### 3.319 $$\int \frac{(d+e x)^2 (2+x+3 x^2-5 x^3+4 x^4)}{(3+2 x+5 x^2)^3} \, dx$$

Optimal. Leaf size=134 $\frac{\left (211875 d^2-342070 d e+14817 e^2\right ) \tan ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{980000 \sqrt{14}}+\frac{(d+e x) (5 x (2203 d+8553 e)+34347 d-6413 e)}{196000 \left (5 x^2+2 x+3\right )}-\frac{(423 x+1367) (d+e x)^2}{7000 \left (5 x^2+2 x+3\right )^2}+\frac{e (40 d-49 e) \log \left (5 x^2+2 x+3\right )}{1250}+\frac{4 e^2 x}{125}$

[Out]

(4*e^2*x)/125 - ((1367 + 423*x)*(d + e*x)^2)/(7000*(3 + 2*x + 5*x^2)^2) + ((d + e*x)*(34347*d - 6413*e + 5*(22
03*d + 8553*e)*x))/(196000*(3 + 2*x + 5*x^2)) + ((211875*d^2 - 342070*d*e + 14817*e^2)*ArcTan[(1 + 5*x)/Sqrt[1
4]])/(980000*Sqrt[14]) + ((40*d - 49*e)*e*Log[3 + 2*x + 5*x^2])/1250

________________________________________________________________________________________

Rubi [A]  time = 0.238742, antiderivative size = 134, 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 = {1644, 1657, 634, 618, 204, 628} $\frac{\left (211875 d^2-342070 d e+14817 e^2\right ) \tan ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{980000 \sqrt{14}}+\frac{(d+e x) (5 x (2203 d+8553 e)+34347 d-6413 e)}{196000 \left (5 x^2+2 x+3\right )}-\frac{(423 x+1367) (d+e x)^2}{7000 \left (5 x^2+2 x+3\right )^2}+\frac{e (40 d-49 e) \log \left (5 x^2+2 x+3\right )}{1250}+\frac{4 e^2 x}{125}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(4*e^2*x)/125 - ((1367 + 423*x)*(d + e*x)^2)/(7000*(3 + 2*x + 5*x^2)^2) + ((d + e*x)*(34347*d - 6413*e + 5*(22
03*d + 8553*e)*x))/(196000*(3 + 2*x + 5*x^2)) + ((211875*d^2 - 342070*d*e + 14817*e^2)*ArcTan[(1 + 5*x)/Sqrt[1
4]])/(980000*Sqrt[14]) + ((40*d - 49*e)*e*Log[3 + 2*x + 5*x^2])/1250

Rule 1644

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = Polynomi
alQuotient[Pq, a + b*x + c*x^2, x], f = Coeff[PolynomialRemainder[Pq, a + b*x + c*x^2, x], x, 0], g = Coeff[Po
lynomialRemainder[Pq, a + b*x + c*x^2, x], x, 1]}, Simp[((d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*(f*b - 2*a*g +
(2*c*f - b*g)*x))/((p + 1)*(b^2 - 4*a*c)), x] + Dist[1/((p + 1)*(b^2 - 4*a*c)), Int[(d + e*x)^(m - 1)*(a + b*x
+ c*x^2)^(p + 1)*ExpandToSum[(p + 1)*(b^2 - 4*a*c)*(d + e*x)*Q + g*(2*a*e*m + b*d*(2*p + 3)) - f*(b*e*m + 2*c
*d*(2*p + 3)) - e*(2*c*f - b*g)*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && PolyQ[Pq, x] && N
eQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 0] && (IntegerQ[p] ||  !IntegerQ[m
] ||  !RationalQ[a, b, c, d, e]) &&  !(IGtQ[m, 0] && RationalQ[a, b, c, d, e] && (IntegerQ[p] || ILtQ[p + 1/2,
0]))

