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

[Out]

1/980000*(83065*d-126009*e)*e^2*x+2/125*e^3*x^2-1/7000*(1367+423*x)*(e*x+d)^3/(5*x^2+2*x+3)^2+1/196000*(e*x+d)
^2*(34347*d-6315*e+(11015*d+49177*e)*x)/(5*x^2+2*x+3)+3/6250*e*(100*d^2-245*d*e+47*e^2)*ln(5*x^2+2*x+3)+3/6860
0000*(353125*d^3-855175*d^2*e+74085*d*e^2+556349*e^3)*arctan(1/14*(1+5*x)*14^(1/2))*14^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.34, antiderivative size = 171, normalized size of antiderivative = 1.00, 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 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 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]))

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 = 0.20, size = 209, normalized size = 1.22 \[ \frac {164640 e \left (100 d^2-245 d e+47 e^2\right ) \log \left (5 x^2+2 x+3\right )-\frac {392 \left (125 d^3 (423 x+1367)+75 d^2 e (5989 x-1269)-15 d e^2 (18323 x+17967)+e^3 (54969-53189 x)\right )}{\left (5 x^2+2 x+3\right )^2}+\frac {14 \left (125 d^3 (11015 x+34347)+75 d^2 e (181765 x-44399)-15 d e^2 (647195 x+809167)+e^3 (2639639-3109005 x)\right )}{5 x^2+2 x+3}+15 \sqrt {14} \left (353125 d^3-855175 d^2 e+74085 d e^2+556349 e^3\right ) \tan ^{-1}\left (\frac {5 x+1}{\sqrt {14}}\right )+548800 e^2 x (60 d-49 e)+5488000 e^3 x^2}{343000000} \]

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

________________________________________________________________________________________

fricas [B]  time = 0.84, size = 441, normalized size = 2.58 \[ \frac {27440000 \, e^{3} x^{6} + 2744000 \, {\left (60 \, d e^{2} - 41 \, e^{3}\right )} x^{5} + 8780800 \, {\left (15 \, d e^{2} - 8 \, e^{3}\right )} x^{4} + 70 \, {\left (275375 \, d^{3} + 2726475 \, d^{2} e + 1257135 \, d e^{2} - 3045929 \, e^{3}\right )} x^{3} + 22667750 \, d^{3} - 20509650 \, d^{2} e - 80825850 \, d e^{2} + 17863398 \, e^{3} + 14 \, {\left (4844125 \, d^{3} + 2123025 \, d^{2} e - 10375875 \, d e^{2} - 2508283 \, e^{3}\right )} x^{2} + 3 \, \sqrt {14} {\left (25 \, {\left (353125 \, d^{3} - 855175 \, d^{2} e + 74085 \, d e^{2} + 556349 \, e^{3}\right )} x^{4} + 20 \, {\left (353125 \, d^{3} - 855175 \, d^{2} e + 74085 \, d e^{2} + 556349 \, e^{3}\right )} x^{3} + 3178125 \, d^{3} - 7696575 \, d^{2} e + 666765 \, d e^{2} + 5007141 \, e^{3} + 34 \, {\left (353125 \, d^{3} - 855175 \, d^{2} e + 74085 \, d e^{2} + 556349 \, e^{3}\right )} x^{2} + 12 \, {\left (353125 \, d^{3} - 855175 \, d^{2} e + 74085 \, d e^{2} + 556349 \, e^{3}\right )} x\right )} \arctan \left (\frac {1}{14} \, \sqrt {14} {\left (5 \, x + 1\right )}\right ) + 42 \, {\left (749125 \, d^{3} + 1444025 \, d^{2} e - 1635675 \, d e^{2} - 1323043 \, e^{3}\right )} x + 32928 \, {\left (25 \, {\left (100 \, d^{2} e - 245 \, d e^{2} + 47 \, e^{3}\right )} x^{4} + 20 \, {\left (100 \, d^{2} e - 245 \, d e^{2} + 47 \, e^{3}\right )} x^{3} + 900 \, d^{2} e - 2205 \, d e^{2} + 423 \, e^{3} + 34 \, {\left (100 \, d^{2} e - 245 \, d e^{2} + 47 \, e^{3}\right )} x^{2} + 12 \, {\left (100 \, d^{2} e - 245 \, d e^{2} + 47 \, e^{3}\right )} x\right )} \log \left (5 \, x^{2} + 2 \, x + 3\right )}{68600000 \, {\left (25 \, x^{4} + 20 \, x^{3} + 34 \, x^{2} + 12 \, x + 9\right )}} \]

