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

Optimal. Leaf size=221 \[ \frac{3}{500} e x^4 \left (100 d^2-165 d e+27 e^2\right )+\frac{x^3 \left (-2475 d^2 e+500 d^3+1215 d e^2+458 e^3\right )}{1875}-\frac{x^2 \left (-6075 d^2 e+4125 d^3-6870 d e^2+881 e^3\right )}{6250}+\frac{\left (-66075 d^2 e+57250 d^3-76620 d e^2+23431 e^3\right ) \log \left (5 x^2+2 x+3\right )}{156250}+\frac{x \left (34350 d^2 e+10125 d^3-13215 d e^2-5108 e^3\right )}{15625}-\frac{\left (449175 d^2 e+52875 d^3-274845 d e^2-53189 e^3\right ) \tan ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{78125 \sqrt{14}}+\frac{3}{125} e^2 x^5 (20 d-11 e)+\frac{2 e^3 x^6}{15} \]

[Out]

((10125*d^3 + 34350*d^2*e - 13215*d*e^2 - 5108*e^3)*x)/15625 - ((4125*d^3 - 6075*d^2*e - 6870*d*e^2 + 881*e^3)
*x^2)/6250 + ((500*d^3 - 2475*d^2*e + 1215*d*e^2 + 458*e^3)*x^3)/1875 + (3*e*(100*d^2 - 165*d*e + 27*e^2)*x^4)
/500 + (3*(20*d - 11*e)*e^2*x^5)/125 + (2*e^3*x^6)/15 - ((52875*d^3 + 449175*d^2*e - 274845*d*e^2 - 53189*e^3)
*ArcTan[(1 + 5*x)/Sqrt[14]])/(78125*Sqrt[14]) + ((57250*d^3 - 66075*d^2*e - 76620*d*e^2 + 23431*e^3)*Log[3 + 2
*x + 5*x^2])/156250

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Rubi [A]  time = 0.190129, antiderivative size = 221, 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} \[ \frac{3}{500} e x^4 \left (100 d^2-165 d e+27 e^2\right )+\frac{x^3 \left (-2475 d^2 e+500 d^3+1215 d e^2+458 e^3\right )}{1875}-\frac{x^2 \left (-6075 d^2 e+4125 d^3-6870 d e^2+881 e^3\right )}{6250}+\frac{\left (-66075 d^2 e+57250 d^3-76620 d e^2+23431 e^3\right ) \log \left (5 x^2+2 x+3\right )}{156250}+\frac{x \left (34350 d^2 e+10125 d^3-13215 d e^2-5108 e^3\right )}{15625}-\frac{\left (449175 d^2 e+52875 d^3-274845 d e^2-53189 e^3\right ) \tan ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{78125 \sqrt{14}}+\frac{3}{125} e^2 x^5 (20 d-11 e)+\frac{2 e^3 x^6}{15} \]

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),x]

[Out]

((10125*d^3 + 34350*d^2*e - 13215*d*e^2 - 5108*e^3)*x)/15625 - ((4125*d^3 - 6075*d^2*e - 6870*d*e^2 + 881*e^3)
*x^2)/6250 + ((500*d^3 - 2475*d^2*e + 1215*d*e^2 + 458*e^3)*x^3)/1875 + (3*e*(100*d^2 - 165*d*e + 27*e^2)*x^4)
/500 + (3*(20*d - 11*e)*e^2*x^5)/125 + (2*e^3*x^6)/15 - ((52875*d^3 + 449175*d^2*e - 274845*d*e^2 - 53189*e^3)
*ArcTan[(1 + 5*x)/Sqrt[14]])/(78125*Sqrt[14]) + ((57250*d^3 - 66075*d^2*e - 76620*d*e^2 + 23431*e^3)*Log[3 + 2
*x + 5*x^2])/156250

