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

Optimal. Leaf size=233 \[ \frac{\left (229 d^2-7 d e-136 e^2\right ) \log \left (5 x^2+2 x+3\right )}{25 \left (5 d^2-2 d e+3 e^2\right )^2}-\frac{3 d^2 e^2+5 d^3 e+4 d^4-d e^3+2 e^4}{e^3 \left (5 d^2-2 d e+3 e^2\right ) (d+e x)}-\frac{\left (28 d^3 e^2+44 d^2 e^3+d^4 e+40 d^5-2 d e^4+e^5\right ) \log (d+e x)}{e^3 \left (5 d^2-2 d e+3 e^2\right )^2}-\frac{\left (423 d^2-2734 d e+293 e^2\right ) \tan ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{25 \sqrt{14} \left (5 d^2-2 d e+3 e^2\right )^2}+\frac{4 x}{5 e^2} \]

[Out]

(4*x)/(5*e^2) - (4*d^4 + 5*d^3*e + 3*d^2*e^2 - d*e^3 + 2*e^4)/(e^3*(5*d^2 - 2*d*e + 3*e^2)*(d + e*x)) - ((423*
d^2 - 2734*d*e + 293*e^2)*ArcTan[(1 + 5*x)/Sqrt[14]])/(25*Sqrt[14]*(5*d^2 - 2*d*e + 3*e^2)^2) - ((40*d^5 + d^4
*e + 28*d^3*e^2 + 44*d^2*e^3 - 2*d*e^4 + e^5)*Log[d + e*x])/(e^3*(5*d^2 - 2*d*e + 3*e^2)^2) + ((229*d^2 - 7*d*
e - 136*e^2)*Log[3 + 2*x + 5*x^2])/(25*(5*d^2 - 2*d*e + 3*e^2)^2)

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Rubi [A]  time = 0.251182, antiderivative size = 233, 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} \[ \frac{\left (229 d^2-7 d e-136 e^2\right ) \log \left (5 x^2+2 x+3\right )}{25 \left (5 d^2-2 d e+3 e^2\right )^2}-\frac{3 d^2 e^2+5 d^3 e+4 d^4-d e^3+2 e^4}{e^3 \left (5 d^2-2 d e+3 e^2\right ) (d+e x)}-\frac{\left (28 d^3 e^2+44 d^2 e^3+d^4 e+40 d^5-2 d e^4+e^5\right ) \log (d+e x)}{e^3 \left (5 d^2-2 d e+3 e^2\right )^2}-\frac{\left (423 d^2-2734 d e+293 e^2\right ) \tan ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{25 \sqrt{14} \left (5 d^2-2 d e+3 e^2\right )^2}+\frac{4 x}{5 e^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*x)/(5*e^2) - (4*d^4 + 5*d^3*e + 3*d^2*e^2 - d*e^3 + 2*e^4)/(e^3*(5*d^2 - 2*d*e + 3*e^2)*(d + e*x)) - ((423*
d^2 - 2734*d*e + 293*e^2)*ArcTan[(1 + 5*x)/Sqrt[14]])/(25*Sqrt[14]*(5*d^2 - 2*d*e + 3*e^2)^2) - ((40*d^5 + d^4
*e + 28*d^3*e^2 + 44*d^2*e^3 - 2*d*e^4 + e^5)*Log[d + e*x])/(e^3*(5*d^2 - 2*d*e + 3*e^2)^2) + ((229*d^2 - 7*d*
e - 136*e^2)*Log[3 + 2*x + 5*x^2])/(25*(5*d^2 - 2*d*e + 3*e^2)^2)

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{2+x+3 x^2-5 x^3+4 x^4}{(d+e x)^2 \left (3+2 x+5 x^2\right )} \, dx &=\int \left (\frac{4}{5 e^2}+\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)^2}+\frac{-40 d^5-d^4 e-28 d^3 e^2-44 d^2 e^3+2 d e^4-e^5}{e^2 \left (5 d^2-2 d e+3 e^2\right )^2 (d+e x)}+\frac{7 d^2+544 d e-113 e^2+2 \left (229 d^2-7 d e-136 e^2\right ) x}{5 \left (5 d^2-2 d e+3 e^2\right )^2 \left (3+2 x+5 x^2\right )}\right ) \, dx\\ &=\frac{4 x}{5 e^2}-\frac{4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4}{e^3 \left (5 d^2-2 d e+3 e^2\right ) (d+e x)}-\frac{\left (40 d^5+d^4 e+28 d^3 e^2+44 d^2 e^3-2 d e^4+e^5\right ) \log (d+e x)}{e^3 \left (5 d^2-2 d e+3 e^2\right )^2}+\frac{\int \frac{7 d^2+544 d e-113 e^2+2 \left (229 d^2-7 d e-136 e^2\right ) x}{3+2 x+5 x^2} \, dx}{5 \left (5 d^2-2 d e+3 e^2\right )^2}\\ &=\frac{4 x}{5 e^2}-\frac{4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4}{e^3 \left (5 d^2-2 d e+3 e^2\right ) (d+e x)}-\frac{\left (40 d^5+d^4 e+28 d^3 e^2+44 d^2 e^3-2 d e^4+e^5\right ) \log (d+e x)}{e^3 \left (5 d^2-2 d e+3 e^2\right )^2}+\frac{\left (229 d^2-7 d e-136 e^2\right ) \int \frac{2+10 x}{3+2 x+5 x^2} \, dx}{25 \left (5 d^2-2 d e+3 e^2\right )^2}-\frac{\left (423 d^2-2734 d e+293 e^2\right ) \int \frac{1}{3+2 x+5 x^2} \, dx}{25 \left (5 d^2-2 d e+3 e^2\right )^2}\\ &=\frac{4 x}{5 e^2}-\frac{4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4}{e^3 \left (5 d^2-2 d e+3 e^2\right ) (d+e x)}-\frac{\left (40 d^5+d^4 e+28 d^3 e^2+44 d^2 e^3-2 d e^4+e^5\right ) \log (d+e x)}{e^3 \left (5 d^2-2 d e+3 e^2\right )^2}+\frac{\left (229 d^2-7 d e-136 e^2\right ) \log \left (3+2 x+5 x^2\right )}{25 \left (5 d^2-2 d e+3 e^2\right )^2}+\frac{\left (2 \left (423 d^2-2734 d e+293 e^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-56-x^2} \, dx,x,2+10 x\right )}{25 \left (5 d^2-2 d e+3 e^2\right )^2}\\ &=\frac{4 x}{5 e^2}-\frac{4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4}{e^3 \left (5 d^2-2 d e+3 e^2\right ) (d+e x)}-\frac{\left (423 d^2-2734 d e+293 e^2\right ) \tan ^{-1}\left (\frac{1+5 x}{\sqrt{14}}\right )}{25 \sqrt{14} \left (5 d^2-2 d e+3 e^2\right )^2}-\frac{\left (40 d^5+d^4 e+28 d^3 e^2+44 d^2 e^3-2 d e^4+e^5\right ) \log (d+e x)}{e^3 \left (5 d^2-2 d e+3 e^2\right )^2}+\frac{\left (229 d^2-7 d e-136 e^2\right ) \log \left (3+2 x+5 x^2\right )}{25 \left (5 d^2-2 d e+3 e^2\right )^2}\\ \end{align*}

