 - 2*(a*b^2 + a^2*c)*d^2*e^4 - 20*(5*c^3*d^2*e^4 - 15*b*c^2*d*e^5
+ 3*(b^2*c + a*c^2)*e^6)*x^4 - 10*(80*c^3*d^3*e^3 - 90*b*c^2*d^2*e^4 + 12*(b^2*c + a*c^2)*d*e^5 + (b^3 + 6*a*
b*c)*e^6)*x^3 - 10*(120*c^3*d^4*e^2 - 110*b*c^2*d^3*e^3 + 12*(b^2*c + a*c^2)*d^2*e^4 + (b^3 + 6*a*b*c)*d*e^5 +
2*(a*b^2 + a^2*c)*e^6)*x^2 - 5*(150*c^3*d^5*e - 125*b*c^2*d^4*e^2 + 3*a^2*b*e^6 + 12*(b^2*c + a*c^2)*d^3*e^3
+ (b^3 + 6*a*b*c)*d^2*e^4 + 2*(a*b^2 + a^2*c)*d*e^5)*x - 60*(2*c^3*d^6 - b*c^2*d^5*e + (2*c^3*d*e^5 - b*c^2*e^
6)*x^5 + 5*(2*c^3*d^2*e^4 - b*c^2*d*e^5)*x^4 + 10*(2*c^3*d^3*e^3 - b*c^2*d^2*e^4)*x^3 + 10*(2*c^3*d^4*e^2 - b*
c^2*d^3*e^3)*x^2 + 5*(2*c^3*d^5*e - b*c^2*d^4*e^2)*x)*log(e*x + d))/(e^12*x^5 + 5*d*e^11*x^4 + 10*d^2*e^10*x^3
+ 10*d^3*e^9*x^2 + 5*d^4*e^8*x + d^5*e^7)

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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((c*x**2+b*x+a)**3/(e*x+d)**6,x)

[Out]

Timed out

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Giac [A]  time = 1.13207, size = 560, normalized size = 2.19 \begin{align*} c^{3} x e^{\left (-6\right )} - 3 \,{\left (2 \, c^{3} d - b c^{2} e\right )} e^{\left (-7\right )} \log \left ({\left | x e + d \right |}\right ) - \frac{{\left (174 \, c^{3} d^{6} - 137 \, b c^{2} d^{5} e + 12 \, b^{2} c d^{4} e^{2} + 12 \, a c^{2} d^{4} e^{2} + b^{3} d^{3} e^{3} + 6 \, a b c d^{3} e^{3} + 2 \, a b^{2} d^{2} e^{4} + 2 \, a^{2} c d^{2} e^{4} + 60 \,{\left (5 \, c^{3} d^{2} e^{4} - 5 \, b c^{2} d e^{5} + b^{2} c e^{6} + a c^{2} e^{6}\right )} x^{4} + 3 \, a^{2} b d e^{5} + 10 \,{\left (100 \, c^{3} d^{3} e^{3} - 90 \, b c^{2} d^{2} e^{4} + 12 \, b^{2} c d e^{5} + 12 \, a c^{2} d e^{5} + b^{3} e^{6} + 6 \, a b c e^{6}\right )} x^{3} + 4 \, a^{3} e^{6} + 10 \,{\left (130 \, c^{3} d^{4} e^{2} - 110 \, b c^{2} d^{3} e^{3} + 12 \, b^{2} c d^{2} e^{4} + 12 \, a c^{2} d^{2} e^{4} + b^{3} d e^{5} + 6 \, a b c d e^{5} + 2 \, a b^{2} e^{6} + 2 \, a^{2} c e^{6}\right )} x^{2} + 5 \,{\left (154 \, c^{3} d^{5} e - 125 \, b c^{2} d^{4} e^{2} + 12 \, b^{2} c d^{3} e^{3} + 12 \, a c^{2} d^{3} e^{3} + b^{3} d^{2} e^{4} + 6 \, a b c d^{2} e^{4} + 2 \, a b^{2} d e^{5} + 2 \, a^{2} c d e^{5} + 3 \, a^{2} b e^{6}\right )} x\right )} e^{\left (-7\right )}}{20 \,{\left (x e + d\right )}^{5}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^3/(e*x+d)^6,x, algorithm="giac")

[Out]

c^3*x*e^(-6) - 3*(2*c^3*d - b*c^2*e)*e^(-7)*log(abs(x*e + d)) - 1/20*(174*c^3*d^6 - 137*b*c^2*d^5*e + 12*b^2*c
*d^4*e^2 + 12*a*c^2*d^4*e^2 + b^3*d^3*e^3 + 6*a*b*c*d^3*e^3 + 2*a*b^2*d^2*e^4 + 2*a^2*c*d^2*e^4 + 60*(5*c^3*d^
2*e^4 - 5*b*c^2*d*e^5 + b^2*c*e^6 + a*c^2*e^6)*x^4 + 3*a^2*b*d*e^5 + 10*(100*c^3*d^3*e^3 - 90*b*c^2*d^2*e^4 +
12*b^2*c*d*e^5 + 12*a*c^2*d*e^5 + b^3*e^6 + 6*a*b*c*e^6)*x^3 + 4*a^3*e^6 + 10*(130*c^3*d^4*e^2 - 110*b*c^2*d^3
*e^3 + 12*b^2*c*d^2*e^4 + 12*a*c^2*d^2*e^4 + b^3*d*e^5 + 6*a*b*c*d*e^5 + 2*a*b^2*e^6 + 2*a^2*c*e^6)*x^2 + 5*(1
54*c^3*d^5*e - 125*b*c^2*d^4*e^2 + 12*b^2*c*d^3*e^3 + 12*a*c^2*d^3*e^3 + b^3*d^2*e^4 + 6*a*b*c*d^2*e^4 + 2*a*b
^2*d*e^5 + 2*a^2*c*d*e^5 + 3*a^2*b*e^6)*x)*e^(-7)/(x*e + d)^5