### 3.310 $$\int \frac{2+x+3 x^2-5 x^3+4 x^4}{(d+e x)^3 (3+2 x+5 x^2)} \, dx$$

Optimal. Leaf size=317 $\frac{\left (-21 d^2 e+458 d^3-816 d e^2+113 e^3\right ) \log \left (5 x^2+2 x+3\right )}{10 \left (5 d^2-2 d e+3 e^2\right )^3}-\frac{3 d^2 e^2+5 d^3 e+4 d^4-d e^3+2 e^4}{2 e^3 \left (5 d^2-2 d e+3 e^2\right ) (d+e x)^2}+\frac{28 d^3 e^2+44 d^2 e^3+d^4 e+40 d^5-2 d e^4+e^5}{e^3 \left (5 d^2-2 d e+3 e^2\right )^2 (d+e x)}+\frac{\left (228 d^4 e^2-242 d^3 e^3+141 d^2 e^4-120 d^5 e+100 d^6+120 d e^5-e^6\right ) \log (d+e x)}{e^3 \left (5 d^2-2 d e+3 e^2\right )^3}-\frac{\left (-4101 d^2 e+423 d^3+879 d e^2+703 e^3\right ) \tan ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{5 \sqrt{14} \left (5 d^2-2 d e+3 e^2\right )^3}$

[Out]

-(4*d^4 + 5*d^3*e + 3*d^2*e^2 - d*e^3 + 2*e^4)/(2*e^3*(5*d^2 - 2*d*e + 3*e^2)*(d + e*x)^2) + (40*d^5 + d^4*e +
28*d^3*e^2 + 44*d^2*e^3 - 2*d*e^4 + e^5)/(e^3*(5*d^2 - 2*d*e + 3*e^2)^2*(d + e*x)) - ((423*d^3 - 4101*d^2*e +
879*d*e^2 + 703*e^3)*ArcTan[(1 + 5*x)/Sqrt[14]])/(5*Sqrt[14]*(5*d^2 - 2*d*e + 3*e^2)^3) + ((100*d^6 - 120*d^5
*e + 228*d^4*e^2 - 242*d^3*e^3 + 141*d^2*e^4 + 120*d*e^5 - e^6)*Log[d + e*x])/(e^3*(5*d^2 - 2*d*e + 3*e^2)^3)
+ ((458*d^3 - 21*d^2*e - 816*d*e^2 + 113*e^3)*Log[3 + 2*x + 5*x^2])/(10*(5*d^2 - 2*d*e + 3*e^2)^3)

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Rubi [A]  time = 0.290028, antiderivative size = 317, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 38, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.132, Rules used = {1628, 634, 618, 204, 628} $\frac{\left (-21 d^2 e+458 d^3-816 d e^2+113 e^3\right ) \log \left (5 x^2+2 x+3\right )}{10 \left (5 d^2-2 d e+3 e^2\right )^3}-\frac{3 d^2 e^2+5 d^3 e+4 d^4-d e^3+2 e^4}{2 e^3 \left (5 d^2-2 d e+3 e^2\right ) (d+e x)^2}+\frac{28 d^3 e^2+44 d^2 e^3+d^4 e+40 d^5-2 d e^4+e^5}{e^3 \left (5 d^2-2 d e+3 e^2\right )^2 (d+e x)}+\frac{\left (228 d^4 e^2-242 d^3 e^3+141 d^2 e^4-120 d^5 e+100 d^6+120 d e^5-e^6\right ) \log (d+e x)}{e^3 \left (5 d^2-2 d e+3 e^2\right )^3}-\frac{\left (-4101 d^2 e+423 d^3+879 d e^2+703 e^3\right ) \tan ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{5 \sqrt{14} \left (5 d^2-2 d e+3 e^2\right )^3}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-(4*d^4 + 5*d^3*e + 3*d^2*e^2 - d*e^3 + 2*e^4)/(2*e^3*(5*d^2 - 2*d*e + 3*e^2)*(d + e*x)^2) + (40*d^5 + d^4*e +
28*d^3*e^2 + 44*d^2*e^3 - 2*d*e^4 + e^5)/(e^3*(5*d^2 - 2*d*e + 3*e^2)^2*(d + e*x)) - ((423*d^3 - 4101*d^2*e +
879*d*e^2 + 703*e^3)*ArcTan[(1 + 5*x)/Sqrt[14]])/(5*Sqrt[14]*(5*d^2 - 2*d*e + 3*e^2)^3) + ((100*d^6 - 120*d^5
*e + 228*d^4*e^2 - 242*d^3*e^3 + 141*d^2*e^4 + 120*d*e^5 - e^6)*Log[d + e*x])/(e^3*(5*d^2 - 2*d*e + 3*e^2)^3)
+ ((458*d^3 - 21*d^2*e - 816*d*e^2 + 113*e^3)*Log[3 + 2*x + 5*x^2])/(10*(5*d^2 - 2*d*e + 3*e^2)^3)

