3.62 \(\int \frac{\sqrt{3-x+2 x^2}}{2+3 x+5 x^2} \, dx\)

Optimal. Leaf size=174 \[ \frac{1}{5} \sqrt{\frac{11}{31} \left (13+10 \sqrt{2}\right )} \tan ^{-1}\left (\frac{\sqrt{\frac{11}{62 \left (13+10 \sqrt{2}\right )}} \left (\left (20+13 \sqrt{2}\right ) x+7 \sqrt{2}+6\right )}{\sqrt{2 x^2-x+3}}\right )-\frac{1}{5} \sqrt{\frac{11}{31} \left (10 \sqrt{2}-13\right )} \tanh ^{-1}\left (\frac{\sqrt{\frac{11}{62 \left (10 \sqrt{2}-13\right )}} \left (\left (20-13 \sqrt{2}\right ) x-7 \sqrt{2}+6\right )}{\sqrt{2 x^2-x+3}}\right )-\frac{1}{5} \sqrt{2} \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right ) \]

[Out]

-(Sqrt[2]*ArcSinh[(1 - 4*x)/Sqrt[23]])/5 + (Sqrt[(11*(13 + 10*Sqrt[2]))/31]*ArcTan[(Sqrt[11/(62*(13 + 10*Sqrt[
2]))]*(6 + 7*Sqrt[2] + (20 + 13*Sqrt[2])*x))/Sqrt[3 - x + 2*x^2]])/5 - (Sqrt[(11*(-13 + 10*Sqrt[2]))/31]*ArcTa
nh[(Sqrt[11/(62*(-13 + 10*Sqrt[2]))]*(6 - 7*Sqrt[2] + (20 - 13*Sqrt[2])*x))/Sqrt[3 - x + 2*x^2]])/5

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Rubi [A]  time = 0.441296, antiderivative size = 174, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.259, Rules used = {989, 619, 215, 1035, 1029, 206, 204} \[ \frac{1}{5} \sqrt{\frac{11}{31} \left (13+10 \sqrt{2}\right )} \tan ^{-1}\left (\frac{\sqrt{\frac{11}{62 \left (13+10 \sqrt{2}\right )}} \left (\left (20+13 \sqrt{2}\right ) x+7 \sqrt{2}+6\right )}{\sqrt{2 x^2-x+3}}\right )-\frac{1}{5} \sqrt{\frac{11}{31} \left (10 \sqrt{2}-13\right )} \tanh ^{-1}\left (\frac{\sqrt{\frac{11}{62 \left (10 \sqrt{2}-13\right )}} \left (\left (20-13 \sqrt{2}\right ) x-7 \sqrt{2}+6\right )}{\sqrt{2 x^2-x+3}}\right )-\frac{1}{5} \sqrt{2} \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[3 - x + 2*x^2]/(2 + 3*x + 5*x^2),x]

[Out]

-(Sqrt[2]*ArcSinh[(1 - 4*x)/Sqrt[23]])/5 + (Sqrt[(11*(13 + 10*Sqrt[2]))/31]*ArcTan[(Sqrt[11/(62*(13 + 10*Sqrt[
2]))]*(6 + 7*Sqrt[2] + (20 + 13*Sqrt[2])*x))/Sqrt[3 - x + 2*x^2]])/5 - (Sqrt[(11*(-13 + 10*Sqrt[2]))/31]*ArcTa
nh[(Sqrt[11/(62*(-13 + 10*Sqrt[2]))]*(6 - 7*Sqrt[2] + (20 - 13*Sqrt[2])*x))/Sqrt[3 - x + 2*x^2]])/5

Rule 989

Int[Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2]/((d_) + (e_.)*(x_) + (f_.)*(x_)^2), x_Symbol] :> Dist[c/f, Int[1/Sq
rt[a + b*x + c*x^2], x], x] - Dist[1/f, Int[(c*d - a*f + (c*e - b*f)*x)/(Sqrt[a + b*x + c*x^2]*(d + e*x + f*x^
2)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0]

