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

Optimal. Leaf size=99 \[ \frac {1}{75} x^3 (20 d-33 e)-\frac {3}{250} x^2 (55 d-27 e)+\frac {(2290 d-881 e) \log \left (5 x^2+2 x+3\right )}{6250}+\frac {1}{625} x (405 d+458 e)-\frac {(2115 d+5989 e) \tan ^{-1}\left (\frac {5 x+1}{\sqrt {14}}\right )}{3125 \sqrt {14}}+\frac {e x^4}{5} \]

[Out]

1/625*(405*d+458*e)*x-3/250*(55*d-27*e)*x^2+1/75*(20*d-33*e)*x^3+1/5*e*x^4+1/6250*(2290*d-881*e)*ln(5*x^2+2*x+
3)-1/43750*(2115*d+5989*e)*arctan(1/14*(1+5*x)*14^(1/2))*14^(1/2)

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Rubi [A]  time = 0.11, antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.139, Rules used = {1628, 634, 618, 204, 628} \[ \frac {1}{75} x^3 (20 d-33 e)-\frac {3}{250} x^2 (55 d-27 e)+\frac {(2290 d-881 e) \log \left (5 x^2+2 x+3\right )}{6250}+\frac {1}{625} x (405 d+458 e)-\frac {(2115 d+5989 e) \tan ^{-1}\left (\frac {5 x+1}{\sqrt {14}}\right )}{3125 \sqrt {14}}+\frac {e x^4}{5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((405*d + 458*e)*x)/625 - (3*(55*d - 27*e)*x^2)/250 + ((20*d - 33*e)*x^3)/75 + (e*x^4)/5 - ((2115*d + 5989*e)*
ArcTan[(1 + 5*x)/Sqrt[14]])/(3125*Sqrt[14]) + ((2290*d - 881*e)*Log[3 + 2*x + 5*x^2])/6250

Rule 204

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

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 1628

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegra
nd[(d + e*x)^m*Pq*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rubi steps

\begin {align*} \int \frac {(d+e x) \left (2+x+3 x^2-5 x^3+4 x^4\right )}{3+2 x+5 x^2} \, dx &=\int \left (\frac {1}{625} (405 d+458 e)-\frac {3}{125} (55 d-27 e) x+\frac {1}{25} (20 d-33 e) x^2+\frac {4 e x^3}{5}+\frac {35 d-1374 e+(2290 d-881 e) x}{625 \left (3+2 x+5 x^2\right )}\right ) \, dx\\ &=\frac {1}{625} (405 d+458 e) x-\frac {3}{250} (55 d-27 e) x^2+\frac {1}{75} (20 d-33 e) x^3+\frac {e x^4}{5}+\frac {1}{625} \int \frac {35 d-1374 e+(2290 d-881 e) x}{3+2 x+5 x^2} \, dx\\ &=\frac {1}{625} (405 d+458 e) x-\frac {3}{250} (55 d-27 e) x^2+\frac {1}{75} (20 d-33 e) x^3+\frac {e x^4}{5}+\frac {(-2115 d-5989 e) \int \frac {1}{3+2 x+5 x^2} \, dx}{3125}+\frac {(2290 d-881 e) \int \frac {2+10 x}{3+2 x+5 x^2} \, dx}{6250}\\ &=\frac {1}{625} (405 d+458 e) x-\frac {3}{250} (55 d-27 e) x^2+\frac {1}{75} (20 d-33 e) x^3+\frac {e x^4}{5}+\frac {(2290 d-881 e) \log \left (3+2 x+5 x^2\right )}{6250}+\frac {(2 (2115 d+5989 e)) \operatorname {Subst}\left (\int \frac {1}{-56-x^2} \, dx,x,2+10 x\right )}{3125}\\ &=\frac {1}{625} (405 d+458 e) x-\frac {3}{250} (55 d-27 e) x^2+\frac {1}{75} (20 d-33 e) x^3+\frac {e x^4}{5}-\frac {(2115 d+5989 e) \tan ^{-1}\left (\frac {1+5 x}{\sqrt {14}}\right )}{3125 \sqrt {14}}+\frac {(2290 d-881 e) \log \left (3+2 x+5 x^2\right )}{6250}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 86, normalized size = 0.87 \[ \frac {21 (2290 d-881 e) \log \left (5 x^2+2 x+3\right )+35 x \left (5 d \left (200 x^2-495 x+486\right )+3 e \left (250 x^3-550 x^2+405 x+916\right )\right )-3 \sqrt {14} (2115 d+5989 e) \tan ^{-1}\left (\frac {5 x+1}{\sqrt {14}}\right )}{131250} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(35*x*(5*d*(486 - 495*x + 200*x^2) + 3*e*(916 + 405*x - 550*x^2 + 250*x^3)) - 3*Sqrt[14]*(2115*d + 5989*e)*Arc
Tan[(1 + 5*x)/Sqrt[14]] + 21*(2290*d - 881*e)*Log[3 + 2*x + 5*x^2])/131250