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 )} \, dx &=\int \left (\frac{4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4}{e^2 \left (5 d^2-2 d e+3 e^2\right ) (d+e x)^3}+\frac{-40 d^5-d^4 e-28 d^3 e^2-44 d^2 e^3+2 d e^4-e^5}{e^2 \left (5 d^2-2 d e+3 e^2\right )^2 (d+e x)^2}+\frac{100 d^6-120 d^5 e+228 d^4 e^2-242 d^3 e^3+141 d^2 e^4+120 d e^5-e^6}{e^2 \left (5 d^2-2 d e+3 e^2\right )^3 (d+e x)}+\frac{7 d^3+816 d^2 e-339 d e^2-118 e^3+\left (458 d^3-21 d^2 e-816 d e^2+113 e^3\right ) x}{\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}{2 e^3 \left (5 d^2-2 d e+3 e^2\right ) (d+e x)^2}+\frac{40 d^5+d^4 e+28 d^3 e^2+44 d^2 e^3-2 d e^4+e^5}{e^3 \left (5 d^2-2 d e+3 e^2\right )^2 (d+e x)}+\frac{\left (100 d^6-120 d^5 e+228 d^4 e^2-242 d^3 e^3+141 d^2 e^4+120 d e^5-e^6\right ) \log (d+e x)}{e^3 \left (5 d^2-2 d e+3 e^2\right )^3}+\frac{\int \frac{7 d^3+816 d^2 e-339 d e^2-118 e^3+\left (458 d^3-21 d^2 e-816 d e^2+113 e^3\right ) x}{3+2 x+5 x^2} \, dx}{\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}{2 e^3 \left (5 d^2-2 d e+3 e^2\right ) (d+e x)^2}+\frac{40 d^5+d^4 e+28 d^3 e^2+44 d^2 e^3-2 d e^4+e^5}{e^3 \left (5 d^2-2 d e+3 e^2\right )^2 (d+e x)}+\frac{\left (100 d^6-120 d^5 e+228 d^4 e^2-242 d^3 e^3+141 d^2 e^4+120 d e^5-e^6\right ) \log (d+e x)}{e^3 \left (5 d^2-2 d e+3 e^2\right )^3}+\frac{\left (458 d^3-21 d^2 e-816 d e^2+113 e^3\right ) \int \frac{2+10 x}{3+2 x+5 x^2} \, dx}{10 \left (5 d^2-2 d e+3 e^2\right )^3}-\frac{\left (423 d^3-4101 d^2 e+879 d e^2+703 e^3\right ) \int \frac{1}{3+2 x+5 x^2} \, dx}{5 \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}{2 e^3 \left (5 d^2-2 d e+3 e^2\right ) (d+e x)^2}+\frac{40 d^5+d^4 e+28 d^3 e^2+44 d^2 e^3-2 d e^4+e^5}{e^3 \left (5 d^2-2 d e+3 e^2\right )^2 (d+e x)}+\frac{\left (100 d^6-120 d^5 e+228 d^4 e^2-242 d^3 e^3+141 d^2 e^4+120 d e^5-e^6\right ) \log (d+e x)}{e^3 \left (5 d^2-2 d e+3 e^2\right )^3}+\frac{\left (458 d^3-21 d^2 e-816 d e^2+113 e^3\right ) \log \left (3+2 x+5 x^2\right )}{10 \left (5 d^2-2 d e+3 e^2\right )^3}+\frac{\left (2 \left (423 d^3-4101 d^2 e+879 d e^2+703 e^3\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-56-x^2} \, dx,x,2+10 x\right )}{5 \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}{2 e^3 \left (5 d^2-2 d e+3 e^2\right ) (d+e x)^2}+\frac{40 d^5+d^4 e+28 d^3 e^2+44 d^2 e^3-2 d e^4+e^5}{e^3 \left (5 d^2-2 d e+3 e^2\right )^2 (d+e x)}-\frac{\left (423 d^3-4101 d^2 e+879 d e^2+703 e^3\right ) \tan ^{-1}\left (\frac{1+5 x}{\sqrt{14}}\right )}{5 \sqrt{14} \left (5 d^2-2 d e+3 e^2\right )^3}+\frac{\left (100 d^6-120 d^5 e+228 d^4 e^2-242 d^3 e^3+141 d^2 e^4+120 d e^5-e^6\right ) \log (d+e x)}{e^3 \left (5 d^2-2 d e+3 e^2\right )^3}+\frac{\left (458 d^3-21 d^2 e-816 d e^2+113 e^3\right ) \log \left (3+2 x+5 x^2\right )}{10 \left (5 d^2-2 d e+3 e^2\right )^3}\\ \end{align*}

