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

Optimal. Leaf size=313 \[ -\frac{x \left (423 d^2-2734 d e+293 e^2\right )+1367 d^2-586 d e-703 e^2}{140 \left (5 x^2+2 x+3\right ) \left (5 d^2-2 d e+3 e^2\right )^2}-\frac{\left (-60 d^2 e^2-8 d^3 e+41 d^4+24 d e^3-5 e^4\right ) \log \left (5 x^2+2 x+3\right )}{2 \left (5 d^2-2 d e+3 e^2\right )^3}-\frac{3 d^2 e^2+5 d^3 e+4 d^4-d e^3+2 e^4}{e \left (5 d^2-2 d e+3 e^2\right )^2 (d+e x)}+\frac{\left (-60 d^2 e^2-8 d^3 e+41 d^4+24 d e^3-5 e^4\right ) \log (d+e x)}{\left (5 d^2-2 d e+3 e^2\right )^3}+\frac{\left (4290 d^2 e^2-10044 d^3 e+1313 d^4+156 d e^3-271 e^4\right ) \tan ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{28 \sqrt{14} \left (5 d^2-2 d e+3 e^2\right )^3} \]

[Out]

-((4*d^4 + 5*d^3*e + 3*d^2*e^2 - d*e^3 + 2*e^4)/(e*(5*d^2 - 2*d*e + 3*e^2)^2*(d + e*x))) - (1367*d^2 - 586*d*e
- 703*e^2 + (423*d^2 - 2734*d*e + 293*e^2)*x)/(140*(5*d^2 - 2*d*e + 3*e^2)^2*(3 + 2*x + 5*x^2)) + ((1313*d^4
- 10044*d^3*e + 4290*d^2*e^2 + 156*d*e^3 - 271*e^4)*ArcTan[(1 + 5*x)/Sqrt[14]])/(28*Sqrt[14]*(5*d^2 - 2*d*e +
3*e^2)^3) + ((41*d^4 - 8*d^3*e - 60*d^2*e^2 + 24*d*e^3 - 5*e^4)*Log[d + e*x])/(5*d^2 - 2*d*e + 3*e^2)^3 - ((41
*d^4 - 8*d^3*e - 60*d^2*e^2 + 24*d*e^3 - 5*e^4)*Log[3 + 2*x + 5*x^2])/(2*(5*d^2 - 2*d*e + 3*e^2)^3)

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Rubi [A]  time = 0.497972, antiderivative size = 313, 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 = {1646, 1628, 634, 618, 204, 628} \[ -\frac{x \left (423 d^2-2734 d e+293 e^2\right )+1367 d^2-586 d e-703 e^2}{140 \left (5 x^2+2 x+3\right ) \left (5 d^2-2 d e+3 e^2\right )^2}-\frac{\left (-60 d^2 e^2-8 d^3 e+41 d^4+24 d e^3-5 e^4\right ) \log \left (5 x^2+2 x+3\right )}{2 \left (5 d^2-2 d e+3 e^2\right )^3}-\frac{3 d^2 e^2+5 d^3 e+4 d^4-d e^3+2 e^4}{e \left (5 d^2-2 d e+3 e^2\right )^2 (d+e x)}+\frac{\left (-60 d^2 e^2-8 d^3 e+41 d^4+24 d e^3-5 e^4\right ) \log (d+e x)}{\left (5 d^2-2 d e+3 e^2\right )^3}+\frac{\left (4290 d^2 e^2-10044 d^3 e+1313 d^4+156 d e^3-271 e^4\right ) \tan ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{28 \sqrt{14} \left (5 d^2-2 d e+3 e^2\right )^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((4*d^4 + 5*d^3*e + 3*d^2*e^2 - d*e^3 + 2*e^4)/(e*(5*d^2 - 2*d*e + 3*e^2)^2*(d + e*x))) - (1367*d^2 - 586*d*e
- 703*e^2 + (423*d^2 - 2734*d*e + 293*e^2)*x)/(140*(5*d^2 - 2*d*e + 3*e^2)^2*(3 + 2*x + 5*x^2)) + ((1313*d^4
- 10044*d^3*e + 4290*d^2*e^2 + 156*d*e^3 - 271*e^4)*ArcTan[(1 + 5*x)/Sqrt[14]])/(28*Sqrt[14]*(5*d^2 - 2*d*e +
3*e^2)^3) + ((41*d^4 - 8*d^3*e - 60*d^2*e^2 + 24*d*e^3 - 5*e^4)*Log[d + e*x])/(5*d^2 - 2*d*e + 3*e^2)^3 - ((41
*d^4 - 8*d^3*e - 60*d^2*e^2 + 24*d*e^3 - 5*e^4)*Log[3 + 2*x + 5*x^2])/(2*(5*d^2 - 2*d*e + 3*e^2)^3)

Rule 1646

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = Polynomi
alQuotient[(d + e*x)^m*Pq, a + b*x + c*x^2, x], f = Coeff[PolynomialRemainder[(d + e*x)^m*Pq, a + b*x + c*x^2,
x], x, 0], g = Coeff[PolynomialRemainder[(d + e*x)^m*Pq, a + b*x + c*x^2, x], x, 1]}, Simp[((b*f - 2*a*g + (2
*c*f - b*g)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)), x] + Dist[1/((p + 1)*(b^2 - 4*a*c)), Int[(d
+ e*x)^m*(a + b*x + c*x^2)^(p + 1)*ExpandToSum[((p + 1)*(b^2 - 4*a*c)*Q)/(d + e*x)^m - ((2*p + 3)*(2*c*f - b*
g))/(d + e*x)^m, x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2
- b*d*e + a*e^2, 0] && LtQ[p, -1] && ILtQ[m, 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{2+x+3 x^2-5 x^3+4 x^4}{(d+e x)^2 \left (3+2 x+5 x^2\right )^2} \, dx &=-\frac{1367 d^2-586 d e-703 e^2+\left (423 d^2-2734 d e+293 e^2\right ) x}{140 \left (5 d^2-2 d e+3 e^2\right )^2 \left (3+2 x+5 x^2\right )}+\frac{1}{56} \int \frac{\frac{2 \left (369 d^4-842 d^3 e+787 d^2 e^2-224 d e^3+168 e^4\right )}{\left (5 d^2-2 d e+3 e^2\right )^2}-\frac{4 \left (462 d^4-285 d^3 e+338 d^2 e^2-171 d e^3+14 e^4\right ) x}{\left (5 d^2-2 d e+3 e^2\right )^2}+\frac{2 \left (560 d^4-448 d^3 e+677 d^2 e^2+278 d e^3+143 e^4\right ) x^2}{\left (5 d^2-2 d e+3 e^2\right )^2}}{(d+e x)^2 \left (3+2 x+5 x^2\right )} \, dx\\ &=-\frac{1367 d^2-586 d e-703 e^2+\left (423 d^2-2734 d e+293 e^2\right ) x}{140 \left (5 d^2-2 d e+3 e^2\right )^2 \left (3+2 x+5 x^2\right )}+\frac{1}{56} \int \left (\frac{56 \left (4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4\right )}{\left (5 d^2-2 d e+3 e^2\right )^2 (d+e x)^2}-\frac{56 e \left (-41 d^4+8 d^3 e+60 d^2 e^2-24 d e^3+5 e^4\right )}{\left (5 d^2-2 d e+3 e^2\right )^3 (d+e x)}+\frac{2 \left (165 d^4-9820 d^3 e+5970 d^2 e^2-516 d e^3-131 e^4-140 \left (41 d^4-8 d^3 e-60 d^2 e^2+24 d e^3-5 e^4\right ) x\right )}{\left (5 d^2-2 d e+3 e^2\right )^3 \left (3+2 x+5 x^2\right )}\right ) \, dx\\ &=-\frac{4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4}{e \left (5 d^2-2 d e+3 e^2\right )^2 (d+e x)}-\frac{1367 d^2-586 d e-703 e^2+\left (423 d^2-2734 d e+293 e^2\right ) x}{140 \left (5 d^2-2 d e+3 e^2\right )^2 \left (3+2 x+5 x^2\right )}+\frac{\left (41 d^4-8 d^3 e-60 d^2 e^2+24 d e^3-5 e^4\right ) \log (d+e x)}{\left (5 d^2-2 d e+3 e^2\right )^3}+\frac{\int \frac{165 d^4-9820 d^3 e+5970 d^2 e^2-516 d e^3-131 e^4-140 \left (41 d^4-8 d^3 e-60 d^2 e^2+24 d e^3-5 e^4\right ) x}{3+2 x+5 x^2} \, dx}{28 \left (5 d^2-2 d e+3 e^2\right )^3}\\ &=-\frac{4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4}{e \left (5 d^2-2 d e+3 e^2\right )^2 (d+e x)}-\frac{1367 d^2-586 d e-703 e^2+\left (423 d^2-2734 d e+293 e^2\right ) x}{140 \left (5 d^2-2 d e+3 e^2\right )^2 \left (3+2 x+5 x^2\right )}+\frac{\left (41 d^4-8 d^3 e-60 d^2 e^2+24 d e^3-5 e^4\right ) \log (d+e x)}{\left (5 d^2-2 d e+3 e^2\right )^3}+\frac{\left (1313 d^4-10044 d^3 e+4290 d^2 e^2+156 d e^3-271 e^4\right ) \int \frac{1}{3+2 x+5 x^2} \, dx}{28 \left (5 d^2-2 d e+3 e^2\right )^3}-\frac{\left (41 d^4-8 d^3 e-60 d^2 e^2+24 d e^3-5 e^4\right ) \int \frac{2+10 x}{3+2 x+5 x^2} \, dx}{2 \left (5 d^2-2 d e+3 e^2\right )^3}\\ &=-\frac{4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4}{e \left (5 d^2-2 d e+3 e^2\right )^2 (d+e x)}-\frac{1367 d^2-586 d e-703 e^2+\left (423 d^2-2734 d e+293 e^2\right ) x}{140 \left (5 d^2-2 d e+3 e^2\right )^2 \left (3+2 x+5 x^2\right )}+\frac{\left (41 d^4-8 d^3 e-60 d^2 e^2+24 d e^3-5 e^4\right ) \log (d+e x)}{\left (5 d^2-2 d e+3 e^2\right )^3}-\frac{\left (41 d^4-8 d^3 e-60 d^2 e^2+24 d e^3-5 e^4\right ) \log \left (3+2 x+5 x^2\right )}{2 \left (5 d^2-2 d e+3 e^2\right )^3}-\frac{\left (1313 d^4-10044 d^3 e+4290 d^2 e^2+156 d e^3-271 e^4\right ) \operatorname{Subst}\left (\int \frac{1}{-56-x^2} \, dx,x,2+10 x\right )}{14 \left (5 d^2-2 d e+3 e^2\right )^3}\\ &=-\frac{4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4}{e \left (5 d^2-2 d e+3 e^2\right )^2 (d+e x)}-\frac{1367 d^2-586 d e-703 e^2+\left (423 d^2-2734 d e+293 e^2\right ) x}{140 \left (5 d^2-2 d e+3 e^2\right )^2 \left (3+2 x+5 x^2\right )}+\frac{\left (1313 d^4-10044 d^3 e+4290 d^2 e^2+156 d e^3-271 e^4\right ) \tan ^{-1}\left (\frac{1+5 x}{\sqrt{14}}\right )}{28 \sqrt{14} \left (5 d^2-2 d e+3 e^2\right )^3}+\frac{\left (41 d^4-8 d^3 e-60 d^2 e^2+24 d e^3-5 e^4\right ) \log (d+e x)}{\left (5 d^2-2 d e+3 e^2\right )^3}-\frac{\left (41 d^4-8 d^3 e-60 d^2 e^2+24 d e^3-5 e^4\right ) \log \left (3+2 x+5 x^2\right )}{2 \left (5 d^2-2 d e+3 e^2\right )^3}\\ \end{align*}

Mathematica [A]  time = 0.254812, size = 270, normalized size = 0.86 \[ \frac{-\frac{14 \left (5 d^2-2 d e+3 e^2\right ) \left (d^2 (423 x+1367)-2 d e (1367 x+293)+e^2 (293 x-703)\right )}{5 x^2+2 x+3}+980 \left (60 d^2 e^2+8 d^3 e-41 d^4-24 d e^3+5 e^4\right ) \log \left (5 x^2+2 x+3\right )-\frac{1960 \left (5 d^2-2 d e+3 e^2\right ) \left (3 d^2 e^2+5 d^3 e+4 d^4-d e^3+2 e^4\right )}{e (d+e x)}+1960 \left (-60 d^2 e^2-8 d^3 e+41 d^4+24 d e^3-5 e^4\right ) \log (d+e x)+5 \sqrt{14} \left (4290 d^2 e^2-10044 d^3 e+1313 d^4+156 d e^3-271 e^4\right ) \tan ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{1960 \left (5 d^2-2 d e+3 e^2\right )^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-1960*(5*d^2 - 2*d*e + 3*e^2)*(4*d^4 + 5*d^3*e + 3*d^2*e^2 - d*e^3 + 2*e^4))/(e*(d + e*x)) - (14*(5*d^2 - 2*
d*e + 3*e^2)*(e^2*(-703 + 293*x) + d^2*(1367 + 423*x) - 2*d*e*(293 + 1367*x)))/(3 + 2*x + 5*x^2) + 5*Sqrt[14]*
(1313*d^4 - 10044*d^3*e + 4290*d^2*e^2 + 156*d*e^3 - 271*e^4)*ArcTan[(1 + 5*x)/Sqrt[14]] + 1960*(41*d^4 - 8*d^
3*e - 60*d^2*e^2 + 24*d*e^3 - 5*e^4)*Log[d + e*x] + 980*(-41*d^4 + 8*d^3*e + 60*d^2*e^2 - 24*d*e^3 + 5*e^4)*Lo
g[3 + 2*x + 5*x^2])/(1960*(5*d^2 - 2*d*e + 3*e^2)^3)

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Maple [B]  time = 0.072, size = 986, normalized size = 3.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-8/(5*d^2-2*d*e+3*e^2)^3*ln(e*x+d)*d^3*e-4/(5*d^2-2*d*e+3*e^2)^2/e/(e*x+d)*d^4-3/(5*d^2-2*d*e+3*e^2)^2*e/(e*x+
d)*d^2-879/350/(5*d^2-2*d*e+3*e^2)^3/(x^2+2/5*x+3/5)*d^2*e^2-423/140/(5*d^2-2*d*e+3*e^2)^3/(x^2+2/5*x+3/5)*d^4
*x-879/700/(5*d^2-2*d*e+3*e^2)^3/(x^2+2/5*x+3/5)*x*e^4-12/(5*d^2-2*d*e+3*e^2)^3*ln(5*x^2+2*x+3)*d*e^3+1313/392
/(5*d^2-2*d*e+3*e^2)^3*14^(1/2)*arctan(1/28*(10*x+2)*14^(1/2))*d^4-5/(5*d^2-2*d*e+3*e^2)^2/(e*x+d)*d^3-5/(5*d^
2-2*d*e+3*e^2)^3*ln(e*x+d)*e^4-1367/140/(5*d^2-2*d*e+3*e^2)^3/(x^2+2/5*x+3/5)*d^4+2109/700/(5*d^2-2*d*e+3*e^2)
^3/(x^2+2/5*x+3/5)*e^4-41/2/(5*d^2-2*d*e+3*e^2)^3*ln(5*x^2+2*x+3)*d^4+5/2/(5*d^2-2*d*e+3*e^2)^3*ln(5*x^2+2*x+3
)*e^4-2/(5*d^2-2*d*e+3*e^2)^2*e^3/(e*x+d)+41/(5*d^2-2*d*e+3*e^2)^3*ln(e*x+d)*d^4-4101/350/(5*d^2-2*d*e+3*e^2)^
3/(x^2+2/5*x+3/5)*x*d^2*e^2+2197/175/(5*d^2-2*d*e+3*e^2)^3/(x^2+2/5*x+3/5)*x*d*e^3+3629/175/(5*d^2-2*d*e+3*e^2
)^3/(x^2+2/5*x+3/5)*x*d^3*e-2511/98/(5*d^2-2*d*e+3*e^2)^3*14^(1/2)*arctan(1/28*(10*x+2)*14^(1/2))*d^3*e+2145/1
96/(5*d^2-2*d*e+3*e^2)^3*14^(1/2)*arctan(1/28*(10*x+2)*14^(1/2))*d^2*e^2+39/98/(5*d^2-2*d*e+3*e^2)^3*14^(1/2)*
arctan(1/28*(10*x+2)*14^(1/2))*d*e^3-60/(5*d^2-2*d*e+3*e^2)^3*ln(e*x+d)*d^2*e^2+24/(5*d^2-2*d*e+3*e^2)^3*ln(e*
x+d)*d*e^3-271/392/(5*d^2-2*d*e+3*e^2)^3*14^(1/2)*arctan(1/28*(10*x+2)*14^(1/2))*e^4+1/(5*d^2-2*d*e+3*e^2)^2*e
^2/(e*x+d)*d+4/(5*d^2-2*d*e+3*e^2)^3*ln(5*x^2+2*x+3)*d^3*e+30/(5*d^2-2*d*e+3*e^2)^3*ln(5*x^2+2*x+3)*d^2*e^2+88
/175/(5*d^2-2*d*e+3*e^2)^3/(x^2+2/5*x+3/5)*d*e^3+1416/175/(5*d^2-2*d*e+3*e^2)^3/(x^2+2/5*x+3/5)*d^3*e

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Maxima [A]  time = 1.5756, size = 740, normalized size = 2.36 \begin{align*} \frac{\sqrt{14}{\left (1313 \, d^{4} - 10044 \, d^{3} e + 4290 \, d^{2} e^{2} + 156 \, d e^{3} - 271 \, e^{4}\right )} \arctan \left (\frac{1}{14} \, \sqrt{14}{\left (5 \, x + 1\right )}\right )}{392 \,{\left (125 \, d^{6} - 150 \, d^{5} e + 285 \, d^{4} e^{2} - 188 \, d^{3} e^{3} + 171 \, d^{2} e^{4} - 54 \, d e^{5} + 27 \, e^{6}\right )}} + \frac{{\left (41 \, d^{4} - 8 \, d^{3} e - 60 \, d^{2} e^{2} + 24 \, d e^{3} - 5 \, e^{4}\right )} \log \left (e x + d\right )}{125 \, d^{6} - 150 \, d^{5} e + 285 \, d^{4} e^{2} - 188 \, d^{3} e^{3} + 171 \, d^{2} e^{4} - 54 \, d e^{5} + 27 \, e^{6}} - \frac{{\left (41 \, d^{4} - 8 \, d^{3} e - 60 \, d^{2} e^{2} + 24 \, d e^{3} - 5 \, e^{4}\right )} \log \left (5 \, x^{2} + 2 \, x + 3\right )}{2 \,{\left (125 \, d^{6} - 150 \, d^{5} e + 285 \, d^{4} e^{2} - 188 \, d^{3} e^{3} + 171 \, d^{2} e^{4} - 54 \, d e^{5} + 27 \, e^{6}\right )}} - \frac{1680 \, d^{4} + 3467 \, d^{3} e + 674 \, d^{2} e^{2} - 1123 \, d e^{3} + 840 \, e^{4} +{\left (2800 \, d^{4} + 3500 \, d^{3} e + 2523 \, d^{2} e^{2} - 3434 \, d e^{3} + 1693 \, e^{4}\right )} x^{2} +{\left (1120 \, d^{4} + 1823 \, d^{3} e - 527 \, d^{2} e^{2} - 573 \, d e^{3} - 143 \, e^{4}\right )} x}{140 \,{\left (75 \, d^{5} e - 60 \, d^{4} e^{2} + 102 \, d^{3} e^{3} - 36 \, d^{2} e^{4} + 27 \, d e^{5} + 5 \,{\left (25 \, d^{4} e^{2} - 20 \, d^{3} e^{3} + 34 \, d^{2} e^{4} - 12 \, d e^{5} + 9 \, e^{6}\right )} x^{3} +{\left (125 \, d^{5} e - 50 \, d^{4} e^{2} + 130 \, d^{3} e^{3} + 8 \, d^{2} e^{4} + 21 \, d e^{5} + 18 \, e^{6}\right )} x^{2} +{\left (50 \, d^{5} e + 35 \, d^{4} e^{2} + 8 \, d^{3} e^{3} + 78 \, d^{2} e^{4} - 18 \, d e^{5} + 27 \, e^{6}\right )} x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/392*sqrt(14)*(1313*d^4 - 10044*d^3*e + 4290*d^2*e^2 + 156*d*e^3 - 271*e^4)*arctan(1/14*sqrt(14)*(5*x + 1))/(
