3.496 \(\int \frac{(a+c x^2)^4}{(d+e x)^2} \, dx\)

Optimal. Leaf size=255 \[ \frac{c^2 x^3 \left (6 a^2 e^4+12 a c d^2 e^2+5 c^2 d^4\right )}{3 e^6}-\frac{c^2 d x^2 \left (6 a^2 e^4+8 a c d^2 e^2+3 c^2 d^4\right )}{e^7}+\frac{c x \left (18 a^2 c d^2 e^4+4 a^3 e^6+20 a c^2 d^4 e^2+7 c^3 d^6\right )}{e^8}+\frac{c^3 x^5 \left (4 a e^2+3 c d^2\right )}{5 e^4}-\frac{c^3 d x^4 \left (2 a e^2+c d^2\right )}{e^5}-\frac{\left (a e^2+c d^2\right )^4}{e^9 (d+e x)}-\frac{8 c d \left (a e^2+c d^2\right )^3 \log (d+e x)}{e^9}-\frac{c^4 d x^6}{3 e^3}+\frac{c^4 x^7}{7 e^2} \]

[Out]

(c*(7*c^3*d^6 + 20*a*c^2*d^4*e^2 + 18*a^2*c*d^2*e^4 + 4*a^3*e^6)*x)/e^8 - (c^2*d*(3*c^2*d^4 + 8*a*c*d^2*e^2 +
6*a^2*e^4)*x^2)/e^7 + (c^2*(5*c^2*d^4 + 12*a*c*d^2*e^2 + 6*a^2*e^4)*x^3)/(3*e^6) - (c^3*d*(c*d^2 + 2*a*e^2)*x^
4)/e^5 + (c^3*(3*c*d^2 + 4*a*e^2)*x^5)/(5*e^4) - (c^4*d*x^6)/(3*e^3) + (c^4*x^7)/(7*e^2) - (c*d^2 + a*e^2)^4/(
e^9*(d + e*x)) - (8*c*d*(c*d^2 + a*e^2)^3*Log[d + e*x])/e^9

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Rubi [A]  time = 0.255321, antiderivative size = 255, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {697} \[ \frac{c^2 x^3 \left (6 a^2 e^4+12 a c d^2 e^2+5 c^2 d^4\right )}{3 e^6}-\frac{c^2 d x^2 \left (6 a^2 e^4+8 a c d^2 e^2+3 c^2 d^4\right )}{e^7}+\frac{c x \left (18 a^2 c d^2 e^4+4 a^3 e^6+20 a c^2 d^4 e^2+7 c^3 d^6\right )}{e^8}+\frac{c^3 x^5 \left (4 a e^2+3 c d^2\right )}{5 e^4}-\frac{c^3 d x^4 \left (2 a e^2+c d^2\right )}{e^5}-\frac{\left (a e^2+c d^2\right )^4}{e^9 (d+e x)}-\frac{8 c d \left (a e^2+c d^2\right )^3 \log (d+e x)}{e^9}-\frac{c^4 d x^6}{3 e^3}+\frac{c^4 x^7}{7 e^2} \]

Antiderivative was successfully verified.

[In]

Int[(a + c*x^2)^4/(d + e*x)^2,x]

[Out]

(c*(7*c^3*d^6 + 20*a*c^2*d^4*e^2 + 18*a^2*c*d^2*e^4 + 4*a^3*e^6)*x)/e^8 - (c^2*d*(3*c^2*d^4 + 8*a*c*d^2*e^2 +
6*a^2*e^4)*x^2)/e^7 + (c^2*(5*c^2*d^4 + 12*a*c*d^2*e^2 + 6*a^2*e^4)*x^3)/(3*e^6) - (c^3*d*(c*d^2 + 2*a*e^2)*x^
4)/e^5 + (c^3*(3*c*d^2 + 4*a*e^2)*x^5)/(5*e^4) - (c^4*d*x^6)/(3*e^3) + (c^4*x^7)/(7*e^2) - (c*d^2 + a*e^2)^4/(
e^9*(d + e*x)) - (8*c*d*(c*d^2 + a*e^2)^3*Log[d + e*x])/e^9