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 )}{3+2 x+5 x^2} \, dx &=\int \left (\frac{10125 d^3+34350 d^2 e-13215 d e^2-5108 e^3}{15625}-\frac{\left (4125 d^3-6075 d^2 e-6870 d e^2+881 e^3\right ) x}{3125}+\frac{1}{625} \left (500 d^3-2475 d^2 e+1215 d e^2+458 e^3\right ) x^2+\frac{3}{125} e \left (100 d^2-165 d e+27 e^2\right ) x^3+\frac{3}{25} (20 d-11 e) e^2 x^4+\frac{4 e^3 x^5}{5}+\frac{875 d^3-103050 d^2 e+39645 d e^2+15324 e^3+\left (57250 d^3-66075 d^2 e-76620 d e^2+23431 e^3\right ) x}{15625 \left (3+2 x+5 x^2\right )}\right ) \, dx\\ &=\frac{\left (10125 d^3+34350 d^2 e-13215 d e^2-5108 e^3\right ) x}{15625}-\frac{\left (4125 d^3-6075 d^2 e-6870 d e^2+881 e^3\right ) x^2}{6250}+\frac{\left (500 d^3-2475 d^2 e+1215 d e^2+458 e^3\right ) x^3}{1875}+\frac{3}{500} e \left (100 d^2-165 d e+27 e^2\right ) x^4+\frac{3}{125} (20 d-11 e) e^2 x^5+\frac{2 e^3 x^6}{15}+\frac{\int \frac{875 d^3-103050 d^2 e+39645 d e^2+15324 e^3+\left (57250 d^3-66075 d^2 e-76620 d e^2+23431 e^3\right ) x}{3+2 x+5 x^2} \, dx}{15625}\\ &=\frac{\left (10125 d^3+34350 d^2 e-13215 d e^2-5108 e^3\right ) x}{15625}-\frac{\left (4125 d^3-6075 d^2 e-6870 d e^2+881 e^3\right ) x^2}{6250}+\frac{\left (500 d^3-2475 d^2 e+1215 d e^2+458 e^3\right ) x^3}{1875}+\frac{3}{500} e \left (100 d^2-165 d e+27 e^2\right ) x^4+\frac{3}{125} (20 d-11 e) e^2 x^5+\frac{2 e^3 x^6}{15}+\frac{\left (57250 d^3-66075 d^2 e-76620 d e^2+23431 e^3\right ) \int \frac{2+10 x}{3+2 x+5 x^2} \, dx}{156250}+\frac{\left (-52875 d^3-449175 d^2 e+274845 d e^2+53189 e^3\right ) \int \frac{1}{3+2 x+5 x^2} \, dx}{78125}\\ &=\frac{\left (10125 d^3+34350 d^2 e-13215 d e^2-5108 e^3\right ) x}{15625}-\frac{\left (4125 d^3-6075 d^2 e-6870 d e^2+881 e^3\right ) x^2}{6250}+\frac{\left (500 d^3-2475 d^2 e+1215 d e^2+458 e^3\right ) x^3}{1875}+\frac{3}{500} e \left (100 d^2-165 d e+27 e^2\right ) x^4+\frac{3}{125} (20 d-11 e) e^2 x^5+\frac{2 e^3 x^6}{15}+\frac{\left (57250 d^3-66075 d^2 e-76620 d e^2+23431 e^3\right ) \log \left (3+2 x+5 x^2\right )}{156250}+\frac{\left (2 \left (52875 d^3+449175 d^2 e-274845 d e^2-53189 e^3\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-56-x^2} \, dx,x,2+10 x\right )}{78125}\\ &=\frac{\left (10125 d^3+34350 d^2 e-13215 d e^2-5108 e^3\right ) x}{15625}-\frac{\left (4125 d^3-6075 d^2 e-6870 d e^2+881 e^3\right ) x^2}{6250}+\frac{\left (500 d^3-2475 d^2 e+1215 d e^2+458 e^3\right ) x^3}{1875}+\frac{3}{500} e \left (100 d^2-165 d e+27 e^2\right ) x^4+\frac{3}{125} (20 d-11 e) e^2 x^5+\frac{2 e^3 x^6}{15}-\frac{\left (52875 d^3+449175 d^2 e-274845 d e^2-53189 e^3\right ) \tan ^{-1}\left (\frac{1+5 x}{\sqrt{14}}\right )}{78125 \sqrt{14}}+\frac{\left (57250 d^3-66075 d^2 e-76620 d e^2+23431 e^3\right ) \log \left (3+2 x+5 x^2\right )}{156250}\\ \end{align*}

