3.4.9 \(\int \frac {2+x+3 x^2-5 x^3+4 x^4}{(d+e x)^2 (3+2 x+5 x^2)} \, dx\) [309]

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

[Out]

4/5*x/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)/(e*x+d)-(40*d^5+d^4*e+28*d^3*e^2+44*d
^2*e^3-2*d*e^4+e^5)*ln(e*x+d)/e^3/(5*d^2-2*d*e+3*e^2)^2+1/25*(229*d^2-7*d*e-136*e^2)*ln(5*x^2+2*x+3)/(5*d^2-2*
d*e+3*e^2)^2-1/350*(423*d^2-2734*d*e+293*e^2)*arctan(1/14*(1+5*x)*14^(1/2))/(5*d^2-2*d*e+3*e^2)^2*14^(1/2)

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Rubi [A]
time = 0.15, antiderivative size = 233, 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 = {1642, 648, 632, 210, 642} \begin {gather*} -\frac {\text {ArcTan}\left (\frac {5 x+1}{\sqrt {14}}\right ) \left (423 d^2-2734 d e+293 e^2\right )}{25 \sqrt {14} \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 (5 x^2+2 x+3\right )}{25 \left (5 d^2-2 d e+3 e^2\right )^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 {4 x}{5 e^2} \end {gather*}

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 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 632

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 642

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 648

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 1642

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)^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 ) \text {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*}

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Mathematica [A]
time = 0.11, size = 233, normalized size = 1.00 \begin {gather*} \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 {gather*}

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 [A]
time = 0.26, size = 213, normalized size = 0.91

method result size
default \(-\frac {4 d^{4}+5 d^{3} e +3 d^{2} e^{2}-d \,e^{3}+2 e^{4}}{e^{3} \left (5 d^{2}-2 d e +3 e^{2}\right ) \left (e x +d \right )}+\frac {\left (-40 d^{5}-d^{4} e -28 d^{3} e^{2}-44 d^{2} e^{3}+2 d \,e^{4}-e^{5}\right ) \ln \left (e x +d \right )}{e^{3} \left (5 d^{2}-2 d e +3 e^{2}\right )^{2}}+\frac {4 x}{5 e^{2}}+\frac {\frac {\left (458 d^{2}-14 d e -272 e^{2}\right ) \ln \left (5 x^{2}+2 x +3\right )}{10}+\frac {\left (-\frac {423}{5} d^{2}+\frac {2734}{5} d e -\frac {293}{5} e^{2}\right ) \sqrt {14}\, \arctan \left (\frac {\left (10 x +2\right ) \sqrt {14}}{28}\right )}{14}}{5 \left (5 d^{2}-2 d e +3 e^{2}\right )^{2}}\) \(213\)
risch \(\text {Expression too large to display}\) \(14526\)

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,method=_RETURNVERBOSE)

[Out]

-1/e^3*(4*d^4+5*d^3*e+3*d^2*e^2-d*e^3+2*e^4)/(5*d^2-2*d*e+3*e^2)/(e*x+d)+(-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*ln(e*x+d)+4/5*x/e^2+1/5/(5*d^2-2*d*e+3*e^2)^2*(1/10*(458*d^2-14*d*e-272
*e^2)*ln(5*x^2+2*x+3)+1/14*(-423/5*d^2+2734/5*d*e-293/5*e^2)*14^(1/2)*arctan(1/28*(10*x+2)*14^(1/2)))

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Maxima [A]
time = 0.54, size = 274, normalized size = 1.18 \begin {gather*} \frac {4}{5} \, x e^{\left (-2\right )} - \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 (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 {{\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 (x e + 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 {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} + {\left (5 \, d^{2} e^{4} - 2 \, d e^{5} + 3 \, e^{6}\right )} x + 3 \, d e^{5}} \end {gather*}

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]

4/5*x*e^(-2) - 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) + 1/25*(229*d^2 - 7*d*e - 136*e^2)*log(5*x^2 + 2*x + 3)/(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(x*e + d)/(25
*d^4*e^3 - 20*d^3*e^4 + 34*d^2*e^5 - 12*d*e^6 + 9*e^7) - (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 + (5*d^2*e^4 - 2*d*e^5 + 3*e^6)*x + 3*d*e^5)

