### 3.317 $$\int \frac{2+x+3 x^2-5 x^3+4 x^4}{(d+e x)^3 (3+2 x+5 x^2)^2} \, dx$$

Optimal. Leaf size=412 $-\frac{x \left (-4101 d^2 e+423 d^3+879 d e^2+703 e^3\right )-879 d^2 e+1367 d^3-2109 d e^2+457 e^3}{28 \left (5 x^2+2 x+3\right ) \left (5 d^2-2 d e+3 e^2\right )^3}-\frac{\left (-846 d^3 e^2+396 d^2 e^3-19 d^4 e+205 d^5+57 d e^4-21 e^5\right ) \log \left (5 x^2+2 x+3\right )}{2 \left (5 d^2-2 d e+3 e^2\right )^4}-\frac{-60 d^2 e^2-8 d^3 e+41 d^4+24 d e^3-5 e^4}{\left (5 d^2-2 d e+3 e^2\right )^3 (d+e x)}-\frac{3 d^2 e^2+5 d^3 e+4 d^4-d e^3+2 e^4}{2 e \left (5 d^2-2 d e+3 e^2\right )^2 (d+e x)^2}+\frac{\left (-846 d^3 e^2+396 d^2 e^3-19 d^4 e+205 d^5+57 d e^4-21 e^5\right ) \log (d+e x)}{\left (5 d^2-2 d e+3 e^2\right )^4}+\frac{\left (35022 d^3 e^2+42858 d^2 e^3-74017 d^4 e+6565 d^5-17247 d e^4+579 e^5\right ) \tan ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{28 \sqrt{14} \left (5 d^2-2 d e+3 e^2\right )^4}$

[Out]

-(4*d^4 + 5*d^3*e + 3*d^2*e^2 - d*e^3 + 2*e^4)/(2*e*(5*d^2 - 2*d*e + 3*e^2)^2*(d + e*x)^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*(d + e*x)) - (1367*d^3 - 879*d^2*e - 2109*d*e^2 +
457*e^3 + (423*d^3 - 4101*d^2*e + 879*d*e^2 + 703*e^3)*x)/(28*(5*d^2 - 2*d*e + 3*e^2)^3*(3 + 2*x + 5*x^2)) +
((6565*d^5 - 74017*d^4*e + 35022*d^3*e^2 + 42858*d^2*e^3 - 17247*d*e^4 + 579*e^5)*ArcTan[(1 + 5*x)/Sqrt[14]])/
(28*Sqrt[14]*(5*d^2 - 2*d*e + 3*e^2)^4) + ((205*d^5 - 19*d^4*e - 846*d^3*e^2 + 396*d^2*e^3 + 57*d*e^4 - 21*e^5
)*Log[d + e*x])/(5*d^2 - 2*d*e + 3*e^2)^4 - ((205*d^5 - 19*d^4*e - 846*d^3*e^2 + 396*d^2*e^3 + 57*d*e^4 - 21*e
^5)*Log[3 + 2*x + 5*x^2])/(2*(5*d^2 - 2*d*e + 3*e^2)^4)

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Rubi [A]  time = 0.714754, antiderivative size = 412, 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} $-\frac{x \left (-4101 d^2 e+423 d^3+879 d e^2+703 e^3\right )-879 d^2 e+1367 d^3-2109 d e^2+457 e^3}{28 \left (5 x^2+2 x+3\right ) \left (5 d^2-2 d e+3 e^2\right )^3}-\frac{\left (-846 d^3 e^2+396 d^2 e^3-19 d^4 e+205 d^5+57 d e^4-21 e^5\right ) \log \left (5 x^2+2 x+3\right )}{2 \left (5 d^2-2 d e+3 e^2\right )^4}-\frac{-60 d^2 e^2-8 d^3 e+41 d^4+24 d e^3-5 e^4}{\left (5 d^2-2 d e+3 e^2\right )^3 (d+e x)}-\frac{3 d^2 e^2+5 d^3 e+4 d^4-d e^3+2 e^4}{2 e \left (5 d^2-2 d e+3 e^2\right )^2 (d+e x)^2}+\frac{\left (-846 d^3 e^2+396 d^2 e^3-19 d^4 e+205 d^5+57 d e^4-21 e^5\right ) \log (d+e x)}{\left (5 d^2-2 d e+3 e^2\right )^4}+\frac{\left (35022 d^3 e^2+42858 d^2 e^3-74017 d^4 e+6565 d^5-17247 d e^4+579 e^5\right ) \tan ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{28 \sqrt{14} \left (5 d^2-2 d e+3 e^2\right )^4}$

Antiderivative was successfully veriﬁed.

