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

Optimal. Leaf size=329 \[ -\frac{x (423 d-1367 e)+1367 d-293 e}{1400 \left (5 x^2+2 x+3\right )^2 \left (5 d^2-2 d e+3 e^2\right )}+\frac{25 x \left (-9033 d^2 e+2203 d^3+3635 d e^2-1829 e^3\right )-92989 d^2 e+171735 d^3+36207 d e^2+1831 e^3}{39200 \left (5 x^2+2 x+3\right ) \left (5 d^2-2 d e+3 e^2\right )^2}-\frac{e \left (3 d^2 e^2+5 d^3 e+4 d^4-d e^3+2 e^4\right ) \log \left (5 x^2+2 x+3\right )}{2 \left (5 d^2-2 d e+3 e^2\right )^3}+\frac{e \left (3 d^2 e^2+5 d^3 e+4 d^4-d e^3+2 e^4\right ) \log (d+e x)}{\left (5 d^2-2 d e+3 e^2\right )^3}+\frac{\left (58530 d^3 e^2-56058 d^2 e^3-16643 d^4 e+42375 d^5+31811 d e^4-8623 e^5\right ) \tan ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{1568 \sqrt{14} \left (5 d^2-2 d e+3 e^2\right )^3} \]

[Out]

-(1367*d - 293*e + (423*d - 1367*e)*x)/(1400*(5*d^2 - 2*d*e + 3*e^2)*(3 + 2*x + 5*x^2)^2) + (171735*d^3 - 9298
9*d^2*e + 36207*d*e^2 + 1831*e^3 + 25*(2203*d^3 - 9033*d^2*e + 3635*d*e^2 - 1829*e^3)*x)/(39200*(5*d^2 - 2*d*e
+ 3*e^2)^2*(3 + 2*x + 5*x^2)) + ((42375*d^5 - 16643*d^4*e + 58530*d^3*e^2 - 56058*d^2*e^3 + 31811*d*e^4 - 862
3*e^5)*ArcTan[(1 + 5*x)/Sqrt[14]])/(1568*Sqrt[14]*(5*d^2 - 2*d*e + 3*e^2)^3) + (e*(4*d^4 + 5*d^3*e + 3*d^2*e^2
- d*e^3 + 2*e^4)*Log[d + e*x])/(5*d^2 - 2*d*e + 3*e^2)^3 - (e*(4*d^4 + 5*d^3*e + 3*d^2*e^2 - d*e^3 + 2*e^4)*L
og[3 + 2*x + 5*x^2])/(2*(5*d^2 - 2*d*e + 3*e^2)^3)

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Rubi [A]  time = 0.496008, antiderivative size = 329, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {1646, 800, 634, 618, 204, 628} \[ -\frac{x (423 d-1367 e)+1367 d-293 e}{1400 \left (5 x^2+2 x+3\right )^2 \left (5 d^2-2 d e+3 e^2\right )}+\frac{25 x \left (-9033 d^2 e+2203 d^3+3635 d e^2-1829 e^3\right )-92989 d^2 e+171735 d^3+36207 d e^2+1831 e^3}{39200 \left (5 x^2+2 x+3\right ) \left (5 d^2-2 d e+3 e^2\right )^2}-\frac{e \left (3 d^2 e^2+5 d^3 e+4 d^4-d e^3+2 e^4\right ) \log \left (5 x^2+2 x+3\right )}{2 \left (5 d^2-2 d e+3 e^2\right )^3}+\frac{e \left (3 d^2 e^2+5 d^3 e+4 d^4-d e^3+2 e^4\right ) \log (d+e x)}{\left (5 d^2-2 d e+3 e^2\right )^3}+\frac{\left (58530 d^3 e^2-56058 d^2 e^3-16643 d^4 e+42375 d^5+31811 d e^4-8623 e^5\right ) \tan ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{1568 \sqrt{14} \left (5 d^2-2 d e+3 e^2\right )^3} \]

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)^3),x]

[Out]

