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

Optimal. Leaf size=168 \[ \frac {(458 d-7 e) \log \left (5 x^2+2 x+3\right )}{250 \left (5 d^2-2 d e+3 e^2\right )}-\frac {(423 d-1367 e) \tan ^{-1}\left (\frac {5 x+1}{\sqrt {14}}\right )}{125 \sqrt {14} \left (5 d^2-2 d e+3 e^2\right )}+\frac {\left (4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4\right ) \log (d+e x)}{e^3 \left (5 d^2-2 d e+3 e^2\right )}-\frac {x (20 d+33 e)}{25 e^2}+\frac {2 x^2}{5 e} \]

[Out]

-1/25*(20*d+33*e)*x/e^2+2/5*x^2/e+(4*d^4+5*d^3*e+3*d^2*e^2-d*e^3+2*e^4)*ln(e*x+d)/e^3/(5*d^2-2*d*e+3*e^2)+1/25
0*(458*d-7*e)*ln(5*x^2+2*x+3)/(5*d^2-2*d*e+3*e^2)-1/1750*(423*d-1367*e)*arctan(1/14*(1+5*x)*14^(1/2))/(5*d^2-2
*d*e+3*e^2)*14^(1/2)

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Rubi [A]  time = 0.19, antiderivative size = 168, normalized size of antiderivative = 1.00, 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} \[ \frac {(458 d-7 e) \log \left (5 x^2+2 x+3\right )}{250 \left (5 d^2-2 d e+3 e^2\right )}+\frac {\left (3 d^2 e^2+5 d^3 e+4 d^4-d e^3+2 e^4\right ) \log (d+e x)}{e^3 \left (5 d^2-2 d e+3 e^2\right )}-\frac {(423 d-1367 e) \tan ^{-1}\left (\frac {5 x+1}{\sqrt {14}}\right )}{125 \sqrt {14} \left (5 d^2-2 d e+3 e^2\right )}-\frac {x (20 d+33 e)}{25 e^2}+\frac {2 x^2}{5 e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((20*d + 33*e)*x)/(25*e^2) + (2*x^2)/(5*e) - ((423*d - 1367*e)*ArcTan[(1 + 5*x)/Sqrt[14]])/(125*Sqrt[14]*(5*d
^2 - 2*d*e + 3*e^2)) + ((4*d^4 + 5*d^3*e + 3*d^2*e^2 - d*e^3 + 2*e^4)*Log[d + e*x])/(e^3*(5*d^2 - 2*d*e + 3*e^
2)) + ((458*d - 7*e)*Log[3 + 2*x + 5*x^2])/(250*(5*d^2 - 2*d*e + 3*e^2))

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]

Rubi steps

\begin {align*} \int \frac {2+x+3 x^2-5 x^3+4 x^4}{(d+e x) \left (3+2 x+5 x^2\right )} \, dx &=\int \left (\frac {-20 d-33 e}{25 e^2}+\frac {4 x}{5 e}+\frac {4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4}{e^2 \left (5 d^2-2 d e+3 e^2\right ) (d+e x)}+\frac {7 d+272 e+(458 d-7 e) x}{25 \left (5 d^2-2 d e+3 e^2\right ) \left (3+2 x+5 x^2\right )}\right ) \, dx\\ &=-\frac {(20 d+33 e) x}{25 e^2}+\frac {2 x^2}{5 e}+\frac {\left (4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4\right ) \log (d+e x)}{e^3 \left (5 d^2-2 d e+3 e^2\right )}+\frac {\int \frac {7 d+272 e+(458 d-7 e) x}{3+2 x+5 x^2} \, dx}{25 \left (5 d^2-2 d e+3 e^2\right )}\\ &=-\frac {(20 d+33 e) x}{25 e^2}+\frac {2 x^2}{5 e}+\frac {\left (4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4\right ) \log (d+e x)}{e^3 \left (5 d^2-2 d e+3 e^2\right )}-\frac {(423 d-1367 e) \int \frac {1}{3+2 x+5 x^2} \, dx}{125 \left (5 d^2-2 d e+3 e^2\right )}+\frac {(458 d-7 e) \int \frac {2+10 x}{3+2 x+5 x^2} \, dx}{250 \left (5 d^2-2 d e+3 e^2\right )}\\ &=-\frac {(20 d+33 e) x}{25 e^2}+\frac {2 x^2}{5 e}+\frac {\left (4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4\right ) \log (d+e x)}{e^3 \left (5 d^2-2 d e+3 e^2\right )}+\frac {(458 d-7 e) \log \left (3+2 x+5 x^2\right )}{250 \left (5 d^2-2 d e+3 e^2\right )}+\frac {(2 (423 d-1367 e)) \operatorname {Subst}\left (\int \frac {1}{-56-x^2} \, dx,x,2+10 x\right )}{125 \left (5 d^2-2 d e+3 e^2\right )}\\ &=-\frac {(20 d+33 e) x}{25 e^2}+\frac {2 x^2}{5 e}-\frac {(423 d-1367 e) \tan ^{-1}\left (\frac {1+5 x}{\sqrt {14}}\right )}{125 \sqrt {14} \left (5 d^2-2 d e+3 e^2\right )}+\frac {\left (4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4\right ) \log (d+e x)}{e^3 \left (5 d^2-2 d e+3 e^2\right )}+\frac {(458 d-7 e) \log \left (3+2 x+5 x^2\right )}{250 \left (5 d^2-2 d e+3 e^2\right )}\\ \end {align*}

