### 3.318 $$\int \frac{(d+e x)^3 (2+x+3 x^2-5 x^3+4 x^4)}{(3+2 x+5 x^2)^3} \, dx$$

Optimal. Leaf size=171 $\frac{3 e \left (100 d^2-245 d e+47 e^2\right ) \log \left (5 x^2+2 x+3\right )}{6250}+\frac{3 \left (-855175 d^2 e+353125 d^3+74085 d e^2+556349 e^3\right ) \tan ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{4900000 \sqrt{14}}+\frac{e^2 x (83065 d-126009 e)}{980000}+\frac{(d+e x)^2 (x (11015 d+49177 e)+3 (11449 d-2105 e))}{196000 \left (5 x^2+2 x+3\right )}-\frac{(423 x+1367) (d+e x)^3}{7000 \left (5 x^2+2 x+3\right )^2}+\frac{2 e^3 x^2}{125}$

[Out]

((83065*d - 126009*e)*e^2*x)/980000 + (2*e^3*x^2)/125 - ((1367 + 423*x)*(d + e*x)^3)/(7000*(3 + 2*x + 5*x^2)^2
) + ((d + e*x)^2*(3*(11449*d - 2105*e) + (11015*d + 49177*e)*x))/(196000*(3 + 2*x + 5*x^2)) + (3*(353125*d^3 -
855175*d^2*e + 74085*d*e^2 + 556349*e^3)*ArcTan[(1 + 5*x)/Sqrt[14]])/(4900000*Sqrt[14]) + (3*e*(100*d^2 - 245
*d*e + 47*e^2)*Log[3 + 2*x + 5*x^2])/6250

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Rubi [A]  time = 0.336016, antiderivative size = 171, 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, 1628, 634, 618, 204, 628} $\frac{3 e \left (100 d^2-245 d e+47 e^2\right ) \log \left (5 x^2+2 x+3\right )}{6250}+\frac{3 \left (-855175 d^2 e+353125 d^3+74085 d e^2+556349 e^3\right ) \tan ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{4900000 \sqrt{14}}+\frac{e^2 x (83065 d-126009 e)}{980000}+\frac{(d+e x)^2 (x (11015 d+49177 e)+3 (11449 d-2105 e))}{196000 \left (5 x^2+2 x+3\right )}-\frac{(423 x+1367) (d+e x)^3}{7000 \left (5 x^2+2 x+3\right )^2}+\frac{2 e^3 x^2}{125}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

((83065*d - 126009*e)*e^2*x)/980000 + (2*e^3*x^2)/125 - ((1367 + 423*x)*(d + e*x)^3)/(7000*(3 + 2*x + 5*x^2)^2
) + ((d + e*x)^2*(3*(11449*d - 2105*e) + (11015*d + 49177*e)*x))/(196000*(3 + 2*x + 5*x^2)) + (3*(353125*d^3 -
855175*d^2*e + 74085*d*e^2 + 556349*e^3)*ArcTan[(1 + 5*x)/Sqrt[14]])/(4900000*Sqrt[14]) + (3*e*(100*d^2 - 245
*d*e + 47*e^2)*Log[3 + 2*x + 5*x^2])/6250

