3.1.25 \(\int \frac {2+x}{(2+4 x-3 x^2) \sqrt {1+3 x-2 x^2}} \, dx\)

Optimal. Leaf size=139 \[ \frac {1}{2} \sqrt {\sqrt {10}-\frac {13}{5}} \tan ^{-1}\left (\frac {\left (1+4 \sqrt {10}\right ) x+3 \left (4-\sqrt {10}\right )}{2 \sqrt {1+\sqrt {10}} \sqrt {-2 x^2+3 x+1}}\right )+\frac {1}{2} \sqrt {\frac {13}{5}+\sqrt {10}} \tanh ^{-1}\left (\frac {\left (1-4 \sqrt {10}\right ) x+3 \left (4+\sqrt {10}\right )}{2 \sqrt {\sqrt {10}-1} \sqrt {-2 x^2+3 x+1}}\right ) \]

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Rubi [A]  time = 0.22, antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {1032, 724, 204, 206} \begin {gather*} \frac {1}{2} \sqrt {\sqrt {10}-\frac {13}{5}} \tan ^{-1}\left (\frac {\left (1+4 \sqrt {10}\right ) x+3 \left (4-\sqrt {10}\right )}{2 \sqrt {1+\sqrt {10}} \sqrt {-2 x^2+3 x+1}}\right )+\frac {1}{2} \sqrt {\frac {13}{5}+\sqrt {10}} \tanh ^{-1}\left (\frac {\left (1-4 \sqrt {10}\right ) x+3 \left (4+\sqrt {10}\right )}{2 \sqrt {\sqrt {10}-1} \sqrt {-2 x^2+3 x+1}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2 + x)/((2 + 4*x - 3*x^2)*Sqrt[1 + 3*x - 2*x^2]),x]

[Out]

(Sqrt[-13/5 + Sqrt[10]]*ArcTan[(3*(4 - Sqrt[10]) + (1 + 4*Sqrt[10])*x)/(2*Sqrt[1 + Sqrt[10]]*Sqrt[1 + 3*x - 2*
x^2])])/2 + (Sqrt[13/5 + Sqrt[10]]*ArcTanh[(3*(4 + Sqrt[10]) + (1 - 4*Sqrt[10])*x)/(2*Sqrt[-1 + Sqrt[10]]*Sqrt
[1 + 3*x - 2*x^2])])/2

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 206

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

Rule 724

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

Rule 1032

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

Rubi steps

\begin {align*} \int \frac {2+x}{\left (2+4 x-3 x^2\right ) \sqrt {1+3 x-2 x^2}} \, dx &=\frac {1}{5} \left (5-4 \sqrt {10}\right ) \int \frac {1}{\left (4-2 \sqrt {10}-6 x\right ) \sqrt {1+3 x-2 x^2}} \, dx+\frac {1}{5} \left (5+4 \sqrt {10}\right ) \int \frac {1}{\left (4+2 \sqrt {10}-6 x\right ) \sqrt {1+3 x-2 x^2}} \, dx\\ &=-\left (\frac {1}{5} \left (2 \left (5-4 \sqrt {10}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{144+72 \left (4-2 \sqrt {10}\right )-8 \left (4-2 \sqrt {10}\right )^2-x^2} \, dx,x,\frac {-12-3 \left (4-2 \sqrt {10}\right )-\left (18-4 \left (4-2 \sqrt {10}\right )\right ) x}{\sqrt {1+3 x-2 x^2}}\right )\right )-\frac {1}{5} \left (2 \left (5+4 \sqrt {10}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{144+72 \left (4+2 \sqrt {10}\right )-8 \left (4+2 \sqrt {10}\right )^2-x^2} \, dx,x,\frac {-12-3 \left (4+2 \sqrt {10}\right )-\left (18-4 \left (4+2 \sqrt {10}\right )\right ) x}{\sqrt {1+3 x-2 x^2}}\right )\\ &=\frac {1}{10} \sqrt {-65+25 \sqrt {10}} \tan ^{-1}\left (\frac {3 \left (4-\sqrt {10}\right )+\left (1+4 \sqrt {10}\right ) x}{2 \sqrt {1+\sqrt {10}} \sqrt {1+3 x-2 x^2}}\right )+\frac {1}{10} \sqrt {65+25 \sqrt {10}} \tanh ^{-1}\left (\frac {3 \left (4+\sqrt {10}\right )+\left (1-4 \sqrt {10}\right ) x}{2 \sqrt {-1+\sqrt {10}} \sqrt {1+3 x-2 x^2}}\right )\\ \end {align*}