Mathematica [A]  time = 0.128532, size = 178, normalized size = 0.81 \[ \frac{35 x \left (450 d^2 e \left (250 x^3-550 x^2+405 x+916\right )+250 d^3 \left (200 x^2-495 x+486\right )+45 d e^2 \left (2000 x^4-4125 x^3+2700 x^2+4580 x-3524\right )+e^3 \left (25000 x^5-49500 x^4+30375 x^3+45800 x^2-26430 x-61296\right )\right )+42 \left (-66075 d^2 e+57250 d^3-76620 d e^2+23431 e^3\right ) \log \left (5 x^2+2 x+3\right )-6 \sqrt{14} \left (449175 d^2 e+52875 d^3-274845 d e^2-53189 e^3\right ) \tan ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{6562500} \]

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),x]

[Out]

(35*x*(250*d^3*(486 - 495*x + 200*x^2) + 450*d^2*e*(916 + 405*x - 550*x^2 + 250*x^3) + 45*d*e^2*(-3524 + 4580*
x + 2700*x^2 - 4125*x^3 + 2000*x^4) + e^3*(-61296 - 26430*x + 45800*x^2 + 30375*x^3 - 49500*x^4 + 25000*x^5))
- 6*Sqrt[14]*(52875*d^3 + 449175*d^2*e - 274845*d*e^2 - 53189*e^3)*ArcTan[(1 + 5*x)/Sqrt[14]] + 42*(57250*d^3
- 66075*d^2*e - 76620*d*e^2 + 23431*e^3)*Log[3 + 2*x + 5*x^2])/6562500

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Maple [A]  time = 0.051, size = 291, normalized size = 1.3 \begin{align*} -{\frac{17967\,\sqrt{14}{d}^{2}e}{43750}\arctan \left ({\frac{ \left ( 10\,x+2 \right ) \sqrt{14}}{28}} \right ) }+{\frac{54969\,\sqrt{14}d{e}^{2}}{218750}\arctan \left ({\frac{ \left ( 10\,x+2 \right ) \sqrt{14}}{28}} \right ) }+{\frac{81\,{d}^{3}x}{125}}+{\frac{23431\,\ln \left ( 5\,{x}^{2}+2\,x+3 \right ){e}^{3}}{156250}}+{\frac{229\,\ln \left ( 5\,{x}^{2}+2\,x+3 \right ){d}^{3}}{625}}+{\frac{458\,{x}^{3}{e}^{3}}{1875}}-{\frac{33\,{x}^{2}{d}^{3}}{50}}-{\frac{33\,{x}^{5}{e}^{3}}{125}}+{\frac{4\,{x}^{3}{d}^{3}}{15}}-{\frac{99\,d{e}^{2}{x}^{4}}{100}}+{\frac{687\,{x}^{2}{e}^{2}d}{625}}-{\frac{7662\,\ln \left ( 5\,{x}^{2}+2\,x+3 \right ) d{e}^{2}}{15625}}-{\frac{2643\,\ln \left ( 5\,{x}^{2}+2\,x+3 \right ){d}^{2}e}{6250}}+{\frac{53189\,\sqrt{14}{e}^{3}}{1093750}\arctan \left ({\frac{ \left ( 10\,x+2 \right ) \sqrt{14}}{28}} \right ) }-{\frac{423\,\sqrt{14}{d}^{3}}{8750}\arctan \left ({\frac{ \left ( 10\,x+2 \right ) \sqrt{14}}{28}} \right ) }-{\frac{33\,{x}^{3}{d}^{2}e}{25}}+{\frac{12\,{x}^{5}d{e}^{2}}{25}}+{\frac{81\,{x}^{3}{e}^{2}d}{125}}-{\frac{2643\,xd{e}^{2}}{3125}}+{\frac{3\,{x}^{4}{d}^{2}e}{5}}+{\frac{1374\,x{d}^{2}e}{625}}+{\frac{243\,{x}^{2}{d}^{2}e}{250}}-{\frac{881\,{e}^{3}{x}^{2}}{6250}}+{\frac{81\,{e}^{3}{x}^{4}}{500}}+{\frac{2\,{e}^{3}{x}^{6}}{15}}-{\frac{5108\,{e}^{3}x}{15625}} \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),x)

[Out]