125*d^6 - 150*d^5*e + 285*d^4*e^2 - 188*d^3*e^3 + 171*d^2*e^4 - 54*d*e^5 + 27*e^6) + (41*d^4 - 8*d^3*e - 60*d^
2*e^2 + 24*d*e^3 - 5*e^4)*log(e*x + d)/(125*d^6 - 150*d^5*e + 285*d^4*e^2 - 188*d^3*e^3 + 171*d^2*e^4 - 54*d*e
^5 + 27*e^6) - 1/2*(41*d^4 - 8*d^3*e - 60*d^2*e^2 + 24*d*e^3 - 5*e^4)*log(5*x^2 + 2*x + 3)/(125*d^6 - 150*d^5*
e + 285*d^4*e^2 - 188*d^3*e^3 + 171*d^2*e^4 - 54*d*e^5 + 27*e^6) - 1/140*(1680*d^4 + 3467*d^3*e + 674*d^2*e^2
- 1123*d*e^3 + 840*e^4 + (2800*d^4 + 3500*d^3*e + 2523*d^2*e^2 - 3434*d*e^3 + 1693*e^4)*x^2 + (1120*d^4 + 1823
*d^3*e - 527*d^2*e^2 - 573*d*e^3 - 143*e^4)*x)/(75*d^5*e - 60*d^4*e^2 + 102*d^3*e^3 - 36*d^2*e^4 + 27*d*e^5 +
5*(25*d^4*e^2 - 20*d^3*e^3 + 34*d^2*e^4 - 12*d*e^5 + 9*e^6)*x^3 + (125*d^5*e - 50*d^4*e^2 + 130*d^3*e^3 + 8*d^
2*e^4 + 21*d*e^5 + 18*e^6)*x^2 + (50*d^5*e + 35*d^4*e^2 + 8*d^3*e^3 + 78*d^2*e^4 - 18*d*e^5 + 27*e^6)*x)

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Fricas [B]  time = 1.91916, size = 2264, normalized size = 7.23 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1960*(117600*d^6 + 195650*d^5*e + 20664*d^4*e^2 + 48132*d^3*e^3 + 118552*d^2*e^4 - 70686*d*e^5 + 35280*e^6
+ 14*(14000*d^6 + 11900*d^5*e + 14015*d^4*e^2 - 11716*d^3*e^3 + 22902*d^2*e^4 - 13688*d*e^5 + 5079*e^6)*x^2 -
5*sqrt(14)*(3939*d^5*e - 30132*d^4*e^2 + 12870*d^3*e^3 + 468*d^2*e^4 - 813*d*e^5 + 5*(1313*d^4*e^2 - 10044*d^3
*e^3 + 4290*d^2*e^4 + 156*d*e^5 - 271*e^6)*x^3 + (6565*d^5*e - 47594*d^4*e^2 + 1362*d^3*e^3 + 9360*d^2*e^4 - 1
043*d*e^5 - 542*e^6)*x^2 + (2626*d^5*e - 16149*d^4*e^2 - 21552*d^3*e^3 + 13182*d^2*e^4 - 74*d*e^5 - 813*e^6)*x
)*arctan(1/14*sqrt(14)*(5*x + 1)) + 14*(5600*d^6 + 6875*d^5*e - 2921*d^4*e^2 + 3658*d^3*e^3 - 1150*d^2*e^4 - 1
433*d*e^5 - 429*e^6)*x - 1960*(123*d^5*e - 24*d^4*e^2 - 180*d^3*e^3 + 72*d^2*e^4 - 15*d*e^5 + 5*(41*d^4*e^2 -
8*d^3*e^3 - 60*d^2*e^4 + 24*d*e^5 - 5*e^6)*x^3 + (205*d^5*e + 42*d^4*e^2 - 316*d^3*e^3 + 23*d*e^5 - 10*e^6)*x^
2 + (82*d^5*e + 107*d^4*e^2 - 144*d^3*e^3 - 132*d^2*e^4 + 62*d*e^5 - 15*e^6)*x)*log(e*x + d) + 980*(123*d^5*e
- 24*d^4*e^2 - 180*d^3*e^3 + 72*d^2*e^4 - 15*d*e^5 + 5*(41*d^4*e^2 - 8*d^3*e^3 - 60*d^2*e^4 + 24*d*e^5 - 5*e^6
)*x^3 + (205*d^5*e + 42*d^4*e^2 - 316*d^3*e^3 + 23*d*e^5 - 10*e^6)*x^2 + (82*d^5*e + 107*d^4*e^2 - 144*d^3*e^3
- 132*d^2*e^4 + 62*d*e^5 - 15*e^6)*x)*log(5*x^2 + 2*x + 3))/(375*d^7*e - 450*d^6*e^2 + 855*d^5*e^3 - 564*d^4*
e^4 + 513*d^3*e^5 - 162*d^2*e^6 + 81*d*e^7 + 5*(125*d^6*e^2 - 150*d^5*e^3 + 285*d^4*e^4 - 188*d^3*e^5 + 171*d^
2*e^6 - 54*d*e^7 + 27*e^8)*x^3 + (625*d^7*e - 500*d^6*e^2 + 1125*d^5*e^3 - 370*d^4*e^4 + 479*d^3*e^5 + 72*d^2*
e^6 + 27*d*e^7 + 54*e^8)*x^2 + (250*d^7*e + 75*d^6*e^2 + 120*d^5*e^3 + 479*d^4*e^4 - 222*d^3*e^5 + 405*d^2*e^6
- 108*d*e^7 + 81*e^8)*x)

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Sympy [C]  time = 27.3404, size = 13362, normalized size = 42.69 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-sqrt(14)*I*(1313*d**4 - 10044*d**3*e + 4290*d**2*e**2 + 156*d*e**3 - 271*e**4)/(784*(125*d**6 - 150*d**5*e +
285*d**4*e**2 - 188*d**3*e**3 + 171*d**2*e**4 - 54*d*e**5 + 27*e**6)) - (41*d**4 - 8*d**3*e - 60*d**2*e**2 +
24*d*e**3 - 5*e**4)/(2*(5*d**2 - 2*d*e + 3*e**2)**3))*log(x + (4503590000000*d**17*(-sqrt(14)*I*(1313*d**4 - 1
0044*d**3*e + 4290*d**2*e**2 + 156*d*e**3 - 271*e**4)/(784*(125*d**6 - 150*d**5*e + 285*d**4*e**2 - 188*d**3*e
**3 + 171*d**2*e**4 - 54*d*e**5 + 27*e**6)) - (41*d**4 - 8*d**3*e - 60*d**2*e**2 + 24*d*e**3 - 5*e**4)/(2*(5*d
**2 - 2*d*e + 3*e**2)**3))**2 - 79236430000000*d**16*e*(-sqrt(14)*I*(1313*d**4 - 10044*d**3*e + 4290*d**2*e**2
+ 156*d*e**3 - 271*e**4)/(784*(125*d**6 - 150*d**5*e + 285*d**4*e**2 - 188*d**3*e**3 + 171*d**2*e**4 - 54*d*e
**5 + 27*e**6)) - (41*d**4 - 8*d**3*e - 60*d**2*e**2 + 24*d*e**3 - 5*e**4)/(2*(5*d**2 - 2*d*e + 3*e**2)**3))**
2 + 219307065600000*d**15*e**2*(-sqrt(14)*I*(1313*d**4 - 10044*d**3*e + 4290*d**2*e**2 + 156*d*e**3 - 271*e**4
)/(784*(125*d**6 - 150*d**5*e + 285*d**4*e**2 - 188*d**3*e**3 + 171*d**2*e**4 - 54*d*e**5 + 27*e**6)) - (41*d*
*4 - 8*d**3*e - 60*d**2*e**2 + 24*d*e**3 - 5*e**4)/(2*(5*d**2 - 2*d*e + 3*e**2)**3))**2 + 1477177520000*d**15*
(-sqrt(14)*I*(1313*d**4 - 10044*d**3*e + 4290*d**2*e**2 + 156*d*e**3 - 271*e**4)/(784*(125*d**6 - 150*d**5*e +
285*d**4*e**2 - 188*d**3*e**3 + 171*d**2*e**4 - 54*d*e**5 + 27*e**6)) - (41*d**4 - 8*d**3*e - 60*d**2*e**2 +
24*d*e**3 - 5*e**4)/(2*(5*d**2 - 2*d*e + 3*e**2)**3)) - 470102080000000*d**14*e**3*(-sqrt(14)*I*(1313*d**4 - 1
0044*d**3*e + 4290*d**2*e**2 + 156*d*e**3 - 271*e**4)/(784*(125*d**6 - 150*d**5*e + 285*d**4*e**2 - 188*d**3*e
**3 + 171*d**2*e**4 - 54*d*e**5 + 27*e**6)) - (41*d**4 - 8*d**3*e - 60*d**2*e**2 + 24*d*e**3 - 5*e**4)/(2*(5*d
**2 - 2*d*e + 3*e**2)**3))**2 - 8062738222000*d**14*e*(-sqrt(14)*I*(1313*d**4 - 10044*d**3*e + 4290*d**2*e**2
+ 156*d*e**3 - 271*e**4)/(784*(125*d**6 - 150*d**5*e + 285*d**4*e**2 - 188*d**3*e**3 + 171*d**2*e**4 - 54*d*e*
*5 + 27*e**6)) - (41*d**4 - 8*d**3*e - 60*d**2*e**2 + 24*d*e**3 - 5*e**4)/(2*(5*d**2 - 2*d*e + 3*e**2)**3)) +
669820607680000*d**13*e**4*(-sqrt(14)*I*(1313*d**4 - 10044*d**3*e + 4290*d**2*e**2 + 156*d*e**3 - 271*e**4)/(7
