3.8 \(\int \frac{a+b \tan ^{-1}(c x)}{(d+e x)^4} \, dx\)

Optimal. Leaf size=206 \[ -\frac{a+b \tan ^{-1}(c x)}{3 e (d+e x)^3}-\frac{b c^3 \left (3 c^2 d^2-e^2\right ) \log \left (c^2 x^2+1\right )}{6 \left (c^2 d^2+e^2\right )^3}-\frac{2 b c^3 d}{3 \left (c^2 d^2+e^2\right )^2 (d+e x)}-\frac{b c}{6 \left (c^2 d^2+e^2\right ) (d+e x)^2}+\frac{b c^3 \left (3 c^2 d^2-e^2\right ) \log (d+e x)}{3 \left (c^2 d^2+e^2\right )^3}+\frac{b c^4 d \left (c^2 d^2-3 e^2\right ) \tan ^{-1}(c x)}{3 e \left (c^2 d^2+e^2\right )^3} \]

[Out]

-(b*c)/(6*(c^2*d^2 + e^2)*(d + e*x)^2) - (2*b*c^3*d)/(3*(c^2*d^2 + e^2)^2*(d + e*x)) + (b*c^4*d*(c^2*d^2 - 3*e
^2)*ArcTan[c*x])/(3*e*(c^2*d^2 + e^2)^3) - (a + b*ArcTan[c*x])/(3*e*(d + e*x)^3) + (b*c^3*(3*c^2*d^2 - e^2)*Lo
g[d + e*x])/(3*(c^2*d^2 + e^2)^3) - (b*c^3*(3*c^2*d^2 - e^2)*Log[1 + c^2*x^2])/(6*(c^2*d^2 + e^2)^3)

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Rubi [A]  time = 0.182223, antiderivative size = 206, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {4862, 710, 801, 635, 203, 260} \[ -\frac{a+b \tan ^{-1}(c x)}{3 e (d+e x)^3}-\frac{b c^3 \left (3 c^2 d^2-e^2\right ) \log \left (c^2 x^2+1\right )}{6 \left (c^2 d^2+e^2\right )^3}-\frac{2 b c^3 d}{3 \left (c^2 d^2+e^2\right )^2 (d+e x)}-\frac{b c}{6 \left (c^2 d^2+e^2\right ) (d+e x)^2}+\frac{b c^3 \left (3 c^2 d^2-e^2\right ) \log (d+e x)}{3 \left (c^2 d^2+e^2\right )^3}+\frac{b c^4 d \left (c^2 d^2-3 e^2\right ) \tan ^{-1}(c x)}{3 e \left (c^2 d^2+e^2\right )^3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcTan[c*x])/(d + e*x)^4,x]

[Out]

-(b*c)/(6*(c^2*d^2 + e^2)*(d + e*x)^2) - (2*b*c^3*d)/(3*(c^2*d^2 + e^2)^2*(d + e*x)) + (b*c^4*d*(c^2*d^2 - 3*e
^2)*ArcTan[c*x])/(3*e*(c^2*d^2 + e^2)^3) - (a + b*ArcTan[c*x])/(3*e*(d + e*x)^3) + (b*c^3*(3*c^2*d^2 - e^2)*Lo
g[d + e*x])/(3*(c^2*d^2 + e^2)^3) - (b*c^3*(3*c^2*d^2 - e^2)*Log[1 + c^2*x^2])/(6*(c^2*d^2 + e^2)^3)

Rule 4862

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Simp[((d + e*x)^(q + 1)*(a + b*
ArcTan[c*x]))/(e*(q + 1)), x] - Dist[(b*c)/(e*(q + 1)), Int[(d + e*x)^(q + 1)/(1 + c^2*x^2), x], x] /; FreeQ[{
a, b, c, d, e, q}, x] && NeQ[q, -1]

Rule 710

Int[((d_) + (e_.)*(x_))^(m_)/((a_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(e*(d + e*x)^(m + 1))/((m + 1)*(c*d^2 +
a*e^2)), x] + Dist[c/(c*d^2 + a*e^2), Int[((d + e*x)^(m + 1)*(d - e*x))/(a + c*x^2), x], x] /; FreeQ[{a, c, d,
e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[m, -1]

