3.1681 \(\int \frac{(A+B x) (a^2+2 a b x+b^2 x^2)^2}{d+e x} \, dx\)

Optimal. Leaf size=156 \[ -\frac{(a+b x)^4 (B d-A e)}{4 e^2}+\frac{(a+b x)^3 (b d-a e) (B d-A e)}{3 e^3}-\frac{(a+b x)^2 (b d-a e)^2 (B d-A e)}{2 e^4}+\frac{b x (b d-a e)^3 (B d-A e)}{e^5}-\frac{(b d-a e)^4 (B d-A e) \log (d+e x)}{e^6}+\frac{B (a+b x)^5}{5 b e} \]

[Out]

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

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Rubi [A]  time = 0.118998, antiderivative size = 156, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used = {27, 77} \[ -\frac{(a+b x)^4 (B d-A e)}{4 e^2}+\frac{(a+b x)^3 (b d-a e) (B d-A e)}{3 e^3}-\frac{(a+b x)^2 (b d-a e)^2 (B d-A e)}{2 e^4}+\frac{b x (b d-a e)^3 (B d-A e)}{e^5}-\frac{(b d-a e)^4 (B d-A e) \log (d+e x)}{e^6}+\frac{B (a+b x)^5}{5 b e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

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

Mathematica [A]  time = 0.140481, size = 258, normalized size = 1.65 \[ \frac{e x \left (60 a^2 b^2 e^2 \left (3 A e (e x-2 d)+B \left (6 d^2-3 d e x+2 e^2 x^2\right )\right )+120 a^3 b e^3 (2 A e-2 B d+B e x)+60 a^4 B e^4+20 a b^3 e \left (2 A e \left (6 d^2-3 d e x+2 e^2 x^2\right )+B \left (6 d^2 e x-12 d^3-4 d e^2 x^2+3 e^3 x^3\right )\right )+b^4 \left (5 A e \left (6 d^2 e x-12 d^3-4 d e^2 x^2+3 e^3 x^3\right )+B \left (20 d^2 e^2 x^2-30 d^3 e x+60 d^4-15 d e^3 x^3+12 e^4 x^4\right )\right )\right )-60 (b d-a e)^4 (B d-A e) \log (d+e x)}{60 e^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(e*x*(60*a^4*B*e^4 + 120*a^3*b*e^3*(-2*B*d + 2*A*e + B*e*x) + 60*a^2*b^2*e^2*(3*A*e*(-2*d + e*x) + B*(6*d^2 -
3*d*e*x + 2*e^2*x^2)) + 20*a*b^3*e*(2*A*e*(6*d^2 - 3*d*e*x + 2*e^2*x^2) + B*(-12*d^3 + 6*d^2*e*x - 4*d*e^2*x^2
+ 3*e^3*x^3)) + b^4*(5*A*e*(-12*d^3 + 6*d^2*e*x - 4*d*e^2*x^2 + 3*e^3*x^3) + B*(60*d^4 - 30*d^3*e*x + 20*d^2*
e^2*x^2 - 15*d*e^3*x^3 + 12*e^4*x^4))) - 60*(b*d - a*e)^4*(B*d - A*e)*Log[d + e*x])/(60*e^6)

