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

Optimal. Leaf size=443 \[ \frac{5 x \left (34698 d^2 e^2-85924 d^3 e+11015 d^4+10348 d e^3-3589 e^4\right )-200502 d^2 e^2-117284 d^3 e+171735 d^4+104428 d e^3-23189 e^4}{7840 \left (5 x^2+2 x+3\right ) \left (5 d^2-2 d e+3 e^2\right )^3}-\frac{x \left (423 d^2-2734 d e+293 e^2\right )+1367 d^2-586 d e-703 e^2}{280 \left (5 x^2+2 x+3\right )^2 \left (5 d^2-2 d e+3 e^2\right )^2}-\frac{e \left (12 d^3 e^2-76 d^2 e^3+83 d^4 e+40 d^5+46 d e^4-9 e^5\right ) \log \left (5 x^2+2 x+3\right )}{2 \left (5 d^2-2 d e+3 e^2\right )^4}-\frac{e \left (3 d^2 e^2+5 d^3 e+4 d^4-d e^3+2 e^4\right )}{\left (5 d^2-2 d e+3 e^2\right )^3 (d+e x)}+\frac{e \left (12 d^3 e^2-76 d^2 e^3+83 d^4 e+40 d^5+46 d e^4-9 e^5\right ) \log (d+e x)}{\left (5 d^2-2 d e+3 e^2\right )^4}+\frac{\left (209039 d^4 e^2-921444 d^3 e^3+380621 d^2 e^4+3070 d^5 e+211875 d^6-49586 d e^5-43695 e^6\right ) \tan ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{1568 \sqrt{14} \left (5 d^2-2 d e+3 e^2\right )^4} \]

[Out]

-((e*(4*d^4 + 5*d^3*e + 3*d^2*e^2 - d*e^3 + 2*e^4))/((5*d^2 - 2*d*e + 3*e^2)^3*(d + e*x))) - (1367*d^2 - 586*d
*e - 703*e^2 + (423*d^2 - 2734*d*e + 293*e^2)*x)/(280*(5*d^2 - 2*d*e + 3*e^2)^2*(3 + 2*x + 5*x^2)^2) + (171735
*d^4 - 117284*d^3*e - 200502*d^2*e^2 + 104428*d*e^3 - 23189*e^4 + 5*(11015*d^4 - 85924*d^3*e + 34698*d^2*e^2 +
10348*d*e^3 - 3589*e^4)*x)/(7840*(5*d^2 - 2*d*e + 3*e^2)^3*(3 + 2*x + 5*x^2)) + ((211875*d^6 + 3070*d^5*e + 2
09039*d^4*e^2 - 921444*d^3*e^3 + 380621*d^2*e^4 - 49586*d*e^5 - 43695*e^6)*ArcTan[(1 + 5*x)/Sqrt[14]])/(1568*S
qrt[14]*(5*d^2 - 2*d*e + 3*e^2)^4) + (e*(40*d^5 + 83*d^4*e + 12*d^3*e^2 - 76*d^2*e^3 + 46*d*e^4 - 9*e^5)*Log[d
+ e*x])/(5*d^2 - 2*d*e + 3*e^2)^4 - (e*(40*d^5 + 83*d^4*e + 12*d^3*e^2 - 76*d^2*e^3 + 46*d*e^4 - 9*e^5)*Log[3
+ 2*x + 5*x^2])/(2*(5*d^2 - 2*d*e + 3*e^2)^4)

