Integrand size = 27, antiderivative size = 174 \[ \int \frac {\sqrt {3-x+2 x^2}}{2+3 x+5 x^2} \, dx=-\frac {1}{5} \sqrt {2} \text {arcsinh}\left (\frac {1-4 x}{\sqrt {23}}\right )+\frac {1}{5} \sqrt {\frac {11}{31} \left (13+10 \sqrt {2}\right )} \arctan \left (\frac {\sqrt {\frac {11}{62 \left (13+10 \sqrt {2}\right )}} \left (6+7 \sqrt {2}+\left (20+13 \sqrt {2}\right ) x\right )}{\sqrt {3-x+2 x^2}}\right )-\frac {1}{5} \sqrt {\frac {11}{31} \left (-13+10 \sqrt {2}\right )} \text {arctanh}\left (\frac {\sqrt {\frac {11}{62 \left (-13+10 \sqrt {2}\right )}} \left (6-7 \sqrt {2}+\left (20-13 \sqrt {2}\right ) x\right )}{\sqrt {3-x+2 x^2}}\right ) \] Output:
-1/5*arcsinh(1/23*(1-4*x)*23^(1/2))*2^(1/2)+1/155*(4433+3410*2^(1/2))^(1/2 )*arctan(11^(1/2)/(806+620*2^(1/2))^(1/2)*(6+7*2^(1/2)+(20+13*2^(1/2))*x)/ (2*x^2-x+3)^(1/2))-1/155*(-4433+3410*2^(1/2))^(1/2)*arctanh(11^(1/2)/(-806 +620*2^(1/2))^(1/2)*(6-7*2^(1/2)+(20-13*2^(1/2))*x)/(2*x^2-x+3)^(1/2))
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.33 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.18 \[ \int \frac {\sqrt {3-x+2 x^2}}{2+3 x+5 x^2} \, dx=\frac {1}{5} \left (-\sqrt {2} \log \left (1-4 x+2 \sqrt {6-2 x+4 x^2}\right )+11 \text {RootSum}\left [-56-26 \sqrt {2} \text {$\#$1}+17 \text {$\#$1}^2+6 \sqrt {2} \text {$\#$1}^3-5 \text {$\#$1}^4\&,\frac {-2 \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right )+2 \sqrt {2} \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}+\log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{-13 \sqrt {2}+17 \text {$\#$1}+9 \sqrt {2} \text {$\#$1}^2-10 \text {$\#$1}^3}\&\right ]\right ) \] Input:
Integrate[Sqrt[3 - x + 2*x^2]/(2 + 3*x + 5*x^2),x]
Output:
(-(Sqrt[2]*Log[1 - 4*x + 2*Sqrt[6 - 2*x + 4*x^2]]) + 11*RootSum[-56 - 26*S qrt[2]*#1 + 17*#1^2 + 6*Sqrt[2]*#1^3 - 5*#1^4 & , (-2*Log[-(Sqrt[2]*x) + S qrt[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
Time = 0.52 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.03, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1320, 27, 1090, 222, 1368, 27, 1362, 217, 219}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\sqrt {2 x^2-x+3}}{5 x^2+3 x+2} \, dx\) |
| \(\Big \downarrow \) 1320 |
| \(\displaystyle \frac {2}{5} \int \frac {1}{\sqrt {2 x^2-x+3}}dx-\frac {1}{5} \int -\frac {11 (1-x)}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{5} \int \frac {1}{\sqrt {2 x^2-x+3}}dx+\frac {11}{5} \int \frac {1-x}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx\) |
| \(\Big \downarrow \) 1090 |
| \(\displaystyle \frac {11}{5} \int \frac {1-x}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx+\frac {1}{5} \sqrt {\frac {2}{23}} \int \frac {1}{\sqrt {\frac {1}{23} (4 x-1)^2+1}}d(4 x-1)\) |
| \(\Big \downarrow \) 222 |
| \(\displaystyle \frac {11}{5} \int \frac {1-x}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx+\frac {1}{5} \sqrt {2} \text {arcsinh}\left (\frac {4 x-1}{\sqrt {23}}\right )\) |
| \(\Big \downarrow \) 1368 |