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Mathematica [A]  time = 0.11, size = 146, normalized size = 0.87 \[ \frac {70 e x \left (5 d^2-2 d e+3 e^2\right ) (e (10 x-33)-20 d)+1750 \left (4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4\right ) \log (d+e x)+7 e^3 (458 d-7 e) \log \left (5 x^2+2 x+3\right )-\sqrt {14} e^3 (423 d-1367 e) \tan ^{-1}\left (\frac {5 x+1}{\sqrt {14}}\right )}{1750 e^3 \left (5 d^2-2 d e+3 e^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(70*e*(5*d^2 - 2*d*e + 3*e^2)*x*(-20*d + e*(-33 + 10*x)) - Sqrt[14]*(423*d - 1367*e)*e^3*ArcTan[(1 + 5*x)/Sqrt
[14]] + 1750*(4*d^4 + 5*d^3*e + 3*d^2*e^2 - d*e^3 + 2*e^4)*Log[d + e*x] + 7*(458*d - 7*e)*e^3*Log[3 + 2*x + 5*
x^2])/(1750*e^3*(5*d^2 - 2*d*e + 3*e^2))

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fricas [A]  time = 1.00, size = 171, normalized size = 1.02 \[ \frac {700 \, {\left (5 \, d^{2} e^{2} - 2 \, d e^{3} + 3 \, e^{4}\right )} x^{2} - \sqrt {14} {\left (423 \, d e^{3} - 1367 \, e^{4}\right )} \arctan \left (\frac {1}{14} \, \sqrt {14} {\left (5 \, x + 1\right )}\right ) - 70 \, {\left (100 \, d^{3} e + 125 \, d^{2} e^{2} - 6 \, d e^{3} + 99 \, e^{4}\right )} x + 1750 \, {\left (4 \, d^{4} + 5 \, d^{3} e + 3 \, d^{2} e^{2} - d e^{3} + 2 \, e^{4}\right )} \log \left (e x + d\right ) + 7 \, {\left (458 \, d e^{3} - 7 \, e^{4}\right )} \log \left (5 \, x^{2} + 2 \, x + 3\right )}{1750 \, {\left (5 \, d^{2} e^{3} - 2 \, d e^{4} + 3 \, e^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1750*(700*(5*d^2*e^2 - 2*d*e^3 + 3*e^4)*x^2 - sqrt(14)*(423*d*e^3 - 1367*e^4)*arctan(1/14*sqrt(14)*(5*x + 1)
) - 70*(100*d^3*e + 125*d^2*e^2 - 6*d*e^3 + 99*e^4)*x + 1750*(4*d^4 + 5*d^3*e + 3*d^2*e^2 - d*e^3 + 2*e^4)*log
(e*x + d) + 7*(458*d*e^3 - 7*e^4)*log(5*x^2 + 2*x + 3))/(5*d^2*e^3 - 2*d*e^4 + 3*e^5)