________________________________________________________________________________________

Rubi [A]  time = 0.89268, antiderivative size = 443, normalized size of antiderivative = 1., number of steps used = 8, 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} \[ \frac{5 x \left (34698 d^2 e^2-85924 d^3 e+11015 d^4+10348 d e^3-3589 e^4\right )-200502 d^2 e^2-117284 d^3 e+171735 d^4+104428 d e^3-23189 e^4}{7840 \left (5 x^2+2 x+3\right ) \left (5 d^2-2 d e+3 e^2\right )^3}-\frac{x \left (423 d^2-2734 d e+293 e^2\right )+1367 d^2-586 d e-703 e^2}{280 \left (5 x^2+2 x+3\right )^2 \left (5 d^2-2 d e+3 e^2\right )^2}-\frac{e \left (12 d^3 e^2-76 d^2 e^3+83 d^4 e+40 d^5+46 d e^4-9 e^5\right ) \log \left (5 x^2+2 x+3\right )}{2 \left (5 d^2-2 d e+3 e^2\right )^4}-\frac{e \left (3 d^2 e^2+5 d^3 e+4 d^4-d e^3+2 e^4\right )}{\left (5 d^2-2 d e+3 e^2\right )^3 (d+e x)}+\frac{e \left (12 d^3 e^2-76 d^2 e^3+83 d^4 e+40 d^5+46 d e^4-9 e^5\right ) \log (d+e x)}{\left (5 d^2-2 d e+3 e^2\right )^4}+\frac{\left (209039 d^4 e^2-921444 d^3 e^3+380621 d^2 e^4+3070 d^5 e+211875 d^6-49586 d e^5-43695 e^6\right ) \tan ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{1568 \sqrt{14} \left (5 d^2-2 d e+3 e^2\right )^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((e*(4*d^4 + 5*d^3*e + 3*d^2*e^2 - d*e^3 + 2*e^4))/((5*d^2 - 2*d*e + 3*e^2)^3*(d + e*x))) - (1367*d^2 - 586*d
*e - 703*e^2 + (423*d^2 - 2734*d*e + 293*e^2)*x)/(280*(5*d^2 - 2*d*e + 3*e^2)^2*(3 + 2*x + 5*x^2)^2) + (171735
*d^4 - 117284*d^3*e - 200502*d^2*e^2 + 104428*d*e^3 - 23189*e^4 + 5*(11015*d^4 - 85924*d^3*e + 34698*d^2*e^2 +
10348*d*e^3 - 3589*e^4)*x)/(7840*(5*d^2 - 2*d*e + 3*e^2)^3*(3 + 2*x + 5*x^2)) + ((211875*d^6 + 3070*d^5*e + 2
09039*d^4*e^2 - 921444*d^3*e^3 + 380621*d^2*e^4 - 49586*d*e^5 - 43695*e^6)*ArcTan[(1 + 5*x)/Sqrt[14]])/(1568*S
qrt[14]*(5*d^2 - 2*d*e + 3*e^2)^4) + (e*(40*d^5 + 83*d^4*e + 12*d^3*e^2 - 76*d^2*e^3 + 46*d*e^4 - 9*e^5)*Log[d
+ e*x])/(5*d^2 - 2*d*e + 3*e^2)^4 - (e*(40*d^5 + 83*d^4*e + 12*d^3*e^2 - 76*d^2*e^3 + 46*d*e^4 - 9*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)^2 \left (3+2 x+5 x^2\right )^3} \, dx &=-\frac{1367 d^2-586 d e-703 e^2+\left (423 d^2-2734 d e+293 e^2\right ) x}{280 \left (5 d^2-2 d e+3 e^2\right )^2 \left (3+2 x+5 x^2\right )^2}+\frac{1}{112} \int \frac{\frac{2 \left (3267 d^4-5686 d^3 e+7577 d^2 e^2-2240 d e^3+1680 e^4\right )}{5 \left (5 d^2-2 d e+3 e^2\right )^2}-\frac{4 \left (4620 d^4-2427 d^3 e+646 d^2 e^2-1417 d e^3+140 e^4\right ) x}{5 \left (5 d^2-2 d e+3 e^2\right )^2}+\frac{2 \left (5600 d^4-4480 d^3 e+6347 d^2 e^2+5514 d e^3+1137 e^4\right ) x^2}{5 \left (5 d^2-2 d e+3 e^2\right )^2}}{(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}{280 \left (5 d^2-2 d e+3 e^2\right )^2 \left (3+2 x+5 x^2\right )^2}+\frac{171735 d^4-117284 d^3 e-200502 d^2 e^2+104428 d e^3-23189 e^4+5 \left (11015 d^4-85924 d^3 e+34698 d^2 e^2+10348 d e^3-3589 e^4\right ) x}{7840 \left (5 d^2-2 d e+3 e^2\right )^3 \left (3+2 x+5 x^2\right )}+\frac{\int \frac{\frac{4 \left (42375 d^6+5020 d^5 e+48810 d^4 e^2-77460 d^3 e^3+66971 d^2 e^4-18816 d e^5+9408 e^6\right )}{\left (5 