Mathematica [A]  time = 0.432427, size = 278, normalized size = 0.88 $-\frac{-7 \left (-21 d^2 e+458 d^3-816 d e^2+113 e^3\right ) \log \left (5 x^2+2 x+3\right )+\frac{35 \left (3 d^2 e^2+5 d^3 e+4 d^4-d e^3+2 e^4\right ) \left (5 d^2-2 d e+3 e^2\right )^2}{e^3 (d+e x)^2}-\frac{70 \left (28 d^3 e^2+44 d^2 e^3+d^4 e+40 d^5-2 d e^4+e^5\right ) \left (5 d^2-2 d e+3 e^2\right )}{e^3 (d+e x)}+\frac{70 \left (-228 d^4 e^2+242 d^3 e^3-141 d^2 e^4+120 d^5 e-100 d^6-120 d e^5+e^6\right ) \log (d+e x)}{e^3}+\sqrt{14} \left (-4101 d^2 e+423 d^3+879 d e^2+703 e^3\right ) \tan ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{70 \left (5 d^2-2 d e+3 e^2\right )^3}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-((35*(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^3*(d + e*x)^2) - (70*(5*d^2
- 2*d*e + 3*e^2)*(40*d^5 + d^4*e + 28*d^3*e^2 + 44*d^2*e^3 - 2*d*e^4 + e^5))/(e^3*(d + e*x)) + Sqrt[14]*(423*d
^3 - 4101*d^2*e + 879*d*e^2 + 703*e^3)*ArcTan[(1 + 5*x)/Sqrt[14]] + (70*(-100*d^6 + 120*d^5*e - 228*d^4*e^2 +
242*d^3*e^3 - 141*d^2*e^4 - 120*d*e^5 + e^6)*Log[d + e*x])/e^3 - 7*(458*d^3 - 21*d^2*e - 816*d*e^2 + 113*e^3)*
Log[3 + 2*x + 5*x^2])/(70*(5*d^2 - 2*d*e + 3*e^2)^3)

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Maple [B]  time = 0.062, size = 819, normalized size = 2.6 \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),x)

[Out]