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 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{(d+e x)^3 \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)^3}{7000 \left (3+2 x+5 x^2\right )^2}+\frac{1}{112} \int \frac{(d+e x)^2 \left (\frac{6}{125} (1089 d+1367 e)-\frac{336}{125} (55 d-27 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)^3}{7000 \left (3+2 x+5 x^2\right )^2}+\frac{(d+e x)^2 (3 (11449 d-2105 e)+(11015 d+49177 e) x)}{196000 \left (3+2 x+5 x^2\right )}+\frac{\int \frac{(d+e x) \left (\frac{12}{25} \left (2825 d^2-5587 d e+842 e^2\right )+\frac{4}{25} (10341 d-22693 e) e x+\frac{25088 e^2 x^2}{25}\right )}{3+2 x+5 x^2} \, dx}{6272}\\ &=-\frac{(1367+423 x) (d+e x)^3}{7000 \left (3+2 x+5 x^2\right )^2}+\frac{(d+e x)^2 (3 (11449 d-2105 e)+(11015 d+49177 e) x)}{196000 \left (3+2 x+5 x^2\right )}+\frac{\int \left (\frac{4}{625} (83065 d-126009 e) e^2+\frac{25088 e^3 x}{125}+\frac{12 \left (70625 d^3-139675 d^2 e-62015 d e^2+126009 e^3+1568 e \left (100 d^2-245 d e+47 e^2\right ) x\right )}{625 \left (3+2 x+5 x^2\right )}\right ) \, dx}{6272}\\ &=\frac{(83065 d-126009 e) e^2 x}{980000}+\frac{2 e^3 x^2}{125}-\frac{(1367+423 x) (d+e x)^3}{7000 \left (3+2 x+5 x^2\right )^2}+\frac{(d+e x)^2 (3 (11449 d-2105 e)+(11015 d+49177 e) x)}{196000 \left (3+2 x+5 x^2\right )}+\frac{3 \int \frac{70625 d^3-139675 d^2 e-62015 d e^2+126009 e^3+1568 e \left (100 d^2-245 d e+47 e^2\right ) x}{3+2 x+5 x^2} \, dx}{980000}\\ &=\frac{(83065 d-126009 e) e^2 x}{980000}+\frac{2 e^3 x^2}{125}-\frac{(1367+423 x) (d+e x)^3}{7000 \left (3+2 x+5 x^2\right )^2}+\frac{(d+e x)^2 (3 (11449 d-2105 e)+(11015 d+49177 e) x)}{196000 \left (3+2 x+5 x^2\right )}+\frac{\left (3 e \left (100 d^2-245 d e+47 e^2\right )\right ) \int \frac{2+10 x}{3+2 x+5 x^2} \, dx}{6250}+\frac{\left (3 \left (353125 d^3-855175 d^2 e+74085 d e^2+556349 e^3\right )\right ) \int \frac{1}{3+2 x+5 x^2} \, dx}{4900000}\\ &=\frac{(83065 d-126009 e) e^2 x}{980000}+\frac{2 e^3 x^2}{125}-\frac{(1367+423 x) (d+e x)^3}{7000 \left (3+2 x+5 x^2\right )^2}+\frac{(d+e x)^2 (3 (11449 d-2105 e)+(11015 d+49177 e) x)}{196000 \left (3+2 x+5 x^2\right )}+\frac{3 e \left (100 d^2-245 d e+47 e^2\right ) \log \left (3+2 x+5 x^2\right )}{6250}-\frac{\left (3 \left (353125 d^3-855175 d^2 e+74085 d e^2+556349 e^3\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-56-x^2} \, dx,x,2+10 x\right )}{2450000}\\ &=\frac{(83065 d-126009 e) e^2 x}{980000}+\frac{2 e^3 x^2}{125}-\frac{(1367+423 x) (d+e x)^3}{7000 \left (3+2 x+5 x^2\right )^2}+\frac{(d+e x)^2 (3 (11449 d-2105 e)+(11015 d+49177 e) x)}{196000 \left (3+2 x+5 x^2\right )}+\frac{3 \left (353125 d^3-855175 d^2 e+74085 d e^2+556349 e^3\right ) \tan ^{-1}\left (\frac{1+5 x}{\sqrt{14}}\right )}{4900000 \sqrt{14}}+\frac{3 e \left (100 d^2-245 d e+47 e^2\right ) \log \left (3+2 x+5 x^2\right )}{6250}\\ \end{align*}