d^2-2 d e+3 e^2\right )^3}+\frac{8 e \left (11015 d^5-53780 d^4 e+28426 d^3 e^2-36692 d^2 e^3+15227 d e^4-3920 e^5\right ) x}{\left (5 d^2-2 d e+3 e^2\right )^3}+\frac{4 e^2 \left (11015 d^4-85924 d^3 e+34698 d^2 e^2+10348 d e^3-3589 e^4\right ) x^2}{\left (5 d^2-2 d e+3 e^2\right )^3}}{(d+e x)^2 \left (3+2 x+5 x^2\right )} \, dx}{6272}\\ &=-\frac{1367 d^2-586 d e-703 e^2+\left (423 d^2-2734 d e+293 e^2\right ) x}{280 \left (5 d^2-2 d e+3 e^2\right )^2 \left (3+2 x+5 x^2\right )^2}+\frac{171735 d^4-117284 d^3 e-200502 d^2 e^2+104428 d e^3-23189 e^4+5 \left (11015 d^4-85924 d^3 e+34698 d^2 e^2+10348 d e^3-3589 e^4\right ) x}{7840 \left (5 d^2-2 d e+3 e^2\right )^3 \left (3+2 x+5 x^2\right )}+\frac{\int \left (\frac{6272 e^2 \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 )^3 (d+e x)^2}-\frac{6272 e^2 \left (-40 d^5-83 d^4 e-12 d^3 e^2+76 d^2 e^3-46 d e^4+9 e^5\right )}{\left (5 d^2-2 d e+3 e^2\right )^4 (d+e x)}+\frac{4 \left (211875 d^6-59650 d^5 e+78895 d^4 e^2-940260 d^3 e^3+499789 d^2 e^4-121714 d e^5-29583 e^6-7840 e \left (40 d^5+83 d^4 e+12 d^3 e^2-76 d^2 e^3+46 d e^4-9 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}{6272}\\ &=-\frac{e \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 )^3 (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}{280 \left (5 d^2-2 d e+3 e^2\right )^2 \left (3+2 x+5 x^2\right )^2}+\frac{171735 d^4-117284 d^3 e-200502 d^2 e^2+104428 d e^3-23189 e^4+5 \left (11015 d^4-85924 d^3 e+34698 d^2 e^2+10348 d e^3-3589 e^4\right ) x}{7840 \left (5 d^2-2 d e+3 e^2\right )^3 \left (3+2 x+5 x^2\right )}+\frac{e \left (40 d^5+83 d^4 e+12 d^3 e^2-76 d^2 e^3+46 d e^4-9 e^5\right ) \log (d+e x)}{\left (5 d^2-2 d e+3 e^2\right )^4}+\frac{\int \frac{211875 d^6-59650 d^5 e+78895 d^4 e^2-940260 d^3 e^3+499789 d^2 e^4-121714 d e^5-29583 e^6-7840 e \left (40 d^5+83 d^4 e+12 d^3 e^2-76 d^2 e^3+46 d e^4-9 e^5\right ) x}{3+2 x+5 x^2} \, dx}{1568 \left (5 d^2-2 d e+3 e^2\right )^4}\\ &=-\frac{e \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 )^3 (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}{280 \left (5 d^2-2 d e+3 e^2\right )^2 \left (3+2 x+5 x^2\right )^2}+\frac{171735 d^4-117284 d^3 e-200502 d^2 e^2+104428 d e^3-23189 e^4+5 \left (11015 d^4-85924 d^3 e+34698 d^2 e^2+10348 d e^3-3589 e^4\right ) x}{7840 \left (5 d^2-2 d e+3 e^2\right )^3 \left (3+2 x+5 x^2\right )}+\frac{e \left (40 d^5+83 d^4 e+12 d^3 e^2-76 d^2 e^3+46 d e^4-9 e^5\right ) \log (d+e x)}{\left (5 d^2-2 d e+3 e^2\right )^4}-\frac{\left (e \left (40 d^5+83 d^4 e+12 d^3 e^2-76 d^2 e^3+46 d e^4-9 e^5\right )\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 (211875 d^6+3070 d^5 e+209039 d^4 e^2-921444 d^3 e^3+380621 d^2 e^4-49586 d e^5-43695 e^6\right ) \int \frac{1}{3+2 x+5 x^2} \, dx}{1568 \left (5 d^2-2 d e+3 e^2\right )^4}\\ &=-\frac{e \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 )^3 (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}{280 \left (5 d^2-2 d e+3 e^2\right )^2 \left (3+2 x+5 x^2\right )^2}+\frac{171735 d^4-117284 d^3 e-200502 d^2 e^2+104428 d e^3-23189 e^4+5 \left (11015 d^4-85924 d^3 e+34698 d^2 e^2+10348 d e^3-3589 