Rule 801

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

Rule 635

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Dist[d, Int[1/(a + c*x^2), x], x] + Dist[e, Int[x/
(a + c*x^2), x], x] /; FreeQ[{a, c, d, e}, x] &&  !NiceSqrtQ[-(a*c)]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rubi steps

\begin{align*} \int \frac{a+b \tan ^{-1}(c x)}{(d+e x)^4} \, dx &=-\frac{a+b \tan ^{-1}(c x)}{3 e (d+e x)^3}+\frac{(b c) \int \frac{1}{(d+e x)^3 \left (1+c^2 x^2\right )} \, dx}{3 e}\\ &=-\frac{b c}{6 \left (c^2 d^2+e^2\right ) (d+e x)^2}-\frac{a+b \tan ^{-1}(c x)}{3 e (d+e x)^3}+\frac{\left (b c^3\right ) \int \frac{d-e x}{(d+e x)^2 \left (1+c^2 x^2\right )} \, dx}{3 e \left (c^2 d^2+e^2\right )}\\ &=-\frac{b c}{6 \left (c^2 d^2+e^2\right ) (d+e x)^2}-\frac{a+b \tan ^{-1}(c x)}{3 e (d+e x)^3}+\frac{\left (b c^3\right ) \int \left (\frac{2 d e^2}{\left (c^2 d^2+e^2\right ) (d+e x)^2}-\frac{e^2 \left (-3 c^2 d^2+e^2\right )}{\left (c^2 d^2+e^2\right )^2 (d+e x)}+\frac{c^2 d \left (c^2 d^2-3 e^2\right )-c^2 e \left (3 c^2 d^2-e^2\right ) x}{\left (c^2 d^2+e^2\right )^2 \left (1+c^2 x^2\right )}\right ) \, dx}{3 e \left (c^2 d^2+e^2\right )}\\ &=-\frac{b c}{6 \left (c^2 d^2+e^2\right ) (d+e x)^2}-\frac{2 b c^3 d}{3 \left (c^2 d^2+e^2\right )^2 (d+e x)}-\frac{a+b \tan ^{-1}(c x)}{3 e (d+e x)^3}+\frac{b c^3 \left (3 c^2 d^2-e^2\right ) \log (d+e x)}{3 \left (c^2 d^2+e^2\right )^3}+\frac{\left (b c^3\right ) \int \frac{c^2 d \left (c^2 d^2-3 e^2\right )-c^2 e \left (3 c^2 d^2-e^2\right ) x}{1+c^2 x^2} \, dx}{3 e \left (c^2 d^2+e^2\right )^3}\\ &=-\frac{b c}{6 \left (c^2 d^2+e^2\right ) (d+e x)^2}-\frac{2 b c^3 d}{3 \left (c^2 d^2+e^2\right )^2 (d+e x)}-\frac{a+b \tan ^{-1}(c x)}{3 e (d+e x)^3}+\frac{b c^3 \left (3 c^2 d^2-e^2\right ) \log (d+e x)}{3 \left (c^2 d^2+e^2\right )^3}+\frac{\left (b c^5 d \left (c^2 d^2-3 e^2\right )\right ) \int \frac{1}{1+c^2 x^2} \, dx}{3 e \left (c^2 d^2+e^2\right )^3}-\frac{\left (b c^5 \left (3 c^2 d^2-e^2\right )\right ) \int \frac{x}{1+c^2 x^2} \, dx}{3 \left (c^2 d^2+e^2\right )^3}\\ &=-\frac{b c}{6 \left (c^2 d^2+e^2\right ) (d+e x)^2}-\frac{2 b c^3 d}{3 \left (c^2 d^2+e^2\right )^2 (d+e x)}+\frac{b c^4 d \left (c^2 d^2-3 e^2\right ) \tan ^{-1}(c x)}{3 e \left (c^2 d^2+e^2\right )^3}-\frac{a+b \tan ^{-1}(c x)}{3 e (d+e x)^3}+\frac{b c^3 \left (3 c^2 d^2-e^2\right ) \log (d+e x)}{3 \left (c^2 d^2+e^2\right )^3}-\frac{b c^3 \left (3 c^2 d^2-e^2\right ) \log \left (1+c^2 x^2\right )}{6 \left (c^2 d^2+e^2\right )^3}\\ \end{align*}

