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

Optimal. Leaf size=174 \[ \frac {2 (22 x+21)}{5 \sqrt {2 x^2+3 x+1}}-\frac {1}{10} \sqrt {\frac {3}{5} \left (2065+653 \sqrt {10}\right )} \tanh ^{-1}\left (\frac {\left (17-4 \sqrt {10}\right ) x+3 \left (4-\sqrt {10}\right )}{2 \sqrt {55-17 \sqrt {10}} \sqrt {2 x^2+3 x+1}}\right )+\frac {1}{10} \sqrt {\frac {3}{5} \left (2065-653 \sqrt {10}\right )} \tanh ^{-1}\left (\frac {\left (17+4 \sqrt {10}\right ) x+3 \left (4+\sqrt {10}\right )}{2 \sqrt {55+17 \sqrt {10}} \sqrt {2 x^2+3 x+1}}\right ) \]

________________________________________________________________________________________

Rubi [A]  time = 0.25, antiderivative size = 174, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {1016, 1032, 724, 206} \begin {gather*} \frac {2 (22 x+21)}{5 \sqrt {2 x^2+3 x+1}}-\frac {1}{10} \sqrt {\frac {3}{5} \left (2065+653 \sqrt {10}\right )} \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}{10} \sqrt {\frac {3}{5} \left (2065-653 \sqrt {10}\right )} \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)*(1 + 3*x + 2*x^2)^(3/2)),x]

[Out]

(2*(21 + 22*x))/(5*Sqrt[1 + 3*x + 2*x^2]) - (Sqrt[(3*(2065 + 653*Sqrt[10]))/5]*ArcTanh[(3*(4 - Sqrt[10]) + (17
- 4*Sqrt[10])*x)/(2*Sqrt[55 - 17*Sqrt[10]]*Sqrt[1 + 3*x + 2*x^2])])/10 + (Sqrt[(3*(2065 - 653*Sqrt[10]))/5]*A
rcTanh[(3*(4 + Sqrt[10]) + (17 + 4*Sqrt[10])*x)/(2*Sqrt[55 + 17*Sqrt[10]]*Sqrt[1 + 3*x + 2*x^2])])/10

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 1016

Int[((g_.) + (h_.)*(x_))*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_) + (e_.)*(x_) + (f_.)*(x_)^2)^(q_), x_Sy
mbol] :> Simp[((a + b*x + c*x^2)^(p + 1)*(d + e*x + f*x^2)^(q + 1)*(g*c*(2*a*c*e - b*(c*d + a*f)) + (g*b - a*h
)*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)) + c*(g*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)) - h*(b*c*d - 2*a*c*e + a*b*f)
)*x))/((b^2 - 4*a*c)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1)), x] + Dist[1/((b^2 - 4*a*c)*((c*d - a*
f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1)), Int[(a + b*x + c*x^2)^(p + 1)*(d + e*x + f*x^2)^q*Simp[(b*h - 2*g*c)
*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1) + (b^2*(g*f) - b*(h*c*d + g*c*e + a*h*f) + 2*(g*c*(c*d - a*
f) - a*(-(h*c*e))))*(a*f*(p + 1) - c*d*(p + 2)) - e*((g*c)*(2*a*c*e - b*(c*d + a*f)) + (g*b - a*h)*(2*c^2*d +
b^2*f - c*(b*e + 2*a*f)))*(p + q + 2) - (2*f*((g*c)*(2*a*c*e - b*(c*d + a*f)) + (g*b - a*h)*(2*c^2*d + b^2*f -
c*(b*e + 2*a*f)))*(p + q + 2) - (b^2*g*f - b*(h*c*d + g*c*e + a*h*f) + 2*(g*c*(c*d - a*f) - a*(-(h*c*e))))*(b
*f*(p + 1) - c*e*(2*p + q + 4)))*x - c*f*(b^2*(g*f) - b*(h*c*d + g*c*e + a*h*f) + 2*(g*c*(c*d - a*f) + a*h*c*e
))*(2*p + 2*q + 5)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, q}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2
- 4*d*f, 0] && LtQ[p, -1] && NeQ[(c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f), 0] &&  !( !IntegerQ[p] && ILtQ[q, -1
])