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fricas [A]  time = 0.78, size = 84, normalized size = 0.85 \[ \frac {1}{5} \, e x^{4} + \frac {1}{75} \, {\left (20 \, d - 33 \, e\right )} x^{3} - \frac {3}{250} \, {\left (55 \, d - 27 \, e\right )} x^{2} - \frac {1}{43750} \, \sqrt {14} {\left (2115 \, d + 5989 \, e\right )} \arctan \left (\frac {1}{14} \, \sqrt {14} {\left (5 \, x + 1\right )}\right ) + \frac {1}{625} \, {\left (405 \, d + 458 \, e\right )} x + \frac {1}{6250} \, {\left (2290 \, d - 881 \, e\right )} \log \left (5 \, x^{2} + 2 \, x + 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*e*x^4 + 1/75*(20*d - 33*e)*x^3 - 3/250*(55*d - 27*e)*x^2 - 1/43750*sqrt(14)*(2115*d + 5989*e)*arctan(1/14*
sqrt(14)*(5*x + 1)) + 1/625*(405*d + 458*e)*x + 1/6250*(2290*d - 881*e)*log(5*x^2 + 2*x + 3)

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giac [A]  time = 0.16, size = 88, normalized size = 0.89 \[ \frac {1}{5} \, x^{4} e + \frac {4}{15} \, d x^{3} - \frac {11}{25} \, x^{3} e - \frac {33}{50} \, d x^{2} + \frac {81}{250} \, x^{2} e - \frac {1}{43750} \, \sqrt {14} {\left (2115 \, d + 5989 \, e\right )} \arctan \left (\frac {1}{14} \, \sqrt {14} {\left (5 \, x + 1\right )}\right ) + \frac {81}{125} \, d x + \frac {458}{625} \, x e + \frac {1}{6250} \, {\left (2290 \, d - 881 \, e\right )} \log \left (5 \, x^{2} + 2 \, x + 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*x^4*e + 4/15*d*x^3 - 11/25*x^3*e - 33/50*d*x^2 + 81/250*x^2*e - 1/43750*sqrt(14)*(2115*d + 5989*e)*arctan(
1/14*sqrt(14)*(5*x + 1)) + 81/125*d*x + 458/625*x*e + 1/6250*(2290*d - 881*e)*log(5*x^2 + 2*x + 3)

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maple [A]  time = 0.00, size = 102, normalized size = 1.03 \[ \frac {e \,x^{4}}{5}+\frac {4 d \,x^{3}}{15}-\frac {11 e \,x^{3}}{25}-\frac {33 d \,x^{2}}{50}+\frac {81 e \,x^{2}}{250}+\frac {81 d x}{125}-\frac {423 \sqrt {14}\, d \arctan \left (\frac {\left (10 x +2\right ) \sqrt {14}}{28}\right )}{8750}+\frac {229 d \ln \left (5 x^{2}+2 x +3\right )}{625}+\frac {458 e x}{625}-\frac {5989 \sqrt {14}\, e \arctan \left (\frac {\left (10 x +2\right ) \sqrt {14}}{28}\right )}{43750}-\frac {881 e \ln \left (5 x^{2}+2 x +3\right )}{6250} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*e*x^4+4/15*x^3*d-11/25*x^3*e-33/50*x^2*d+81/250*e*x^2+81/125*d*x+458/625*e*x+229/625*ln(5*x^2+2*x+3)*d-881
/6250*e*ln(5*x^2+2*x+3)-423/8750*14^(1/2)*arctan(1/28*(10*x+2)*14^(1/2))*d-5989/43750*14^(1/2)*arctan(1/28*(10
*x+2)*14^(1/2))*e

