3.1.62 \(\int \frac {\sqrt {3-x+2 x^2}}{2+3 x+5 x^2} \, dx\) [62]
Optimal. Leaf size=174 \[ -\frac {1}{5} \sqrt {2} \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )+\frac {1}{5} \sqrt {\frac {11}{31} \left (13+10 \sqrt {2}\right )} \tan ^{-1}\left (\frac {\sqrt {\frac {11}{62 \left (13+10 \sqrt {2}\right )}} \left (6+7 \sqrt {2}+\left (20+13 \sqrt {2}\right ) x\right )}{\sqrt {3-x+2 x^2}}\right )-\frac {1}{5} \sqrt {\frac {11}{31} \left (-13+10 \sqrt {2}\right )} \tanh ^{-1}\left (\frac {\sqrt {\frac {11}{62 \left (-13+10 \sqrt {2}\right )}} \left (6-7 \sqrt {2}+\left (20-13 \sqrt {2}\right ) x\right )}{\sqrt {3-x+2 x^2}}\right ) \]
[Out]
-1/5*arcsinh(1/23*(1-4*x)*23^(1/2))*2^(1/2)-1/155*arctanh(1/62*(6+x*(20-13*2^(1/2))-7*2^(1/2))*682^(1/2)/(-13+
10*2^(1/2))^(1/2)/(2*x^2-x+3)^(1/2))*(-4433+3410*2^(1/2))^(1/2)+1/155*arctan(1/62*(6+7*2^(1/2)+x*(20+13*2^(1/2
)))*682^(1/2)/(13+10*2^(1/2))^(1/2)/(2*x^2-x+3)^(1/2))*(4433+3410*2^(1/2))^(1/2)
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Rubi [A]
time = 0.28, antiderivative size = 174, normalized size of antiderivative = 1.00, 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 = {1003, 633, 221,
1049, 1043, 212, 210} \begin {gather*} \frac {1}{5} \sqrt {\frac {11}{31} \left (13+10 \sqrt {2}\right )} \text {ArcTan}\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 ) \end {gather*}
Antiderivative was successfully verified.
[In]
Int[Sqrt[3 - x + 2*x^2]/(2 + 3*x + 5*x^2),x]
[Out]
-1/5*(Sqrt[2]*ArcSinh[(1 - 4*x)/Sqrt[23]]) + (Sqrt[(11*(13 + 10*Sqrt[2]))/31]*ArcTan[(Sqrt[11/(62*(13 + 10*Sqr
t[2]))]*(6 + 7*Sqrt[2] + (20 + 13*Sqrt[2])*x))/Sqrt[3 - x + 2*x^2]])/5 - (Sqrt[(11*(-13 + 10*Sqrt[2]))/31]*Arc
Tanh[(Sqrt[11/(62*(-13 + 10*Sqrt[2]))]*(6 - 7*Sqrt[2] + (20 - 13*Sqrt[2])*x))/Sqrt[3 - x + 2*x^2]])/5
Rule 210
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])
Rule 212
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])
Rule 221
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 633
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 1003
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 1043
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 1049
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]
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}} \text {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 ) \text {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 ) \text {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*}
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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 0.20, size = 206, normalized size = 1.18 \begin {gather*} \frac {1}{5} \left (-\sqrt {2} \log \left (1-4 x+2 \sqrt {6-2 x+4 x^2}\right )+11 \text {RootSum}\left [-56-26 \sqrt {2} \text {$\#$1}+17 \text {$\#$1}^2+6 \sqrt {2} \text {$\#$1}^3-5 \text {$\#$1}^4\&,\frac {-2 \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right )+2 \sqrt {2} \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}+\log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{-13 \sqrt {2}+17 \text {$\#$1}+9 \sqrt {2} \text {$\#$1}^2-10 \text {$\#$1}^3}\&\right ]\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[Sqrt[3 - x + 2*x^2]/(2 + 3*x + 5*x^2),x]
[Out]
(-(Sqrt[2]*Log[1 - 4*x + 2*Sqrt[6 - 2*x + 4*x^2]]) + 11*RootSum[-56 - 26*Sqrt[2]*#1 + 17*#1^2 + 6*Sqrt[2]*#1^3
- 5*#1^4 & , (-2*Log[-(Sqrt[2]*x) + Sqrt[3 - x + 2*x^2] - #1] + 2*Sqrt[2]*Log[-(Sqrt[2]*x) + Sqrt[3 - x + 2*x
^2] - #1]*#1 + Log[-(Sqrt[2]*x) + Sqrt[3 - x + 2*x^2] - #1]*#1^2)/(-13*Sqrt[2] + 17*#1 + 9*Sqrt[2]*#1^2 - 10*#
1^3) & ])/5
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(2064\) vs.