Mathematica [A]  time = 0.20124, size = 209, normalized size = 1.22 $\frac{-\frac{392 \left (75 d^2 e (5989 x-1269)+125 d^3 (423 x+1367)-15 d e^2 (18323 x+17967)+e^3 (54969-53189 x)\right )}{\left (5 x^2+2 x+3\right )^2}+\frac{14 \left (75 d^2 e (181765 x-44399)+125 d^3 (11015 x+34347)-15 d e^2 (647195 x+809167)+e^3 (2639639-3109005 x)\right )}{5 x^2+2 x+3}+164640 e \left (100 d^2-245 d e+47 e^2\right ) \log \left (5 x^2+2 x+3\right )+15 \sqrt{14} \left (-855175 d^2 e+353125 d^3+74085 d e^2+556349 e^3\right ) \tan ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )+548800 e^2 x (60 d-49 e)+5488000 e^3 x^2}{343000000}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(548800*(60*d - 49*e)*e^2*x + 5488000*e^3*x^2 - (392*(e^3*(54969 - 53189*x) + 125*d^3*(1367 + 423*x) + 75*d^2*
e*(-1269 + 5989*x) - 15*d*e^2*(17967 + 18323*x)))/(3 + 2*x + 5*x^2)^2 + (14*(e^3*(2639639 - 3109005*x) + 125*d
^3*(34347 + 11015*x) + 75*d^2*e*(-44399 + 181765*x) - 15*d*e^2*(809167 + 647195*x)))/(3 + 2*x + 5*x^2) + 15*Sq
rt[14]*(353125*d^3 - 855175*d^2*e + 74085*d*e^2 + 556349*e^3)*ArcTan[(1 + 5*x)/Sqrt[14]] + 164640*e*(100*d^2 -
245*d*e + 47*e^2)*Log[3 + 2*x + 5*x^2])/343000000

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Maple [A]  time = 0.059, size = 267, normalized size = 1.6 \begin{align*}{\frac{2\,{e}^{3}{x}^{2}}{125}}+{\frac{12\,xd{e}^{2}}{125}}-{\frac{49\,{e}^{3}x}{625}}+{\frac{1}{25\, \left ( 5\,{x}^{2}+2\,x+3 \right ) ^{2}} \left ( \left ({\frac{11015\,{d}^{3}}{1568}}+{\frac{109059\,{d}^{2}e}{1568}}-{\frac{388317\,d{e}^{2}}{7840}}-{\frac{621801\,{e}^{3}}{39200}} \right ){x}^{3}+ \left ({\frac{38753\,{d}^{3}}{1568}}+{\frac{84921\,{d}^{2}e}{7840}}-{\frac{640827\,d{e}^{2}}{7840}}+{\frac{1396037\,{e}^{3}}{196000}} \right ){x}^{2}+ \left ({\frac{17979\,{d}^{3}}{1568}}+{\frac{173283\,{d}^{2}e}{7840}}-{\frac{73125\,d{e}^{2}}{1568}}-{\frac{511689\,{e}^{3}}{196000}} \right ) x+{\frac{12953\,{d}^{3}}{1568}}-{\frac{58599\,{d}^{2}e}{7840}}-{\frac{230931\,d{e}^{2}}{7840}}+{\frac{1275957\,{e}^{3}}{196000}} \right ) }+{\frac{6\,\ln \left ( 5\,{x}^{2}+2\,x+3 \right ){d}^{2}e}{125}}-{\frac{147\,\ln \left ( 5\,{x}^{2}+2\,x+3 \right ) d{e}^{2}}{1250}}+{\frac{141\,\ln \left ( 5\,{x}^{2}+2\,x+3 \right ){e}^{3}}{6250}}+{\frac{339\,\sqrt{14}{d}^{3}}{21952}\arctan \left ({\frac{ \left ( 10\,x+2 \right ) \sqrt{14}}{28}} \right ) }-{\frac{102621\,\sqrt{14}{d}^{2}e}{2744000}\arctan \left ({\frac{ \left ( 10\,x+2 \right ) \sqrt{14}}{28}} \right ) }+{\frac{44451\,\sqrt{14}d{e}^{2}}{13720000}\arctan \left ({\frac{ \left ( 10\,x+2 \right ) \sqrt{14}}{28}} \right ) }+{\frac{1669047\,\sqrt{14}{e}^{3}}{68600000}\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)^3*(4*x^4-5*x^3+3*x^2+x+2)/(5*x^2+2*x+3)^3,x)