-(1367*d - 293*e + (423*d - 1367*e)*x)/(1400*(5*d^2 - 2*d*e + 3*e^2)*(3 + 2*x + 5*x^2)^2) + (171735*d^3 - 9298
9*d^2*e + 36207*d*e^2 + 1831*e^3 + 25*(2203*d^3 - 9033*d^2*e + 3635*d*e^2 - 1829*e^3)*x)/(39200*(5*d^2 - 2*d*e
+ 3*e^2)^2*(3 + 2*x + 5*x^2)) + ((42375*d^5 - 16643*d^4*e + 58530*d^3*e^2 - 56058*d^2*e^3 + 31811*d*e^4 - 862
3*e^5)*ArcTan[(1 + 5*x)/Sqrt[14]])/(1568*Sqrt[14]*(5*d^2 - 2*d*e + 3*e^2)^3) + (e*(4*d^4 + 5*d^3*e + 3*d^2*e^2
- d*e^3 + 2*e^4)*Log[d + e*x])/(5*d^2 - 2*d*e + 3*e^2)^3 - (e*(4*d^4 + 5*d^3*e + 3*d^2*e^2 - d*e^3 + 2*e^4)*L
og[3 + 2*x + 5*x^2])/(2*(5*d^2 - 2*d*e + 3*e^2)^3)

Rule 1646

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = Polynomi
alQuotient[(d + e*x)^m*Pq, a + b*x + c*x^2, x], f = Coeff[PolynomialRemainder[(d + e*x)^m*Pq, a + b*x + c*x^2,
x], x, 0], g = Coeff[PolynomialRemainder[(d + e*x)^m*Pq, a + b*x + c*x^2, x], x, 1]}, Simp[((b*f - 2*a*g + (2
*c*f - b*g)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)), x] + Dist[1/((p + 1)*(b^2 - 4*a*c)), Int[(d
+ e*x)^m*(a + b*x + c*x^2)^(p + 1)*ExpandToSum[((p + 1)*(b^2 - 4*a*c)*Q)/(d + e*x)^m - ((2*p + 3)*(2*c*f - b*
g))/(d + e*x)^m, x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2
- b*d*e + a*e^2, 0] && LtQ[p, -1] && ILtQ[m, 0]

Rule 800

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[Exp
andIntegrand[((d + e*x)^m*(f + g*x))/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 -
4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[m]