Rule 1657

Int[(Pq_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x + c*x^2)^p, x
], x] /; FreeQ[{a, b, c}, 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{(d+e x)^2 \left (2+x+3 x^2-5 x^3+4 x^4\right )}{\left (3+2 x+5 x^2\right )^3} \, dx &=-\frac{(1367+423 x) (d+e x)^2}{7000 \left (3+2 x+5 x^2\right )^2}+\frac{1}{112} \int \frac{(d+e x) \left (\frac{2}{125} (3267 d+2734 e)-\frac{6}{125} (3080 d-1371 e) x+\frac{112}{25} (20 d-33 e) x^2+\frac{448 e x^3}{5}\right )}{\left (3+2 x+5 x^2\right )^2} \, dx\\ &=-\frac{(1367+423 x) (d+e x)^2}{7000 \left (3+2 x+5 x^2\right )^2}+\frac{(d+e x) (34347 d-6413 e+5 (2203 d+8553 e) x)}{196000 \left (3+2 x+5 x^2\right )}+\frac{\int \frac{\frac{4}{125} \left (42375 d^2-55870 d e+6413 e^2\right )+\frac{6272}{125} (40 d-41 e) e x+\frac{25088 e^2 x^2}{25}}{3+2 x+5 x^2} \, dx}{6272}\\ &=-\frac{(1367+423 x) (d+e x)^2}{7000 \left (3+2 x+5 x^2\right )^2}+\frac{(d+e x) (34347 d-6413 e+5 (2203 d+8553 e) x)}{196000 \left (3+2 x+5 x^2\right )}+\frac{\int \left (\frac{25088 e^2}{125}+\frac{4 \left (42375 d^2-55870 d e-12403 e^2+1568 (40 d-49 e) e x\right )}{125 \left (3+2 x+5 x^2\right )}\right ) \, dx}{6272}\\ &=\frac{4 e^2 x}{125}-\frac{(1367+423 x) (d+e x)^2}{7000 \left (3+2 x+5 x^2\right )^2}+\frac{(d+e x) (34347 d-6413 e+5 (2203 d+8553 e) x)}{196000 \left (3+2 x+5 x^2\right )}+\frac{\int \frac{42375 d^2-55870 d e-12403 e^2+1568 (40 d-49 e) e x}{3+2 x+5 x^2} \, dx}{196000}\\ &=\frac{4 e^2 x}{125}-\frac{(1367+423 x) (d+e x)^2}{7000 \left (3+2 x+5 x^2\right )^2}+\frac{(d+e x) (34347 d-6413 e+5 (2203 d+8553 e) x)}{196000 \left (3+2 x+5 x^2\right )}+\frac{((40 d-49 e) e) \int \frac{2+10 x}{3+2 x+5 x^2} \, dx}{1250}+\frac{\left (211875 d^2-342070 d e+14817 e^2\right ) \int \frac{1}{3+2 x+5 x^2} \, dx}{980000}\\ &=\frac{4 e^2 x}{125}-\frac{(1367+423 x) (d+e x)^2}{7000 \left (3+2 x+5 x^2\right )^2}+\frac{(d+e x) (34347 d-6413 e+5 (2203 d+8553 e) x)}{196000 \left (3+2 x+5 x^2\right )}+\frac{(40 d-49 e) e \log \left (3+2 x+5 x^2\right )}{1250}+\frac{\left (-211875 d^2+342070 d e-14817 e^2\right ) \operatorname{Subst}\left (\int \frac{1}{-56-x^2} \, dx,x,2+10 x\right )}{490000}\\ &=\frac{4 e^2 x}{125}-\frac{(1367+423 x) (d+e x)^2}{7000 \left (3+2 x+5 x^2\right )^2}+\frac{(d+e x) (34347 d-6413 e+5 (2203 d+8553 e) x)}{196000 \left (3+2 x+5 x^2\right )}+\frac{\left (211875 d^2-342070 d e+14817 e^2\right ) \tan ^{-1}\left (\frac{1+5 x}{\sqrt{14}}\right )}{980000 \sqrt{14}}+\frac{(40 d-49 e) e \log \left (3+2 x+5 x^2\right )}{1250}\\ \end{align*}

Mathematica [A]  time = 0.176357, size = 146, normalized size = 1.09 $\frac{70 \left (\frac{5 \left (5 d^2 \left (11015 x^3+38753 x^2+17979 x+12953\right )+2 d e \left (181765 x^3+28307 x^2+57761 x-19533\right )+e^2 \left (156800 x^5+125440 x^4+83809 x^3-138345 x^2-65427 x-76977\right )\right )}{\left (5 x^2+2 x+3\right )^2}+784 e (40 d-49 e) \log \left (5 x^2+2 x+3\right )\right )+5 \sqrt{14} \left (211875 d^2-342070 d e+14817 e^2\right ) \tan ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{68600000}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(5*Sqrt[14]*(211875*d^2 - 342070*d*e + 14817*e^2)*ArcTan[(1 + 5*x)/Sqrt[14]] + 70*((5*(5*d^2*(12953 + 17979*x
+ 38753*x^2 + 11015*x^3) + 2*d*e*(-19533 + 57761*x + 28307*x^2 + 181765*x^3) + e^2*(-76977 - 65427*x - 138345*
x^2 + 83809*x^3 + 125440*x^4 + 156800*x^5)))/(3 + 2*x + 5*x^2)^2 + 784*(40*d - 49*e)*e*Log[3 + 2*x + 5*x^2]))/
68600000