Mathematica [A]  time = 0.149533, size = 233, normalized size = 1. \[ \frac{\left (229 d^2-7 d e-136 e^2\right ) \log \left (5 x^2+2 x+3\right )}{25 \left (5 d^2-2 d e+3 e^2\right )^2}+\frac{-3 d^2 e^2-5 d^3 e-4 d^4+d e^3-2 e^4}{e^3 \left (5 d^2-2 d e+3 e^2\right ) (d+e x)}+\frac{\left (-28 d^3 e^2-44 d^2 e^3-d^4 e-40 d^5+2 d e^4-e^5\right ) \log (d+e x)}{e^3 \left (5 d^2-2 d e+3 e^2\right )^2}+\frac{\left (-423 d^2+2734 d e-293 e^2\right ) \tan ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{25 \sqrt{14} \left (5 d^2-2 d e+3 e^2\right )^2}+\frac{4 x}{5 e^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*x)/(5*e^2) + (-4*d^4 - 5*d^3*e - 3*d^2*e^2 + d*e^3 - 2*e^4)/(e^3*(5*d^2 - 2*d*e + 3*e^2)*(d + e*x)) + ((-42
3*d^2 + 2734*d*e - 293*e^2)*ArcTan[(1 + 5*x)/Sqrt[14]])/(25*Sqrt[14]*(5*d^2 - 2*d*e + 3*e^2)^2) + ((-40*d^5 -
d^4*e - 28*d^3*e^2 - 44*d^2*e^3 + 2*d*e^4 - e^5)*Log[d + e*x])/(e^3*(5*d^2 - 2*d*e + 3*e^2)^2) + ((229*d^2 - 7
*d*e - 136*e^2)*Log[3 + 2*x + 5*x^2])/(25*(5*d^2 - 2*d*e + 3*e^2)^2)

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Maple [B]  time = 0.063, size = 538, normalized size = 2.3 \begin{align*}{\frac{4\,x}{5\,{e}^{2}}}+{\frac{229\,\ln \left ( 5\,{x}^{2}+2\,x+3 \right ){d}^{2}}{25\, \left ( 5\,{d}^{2}-2\,de+3\,{e}^{2} \right ) ^{2}}}-{\frac{7\,\ln \left ( 5\,{x}^{2}+2\,x+3 \right ) de}{25\, \left ( 5\,{d}^{2}-2\,de+3\,{e}^{2} \right ) ^{2}}}-{\frac{136\,\ln \left ( 5\,{x}^{2}+2\,x+3 \right ){e}^{2}}{25\, \left ( 5\,{d}^{2}-2\,de+3\,{e}^{2} \right ) ^{2}}}-{\frac{423\,\sqrt{14}{d}^{2}}{350\, \left ( 5\,{d}^{2}-2\,de+3\,{e}^{2} \right ) ^{2}}\arctan \left ({\frac{ \left ( 10\,x+2 \right ) \sqrt{14}}{28}} \right ) }+{\frac{1367\,\sqrt{14}de}{175\, \left ( 5\,{d}^{2}-2\,de+3\,{e}^{2} \right ) ^{2}}\arctan \left ({\frac{ \left ( 10\,x+2 \right ) \sqrt{14}}{28}} \right ) }-{\frac{293\,\sqrt{14}{e}^{2}}{350\, \left ( 5\,{d}^{2}-2\,de+3\,{e}^{2} \right ) ^{2}}\arctan \left ({\frac{ \left ( 10\,x+2 \right ) \sqrt{14}}{28}} \right ) }-4\,{\frac{{d}^{4}}{{e}^{3} \left ( 5\,{d}^{2}-2\,de+3\,{e}^{2} \right ) \left ( ex+d \right ) }}-5\,{\frac{{d}^{3}}{{e}^{2} \left ( 5\,{d}^{2}-2\,de+3\,{e}^{2} \right ) \left ( ex+d \right ) }}-3\,{\frac{{d}^{2}}{ \left ( 5\,{d}^{2}-2\,de+3\,{e}^{2} \right ) e \left ( ex+d \right ) }}+{\frac{d}{ \left ( 5\,{d}^{2}-2\,de+3\,{e}^{2} \right ) \left ( ex+d \right ) }}-2\,{\frac{e}{ \left ( 5\,{d}^{2}-2\,de+3\,{e}^{2} \right ) \left ( ex+d \right ) }}-40\,{\frac{\ln \left ( ex+d \right ){d}^{5}}{{e}^{3} \left ( 5\,{d}^{2}-2\,de+3\,{e}^{2} \right ) ^{2}}}-{\frac{\ln \left ( ex+d \right ){d}^{4}}{{e}^{2} \left ( 5\,{d}^{2}-2\,de+3\,{e}^{2} \right ) ^{2}}}-28\,{\frac{\ln \left ( ex+d \right ){d}^{3}}{ \left ( 5\,{d}^{2}-2\,de+3\,{e}^{2} \right ) ^{2}e}}-44\,{\frac{\ln \left ( ex+d \right ){d}^{2}}{ \left ( 5\,{d}^{2}-2\,de+3\,{e}^{2} \right ) ^{2}}}+2\,{\frac{e\ln \left ( ex+d \right ) d}{ \left ( 5\,{d}^{2}-2\,de+3\,{e}^{2} \right ) ^{2}}}-{\frac{{e}^{2}\ln \left ( ex+d \right ) }{ \left ( 5\,{d}^{2}-2\,de+3\,{e}^{2} \right ) ^{2}}} \end{align*}

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)^2/(5*x^2+2*x+3),x)

[Out]

4/5*x/e^2+229/25/(5*d^2-2*d*e+3*e^2)^2*ln(5*x^2+2*x+3)*d^2-7/25/(5*d^2-2*d*e+3*e^2)^2*ln(5*x^2+2*x+3)*d*e-136/
25/(5*d^2-2*d*e+3*e^2)^2*ln(5*x^2+2*x+3)*e^2-423/350/(5*d^2-2*d*e+3*e^2)^2*14^(1/2)*arctan(1/28*(10*x+2)*14^(1
/2))*d^2+1367/175/(5*d^2-2*d*e+3*e^2)^2*14^(1/2)*arctan(1/28*(10*x+2)*14^(1/2))*d*e-293/350/(5*d^2-2*d*e+3*e^2
)^2*14^(1/2)*arctan(1/28*(10*x+2)*14^(1/2))*e^2-4/e^3/(5*d^2-2*d*e+3*e^2)/(e*x+d)*d^4-5/e^2/(5*d^2-2*d*e+3*e^2
)/(e*x+d)*d^3-3/e/(5*d^2-2*d*e+3*e^2)/(e*x+d)*d^2+1/(5*d^2-2*d*e+3*e^2)/(e*x+d)*d-2*e/(5*d^2-2*d*e+3*e^2)/(e*x
+d)-40/e^3/(5*d^2-2*d*e+3*e^2)^2*ln(e*x+d)*d^5-1/e^2/(5*d^2-2*d*e+3*e^2)^2*ln(e*x+d)*d^4-28/e/(5*d^2-2*d*e+3*e
^2)^2*ln(e*x+d)*d^3-44/(5*d^2-2*d*e+3*e^2)^2*ln(e*x+d)*d^2+2*e/(5*d^2-2*d*e+3*e^2)^2*ln(e*x+d)*d-e^2/(5*d^2-2*
d*e+3*e^2)^2*ln(e*x+d)