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) \left (3+2 x+5 x^2\right )^3} \, dx &=-\frac{1367 d-293 e+(423 d-1367 e) x}{1400 \left (5 d^2-2 d e+3 e^2\right ) \left (3+2 x+5 x^2\right )^2}+\frac{1}{112} \int \frac{\frac{2 \left (3267 d^2-2843 d e+2800 e^2\right )}{25 \left (5 d^2-2 d e+3 e^2\right )}-\frac{6 \left (3080 d^2-809 d e+481 e^2\right ) x}{25 \left (5 d^2-2 d e+3 e^2\right )}+\frac{448 x^2}{5}}{(d+e x) \left (3+2 x+5 x^2\right )^2} \, dx\\ &=-\frac{1367 d-293 e+(423 d-1367 e) x}{1400 \left (5 d^2-2 d e+3 e^2\right ) \left (3+2 x+5 x^2\right )^2}+\frac{171735 d^3-92989 d^2 e+36207 d e^2+1831 e^3+25 \left (2203 d^3-9033 d^2 e+3635 d e^2-1829 e^3\right ) x}{39200 \left (5 d^2-2 d e+3 e^2\right )^2 \left (3+2 x+5 x^2\right )}+\frac{\int \frac{\frac{4 \left (8475 d^4-1193 d^3 e+8339 d^2 e^2-3397 d e^3+3136 e^4\right )}{\left (5 d^2-2 d e+3 e^2\right )^2}+\frac{4 e \left (2203 d^3-9033 d^2 e+3635 d e^2-1829 e^3\right ) x}{\left (5 d^2-2 d e+3 e^2\right )^2}}{(d+e x) \left (3+2 x+5 x^2\right )} \, dx}{6272}\\ &=-\frac{1367 d-293 e+(423 d-1367 e) x}{1400 \left (5 d^2-2 d e+3 e^2\right ) \left (3+2 x+5 x^2\right )^2}+\frac{171735 d^3-92989 d^2 e+36207 d e^2+1831 e^3+25 \left (2203 d^3-9033 d^2 e+3635 d e^2-1829 e^3\right ) x}{39200 \left (5 d^2-2 d e+3 e^2\right )^2 \left (3+2 x+5 x^2\right )}+\frac{\int \left (\frac{6272 e^2 \left (4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4\right )}{\left (5 d^2-2 d e+3 e^2\right )^3 (d+e x)}+\frac{4 \left (42375 d^5-22915 d^4 e+50690 d^3 e^2-60762 d^2 e^3+33379 d e^4-11759 e^5-7840 e \left (4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4\right ) x\right )}{\left (5 d^2-2 d e+3 e^2\right )^3 \left (3+2 x+5 x^2\right )}\right ) \, dx}{6272}\\ &=-\frac{1367 d-293 e+(423 d-1367 e) x}{1400 \left (5 d^2-2 d e+3 e^2\right ) \left (3+2 x+5 x^2\right )^2}+\frac{171735 d^3-92989 d^2 e+36207 d e^2+1831 e^3+25 \left (2203 d^3-9033 d^2 e+3635 d e^2-1829 e^3\right ) x}{39200 \left (5 d^2-2 d e+3 e^2\right )^2 \left (3+2 x+5 x^2\right )}+\frac{e \left (4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4\right ) \log (d+e x)}{\left (5 d^2-2 d e+3 e^2\right )^3}+\frac{\int \frac{42375 d^5-22915 d^4 e+50690 d^3 e^2-60762 d^2 e^3+33379 d e^4-11759 e^5-7840 e \left (4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4\right ) x}{3+2 x+5 x^2} \, dx}{1568 \left (5 d^2-2 d e+3 e^2\right )^3}\\ &=-\frac{1367 d-293 e+(423 d-1367 e) x}{1400 \left (5 d^2-2 d e+3 e^2\right ) \left (3+2 x+5 x^2\right )^2}+\frac{171735 d^3-92989 d^2 e+36207 d e^2+1831 e^3+25 \left (2203 d^3-9033 d^2 e+3635 d e^2-1829 e^3\right ) x}{39200 \left (5 d^2-2 d e+3 e^2\right )^2 \left (3+2 x+5 x^2\right )}+\frac{e \left (4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4\right ) \log (d+e x)}{\left (5 d^2-2 d e+3 e^2\right )^3}-\frac{\left (e \left (4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4\right )\right ) \int \frac{2+10 x}{3+2 x+5 x^2} \, dx}{2 \left (5 d^2-2 d e+3 e^2\right )^3}+\frac{\left (42375 d^5-16643 d^4 e+58530 d^3 e^2-56058 d^2 e^3+31811 d e^4-8623 e^5\right ) \int \frac{1}{3+2 x+5 x^2} \, dx}{1568 \left (5 d^2-2 d e+3 e^2\right )^3}\\ &=-\frac{1367 d-293 e+(423 d-1367 e) x}{1400 \left (5 d^2-2 d e+3 e^2\right ) \left (3+2 x+5 x^2\right )^2}+\frac{171735 d^3-92989 d^2 e+36207 d e^2+1831 e^3+25 \left (2203 d^3-9033 d^2 e+3635 d e^2-1829 e^3\right ) x}{39200 \left (5 d^2-2 d e+3 e^2\right )^2 \left (3+2 x+5 x^2\right )}+\frac{e \left (4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4\right ) \log (d+e x)}{\left (5 d^2-2 d e+3 e^2\right )^3}-\frac{e \left (4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4\right ) \log \left (3+2 x+5 x^2\right )}{2 \left (5 d^2-2 d e+3 e^2\right )^3}-\frac{\left (42375 d^5-16643 d^4 e+58530 d^3 e^2-56058 d^2 e^3+31811 d e^4-8623 e^5\right ) \operatorname{Subst}\left (\int \frac{1}{-56-x^2} \, dx,x,2+10 x\right )}{784 \left (5 d^2-2 d e+3 e^2\right )^3}\\ &=-\frac{1367 d-293 e+(423 d-1367 e) x}{1400 \left (5 d^2-2 d e+3 e^2\right ) \left (3+2 x+5 x^2\right )^2}+\frac{171735 d^3-92989 d^2 e+36207 d e^2+1831 e^3+25 \left (2203 d^3-9033 d^2 e+3635 d e^2-1829 e^3\right ) x}{39200 \left (5 d^2-2 d e+3 e^2\right )^2 \left (3+2 x+5 x^2\right )}+\frac{\left (42375 d^5-16643 d^4 e+58530 d^3 e^2-56058 d^2 e^3+31811 d e^4-8623 e^5\right ) \tan ^{-1}\left (\frac{1+5 x}{\sqrt{14}}\right )}{1568 \sqrt{14} \left (5 d^2-2 d e+3 e^2\right )^3}+\frac{e \left (4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4\right ) \log (d+e x)}{\left (5 d^2-2 d e+3 e^2\right )^3}-\frac{e \left (4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4\right ) \log \left (3+2 x+5 x^2\right )}{2 \left (5 d^2-2 d e+3 e^2\right )^3}\\ \end{align*}