Rule 697

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

Rubi steps

\begin{align*} \int \frac{\left (a+c x^2\right )^4}{(d+e x)^2} \, dx &=\int \left (\frac{c \left (7 c^3 d^6+20 a c^2 d^4 e^2+18 a^2 c d^2 e^4+4 a^3 e^6\right )}{e^8}-\frac{2 c^2 d \left (3 c^2 d^4+8 a c d^2 e^2+6 a^2 e^4\right ) x}{e^7}+\frac{c^2 \left (5 c^2 d^4+12 a c d^2 e^2+6 a^2 e^4\right ) x^2}{e^6}-\frac{4 c^3 d \left (c d^2+2 a e^2\right ) x^3}{e^5}+\frac{c^3 \left (3 c d^2+4 a e^2\right ) x^4}{e^4}-\frac{2 c^4 d x^5}{e^3}+\frac{c^4 x^6}{e^2}+\frac{\left (c d^2+a e^2\right )^4}{e^8 (d+e x)^2}-\frac{8 c d \left (c d^2+a e^2\right )^3}{e^8 (d+e x)}\right ) \, dx\\ &=\frac{c \left (7 c^3 d^6+20 a c^2 d^4 e^2+18 a^2 c d^2 e^4+4 a^3 e^6\right ) x}{e^8}-\frac{c^2 d \left (3 c^2 d^4+8 a c d^2 e^2+6 a^2 e^4\right ) x^2}{e^7}+\frac{c^2 \left (5 c^2 d^4+12 a c d^2 e^2+6 a^2 e^4\right ) x^3}{3 e^6}-\frac{c^3 d \left (c d^2+2 a e^2\right ) x^4}{e^5}+\frac{c^3 \left (3 c d^2+4 a e^2\right ) x^5}{5 e^4}-\frac{c^4 d x^6}{3 e^3}+\frac{c^4 x^7}{7 e^2}-\frac{\left (c d^2+a e^2\right )^4}{e^9 (d+e x)}-\frac{8 c d \left (c d^2+a e^2\right )^3 \log (d+e x)}{e^9}\\ \end{align*}

Mathematica [A]  time = 0.0938536, size = 289, normalized size = 1.13 \[ \frac{210 a^2 c^2 e^4 \left (6 d^2 e^2 x^2+9 d^3 e x-3 d^4-2 d e^3 x^3+e^4 x^4\right )+420 a^3 c e^6 \left (-d^2+d e x+e^2 x^2\right )-105 a^4 e^8+42 a c^3 e^2 \left (30 d^4 e^2 x^2-10 d^3 e^3 x^3+5 d^2 e^4 x^4+50 d^5 e x-10 d^6-3 d e^5 x^5+2 e^6 x^6\right )-840 c d (d+e x) \left (a e^2+c d^2\right )^3 \log (d+e x)+c^4 \left (420 d^6 e^2 x^2-140 d^5 e^3 x^3+70 d^4 e^4 x^4-42 d^3 e^5 x^5+28 d^2 e^6 x^6+735 d^7 e x-105 d^8-20 d e^7 x^7+15 e^8 x^8\right )}{105 e^9 (d+e x)} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + c*x^2)^4/(d + e*x)^2,x]

[Out]

(-105*a^4*e^8 + 420*a^3*c*e^6*(-d^2 + d*e*x + e^2*x^2) + 210*a^2*c^2*e^4*(-3*d^4 + 9*d^3*e*x + 6*d^2*e^2*x^2 -
2*d*e^3*x^3 + e^4*x^4) + 42*a*c^3*e^2*(-10*d^6 + 50*d^5*e*x + 30*d^4*e^2*x^2 - 10*d^3*e^3*x^3 + 5*d^2*e^4*x^4
- 3*d*e^5*x^5 + 2*e^6*x^6) + c^4*(-105*d^8 + 735*d^7*e*x + 420*d^6*e^2*x^2 - 140*d^5*e^3*x^3 + 70*d^4*e^4*x^4
- 42*d^3*e^5*x^5 + 28*d^2*e^6*x^6 - 20*d*e^7*x^7 + 15*e^8*x^8) - 840*c*d*(c*d^2 + a*e^2)^3*(d + e*x)*Log[d +
e*x])/(105*e^9*(d + e*x))

