3.4.10 \(\int \frac {2+x+3 x^2-5 x^3+4 x^4}{(d+e x)^3 (3+2 x+5 x^2)} \, dx\) [310]

Optimal. Leaf size=317 \[ -\frac {4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4}{2 e^3 \left (5 d^2-2 d e+3 e^2\right ) (d+e x)^2}+\frac {40 d^5+d^4 e+28 d^3 e^2+44 d^2 e^3-2 d e^4+e^5}{e^3 \left (5 d^2-2 d e+3 e^2\right )^2 (d+e x)}-\frac {\left (423 d^3-4101 d^2 e+879 d e^2+703 e^3\right ) \tan ^{-1}\left (\frac {1+5 x}{\sqrt {14}}\right )}{5 \sqrt {14} \left (5 d^2-2 d e+3 e^2\right )^3}+\frac {\left (100 d^6-120 d^5 e+228 d^4 e^2-242 d^3 e^3+141 d^2 e^4+120 d e^5-e^6\right ) \log (d+e x)}{e^3 \left (5 d^2-2 d e+3 e^2\right )^3}+\frac {\left (458 d^3-21 d^2 e-816 d e^2+113 e^3\right ) \log \left (3+2 x+5 x^2\right )}{10 \left (5 d^2-2 d e+3 e^2\right )^3} \]

[Out]

1/2*(-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)^2+(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/(e*x+d)+(100*d^6-120*d^5*e+228*d^4*e^2-242*d^3*e^3+141*d^2*e^4+120*d
*e^5-e^6)*ln(e*x+d)/e^3/(5*d^2-2*d*e+3*e^2)^3+1/10*(458*d^3-21*d^2*e-816*d*e^2+113*e^3)*ln(5*x^2+2*x+3)/(5*d^2
-2*d*e+3*e^2)^3-1/70*(423*d^3-4101*d^2*e+879*d*e^2+703*e^3)*arctan(1/14*(1+5*x)*14^(1/2))/(5*d^2-2*d*e+3*e^2)^
3*14^(1/2)

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Rubi [A]
time = 0.18, antiderivative size = 317, 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^3-4101 d^2 e+879 d e^2+703 e^3\right )}{5 \sqrt {14} \left (5 d^2-2 d e+3 e^2\right )^3}+\frac {\left (458 d^3-21 d^2 e-816 d e^2+113 e^3\right ) \log \left (5 x^2+2 x+3\right )}{10 \left (5 d^2-2 d e+3 e^2\right )^3}-\frac {4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4}{2 e^3 \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^3 \left (5 d^2-2 d e+3 e^2\right )^2 (d+e x)}+\frac {\left (100 d^6-120 d^5 e+228 d^4 e^2-242 d^3 e^3+141 d^2 e^4+120 d e^5-e^6\right ) \log (d+e x)}{e^3 \left (5 d^2-2 d e+3 e^2\right )^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*(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)^2) + (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*(d + e*x)) - ((423*d^3 - 4101*d^2*e
+ 879*d*e^2 + 703*e^3)*ArcTan[(1 + 5*x)/Sqrt[14]])/(5*Sqrt[14]*(5*d^2 - 2*d*e + 3*e^2)^3) + ((100*d^6 - 120*d
^5*e + 228*d^4*e^2 - 242*d^3*e^3 + 141*d^2*e^4 + 120*d*e^5 - e^6)*Log[d + e*x])/(e^3*(5*d^2 - 2*d*e + 3*e^2)^3
) + ((458*d^3 - 21*d^2*e - 816*d*e^2 + 113*e^3)*Log[3 + 2*x + 5*x^2])/(10*(5*d^2 - 2*d*e + 3*e^2)^3)