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Mathematica [A]  time = 0.30, size = 140, normalized size = 1.01 \begin {gather*} \frac {\left (4 \sqrt {10}-5\right ) \tan ^{-1}\left (\frac {4 \sqrt {10} x+x-3 \sqrt {10}+12}{2 \sqrt {1+\sqrt {10}} \sqrt {-2 x^2+3 x+1}}\right )+3 \sqrt {5 \left (7+2 \sqrt {10}\right )} \tanh ^{-1}\left (\frac {-4 \sqrt {10} x+x+3 \left (4+\sqrt {10}\right )}{2 \sqrt {\sqrt {10}-1} \sqrt {-2 x^2+3 x+1}}\right )}{10 \sqrt {1+\sqrt {10}}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[(2 + x)/((2 + 4*x - 3*x^2)*Sqrt[1 + 3*x - 2*x^2]),x]

[Out]

((-5 + 4*Sqrt[10])*ArcTan[(12 - 3*Sqrt[10] + x + 4*Sqrt[10]*x)/(2*Sqrt[1 + Sqrt[10]]*Sqrt[1 + 3*x - 2*x^2])] +
3*Sqrt[5*(7 + 2*Sqrt[10])]*ArcTanh[(3*(4 + Sqrt[10]) + x - 4*Sqrt[10]*x)/(2*Sqrt[-1 + Sqrt[10]]*Sqrt[1 + 3*x
- 2*x^2])])/(10*Sqrt[1 + Sqrt[10]])

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IntegrateAlgebraic [C]  time = 0.34, size = 149, normalized size = 1.07 \begin {gather*} -\frac {1}{2} \text {RootSum}\left [2 \text {$\#$1}^4-8 \text {$\#$1}^3+8 \text {$\#$1}^2+20 \text {$\#$1}+5\&,\frac {2 \text {$\#$1}^2 \log \left (\text {$\#$1} (-x)+\sqrt {-2 x^2+3 x+1}-1\right )-2 \text {$\#$1}^2 \log (x)-2 \text {$\#$1} \log \left (\text {$\#$1} (-x)+\sqrt {-2 x^2+3 x+1}-1\right )+7 \log \left (\text {$\#$1} (-x)+\sqrt {-2 x^2+3 x+1}-1\right )+2 \text {$\#$1} \log (x)-7 \log (x)}{2 \text {$\#$1}^3-6 \text {$\#$1}^2+4 \text {$\#$1}+5}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(2 + x)/((2 + 4*x - 3*x^2)*Sqrt[1 + 3*x - 2*x^2]),x]

[Out]

-1/2*RootSum[5 + 20*#1 + 8*#1^2 - 8*#1^3 + 2*#1^4 & , (-7*Log[x] + 7*Log[-1 + Sqrt[1 + 3*x - 2*x^2] - x*#1] +
2*Log[x]*#1 - 2*Log[-1 + Sqrt[1 + 3*x - 2*x^2] - x*#1]*#1 - 2*Log[x]*#1^2 + 2*Log[-1 + Sqrt[1 + 3*x - 2*x^2] -
x*#1]*#1^2)/(5 + 4*#1 - 6*#1^2 + 2*#1^3) & ]

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fricas [B]  time = 0.44, size = 322, normalized size = 2.32 \begin {gather*} \frac {2}{5} \, \sqrt {5} \sqrt {5 \, \sqrt {5} \sqrt {2} - 13} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {5} x - \sqrt {2} x\right )} \sqrt {5 \, \sqrt {5} \sqrt {2} - 13} \sqrt {\frac {\sqrt {5} \sqrt {2} {\left (3 \, x^{2} + 2 \, x\right )} + 6 \, x^{2} - 2 \, {\left (\sqrt {5} \sqrt {2} x + 2 \, x + 2\right )} \sqrt {-2 \, x^{2} + 3 \, x + 1} + 10 \, x + 4}{x^{2}}} + 2 \, {\left (\sqrt {2} {\left (4 \, x - 1\right )} + \sqrt {5} {\left (x + 2\right )} - \sqrt {-2 \, x^{2} + 3 \, x + 1} {\left (2 \, \sqrt {5} - \sqrt {2}\right )}\right )} \sqrt {5 \, \sqrt {5} \sqrt {2} - 13}}{18 \, x}\right ) - \frac {1}{10} \, \sqrt {5} \sqrt {5 \, \sqrt {5} \sqrt {2} + 13} \log \left (\frac {9 \, \sqrt {5} \sqrt {2} x + {\left (4 \, \sqrt {5} x - 7 \, \sqrt {2} x\right )} \sqrt {5 \, \sqrt {5} \sqrt {2} + 13} - 18 \, x + 18 \, \sqrt {-2 \, x^{2} + 3 \, x + 1} - 18}{x}\right ) + \frac {1}{10} \, \sqrt {5} \sqrt {5 \, \sqrt {5} \sqrt {2} + 13} \log \left (\frac {9 \, \sqrt {5} \sqrt {2} x - {\left (4 \, \sqrt {5} x - 7 \, \sqrt {2} x\right )} \sqrt {5 \, \sqrt {5} \sqrt {2} + 13} - 18 \, x + 18 \, \sqrt {-2 \, x^{2} + 3 \, x + 1} - 18}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+x)/(-3*x^2+4*x+2)/(-2*x^2+3*x+1)^(1/2),x, algorithm="fricas")