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Maple [B]  time = 0.006, size = 521, normalized size = 3.3 \begin{align*} 2\,{\frac{{a}^{3}B{x}^{2}b}{e}}+{\frac{aB{x}^{4}{b}^{3}}{e}}+{\frac{A{x}^{2}{b}^{4}{d}^{2}}{2\,{e}^{3}}}-{\frac{A{d}^{3}{b}^{4}x}{{e}^{4}}}+3\,{\frac{{a}^{2}A{b}^{2}{x}^{2}}{e}}-{\frac{{b}^{4}B{x}^{2}{d}^{3}}{2\,{e}^{4}}}+4\,{\frac{{a}^{3}Abx}{e}}+{\frac{4\,aA{b}^{3}{x}^{3}}{3\,e}}+{\frac{{b}^{4}B{x}^{3}{d}^{2}}{3\,{e}^{3}}}+2\,{\frac{{a}^{2}B{x}^{3}{b}^{2}}{e}}+{\frac{\ln \left ( ex+d \right ) A{b}^{4}{d}^{4}}{{e}^{5}}}-{\frac{\ln \left ( ex+d \right ) B{a}^{4}d}{{e}^{2}}}-{\frac{\ln \left ( ex+d \right ) B{b}^{4}{d}^{5}}{{e}^{6}}}-{\frac{A{x}^{3}{b}^{4}d}{3\,{e}^{2}}}-{\frac{{b}^{4}B{x}^{4}d}{4\,{e}^{2}}}+{\frac{B{b}^{4}{d}^{4}x}{{e}^{5}}}-4\,{\frac{{a}^{3}bBdx}{{e}^{2}}}-3\,{\frac{B{x}^{2}{a}^{2}{b}^{2}d}{{e}^{2}}}+2\,{\frac{aB{x}^{2}{b}^{3}{d}^{2}}{{e}^{3}}}-6\,{\frac{Ad{a}^{2}{b}^{2}x}{{e}^{2}}}-{\frac{4\,B{x}^{3}a{b}^{3}d}{3\,{e}^{2}}}+6\,{\frac{{b}^{2}B{a}^{2}{d}^{2}x}{{e}^{3}}}-4\,{\frac{Ba{b}^{3}{d}^{3}x}{{e}^{4}}}+6\,{\frac{\ln \left ( ex+d \right ) A{a}^{2}{b}^{2}{d}^{2}}{{e}^{3}}}+4\,{\frac{A{d}^{2}a{b}^{3}x}{{e}^{3}}}-2\,{\frac{A{x}^{2}a{b}^{3}d}{{e}^{2}}}-4\,{\frac{\ln \left ( ex+d \right ) Aa{b}^{3}{d}^{3}}{{e}^{4}}}+4\,{\frac{\ln \left ( ex+d \right ) B{a}^{3}b{d}^{2}}{{e}^{3}}}-4\,{\frac{\ln \left ( ex+d \right ) A{a}^{3}bd}{{e}^{2}}}-6\,{\frac{\ln \left ( ex+d \right ) B{a}^{2}{b}^{2}{d}^{3}}{{e}^{4}}}+4\,{\frac{\ln \left ( ex+d \right ) Ba{b}^{3}{d}^{4}}{{e}^{5}}}+{\frac{{a}^{4}Bx}{e}}+{\frac{A{b}^{4}{x}^{4}}{4\,e}}+{\frac{{b}^{4}B{x}^{5}}{5\,e}}+{\frac{\ln \left ( ex+d \right ) A{a}^{4}}{e}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/e*B*x^2*a^3*b+1/e*B*x^4*a*b^3+1/2/e^3*A*x^2*b^4*d^2-1/e^4*A*b^4*d^3*x+3/e*A*x^2*a^2*b^2-1/2/e^4*B*x^2*b^4*d^
3+4/e*A*a^3*b*x+4/3/e*A*x^3*a*b^3+1/3/e^3*B*x^3*b^4*d^2+2/e*B*x^3*a^2*b^2+1/e^5*ln(e*x+d)*A*b^4*d^4-1/e^2*ln(e
*x+d)*B*a^4*d-1/e^6*ln(e*x+d)*B*b^4*d^5-1/3/e^2*A*x^3*b^4*d-1/4/e^2*B*x^4*b^4*d+1/e^5*B*b^4*d^4*x-4/e^2*B*a^3*
b*d*x-3/e^2*B*x^2*a^2*b^2*d+2/e^3*B*x^2*a*b^3*d^2-6/e^2*A*a^2*b^2*d*x-4/3/e^2*B*x^3*a*b^3*d+6/e^3*B*a^2*b^2*d^
2*x-4/e^4*B*a*b^3*d^3*x+6/e^3*ln(e*x+d)*A*a^2*b^2*d^2+4/e^3*A*a*b^3*d^2*x-2/e^2*A*x^2*a*b^3*d-4/e^4*ln(e*x+d)*
A*a*b^3*d^3+4/e^3*ln(e*x+d)*B*a^3*b*d^2-4/e^2*ln(e*x+d)*A*a^3*b*d-6/e^4*ln(e*x+d)*B*a^2*b^2*d^3+4/e^5*ln(e*x+d
)*B*a*b^3*d^4+1/e*B*a^4*x+1/4/e*A*x^4*b^4+1/5/e*b^4*B*x^5+1/e*ln(e*x+d)*A*a^4