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)^3 \left (3+2 x+5 x^2\right )} \, dx &=\int \left (\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)^3}+\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)^2}+\frac {100 d^6-120 d^5 e+228 d^4 e^2-242 d^3 e^3+141 d^2 e^4+120 d e^5-e^6}{e^2 \left (5 d^2-2 d e+3 e^2\right )^3 (d+e x)}+\frac {7 d^3+816 d^2 e-339 d e^2-118 e^3+\left (458 d^3-21 d^2 e-816 d e^2+113 e^3\right ) x}{\left (5 d^2-2 d e+3 e^2\right )^3 \left (3+2 x+5 x^2\right )}\right ) \, dx\\ &=-\frac {4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4}{2 e^3 \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^3 \left (5 d^2-2 d e+3 e^2\right )^2 (d+e x)}+\frac {\left (100 d^6-120 d^5 e+228 d^4 e^2-242 d^3 e^3+141 d^2 e^4+120 d e^5-e^6\right ) \log (d+e x)}{e^3 \left (5 d^2-2 d e+3 e^2\right )^3}+\frac {\int \frac {7 d^3+816 d^2 e-339 d e^2-118 e^3+\left (458 d^3-21 d^2 e-816 d e^2+113 e^3\right ) x}{3+2 x+5 x^2} \, dx}{\left (5 d^2-2 d e+3 e^2\right )^3}\\ &=-\frac {4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4}{2 e^3 \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^3 \left (5 d^2-2 d e+3 e^2\right )^2 (d+e x)}+\frac {\left (100 d^6-120 d^5 e+228 d^4 e^2-242 d^3 e^3+141 d^2 e^4+120 d e^5-e^6\right ) \log (d+e x)}{e^3 \left (5 d^2-2 d e+3 e^2\right )^3}+\frac {\left (458 d^3-21 d^2 e-816 d e^2+113 e^3\right ) \int \frac {2+10 x}{3+2 x+5 x^2} \, dx}{10 \left (5 d^2-2 d e+3 e^2\right )^3}-\frac {\left (423 d^3-4101 d^2 e+879 d e^2+703 e^3\right ) \int \frac {1}{3+2 x+5 x^2} \, dx}{5 \left (5 d^2-2 d e+3 e^2\right )^3}\\ &=-\frac {4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4}{2 e^3 \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^3 \left (5 d^2-2 d e+3 e^2\right )^2 (d+e x)}+\frac {\left (100 d^6-120 d^5 e+228 d^4 e^2-242 d^3 e^3+141 d^2 e^4+120 d e^5-e^6\right ) \log (d+e x)}{e^3 \left (5 d^2-2 d e+3 e^2\right )^3}+\frac {\left (458 d^3-21 d^2 e-816 d e^2+113 e^3\right ) \log \left (3+2 x+5 x^2\right )}{10 \left (5 d^2-2 d e+3 e^2\right )^3}+\frac {\left (2 \left (423 d^3-4101 d^2 e+879 d e^2+703 e^3\right )\right ) \text {Subst}\left (\int \frac {1}{-56-x^2} \, dx,x,2+10 x\right )}{5 \left (5 d^2-2 d e+3 e^2\right )^3}\\ &=-\frac {4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4}{2 e^3 \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^3 \left (5 d^2-2 d e+3 e^2\right )^2 (d+e x)}-\frac {\left (423 d^3-4101 d^2 e+879 d e^2+703 e^3\right ) \tan ^{-1}\left (\frac {1+5 x}{\sqrt {14}}\right )}{5 \sqrt {14} \left (5 d^2-2 d e+3 e^2\right )^3}+\frac {\left (100 d^6-120 d^5 e+228 d^4 e^2-242 d^3 e^3+141 d^2 e^4+120 d e^5-e^6\right ) \log (d+e x)}{e^3 \left (5 d^2-2 d e+3 e^2\right )^3}+\frac {\left (458 d^3-21 d^2 e-816 d e^2+113 e^3\right ) \log \left (3+2 x+5 x^2\right )}{10 \left (5 d^2-2 d e+3 e^2\right )^3}\\ \end {align*}