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)

________________________________________________________________________________________

giac [A]  time = 0.21, size = 201, normalized size = 1.18 \[ \frac {2}{125} \, x^{2} e^{3} + \frac {12}{125} \, d x e^{2} + \frac {3}{68600000} \, \sqrt {14} {\left (353125 \, d^{3} - 855175 \, d^{2} e + 74085 \, d e^{2} + 556349 \, e^{3}\right )} \arctan \left (\frac {1}{14} \, \sqrt {14} {\left (5 \, x + 1\right )}\right ) - \frac {49}{625} \, x e^{3} + \frac {3}{6250} \, {\left (100 \, d^{2} e - 245 \, d e^{2} + 47 \, e^{3}\right )} \log \left (5 \, x^{2} + 2 \, x + 3\right ) + \frac {5 \, {\left (275375 \, d^{3} + 2726475 \, d^{2} e - 1941585 \, d e^{2} - 621801 \, e^{3}\right )} x^{3} + 1619125 \, d^{3} + {\left (4844125 \, d^{3} + 2123025 \, d^{2} e - 16020675 \, d e^{2} + 1396037 \, e^{3}\right )} x^{2} - 1464975 \, d^{2} e + 3 \, {\left (749125 \, d^{3} + 1444025 \, d^{2} e - 3046875 \, d e^{2} - 170563 \, e^{3}\right )} x - 5773275 \, d e^{2} + 1275957 \, e^{3}}{4900000 \, {\left (5 \, x^{2} + 2 \, x + 3\right )}^{2}} \]

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

________________________________________________________________________________________

maple [A]  time = 0.02, size = 267, normalized size = 1.56 \[ \frac {2 e^{3} x^{2}}{125}+\frac {339 \sqrt {14}\, d^{3} \arctan \left (\frac {\left (10 x +2\right ) \sqrt {14}}{28}\right )}{21952}-\frac {102621 \sqrt {14}\, d^{2} e \arctan \left (\frac {\left (10 x +2\right ) \sqrt {14}}{28}\right )}{2744000}+\frac {6 d^{2} e \ln \left (5 x^{2}+2 x +3\right )}{125}+\frac {12 d \,e^{2} x}{125}+\frac {44451 \sqrt {14}\, d \,e^{2} \arctan \left (\frac {\left (10 x +2\right ) \sqrt {14}}{28}\right )}{13720000}-\frac {147 d \,e^{2} \ln \left (5 x^{2}+2 x +3\right )}{1250}-\frac {49 e^{3} x}{625}+\frac {1669047 \sqrt {14}\, e^{3} \arctan \left (\frac {\left (10 x +2\right ) \sqrt {14}}{28}\right )}{68600000}+\frac {141 e^{3} \ln \left (5 x^{2}+2 x +3\right )}{6250}+\frac {\frac {12953 d^{3}}{1568}-\frac {58599 d^{2} e}{7840}-\frac {230931 d \,e^{2}}{7840}+\frac {1275957 e^{3}}{196000}+\left (\frac {11015}{1568} d^{3}+\frac {109059}{1568} d^{2} e -\frac {388317}{7840} d \,e^{2}-\frac {621801}{39200} e^{3}\right ) x^{3}+\left (\frac {38753}{1568} d^{3}+\frac {84921}{7840} d^{2} e -\frac {640827}{7840} d \,e^{2}+\frac {1396037}{196000} e^{3}\right ) x^{2}+\left (\frac {17979}{1568} d^{3}+\frac {173283}{7840} d^{2} e -\frac {73125}{1568} d \,e^{2}-\frac {511689}{196000} e^{3}\right ) x}{25 \left (5 x^{2}+2 x +3\right )^{2}} \]