Mathematica [A]  time = 0.640897, size = 254, normalized size = 1.23 \[ -\frac{2 \left (a+b \tan ^{-1}(c x)\right )+\frac{b c (d+e x) \left (4 c^2 d e \left (c^2 d^2+e^2\right ) (d+e x)-c^2 \left (c^2 d^2 \left (\sqrt{-c^2} d-3 e\right )+e^2 \left (e-3 \sqrt{-c^2} d\right )\right ) \log \left (1-\sqrt{-c^2} x\right ) (d+e x)^2-c^2 \left (e^2 \left (3 \sqrt{-c^2} d+e\right )-c^2 d^2 \left (\sqrt{-c^2} d+3 e\right )\right ) \log \left (\sqrt{-c^2} x+1\right ) (d+e x)^2-2 c^2 e \left (3 c^2 d^2-e^2\right ) (d+e x)^2 \log (d+e x)+e \left (c^2 d^2+e^2\right )^2\right )}{\left (c^2 d^2+e^2\right )^3}}{6 e (d+e x)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(2*(a + b*ArcTan[c*x]) + (b*c*(d + e*x)*(e*(c^2*d^2 + e^2)^2 + 4*c^2*d*e*(c^2*d^2 + e^2)*(d + e*x) - c^2*(c^2
*d^2*(Sqrt[-c^2]*d - 3*e) + e^2*(-3*Sqrt[-c^2]*d + e))*(d + e*x)^2*Log[1 - Sqrt[-c^2]*x] - c^2*(e^2*(3*Sqrt[-c
^2]*d + e) - c^2*d^2*(Sqrt[-c^2]*d + 3*e))*(d + e*x)^2*Log[1 + Sqrt[-c^2]*x] - 2*c^2*e*(3*c^2*d^2 - e^2)*(d +
e*x)^2*Log[d + e*x]))/(c^2*d^2 + e^2)^3)/(6*e*(d + e*x)^3)