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Mathematica [A]
time = 0.28, size = 278, normalized size = 0.88 \begin {gather*} -\frac {\frac {35 \left (5 d^2-2 d e+3 e^2\right )^2 \left (4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4\right )}{e^3 (d+e x)^2}-\frac {70 \left (5 d^2-2 d e+3 e^2\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 (d+e x)}+\sqrt {14} \left (423 d^3-4101 d^2 e+879 d e^2+703 e^3\right ) \tan ^{-1}\left (\frac {1+5 x}{\sqrt {14}}\right )+\frac {70 \left (-100 d^6+120 d^5 e-228 d^4 e^2+242 d^3 e^3-141 d^2 e^4-120 d e^5+e^6\right ) \log (d+e x)}{e^3}-7 \left (458 d^3-21 d^2 e-816 d e^2+113 e^3\right ) \log \left (3+2 x+5 x^2\right )}{70 \left (5 d^2-2 d e+3 e^2\right )^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/70*((35*(5*d^2 - 2*d*e + 3*e^2)^2*(4*d^4 + 5*d^3*e + 3*d^2*e^2 - d*e^3 + 2*e^4))/(e^3*(d + e*x)^2) - (70*(5
*d^2 - 2*d*e + 3*e^2)*(40*d^5 + d^4*e + 28*d^3*e^2 + 44*d^2*e^3 - 2*d*e^4 + e^5))/(e^3*(d + e*x)) + Sqrt[14]*(
423*d^3 - 4101*d^2*e + 879*d*e^2 + 703*e^3)*ArcTan[(1 + 5*x)/Sqrt[14]] + (70*(-100*d^6 + 120*d^5*e - 228*d^4*e
^2 + 242*d^3*e^3 - 141*d^2*e^4 - 120*d*e^5 + e^6)*Log[d + e*x])/e^3 - 7*(458*d^3 - 21*d^2*e - 816*d*e^2 + 113*
e^3)*Log[3 + 2*x + 5*x^2])/(5*d^2 - 2*d*e + 3*e^2)^3

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Maple [A]
time = 0.25, size = 298, normalized size = 0.94

method	result	size
default	\(-\frac {4 d^{4}+5 d^{3} e +3 d^{2} e^{2}-d \,e^{3}+2 e^{4}}{2 e^{3} \left (5 d^{2}-2 d e +3 e^{2}\right ) \left (e x +d \right )^{2}}-\frac {-40 d^{5}-d^{4} e -28 d^{3} e^{2}-44 d^{2} e^{3}+2 d \,e^{4}-e^{5}}{e^{3} \left (5 d^{2}-2 d e +3 e^{2}\right )^{2} \left (e x +d \right )}+\frac {\left (100 d^{6}-120 d^{5} e +228 d^{4} e^{2}-242 d^{3} e^{3}+141 d^{2} e^{4}+120 d \,e^{5}-e^{6}\right ) \ln \left (e x +d \right )}{e^{3} \left (5 d^{2}-2 d e +3 e^{2}\right )^{3}}+\frac {\frac {\left (458 d^{3}-21 d^{2} e -816 d \,e^{2}+113 e^{3}\right ) \ln \left (5 x^{2}+2 x +3\right )}{10}+\frac {\left (-\frac {423}{5} d^{3}+\frac {4101}{5} d^{2} e -\frac {879}{5} d \,e^{2}-\frac {703}{5} e^{3}\right ) \sqrt {14}\, \arctan \left (\frac {\left (10 x +2\right ) \sqrt {14}}{28}\right )}{14}}{\left (5 d^{2}-2 d e +3 e^{2}\right )^{3}}\)	\(298\)
risch	\(\text {Expression too large to display}\)	\(24519\)

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)^3/(5*x^2+2*x+3),x,method=_RETURNVERBOSE)

[Out]

-1/2*(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)^2-(-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/(e*x+d)+(100*d^6-120*d^5*e+228*d^4*e^2-242*d^3*e^3+141*d^2*e^4+120*
d*e^5-e^6)*ln(e*x+d)/e^3/(5*d^2-2*d*e+3*e^2)^3+1/(5*d^2-2*d*e+3*e^2)^3*(1/10*(458*d^3-21*d^2*e-816*d*e^2+113*e
^3)*ln(5*x^2+2*x+3)+1/14*(-423/5*d^3+4101/5*d^2*e-879/5*d*e^2-703/5*e^3)*14^(1/2)*arctan(1/28*(10*x+2)*14^(1/2
)))