84*(125*d**6 - 150*d**5*e + 285*d**4*e**2 - 188*d**3*e**3 + 171*d**2*e**4 - 54*d*e**5 + 27*e**6)) - (41*d**4 -
8*d**3*e - 60*d**2*e**2 + 24*d*e**3 - 5*e**4)/(2*(5*d**2 - 2*d*e + 3*e**2)**3))**2 + 13401619991200*d**13*e**
2*(-sqrt(14)*I*(1313*d**4 - 10044*d**3*e + 4290*d**2*e**2 + 156*d*e**3 - 271*e**4)/(784*(125*d**6 - 150*d**5*e
+ 285*d**4*e**2 - 188*d**3*e**3 + 171*d**2*e**4 - 54*d*e**5 + 27*e**6)) - (41*d**4 - 8*d**3*e - 60*d**2*e**2
+ 24*d*e**3 - 5*e**4)/(2*(5*d**2 - 2*d*e + 3*e**2)**3)) + 132446413125*d**13 - 748279970905600*d**12*e**5*(-sq
rt(14)*I*(1313*d**4 - 10044*d**3*e + 4290*d**2*e**2 + 156*d*e**3 - 271*e**4)/(784*(125*d**6 - 150*d**5*e + 285
*d**4*e**2 - 188*d**3*e**3 + 171*d**2*e**4 - 54*d*e**5 + 27*e**6)) - (41*d**4 - 8*d**3*e - 60*d**2*e**2 + 24*d
*e**3 - 5*e**4)/(2*(5*d**2 - 2*d*e + 3*e**2)**3))**2 - 8564369003120*d**12*e**3*(-sqrt(14)*I*(1313*d**4 - 1004
4*d**3*e + 4290*d**2*e**2 + 156*d*e**3 - 271*e**4)/(784*(125*d**6 - 150*d**5*e + 285*d**4*e**2 - 188*d**3*e**3
+ 171*d**2*e**4 - 54*d*e**5 + 27*e**6)) - (41*d**4 - 8*d**3*e - 60*d**2*e**2 + 24*d*e**3 - 5*e**4)/(2*(5*d**2
- 2*d*e + 3*e**2)**3)) + 684029295980*d**12*e + 599319595212800*d**11*e**6*(-sqrt(14)*I*(1313*d**4 - 10044*d*
*3*e + 4290*d**2*e**2 + 156*d*e**3 - 271*e**4)/(784*(125*d**6 - 150*d**5*e + 285*d**4*e**2 - 188*d**3*e**3 + 1
71*d**2*e**4 - 54*d*e**5 + 27*e**6)) - (41*d**4 - 8*d**3*e - 60*d**2*e**2 + 24*d*e**3 - 5*e**4)/(2*(5*d**2 - 2
*d*e + 3*e**2)**3))**2 - 12510243478208*d**11*e**4*(-sqrt(14)*I*(1313*d**4 - 10044*d**3*e + 4290*d**2*e**2 + 1
56*d*e**3 - 271*e**4)/(784*(125*d**6 - 150*d**5*e + 285*d**4*e**2 - 188*d**3*e**3 + 171*d**2*e**4 - 54*d*e**5
+ 27*e**6)) - (41*d**4 - 8*d**3*e - 60*d**2*e**2 + 24*d*e**3 - 5*e**4)/(2*(5*d**2 - 2*d*e + 3*e**2)**3)) + 764
206623630*d**11*e**2 - 291411662710784*d**10*e**7*(-sqrt(14)*I*(1313*d**4 - 10044*d**3*e + 4290*d**2*e**2 + 15
6*d*e**3 - 271*e**4)/(784*(125*d**6 - 150*d**5*e + 285*d**4*e**2 - 188*d**3*e**3 + 171*d**2*e**4 - 54*d*e**5 +
27*e**6)) - (41*d**4 - 8*d**3*e - 60*d**2*e**2 + 24*d*e**3 - 5*e**4)/(2*(5*d**2 - 2*d*e + 3*e**2)**3))**2 + 4
9159980986704*d**10*e**5*(-sqrt(14)*I*(1313*d**4 - 10044*d**3*e + 4290*d**2*e**2 + 156*d*e**3 - 271*e**4)/(784
*(125*d**6 - 150*d**5*e + 285*d**4*e**2 - 188*d**3*e**3 + 171*d**2*e**4 - 54*d*e**5 + 27*e**6)) - (41*d**4 - 8
*d**3*e - 60*d**2*e**2 + 24*d*e**3 - 5*e**4)/(2*(5*d**2 - 2*d*e + 3*e**2)**3)) - 14657220189100*d**10*e**3 - 2
7190445185792*d**9*e**8*(-sqrt(14)*I*(1313*d**4 - 10044*d**3*e + 4290*d**2*e**2 + 156*d*e**3 - 271*e**4)/(784*
(125*d**6 - 150*d**5*e + 285*d**4*e**2 - 188*d**3*e**3 + 171*d**2*e**4 - 54*d*e**5 + 27*e**6)) - (41*d**4 - 8*
d**3*e - 60*d**2*e**2 + 24*d*e**3 - 5*e**4)/(2*(5*d**2 - 2*d*e + 3*e**2)**3))**2 - 77659175364512*d**9*e**6*(-
sqrt(14)*I*(1313*d**4 - 10044*d**3*e + 4290*d**2*e**2 + 156*d*e**3 - 271*e**4)/(784*(125*d**6 - 150*d**5*e + 2
85*d**4*e**2 - 188*d**3*e**3 + 171*d**2*e**4 - 54*d*e**5 + 27*e**6)) - (41*d**4 - 8*d**3*e - 60*d**2*e**2 + 24
*d*e**3 - 5*e**4)/(2*(5*d**2 - 2*d*e + 3*e**2)**3)) + 16942805253691*d**9*e**4 + 253830846834432*d**8*e**9*(-s
qrt(14)*I*(1313*d**4 - 10044*d**3*e + 4290*d**2*e**2 + 156*d*e**3 - 271*e**4)/(784*(125*d**6 - 150*d**5*e + 28
5*d**4*e**2 - 188*d**3*e**3 + 171*d**2*e**4 - 54*d*e**5 + 27*e**6)) - (41*d**4 - 8*d**3*e - 60*d**2*e**2 + 24*
d*e**3 - 5*e**4)/(2*(5*d**2 - 2*d*e + 3*e**2)**3))**2 + 91313688339216*d**8*e**7*(-sqrt(14)*I*(1313*d**4 - 100
44*d**3*e + 4290*d**2*e**2 + 156*d*e**3 - 271*e**4)/(784*(125*d**6 - 150*d**5*e + 285*d**4*e**2 - 188*d**3*e**
3 + 171*d**2*e**4 - 54*d*e**5 + 27*e**6)) - (41*d**4 - 8*d**3*e - 60*d**2*e**2 + 24*d*e**3 - 5*e**4)/(2*(5*d**
2 - 2*d*e + 3*e**2)**3)) + 6404919470120*d**8*e**5 - 308064129587200*d**7*e**10*(-sqrt(14)*I*(1313*d**4 - 1004
4*d**3*e + 4290*d**2*e**2 + 156*d*e**3 - 271*e**4)/(784*(125*d**6 - 150*d**5*e + 285*d**4*e**2 - 188*d**3*e**3
+ 171*d**2*e**4 - 54*d*e**5 + 27*e**6)) - (41*d**4 - 8*d**3*e - 60*d**2*e**2 + 24*d*e**3 - 5*e**4)/(2*(5*d**2
- 2*d*e + 3*e**2)**3))**2 - 78573287795968*d**7*e**8*(-sqrt(14)*I*(1313*d**4 - 10044*d**3*e + 4290*d**2*e**2
+ 156*d*e**3 - 271*e**4)/(784*(125*d**6 - 150*d**5*e + 285*d**4*e**2 - 188*d**3*e**3 + 171*d**2*e**4 - 54*d*e*
*5 + 27*e**6)) - (41*d**4 - 8*d**3*e - 60*d**2*e**2 + 24*d*e**3 - 5*e**4)/(2*(5*d**2 - 2*d*e + 3*e**2)**3)) -
16998879119292*d**7*e**6 + 262468005502976*d**6*e**11*(-sqrt(14)*I*(1313*d**4 - 10044*d**3*e + 4290*d**2*e**2
+ 156*d*e**3 - 271*e**4)/(784*(125*d**6 - 150*d**5*e + 285*d**4*e**2 - 188*d**3*e**3 + 171*d**2*e**4 - 54*d*e*
*5 + 27*e**6)) - (41*d**4 - 8*d**3*e - 60*d**2*e**2 + 24*d*e**3 - 5*e**4)/(2*(5*d**2 - 2*d*e + 3*e**2)**3))**2
+ 55676827575152*d**6*e**9*(-sqrt(14)*I*(1313*d**4 - 10044*d**3*e + 4290*d**2*e**2 + 156*d*e**3 - 271*e**4)/(
784*(125*d**6 - 150*d**5*e + 285*d**4*e**2 - 188*d**3*e**3 + 171*d**2*e**4 - 54*d*e**5 + 27*e**6)) - (41*d**4
- 8*d**3*e - 60*d**2*e**2 + 24*d*e**3 - 5*e**4)/(2*(5*d**2 - 2*d*e + 3*e**2)**3)) + 5633839731848*d**6*e**7 -
162086347196928*d**5*e**12*(-sqrt(14)*I*(1313*d**4 - 10044*d**3*e + 4290*d**2*e**2 + 156*d*e**3 - 271*e**4)/(7
84*(125*d**6 - 150*d**5*e + 285*d**4*e**2 - 188*d**3*e**3 + 171*d**2*e**4 - 54*d*e**5 + 27*e**6)) - (41*d**4 -
8*d**3*e - 60*d**2*e**2 + 24*d*e**3 - 5*e**4)/(2*(5*d**2 - 2*d*e + 3*e**2)**3))**2 - 30431528150688*d**5*e**1
0*(-sqrt(14)*I*(1313*d**4 - 10044*d**3*e + 4290*d**2*e**2 + 156*d*e**3 - 271*e**4)/(784*(125*d**6 - 150*d**5*e
+ 285*d**4*e**2 - 188*d**3*e**3 + 171*d**2*e**4 - 54*d*e**5 + 27*e**6)) - (41*d**4 - 8*d**3*e - 60*d**2*e**2
+ 24*d*e**3 - 5*e**4)/(2*(5*d**2 - 2*d*e + 3*e**2)**3)) + 3033254622763*d**5*e**8 + 82236632099328*d**4*e**13*
(-sqrt(14)*I*(1313*d**4 - 10044*d**3*e + 4290*d**2*e**2 + 156*d*e**3 - 271*e**4)/(784*(125*d**6 - 150*d**5*e +