e^4\right ) x}{7840 \left (5 d^2-2 d e+3 e^2\right )^3 \left (3+2 x+5 x^2\right )}+\frac{e \left (40 d^5+83 d^4 e+12 d^3 e^2-76 d^2 e^3+46 d e^4-9 e^5\right ) \log (d+e x)}{\left (5 d^2-2 d e+3 e^2\right )^4}-\frac{e \left (40 d^5+83 d^4 e+12 d^3 e^2-76 d^2 e^3+46 d e^4-9 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 (211875 d^6+3070 d^5 e+209039 d^4 e^2-921444 d^3 e^3+380621 d^2 e^4-49586 d e^5-43695 e^6\right ) \operatorname{Subst}\left (\int \frac{1}{-56-x^2} \, dx,x,2+10 x\right )}{784 \left (5 d^2-2 d e+3 e^2\right )^4}\\ &=-\frac{e \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 )^3 (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}{280 \left (5 d^2-2 d e+3 e^2\right )^2 \left (3+2 x+5 x^2\right )^2}+\frac{171735 d^4-117284 d^3 e-200502 d^2 e^2+104428 d e^3-23189 e^4+5 \left (11015 d^4-85924 d^3 e+34698 d^2 e^2+10348 d e^3-3589 e^4\right ) x}{7840 \left (5 d^2-2 d e+3 e^2\right )^3 \left (3+2 x+5 x^2\right )}+\frac{\left (211875 d^6+3070 d^5 e+209039 d^4 e^2-921444 d^3 e^3+380621 d^2 e^4-49586 d e^5-43695 e^6\right ) \tan ^{-1}\left (\frac{1+5 x}{\sqrt{14}}\right )}{1568 \sqrt{14} \left (5 d^2-2 d e+3 e^2\right )^4}+\frac{e \left (40 d^5+83 d^4 e+12 d^3 e^2-76 d^2 e^3+46 d e^4-9 e^5\right ) \log (d+e x)}{\left (5 d^2-2 d e+3 e^2\right )^4}-\frac{e \left (40 d^5+83 d^4 e+12 d^3 e^2-76 d^2 e^3+46 d e^4-9 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 = 0.480602, size = 389, normalized size = 0.88 \[ \frac{-\frac{392 \left (5 d^2-2 d e+3 e^2\right )^2 \left (d^2 (423 x+1367)-2 d e (1367 x+293)+e^2 (293 x-703)\right )}{\left (5 x^2+2 x+3\right )^2}+\frac{14 \left (5 d^2-2 d e+3 e^2\right ) \left (6 d^2 e^2 (28915 x-33417)-4 d^3 e (107405 x+29321)+5 d^4 (11015 x+34347)+4 d e^3 (12935 x+26107)-e^4 (17945 x+23189)\right )}{5 x^2+2 x+3}-54880 e \left (12 d^3 e^2-76 d^2 e^3+83 d^4 e+40 d^5+46 d e^4-9 e^5\right ) \log \left (5 x^2+2 x+3\right )-\frac{109760 e \left (3 d^2 e^2+5 d^3 e+4 d^4-d e^3+2 e^4\right ) \left (5 d^2-2 d e+3 e^2\right )}{d+e x}+109760 e \left (12 d^3 e^2-76 d^2 e^3+83 d^4 e+40 d^5+46 d e^4-9 e^5\right ) \log (d+e x)+5 \sqrt{14} \left (209039 d^4 e^2-921444 d^3 e^3+380621 d^2 e^4+3070 d^5 e+211875 d^6-49586 d e^5-43695 e^6\right ) \tan ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{109760 \left (5 d^2-2 d e+3 e^2\right )^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-109760*e*(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))/(d + e*x) - (392*(5*d^2 - 2
*d*e + 3*e^2)^2*(e^2*(-703 + 293*x) + d^2*(1367 + 423*x) - 2*d*e*(293 + 1367*x)))/(3 + 2*x + 5*x^2)^2 + (14*(5
*d^2 - 2*d*e + 3*e^2)*(5*d^4*(34347 + 11015*x) + 4*d*e^3*(26107 + 12935*x) - e^4*(23189 + 17945*x) + 6*d^2*e^2
*(-33417 + 28915*x) - 4*d^3*e*(29321 + 107405*x)))/(3 + 2*x + 5*x^2) + 5*Sqrt[14]*(211875*d^6 + 3070*d^5*e + 2
09039*d^4*e^2 - 921444*d^3*e^3 + 380621*d^2*e^4 - 49586*d*e^5 - 43695*e^6)*ArcTan[(1 + 5*x)/Sqrt[14]] + 109760
*e*(40*d^5 + 83*d^4*e + 12*d^3*e^2 - 76*d^2*e^3 + 46*d*e^4 - 9*e^5)*Log[d + e*x] - 54880*e*(40*d^5 + 83*d^4*e
+ 12*d^3*e^2 - 76*d^2*e^3 + 46*d*e^4 - 9*e^5)*Log[3 + 2*x + 5*x^2])/(109760*(5*d^2 - 2*d*e + 3*e^2)^4)

