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

Optimal. Leaf size=189 \[ \frac{e x^2 \left (840 d^2-1722 d e+373 e^2\right )}{3500}-\frac{\left (-1545 d^2 e+1025 d^3-2601 d e^2+832 e^3\right ) \log \left (5 x^2+2 x+3\right )}{6250}+\frac{x \left (-17220 d^2 e+2800 d^3+9921 d e^2+6053 e^3\right )}{17500}+\frac{\left (317565 d^2 e+32825 d^3-221643 d e^2-67499 e^3\right ) \tan ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{87500 \sqrt{14}}+\frac{1}{375} e^2 x^3 (60 d-41 e)-\frac{(423 x+1367) (d+e x)^3}{3500 \left (5 x^2+2 x+3\right )}+\frac{e^3 x^4}{25} \]

[Out]

((2800*d^3 - 17220*d^2*e + 9921*d*e^2 + 6053*e^3)*x)/17500 + (e*(840*d^2 - 1722*d*e + 373*e^2)*x^2)/3500 + ((6
0*d - 41*e)*e^2*x^3)/375 + (e^3*x^4)/25 - ((1367 + 423*x)*(d + e*x)^3)/(3500*(3 + 2*x + 5*x^2)) + ((32825*d^3
+ 317565*d^2*e - 221643*d*e^2 - 67499*e^3)*ArcTan[(1 + 5*x)/Sqrt[14]])/(87500*Sqrt[14]) - ((1025*d^3 - 1545*d^
2*e - 2601*d*e^2 + 832*e^3)*Log[3 + 2*x + 5*x^2])/6250

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Rubi [A]  time = 0.255334, antiderivative size = 189, normalized size of antiderivative = 1., number of steps used = 7, 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} \[ \frac{e x^2 \left (840 d^2-1722 d e+373 e^2\right )}{3500}-\frac{\left (-1545 d^2 e+1025 d^3-2601 d e^2+832 e^3\right ) \log \left (5 x^2+2 x+3\right )}{6250}+\frac{x \left (-17220 d^2 e+2800 d^3+9921 d e^2+6053 e^3\right )}{17500}+\frac{\left (317565 d^2 e+32825 d^3-221643 d e^2-67499 e^3\right ) \tan ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{87500 \sqrt{14}}+\frac{1}{375} e^2 x^3 (60 d-41 e)-\frac{(423 x+1367) (d+e x)^3}{3500 \left (5 x^2+2 x+3\right )}+\frac{e^3 x^4}{25} \]

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)^2,x]

[Out]