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 ) \left (1+3 x+2 x^2\right )^{3/2}} \, dx &=\frac {2 (21+22 x)}{5 \sqrt {1+3 x+2 x^2}}-\frac {2}{15} \int \frac {-72+\frac {81 x}{2}}{\left (2+4 x-3 x^2\right ) \sqrt {1+3 x+2 x^2}} \, dx\\ &=\frac {2 (21+22 x)}{5 \sqrt {1+3 x+2 x^2}}-\frac {1}{5} \left (9 \left (3-\sqrt {10}\right )\right ) \int \frac {1}{\left (4+2 \sqrt {10}-6 x\right ) \sqrt {1+3 x+2 x^2}} \, dx-\frac {1}{5} \left (9 \left (3+\sqrt {10}\right )\right ) \int \frac {1}{\left (4-2 \sqrt {10}-6 x\right ) \sqrt {1+3 x+2 x^2}} \, dx\\ &=\frac {2 (21+22 x)}{5 \sqrt {1+3 x+2 x^2}}+\frac {1}{5} \left (18 \left (3-\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}{5} \left (18 \left (3+\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 {2 (21+22 x)}{5 \sqrt {1+3 x+2 x^2}}-\frac {1}{10} \sqrt {\frac {3}{5} \left (2065+653 \sqrt {10}\right )} \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 {\frac {3}{5} \left (2065-653 \sqrt {10}\right )} \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 = 0.57, size = 172, normalized size = 0.99 \begin {gather*} \frac {1}{50} \left (\frac {\sqrt {30975-9795 \sqrt {10}} \sqrt {2 x^2+3 x+1} \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 )+440 x+420}{\sqrt {2 x^2+3 x+1}}-\sqrt {30975+9795 \sqrt {10}} \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 )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-(Sqrt[30975 + 9795*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])]) + (420 + 440*x + Sqrt[30975 - 9795*Sqrt[10]]*Sqrt[1 + 3*x + 2*x^2]*ArcTanh[(12 + 3*Sqrt[10
] + 17*x + 4*Sqrt[10]*x)/(2*Sqrt[55 + 17*Sqrt[10]]*Sqrt[1 + 3*x + 2*x^2])])/Sqrt[1 + 3*x + 2*x^2])/50

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 1.02, size = 148, normalized size = 0.85 \begin {gather*} \frac {2 \sqrt {2 x^2+3 x+1} (22 x+21)}{5 (x+1) (2 x+1)}-\frac {1}{5} \sqrt {\frac {1}{5} \left (6195+1959 \sqrt {10}\right )} \tanh ^{-1}\left (\frac {\sqrt {1-\sqrt {\frac {2}{5}}} \sqrt {2 x^2+3 x+1}}{2 x+1}\right )+\frac {9 \tanh ^{-1}\left (\frac {\sqrt {1+\sqrt {\frac {2}{5}}} \sqrt {2 x^2+3 x+1}}{2 x+1}\right )}{5 \sqrt {2065+653 \sqrt {10}}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(21 + 22*x)*Sqrt[1 + 3*x + 2*x^2])/(5*(1 + x)*(1 + 2*x)) - (Sqrt[(6195 + 1959*Sqrt[10])/5]*ArcTanh[(Sqrt[1
- Sqrt[2/5]]*Sqrt[1 + 3*x + 2*x^2])/(1 + 2*x)])/5 + (9*ArcTanh[(Sqrt[1 + Sqrt[2/5]]*Sqrt[1 + 3*x + 2*x^2])/(1
+ 2*x)])/(5*Sqrt[2065 + 653*Sqrt[10]])

