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

Optimal. Leaf size=151 \[ \frac{1}{2} \sqrt{1-\frac{7 \sqrt{\frac{2}{5}}}{5}} \tanh ^{-1}\left (\frac{\left (17+4 \sqrt{10}\right ) x+3 \left (4+\sqrt{10}\right )}{2 \sqrt{55+17 \sqrt{10}} \sqrt{2 x^2+3 x+1}}\right )-\frac{1}{2} \sqrt{1+\frac{7 \sqrt{\frac{2}{5}}}{5}} \tanh ^{-1}\left (\frac{\left (17-4 \sqrt{10}\right ) x+3 \left (4-\sqrt{10}\right )}{2 \sqrt{55-17 \sqrt{10}} \sqrt{2 x^2+3 x+1}}\right ) \]

[Out]

-(Sqrt[1 + (7*Sqrt[2/5])/5]*ArcTanh[(3*(4 - Sqrt[10]) + (17 - 4*Sqrt[10])*x)/(2*Sqrt[55 - 17*Sqrt[10]]*Sqrt[1
+ 3*x + 2*x^2])])/2 + (Sqrt[1 - (7*Sqrt[2/5])/5]*ArcTanh[(3*(4 + Sqrt[10]) + (17 + 4*Sqrt[10])*x)/(2*Sqrt[55 +
17*Sqrt[10]]*Sqrt[1 + 3*x + 2*x^2])])/2

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Rubi [A]  time = 0.228555, antiderivative size = 151, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {1032, 724, 206} \[ \frac{1}{2} \sqrt{1-\frac{7 \sqrt{\frac{2}{5}}}{5}} \tanh ^{-1}\left (\frac{\left (17+4 \sqrt{10}\right ) x+3 \left (4+\sqrt{10}\right )}{2 \sqrt{55+17 \sqrt{10}} \sqrt{2 x^2+3 x+1}}\right )-\frac{1}{2} \sqrt{1+\frac{7 \sqrt{\frac{2}{5}}}{5}} \tanh ^{-1}\left (\frac{\left (17-4 \sqrt{10}\right ) x+3 \left (4-\sqrt{10}\right )}{2 \sqrt{55-17 \sqrt{10}} \sqrt{2 x^2+3 x+1}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[1 + (7*Sqrt[2/5])/5]*ArcTanh[(3*(4 - Sqrt[10]) + (17 - 4*Sqrt[10])*x)/(2*Sqrt[55 - 17*Sqrt[10]]*Sqrt[1
+ 3*x + 2*x^2])])/2 + (Sqrt[1 - (7*Sqrt[2/5])/5]*ArcTanh[(3*(4 + Sqrt[10]) + (17 + 4*Sqrt[10])*x)/(2*Sqrt[55 +
17*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 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{25+7 \sqrt{10}} \tanh ^{-1}\left (\frac{3 \left (4-\sqrt{10}\right )+\left (17-4 \sqrt{10}\right ) x}{2 \sqrt{55-17 \sqrt{10}} \sqrt{1+3 x+2 x^2}}\right )+\frac{1}{10} \sqrt{25-7 \sqrt{10}} \tanh ^{-1}\left (\frac{3 \left (4+\sqrt{10}\right )+\left (17+4 \sqrt{10}\right ) x}{2 \sqrt{55+17 \sqrt{10}} \sqrt{1+3 x+2 x^2}}\right )\\ \end{align*}

Mathematica [A]  time = 0.354072, size = 148, normalized size = 0.98 \[ \frac{\left (5-4 \sqrt{10}\right ) \tanh ^{-1}\left (\frac{-4 \sqrt{10} x+17 x-3 \sqrt{10}+12}{2 \sqrt{55-17 \sqrt{10}} \sqrt{2 x^2+3 x+1}}\right )+3 \sqrt{285-90 \sqrt{10}} \tanh ^{-1}\left (\frac{\left (17+4 \sqrt{10}\right ) x+3 \left (4+\sqrt{10}\right )}{2 \sqrt{55+17 \sqrt{10}} \sqrt{2 x^2+3 x+1}}\right )}{10 \sqrt{55-17 \sqrt{10}}} \]

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])*ArcTanh[(12 - 3*Sqrt[10] + 17*x - 4*Sqrt[10]*x)/(2*Sqrt[55 - 17*Sqrt[10]]*Sqrt[1 + 3*x + 2*x
^2])] + 3*Sqrt[285 - 90*Sqrt[10]]*ArcTanh[(3*(4 + Sqrt[10]) + (17 + 4*Sqrt[10])*x)/(2*Sqrt[55 + 17*Sqrt[10]]*S
qrt[1 + 3*x + 2*x^2])])/(10*Sqrt[55 - 17*Sqrt[10]])