[Out]

2/125*e^3*x^2+12/125*x*d*e^2-49/625*e^3*x+1/25*((11015/1568*d^3+109059/1568*d^2*e-388317/7840*d*e^2-621801/392
00*e^3)*x^3+(38753/1568*d^3+84921/7840*d^2*e-640827/7840*d*e^2+1396037/196000*e^3)*x^2+(17979/1568*d^3+173283/
7840*d^2*e-73125/1568*d*e^2-511689/196000*e^3)*x+12953/1568*d^3-58599/7840*d^2*e-230931/7840*d*e^2+1275957/196
000*e^3)/(5*x^2+2*x+3)^2+6/125*ln(5*x^2+2*x+3)*d^2*e-147/1250*ln(5*x^2+2*x+3)*d*e^2+141/6250*ln(5*x^2+2*x+3)*e
^3+339/21952*14^(1/2)*arctan(1/28*(10*x+2)*14^(1/2))*d^3-102621/2744000*14^(1/2)*arctan(1/28*(10*x+2)*14^(1/2)
)*d^2*e+44451/13720000*14^(1/2)*arctan(1/28*(10*x+2)*14^(1/2))*d*e^2+1669047/68600000*14^(1/2)*arctan(1/28*(10
*x+2)*14^(1/2))*e^3

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Maxima [A]  time = 1.55006, size = 300, normalized size = 1.75 \begin{align*} \frac{2}{125} \, e^{3} x^{2} + \frac{3}{68600000} \, \sqrt{14}{\left (353125 \, d^{3} - 855175 \, d^{2} e + 74085 \, d e^{2} + 556349 \, e^{3}\right )} \arctan \left (\frac{1}{14} \, \sqrt{14}{\left (5 \, x + 1\right )}\right ) + \frac{1}{625} \,{\left (60 \, d e^{2} - 49 \, e^{3}\right )} x + \frac{3}{6250} \,{\left (100 \, d^{2} e - 245 \, d e^{2} + 47 \, e^{3}\right )} \log \left (5 \, x^{2} + 2 \, x + 3\right ) + \frac{5 \,{\left (275375 \, d^{3} + 2726475 \, d^{2} e - 1941585 \, d e^{2} - 621801 \, e^{3}\right )} x^{3} + 1619125 \, d^{3} - 1464975 \, d^{2} e - 5773275 \, d e^{2} + 1275957 \, e^{3} +{\left (4844125 \, d^{3} + 2123025 \, d^{2} e - 16020675 \, d e^{2} + 1396037 \, e^{3}\right )} x^{2} + 3 \,{\left (749125 \, d^{3} + 1444025 \, d^{2} e - 3046875 \, d e^{2} - 170563 \, e^{3}\right )} x}{4900000 \,{\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)^3*(4*x^4-5*x^3+3*x^2+x+2)/(5*x^2+2*x+3)^3,x, algorithm="maxima")

[Out]

2/125*e^3*x^2 + 3/68600000*sqrt(14)*(353125*d^3 - 855175*d^2*e + 74085*d*e^2 + 556349*e^3)*arctan(1/14*sqrt(14
)*(5*x + 1)) + 1/625*(60*d*e^2 - 49*e^3)*x + 3/6250*(100*d^2*e - 245*d*e^2 + 47*e^3)*log(5*x^2 + 2*x + 3) + 1/
4900000*(5*(275375*d^3 + 2726475*d^2*e - 1941585*d*e^2 - 621801*e^3)*x^3 + 1619125*d^3 - 1464975*d^2*e - 57732
75*d*e^2 + 1275957*e^3 + (4844125*d^3 + 2123025*d^2*e - 16020675*d*e^2 + 1396037*e^3)*x^2 + 3*(749125*d^3 + 14
44025*d^2*e - 3046875*d*e^2 - 170563*e^3)*x)/(25*x^4 + 20*x^3 + 34*x^2 + 12*x + 9)

