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

Optimal. Leaf size=166 \[ -\frac {2 (14 x+15)}{17 \sqrt {-2 x^2+3 x+1}}-\frac {9}{2} \sqrt {\frac {1}{5} \left (\sqrt {10}-3\right )} \tan ^{-1}\left (\frac {\left (1+4 \sqrt {10}\right ) x+3 \left (4-\sqrt {10}\right )}{2 \sqrt {1+\sqrt {10}} \sqrt {-2 x^2+3 x+1}}\right )+\frac {9}{2} \sqrt {\frac {1}{5} \left (3+\sqrt {10}\right )} \tanh ^{-1}\left (\frac {\left (1-4 \sqrt {10}\right ) x+3 \left (4+\sqrt {10}\right )}{2 \sqrt {\sqrt {10}-1} \sqrt {-2 x^2+3 x+1}}\right ) \]

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Rubi [A]  time = 0.22, antiderivative size = 166, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1016, 12, 1032, 724, 204, 206} \begin {gather*} -\frac {2 (14 x+15)}{17 \sqrt {-2 x^2+3 x+1}}-\frac {9}{2} \sqrt {\frac {1}{5} \left (\sqrt {10}-3\right )} \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 {9}{2} \sqrt {\frac {1}{5} \left (3+\sqrt {10}\right )} \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)*(1 + 3*x - 2*x^2)^(3/2)),x]

[Out]

(-2*(15 + 14*x))/(17*Sqrt[1 + 3*x - 2*x^2]) - (9*Sqrt[(-3 + Sqrt[10])/5]*ArcTan[(3*(4 - Sqrt[10]) + (1 + 4*Sqr
t[10])*x)/(2*Sqrt[1 + Sqrt[10]]*Sqrt[1 + 3*x - 2*x^2])])/2 + (9*Sqrt[(3 + Sqrt[10])/5]*ArcTanh[(3*(4 + Sqrt[10
]) + (1 - 4*Sqrt[10])*x)/(2*Sqrt[-1 + Sqrt[10]]*Sqrt[1 + 3*x - 2*x^2])])/2

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 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 (15+14 x)}{17 \sqrt {1+3 x-2 x^2}}+\frac {2}{17} \int \frac {153 x}{2 \left (2+4 x-3 x^2\right ) \sqrt {1+3 x-2 x^2}} \, dx\\ &=-\frac {2 (15+14 x)}{17 \sqrt {1+3 x-2 x^2}}+9 \int \frac {x}{\left (2+4 x-3 x^2\right ) \sqrt {1+3 x-2 x^2}} \, dx\\ &=-\frac {2 (15+14 x)}{17 \sqrt {1+3 x-2 x^2}}+\frac {1}{5} \left (9 \left (5-\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 (5+\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 (15+14 x)}{17 \sqrt {1+3 x-2 x^2}}-\frac {1}{5} \left (18 \left (5-\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 (5+\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 (15+14 x)}{17 \sqrt {1+3 x-2 x^2}}-\frac {9}{2} \sqrt {\frac {1}{5} \left (-3+\sqrt {10}\right )} \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 {9}{2} \sqrt {\frac {1}{5} \left (3+\sqrt {10}\right )} \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 = 0.38, size = 167, normalized size = 1.01 \begin {gather*} \frac {1}{170} \left (153 \sqrt {5 \left (3+\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 )-\frac {153 \sqrt {5 \left (\sqrt {10}-3\right )} \sqrt {-2 x^2+3 x+1} \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 )+280 x+300}{\sqrt {-2 x^2+3 x+1}}\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]

(-((300 + 280*x + 153*Sqrt[5*(-3 + Sqrt[10])]*Sqrt[1 + 3*x - 2*x^2]*ArcTan[(12 - 3*Sqrt[10] + x + 4*Sqrt[10]*x
)/(2*Sqrt[1 + Sqrt[10]]*Sqrt[1 + 3*x - 2*x^2])])/Sqrt[1 + 3*x - 2*x^2]) + 153*Sqrt[5*(3 + Sqrt[10])]*ArcTanh[(
3*(4 + Sqrt[10]) + x - 4*Sqrt[10]*x)/(2*Sqrt[-1 + Sqrt[10]]*Sqrt[1 + 3*x - 2*x^2])])/170