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Fricas [A]
time = 0.44, size = 398, normalized size = 1.71 \begin {gather*} -\frac {7000 \, d^{6} + \sqrt {14} {\left (423 \, d^{3} e^{3} + 293 \, x e^{6} - {\left (2734 \, d x - 293 \, d\right )} e^{5} + {\left (423 \, d^{2} x - 2734 \, d^{2}\right )} e^{4}\right )} \arctan \left (\frac {1}{14} \, \sqrt {14} {\left (5 \, x + 1\right )}\right ) - 420 \, {\left (6 \, x^{2} - 5\right )} e^{6} + 70 \, {\left (48 \, d x^{2} - 36 \, d x - 35 \, d\right )} e^{5} - 70 \, {\left (136 \, d^{2} x^{2} - 48 \, d^{2} x - 105 \, d^{2}\right )} e^{4} + 280 \, {\left (20 \, d^{3} x^{2} - 34 \, d^{3} x + 5 \, d^{3}\right )} e^{3} - 350 \, {\left (20 \, d^{4} x^{2} - 16 \, d^{4} x - 17 \, d^{4}\right )} e^{2} - 350 \, {\left (20 \, d^{5} x - 17 \, d^{5}\right )} e - 14 \, {\left (229 \, d^{3} e^{3} - 136 \, x e^{6} - {\left (7 \, d x + 136 \, d\right )} e^{5} + {\left (229 \, d^{2} x - 7 \, d^{2}\right )} e^{4}\right )} \log \left (5 \, x^{2} + 2 \, x + 3\right ) + 350 \, {\left (40 \, d^{6} + x e^{6} - {\left (2 \, d x - d\right )} e^{5} + 2 \, {\left (22 \, d^{2} x - d^{2}\right )} e^{4} + 4 \, {\left (7 \, d^{3} x + 11 \, d^{3}\right )} e^{3} + {\left (d^{4} x + 28 \, d^{4}\right )} e^{2} + {\left (40 \, d^{5} x + d^{5}\right )} e\right )} \log \left (x e + d\right )}{350 \, {\left (25 \, d^{5} e^{3} + 9 \, x e^{8} - 3 \, {\left (4 \, d x - 3 \, d\right )} e^{7} + 2 \, {\left (17 \, d^{2} x - 6 \, d^{2}\right )} e^{6} - 2 \, {\left (10 \, d^{3} x - 17 \, d^{3}\right )} e^{5} + 5 \, {\left (5 \, d^{4} x - 4 \, d^{4}\right )} e^{4}\right )}} \end {gather*}

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 + sqrt(14)*(423*d^3*e^3 + 293*x*e^6 - (2734*d*x - 293*d)*e^5 + (423*d^2*x - 2734*d^2)*e^4)*ar
ctan(1/14*sqrt(14)*(5*x + 1)) - 420*(6*x^2 - 5)*e^6 + 70*(48*d*x^2 - 36*d*x - 35*d)*e^5 - 70*(136*d^2*x^2 - 48
*d^2*x - 105*d^2)*e^4 + 280*(20*d^3*x^2 - 34*d^3*x + 5*d^3)*e^3 - 350*(20*d^4*x^2 - 16*d^4*x - 17*d^4)*e^2 - 3
50*(20*d^5*x - 17*d^5)*e - 14*(229*d^3*e^3 - 136*x*e^6 - (7*d*x + 136*d)*e^5 + (229*d^2*x - 7*d^2)*e^4)*log(5*
x^2 + 2*x + 3) + 350*(40*d^6 + x*e^6 - (2*d*x - d)*e^5 + 2*(22*d^2*x - d^2)*e^4 + 4*(7*d^3*x + 11*d^3)*e^3 + (
d^4*x + 28*d^4)*e^2 + (40*d^5*x + d^5)*e)*log(x*e + d))/(25*d^5*e^3 + 9*x*e^8 - 3*(4*d*x - 3*d)*e^7 + 2*(17*d^
2*x - 6*d^2)*e^6 - 2*(10*d^3*x - 17*d^3)*e^5 + 5*(5*d^4*x - 4*d^4)*e^4)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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]