285*d**4*e**2 - 188*d**3*e**3 + 171*d**2*e**4 - 54*d*e**5 + 27*e**6)) - (41*d**4 - 8*d**3*e - 60*d**2*e**2 +
24*d*e**3 - 5*e**4)/(2*(5*d**2 - 2*d*e + 3*e**2)**3))**2 + 13587008752688*d**4*e**11*(-sqrt(14)*I*(1313*d**4 -
10044*d**3*e + 4290*d**2*e**2 + 156*d*e**3 - 271*e**4)/(784*(125*d**6 - 150*d**5*e + 285*d**4*e**2 - 188*d**3
*e**3 + 171*d**2*e**4 - 54*d*e**5 + 27*e**6)) - (41*d**4 - 8*d**3*e - 60*d**2*e**2 + 24*d*e**3 - 5*e**4)/(2*(5
*d**2 - 2*d*e + 3*e**2)**3)) - 3506827379684*d**4*e**9 - 30865482805248*d**3*e**14*(-sqrt(14)*I*(1313*d**4 - 1
0044*d**3*e + 4290*d**2*e**2 + 156*d*e**3 - 271*e**4)/(784*(125*d**6 - 150*d**5*e + 285*d**4*e**2 - 188*d**3*e
**3 + 171*d**2*e**4 - 54*d*e**5 + 27*e**6)) - (41*d**4 - 8*d**3*e - 60*d**2*e**2 + 24*d*e**3 - 5*e**4)/(2*(5*d
**2 - 2*d*e + 3*e**2)**3))**2 - 4535008734144*d**3*e**12*(-sqrt(14)*I*(1313*d**4 - 10044*d**3*e + 4290*d**2*e*
*2 + 156*d*e**3 - 271*e**4)/(784*(125*d**6 - 150*d**5*e + 285*d**4*e**2 - 188*d**3*e**3 + 171*d**2*e**4 - 54*d
*e**5 + 27*e**6)) - (41*d**4 - 8*d**3*e - 60*d**2*e**2 + 24*d*e**3 - 5*e**4)/(2*(5*d**2 - 2*d*e + 3*e**2)**3))
+ 1484229456462*d**3*e**10 + 9233948989440*d**2*e**15*(-sqrt(14)*I*(1313*d**4 - 10044*d**3*e + 4290*d**2*e**2
+ 156*d*e**3 - 271*e**4)/(784*(125*d**6 - 150*d**5*e + 285*d**4*e**2 - 188*d**3*e**3 + 171*d**2*e**4 - 54*d*e
**5 + 27*e**6)) - (41*d**4 - 8*d**3*e - 60*d**2*e**2 + 24*d*e**3 - 5*e**4)/(2*(5*d**2 - 2*d*e + 3*e**2)**3))**
2 + 1144385029872*d**2*e**13*(-sqrt(14)*I*(1313*d**4 - 10044*d**3*e + 4290*d**2*e**2 + 156*d*e**3 - 271*e**4)/
(784*(125*d**6 - 150*d**5*e + 285*d**4*e**2 - 188*d**3*e**3 + 171*d**2*e**4 - 54*d*e**5 + 27*e**6)) - (41*d**4
- 8*d**3*e - 60*d**2*e**2 + 24*d*e**3 - 5*e**4)/(2*(5*d**2 - 2*d*e + 3*e**2)**3)) - 361088969436*d**2*e**11 -
1739174903424*d*e**16*(-sqrt(14)*I*(1313*d**4 - 10044*d**3*e + 4290*d**2*e**2 + 156*d*e**3 - 271*e**4)/(784*(
125*d**6 - 150*d**5*e + 285*d**4*e**2 - 188*d**3*e**3 + 171*d**2*e**4 - 54*d*e**5 + 27*e**6)) - (41*d**4 - 8*d
**3*e - 60*d**2*e**2 + 24*d*e**3 - 5*e**4)/(2*(5*d**2 - 2*d*e + 3*e**2)**3))**2 - 187156660320*d*e**14*(-sqrt(
14)*I*(1313*d**4 - 10044*d**3*e + 4290*d**2*e**2 + 156*d*e**3 - 271*e**4)/(784*(125*d**6 - 150*d**5*e + 285*d*
*4*e**2 - 188*d**3*e**3 + 171*d**2*e**4 - 54*d*e**5 + 27*e**6)) - (41*d**4 - 8*d**3*e - 60*d**2*e**2 + 24*d*e*
*3 - 5*e**4)/(2*(5*d**2 - 2*d*e + 3*e**2)**3)) + 50336842869*d*e**12 + 196869004416*e**17*(-sqrt(14)*I*(1313*d
**4 - 10044*d**3*e + 4290*d**2*e**2 + 156*d*e**3 - 271*e**4)/(784*(125*d**6 - 150*d**5*e + 285*d**4*e**2 - 188
*d**3*e**3 + 171*d**2*e**4 - 54*d*e**5 + 27*e**6)) - (41*d**4 - 8*d**3*e - 60*d**2*e**2 + 24*d*e**3 - 5*e**4)/
(2*(5*d**2 - 2*d*e + 3*e**2)**3))**2 + 17373868848*e**15*(-sqrt(14)*I*(1313*d**4 - 10044*d**3*e + 4290*d**2*e*
*2 + 156*d*e**3 - 271*e**4)/(784*(125*d**6 - 150*d**5*e + 285*d**4*e**2 - 188*d**3*e**3 + 171*d**2*e**4 - 54*d
*e**5 + 27*e**6)) - (41*d**4 - 8*d**3*e - 60*d**2*e**2 + 24*d*e**3 - 5*e**4)/(2*(5*d**2 - 2*d*e + 3*e**2)**3))
- 3533954480*e**13)/(1101474866245*d**12*e - 9024487794180*d**11*e**2 + 5764879624590*d**10*e**3 + 1796913697
1220*d**9*e**4 - 16485388615365*d**8*e**5 - 12221510721480*d**7*e**6 + 21212253502020*d**6*e**7 - 117103352353
20*d**5*e**8 + 3048287389995*d**4*e**9 - 183650820660*d**3*e**10 - 118302770610*d**2*e**11 + 34222696740*d*e**
12 - 3445820555*e**13)) + (sqrt(14)*I*(1313*d**4 - 10044*d**3*e + 4290*d**2*e**2 + 156*d*e**3 - 271*e**4)/(784
*(125*d**6 - 150*d**5*e + 285*d**4*e**2 - 188*d**3*e**3 + 171*d**2*e**4 - 54*d*e**5 + 27*e**6)) - (41*d**4 - 8
*d**3*e - 60*d**2*e**2 + 24*d*e**3 - 5*e**4)/(2*(5*d**2 - 2*d*e + 3*e**2)**3))*log(x + (4503590000000*d**17*(s
qrt(14)*I*(1313*d**4 - 10044*d**3*e + 4290*d**2*e**2 + 156*d*e**3 - 271*e**4)/(784*(125*d**6 - 150*d**5*e + 28
5*d**4*e**2 - 188*d**3*e**3 + 171*d**2*e**4 - 54*d*e**5 + 27*e**6)) - (41*d**4 - 8*d**3*e - 60*d**2*e**2 + 24*
d*e**3 - 5*e**4)/(2*(5*d**2 - 2*d*e + 3*e**2)**3))**2 - 79236430000000*d**16*e*(sqrt(14)*I*(1313*d**4 - 10044*
d**3*e + 4290*d**2*e**2 + 156*d*e**3 - 271*e**4)/(784*(125*d**6 - 150*d**5*e + 285*d**4*e**2 - 188*d**3*e**3 +
171*d**2*e**4 - 54*d*e**5 + 27*e**6)) - (41*d**4 - 8*d**3*e - 60*d**2*e**2 + 24*d*e**3 - 5*e**4)/(2*(5*d**2 -
2*d*e + 3*e**2)**3))**2 + 219307065600000*d**15*e**2*(sqrt(14)*I*(1313*d**4 - 10044*d**3*e + 4290*d**2*e**2 +
156*d*e**3 - 271*e**4)/(784*(125*d**6 - 150*d**5*e + 285*d**4*e**2 - 188*d**3*e**3 + 171*d**2*e**4 - 54*d*e**
5 + 27*e**6)) - (41*d**4 - 8*d**3*e - 60*d**2*e**2 + 24*d*e**3 - 5*e**4)/(2*(5*d**2 - 2*d*e + 3*e**2)**3))**2
+ 1477177520000*d**15*(sqrt(14)*I*(1313*d**4 - 10044*d**3*e + 4290*d**2*e**2 + 156*d*e**3 - 271*e**4)/(784*(12
5*d**6 - 150*d**5*e + 285*d**4*e**2 - 188*d**3*e**3 + 171*d**2*e**4 - 54*d*e**5 + 27*e**6)) - (41*d**4 - 8*d**
3*e - 60*d**2*e**2 + 24*d*e**3 - 5*e**4)/(2*(5*d**2 - 2*d*e + 3*e**2)**3)) - 470102080000000*d**14*e**3*(sqrt(
14)*I*(1313*d**4 - 10044*d**3*e + 4290*d**2*e**2 + 156*d*e**3 - 271*e**4)/(784*(125*d**6 - 150*d**5*e + 285*d*
*4*e**2 - 188*d**3*e**3 + 171*d**2*e**4 - 54*d*e**5 + 27*e**6)) - (41*d**4 - 8*d**3*e - 60*d**2*e**2 + 24*d*e*
*3 - 5*e**4)/(2*(5*d**2 - 2*d*e + 3*e**2)**3))**2 - 8062738222000*d**14*e*(sqrt(14)*I*(1313*d**4 - 10044*d**3*
e + 4290*d**2*e**2 + 156*d*e**3 - 271*e**4)/(784*(125*d**6 - 150*d**5*e + 285*d**4*e**2 - 188*d**3*e**3 + 171*
d**2*e**4 - 54*d*e**5 + 27*e**6)) - (41*d**4 - 8*d**3*e - 60*d**2*e**2 + 24*d*e**3 - 5*e**4)/(2*(5*d**2 - 2*d*
e + 3*e**2)**3)) + 669820607680000*d**13*e**4*(sqrt(14)*I*(1313*d**4 - 10044*d**3*e + 4290*d**2*e**2 + 156*d*e
