### 3.308 $$\int \frac{2+x+3 x^2-5 x^3+4 x^4}{(d+e x) (3+2 x+5 x^2)} \, dx$$

Optimal. Leaf size=168 $\frac{(458 d-7 e) \log \left (5 x^2+2 x+3\right )}{250 \left (5 d^2-2 d e+3 e^2\right )}+\frac{\left (3 d^2 e^2+5 d^3 e+4 d^4-d e^3+2 e^4\right ) \log (d+e x)}{e^3 \left (5 d^2-2 d e+3 e^2\right )}-\frac{(423 d-1367 e) \tan ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{125 \sqrt{14} \left (5 d^2-2 d e+3 e^2\right )}-\frac{x (20 d+33 e)}{25 e^2}+\frac{2 x^2}{5 e}$

[Out]

-((20*d + 33*e)*x)/(25*e^2) + (2*x^2)/(5*e) - ((423*d - 1367*e)*ArcTan[(1 + 5*x)/Sqrt[14]])/(125*Sqrt[14]*(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)*Log[d + e*x])/(e^3*(5*d^2 - 2*d*e + 3*e^
2)) + ((458*d - 7*e)*Log[3 + 2*x + 5*x^2])/(250*(5*d^2 - 2*d*e + 3*e^2))

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Rubi [A]  time = 0.193678, antiderivative size = 168, 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} $\frac{(458 d-7 e) \log \left (5 x^2+2 x+3\right )}{250 \left (5 d^2-2 d e+3 e^2\right )}+\frac{\left (3 d^2 e^2+5 d^3 e+4 d^4-d e^3+2 e^4\right ) \log (d+e x)}{e^3 \left (5 d^2-2 d e+3 e^2\right )}-\frac{(423 d-1367 e) \tan ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{125 \sqrt{14} \left (5 d^2-2 d e+3 e^2\right )}-\frac{x (20 d+33 e)}{25 e^2}+\frac{2 x^2}{5 e}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-((20*d + 33*e)*x)/(25*e^2) + (2*x^2)/(5*e) - ((423*d - 1367*e)*ArcTan[(1 + 5*x)/Sqrt[14]])/(125*Sqrt[14]*(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)*Log[d + e*x])/(e^3*(5*d^2 - 2*d*e + 3*e^
2)) + ((458*d - 7*e)*Log[3 + 2*x + 5*x^2])/(250*(5*d^2 - 2*d*e + 3*e^2))

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) \left (3+2 x+5 x^2\right )} \, dx &=\int \left (\frac{-20 d-33 e}{25 e^2}+\frac{4 x}{5 e}+\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)}+\frac{7 d+272 e+(458 d-7 e) x}{25 \left (5 d^2-2 d e+3 e^2\right ) \left (3+2 x+5 x^2\right )}\right ) \, dx\\ &=-\frac{(20 d+33 e) x}{25 e^2}+\frac{2 x^2}{5 e}+\frac{\left (4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4\right ) \log (d+e x)}{e^3 \left (5 d^2-2 d e+3 e^2\right )}+\frac{\int \frac{7 d+272 e+(458 d-7 e) x}{3+2 x+5 x^2} \, dx}{25 \left (5 d^2-2 d e+3 e^2\right )}\\ &=-\frac{(20 d+33 e) x}{25 e^2}+\frac{2 x^2}{5 e}+\frac{\left (4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4\right ) \log (d+e x)}{e^3 \left (5 d^2-2 d e+3 e^2\right )}-\frac{(423 d-1367 e) \int \frac{1}{3+2 x+5 x^2} \, dx}{125 \left (5 d^2-2 d e+3 e^2\right )}+\frac{(458 d-7 e) \int \frac{2+10 x}{3+2 x+5 x^2} \, dx}{250 \left (5 d^2-2 d e+3 e^2\right )}\\ &=-\frac{(20 d+33 e) x}{25 e^2}+\frac{2 x^2}{5 e}+\frac{\left (4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4\right ) \log (d+e x)}{e^3 \left (5 d^2-2 d e+3 e^2\right )}+\frac{(458 d-7 e) \log \left (3+2 x+5 x^2\right )}{250 \left (5 d^2-2 d e+3 e^2\right )}+\frac{(2 (423 d-1367 e)) \operatorname{Subst}\left (\int \frac{1}{-56-x^2} \, dx,x,2+10 x\right )}{125 \left (5 d^2-2 d e+3 e^2\right )}\\ &=-\frac{(20 d+33 e) x}{25 e^2}+\frac{2 x^2}{5 e}-\frac{(423 d-1367 e) \tan ^{-1}\left (\frac{1+5 x}{\sqrt{14}}\right )}{125 \sqrt{14} \left (5 d^2-2 d e+3 e^2\right )}+\frac{\left (4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4\right ) \log (d+e x)}{e^3 \left (5 d^2-2 d e+3 e^2\right )}+\frac{(458 d-7 e) \log \left (3+2 x+5 x^2\right )}{250 \left (5 d^2-2 d e+3 e^2\right )}\\ \end{align*}