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Maxima [B]  time = 1.09795, size = 544, normalized size = 3.49 \begin{align*} \frac{12 \, B b^{4} e^{4} x^{5} - 15 \,{\left (B b^{4} d e^{3} -{\left (4 \, B a b^{3} + A b^{4}\right )} e^{4}\right )} x^{4} + 20 \,{\left (B b^{4} d^{2} e^{2} -{\left (4 \, B a b^{3} + A b^{4}\right )} d e^{3} + 2 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{4}\right )} x^{3} - 30 \,{\left (B b^{4} d^{3} e -{\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e^{2} + 2 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{3} - 2 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{4}\right )} x^{2} + 60 \,{\left (B b^{4} d^{4} -{\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e + 2 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{2} - 2 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{3} +{\left (B a^{4} + 4 \, A a^{3} b\right )} e^{4}\right )} x}{60 \, e^{5}} - \frac{{\left (B b^{4} d^{5} - A a^{4} e^{5} -{\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e + 2 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{2} - 2 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e^{3} +{\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{4}\right )} \log \left (e x + d\right )}{e^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/60*(12*B*b^4*e^4*x^5 - 15*(B*b^4*d*e^3 - (4*B*a*b^3 + A*b^4)*e^4)*x^4 + 20*(B*b^4*d^2*e^2 - (4*B*a*b^3 + A*b
^4)*d*e^3 + 2*(3*B*a^2*b^2 + 2*A*a*b^3)*e^4)*x^3 - 30*(B*b^4*d^3*e - (4*B*a*b^3 + A*b^4)*d^2*e^2 + 2*(3*B*a^2*
b^2 + 2*A*a*b^3)*d*e^3 - 2*(2*B*a^3*b + 3*A*a^2*b^2)*e^4)*x^2 + 60*(B*b^4*d^4 - (4*B*a*b^3 + A*b^4)*d^3*e + 2*
(3*B*a^2*b^2 + 2*A*a*b^3)*d^2*e^2 - 2*(2*B*a^3*b + 3*A*a^2*b^2)*d*e^3 + (B*a^4 + 4*A*a^3*b)*e^4)*x)/e^5 - (B*b
^4*d^5 - A*a^4*e^5 - (4*B*a*b^3 + A*b^4)*d^4*e + 2*(3*B*a^2*b^2 + 2*A*a*b^3)*d^3*e^2 - 2*(2*B*a^3*b + 3*A*a^2*
b^2)*d^2*e^3 + (B*a^4 + 4*A*a^3*b)*d*e^4)*log(e*x + d)/e^6