((2800*d^3 - 17220*d^2*e + 9921*d*e^2 + 6053*e^3)*x)/17500 + (e*(840*d^2 - 1722*d*e + 373*e^2)*x^2)/3500 + ((6
0*d - 41*e)*e^2*x^3)/375 + (e^3*x^4)/25 - ((1367 + 423*x)*(d + e*x)^3)/(3500*(3 + 2*x + 5*x^2)) + ((32825*d^3
+ 317565*d^2*e - 221643*d*e^2 - 67499*e^3)*ArcTan[(1 + 5*x)/Sqrt[14]])/(87500*Sqrt[14]) - ((1025*d^3 - 1545*d^
2*e - 2601*d*e^2 + 832*e^3)*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 )^2} \, dx &=-\frac{(1367+423 x) (d+e x)^3}{3500 \left (3+2 x+5 x^2\right )}+\frac{1}{56} \int \frac{(d+e x)^2 \left (\frac{6}{125} (615 d+1367 e)-\frac{12}{125} (770 d-519 e) x+\frac{56}{25} (20 d-33 e) x^2+\frac{224 e x^3}{5}\right )}{3+2 x+5 x^2} \, dx\\ &=-\frac{(1367+423 x) (d+e x)^3}{3500 \left (3+2 x+5 x^2\right )}+\frac{1}{56} \int \left (\frac{2}{625} \left (2800 d^3-17220 d^2 e+9921 d e^2+6053 e^3\right )+\frac{4}{125} e \left (840 d^2-1722 d e+373 e^2\right ) x+\frac{56}{125} (60 d-41 e) e^2 x^2+\frac{224 e^3 x^3}{25}+\frac{2 \left (3 \left (275 d^3+24055 d^2 e-9921 d e^2-6053 e^3\right )-28 \left (1025 d^3-1545 d^2 e-2601 d e^2+832 e^3\right ) x\right )}{625 \left (3+2 x+5 x^2\right )}\right ) \, dx\\ &=\frac{\left (2800 d^3-17220 d^2 e+9921 d e^2+6053 e^3\right ) x}{17500}+\frac{e \left (840 d^2-1722 d e+373 e^2\right ) x^2}{3500}+\frac{1}{375} (60 d-41 e) e^2 x^3+\frac{e^3 x^4}{25}-\frac{(1367+423 x) (d+e x)^3}{3500 \left (3+2 x+5 x^2\right )}+\frac{\int \frac{3 \left (275 d^3+24055 d^2 e-9921 d e^2-6053 e^3\right )-28 \left (1025 d^3-1545 d^2 e-2601 d e^2+832 e^3\right ) x}{3+2 x+5 x^2} \, dx}{17500}\\ &=\frac{\left (2800 d^3-17220 d^2 e+9921 d e^2+6053 e^3\right ) x}{17500}+\frac{e \left (840 d^2-1722 d e+373 e^2\right ) x^2}{3500}+\frac{1}{375} (60 d-41 e) e^2 x^3+\frac{e^3 x^4}{25}-\frac{(1367+423 x) (d+e x)^3}{3500 \left (3+2 x+5 x^2\right )}+\frac{\left (32825 d^3+317565 d^2 e-221643 d e^2-67499 e^3\right ) \int \frac{1}{3+2 x+5 x^2} \, dx}{87500}+\frac{\left (-1025 d^3+1545 d^2 e+2601 d e^2-832 e^3\right ) \int \frac{2+10 x}{3+2 x+5 x^2} \, dx}{6250}\\ &=\frac{\left (2800 d^3-17220 d^2 e+9921 d e^2+6053 e^3\right ) x}{17500}+\frac{e \left (840 d^2-1722 d e+373 e^2\right ) x^2}{3500}+\frac{1}{375} (60 d-41 e) e^2 x^3+\frac{e^3 x^4}{25}-\frac{(1367+423 x) (d+e x)^3}{3500 \left (3+2 x+5 x^2\right )}-\frac{\left (1025 d^3-1545 d^2 e-2601 d e^2+832 e^3\right ) \log \left (3+2 x+5 x^2\right )}{6250}+\frac{\left (-32825 d^3-317565 d^2 e+221643 d e^2+67499 e^3\right ) \operatorname{Subst}\left (\int \frac{1}{-56-x^2} \, dx,x,2+10 x\right )}{43750}\\ &=\frac{\left (2800 d^3-17220 d^2 e+9921 d e^2+6053 e^3\right ) x}{17500}+\frac{e \left (840 d^2-1722 d e+373 e^2\right ) x^2}{3500}+\frac{1}{375} (60 d-41 e) e^2 x^3+\frac{e^3 x^4}{25}-\frac{(1367+423 x) (d+e x)^3}{3500 \left (3+2 x+5 x^2\right )}+\frac{\left (32825 d^3+317565 d^2 e-221643 d e^2-67499 e^3\right ) \tan ^{-1}\left (\frac{1+5 x}{\sqrt{14}}\right )}{87500 \sqrt{14}}-\frac{\left (1025 d^3-1545 d^2 e-2601 d e^2+832 e^3\right ) \log \left (3+2 x+5 x^2\right )}{6250}\\ \end{align*}

Mathematica [A]  time = 0.155091, size = 209, normalized size = 1.11 \[ \frac{14700 e x^2 \left (300 d^2-615 d e+103 e^2\right )-\frac{42 \left (75 d^2 e (5989 x-1269)+125 d^3 (423 x+1367)-15 d e^2 (18323 x+17967)+e^3 (54969-53189 x)\right )}{5 x^2+2 x+3}+2940 \left (1545 d^2 e-1025 d^3+2601 d e^2-832 e^3\right ) \log \left (5 x^2+2 x+3\right )+5880 x \left (-3075 d^2 e+500 d^3+1545 d e^2+867 e^3\right )+15 \sqrt{14} \left (317565 d^2 e+32825 d^3-221643 d e^2-67499 e^3\right ) \tan ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )+49000 e^2 x^3 (60 d-41 e)+735000 e^3 x^4}{18375000} \]

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)^2,x]

[Out]