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Fricas [B]  time = 1.25751, size = 1339, normalized size = 7.83 \begin{align*} \frac{27440000 \, e^{3} x^{6} + 2744000 \,{\left (60 \, d e^{2} - 41 \, e^{3}\right )} x^{5} + 8780800 \,{\left (15 \, d e^{2} - 8 \, e^{3}\right )} x^{4} + 70 \,{\left (275375 \, d^{3} + 2726475 \, d^{2} e + 1257135 \, d e^{2} - 3045929 \, e^{3}\right )} x^{3} + 22667750 \, d^{3} - 20509650 \, d^{2} e - 80825850 \, d e^{2} + 17863398 \, e^{3} + 14 \,{\left (4844125 \, d^{3} + 2123025 \, d^{2} e - 10375875 \, d e^{2} - 2508283 \, e^{3}\right )} x^{2} + 3 \, \sqrt{14}{\left (25 \,{\left (353125 \, d^{3} - 855175 \, d^{2} e + 74085 \, d e^{2} + 556349 \, e^{3}\right )} x^{4} + 20 \,{\left (353125 \, d^{3} - 855175 \, d^{2} e + 74085 \, d e^{2} + 556349 \, e^{3}\right )} x^{3} + 3178125 \, d^{3} - 7696575 \, d^{2} e + 666765 \, d e^{2} + 5007141 \, e^{3} + 34 \,{\left (353125 \, d^{3} - 855175 \, d^{2} e + 74085 \, d e^{2} + 556349 \, e^{3}\right )} x^{2} + 12 \,{\left (353125 \, d^{3} - 855175 \, d^{2} e + 74085 \, d e^{2} + 556349 \, e^{3}\right )} x\right )} \arctan \left (\frac{1}{14} \, \sqrt{14}{\left (5 \, x + 1\right )}\right ) + 42 \,{\left (749125 \, d^{3} + 1444025 \, d^{2} e - 1635675 \, d e^{2} - 1323043 \, e^{3}\right )} x + 32928 \,{\left (25 \,{\left (100 \, d^{2} e - 245 \, d e^{2} + 47 \, e^{3}\right )} x^{4} + 20 \,{\left (100 \, d^{2} e - 245 \, d e^{2} + 47 \, e^{3}\right )} x^{3} + 900 \, d^{2} e - 2205 \, d e^{2} + 423 \, e^{3} + 34 \,{\left (100 \, d^{2} e - 245 \, d e^{2} + 47 \, e^{3}\right )} x^{2} + 12 \,{\left (100 \, d^{2} e - 245 \, d e^{2} + 47 \, e^{3}\right )} x\right )} \log \left (5 \, x^{2} + 2 \, x + 3\right )}{68600000 \,{\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)^3*(4*x^4-5*x^3+3*x^2+x+2)/(5*x^2+2*x+3)^3,x, algorithm="fricas")

[Out]

1/68600000*(27440000*e^3*x^6 + 2744000*(60*d*e^2 - 41*e^3)*x^5 + 8780800*(15*d*e^2 - 8*e^3)*x^4 + 70*(275375*d
^3 + 2726475*d^2*e + 1257135*d*e^2 - 3045929*e^3)*x^3 + 22667750*d^3 - 20509650*d^2*e - 80825850*d*e^2 + 17863
398*e^3 + 14*(4844125*d^3 + 2123025*d^2*e - 10375875*d*e^2 - 2508283*e^3)*x^2 + 3*sqrt(14)*(25*(353125*d^3 - 8
55175*d^2*e + 74085*d*e^2 + 556349*e^3)*x^4 + 20*(353125*d^3 - 855175*d^2*e + 74085*d*e^2 + 556349*e^3)*x^3 +
3178125*d^3 - 7696575*d^2*e + 666765*d*e^2 + 5007141*e^3 + 34*(353125*d^3 - 855175*d^2*e + 74085*d*e^2 + 55634
9*e^3)*x^2 + 12*(353125*d^3 - 855175*d^2*e + 74085*d*e^2 + 556349*e^3)*x)*arctan(1/14*sqrt(14)*(5*x + 1)) + 42
*(749125*d^3 + 1444025*d^2*e - 1635675*d*e^2 - 1323043*e^3)*x + 32928*(25*(100*d^2*e - 245*d*e^2 + 47*e^3)*x^4
+ 20*(100*d^2*e - 245*d*e^2 + 47*e^3)*x^3 + 900*d^2*e - 2205*d*e^2 + 423*e^3 + 34*(100*d^2*e - 245*d*e^2 + 47
*e^3)*x^2 + 12*(100*d^2*e - 245*d*e^2 + 47*e^3)*x)*log(5*x^2 + 2*x + 3))/(25*x^4 + 20*x^3 + 34*x^2 + 12*x + 9)