Rule 619

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[1/(2*c*((-4*c)/(b^2 - 4*a*c))^p), Subst[Int[Si
mp[1 - x^2/(b^2 - 4*a*c), x]^p, x], x, b + 2*c*x], x] /; FreeQ[{a, b, c, p}, x] && GtQ[4*a - b^2/c, 0]

Rule 215

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rule 1035

Int[((g_.) + (h_.)*(x_))/(((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symb
ol] :> With[{q = Rt[(c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f), 2]}, Dist[1/(2*q), Int[Simp[h*(b*d - a*e) - g*(c*
d - a*f - q) - (g*(c*e - b*f) - h*(c*d - a*f + q))*x, x]/((a + b*x + c*x^2)*Sqrt[d + e*x + f*x^2]), x], x] - D
ist[1/(2*q), Int[Simp[h*(b*d - a*e) - g*(c*d - a*f + q) - (g*(c*e - b*f) - h*(c*d - a*f - q))*x, x]/((a + b*x
+ c*x^2)*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] && NeQ[b*d - a*e, 0] && NegQ[b^2 - 4*a*c]

Rule 1029

Int[((g_.) + (h_.)*(x_))/(((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symb
ol] :> Dist[-2*g*(g*b - 2*a*h), Subst[Int[1/Simp[g*(g*b - 2*a*h)*(b^2 - 4*a*c) - (b*d - a*e)*x^2, x], x], x, S
imp[g*b - 2*a*h - (b*h - 2*g*c)*x, x]/Sqrt[d + e*x + f*x^2]], 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] && NeQ[b*d - a*e, 0] && EqQ[h^2*(b*d - a*e) - 2*g*h*(c*d - a*f) + g^2*(
c*e - b*f), 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 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])

Rubi steps

\begin{align*} \int \frac{\sqrt{3-x+2 x^2}}{2+3 x+5 x^2} \, dx &=-\left (\frac{1}{5} \int \frac{-11+11 x}{\sqrt{3-x+2 x^2} \left (2+3 x+5 x^2\right )} \, dx\right )+\frac{2}{5} \int \frac{1}{\sqrt{3-x+2 x^2}} \, dx\\ &=\frac{1}{5} \sqrt{\frac{2}{23}} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{23}}} \, dx,x,-1+4 x\right )+\frac{\int \frac{121 \left (2+\sqrt{2}\right )-121 \sqrt{2} x}{\sqrt{3-x+2 x^2} \left (2+3 x+5 x^2\right )} \, dx}{110 \sqrt{2}}-\frac{\int \frac{121 \left (2-\sqrt{2}\right )+121 \sqrt{2} x}{\sqrt{3-x+2 x^2} \left (2+3 x+5 x^2\right )} \, dx}{110 \sqrt{2}}\\ &=-\frac{1}{5} \sqrt{2} \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )-\frac{1}{5} \left (1331 \left (20-13 \sqrt{2}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-907742 \left (13-10 \sqrt{2}\right )-11 x^2} \, dx,x,\frac{121 \left (6-7 \sqrt{2}\right )+121 \left (20-13 \sqrt{2}\right ) x}{\sqrt{3-x+2 x^2}}\right )-\frac{1}{5} \left (1331 \left (20+13 \sqrt{2}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-907742 \left (13+10 \sqrt{2}\right )-11 x^2} \, dx,x,\frac{121 \left (6+7 \sqrt{2}\right )+121 \left (20+13 \sqrt{2}\right ) x}{\sqrt{3-x+2 x^2}}\right )\\ &=-\frac{1}{5} \sqrt{2} \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )+\frac{1}{5} \sqrt{\frac{11}{31} \left (13+10 \sqrt{2}\right )} \tan ^{-1}\left (\frac{\sqrt{\frac{11}{62 \left (13+10 \sqrt{2}\right )}} \left (6+7 \sqrt{2}+\left (20+13 \sqrt{2}\right ) x\right )}{\sqrt{3-x+2 x^2}}\right )-\frac{1}{5} \sqrt{\frac{11}{31} \left (-13+10 \sqrt{2}\right )} \tanh ^{-1}\left (\frac{\sqrt{\frac{11}{62 \left (-13+10 \sqrt{2}\right )}} \left (6-7 \sqrt{2}+\left (20-13 \sqrt{2}\right ) x\right )}{\sqrt{3-x+2 x^2}}\right )\\ \end{align*}