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Maxima [A]  time = 1.47393, size = 397, normalized size = 1.7 \begin{align*} -\frac{\sqrt{14}{\left (423 \, d^{2} - 2734 \, d e + 293 \, e^{2}\right )} \arctan \left (\frac{1}{14} \, \sqrt{14}{\left (5 \, x + 1\right )}\right )}{350 \,{\left (25 \, d^{4} - 20 \, d^{3} e + 34 \, d^{2} e^{2} - 12 \, d e^{3} + 9 \, e^{4}\right )}} - \frac{{\left (40 \, d^{5} + d^{4} e + 28 \, d^{3} e^{2} + 44 \, d^{2} e^{3} - 2 \, d e^{4} + e^{5}\right )} \log \left (e x + d\right )}{25 \, d^{4} e^{3} - 20 \, d^{3} e^{4} + 34 \, d^{2} e^{5} - 12 \, d e^{6} + 9 \, e^{7}} + \frac{{\left (229 \, d^{2} - 7 \, d e - 136 \, e^{2}\right )} \log \left (5 \, x^{2} + 2 \, x + 3\right )}{25 \,{\left (25 \, d^{4} - 20 \, d^{3} e + 34 \, d^{2} e^{2} - 12 \, d e^{3} + 9 \, e^{4}\right )}} - \frac{4 \, d^{4} + 5 \, d^{3} e + 3 \, d^{2} e^{2} - d e^{3} + 2 \, e^{4}}{5 \, d^{3} e^{3} - 2 \, d^{2} e^{4} + 3 \, d e^{5} +{\left (5 \, d^{2} e^{4} - 2 \, d e^{5} + 3 \, e^{6}\right )} x} + \frac{4 \, x}{5 \, e^{2}} \end{align*}

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)^2/(5*x^2+2*x+3),x, algorithm="maxima")

[Out]

-1/350*sqrt(14)*(423*d^2 - 2734*d*e + 293*e^2)*arctan(1/14*sqrt(14)*(5*x + 1))/(25*d^4 - 20*d^3*e + 34*d^2*e^2
- 12*d*e^3 + 9*e^4) - (40*d^5 + d^4*e + 28*d^3*e^2 + 44*d^2*e^3 - 2*d*e^4 + e^5)*log(e*x + d)/(25*d^4*e^3 - 2
0*d^3*e^4 + 34*d^2*e^5 - 12*d*e^6 + 9*e^7) + 1/25*(229*d^2 - 7*d*e - 136*e^2)*log(5*x^2 + 2*x + 3)/(25*d^4 - 2
0*d^3*e + 34*d^2*e^2 - 12*d*e^3 + 9*e^4) - (4*d^4 + 5*d^3*e + 3*d^2*e^2 - d*e^3 + 2*e^4)/(5*d^3*e^3 - 2*d^2*e^
4 + 3*d*e^5 + (5*d^2*e^4 - 2*d*e^5 + 3*e^6)*x) + 4/5*x/e^2

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Fricas [A]  time = 1.82684, size = 995, normalized size = 4.27 \begin{align*} -\frac{7000 \, d^{6} + 5950 \, d^{5} e + 5950 \, d^{4} e^{2} + 1400 \, d^{3} e^{3} + 7350 \, d^{2} e^{4} - 2450 \, d e^{5} + 2100 \, e^{6} - 280 \,{\left (25 \, d^{4} e^{2} - 20 \, d^{3} e^{3} + 34 \, d^{2} e^{4} - 12 \, d e^{5} + 9 \, e^{6}\right )} x^{2} + \sqrt{14}{\left (423 \, d^{3} e^{3} - 2734 \, d^{2} e^{4} + 293 \, d e^{5} +{\left (423 \, d^{2} e^{4} - 2734 \, d e^{5} + 293 \, e^{6}\right )} x\right )} \arctan \left (\frac{1}{14} \, \sqrt{14}{\left (5 \, x + 1\right )}\right ) - 280 \,{\left (25 \, d^{5} e - 20 \, d^{4} e^{2} + 34 \, d^{3} e^{3} - 12 \, d^{2} e^{4} + 9 \, d e^{5}\right )} x + 350 \,{\left (40 \, d^{6} + d^{5} e + 28 \, d^{4} e^{2} + 44 \, d^{3} e^{3} - 2 \, d^{2} e^{4} + d e^{5} +{\left (40 \, d^{5} e + d^{4} e^{2} + 28 \, d^{3} e^{3} + 44 \, d^{2} e^{4} - 2 \, d e^{5} + e^{6}\right )} x\right )} \log \left (e x + d\right ) - 14 \,{\left (229 \, d^{3} e^{3} - 7 \, d^{2} e^{4} - 136 \, d e^{5} +{\left (229 \, d^{2} e^{4} - 7 \, d e^{5} - 136 \, e^{6}\right )} x\right )} \log \left (5 \, x^{2} + 2 \, x + 3\right )}{350 \,{\left (25 \, d^{5} e^{3} - 20 \, d^{4} e^{4} + 34 \, d^{3} e^{5} - 12 \, d^{2} e^{6} + 9 \, d e^{7} +{\left (25 \, d^{4} e^{4} - 20 \, d^{3} e^{5} + 34 \, d^{2} e^{6} - 12 \, d e^{7} + 9 \, e^{8}\right )} x\right )}} \end{align*}

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)^2/(5*x^2+2*x+3),x, algorithm="fricas")

[Out]

-1/350*(7000*d^6 + 5950*d^5*e + 5950*d^4*e^2 + 1400*d^3*e^3 + 7350*d^2*e^4 - 2450*d*e^5 + 2100*e^6 - 280*(25*d
^4*e^2 - 20*d^3*e^3 + 34*d^2*e^4 - 12*d*e^5 + 9*e^6)*x^2 + sqrt(14)*(423*d^3*e^3 - 2734*d^2*e^4 + 293*d*e^5 +
(423*d^2*e^4 - 2734*d*e^5 + 293*e^6)*x)*arctan(1/14*sqrt(14)*(5*x + 1)) - 280*(25*d^5*e - 20*d^4*e^2 + 34*d^3*
e^3 - 12*d^2*e^4 + 9*d*e^5)*x + 350*(40*d^6 + d^5*e + 28*d^4*e^2 + 44*d^3*e^3 - 2*d^2*e^4 + d*e^5 + (40*d^5*e
+ d^4*e^2 + 28*d^3*e^3 + 44*d^2*e^4 - 2*d*e^5 + e^6)*x)*log(e*x + d) - 14*(229*d^3*e^3 - 7*d^2*e^4 - 136*d*e^5
+ (229*d^2*e^4 - 7*d*e^5 - 136*e^6)*x)*log(5*x^2 + 2*x + 3))/(25*d^5*e^3 - 20*d^4*e^4 + 34*d^3*e^5 - 12*d^2*e
^6 + 9*d*e^7 + (25*d^4*e^4 - 20*d^3*e^5 + 34*d^2*e^6 - 12*d*e^7 + 9*e^8)*x)

________________________________________________________________________________________

Sympy [C]  time = 20.7065, size = 8391, normalized size = 36.01 \begin{align*} \text{result too large to display} \end{align*}

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)**2/(5*x**2+2*x+3),x)

[Out]