**3 - 271*e**4)/(784*(125*d**6 - 150*d**5*e + 285*d**4*e**2 - 188*d**3*e**3 + 171*d**2*e**4 - 54*d*e**5 + 27*e
**6)) - (41*d**4 - 8*d**3*e - 60*d**2*e**2 + 24*d*e**3 - 5*e**4)/(2*(5*d**2 - 2*d*e + 3*e**2)**3))**2 + 134016
19991200*d**13*e**2*(sqrt(14)*I*(1313*d**4 - 10044*d**3*e + 4290*d**2*e**2 + 156*d*e**3 - 271*e**4)/(784*(125*
d**6 - 150*d**5*e + 285*d**4*e**2 - 188*d**3*e**3 + 171*d**2*e**4 - 54*d*e**5 + 27*e**6)) - (41*d**4 - 8*d**3*
e - 60*d**2*e**2 + 24*d*e**3 - 5*e**4)/(2*(5*d**2 - 2*d*e + 3*e**2)**3)) + 132446413125*d**13 - 74827997090560
0*d**12*e**5*(sqrt(14)*I*(1313*d**4 - 10044*d**3*e + 4290*d**2*e**2 + 156*d*e**3 - 271*e**4)/(784*(125*d**6 -
150*d**5*e + 285*d**4*e**2 - 188*d**3*e**3 + 171*d**2*e**4 - 54*d*e**5 + 27*e**6)) - (41*d**4 - 8*d**3*e - 60*
d**2*e**2 + 24*d*e**3 - 5*e**4)/(2*(5*d**2 - 2*d*e + 3*e**2)**3))**2 - 8564369003120*d**12*e**3*(sqrt(14)*I*(1
313*d**4 - 10044*d**3*e + 4290*d**2*e**2 + 156*d*e**3 - 271*e**4)/(784*(125*d**6 - 150*d**5*e + 285*d**4*e**2
- 188*d**3*e**3 + 171*d**2*e**4 - 54*d*e**5 + 27*e**6)) - (41*d**4 - 8*d**3*e - 60*d**2*e**2 + 24*d*e**3 - 5*e
**4)/(2*(5*d**2 - 2*d*e + 3*e**2)**3)) + 684029295980*d**12*e + 599319595212800*d**11*e**6*(sqrt(14)*I*(1313*d
**4 - 10044*d**3*e + 4290*d**2*e**2 + 156*d*e**3 - 271*e**4)/(784*(125*d**6 - 150*d**5*e + 285*d**4*e**2 - 188
*d**3*e**3 + 171*d**2*e**4 - 54*d*e**5 + 27*e**6)) - (41*d**4 - 8*d**3*e - 60*d**2*e**2 + 24*d*e**3 - 5*e**4)/
(2*(5*d**2 - 2*d*e + 3*e**2)**3))**2 - 12510243478208*d**11*e**4*(sqrt(14)*I*(1313*d**4 - 10044*d**3*e + 4290*
d**2*e**2 + 156*d*e**3 - 271*e**4)/(784*(125*d**6 - 150*d**5*e + 285*d**4*e**2 - 188*d**3*e**3 + 171*d**2*e**4
- 54*d*e**5 + 27*e**6)) - (41*d**4 - 8*d**3*e - 60*d**2*e**2 + 24*d*e**3 - 5*e**4)/(2*(5*d**2 - 2*d*e + 3*e**
2)**3)) + 764206623630*d**11*e**2 - 291411662710784*d**10*e**7*(sqrt(14)*I*(1313*d**4 - 10044*d**3*e + 4290*d*
*2*e**2 + 156*d*e**3 - 271*e**4)/(784*(125*d**6 - 150*d**5*e + 285*d**4*e**2 - 188*d**3*e**3 + 171*d**2*e**4 -
54*d*e**5 + 27*e**6)) - (41*d**4 - 8*d**3*e - 60*d**2*e**2 + 24*d*e**3 - 5*e**4)/(2*(5*d**2 - 2*d*e + 3*e**2)
**3))**2 + 49159980986704*d**10*e**5*(sqrt(14)*I*(1313*d**4 - 10044*d**3*e + 4290*d**2*e**2 + 156*d*e**3 - 271
*e**4)/(784*(125*d**6 - 150*d**5*e + 285*d**4*e**2 - 188*d**3*e**3 + 171*d**2*e**4 - 54*d*e**5 + 27*e**6)) - (
41*d**4 - 8*d**3*e - 60*d**2*e**2 + 24*d*e**3 - 5*e**4)/(2*(5*d**2 - 2*d*e + 3*e**2)**3)) - 14657220189100*d**
10*e**3 - 27190445185792*d**9*e**8*(sqrt(14)*I*(1313*d**4 - 10044*d**3*e + 4290*d**2*e**2 + 156*d*e**3 - 271*e
**4)/(784*(125*d**6 - 150*d**5*e + 285*d**4*e**2 - 188*d**3*e**3 + 171*d**2*e**4 - 54*d*e**5 + 27*e**6)) - (41
*d**4 - 8*d**3*e - 60*d**2*e**2 + 24*d*e**3 - 5*e**4)/(2*(5*d**2 - 2*d*e + 3*e**2)**3))**2 - 77659175364512*d*
*9*e**6*(sqrt(14)*I*(1313*d**4 - 10044*d**3*e + 4290*d**2*e**2 + 156*d*e**3 - 271*e**4)/(784*(125*d**6 - 150*d
**5*e + 285*d**4*e**2 - 188*d**3*e**3 + 171*d**2*e**4 - 54*d*e**5 + 27*e**6)) - (41*d**4 - 8*d**3*e - 60*d**2*
e**2 + 24*d*e**3 - 5*e**4)/(2*(5*d**2 - 2*d*e + 3*e**2)**3)) + 16942805253691*d**9*e**4 + 253830846834432*d**8
*e**9*(sqrt(14)*I*(1313*d**4 - 10044*d**3*e + 4290*d**2*e**2 + 156*d*e**3 - 271*e**4)/(784*(125*d**6 - 150*d**
5*e + 285*d**4*e**2 - 188*d**3*e**3 + 171*d**2*e**4 - 54*d*e**5 + 27*e**6)) - (41*d**4 - 8*d**3*e - 60*d**2*e*
*2 + 24*d*e**3 - 5*e**4)/(2*(5*d**2 - 2*d*e + 3*e**2)**3))**2 + 91313688339216*d**8*e**7*(sqrt(14)*I*(1313*d**
4 - 10044*d**3*e + 4290*d**2*e**2 + 156*d*e**3 - 271*e**4)/(784*(125*d**6 - 150*d**5*e + 285*d**4*e**2 - 188*d
**3*e**3 + 171*d**2*e**4 - 54*d*e**5 + 27*e**6)) - (41*d**4 - 8*d**3*e - 60*d**2*e**2 + 24*d*e**3 - 5*e**4)/(2
*(5*d**2 - 2*d*e + 3*e**2)**3)) + 6404919470120*d**8*e**5 - 308064129587200*d**7*e**10*(sqrt(14)*I*(1313*d**4
- 10044*d**3*e + 4290*d**2*e**2 + 156*d*e**3 - 271*e**4)/(784*(125*d**6 - 150*d**5*e + 285*d**4*e**2 - 188*d**
3*e**3 + 171*d**2*e**4 - 54*d*e**5 + 27*e**6)) - (41*d**4 - 8*d**3*e - 60*d**2*e**2 + 24*d*e**3 - 5*e**4)/(2*(
5*d**2 - 2*d*e + 3*e**2)**3))**2 - 78573287795968*d**7*e**8*(sqrt(14)*I*(1313*d**4 - 10044*d**3*e + 4290*d**2*
e**2 + 156*d*e**3 - 271*e**4)/(784*(125*d**6 - 150*d**5*e + 285*d**4*e**2 - 188*d**3*e**3 + 171*d**2*e**4 - 54
*d*e**5 + 27*e**6)) - (41*d**4 - 8*d**3*e - 60*d**2*e**2 + 24*d*e**3 - 5*e**4)/(2*(5*d**2 - 2*d*e + 3*e**2)**3
)) - 16998879119292*d**7*e**6 + 262468005502976*d**6*e**11*(sqrt(14)*I*(1313*d**4 - 10044*d**3*e + 4290*d**2*e
**2 + 156*d*e**3 - 271*e**4)/(784*(125*d**6 - 150*d**5*e + 285*d**4*e**2 - 188*d**3*e**3 + 171*d**2*e**4 - 54*
d*e**5 + 27*e**6)) - (41*d**4 - 8*d**3*e - 60*d**2*e**2 + 24*d*e**3 - 5*e**4)/(2*(5*d**2 - 2*d*e + 3*e**2)**3)
)**2 + 55676827575152*d**6*e**9*(sqrt(14)*I*(1313*d**4 - 10044*d**3*e + 4290*d**2*e**2 + 156*d*e**3 - 271*e**4
)/(784*(125*d**6 - 150*d**5*e + 285*d**4*e**2 - 188*d**3*e**3 + 171*d**2*e**4 - 54*d*e**5 + 27*e**6)) - (41*d*
*4 - 8*d**3*e - 60*d**2*e**2 + 24*d*e**3 - 5*e**4)/(2*(5*d**2 - 2*d*e + 3*e**2)**3)) + 5633839731848*d**6*e**7
- 162086347196928*d**5*e**12*(sqrt(14)*I*(1313*d**4 - 10044*d**3*e + 4290*d**2*e**2 + 156*d*e**3 - 271*e**4)/
(784*(125*d**6 - 150*d**5*e + 285*d**4*e**2 - 188*d**3*e**3 + 171*d**2*e**4 - 54*d*e**5 + 27*e**6)) - (41*d**4
- 8*d**3*e - 60*d**2*e**2 + 24*d*e**3 - 5*e**4)/(2*(5*d**2 - 2*d*e + 3*e**2)**3))**2 - 30431528150688*d**5*e*
*10*(sqrt(14)*I*(1313*d**4 - 10044*d**3*e + 4290*d**2*e**2 + 156*d*e**3 - 271*e**4)/(784*(125*d**6 - 150*d**5*
e + 285*d**4*e**2 - 188*d**3*e**3 + 171*d**2*e**4 - 54*d*e**5 + 27*e**6)) - (41*d**4 - 8*d**3*e - 60*d**2*e**2
+ 24*d*e**3 - 5*e**4)/(2*(5*d**2 - 2*d*e + 3*e**2)**3)) + 3033254622763*d**5*e**8 + 82236632099328*d**4*e**13
*(sqrt(14)*I*(1313*d**4 - 10044*d**3*e + 4290*d**2*e**2 + 156*d*e**3 - 271*e**4)/(784*(125*d**6 - 150*d**5*e +
285*d**4*e**2 - 188*d**3*e**3 + 171*d**2*e**4 - 54*d*e**5 + 27*e**6)) - (41*d**4 - 8*d**3*e - 60*d**2*e**2 +