-17967/43750*14^(1/2)*arctan(1/28*(10*x+2)*14^(1/2))*d^2*e+54969/218750*14^(1/2)*arctan(1/28*(10*x+2)*14^(1/2)
)*d*e^2+81/125*d^3*x+23431/156250*ln(5*x^2+2*x+3)*e^3+229/625*ln(5*x^2+2*x+3)*d^3+458/1875*x^3*e^3-33/50*x^2*d
^3-33/125*x^5*e^3+4/15*x^3*d^3-99/100*d*e^2*x^4+687/625*x^2*e^2*d-7662/15625*ln(5*x^2+2*x+3)*d*e^2-2643/6250*l
n(5*x^2+2*x+3)*d^2*e+53189/1093750*14^(1/2)*arctan(1/28*(10*x+2)*14^(1/2))*e^3-423/8750*14^(1/2)*arctan(1/28*(
10*x+2)*14^(1/2))*d^3-33/25*x^3*d^2*e+12/25*x^5*d*e^2+81/125*x^3*e^2*d-2643/3125*x*d*e^2+3/5*x^4*d^2*e+1374/62
5*x*d^2*e+243/250*x^2*d^2*e-881/6250*e^3*x^2+81/500*e^3*x^4+2/15*e^3*x^6-5108/15625*e^3*x

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Maxima [A]  time = 1.49614, size = 278, normalized size = 1.26 \begin{align*} \frac{2}{15} \, e^{3} x^{6} + \frac{3}{125} \,{\left (20 \, d e^{2} - 11 \, e^{3}\right )} x^{5} + \frac{3}{500} \,{\left (100 \, d^{2} e - 165 \, d e^{2} + 27 \, e^{3}\right )} x^{4} + \frac{1}{1875} \,{\left (500 \, d^{3} - 2475 \, d^{2} e + 1215 \, d e^{2} + 458 \, e^{3}\right )} x^{3} - \frac{1}{6250} \,{\left (4125 \, d^{3} - 6075 \, d^{2} e - 6870 \, d e^{2} + 881 \, e^{3}\right )} x^{2} - \frac{1}{1093750} \, \sqrt{14}{\left (52875 \, d^{3} + 449175 \, d^{2} e - 274845 \, d e^{2} - 53189 \, e^{3}\right )} \arctan \left (\frac{1}{14} \, \sqrt{14}{\left (5 \, x + 1\right )}\right ) + \frac{1}{15625} \,{\left (10125 \, d^{3} + 34350 \, d^{2} e - 13215 \, d e^{2} - 5108 \, e^{3}\right )} x + \frac{1}{156250} \,{\left (57250 \, d^{3} - 66075 \, d^{2} e - 76620 \, d e^{2} + 23431 \, e^{3}\right )} \log \left (5 \, x^{2} + 2 \, x + 3\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),x, algorithm="maxima")

[Out]

2/15*e^3*x^6 + 3/125*(20*d*e^2 - 11*e^3)*x^5 + 3/500*(100*d^2*e - 165*d*e^2 + 27*e^3)*x^4 + 1/1875*(500*d^3 -
2475*d^2*e + 1215*d*e^2 + 458*e^3)*x^3 - 1/6250*(4125*d^3 - 6075*d^2*e - 6870*d*e^2 + 881*e^3)*x^2 - 1/1093750
*sqrt(14)*(52875*d^3 + 449175*d^2*e - 274845*d*e^2 - 53189*e^3)*arctan(1/14*sqrt(14)*(5*x + 1)) + 1/15625*(101
25*d^3 + 34350*d^2*e - 13215*d*e^2 - 5108*e^3)*x + 1/156250*(57250*d^3 - 66075*d^2*e - 76620*d*e^2 + 23431*e^3
)*log(5*x^2 + 2*x + 3)

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Fricas [A]  time = 1.00415, size = 626, normalized size = 2.83 \begin{align*} \frac{2}{15} \, e^{3} x^{6} + \frac{3}{125} \,{\left (20 \, d e^{2} - 11 \, e^{3}\right )} x^{5} + \frac{3}{500} \,{\left (100 \, d^{2} e - 165 \, d e^{2} + 27 \, e^{3}\right )} x^{4} + \frac{1}{1875} \,{\left (500 \, d^{3} - 2475 \, d^{2} e + 1215 \, d e^{2} + 458 \, e^{3}\right )} x^{3} - \frac{1}{6250} \,{\left (4125 \, d^{3} - 6075 \, d^{2} e - 6870 \, d e^{2} + 881 \, e^{3}\right )} x^{2} - \frac{1}{1093750} \, \sqrt{14}{\left (52875 \, d^{3} + 449175 \, d^{2} e - 274845 \, d e^{2} - 53189 \, e^{3}\right )} \arctan \left (\frac{1}{14} \, \sqrt{14}{\left (5 \, x + 1\right )}\right ) + \frac{1}{15625} \,{\left (10125 \, d^{3} + 34350 \, d^{2} e - 13215 \, d e^{2} - 5108 \, e^{3}\right )} x + \frac{1}{156250} \,{\left (57250 \, d^{3} - 66075 \, d^{2} e - 76620 \, d e^{2} + 23431 \, e^{3}\right )} \log \left (5 \, x^{2} + 2 \, x + 3\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),x, algorithm="fricas")

