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

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

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Rubi [A]  time = 0.23, antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {1032, 724, 206} \begin {gather*} \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 ) \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[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 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 {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*}

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Mathematica [A]  time = 0.36, size = 148, normalized size = 0.98 \begin {gather*} \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}}} \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])*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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IntegrateAlgebraic [A]  time = 0.78, size = 109, normalized size = 0.72 \begin {gather*} \frac {1}{5} \sqrt {25-7 \sqrt {10}} \tanh ^{-1}\left (\frac {\sqrt {1+\sqrt {\frac {2}{5}}} \sqrt {2 x^2+3 x+1}}{2 x+1}\right )-\frac {1}{5} \sqrt {25+7 \sqrt {10}} \tanh ^{-1}\left (\frac {\sqrt {1-\sqrt {\frac {2}{5}}} \sqrt {2 x^2+3 x+1}}{2 x+1}\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]

-1/5*(Sqrt[25 + 7*Sqrt[10]]*ArcTanh[(Sqrt[1 - Sqrt[2/5]]*Sqrt[1 + 3*x + 2*x^2])/(1 + 2*x)]) + (Sqrt[25 - 7*Sqr
t[10]]*ArcTanh[(Sqrt[1 + Sqrt[2/5]]*Sqrt[1 + 3*x + 2*x^2])/(1 + 2*x)])/5

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fricas [B]  time = 0.43, size = 245, normalized size = 1.62 \begin {gather*} \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 {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]

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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giac [A]  time = 0.48, size = 93, normalized size = 0.62 \begin {gather*} 0.169235232112667 \, \log \left (-\sqrt {2} x + \sqrt {2 \, x^{2} + 3 \, x + 1} + 5.90976932712000\right ) - 0.686556214893333 \, \log \left (-\sqrt {2} x + \sqrt {2 \, x^{2} + 3 \, x + 1} - 0.176527156327000\right ) + 0.686556214893333 \, \log \left (-\sqrt {2} x + \sqrt {2 \, x^{2} + 3 \, x + 1} - 0.919278730509000\right ) - 0.169235232112667 \, \log \left (-\sqrt {2} x + \sqrt {2 \, x^{2} + 3 \, x + 1} - 1.04272727395000\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="giac")

[Out]

0.169235232112667*log(-sqrt(2)*x + sqrt(2*x^2 + 3*x + 1) + 5.90976932712000) - 0.686556214893333*log(-sqrt(2)*
x + sqrt(2*x^2 + 3*x + 1) - 0.176527156327000) + 0.686556214893333*log(-sqrt(2)*x + sqrt(2*x^2 + 3*x + 1) - 0.
919278730509000) - 0.169235232112667*log(-sqrt(2)*x + sqrt(2*x^2 + 3*x + 1) - 1.04272727395000)

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maple [A]  time = 0.05, size = 186, normalized size = 1.23 \begin {gather*} \frac {\left (-8+\sqrt {10}\right ) \sqrt {10}\, \arctanh \left (\frac {55-17 \sqrt {10}+\frac {9 \left (\frac {17}{3}-\frac {4 \sqrt {10}}{3}\right ) \left (x -\frac {2}{3}+\frac {\sqrt {10}}{3}\right )}{2}}{\sqrt {55-17 \sqrt {10}}\, \sqrt {18 \left (x -\frac {2}{3}+\frac {\sqrt {10}}{3}\right )^{2}+9 \left (\frac {17}{3}-\frac {4 \sqrt {10}}{3}\right ) \left (x -\frac {2}{3}+\frac {\sqrt {10}}{3}\right )+55-17 \sqrt {10}}}\right )}{20 \sqrt {55-17 \sqrt {10}}}+\frac {\left (8+\sqrt {10}\right ) \sqrt {10}\, \arctanh \left (\frac {55+17 \sqrt {10}+\frac {9 \left (\frac {17}{3}+\frac {4 \sqrt {10}}{3}\right ) \left (x -\frac {2}{3}-\frac {\sqrt {10}}{3}\right )}{2}}{\sqrt {55+17 \sqrt {10}}\, \sqrt {18 \left (x -\frac {2}{3}-\frac {\sqrt {10}}{3}\right )^{2}+9 \left (\frac {17}{3}+\frac {4 \sqrt {10}}{3}\right ) \left (x -\frac {2}{3}-\frac {\sqrt {10}}{3}\right )+55+17 \sqrt {10}}}\right )}{20 \sqrt {55+17 \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]

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))+5
5-17*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 = 1.07, size = 363, normalized size = 2.40 \begin {gather*} \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 {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]

1/60*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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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) - 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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