________________________________________________________________________________________

Maple [A]  time = 0.053, size = 179, normalized size = 1.3 \begin{align*}{\frac{4\,{e}^{2}x}{125}}+{\frac{1}{5\, \left ( 5\,{x}^{2}+2\,x+3 \right ) ^{2}} \left ( \left ({\frac{2203\,{d}^{2}}{1568}}+{\frac{36353\,de}{3920}}-{\frac{129439\,{e}^{2}}{39200}} \right ){x}^{3}+ \left ({\frac{38753\,{d}^{2}}{7840}}+{\frac{28307\,de}{19600}}-{\frac{213609\,{e}^{2}}{39200}} \right ){x}^{2}+ \left ({\frac{17979\,{d}^{2}}{7840}}+{\frac{57761\,de}{19600}}-{\frac{4875\,{e}^{2}}{1568}} \right ) x+{\frac{12953\,{d}^{2}}{7840}}-{\frac{19533\,de}{19600}}-{\frac{76977\,{e}^{2}}{39200}} \right ) }+{\frac{4\,\ln \left ( 5\,{x}^{2}+2\,x+3 \right ) de}{125}}-{\frac{49\,\ln \left ( 5\,{x}^{2}+2\,x+3 \right ){e}^{2}}{1250}}+{\frac{339\,\sqrt{14}{d}^{2}}{21952}\arctan \left ({\frac{ \left ( 10\,x+2 \right ) \sqrt{14}}{28}} \right ) }-{\frac{34207\,\sqrt{14}de}{1372000}\arctan \left ({\frac{ \left ( 10\,x+2 \right ) \sqrt{14}}{28}} \right ) }+{\frac{14817\,\sqrt{14}{e}^{2}}{13720000}\arctan \left ({\frac{ \left ( 10\,x+2 \right ) \sqrt{14}}{28}} \right ) } \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/125*e^2*x+1/5*((2203/1568*d^2+36353/3920*d*e-129439/39200*e^2)*x^3+(38753/7840*d^2+28307/19600*d*e-213609/39
200*e^2)*x^2+(17979/7840*d^2+57761/19600*d*e-4875/1568*e^2)*x+12953/7840*d^2-19533/19600*d*e-76977/39200*e^2)/
(5*x^2+2*x+3)^2+4/125*ln(5*x^2+2*x+3)*d*e-49/1250*ln(5*x^2+2*x+3)*e^2+339/21952*14^(1/2)*arctan(1/28*(10*x+2)*
14^(1/2))*d^2-34207/1372000*14^(1/2)*arctan(1/28*(10*x+2)*14^(1/2))*d*e+14817/13720000*14^(1/2)*arctan(1/28*(1
0*x+2)*14^(1/2))*e^2

________________________________________________________________________________________

Maxima [A]  time = 1.54316, size = 209, normalized size = 1.56 \begin{align*} \frac{4}{125} \, e^{2} x + \frac{1}{13720000} \, \sqrt{14}{\left (211875 \, d^{2} - 342070 \, d e + 14817 \, e^{2}\right )} \arctan \left (\frac{1}{14} \, \sqrt{14}{\left (5 \, x + 1\right )}\right ) + \frac{1}{1250} \,{\left (40 \, d e - 49 \, e^{2}\right )} \log \left (5 \, x^{2} + 2 \, x + 3\right ) + \frac{{\left (55075 \, d^{2} + 363530 \, d e - 129439 \, e^{2}\right )} x^{3} +{\left (193765 \, d^{2} + 56614 \, d e - 213609 \, e^{2}\right )} x^{2} + 64765 \, d^{2} - 39066 \, d e - 76977 \, e^{2} +{\left (89895 \, d^{2} + 115522 \, d e - 121875 \, e^{2}\right )} x}{196000 \,{\left (25 \, x^{4} + 20 \, x^{3} + 34 \, x^{2} + 12 \, x + 9\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/125*e^2*x + 1/13720000*sqrt(14)*(211875*d^2 - 342070*d*e + 14817*e^2)*arctan(1/14*sqrt(14)*(5*x + 1)) + 1/12
50*(40*d*e - 49*e^2)*log(5*x^2 + 2*x + 3) + 1/196000*((55075*d^2 + 363530*d*e - 129439*e^2)*x^3 + (193765*d^2
+ 56614*d*e - 213609*e^2)*x^2 + 64765*d^2 - 39066*d*e - 76977*e^2 + (89895*d^2 + 115522*d*e - 121875*e^2)*x)/(
25*x^4 + 20*x^3 + 34*x^2 + 12*x + 9)