Mathematica [A]  time = 0.29743, size = 282, normalized size = 0.86 \[ \frac{\frac{392 \left (5 d^2-2 d e+3 e^2\right )^2 (e (1367 x+293)-d (423 x+1367))}{\left (5 x^2+2 x+3\right )^2}+\frac{14 \left (5 d^2-2 d e+3 e^2\right ) \left (-d^2 e (225825 x+92989)+5 d^3 (11015 x+34347)+d e^2 (90875 x+36207)+e^3 (1831-45725 x)\right )}{5 x^2+2 x+3}-274400 e \left (3 d^2 e^2+5 d^3 e+4 d^4-d e^3+2 e^4\right ) \log \left (5 x^2+2 x+3\right )+548800 e \left (3 d^2 e^2+5 d^3 e+4 d^4-d e^3+2 e^4\right ) \log (d+e x)+25 \sqrt{14} \left (58530 d^3 e^2-56058 d^2 e^3-16643 d^4 e+42375 d^5+31811 d e^4-8623 e^5\right ) \tan ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{548800 \left (5 d^2-2 d e+3 e^2\right )^3} \]

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)^3),x]

[Out]

((392*(5*d^2 - 2*d*e + 3*e^2)^2*(-(d*(1367 + 423*x)) + e*(293 + 1367*x)))/(3 + 2*x + 5*x^2)^2 + (14*(5*d^2 - 2
*d*e + 3*e^2)*(e^3*(1831 - 45725*x) + 5*d^3*(34347 + 11015*x) + d*e^2*(36207 + 90875*x) - d^2*e*(92989 + 22582
5*x)))/(3 + 2*x + 5*x^2) + 25*Sqrt[14]*(42375*d^5 - 16643*d^4*e + 58530*d^3*e^2 - 56058*d^2*e^3 + 31811*d*e^4
- 8623*e^5)*ArcTan[(1 + 5*x)/Sqrt[14]] + 548800*e*(4*d^4 + 5*d^3*e + 3*d^2*e^2 - d*e^3 + 2*e^4)*Log[d + e*x] -
274400*e*(4*d^4 + 5*d^3*e + 3*d^2*e^2 - d*e^3 + 2*e^4)*Log[3 + 2*x + 5*x^2])/(548800*(5*d^2 - 2*d*e + 3*e^2)^
3)

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Maple [B]  time = 0.071, size = 1437, normalized size = 4.4 \begin{align*} \text{result too large to display} \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)/(5*x^2+2*x+3)^3,x)

[Out]