Mathematica [C]  time = 0.463022, size = 185, normalized size = 1.06 \[ -\frac{1}{5} \sqrt{2} \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )-\frac{1}{5} i \sqrt{\frac{11}{62}} \left (\sqrt{13+i \sqrt{31}} \tanh ^{-1}\left (\frac{-4 i \sqrt{31} x-22 x+i \sqrt{31}+63}{2 \sqrt{286+22 i \sqrt{31}} \sqrt{2 x^2-x+3}}\right )-\sqrt{13-i \sqrt{31}} \tanh ^{-1}\left (\frac{4 i \sqrt{31} x-22 x-i \sqrt{31}+63}{2 \sqrt{286-22 i \sqrt{31}} \sqrt{2 x^2-x+3}}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[3 - x + 2*x^2]/(2 + 3*x + 5*x^2),x]

[Out]

-(Sqrt[2]*ArcSinh[(1 - 4*x)/Sqrt[23]])/5 - (I/5)*Sqrt[11/62]*(Sqrt[13 + I*Sqrt[31]]*ArcTanh[(63 + I*Sqrt[31] -
22*x - (4*I)*Sqrt[31]*x)/(2*Sqrt[286 + (22*I)*Sqrt[31]]*Sqrt[3 - x + 2*x^2])] - Sqrt[13 - I*Sqrt[31]]*ArcTanh
[(63 - I*Sqrt[31] - 22*x + (4*I)*Sqrt[31]*x)/(2*Sqrt[286 - (22*I)*Sqrt[31]]*Sqrt[3 - x + 2*x^2])])

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Maple [B]  time = 0.276, size = 2065, normalized size = 11.9 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x^2-x+3)^(1/2)/(5*x^2+3*x+2),x)

[Out]