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Maple [A]  time = 0.036, size = 282, normalized size = 1.4 \begin{align*} -{\frac{a{c}^{3}}{3\, \left ( ecx+dc \right ) ^{3}e}}-{\frac{{c}^{3}b\arctan \left ( cx \right ) }{3\, \left ( ecx+dc \right ) ^{3}e}}-{\frac{{c}^{3}b}{ \left ( 6\,{c}^{2}{d}^{2}+6\,{e}^{2} \right ) \left ( ecx+dc \right ) ^{2}}}+{\frac{{c}^{5}b\ln \left ( ecx+dc \right ){d}^{2}}{ \left ({c}^{2}{d}^{2}+{e}^{2} \right ) ^{3}}}-{\frac{{c}^{3}b{e}^{2}\ln \left ( ecx+dc \right ) }{3\, \left ({c}^{2}{d}^{2}+{e}^{2} \right ) ^{3}}}-{\frac{2\,b{c}^{4}d}{3\, \left ({c}^{2}{d}^{2}+{e}^{2} \right ) ^{2} \left ( ecx+dc \right ) }}-{\frac{b{c}^{5}\ln \left ({c}^{2}{x}^{2}+1 \right ){d}^{2}}{2\, \left ({c}^{2}{d}^{2}+{e}^{2} \right ) ^{3}}}+{\frac{{c}^{3}b{e}^{2}\ln \left ({c}^{2}{x}^{2}+1 \right ) }{6\, \left ({c}^{2}{d}^{2}+{e}^{2} \right ) ^{3}}}+{\frac{{c}^{6}b\arctan \left ( cx \right ){d}^{3}}{3\, \left ({c}^{2}{d}^{2}+{e}^{2} \right ) ^{3}e}}-{\frac{b{c}^{4}e\arctan \left ( cx \right ) d}{ \left ({c}^{2}{d}^{2}+{e}^{2} \right ) ^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.55636, size = 505, normalized size = 2.45 \begin{align*} -\frac{1}{6} \,{\left (c{\left (\frac{{\left (3 \, c^{4} d^{2} - c^{2} e^{2}\right )} \log \left (c^{2} x^{2} + 1\right )}{c^{6} d^{6} + 3 \, c^{4} d^{4} e^{2} + 3 \, c^{2} d^{2} e^{4} + e^{6}} - \frac{2 \,{\left (3 \, c^{4} d^{2} - c^{2} e^{2}\right )} \log \left (e x + d\right )}{c^{6} d^{6} + 3 \, c^{4} d^{4} e^{2} + 3 \, c^{2} d^{2} e^{4} + e^{6}} + \frac{4 \, c^{2} d e x + 5 \, c^{2} d^{2} + e^{2}}{c^{4} d^{6} + 2 \, c^{2} d^{4} e^{2} + d^{2} e^{4} +{\left (c^{4} d^{4} e^{2} + 2 \, c^{2} d^{2} e^{4} + e^{6}\right )} x^{2} + 2 \,{\left (c^{4} d^{5} e + 2 \, c^{2} d^{3} e^{3} + d e^{5}\right )} x} - \frac{2 \,{\left (c^{6} d^{3} - 3 \, c^{4} d e^{2}\right )} \arctan \left (c x\right )}{{\left (c^{6} d^{6} e + 3 \, c^{4} d^{4} e^{3} + 3 \, c^{2} d^{2} e^{5} + e^{7}\right )} c}\right )} + \frac{2 \, \arctan \left (c x\right )}{e^{4} x^{3} + 3 \, d e^{3} x^{2} + 3 \, d^{2} e^{2} x + d^{3} e}\right )} b - \frac{a}{3 \,{\left (e^{4} x^{3} + 3 \, d e^{3} x^{2} + 3 \, d^{2} e^{2} x + d^{3} e\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(c*((3*c^4*d^2 - c^2*e^2)*log(c^2*x^2 + 1)/(c^6*d^6 + 3*c^4*d^4*e^2 + 3*c^2*d^2*e^4 + e^6) - 2*(3*c^4*d^2
- c^2*e^2)*log(e*x + d)/(c^6*d^6 + 3*c^4*d^4*e^2 + 3*c^2*d^2*e^4 + e^6) + (4*c^2*d*e*x + 5*c^2*d^2 + e^2)/(c^
4*d^6 + 2*c^2*d^4*e^2 + d^2*e^4 + (c^4*d^4*e^2 + 2*c^2*d^2*e^4 + e^6)*x^2 + 2*(c^4*d^5*e + 2*c^2*d^3*e^3 + d*e
^5)*x) - 2*(c^6*d^3 - 3*c^4*d*e^2)*arctan(c*x)/((c^6*d^6*e + 3*c^4*d^4*e^3 + 3*c^2*d^2*e^5 + e^7)*c)) + 2*arct
an(c*x)/(e^4*x^3 + 3*d*e^3*x^2 + 3*d^2*e^2*x + d^3*e))*b - 1/3*a/(e^4*x^3 + 3*d*e^3*x^2 + 3*d^2*e^2*x + d^3*e)