[In]

Int[(2 + x + 3*x^2 - 5*x^3 + 4*x^4)/((d + e*x)^3*(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)/(2*e*(5*d^2 - 2*d*e + 3*e^2)^2*(d + e*x)^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*(d + e*x)) - (1367*d^3 - 879*d^2*e - 2109*d*e^2 +
457*e^3 + (423*d^3 - 4101*d^2*e + 879*d*e^2 + 703*e^3)*x)/(28*(5*d^2 - 2*d*e + 3*e^2)^3*(3 + 2*x + 5*x^2)) +
((6565*d^5 - 74017*d^4*e + 35022*d^3*e^2 + 42858*d^2*e^3 - 17247*d*e^4 + 579*e^5)*ArcTan[(1 + 5*x)/Sqrt[14]])/
(28*Sqrt[14]*(5*d^2 - 2*d*e + 3*e^2)^4) + ((205*d^5 - 19*d^4*e - 846*d^3*e^2 + 396*d^2*e^3 + 57*d*e^4 - 21*e^5
)*Log[d + e*x])/(5*d^2 - 2*d*e + 3*e^2)^4 - ((205*d^5 - 19*d^4*e - 846*d^3*e^2 + 396*d^2*e^3 + 57*d*e^4 - 21*e
^5)*Log[3 + 2*x + 5*x^2])/(2*(5*d^2 - 2*d*e + 3*e^2)^4)