-2*e/(5*d^2-2*d*e+3*e^2)^2/(e*x+d)*d+40/e^3/(5*d^2-2*d*e+3*e^2)^2/(e*x+d)*d^5+141/(5*d^2-2*d*e+3*e^2)^3*e*ln(e
*x+d)*d^2+120/(5*d^2-2*d*e+3*e^2)^3*e^2*ln(e*x+d)*d-703/70/(5*d^2-2*d*e+3*e^2)^3*14^(1/2)*arctan(1/28*(10*x+2)
*14^(1/2))*e^3+1/e^2/(5*d^2-2*d*e+3*e^2)^2/(e*x+d)*d^4-423/70/(5*d^2-2*d*e+3*e^2)^3*14^(1/2)*arctan(1/28*(10*x
+2)*14^(1/2))*d^3-3/2/e/(5*d^2-2*d*e+3*e^2)/(e*x+d)^2*d^2-879/70/(5*d^2-2*d*e+3*e^2)^3*14^(1/2)*arctan(1/28*(1
0*x+2)*14^(1/2))*d*e^2+4101/70/(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+229/5/(5*d^
2-2*d*e+3*e^2)^3*ln(5*x^2+2*x+3)*d^3+113/10/(5*d^2-2*d*e+3*e^2)^3*ln(5*x^2+2*x+3)*e^3-e/(5*d^2-2*d*e+3*e^2)/(e
*x+d)^2+1/2/(5*d^2-2*d*e+3*e^2)/(e*x+d)^2*d+44/(5*d^2-2*d*e+3*e^2)^2/(e*x+d)*d^2+e^2/(5*d^2-2*d*e+3*e^2)^2/(e*
x+d)-242/(5*d^2-2*d*e+3*e^2)^3*ln(e*x+d)*d^3-1/(5*d^2-2*d*e+3*e^2)^3*e^3*ln(e*x+d)-2/e^3/(5*d^2-2*d*e+3*e^2)/(
e*x+d)^2*d^4-5/2/e^2/(5*d^2-2*d*e+3*e^2)/(e*x+d)^2*d^3+28/e/(5*d^2-2*d*e+3*e^2)^2/(e*x+d)*d^3-21/10/(5*d^2-2*d
*e+3*e^2)^3*ln(5*x^2+2*x+3)*d^2*e-408/5/(5*d^2-2*d*e+3*e^2)^3*ln(5*x^2+2*x+3)*d*e^2+100/(5*d^2-2*d*e+3*e^2)^3/
e^3*ln(e*x+d)*d^6-120/(5*d^2-2*d*e+3*e^2)^3/e^2*ln(e*x+d)*d^5+228/(5*d^2-2*d*e+3*e^2)^3/e*ln(e*x+d)*d^4

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Maxima [A]  time = 1.50774, size = 672, normalized size = 2.12 \begin{align*} -\frac{\sqrt{14}{\left (423 \, d^{3} - 4101 \, d^{2} e + 879 \, d e^{2} + 703 \, e^{3}\right )} \arctan \left (\frac{1}{14} \, \sqrt{14}{\left (5 \, x + 1\right )}\right )}{70 \,{\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 (100 \, d^{6} - 120 \, d^{5} e + 228 \, d^{4} e^{2} - 242 \, d^{3} e^{3} + 141 \, d^{2} e^{4} + 120 \, d e^{5} - e^{6}\right )} \log \left (e x + d\right )}{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}} + \frac{{\left (458 \, d^{3} - 21 \, d^{2} e - 816 \, d e^{2} + 113 \, e^{3}\right )} \log \left (5 \, x^{2} + 2 \, x + 3\right )}{10 \,{\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{60 \, d^{6} - 15 \, d^{5} e + 39 \, d^{4} e^{2} + 84 \, d^{3} e^{3} - 25 \, d^{2} e^{4} + 9 \, d e^{5} - 6 \, e^{6} + 2 \,{\left (40 \, d^{5} e + d^{4} e^{2} + 28 \, d^{3} e^{3} + 44 \, d^{2} e^{4} - 2 \, d e^{5} + e^{6}\right )} x}{2 \,{\left (25 \, d^{6} e^{3} - 20 \, d^{5} e^{4} + 34 \, d^{4} e^{5} - 12 \, d^{3} e^{6} + 9 \, d^{2} e^{7} +{\left (25 \, d^{4} e^{5} - 20 \, d^{3} e^{6} + 34 \, d^{2} e^{7} - 12 \, d e^{8} + 9 \, e^{9}\right )} x^{2} + 2 \,{\left (25 \, d^{5} e^{4} - 20 \, d^{4} e^{5} + 34 \, d^{3} e^{6} - 12 \, d^{2} e^{7} + 9 \, d e^{8}\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)^3/(5*x^2+2*x+3),x, algorithm="maxima")