________________________________________________________________________________________

fricas [B]  time = 0.45, size = 365, normalized size = 2.10 \begin {gather*} \frac {\sqrt {5} {\left (2 \, x^{2} + 3 \, x + 1\right )} \sqrt {1959 \, \sqrt {10} + 6195} \log \left (-\frac {45 \, \sqrt {10} x + {\left (41 \, \sqrt {10} \sqrt {5} x - 130 \, \sqrt {5} x\right )} \sqrt {1959 \, \sqrt {10} + 6195} + 90 \, x - 90 \, \sqrt {2 \, x^{2} + 3 \, x + 1} + 90}{x}\right ) - \sqrt {5} {\left (2 \, x^{2} + 3 \, x + 1\right )} \sqrt {1959 \, \sqrt {10} + 6195} \log \left (-\frac {45 \, \sqrt {10} x - {\left (41 \, \sqrt {10} \sqrt {5} x - 130 \, \sqrt {5} x\right )} \sqrt {1959 \, \sqrt {10} + 6195} + 90 \, x - 90 \, \sqrt {2 \, x^{2} + 3 \, x + 1} + 90}{x}\right ) + \sqrt {5} {\left (2 \, x^{2} + 3 \, x + 1\right )} \sqrt {-1959 \, \sqrt {10} + 6195} \log \left (\frac {45 \, \sqrt {10} x + {\left (41 \, \sqrt {10} \sqrt {5} x + 130 \, \sqrt {5} x\right )} \sqrt {-1959 \, \sqrt {10} + 6195} - 90 \, x + 90 \, \sqrt {2 \, x^{2} + 3 \, x + 1} - 90}{x}\right ) - \sqrt {5} {\left (2 \, x^{2} + 3 \, x + 1\right )} \sqrt {-1959 \, \sqrt {10} + 6195} \log \left (\frac {45 \, \sqrt {10} x - {\left (41 \, \sqrt {10} \sqrt {5} x + 130 \, \sqrt {5} x\right )} \sqrt {-1959 \, \sqrt {10} + 6195} - 90 \, x + 90 \, \sqrt {2 \, x^{2} + 3 \, x + 1} - 90}{x}\right ) + 840 \, x^{2} + 20 \, \sqrt {2 \, x^{2} + 3 \, x + 1} {\left (22 \, x + 21\right )} + 1260 \, x + 420}{50 \, {\left (2 \, x^{2} + 3 \, x + 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)^(3/2),x, algorithm="fricas")

[Out]

1/50*(sqrt(5)*(2*x^2 + 3*x + 1)*sqrt(1959*sqrt(10) + 6195)*log(-(45*sqrt(10)*x + (41*sqrt(10)*sqrt(5)*x - 130*
sqrt(5)*x)*sqrt(1959*sqrt(10) + 6195) + 90*x - 90*sqrt(2*x^2 + 3*x + 1) + 90)/x) - sqrt(5)*(2*x^2 + 3*x + 1)*s
qrt(1959*sqrt(10) + 6195)*log(-(45*sqrt(10)*x - (41*sqrt(10)*sqrt(5)*x - 130*sqrt(5)*x)*sqrt(1959*sqrt(10) + 6
195) + 90*x - 90*sqrt(2*x^2 + 3*x + 1) + 90)/x) + sqrt(5)*(2*x^2 + 3*x + 1)*sqrt(-1959*sqrt(10) + 6195)*log((4
5*sqrt(10)*x + (41*sqrt(10)*sqrt(5)*x + 130*sqrt(5)*x)*sqrt(-1959*sqrt(10) + 6195) - 90*x + 90*sqrt(2*x^2 + 3*
x + 1) - 90)/x) - sqrt(5)*(2*x^2 + 3*x + 1)*sqrt(-1959*sqrt(10) + 6195)*log((45*sqrt(10)*x - (41*sqrt(10)*sqrt
(5)*x + 130*sqrt(5)*x)*sqrt(-1959*sqrt(10) + 6195) - 90*x + 90*sqrt(2*x^2 + 3*x + 1) - 90)/x) + 840*x^2 + 20*s
qrt(2*x^2 + 3*x + 1)*(22*x + 21) + 1260*x + 420)/(2*x^2 + 3*x + 1)