1/5*2^(1/2)*arcsinh(4/23*23^(1/2)*(x-1/4))-1/52855*(8*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)^2+3*(2^(1/2)-1+x)^2/(2^(1/
2)+1-x)^2*2^(1/2)+8-3*2^(1/2))^(1/2)*2^(1/2)*(285*(-775687+549362*2^(1/2))^(1/2)*2^(1/2)*(-8866+6820*2^(1/2))^
(1/2)*arctan(1/11692487*(-775687+549362*2^(1/2))^(1/2)*(-23*(8+3*2^(1/2))*(-23*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)^2
+24*2^(1/2)-41))^(1/2)*(6485*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)^2*2^(1/2)+10368*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)^2+223
79*2^(1/2)+32016)/(23*(2^(1/2)-1+x)^4/(2^(1/2)+1-x)^4+82*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)^2+23)*(8+3*2^(1/2))*(2^
(1/2)-1+x)/(2^(1/2)+1-x))+386*(-775687+549362*2^(1/2))^(1/2)*(-8866+6820*2^(1/2))^(1/2)*arctan(1/11692487*(-77
5687+549362*2^(1/2))^(1/2)*(-23*(8+3*2^(1/2))*(-23*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)^2+24*2^(1/2)-41))^(1/2)*(6485
*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)^2*2^(1/2)+10368*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)^2+22379*2^(1/2)+32016)/(23*(2^(1/
2)-1+x)^4/(2^(1/2)+1-x)^4+82*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)^2+23)*(8+3*2^(1/2))*(2^(1/2)-1+x)/(2^(1/2)+1-x))-27
4846*arctanh(31/2*(8*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)^2+3*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)^2*2^(1/2)+8-3*2^(1/2))^(1
/2)/(-8866+6820*2^(1/2))^(1/2))*2^(1/2)-1543366*arctanh(31/2*(8*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)^2+3*(2^(1/2)-1+x
)^2/(2^(1/2)+1-x)^2*2^(1/2)+8-3*2^(1/2))^(1/2)/(-8866+6820*2^(1/2))^(1/2)))/((8*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)^
2+3*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)^2*2^(1/2)+8-3*2^(1/2))/(1+(2^(1/2)-1+x)/(2^(1/2)+1-x))^2)^(1/2)/(1+(2^(1/2)-
1+x)/(2^(1/2)+1-x))/(8+3*2^(1/2))/(-8866+6820*2^(1/2))^(1/2)+1/21142*(8*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)^2+3*(2^(
1/2)-1+x)^2/(2^(1/2)+1-x)^2*2^(1/2)+8-3*2^(1/2))^(1/2)*2^(1/2)*(151*(-775687+549362*2^(1/2))^(1/2)*2^(1/2)*(-8
866+6820*2^(1/2))^(1/2)*arctan(1/11692487*(-775687+549362*2^(1/2))^(1/2)*(-23*(8+3*2^(1/2))*(-23*(2^(1/2)-1+x)
^2/(2^(1/2)+1-x)^2+24*2^(1/2)-41))^(1/2)*(6485*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)^2*2^(1/2)+10368*(2^(1/2)-1+x)^2/(
2^(1/2)+1-x)^2+22379*2^(1/2)+32016)/(23*(2^(1/2)-1+x)^4/(2^(1/2)+1-x)^4+82*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)^2+23)
*(8+3*2^(1/2))*(2^(1/2)-1+x)/(2^(1/2)+1-x))+218*(-775687+549362*2^(1/2))^(1/2)*(-8866+6820*2^(1/2))^(1/2)*arct
an(1/11692487*(-775687+549362*2^(1/2))^(1/2)*(-23*(8+3*2^(1/2))*(-23*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)^2+24*2^(1/2
)-41))^(1/2)*(6485*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)^2*2^(1/2)+10368*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)^2+22379*2^(1/2)