(-sqrt(14)*I*(423*d**2 - 2734*d*e + 293*e**2)/(700*(25*d**4 - 20*d**3*e + 34*d**2*e**2 - 12*d*e**3 + 9*e**4))
+ (229*d**2 - 7*d*e - 136*e**2)/(25*(5*d**2 - 2*d*e + 3*e**2)**2))*log(x + (-7840000000*d**14*(-sqrt(14)*I*(42
3*d**2 - 2734*d*e + 293*e**2)/(700*(25*d**4 - 20*d**3*e + 34*d**2*e**2 - 12*d*e**3 + 9*e**4)) + (229*d**2 - 7*
d*e - 136*e**2)/(25*(5*d**2 - 2*d*e + 3*e**2)**2)) - 4900000000*d**13*e**3*(-sqrt(14)*I*(423*d**2 - 2734*d*e +
293*e**2)/(700*(25*d**4 - 20*d**3*e + 34*d**2*e**2 - 12*d*e**3 + 9*e**4)) + (229*d**2 - 7*d*e - 136*e**2)/(25
*(5*d**2 - 2*d*e + 3*e**2)**2))**2 + 5880000000*d**13*e*(-sqrt(14)*I*(423*d**2 - 2734*d*e + 293*e**2)/(700*(25
*d**4 - 20*d**3*e + 34*d**2*e**2 - 12*d*e**3 + 9*e**4)) + (229*d**2 - 7*d*e - 136*e**2)/(25*(5*d**2 - 2*d*e +
3*e**2)**2)) + 7717500000*d**12*e**4*(-sqrt(14)*I*(423*d**2 - 2734*d*e + 293*e**2)/(700*(25*d**4 - 20*d**3*e +
34*d**2*e**2 - 12*d*e**3 + 9*e**4)) + (229*d**2 - 7*d*e - 136*e**2)/(25*(5*d**2 - 2*d*e + 3*e**2)**2))**2 - 2
1329700000*d**12*e**2*(-sqrt(14)*I*(423*d**2 - 2734*d*e + 293*e**2)/(700*(25*d**4 - 20*d**3*e + 34*d**2*e**2 -
12*d*e**3 + 9*e**4)) + (229*d**2 - 7*d*e - 136*e**2)/(25*(5*d**2 - 2*d*e + 3*e**2)**2)) + 3062080000*d**12 -
19327875000*d**11*e**5*(-sqrt(14)*I*(423*d**2 - 2734*d*e + 293*e**2)/(700*(25*d**4 - 20*d**3*e + 34*d**2*e**2
- 12*d*e**3 + 9*e**4)) + (229*d**2 - 7*d*e - 136*e**2)/(25*(5*d**2 - 2*d*e + 3*e**2)**2))**2 - 5507600000*d**1
1*e**3*(-sqrt(14)*I*(423*d**2 - 2734*d*e + 293*e**2)/(700*(25*d**4 - 20*d**3*e + 34*d**2*e**2 - 12*d*e**3 + 9*
e**4)) + (229*d**2 - 7*d*e - 136*e**2)/(25*(5*d**2 - 2*d*e + 3*e**2)**2)) - 1159536000*d**11*e + 10872225000*d
**10*e**6*(-sqrt(14)*I*(423*d**2 - 2734*d*e + 293*e**2)/(700*(25*d**4 - 20*d**3*e + 34*d**2*e**2 - 12*d*e**3 +
9*e**4)) + (229*d**2 - 7*d*e - 136*e**2)/(25*(5*d**2 - 2*d*e + 3*e**2)**2))**2 - 7039144000*d**10*e**4*(-sqrt
(14)*I*(423*d**2 - 2734*d*e + 293*e**2)/(700*(25*d**4 - 20*d**3*e + 34*d**2*e**2 - 12*d*e**3 + 9*e**4)) + (229
*d**2 - 7*d*e - 136*e**2)/(25*(5*d**2 - 2*d*e + 3*e**2)**2)) + 2648473800*d**10*e**2 - 10871735000*d**9*e**7*(
-sqrt(14)*I*(423*d**2 - 2734*d*e + 293*e**2)/(700*(25*d**4 - 20*d**3*e + 34*d**2*e**2 - 12*d*e**3 + 9*e**4)) +
(229*d**2 - 7*d*e - 136*e**2)/(25*(5*d**2 - 2*d*e + 3*e**2)**2))**2 - 28626939600*d**9*e**5*(-sqrt(14)*I*(423
*d**2 - 2734*d*e + 293*e**2)/(700*(25*d**4 - 20*d**3*e + 34*d**2*e**2 - 12*d*e**3 + 9*e**4)) + (229*d**2 - 7*d
*e - 136*e**2)/(25*(5*d**2 - 2*d*e + 3*e**2)**2)) + 5631029040*d**9*e**3 - 12890563000*d**8*e**8*(-sqrt(14)*I*
(423*d**2 - 2734*d*e + 293*e**2)/(700*(25*d**4 - 20*d**3*e + 34*d**2*e**2 - 12*d*e**3 + 9*e**4)) + (229*d**2 -
7*d*e - 136*e**2)/(25*(5*d**2 - 2*d*e + 3*e**2)**2))**2 + 3140906580*d**8*e**6*(-sqrt(14)*I*(423*d**2 - 2734*
d*e + 293*e**2)/(700*(25*d**4 - 20*d**3*e + 34*d**2*e**2 - 12*d*e**3 + 9*e**4)) + (229*d**2 - 7*d*e - 136*e**2
)/(25*(5*d**2 - 2*d*e + 3*e**2)**2)) - 3844841924*d**8*e**4 + 14866261200*d**7*e**9*(-sqrt(14)*I*(423*d**2 - 2
734*d*e + 293*e**2)/(700*(25*d**4 - 20*d**3*e + 34*d**2*e**2 - 12*d*e**3 + 9*e**4)) + (229*d**2 - 7*d*e - 136*
e**2)/(25*(5*d**2 - 2*d*e + 3*e**2)**2))**2 - 16078247136*d**7*e**7*(-sqrt(14)*I*(423*d**2 - 2734*d*e + 293*e*
*2)/(700*(25*d**4 - 20*d**3*e + 34*d**2*e**2 - 12*d*e**3 + 9*e**4)) + (229*d**2 - 7*d*e - 136*e**2)/(25*(5*d**
2 - 2*d*e + 3*e**2)**2)) - 1183700793*d**7*e**5 - 24188575600*d**6*e**10*(-sqrt(14)*I*(423*d**2 - 2734*d*e + 2
93*e**2)/(700*(25*d**4 - 20*d**3*e + 34*d**2*e**2 - 12*d*e**3 + 9*e**4)) + (229*d**2 - 7*d*e - 136*e**2)/(25*(
5*d**2 - 2*d*e + 3*e**2)**2))**2 - 7728337232*d**6*e**8*(-sqrt(14)*I*(423*d**2 - 2734*d*e + 293*e**2)/(700*(25
*d**4 - 20*d**3*e + 34*d**2*e**2 - 12*d*e**3 + 9*e**4)) + (229*d**2 - 7*d*e - 136*e**2)/(25*(5*d**2 - 2*d*e +
3*e**2)**2)) + 1694057982*d**6*e**6 + 14439653200*d**5*e**11*(-sqrt(14)*I*(423*d**2 - 2734*d*e + 293*e**2)/(70
0*(25*d**4 - 20*d**3*e + 34*d**2*e**2 - 12*d*e**3 + 9*e**4)) + (229*d**2 - 7*d*e - 136*e**2)/(25*(5*d**2 - 2*d
*e + 3*e**2)**2))**2 + 2286078144*d**5*e**9*(-sqrt(14)*I*(423*d**2 - 2734*d*e + 293*e**2)/(700*(25*d**4 - 20*d
**3*e + 34*d**2*e**2 - 12*d*e**3 + 9*e**4)) + (229*d**2 - 7*d*e - 136*e**2)/(25*(5*d**2 - 2*d*e + 3*e**2)**2))
- 5520804349*d**5*e**7 - 10082618000*d**4*e**12*(-sqrt(14)*I*(423*d**2 - 2734*d*e + 293*e**2)/(700*(25*d**4 -
20*d**3*e + 34*d**2*e**2 - 12*d*e**3 + 9*e**4)) + (229*d**2 - 7*d*e - 136*e**2)/(25*(5*d**2 - 2*d*e + 3*e**2)
**2))**2 - 7135930760*d**4*e**10*(-sqrt(14)*I*(423*d**2 - 2734*d*e + 293*e**2)/(700*(25*d**4 - 20*d**3*e + 34*
d**2*e**2 - 12*d*e**3 + 9*e**4)) + (229*d**2 - 7*d*e - 136*e**2)/(25*(5*d**2 - 2*d*e + 3*e**2)**2)) - 42477147
00*d**4*e**8 + 3006129000*d**3*e**13*(-sqrt(14)*I*(423*d**2 - 2734*d*e + 293*e**2)/(700*(25*d**4 - 20*d**3*e +
34*d**2*e**2 - 12*d*e**3 + 9*e**4)) + (229*d**2 - 7*d*e - 136*e**2)/(25*(5*d**2 - 2*d*e + 3*e**2)**2))**2 + 2
323015520*d**3*e**11*(-sqrt(14)*I*(423*d**2 - 2734*d*e + 293*e**2)/(700*(25*d**4 - 20*d**3*e + 34*d**2*e**2 -
12*d*e**3 + 9*e**4)) + (229*d**2 - 7*d*e - 136*e**2)/(25*(5*d**2 - 2*d*e + 3*e**2)**2)) + 1298698281*d**3*e**9
- 918199800*d**2*e**14*(-sqrt(14)*I*(423*d**2 - 2734*d*e + 293*e**2)/(700*(25*d**4 - 20*d**3*e + 34*d**2*e**2
- 12*d*e**3 + 9*e**4)) + (229*d**2 - 7*d*e - 136*e**2)/(25*(5*d**2 - 2*d*e + 3*e**2)**2))**2 - 1227448656*d**
2*e**12*(-sqrt(14)*I*(423*d**2 - 2734*d*e + 293*e**2)/(700*(25*d**4 - 20*d**3*e + 34*d**2*e**2 - 12*d*e**3 + 9
*e**4)) + (229*d**2 - 7*d*e - 136*e**2)/(25*(5*d**2 - 2*d*e + 3*e**2)**2)) - 128577018*d**2*e**10 - 38820600*d
*e**15*(-sqrt(14)*I*(423*d**2 - 2734*d*e + 293*e**2)/(700*(25*d**4 - 20*d**3*e + 34*d**2*e**2 - 12*d*e**3 + 9*
e**4)) + (229*d**2 - 7*d*e - 136*e**2)/(25*(5*d**2 - 2*d*e + 3*e**2)**2))**2 + 157117968*d*e**13*(-sqrt(14)*I*
(423*d**2 - 2734*d*e + 293*e**2)/(700*(25*d**4 - 20*d**3*e + 34*d**2*e**2 - 12*d*e**3 + 9*e**4)) + (229*d**2 -