________________________________________________________________________________________

giac [A]  time = 0.50, size = 112, normalized size = 0.64 \begin {gather*} \frac {2 \, {\left (22 \, x + 21\right )}}{5 \, \sqrt {2 \, x^{2} + 3 \, x + 1}} + 0.0140045514133333 \, \log \left (-\sqrt {2} x + \sqrt {2 \, x^{2} + 3 \, x + 1} + 5.90976932712000\right ) - 4.97793168620000 \, \log \left (-\sqrt {2} x + \sqrt {2 \, x^{2} + 3 \, x + 1} - 0.176527156327000\right ) + 4.97793168620000 \, \log \left (-\sqrt {2} x + \sqrt {2 \, x^{2} + 3 \, x + 1} - 0.919278730509000\right ) - 0.0140045514125333 \, \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)^(3/2),x, algorithm="giac")

[Out]

2/5*(22*x + 21)/sqrt(2*x^2 + 3*x + 1) + 0.0140045514133333*log(-sqrt(2)*x + sqrt(2*x^2 + 3*x + 1) + 5.90976932
712000) - 4.97793168620000*log(-sqrt(2)*x + sqrt(2*x^2 + 3*x + 1) - 0.176527156327000) + 4.97793168620000*log(
-sqrt(2)*x + sqrt(2*x^2 + 3*x + 1) - 0.919278730509000) - 0.0140045514125333*log(-sqrt(2)*x + sqrt(2*x^2 + 3*x
+ 1) - 1.04272727395000)

________________________________________________________________________________________

maple [B]  time = 0.02, size = 466, normalized size = 2.68 \begin {gather*} -\frac {\left (-8+\sqrt {10}\right ) \sqrt {10}\, \left (-\frac {\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 )}{\left (\frac {55}{9}-\frac {17 \sqrt {10}}{9}\right ) \sqrt {55-17 \sqrt {10}}}+\frac {1}{3 \left (\frac {55}{9}-\frac {17 \sqrt {10}}{9}\right ) \sqrt {2 \left (x -\frac {2}{3}+\frac {\sqrt {10}}{3}\right )^{2}+\left (\frac {17}{3}-\frac {4 \sqrt {10}}{3}\right ) \left (x -\frac {2}{3}+\frac {\sqrt {10}}{3}\right )+\frac {55}{9}-\frac {17 \sqrt {10}}{9}}}-\frac {\left (\frac {17}{3}-\frac {4 \sqrt {10}}{3}\right ) \left (4 x +3\right )}{3 \left (\frac {55}{9}-\frac {17 \sqrt {10}}{9}\right ) \left (\frac {440}{9}-\frac {136 \sqrt {10}}{9}-\left (\frac {17}{3}-\frac {4 \sqrt {10}}{3}\right )^{2}\right ) \sqrt {2 \left (x -\frac {2}{3}+\frac {\sqrt {10}}{3}\right )^{2}+\left (\frac {17}{3}-\frac {4 \sqrt {10}}{3}\right ) \left (x -\frac {2}{3}+\frac {\sqrt {10}}{3}\right )+\frac {55}{9}-\frac {17 \sqrt {10}}{9}}}\right )}{20}-\frac {\left (8+\sqrt {10}\right ) \sqrt {10}\, \left (-\frac {\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 )}{\left (\frac {55}{9}+\frac {17 \sqrt {10}}{9}\right ) \sqrt {55+17 \sqrt {10}}}+\frac {1}{3 \left (\frac {55}{9}+\frac {17 \sqrt {10}}{9}\right ) \sqrt {2 \left (x -\frac {2}{3}-\frac {\sqrt {10}}{3}\right )^{2}+\left (\frac {17}{3}+\frac {4 \sqrt {10}}{3}\right ) \left (x -\frac {2}{3}-\frac {\sqrt {10}}{3}\right )+\frac {55}{9}+\frac {17 \sqrt {10}}{9}}}-\frac {\left (\frac {17}{3}+\frac {4 \sqrt {10}}{3}\right ) \left (4 x +3\right )}{3 \left (\frac {55}{9}+\frac {17 \sqrt {10}}{9}\right ) \left (\frac {440}{9}+\frac {136 \sqrt {10}}{9}-\left (\frac {17}{3}+\frac {4 \sqrt {10}}{3}\right )^{2}\right ) \sqrt {2 \left (x -\frac {2}{3}-\frac {\sqrt {10}}{3}\right )^{2}+\left (\frac {17}{3}+\frac {4 \sqrt {10}}{3}\right ) \left (x -\frac {2}{3}-\frac {\sqrt {10}}{3}\right )+\frac {55}{9}+\frac {17 \sqrt {10}}{9}}}\right )}{20} \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)^(3/2),x)