| \(\displaystyle \frac {11}{5} \left (\frac {\int -\frac {11 \left (\sqrt {2} x-\sqrt {2}+2\right )}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx}{22 \sqrt {2}}-\frac {\int -\frac {11 \left (-\sqrt {2} x+\sqrt {2}+2\right )}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx}{22 \sqrt {2}}\right )+\frac {1}{5} \sqrt {2} \text {arcsinh}\left (\frac {4 x-1}{\sqrt {23}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {11}{5} \left (\frac {\int \frac {-\sqrt {2} x+\sqrt {2}+2}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx}{2 \sqrt {2}}-\frac {\int \frac {\sqrt {2} x-\sqrt {2}+2}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx}{2 \sqrt {2}}\right )+\frac {1}{5} \sqrt {2} \text {arcsinh}\left (\frac {4 x-1}{\sqrt {23}}\right )\) |
| \(\Big \downarrow \) 1362 |
| \(\displaystyle \frac {11}{5} \left (\sqrt {2} \left (13-10 \sqrt {2}\right ) \int \frac {1}{-\frac {11 \left (\left (20-13 \sqrt {2}\right ) x-7 \sqrt {2}+6\right )^2}{2 x^2-x+3}-62 \left (13-10 \sqrt {2}\right )}d\frac {\left (20-13 \sqrt {2}\right ) x-7 \sqrt {2}+6}{\sqrt {2 x^2-x+3}}-\sqrt {2} \left (13+10 \sqrt {2}\right ) \int \frac {1}{-\frac {11 \left (\left (20+13 \sqrt {2}\right ) x+7 \sqrt {2}+6\right )^2}{2 x^2-x+3}-62 \left (13+10 \sqrt {2}\right )}d\frac {\left (20+13 \sqrt {2}\right ) x+7 \sqrt {2}+6}{\sqrt {2 x^2-x+3}}\right )+\frac {1}{5} \sqrt {2} \text {arcsinh}\left (\frac {4 x-1}{\sqrt {23}}\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {11}{5} \left (\sqrt {2} \left (13-10 \sqrt {2}\right ) \int \frac {1}{-\frac {11 \left (\left (20-13 \sqrt {2}\right ) x-7 \sqrt {2}+6\right )^2}{2 x^2-x+3}-62 \left (13-10 \sqrt {2}\right )}d\frac {\left (20-13 \sqrt {2}\right ) x-7 \sqrt {2}+6}{\sqrt {2 x^2-x+3}}+\sqrt {\frac {1}{341} \left (13+10 \sqrt {2}\right )} \arctan \left (\frac {\sqrt {\frac {11}{62 \left (13+10 \sqrt {2}\right )}} \left (\left (20+13 \sqrt {2}\right ) x+7 \sqrt {2}+6\right )}{\sqrt {2 x^2-x+3}}\right )\right )+\frac {1}{5} \sqrt {2} \text {arcsinh}\left (\frac {4 x-1}{\sqrt {23}}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{5} \sqrt {2} \text {arcsinh}\left (\frac {4 x-1}{\sqrt {23}}\right )+\frac {11}{5} \left (\sqrt {\frac {1}{341} \left (13+10 \sqrt {2}\right )} \arctan \left (\frac {\sqrt {\frac {11}{62 \left (13+10 \sqrt {2}\right )}} \left (\left (20+13 \sqrt {2}\right ) x+7 \sqrt {2}+6\right )}{\sqrt {2 x^2-x+3}}\right )+\frac {\left (13-10 \sqrt {2}\right ) \text {arctanh}\left (\frac {\sqrt {\frac {11}{62 \left (10 \sqrt {2}-13\right )}} \left (\left (20-13 \sqrt {2}\right ) x-7 \sqrt {2}+6\right )}{\sqrt {2 x^2-x+3}}\right )}{\sqrt {341 \left (10 \sqrt {2}-13\right )}}\right )\) |
Input:
Int[Sqrt[3 - x + 2*x^2]/(2 + 3*x + 5*x^2),x]
Output:
(Sqrt[2]*ArcSinh[(-1 + 4*x)/Sqrt[23]])/5 + (11*(Sqrt[(13 + 10*Sqrt[2])/341 ]*ArcTan[(Sqrt[11/(62*(13 + 10*Sqrt[2]))]*(6 + 7*Sqrt[2] + (20 + 13*Sqrt[2 ])*x))/Sqrt[3 - x + 2*x^2]] + ((13 - 10*Sqrt[2])*ArcTanh[(Sqrt[11/(62*(-13 + 10*Sqrt[2]))]*(6 - 7*Sqrt[2] + (20 - 13*Sqrt[2])*x))/Sqrt[3 - x + 2*x^2 ]])/Sqrt[341*(-13 + 10*Sqrt[2])]))/5