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*d*e^2*x-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*d^2*e*ln(5*x^2+2*x+3)-147/1250*d*e^2*ln(5*x^2+2*x+3)+141/6250*e^3*ln(5*x^2+2*x+
3)+339/21952*14^(1/2)*d^3*arctan(1/28*(10*x+2)*14^(1/2))-102621/2744000*14^(1/2)*d^2*e*arctan(1/28*(10*x+2)*14
^(1/2))+44451/13720000*14^(1/2)*d*e^2*arctan(1/28*(10*x+2)*14^(1/2))+1669047/68600000*14^(1/2)*e^3*arctan(1/28
*(10*x+2)*14^(1/2))

________________________________________________________________________________________

maxima [A]  time = 0.97, size = 222, normalized size = 1.30 \[ \frac {2}{125} \, e^{3} x^{2} + \frac {3}{68600000} \, \sqrt {14} {\left (353125 \, d^{3} - 855175 \, d^{2} e + 74085 \, d e^{2} + 556349 \, e^{3}\right )} \arctan \left (\frac {1}{14} \, \sqrt {14} {\left (5 \, x + 1\right )}\right ) + \frac {1}{625} \, {\left (60 \, d e^{2} - 49 \, e^{3}\right )} x + \frac {3}{6250} \, {\left (100 \, d^{2} e - 245 \, d e^{2} + 47 \, e^{3}\right )} \log \left (5 \, x^{2} + 2 \, x + 3\right ) + \frac {5 \, {\left (275375 \, d^{3} + 2726475 \, d^{2} e - 1941585 \, d e^{2} - 621801 \, e^{3}\right )} x^{3} + 1619125 \, d^{3} - 1464975 \, d^{2} e - 5773275 \, d e^{2} + 1275957 \, e^{3} + {\left (4844125 \, d^{3} + 2123025 \, d^{2} e - 16020675 \, d e^{2} + 1396037 \, e^{3}\right )} x^{2} + 3 \, {\left (749125 \, d^{3} + 1444025 \, d^{2} e - 3046875 \, d e^{2} - 170563 \, e^{3}\right )} x}{4900000 \, {\left (25 \, x^{4} + 20 \, x^{3} + 34 \, x^{2} + 12 \, x + 9\right )}} \]

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)

________________________________________________________________________________________

mupad [B]  time = 0.15, size = 299, normalized size = 1.75 \[ x\,\left (\frac {e^2\,\left (12\,d-5\,e\right )}{125}-\frac {24\,e^3}{625}\right )-\frac {\frac {1154655\,d\,e^2}{1568}+\frac {292995\,d^2\,e}{1568}+x\,\left (-\frac {449475\,d^3}{1568}-\frac {866415\,d^2\,e}{1568}+\frac {1828125\,d\,e^2}{1568}+\frac {511689\,e^3}{7840}\right )-\frac {323825\,d^3}{1568}-\frac {1275957\,e^3}{7840}+x^3\,\left (-\frac {275375\,d^3}{1568}-\frac {2726475\,d^2\,e}{1568}+\frac {1941585\,d\,e^2}{1568}+\frac {621801\,e^3}{1568}\right )-x^2\,\left (\frac {968825\,d^3}{1568}+\frac {424605\,d^2\,e}{1568}-\frac {3204135\,d\,e^2}{1568}+\frac {1396037\,e^3}{7840}\right )}{15625\,x^4+12500\,x^3+21250\,x^2+7500\,x+5625}+\ln \left (5\,x^2+2\,x+3\right )\,\left (\frac {6\,d^2\,e}{125}-\frac {147\,d\,e^2}{1250}+\frac {141\,e^3}{6250}\right )+\frac {2\,e^3\,x^2}{125}+\frac {3\,\sqrt {14}\,\mathrm {atan}\left (\frac {\frac {3\,\sqrt {14}\,\left (353125\,d^3-855175\,d^2\,e+74085\,d\,e^2+556349\,e^3\right )}{68600000}+\frac {3\,\sqrt {14}\,x\,\left (353125\,d^3-855175\,d^2\,e+74085\,d\,e^2+556349\,e^3\right )}{13720000}}{\frac {339\,d^3}{1568}-\frac {102621\,d^2\,e}{196000}+\frac {44451\,d\,e^2}{980000}+\frac {1669047\,e^3}{4900000}}\right )\,\left (353125\,d^3-855175\,d^2\,e+74085\,d\,e^2+556349\,e^3\right )}{68600000} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*((e^2*(12*d - 5*e))/125 - (24*e^3)/625) - ((1154655*d*e^2)/1568 + (292995*d^2*e)/1568 + x*((1828125*d*e^2)/1
568 - (866415*d^2*e)/1568 - (449475*d^3)/1568 + (511689*e^3)/7840) - (323825*d^3)/1568 - (1275957*e^3)/7840 +
x^3*((1941585*d*e^2)/1568 - (2726475*d^2*e)/1568 - (275375*d^3)/1568 + (621801*e^3)/1568) - x^2*((424605*d^2*e
)/1568 - (3204135*d*e^2)/1568 + (968825*d^3)/1568 + (1396037*e^3)/7840))/(7500*x + 21250*x^2 + 12500*x^3 + 156
25*x^4 + 5625) + log(2*x + 5*x^2 + 3)*((6*d^2*e)/125 - (147*d*e^2)/1250 + (141*e^3)/6250) + (2*e^3*x^2)/125 +
(3*14^(1/2)*atan(((3*14^(1/2)*(74085*d*e^2 - 855175*d^2*e + 353125*d^3 + 556349*e^3))/68600000 + (3*14^(1/2)*x
*(74085*d*e^2 - 855175*d^2*e + 353125*d^3 + 556349*e^3))/13720000)/((44451*d*e^2)/980000 - (102621*d^2*e)/1960
00 + (339*d^3)/1568 + (1669047*e^3)/4900000))*(74085*d*e^2 - 855175*d^2*e + 353125*d^3 + 556349*e^3))/68600000