4*e/(5*d^2-2*d*e+3*e^2)^3*ln(e*x+d)*d^4-27435/1568/(5*d^2-2*d*e+3*e^2)^3/(5*x^2+2*x+3)^2*x^3*e^5+193765/1568/(
5*d^2-2*d*e+3*e^2)^3/(5*x^2+2*x+3)^2*x^2*d^5-49377/7840/(5*d^2-2*d*e+3*e^2)^3/(5*x^2+2*x+3)^2*x^2*e^5-8623/219
52/(5*d^2-2*d*e+3*e^2)^3*14^(1/2)*arctan(1/28*(10*x+2)*14^(1/2))*e^5+3*e^3/(5*d^2-2*d*e+3*e^2)^3*ln(e*x+d)*d^2
-e^4/(5*d^2-2*d*e+3*e^2)^3*ln(e*x+d)*d+5*e^2/(5*d^2-2*d*e+3*e^2)^3*ln(e*x+d)*d^3+64765/1568/(5*d^2-2*d*e+3*e^2
)^3/(5*x^2+2*x+3)^2*d^5+18063/7840/(5*d^2-2*d*e+3*e^2)^3/(5*x^2+2*x+3)^2*e^5-1/(5*d^2-2*d*e+3*e^2)^3*ln(5*x^2+
2*x+3)*e^5+2*e^5/(5*d^2-2*d*e+3*e^2)^3*ln(e*x+d)-11211/7840/(5*d^2-2*d*e+3*e^2)^3/(5*x^2+2*x+3)^2*x*e^5-25611/
7840/(5*d^2-2*d*e+3*e^2)^3/(5*x^2+2*x+3)^2*d*e^4-3/2/(5*d^2-2*d*e+3*e^2)^3*ln(5*x^2+2*x+3)*d^2*e^3-5/2/(5*d^2-
2*d*e+3*e^2)^3*ln(5*x^2+2*x+3)*d^3*e^2-58185/1568/(5*d^2-2*d*e+3*e^2)^3/(5*x^2+2*x+3)^2*d^4*e+118119/3920/(5*d
^2-2*d*e+3*e^2)^3/(5*x^2+2*x+3)^2*d^3*e^2-28843/3920/(5*d^2-2*d*e+3*e^2)^3/(5*x^2+2*x+3)^2*d^2*e^3+42375/21952
/(5*d^2-2*d*e+3*e^2)^3*14^(1/2)*arctan(1/28*(10*x+2)*14^(1/2))*d^5-250449/3920/(5*d^2-2*d*e+3*e^2)^3/(5*x^2+2*
x+3)^2*x*d^2*e^3+147247/7840/(5*d^2-2*d*e+3*e^2)^3/(5*x^2+2*x+3)^2*x*d*e^4+1/2/(5*d^2-2*d*e+3*e^2)^3*ln(5*x^2+
2*x+3)*d*e^4-2/(5*d^2-2*d*e+3*e^2)^3*ln(5*x^2+2*x+3)*d^4*e+55075/1568/(5*d^2-2*d*e+3*e^2)^3/(5*x^2+2*x+3)^2*x^
3*d^5+89895/1568/(5*d^2-2*d*e+3*e^2)^3/(5*x^2+2*x+3)^2*d^5*x+31811/21952/(5*d^2-2*d*e+3*e^2)^3*14^(1/2)*arctan
(1/28*(10*x+2)*14^(1/2))*d*e^4+29265/10976/(5*d^2-2*d*e+3*e^2)^3*14^(1/2)*arctan(1/28*(10*x+2)*14^(1/2))*d^3*e
^2-247855/1568/(5*d^2-2*d*e+3*e^2)^3/(5*x^2+2*x+3)^2*x^3*d^4*e+107125/784/(5*d^2-2*d*e+3*e^2)^3/(5*x^2+2*x+3)^
2*x^3*d^3*e^2-108785/784/(5*d^2-2*d*e+3*e^2)^3/(5*x^2+2*x+3)^2*x^3*d^2*e^3+72815/1568/(5*d^2-2*d*e+3*e^2)^3/(5
*x^2+2*x+3)^2*x^3*d*e^4-388683/3920/(5*d^2-2*d*e+3*e^2)^3/(5*x^2+2*x+3)^2*x^2*d^2*e^3+250589/7840/(5*d^2-2*d*e
+3*e^2)^3/(5*x^2+2*x+3)^2*x^2*d*e^4-16643/21952/(5*d^2-2*d*e+3*e^2)^3*14^(1/2)*arctan(1/28*(10*x+2)*14^(1/2))*
d^4*e-28029/10976/(5*d^2-2*d*e+3*e^2)^3*14^(1/2)*arctan(1/28*(10*x+2)*14^(1/2))*d^2*e^3-260825/1568/(5*d^2-2*d
*e+3*e^2)^3/(5*x^2+2*x+3)^2*x^2*d^4*e+655359/3920/(5*d^2-2*d*e+3*e^2)^3/(5*x^2+2*x+3)^2*x^2*d^3*e^2+380997/392
0/(5*d^2-2*d*e+3*e^2)^3/(5*x^2+2*x+3)^2*x*d^3*e^2-165635/1568/(5*d^2-2*d*e+3*e^2)^3/(5*x^2+2*x+3)^2*x*d^4*e