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)^3 \left (3+2 x+5 x^2\right )^2} \, dx &=-\frac{1367 d^3-879 d^2 e-2109 d e^2+457 e^3+\left (423 d^3-4101 d^2 e+879 d e^2+703 e^3\right ) x}{28 \left (5 d^2-2 d e+3 e^2\right )^3 \left (3+2 x+5 x^2\right )}+\frac{1}{56} \int \frac{\frac{6 \left (615 d^6-2105 d^5 e+2535 d^4 e^2-1037 d^3 e^3+1064 d^2 e^4-336 d e^5+168 e^6\right )}{\left (5 d^2-2 d e+3 e^2\right )^3}-\frac{2 \left (4620 d^6-4275 d^5 e+5925 d^4 e^2-5651 d^3 e^3-663 d^2 e^4-168 d e^5+84 e^6\right ) x}{\left (5 d^2-2 d e+3 e^2\right )^3}+\frac{2 \left (2800 d^6-3360 d^5 e+5115 d^4 e^2+5527 d^3 e^3+1311 d^2 e^4+1251 d e^5-28 e^6\right ) x^2}{\left (5 d^2-2 d e+3 e^2\right )^3}-\frac{2 e^3 \left (423 d^3-4101 d^2 e+879 d e^2+703 e^3\right ) x^3}{\left (5 d^2-2 d e+3 e^2\right )^3}}{(d+e x)^3 \left (3+2 x+5 x^2\right )} \, dx\\ &=-\frac{1367 d^3-879 d^2 e-2109 d e^2+457 e^3+\left (423 d^3-4101 d^2 e+879 d e^2+703 e^3\right ) x}{28 \left (5 d^2-2 d e+3 e^2\right )^3 \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)^3}-\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)^2}-\frac{56 e \left (-205 d^5+19 d^4 e+846 d^3 e^2-396 d^2 e^3-57 d e^4+21 e^5\right )}{\left (5 d^2-2 d e+3 e^2\right )^4 (d+e x)}+\frac{2 \left (3 \left (275 d^5-24495 d^4 e+19570 d^3 e^2+10590 d^2 e^3-6281 d e^4+389 e^5\right )-140 \left (205 d^5-19 d^4 e-846 d^3 e^2+396 d^2 e^3+57 d e^4-21 e^5\right ) x\right )}{\left (5 d^2-2 d e+3 e^2\right )^4 \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}{2 e \left (5 d^2-2 d e+3 e^2\right )^2 (d+e x)^2}-\frac{41 d^4-8 d^3 e-60 d^2 e^2+24 d e^3-5 e^4}{\left (5 d^2-2 d e+3 e^2\right )^3 (d+e x)}-\frac{1367 d^3-879 d^2 e-2109 d e^2+457 e^3+\left (423 d^3-4101 d^2 e+879 d e^2+703 e^3\right ) x}{28 \left (5 d^2-2 d e+3 e^2\right )^3 \left (3+2 x+5 x^2\right )}+\frac{\left (205 d^5-19 d^4 e-846 d^3 e^2+396 d^2 e^3+57 d e^4-21 e^5\right ) \log (d+e x)}{\left (5 d^2-2 d e+3 e^2\right )^4}+\frac{\int \frac{3 \left (275 d^5-24495 d^4 e+19570 d^3 e^2+10590 d^2 e^3-6281 d e^4+389 e^5\right )-140 \left (205 d^5-19 d^4 e-846 d^3 e^2+396 d^2 e^3+57 d e^4-21 e^5\right ) x}{3+2 x+5 x^2} \, dx}{28 \left (5 d^2-2 d e+3 e^2\right )^4}\\ &=-\frac{4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4}{2 e \left (5 d^2-2 d e+3 e^2\right )^2 (d+e x)^2}-\frac{41 d^4-8 d^3 e-60 d^2 e^2+24 d e^3-5 e^4}{\left (5 d^2-2 d e+3 e^2\right )^3 (d+e x)}-\frac{1367 d^3-879 d^2 e-2109 d e^2+457 e^3+\left (423 d^3-4101 d^2 e+879 d e^2+703 e^3\right ) x}{28 \left (5 d^2-2 d e+3 e^2\right )^3 \left (3+2 x+5 x^2\right )}+\frac{\left (205 d^5-19 d^4 e-846 d^3 e^2+396 d^2 e^3+57 d e^4-21 e^5\right ) \log (d+e x)}{\left (5 d^2-2 d e+3 e^2\right )^4}-\frac{\left (205 d^5-19 d^4 e-846 d^3 e^2+396 d^2 e^3+57 d e^4-21 e^5\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 )^4}+\frac{\left (6565 d^5-74017 d^4 e+35022 d^3 e^2+42858 d^2 e^3-17247 d e^4+579 e^5\right ) \int \frac{1}{3+2 x+5 x^2} \, dx}{28 \left (5 d^2-2 d e+3 e^2\right )^4}\\ &=-\frac{4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4}{2 e \left (5 d^2-2 d e+3 e^2\right )^2 (d+e x)^2}-\frac{41 d^4-8 d^3 e-60 d^2 e^2+24 d e^3-5 e^4}{\left (5 d^2-2 d e+3 e^2\right )^3 (d+e x)}-\frac{1367 d^3-879 d^2 e-2109 d e^2+457 e^3+\left (423 d^3-4101 d^2 e+879 d e^2+703 e^3\right ) x}{28 \left (5 d^2-2 d e+3 e^2\right )^3 \left (3+2 x+5 x^2\right )}+\frac{\left (205 d^5-19 d^4 e-846 d^3 e^2+396 d^2 e^3+57 d e^4-21 e^5\right ) \log (d+e x)}{\left (5 d^2-2 d e+3 e^2\right )^4}-\frac{\left (205 d^5-19 d^4 e-846 d^3 e^2+396 d^2 e^3+57 d e^4-21 e^5\right ) \log \left (3+2 x+5 x^2\right )}{2 \left (5 d^2-2 d e+3 e^2\right )^4}-\frac{\left (6565 d^5-74017 d^4 e+35022 d^3 e^2+42858 d^2 e^3-17247 d e^4+579 e^5\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 )^4}\\ &=-\frac{4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4}{2 e \left (5 d^2-2 d e+3 e^2\right )^2 (d+e x)^2}-\frac{41 d^4-8 d^3 e-60 d^2 e^2+24 d e^3-5 e^4}{\left (5 d^2-2 d e+3 e^2\right )^3 (d+e x)}-\frac{1367 d^3-879 d^2 e-2109 d e^2+457 e^3+\left (423 d^3-4101 d^2 e+879 d e^2+703 e^3\right ) x}{28 \left (5 d^2-2 d e+3 e^2\right )^3 \left (3+2 x+5 x^2\right )}+\frac{\left (6565 d^5-74017 d^4 e+35022 d^3 e^2+42858 d^2 e^3-17247 d e^4+579 e^5\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 )^4}+\frac{\left (205 d^5-19 d^4 e-846 d^3 e^2+396 d^2 e^3+57 d e^4-21 e^5\right ) \log (d+e x)}{\left (5 d^2-2 d e+3 e^2\right )^4}-\frac{\left (205 d^5-19 d^4 e-846 d^3 e^2+396 d^2 e^3+57 d e^4-21 e^5\right ) \log \left (3+2 x+5 x^2\right )}{2 \left (5 d^2-2 d e+3 e^2\right )^4}\\ \end{align*}

