3.4.5 \(\int \frac {(d+e x)^2 (2+x+3 x^2-5 x^3+4 x^4)}{3+2 x+5 x^2} \, dx\) [305]

Optimal. Leaf size=156 \[ \frac {\left (2025 d^2+4580 d e-881 e^2\right ) x}{3125}-\frac {\left (825 d^2-810 d e-458 e^2\right ) x^2}{1250}+\frac {1}{375} \left (100 d^2-330 d e+81 e^2\right ) x^3+\frac {1}{100} (40 d-33 e) e x^4+\frac {4 e^2 x^5}{25}-\frac {\left (10575 d^2+59890 d e-18323 e^2\right ) \tan ^{-1}\left (\frac {1+5 x}{\sqrt {14}}\right )}{15625 \sqrt {14}}+\frac {\left (5725 d^2-4405 d e-2554 e^2\right ) \log \left (3+2 x+5 x^2\right )}{15625} \]

[Out]

1/3125*(2025*d^2+4580*d*e-881*e^2)*x-1/1250*(825*d^2-810*d*e-458*e^2)*x^2+1/375*(100*d^2-330*d*e+81*e^2)*x^3+1
/100*(40*d-33*e)*e*x^4+4/25*e^2*x^5+1/15625*(5725*d^2-4405*d*e-2554*e^2)*ln(5*x^2+2*x+3)-1/218750*(10575*d^2+5
9890*d*e-18323*e^2)*arctan(1/14*(1+5*x)*14^(1/2))*14^(1/2)

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Rubi [A]
time = 0.10, antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.132, Rules used = {1642, 648, 632, 210, 642} \begin {gather*} -\frac {\text {ArcTan}\left (\frac {5 x+1}{\sqrt {14}}\right ) \left (10575 d^2+59890 d e-18323 e^2\right )}{15625 \sqrt {14}}+\frac {1}{375} x^3 \left (100 d^2-330 d e+81 e^2\right )-\frac {x^2 \left (825 d^2-810 d e-458 e^2\right )}{1250}+\frac {\left (5725 d^2-4405 d e-2554 e^2\right ) \log \left (5 x^2+2 x+3\right )}{15625}+\frac {x \left (2025 d^2+4580 d e-881 e^2\right )}{3125}+\frac {1}{100} e x^4 (40 d-33 e)+\frac {4 e^2 x^5}{25} \end {gather*}

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),x]

[Out]

((2025*d^2 + 4580*d*e - 881*e^2)*x)/3125 - ((825*d^2 - 810*d*e - 458*e^2)*x^2)/1250 + ((100*d^2 - 330*d*e + 81
*e^2)*x^3)/375 + ((40*d - 33*e)*e*x^4)/100 + (4*e^2*x^5)/25 - ((10575*d^2 + 59890*d*e - 18323*e^2)*ArcTan[(1 +
5*x)/Sqrt[14]])/(15625*Sqrt[14]) + ((5725*d^2 - 4405*d*e - 2554*e^2)*Log[3 + 2*x + 5*x^2])/15625

Rule 210

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

Rule 632

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 642

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]

Rule 648

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 1642

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]