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Maxima [A]  time = 1.57757, size = 771, normalized size = 2.34 \begin{align*} \frac{\sqrt{14}{\left (42375 \, d^{5} - 16643 \, d^{4} e + 58530 \, d^{3} e^{2} - 56058 \, d^{2} e^{3} + 31811 \, d e^{4} - 8623 \, e^{5}\right )} \arctan \left (\frac{1}{14} \, \sqrt{14}{\left (5 \, x + 1\right )}\right )}{21952 \,{\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 (4 \, d^{4} e + 5 \, d^{3} e^{2} + 3 \, d^{2} e^{3} - d e^{4} + 2 \, e^{5}\right )} \log \left (e x + d\right )}{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}} - \frac{{\left (4 \, d^{4} e + 5 \, d^{3} e^{2} + 3 \, d^{2} e^{3} - d e^{4} + 2 \, e^{5}\right )} \log \left (5 \, x^{2} + 2 \, x + 3\right )}{2 \,{\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{25 \,{\left (2203 \, d^{3} - 9033 \, d^{2} e + 3635 \, d e^{2} - 1829 \, e^{3}\right )} x^{3} + 64765 \, d^{3} - 32279 \, d^{2} e - 4523 \, d e^{2} + 6021 \, e^{3} +{\left (193765 \, d^{3} - 183319 \, d^{2} e + 72557 \, d e^{2} - 16459 \, e^{3}\right )} x^{2} +{\left (89895 \, d^{3} - 129677 \, d^{2} e + 46591 \, d e^{2} - 3737 \, e^{3}\right )} x}{7840 \,{\left (25 \,{\left (25 \, d^{4} - 20 \, d^{3} e + 34 \, d^{2} e^{2} - 12 \, d e^{3} + 9 \, e^{4}\right )} x^{4} + 225 \, d^{4} - 180 \, d^{3} e + 306 \, d^{2} e^{2} - 108 \, d e^{3} + 81 \, e^{4} + 20 \,{\left (25 \, d^{4} - 20 \, d^{3} e + 34 \, d^{2} e^{2} - 12 \, d e^{3} + 9 \, e^{4}\right )} x^{3} + 34 \,{\left (25 \, d^{4} - 20 \, d^{3} e + 34 \, d^{2} e^{2} - 12 \, d e^{3} + 9 \, e^{4}\right )} x^{2} + 12 \,{\left (25 \, d^{4} - 20 \, d^{3} e + 34 \, d^{2} e^{2} - 12 \, d e^{3} + 9 \, e^{4}\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)/(5*x^2+2*x+3)^3,x, algorithm="maxima")

[Out]

1/21952*sqrt(14)*(42375*d^5 - 16643*d^4*e + 58530*d^3*e^2 - 56058*d^2*e^3 + 31811*d*e^4 - 8623*e^5)*arctan(1/1
4*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) + (4
*d^4*e + 5*d^3*e^2 + 3*d^2*e^3 - d*e^4 + 2*e^5)*log(e*x + d)/(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) - 1/2*(4*d^4*e + 5*d^3*e^2 + 3*d^2*e^3 - d*e^4 + 2*e^5)*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) + 1/7840*(25*(2203*d^3 -
9033*d^2*e + 3635*d*e^2 - 1829*e^3)*x^3 + 64765*d^3 - 32279*d^2*e - 4523*d*e^2 + 6021*e^3 + (193765*d^3 - 183
319*d^2*e + 72557*d*e^2 - 16459*e^3)*x^2 + (89895*d^3 - 129677*d^2*e + 46591*d*e^2 - 3737*e^3)*x)/(25*(25*d^4
- 20*d^3*e + 34*d^2*e^2 - 12*d*e^3 + 9*e^4)*x^4 + 225*d^4 - 180*d^3*e + 306*d^2*e^2 - 108*d*e^3 + 81*e^4 + 20*
(25*d^4 - 20*d^3*e + 34*d^2*e^2 - 12*d*e^3 + 9*e^4)*x^3 + 34*(25*d^4 - 20*d^3*e + 34*d^2*e^2 - 12*d*e^3 + 9*e^
4)*x^2 + 12*(25*d^4 - 20*d^3*e + 34*d^2*e^2 - 12*d*e^3 + 9*e^4)*x)