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Maxima [A]
time = 0.54, size = 455, normalized size = 1.44 \begin {gather*} -\frac {\sqrt {14} {\left (423 \, d^{3} - 4101 \, d^{2} e + 879 \, d e^{2} + 703 \, e^{3}\right )} \arctan \left (\frac {1}{14} \, \sqrt {14} {\left (5 \, x + 1\right )}\right )}{70 \, {\left (125 \, d^{6} - 150 \, d^{5} e + 285 \, d^{4} e^{2} - 188 \, d^{3} e^{3} + 171 \, d^{2} e^{4} - 54 \, d e^{5} + 27 \, e^{6}\right )}} + \frac {{\left (458 \, d^{3} - 21 \, d^{2} e - 816 \, d e^{2} + 113 \, e^{3}\right )} \log \left (5 \, x^{2} + 2 \, x + 3\right )}{10 \, {\left (125 \, d^{6} - 150 \, d^{5} e + 285 \, d^{4} e^{2} - 188 \, d^{3} e^{3} + 171 \, d^{2} e^{4} - 54 \, d e^{5} + 27 \, e^{6}\right )}} + \frac {{\left (100 \, d^{6} - 120 \, d^{5} e + 228 \, d^{4} e^{2} - 242 \, d^{3} e^{3} + 141 \, d^{2} e^{4} + 120 \, d e^{5} - e^{6}\right )} \log \left (x e + d\right )}{125 \, d^{6} e^{3} - 150 \, d^{5} e^{4} + 285 \, d^{4} e^{5} - 188 \, d^{3} e^{6} + 171 \, d^{2} e^{7} - 54 \, d e^{8} + 27 \, e^{9}} + \frac {60 \, d^{6} - 15 \, d^{5} e + 39 \, d^{4} e^{2} + 84 \, d^{3} e^{3} - 25 \, d^{2} e^{4} + 2 \, {\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 + 9 \, d e^{5} - 6 \, e^{6}}{2 \, {\left (25 \, d^{6} e^{3} - 20 \, d^{5} e^{4} + 34 \, d^{4} e^{5} - 12 \, d^{3} e^{6} + {\left (25 \, d^{4} e^{5} - 20 \, d^{3} e^{6} + 34 \, d^{2} e^{7} - 12 \, d e^{8} + 9 \, e^{9}\right )} x^{2} + 9 \, d^{2} e^{7} + 2 \, {\left (25 \, d^{5} e^{4} - 20 \, d^{4} e^{5} + 34 \, d^{3} e^{6} - 12 \, d^{2} e^{7} + 9 \, d e^{8}\right )} x\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)^3/(5*x^2+2*x+3),x, algorithm="maxima")

[Out]