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Fricas [B]  time = 8.10537, size = 1280, normalized size = 6.21 \begin{align*} -\frac{2 \, a c^{6} d^{6} + 5 \, b c^{5} d^{5} e + 6 \, a c^{4} d^{4} e^{2} + 6 \, b c^{3} d^{3} e^{3} + 6 \, a c^{2} d^{2} e^{4} + b c d e^{5} + 2 \, a e^{6} + 4 \,{\left (b c^{5} d^{3} e^{3} + b c^{3} d e^{5}\right )} x^{2} +{\left (9 \, b c^{5} d^{4} e^{2} + 10 \, b c^{3} d^{2} e^{4} + b c e^{6}\right )} x + 2 \,{\left (6 \, b c^{4} d^{4} e^{2} + 3 \, b c^{2} d^{2} e^{4} + b e^{6} -{\left (b c^{6} d^{3} e^{3} - 3 \, b c^{4} d e^{5}\right )} x^{3} - 3 \,{\left (b c^{6} d^{4} e^{2} - 3 \, b c^{4} d^{2} e^{4}\right )} x^{2} - 3 \,{\left (b c^{6} d^{5} e - 3 \, b c^{4} d^{3} e^{3}\right )} x\right )} \arctan \left (c x\right ) +{\left (3 \, b c^{5} d^{5} e - b c^{3} d^{3} e^{3} +{\left (3 \, b c^{5} d^{2} e^{4} - b c^{3} e^{6}\right )} x^{3} + 3 \,{\left (3 \, b c^{5} d^{3} e^{3} - b c^{3} d e^{5}\right )} x^{2} + 3 \,{\left (3 \, b c^{5} d^{4} e^{2} - b c^{3} d^{2} e^{4}\right )} x\right )} \log \left (c^{2} x^{2} + 1\right ) - 2 \,{\left (3 \, b c^{5} d^{5} e - b c^{3} d^{3} e^{3} +{\left (3 \, b c^{5} d^{2} e^{4} - b c^{3} e^{6}\right )} x^{3} + 3 \,{\left (3 \, b c^{5} d^{3} e^{3} - b c^{3} d e^{5}\right )} x^{2} + 3 \,{\left (3 \, b c^{5} d^{4} e^{2} - b c^{3} d^{2} e^{4}\right )} x\right )} \log \left (e x + d\right )}{6 \,{\left (c^{6} d^{9} e + 3 \, c^{4} d^{7} e^{3} + 3 \, c^{2} d^{5} e^{5} + d^{3} e^{7} +{\left (c^{6} d^{6} e^{4} + 3 \, c^{4} d^{4} e^{6} + 3 \, c^{2} d^{2} e^{8} + e^{10}\right )} x^{3} + 3 \,{\left (c^{6} d^{7} e^{3} + 3 \, c^{4} d^{5} e^{5} + 3 \, c^{2} d^{3} e^{7} + d e^{9}\right )} x^{2} + 3 \,{\left (c^{6} d^{8} e^{2} + 3 \, c^{4} d^{6} e^{4} + 3 \, c^{2} d^{4} e^{6} + d^{2} e^{8}\right )} x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(2*a*c^6*d^6 + 5*b*c^5*d^5*e + 6*a*c^4*d^4*e^2 + 6*b*c^3*d^3*e^3 + 6*a*c^2*d^2*e^4 + b*c*d*e^5 + 2*a*e^6
+ 4*(b*c^5*d^3*e^3 + b*c^3*d*e^5)*x^2 + (9*b*c^5*d^4*e^2 + 10*b*c^3*d^2*e^4 + b*c*e^6)*x + 2*(6*b*c^4*d^4*e^2
+ 3*b*c^2*d^2*e^4 + b*e^6 - (b*c^6*d^3*e^3 - 3*b*c^4*d*e^5)*x^3 - 3*(b*c^6*d^4*e^2 - 3*b*c^4*d^2*e^4)*x^2 - 3*
(b*c^6*d^5*e - 3*b*c^4*d^3*e^3)*x)*arctan(c*x) + (3*b*c^5*d^5*e - b*c^3*d^3*e^3 + (3*b*c^5*d^2*e^4 - b*c^3*e^6
)*x^3 + 3*(3*b*c^5*d^3*e^3 - b*c^3*d*e^5)*x^2 + 3*(3*b*c^5*d^4*e^2 - b*c^3*d^2*e^4)*x)*log(c^2*x^2 + 1) - 2*(3
*b*c^5*d^5*e - b*c^3*d^3*e^3 + (3*b*c^5*d^2*e^4 - b*c^3*e^6)*x^3 + 3*(3*b*c^5*d^3*e^3 - b*c^3*d*e^5)*x^2 + 3*(
3*b*c^5*d^4*e^2 - b*c^3*d^2*e^4)*x)*log(e*x + d))/(c^6*d^9*e + 3*c^4*d^7*e^3 + 3*c^2*d^5*e^5 + d^3*e^7 + (c^6*
d^6*e^4 + 3*c^4*d^4*e^6 + 3*c^2*d^2*e^8 + e^10)*x^3 + 3*(c^6*d^7*e^3 + 3*c^4*d^5*e^5 + 3*c^2*d^3*e^7 + d*e^9)*
x^2 + 3*(c^6*d^8*e^2 + 3*c^4*d^6*e^4 + 3*c^2*d^4*e^6 + d^2*e^8)*x)