________________________________________________________________________________________

Maple [B]  time = 0.077, size = 1850, normalized size = 4.2 \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)^3,x)

[Out]

-9*e^6/(5*d^2-2*d*e+3*e^2)^4*ln(e*x+d)-2*e^5/(5*d^2-2*d*e+3*e^2)^3/(e*x+d)-6309/1568/(5*d^2-2*d*e+3*e^2)^4/(5*
x^2+2*x+3)^2*e^6+323825/1568/(5*d^2-2*d*e+3*e^2)^4/(5*x^2+2*x+3)^2*d^6+9/2/(5*d^2-2*d*e+3*e^2)^4*ln(5*x^2+2*x+
3)*e^6+116869/392/(5*d^2-2*d*e+3*e^2)^4/(5*x^2+2*x+3)^2*d^3*e^3-74895/1568/(5*d^2-2*d*e+3*e^2)^4/(5*x^2+2*x+3)
^2*x*e^6-91101/1568/(5*d^2-2*d*e+3*e^2)^4/(5*x^2+2*x+3)^2*x^2*e^6-53835/1568/(5*d^2-2*d*e+3*e^2)^4/(5*x^2+2*x+
3)^2*x^3*e^6+275375/1568/(5*d^2-2*d*e+3*e^2)^4/(5*x^2+2*x+3)^2*x^3*d^6+449475/1568/(5*d^2-2*d*e+3*e^2)^4/(5*x^
2+2*x+3)^2*d^6*x+99045/784/(5*d^2-2*d*e+3*e^2)^4/(5*x^2+2*x+3)^2*d*e^5-530209/1568/(5*d^2-2*d*e+3*e^2)^4/(5*x^
2+2*x+3)^2*d^2*e^4-161395/784/(5*d^2-2*d*e+3*e^2)^4/(5*x^2+2*x+3)^2*d^5*e-379131/1568/(5*d^2-2*d*e+3*e^2)^4/(5
*x^2+2*x+3)^2*d^4*e^2+40*e/(5*d^2-2*d*e+3*e^2)^4*ln(e*x+d)*d^5+83*e^2/(5*d^2-2*d*e+3*e^2)^4*ln(e*x+d)*d^4+12*e
^3/(5*d^2-2*d*e+3*e^2)^4*ln(e*x+d)*d^3+327265/1568/(5*d^2-2*d*e+3*e^2)^4/(5*x^2+2*x+3)^2*x^3*d^2*e^4+95555/784
/(5*d^2-2*d*e+3*e^2)^4/(5*x^2+2*x+3)^2*x^3*d*e^5-648385/784/(5*d^2-2*d*e+3*e^2)^4/(5*x^2+2*x+3)^2*x*d^5*e+2180
53/784/(5*d^2-2*d*e+3*e^2)^4/(5*x^2+2*x+3)^2*x^2*d*e^5+208007/784/(5*d^2-2*d*e+3*e^2)^4/(5*x^2+2*x+3)^2*x*d*e^
5-916595/784/(5*d^2-2*d*e+3*e^2)^4/(5*x^2+2*x+3)^2*x^2*d^5*e+606287/1568/(5*d^2-2*d*e+3*e^2)^4/(5*x^2+2*x+3)^2
*x*d^4*e^2-3993/392/(5*d^2-2*d*e+3*e^2)^4/(5*x^2+2*x+3)^2*x*d^3*e^3+380621/21952/(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^4-24793/10976/(5*d^2-2*d*e+3*e^2)^4*14^(1/2)*arctan(1/28*(10*x+2)*14^(1
/2))*d*e^5+968825/1568/(5*d^2-2*d*e+3*e^2)^4/(5*x^2+2*x+3)^2*x^2*d^6-344285/392/(5*d^2-2*d*e+3*e^2)^4/(5*x^2+2
*x+3)^2*x^3*d^3*e^3-76*e^4/(5*d^2-2*d*e+3*e^2)^4*ln(e*x+d)*d^2+46*e^5/(5*d^2-2*d*e+3*e^2)^4*ln(e*x+d)*d-4*e/(5
*d^2-2*d*e+3*e^2)^3/(e*x+d)*d^4-5*e^2/(5*d^2-2*d*e+3*e^2)^3/(e*x+d)*d^3+211875/21952/(5*d^2-2*d*e+3*e^2)^4*14^
(1/2)*arctan(1/28*(10*x+2)*14^(1/2))*d^6-43695/21952/(5*d^2-2*d*e+3*e^2)^4*14^(1/2)*arctan(1/28*(10*x+2)*14^(1
/2))*e^6-6/(5*d^2-2*d*e+3*e^2)^4*ln(5*x^2+2*x+3)*d^3*e^3+38/(5*d^2-2*d*e+3*e^2)^4*ln(5*x^2+2*x+3)*d^2*e^4-23/(
5*d^2-2*d*e+3*e^2)^4*ln(5*x^2+2*x+3)*d*e^5-20/(5*d^2-2*d*e+3*e^2)^4*ln(5*x^2+2*x+3)*d^5*e-83/2/(5*d^2-2*d*e+3*
e^2)^4*ln(5*x^2+2*x+3)*d^4*e^2-3*e^3/(5*d^2-2*d*e+3*e^2)^3/(e*x+d)*d^2+e^4/(5*d^2-2*d*e+3*e^2)^3/(e*x+d)*d-434
995/1568/(5*d^2-2*d*e+3*e^2)^4/(5*x^2+2*x+3)^2*x*d^2*e^4-1129125/784/(5*d^2-2*d*e+3*e^2)^4/(5*x^2+2*x+3)^2*x^3
*d^5*e+504029/1568/(5*d^2-2*d*e+3*e^2)^4/(5*x^2+2*x+3)^2*x^2*d^4*e^2+1891915/1568/(5*d^2-2*d*e+3*e^2)^4/(5*x^2
+2*x+3)^2*x^3*d^4*e^2+5109/392/(5*d^2-2*d*e+3*e^2)^4/(5*x^2+2*x+3)^2*x^2*d^3*e^3-795401/1568/(5*d^2-2*d*e+3*e^
2)^4/(5*x^2+2*x+3)^2*x^2*d^2*e^4+1535/10976/(5*d^2-2*d*e+3*e^2)^4*14^(1/2)*arctan(1/28*(10*x+2)*14^(1/2))*d^5*
e+209039/21952/(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^2-230361/5488/(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^3