(5880*(500*d^3 - 3075*d^2*e + 1545*d*e^2 + 867*e^3)*x + 14700*e*(300*d^2 - 615*d*e + 103*e^2)*x^2 + 49000*(60*
d - 41*e)*e^2*x^3 + 735000*e^3*x^4 - (42*(e^3*(54969 - 53189*x) + 125*d^3*(1367 + 423*x) + 75*d^2*e*(-1269 + 5
989*x) - 15*d*e^2*(17967 + 18323*x)))/(3 + 2*x + 5*x^2) + 15*Sqrt[14]*(32825*d^3 + 317565*d^2*e - 221643*d*e^2
- 67499*e^3)*ArcTan[(1 + 5*x)/Sqrt[14]] + 2940*(-1025*d^3 + 1545*d^2*e + 2601*d*e^2 - 832*e^3)*Log[3 + 2*x +
5*x^2])/18375000

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Maple [A]  time = 0.055, size = 283, normalized size = 1.5 \begin{align*}{\frac{{e}^{3}{x}^{4}}{25}}+{\frac{4\,{x}^{3}{e}^{2}d}{25}}-{\frac{41\,{x}^{3}{e}^{3}}{375}}+{\frac{6\,{x}^{2}{d}^{2}e}{25}}-{\frac{123\,{x}^{2}{e}^{2}d}{250}}+{\frac{103\,{e}^{3}{x}^{2}}{1250}}+{\frac{4\,{d}^{3}x}{25}}-{\frac{123\,x{d}^{2}e}{125}}+{\frac{309\,xd{e}^{2}}{625}}+{\frac{867\,{e}^{3}x}{3125}}-{\frac{1}{3125} \left ( \left ({\frac{2115\,{d}^{3}}{28}}+{\frac{17967\,{d}^{2}e}{28}}-{\frac{54969\,d{e}^{2}}{140}}-{\frac{53189\,{e}^{3}}{700}} \right ) x+{\frac{6835\,{d}^{3}}{28}}-{\frac{3807\,{d}^{2}e}{28}}-{\frac{53901\,d{e}^{2}}{140}}+{\frac{54969\,{e}^{3}}{700}} \right ) \left ({x}^{2}+{\frac{2\,x}{5}}+{\frac{3}{5}} \right ) ^{-1}}-{\frac{41\,\ln \left ( 5\,{x}^{2}+2\,x+3 \right ){d}^{3}}{250}}+{\frac{309\,\ln \left ( 5\,{x}^{2}+2\,x+3 \right ){d}^{2}e}{1250}}+{\frac{2601\,\ln \left ( 5\,{x}^{2}+2\,x+3 \right ) d{e}^{2}}{6250}}-{\frac{416\,\ln \left ( 5\,{x}^{2}+2\,x+3 \right ){e}^{3}}{3125}}+{\frac{1313\,\sqrt{14}{d}^{3}}{49000}\arctan \left ({\frac{ \left ( 10\,x+2 \right ) \sqrt{14}}{28}} \right ) }+{\frac{63513\,\sqrt{14}{d}^{2}e}{245000}\arctan \left ({\frac{ \left ( 10\,x+2 \right ) \sqrt{14}}{28}} \right ) }-{\frac{221643\,\sqrt{14}d{e}^{2}}{1225000}\arctan \left ({\frac{ \left ( 10\,x+2 \right ) \sqrt{14}}{28}} \right ) }-{\frac{67499\,\sqrt{14}{e}^{3}}{1225000}\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)^2,x)

[Out]

1/25*e^3*x^4+4/25*x^3*e^2*d-41/375*x^3*e^3+6/25*x^2*d^2*e-123/250*x^2*e^2*d+103/1250*e^3*x^2+4/25*d^3*x-123/12
5*x*d^2*e+309/625*x*d*e^2+867/3125*e^3*x-1/3125*((2115/28*d^3+17967/28*d^2*e-54969/140*d*e^2-53189/700*e^3)*x+
6835/28*d^3-3807/28*d^2*e-53901/140*d*e^2+54969/700*e^3)/(x^2+2/5*x+3/5)-41/250*ln(5*x^2+2*x+3)*d^3+309/1250*l
n(5*x^2+2*x+3)*d^2*e+2601/6250*ln(5*x^2+2*x+3)*d*e^2-416/3125*ln(5*x^2+2*x+3)*e^3+1313/49000*14^(1/2)*arctan(1
/28*(10*x+2)*14^(1/2))*d^3+63513/245000*14^(1/2)*arctan(1/28*(10*x+2)*14^(1/2))*d^2*e-221643/1225000*14^(1/2)*
arctan(1/28*(10*x+2)*14^(1/2))*d*e^2-67499/1225000*14^(1/2)*arctan(1/28*(10*x+2)*14^(1/2))*e^3