Mathematica [A]  time = 0.10388, size = 146, normalized size = 0.87 $\frac{70 e x \left (5 d^2-2 d e+3 e^2\right ) (e (10 x-33)-20 d)+1750 \left (3 d^2 e^2+5 d^3 e+4 d^4-d e^3+2 e^4\right ) \log (d+e x)+7 e^3 (458 d-7 e) \log \left (5 x^2+2 x+3\right )-\sqrt{14} e^3 (423 d-1367 e) \tan ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{1750 e^3 \left (5 d^2-2 d e+3 e^2\right )}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(70*e*(5*d^2 - 2*d*e + 3*e^2)*x*(-20*d + e*(-33 + 10*x)) - Sqrt[14]*(423*d - 1367*e)*e^3*ArcTan[(1 + 5*x)/Sqrt
[14]] + 1750*(4*d^4 + 5*d^3*e + 3*d^2*e^2 - d*e^3 + 2*e^4)*Log[d + e*x] + 7*(458*d - 7*e)*e^3*Log[3 + 2*x + 5*
x^2])/(1750*e^3*(5*d^2 - 2*d*e + 3*e^2))

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Maple [A]  time = 0.058, size = 298, normalized size = 1.8 \begin{align*}{\frac{2\,{x}^{2}}{5\,e}}-{\frac{4\,dx}{5\,{e}^{2}}}-{\frac{33\,x}{25\,e}}+{\frac{229\,\ln \left ( 5\,{x}^{2}+2\,x+3 \right ) d}{625\,{d}^{2}-250\,de+375\,{e}^{2}}}-{\frac{7\,e\ln \left ( 5\,{x}^{2}+2\,x+3 \right ) }{1250\,{d}^{2}-500\,de+750\,{e}^{2}}}-{\frac{423\,\sqrt{14}d}{8750\,{d}^{2}-3500\,de+5250\,{e}^{2}}\arctan \left ({\frac{ \left ( 10\,x+2 \right ) \sqrt{14}}{28}} \right ) }+{\frac{1367\,\sqrt{14}e}{8750\,{d}^{2}-3500\,de+5250\,{e}^{2}}\arctan \left ({\frac{ \left ( 10\,x+2 \right ) \sqrt{14}}{28}} \right ) }+4\,{\frac{\ln \left ( ex+d \right ){d}^{4}}{{e}^{3} \left ( 5\,{d}^{2}-2\,de+3\,{e}^{2} \right ) }}+5\,{\frac{\ln \left ( ex+d \right ){d}^{3}}{{e}^{2} \left ( 5\,{d}^{2}-2\,de+3\,{e}^{2} \right ) }}+3\,{\frac{\ln \left ( ex+d \right ){d}^{2}}{e \left ( 5\,{d}^{2}-2\,de+3\,{e}^{2} \right ) }}-{\frac{\ln \left ( ex+d \right ) d}{5\,{d}^{2}-2\,de+3\,{e}^{2}}}+2\,{\frac{e\ln \left ( ex+d \right ) }{5\,{d}^{2}-2\,de+3\,{e}^{2}}} \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)/(5*x^2+2*x+3),x)

[Out]