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Maple [A]  time = 0.125, size = 186, normalized size = 1.2 \begin{align*}{\frac{ \left ( 8+\sqrt{10} \right ) \sqrt{10}}{20\,\sqrt{55+17\,\sqrt{10}}}{\it Artanh} \left ({\frac{9}{2\,\sqrt{55+17\,\sqrt{10}}} \left ({\frac{110}{9}}+{\frac{34\,\sqrt{10}}{9}}+ \left ({\frac{17}{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 ({\frac{17}{3}}+4/3\,\sqrt{10} \right ) \left ( x-2/3-1/3\,\sqrt{10} \right ) +55+17\,\sqrt{10}}}}} \right ) }+{\frac{ \left ( -8+\sqrt{10} \right ) \sqrt{10}}{20\,\sqrt{55-17\,\sqrt{10}}}{\it Artanh} \left ({\frac{9}{2\,\sqrt{55-17\,\sqrt{10}}} \left ({\frac{110}{9}}-{\frac{34\,\sqrt{10}}{9}}+ \left ({\frac{17}{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 ({\frac{17}{3}}-4/3\,\sqrt{10} \right ) \left ( x-2/3+1/3\,\sqrt{10} \right ) +55-17\,\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]

1/20*(8+10^(1/2))*10^(1/2)/(55+17*10^(1/2))^(1/2)*arctanh(9/2*(110/9+34/9*10^(1/2)+(17/3+4/3*10^(1/2))*(x-2/3-
1/3*10^(1/2)))/(55+17*10^(1/2))^(1/2)/(18*(x-2/3-1/3*10^(1/2))^2+9*(17/3+4/3*10^(1/2))*(x-2/3-1/3*10^(1/2))+55
+17*10^(1/2))^(1/2))+1/20*(-8+10^(1/2))*10^(1/2)/(55-17*10^(1/2))^(1/2)*arctanh(9/2*(110/9-34/9*10^(1/2)+(17/3
-4/3*10^(1/2))*(x-2/3+1/3*10^(1/2)))/(55-17*10^(1/2))^(1/2)/(18*(x-2/3+1/3*10^(1/2))^2+9*(17/3-4/3*10^(1/2))*(
x-2/3+1/3*10^(1/2))+55-17*10^(1/2))^(1/2))

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Maxima [B]  time = 1.53812, size = 490, normalized size = 3.25 \begin{align*} \frac{1}{60} \, \sqrt{10}{\left (\frac{3 \, \sqrt{10} \log \left (\frac{2}{9} \, \sqrt{10} + \frac{2 \, \sqrt{2 \, x^{2} + 3 \, x + 1} \sqrt{17 \, \sqrt{10} + 55}}{3 \,{\left | 6 \, x - 2 \, \sqrt{10} - 4 \right |}} + \frac{34 \, \sqrt{10}}{9 \,{\left | 6 \, x - 2 \, \sqrt{10} - 4 \right |}} + \frac{110}{9 \,{\left | 6 \, x - 2 \, \sqrt{10} - 4 \right |}} + \frac{17}{18}\right )}{\sqrt{17 \, \sqrt{10} + 55}} + \frac{\sqrt{10} \log \left (-\frac{2}{9} \, \sqrt{10} + \frac{2 \, \sqrt{2 \, x^{2} + 3 \, x + 1} \sqrt{-\frac{17}{9} \, \sqrt{10} + \frac{55}{9}}}{{\left | 6 \, x + 2 \, \sqrt{10} - 4 \right |}} - \frac{34 \, \sqrt{10}}{9 \,{\left | 6 \, x + 2 \, \sqrt{10} - 4 \right |}} + \frac{110}{9 \,{\left | 6 \, x + 2 \, \sqrt{10} - 4 \right |}} + \frac{17}{18}\right )}{\sqrt{-\frac{17}{9} \, \sqrt{10} + \frac{55}{9}}} + \frac{24 \, \log \left (\frac{2}{9} \, \sqrt{10} + \frac{2 \, \sqrt{2 \, x^{2} + 3 \, x + 1} \sqrt{17 \, \sqrt{10} + 55}}{3 \,{\left | 6 \, x - 2 \, \sqrt{10} - 4 \right |}} + \frac{34 \, \sqrt{10}}{9 \,{\left | 6 \, x - 2 \, \sqrt{10} - 4 \right |}} + \frac{110}{9 \,{\left | 6 \, x - 2 \, \sqrt{10} - 4 \right |}} + \frac{17}{18}\right )}{\sqrt{17 \, \sqrt{10} + 55}} - \frac{8 \, \log \left (-\frac{2}{9} \, \sqrt{10} + \frac{2 \, \sqrt{2 \, x^{2} + 3 \, x + 1} \sqrt{-\frac{17}{9} \, \sqrt{10} + \frac{55}{9}}}{{\left | 6 \, x + 2 \, \sqrt{10} - 4 \right |}} - \frac{34 \, \sqrt{10}}{9 \,{\left | 6 \, x + 2 \, \sqrt{10} - 4 \right |}} + \frac{110}{9 \,{\left | 6 \, x + 2 \, \sqrt{10} - 4 \right |}} + \frac{17}{18}\right )}{\sqrt{-\frac{17}{9} \, \sqrt{10} + \frac{55}{9}}}\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]