-1/70*sqrt(14)*(423*d^3 - 4101*d^2*e + 879*d*e^2 + 703*e^3)*arctan(1/14*sqrt(14)*(5*x + 1))/(125*d^6 - 150*d^5
*e + 285*d^4*e^2 - 188*d^3*e^3 + 171*d^2*e^4 - 54*d*e^5 + 27*e^6) + 1/10*(458*d^3 - 21*d^2*e - 816*d*e^2 + 113
*e^3)*log(5*x^2 + 2*x + 3)/(125*d^6 - 150*d^5*e + 285*d^4*e^2 - 188*d^3*e^3 + 171*d^2*e^4 - 54*d*e^5 + 27*e^6)
+ (100*d^6 - 120*d^5*e + 228*d^4*e^2 - 242*d^3*e^3 + 141*d^2*e^4 + 120*d*e^5 - e^6)*log(x*e + d)/(125*d^6*e^3
- 150*d^5*e^4 + 285*d^4*e^5 - 188*d^3*e^6 + 171*d^2*e^7 - 54*d*e^8 + 27*e^9) + 1/2*(60*d^6 - 15*d^5*e + 39*d^
4*e^2 + 84*d^3*e^3 - 25*d^2*e^4 + 2*(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 + 9*d*e^5
- 6*e^6)/(25*d^6*e^3 - 20*d^5*e^4 + 34*d^4*e^5 - 12*d^3*e^6 + (25*d^4*e^5 - 20*d^3*e^6 + 34*d^2*e^7 - 12*d*e^
8 + 9*e^9)*x^2 + 9*d^2*e^7 + 2*(25*d^5*e^4 - 20*d^4*e^5 + 34*d^3*e^6 - 12*d^2*e^7 + 9*d*e^8)*x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 675 vs. \(2 (297) = 594\).
time = 0.55, size = 675, normalized size = 2.13 \begin {gather*} \frac {10500 \, d^{8} - \sqrt {14} {\left (423 \, d^{5} e^{3} + 703 \, x^{2} e^{8} + {\left (879 \, d x^{2} + 1406 \, d x\right )} e^{7} - {\left (4101 \, d^{2} x^{2} - 1758 \, d^{2} x - 703 \, d^{2}\right )} e^{6} + 3 \, {\left (141 \, d^{3} x^{2} - 2734 \, d^{3} x + 293 \, d^{3}\right )} e^{5} + 3 \, {\left (282 \, d^{4} x - 1367 \, d^{4}\right )} e^{4}\right )} \arctan \left (\frac {1}{14} \, \sqrt {14} {\left (5 \, x + 1\right )}\right ) + 210 \, {\left (x - 3\right )} e^{8} - 35 \, {\left (16 \, d x - 39 \, d\right )} e^{7} + 105 \, {\left (94 \, d^{2} x - 41 \, d^{2}\right )} e^{6} - 35 \, {\left (28 \, d^{3} x - 347 \, d^{3}\right )} e^{5} + 70 \, {\left (167 \, d^{4} x - 88 \, d^{4}\right )} e^{4} + 105 \, {\left (172 \, d^{5} x + 99 \, d^{5}\right )} e^{3} - 525 \, {\left (10 \, d^{6} x - 27 \, d^{6}\right )} e^{2} + 175 \, {\left (80 \, d^{7} x - 39 \, d^{7}\right )} e + 7 \, {\left (458 \, d^{5} e^{3} + 113 \, x^{2} e^{8} - 2 \, {\left (408 \, d x^{2} - 113 \, d x\right )} e^{7} - {\left (21 \, d^{2} x^{2} + 1632 \, d^{2} x - 113 \, d^{2}\right )} e^{6} + 2 \, {\left (229 \, d^{3} x^{2} - 21 \, d^{3} x - 408 \, d^{3}\right )} e^{5} + {\left (916 \, d^{4} x - 21 \, d^{4}\right )} e^{4}\right )} \log \left (5 \, x^{2} + 2 \, x + 3\right ) + 70 \, {\left (100 \, d^{8} - x^{2} e^{8} + 2 \, {\left (60 \, d x^{2} - d x\right )} e^{7} + {\left (141 \, d^{2} x^{2} + 240 \, d^{2} x - d^{2}\right )} e^{6} - 2 \, {\left (121 \, d^{3} x^{2} - 141 \, d^{3} x - 60 \, d^{3}\right )} e^{5} + {\left (228 \, d^{4} x^{2} - 484 \, d^{4} x + 141 \, d^{4}\right )} e^{4} - 2 \, {\left (60 \, d^{5} x^{2} - 228 \, d^{5} x + 121 \, d^{5}\right )} e^{3} + 4 \, {\left (25 \, d^{6} x^{2} - 60 \, d^{6} x + 57 \, d^{6}\right )} e^{2} + 40 \, {\left (5 \, d^{7} x - 3 \, d^{7}\right )} e\right )} \log \left (x e + d\right )}{70 \, {\left (125 \, d^{8} e^{3} + 27 \, x^{2} e^{11} - 54 \, {\left (d x^{2} - d x\right )} e^{10} + 9 \, {\left (19 \, d^{2} x^{2} - 12 \, d^{2} x + 3 \, d^{2}\right )} e^{9} - 2 \, {\left (94 \, d^{3} x^{2} - 171 \, d^{3} x + 27 \, d^{3}\right )} e^{8} + {\left (285 \, d^{4} x^{2} - 376 \, d^{4} x + 171 \, d^{4}\right )} e^{7} - 2 \, {\left (75 \, d^{5} x^{2} - 285 \, d^{5} x + 94 \, d^{5}\right )} e^{6} + 5 \, {\left (25 \, d^{6} x^{2} - 60 \, d^{6} x + 57 \, d^{6}\right )} e^{5} + 50 \, {\left (5 \, d^{7} x - 3 \, d^{7}\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)^3/(5*x^2+2*x+3),x, algorithm="fricas")