2/5*x^2/e-4/5/e^2*x*d-33/25/e*x+229/5/(125*d^2-50*d*e+75*e^2)*ln(5*x^2+2*x+3)*d-7/10/(125*d^2-50*d*e+75*e^2)*l
n(5*x^2+2*x+3)*e-423/70/(125*d^2-50*d*e+75*e^2)*14^(1/2)*arctan(1/28*(10*x+2)*14^(1/2))*d+1367/70/(125*d^2-50*
d*e+75*e^2)*14^(1/2)*arctan(1/28*(10*x+2)*14^(1/2))*e+4/e^3/(5*d^2-2*d*e+3*e^2)*ln(e*x+d)*d^4+5/e^2/(5*d^2-2*d
*e+3*e^2)*ln(e*x+d)*d^3+3/e/(5*d^2-2*d*e+3*e^2)*ln(e*x+d)*d^2-1/(5*d^2-2*d*e+3*e^2)*ln(e*x+d)*d+2*e/(5*d^2-2*d
*e+3*e^2)*ln(e*x+d)

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Maxima [A]  time = 1.4724, size = 216, normalized size = 1.29 \begin{align*} -\frac{\sqrt{14}{\left (423 \, d - 1367 \, e\right )} \arctan \left (\frac{1}{14} \, \sqrt{14}{\left (5 \, x + 1\right )}\right )}{1750 \,{\left (5 \, d^{2} - 2 \, d e + 3 \, e^{2}\right )}} + \frac{{\left (4 \, d^{4} + 5 \, d^{3} e + 3 \, d^{2} e^{2} - d e^{3} + 2 \, e^{4}\right )} \log \left (e x + d\right )}{5 \, d^{2} e^{3} - 2 \, d e^{4} + 3 \, e^{5}} + \frac{{\left (458 \, d - 7 \, e\right )} \log \left (5 \, x^{2} + 2 \, x + 3\right )}{250 \,{\left (5 \, d^{2} - 2 \, d e + 3 \, e^{2}\right )}} + \frac{10 \, e x^{2} -{\left (20 \, d + 33 \, e\right )} x}{25 \, e^{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)/(5*x^2+2*x+3),x, algorithm="maxima")

[Out]

-1/1750*sqrt(14)*(423*d - 1367*e)*arctan(1/14*sqrt(14)*(5*x + 1))/(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)*log(e*x + d)/(5*d^2*e^3 - 2*d*e^4 + 3*e^5) + 1/250*(458*d - 7*e)*log(5*x^2 + 2*x +
3)/(5*d^2 - 2*d*e + 3*e^2) + 1/25*(10*e*x^2 - (20*d + 33*e)*x)/e^2

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Fricas [A]  time = 1.52551, size = 414, normalized size = 2.46 \begin{align*} \frac{700 \,{\left (5 \, d^{2} e^{2} - 2 \, d e^{3} + 3 \, e^{4}\right )} x^{2} - \sqrt{14}{\left (423 \, d e^{3} - 1367 \, e^{4}\right )} \arctan \left (\frac{1}{14} \, \sqrt{14}{\left (5 \, x + 1\right )}\right ) - 70 \,{\left (100 \, d^{3} e + 125 \, d^{2} e^{2} - 6 \, d e^{3} + 99 \, e^{4}\right )} x + 1750 \,{\left (4 \, d^{4} + 5 \, d^{3} e + 3 \, d^{2} e^{2} - d e^{3} + 2 \, e^{4}\right )} \log \left (e x + d\right ) + 7 \,{\left (458 \, d e^{3} - 7 \, e^{4}\right )} \log \left (5 \, x^{2} + 2 \, x + 3\right )}{1750 \,{\left (5 \, d^{2} e^{3} - 2 \, d e^{4} + 3 \, e^{5}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1750*(700*(5*d^2*e^2 - 2*d*e^3 + 3*e^4)*x^2 - sqrt(14)*(423*d*e^3 - 1367*e^4)*arctan(1/14*sqrt(14)*(5*x + 1)
) - 70*(100*d^3*e + 125*d^2*e^2 - 6*d*e^3 + 99*e^4)*x + 1750*(4*d^4 + 5*d^3*e + 3*d^2*e^2 - d*e^3 + 2*e^4)*log
(e*x + d) + 7*(458*d*e^3 - 7*e^4)*log(5*x^2 + 2*x + 3))/(5*d^2*e^3 - 2*d*e^4 + 3*e^5)

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Sympy [C]  time = 11.239, size = 4106, normalized size = 24.44 \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)/(5*x**2+2*x+3),x)

[Out]