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giac [A]  time = 0.22, size = 158, normalized size = 0.94 \[ \frac {1}{25} \, {\left (10 \, x^{2} e - 20 \, d x - 33 \, x e\right )} e^{\left (-2\right )} - \frac {\sqrt {14} {\left (423 \, d - 1367 \, e\right )} \arctan \left (\frac {1}{14} \, \sqrt {14} {\left (5 \, x + 1\right )}\right )}{1750 \, {\left (5 \, d^{2} - 2 \, d e + 3 \, e^{2}\right )}} + \frac {{\left (458 \, d - 7 \, e\right )} \log \left (5 \, x^{2} + 2 \, x + 3\right )}{250 \, {\left (5 \, d^{2} - 2 \, d e + 3 \, e^{2}\right )}} + \frac {{\left (4 \, d^{4} + 5 \, d^{3} e + 3 \, d^{2} e^{2} - d e^{3} + 2 \, e^{4}\right )} \log \left ({\left | x e + d \right |}\right )}{5 \, d^{2} e^{3} - 2 \, d e^{4} + 3 \, e^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/25*(10*x^2*e - 20*d*x - 33*x*e)*e^(-2) - 1/1750*sqrt(14)*(423*d - 1367*e)*arctan(1/14*sqrt(14)*(5*x + 1))/(5
*d^2 - 2*d*e + 3*e^2) + 1/250*(458*d - 7*e)*log(5*x^2 + 2*x + 3)/(5*d^2 - 2*d*e + 3*e^2) + (4*d^4 + 5*d^3*e +
3*d^2*e^2 - d*e^3 + 2*e^4)*log(abs(x*e + d))/(5*d^2*e^3 - 2*d*e^4 + 3*e^5)

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maple [A]  time = 0.01, size = 298, normalized size = 1.77 \[ \frac {4 d^{4} \ln \left (e x +d \right )}{\left (5 d^{2}-2 d e +3 e^{2}\right ) e^{3}}+\frac {5 d^{3} \ln \left (e x +d \right )}{\left (5 d^{2}-2 d e +3 e^{2}\right ) e^{2}}+\frac {3 d^{2} \ln \left (e x +d \right )}{\left (5 d^{2}-2 d e +3 e^{2}\right ) e}-\frac {423 \sqrt {14}\, d \arctan \left (\frac {\left (10 x +2\right ) \sqrt {14}}{28}\right )}{70 \left (125 d^{2}-50 d e +75 e^{2}\right )}-\frac {d \ln \left (e x +d \right )}{5 d^{2}-2 d e +3 e^{2}}+\frac {229 d \ln \left (5 x^{2}+2 x +3\right )}{5 \left (125 d^{2}-50 d e +75 e^{2}\right )}+\frac {1367 \sqrt {14}\, e \arctan \left (\frac {\left (10 x +2\right ) \sqrt {14}}{28}\right )}{70 \left (125 d^{2}-50 d e +75 e^{2}\right )}+\frac {2 e \ln \left (e x +d \right )}{5 d^{2}-2 d e +3 e^{2}}-\frac {7 e \ln \left (5 x^{2}+2 x +3\right )}{10 \left (125 d^{2}-50 d e +75 e^{2}\right )}+\frac {2 x^{2}}{5 e}-\frac {4 d x}{5 e^{2}}-\frac {33 x}{25 e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5/e*x^2-4/5*d/e^2*x-33/25/e*x+229/5/(125*d^2-50*d*e+75*e^2)*ln(5*x^2+2*x+3)*d-7/10/(125*d^2-50*d*e+75*e^2)*l
n(5*x^2+2*x+3)*e-423/70/(125*d^2-50*d*e+75*e^2)*14^(1/2)*arctan(1/28*(10*x+2)*14^(1/2))*d+1367/70/(125*d^2-50*
d*e+75*e^2)*14^(1/2)*arctan(1/28*(10*x+2)*14^(1/2))*e+4/e^3/(5*d^2-2*d*e+3*e^2)*ln(e*x+d)*d^4+5/e^2/(5*d^2-2*d
*e+3*e^2)*ln(e*x+d)*d^3+3/e/(5*d^2-2*d*e+3*e^2)*ln(e*x+d)*d^2-1/(5*d^2-2*d*e+3*e^2)*ln(e*x+d)*d+2*e/(5*d^2-2*d
*e+3*e^2)*ln(e*x+d)