7*d*e - 136*e**2)/(25*(5*d**2 - 2*d*e + 3*e**2)**2)) + 25259757*d*e**11 + 63844200*e**16*(-sqrt(14)*I*(423*d*
*2 - 2734*d*e + 293*e**2)/(700*(25*d**4 - 20*d**3*e + 34*d**2*e**2 - 12*d*e**3 + 9*e**4)) + (229*d**2 - 7*d*e
- 136*e**2)/(25*(5*d**2 - 2*d*e + 3*e**2)**2))**2 + 38078964*e**14*(-sqrt(14)*I*(423*d**2 - 2734*d*e + 293*e**
2)/(700*(25*d**4 - 20*d**3*e + 34*d**2*e**2 - 12*d*e**3 + 9*e**4)) + (229*d**2 - 7*d*e - 136*e**2)/(25*(5*d**2
- 2*d*e + 3*e**2)**2)) + 3442796*e**12)/(947520000*d**12 - 6076784000*d**11*e + 1677232200*d**10*e**2 - 59931
64240*d**9*e**3 - 15153874456*d**8*e**4 + 607741008*d**7*e**5 - 8131500617*d**6*e**6 - 9569972586*d**5*e**7 +
3091977675*d**4*e**8 + 698760764*d**3*e**9 + 9842433*d**2*e**10 - 95316042*d*e**11 + 9092669*e**12)) + (sqrt(1
4)*I*(423*d**2 - 2734*d*e + 293*e**2)/(700*(25*d**4 - 20*d**3*e + 34*d**2*e**2 - 12*d*e**3 + 9*e**4)) + (229*d
**2 - 7*d*e - 136*e**2)/(25*(5*d**2 - 2*d*e + 3*e**2)**2))*log(x + (-7840000000*d**14*(sqrt(14)*I*(423*d**2 -
2734*d*e + 293*e**2)/(700*(25*d**4 - 20*d**3*e + 34*d**2*e**2 - 12*d*e**3 + 9*e**4)) + (229*d**2 - 7*d*e - 136
*e**2)/(25*(5*d**2 - 2*d*e + 3*e**2)**2)) - 4900000000*d**13*e**3*(sqrt(14)*I*(423*d**2 - 2734*d*e + 293*e**2)
/(700*(25*d**4 - 20*d**3*e + 34*d**2*e**2 - 12*d*e**3 + 9*e**4)) + (229*d**2 - 7*d*e - 136*e**2)/(25*(5*d**2 -
2*d*e + 3*e**2)**2))**2 + 5880000000*d**13*e*(sqrt(14)*I*(423*d**2 - 2734*d*e + 293*e**2)/(700*(25*d**4 - 20*
d**3*e + 34*d**2*e**2 - 12*d*e**3 + 9*e**4)) + (229*d**2 - 7*d*e - 136*e**2)/(25*(5*d**2 - 2*d*e + 3*e**2)**2)
) + 7717500000*d**12*e**4*(sqrt(14)*I*(423*d**2 - 2734*d*e + 293*e**2)/(700*(25*d**4 - 20*d**3*e + 34*d**2*e**
2 - 12*d*e**3 + 9*e**4)) + (229*d**2 - 7*d*e - 136*e**2)/(25*(5*d**2 - 2*d*e + 3*e**2)**2))**2 - 21329700000*d
**12*e**2*(sqrt(14)*I*(423*d**2 - 2734*d*e + 293*e**2)/(700*(25*d**4 - 20*d**3*e + 34*d**2*e**2 - 12*d*e**3 +
9*e**4)) + (229*d**2 - 7*d*e - 136*e**2)/(25*(5*d**2 - 2*d*e + 3*e**2)**2)) + 3062080000*d**12 - 19327875000*d
**11*e**5*(sqrt(14)*I*(423*d**2 - 2734*d*e + 293*e**2)/(700*(25*d**4 - 20*d**3*e + 34*d**2*e**2 - 12*d*e**3 +
9*e**4)) + (229*d**2 - 7*d*e - 136*e**2)/(25*(5*d**2 - 2*d*e + 3*e**2)**2))**2 - 5507600000*d**11*e**3*(sqrt(1
4)*I*(423*d**2 - 2734*d*e + 293*e**2)/(700*(25*d**4 - 20*d**3*e + 34*d**2*e**2 - 12*d*e**3 + 9*e**4)) + (229*d
**2 - 7*d*e - 136*e**2)/(25*(5*d**2 - 2*d*e + 3*e**2)**2)) - 1159536000*d**11*e + 10872225000*d**10*e**6*(sqrt
(14)*I*(423*d**2 - 2734*d*e + 293*e**2)/(700*(25*d**4 - 20*d**3*e + 34*d**2*e**2 - 12*d*e**3 + 9*e**4)) + (229
*d**2 - 7*d*e - 136*e**2)/(25*(5*d**2 - 2*d*e + 3*e**2)**2))**2 - 7039144000*d**10*e**4*(sqrt(14)*I*(423*d**2
- 2734*d*e + 293*e**2)/(700*(25*d**4 - 20*d**3*e + 34*d**2*e**2 - 12*d*e**3 + 9*e**4)) + (229*d**2 - 7*d*e - 1
36*e**2)/(25*(5*d**2 - 2*d*e + 3*e**2)**2)) + 2648473800*d**10*e**2 - 10871735000*d**9*e**7*(sqrt(14)*I*(423*d
**2 - 2734*d*e + 293*e**2)/(700*(25*d**4 - 20*d**3*e + 34*d**2*e**2 - 12*d*e**3 + 9*e**4)) + (229*d**2 - 7*d*e
- 136*e**2)/(25*(5*d**2 - 2*d*e + 3*e**2)**2))**2 - 28626939600*d**9*e**5*(sqrt(14)*I*(423*d**2 - 2734*d*e +
293*e**2)/(700*(25*d**4 - 20*d**3*e + 34*d**2*e**2 - 12*d*e**3 + 9*e**4)) + (229*d**2 - 7*d*e - 136*e**2)/(25*
(5*d**2 - 2*d*e + 3*e**2)**2)) + 5631029040*d**9*e**3 - 12890563000*d**8*e**8*(sqrt(14)*I*(423*d**2 - 2734*d*e
+ 293*e**2)/(700*(25*d**4 - 20*d**3*e + 34*d**2*e**2 - 12*d*e**3 + 9*e**4)) + (229*d**2 - 7*d*e - 136*e**2)/(
25*(5*d**2 - 2*d*e + 3*e**2)**2))**2 + 3140906580*d**8*e**6*(sqrt(14)*I*(423*d**2 - 2734*d*e + 293*e**2)/(700*
(25*d**4 - 20*d**3*e + 34*d**2*e**2 - 12*d*e**3 + 9*e**4)) + (229*d**2 - 7*d*e - 136*e**2)/(25*(5*d**2 - 2*d*e
+ 3*e**2)**2)) - 3844841924*d**8*e**4 + 14866261200*d**7*e**9*(sqrt(14)*I*(423*d**2 - 2734*d*e + 293*e**2)/(7
00*(25*d**4 - 20*d**3*e + 34*d**2*e**2 - 12*d*e**3 + 9*e**4)) + (229*d**2 - 7*d*e - 136*e**2)/(25*(5*d**2 - 2*
d*e + 3*e**2)**2))**2 - 16078247136*d**7*e**7*(sqrt(14)*I*(423*d**2 - 2734*d*e + 293*e**2)/(700*(25*d**4 - 20*
d**3*e + 34*d**2*e**2 - 12*d*e**3 + 9*e**4)) + (229*d**2 - 7*d*e - 136*e**2)/(25*(5*d**2 - 2*d*e + 3*e**2)**2)
) - 1183700793*d**7*e**5 - 24188575600*d**6*e**10*(sqrt(14)*I*(423*d**2 - 2734*d*e + 293*e**2)/(700*(25*d**4 -
20*d**3*e + 34*d**2*e**2 - 12*d*e**3 + 9*e**4)) + (229*d**2 - 7*d*e - 136*e**2)/(25*(5*d**2 - 2*d*e + 3*e**2)
**2))**2 - 7728337232*d**6*e**8*(sqrt(14)*I*(423*d**2 - 2734*d*e + 293*e**2)/(700*(25*d**4 - 20*d**3*e + 34*d*
*2*e**2 - 12*d*e**3 + 9*e**4)) + (229*d**2 - 7*d*e - 136*e**2)/(25*(5*d**2 - 2*d*e + 3*e**2)**2)) + 1694057982
*d**6*e**6 + 14439653200*d**5*e**11*(sqrt(14)*I*(423*d**2 - 2734*d*e + 293*e**2)/(700*(25*d**4 - 20*d**3*e + 3
4*d**2*e**2 - 12*d*e**3 + 9*e**4)) + (229*d**2 - 7*d*e - 136*e**2)/(25*(5*d**2 - 2*d*e + 3*e**2)**2))**2 + 228
6078144*d**5*e**9*(sqrt(14)*I*(423*d**2 - 2734*d*e + 293*e**2)/(700*(25*d**4 - 20*d**3*e + 34*d**2*e**2 - 12*d
*e**3 + 9*e**4)) + (229*d**2 - 7*d*e - 136*e**2)/(25*(5*d**2 - 2*d*e + 3*e**2)**2)) - 5520804349*d**5*e**7 - 1
0082618000*d**4*e**12*(sqrt(14)*I*(423*d**2 - 2734*d*e + 293*e**2)/(700*(25*d**4 - 20*d**3*e + 34*d**2*e**2 -
12*d*e**3 + 9*e**4)) + (229*d**2 - 7*d*e - 136*e**2)/(25*(5*d**2 - 2*d*e + 3*e**2)**2))**2 - 7135930760*d**4*e
**10*(sqrt(14)*I*(423*d**2 - 2734*d*e + 293*e**2)/(700*(25*d**4 - 20*d**3*e + 34*d**2*e**2 - 12*d*e**3 + 9*e**
4)) + (229*d**2 - 7*d*e - 136*e**2)/(25*(5*d**2 - 2*d*e + 3*e**2)**2)) - 4247714700*d**4*e**8 + 3006129000*d**
3*e**13*(sqrt(14)*I*(423*d**2 - 2734*d*e + 293*e**2)/(700*(25*d**4 - 20*d**3*e + 34*d**2*e**2 - 12*d*e**3 + 9*
e**4)) + (229*d**2 - 7*d*e - 136*e**2)/(25*(5*d**2 - 2*d*e + 3*e**2)**2))**2 + 2323015520*d**3*e**11*(sqrt(14)
*I*(423*d**2 - 2734*d*e + 293*e**2)/(700*(25*d**4 - 20*d**3*e + 34*d**2*e**2 - 12*d*e**3 + 9*e**4)) + (229*d**
2 - 7*d*e - 136*e**2)/(25*(5*d**2 - 2*d*e + 3*e**2)**2)) + 1298698281*d**3*e**9 - 918199800*d**2*e**14*(sqrt(1