\(2(125)=250\).
time = 0.69, size = 2065, normalized size = 11.87
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| method |
result |
size |
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| trager |
\(-\frac {\RootOf \left (\textit {\_Z}^{2}-2\right ) \ln \left (-4 \RootOf \left (\textit {\_Z}^{2}-2\right ) x +4 \sqrt {2 x^{2}-x +3}+\RootOf \left (\textit {\_Z}^{2}-2\right )\right )}{5}+\RootOf \left (24025 \textit {\_Z}^{4}+4433 \textit {\_Z}^{2}+242\right ) \ln \left (\frac {649275625 x \RootOf \left (24025 \textit {\_Z}^{4}+4433 \textit {\_Z}^{2}+242\right )^{5}+183764900 \RootOf \left (24025 \textit {\_Z}^{4}+4433 \textit {\_Z}^{2}+242\right )^{3} x +53826850 \sqrt {2 x^{2}-x +3}\, \RootOf \left (24025 \textit {\_Z}^{4}+4433 \textit {\_Z}^{2}+242\right )^{2}+80135000 \RootOf \left (24025 \textit {\_Z}^{4}+4433 \textit {\_Z}^{2}+242\right )^{3}+7037844 \RootOf \left (24025 \textit {\_Z}^{4}+4433 \textit {\_Z}^{2}+242\right ) x +5073772 \sqrt {2 x^{2}-x +3}+19021200 \RootOf \left (24025 \textit {\_Z}^{4}+4433 \textit {\_Z}^{2}+242\right )}{775 x \RootOf \left (24025 \textit {\_Z}^{4}+4433 \textit {\_Z}^{2}+242\right )^{2}+88 x +22}\right )+\frac {\RootOf \left (\textit {\_Z}^{2}+24025 \RootOf \left (24025 \textit {\_Z}^{4}+4433 \textit {\_Z}^{2}+242\right )^{2}+4433\right ) \ln \left (\frac {5194205 \RootOf \left (\textit {\_Z}^{2}+24025 \RootOf \left (24025 \textit {\_Z}^{4}+4433 \textit {\_Z}^{2}+242\right )^{2}+4433\right ) \RootOf \left (24025 \textit {\_Z}^{4}+4433 \textit {\_Z}^{2}+242\right )^{4} x +446710 \RootOf \left (24025 \textit {\_Z}^{4}+4433 \textit {\_Z}^{2}+242\right )^{2} \RootOf \left (\textit {\_Z}^{2}+24025 \RootOf \left (24025 \textit {\_Z}^{4}+4433 \textit {\_Z}^{2}+242\right )^{2}+4433\right ) x -641080 \RootOf \left (24025 \textit {\_Z}^{4}+4433 \textit {\_Z}^{2}+242\right )^{2} \RootOf \left (\textit {\_Z}^{2}+24025 \RootOf \left (24025 \textit {\_Z}^{4}+4433 \textit {\_Z}^{2}+242\right )^{2}+4433\right )-66745294 \sqrt {2 x^{2}-x +3}\, \RootOf \left (24025 \textit {\_Z}^{4}+4433 \textit {\_Z}^{2}+242\right )^{2}-38115 \RootOf \left (\textit {\_Z}^{2}+24025 \RootOf \left (24025 \textit {\_Z}^{4}+4433 \textit {\_Z}^{2}+242\right )^{2}+4433\right ) x +33880 \RootOf \left (\textit {\_Z}^{2}+24025 \RootOf \left (24025 \textit {\_Z}^{4}+4433 \textit {\_Z}^{2}+242\right )^{2}+4433\right )-6024106 \sqrt {2 x^{2}-x +3}}{775 x \RootOf \left (24025 \textit {\_Z}^{4}+4433 \textit {\_Z}^{2}+242\right )^{2}+55 x -22}\right )}{155}\) |
\(479\) |
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\(\text {Expression too large to display}\) |
\(2065\) |
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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,method=_RETURNVERBOSE)
[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)*(2^(1/2)-1+x)^
2/(2^(1/2)+1-x)^2+8-3*2^(1/2))^(1/2)*2^(1/2)*(285*2^(1/2)*(-8866+6820*2^(1/2))^(1/2)*arctan(1/11692487*(-77568
7+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)*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)^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)*(2^(1/2)-1+x)/(2^(1/2)+1-x)*(8+3*2^(1/2)))*(-775
687+549362*2^(1/2))^(1/2)+386*(-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)*(2^(1/2)-1+x)^2/(2^(1/
2)+1-x)^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)*(2^(1/2)-1+x)/(2^(1/2)+1-x)*(8+3*2^(1/2)))*(-775687+549362*2^(1/2))^(1/2)-27