[Out]

1/70*(10500*d^8 - sqrt(14)*(423*d^5*e^3 + 703*x^2*e^8 + (879*d*x^2 + 1406*d*x)*e^7 - (4101*d^2*x^2 - 1758*d^2*
x - 703*d^2)*e^6 + 3*(141*d^3*x^2 - 2734*d^3*x + 293*d^3)*e^5 + 3*(282*d^4*x - 1367*d^4)*e^4)*arctan(1/14*sqrt
(14)*(5*x + 1)) + 210*(x - 3)*e^8 - 35*(16*d*x - 39*d)*e^7 + 105*(94*d^2*x - 41*d^2)*e^6 - 35*(28*d^3*x - 347*
d^3)*e^5 + 70*(167*d^4*x - 88*d^4)*e^4 + 105*(172*d^5*x + 99*d^5)*e^3 - 525*(10*d^6*x - 27*d^6)*e^2 + 175*(80*
d^7*x - 39*d^7)*e + 7*(458*d^5*e^3 + 113*x^2*e^8 - 2*(408*d*x^2 - 113*d*x)*e^7 - (21*d^2*x^2 + 1632*d^2*x - 11
3*d^2)*e^6 + 2*(229*d^3*x^2 - 21*d^3*x - 408*d^3)*e^5 + (916*d^4*x - 21*d^4)*e^4)*log(5*x^2 + 2*x + 3) + 70*(1
00*d^8 - x^2*e^8 + 2*(60*d*x^2 - d*x)*e^7 + (141*d^2*x^2 + 240*d^2*x - d^2)*e^6 - 2*(121*d^3*x^2 - 141*d^3*x -
60*d^3)*e^5 + (228*d^4*x^2 - 484*d^4*x + 141*d^4)*e^4 - 2*(60*d^5*x^2 - 228*d^5*x + 121*d^5)*e^3 + 4*(25*d^6*
x^2 - 60*d^6*x + 57*d^6)*e^2 + 40*(5*d^7*x - 3*d^7)*e)*log(x*e + d))/(125*d^8*e^3 + 27*x^2*e^11 - 54*(d*x^2 -
d*x)*e^10 + 9*(19*d^2*x^2 - 12*d^2*x + 3*d^2)*e^9 - 2*(94*d^3*x^2 - 171*d^3*x + 27*d^3)*e^8 + (285*d^4*x^2 - 3
76*d^4*x + 171*d^4)*e^7 - 2*(75*d^5*x^2 - 285*d^5*x + 94*d^5)*e^6 + 5*(25*d^6*x^2 - 60*d^6*x + 57*d^6)*e^5 + 5
0*(5*d^7*x - 3*d^7)*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)**3/(5*x**2+2*x+3),x)

[Out]