4)*I*(423*d**2 - 2734*d*e + 293*e**2)/(700*(25*d**4 - 20*d**3*e + 34*d**2*e**2 - 12*d*e**3 + 9*e**4)) + (229*d
**2 - 7*d*e - 136*e**2)/(25*(5*d**2 - 2*d*e + 3*e**2)**2))**2 - 1227448656*d**2*e**12*(sqrt(14)*I*(423*d**2 -
2734*d*e + 293*e**2)/(700*(25*d**4 - 20*d**3*e + 34*d**2*e**2 - 12*d*e**3 + 9*e**4)) + (229*d**2 - 7*d*e - 136
*e**2)/(25*(5*d**2 - 2*d*e + 3*e**2)**2)) - 128577018*d**2*e**10 - 38820600*d*e**15*(sqrt(14)*I*(423*d**2 - 27
34*d*e + 293*e**2)/(700*(25*d**4 - 20*d**3*e + 34*d**2*e**2 - 12*d*e**3 + 9*e**4)) + (229*d**2 - 7*d*e - 136*e
**2)/(25*(5*d**2 - 2*d*e + 3*e**2)**2))**2 + 157117968*d*e**13*(sqrt(14)*I*(423*d**2 - 2734*d*e + 293*e**2)/(7
00*(25*d**4 - 20*d**3*e + 34*d**2*e**2 - 12*d*e**3 + 9*e**4)) + (229*d**2 - 7*d*e - 136*e**2)/(25*(5*d**2 - 2*
d*e + 3*e**2)**2)) + 25259757*d*e**11 + 63844200*e**16*(sqrt(14)*I*(423*d**2 - 2734*d*e + 293*e**2)/(700*(25*d
**4 - 20*d**3*e + 34*d**2*e**2 - 12*d*e**3 + 9*e**4)) + (229*d**2 - 7*d*e - 136*e**2)/(25*(5*d**2 - 2*d*e + 3*
e**2)**2))**2 + 38078964*e**14*(sqrt(14)*I*(423*d**2 - 2734*d*e + 293*e**2)/(700*(25*d**4 - 20*d**3*e + 34*d**
2*e**2 - 12*d*e**3 + 9*e**4)) + (229*d**2 - 7*d*e - 136*e**2)/(25*(5*d**2 - 2*d*e + 3*e**2)**2)) + 3442796*e**
12)/(947520000*d**12 - 6076784000*d**11*e + 1677232200*d**10*e**2 - 5993164240*d**9*e**3 - 15153874456*d**8*e*
*4 + 607741008*d**7*e**5 - 8131500617*d**6*e**6 - 9569972586*d**5*e**7 + 3091977675*d**4*e**8 + 698760764*d**3
*e**9 + 9842433*d**2*e**10 - 95316042*d*e**11 + 9092669*e**12)) - (4*d**4 + 5*d**3*e + 3*d**2*e**2 - d*e**3 +
2*e**4)/(5*d**3*e**3 - 2*d**2*e**4 + 3*d*e**5 + x*(5*d**2*e**4 - 2*d*e**5 + 3*e**6)) + 4*x/(5*e**2) - (40*d**5
+ d**4*e + 28*d**3*e**2 + 44*d**2*e**3 - 2*d*e**4 + e**5)*log(x + (7840000000*d**14*(40*d**5 + d**4*e + 28*d*
*3*e**2 + 44*d**2*e**3 - 2*d*e**4 + e**5)/(e**3*(5*d**2 - 2*d*e + 3*e**2)**2) - 5880000000*d**13*(40*d**5 + d*
*4*e + 28*d**3*e**2 + 44*d**2*e**3 - 2*d*e**4 + e**5)/(e**2*(5*d**2 - 2*d*e + 3*e**2)**2) - 4900000000*d**13*(
40*d**5 + d**4*e + 28*d**3*e**2 + 44*d**2*e**3 - 2*d*e**4 + e**5)**2/(e**3*(5*d**2 - 2*d*e + 3*e**2)**4) + 306
2080000*d**12 + 21329700000*d**12*(40*d**5 + d**4*e + 28*d**3*e**2 + 44*d**2*e**3 - 2*d*e**4 + e**5)/(e*(5*d**
2 - 2*d*e + 3*e**2)**2) + 7717500000*d**12*(40*d**5 + d**4*e + 28*d**3*e**2 + 44*d**2*e**3 - 2*d*e**4 + e**5)*
*2/(e**2*(5*d**2 - 2*d*e + 3*e**2)**4) - 1159536000*d**11*e + 5507600000*d**11*(40*d**5 + d**4*e + 28*d**3*e**
2 + 44*d**2*e**3 - 2*d*e**4 + e**5)/(5*d**2 - 2*d*e + 3*e**2)**2 - 19327875000*d**11*(40*d**5 + d**4*e + 28*d*
*3*e**2 + 44*d**2*e**3 - 2*d*e**4 + e**5)**2/(e*(5*d**2 - 2*d*e + 3*e**2)**4) + 2648473800*d**10*e**2 + 703914
4000*d**10*e*(40*d**5 + d**4*e + 28*d**3*e**2 + 44*d**2*e**3 - 2*d*e**4 + e**5)/(5*d**2 - 2*d*e + 3*e**2)**2 +
10872225000*d**10*(40*d**5 + d**4*e + 28*d**3*e**2 + 44*d**2*e**3 - 2*d*e**4 + e**5)**2/(5*d**2 - 2*d*e + 3*e
**2)**4 + 5631029040*d**9*e**3 + 28626939600*d**9*e**2*(40*d**5 + d**4*e + 28*d**3*e**2 + 44*d**2*e**3 - 2*d*e
**4 + e**5)/(5*d**2 - 2*d*e + 3*e**2)**2 - 10871735000*d**9*e*(40*d**5 + d**4*e + 28*d**3*e**2 + 44*d**2*e**3
- 2*d*e**4 + e**5)**2/(5*d**2 - 2*d*e + 3*e**2)**4 - 3844841924*d**8*e**4 - 3140906580*d**8*e**3*(40*d**5 + d*
*4*e + 28*d**3*e**2 + 44*d**2*e**3 - 2*d*e**4 + e**5)/(5*d**2 - 2*d*e + 3*e**2)**2 - 12890563000*d**8*e**2*(40
*d**5 + d**4*e + 28*d**3*e**2 + 44*d**2*e**3 - 2*d*e**4 + e**5)**2/(5*d**2 - 2*d*e + 3*e**2)**4 - 1183700793*d
**7*e**5 + 16078247136*d**7*e**4*(40*d**5 + d**4*e + 28*d**3*e**2 + 44*d**2*e**3 - 2*d*e**4 + e**5)/(5*d**2 -
2*d*e + 3*e**2)**2 + 14866261200*d**7*e**3*(40*d**5 + d**4*e + 28*d**3*e**2 + 44*d**2*e**3 - 2*d*e**4 + e**5)*
*2/(5*d**2 - 2*d*e + 3*e**2)**4 + 1694057982*d**6*e**6 + 7728337232*d**6*e**5*(40*d**5 + d**4*e + 28*d**3*e**2
+ 44*d**2*e**3 - 2*d*e**4 + e**5)/(5*d**2 - 2*d*e + 3*e**2)**2 - 24188575600*d**6*e**4*(40*d**5 + d**4*e + 28
*d**3*e**2 + 44*d**2*e**3 - 2*d*e**4 + e**5)**2/(5*d**2 - 2*d*e + 3*e**2)**4 - 5520804349*d**5*e**7 - 22860781
44*d**5*e**6*(40*d**5 + d**4*e + 28*d**3*e**2 + 44*d**2*e**3 - 2*d*e**4 + e**5)/(5*d**2 - 2*d*e + 3*e**2)**2 +
14439653200*d**5*e**5*(40*d**5 + d**4*e + 28*d**3*e**2 + 44*d**2*e**3 - 2*d*e**4 + e**5)**2/(5*d**2 - 2*d*e +
3*e**2)**4 - 4247714700*d**4*e**8 + 7135930760*d**4*e**7*(40*d**5 + d**4*e + 28*d**3*e**2 + 44*d**2*e**3 - 2*
d*e**4 + e**5)/(5*d**2 - 2*d*e + 3*e**2)**2 - 10082618000*d**4*e**6*(40*d**5 + d**4*e + 28*d**3*e**2 + 44*d**2
*e**3 - 2*d*e**4 + e**5)**2/(5*d**2 - 2*d*e + 3*e**2)**4 + 1298698281*d**3*e**9 - 2323015520*d**3*e**8*(40*d**
5 + d**4*e + 28*d**3*e**2 + 44*d**2*e**3 - 2*d*e**4 + e**5)/(5*d**2 - 2*d*e + 3*e**2)**2 + 3006129000*d**3*e**
7*(40*d**5 + d**4*e + 28*d**3*e**2 + 44*d**2*e**3 - 2*d*e**4 + e**5)**2/(5*d**2 - 2*d*e + 3*e**2)**4 - 1285770
18*d**2*e**10 + 1227448656*d**2*e**9*(40*d**5 + d**4*e + 28*d**3*e**2 + 44*d**2*e**3 - 2*d*e**4 + e**5)/(5*d**
2 - 2*d*e + 3*e**2)**2 - 918199800*d**2*e**8*(40*d**5 + d**4*e + 28*d**3*e**2 + 44*d**2*e**3 - 2*d*e**4 + e**5
)**2/(5*d**2 - 2*d*e + 3*e**2)**4 + 25259757*d*e**11 - 157117968*d*e**10*(40*d**5 + d**4*e + 28*d**3*e**2 + 44
*d**2*e**3 - 2*d*e**4 + e**5)/(5*d**2 - 2*d*e + 3*e**2)**2 - 38820600*d*e**9*(40*d**5 + d**4*e + 28*d**3*e**2
+ 44*d**2*e**3 - 2*d*e**4 + e**5)**2/(5*d**2 - 2*d*e + 3*e**2)**4 + 3442796*e**12 - 38078964*e**11*(40*d**5 +
d**4*e + 28*d**3*e**2 + 44*d**2*e**3 - 2*d*e**4 + e**5)/(5*d**2 - 2*d*e + 3*e**2)**2 + 63844200*e**10*(40*d**5
+ d**4*e + 28*d**3*e**2 + 44*d**2*e**3 - 2*d*e**4 + e**5)**2/(5*d**2 - 2*d*e + 3*e**2)**4)/(947520000*d**12 -
6076784000*d**11*e + 1677232200*d**10*e**2 - 5993164240*d**9*e**3 - 15153874456*d**8*e**4 + 607741008*d**7*e*
*5 - 8131500617*d**6*e**6 - 9569972586*d**5*e**7 + 3091977675*d**4*e**8 + 698760764*d**3*e**9 + 9842433*d**2*e
**10 - 95316042*d*e**11 + 9092669*e**12))/(e**3*(5*d**2 - 2*d*e + 3*e**2)**2)