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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((a+b*atan(c*x))/(e*x+d)**4,x)

[Out]

Timed out

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Giac [B]  time = 10.9004, size = 1042, normalized size = 5.06 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(2*b*c^6*d^3*x^3*arctan(c*x)*e^3 + 6*b*c^6*d^4*x^2*arctan(c*x)*e^2 + 6*b*c^6*d^5*x*arctan(c*x)*e - 2*a*c^6
*d^6 - 3*b*c^5*d^2*x^3*e^4*log(c^2*x^2 + 1) - 9*b*c^5*d^3*x^2*e^3*log(c^2*x^2 + 1) - 9*b*c^5*d^4*x*e^2*log(c^2
*x^2 + 1) - 3*b*c^5*d^5*e*log(c^2*x^2 + 1) + 6*b*c^5*d^2*x^3*e^4*log(abs(x*e + d)) + 18*b*c^5*d^3*x^2*e^3*log(
abs(x*e + d)) + 18*b*c^5*d^4*x*e^2*log(abs(x*e + d)) + 6*b*c^5*d^5*e*log(abs(x*e + d)) - 4*b*c^5*d^3*x^2*e^3 -
9*b*c^5*d^4*x*e^2 - 5*b*c^5*d^5*e - 6*b*c^4*d*x^3*arctan(c*x)*e^5 - 18*b*c^4*d^2*x^2*arctan(c*x)*e^4 - 18*b*c
^4*d^3*x*arctan(c*x)*e^3 - 12*b*c^4*d^4*arctan(c*x)*e^2 - 6*a*c^4*d^4*e^2 + b*c^3*x^3*e^6*log(c^2*x^2 + 1) + 3
*b*c^3*d*x^2*e^5*log(c^2*x^2 + 1) + 3*b*c^3*d^2*x*e^4*log(c^2*x^2 + 1) + b*c^3*d^3*e^3*log(c^2*x^2 + 1) - 2*b*
c^3*x^3*e^6*log(abs(x*e + d)) - 6*b*c^3*d*x^2*e^5*log(abs(x*e + d)) - 6*b*c^3*d^2*x*e^4*log(abs(x*e + d)) - 2*
b*c^3*d^3*e^3*log(abs(x*e + d)) - 4*b*c^3*d*x^2*e^5 - 10*b*c^3*d^2*x*e^4 - 6*b*c^3*d^3*e^3 - 6*b*c^2*d^2*arcta
n(c*x)*e^4 - 6*a*c^2*d^2*e^4 - b*c*x*e^6 - b*c*d*e^5 - 2*b*arctan(c*x)*e^6 - 2*a*e^6)/(c^6*d^6*x^3*e^4 + 3*c^6
*d^7*x^2*e^3 + 3*c^6*d^8*x*e^2 + c^6*d^9*e + 3*c^4*d^4*x^3*e^6 + 9*c^4*d^5*x^2*e^5 + 9*c^4*d^6*x*e^4 + 3*c^4*d
^7*e^3 + 3*c^2*d^2*x^3*e^8 + 9*c^2*d^3*x^2*e^7 + 9*c^2*d^4*x*e^6 + 3*c^2*d^5*e^5 + x^3*e^10 + 3*d*x^2*e^9 + 3*
d^2*x*e^8 + d^3*e^7)