3.312 \(\int \frac{(d+e x)^2 (2+x+3 x^2-5 x^3+4 x^4)}{(3+2 x+5 x^2)^2} \, dx\)

Optimal. Leaf size=140 \[ -\frac{\left (1025 d^2-1030 d e-867 e^2\right ) \log \left (5 x^2+2 x+3\right )}{6250}+\frac{x \left (2800 d^2-11480 d e+3307 e^2\right )}{17500}+\frac{\left (32825 d^2+211710 d e-73881 e^2\right ) \tan ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{87500 \sqrt{14}}+\frac{1}{250} e x^2 (40 d-41 e)-\frac{(423 x+1367) (d+e x)^2}{3500 \left (5 x^2+2 x+3\right )}+\frac{4 e^2 x^3}{75} \]

[Out]

((2800*d^2 - 11480*d*e + 3307*e^2)*x)/17500 + ((40*d - 41*e)*e*x^2)/250 + (4*e^2*x^3)/75 - ((1367 + 423*x)*(d
+ e*x)^2)/(3500*(3 + 2*x + 5*x^2)) + ((32825*d^2 + 211710*d*e - 73881*e^2)*ArcTan[(1 + 5*x)/Sqrt[14]])/(87500*
Sqrt[14]) - ((1025*d^2 - 1030*d*e - 867*e^2)*Log[3 + 2*x + 5*x^2])/6250

________________________________________________________________________________________

Rubi [A]  time = 0.208256, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {1644, 1628, 634, 618, 204, 628} \[ -\frac{\left (1025 d^2-1030 d e-867 e^2\right ) \log \left (5 x^2+2 x+3\right )}{6250}+\frac{x \left (2800 d^2-11480 d e+3307 e^2\right )}{17500}+\frac{\left (32825 d^2+211710 d e-73881 e^2\right ) \tan ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{87500 \sqrt{14}}+\frac{1}{250} e x^2 (40 d-41 e)-\frac{(423 x+1367) (d+e x)^2}{3500 \left (5 x^2+2 x+3\right )}+\frac{4 e^2 x^3}{75} \]

Antiderivative was successfully verified.

[In]

Int[((d + e*x)^2*(2 + x + 3*x^2 - 5*x^3 + 4*x^4))/(3 + 2*x + 5*x^2)^2,x]

[Out]

((2800*d^2 - 11480*d*e + 3307*e^2)*x)/17500 + ((40*d - 41*e)*e*x^2)/250 + (4*e^2*x^3)/75 - ((1367 + 423*x)*(d
+ e*x)^2)/(3500*(3 + 2*x + 5*x^2)) + ((32825*d^2 + 211710*d*e - 73881*e^2)*ArcTan[(1 + 5*x)/Sqrt[14]])/(87500*
Sqrt[14]) - ((1025*d^2 - 1030*d*e - 867*e^2)*Log[3 + 2*x + 5*x^2])/6250

Rule 1644

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = Polynomi
alQuotient[Pq, a + b*x + c*x^2, x], f = Coeff[PolynomialRemainder[Pq, a + b*x + c*x^2, x], x, 0], g = Coeff[Po
lynomialRemainder[Pq, a + b*x + c*x^2, x], x, 1]}, Simp[((d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*(f*b - 2*a*g +
(2*c*f - b*g)*x))/((p + 1)*(b^2 - 4*a*c)), x] + Dist[1/((p + 1)*(b^2 - 4*a*c)), Int[(d + e*x)^(m - 1)*(a + b*x
+ c*x^2)^(p + 1)*ExpandToSum[(p + 1)*(b^2 - 4*a*c)*(d + e*x)*Q + g*(2*a*e*m + b*d*(2*p + 3)) - f*(b*e*m + 2*c
*d*(2*p + 3)) - e*(2*c*f - b*g)*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && PolyQ[Pq, x] && N
eQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 0] && (IntegerQ[p] ||  !IntegerQ[m
] ||  !RationalQ[a, b, c, d, e]) &&  !(IGtQ[m, 0] && RationalQ[a, b, c, d, e] && (IntegerQ[p] || ILtQ[p + 1/2,
0]))

Rule 1628

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegra
nd[(d + e*x)^m*Pq*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