________________________________________________________________________________________

Maxima [B]  time = 1.7046, size = 1237, 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)^2/(5*x^2+2*x+3)^3,x, algorithm="maxima")

[Out]

1/21952*sqrt(14)*(211875*d^6 + 3070*d^5*e + 209039*d^4*e^2 - 921444*d^3*e^3 + 380621*d^2*e^4 - 49586*d*e^5 - 4
3695*e^6)*arctan(1/14*sqrt(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) + (40*d^5*e + 83*d^4*e^2 + 12*d^3*e^3 - 76*d^2*e^4 + 46*d*e^
5 - 9*e^6)*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 + 75
6*d^2*e^6 - 216*d*e^7 + 81*e^8) - 1/2*(40*d^5*e + 83*d^4*e^2 + 12*d^3*e^3 - 76*d^2*e^4 + 46*d*e^5 - 9*e^6)*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) + 1/1568*(64765*d^5 - 95100*d^4*e - 200706*d^3*e^2 + 22292*d^2*e^3 + 12009*d*e^4 - 2
8224*e^5 - 5*(20345*d^4*e + 125124*d^3*e^2 - 11178*d^2*e^3 - 18188*d*e^4 + 19269*e^5)*x^4 + (55075*d^5 - 36129
5*d^4*e - 272442*d^3*e^2 - 173446*d^2*e^3 + 138539*d*e^4 - 93087*e^5)*x^3 + (193765*d^5 - 412485*d^4*e - 62106
2*d^3*e^2 - 56850*d^2*e^3 + 144973*d*e^4 - 131589*e^5)*x^2 + 3*(29965*d^5 - 77965*d^4*e - 51590*d^3*e^2 - 2152
2*d^2*e^3 + 19493*d*e^4 - 13245*e^5)*x)/(1125*d^7 - 1350*d^6*e + 2565*d^5*e^2 - 1692*d^4*e^3 + 1539*d^3*e^4 -
486*d^2*e^5 + 243*d*e^6 + 25*(125*d^6*e - 150*d^5*e^2 + 285*d^4*e^3 - 188*d^3*e^4 + 171*d^2*e^5 - 54*d*e^6 + 2
7*e^7)*x^5 + 5*(625*d^7 - 250*d^6*e + 825*d^5*e^2 + 200*d^4*e^3 + 103*d^3*e^4 + 414*d^2*e^5 - 81*d*e^6 + 108*e
^7)*x^4 + 2*(1250*d^7 + 625*d^6*e + 300*d^5*e^2 + 2965*d^4*e^3 - 1486*d^3*e^4 + 2367*d^2*e^5 - 648*d*e^6 + 459
*e^7)*x^3 + 2*(2125*d^7 - 1800*d^6*e + 3945*d^5*e^2 - 1486*d^4*e^3 + 1779*d^3*e^4 + 108*d^2*e^5 + 135*d*e^6 +
162*e^7)*x^2 + 3*(500*d^7 - 225*d^6*e + 690*d^5*e^2 + 103*d^4*e^3 + 120*d^3*e^4 + 297*d^2*e^5 - 54*d*e^6 + 81*
e^7)*x)