Rubi steps

\begin {align*} \int \frac {(d+e x)^2 \left (2+x+3 x^2-5 x^3+4 x^4\right )}{3+2 x+5 x^2} \, dx &=\int \left (\frac {2025 d^2+4580 d e-881 e^2}{3125}-\frac {1}{625} \left (825 d^2-810 d e-458 e^2\right ) x+\frac {1}{125} \left (100 d^2-330 d e+81 e^2\right ) x^2+\frac {1}{25} (40 d-33 e) e x^3+\frac {4 e^2 x^4}{5}+\frac {175 d^2-13740 d e+2643 e^2+2 \left (5725 d^2-4405 d e-2554 e^2\right ) x}{3125 \left (3+2 x+5 x^2\right )}\right ) \, dx\\ &=\frac {\left (2025 d^2+4580 d e-881 e^2\right ) x}{3125}-\frac {\left (825 d^2-810 d e-458 e^2\right ) x^2}{1250}+\frac {1}{375} \left (100 d^2-330 d e+81 e^2\right ) x^3+\frac {1}{100} (40 d-33 e) e x^4+\frac {4 e^2 x^5}{25}+\frac {\int \frac {175 d^2-13740 d e+2643 e^2+2 \left (5725 d^2-4405 d e-2554 e^2\right ) x}{3+2 x+5 x^2} \, dx}{3125}\\ &=\frac {\left (2025 d^2+4580 d e-881 e^2\right ) x}{3125}-\frac {\left (825 d^2-810 d e-458 e^2\right ) x^2}{1250}+\frac {1}{375} \left (100 d^2-330 d e+81 e^2\right ) x^3+\frac {1}{100} (40 d-33 e) e x^4+\frac {4 e^2 x^5}{25}+\frac {\left (5725 d^2-4405 d e-2554 e^2\right ) \int \frac {2+10 x}{3+2 x+5 x^2} \, dx}{15625}+\frac {\left (-10575 d^2-59890 d e+18323 e^2\right ) \int \frac {1}{3+2 x+5 x^2} \, dx}{15625}\\ &=\frac {\left (2025 d^2+4580 d e-881 e^2\right ) x}{3125}-\frac {\left (825 d^2-810 d e-458 e^2\right ) x^2}{1250}+\frac {1}{375} \left (100 d^2-330 d e+81 e^2\right ) x^3+\frac {1}{100} (40 d-33 e) e x^4+\frac {4 e^2 x^5}{25}+\frac {\left (5725 d^2-4405 d e-2554 e^2\right ) \log \left (3+2 x+5 x^2\right )}{15625}+\frac {\left (2 \left (10575 d^2+59890 d e-18323 e^2\right )\right ) \text {Subst}\left (\int \frac {1}{-56-x^2} \, dx,x,2+10 x\right )}{15625}\\ &=\frac {\left (2025 d^2+4580 d e-881 e^2\right ) x}{3125}-\frac {\left (825 d^2-810 d e-458 e^2\right ) x^2}{1250}+\frac {1}{375} \left (100 d^2-330 d e+81 e^2\right ) x^3+\frac {1}{100} (40 d-33 e) e x^4+\frac {4 e^2 x^5}{25}-\frac {\left (10575 d^2+59890 d e-18323 e^2\right ) \tan ^{-1}\left (\frac {1+5 x}{\sqrt {14}}\right )}{15625 \sqrt {14}}+\frac {\left (5725 d^2-4405 d e-2554 e^2\right ) \log \left (3+2 x+5 x^2\right )}{15625}\\ \end {align*}

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Mathematica [A]
time = 0.06, size = 130, normalized size = 0.83 \begin {gather*} \frac {35 x \left (50 d^2 \left (486-495 x+200 x^2\right )+60 d e \left (916+405 x-550 x^2+250 x^3\right )+3 e^2 \left (-3524+4580 x+2700 x^2-4125 x^3+2000 x^4\right )\right )-6 \sqrt {14} \left (10575 d^2+59890 d e-18323 e^2\right ) \tan ^{-1}\left (\frac {1+5 x}{\sqrt {14}}\right )+84 \left (5725 d^2-4405 d e-2554 e^2\right ) \log \left (3+2 x+5 x^2\right )}{1312500} \end {gather*}

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),x]

[Out]

(35*x*(50*d^2*(486 - 495*x + 200*x^2) + 60*d*e*(916 + 405*x - 550*x^2 + 250*x^3) + 3*e^2*(-3524 + 4580*x + 270
0*x^2 - 4125*x^3 + 2000*x^4)) - 6*Sqrt[14]*(10575*d^2 + 59890*d*e - 18323*e^2)*ArcTan[(1 + 5*x)/Sqrt[14]] + 84
*(5725*d^2 - 4405*d*e - 2554*e^2)*Log[3 + 2*x + 5*x^2])/1312500

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Maple [A]
time = 0.15, size = 147, normalized size = 0.94