Mathematica [A]  time = 0.370097, size = 363, normalized size = 0.88 $\frac{-\frac{14 \left (5 d^2-2 d e+3 e^2\right ) \left (-3 d^2 e (1367 x+293)+d^3 (423 x+1367)+3 d e^2 (293 x-703)+e^3 (703 x+457)\right )}{5 x^2+2 x+3}+196 \left (846 d^3 e^2-396 d^2 e^3+19 d^4 e-205 d^5-57 d e^4+21 e^5\right ) \log \left (5 x^2+2 x+3\right )-\frac{196 \left (3 d^2 e^2+5 d^3 e+4 d^4-d e^3+2 e^4\right ) \left (5 d^2-2 d e+3 e^2\right )^2}{e (d+e x)^2}+\frac{392 \left (60 d^2 e^2+8 d^3 e-41 d^4-24 d e^3+5 e^4\right ) \left (5 d^2-2 d e+3 e^2\right )}{d+e x}+392 \left (-846 d^3 e^2+396 d^2 e^3-19 d^4 e+205 d^5+57 d e^4-21 e^5\right ) \log (d+e x)+\sqrt{14} \left (35022 d^3 e^2+42858 d^2 e^3-74017 d^4 e+6565 d^5-17247 d e^4+579 e^5\right ) \tan ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{392 \left (5 d^2-2 d e+3 e^2\right )^4}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

((-196*(5*d^2 - 2*d*e + 3*e^2)^2*(4*d^4 + 5*d^3*e + 3*d^2*e^2 - d*e^3 + 2*e^4))/(e*(d + e*x)^2) + (392*(5*d^2
- 2*d*e + 3*e^2)*(-41*d^4 + 8*d^3*e + 60*d^2*e^2 - 24*d*e^3 + 5*e^4))/(d + e*x) - (14*(5*d^2 - 2*d*e + 3*e^2)*
(3*d*e^2*(-703 + 293*x) + d^3*(1367 + 423*x) + e^3*(457 + 703*x) - 3*d^2*e*(293 + 1367*x)))/(3 + 2*x + 5*x^2)
+ Sqrt[14]*(6565*d^5 - 74017*d^4*e + 35022*d^3*e^2 + 42858*d^2*e^3 - 17247*d*e^4 + 579*e^5)*ArcTan[(1 + 5*x)/S
qrt[14]] + 392*(205*d^5 - 19*d^4*e - 846*d^3*e^2 + 396*d^2*e^3 + 57*d*e^4 - 21*e^5)*Log[d + e*x] + 196*(-205*d
^5 + 19*d^4*e + 846*d^3*e^2 - 396*d^2*e^3 - 57*d*e^4 + 21*e^5)*Log[3 + 2*x + 5*x^2])/(392*(5*d^2 - 2*d*e + 3*e
^2)^4)

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Maple [B]  time = 0.073, size = 1314, 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)^3/(5*x^2+2*x+3)^2,x)

[Out]