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Maple [A]  time = 0.052, size = 378, normalized size = 1.5 \begin{align*}{\frac{{c}^{4}{x}^{7}}{7\,{e}^{2}}}-4\,{\frac{a{c}^{3}{d}^{6}}{{e}^{7} \left ( ex+d \right ) }}-6\,{\frac{{c}^{2}{x}^{2}{a}^{2}d}{{e}^{3}}}-8\,{\frac{{c}^{3}{x}^{2}a{d}^{3}}{{e}^{5}}}+18\,{\frac{{a}^{2}{c}^{2}{d}^{2}x}{{e}^{4}}}+20\,{\frac{{d}^{4}a{c}^{3}x}{{e}^{6}}}-8\,{\frac{cd\ln \left ( ex+d \right ){a}^{3}}{{e}^{3}}}-24\,{\frac{{c}^{2}{d}^{3}\ln \left ( ex+d \right ){a}^{2}}{{e}^{5}}}-24\,{\frac{{c}^{3}{d}^{5}\ln \left ( ex+d \right ) a}{{e}^{7}}}-4\,{\frac{{a}^{3}c{d}^{2}}{{e}^{3} \left ( ex+d \right ) }}-6\,{\frac{{a}^{2}{c}^{2}{d}^{4}}{{e}^{5} \left ( ex+d \right ) }}-{\frac{{a}^{4}}{e \left ( ex+d \right ) }}-2\,{\frac{{c}^{3}{x}^{4}ad}{{e}^{3}}}+4\,{\frac{{x}^{3}{c}^{3}a{d}^{2}}{{e}^{4}}}-{\frac{{c}^{4}{x}^{4}{d}^{3}}{{e}^{5}}}+2\,{\frac{{c}^{2}{x}^{3}{a}^{2}}{{e}^{2}}}+{\frac{5\,{c}^{4}{x}^{3}{d}^{4}}{3\,{e}^{6}}}-3\,{\frac{{c}^{4}{x}^{2}{d}^{5}}{{e}^{7}}}-8\,{\frac{{c}^{4}{d}^{7}\ln \left ( ex+d \right ) }{{e}^{9}}}-{\frac{{c}^{4}{d}^{8}}{{e}^{9} \left ( ex+d \right ) }}+4\,{\frac{{a}^{3}cx}{{e}^{2}}}+7\,{\frac{{c}^{4}{d}^{6}x}{{e}^{8}}}+{\frac{4\,{c}^{3}{x}^{5}a}{5\,{e}^{2}}}+{\frac{3\,{c}^{4}{x}^{5}{d}^{2}}{5\,{e}^{4}}}-{\frac{{c}^{4}d{x}^{6}}{3\,{e}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+a)^4/(e*x+d)^2,x)

[Out]

1/7*c^4*x^7/e^2-4/e^7/(e*x+d)*a*c^3*d^6-6*c^2/e^3*x^2*a^2*d-8*c^3/e^5*x^2*a*d^3+18*c^2/e^4*a^2*d^2*x+20*c^3/e^
6*d^4*a*x-8*c*d/e^3*ln(e*x+d)*a^3-24*c^2*d^3/e^5*ln(e*x+d)*a^2-24*c^3*d^5/e^7*ln(e*x+d)*a-4/e^3/(e*x+d)*a^3*c*
d^2-6/e^5/(e*x+d)*a^2*c^2*d^4-1/e/(e*x+d)*a^4-2*c^3/e^3*x^4*a*d+4*c^3/e^4*x^3*a*d^2-c^4/e^5*x^4*d^3+2*c^2/e^2*
x^3*a^2+5/3*c^4/e^6*x^3*d^4-3*c^4/e^7*x^2*d^5-8*c^4*d^7/e^9*ln(e*x+d)-1/e^9/(e*x+d)*c^4*d^8+4*c/e^2*a^3*x+7*c^
4/e^8*d^6*x+4/5*c^3/e^2*x^5*a+3/5*c^4/e^4*x^5*d^2-1/3*c^4*d*x^6/e^3