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Fricas [B]  time = 2.32411, size = 2678, normalized size = 8.14 \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)/(5*x^2+2*x+3)^3,x, algorithm="fricas")

[Out]

1/109760*(4533550*d^5 - 4072950*d^4*e + 3307332*d^3*e^2 - 807604*d^2*e^3 - 358554*d*e^4 + 252882*e^5 + 350*(11
015*d^5 - 49571*d^4*e + 42850*d^3*e^2 - 43514*d^2*e^3 + 14563*d*e^4 - 5487*e^5)*x^3 + 14*(968825*d^5 - 1304125
*d^4*e + 1310718*d^3*e^2 - 777366*d^2*e^3 + 250589*d*e^4 - 49377*e^5)*x^2 + 5*sqrt(14)*(381375*d^5 - 149787*d^
4*e + 526770*d^3*e^2 - 504522*d^2*e^3 + 286299*d*e^4 - 77607*e^5 + 25*(42375*d^5 - 16643*d^4*e + 58530*d^3*e^2
- 56058*d^2*e^3 + 31811*d*e^4 - 8623*e^5)*x^4 + 20*(42375*d^5 - 16643*d^4*e + 58530*d^3*e^2 - 56058*d^2*e^3 +
31811*d*e^4 - 8623*e^5)*x^3 + 34*(42375*d^5 - 16643*d^4*e + 58530*d^3*e^2 - 56058*d^2*e^3 + 31811*d*e^4 - 862
3*e^5)*x^2 + 12*(42375*d^5 - 16643*d^4*e + 58530*d^3*e^2 - 56058*d^2*e^3 + 31811*d*e^4 - 8623*e^5)*x)*arctan(1
/14*sqrt(14)*(5*x + 1)) + 14*(449475*d^5 - 828175*d^4*e + 761994*d^3*e^2 - 500898*d^2*e^3 + 147247*d*e^4 - 112
11*e^5)*x + 109760*(36*d^4*e + 45*d^3*e^2 + 27*d^2*e^3 - 9*d*e^4 + 18*e^5 + 25*(4*d^4*e + 5*d^3*e^2 + 3*d^2*e^
3 - d*e^4 + 2*e^5)*x^4 + 20*(4*d^4*e + 5*d^3*e^2 + 3*d^2*e^3 - d*e^4 + 2*e^5)*x^3 + 34*(4*d^4*e + 5*d^3*e^2 +
3*d^2*e^3 - d*e^4 + 2*e^5)*x^2 + 12*(4*d^4*e + 5*d^3*e^2 + 3*d^2*e^3 - d*e^4 + 2*e^5)*x)*log(e*x + d) - 54880*
(36*d^4*e + 45*d^3*e^2 + 27*d^2*e^3 - 9*d*e^4 + 18*e^5 + 25*(4*d^4*e + 5*d^3*e^2 + 3*d^2*e^3 - d*e^4 + 2*e^5)*
x^4 + 20*(4*d^4*e + 5*d^3*e^2 + 3*d^2*e^3 - d*e^4 + 2*e^5)*x^3 + 34*(4*d^4*e + 5*d^3*e^2 + 3*d^2*e^3 - d*e^4 +
2*e^5)*x^2 + 12*(4*d^4*e + 5*d^3*e^2 + 3*d^2*e^3 - d*e^4 + 2*e^5)*x)*log(5*x^2 + 2*x + 3))/(1125*d^6 - 1350*d
^5*e + 2565*d^4*e^2 - 1692*d^3*e^3 + 1539*d^2*e^4 - 486*d*e^5 + 243*e^6 + 25*(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)*x^4 + 20*(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)*x^3 + 34*(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)*x^2 + 12*(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)*
x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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)/(5*x**2+2*x+3)**3,x)