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IntegrateAlgebraic [C]  time = 0.39, size = 149, normalized size = 0.90 \begin {gather*} \frac {9}{2} \text {RootSum}\left [2 \text {$\#$1}^4-8 \text {$\#$1}^3+8 \text {$\#$1}^2+20 \text {$\#$1}+5\&,\frac {-3 \log \left (\text {$\#$1} (-x)+\sqrt {-2 x^2+3 x+1}-1\right )+2 \text {$\#$1} \log \left (\text {$\#$1} (-x)+\sqrt {-2 x^2+3 x+1}-1\right )-2 \text {$\#$1} \log (x)+3 \log (x)}{2 \text {$\#$1}^3-6 \text {$\#$1}^2+4 \text {$\#$1}+5}\&\right ]+\frac {2 \sqrt {-2 x^2+3 x+1} (14 x+15)}{17 \left (2 x^2-3 x-1\right )} \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*(15 + 14*x)*Sqrt[1 + 3*x - 2*x^2])/(17*(-1 - 3*x + 2*x^2)) + (9*RootSum[5 + 20*#1 + 8*#1^2 - 8*#1^3 + 2*#1^
4 & , (3*Log[x] - 3*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)/(5 + 4*#1 - 6*#1^2 + 2*#1^3) & ])/2

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fricas [B]  time = 0.44, size = 344, normalized size = 2.07 \begin {gather*} -\frac {612 \, \sqrt {5} {\left (2 \, x^{2} - 3 \, x - 1\right )} \sqrt {\sqrt {10} - 3} \arctan \left (\frac {\sqrt {10} \sqrt {5} \sqrt {2} x \sqrt {\sqrt {10} - 3} \sqrt {\frac {6 \, x^{2} + \sqrt {10} {\left (3 \, x^{2} + 2 \, x\right )} - 2 \, \sqrt {-2 \, x^{2} + 3 \, x + 1} {\left (\sqrt {10} x + 2 \, x + 2\right )} + 10 \, x + 4}{x^{2}}} + 2 \, {\left (\sqrt {10} \sqrt {5} {\left (x + 1\right )} - \sqrt {10} \sqrt {5} \sqrt {-2 \, x^{2} + 3 \, x + 1} + 5 \, \sqrt {5} x\right )} \sqrt {\sqrt {10} - 3}}{10 \, x}\right ) + 153 \, \sqrt {5} {\left (2 \, x^{2} - 3 \, x - 1\right )} \sqrt {\sqrt {10} + 3} \log \left (\frac {9 \, {\left (5 \, \sqrt {10} x + {\left (3 \, \sqrt {10} \sqrt {5} x - 10 \, \sqrt {5} x\right )} \sqrt {\sqrt {10} + 3} - 10 \, x + 10 \, \sqrt {-2 \, x^{2} + 3 \, x + 1} - 10\right )}}{x}\right ) - 153 \, \sqrt {5} {\left (2 \, x^{2} - 3 \, x - 1\right )} \sqrt {\sqrt {10} + 3} \log \left (\frac {9 \, {\left (5 \, \sqrt {10} x - {\left (3 \, \sqrt {10} \sqrt {5} x - 10 \, \sqrt {5} x\right )} \sqrt {\sqrt {10} + 3} - 10 \, x + 10 \, \sqrt {-2 \, x^{2} + 3 \, x + 1} - 10\right )}}{x}\right ) + 600 \, x^{2} - 20 \, \sqrt {-2 \, x^{2} + 3 \, x + 1} {\left (14 \, x + 15\right )} - 900 \, x - 300}{170 \, {\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]