[Out]

2/15*e^3*x^6 + 3/125*(20*d*e^2 - 11*e^3)*x^5 + 3/500*(100*d^2*e - 165*d*e^2 + 27*e^3)*x^4 + 1/1875*(500*d^3 -
2475*d^2*e + 1215*d*e^2 + 458*e^3)*x^3 - 1/6250*(4125*d^3 - 6075*d^2*e - 6870*d*e^2 + 881*e^3)*x^2 - 1/1093750
*sqrt(14)*(52875*d^3 + 449175*d^2*e - 274845*d*e^2 - 53189*e^3)*arctan(1/14*sqrt(14)*(5*x + 1)) + 1/15625*(101
25*d^3 + 34350*d^2*e - 13215*d*e^2 - 5108*e^3)*x + 1/156250*(57250*d^3 - 66075*d^2*e - 76620*d*e^2 + 23431*e^3
)*log(5*x^2 + 2*x + 3)

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Sympy [C]  time = 1.24721, size = 450, normalized size = 2.04 \begin{align*} \frac{2 e^{3} x^{6}}{15} + x^{5} \left (\frac{12 d e^{2}}{25} - \frac{33 e^{3}}{125}\right ) + x^{4} \left (\frac{3 d^{2} e}{5} - \frac{99 d e^{2}}{100} + \frac{81 e^{3}}{500}\right ) + x^{3} \left (\frac{4 d^{3}}{15} - \frac{33 d^{2} e}{25} + \frac{81 d e^{2}}{125} + \frac{458 e^{3}}{1875}\right ) + x^{2} \left (- \frac{33 d^{3}}{50} + \frac{243 d^{2} e}{250} + \frac{687 d e^{2}}{625} - \frac{881 e^{3}}{6250}\right ) + x \left (\frac{81 d^{3}}{125} + \frac{1374 d^{2} e}{625} - \frac{2643 d e^{2}}{3125} - \frac{5108 e^{3}}{15625}\right ) + \left (\frac{229 d^{3}}{625} - \frac{2643 d^{2} e}{6250} - \frac{7662 d e^{2}}{15625} + \frac{23431 e^{3}}{156250} - \frac{\sqrt{14} i \left (52875 d^{3} + 449175 d^{2} e - 274845 d e^{2} - 53189 e^{3}\right )}{2187500}\right ) \log{\left (x + \frac{10575 d^{3} + 89835 d^{2} e - 54969 d e^{2} - \frac{53189 e^{3}}{5} + \frac{\sqrt{14} i \left (52875 d^{3} + 449175 d^{2} e - 274845 d e^{2} - 53189 e^{3}\right )}{5}}{52875 d^{3} + 449175 d^{2} e - 274845 d e^{2} - 53189 e^{3}} \right )} + \left (\frac{229 d^{3}}{625} - \frac{2643 d^{2} e}{6250} - \frac{7662 d e^{2}}{15625} + \frac{23431 e^{3}}{156250} + \frac{\sqrt{14} i \left (52875 d^{3} + 449175 d^{2} e - 274845 d e^{2} - 53189 e^{3}\right )}{2187500}\right ) \log{\left (x + \frac{10575 d^{3} + 89835 d^{2} e - 54969 d e^{2} - \frac{53189 e^{3}}{5} - \frac{\sqrt{14} i \left (52875 d^{3} + 449175 d^{2} e - 274845 d e^{2} - 53189 e^{3}\right )}{5}}{52875 d^{3} + 449175 d^{2} e - 274845 d e^{2} - 53189 e^{3}} \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),x)

[Out]