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Sympy [C]  time = 5.28771, size = 469, normalized size = 2.74 \begin{align*} \frac{2 e^{3} x^{2}}{125} + x \left (\frac{12 d e^{2}}{125} - \frac{49 e^{3}}{625}\right ) + \left (\frac{3 e \left (100 d^{2} - 245 d e + 47 e^{2}\right )}{6250} - \frac{3 \sqrt{14} i \left (353125 d^{3} - 855175 d^{2} e + 74085 d e^{2} + 556349 e^{3}\right )}{137200000}\right ) \log{\left (x + \frac{211875 d^{3} - 1830225 d^{2} e + 3271395 d e^{2} - 285237 e^{3} + \frac{65856 e \left (100 d^{2} - 245 d e + 47 e^{2}\right )}{5} - \frac{3 \sqrt{14} i \left (353125 d^{3} - 855175 d^{2} e + 74085 d e^{2} + 556349 e^{3}\right )}{5}}{1059375 d^{3} - 2565525 d^{2} e + 222255 d e^{2} + 1669047 e^{3}} \right )} + \left (\frac{3 e \left (100 d^{2} - 245 d e + 47 e^{2}\right )}{6250} + \frac{3 \sqrt{14} i \left (353125 d^{3} - 855175 d^{2} e + 74085 d e^{2} + 556349 e^{3}\right )}{137200000}\right ) \log{\left (x + \frac{211875 d^{3} - 1830225 d^{2} e + 3271395 d e^{2} - 285237 e^{3} + \frac{65856 e \left (100 d^{2} - 245 d e + 47 e^{2}\right )}{5} + \frac{3 \sqrt{14} i \left (353125 d^{3} - 855175 d^{2} e + 74085 d e^{2} + 556349 e^{3}\right )}{5}}{1059375 d^{3} - 2565525 d^{2} e + 222255 d e^{2} + 1669047 e^{3}} \right )} + \frac{1619125 d^{3} - 1464975 d^{2} e - 5773275 d e^{2} + 1275957 e^{3} + x^{3} \left (1376875 d^{3} + 13632375 d^{2} e - 9707925 d e^{2} - 3109005 e^{3}\right ) + x^{2} \left (4844125 d^{3} + 2123025 d^{2} e - 16020675 d e^{2} + 1396037 e^{3}\right ) + x \left (2247375 d^{3} + 4332075 d^{2} e - 9140625 d e^{2} - 511689 e^{3}\right )}{122500000 x^{4} + 98000000 x^{3} + 166600000 x^{2} + 58800000 x + 44100000} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*e**3*x**2/125 + x*(12*d*e**2/125 - 49*e**3/625) + (3*e*(100*d**2 - 245*d*e + 47*e**2)/6250 - 3*sqrt(14)*I*(3
53125*d**3 - 855175*d**2*e + 74085*d*e**2 + 556349*e**3)/137200000)*log(x + (211875*d**3 - 1830225*d**2*e + 32
71395*d*e**2 - 285237*e**3 + 65856*e*(100*d**2 - 245*d*e + 47*e**2)/5 - 3*sqrt(14)*I*(353125*d**3 - 855175*d**
2*e + 74085*d*e**2 + 556349*e**3)/5)/(1059375*d**3 - 2565525*d**2*e + 222255*d*e**2 + 1669047*e**3)) + (3*e*(1
00*d**2 - 245*d*e + 47*e**2)/6250 + 3*sqrt(14)*I*(353125*d**3 - 855175*d**2*e + 74085*d*e**2 + 556349*e**3)/13
7200000)*log(x + (211875*d**3 - 1830225*d**2*e + 3271395*d*e**2 - 285237*e**3 + 65856*e*(100*d**2 - 245*d*e +
47*e**2)/5 + 3*sqrt(14)*I*(353125*d**3 - 855175*d**2*e + 74085*d*e**2 + 556349*e**3)/5)/(1059375*d**3 - 256552
5*d**2*e + 222255*d*e**2 + 1669047*e**3)) + (1619125*d**3 - 1464975*d**2*e - 5773275*d*e**2 + 1275957*e**3 + x
**3*(1376875*d**3 + 13632375*d**2*e - 9707925*d*e**2 - 3109005*e**3) + x**2*(4844125*d**3 + 2123025*d**2*e - 1
6020675*d*e**2 + 1396037*e**3) + x*(2247375*d**3 + 4332075*d**2*e - 9140625*d*e**2 - 511689*e**3))/(122500000*
x**4 + 98000000*x**3 + 166600000*x**2 + 58800000*x + 44100000)