[Out]

-1/20*(-8+10^(1/2))*10^(1/2)*(1/3/(55/9-17/9*10^(1/2))/(2*(x-2/3+1/3*10^(1/2))^2+(17/3-4/3*10^(1/2))*(x-2/3+1/
3*10^(1/2))+55/9-17/9*10^(1/2))^(1/2)-1/3*(17/3-4/3*10^(1/2))/(55/9-17/9*10^(1/2))*(3+4*x)/(440/9-136/9*10^(1/
2)-(17/3-4/3*10^(1/2))^2)/(2*(x-2/3+1/3*10^(1/2))^2+(17/3-4/3*10^(1/2))*(x-2/3+1/3*10^(1/2))+55/9-17/9*10^(1/2
))^(1/2)-1/(55/9-17/9*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)*(1/3/(55/9+17/9*10^(1/2))/(2*(x-2/3-1/3*10^(1/2))^2+(17/
3+4/3*10^(1/2))*(x-2/3-1/3*10^(1/2))+55/9+17/9*10^(1/2))^(1/2)-1/3*(17/3+4/3*10^(1/2))/(55/9+17/9*10^(1/2))*(3
+4*x)/(440/9+136/9*10^(1/2)-(17/3+4/3*10^(1/2))^2)/(2*(x-2/3-1/3*10^(1/2))^2+(17/3+4/3*10^(1/2))*(x-2/3-1/3*10
^(1/2))+55/9+17/9*10^(1/2))^(1/2)-1/(55/9+17/9*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)))

________________________________________________________________________________________

maxima [B]  time = 1.10, size = 668, normalized size = 3.84 \begin {gather*} -\frac {1}{60} \, \sqrt {10} {\left (\frac {588 \, \sqrt {10} x}{17 \, \sqrt {10} \sqrt {2 \, x^{2} + 3 \, x + 1} + 55 \, \sqrt {2 \, x^{2} + 3 \, x + 1}} - \frac {588 \, \sqrt {10} x}{17 \, \sqrt {10} \sqrt {2 \, x^{2} + 3 \, x + 1} - 55 \, \sqrt {2 \, x^{2} + 3 \, x + 1}} + \frac {2112 \, x}{17 \, \sqrt {10} \sqrt {2 \, x^{2} + 3 \, x + 1} + 55 \, \sqrt {2 \, x^{2} + 3 \, x + 1}} + \frac {2112 \, x}{17 \, \sqrt {10} \sqrt {2 \, x^{2} + 3 \, x + 1} - 55 \, \sqrt {2 \, x^{2} + 3 \, x + 1}} - \frac {27 \, \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 )}{{\left (17 \, \sqrt {10} + 55\right )}^{\frac {3}{2}}} - \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 )}{{\left (-\frac {17}{9} \, \sqrt {10} + \frac {55}{9}\right )}^{\frac {3}{2}}} + \frac {450 \, \sqrt {10}}{17 \, \sqrt {10} \sqrt {2 \, x^{2} + 3 \, x + 1} + 55 \, \sqrt {2 \, x^{2} + 3 \, x + 1}} - \frac {450 \, \sqrt {10}}{17 \, \sqrt {10} \sqrt {2 \, x^{2} + 3 \, x + 1} - 55 \, \sqrt {2 \, x^{2} + 3 \, x + 1}} - \frac {216 \, \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 )}{{\left (17 \, \sqrt {10} + 55\right )}^{\frac {3}{2}}} + \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 )}{{\left (-\frac {17}{9} \, \sqrt {10} + \frac {55}{9}\right )}^{\frac {3}{2}}} + \frac {1656}{17 \, \sqrt {10} \sqrt {2 \, x^{2} + 3 \, x + 1} + 55 \, \sqrt {2 \, x^{2} + 3 \, x + 1}} + \frac {1656}{17 \, \sqrt {10} \sqrt {2 \, x^{2} + 3 \, x + 1} - 55 \, \sqrt {2 \, x^{2} + 3 \, x + 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)^(3/2),x, algorithm="maxima")