+32016)/(23*(2^(1/2)-1+x)^4/(2^(1/2)+1-x)^4+82*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)^2+23)*(8+3*2^(1/2))*(2^(1/2)-1+x)
/(2^(1/2)+1-x))+401698*arctanh(31/2*(8*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)^2+3*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)^2*2^(1/
2)+8-3*2^(1/2))^(1/2)/(-8866+6820*2^(1/2))^(1/2))*2^(1/2)-63426*arctanh(31/2*(8*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)^
2+3*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)^2*2^(1/2)+8-3*2^(1/2))^(1/2)/(-8866+6820*2^(1/2))^(1/2)))/((8*(2^(1/2)-1+x)^
2/(2^(1/2)+1-x)^2+3*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)^2*2^(1/2)+8-3*2^(1/2))/(1+(2^(1/2)-1+x)/(2^(1/2)+1-x))^2)^(1
/2)/(1+(2^(1/2)-1+x)/(2^(1/2)+1-x))/(8+3*2^(1/2))/(-8866+6820*2^(1/2))^(1/2)+3/21142*(8*(2^(1/2)-1+x)^2/(2^(1/
2)+1-x)^2+3*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)^2*2^(1/2)+8-3*2^(1/2))^(1/2)*2^(1/2)*(369*(-775687+549362*2^(1/2))^(
1/2)*2^(1/2)*(-8866+6820*2^(1/2))^(1/2)*arctan(1/11692487*(-775687+549362*2^(1/2))^(1/2)*(-23*(8+3*2^(1/2))*(-
23*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)^2+24*2^(1/2)-41))^(1/2)*(6485*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)^2*2^(1/2)+10368*(
2^(1/2)-1+x)^2/(2^(1/2)+1-x)^2+22379*2^(1/2)+32016)/(23*(2^(1/2)-1+x)^4/(2^(1/2)+1-x)^4+82*(2^(1/2)-1+x)^2/(2^
(1/2)+1-x)^2+23)*(8+3*2^(1/2))*(2^(1/2)-1+x)/(2^(1/2)+1-x))+520*(-775687+549362*2^(1/2))^(1/2)*(-8866+6820*2^(
1/2))^(1/2)*arctan(1/11692487*(-775687+549362*2^(1/2))^(1/2)*(-23*(8+3*2^(1/2))*(-23*(2^(1/2)-1+x)^2/(2^(1/2)+
1-x)^2+24*2^(1/2)-41))^(1/2)*(6485*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)^2*2^(1/2)+10368*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)
^2+22379*2^(1/2)+32016)/(23*(2^(1/2)-1+x)^4/(2^(1/2)+1-x)^4+82*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)^2+23)*(8+3*2^(1/2
))*(2^(1/2)-1+x)/(2^(1/2)+1-x))+465124*arctanh(31/2*(8*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)^2+3*(2^(1/2)-1+x)^2/(2^(1
/2)+1-x)^2*2^(1/2)+8-3*2^(1/2))^(1/2)/(-8866+6820*2^(1/2))^(1/2))*2^(1/2)-866822*arctanh(31/2*(8*(2^(1/2)-1+x)
^2/(2^(1/2)+1-x)^2+3*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)^2*2^(1/2)+8-3*2^(1/2))^(1/2)/(-8866+6820*2^(1/2))^(1/2)))/(
(8*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)^2+3*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)^2*2^(1/2)+8-3*2^(1/2))/(1+(2^(1/2)-1+x)/(2^
(1/2)+1-x))^2)^(1/2)/(1+(2^(1/2)-1+x)/(2^(1/2)+1-x))/(8+3*2^(1/2))/(-8866+6820*2^(1/2))^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{2 \, x^{2} - x + 3}}{5 \, x^{2} + 3 \, x + 2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^2-x+3)^(1/2)/(5*x^2+3*x+2),x, algorithm="maxima")