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Giac [A]  time = 1.14852, size = 271, normalized size = 1.58 \begin{align*} \frac{2}{125} \, x^{2} e^{3} + \frac{12}{125} \, d x e^{2} + \frac{3}{68600000} \, \sqrt{14}{\left (353125 \, d^{3} - 855175 \, d^{2} e + 74085 \, d e^{2} + 556349 \, e^{3}\right )} \arctan \left (\frac{1}{14} \, \sqrt{14}{\left (5 \, x + 1\right )}\right ) - \frac{49}{625} \, x e^{3} + \frac{3}{6250} \,{\left (100 \, d^{2} e - 245 \, d e^{2} + 47 \, e^{3}\right )} \log \left (5 \, x^{2} + 2 \, x + 3\right ) + \frac{5 \,{\left (275375 \, d^{3} + 2726475 \, d^{2} e - 1941585 \, d e^{2} - 621801 \, e^{3}\right )} x^{3} + 1619125 \, d^{3} +{\left (4844125 \, d^{3} + 2123025 \, d^{2} e - 16020675 \, d e^{2} + 1396037 \, e^{3}\right )} x^{2} - 1464975 \, d^{2} e + 3 \,{\left (749125 \, d^{3} + 1444025 \, d^{2} e - 3046875 \, d e^{2} - 170563 \, e^{3}\right )} x - 5773275 \, d e^{2} + 1275957 \, e^{3}}{4900000 \,{\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)^3*(4*x^4-5*x^3+3*x^2+x+2)/(5*x^2+2*x+3)^3,x, algorithm="giac")

[Out]

2/125*x^2*e^3 + 12/125*d*x*e^2 + 3/68600000*sqrt(14)*(353125*d^3 - 855175*d^2*e + 74085*d*e^2 + 556349*e^3)*ar
ctan(1/14*sqrt(14)*(5*x + 1)) - 49/625*x*e^3 + 3/6250*(100*d^2*e - 245*d*e^2 + 47*e^3)*log(5*x^2 + 2*x + 3) +
1/4900000*(5*(275375*d^3 + 2726475*d^2*e - 1941585*d*e^2 - 621801*e^3)*x^3 + 1619125*d^3 + (4844125*d^3 + 2123
025*d^2*e - 16020675*d*e^2 + 1396037*e^3)*x^2 - 1464975*d^2*e + 3*(749125*d^3 + 1444025*d^2*e - 3046875*d*e^2
- 170563*e^3)*x - 5773275*d*e^2 + 1275957*e^3)/(5*x^2 + 2*x + 3)^2