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Maxima [A]  time = 1.6389, size = 286, normalized size = 1.51 \begin{align*} \frac{1}{25} \, e^{3} x^{4} + \frac{1}{375} \,{\left (60 \, d e^{2} - 41 \, e^{3}\right )} x^{3} + \frac{1}{1250} \,{\left (300 \, d^{2} e - 615 \, d e^{2} + 103 \, e^{3}\right )} x^{2} + \frac{1}{1225000} \, \sqrt{14}{\left (32825 \, d^{3} + 317565 \, d^{2} e - 221643 \, d e^{2} - 67499 \, e^{3}\right )} \arctan \left (\frac{1}{14} \, \sqrt{14}{\left (5 \, x + 1\right )}\right ) + \frac{1}{3125} \,{\left (500 \, d^{3} - 3075 \, d^{2} e + 1545 \, d e^{2} + 867 \, e^{3}\right )} x - \frac{1}{6250} \,{\left (1025 \, d^{3} - 1545 \, d^{2} e - 2601 \, d e^{2} + 832 \, e^{3}\right )} \log \left (5 \, x^{2} + 2 \, x + 3\right ) - \frac{170875 \, d^{3} - 95175 \, d^{2} e - 269505 \, d e^{2} + 54969 \, e^{3} +{\left (52875 \, d^{3} + 449175 \, d^{2} e - 274845 \, d e^{2} - 53189 \, e^{3}\right )} x}{437500 \,{\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)^2,x, algorithm="maxima")

[Out]

1/25*e^3*x^4 + 1/375*(60*d*e^2 - 41*e^3)*x^3 + 1/1250*(300*d^2*e - 615*d*e^2 + 103*e^3)*x^2 + 1/1225000*sqrt(1
4)*(32825*d^3 + 317565*d^2*e - 221643*d*e^2 - 67499*e^3)*arctan(1/14*sqrt(14)*(5*x + 1)) + 1/3125*(500*d^3 - 3
075*d^2*e + 1545*d*e^2 + 867*e^3)*x - 1/6250*(1025*d^3 - 1545*d^2*e - 2601*d*e^2 + 832*e^3)*log(5*x^2 + 2*x +
3) - 1/437500*(170875*d^3 - 95175*d^2*e - 269505*d*e^2 + 54969*e^3 + (52875*d^3 + 449175*d^2*e - 274845*d*e^2
- 53189*e^3)*x)/(5*x^2 + 2*x + 3)

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Fricas [B]  time = 1.28878, size = 1046, normalized size = 5.53 \begin{align*} \frac{3675000 \, e^{3} x^{6} + 1225000 \,{\left (12 \, d e^{2} - 7 \, e^{3}\right )} x^{5} + 122500 \,{\left (180 \, d^{2} e - 321 \, d e^{2} + 47 \, e^{3}\right )} x^{4} + 147000 \,{\left (100 \, d^{3} - 555 \, d^{2} e + 246 \, d e^{2} + 153 \, e^{3}\right )} x^{3} - 7176750 \, d^{3} + 3997350 \, d^{2} e + 11319210 \, d e^{2} - 2308698 \, e^{3} + 2940 \,{\left (2000 \, d^{3} - 7800 \, d^{2} e - 3045 \, d e^{2} + 5013 \, e^{3}\right )} x^{2} + 15 \, \sqrt{14}{\left (98475 \, d^{3} + 952695 \, d^{2} e - 664929 \, d e^{2} - 202497 \, e^{3} + 5 \,{\left (32825 \, d^{3} + 317565 \, d^{2} e - 221643 \, d e^{2} - 67499 \, e^{3}\right )} x^{2} + 2 \,{\left (32825 \, d^{3} + 317565 \, d^{2} e - 221643 \, d e^{2} - 67499 \, e^{3}\right )} x\right )} \arctan \left (\frac{1}{14} \, \sqrt{14}{\left (5 \, x + 1\right )}\right ) + 42 \,{\left (157125 \, d^{3} - 1740675 \, d^{2} e + 923745 \, d e^{2} + 417329 \, e^{3}\right )} x - 2940 \,{\left (3075 \, d^{3} - 4635 \, d^{2} e - 7803 \, d e^{2} + 2496 \, e^{3} + 5 \,{\left (1025 \, d^{3} - 1545 \, d^{2} e - 2601 \, d e^{2} + 832 \, e^{3}\right )} x^{2} + 2 \,{\left (1025 \, d^{3} - 1545 \, d^{2} e - 2601 \, d e^{2} + 832 \, e^{3}\right )} x\right )} \log \left (5 \, x^{2} + 2 \, x + 3\right )}{18375000 \,{\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)^2,x, algorithm="fricas")