-1/170*(612*sqrt(5)*(2*x^2 - 3*x - 1)*sqrt(sqrt(10) - 3)*arctan(1/10*(sqrt(10)*sqrt(5)*sqrt(2)*x*sqrt(sqrt(10)
- 3)*sqrt((6*x^2 + sqrt(10)*(3*x^2 + 2*x) - 2*sqrt(-2*x^2 + 3*x + 1)*(sqrt(10)*x + 2*x + 2) + 10*x + 4)/x^2)
+ 2*(sqrt(10)*sqrt(5)*(x + 1) - sqrt(10)*sqrt(5)*sqrt(-2*x^2 + 3*x + 1) + 5*sqrt(5)*x)*sqrt(sqrt(10) - 3))/x)
+ 153*sqrt(5)*(2*x^2 - 3*x - 1)*sqrt(sqrt(10) + 3)*log(9*(5*sqrt(10)*x + (3*sqrt(10)*sqrt(5)*x - 10*sqrt(5)*x)
*sqrt(sqrt(10) + 3) - 10*x + 10*sqrt(-2*x^2 + 3*x + 1) - 10)/x) - 153*sqrt(5)*(2*x^2 - 3*x - 1)*sqrt(sqrt(10)
+ 3)*log(9*(5*sqrt(10)*x - (3*sqrt(10)*sqrt(5)*x - 10*sqrt(5)*x)*sqrt(sqrt(10) + 3) - 10*x + 10*sqrt(-2*x^2 +
3*x + 1) - 10)/x) + 600*x^2 - 20*sqrt(-2*x^2 + 3*x + 1)*(14*x + 15) - 900*x - 300)/(2*x^2 - 3*x - 1)

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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)^(3/2),x, algorithm="giac")

[Out]

Timed out

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maple [B]  time = 0.03, size = 760, normalized size = 4.58

result too large to display

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]

26/255*10^(1/2)/(-1/9-1/9*10^(1/2))/(-2*(x-2/3+1/3*10^(1/2))^2+(1/3+4/3*10^(1/2))*(x-2/3+1/3*10^(1/2))-1/9-1/9
*10^(1/2))^(1/2)+32/765/(-1/9-1/9*10^(1/2))/(-2*(x-2/3+1/3*10^(1/2))^2+(1/3+4/3*10^(1/2))*(x-2/3+1/3*10^(1/2))
-1/9-1/9*10^(1/2))^(1/2)*10^(1/2)*x-62/153/(-1/9-1/9*10^(1/2))/(-2*(x-2/3+1/3*10^(1/2))^2+(1/3+4/3*10^(1/2))*(
x-2/3+1/3*10^(1/2))-1/9-1/9*10^(1/2))^(1/2)*x+7/51/(-1/9-1/9*10^(1/2))/(-2*(x-2/3+1/3*10^(1/2))^2+(1/3+4/3*10^
(1/2))*(x-2/3+1/3*10^(1/2))-1/9-1/9*10^(1/2))^(1/2)+2/5*10^(1/2)/(-1/9-1/9*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/(-1/9-1/9*10^(1/2))/(1+10^(1/2))^(1/2)*arct
an(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))-26/255*10^(1/2)/(-1/9+1/9*10^(1/2))/(-2*(x-2/
3-1/3*10^(1/2))^2+(1/3-4/3*10^(1/2))*(x-2/3-1/3*10^(1/2))-1/9+1/9*10^(1/2))^(1/2)-32/765/(-1/9+1/9*10^(1/2))/(
-2*(x-2/3-1/3*10^(1/2))^2+(1/3-4/3*10^(1/2))*(x-2/3-1/3*10^(1/2))-1/9+1/9*10^(1/2))^(1/2)*10^(1/2)*x-62/153/(-
1/9+1/9*10^(1/2))/(-2*(x-2/3-1/3*10^(1/2))^2+(1/3-4/3*10^(1/2))*(x-2/3-1/3*10^(1/2))-1/9+1/9*10^(1/2))^(1/2)*x
+7/51/(-1/9+1/9*10^(1/2))/(-2*(x-2/3-1/3*10^(1/2))^2+(1/3-4/3*10^(1/2))*(x-2/3-1/3*10^(1/2))-1/9+1/9*10^(1/2))
^(1/2)+2/5*10^(1/2)/(-1/9+1/9*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/(-1/9+1/9*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))