[Out]

2/5*sqrt(5)*sqrt(5*sqrt(5)*sqrt(2) - 13)*arctan(1/18*(sqrt(2)*(2*sqrt(5)*x - sqrt(2)*x)*sqrt(5*sqrt(5)*sqrt(2)
- 13)*sqrt((sqrt(5)*sqrt(2)*(3*x^2 + 2*x) + 6*x^2 - 2*(sqrt(5)*sqrt(2)*x + 2*x + 2)*sqrt(-2*x^2 + 3*x + 1) +
10*x + 4)/x^2) + 2*(sqrt(2)*(4*x - 1) + sqrt(5)*(x + 2) - sqrt(-2*x^2 + 3*x + 1)*(2*sqrt(5) - sqrt(2)))*sqrt(5
*sqrt(5)*sqrt(2) - 13))/x) - 1/10*sqrt(5)*sqrt(5*sqrt(5)*sqrt(2) + 13)*log((9*sqrt(5)*sqrt(2)*x + (4*sqrt(5)*x
- 7*sqrt(2)*x)*sqrt(5*sqrt(5)*sqrt(2) + 13) - 18*x + 18*sqrt(-2*x^2 + 3*x + 1) - 18)/x) + 1/10*sqrt(5)*sqrt(5
*sqrt(5)*sqrt(2) + 13)*log((9*sqrt(5)*sqrt(2)*x - (4*sqrt(5)*x - 7*sqrt(2)*x)*sqrt(5*sqrt(5)*sqrt(2) + 13) - 1
8*x + 18*sqrt(-2*x^2 + 3*x + 1) - 18)/x)

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giac [F(-1)]  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((2+x)/(-3*x^2+4*x+2)/(-2*x^2+3*x+1)^(1/2),x, algorithm="giac")

[Out]

Timed out

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maple [B]  time = 0.07, size = 324, normalized size = 2.33 \begin {gather*} \frac {2 \sqrt {10}\, \arctanh \left (\frac {-1+\sqrt {10}+\frac {9 \left (\frac {1}{3}-\frac {4 \sqrt {10}}{3}\right ) \left (x -\frac {2}{3}-\frac {\sqrt {10}}{3}\right )}{2}}{\sqrt {-1+\sqrt {10}}\, \sqrt {-18 \left (x -\frac {2}{3}-\frac {\sqrt {10}}{3}\right )^{2}+9 \left (\frac {1}{3}-\frac {4 \sqrt {10}}{3}\right ) \left (x -\frac {2}{3}-\frac {\sqrt {10}}{3}\right )-1+\sqrt {10}}}\right )}{5 \sqrt {-1+\sqrt {10}}}+\frac {\arctanh \left (\frac {-1+\sqrt {10}+\frac {9 \left (\frac {1}{3}-\frac {4 \sqrt {10}}{3}\right ) \left (x -\frac {2}{3}-\frac {\sqrt {10}}{3}\right )}{2}}{\sqrt {-1+\sqrt {10}}\, \sqrt {-18 \left (x -\frac {2}{3}-\frac {\sqrt {10}}{3}\right )^{2}+9 \left (\frac {1}{3}-\frac {4 \sqrt {10}}{3}\right ) \left (x -\frac {2}{3}-\frac {\sqrt {10}}{3}\right )-1+\sqrt {10}}}\right )}{2 \sqrt {-1+\sqrt {10}}}+\frac {2 \sqrt {10}\, \arctan \left (\frac {-1-\sqrt {10}+\frac {9 \left (\frac {1}{3}+\frac {4 \sqrt {10}}{3}\right ) \left (x -\frac {2}{3}+\frac {\sqrt {10}}{3}\right )}{2}}{\sqrt {1+\sqrt {10}}\, \sqrt {-18 \left (x -\frac {2}{3}+\frac {\sqrt {10}}{3}\right )^{2}+9 \left (\frac {1}{3}+\frac {4 \sqrt {10}}{3}\right ) \left (x -\frac {2}{3}+\frac {\sqrt {10}}{3}\right )-1-\sqrt {10}}}\right )}{5 \sqrt {1+\sqrt {10}}}-\frac {\arctan \left (\frac {-1-\sqrt {10}+\frac {9 \left (\frac {1}{3}+\frac {4 \sqrt {10}}{3}\right ) \left (x -\frac {2}{3}+\frac {\sqrt {10}}{3}\right )}{2}}{\sqrt {1+\sqrt {10}}\, \sqrt {-18 \left (x -\frac {2}{3}+\frac {\sqrt {10}}{3}\right )^{2}+9 \left (\frac {1}{3}+\frac {4 \sqrt {10}}{3}\right ) \left (x -\frac {2}{3}+\frac {\sqrt {10}}{3}\right )-1-\sqrt {10}}}\right )}{2 \sqrt {1+\sqrt {10}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2+x)/(-3*x^2+4*x+2)/(-2*x^2+3*x+1)^(1/2),x)