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maxima [A]  time = 0.96, size = 160, normalized size = 0.95 \[ -\frac {\sqrt {14} {\left (423 \, d - 1367 \, e\right )} \arctan \left (\frac {1}{14} \, \sqrt {14} {\left (5 \, x + 1\right )}\right )}{1750 \, {\left (5 \, d^{2} - 2 \, d e + 3 \, e^{2}\right )}} + \frac {{\left (4 \, d^{4} + 5 \, d^{3} e + 3 \, d^{2} e^{2} - d e^{3} + 2 \, e^{4}\right )} \log \left (e x + d\right )}{5 \, d^{2} e^{3} - 2 \, d e^{4} + 3 \, e^{5}} + \frac {{\left (458 \, d - 7 \, e\right )} \log \left (5 \, x^{2} + 2 \, x + 3\right )}{250 \, {\left (5 \, d^{2} - 2 \, d e + 3 \, e^{2}\right )}} + \frac {10 \, e x^{2} - {\left (20 \, d + 33 \, e\right )} x}{25 \, e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1750*sqrt(14)*(423*d - 1367*e)*arctan(1/14*sqrt(14)*(5*x + 1))/(5*d^2 - 2*d*e + 3*e^2) + (4*d^4 + 5*d^3*e +
3*d^2*e^2 - d*e^3 + 2*e^4)*log(e*x + d)/(5*d^2*e^3 - 2*d*e^4 + 3*e^5) + 1/250*(458*d - 7*e)*log(5*x^2 + 2*x +
3)/(5*d^2 - 2*d*e + 3*e^2) + 1/25*(10*e*x^2 - (20*d + 33*e)*x)/e^2

