3.41 \(\int \frac {1}{(3-x+2 x^2) (2+3 x+5 x^2)} \, dx\)

Optimal. Leaf size=73 \[ -\frac {1}{44} \log \left (2 x^2-x+3\right )+\frac {1}{44} \log \left (5 x^2+3 x+2\right )+\frac {3 \tan ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{22 \sqrt {23}}+\frac {13 \tan ^{-1}\left (\frac {10 x+3}{\sqrt {31}}\right )}{22 \sqrt {31}} \]

[Out]

-1/44*ln(2*x^2-x+3)+1/44*ln(5*x^2+3*x+2)+3/506*arctan(1/23*(1-4*x)*23^(1/2))*23^(1/2)+13/682*arctan(1/31*(3+10
*x)*31^(1/2))*31^(1/2)

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Rubi [A]  time = 0.05, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {980, 634, 618, 204, 628} \[ -\frac {1}{44} \log \left (2 x^2-x+3\right )+\frac {1}{44} \log \left (5 x^2+3 x+2\right )+\frac {3 \tan ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{22 \sqrt {23}}+\frac {13 \tan ^{-1}\left (\frac {10 x+3}{\sqrt {31}}\right )}{22 \sqrt {31}} \]

Antiderivative was successfully verified.

[In]

Int[1/((3 - x + 2*x^2)*(2 + 3*x + 5*x^2)),x]

[Out]

(3*ArcTan[(1 - 4*x)/Sqrt[23]])/(22*Sqrt[23]) + (13*ArcTan[(3 + 10*x)/Sqrt[31]])/(22*Sqrt[31]) - Log[3 - x + 2*
x^2]/44 + Log[2 + 3*x + 5*x^2]/44

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 980

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

Rubi steps

\begin {align*} \int \frac {1}{\left (3-x+2 x^2\right ) \left (2+3 x+5 x^2\right )} \, dx &=\frac {1}{242} \int \frac {-11-22 x}{3-x+2 x^2} \, dx+\frac {1}{242} \int \frac {88+55 x}{2+3 x+5 x^2} \, dx\\ &=-\left (\frac {1}{44} \int \frac {-1+4 x}{3-x+2 x^2} \, dx\right )+\frac {1}{44} \int \frac {3+10 x}{2+3 x+5 x^2} \, dx-\frac {3}{44} \int \frac {1}{3-x+2 x^2} \, dx+\frac {13}{44} \int \frac {1}{2+3 x+5 x^2} \, dx\\ &=-\frac {1}{44} \log \left (3-x+2 x^2\right )+\frac {1}{44} \log \left (2+3 x+5 x^2\right )+\frac {3}{22} \operatorname {Subst}\left (\int \frac {1}{-23-x^2} \, dx,x,-1+4 x\right )-\frac {13}{22} \operatorname {Subst}\left (\int \frac {1}{-31-x^2} \, dx,x,3+10 x\right )\\ &=\frac {3 \tan ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{22 \sqrt {23}}+\frac {13 \tan ^{-1}\left (\frac {3+10 x}{\sqrt {31}}\right )}{22 \sqrt {31}}-\frac {1}{44} \log \left (3-x+2 x^2\right )+\frac {1}{44} \log \left (2+3 x+5 x^2\right )\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 73, normalized size = 1.00 \[ -\frac {1}{44} \log \left (2 x^2-x+3\right )+\frac {1}{44} \log \left (5 x^2+3 x+2\right )-\frac {3 \tan ^{-1}\left (\frac {4 x-1}{\sqrt {23}}\right )}{22 \sqrt {23}}+\frac {13 \tan ^{-1}\left (\frac {10 x+3}{\sqrt {31}}\right )}{22 \sqrt {31}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((3 - x + 2*x^2)*(2 + 3*x + 5*x^2)),x]

[Out]

(-3*ArcTan[(-1 + 4*x)/Sqrt[23]])/(22*Sqrt[23]) + (13*ArcTan[(3 + 10*x)/Sqrt[31]])/(22*Sqrt[31]) - Log[3 - x +
2*x^2]/44 + Log[2 + 3*x + 5*x^2]/44

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fricas [A]  time = 0.81, size = 59, normalized size = 0.81 \[ \frac {13}{682} \, \sqrt {31} \arctan \left (\frac {1}{31} \, \sqrt {31} {\left (10 \, x + 3\right )}\right ) - \frac {3}{506} \, \sqrt {23} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) + \frac {1}{44} \, \log \left (5 \, x^{2} + 3 \, x + 2\right ) - \frac {1}{44} \, \log \left (2 \, x^{2} - x + 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2*x^2-x+3)/(5*x^2+3*x+2),x, algorithm="fricas")

[Out]

13/682*sqrt(31)*arctan(1/31*sqrt(31)*(10*x + 3)) - 3/506*sqrt(23)*arctan(1/23*sqrt(23)*(4*x - 1)) + 1/44*log(5
*x^2 + 3*x + 2) - 1/44*log(2*x^2 - x + 3)