Timed out

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Giac [A]
time = 3.71, size = 406, normalized size = 1.28 \begin {gather*} -\frac {\sqrt {14} {\left (423 \, d^{3} - 4101 \, d^{2} e + 879 \, d e^{2} + 703 \, e^{3}\right )} \arctan \left (\frac {1}{14} \, \sqrt {14} {\left (5 \, x + 1\right )}\right )}{70 \, {\left (125 \, d^{6} - 150 \, d^{5} e + 285 \, d^{4} e^{2} - 188 \, d^{3} e^{3} + 171 \, d^{2} e^{4} - 54 \, d e^{5} + 27 \, e^{6}\right )}} + \frac {{\left (458 \, d^{3} - 21 \, d^{2} e - 816 \, d e^{2} + 113 \, e^{3}\right )} \log \left (5 \, x^{2} + 2 \, x + 3\right )}{10 \, {\left (125 \, d^{6} - 150 \, d^{5} e + 285 \, d^{4} e^{2} - 188 \, d^{3} e^{3} + 171 \, d^{2} e^{4} - 54 \, d e^{5} + 27 \, e^{6}\right )}} + \frac {{\left (100 \, d^{6} - 120 \, d^{5} e + 228 \, d^{4} e^{2} - 242 \, d^{3} e^{3} + 141 \, d^{2} e^{4} + 120 \, d e^{5} - e^{6}\right )} \log \left ({\left | x e + d \right |}\right )}{125 \, d^{6} e^{3} - 150 \, d^{5} e^{4} + 285 \, d^{4} e^{5} - 188 \, d^{3} e^{6} + 171 \, d^{2} e^{7} - 54 \, d e^{8} + 27 \, e^{9}} + \frac {{\left (2 \, {\left (200 \, d^{7} - 75 \, d^{6} e + 258 \, d^{5} e^{2} + 167 \, d^{4} e^{3} - 14 \, d^{3} e^{4} + 141 \, d^{2} e^{5} - 8 \, d e^{6} + 3 \, e^{7}\right )} x + {\left (300 \, d^{8} - 195 \, d^{7} e + 405 \, d^{6} e^{2} + 297 \, d^{5} e^{3} - 176 \, d^{4} e^{4} + 347 \, d^{3} e^{5} - 123 \, d^{2} e^{6} + 39 \, d e^{7} - 18 \, e^{8}\right )} e^{\left (-1\right )}\right )} e^{\left (-2\right )}}{2 \, {\left (5 \, d^{2} - 2 \, d e + 3 \, e^{2}\right )}^{3} {\left (x e + d\right )}^{2}} \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)^3/(5*x^2+2*x+3),x, algorithm="giac")

[Out]

-1/70*sqrt(14)*(423*d^3 - 4101*d^2*e + 879*d*e^2 + 703*e^3)*arctan(1/14*sqrt(14)*(5*x + 1))/(125*d^6 - 150*d^5
*e + 285*d^4*e^2 - 188*d^3*e^3 + 171*d^2*e^4 - 54*d*e^5 + 27*e^6) + 1/10*(458*d^3 - 21*d^2*e - 816*d*e^2 + 113
*e^3)*log(5*x^2 + 2*x + 3)/(125*d^6 - 150*d^5*e + 285*d^4*e^2 - 188*d^3*e^3 + 171*d^2*e^4 - 54*d*e^5 + 27*e^6)
+ (100*d^6 - 120*d^5*e + 228*d^4*e^2 - 242*d^3*e^3 + 141*d^2*e^4 + 120*d*e^5 - e^6)*log(abs(x*e + d))/(125*d^
6*e^3 - 150*d^5*e^4 + 285*d^4*e^5 - 188*d^3*e^6 + 171*d^2*e^7 - 54*d*e^8 + 27*e^9) + 1/2*(2*(200*d^7 - 75*d^6*
e + 258*d^5*e^2 + 167*d^4*e^3 - 14*d^3*e^4 + 141*d^2*e^5 - 8*d*e^6 + 3*e^7)*x + (300*d^8 - 195*d^7*e + 405*d^6
*e^2 + 297*d^5*e^3 - 176*d^4*e^4 + 347*d^3*e^5 - 123*d^2*e^6 + 39*d*e^7 - 18*e^8)*e^(-1))*e^(-2)/((5*d^2 - 2*d
*e + 3*e^2)^3*(x*e + d)^2)

