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

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

[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

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Rubi [A]  time = 0.220049, antiderivative size = 139, normalized size of antiderivative = 1., 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} \[ \frac{1}{2} \sqrt{\sqrt{10}-\frac{13}{5}} \tan ^{-1}\left (\frac{\left (1+4 \sqrt{10}\right ) x+3 \left (4-\sqrt{10}\right )}{2 \sqrt{1+\sqrt{10}} \sqrt{-2 x^2+3 x+1}}\right )+\frac{1}{2} \sqrt{\frac{13}{5}+\sqrt{10}} \tanh ^{-1}\left (\frac{\left (1-4 \sqrt{10}\right ) x+3 \left (4+\sqrt{10}\right )}{2 \sqrt{\sqrt{10}-1} \sqrt{-2 x^2+3 x+1}}\right ) \]

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 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]

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 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])

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*}

Mathematica [A]  time = 0.295987, size = 140, normalized size = 1.01 \[ \frac{\left (4 \sqrt{10}-5\right ) \tan ^{-1}\left (\frac{4 \sqrt{10} x+x-3 \sqrt{10}+12}{2 \sqrt{1+\sqrt{10}} \sqrt{-2 x^2+3 x+1}}\right )+3 \sqrt{5 \left (7+2 \sqrt{10}\right )} \tanh ^{-1}\left (\frac{-4 \sqrt{10} x+x+3 \left (4+\sqrt{10}\right )}{2 \sqrt{\sqrt{10}-1} \sqrt{-2 x^2+3 x+1}}\right )}{10 \sqrt{1+\sqrt{10}}} \]

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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Maple [B]  time = 0.135, size = 324, normalized size = 2.3 \begin{align*}{\frac{2\,\sqrt{10}}{5\,\sqrt{-1+\sqrt{10}}}{\it Artanh} \left ({\frac{9}{2\,\sqrt{-1+\sqrt{10}}} \left ( -{\frac{2}{9}}+{\frac{2\,\sqrt{10}}{9}}+ \left ({\frac{1}{3}}-{\frac{4\,\sqrt{10}}{3}} \right ) \left ( x-{\frac{2}{3}}-{\frac{\sqrt{10}}{3}} \right ) \right ){\frac{1}{\sqrt{-18\, \left ( x-2/3-1/3\,\sqrt{10} \right ) ^{2}+9\, \left ( 1/3-4/3\,\sqrt{10} \right ) \left ( x-2/3-1/3\,\sqrt{10} \right ) -1+\sqrt{10}}}}} \right ) }+{\frac{1}{2\,\sqrt{-1+\sqrt{10}}}{\it Artanh} \left ({\frac{9}{2\,\sqrt{-1+\sqrt{10}}} \left ( -{\frac{2}{9}}+{\frac{2\,\sqrt{10}}{9}}+ \left ({\frac{1}{3}}-{\frac{4\,\sqrt{10}}{3}} \right ) \left ( x-{\frac{2}{3}}-{\frac{\sqrt{10}}{3}} \right ) \right ){\frac{1}{\sqrt{-18\, \left ( x-2/3-1/3\,\sqrt{10} \right ) ^{2}+9\, \left ( 1/3-4/3\,\sqrt{10} \right ) \left ( x-2/3-1/3\,\sqrt{10} \right ) -1+\sqrt{10}}}}} \right ) }+{\frac{2\,\sqrt{10}}{5\,\sqrt{1+\sqrt{10}}}\arctan \left ({\frac{9}{2\,\sqrt{1+\sqrt{10}}} \left ( -{\frac{2}{9}}-{\frac{2\,\sqrt{10}}{9}}+ \left ({\frac{1}{3}}+{\frac{4\,\sqrt{10}}{3}} \right ) \left ( x-{\frac{2}{3}}+{\frac{\sqrt{10}}{3}} \right ) \right ){\frac{1}{\sqrt{-18\, \left ( x-2/3+1/3\,\sqrt{10} \right ) ^{2}+9\, \left ( 1/3+4/3\,\sqrt{10} \right ) \left ( x-2/3+1/3\,\sqrt{10} \right ) -1-\sqrt{10}}}}} \right ) }-{\frac{1}{2\,\sqrt{1+\sqrt{10}}}\arctan \left ({\frac{9}{2\,\sqrt{1+\sqrt{10}}} \left ( -{\frac{2}{9}}-{\frac{2\,\sqrt{10}}{9}}+ \left ({\frac{1}{3}}+{\frac{4\,\sqrt{10}}{3}} \right ) \left ( x-{\frac{2}{3}}+{\frac{\sqrt{10}}{3}} \right ) \right ){\frac{1}{\sqrt{-18\, \left ( x-2/3+1/3\,\sqrt{10} \right ) ^{2}+9\, \left ( 1/3+4/3\,\sqrt{10} \right ) \left ( x-2/3+1/3\,\sqrt{10} \right ) -1-\sqrt{10}}}}} \right ) } \end{align*}

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)*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+1
0^(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))+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))

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Maxima [B]  time = 1.56001, size = 487, normalized size = 3.5 \begin{align*} -\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{align*}

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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Fricas [B]  time = 1.64926, size = 948, normalized size = 6.82 \begin{align*} \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{align*}

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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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \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{align*}

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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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

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]

Exception raised: TypeError