[Out]

1/18375000*(3675000*e^3*x^6 + 1225000*(12*d*e^2 - 7*e^3)*x^5 + 122500*(180*d^2*e - 321*d*e^2 + 47*e^3)*x^4 + 1
47000*(100*d^3 - 555*d^2*e + 246*d*e^2 + 153*e^3)*x^3 - 7176750*d^3 + 3997350*d^2*e + 11319210*d*e^2 - 2308698
*e^3 + 2940*(2000*d^3 - 7800*d^2*e - 3045*d*e^2 + 5013*e^3)*x^2 + 15*sqrt(14)*(98475*d^3 + 952695*d^2*e - 6649
29*d*e^2 - 202497*e^3 + 5*(32825*d^3 + 317565*d^2*e - 221643*d*e^2 - 67499*e^3)*x^2 + 2*(32825*d^3 + 317565*d^
2*e - 221643*d*e^2 - 67499*e^3)*x)*arctan(1/14*sqrt(14)*(5*x + 1)) + 42*(157125*d^3 - 1740675*d^2*e + 923745*d
*e^2 + 417329*e^3)*x - 2940*(3075*d^3 - 4635*d^2*e - 7803*d*e^2 + 2496*e^3 + 5*(1025*d^3 - 1545*d^2*e - 2601*d
*e^2 + 832*e^3)*x^2 + 2*(1025*d^3 - 1545*d^2*e - 2601*d*e^2 + 832*e^3)*x)*log(5*x^2 + 2*x + 3))/(5*x^2 + 2*x +
3)

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Sympy [C]  time = 2.55615, size = 444, normalized size = 2.35 \begin{align*} \frac{e^{3} x^{4}}{25} + x^{3} \left (\frac{4 d e^{2}}{25} - \frac{41 e^{3}}{375}\right ) + x^{2} \left (\frac{6 d^{2} e}{25} - \frac{123 d e^{2}}{250} + \frac{103 e^{3}}{1250}\right ) + x \left (\frac{4 d^{3}}{25} - \frac{123 d^{2} e}{125} + \frac{309 d e^{2}}{625} + \frac{867 e^{3}}{3125}\right ) + \left (- \frac{41 d^{3}}{250} + \frac{309 d^{2} e}{1250} + \frac{2601 d e^{2}}{6250} - \frac{416 e^{3}}{3125} - \frac{\sqrt{14} i \left (32825 d^{3} + 317565 d^{2} e - 221643 d e^{2} - 67499 e^{3}\right )}{2450000}\right ) \log{\left (x + \frac{6565 d^{3} + 63513 d^{2} e - \frac{221643 d e^{2}}{5} - \frac{67499 e^{3}}{5} - \frac{\sqrt{14} i \left (32825 d^{3} + 317565 d^{2} e - 221643 d e^{2} - 67499 e^{3}\right )}{5}}{32825 d^{3} + 317565 d^{2} e - 221643 d e^{2} - 67499 e^{3}} \right )} + \left (- \frac{41 d^{3}}{250} + \frac{309 d^{2} e}{1250} + \frac{2601 d e^{2}}{6250} - \frac{416 e^{3}}{3125} + \frac{\sqrt{14} i \left (32825 d^{3} + 317565 d^{2} e - 221643 d e^{2} - 67499 e^{3}\right )}{2450000}\right ) \log{\left (x + \frac{6565 d^{3} + 63513 d^{2} e - \frac{221643 d e^{2}}{5} - \frac{67499 e^{3}}{5} + \frac{\sqrt{14} i \left (32825 d^{3} + 317565 d^{2} e - 221643 d e^{2} - 67499 e^{3}\right )}{5}}{32825 d^{3} + 317565 d^{2} e - 221643 d e^{2} - 67499 e^{3}} \right )} - \frac{170875 d^{3} - 95175 d^{2} e - 269505 d e^{2} + 54969 e^{3} + x \left (52875 d^{3} + 449175 d^{2} e - 274845 d e^{2} - 53189 e^{3}\right )}{2187500 x^{2} + 875000 x + 1312500} \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)**2,x)