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Mupad [B]
time = 4.76, size = 493, normalized size = 1.56 \begin {gather*} \frac {\frac {60\,d^6-15\,d^5\,e+39\,d^4\,e^2+84\,d^3\,e^3-25\,d^2\,e^4+9\,d\,e^5-6\,e^6}{2\,e^3\,\left (25\,d^4-20\,d^3\,e+34\,d^2\,e^2-12\,d\,e^3+9\,e^4\right )}+\frac {x\,\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^2\,\left (25\,d^4-20\,d^3\,e+34\,d^2\,e^2-12\,d\,e^3+9\,e^4\right )}}{d^2+2\,d\,e\,x+e^2\,x^2}-\frac {\ln \left (x+\frac {1}{5}-\frac {\sqrt {14}\,1{}\mathrm {i}}{5}\right )\,\left (\left (\frac {423\,\sqrt {14}}{140}-\frac {229}{5}{}\mathrm {i}\right )\,d^3+\left (-\frac {4101\,\sqrt {14}}{140}+\frac {21}{10}{}\mathrm {i}\right )\,d^2\,e+\left (\frac {879\,\sqrt {14}}{140}+\frac {408}{5}{}\mathrm {i}\right )\,d\,e^2+\left (\frac {703\,\sqrt {14}}{140}-\frac {113}{10}{}\mathrm {i}\right )\,e^3\right )}{d^6\,125{}\mathrm {i}-d^5\,e\,150{}\mathrm {i}+d^4\,e^2\,285{}\mathrm {i}-d^3\,e^3\,188{}\mathrm {i}+d^2\,e^4\,171{}\mathrm {i}-d\,e^5\,54{}\mathrm {i}+e^6\,27{}\mathrm {i}}+\frac {\ln \left (x+\frac {1}{5}+\frac {\sqrt {14}\,1{}\mathrm {i}}{5}\right )\,\left (\left (\frac {423\,\sqrt {14}}{140}+\frac {229}{5}{}\mathrm {i}\right )\,d^3+\left (-\frac {4101\,\sqrt {14}}{140}-\frac {21}{10}{}\mathrm {i}\right )\,d^2\,e+\left (\frac {879\,\sqrt {14}}{140}-\frac {408}{5}{}\mathrm {i}\right )\,d\,e^2+\left (\frac {703\,\sqrt {14}}{140}+\frac {113}{10}{}\mathrm {i}\right )\,e^3\right )}{d^6\,125{}\mathrm {i}-d^5\,e\,150{}\mathrm {i}+d^4\,e^2\,285{}\mathrm {i}-d^3\,e^3\,188{}\mathrm {i}+d^2\,e^4\,171{}\mathrm {i}-d\,e^5\,54{}\mathrm {i}+e^6\,27{}\mathrm {i}}+\frac {\ln \left (d+e\,x\right )\,\left (100\,d^6-120\,d^5\,e+228\,d^4\,e^2-242\,d^3\,e^3+141\,d^2\,e^4+120\,d\,e^5-e^6\right )}{e^3\,{\left (5\,d^2-2\,d\,e+3\,e^2\right )}^3} \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)^3*(2*x + 5*x^2 + 3)),x)

[Out]

((9*d*e^5 - 15*d^5*e + 60*d^6 - 6*e^6 - 25*d^2*e^4 + 84*d^3*e^3 + 39*d^4*e^2)/(2*e^3*(25*d^4 - 20*d^3*e - 12*d
*e^3 + 9*e^4 + 34*d^2*e^2)) + (x*(d^4*e - 2*d*e^4 + 40*d^5 + e^5 + 44*d^2*e^3 + 28*d^3*e^2))/(e^2*(25*d^4 - 20
*d^3*e - 12*d*e^3 + 9*e^4 + 34*d^2*e^2)))/(d^2 + e^2*x^2 + 2*d*e*x) - (log(x - (14^(1/2)*1i)/5 + 1/5)*(d^3*((4
23*14^(1/2))/140 - 229i/5) + e^3*((703*14^(1/2))/140 - 113i/10) + d*e^2*((879*14^(1/2))/140 + 408i/5) - d^2*e*
((4101*14^(1/2))/140 - 21i/10)))/(d^6*125i - d^5*e*150i - d*e^5*54i + e^6*27i + d^2*e^4*171i - d^3*e^3*188i +
d^4*e^2*285i) + (log(x + (14^(1/2)*1i)/5 + 1/5)*(d^3*((423*14^(1/2))/140 + 229i/5) + e^3*((703*14^(1/2))/140 +
113i/10) + d*e^2*((879*14^(1/2))/140 - 408i/5) - d^2*e*((4101*14^(1/2))/140 + 21i/10)))/(d^6*125i - d^5*e*150
i - d*e^5*54i + e^6*27i + d^2*e^4*171i - d^3*e^3*188i + d^4*e^2*285i) + (log(d + e*x)*(120*d*e^5 - 120*d^5*e +
100*d^6 - e^6 + 141*d^2*e^4 - 242*d^3*e^3 + 228*d^4*e^2))/(e^3*(5*d^2 - 2*d*e + 3*e^2)^3)

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