________________________________________________________________________________________

Fricas [B]  time = 3.37601, size = 4680, normalized size = 10.56 \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)^3,x, algorithm="fricas")

[Out]

1/21952*(4533550*d^7 - 8470420*d^6*e - 8666490*d^5*e^2 + 3186008*d^4*e^3 - 8213198*d^3*e^4 - 1375668*d^2*e^5 +
1294650*d*e^6 - 1185408*e^7 - 70*(101725*d^6*e + 584930*d^5*e^2 - 245103*d^4*e^3 + 306788*d^3*e^4 + 99187*d^2
*e^5 - 93102*d*e^6 + 57807*e^7)*x^4 + 14*(275375*d^7 - 1916625*d^6*e - 474395*d^5*e^2 - 1406231*d^4*e^3 + 2222
61*d^3*e^4 - 1262851*d^2*e^5 + 601791*d*e^6 - 279261*e^7)*x^3 + 14*(968825*d^7 - 2449955*d^6*e - 1699045*d^5*e
^2 - 279581*d^4*e^3 - 1024621*d^3*e^4 - 1118441*d^2*e^5 + 698097*d*e^6 - 394767*e^7)*x^2 + sqrt(14)*(1906875*d
^7 + 27630*d^6*e + 1881351*d^5*e^2 - 8292996*d^4*e^3 + 3425589*d^3*e^4 - 446274*d^2*e^5 - 393255*d*e^6 + 25*(2
11875*d^6*e + 3070*d^5*e^2 + 209039*d^4*e^3 - 921444*d^3*e^4 + 380621*d^2*e^5 - 49586*d*e^6 - 43695*e^7)*x^5 +
5*(1059375*d^7 + 862850*d^6*e + 1057475*d^5*e^2 - 3771064*d^4*e^3 - 1782671*d^3*e^4 + 1274554*d^2*e^5 - 41681
9*d*e^6 - 174780*e^7)*x^4 + 2*(2118750*d^7 + 3632575*d^6*e + 2142580*d^5*e^2 - 5660777*d^4*e^3 - 11858338*d^3*
e^4 + 5974697*d^2*e^5 - 1279912*d*e^6 - 742815*e^7)*x^3 + 2*(3601875*d^7 + 1323440*d^6*e + 3572083*d^5*e^2 - 1
4410314*d^4*e^3 + 941893*d^3*e^4 + 1440764*d^2*e^5 - 1040331*d*e^6 - 262170*e^7)*x^2 + 3*(847500*d^7 + 647905*
d^6*e + 845366*d^5*e^2 - 3058659*d^4*e^3 - 1241848*d^3*e^4 + 943519*d^2*e^5 - 323538*d*e^6 - 131085*e^7)*x)*ar
ctan(1/14*sqrt(14)*(5*x + 1)) + 42*(149825*d^7 - 449755*d^6*e - 12125*d^5*e^2 - 238325*d^4*e^3 - 14261*d^3*e^4
- 169777*d^2*e^5 + 84969*d*e^6 - 39735*e^7)*x + 21952*(360*d^6*e + 747*d^5*e^2 + 108*d^4*e^3 - 684*d^3*e^4 +
414*d^2*e^5 - 81*d*e^6 + 25*(40*d^5*e^2 + 83*d^4*e^3 + 12*d^3*e^4 - 76*d^2*e^5 + 46*d*e^6 - 9*e^7)*x^5 + 5*(20
0*d^6*e + 575*d^5*e^2 + 392*d^4*e^3 - 332*d^3*e^4 - 74*d^2*e^5 + 139*d*e^6 - 36*e^7)*x^4 + 2*(400*d^6*e + 1510
*d^5*e^2 + 1531*d^4*e^3 - 556*d^3*e^4 - 832*d^2*e^5 + 692*d*e^6 - 153*e^7)*x^3 + 2*(680*d^6*e + 1651*d^5*e^2 +
702*d^4*e^3 - 1220*d^3*e^4 + 326*d^2*e^5 + 123*d*e^6 - 54*e^7)*x^2 + 3*(160*d^6*e + 452*d^5*e^2 + 297*d^4*e^3
- 268*d^3*e^4 - 44*d^2*e^5 + 102*d*e^6 - 27*e^7)*x)*log(e*x + d) - 10976*(360*d^6*e + 747*d^5*e^2 + 108*d^4*e
^3 - 684*d^3*e^4 + 414*d^2*e^5 - 81*d*e^6 + 25*(40*d^5*e^2 + 83*d^4*e^3 + 12*d^3*e^4 - 76*d^2*e^5 + 46*d*e^6 -
9*e^7)*x^5 + 5*(200*d^6*e + 575*d^5*e^2 + 392*d^4*e^3 - 332*d^3*e^4 - 74*d^2*e^5 + 139*d*e^6 - 36*e^7)*x^4 +
2*(400*d^6*e + 1510*d^5*e^2 + 1531*d^4*e^3 - 556*d^3*e^4 - 832*d^2*e^5 + 692*d*e^6 - 153*e^7)*x^3 + 2*(680*d^6
*e + 1651*d^5*e^2 + 702*d^4*e^3 - 1220*d^3*e^4 + 326*d^2*e^5 + 123*d*e^6 - 54*e^7)*x^2 + 3*(160*d^6*e + 452*d^
5*e^2 + 297*d^4*e^3 - 268*d^3*e^4 - 44*d^2*e^5 + 102*d*e^6 - 27*e^7)*x)*log(5*x^2 + 2*x + 3))/(5625*d^9 - 9000
*d^8*e + 18900*d^7*e^2 - 17640*d^6*e^3 + 18774*d^5*e^4 - 10584*d^4*e^5 + 6804*d^3*e^6 - 1944*d^2*e^7 + 729*d*e
^8 + 25*(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)*x^5 + 5*(3125*d^9 - 2500*d^8*e + 6500*d^7*e^2 - 1400*d^6*e^3 + 2590*d^5*e^4 + 2464*d^4*e^5
- 924*d^3*e^6 + 1944*d^2*e^7 - 459*d*e^8 + 324*e^9)*x^4 + 2*(6250*d^9 + 625*d^8*e + 4000*d^7*e^2 + 16100*d^6*
e^3 - 12460*d^5*e^4 + 23702*d^4*e^5 - 12432*d^3*e^6 + 10692*d^2*e^7 - 2862*d*e^8 + 1377*e^9)*x^3 + 2*(10625*d^
9 - 13250*d^8*e + 29700*d^7*e^2 - 20720*d^6*e^3 + 23702*d^5*e^4 - 7476*d^4*e^5 + 5796*d^3*e^6 + 864*d^2*e^7 +
81*d*e^8 + 486*e^9)*x^2 + 3*(2500*d^9 - 2125*d^8*e + 5400*d^7*e^2 - 1540*d^6*e^3 + 2464*d^5*e^4 + 1554*d^4*e^5
- 504*d^3*e^6 + 1404*d^2*e^7 - 324*d*e^8 + 243*e^9)*x)