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

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rubi steps

\begin{align*} \int \frac{(d+e x)^2 \left (2+x+3 x^2-5 x^3+4 x^4\right )}{\left (3+2 x+5 x^2\right )^2} \, dx &=-\frac{(1367+423 x) (d+e x)^2}{3500 \left (3+2 x+5 x^2\right )}+\frac{1}{56} \int \frac{(d+e x) \left (\frac{2}{125} (1845 d+2734 e)-\frac{6}{125} (1540 d-897 e) x+\frac{56}{25} (20 d-33 e) x^2+\frac{224 e x^3}{5}\right )}{3+2 x+5 x^2} \, dx\\ &=-\frac{(1367+423 x) (d+e x)^2}{3500 \left (3+2 x+5 x^2\right )}+\frac{1}{56} \int \left (\frac{2}{625} \left (2800 d^2-11480 d e+3307 e^2\right )+\frac{56}{125} (40 d-41 e) e x+\frac{224 e^2 x^2}{25}+\frac{2 \left (825 d^2+48110 d e-9921 e^2-28 \left (1025 d^2-1030 d e-867 e^2\right ) x\right )}{625 \left (3+2 x+5 x^2\right )}\right ) \, dx\\ &=\frac{\left (2800 d^2-11480 d e+3307 e^2\right ) x}{17500}+\frac{1}{250} (40 d-41 e) e x^2+\frac{4 e^2 x^3}{75}-\frac{(1367+423 x) (d+e x)^2}{3500 \left (3+2 x+5 x^2\right )}+\frac{\int \frac{825 d^2+48110 d e-9921 e^2-28 \left (1025 d^2-1030 d e-867 e^2\right ) x}{3+2 x+5 x^2} \, dx}{17500}\\ &=\frac{\left (2800 d^2-11480 d e+3307 e^2\right ) x}{17500}+\frac{1}{250} (40 d-41 e) e x^2+\frac{4 e^2 x^3}{75}-\frac{(1367+423 x) (d+e x)^2}{3500 \left (3+2 x+5 x^2\right )}+\frac{\left (32825 d^2+211710 d e-73881 e^2\right ) \int \frac{1}{3+2 x+5 x^2} \, dx}{87500}+\frac{\left (-1025 d^2+1030 d e+867 e^2\right ) \int \frac{2+10 x}{3+2 x+5 x^2} \, dx}{6250}\\ &=\frac{\left (2800 d^2-11480 d e+3307 e^2\right ) x}{17500}+\frac{1}{250} (40 d-41 e) e x^2+\frac{4 e^2 x^3}{75}-\frac{(1367+423 x) (d+e x)^2}{3500 \left (3+2 x+5 x^2\right )}-\frac{\left (1025 d^2-1030 d e-867 e^2\right ) \log \left (3+2 x+5 x^2\right )}{6250}+\frac{\left (-32825 d^2-211710 d e+73881 e^2\right ) \operatorname{Subst}\left (\int \frac{1}{-56-x^2} \, dx,x,2+10 x\right )}{43750}\\ &=\frac{\left (2800 d^2-11480 d e+3307 e^2\right ) x}{17500}+\frac{1}{250} (40 d-41 e) e x^2+\frac{4 e^2 x^3}{75}-\frac{(1367+423 x) (d+e x)^2}{3500 \left (3+2 x+5 x^2\right )}+\frac{\left (32825 d^2+211710 d e-73881 e^2\right ) \tan ^{-1}\left (\frac{1+5 x}{\sqrt{14}}\right )}{87500 \sqrt{14}}-\frac{\left (1025 d^2-1030 d e-867 e^2\right ) \log \left (3+2 x+5 x^2\right )}{6250}\\ \end{align*}

Mathematica [A]  time = 0.109274, size = 150, normalized size = 1.07 \[ \frac{-\frac{42 \left (25 d^2 (423 x+1367)+10 d e (5989 x-1269)-e^2 (18323 x+17967)\right )}{5 x^2+2 x+3}+588 \left (-1025 d^2+1030 d e+867 e^2\right ) \log \left (5 x^2+2 x+3\right )+5880 x \left (100 d^2-410 d e+103 e^2\right )+3 \sqrt{14} \left (32825 d^2+211710 d e-73881 e^2\right ) \tan ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )+14700 e x^2 (40 d-41 e)+196000 e^2 x^3}{3675000} \]

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x)^2*(2 + x + 3*x^2 - 5*x^3 + 4*x^4))/(3 + 2*x + 5*x^2)^2,x]

[Out]