57/(5*d^2-2*d*e+3*e^2)^4*ln(e*x+d)*d*e^4-19/(5*d^2-2*d*e+3*e^2)^4*ln(e*x+d)*d^4*e-846/(5*d^2-2*d*e+3*e^2)^4*ln
(e*x+d)*d^3*e^2+205/(5*d^2-2*d*e+3*e^2)^4*ln(e*x+d)*d^5-21/(5*d^2-2*d*e+3*e^2)^4*ln(e*x+d)*e^5-1367/28/(5*d^2-
2*d*e+3*e^2)^4/(x^2+2/5*x+3/5)*d^5-1371/140/(5*d^2-2*d*e+3*e^2)^4/(x^2+2/5*x+3/5)*e^5-205/2/(5*d^2-2*d*e+3*e^2
)^4*ln(5*x^2+2*x+3)*d^5+21/2/(5*d^2-2*d*e+3*e^2)^4*ln(5*x^2+2*x+3)*e^5-1/(5*d^2-2*d*e+3*e^2)^2*e^3/(e*x+d)^2-4
1/(5*d^2-2*d*e+3*e^2)^3/(e*x+d)*d^4+5/(5*d^2-2*d*e+3*e^2)^3/(e*x+d)*e^4-5/2/(5*d^2-2*d*e+3*e^2)^2/(e*x+d)^2*d^
3+21351/140/(5*d^2-2*d*e+3*e^2)^4/(x^2+2/5*x+3/5)*x*d^4*e-6933/70/(5*d^2-2*d*e+3*e^2)^4/(x^2+2/5*x+3/5)*x*d^3*
e^2+5273/70/(5*d^2-2*d*e+3*e^2)^4/(x^2+2/5*x+3/5)*x*d^2*e^3+21429/196/(5*d^2-2*d*e+3*e^2)^4*14^(1/2)*arctan(1/
28*(10*x+2)*14^(1/2))*d^2*e^3+1/2/(5*d^2-2*d*e+3*e^2)^2*e^2/(e*x+d)^2*d+8/(5*d^2-2*d*e+3*e^2)^3/(e*x+d)*d^3*e+
60/(5*d^2-2*d*e+3*e^2)^3/(e*x+d)*d^2*e^2-24/(5*d^2-2*d*e+3*e^2)^3/(e*x+d)*d*e^3-423/28/(5*d^2-2*d*e+3*e^2)^4/(
x^2+2/5*x+3/5)*d^5*x-2109/140/(5*d^2-2*d*e+3*e^2)^4/(x^2+2/5*x+3/5)*x*e^5+7129/140/(5*d^2-2*d*e+3*e^2)^4/(x^2+
2/5*x+3/5)*d^4*e+2343/70/(5*d^2-2*d*e+3*e^2)^4/(x^2+2/5*x+3/5)*d^3*e^2-1933/70/(5*d^2-2*d*e+3*e^2)^4/(x^2+2/5*
x+3/5)*d^2*e^3+7241/140/(5*d^2-2*d*e+3*e^2)^4/(x^2+2/5*x+3/5)*d*e^4+19/2/(5*d^2-2*d*e+3*e^2)^4*ln(5*x^2+2*x+3)
*d^4*e+423/(5*d^2-2*d*e+3*e^2)^4*ln(5*x^2+2*x+3)*d^3*e^2+396/(5*d^2-2*d*e+3*e^2)^4*ln(e*x+d)*d^2*e^3-17247/392
/(5*d^2-2*d*e+3*e^2)^4*14^(1/2)*arctan(1/28*(10*x+2)*14^(1/2))*d*e^4-1231/140/(5*d^2-2*d*e+3*e^2)^4/(x^2+2/5*x
+3/5)*x*d*e^4-74017/392/(5*d^2-2*d*e+3*e^2)^4*14^(1/2)*arctan(1/28*(10*x+2)*14^(1/2))*d^4*e+17511/196/(5*d^2-2
*d*e+3*e^2)^4*14^(1/2)*arctan(1/28*(10*x+2)*14^(1/2))*d^3*e^2-198/(5*d^2-2*d*e+3*e^2)^4*ln(5*x^2+2*x+3)*d^2*e^
3-57/2/(5*d^2-2*d*e+3*e^2)^4*ln(5*x^2+2*x+3)*d*e^4+6565/392/(5*d^2-2*d*e+3*e^2)^4*14^(1/2)*arctan(1/28*(10*x+2
)*14^(1/2))*d^5+579/392/(5*d^2-2*d*e+3*e^2)^4*14^(1/2)*arctan(1/28*(10*x+2)*14^(1/2))*e^5-2/(5*d^2-2*d*e+3*e^2
)^2/e/(e*x+d)^2*d^4-3/2/(5*d^2-2*d*e+3*e^2)^2*e/(e*x+d)^2*d^2