[Out]

-1/60*sqrt(10)*(588*sqrt(10)*x/(17*sqrt(10)*sqrt(2*x^2 + 3*x + 1) + 55*sqrt(2*x^2 + 3*x + 1)) - 588*sqrt(10)*x
/(17*sqrt(10)*sqrt(2*x^2 + 3*x + 1) - 55*sqrt(2*x^2 + 3*x + 1)) + 2112*x/(17*sqrt(10)*sqrt(2*x^2 + 3*x + 1) +
55*sqrt(2*x^2 + 3*x + 1)) + 2112*x/(17*sqrt(10)*sqrt(2*x^2 + 3*x + 1) - 55*sqrt(2*x^2 + 3*x + 1)) - 27*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)/(17*sqrt(10) + 55)^(3/2) - 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)/(-17/9*sqrt(10) + 55/9)^(3/2) + 450*sqrt
(10)/(17*sqrt(10)*sqrt(2*x^2 + 3*x + 1) + 55*sqrt(2*x^2 + 3*x + 1)) - 450*sqrt(10)/(17*sqrt(10)*sqrt(2*x^2 + 3
*x + 1) - 55*sqrt(2*x^2 + 3*x + 1)) - 216*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)
/(17*sqrt(10) + 55)^(3/2) + 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)/(-17/9*
sqrt(10) + 55/9)^(3/2) + 1656/(17*sqrt(10)*sqrt(2*x^2 + 3*x + 1) + 55*sqrt(2*x^2 + 3*x + 1)) + 1656/(17*sqrt(1
0)*sqrt(2*x^2 + 3*x + 1) - 55*sqrt(2*x^2 + 3*x + 1)))

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {x}{6 x^{4} \sqrt {2 x^{2} + 3 x + 1} + x^{3} \sqrt {2 x^{2} + 3 x + 1} - 13 x^{2} \sqrt {2 x^{2} + 3 x + 1} - 10 x \sqrt {2 x^{2} + 3 x + 1} - 2 \sqrt {2 x^{2} + 3 x + 1}}\, dx - \int \frac {2}{6 x^{4} \sqrt {2 x^{2} + 3 x + 1} + x^{3} \sqrt {2 x^{2} + 3 x + 1} - 13 x^{2} \sqrt {2 x^{2} + 3 x + 1} - 10 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)**(3/2),x)

[Out]

-Integral(x/(6*x**4*sqrt(2*x**2 + 3*x + 1) + x**3*sqrt(2*x**2 + 3*x + 1) - 13*x**2*sqrt(2*x**2 + 3*x + 1) - 10
*x*sqrt(2*x**2 + 3*x + 1) - 2*sqrt(2*x**2 + 3*x + 1)), x) - Integral(2/(6*x**4*sqrt(2*x**2 + 3*x + 1) + x**3*s
qrt(2*x**2 + 3*x + 1) - 13*x**2*sqrt(2*x**2 + 3*x + 1) - 10*x*sqrt(2*x**2 + 3*x + 1) - 2*sqrt(2*x**2 + 3*x + 1
)), x)

________________________________________________________________________________________