(5880*(100*d^2 - 410*d*e + 103*e^2)*x + 14700*(40*d - 41*e)*e*x^2 + 196000*e^2*x^3 - (42*(25*d^2*(1367 + 423*x
) + 10*d*e*(-1269 + 5989*x) - e^2*(17967 + 18323*x)))/(3 + 2*x + 5*x^2) + 3*Sqrt[14]*(32825*d^2 + 211710*d*e -
73881*e^2)*ArcTan[(1 + 5*x)/Sqrt[14]] + 588*(-1025*d^2 + 1030*d*e + 867*e^2)*Log[3 + 2*x + 5*x^2])/3675000

________________________________________________________________________________________

Maple [A]  time = 0.056, size = 189, normalized size = 1.4 \begin{align*}{\frac{4\,{e}^{2}{x}^{3}}{75}}+{\frac{4\,{x}^{2}de}{25}}-{\frac{41\,{x}^{2}{e}^{2}}{250}}+{\frac{4\,{d}^{2}x}{25}}-{\frac{82\,xde}{125}}+{\frac{103\,{e}^{2}x}{625}}-{\frac{1}{625} \left ( \left ({\frac{423\,{d}^{2}}{28}}+{\frac{5989\,de}{70}}-{\frac{18323\,{e}^{2}}{700}} \right ) x+{\frac{1367\,{d}^{2}}{28}}-{\frac{1269\,de}{70}}-{\frac{17967\,{e}^{2}}{700}} \right ) \left ({x}^{2}+{\frac{2\,x}{5}}+{\frac{3}{5}} \right ) ^{-1}}-{\frac{41\,\ln \left ( 5\,{x}^{2}+2\,x+3 \right ){d}^{2}}{250}}+{\frac{103\,\ln \left ( 5\,{x}^{2}+2\,x+3 \right ) de}{625}}+{\frac{867\,\ln \left ( 5\,{x}^{2}+2\,x+3 \right ){e}^{2}}{6250}}+{\frac{1313\,\sqrt{14}{d}^{2}}{49000}\arctan \left ({\frac{ \left ( 10\,x+2 \right ) \sqrt{14}}{28}} \right ) }+{\frac{21171\,\sqrt{14}de}{122500}\arctan \left ({\frac{ \left ( 10\,x+2 \right ) \sqrt{14}}{28}} \right ) }-{\frac{73881\,\sqrt{14}{e}^{2}}{1225000}\arctan \left ({\frac{ \left ( 10\,x+2 \right ) \sqrt{14}}{28}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^2*(4*x^4-5*x^3+3*x^2+x+2)/(5*x^2+2*x+3)^2,x)

[Out]

4/75*e^2*x^3+4/25*x^2*d*e-41/250*x^2*e^2+4/25*d^2*x-82/125*x*d*e+103/625*e^2*x-1/625*((423/28*d^2+5989/70*d*e-
18323/700*e^2)*x+1367/28*d^2-1269/70*d*e-17967/700*e^2)/(x^2+2/5*x+3/5)-41/250*ln(5*x^2+2*x+3)*d^2+103/625*ln(
5*x^2+2*x+3)*d*e+867/6250*ln(5*x^2+2*x+3)*e^2+1313/49000*14^(1/2)*arctan(1/28*(10*x+2)*14^(1/2))*d^2+21171/122
500*14^(1/2)*arctan(1/28*(10*x+2)*14^(1/2))*d*e-73881/1225000*14^(1/2)*arctan(1/28*(10*x+2)*14^(1/2))*e^2

________________________________________________________________________________________

Maxima [A]  time = 1.5442, size = 198, normalized size = 1.41 \begin{align*} \frac{4}{75} \, e^{2} x^{3} + \frac{1}{250} \,{\left (40 \, d e - 41 \, e^{2}\right )} x^{2} + \frac{1}{1225000} \, \sqrt{14}{\left (32825 \, d^{2} + 211710 \, d e - 73881 \, e^{2}\right )} \arctan \left (\frac{1}{14} \, \sqrt{14}{\left (5 \, x + 1\right )}\right ) + \frac{1}{625} \,{\left (100 \, d^{2} - 410 \, d e + 103 \, e^{2}\right )} x - \frac{1}{6250} \,{\left (1025 \, d^{2} - 1030 \, d e - 867 \, e^{2}\right )} \log \left (5 \, x^{2} + 2 \, x + 3\right ) - \frac{34175 \, d^{2} - 12690 \, d e - 17967 \, e^{2} +{\left (10575 \, d^{2} + 59890 \, d e - 18323 \, e^{2}\right )} x}{87500 \,{\left (5 \, x^{2} + 2 \, x + 3\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^2*(4*x^4-5*x^3+3*x^2+x+2)/(5*x^2+2*x+3)^2,x, algorithm="maxima")