[Out]

integrate(sqrt(2*x^2 - x + 3)/(5*x^2 + 3*x + 2), x)

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Fricas [B]  time = 4.39329, size = 7056, normalized size = 40.55 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^2-x+3)^(1/2)/(5*x^2+3*x+2),x, algorithm="fricas")

[Out]

1/1550*6050^(1/4)*sqrt(31)*sqrt(5)*sqrt(2)*sqrt(13*sqrt(2) + 20)*arctan(1/89125*(460*sqrt(5)*(4*6050^(3/4)*sqr
t(31)*(4702*x^7 - 19541*x^6 + 40352*x^5 - 68777*x^4 + 35480*x^3 - 19080*x^2 - sqrt(2)*(4028*x^7 - 14488*x^6 +
30919*x^5 - 46671*x^4 + 22688*x^3 - 9144*x^2 - 27648*x + 17280) - 34560*x + 27648) + 5*6050^(1/4)*sqrt(31)*(22
836*x^7 - 355266*x^6 + 1914360*x^5 - 4475096*x^4 + 5840640*x^3 - 4011840*x^2 - sqrt(2)*(18463*x^7 - 280047*x^6
+ 1453472*x^5 - 3238500*x^4 + 4140576*x^3 - 2378592*x^2 - 3068928*x + 1990656) - 3981312*x + 3068928))*sqrt(2
*x^2 - x + 3)*sqrt(13*sqrt(2) + 20) + 253000*sqrt(31)*sqrt(2)*(28180*x^8 - 254666*x^7 + 704270*x^6 - 1385256*x
^5 + 1549144*x^4 - 642048*x^3 - 98496*x^2 - sqrt(2)*(8746*x^8 - 102335*x^7 + 396104*x^6 - 783113*x^5 + 1320710
*x^4 - 752088*x^3 + 396144*x^2 + 546048*x - 539136) + 1154304*x - 456192) - 2*sqrt(10)*(sqrt(5)*(4*6050^(3/4)*
sqrt(31)*(15454*x^7 - 22399*x^6 + 73509*x^5 - 37360*x^4 + 52200*x^3 + 13824*x^2 - sqrt(2)*(15438*x^7 - 22007*x
^6 + 69837*x^5 - 21232*x^4 + 19368*x^3 + 44928*x^2 - 44928*x) - 13824*x) + 5*6050^(1/4)*sqrt(31)*(77254*x^7 -
1000024*x^6 + 3868360*x^5 - 5120640*x^4 + 7012800*x^3 + 2405376*x^2 - sqrt(2)*(69479*x^7 - 898236*x^6 + 345474
0*x^5 - 4394304*x^4 + 5347296*x^3 + 4478976*x^2 - 4478976*x) - 2405376*x))*sqrt(2*x^2 - x + 3)*sqrt(13*sqrt(2)
+ 20) + 550*sqrt(31)*sqrt(2)*(123408*x^8 - 914152*x^7 + 1578888*x^6 - 3293072*x^5 + 396480*x^4 + 798336*x^3 -
3822336*x^2 - sqrt(2)*(15550*x^8 - 118051*x^7 + 244047*x^6 - 707374*x^5 + 1053960*x^4 - 1667952*x^3 + 1209600
*x^2 - 1036800*x) + 3276288*x) + 25*sqrt(31)*(254591*x^8 - 4815126*x^7 + 32303580*x^6 - 90866808*x^5 + 1087819
20*x^4 - 74219328*x^3 - 168956928*x^2 - 15488*sqrt(2)*(4*x^8 - 76*x^7 + 517*x^6 - 1536*x^5 + 2385*x^4 - 3618*x
^3 + 2268*x^2 - 1944*x) + 144820224*x))*sqrt(-(6050^(1/4)*sqrt(5)*sqrt(2*x^2 - x + 3)*(sqrt(2)*(3*x + 5) - 8*x
+ 2)*sqrt(13*sqrt(2) + 20) - 245*x^2 - 220*sqrt(2)*(2*x^2 - x + 3) + 755*x - 1000)/x^2) + 2875*sqrt(31)*(2828
123*x^8 - 9696916*x^7 + 53385560*x^6 - 142835344*x^5 + 254146592*x^4 - 249300096*x^3 + 37981440*x^2 - 7744*sqr
t(2)*(1348*x^8 - 2692*x^7 + 9789*x^6 - 10070*x^5 + 15569*x^4 - 5568*x^3 + 1080*x^2 + 4320*x - 5184) + 22306406
4*x - 94887936))/(2585191*x^8 - 4661200*x^7 + 14191920*x^6 + 490880*x^5 - 13562944*x^4 + 44249088*x^3 - 346152
96*x^2 - 24772608*x + 18579456)) + 1/1550*6050^(1/4)*sqrt(31)*sqrt(5)*sqrt(2)*sqrt(13*sqrt(2) + 20)*arctan(1/8
9125*(460*sqrt(5)*(4*6050^(3/4)*sqrt(31)*(4702*x^7 - 19541*x^6 + 40352*x^5 - 68777*x^4 + 35480*x^3 - 19080*x^2
- sqrt(2)*(4028*x^7 - 14488*x^6 + 30919*x^5 - 46671*x^4 + 22688*x^3 - 9144*x^2 - 27648*x + 17280) - 34560*x +