________________________________________________________________________________________

Fricas [B]  time = 1.25355, size = 933, normalized size = 6.96 \begin{align*} \frac{10976000 \, e^{2} x^{5} + 8780800 \, e^{2} x^{4} + 70 \,{\left (55075 \, d^{2} + 363530 \, d e + 83809 \, e^{2}\right )} x^{3} + 70 \,{\left (193765 \, d^{2} + 56614 \, d e - 138345 \, e^{2}\right )} x^{2} + \sqrt{14}{\left (25 \,{\left (211875 \, d^{2} - 342070 \, d e + 14817 \, e^{2}\right )} x^{4} + 20 \,{\left (211875 \, d^{2} - 342070 \, d e + 14817 \, e^{2}\right )} x^{3} + 34 \,{\left (211875 \, d^{2} - 342070 \, d e + 14817 \, e^{2}\right )} x^{2} + 1906875 \, d^{2} - 3078630 \, d e + 133353 \, e^{2} + 12 \,{\left (211875 \, d^{2} - 342070 \, d e + 14817 \, e^{2}\right )} x\right )} \arctan \left (\frac{1}{14} \, \sqrt{14}{\left (5 \, x + 1\right )}\right ) + 4533550 \, d^{2} - 2734620 \, d e - 5388390 \, e^{2} + 70 \,{\left (89895 \, d^{2} + 115522 \, d e - 65427 \, e^{2}\right )} x + 10976 \,{\left (25 \,{\left (40 \, d e - 49 \, e^{2}\right )} x^{4} + 20 \,{\left (40 \, d e - 49 \, e^{2}\right )} x^{3} + 34 \,{\left (40 \, d e - 49 \, e^{2}\right )} x^{2} + 360 \, d e - 441 \, e^{2} + 12 \,{\left (40 \, d e - 49 \, e^{2}\right )} x\right )} \log \left (5 \, x^{2} + 2 \, x + 3\right )}{13720000 \,{\left (25 \, x^{4} + 20 \, x^{3} + 34 \, x^{2} + 12 \, x + 9\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/13720000*(10976000*e^2*x^5 + 8780800*e^2*x^4 + 70*(55075*d^2 + 363530*d*e + 83809*e^2)*x^3 + 70*(193765*d^2
+ 56614*d*e - 138345*e^2)*x^2 + sqrt(14)*(25*(211875*d^2 - 342070*d*e + 14817*e^2)*x^4 + 20*(211875*d^2 - 3420
70*d*e + 14817*e^2)*x^3 + 34*(211875*d^2 - 342070*d*e + 14817*e^2)*x^2 + 1906875*d^2 - 3078630*d*e + 133353*e^
2 + 12*(211875*d^2 - 342070*d*e + 14817*e^2)*x)*arctan(1/14*sqrt(14)*(5*x + 1)) + 4533550*d^2 - 2734620*d*e -
5388390*e^2 + 70*(89895*d^2 + 115522*d*e - 65427*e^2)*x + 10976*(25*(40*d*e - 49*e^2)*x^4 + 20*(40*d*e - 49*e^
2)*x^3 + 34*(40*d*e - 49*e^2)*x^2 + 360*d*e - 441*e^2 + 12*(40*d*e - 49*e^2)*x)*log(5*x^2 + 2*x + 3))/(25*x^4
+ 20*x^3 + 34*x^2 + 12*x + 9)