(-sqrt(14)*I*(423*d - 1367*e)/(3500*(5*d**2 - 2*d*e + 3*e**2)) + (458*d - 7*e)/(250*(5*d**2 - 2*d*e + 3*e**2))
)*log(x + (-392000000*d**10*(-sqrt(14)*I*(423*d - 1367*e)/(3500*(5*d**2 - 2*d*e + 3*e**2)) + (458*d - 7*e)/(25
0*(5*d**2 - 2*d*e + 3*e**2))) - 823200000*d**9*e*(-sqrt(14)*I*(423*d - 1367*e)/(3500*(5*d**2 - 2*d*e + 3*e**2)
) + (458*d - 7*e)/(250*(5*d**2 - 2*d*e + 3*e**2))) + 153104000*d**9 + 490000000*d**8*e**3*(-sqrt(14)*I*(423*d
- 1367*e)/(3500*(5*d**2 - 2*d*e + 3*e**2)) + (458*d - 7*e)/(250*(5*d**2 - 2*d*e + 3*e**2)))**2 - 1043700000*d*
*8*e**2*(-sqrt(14)*I*(423*d - 1367*e)/(3500*(5*d**2 - 2*d*e + 3*e**2)) + (458*d - 7*e)/(250*(5*d**2 - 2*d*e +
3*e**2))) + 349944000*d**8*e + 220500000*d**7*e**4*(-sqrt(14)*I*(423*d - 1367*e)/(3500*(5*d**2 - 2*d*e + 3*e**
2)) + (458*d - 7*e)/(250*(5*d**2 - 2*d*e + 3*e**2)))**2 - 646800000*d**7*e**3*(-sqrt(14)*I*(423*d - 1367*e)/(3
500*(5*d**2 - 2*d*e + 3*e**2)) + (458*d - 7*e)/(250*(5*d**2 - 2*d*e + 3*e**2))) + 386841000*d**7*e**2 + 617925
000*d**6*e**5*(-sqrt(14)*I*(423*d - 1367*e)/(3500*(5*d**2 - 2*d*e + 3*e**2)) + (458*d - 7*e)/(250*(5*d**2 - 2*
d*e + 3*e**2)))**2 - 872200000*d**6*e**4*(-sqrt(14)*I*(423*d - 1367*e)/(3500*(5*d**2 - 2*d*e + 3*e**2)) + (458
*d - 7*e)/(250*(5*d**2 - 2*d*e + 3*e**2))) + 39565736*d**6*e**3 - 356370000*d**5*e**6*(-sqrt(14)*I*(423*d - 13
67*e)/(3500*(5*d**2 - 2*d*e + 3*e**2)) + (458*d - 7*e)/(250*(5*d**2 - 2*d*e + 3*e**2)))**2 - 573645520*d**5*e*
*5*(-sqrt(14)*I*(423*d - 1367*e)/(3500*(5*d**2 - 2*d*e + 3*e**2)) + (458*d - 7*e)/(250*(5*d**2 - 2*d*e + 3*e**
2))) + 14633332*d**5*e**4 + 1259909000*d**4*e**7*(-sqrt(14)*I*(423*d - 1367*e)/(3500*(5*d**2 - 2*d*e + 3*e**2)
) + (458*d - 7*e)/(250*(5*d**2 - 2*d*e + 3*e**2)))**2 - 115902052*d**4*e**6*(-sqrt(14)*I*(423*d - 1367*e)/(350
0*(5*d**2 - 2*d*e + 3*e**2)) + (458*d - 7*e)/(250*(5*d**2 - 2*d*e + 3*e**2))) + 107677543*d**4*e**5 - 10457440
00*d**3*e**8*(-sqrt(14)*I*(423*d - 1367*e)/(3500*(5*d**2 - 2*d*e + 3*e**2)) + (458*d - 7*e)/(250*(5*d**2 - 2*d
*e + 3*e**2)))**2 - 126665168*d**3*e**7*(-sqrt(14)*I*(423*d - 1367*e)/(3500*(5*d**2 - 2*d*e + 3*e**2)) + (458*
d - 7*e)/(250*(5*d**2 - 2*d*e + 3*e**2))) + 129989935*d**3*e**6 + 850339000*d**2*e**9*(-sqrt(14)*I*(423*d - 13
67*e)/(3500*(5*d**2 - 2*d*e + 3*e**2)) + (458*d - 7*e)/(250*(5*d**2 - 2*d*e + 3*e**2)))**2 - 218333192*d**2*e*
*8*(-sqrt(14)*I*(423*d - 1367*e)/(3500*(5*d**2 - 2*d*e + 3*e**2)) + (458*d - 7*e)/(250*(5*d**2 - 2*d*e + 3*e**
2))) - 50221473*d**2*e**7 - 358554000*d*e**10*(-sqrt(14)*I*(423*d - 1367*e)/(3500*(5*d**2 - 2*d*e + 3*e**2)) +
(458*d - 7*e)/(250*(5*d**2 - 2*d*e + 3*e**2)))**2 + 113884512*d*e**9*(-sqrt(14)*I*(423*d - 1367*e)/(3500*(5*d
**2 - 2*d*e + 3*e**2)) + (458*d - 7*e)/(250*(5*d**2 - 2*d*e + 3*e**2))) + 17826327*d*e**8 + 106659000*e**11*(-
sqrt(14)*I*(423*d - 1367*e)/(3500*(5*d**2 - 2*d*e + 3*e**2)) + (458*d - 7*e)/(250*(5*d**2 - 2*d*e + 3*e**2)))*