[Out]

-1/70*sqrt(14)*(423*d^3 - 4101*d^2*e + 879*d*e^2 + 703*e^3)*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) + (100*d^6 - 120*d^5*e + 228*d^4*e^2 - 242*d
^3*e^3 + 141*d^2*e^4 + 120*d*e^5 - e^6)*log(e*x + d)/(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) + 1/10*(458*d^3 - 21*d^2*e - 816*d*e^2 + 113*e^3)*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) + 1/2*(60*d^6 - 15*d^5*e + 39*d^
4*e^2 + 84*d^3*e^3 - 25*d^2*e^4 + 9*d*e^5 - 6*e^6 + 2*(40*d^5*e + d^4*e^2 + 28*d^3*e^3 + 44*d^2*e^4 - 2*d*e^5
+ e^6)*x)/(25*d^6*e^3 - 20*d^5*e^4 + 34*d^4*e^5 - 12*d^3*e^6 + 9*d^2*e^7 + (25*d^4*e^5 - 20*d^3*e^6 + 34*d^2*e
^7 - 12*d*e^8 + 9*e^9)*x^2 + 2*(25*d^5*e^4 - 20*d^4*e^5 + 34*d^3*e^6 - 12*d^2*e^7 + 9*d*e^8)*x)

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Fricas [B]  time = 2.30544, size = 1674, normalized size = 5.28 \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),x, algorithm="fricas")

[Out]

1/70*(10500*d^8 - 6825*d^7*e + 14175*d^6*e^2 + 10395*d^5*e^3 - 6160*d^4*e^4 + 12145*d^3*e^5 - 4305*d^2*e^6 + 1
365*d*e^7 - 630*e^8 - sqrt(14)*(423*d^5*e^3 - 4101*d^4*e^4 + 879*d^3*e^5 + 703*d^2*e^6 + (423*d^3*e^5 - 4101*d
^2*e^6 + 879*d*e^7 + 703*e^8)*x^2 + 2*(423*d^4*e^4 - 4101*d^3*e^5 + 879*d^2*e^6 + 703*d*e^7)*x)*arctan(1/14*sq
rt(14)*(5*x + 1)) + 70*(200*d^7*e - 75*d^6*e^2 + 258*d^5*e^3 + 167*d^4*e^4 - 14*d^3*e^5 + 141*d^2*e^6 - 8*d*e^
7 + 3*e^8)*x + 70*(100*d^8 - 120*d^7*e + 228*d^6*e^2 - 242*d^5*e^3 + 141*d^4*e^4 + 120*d^3*e^5 - d^2*e^6 + (10
0*d^6*e^2 - 120*d^5*e^3 + 228*d^4*e^4 - 242*d^3*e^5 + 141*d^2*e^6 + 120*d*e^7 - e^8)*x^2 + 2*(100*d^7*e - 120*
d^6*e^2 + 228*d^5*e^3 - 242*d^4*e^4 + 141*d^3*e^5 + 120*d^2*e^6 - d*e^7)*x)*log(e*x + d) + 7*(458*d^5*e^3 - 21
*d^4*e^4 - 816*d^3*e^5 + 113*d^2*e^6 + (458*d^3*e^5 - 21*d^2*e^6 - 816*d*e^7 + 113*e^8)*x^2 + 2*(458*d^4*e^4 -
21*d^3*e^5 - 816*d^2*e^6 + 113*d*e^7)*x)*log(5*x^2 + 2*x + 3))/(125*d^8*e^3 - 150*d^7*e^4 + 285*d^6*e^5 - 188
*d^5*e^6 + 171*d^4*e^7 - 54*d^3*e^8 + 27*d^2*e^9 + (125*d^6*e^5 - 150*d^5*e^6 + 285*d^4*e^7 - 188*d^3*e^8 + 17
1*d^2*e^9 - 54*d*e^10 + 27*e^11)*x^2 + 2*(125*d^7*e^4 - 150*d^6*e^5 + 285*d^5*e^6 - 188*d^4*e^7 + 171*d^3*e^8
- 54*d^2*e^9 + 27*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),x)