24*d*e**3 - 5*e**4)/(2*(5*d**2 - 2*d*e + 3*e**2)**3))**2 + 13587008752688*d**4*e**11*(sqrt(14)*I*(1313*d**4 -
10044*d**3*e + 4290*d**2*e**2 + 156*d*e**3 - 271*e**4)/(784*(125*d**6 - 150*d**5*e + 285*d**4*e**2 - 188*d**3*
e**3 + 171*d**2*e**4 - 54*d*e**5 + 27*e**6)) - (41*d**4 - 8*d**3*e - 60*d**2*e**2 + 24*d*e**3 - 5*e**4)/(2*(5*
d**2 - 2*d*e + 3*e**2)**3)) - 3506827379684*d**4*e**9 - 30865482805248*d**3*e**14*(sqrt(14)*I*(1313*d**4 - 100
44*d**3*e + 4290*d**2*e**2 + 156*d*e**3 - 271*e**4)/(784*(125*d**6 - 150*d**5*e + 285*d**4*e**2 - 188*d**3*e**
3 + 171*d**2*e**4 - 54*d*e**5 + 27*e**6)) - (41*d**4 - 8*d**3*e - 60*d**2*e**2 + 24*d*e**3 - 5*e**4)/(2*(5*d**
2 - 2*d*e + 3*e**2)**3))**2 - 4535008734144*d**3*e**12*(sqrt(14)*I*(1313*d**4 - 10044*d**3*e + 4290*d**2*e**2
+ 156*d*e**3 - 271*e**4)/(784*(125*d**6 - 150*d**5*e + 285*d**4*e**2 - 188*d**3*e**3 + 171*d**2*e**4 - 54*d*e*
*5 + 27*e**6)) - (41*d**4 - 8*d**3*e - 60*d**2*e**2 + 24*d*e**3 - 5*e**4)/(2*(5*d**2 - 2*d*e + 3*e**2)**3)) +
1484229456462*d**3*e**10 + 9233948989440*d**2*e**15*(sqrt(14)*I*(1313*d**4 - 10044*d**3*e + 4290*d**2*e**2 + 1
56*d*e**3 - 271*e**4)/(784*(125*d**6 - 150*d**5*e + 285*d**4*e**2 - 188*d**3*e**3 + 171*d**2*e**4 - 54*d*e**5
+ 27*e**6)) - (41*d**4 - 8*d**3*e - 60*d**2*e**2 + 24*d*e**3 - 5*e**4)/(2*(5*d**2 - 2*d*e + 3*e**2)**3))**2 +
1144385029872*d**2*e**13*(sqrt(14)*I*(1313*d**4 - 10044*d**3*e + 4290*d**2*e**2 + 156*d*e**3 - 271*e**4)/(784*
(125*d**6 - 150*d**5*e + 285*d**4*e**2 - 188*d**3*e**3 + 171*d**2*e**4 - 54*d*e**5 + 27*e**6)) - (41*d**4 - 8*
d**3*e - 60*d**2*e**2 + 24*d*e**3 - 5*e**4)/(2*(5*d**2 - 2*d*e + 3*e**2)**3)) - 361088969436*d**2*e**11 - 1739
174903424*d*e**16*(sqrt(14)*I*(1313*d**4 - 10044*d**3*e + 4290*d**2*e**2 + 156*d*e**3 - 271*e**4)/(784*(125*d*
*6 - 150*d**5*e + 285*d**4*e**2 - 188*d**3*e**3 + 171*d**2*e**4 - 54*d*e**5 + 27*e**6)) - (41*d**4 - 8*d**3*e
- 60*d**2*e**2 + 24*d*e**3 - 5*e**4)/(2*(5*d**2 - 2*d*e + 3*e**2)**3))**2 - 187156660320*d*e**14*(sqrt(14)*I*(
1313*d**4 - 10044*d**3*e + 4290*d**2*e**2 + 156*d*e**3 - 271*e**4)/(784*(125*d**6 - 150*d**5*e + 285*d**4*e**2
- 188*d**3*e**3 + 171*d**2*e**4 - 54*d*e**5 + 27*e**6)) - (41*d**4 - 8*d**3*e - 60*d**2*e**2 + 24*d*e**3 - 5*
e**4)/(2*(5*d**2 - 2*d*e + 3*e**2)**3)) + 50336842869*d*e**12 + 196869004416*e**17*(sqrt(14)*I*(1313*d**4 - 10
044*d**3*e + 4290*d**2*e**2 + 156*d*e**3 - 271*e**4)/(784*(125*d**6 - 150*d**5*e + 285*d**4*e**2 - 188*d**3*e*
*3 + 171*d**2*e**4 - 54*d*e**5 + 27*e**6)) - (41*d**4 - 8*d**3*e - 60*d**2*e**2 + 24*d*e**3 - 5*e**4)/(2*(5*d*
*2 - 2*d*e + 3*e**2)**3))**2 + 17373868848*e**15*(sqrt(14)*I*(1313*d**4 - 10044*d**3*e + 4290*d**2*e**2 + 156*
d*e**3 - 271*e**4)/(784*(125*d**6 - 150*d**5*e + 285*d**4*e**2 - 188*d**3*e**3 + 171*d**2*e**4 - 54*d*e**5 + 2
7*e**6)) - (41*d**4 - 8*d**3*e - 60*d**2*e**2 + 24*d*e**3 - 5*e**4)/(2*(5*d**2 - 2*d*e + 3*e**2)**3)) - 353395
4480*e**13)/(1101474866245*d**12*e - 9024487794180*d**11*e**2 + 5764879624590*d**10*e**3 + 17969136971220*d**9
*e**4 - 16485388615365*d**8*e**5 - 12221510721480*d**7*e**6 + 21212253502020*d**6*e**7 - 11710335235320*d**5*e
**8 + 3048287389995*d**4*e**9 - 183650820660*d**3*e**10 - 118302770610*d**2*e**11 + 34222696740*d*e**12 - 3445
820555*e**13)) - (1680*d**4 + 3467*d**3*e + 674*d**2*e**2 - 1123*d*e**3 + 840*e**4 + x**2*(2800*d**4 + 3500*d*
*3*e + 2523*d**2*e**2 - 3434*d*e**3 + 1693*e**4) + x*(1120*d**4 + 1823*d**3*e - 527*d**2*e**2 - 573*d*e**3 - 1
43*e**4))/(10500*d**5*e - 8400*d**4*e**2 + 14280*d**3*e**3 - 5040*d**2*e**4 + 3780*d*e**5 + x**3*(17500*d**4*e
**2 - 14000*d**3*e**3 + 23800*d**2*e**4 - 8400*d*e**5 + 6300*e**6) + x**2*(17500*d**5*e - 7000*d**4*e**2 + 182
00*d**3*e**3 + 1120*d**2*e**4 + 2940*d*e**5 + 2520*e**6) + x*(7000*d**5*e + 4900*d**4*e**2 + 1120*d**3*e**3 +
10920*d**2*e**4 - 2520*d*e**5 + 3780*e**6)) + (41*d**4 - 8*d**3*e - 60*d**2*e**2 + 24*d*e**3 - 5*e**4)*log(x +
(4503590000000*d**17*(41*d**4 - 8*d**3*e - 60*d**2*e**2 + 24*d*e**3 - 5*e**4)**2/(5*d**2 - 2*d*e + 3*e**2)**6
- 79236430000000*d**16*e*(41*d**4 - 8*d**3*e - 60*d**2*e**2 + 24*d*e**3 - 5*e**4)**2/(5*d**2 - 2*d*e + 3*e**2
)**6 + 219307065600000*d**15*e**2*(41*d**4 - 8*d**3*e - 60*d**2*e**2 + 24*d*e**3 - 5*e**4)**2/(5*d**2 - 2*d*e
+ 3*e**2)**6 + 1477177520000*d**15*(41*d**4 - 8*d**3*e - 60*d**2*e**2 + 24*d*e**3 - 5*e**4)/(5*d**2 - 2*d*e +
3*e**2)**3 - 470102080000000*d**14*e**3*(41*d**4 - 8*d**3*e - 60*d**2*e**2 + 24*d*e**3 - 5*e**4)**2/(5*d**2 -
2*d*e + 3*e**2)**6 - 8062738222000*d**14*e*(41*d**4 - 8*d**3*e - 60*d**2*e**2 + 24*d*e**3 - 5*e**4)/(5*d**2 -
2*d*e + 3*e**2)**3 + 669820607680000*d**13*e**4*(41*d**4 - 8*d**3*e - 60*d**2*e**2 + 24*d*e**3 - 5*e**4)**2/(5
*d**2 - 2*d*e + 3*e**2)**6 + 13401619991200*d**13*e**2*(41*d**4 - 8*d**3*e - 60*d**2*e**2 + 24*d*e**3 - 5*e**4
)/(5*d**2 - 2*d*e + 3*e**2)**3 + 132446413125*d**13 - 748279970905600*d**12*e**5*(41*d**4 - 8*d**3*e - 60*d**2
*e**2 + 24*d*e**3 - 5*e**4)**2/(5*d**2 - 2*d*e + 3*e**2)**6 - 8564369003120*d**12*e**3*(41*d**4 - 8*d**3*e - 6
0*d**2*e**2 + 24*d*e**3 - 5*e**4)/(5*d**2 - 2*d*e + 3*e**2)**3 + 684029295980*d**12*e + 599319595212800*d**11*
e**6*(41*d**4 - 8*d**3*e - 60*d**2*e**2 + 24*d*e**3 - 5*e**4)**2/(5*d**2 - 2*d*e + 3*e**2)**6 - 12510243478208
*d**11*e**4*(41*d**4 - 8*d**3*e - 60*d**2*e**2 + 24*d*e**3 - 5*e**4)/(5*d**2 - 2*d*e + 3*e**2)**3 + 7642066236
30*d**11*e**2 - 291411662710784*d**10*e**7*(41*d**4 - 8*d**3*e - 60*d**2*e**2 + 24*d*e**3 - 5*e**4)**2/(5*d**2
- 2*d*e + 3*e**2)**6 + 49159980986704*d**10*e**5*(41*d**4 - 8*d**3*e - 60*d**2*e**2 + 24*d*e**3 - 5*e**4)/(5*
d**2 - 2*d*e + 3*e**2)**3 - 14657220189100*d**10*e**3 - 27190445185792*d**9*e**8*(41*d**4 - 8*d**3*e - 60*d**2
*e**2 + 24*d*e**3 - 5*e**4)**2/(5*d**2 - 2*d*e + 3*e**2)**6 - 77659175364512*d**9*e**6*(41*d**4 - 8*d**3*e - 6