method	result	size
default	\(\frac {4 e^{2} x^{5}}{25}+\frac {2 x^{4} d e}{5}-\frac {33 x^{4} e^{2}}{100}+\frac {4 x^{3} d^{2}}{15}-\frac {22 x^{3} d e}{25}+\frac {27 e^{2} x^{3}}{125}-\frac {33 d^{2} x^{2}}{50}+\frac {81 d e \,x^{2}}{125}+\frac {229 e^{2} x^{2}}{625}+\frac {81 d^{2} x}{125}+\frac {916 d e x}{625}-\frac {881 e^{2} x}{3125}+\frac {\left (11450 d^{2}-8810 d e -5108 e^{2}\right ) \ln \left (5 x^{2}+2 x +3\right )}{31250}+\frac {\left (-2115 d^{2}-11978 d e +\frac {18323}{5} e^{2}\right ) \sqrt {14}\, \arctan \left (\frac {\left (10 x +2\right ) \sqrt {14}}{28}\right )}{43750}\)	\(147\)
risch	\(-\frac {33 x^{4} e^{2}}{100}+\frac {4 x^{3} d^{2}}{15}-\frac {33 d^{2} x^{2}}{50}-\frac {22 x^{3} d e}{25}+\frac {2 x^{4} d e}{5}+\frac {229 d^{2} \ln \left (350 x^{2}+140 x +210\right )}{625}-\frac {2554 e^{2} \ln \left (350 x^{2}+140 x +210\right )}{15625}+\frac {916 d e x}{625}-\frac {5989 \sqrt {14}\, d e \arctan \left (\frac {5 \sqrt {14}\, x}{14}+\frac {\sqrt {14}}{14}\right )}{21875}+\frac {27 e^{2} x^{3}}{125}+\frac {81 d^{2} x}{125}+\frac {229 e^{2} x^{2}}{625}+\frac {81 d e \,x^{2}}{125}-\frac {881 d e \ln \left (350 x^{2}+140 x +210\right )}{3125}-\frac {423 \sqrt {14}\, d^{2} \arctan \left (\frac {5 \sqrt {14}\, x}{14}+\frac {\sqrt {14}}{14}\right )}{8750}+\frac {18323 \sqrt {14}\, e^{2} \arctan \left (\frac {5 \sqrt {14}\, x}{14}+\frac {\sqrt {14}}{14}\right )}{218750}-\frac {881 e^{2} x}{3125}+\frac {4 e^{2} x^{5}}{25}\)	\(197\)

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),x,method=_RETURNVERBOSE)

[Out]

4/25*e^2*x^5+2/5*x^4*d*e-33/100*x^4*e^2+4/15*x^3*d^2-22/25*x^3*d*e+27/125*e^2*x^3-33/50*d^2*x^2+81/125*d*e*x^2
+229/625*e^2*x^2+81/125*d^2*x+916/625*d*e*x-881/3125*e^2*x+1/31250*(11450*d^2-8810*d*e-5108*e^2)*ln(5*x^2+2*x+
3)+1/43750*(-2115*d^2-11978*d*e+18323/5*e^2)*14^(1/2)*arctan(1/28*(10*x+2)*14^(1/2))

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Maxima [A]
time = 0.56, size = 140, normalized size = 0.90 \begin {gather*} \frac {4}{25} \, x^{5} e^{2} + \frac {1}{100} \, {\left (40 \, d e - 33 \, e^{2}\right )} x^{4} + \frac {1}{375} \, {\left (100 \, d^{2} - 330 \, d e + 81 \, e^{2}\right )} x^{3} - \frac {1}{1250} \, {\left (825 \, d^{2} - 810 \, d e - 458 \, e^{2}\right )} x^{2} - \frac {1}{218750} \, \sqrt {14} {\left (10575 \, d^{2} + 59890 \, d e - 18323 \, e^{2}\right )} \arctan \left (\frac {1}{14} \, \sqrt {14} {\left (5 \, x + 1\right )}\right ) + \frac {1}{3125} \, {\left (2025 \, d^{2} + 4580 \, d e - 881 \, e^{2}\right )} x + \frac {1}{15625} \, {\left (5725 \, d^{2} - 4405 \, d e - 2554 \, e^{2}\right )} \log \left (5 \, x^{2} + 2 \, x + 3\right ) \end {gather*}

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),x, algorithm="maxima")

[Out]

4/25*x^5*e^2 + 1/100*(40*d*e - 33*e^2)*x^4 + 1/375*(100*d^2 - 330*d*e + 81*e^2)*x^3 - 1/1250*(825*d^2 - 810*d*
e - 458*e^2)*x^2 - 1/218750*sqrt(14)*(10575*d^2 + 59890*d*e - 18323*e^2)*arctan(1/14*sqrt(14)*(5*x + 1)) + 1/3
125*(2025*d^2 + 4580*d*e - 881*e^2)*x + 1/15625*(5725*d^2 - 4405*d*e - 2554*e^2)*log(5*x^2 + 2*x + 3)

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Fricas [A]
time = 0.39, size = 137, normalized size = 0.88 \begin {gather*} \frac {4}{15} \, d^{2} x^{3} - \frac {33}{50} \, d^{2} x^{2} + \frac {81}{125} \, d^{2} x - \frac {1}{218750} \, \sqrt {14} {\left (10575 \, d^{2} + 59890 \, d e - 18323 \, e^{2}\right )} \arctan \left (\frac {1}{14} \, \sqrt {14} {\left (5 \, x + 1\right )}\right ) + \frac {1}{12500} \, {\left (2000 \, x^{5} - 4125 \, x^{4} + 2700 \, x^{3} + 4580 \, x^{2} - 3524 \, x\right )} e^{2} + \frac {1}{625} \, {\left (250 \, d x^{4} - 550 \, d x^{3} + 405 \, d x^{2} + 916 \, d x\right )} e + \frac {1}{15625} \, {\left (5725 \, d^{2} - 4405 \, d e - 2554 \, e^{2}\right )} \log \left (5 \, x^{2} + 2 \, x + 3\right ) \end {gather*}