*2 - 89860932*e**10*(-sqrt(14)*I*(423*d - 1367*e)/(3500*(5*d**2 - 2*d*e + 3*e**2)) + (458*d - 7*e)/(250*(5*d**
2 - 2*d*e + 3*e**2))) + 12503288*e**9)/(47376000*d**9 - 34664000*d**8*e - 237671000*d**7*e**2 - 447135416*d**6
*e**3 - 79441992*d**5*e**4 + 39361392*d**4*e**5 + 28919955*d**3*e**6 - 233063217*d**2*e**7 + 141064083*d*e**8
- 59791213*e**9)) + (sqrt(14)*I*(423*d - 1367*e)/(3500*(5*d**2 - 2*d*e + 3*e**2)) + (458*d - 7*e)/(250*(5*d**2
- 2*d*e + 3*e**2)))*log(x + (-392000000*d**10*(sqrt(14)*I*(423*d - 1367*e)/(3500*(5*d**2 - 2*d*e + 3*e**2)) +
(458*d - 7*e)/(250*(5*d**2 - 2*d*e + 3*e**2))) - 823200000*d**9*e*(sqrt(14)*I*(423*d - 1367*e)/(3500*(5*d**2
- 2*d*e + 3*e**2)) + (458*d - 7*e)/(250*(5*d**2 - 2*d*e + 3*e**2))) + 153104000*d**9 + 490000000*d**8*e**3*(sq
rt(14)*I*(423*d - 1367*e)/(3500*(5*d**2 - 2*d*e + 3*e**2)) + (458*d - 7*e)/(250*(5*d**2 - 2*d*e + 3*e**2)))**2
- 1043700000*d**8*e**2*(sqrt(14)*I*(423*d - 1367*e)/(3500*(5*d**2 - 2*d*e + 3*e**2)) + (458*d - 7*e)/(250*(5*
d**2 - 2*d*e + 3*e**2))) + 349944000*d**8*e + 220500000*d**7*e**4*(sqrt(14)*I*(423*d - 1367*e)/(3500*(5*d**2 -
2*d*e + 3*e**2)) + (458*d - 7*e)/(250*(5*d**2 - 2*d*e + 3*e**2)))**2 - 646800000*d**7*e**3*(sqrt(14)*I*(423*d
- 1367*e)/(3500*(5*d**2 - 2*d*e + 3*e**2)) + (458*d - 7*e)/(250*(5*d**2 - 2*d*e + 3*e**2))) + 386841000*d**7*
e**2 + 617925000*d**6*e**5*(sqrt(14)*I*(423*d - 1367*e)/(3500*(5*d**2 - 2*d*e + 3*e**2)) + (458*d - 7*e)/(250*
(5*d**2 - 2*d*e + 3*e**2)))**2 - 872200000*d**6*e**4*(sqrt(14)*I*(423*d - 1367*e)/(3500*(5*d**2 - 2*d*e + 3*e*
*2)) + (458*d - 7*e)/(250*(5*d**2 - 2*d*e + 3*e**2))) + 39565736*d**6*e**3 - 356370000*d**5*e**6*(sqrt(14)*I*(
423*d - 1367*e)/(3500*(5*d**2 - 2*d*e + 3*e**2)) + (458*d - 7*e)/(250*(5*d**2 - 2*d*e + 3*e**2)))**2 - 5736455
20*d**5*e**5*(sqrt(14)*I*(423*d - 1367*e)/(3500*(5*d**2 - 2*d*e + 3*e**2)) + (458*d - 7*e)/(250*(5*d**2 - 2*d*
e + 3*e**2))) + 14633332*d**5*e**4 + 1259909000*d**4*e**7*(sqrt(14)*I*(423*d - 1367*e)/(3500*(5*d**2 - 2*d*e +
3*e**2)) + (458*d - 7*e)/(250*(5*d**2 - 2*d*e + 3*e**2)))**2 - 115902052*d**4*e**6*(sqrt(14)*I*(423*d - 1367*
e)/(3500*(5*d**2 - 2*d*e + 3*e**2)) + (458*d - 7*e)/(250*(5*d**2 - 2*d*e + 3*e**2))) + 107677543*d**4*e**5 - 1
045744000*d**3*e**8*(sqrt(14)*I*(423*d - 1367*e)/(3500*(5*d**2 - 2*d*e + 3*e**2)) + (458*d - 7*e)/(250*(5*d**2
- 2*d*e + 3*e**2)))**2 - 126665168*d**3*e**7*(sqrt(14)*I*(423*d - 1367*e)/(3500*(5*d**2 - 2*d*e + 3*e**2)) +
(458*d - 7*e)/(250*(5*d**2 - 2*d*e + 3*e**2))) + 129989935*d**3*e**6 + 850339000*d**2*e**9*(sqrt(14)*I*(423*d
- 1367*e)/(3500*(5*d**2 - 2*d*e + 3*e**2)) + (458*d - 7*e)/(250*(5*d**2 - 2*d*e + 3*e**2)))**2 - 218333192*d**
2*e**8*(sqrt(14)*I*(423*d - 1367*e)/(3500*(5*d**2 - 2*d*e + 3*e**2)) + (458*d - 7*e)/(250*(5*d**2 - 2*d*e + 3*
e**2))) - 50221473*d**2*e**7 - 358554000*d*e**10*(sqrt(14)*I*(423*d - 1367*e)/(3500*(5*d**2 - 2*d*e + 3*e**2))
+ (458*d - 7*e)/(250*(5*d**2 - 2*d*e + 3*e**2)))**2 + 113884512*d*e**9*(sqrt(14)*I*(423*d - 1367*e)/(3500*(5*