[Out]

2/5*10^(1/2)/(1+10^(1/2))^(1/2)*arctan(9/2*(-2/9-2/9*10^(1/2)+(1/3+4/3*10^(1/2))*(x-2/3+1/3*10^(1/2)))/(1+10^(
1/2))^(1/2)/(-18*(x-2/3+1/3*10^(1/2))^2+9*(1/3+4/3*10^(1/2))*(x-2/3+1/3*10^(1/2))-1-10^(1/2))^(1/2))-1/2/(1+10
^(1/2))^(1/2)*arctan(9/2*(-2/9-2/9*10^(1/2)+(1/3+4/3*10^(1/2))*(x-2/3+1/3*10^(1/2)))/(1+10^(1/2))^(1/2)/(-18*(
x-2/3+1/3*10^(1/2))^2+9*(1/3+4/3*10^(1/2))*(x-2/3+1/3*10^(1/2))-1-10^(1/2))^(1/2))+2/5*10^(1/2)/(-1+10^(1/2))^
(1/2)*arctanh(9/2*(-2/9+2/9*10^(1/2)+(1/3-4/3*10^(1/2))*(x-2/3-1/3*10^(1/2)))/(-1+10^(1/2))^(1/2)/(-18*(x-2/3-
1/3*10^(1/2))^2+9*(1/3-4/3*10^(1/2))*(x-2/3-1/3*10^(1/2))-1+10^(1/2))^(1/2))+1/2/(-1+10^(1/2))^(1/2)*arctanh(9
/2*(-2/9+2/9*10^(1/2)+(1/3-4/3*10^(1/2))*(x-2/3-1/3*10^(1/2)))/(-1+10^(1/2))^(1/2)/(-18*(x-2/3-1/3*10^(1/2))^2
+9*(1/3-4/3*10^(1/2))*(x-2/3-1/3*10^(1/2))-1+10^(1/2))^(1/2))

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maxima [B]  time = 1.07, size = 361, normalized size = 2.60 \begin {gather*} -\frac {1}{20} \, \sqrt {10} {\left (\frac {\sqrt {10} \arcsin \left (\frac {8 \, \sqrt {17} \sqrt {10} x}{17 \, {\left | 6 \, x + 2 \, \sqrt {10} - 4 \right |}} + \frac {2 \, \sqrt {17} x}{17 \, {\left | 6 \, x + 2 \, \sqrt {10} - 4 \right |}} - \frac {6 \, \sqrt {17} \sqrt {10}}{17 \, {\left | 6 \, x + 2 \, \sqrt {10} - 4 \right |}} + \frac {24 \, \sqrt {17}}{17 \, {\left | 6 \, x + 2 \, \sqrt {10} - 4 \right |}}\right )}{\sqrt {\sqrt {10} + 1}} - \frac {\sqrt {10} \log \left (-\frac {2}{9} \, \sqrt {10} + \frac {2 \, \sqrt {-2 \, x^{2} + 3 \, x + 1} \sqrt {\sqrt {10} - 1}}{3 \, {\left | 6 \, x - 2 \, \sqrt {10} - 4 \right |}} + \frac {2 \, \sqrt {10}}{9 \, {\left | 6 \, x - 2 \, \sqrt {10} - 4 \right |}} - \frac {2}{9 \, {\left | 6 \, x - 2 \, \sqrt {10} - 4 \right |}} + \frac {1}{18}\right )}{\sqrt {\sqrt {10} - 1}} - \frac {8 \, \arcsin \left (\frac {8 \, \sqrt {17} \sqrt {10} x}{17 \, {\left | 6 \, x + 2 \, \sqrt {10} - 4 \right |}} + \frac {2 \, \sqrt {17} x}{17 \, {\left | 6 \, x + 2 \, \sqrt {10} - 4 \right |}} - \frac {6 \, \sqrt {17} \sqrt {10}}{17 \, {\left | 6 \, x + 2 \, \sqrt {10} - 4 \right |}} + \frac {24 \, \sqrt {17}}{17 \, {\left | 6 \, x + 2 \, \sqrt {10} - 4 \right |}}\right )}{\sqrt {\sqrt {10} + 1}} - \frac {8 \, \log \left (-\frac {2}{9} \, \sqrt {10} + \frac {2 \, \sqrt {-2 \, x^{2} + 3 \, x + 1} \sqrt {\sqrt {10} - 1}}{3 \, {\left | 6 \, x - 2 \, \sqrt {10} - 4 \right |}} + \frac {2 \, \sqrt {10}}{9 \, {\left | 6 \, x - 2 \, \sqrt {10} - 4 \right |}} - \frac {2}{9 \, {\left | 6 \, x - 2 \, \sqrt {10} - 4 \right |}} + \frac {1}{18}\right )}{\sqrt {\sqrt {10} - 1}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+x)/(-3*x^2+4*x+2)/(-2*x^2+3*x+1)^(1/2),x, algorithm="maxima")