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),x, algorithm="fricas")

[Out]

4/15*d^2*x^3 - 33/50*d^2*x^2 + 81/125*d^2*x - 1/218750*sqrt(14)*(10575*d^2 + 59890*d*e - 18323*e^2)*arctan(1/1
4*sqrt(14)*(5*x + 1)) + 1/12500*(2000*x^5 - 4125*x^4 + 2700*x^3 + 4580*x^2 - 3524*x)*e^2 + 1/625*(250*d*x^4 -
550*d*x^3 + 405*d*x^2 + 916*d*x)*e + 1/15625*(5725*d^2 - 4405*d*e - 2554*e^2)*log(5*x^2 + 2*x + 3)

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Sympy [C] Result contains complex when optimal does not.
time = 0.52, size = 303, normalized size = 1.94 \begin {gather*} \frac {4 e^{2} x^{5}}{25} + x^{4} \cdot \left (\frac {2 d e}{5} - \frac {33 e^{2}}{100}\right ) + x^{3} \cdot \left (\frac {4 d^{2}}{15} - \frac {22 d e}{25} + \frac {27 e^{2}}{125}\right ) + x^{2} \left (- \frac {33 d^{2}}{50} + \frac {81 d e}{125} + \frac {229 e^{2}}{625}\right ) + x \left (\frac {81 d^{2}}{125} + \frac {916 d e}{625} - \frac {881 e^{2}}{3125}\right ) + \left (\frac {229 d^{2}}{625} - \frac {881 d e}{3125} - \frac {2554 e^{2}}{15625} - \frac {\sqrt {14} i \left (10575 d^{2} + 59890 d e - 18323 e^{2}\right )}{437500}\right ) \log {\left (x + \frac {2115 d^{2} + 11978 d e - \frac {18323 e^{2}}{5} + \frac {\sqrt {14} i \left (10575 d^{2} + 59890 d e - 18323 e^{2}\right )}{5}}{10575 d^{2} + 59890 d e - 18323 e^{2}} \right )} + \left (\frac {229 d^{2}}{625} - \frac {881 d e}{3125} - \frac {2554 e^{2}}{15625} + \frac {\sqrt {14} i \left (10575 d^{2} + 59890 d e - 18323 e^{2}\right )}{437500}\right ) \log {\left (x + \frac {2115 d^{2} + 11978 d e - \frac {18323 e^{2}}{5} - \frac {\sqrt {14} i \left (10575 d^{2} + 59890 d e - 18323 e^{2}\right )}{5}}{10575 d^{2} + 59890 d e - 18323 e^{2}} \right )} \end {gather*}

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),x)

[Out]

4*e**2*x**5/25 + x**4*(2*d*e/5 - 33*e**2/100) + x**3*(4*d**2/15 - 22*d*e/25 + 27*e**2/125) + x**2*(-33*d**2/50
+ 81*d*e/125 + 229*e**2/625) + x*(81*d**2/125 + 916*d*e/625 - 881*e**2/3125) + (229*d**2/625 - 881*d*e/3125 -
2554*e**2/15625 - sqrt(14)*I*(10575*d**2 + 59890*d*e - 18323*e**2)/437500)*log(x + (2115*d**2 + 11978*d*e - 1
8323*e**2/5 + sqrt(14)*I*(10575*d**2 + 59890*d*e - 18323*e**2)/5)/(10575*d**2 + 59890*d*e - 18323*e**2)) + (22
9*d**2/625 - 881*d*e/3125 - 2554*e**2/15625 + sqrt(14)*I*(10575*d**2 + 59890*d*e - 18323*e**2)/437500)*log(x +
(2115*d**2 + 11978*d*e - 18323*e**2/5 - sqrt(14)*I*(10575*d**2 + 59890*d*e - 18323*e**2)/5)/(10575*d**2 + 598
90*d*e - 18323*e**2))