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Fricas [B]  time = 1.54367, size = 830, normalized size = 5.32 \begin{align*} \frac{12 \, B b^{4} e^{5} x^{5} - 15 \,{\left (B b^{4} d e^{4} -{\left (4 \, B a b^{3} + A b^{4}\right )} e^{5}\right )} x^{4} + 20 \,{\left (B b^{4} d^{2} e^{3} -{\left (4 \, B a b^{3} + A b^{4}\right )} d e^{4} + 2 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{5}\right )} x^{3} - 30 \,{\left (B b^{4} d^{3} e^{2} -{\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e^{3} + 2 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{4} - 2 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{5}\right )} x^{2} + 60 \,{\left (B b^{4} d^{4} e -{\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e^{2} + 2 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{3} - 2 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{4} +{\left (B a^{4} + 4 \, A a^{3} b\right )} e^{5}\right )} x - 60 \,{\left (B b^{4} d^{5} - A a^{4} e^{5} -{\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e + 2 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{2} - 2 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e^{3} +{\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{4}\right )} \log \left (e x + d\right )}{60 \, e^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/60*(12*B*b^4*e^5*x^5 - 15*(B*b^4*d*e^4 - (4*B*a*b^3 + A*b^4)*e^5)*x^4 + 20*(B*b^4*d^2*e^3 - (4*B*a*b^3 + A*b
^4)*d*e^4 + 2*(3*B*a^2*b^2 + 2*A*a*b^3)*e^5)*x^3 - 30*(B*b^4*d^3*e^2 - (4*B*a*b^3 + A*b^4)*d^2*e^3 + 2*(3*B*a^
2*b^2 + 2*A*a*b^3)*d*e^4 - 2*(2*B*a^3*b + 3*A*a^2*b^2)*e^5)*x^2 + 60*(B*b^4*d^4*e - (4*B*a*b^3 + A*b^4)*d^3*e^
2 + 2*(3*B*a^2*b^2 + 2*A*a*b^3)*d^2*e^3 - 2*(2*B*a^3*b + 3*A*a^2*b^2)*d*e^4 + (B*a^4 + 4*A*a^3*b)*e^5)*x - 60*
(B*b^4*d^5 - A*a^4*e^5 - (4*B*a*b^3 + A*b^4)*d^4*e + 2*(3*B*a^2*b^2 + 2*A*a*b^3)*d^3*e^2 - 2*(2*B*a^3*b + 3*A*
a^2*b^2)*d^2*e^3 + (B*a^4 + 4*A*a^3*b)*d*e^4)*log(e*x + d))/e^6

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Sympy [B]  time = 1.46927, size = 345, normalized size = 2.21 \begin{align*} \frac{B b^{4} x^{5}}{5 e} + \frac{x^{4} \left (A b^{4} e + 4 B a b^{3} e - B b^{4} d\right )}{4 e^{2}} + \frac{x^{3} \left (4 A a b^{3} e^{2} - A b^{4} d e + 6 B a^{2} b^{2} e^{2} - 4 B a b^{3} d e + B b^{4} d^{2}\right )}{3 e^{3}} + \frac{x^{2} \left (6 A a^{2} b^{2} e^{3} - 4 A a b^{3} d e^{2} + A b^{4} d^{2} e + 4 B a^{3} b e^{3} - 6 B a^{2} b^{2} d e^{2} + 4 B a b^{3} d^{2} e - B b^{4} d^{3}\right )}{2 e^{4}} + \frac{x \left (4 A a^{3} b e^{4} - 6 A a^{2} b^{2} d e^{3} + 4 A a b^{3} d^{2} e^{2} - A b^{4} d^{3} e + B a^{4} e^{4} - 4 B a^{3} b d e^{3} + 6 B a^{2} b^{2} d^{2} e^{2} - 4 B a b^{3} d^{3} e + B b^{4} d^{4}\right )}{e^{5}} - \frac{\left (- A e + B d\right ) \left (a e - b d\right )^{4} \log{\left (d + e x \right )}}{e^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b**2*x**2+2*a*b*x+a**2)**2/(e*x+d),x)

[Out]

B*b**4*x**5/(5*e) + x**4*(A*b**4*e + 4*B*a*b**3*e - B*b**4*d)/(4*e**2) + x**3*(4*A*a*b**3*e**2 - A*b**4*d*e +
6*B*a**2*b**2*e**2 - 4*B*a*b**3*d*e + B*b**4*d**2)/(3*e**3) + x**2*(6*A*a**2*b**2*e**3 - 4*A*a*b**3*d*e**2 + A
*b**4*d**2*e + 4*B*a**3*b*e**3 - 6*B*a**2*b**2*d*e**2 + 4*B*a*b**3*d**2*e - B*b**4*d**3)/(2*e**4) + x*(4*A*a**
3*b*e**4 - 6*A*a**2*b**2*d*e**3 + 4*A*a*b**3*d**2*e**2 - A*b**4*d**3*e + B*a**4*e**4 - 4*B*a**3*b*d*e**3 + 6*B
*a**2*b**2*d**2*e**2 - 4*B*a*b**3*d**3*e + B*b**4*d**4)/e**5 - (-A*e + B*d)*(a*e - b*d)**4*log(d + e*x)/e**6