[Out]

4/75*e^2*x^3 + 1/250*(40*d*e - 41*e^2)*x^2 + 1/1225000*sqrt(14)*(32825*d^2 + 211710*d*e - 73881*e^2)*arctan(1/
14*sqrt(14)*(5*x + 1)) + 1/625*(100*d^2 - 410*d*e + 103*e^2)*x - 1/6250*(1025*d^2 - 1030*d*e - 867*e^2)*log(5*
x^2 + 2*x + 3) - 1/87500*(34175*d^2 - 12690*d*e - 17967*e^2 + (10575*d^2 + 59890*d*e - 18323*e^2)*x)/(5*x^2 +
2*x + 3)

________________________________________________________________________________________

Fricas [A]  time = 1.21679, size = 747, normalized size = 5.34 \begin{align*} \frac{980000 \, e^{2} x^{5} + 24500 \,{\left (120 \, d e - 107 \, e^{2}\right )} x^{4} + 58800 \,{\left (50 \, d^{2} - 185 \, d e + 41 \, e^{2}\right )} x^{3} + 2940 \,{\left (400 \, d^{2} - 1040 \, d e - 203 \, e^{2}\right )} x^{2} + 3 \, \sqrt{14}{\left (5 \,{\left (32825 \, d^{2} + 211710 \, d e - 73881 \, e^{2}\right )} x^{2} + 98475 \, d^{2} + 635130 \, d e - 221643 \, e^{2} + 2 \,{\left (32825 \, d^{2} + 211710 \, d e - 73881 \, e^{2}\right )} x\right )} \arctan \left (\frac{1}{14} \, \sqrt{14}{\left (5 \, x + 1\right )}\right ) - 1435350 \, d^{2} + 532980 \, d e + 754614 \, e^{2} + 42 \,{\left (31425 \, d^{2} - 232090 \, d e + 61583 \, e^{2}\right )} x - 588 \,{\left (5 \,{\left (1025 \, d^{2} - 1030 \, d e - 867 \, e^{2}\right )} x^{2} + 3075 \, d^{2} - 3090 \, d e - 2601 \, e^{2} + 2 \,{\left (1025 \, d^{2} - 1030 \, d e - 867 \, e^{2}\right )} x\right )} \log \left (5 \, x^{2} + 2 \, x + 3\right )}{3675000 \,{\left (5 \, x^{2} + 2 \, x + 3\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^2*(4*x^4-5*x^3+3*x^2+x+2)/(5*x^2+2*x+3)^2,x, algorithm="fricas")

[Out]

1/3675000*(980000*e^2*x^5 + 24500*(120*d*e - 107*e^2)*x^4 + 58800*(50*d^2 - 185*d*e + 41*e^2)*x^3 + 2940*(400*
d^2 - 1040*d*e - 203*e^2)*x^2 + 3*sqrt(14)*(5*(32825*d^2 + 211710*d*e - 73881*e^2)*x^2 + 98475*d^2 + 635130*d*
e - 221643*e^2 + 2*(32825*d^2 + 211710*d*e - 73881*e^2)*x)*arctan(1/14*sqrt(14)*(5*x + 1)) - 1435350*d^2 + 532
980*d*e + 754614*e^2 + 42*(31425*d^2 - 232090*d*e + 61583*e^2)*x - 588*(5*(1025*d^2 - 1030*d*e - 867*e^2)*x^2
+ 3075*d^2 - 3090*d*e - 2601*e^2 + 2*(1025*d^2 - 1030*d*e - 867*e^2)*x)*log(5*x^2 + 2*x + 3))/(5*x^2 + 2*x + 3
)