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Maxima [B]  time = 1.66047, size = 1149, normalized size = 2.79 \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)^3/(5*x^2+2*x+3)^2,x, algorithm="maxima")

[Out]

1/392*sqrt(14)*(6565*d^5 - 74017*d^4*e + 35022*d^3*e^2 + 42858*d^2*e^3 - 17247*d*e^4 + 579*e^5)*arctan(1/14*sq
rt(14)*(5*x + 1))/(625*d^8 - 1000*d^7*e + 2100*d^6*e^2 - 1960*d^5*e^3 + 2086*d^4*e^4 - 1176*d^3*e^5 + 756*d^2*
e^6 - 216*d*e^7 + 81*e^8) + (205*d^5 - 19*d^4*e - 846*d^3*e^2 + 396*d^2*e^3 + 57*d*e^4 - 21*e^5)*log(e*x + d)/
(625*d^8 - 1000*d^7*e + 2100*d^6*e^2 - 1960*d^5*e^3 + 2086*d^4*e^4 - 1176*d^3*e^5 + 756*d^2*e^6 - 216*d*e^7 +
81*e^8) - 1/2*(205*d^5 - 19*d^4*e - 846*d^3*e^2 + 396*d^2*e^3 + 57*d*e^4 - 21*e^5)*log(5*x^2 + 2*x + 3)/(625*d
^8 - 1000*d^7*e + 2100*d^6*e^2 - 1960*d^5*e^3 + 2086*d^4*e^4 - 1176*d^3*e^5 + 756*d^2*e^6 - 216*d*e^7 + 81*e^8
) - 1/28*(840*d^6 + 5525*d^5*e - 837*d^4*e^2 - 6981*d^3*e^3 + 3355*d^2*e^4 - 714*d*e^5 + 252*e^6 + (5740*d^4*e
^2 - 697*d^3*e^3 - 12501*d^2*e^4 + 4239*d*e^5 + 3*e^6)*x^3 + (1400*d^6 + 6930*d^5*e + 3212*d^4*e^2 - 15403*d^3
*e^3 + 2349*d^2*e^4 - 549*d*e^5 + 597*e^6)*x^2 + (560*d^6 + 3195*d^5*e + 2105*d^4*e^2 - 4799*d^3*e^3 - 6623*d^
2*e^4 + 2454*d*e^5 - 252*e^6)*x)/(375*d^8*e - 450*d^7*e^2 + 855*d^6*e^3 - 564*d^5*e^4 + 513*d^4*e^5 - 162*d^3*
e^6 + 81*d^2*e^7 + 5*(125*d^6*e^3 - 150*d^5*e^4 + 285*d^4*e^5 - 188*d^3*e^6 + 171*d^2*e^7 - 54*d*e^8 + 27*e^9)
*x^4 + 2*(625*d^7*e^2 - 625*d^6*e^3 + 1275*d^5*e^4 - 655*d^4*e^5 + 667*d^3*e^6 - 99*d^2*e^7 + 81*d*e^8 + 27*e^
9)*x^3 + (625*d^8*e - 250*d^7*e^2 + 1200*d^6*e^3 - 250*d^5*e^4 + 958*d^4*e^5 - 150*d^3*e^6 + 432*d^2*e^7 - 54*
d*e^8 + 81*e^9)*x^2 + 2*(125*d^8*e + 225*d^7*e^2 - 165*d^6*e^3 + 667*d^5*e^4 - 393*d^4*e^5 + 459*d^3*e^6 - 135
*d^2*e^7 + 81*d*e^8)*x)

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Fricas [B]  time = 2.90437, size = 3853, normalized size = 9.35 \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)^3/(5*x^2+2*x+3)^2,x, algorithm="fricas")

[Out]

