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

[Out]

1/2*(-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/(e*x+d)^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/(e*x+d)+1/28*(-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)/(5*d^2-2*d*e+3*e^2)^3/(5*x^2+2*x+3)+(205*d^5-19*d^4*e-846*d^3*e^2+396*d^2*e^3+57*d*e^4-21*e^
5)*ln(e*x+d)/(5*d^2-2*d*e+3*e^2)^4-1/2*(205*d^5-19*d^4*e-846*d^3*e^2+396*d^2*e^3+57*d*e^4-21*e^5)*ln(5*x^2+2*x
+3)/(5*d^2-2*d*e+3*e^2)^4+1/392*(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*(1+5*x)*14^(1/2))/(5*d^2-2*d*e+3*e^2)^4*14^(1/2)

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Rubi [A]  time = 0.71, antiderivative size = 412, normalized size of antiderivative = 1.00, 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 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 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 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]

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 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 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]

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*}

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Mathematica [A]  time = 0.40, size = 363, normalized size = 0.88 \[ \frac {-\frac {14 \left (5 d^2-2 d e+3 e^2\right ) \left (d^3 (423 x+1367)-3 d^2 e (1367 x+293)+3 d e^2 (293 x-703)+e^3 (703 x+457)\right )}{5 x^2+2 x+3}-\frac {196 \left (4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4\right ) \left (5 d^2-2 d e+3 e^2\right )^2}{e (d+e x)^2}+\frac {392 \left (-41 d^4+8 d^3 e+60 d^2 e^2-24 d e^3+5 e^4\right ) \left (5 d^2-2 d e+3 e^2\right )}{d+e x}+196 \left (-205 d^5+19 d^4 e+846 d^3 e^2-396 d^2 e^3-57 d e^4+21 e^5\right ) \log \left (5 x^2+2 x+3\right )+392 \left (205 d^5-19 d^4 e-846 d^3 e^2+396 d^2 e^3+57 d e^4-21 e^5\right ) \log (d+e x)+\sqrt {14} \left (6565 d^5-74017 d^4 e+35022 d^3 e^2+42858 d^2 e^3-17247 d e^4+579 e^5\right ) \tan ^{-1}\left (\frac {5 x+1}{\sqrt {14}}\right )}{392 \left (5 d^2-2 d e+3 e^2\right )^4} \]

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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fricas [B]  time = 1.46, size = 1499, normalized size = 3.64 \[ \text {result too large to display} \]

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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giac [A]  time = 0.20, size = 595, normalized size = 1.44 \[ \frac {\sqrt {14} {\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 )} \arctan \left (\frac {1}{14} \, \sqrt {14} {\left (5 \, x + 1\right )}\right )}{392 \, {\left (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}\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 \left (5 \, x^{2} + 2 \, x + 3\right )}{2 \, {\left (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}\right )}} + \frac {{\left (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}\right )} \log \left ({\left | x e + d \right |}\right )}{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}} - \frac {{\left (4200 \, d^{8} + 25945 \, d^{7} e - 12715 \, d^{6} e^{2} - 16656 \, d^{5} e^{3} + 28226 \, d^{4} e^{4} + {\left (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}\right )} x^{3} - 31223 \, d^{3} e^{5} + {\left (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}\right )} x^{2} + 12753 \, d^{2} e^{6} + {\left (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}\right )} x - 2646 \, d e^{7} + 756 \, e^{8}\right )} e^{\left (-1\right )}}{28 \, {\left (5 \, d^{2} - 2 \, d e + 3 \, e^{2}\right )}^{4} {\left (5 \, x^{2} + 2 \, x + 3\right )} {\left (x e + d\right )}^{2}} \]

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)

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maple [B]  time = 0.03, size = 1314, normalized size = 3.19 \[ \text {result too large to display} \]

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]

-3/2/(5*d^2-2*d*e+3*e^2)^2*e/(e*x+d)^2*d^2+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^3-19/(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+396/(5*d^2-2*d*e+3*e^2)^4*ln(e*x+d)*d^2*e^
3+57/(5*d^2-2*d*e+3*e^2)^4*ln(e*x+d)*d*e^4-423/28/(5*d^2-2*d*e+3*e^2)^4/(x^2+2/5*x+3/5)*x*d^5-2109/140/(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^2-198/(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^4-5/2/(5*d^2-2*d*e+3*e^2
)^2/(e*x+d)^2*d^3+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
-41/(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+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^3-17247/392/(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+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^3-1231/140/(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