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Maxima [A]  time = 1.14774, size = 446, normalized size = 1.75 \begin{align*} -\frac{c^{4} d^{8} + 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} + 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}}{e^{10} x + d e^{9}} + \frac{15 \, c^{4} e^{6} x^{7} - 35 \, c^{4} d e^{5} x^{6} + 21 \,{\left (3 \, c^{4} d^{2} e^{4} + 4 \, a c^{3} e^{6}\right )} x^{5} - 105 \,{\left (c^{4} d^{3} e^{3} + 2 \, a c^{3} d e^{5}\right )} x^{4} + 35 \,{\left (5 \, c^{4} d^{4} e^{2} + 12 \, a c^{3} d^{2} e^{4} + 6 \, a^{2} c^{2} e^{6}\right )} x^{3} - 105 \,{\left (3 \, c^{4} d^{5} e + 8 \, a c^{3} d^{3} e^{3} + 6 \, a^{2} c^{2} d e^{5}\right )} x^{2} + 105 \,{\left (7 \, c^{4} d^{6} + 20 \, a c^{3} d^{4} e^{2} + 18 \, a^{2} c^{2} d^{2} e^{4} + 4 \, a^{3} c e^{6}\right )} x}{105 \, e^{8}} - \frac{8 \,{\left (c^{4} d^{7} + 3 \, a c^{3} d^{5} e^{2} + 3 \, a^{2} c^{2} d^{3} e^{4} + a^{3} c d e^{6}\right )} \log \left (e x + d\right )}{e^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)^4/(e*x+d)^2,x, algorithm="maxima")

[Out]

-(c^4*d^8 + 4*a*c^3*d^6*e^2 + 6*a^2*c^2*d^4*e^4 + 4*a^3*c*d^2*e^6 + a^4*e^8)/(e^10*x + d*e^9) + 1/105*(15*c^4*
e^6*x^7 - 35*c^4*d*e^5*x^6 + 21*(3*c^4*d^2*e^4 + 4*a*c^3*e^6)*x^5 - 105*(c^4*d^3*e^3 + 2*a*c^3*d*e^5)*x^4 + 35
*(5*c^4*d^4*e^2 + 12*a*c^3*d^2*e^4 + 6*a^2*c^2*e^6)*x^3 - 105*(3*c^4*d^5*e + 8*a*c^3*d^3*e^3 + 6*a^2*c^2*d*e^5
)*x^2 + 105*(7*c^4*d^6 + 20*a*c^3*d^4*e^2 + 18*a^2*c^2*d^2*e^4 + 4*a^3*c*e^6)*x)/e^8 - 8*(c^4*d^7 + 3*a*c^3*d^
5*e^2 + 3*a^2*c^2*d^3*e^4 + a^3*c*d*e^6)*log(e*x + d)/e^9

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Fricas [A]  time = 1.86017, size = 872, normalized size = 3.42 \begin{align*} \frac{15 \, c^{4} e^{8} x^{8} - 20 \, c^{4} d e^{7} x^{7} - 105 \, c^{4} d^{8} - 420 \, a c^{3} d^{6} e^{2} - 630 \, a^{2} c^{2} d^{4} e^{4} - 420 \, a^{3} c d^{2} e^{6} - 105 \, a^{4} e^{8} + 28 \,{\left (c^{4} d^{2} e^{6} + 3 \, a c^{3} e^{8}\right )} x^{6} - 42 \,{\left (c^{4} d^{3} e^{5} + 3 \, a c^{3} d e^{7}\right )} x^{5} + 70 \,{\left (c^{4} d^{4} e^{4} + 3 \, a c^{3} d^{2} e^{6} + 3 \, a^{2} c^{2} e^{8}\right )} x^{4} - 140 \,{\left (c^{4} d^{5} e^{3} + 3 \, a c^{3} d^{3} e^{5} + 3 \, a^{2} c^{2} d e^{7}\right )} x^{3} + 420 \,{\left (c^{4} d^{6} e^{2} + 3 \, a c^{3} d^{4} e^{4} + 3 \, a^{2} c^{2} d^{2} e^{6} + a^{3} c e^{8}\right )} x^{2} + 105 \,{\left (7 \, c^{4} d^{7} e + 20 \, a c^{3} d^{5} e^{3} + 18 \, a^{2} c^{2} d^{3} e^{5} + 4 \, a^{3} c d e^{7}\right )} x - 840 \,{\left (c^{4} d^{8} + 3 \, a c^{3} d^{6} e^{2} + 3 \, a^{2} c^{2} d^{4} e^{4} + a^{3} c d^{2} e^{6} +{\left (c^{4} d^{7} e + 3 \, a c^{3} d^{5} e^{3} + 3 \, a^{2} c^{2} d^{3} e^{5} + a^{3} c d e^{7}\right )} x\right )} \log \left (e x + d\right )}{105 \,{\left (e^{10} x + d e^{9}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)^4/(e*x+d)^2,x, algorithm="fricas")