-1/392*(58800*d^8 + 363230*d^7*e - 178010*d^6*e^2 - 233184*d^5*e^3 + 395164*d^4*e^4 - 437122*d^3*e^5 + 178542*
d^2*e^6 - 37044*d*e^7 + 10584*e^8 + 14*(28700*d^6*e^2 - 14965*d^5*e^3 - 43891*d^4*e^4 + 44106*d^3*e^5 - 45966*
d^2*e^6 + 12711*d*e^7 + 9*e^8)*x^3 + 14*(7000*d^8 + 31850*d^7*e + 6400*d^6*e^2 - 62649*d^5*e^3 + 52187*d^4*e^4
- 53652*d^3*e^5 + 11130*d^2*e^6 - 2841*d*e^7 + 1791*e^8)*x^2 - sqrt(14)*(19695*d^7*e - 222051*d^6*e^2 + 10506
6*d^5*e^3 + 128574*d^4*e^4 - 51741*d^3*e^5 + 1737*d^2*e^6 + 5*(6565*d^5*e^3 - 74017*d^4*e^4 + 35022*d^3*e^5 +
42858*d^2*e^6 - 17247*d*e^7 + 579*e^8)*x^4 + 2*(32825*d^6*e^2 - 363520*d^5*e^3 + 101093*d^4*e^4 + 249312*d^3*e
^5 - 43377*d^2*e^6 - 14352*d*e^7 + 579*e^8)*x^3 + (32825*d^7*e - 343825*d^6*e^2 - 101263*d^5*e^3 + 132327*d^4*
e^4 + 190263*d^3*e^5 + 62481*d^2*e^6 - 49425*d*e^7 + 1737*e^8)*x^2 + 2*(6565*d^7*e - 54322*d^6*e^2 - 187029*d^
5*e^3 + 147924*d^4*e^4 + 111327*d^3*e^5 - 51162*d^2*e^6 + 1737*d*e^7)*x)*arctan(1/14*sqrt(14)*(5*x + 1)) + 14*
(2800*d^8 + 14855*d^7*e + 5815*d^6*e^2 - 18620*d^5*e^3 - 17202*d^4*e^4 + 11119*d^3*e^5 - 26037*d^2*e^6 + 7866*
d*e^7 - 756*e^8)*x - 392*(615*d^7*e - 57*d^6*e^2 - 2538*d^5*e^3 + 1188*d^4*e^4 + 171*d^3*e^5 - 63*d^2*e^6 + 5*
(205*d^5*e^3 - 19*d^4*e^4 - 846*d^3*e^5 + 396*d^2*e^6 + 57*d*e^7 - 21*e^8)*x^4 + 2*(1025*d^6*e^2 + 110*d^5*e^3
- 4249*d^4*e^4 + 1134*d^3*e^5 + 681*d^2*e^6 - 48*d*e^7 - 21*e^8)*x^3 + (1025*d^7*e + 725*d^6*e^2 - 3691*d^5*e
^3 - 1461*d^4*e^4 - 669*d^3*e^5 + 1311*d^2*e^6 + 87*d*e^7 - 63*e^8)*x^2 + 2*(205*d^7*e + 596*d^6*e^2 - 903*d^5
*e^3 - 2142*d^4*e^4 + 1245*d^3*e^5 + 150*d^2*e^6 - 63*d*e^7)*x)*log(e*x + d) + 196*(615*d^7*e - 57*d^6*e^2 - 2
538*d^5*e^3 + 1188*d^4*e^4 + 171*d^3*e^5 - 63*d^2*e^6 + 5*(205*d^5*e^3 - 19*d^4*e^4 - 846*d^3*e^5 + 396*d^2*e^
6 + 57*d*e^7 - 21*e^8)*x^4 + 2*(1025*d^6*e^2 + 110*d^5*e^3 - 4249*d^4*e^4 + 1134*d^3*e^5 + 681*d^2*e^6 - 48*d*
e^7 - 21*e^8)*x^3 + (1025*d^7*e + 725*d^6*e^2 - 3691*d^5*e^3 - 1461*d^4*e^4 - 669*d^3*e^5 + 1311*d^2*e^6 + 87*
d*e^7 - 63*e^8)*x^2 + 2*(205*d^7*e + 596*d^6*e^2 - 903*d^5*e^3 - 2142*d^4*e^4 + 1245*d^3*e^5 + 150*d^2*e^6 - 6
3*d*e^7)*x)*log(5*x^2 + 2*x + 3))/(1875*d^10*e - 3000*d^9*e^2 + 6300*d^8*e^3 - 5880*d^7*e^4 + 6258*d^6*e^5 - 3
528*d^5*e^6 + 2268*d^4*e^7 - 648*d^3*e^8 + 243*d^2*e^9 + 5*(625*d^8*e^3 - 1000*d^7*e^4 + 2100*d^6*e^5 - 1960*d
^5*e^6 + 2086*d^4*e^7 - 1176*d^3*e^8 + 756*d^2*e^9 - 216*d*e^10 + 81*e^11)*x^4 + 2*(3125*d^9*e^2 - 4375*d^8*e^
3 + 9500*d^7*e^4 - 7700*d^6*e^5 + 8470*d^5*e^6 - 3794*d^4*e^7 + 2604*d^3*e^8 - 324*d^2*e^9 + 189*d*e^10 + 81*e
^11)*x^3 + (3125*d^10*e - 2500*d^9*e^2 + 8375*d^8*e^3 - 4400*d^7*e^4 + 8890*d^6*e^5 - 3416*d^5*e^6 + 5334*d^4*
e^7 - 1584*d^3*e^8 + 1809*d^2*e^9 - 324*d*e^10 + 243*e^11)*x^2 + 2*(625*d^10*e + 875*d^9*e^2 - 900*d^8*e^3 + 4
340*d^7*e^4 - 3794*d^6*e^5 + 5082*d^5*e^6 - 2772*d^4*e^7 + 2052*d^3*e^8 - 567*d^2*e^9 + 243*d*e^10)*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((4*x**4-5*x**3+3*x**2+x+2)/(e*x+d)**3/(5*x**2+2*x+3)**2,x)