[Out]

Timed out

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Giac [A]  time = 1.16568, size = 548, normalized size = 1.73 \begin{align*} -\frac{\sqrt{14}{\left (423 \, d^{3} - 4101 \, d^{2} e + 879 \, d e^{2} + 703 \, e^{3}\right )} \arctan \left (\frac{1}{14} \, \sqrt{14}{\left (5 \, x + 1\right )}\right )}{70 \,{\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 (458 \, d^{3} - 21 \, d^{2} e - 816 \, d e^{2} + 113 \, e^{3}\right )} \log \left (5 \, x^{2} + 2 \, x + 3\right )}{10 \,{\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 (100 \, d^{6} - 120 \, d^{5} e + 228 \, d^{4} e^{2} - 242 \, d^{3} e^{3} + 141 \, d^{2} e^{4} + 120 \, d e^{5} - e^{6}\right )} \log \left ({\left | x e + d \right |}\right )}{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}} + \frac{{\left (2 \,{\left (200 \, d^{7} - 75 \, d^{6} e + 258 \, d^{5} e^{2} + 167 \, d^{4} e^{3} - 14 \, d^{3} e^{4} + 141 \, d^{2} e^{5} - 8 \, d e^{6} + 3 \, e^{7}\right )} x +{\left (300 \, d^{8} - 195 \, d^{7} e + 405 \, d^{6} e^{2} + 297 \, d^{5} e^{3} - 176 \, d^{4} e^{4} + 347 \, d^{3} e^{5} - 123 \, d^{2} e^{6} + 39 \, d e^{7} - 18 \, e^{8}\right )} e^{\left (-1\right )}\right )} e^{\left (-2\right )}}{2 \,{\left (5 \, d^{2} - 2 \, d e + 3 \, e^{2}\right )}^{3}{\left (x e + d\right )}^{2}} \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),x, algorithm="giac")

[Out]

-1/70*sqrt(14)*(423*d^3 - 4101*d^2*e + 879*d*e^2 + 703*e^3)*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) + 1/10*(458*d^3 - 21*d^2*e - 816*d*e^2 + 113
*e^3)*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)
+ (100*d^6 - 120*d^5*e + 228*d^4*e^2 - 242*d^3*e^3 + 141*d^2*e^4 + 120*d*e^5 - e^6)*log(abs(x*e + d))/(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) + 1/2*(2*(200*d^7 - 75*d^6*
e + 258*d^5*e^2 + 167*d^4*e^3 - 14*d^3*e^4 + 141*d^2*e^5 - 8*d*e^6 + 3*e^7)*x + (300*d^8 - 195*d^7*e + 405*d^6
*e^2 + 297*d^5*e^3 - 176*d^4*e^4 + 347*d^3*e^5 - 123*d^2*e^6 + 39*d*e^7 - 18*e^8)*e^(-1))*e^(-2)/((5*d^2 - 2*d
*e + 3*e^2)^3*(x*e + d)^2)