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Giac [A]  time = 1.17144, size = 479, normalized size = 2.06 \begin{align*} \frac{1}{25} \,{\left (40 \, d + 33 \, e\right )} e^{\left (-3\right )} \log \left (\frac{{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) - \frac{\sqrt{14}{\left (423 \, d^{2} e^{2} - 2734 \, d e^{3} + 293 \, e^{4}\right )} \arctan \left (\frac{1}{14} \, \sqrt{14}{\left (5 \, d - \frac{5 \, d^{2}}{x e + d} + \frac{2 \, d e}{x e + d} - \frac{3 \, e^{2}}{x e + d} - e\right )} e^{\left (-1\right )}\right ) e^{\left (-2\right )}}{350 \,{\left (25 \, d^{4} - 20 \, d^{3} e + 34 \, d^{2} e^{2} - 12 \, d e^{3} + 9 \, e^{4}\right )}} + \frac{4}{5} \,{\left (x e + d\right )} e^{\left (-3\right )} + \frac{{\left (229 \, d^{2} - 7 \, d e - 136 \, e^{2}\right )} \log \left (-\frac{10 \, d}{x e + d} + \frac{5 \, d^{2}}{{\left (x e + d\right )}^{2}} + \frac{2 \, e}{x e + d} - \frac{2 \, d e}{{\left (x e + d\right )}^{2}} + \frac{3 \, e^{2}}{{\left (x e + d\right )}^{2}} + 5\right )}{25 \,{\left (25 \, d^{4} - 20 \, d^{3} e + 34 \, d^{2} e^{2} - 12 \, d e^{3} + 9 \, e^{4}\right )}} - \frac{\frac{4 \, d^{4} e^{3}}{x e + d} + \frac{5 \, d^{3} e^{4}}{x e + d} + \frac{3 \, d^{2} e^{5}}{x e + d} - \frac{d e^{6}}{x e + d} + \frac{2 \, e^{7}}{x e + d}}{5 \, d^{2} e^{6} - 2 \, d e^{7} + 3 \, e^{8}} \end{align*}