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giac [A]  time = 0.19, size = 59, normalized size = 0.81 \[ \frac {13}{682} \, \sqrt {31} \arctan \left (\frac {1}{31} \, \sqrt {31} {\left (10 \, x + 3\right )}\right ) - \frac {3}{506} \, \sqrt {23} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) + \frac {1}{44} \, \log \left (5 \, x^{2} + 3 \, x + 2\right ) - \frac {1}{44} \, \log \left (2 \, x^{2} - x + 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2*x^2-x+3)/(5*x^2+3*x+2),x, algorithm="giac")

[Out]

13/682*sqrt(31)*arctan(1/31*sqrt(31)*(10*x + 3)) - 3/506*sqrt(23)*arctan(1/23*sqrt(23)*(4*x - 1)) + 1/44*log(5
*x^2 + 3*x + 2) - 1/44*log(2*x^2 - x + 3)

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maple [A]  time = 0.01, size = 60, normalized size = 0.82 \[ \frac {13 \sqrt {31}\, \arctan \left (\frac {\left (10 x +3\right ) \sqrt {31}}{31}\right )}{682}-\frac {3 \sqrt {23}\, \arctan \left (\frac {\left (4 x -1\right ) \sqrt {23}}{23}\right )}{506}-\frac {\ln \left (2 x^{2}-x +3\right )}{44}+\frac {\ln \left (5 x^{2}+3 x +2\right )}{44} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(2*x^2-x+3)/(5*x^2+3*x+2),x)

[Out]

1/44*ln(5*x^2+3*x+2)+13/682*31^(1/2)*arctan(1/31*(10*x+3)*31^(1/2))-1/44*ln(2*x^2-x+3)-3/506*23^(1/2)*arctan(1
/23*(4*x-1)*23^(1/2))

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maxima [A]  time = 0.96, size = 59, normalized size = 0.81 \[ \frac {13}{682} \, \sqrt {31} \arctan \left (\frac {1}{31} \, \sqrt {31} {\left (10 \, x + 3\right )}\right ) - \frac {3}{506} \, \sqrt {23} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) + \frac {1}{44} \, \log \left (5 \, x^{2} + 3 \, x + 2\right ) - \frac {1}{44} \, \log \left (2 \, x^{2} - x + 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2*x^2-x+3)/(5*x^2+3*x+2),x, algorithm="maxima")

[Out]

13/682*sqrt(31)*arctan(1/31*sqrt(31)*(10*x + 3)) - 3/506*sqrt(23)*arctan(1/23*sqrt(23)*(4*x - 1)) + 1/44*log(5
*x^2 + 3*x + 2) - 1/44*log(2*x^2 - x + 3)

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mupad [B]  time = 0.19, size = 79, normalized size = 1.08 \[ \ln \left (x-\frac {1}{4}-\frac {\sqrt {23}\,1{}\mathrm {i}}{4}\right )\,\left (-\frac {1}{44}+\frac {\sqrt {23}\,3{}\mathrm {i}}{1012}\right )-\ln \left (x-\frac {1}{4}+\frac {\sqrt {23}\,1{}\mathrm {i}}{4}\right )\,\left (\frac {1}{44}+\frac {\sqrt {23}\,3{}\mathrm {i}}{1012}\right )-\ln \left (x+\frac {3}{10}-\frac {\sqrt {31}\,1{}\mathrm {i}}{10}\right )\,\left (-\frac {1}{44}+\frac {\sqrt {31}\,13{}\mathrm {i}}{1364}\right )+\ln \left (x+\frac {3}{10}+\frac {\sqrt {31}\,1{}\mathrm {i}}{10}\right )\,\left (\frac {1}{44}+\frac {\sqrt {31}\,13{}\mathrm {i}}{1364}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((2*x^2 - x + 3)*(3*x + 5*x^2 + 2)),x)

[Out]

log(x - (23^(1/2)*1i)/4 - 1/4)*((23^(1/2)*3i)/1012 - 1/44) - log(x + (23^(1/2)*1i)/4 - 1/4)*((23^(1/2)*3i)/101
2 + 1/44) - log(x - (31^(1/2)*1i)/10 + 3/10)*((31^(1/2)*13i)/1364 - 1/44) + log(x + (31^(1/2)*1i)/10 + 3/10)*(
(31^(1/2)*13i)/1364 + 1/44)

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sympy [A]  time = 0.24, size = 83, normalized size = 1.14 \[ - \frac {\log {\left (x^{2} - \frac {x}{2} + \frac {3}{2} \right )}}{44} + \frac {\log {\left (x^{2} + \frac {3 x}{5} + \frac {2}{5} \right )}}{44} - \frac {3 \sqrt {23} \operatorname {atan}{\left (\frac {4 \sqrt {23} x}{23} - \frac {\sqrt {23}}{23} \right )}}{506} + \frac {13 \sqrt {31} \operatorname {atan}{\left (\frac {10 \sqrt {31} x}{31} + \frac {3 \sqrt {31}}{31} \right )}}{682} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2*x**2-x+3)/(5*x**2+3*x+2),x)

[Out]

-log(x**2 - x/2 + 3/2)/44 + log(x**2 + 3*x/5 + 2/5)/44 - 3*sqrt(23)*atan(4*sqrt(23)*x/23 - sqrt(23)/23)/506 +
13*sqrt(31)*atan(10*sqrt(31)*x/31 + 3*sqrt(31)/31)/682

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