Timed out

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Giac [A]
time = 4.43, size = 355, normalized size = 1.52 \begin {gather*} \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 {gather*}

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)

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Mupad [B]
time = 4.67, size = 312, normalized size = 1.34 \begin {gather*} \frac {4\,x}{5\,e^2}-\frac {\ln \left (x+\frac {1}{5}-\frac {\sqrt {14}\,1{}\mathrm {i}}{5}\right )\,\left (\left (\frac {423\,\sqrt {14}}{700}-\frac {229}{25}{}\mathrm {i}\right )\,d^2+\left (-\frac {1367\,\sqrt {14}}{350}+\frac {7}{25}{}\mathrm {i}\right )\,d\,e+\left (\frac {293\,\sqrt {14}}{700}+\frac {136}{25}{}\mathrm {i}\right )\,e^2\right )}{d^4\,25{}\mathrm {i}-d^3\,e\,20{}\mathrm {i}+d^2\,e^2\,34{}\mathrm {i}-d\,e^3\,12{}\mathrm {i}+e^4\,9{}\mathrm {i}}+\frac {\ln \left (x+\frac {1}{5}+\frac {\sqrt {14}\,1{}\mathrm {i}}{5}\right )\,\left (\left (\frac {423\,\sqrt {14}}{700}+\frac {229}{25}{}\mathrm {i}\right )\,d^2+\left (-\frac {1367\,\sqrt {14}}{350}-\frac {7}{25}{}\mathrm {i}\right )\,d\,e+\left (\frac {293\,\sqrt {14}}{700}-\frac {136}{25}{}\mathrm {i}\right )\,e^2\right )}{d^4\,25{}\mathrm {i}-d^3\,e\,20{}\mathrm {i}+d^2\,e^2\,34{}\mathrm {i}-d\,e^3\,12{}\mathrm {i}+e^4\,9{}\mathrm {i}}-\frac {5\,\left (4\,d^4+5\,d^3\,e+3\,d^2\,e^2-d\,e^3+2\,e^4\right )}{e\,\left (5\,x\,e^3+5\,d\,e^2\right )\,\left (5\,d^2-2\,d\,e+3\,e^2\right )}-\frac {\ln \left (d+e\,x\right )\,\left (40\,d^5+d^4\,e+28\,d^3\,e^2+44\,d^2\,e^3-2\,d\,e^4+e^5\right )}{e^3\,{\left (5\,d^2-2\,d\,e+3\,e^2\right )}^2} \end {gather*}

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*(2*x + 5*x^2 + 3)),x)

[Out]

(4*x)/(5*e^2) - (log(x - (14^(1/2)*1i)/5 + 1/5)*(d^2*((423*14^(1/2))/700 - 229i/25) + e^2*((293*14^(1/2))/700
+ 136i/25) - d*e*((1367*14^(1/2))/350 - 7i/25)))/(d^4*25i - d^3*e*20i - d*e^3*12i + e^4*9i + d^2*e^2*34i) + (l
og(x + (14^(1/2)*1i)/5 + 1/5)*(d^2*((423*14^(1/2))/700 + 229i/25) + e^2*((293*14^(1/2))/700 - 136i/25) - d*e*(
(1367*14^(1/2))/350 + 7i/25)))/(d^4*25i - d^3*e*20i - d*e^3*12i + e^4*9i + d^2*e^2*34i) - (5*(5*d^3*e - d*e^3
+ 4*d^4 + 2*e^4 + 3*d^2*e^2))/(e*(5*d*e^2 + 5*e^3*x)*(5*d^2 - 2*d*e + 3*e^2)) - (log(d + e*x)*(d^4*e - 2*d*e^4
 + 40*d^5 + e^5 + 44*d^2*e^3 + 28*d^3*e^2))/(e^3*(5*d^2 - 2*d*e + 3*e^2)^2)

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