27648) + 5*6050^(1/4)*sqrt(31)*(22836*x^7 - 355266*x^6 + 1914360*x^5 - 4475096*x^4 + 5840640*x^3 - 4011840*x^
2 - sqrt(2)*(18463*x^7 - 280047*x^6 + 1453472*x^5 - 3238500*x^4 + 4140576*x^3 - 2378592*x^2 - 3068928*x + 1990
656) - 3981312*x + 3068928))*sqrt(2*x^2 - x + 3)*sqrt(13*sqrt(2) + 20) - 253000*sqrt(31)*sqrt(2)*(28180*x^8 -
254666*x^7 + 704270*x^6 - 1385256*x^5 + 1549144*x^4 - 642048*x^3 - 98496*x^2 - sqrt(2)*(8746*x^8 - 102335*x^7
+ 396104*x^6 - 783113*x^5 + 1320710*x^4 - 752088*x^3 + 396144*x^2 + 546048*x - 539136) + 1154304*x - 456192) -
2*sqrt(10)*(sqrt(5)*(4*6050^(3/4)*sqrt(31)*(15454*x^7 - 22399*x^6 + 73509*x^5 - 37360*x^4 + 52200*x^3 + 13824
*x^2 - sqrt(2)*(15438*x^7 - 22007*x^6 + 69837*x^5 - 21232*x^4 + 19368*x^3 + 44928*x^2 - 44928*x) - 13824*x) +
5*6050^(1/4)*sqrt(31)*(77254*x^7 - 1000024*x^6 + 3868360*x^5 - 5120640*x^4 + 7012800*x^3 + 2405376*x^2 - sqrt(
2)*(69479*x^7 - 898236*x^6 + 3454740*x^5 - 4394304*x^4 + 5347296*x^3 + 4478976*x^2 - 4478976*x) - 2405376*x))*
sqrt(2*x^2 - x + 3)*sqrt(13*sqrt(2) + 20) - 550*sqrt(31)*sqrt(2)*(123408*x^8 - 914152*x^7 + 1578888*x^6 - 3293
072*x^5 + 396480*x^4 + 798336*x^3 - 3822336*x^2 - sqrt(2)*(15550*x^8 - 118051*x^7 + 244047*x^6 - 707374*x^5 +
1053960*x^4 - 1667952*x^3 + 1209600*x^2 - 1036800*x) + 3276288*x) - 25*sqrt(31)*(254591*x^8 - 4815126*x^7 + 32
303580*x^6 - 90866808*x^5 + 108781920*x^4 - 74219328*x^3 - 168956928*x^2 - 15488*sqrt(2)*(4*x^8 - 76*x^7 + 517
*x^6 - 1536*x^5 + 2385*x^4 - 3618*x^3 + 2268*x^2 - 1944*x) + 144820224*x))*sqrt((6050^(1/4)*sqrt(5)*sqrt(2*x^2
- x + 3)*(sqrt(2)*(3*x + 5) - 8*x + 2)*sqrt(13*sqrt(2) + 20) + 245*x^2 + 220*sqrt(2)*(2*x^2 - x + 3) - 755*x
+ 1000)/x^2) - 2875*sqrt(31)*(2828123*x^8 - 9696916*x^7 + 53385560*x^6 - 142835344*x^5 + 254146592*x^4 - 24930
0096*x^3 + 37981440*x^2 - 7744*sqrt(2)*(1348*x^8 - 2692*x^7 + 9789*x^6 - 10070*x^5 + 15569*x^4 - 5568*x^3 + 10
80*x^2 + 4320*x - 5184) + 223064064*x - 94887936))/(2585191*x^8 - 4661200*x^7 + 14191920*x^6 + 490880*x^5 - 13
562944*x^4 + 44249088*x^3 - 34615296*x^2 - 24772608*x + 18579456)) - 1/6200*6050^(1/4)*sqrt(5)*sqrt(13*sqrt(2)
+ 20)*(13*sqrt(2) - 20)*log(40*(6050^(1/4)*sqrt(5)*sqrt(2*x^2 - x + 3)*(sqrt(2)*(3*x + 5) - 8*x + 2)*sqrt(13*
sqrt(2) + 20) + 245*x^2 + 220*sqrt(2)*(2*x^2 - x + 3) - 755*x + 1000)/x^2) + 1/6200*6050^(1/4)*sqrt(5)*sqrt(13
*sqrt(2) + 20)*(13*sqrt(2) - 20)*log(-40*(6050^(1/4)*sqrt(5)*sqrt(2*x^2 - x + 3)*(sqrt(2)*(3*x + 5) - 8*x + 2)
*sqrt(13*sqrt(2) + 20) - 245*x^2 - 220*sqrt(2)*(2*x^2 - x + 3) + 755*x - 1000)/x^2) + 1/10*sqrt(2)*log(-4*sqrt
(2)*sqrt(2*x^2 - x + 3)*(4*x - 1) - 32*x^2 + 16*x - 25)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{2 x^{2} - x + 3}}{5 x^{2} + 3 x + 2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x**2-x+3)**(1/2)/(5*x**2+3*x+2),x)

[Out]

Integral(sqrt(2*x**2 - x + 3)/(5*x**2 + 3*x + 2), 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^2-x+3)^(1/2)/(5*x^2+3*x+2),x, algorithm="giac")

[Out]

Exception raised: TypeError