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Giac [B]  time = 1.14473, size = 597, normalized size = 3.83 \begin{align*} -{\left (B b^{4} d^{5} - 4 \, B a b^{3} d^{4} e - A b^{4} d^{4} e + 6 \, B a^{2} b^{2} d^{3} e^{2} + 4 \, A a b^{3} d^{3} e^{2} - 4 \, B a^{3} b d^{2} e^{3} - 6 \, A a^{2} b^{2} d^{2} e^{3} + B a^{4} d e^{4} + 4 \, A a^{3} b d e^{4} - A a^{4} e^{5}\right )} e^{\left (-6\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{1}{60} \,{\left (12 \, B b^{4} x^{5} e^{4} - 15 \, B b^{4} d x^{4} e^{3} + 20 \, B b^{4} d^{2} x^{3} e^{2} - 30 \, B b^{4} d^{3} x^{2} e + 60 \, B b^{4} d^{4} x + 60 \, B a b^{3} x^{4} e^{4} + 15 \, A b^{4} x^{4} e^{4} - 80 \, B a b^{3} d x^{3} e^{3} - 20 \, A b^{4} d x^{3} e^{3} + 120 \, B a b^{3} d^{2} x^{2} e^{2} + 30 \, A b^{4} d^{2} x^{2} e^{2} - 240 \, B a b^{3} d^{3} x e - 60 \, A b^{4} d^{3} x e + 120 \, B a^{2} b^{2} x^{3} e^{4} + 80 \, A a b^{3} x^{3} e^{4} - 180 \, B a^{2} b^{2} d x^{2} e^{3} - 120 \, A a b^{3} d x^{2} e^{3} + 360 \, B a^{2} b^{2} d^{2} x e^{2} + 240 \, A a b^{3} d^{2} x e^{2} + 120 \, B a^{3} b x^{2} e^{4} + 180 \, A a^{2} b^{2} x^{2} e^{4} - 240 \, B a^{3} b d x e^{3} - 360 \, A a^{2} b^{2} d x e^{3} + 60 \, B a^{4} x e^{4} + 240 \, A a^{3} b x e^{4}\right )} e^{\left (-5\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(B*b^4*d^5 - 4*B*a*b^3*d^4*e - A*b^4*d^4*e + 6*B*a^2*b^2*d^3*e^2 + 4*A*a*b^3*d^3*e^2 - 4*B*a^3*b*d^2*e^3 - 6*
A*a^2*b^2*d^2*e^3 + B*a^4*d*e^4 + 4*A*a^3*b*d*e^4 - A*a^4*e^5)*e^(-6)*log(abs(x*e + d)) + 1/60*(12*B*b^4*x^5*e
^4 - 15*B*b^4*d*x^4*e^3 + 20*B*b^4*d^2*x^3*e^2 - 30*B*b^4*d^3*x^2*e + 60*B*b^4*d^4*x + 60*B*a*b^3*x^4*e^4 + 15
*A*b^4*x^4*e^4 - 80*B*a*b^3*d*x^3*e^3 - 20*A*b^4*d*x^3*e^3 + 120*B*a*b^3*d^2*x^2*e^2 + 30*A*b^4*d^2*x^2*e^2 -
240*B*a*b^3*d^3*x*e - 60*A*b^4*d^3*x*e + 120*B*a^2*b^2*x^3*e^4 + 80*A*a*b^3*x^3*e^4 - 180*B*a^2*b^2*d*x^2*e^3
- 120*A*a*b^3*d*x^2*e^3 + 360*B*a^2*b^2*d^2*x*e^2 + 240*A*a*b^3*d^2*x*e^2 + 120*B*a^3*b*x^2*e^4 + 180*A*a^2*b^
2*x^2*e^4 - 240*B*a^3*b*d*x*e^3 - 360*A*a^2*b^2*d*x*e^3 + 60*B*a^4*x*e^4 + 240*A*a^3*b*x*e^4)*e^(-5)