[Out]

e**3*x**4/25 + x**3*(4*d*e**2/25 - 41*e**3/375) + x**2*(6*d**2*e/25 - 123*d*e**2/250 + 103*e**3/1250) + x*(4*d
**3/25 - 123*d**2*e/125 + 309*d*e**2/625 + 867*e**3/3125) + (-41*d**3/250 + 309*d**2*e/1250 + 2601*d*e**2/6250
- 416*e**3/3125 - sqrt(14)*I*(32825*d**3 + 317565*d**2*e - 221643*d*e**2 - 67499*e**3)/2450000)*log(x + (6565
*d**3 + 63513*d**2*e - 221643*d*e**2/5 - 67499*e**3/5 - sqrt(14)*I*(32825*d**3 + 317565*d**2*e - 221643*d*e**2
- 67499*e**3)/5)/(32825*d**3 + 317565*d**2*e - 221643*d*e**2 - 67499*e**3)) + (-41*d**3/250 + 309*d**2*e/1250
+ 2601*d*e**2/6250 - 416*e**3/3125 + sqrt(14)*I*(32825*d**3 + 317565*d**2*e - 221643*d*e**2 - 67499*e**3)/245
0000)*log(x + (6565*d**3 + 63513*d**2*e - 221643*d*e**2/5 - 67499*e**3/5 + sqrt(14)*I*(32825*d**3 + 317565*d**
2*e - 221643*d*e**2 - 67499*e**3)/5)/(32825*d**3 + 317565*d**2*e - 221643*d*e**2 - 67499*e**3)) - (170875*d**3
- 95175*d**2*e - 269505*d*e**2 + 54969*e**3 + x*(52875*d**3 + 449175*d**2*e - 274845*d*e**2 - 53189*e**3))/(2
187500*x**2 + 875000*x + 1312500)

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Giac [A]  time = 1.13359, size = 278, normalized size = 1.47 \begin{align*} \frac{1}{25} \, x^{4} e^{3} + \frac{4}{25} \, d x^{3} e^{2} + \frac{6}{25} \, d^{2} x^{2} e + \frac{4}{25} \, d^{3} x - \frac{41}{375} \, x^{3} e^{3} - \frac{123}{250} \, d x^{2} e^{2} - \frac{123}{125} \, d^{2} x e + \frac{103}{1250} \, x^{2} e^{3} + \frac{309}{625} \, d x e^{2} + \frac{1}{1225000} \, \sqrt{14}{\left (32825 \, d^{3} + 317565 \, d^{2} e - 221643 \, d e^{2} - 67499 \, e^{3}\right )} \arctan \left (\frac{1}{14} \, \sqrt{14}{\left (5 \, x + 1\right )}\right ) + \frac{867}{3125} \, x e^{3} - \frac{1}{6250} \,{\left (1025 \, d^{3} - 1545 \, d^{2} e - 2601 \, d e^{2} + 832 \, e^{3}\right )} \log \left (5 \, x^{2} + 2 \, x + 3\right ) - \frac{170875 \, d^{3} - 95175 \, d^{2} e +{\left (52875 \, d^{3} + 449175 \, d^{2} e - 274845 \, d e^{2} - 53189 \, e^{3}\right )} x - 269505 \, d e^{2} + 54969 \, e^{3}}{437500 \,{\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)^2,x, algorithm="giac")

[Out]

1/25*x^4*e^3 + 4/25*d*x^3*e^2 + 6/25*d^2*x^2*e + 4/25*d^3*x - 41/375*x^3*e^3 - 123/250*d*x^2*e^2 - 123/125*d^2
*x*e + 103/1250*x^2*e^3 + 309/625*d*x*e^2 + 1/1225000*sqrt(14)*(32825*d^3 + 317565*d^2*e - 221643*d*e^2 - 6749
9*e^3)*arctan(1/14*sqrt(14)*(5*x + 1)) + 867/3125*x*e^3 - 1/6250*(1025*d^3 - 1545*d^2*e - 2601*d*e^2 + 832*e^3
)*log(5*x^2 + 2*x + 3) - 1/437500*(170875*d^3 - 95175*d^2*e + (52875*d^3 + 449175*d^2*e - 274845*d*e^2 - 53189
*e^3)*x - 269505*d*e^2 + 54969*e^3)/(5*x^2 + 2*x + 3)