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)^2/(5*x^2+2*x+3),x, algorithm="giac")

[Out]

1/25*(40*d + 33*e)*e^(-3)*log(abs(x*e + d)*e^(-1)/(x*e + d)^2) - 1/350*sqrt(14)*(423*d^2*e^2 - 2734*d*e^3 + 29
3*e^4)*arctan(1/14*sqrt(14)*(5*d - 5*d^2/(x*e + d) + 2*d*e/(x*e + d) - 3*e^2/(x*e + d) - e)*e^(-1))*e^(-2)/(25
*d^4 - 20*d^3*e + 34*d^2*e^2 - 12*d*e^3 + 9*e^4) + 4/5*(x*e + d)*e^(-3) + 1/25*(229*d^2 - 7*d*e - 136*e^2)*log
(-10*d/(x*e + d) + 5*d^2/(x*e + d)^2 + 2*e/(x*e + d) - 2*d*e/(x*e + d)^2 + 3*e^2/(x*e + d)^2 + 5)/(25*d^4 - 20
*d^3*e + 34*d^2*e^2 - 12*d*e^3 + 9*e^4) - (4*d^4*e^3/(x*e + d) + 5*d^3*e^4/(x*e + d) + 3*d^2*e^5/(x*e + d) - d
*e^6/(x*e + d) + 2*e^7/(x*e + d))/(5*d^2*e^6 - 2*d*e^7 + 3*e^8)