0*d**2*e**2 + 24*d*e**3 - 5*e**4)/(5*d**2 - 2*d*e + 3*e**2)**3 + 16942805253691*d**9*e**4 + 253830846834432*d*
*8*e**9*(41*d**4 - 8*d**3*e - 60*d**2*e**2 + 24*d*e**3 - 5*e**4)**2/(5*d**2 - 2*d*e + 3*e**2)**6 + 91313688339
216*d**8*e**7*(41*d**4 - 8*d**3*e - 60*d**2*e**2 + 24*d*e**3 - 5*e**4)/(5*d**2 - 2*d*e + 3*e**2)**3 + 64049194
70120*d**8*e**5 - 308064129587200*d**7*e**10*(41*d**4 - 8*d**3*e - 60*d**2*e**2 + 24*d*e**3 - 5*e**4)**2/(5*d*
*2 - 2*d*e + 3*e**2)**6 - 78573287795968*d**7*e**8*(41*d**4 - 8*d**3*e - 60*d**2*e**2 + 24*d*e**3 - 5*e**4)/(5
*d**2 - 2*d*e + 3*e**2)**3 - 16998879119292*d**7*e**6 + 262468005502976*d**6*e**11*(41*d**4 - 8*d**3*e - 60*d*
*2*e**2 + 24*d*e**3 - 5*e**4)**2/(5*d**2 - 2*d*e + 3*e**2)**6 + 55676827575152*d**6*e**9*(41*d**4 - 8*d**3*e -
60*d**2*e**2 + 24*d*e**3 - 5*e**4)/(5*d**2 - 2*d*e + 3*e**2)**3 + 5633839731848*d**6*e**7 - 162086347196928*d
**5*e**12*(41*d**4 - 8*d**3*e - 60*d**2*e**2 + 24*d*e**3 - 5*e**4)**2/(5*d**2 - 2*d*e + 3*e**2)**6 - 304315281
50688*d**5*e**10*(41*d**4 - 8*d**3*e - 60*d**2*e**2 + 24*d*e**3 - 5*e**4)/(5*d**2 - 2*d*e + 3*e**2)**3 + 30332
54622763*d**5*e**8 + 82236632099328*d**4*e**13*(41*d**4 - 8*d**3*e - 60*d**2*e**2 + 24*d*e**3 - 5*e**4)**2/(5*
d**2 - 2*d*e + 3*e**2)**6 + 13587008752688*d**4*e**11*(41*d**4 - 8*d**3*e - 60*d**2*e**2 + 24*d*e**3 - 5*e**4)
/(5*d**2 - 2*d*e + 3*e**2)**3 - 3506827379684*d**4*e**9 - 30865482805248*d**3*e**14*(41*d**4 - 8*d**3*e - 60*d
**2*e**2 + 24*d*e**3 - 5*e**4)**2/(5*d**2 - 2*d*e + 3*e**2)**6 - 4535008734144*d**3*e**12*(41*d**4 - 8*d**3*e
- 60*d**2*e**2 + 24*d*e**3 - 5*e**4)/(5*d**2 - 2*d*e + 3*e**2)**3 + 1484229456462*d**3*e**10 + 9233948989440*d
**2*e**15*(41*d**4 - 8*d**3*e - 60*d**2*e**2 + 24*d*e**3 - 5*e**4)**2/(5*d**2 - 2*d*e + 3*e**2)**6 + 114438502
9872*d**2*e**13*(41*d**4 - 8*d**3*e - 60*d**2*e**2 + 24*d*e**3 - 5*e**4)/(5*d**2 - 2*d*e + 3*e**2)**3 - 361088
969436*d**2*e**11 - 1739174903424*d*e**16*(41*d**4 - 8*d**3*e - 60*d**2*e**2 + 24*d*e**3 - 5*e**4)**2/(5*d**2
- 2*d*e + 3*e**2)**6 - 187156660320*d*e**14*(41*d**4 - 8*d**3*e - 60*d**2*e**2 + 24*d*e**3 - 5*e**4)/(5*d**2 -
2*d*e + 3*e**2)**3 + 50336842869*d*e**12 + 196869004416*e**17*(41*d**4 - 8*d**3*e - 60*d**2*e**2 + 24*d*e**3
- 5*e**4)**2/(5*d**2 - 2*d*e + 3*e**2)**6 + 17373868848*e**15*(41*d**4 - 8*d**3*e - 60*d**2*e**2 + 24*d*e**3 -
5*e**4)/(5*d**2 - 2*d*e + 3*e**2)**3 - 3533954480*e**13)/(1101474866245*d**12*e - 9024487794180*d**11*e**2 +
5764879624590*d**10*e**3 + 17969136971220*d**9*e**4 - 16485388615365*d**8*e**5 - 12221510721480*d**7*e**6 + 21
212253502020*d**6*e**7 - 11710335235320*d**5*e**8 + 3048287389995*d**4*e**9 - 183650820660*d**3*e**10 - 118302
770610*d**2*e**11 + 34222696740*d*e**12 - 3445820555*e**13))/(5*d**2 - 2*d*e + 3*e**2)**3

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Giac [A]  time = 1.22489, size = 771, normalized size = 2.46 \begin{align*} \frac{\sqrt{14}{\left (1313 \, d^{4} e^{2} - 10044 \, d^{3} e^{3} + 4290 \, d^{2} e^{4} + 156 \, d e^{5} - 271 \, e^{6}\right )} \arctan \left (\frac{1}{14} \, \sqrt{14}{\left (5 \, d - \frac{5 \, d^{2}}{x e + d} + \frac{2 \, d e}{x e + d} - \frac{3 \, e^{2}}{x e + d} - e\right )} e^{\left (-1\right )}\right ) e^{\left (-2\right )}}{392 \,{\left (125 \, d^{6} - 150 \, d^{5} e + 285 \, d^{4} e^{2} - 188 \, d^{3} e^{3} + 171 \, d^{2} e^{4} - 54 \, d e^{5} + 27 \, e^{6}\right )}} - \frac{{\left (41 \, d^{4} - 8 \, d^{3} e - 60 \, d^{2} e^{2} + 24 \, d e^{3} - 5 \, e^{4}\right )} \log \left (-\frac{10 \, d}{x e + d} + \frac{5 \, d^{2}}{{\left (x e + d\right )}^{2}} + \frac{2 \, e}{x e + d} - \frac{2 \, d e}{{\left (x e + d\right )}^{2}} + \frac{3 \, e^{2}}{{\left (x e + d\right )}^{2}} + 5\right )}{2 \,{\left (125 \, d^{6} - 150 \, d^{5} e + 285 \, d^{4} e^{2} - 188 \, d^{3} e^{3} + 171 \, d^{2} e^{4} - 54 \, d e^{5} + 27 \, e^{6}\right )}} - \frac{\frac{4 \, d^{4} e^{3}}{x e + d} + \frac{5 \, d^{3} e^{4}}{x e + d} + \frac{3 \, d^{2} e^{5}}{x e + d} - \frac{d e^{6}}{x e + d} + \frac{2 \, e^{7}}{x e + d}}{25 \, d^{4} e^{4} - 20 \, d^{3} e^{5} + 34 \, d^{2} e^{6} - 12 \, d e^{7} + 9 \, e^{8}} + \frac{\frac{423 \, d^{3} e - 4101 \, d^{2} e^{2} + 879 \, d e^{3} + 703 \, e^{4}}{5 \, d^{2} - 2 \, d e + 3 \, e^{2}} - \frac{{\left (423 \, d^{4} e^{2} - 5468 \, d^{3} e^{3} + 1758 \, d^{2} e^{4} + 2812 \, d e^{5} - 457 \, e^{6}\right )} e^{\left (-1\right )}}{{\left (5 \, d^{2} - 2 \, d e + 3 \, e^{2}\right )}{\left (x e + d\right )}}}{28 \,{\left (5 \, d^{2} - 2 \, d e + 3 \, e^{2}\right )}^{2}{\left (\frac{10 \, d}{x e + d} - \frac{5 \, d^{2}}{{\left (x e + d\right )}^{2}} - \frac{2 \, e}{x e + d} + \frac{2 \, d e}{{\left (x e + d\right )}^{2}} - \frac{3 \, e^{2}}{{\left (x e + d\right )}^{2}} - 5\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/392*sqrt(14)*(1313*d^4*e^2 - 10044*d^3*e^3 + 4290*d^2*e^4 + 156*d*e^5 - 271*e^6)*arctan(1/14*sqrt(14)*(5*d -
5*d^2/(x*e + d) + 2*d*e/(x*e + d) - 3*e^2/(x*e + d) - e)*e^(-1))*e^(-2)/(125*d^6 - 150*d^5*e + 285*d^4*e^2 -
188*d^3*e^3 + 171*d^2*e^4 - 54*d*e^5 + 27*e^6) - 1/2*(41*d^4 - 8*d^3*e - 60*d^2*e^2 + 24*d*e^3 - 5*e^4)*log(-1
0*d/(x*e + d) + 5*d^2/(x*e + d)^2 + 2*e/(x*e + d) - 2*d*e/(x*e + d)^2 + 3*e^2/(x*e + d)^2 + 5)/(125*d^6 - 150*
d^5*e + 285*d^4*e^2 - 188*d^3*e^3 + 171*d^2*e^4 - 54*d*e^5 + 27*e^6) - (4*d^4*e^3/(x*e + d) + 5*d^3*e^4/(x*e +
d) + 3*d^2*e^5/(x*e + d) - d*e^6/(x*e + d) + 2*e^7/(x*e + d))/(25*d^4*e^4 - 20*d^3*e^5 + 34*d^2*e^6 - 12*d*e^
7 + 9*e^8) + 1/28*((423*d^3*e - 4101*d^2*e^2 + 879*d*e^3 + 703*e^4)/(5*d^2 - 2*d*e + 3*e^2) - (423*d^4*e^2 - 5
468*d^3*e^3 + 1758*d^2*e^4 + 2812*d*e^5 - 457*e^6)*e^(-1)/((5*d^2 - 2*d*e + 3*e^2)*(x*e + d)))/((5*d^2 - 2*d*e
+ 3*e^2)^2*(10*d/(x*e + d) - 5*d^2/(x*e + d)^2 - 2*e/(x*e + d) + 2*d*e/(x*e + d)^2 - 3*e^2/(x*e + d)^2 - 5))