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maxima [B]  time = 1.11, size = 678, normalized size = 4.08 \begin {gather*} \frac {1}{340} \, \sqrt {10} {\left (\frac {124 \, \sqrt {10} x}{\sqrt {10} \sqrt {-2 \, x^{2} + 3 \, x + 1} + \sqrt {-2 \, x^{2} + 3 \, x + 1}} - \frac {124 \, \sqrt {10} x}{\sqrt {10} \sqrt {-2 \, x^{2} + 3 \, x + 1} - \sqrt {-2 \, x^{2} + 3 \, x + 1}} + \frac {153 \, \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 {10} \sqrt {\sqrt {10} + 1} + \sqrt {\sqrt {10} + 1}} - \frac {128 \, x}{\sqrt {10} \sqrt {-2 \, x^{2} + 3 \, x + 1} + \sqrt {-2 \, x^{2} + 3 \, x + 1}} - \frac {128 \, x}{\sqrt {10} \sqrt {-2 \, x^{2} + 3 \, x + 1} - \sqrt {-2 \, x^{2} + 3 \, x + 1}} - \frac {1224 \, \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 {10} \sqrt {\sqrt {10} + 1} + \sqrt {\sqrt {10} + 1}} + \frac {153 \, \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 )}{{\left (\sqrt {10} - 1\right )}^{\frac {3}{2}}} - \frac {42 \, \sqrt {10}}{\sqrt {10} \sqrt {-2 \, x^{2} + 3 \, x + 1} + \sqrt {-2 \, x^{2} + 3 \, x + 1}} + \frac {42 \, \sqrt {10}}{\sqrt {10} \sqrt {-2 \, x^{2} + 3 \, x + 1} - \sqrt {-2 \, x^{2} + 3 \, x + 1}} + \frac {1224 \, \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 )}{{\left (\sqrt {10} - 1\right )}^{\frac {3}{2}}} - \frac {312}{\sqrt {10} \sqrt {-2 \, x^{2} + 3 \, x + 1} + \sqrt {-2 \, x^{2} + 3 \, x + 1}} - \frac {312}{\sqrt {10} \sqrt {-2 \, x^{2} + 3 \, x + 1} - \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]

1/340*sqrt(10)*(124*sqrt(10)*x/(sqrt(10)*sqrt(-2*x^2 + 3*x + 1) + sqrt(-2*x^2 + 3*x + 1)) - 124*sqrt(10)*x/(sq
rt(10)*sqrt(-2*x^2 + 3*x + 1) - sqrt(-2*x^2 + 3*x + 1)) + 153*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(10)*sqrt(sqrt(10) + 1) + sqrt(sqrt(10) + 1)) - 128*x/(sq
rt(10)*sqrt(-2*x^2 + 3*x + 1) + sqrt(-2*x^2 + 3*x + 1)) - 128*x/(sqrt(10)*sqrt(-2*x^2 + 3*x + 1) - sqrt(-2*x^2
+ 3*x + 1)) - 1224*arcsin(8/17*sqrt(17)*sqrt(10)*x/abs(6*x + 2*sqrt(10) - 4) + 2/17*sqrt(17)*x/abs(6*x + 2*sq
rt(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))/(sq
rt(10)*sqrt(sqrt(10) + 1) + sqrt(sqrt(10) + 1)) + 153*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(1
0) - 4) + 1/18)/(sqrt(10) - 1)^(3/2) - 42*sqrt(10)/(sqrt(10)*sqrt(-2*x^2 + 3*x + 1) + sqrt(-2*x^2 + 3*x + 1))
+ 42*sqrt(10)/(sqrt(10)*sqrt(-2*x^2 + 3*x + 1) - sqrt(-2*x^2 + 3*x + 1)) + 1224*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/a
bs(6*x - 2*sqrt(10) - 4) + 1/18)/(sqrt(10) - 1)^(3/2) - 312/(sqrt(10)*sqrt(-2*x^2 + 3*x + 1) + sqrt(-2*x^2 + 3
*x + 1)) - 312/(sqrt(10)*sqrt(-2*x^2 + 3*x + 1) - sqrt(-2*x^2 + 3*x + 1)))

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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)

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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} + 17 x^{3} \sqrt {- 2 x^{2} + 3 x + 1} - 5 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} + 17 x^{3} \sqrt {- 2 x^{2} + 3 x + 1} - 5 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) + 17*x**3*sqrt(-2*x**2 + 3*x + 1) - 5*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
) + 17*x**3*sqrt(-2*x**2 + 3*x + 1) - 5*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)

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