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maxima [B]  time = 1.09, size = 851, normalized size = 2.07 \[ \frac {\sqrt {14} {\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 )} \arctan \left (\frac {1}{14} \, \sqrt {14} {\left (5 \, x + 1\right )}\right )}{392 \, {\left (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}\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 \left (e x + d\right )}{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}} - \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 (5 \, x^{2} + 2 \, x + 3\right )}{2 \, {\left (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}\right )}} - \frac {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} + {\left (5740 \, d^{4} e^{2} - 697 \, d^{3} e^{3} - 12501 \, d^{2} e^{4} + 4239 \, d e^{5} + 3 \, e^{6}\right )} x^{3} + {\left (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}\right )} x^{2} + {\left (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}\right )} x}{28 \, {\left (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 \, {\left (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}\right )} x^{4} + 2 \, {\left (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}\right )} x^{3} + {\left (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}\right )} x^{2} + 2 \, {\left (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}\right )} x\right )}} \]

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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mupad [B]  time = 4.94, size = 887, normalized size = 2.15 \[ \ln \left (d+e\,x\right )\,\left (\frac {\frac {41\,d}{5}+\frac {29\,e}{5}}{{\left (5\,d^2-2\,d\,e+3\,e^2\right )}^2}+\frac {168\,e^4\,\left (458\,d-7\,e\right )}{125\,{\left (5\,d^2-2\,d\,e+3\,e^2\right )}^4}-\frac {2\,e^2\,\left (12610\,d+1329\,e\right )}{125\,{\left (5\,d^2-2\,d\,e+3\,e^2\right )}^3}\right )-\frac {\frac {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}{28\,e\,\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 {x^3\,\left (5740\,d^4\,e-697\,d^3\,e^2-12501\,d^2\,e^3+4239\,d\,e^4+3\,e^5\right )}{28\,\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 {x^2\,\left (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\right )}{28\,e\,\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 {x\,\left (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\right )}{28\,e\,\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 )}}{x^2\,\left (5\,d^2+4\,d\,e+3\,e^2\right )+x\,\left (2\,d^2+6\,e\,d\right )+3\,d^2+x^3\,\left (2\,e^2+10\,d\,e\right )+5\,e^2\,x^4}+\frac {\ln \left (x+\frac {1}{5}-\frac {\sqrt {14}\,1{}\mathrm {i}}{5}\right )\,\left (\left (\frac {6565\,\sqrt {14}}{784}-\frac {205}{2}{}\mathrm {i}\right )\,d^5+\left (-\frac {74017\,\sqrt {14}}{784}+\frac {19}{2}{}\mathrm {i}\right )\,d^4\,e+\left (\frac {17511\,\sqrt {14}}{392}+423{}\mathrm {i}\right )\,d^3\,e^2+\left (\frac {21429\,\sqrt {14}}{392}-198{}\mathrm {i}\right )\,d^2\,e^3+\left (-\frac {17247\,\sqrt {14}}{784}-\frac {57}{2}{}\mathrm {i}\right )\,d\,e^4+\left (\frac {579\,\sqrt {14}}{784}+\frac {21}{2}{}\mathrm {i}\right )\,e^5\right )}{d^8\,625{}\mathrm {i}-d^7\,e\,1000{}\mathrm {i}+d^6\,e^2\,2100{}\mathrm {i}-d^5\,e^3\,1960{}\mathrm {i}+d^4\,e^4\,2086{}\mathrm {i}-d^3\,e^5\,1176{}\mathrm {i}+d^2\,e^6\,756{}\mathrm {i}-d\,e^7\,216{}\mathrm {i}+e^8\,81{}\mathrm {i}}-\frac {\ln \left (x+\frac {1}{5}+\frac {\sqrt {14}\,1{}\mathrm {i}}{5}\right )\,\left (\left (\frac {6565\,\sqrt {14}}{784}+\frac {205}{2}{}\mathrm {i}\right )\,d^5+\left (-\frac {74017\,\sqrt {14}}{784}-\frac {19}{2}{}\mathrm {i}\right )\,d^4\,e+\left (\frac {17511\,\sqrt {14}}{392}-423{}\mathrm {i}\right )\,d^3\,e^2+\left (\frac {21429\,\sqrt {14}}{392}+198{}\mathrm {i}\right )\,d^2\,e^3+\left (-\frac {17247\,\sqrt {14}}{784}+\frac {57}{2}{}\mathrm {i}\right )\,d\,e^4+\left (\frac {579\,\sqrt {14}}{784}-\frac {21}{2}{}\mathrm {i}\right )\,e^5\right )}{d^8\,625{}\mathrm {i}-d^7\,e\,1000{}\mathrm {i}+d^6\,e^2\,2100{}\mathrm {i}-d^5\,e^3\,1960{}\mathrm {i}+d^4\,e^4\,2086{}\mathrm {i}-d^3\,e^5\,1176{}\mathrm {i}+d^2\,e^6\,756{}\mathrm {i}-d\,e^7\,216{}\mathrm {i}+e^8\,81{}\mathrm {i}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(d + e*x)*(((41*d)/5 + (29*e)/5)/(5*d^2 - 2*d*e + 3*e^2)^2 + (168*e^4*(458*d - 7*e))/(125*(5*d^2 - 2*d*e +
3*e^2)^4) - (2*e^2*(12610*d + 1329*e))/(125*(5*d^2 - 2*d*e + 3*e^2)^3)) - ((5525*d^5*e - 714*d*e^5 + 840*d^6 +
252*e^6 + 3355*d^2*e^4 - 6981*d^3*e^3 - 837*d^4*e^2)/(28*e*(125*d^6 - 150*d^5*e - 54*d*e^5 + 27*e^6 + 171*d^2
*e^4 - 188*d^3*e^3 + 285*d^4*e^2)) + (x^3*(4239*d*e^4 + 5740*d^4*e + 3*e^5 - 12501*d^2*e^3 - 697*d^3*e^2))/(28
*(125*d^6 - 150*d^5*e - 54*d*e^5 + 27*e^6 + 171*d^2*e^4 - 188*d^3*e^3 + 285*d^4*e^2)) + (x^2*(6930*d^5*e - 549
*d*e^5 + 1400*d^6 + 597*e^6 + 2349*d^2*e^4 - 15403*d^3*e^3 + 3212*d^4*e^2))/(28*e*(125*d^6 - 150*d^5*e - 54*d*
e^5 + 27*e^6 + 171*d^2*e^4 - 188*d^3*e^3 + 285*d^4*e^2)) + (x*(2454*d*e^5 + 3195*d^5*e + 560*d^6 - 252*e^6 - 6
623*d^2*e^4 - 4799*d^3*e^3 + 2105*d^4*e^2))/(28*e*(125*d^6 - 150*d^5*e - 54*d*e^5 + 27*e^6 + 171*d^2*e^4 - 188
*d^3*e^3 + 285*d^4*e^2)))/(x^2*(4*d*e + 5*d^2 + 3*e^2) + x*(6*d*e + 2*d^2) + 3*d^2 + x^3*(10*d*e + 2*e^2) + 5*
e^2*x^4) + (log(x - (14^(1/2)*1i)/5 + 1/5)*(d^5*((6565*14^(1/2))/784 - 205i/2) + e^5*((579*14^(1/2))/784 + 21i
/2) + d^3*e^2*((17511*14^(1/2))/392 + 423i) + d^2*e^3*((21429*14^(1/2))/392 - 198i) - d*e^4*((17247*14^(1/2))/
784 + 57i/2) - d^4*e*((74017*14^(1/2))/784 - 19i/2)))/(d^8*625i - d^7*e*1000i - d*e^7*216i + e^8*81i + d^2*e^6
*756i - d^3*e^5*1176i + d^4*e^4*2086i - d^5*e^3*1960i + d^6*e^2*2100i) - (log(x + (14^(1/2)*1i)/5 + 1/5)*(d^5*
((6565*14^(1/2))/784 + 205i/2) + e^5*((579*14^(1/2))/784 - 21i/2) + d^3*e^2*((17511*14^(1/2))/392 - 423i) + d^
2*e^3*((21429*14^(1/2))/392 + 198i) - d*e^4*((17247*14^(1/2))/784 - 57i/2) - d^4*e*((74017*14^(1/2))/784 + 19i
/2)))/(d^8*625i - d^7*e*1000i - d*e^7*216i + e^8*81i + d^2*e^6*756i - d^3*e^5*1176i + d^4*e^4*2086i - d^5*e^3*
1960i + d^6*e^2*2100i)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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