[Out]

1/105*(15*c^4*e^8*x^8 - 20*c^4*d*e^7*x^7 - 105*c^4*d^8 - 420*a*c^3*d^6*e^2 - 630*a^2*c^2*d^4*e^4 - 420*a^3*c*d
^2*e^6 - 105*a^4*e^8 + 28*(c^4*d^2*e^6 + 3*a*c^3*e^8)*x^6 - 42*(c^4*d^3*e^5 + 3*a*c^3*d*e^7)*x^5 + 70*(c^4*d^4
*e^4 + 3*a*c^3*d^2*e^6 + 3*a^2*c^2*e^8)*x^4 - 140*(c^4*d^5*e^3 + 3*a*c^3*d^3*e^5 + 3*a^2*c^2*d*e^7)*x^3 + 420*
(c^4*d^6*e^2 + 3*a*c^3*d^4*e^4 + 3*a^2*c^2*d^2*e^6 + a^3*c*e^8)*x^2 + 105*(7*c^4*d^7*e + 20*a*c^3*d^5*e^3 + 18
*a^2*c^2*d^3*e^5 + 4*a^3*c*d*e^7)*x - 840*(c^4*d^8 + 3*a*c^3*d^6*e^2 + 3*a^2*c^2*d^4*e^4 + a^3*c*d^2*e^6 + (c^
4*d^7*e + 3*a*c^3*d^5*e^3 + 3*a^2*c^2*d^3*e^5 + a^3*c*d*e^7)*x)*log(e*x + d))/(e^10*x + d*e^9)

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Sympy [A]  time = 1.39615, size = 308, normalized size = 1.21 \begin{align*} - \frac{c^{4} d x^{6}}{3 e^{3}} + \frac{c^{4} x^{7}}{7 e^{2}} - \frac{8 c d \left (a e^{2} + c d^{2}\right )^{3} \log{\left (d + e x \right )}}{e^{9}} - \frac{a^{4} e^{8} + 4 a^{3} c d^{2} e^{6} + 6 a^{2} c^{2} d^{4} e^{4} + 4 a c^{3} d^{6} e^{2} + c^{4} d^{8}}{d e^{9} + e^{10} x} + \frac{x^{5} \left (4 a c^{3} e^{2} + 3 c^{4} d^{2}\right )}{5 e^{4}} - \frac{x^{4} \left (2 a c^{3} d e^{2} + c^{4} d^{3}\right )}{e^{5}} + \frac{x^{3} \left (6 a^{2} c^{2} e^{4} + 12 a c^{3} d^{2} e^{2} + 5 c^{4} d^{4}\right )}{3 e^{6}} - \frac{x^{2} \left (6 a^{2} c^{2} d e^{4} + 8 a c^{3} d^{3} e^{2} + 3 c^{4} d^{5}\right )}{e^{7}} + \frac{x \left (4 a^{3} c e^{6} + 18 a^{2} c^{2} d^{2} e^{4} + 20 a c^{3} d^{4} e^{2} + 7 c^{4} d^{6}\right )}{e^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+a)**4/(e*x+d)**2,x)

[Out]

-c**4*d*x**6/(3*e**3) + c**4*x**7/(7*e**2) - 8*c*d*(a*e**2 + c*d**2)**3*log(d + e*x)/e**9 - (a**4*e**8 + 4*a**
3*c*d**2*e**6 + 6*a**2*c**2*d**4*e**4 + 4*a*c**3*d**6*e**2 + c**4*d**8)/(d*e**9 + e**10*x) + x**5*(4*a*c**3*e*
*2 + 3*c**4*d**2)/(5*e**4) - x**4*(2*a*c**3*d*e**2 + c**4*d**3)/e**5 + x**3*(6*a**2*c**2*e**4 + 12*a*c**3*d**2
*e**2 + 5*c**4*d**4)/(3*e**6) - x**2*(6*a**2*c**2*d*e**4 + 8*a*c**3*d**3*e**2 + 3*c**4*d**5)/e**7 + x*(4*a**3*
c*e**6 + 18*a**2*c**2*d**2*e**4 + 20*a*c**3*d**4*e**2 + 7*c**4*d**6)/e**8