________________________________________________________________________________________

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)**2/(5*x**2+2*x+3)**3,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.26386, size = 1029, normalized size = 2.32 \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)^3,x, algorithm="giac")

[Out]

1/21952*sqrt(14)*(211875*d^6*e^2 + 3070*d^5*e^3 + 209039*d^4*e^4 - 921444*d^3*e^5 + 380621*d^2*e^6 - 49586*d*e
^7 - 43695*e^8)*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)/(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*(40*d^5*e + 83*d^4*e^2 + 12*d^3*e^3 - 76*d^2*e^4 + 46*d*e^5 - 9*e^6)*log(-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)/(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) - (4*d^4*e^7/(x*e + d)
+ 5*d^3*e^8/(x*e + d) + 3*d^2*e^9/(x*e + d) - d*e^10/(x*e + d) + 2*e^11/(x*e + d))/(125*d^6*e^6 - 150*d^5*e^7
+ 285*d^4*e^8 - 188*d^3*e^9 + 171*d^2*e^10 - 54*d*e^11 + 27*e^12) + 1/1568*(275375*d^5*e - 3006775*d^4*e^2 +
1394650*d^3*e^3 + 1835350*d^2*e^4 - 734925*d*e^5 - 5*(165225*d^6*e^2 - 1997830*d^5*e^3 + 1218421*d^4*e^4 + 152
0564*d^3*e^5 - 947049*d^2*e^6 + 93386*d*e^7 + 7963*e^8)*e^(-1)/(x*e + d) + (826125*d^7*e^3 - 10957975*d^6*e^4
+ 8449735*d^5*e^5 + 8211175*d^4*e^6 - 7879025*d^3*e^7 + 2996315*d^2*e^8 - 443947*d*e^9 - 67267*e^10)*e^(-2)/(x
*e + d)^2 - (275375*d^8*e^4 - 3975600*d^7*e^5 + 3752280*d^6*e^6 + 2119880*d^5*e^7 - 3655050*d^4*e^8 + 4008480*
d^3*e^9 - 1453312*d^2*e^10 - 197784*d*e^11 + 66483*e^12)*e^(-3)/(x*e + d)^3 + 17525*e^6)/((5*d^2 - 2*d*e + 3*e
^2)^4*(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)^2)