2*e**3*x**6/15 + x**5*(12*d*e**2/25 - 33*e**3/125) + x**4*(3*d**2*e/5 - 99*d*e**2/100 + 81*e**3/500) + x**3*(4
*d**3/15 - 33*d**2*e/25 + 81*d*e**2/125 + 458*e**3/1875) + x**2*(-33*d**3/50 + 243*d**2*e/250 + 687*d*e**2/625
- 881*e**3/6250) + x*(81*d**3/125 + 1374*d**2*e/625 - 2643*d*e**2/3125 - 5108*e**3/15625) + (229*d**3/625 - 2
643*d**2*e/6250 - 7662*d*e**2/15625 + 23431*e**3/156250 - sqrt(14)*I*(52875*d**3 + 449175*d**2*e - 274845*d*e*
*2 - 53189*e**3)/2187500)*log(x + (10575*d**3 + 89835*d**2*e - 54969*d*e**2 - 53189*e**3/5 + sqrt(14)*I*(52875
*d**3 + 449175*d**2*e - 274845*d*e**2 - 53189*e**3)/5)/(52875*d**3 + 449175*d**2*e - 274845*d*e**2 - 53189*e**
3)) + (229*d**3/625 - 2643*d**2*e/6250 - 7662*d*e**2/15625 + 23431*e**3/156250 + sqrt(14)*I*(52875*d**3 + 4491
75*d**2*e - 274845*d*e**2 - 53189*e**3)/2187500)*log(x + (10575*d**3 + 89835*d**2*e - 54969*d*e**2 - 53189*e**
3/5 - sqrt(14)*I*(52875*d**3 + 449175*d**2*e - 274845*d*e**2 - 53189*e**3)/5)/(52875*d**3 + 449175*d**2*e - 27
4845*d*e**2 - 53189*e**3))

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Giac [A]  time = 1.1473, size = 286, normalized size = 1.29 \begin{align*} \frac{2}{15} \, x^{6} e^{3} + \frac{12}{25} \, d x^{5} e^{2} + \frac{3}{5} \, d^{2} x^{4} e + \frac{4}{15} \, d^{3} x^{3} - \frac{33}{125} \, x^{5} e^{3} - \frac{99}{100} \, d x^{4} e^{2} - \frac{33}{25} \, d^{2} x^{3} e - \frac{33}{50} \, d^{3} x^{2} + \frac{81}{500} \, x^{4} e^{3} + \frac{81}{125} \, d x^{3} e^{2} + \frac{243}{250} \, d^{2} x^{2} e + \frac{81}{125} \, d^{3} x + \frac{458}{1875} \, x^{3} e^{3} + \frac{687}{625} \, d x^{2} e^{2} + \frac{1374}{625} \, d^{2} x e - \frac{881}{6250} \, x^{2} e^{3} - \frac{2643}{3125} \, d x e^{2} - \frac{1}{1093750} \, \sqrt{14}{\left (52875 \, d^{3} + 449175 \, d^{2} e - 274845 \, d e^{2} - 53189 \, e^{3}\right )} \arctan \left (\frac{1}{14} \, \sqrt{14}{\left (5 \, x + 1\right )}\right ) - \frac{5108}{15625} \, x e^{3} + \frac{1}{156250} \,{\left (57250 \, d^{3} - 66075 \, d^{2} e - 76620 \, d e^{2} + 23431 \, e^{3}\right )} \log \left (5 \, x^{2} + 2 \, x + 3\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),x, algorithm="giac")

[Out]

2/15*x^6*e^3 + 12/25*d*x^5*e^2 + 3/5*d^2*x^4*e + 4/15*d^3*x^3 - 33/125*x^5*e^3 - 99/100*d*x^4*e^2 - 33/25*d^2*
x^3*e - 33/50*d^3*x^2 + 81/500*x^4*e^3 + 81/125*d*x^3*e^2 + 243/250*d^2*x^2*e + 81/125*d^3*x + 458/1875*x^3*e^
3 + 687/625*d*x^2*e^2 + 1374/625*d^2*x*e - 881/6250*x^2*e^3 - 2643/3125*d*x*e^2 - 1/1093750*sqrt(14)*(52875*d^
3 + 449175*d^2*e - 274845*d*e^2 - 53189*e^3)*arctan(1/14*sqrt(14)*(5*x + 1)) - 5108/15625*x*e^3 + 1/156250*(57
250*d^3 - 66075*d^2*e - 76620*d*e^2 + 23431*e^3)*log(5*x^2 + 2*x + 3)