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maxima [A]  time = 0.96, size = 84, normalized size = 0.85 \[ \frac {1}{5} \, e x^{4} + \frac {1}{75} \, {\left (20 \, d - 33 \, e\right )} x^{3} - \frac {3}{250} \, {\left (55 \, d - 27 \, e\right )} x^{2} - \frac {1}{43750} \, \sqrt {14} {\left (2115 \, d + 5989 \, e\right )} \arctan \left (\frac {1}{14} \, \sqrt {14} {\left (5 \, x + 1\right )}\right ) + \frac {1}{625} \, {\left (405 \, d + 458 \, e\right )} x + \frac {1}{6250} \, {\left (2290 \, d - 881 \, e\right )} \log \left (5 \, x^{2} + 2 \, x + 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*e*x^4 + 1/75*(20*d - 33*e)*x^3 - 3/250*(55*d - 27*e)*x^2 - 1/43750*sqrt(14)*(2115*d + 5989*e)*arctan(1/14*
sqrt(14)*(5*x + 1)) + 1/625*(405*d + 458*e)*x + 1/6250*(2290*d - 881*e)*log(5*x^2 + 2*x + 3)

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mupad [B]  time = 0.07, size = 107, normalized size = 1.08 \[ x^3\,\left (\frac {4\,d}{15}-\frac {11\,e}{25}\right )-x^2\,\left (\frac {33\,d}{50}-\frac {81\,e}{250}\right )+\ln \left (5\,x^2+2\,x+3\right )\,\left (\frac {229\,d}{625}-\frac {881\,e}{6250}\right )+\frac {e\,x^4}{5}+x\,\left (\frac {81\,d}{125}+\frac {458\,e}{625}\right )-\frac {\sqrt {14}\,\mathrm {atan}\left (\frac {\frac {\sqrt {14}\,\left (2115\,d+5989\,e\right )}{43750}+\frac {\sqrt {14}\,x\,\left (2115\,d+5989\,e\right )}{8750}}{\frac {423\,d}{625}+\frac {5989\,e}{3125}}\right )\,\left (2115\,d+5989\,e\right )}{43750} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^3*((4*d)/15 - (11*e)/25) - x^2*((33*d)/50 - (81*e)/250) + log(2*x + 5*x^2 + 3)*((229*d)/625 - (881*e)/6250)
+ (e*x^4)/5 + x*((81*d)/125 + (458*e)/625) - (14^(1/2)*atan(((14^(1/2)*(2115*d + 5989*e))/43750 + (14^(1/2)*x*
(2115*d + 5989*e))/8750)/((423*d)/625 + (5989*e)/3125))*(2115*d + 5989*e))/43750

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sympy [C]  time = 0.85, size = 163, normalized size = 1.65 \[ \frac {e x^{4}}{5} + x^{3} \left (\frac {4 d}{15} - \frac {11 e}{25}\right ) + x^{2} \left (- \frac {33 d}{50} + \frac {81 e}{250}\right ) + x \left (\frac {81 d}{125} + \frac {458 e}{625}\right ) + \left (\frac {229 d}{625} - \frac {881 e}{6250} - \frac {\sqrt {14} i \left (2115 d + 5989 e\right )}{87500}\right ) \log {\left (x + \frac {423 d + \frac {5989 e}{5} + \frac {\sqrt {14} i \left (2115 d + 5989 e\right )}{5}}{2115 d + 5989 e} \right )} + \left (\frac {229 d}{625} - \frac {881 e}{6250} + \frac {\sqrt {14} i \left (2115 d + 5989 e\right )}{87500}\right ) \log {\left (x + \frac {423 d + \frac {5989 e}{5} - \frac {\sqrt {14} i \left (2115 d + 5989 e\right )}{5}}{2115 d + 5989 e} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e*x**4/5 + x**3*(4*d/15 - 11*e/25) + x**2*(-33*d/50 + 81*e/250) + x*(81*d/125 + 458*e/625) + (229*d/625 - 881*
e/6250 - sqrt(14)*I*(2115*d + 5989*e)/87500)*log(x + (423*d + 5989*e/5 + sqrt(14)*I*(2115*d + 5989*e)/5)/(2115
*d + 5989*e)) + (229*d/625 - 881*e/6250 + sqrt(14)*I*(2115*d + 5989*e)/87500)*log(x + (423*d + 5989*e/5 - sqrt
(14)*I*(2115*d + 5989*e)/5)/(2115*d + 5989*e))

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