________________________________________________________________________________________

Sympy [C]  time = 3.38472, size = 304, normalized size = 2.27 \begin{align*} \frac{4 e^{2} x}{125} + \left (\frac{e \left (40 d - 49 e\right )}{1250} - \frac{\sqrt{14} i \left (211875 d^{2} - 342070 d e + 14817 e^{2}\right )}{27440000}\right ) \log{\left (x + \frac{42375 d^{2} - 244030 d e + 218093 e^{2} + \frac{21952 e \left (40 d - 49 e\right )}{5} - \frac{\sqrt{14} i \left (211875 d^{2} - 342070 d e + 14817 e^{2}\right )}{5}}{211875 d^{2} - 342070 d e + 14817 e^{2}} \right )} + \left (\frac{e \left (40 d - 49 e\right )}{1250} + \frac{\sqrt{14} i \left (211875 d^{2} - 342070 d e + 14817 e^{2}\right )}{27440000}\right ) \log{\left (x + \frac{42375 d^{2} - 244030 d e + 218093 e^{2} + \frac{21952 e \left (40 d - 49 e\right )}{5} + \frac{\sqrt{14} i \left (211875 d^{2} - 342070 d e + 14817 e^{2}\right )}{5}}{211875 d^{2} - 342070 d e + 14817 e^{2}} \right )} + \frac{64765 d^{2} - 39066 d e - 76977 e^{2} + x^{3} \left (55075 d^{2} + 363530 d e - 129439 e^{2}\right ) + x^{2} \left (193765 d^{2} + 56614 d e - 213609 e^{2}\right ) + x \left (89895 d^{2} + 115522 d e - 121875 e^{2}\right )}{4900000 x^{4} + 3920000 x^{3} + 6664000 x^{2} + 2352000 x + 1764000} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4*e**2*x/125 + (e*(40*d - 49*e)/1250 - sqrt(14)*I*(211875*d**2 - 342070*d*e + 14817*e**2)/27440000)*log(x + (4
2375*d**2 - 244030*d*e + 218093*e**2 + 21952*e*(40*d - 49*e)/5 - sqrt(14)*I*(211875*d**2 - 342070*d*e + 14817*
e**2)/5)/(211875*d**2 - 342070*d*e + 14817*e**2)) + (e*(40*d - 49*e)/1250 + sqrt(14)*I*(211875*d**2 - 342070*d
*e + 14817*e**2)/27440000)*log(x + (42375*d**2 - 244030*d*e + 218093*e**2 + 21952*e*(40*d - 49*e)/5 + sqrt(14)
*I*(211875*d**2 - 342070*d*e + 14817*e**2)/5)/(211875*d**2 - 342070*d*e + 14817*e**2)) + (64765*d**2 - 39066*d
*e - 76977*e**2 + x**3*(55075*d**2 + 363530*d*e - 129439*e**2) + x**2*(193765*d**2 + 56614*d*e - 213609*e**2)
+ x*(89895*d**2 + 115522*d*e - 121875*e**2))/(4900000*x**4 + 3920000*x**3 + 6664000*x**2 + 2352000*x + 1764000
)

________________________________________________________________________________________

Giac [A]  time = 1.14012, size = 194, normalized size = 1.45 \begin{align*} \frac{1}{13720000} \, \sqrt{14}{\left (211875 \, d^{2} - 342070 \, d e + 14817 \, e^{2}\right )} \arctan \left (\frac{1}{14} \, \sqrt{14}{\left (5 \, x + 1\right )}\right ) + \frac{4}{125} \, x e^{2} + \frac{1}{1250} \,{\left (40 \, d e - 49 \, e^{2}\right )} \log \left (5 \, x^{2} + 2 \, x + 3\right ) + \frac{{\left (55075 \, d^{2} + 363530 \, d e - 129439 \, e^{2}\right )} x^{3} +{\left (193765 \, d^{2} + 56614 \, d e - 213609 \, e^{2}\right )} x^{2} + 64765 \, d^{2} +{\left (89895 \, d^{2} + 115522 \, d e - 121875 \, e^{2}\right )} x - 39066 \, d e - 76977 \, e^{2}}{196000 \,{\left (5 \, x^{2} + 2 \, x + 3\right )}^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/13720000*sqrt(14)*(211875*d^2 - 342070*d*e + 14817*e^2)*arctan(1/14*sqrt(14)*(5*x + 1)) + 4/125*x*e^2 + 1/12
50*(40*d*e - 49*e^2)*log(5*x^2 + 2*x + 3) + 1/196000*((55075*d^2 + 363530*d*e - 129439*e^2)*x^3 + (193765*d^2
+ 56614*d*e - 213609*e^2)*x^2 + 64765*d^2 + (89895*d^2 + 115522*d*e - 121875*e^2)*x - 39066*d*e - 76977*e^2)/(
5*x^2 + 2*x + 3)^2