1/60*sqrt(10)*(3*sqrt(10)*log(2/9*sqrt(10) + 2/3*sqrt(2*x^2 + 3*x + 1)*sqrt(17*sqrt(10) + 55)/abs(6*x - 2*sqrt
(10) - 4) + 34/9*sqrt(10)/abs(6*x - 2*sqrt(10) - 4) + 110/9/abs(6*x - 2*sqrt(10) - 4) + 17/18)/sqrt(17*sqrt(10
) + 55) + sqrt(10)*log(-2/9*sqrt(10) + 2*sqrt(2*x^2 + 3*x + 1)*sqrt(-17/9*sqrt(10) + 55/9)/abs(6*x + 2*sqrt(10
) - 4) - 34/9*sqrt(10)/abs(6*x + 2*sqrt(10) - 4) + 110/9/abs(6*x + 2*sqrt(10) - 4) + 17/18)/sqrt(-17/9*sqrt(10
) + 55/9) + 24*log(2/9*sqrt(10) + 2/3*sqrt(2*x^2 + 3*x + 1)*sqrt(17*sqrt(10) + 55)/abs(6*x - 2*sqrt(10) - 4) +
34/9*sqrt(10)/abs(6*x - 2*sqrt(10) - 4) + 110/9/abs(6*x - 2*sqrt(10) - 4) + 17/18)/sqrt(17*sqrt(10) + 55) - 8
*log(-2/9*sqrt(10) + 2*sqrt(2*x^2 + 3*x + 1)*sqrt(-17/9*sqrt(10) + 55/9)/abs(6*x + 2*sqrt(10) - 4) - 34/9*sqrt
(10)/abs(6*x + 2*sqrt(10) - 4) + 110/9/abs(6*x + 2*sqrt(10) - 4) + 17/18)/sqrt(-17/9*sqrt(10) + 55/9))

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Fricas [B]  time = 1.28405, size = 709, normalized size = 4.7 \begin{align*} \frac{1}{10} \, \sqrt{7 \, \sqrt{10} + 25} \log \left (-\frac{3 \, \sqrt{10} x +{\left (\sqrt{10} x - 4 \, x\right )} \sqrt{7 \, \sqrt{10} + 25} + 6 \, x - 6 \, \sqrt{2 \, x^{2} + 3 \, x + 1} + 6}{x}\right ) - \frac{1}{10} \, \sqrt{7 \, \sqrt{10} + 25} \log \left (-\frac{3 \, \sqrt{10} x -{\left (\sqrt{10} x - 4 \, x\right )} \sqrt{7 \, \sqrt{10} + 25} + 6 \, x - 6 \, \sqrt{2 \, x^{2} + 3 \, x + 1} + 6}{x}\right ) + \frac{1}{10} \, \sqrt{-7 \, \sqrt{10} + 25} \log \left (\frac{3 \, \sqrt{10} x +{\left (\sqrt{10} x + 4 \, x\right )} \sqrt{-7 \, \sqrt{10} + 25} - 6 \, x + 6 \, \sqrt{2 \, x^{2} + 3 \, x + 1} - 6}{x}\right ) - \frac{1}{10} \, \sqrt{-7 \, \sqrt{10} + 25} \log \left (\frac{3 \, \sqrt{10} x -{\left (\sqrt{10} x + 4 \, x\right )} \sqrt{-7 \, \sqrt{10} + 25} - 6 \, x + 6 \, \sqrt{2 \, x^{2} + 3 \, x + 1} - 6}{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]

1/10*sqrt(7*sqrt(10) + 25)*log(-(3*sqrt(10)*x + (sqrt(10)*x - 4*x)*sqrt(7*sqrt(10) + 25) + 6*x - 6*sqrt(2*x^2
+ 3*x + 1) + 6)/x) - 1/10*sqrt(7*sqrt(10) + 25)*log(-(3*sqrt(10)*x - (sqrt(10)*x - 4*x)*sqrt(7*sqrt(10) + 25)
+ 6*x - 6*sqrt(2*x^2 + 3*x + 1) + 6)/x) + 1/10*sqrt(-7*sqrt(10) + 25)*log((3*sqrt(10)*x + (sqrt(10)*x + 4*x)*s
qrt(-7*sqrt(10) + 25) - 6*x + 6*sqrt(2*x^2 + 3*x + 1) - 6)/x) - 1/10*sqrt(-7*sqrt(10) + 25)*log((3*sqrt(10)*x
- (sqrt(10)*x + 4*x)*sqrt(-7*sqrt(10) + 25) - 6*x + 6*sqrt(2*x^2 + 3*x + 1) - 6)/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) - Inte
gral(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