d**2 - 2*d*e + 3*e**2)) + (458*d - 7*e)/(250*(5*d**2 - 2*d*e + 3*e**2))) + 17826327*d*e**8 + 106659000*e**11*(
sqrt(14)*I*(423*d - 1367*e)/(3500*(5*d**2 - 2*d*e + 3*e**2)) + (458*d - 7*e)/(250*(5*d**2 - 2*d*e + 3*e**2)))*
*2 - 89860932*e**10*(sqrt(14)*I*(423*d - 1367*e)/(3500*(5*d**2 - 2*d*e + 3*e**2)) + (458*d - 7*e)/(250*(5*d**2
- 2*d*e + 3*e**2))) + 12503288*e**9)/(47376000*d**9 - 34664000*d**8*e - 237671000*d**7*e**2 - 447135416*d**6*
e**3 - 79441992*d**5*e**4 + 39361392*d**4*e**5 + 28919955*d**3*e**6 - 233063217*d**2*e**7 + 141064083*d*e**8 -
59791213*e**9)) + 2*x**2/(5*e) - x*(20*d + 33*e)/(25*e**2) + (4*d**4 + 5*d**3*e + 3*d**2*e**2 - d*e**3 + 2*e*
*4)*log(x + (-392000000*d**10*(4*d**4 + 5*d**3*e + 3*d**2*e**2 - d*e**3 + 2*e**4)/(e**3*(5*d**2 - 2*d*e + 3*e*
*2)) + 153104000*d**9 - 823200000*d**9*(4*d**4 + 5*d**3*e + 3*d**2*e**2 - d*e**3 + 2*e**4)/(e**2*(5*d**2 - 2*d
*e + 3*e**2)) + 349944000*d**8*e - 1043700000*d**8*(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)) + 490000000*d**8*(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)**2) + 386841000*d**7*e**2 - 646800000*d**7*(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) + 220500000*d**7*(4*d**4 + 5*d**3*e + 3*d**2*e**2 - d*e**3 + 2*e**4)**2/(e**2*(5*d
**2 - 2*d*e + 3*e**2)**2) + 39565736*d**6*e**3 - 872200000*d**6*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) + 617925000*d**6*(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) + 14633332*d**5*e**4 - 573645520*d**5*e**2*(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) - 356370000*d**5*(4*d**4 + 5*d**3*e + 3*d**2*e**2 - d*e**3 + 2*e**4)**
2/(5*d**2 - 2*d*e + 3*e**2)**2 + 107677543*d**4*e**5 - 115902052*d**4*e**3*(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) + 1259909000*d**4*e*(4*d**4 + 5*d**3*e + 3*d**2*e**2 - d*e**3 + 2*e
**4)**2/(5*d**2 - 2*d*e + 3*e**2)**2 + 129989935*d**3*e**6 - 126665168*d**3*e**4*(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) - 1045744000*d**3*e**2*(4*d**4 + 5*d**3*e + 3*d**2*e**2 - d*e
**3 + 2*e**4)**2/(5*d**2 - 2*d*e + 3*e**2)**2 - 50221473*d**2*e**7 - 218333192*d**2*e**5*(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) + 850339000*d**2*e**3*(4*d**4 + 5*d**3*e + 3*d**2*e**
2 - d*e**3 + 2*e**4)**2/(5*d**2 - 2*d*e + 3*e**2)**2 + 17826327*d*e**8 + 113884512*d*e**6*(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) - 358554000*d*e**4*(4*d**4 + 5*d**3*e + 3*d**2*e**2
- d*e**3 + 2*e**4)**2/(5*d**2 - 2*d*e + 3*e**2)**2 + 12503288*e**9 - 89860932*e**7*(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) + 106659000*e**5*(4*d**4 + 5*d**3*e + 3*d**2*e**2 - d*e**3
+ 2*e**4)**2/(5*d**2 - 2*d*e + 3*e**2)**2)/(47376000*d**9 - 34664000*d**8*e - 237671000*d**7*e**2 - 447135416*
d**6*e**3 - 79441992*d**5*e**4 + 39361392*d**4*e**5 + 28919955*d**3*e**6 - 233063217*d**2*e**7 + 141064083*d*e
**8 - 59791213*e**9))/(e**3*(5*d**2 - 2*d*e + 3*e**2))