________________________________________________________________________________________

Sympy [C]  time = 1.74327, size = 298, normalized size = 2.13 \begin{align*} \frac{4 e^{2} x^{3}}{75} + x^{2} \left (\frac{4 d e}{25} - \frac{41 e^{2}}{250}\right ) + x \left (\frac{4 d^{2}}{25} - \frac{82 d e}{125} + \frac{103 e^{2}}{625}\right ) + \left (- \frac{41 d^{2}}{250} + \frac{103 d e}{625} + \frac{867 e^{2}}{6250} - \frac{\sqrt{14} i \left (32825 d^{2} + 211710 d e - 73881 e^{2}\right )}{2450000}\right ) \log{\left (x + \frac{6565 d^{2} + 42342 d e - \frac{73881 e^{2}}{5} - \frac{\sqrt{14} i \left (32825 d^{2} + 211710 d e - 73881 e^{2}\right )}{5}}{32825 d^{2} + 211710 d e - 73881 e^{2}} \right )} + \left (- \frac{41 d^{2}}{250} + \frac{103 d e}{625} + \frac{867 e^{2}}{6250} + \frac{\sqrt{14} i \left (32825 d^{2} + 211710 d e - 73881 e^{2}\right )}{2450000}\right ) \log{\left (x + \frac{6565 d^{2} + 42342 d e - \frac{73881 e^{2}}{5} + \frac{\sqrt{14} i \left (32825 d^{2} + 211710 d e - 73881 e^{2}\right )}{5}}{32825 d^{2} + 211710 d e - 73881 e^{2}} \right )} - \frac{34175 d^{2} - 12690 d e - 17967 e^{2} + x \left (10575 d^{2} + 59890 d e - 18323 e^{2}\right )}{437500 x^{2} + 175000 x + 262500} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**2*(4*x**4-5*x**3+3*x**2+x+2)/(5*x**2+2*x+3)**2,x)

[Out]

4*e**2*x**3/75 + x**2*(4*d*e/25 - 41*e**2/250) + x*(4*d**2/25 - 82*d*e/125 + 103*e**2/625) + (-41*d**2/250 + 1
03*d*e/625 + 867*e**2/6250 - sqrt(14)*I*(32825*d**2 + 211710*d*e - 73881*e**2)/2450000)*log(x + (6565*d**2 + 4
2342*d*e - 73881*e**2/5 - sqrt(14)*I*(32825*d**2 + 211710*d*e - 73881*e**2)/5)/(32825*d**2 + 211710*d*e - 7388
1*e**2)) + (-41*d**2/250 + 103*d*e/625 + 867*e**2/6250 + sqrt(14)*I*(32825*d**2 + 211710*d*e - 73881*e**2)/245
0000)*log(x + (6565*d**2 + 42342*d*e - 73881*e**2/5 + sqrt(14)*I*(32825*d**2 + 211710*d*e - 73881*e**2)/5)/(32
825*d**2 + 211710*d*e - 73881*e**2)) - (34175*d**2 - 12690*d*e - 17967*e**2 + x*(10575*d**2 + 59890*d*e - 1832
3*e**2))/(437500*x**2 + 175000*x + 262500)

________________________________________________________________________________________

Giac [A]  time = 1.16566, size = 196, normalized size = 1.4 \begin{align*} \frac{4}{75} \, x^{3} e^{2} + \frac{4}{25} \, d x^{2} e + \frac{4}{25} \, d^{2} x - \frac{41}{250} \, x^{2} e^{2} - \frac{82}{125} \, d x e + \frac{1}{1225000} \, \sqrt{14}{\left (32825 \, d^{2} + 211710 \, d e - 73881 \, e^{2}\right )} \arctan \left (\frac{1}{14} \, \sqrt{14}{\left (5 \, x + 1\right )}\right ) + \frac{103}{625} \, x e^{2} - \frac{1}{6250} \,{\left (1025 \, d^{2} - 1030 \, d e - 867 \, e^{2}\right )} \log \left (5 \, x^{2} + 2 \, x + 3\right ) - \frac{34175 \, d^{2} +{\left (10575 \, d^{2} + 59890 \, d e - 18323 \, e^{2}\right )} x - 12690 \, d e - 17967 \, e^{2}}{87500 \,{\left (5 \, x^{2} + 2 \, x + 3\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^2*(4*x^4-5*x^3+3*x^2+x+2)/(5*x^2+2*x+3)^2,x, algorithm="giac")

[Out]

4/75*x^3*e^2 + 4/25*d*x^2*e + 4/25*d^2*x - 41/250*x^2*e^2 - 82/125*d*x*e + 1/1225000*sqrt(14)*(32825*d^2 + 211
710*d*e - 73881*e^2)*arctan(1/14*sqrt(14)*(5*x + 1)) + 103/625*x*e^2 - 1/6250*(1025*d^2 - 1030*d*e - 867*e^2)*
log(5*x^2 + 2*x + 3) - 1/87500*(34175*d^2 + (10575*d^2 + 59890*d*e - 18323*e^2)*x - 12690*d*e - 17967*e^2)/(5*
x^2 + 2*x + 3)