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Giac [A]  time = 1.29118, size = 535, normalized size = 2.1 \begin{align*} \frac{1}{105} \,{\left (15 \, c^{4} - \frac{140 \, c^{4} d}{x e + d} + \frac{84 \,{\left (7 \, c^{4} d^{2} e^{2} + a c^{3} e^{4}\right )} e^{\left (-2\right )}}{{\left (x e + d\right )}^{2}} - \frac{210 \,{\left (7 \, c^{4} d^{3} e^{3} + 3 \, a c^{3} d e^{5}\right )} e^{\left (-3\right )}}{{\left (x e + d\right )}^{3}} + \frac{70 \,{\left (35 \, c^{4} d^{4} e^{4} + 30 \, a c^{3} d^{2} e^{6} + 3 \, a^{2} c^{2} e^{8}\right )} e^{\left (-4\right )}}{{\left (x e + d\right )}^{4}} - \frac{420 \,{\left (7 \, c^{4} d^{5} e^{5} + 10 \, a c^{3} d^{3} e^{7} + 3 \, a^{2} c^{2} d e^{9}\right )} e^{\left (-5\right )}}{{\left (x e + d\right )}^{5}} + \frac{420 \,{\left (7 \, c^{4} d^{6} e^{6} + 15 \, a c^{3} d^{4} e^{8} + 9 \, a^{2} c^{2} d^{2} e^{10} + a^{3} c e^{12}\right )} e^{\left (-6\right )}}{{\left (x e + d\right )}^{6}}\right )}{\left (x e + d\right )}^{7} e^{\left (-9\right )} + 8 \,{\left (c^{4} d^{7} + 3 \, a c^{3} d^{5} e^{2} + 3 \, a^{2} c^{2} d^{3} e^{4} + a^{3} c d e^{6}\right )} e^{\left (-9\right )} \log \left (\frac{{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) -{\left (\frac{c^{4} d^{8} e^{7}}{x e + d} + \frac{4 \, a c^{3} d^{6} e^{9}}{x e + d} + \frac{6 \, a^{2} c^{2} d^{4} e^{11}}{x e + d} + \frac{4 \, a^{3} c d^{2} e^{13}}{x e + d} + \frac{a^{4} e^{15}}{x e + d}\right )} e^{\left (-16\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)^4/(e*x+d)^2,x, algorithm="giac")

[Out]

1/105*(15*c^4 - 140*c^4*d/(x*e + d) + 84*(7*c^4*d^2*e^2 + a*c^3*e^4)*e^(-2)/(x*e + d)^2 - 210*(7*c^4*d^3*e^3 +
3*a*c^3*d*e^5)*e^(-3)/(x*e + d)^3 + 70*(35*c^4*d^4*e^4 + 30*a*c^3*d^2*e^6 + 3*a^2*c^2*e^8)*e^(-4)/(x*e + d)^4
- 420*(7*c^4*d^5*e^5 + 10*a*c^3*d^3*e^7 + 3*a^2*c^2*d*e^9)*e^(-5)/(x*e + d)^5 + 420*(7*c^4*d^6*e^6 + 15*a*c^3
*d^4*e^8 + 9*a^2*c^2*d^2*e^10 + a^3*c*e^12)*e^(-6)/(x*e + d)^6)*(x*e + d)^7*e^(-9) + 8*(c^4*d^7 + 3*a*c^3*d^5*
e^2 + 3*a^2*c^2*d^3*e^4 + a^3*c*d*e^6)*e^(-9)*log(abs(x*e + d)*e^(-1)/(x*e + d)^2) - (c^4*d^8*e^7/(x*e + d) +
4*a*c^3*d^6*e^9/(x*e + d) + 6*a^2*c^2*d^4*e^11/(x*e + d) + 4*a^3*c*d^2*e^13/(x*e + d) + a^4*e^15/(x*e + d))*e^
(-16)