4846*arctanh(31/2*(8*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)^2+3*2^(1/2)*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)^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)*(2^(
1/2)-1+x)^2/(2^(1/2)+1-x)^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)*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)^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)*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)^2+8-3*2^(1/2))^(1/2)*2^(1/2)*(151*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)-4
1))^(1/2)*(6485*2^(1/2)*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)^2+10368*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)^2+22379*2^(1/2)+32
016)/(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)*(2^(1/2)-1+x)/(2^(1/2)+1-x)*(8
+3*2^(1/2)))*(-775687+549362*2^(1/2))^(1/2)+218*(-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)*(2^(
1/2)-1+x)^2/(2^(1/2)+1-x)^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)*(2^(1/2)-1+x)/(2^(1/2)+1-x)*(8+3*2^(1/2)))*(-775687+549362
*2^(1/2))^(1/2)+401698*arctanh(31/2*(8*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)^2+3*2^(1/2)*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)
^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)*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)^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)*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)^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)*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)^2+8-3*2^(1/2))^(1/2)*2^(1/2)*(369*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)*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)^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)*(2^(1/2)-1+x)/
(2^(1/2)+1-x)*(8+3*2^(1/2)))*(-775687+549362*2^(1/2))^(1/2)+520*(-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)*(6
485*2^(1/2)*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)^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)*(2^(1/2)-1+x)/(2^(1/2)+1-x)*(8+3*2^(1/2)))
*(-775687+549362*2^(1/2))^(1/2)+465124*arctanh(31/2*(8*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)^2+3*2^(1/2)*(2^(1/2)-1+x)
^2/(2^(1/2)+1-x)^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)*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)^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)*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)^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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
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] Leaf count of result is larger than twice the leaf count of optimal. 2016 vs.
\(2 (125) = 250\).
time = 3.97, size = 2016, normalized size = 11.59 \begin {gather*} \text {Too large to display} \end {gather*}
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^...
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {2 x^{2} - x + 3}}{5 x^{2} + 3 x + 2}\, dx \end {gather*}
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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
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 >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Francis algorithm failure for[-1.0,infinity,infinity,infinity,infinity]proot error [1.0,infinity,infinity,i
nfinity,inf
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sqrt {2\,x^2-x+3}}{5\,x^2+3\,x+2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((2*x^2 - x + 3)^(1/2)/(3*x + 5*x^2 + 2),x)
[Out]
int((2*x^2 - x + 3)^(1/2)/(3*x + 5*x^2 + 2), x)
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