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Giac [A]  time = 1.12489, size = 213, normalized size = 1.27 \begin{align*} \frac{1}{25} \,{\left (10 \, x^{2} e - 20 \, d x - 33 \, x e\right )} e^{\left (-2\right )} - \frac{\sqrt{14}{\left (423 \, d - 1367 \, e\right )} \arctan \left (\frac{1}{14} \, \sqrt{14}{\left (5 \, x + 1\right )}\right )}{1750 \,{\left (5 \, d^{2} - 2 \, d e + 3 \, e^{2}\right )}} + \frac{{\left (458 \, d - 7 \, e\right )} \log \left (5 \, x^{2} + 2 \, x + 3\right )}{250 \,{\left (5 \, d^{2} - 2 \, d e + 3 \, e^{2}\right )}} + \frac{{\left (4 \, d^{4} + 5 \, d^{3} e + 3 \, d^{2} e^{2} - d e^{3} + 2 \, e^{4}\right )} \log \left ({\left | x e + d \right |}\right )}{5 \, d^{2} e^{3} - 2 \, d e^{4} + 3 \, e^{5}} \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)/(5*x^2+2*x+3),x, algorithm="giac")

[Out]

1/25*(10*x^2*e - 20*d*x - 33*x*e)*e^(-2) - 1/1750*sqrt(14)*(423*d - 1367*e)*arctan(1/14*sqrt(14)*(5*x + 1))/(5
*d^2 - 2*d*e + 3*e^2) + 1/250*(458*d - 7*e)*log(5*x^2 + 2*x + 3)/(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)*log(abs(x*e + d))/(5*d^2*e^3 - 2*d*e^4 + 3*e^5)