________________________________________________________________________________________

sympy [C]  time = 8.08, size = 469, normalized size = 2.74 \[ \frac {2 e^{3} x^{2}}{125} + x \left (\frac {12 d e^{2}}{125} - \frac {49 e^{3}}{625}\right ) + \left (\frac {3 e \left (100 d^{2} - 245 d e + 47 e^{2}\right )}{6250} - \frac {3 \sqrt {14} i \left (353125 d^{3} - 855175 d^{2} e + 74085 d e^{2} + 556349 e^{3}\right )}{137200000}\right ) \log {\left (x + \frac {211875 d^{3} - 1830225 d^{2} e + 3271395 d e^{2} - 285237 e^{3} + \frac {65856 e \left (100 d^{2} - 245 d e + 47 e^{2}\right )}{5} - \frac {3 \sqrt {14} i \left (353125 d^{3} - 855175 d^{2} e + 74085 d e^{2} + 556349 e^{3}\right )}{5}}{1059375 d^{3} - 2565525 d^{2} e + 222255 d e^{2} + 1669047 e^{3}} \right )} + \left (\frac {3 e \left (100 d^{2} - 245 d e + 47 e^{2}\right )}{6250} + \frac {3 \sqrt {14} i \left (353125 d^{3} - 855175 d^{2} e + 74085 d e^{2} + 556349 e^{3}\right )}{137200000}\right ) \log {\left (x + \frac {211875 d^{3} - 1830225 d^{2} e + 3271395 d e^{2} - 285237 e^{3} + \frac {65856 e \left (100 d^{2} - 245 d e + 47 e^{2}\right )}{5} + \frac {3 \sqrt {14} i \left (353125 d^{3} - 855175 d^{2} e + 74085 d e^{2} + 556349 e^{3}\right )}{5}}{1059375 d^{3} - 2565525 d^{2} e + 222255 d e^{2} + 1669047 e^{3}} \right )} + \frac {1619125 d^{3} - 1464975 d^{2} e - 5773275 d e^{2} + 1275957 e^{3} + x^{3} \left (1376875 d^{3} + 13632375 d^{2} e - 9707925 d e^{2} - 3109005 e^{3}\right ) + x^{2} \left (4844125 d^{3} + 2123025 d^{2} e - 16020675 d e^{2} + 1396037 e^{3}\right ) + x \left (2247375 d^{3} + 4332075 d^{2} e - 9140625 d e^{2} - 511689 e^{3}\right )}{122500000 x^{4} + 98000000 x^{3} + 166600000 x^{2} + 58800000 x + 44100000} \]

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)

________________________________________________________________________________________