[Out]

-1/20*sqrt(10)*(sqrt(10)*arcsin(8/17*sqrt(17)*sqrt(10)*x/abs(6*x + 2*sqrt(10) - 4) + 2/17*sqrt(17)*x/abs(6*x +
2*sqrt(10) - 4) - 6/17*sqrt(17)*sqrt(10)/abs(6*x + 2*sqrt(10) - 4) + 24/17*sqrt(17)/abs(6*x + 2*sqrt(10) - 4)
)/sqrt(sqrt(10) + 1) - sqrt(10)*log(-2/9*sqrt(10) + 2/3*sqrt(-2*x^2 + 3*x + 1)*sqrt(sqrt(10) - 1)/abs(6*x - 2*
sqrt(10) - 4) + 2/9*sqrt(10)/abs(6*x - 2*sqrt(10) - 4) - 2/9/abs(6*x - 2*sqrt(10) - 4) + 1/18)/sqrt(sqrt(10) -
1) - 8*arcsin(8/17*sqrt(17)*sqrt(10)*x/abs(6*x + 2*sqrt(10) - 4) + 2/17*sqrt(17)*x/abs(6*x + 2*sqrt(10) - 4)
- 6/17*sqrt(17)*sqrt(10)/abs(6*x + 2*sqrt(10) - 4) + 24/17*sqrt(17)/abs(6*x + 2*sqrt(10) - 4))/sqrt(sqrt(10) +
1) - 8*log(-2/9*sqrt(10) + 2/3*sqrt(-2*x^2 + 3*x + 1)*sqrt(sqrt(10) - 1)/abs(6*x - 2*sqrt(10) - 4) + 2/9*sqrt
(10)/abs(6*x - 2*sqrt(10) - 4) - 2/9/abs(6*x - 2*sqrt(10) - 4) + 1/18)/sqrt(sqrt(10) - 1))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x+2}{\sqrt {-2\,x^2+3\,x+1}\,\left (-3\,x^2+4\,x+2\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x + 2)/((3*x - 2*x^2 + 1)^(1/2)*(4*x - 3*x^2 + 2)),x)

[Out]

int((x + 2)/((3*x - 2*x^2 + 1)^(1/2)*(4*x - 3*x^2 + 2)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {x}{3 x^{2} \sqrt {- 2 x^{2} + 3 x + 1} - 4 x \sqrt {- 2 x^{2} + 3 x + 1} - 2 \sqrt {- 2 x^{2} + 3 x + 1}}\, dx - \int \frac {2}{3 x^{2} \sqrt {- 2 x^{2} + 3 x + 1} - 4 x \sqrt {- 2 x^{2} + 3 x + 1} - 2 \sqrt {- 2 x^{2} + 3 x + 1}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+x)/(-3*x**2+4*x+2)/(-2*x**2+3*x+1)**(1/2),x)

[Out]

-Integral(x/(3*x**2*sqrt(-2*x**2 + 3*x + 1) - 4*x*sqrt(-2*x**2 + 3*x + 1) - 2*sqrt(-2*x**2 + 3*x + 1)), x) - I
ntegral(2/(3*x**2*sqrt(-2*x**2 + 3*x + 1) - 4*x*sqrt(-2*x**2 + 3*x + 1) - 2*sqrt(-2*x**2 + 3*x + 1)), x)

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