[Out]

Timed out

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Giac [A]  time = 1.25734, size = 803, normalized size = 1.95 \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)^3/(5*x^2+2*x+3)^2,x, algorithm="giac")

[Out]

1/392*sqrt(14)*(6565*d^5 - 74017*d^4*e + 35022*d^3*e^2 + 42858*d^2*e^3 - 17247*d*e^4 + 579*e^5)*arctan(1/14*sq
rt(14)*(5*x + 1))/(625*d^8 - 1000*d^7*e + 2100*d^6*e^2 - 1960*d^5*e^3 + 2086*d^4*e^4 - 1176*d^3*e^5 + 756*d^2*
e^6 - 216*d*e^7 + 81*e^8) - 1/2*(205*d^5 - 19*d^4*e - 846*d^3*e^2 + 396*d^2*e^3 + 57*d*e^4 - 21*e^5)*log(5*x^2
+ 2*x + 3)/(625*d^8 - 1000*d^7*e + 2100*d^6*e^2 - 1960*d^5*e^3 + 2086*d^4*e^4 - 1176*d^3*e^5 + 756*d^2*e^6 -
216*d*e^7 + 81*e^8) + (205*d^5*e - 19*d^4*e^2 - 846*d^3*e^3 + 396*d^2*e^4 + 57*d*e^5 - 21*e^6)*log(abs(x*e + d
))/(625*d^8*e - 1000*d^7*e^2 + 2100*d^6*e^3 - 1960*d^5*e^4 + 2086*d^4*e^5 - 1176*d^3*e^6 + 756*d^2*e^7 - 216*d
*e^8 + 81*e^9) - 1/28*(4200*d^8 + 25945*d^7*e - 12715*d^6*e^2 - 16656*d^5*e^3 + 28226*d^4*e^4 + (28700*d^6*e^2
- 14965*d^5*e^3 - 43891*d^4*e^4 + 44106*d^3*e^5 - 45966*d^2*e^6 + 12711*d*e^7 + 9*e^8)*x^3 - 31223*d^3*e^5 +
(7000*d^8 + 31850*d^7*e + 6400*d^6*e^2 - 62649*d^5*e^3 + 52187*d^4*e^4 - 53652*d^3*e^5 + 11130*d^2*e^6 - 2841*
d*e^7 + 1791*e^8)*x^2 + 12753*d^2*e^6 + (2800*d^8 + 14855*d^7*e + 5815*d^6*e^2 - 18620*d^5*e^3 - 17202*d^4*e^4
+ 11119*d^3*e^5 - 26037*d^2*e^6 + 7866*d*e^7 - 756*e^8)*x - 2646*d*e^7 + 756*e^8)*e^(-1)/((5*d^2 - 2*d*e + 3*
e^2)^4*(5*x^2 + 2*x + 3)*(x*e + d)^2)