[Out]

Timed out

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Giac [A]  time = 1.15806, size = 621, normalized size = 1.89 \begin{align*} \frac{\sqrt{14}{\left (42375 \, d^{5} - 16643 \, d^{4} e + 58530 \, d^{3} e^{2} - 56058 \, d^{2} e^{3} + 31811 \, d e^{4} - 8623 \, e^{5}\right )} \arctan \left (\frac{1}{14} \, \sqrt{14}{\left (5 \, x + 1\right )}\right )}{21952 \,{\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 (4 \, d^{4} e + 5 \, d^{3} e^{2} + 3 \, d^{2} e^{3} - d e^{4} + 2 \, e^{5}\right )} \log \left (5 \, x^{2} + 2 \, x + 3\right )}{2 \,{\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 (4 \, d^{4} e^{2} + 5 \, d^{3} e^{3} + 3 \, d^{2} e^{4} - d e^{5} + 2 \, e^{6}\right )} \log \left ({\left | x e + d \right |}\right )}{125 \, d^{6} e - 150 \, d^{5} e^{2} + 285 \, d^{4} e^{3} - 188 \, d^{3} e^{4} + 171 \, d^{2} e^{5} - 54 \, d e^{6} + 27 \, e^{7}} + \frac{323825 \, d^{5} - 290925 \, d^{4} e + 25 \,{\left (11015 \, d^{5} - 49571 \, d^{4} e + 42850 \, d^{3} e^{2} - 43514 \, d^{2} e^{3} + 14563 \, d e^{4} - 5487 \, e^{5}\right )} x^{3} + 236238 \, d^{3} e^{2} +{\left (968825 \, d^{5} - 1304125 \, d^{4} e + 1310718 \, d^{3} e^{2} - 777366 \, d^{2} e^{3} + 250589 \, d e^{4} - 49377 \, e^{5}\right )} x^{2} - 57686 \, d^{2} e^{3} +{\left (449475 \, d^{5} - 828175 \, d^{4} e + 761994 \, d^{3} e^{2} - 500898 \, d^{2} e^{3} + 147247 \, d e^{4} - 11211 \, e^{5}\right )} x - 25611 \, d e^{4} + 18063 \, e^{5}}{7840 \,{\left (5 \, d^{2} - 2 \, d e + 3 \, e^{2}\right )}^{3}{\left (5 \, x^{2} + 2 \, x + 3\right )}^{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)/(5*x^2+2*x+3)^3,x, algorithm="giac")

[Out]

1/21952*sqrt(14)*(42375*d^5 - 16643*d^4*e + 58530*d^3*e^2 - 56058*d^2*e^3 + 31811*d*e^4 - 8623*e^5)*arctan(1/1
4*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/
2*(4*d^4*e + 5*d^3*e^2 + 3*d^2*e^3 - d*e^4 + 2*e^5)*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) + (4*d^4*e^2 + 5*d^3*e^3 + 3*d^2*e^4 - d*e^5 + 2*e^6)*log(abs(x
*e + d))/(125*d^6*e - 150*d^5*e^2 + 285*d^4*e^3 - 188*d^3*e^4 + 171*d^2*e^5 - 54*d*e^6 + 27*e^7) + 1/7840*(323
825*d^5 - 290925*d^4*e + 25*(11015*d^5 - 49571*d^4*e + 42850*d^3*e^2 - 43514*d^2*e^3 + 14563*d*e^4 - 5487*e^5)
*x^3 + 236238*d^3*e^2 + (968825*d^5 - 1304125*d^4*e + 1310718*d^3*e^2 - 777366*d^2*e^3 + 250589*d*e^4 - 49377*
e^5)*x^2 - 57686*d^2*e^3 + (449475*d^5 - 828175*d^4*e + 761994*d^3*e^2 - 500898*d^2*e^3 + 147247*d*e^4 - 11211
*e^5)*x - 25611*d*e^4 + 18063*e^5)/((5*d^2 - 2*d*e + 3*e^2)^3*(5*x^2 + 2*x + 3)^2)