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mupad [B]  time = 6.39, size = 713, normalized size = 4.24 \[ \frac {2\,x^2}{5\,e}-\ln \left (d+e\,x\right )\,\left (\frac {\frac {458\,d}{125}-\frac {7\,e}{125}}{5\,d^2-2\,d\,e+3\,e^2}-\frac {100\,d^2+165\,d\,e+81\,e^2}{125\,e^3}\right )-x\,\left (\frac {4\,\left (5\,d+2\,e\right )}{25\,e^2}+\frac {1}{e}\right )-\frac {\ln \left (\frac {-28\,d^3+1053\,d^2\,e+1791\,d\,e^2+916\,e^3}{25\,e^2}-\frac {x\,\left (1832\,d^3+2318\,d^2\,e+321\,d\,e^2-2249\,e^3\right )}{25\,e^2}+\frac {\left (d\,\left (\frac {423\,\sqrt {14}}{3500}-\frac {229}{125}{}\mathrm {i}\right )-e\,\left (\frac {1367\,\sqrt {14}}{3500}-\frac {7}{250}{}\mathrm {i}\right )\right )\,\left (\frac {-1000\,d^4+4350\,d^3\,e+8490\,d^2\,e^2+4751\,d\,e^3+874\,e^4}{25\,e^2}+\frac {x\,\left (-5000\,d^4-6250\,d^3\,e+1850\,d^2\,e^2+8200\,d\,e^3+2917\,e^4\right )}{25\,e^2}-\frac {\left (\frac {1250\,d^2\,e^3-14500\,d\,e^4+750\,e^5}{25\,e^2}-\frac {x\,\left (-6250\,d^2\,e^3+2500\,d\,e^4+10250\,e^5\right )}{25\,e^2}\right )\,\left (d\,\left (\frac {423\,\sqrt {14}}{3500}-\frac {229}{125}{}\mathrm {i}\right )-e\,\left (\frac {1367\,\sqrt {14}}{3500}-\frac {7}{250}{}\mathrm {i}\right )\right )}{d^2\,5{}\mathrm {i}-d\,e\,2{}\mathrm {i}+e^2\,3{}\mathrm {i}}\right )}{d^2\,5{}\mathrm {i}-d\,e\,2{}\mathrm {i}+e^2\,3{}\mathrm {i}}\right )\,\left (d\,\left (\frac {423\,\sqrt {14}}{3500}-\frac {229}{125}{}\mathrm {i}\right )-e\,\left (\frac {1367\,\sqrt {14}}{3500}-\frac {7}{250}{}\mathrm {i}\right )\right )}{d^2\,5{}\mathrm {i}-d\,e\,2{}\mathrm {i}+e^2\,3{}\mathrm {i}}+\frac {\ln \left (\frac {-28\,d^3+1053\,d^2\,e+1791\,d\,e^2+916\,e^3}{25\,e^2}-\frac {x\,\left (1832\,d^3+2318\,d^2\,e+321\,d\,e^2-2249\,e^3\right )}{25\,e^2}-\frac {\left (d\,\left (\frac {423\,\sqrt {14}}{3500}+\frac {229}{125}{}\mathrm {i}\right )-e\,\left (\frac {1367\,\sqrt {14}}{3500}+\frac {7}{250}{}\mathrm {i}\right )\right )\,\left (\frac {-1000\,d^4+4350\,d^3\,e+8490\,d^2\,e^2+4751\,d\,e^3+874\,e^4}{25\,e^2}+\frac {x\,\left (-5000\,d^4-6250\,d^3\,e+1850\,d^2\,e^2+8200\,d\,e^3+2917\,e^4\right )}{25\,e^2}+\frac {\left (\frac {1250\,d^2\,e^3-14500\,d\,e^4+750\,e^5}{25\,e^2}-\frac {x\,\left (-6250\,d^2\,e^3+2500\,d\,e^4+10250\,e^5\right )}{25\,e^2}\right )\,\left (d\,\left (\frac {423\,\sqrt {14}}{3500}+\frac {229}{125}{}\mathrm {i}\right )-e\,\left (\frac {1367\,\sqrt {14}}{3500}+\frac {7}{250}{}\mathrm {i}\right )\right )}{d^2\,5{}\mathrm {i}-d\,e\,2{}\mathrm {i}+e^2\,3{}\mathrm {i}}\right )}{d^2\,5{}\mathrm {i}-d\,e\,2{}\mathrm {i}+e^2\,3{}\mathrm {i}}\right )\,\left (d\,\left (\frac {423\,\sqrt {14}}{3500}+\frac {229}{125}{}\mathrm {i}\right )-e\,\left (\frac {1367\,\sqrt {14}}{3500}+\frac {7}{250}{}\mathrm {i}\right )\right )}{d^2\,5{}\mathrm {i}-d\,e\,2{}\mathrm {i}+e^2\,3{}\mathrm {i}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*x^2)/(5*e) - log(d + e*x)*(((458*d)/125 - (7*e)/125)/(5*d^2 - 2*d*e + 3*e^2) - (165*d*e + 100*d^2 + 81*e^2)
/(125*e^3)) - x*((4*(5*d + 2*e))/(25*e^2) + 1/e) - (log((1791*d*e^2 + 1053*d^2*e - 28*d^3 + 916*e^3)/(25*e^2)
- (x*(321*d*e^2 + 2318*d^2*e + 1832*d^3 - 2249*e^3))/(25*e^2) + ((d*((423*14^(1/2))/3500 - 229i/125) - e*((136
7*14^(1/2))/3500 - 7i/250))*((4751*d*e^3 + 4350*d^3*e - 1000*d^4 + 874*e^4 + 8490*d^2*e^2)/(25*e^2) + (x*(8200
*d*e^3 - 6250*d^3*e - 5000*d^4 + 2917*e^4 + 1850*d^2*e^2))/(25*e^2) - (((750*e^5 - 14500*d*e^4 + 1250*d^2*e^3)
/(25*e^2) - (x*(2500*d*e^4 + 10250*e^5 - 6250*d^2*e^3))/(25*e^2))*(d*((423*14^(1/2))/3500 - 229i/125) - e*((13
67*14^(1/2))/3500 - 7i/250)))/(d^2*5i - d*e*2i + e^2*3i)))/(d^2*5i - d*e*2i + e^2*3i))*(d*((423*14^(1/2))/3500
- 229i/125) - e*((1367*14^(1/2))/3500 - 7i/250)))/(d^2*5i - d*e*2i + e^2*3i) + (log((1791*d*e^2 + 1053*d^2*e
- 28*d^3 + 916*e^3)/(25*e^2) - (x*(321*d*e^2 + 2318*d^2*e + 1832*d^3 - 2249*e^3))/(25*e^2) - ((d*((423*14^(1/2
))/3500 + 229i/125) - e*((1367*14^(1/2))/3500 + 7i/250))*((4751*d*e^3 + 4350*d^3*e - 1000*d^4 + 874*e^4 + 8490
*d^2*e^2)/(25*e^2) + (x*(8200*d*e^3 - 6250*d^3*e - 5000*d^4 + 2917*e^4 + 1850*d^2*e^2))/(25*e^2) + (((750*e^5
- 14500*d*e^4 + 1250*d^2*e^3)/(25*e^2) - (x*(2500*d*e^4 + 10250*e^5 - 6250*d^2*e^3))/(25*e^2))*(d*((423*14^(1/
2))/3500 + 229i/125) - e*((1367*14^(1/2))/3500 + 7i/250)))/(d^2*5i - d*e*2i + e^2*3i)))/(d^2*5i - d*e*2i + e^2
*3i))*(d*((423*14^(1/2))/3500 + 229i/125) - e*((1367*14^(1/2))/3500 + 7i/250)))/(d^2*5i - d*e*2i + e^2*3i)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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