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Giac [A]
time = 3.30, size = 145, normalized size = 0.93 \begin {gather*} \frac {4}{25} \, x^{5} e^{2} + \frac {2}{5} \, d x^{4} e + \frac {4}{15} \, d^{2} x^{3} - \frac {33}{100} \, x^{4} e^{2} - \frac {22}{25} \, d x^{3} e - \frac {33}{50} \, d^{2} x^{2} + \frac {27}{125} \, x^{3} e^{2} + \frac {81}{125} \, d x^{2} e + \frac {81}{125} \, d^{2} x + \frac {229}{625} \, x^{2} e^{2} + \frac {916}{625} \, d x e - \frac {1}{218750} \, \sqrt {14} {\left (10575 \, d^{2} + 59890 \, d e - 18323 \, e^{2}\right )} \arctan \left (\frac {1}{14} \, \sqrt {14} {\left (5 \, x + 1\right )}\right ) - \frac {881}{3125} \, x e^{2} + \frac {1}{15625} \, {\left (5725 \, d^{2} - 4405 \, d e - 2554 \, e^{2}\right )} \log \left (5 \, x^{2} + 2 \, x + 3\right ) \end {gather*}

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),x, algorithm="giac")

[Out]

4/25*x^5*e^2 + 2/5*d*x^4*e + 4/15*d^2*x^3 - 33/100*x^4*e^2 - 22/25*d*x^3*e - 33/50*d^2*x^2 + 27/125*x^3*e^2 +
81/125*d*x^2*e + 81/125*d^2*x + 229/625*x^2*e^2 + 916/625*d*x*e - 1/218750*sqrt(14)*(10575*d^2 + 59890*d*e - 1
8323*e^2)*arctan(1/14*sqrt(14)*(5*x + 1)) - 881/3125*x*e^2 + 1/15625*(5725*d^2 - 4405*d*e - 2554*e^2)*log(5*x^
2 + 2*x + 3)

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Mupad [B]
time = 0.10, size = 223, normalized size = 1.43 \begin {gather*} x\,\left (\frac {4\,d\,e}{5}+\frac {52\,e\,\left (8\,d-5\,e\right )}{625}+\frac {81\,d^2}{125}+\frac {419\,e^2}{3125}\right )-\ln \left (5\,x^2+2\,x+3\right )\,\left (-\frac {229\,d^2}{625}+\frac {881\,d\,e}{3125}+\frac {2554\,e^2}{15625}\right )+x^4\,\left (\frac {e\,\left (8\,d-5\,e\right )}{20}-\frac {2\,e^2}{25}\right )-x^3\,\left (\frac {2\,d\,e}{3}+\frac {2\,e\,\left (8\,d-5\,e\right )}{75}-\frac {4\,d^2}{15}-\frac {31\,e^2}{375}\right )+x^2\,\left (d\,e-\frac {11\,e\,\left (8\,d-5\,e\right )}{250}-\frac {33\,d^2}{50}+\frac {183\,e^2}{1250}\right )+\frac {4\,e^2\,x^5}{25}-\frac {\sqrt {14}\,\mathrm {atan}\left (\frac {\frac {\sqrt {14}\,\left (10575\,d^2+59890\,d\,e-18323\,e^2\right )}{218750}+\frac {\sqrt {14}\,x\,\left (10575\,d^2+59890\,d\,e-18323\,e^2\right )}{43750}}{\frac {423\,d^2}{625}+\frac {11978\,d\,e}{3125}-\frac {18323\,e^2}{15625}}\right )\,\left (10575\,d^2+59890\,d\,e-18323\,e^2\right )}{218750} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*((4*d*e)/5 + (52*e*(8*d - 5*e))/625 + (81*d^2)/125 + (419*e^2)/3125) - log(2*x + 5*x^2 + 3)*((881*d*e)/3125
- (229*d^2)/625 + (2554*e^2)/15625) + x^4*((e*(8*d - 5*e))/20 - (2*e^2)/25) - x^3*((2*d*e)/3 + (2*e*(8*d - 5*e
))/75 - (4*d^2)/15 - (31*e^2)/375) + x^2*(d*e - (11*e*(8*d - 5*e))/250 - (33*d^2)/50 + (183*e^2)/1250) + (4*e^
2*x^5)/25 - (14^(1/2)*atan(((14^(1/2)*(59890*d*e + 10575*d^2 - 18323*e^2))/218750 + (14^(1/2)*x*(59890*d*e + 1
0575*d^2 - 18323*e^2))/43750)/((11978*d*e)/3125 + (423*d^2)/625 - (18323*e^2)/15625))*(59890*d*e + 10575*d^2 -
18323*e^2))/218750

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