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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])
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] && (Gt Q[a, 0] || LtQ[b, 0])
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]
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/(2*c*(-4* (c/(b^2 - 4*a*c)))^p) Subst[Int[Simp[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]
Int[Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2]/((d_) + (e_.)*(x_) + (f_.)*(x_)^ 2), x_Symbol] :> Simp[c/f Int[1/Sqrt[a + b*x + c*x^2], x], x] - Simp[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]
Int[((g_.) + (h_.)*(x_))/(((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[-2*g*(g*b - 2*a*h) Subst[I nt[1/Simp[g*(g*b - 2*a*h)*(b^2 - 4*a*c) - (b*d - a*e)*x^2, x], x], x, Simp[ 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] && Ne Q[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]
Int[((g_.) + (h_.)*(x_))/(((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symbol] :> With[{q = Rt[(c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f), 2]}, Simp[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)*Sqr t[d + e*x + f*x^2]), x], x] - Simp[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] && N eQ[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]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 4.69 (sec) , antiderivative size = 483, normalized size of antiderivative = 2.78
| method | result | size |
| trager | \(\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \ln \left (4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x +4 \sqrt {2 x^{2}-x +3}-\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right )\right )}{5}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+24025 \operatorname {RootOf}\left (24025 \textit {\_Z}^{4}+4433 \textit {\_Z}^{2}+242\right )^{2}+4433\right ) \ln \left (-\frac {5194205 \operatorname {RootOf}\left (\textit {\_Z}^{2}+24025 \operatorname {RootOf}\left (24025 \textit {\_Z}^{4}+4433 \textit {\_Z}^{2}+242\right )^{2}+4433\right ) \operatorname {RootOf}\left (24025 \textit {\_Z}^{4}+4433 \textit {\_Z}^{2}+242\right )^{4} x +446710 \operatorname {RootOf}\left (24025 \textit {\_Z}^{4}+4433 \textit {\_Z}^{2}+242\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+24025 \operatorname {RootOf}\left (24025 \textit {\_Z}^{4}+4433 \textit {\_Z}^{2}+242\right )^{2}+4433\right ) x +66745294 \sqrt {2 x^{2}-x +3}\, \operatorname {RootOf}\left (24025 \textit {\_Z}^{4}+4433 \textit {\_Z}^{2}+242\right )^{2}-641080 \operatorname {RootOf}\left (24025 \textit {\_Z}^{4}+4433 \textit {\_Z}^{2}+242\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+24025 \operatorname {RootOf}\left (24025 \textit {\_Z}^{4}+4433 \textit {\_Z}^{2}+242\right )^{2}+4433\right )-38115 \operatorname {RootOf}\left (\textit {\_Z}^{2}+24025 \operatorname {RootOf}\left (24025 \textit {\_Z}^{4}+4433 \textit {\_Z}^{2}+242\right )^{2}+4433\right ) x +6024106 \sqrt {2 x^{2}-x +3}+33880 \operatorname {RootOf}\left (\textit {\_Z}^{2}+24025 \operatorname {RootOf}\left (24025 \textit {\_Z}^{4}+4433 \textit {\_Z}^{2}+242\right )^{2}+4433\right )}{775 x \operatorname {RootOf}\left (24025 \textit {\_Z}^{4}+4433 \textit {\_Z}^{2}+242\right )^{2}+55 x -22}\right )}{155}-\operatorname {RootOf}\left (24025 \textit {\_Z}^{4}+4433 \textit {\_Z}^{2}+242\right ) \ln \left (\frac {-649275625 x \operatorname {RootOf}\left (24025 \textit {\_Z}^{4}+4433 \textit {\_Z}^{2}+242\right )^{5}-183764900 \operatorname {RootOf}\left (24025 \textit {\_Z}^{4}+4433 \textit {\_Z}^{2}+242\right )^{3} x +53826850 \sqrt {2 x^{2}-x +3}\, \operatorname {RootOf}\left (24025 \textit {\_Z}^{4}+4433 \textit {\_Z}^{2}+242\right )^{2}-80135000 \operatorname {RootOf}\left (24025 \textit {\_Z}^{4}+4433 \textit {\_Z}^{2}+242\right )^{3}-7037844 x \operatorname {RootOf}\left (24025 \textit {\_Z}^{4}+4433 \textit {\_Z}^{2}+242\right )+5073772 \sqrt {2 x^{2}-x +3}-19021200 \operatorname {RootOf}\left (24025 \textit {\_Z}^{4}+4433 \textit {\_Z}^{2}+242\right )}{775 x \operatorname {RootOf}\left (24025 \textit {\_Z}^{4}+4433 \textit {\_Z}^{2}+242\right )^{2}+88 x +22}\right )\) | \(483\) |
| default | \(\text {Expression too large to display}\) | \(2065\) |
Input:
int((2*x^2-x+3)^(1/2)/(5*x^2+3*x+2),x,method=_RETURNVERBOSE)
Output:
1/5*RootOf(_Z^2-2)*ln(4*RootOf(_Z^2-2)*x+4*(2*x^2-x+3)^(1/2)-RootOf(_Z^2-2 ))-1/155*RootOf(_Z^2+24025*RootOf(24025*_Z^4+4433*_Z^2+242)^2+4433)*ln(-(5 194205*RootOf(_Z^2+24025*RootOf(24025*_Z^4+4433*_Z^2+242)^2+4433)*RootOf(2 4025*_Z^4+4433*_Z^2+242)^4*x+446710*RootOf(24025*_Z^4+4433*_Z^2+242)^2*Roo tOf(_Z^2+24025*RootOf(24025*_Z^4+4433*_Z^2+242)^2+4433)*x+66745294*(2*x^2- x+3)^(1/2)*RootOf(24025*_Z^4+4433*_Z^2+242)^2-641080*RootOf(24025*_Z^4+443 3*_Z^2+242)^2*RootOf(_Z^2+24025*RootOf(24025*_Z^4+4433*_Z^2+242)^2+4433)-3 8115*RootOf(_Z^2+24025*RootOf(24025*_Z^4+4433*_Z^2+242)^2+4433)*x+6024106* (2*x^2-x+3)^(1/2)+33880*RootOf(_Z^2+24025*RootOf(24025*_Z^4+4433*_Z^2+242) ^2+4433))/(775*x*RootOf(24025*_Z^4+4433*_Z^2+242)^2+55*x-22))-RootOf(24025 *_Z^4+4433*_Z^2+242)*ln((-649275625*x*RootOf(24025*_Z^4+4433*_Z^2+242)^5-1 83764900*RootOf(24025*_Z^4+4433*_Z^2+242)^3*x+53826850*(2*x^2-x+3)^(1/2)*R ootOf(24025*_Z^4+4433*_Z^2+242)^2-80135000*RootOf(24025*_Z^4+4433*_Z^2+242 )^3-7037844*x*RootOf(24025*_Z^4+4433*_Z^2+242)+5073772*(2*x^2-x+3)^(1/2)-1 9021200*RootOf(24025*_Z^4+4433*_Z^2+242))/(775*x*RootOf(24025*_Z^4+4433*_Z ^2+242)^2+88*x+22))
Leaf count of result is larger than twice the leaf count of optimal. 492 vs. \(2 (124) = 248\).
Time = 0.09 (sec) , antiderivative size = 492, normalized size of antiderivative = 2.83 \[ \int \frac {\sqrt {3-x+2 x^2}}{2+3 x+5 x^2} \, dx =\text {Too large to display} \] Input:
integrate((2*x^2-x+3)^(1/2)/(5*x^2+3*x+2),x, algorithm="fricas")
Output:
-1/10*sqrt(110/31*sqrt(2) + 143/31)*arctan(-1/11*(88*(25*x^3 - 61*x^2 - sq rt(2)*(22*x^3 - 47*x^2 - 8*x + 24) - 32*x + 24)*sqrt(2*x^2 - x + 3) + (171 *x^4 + 1212*x^3 - 1640*x^2 - 176*sqrt(2)*(6*x^4 + 5*x^3 + 5*x^2 + 12*x) - 3936*x)*sqrt(110/31*sqrt(2) - 143/31))*sqrt(110/31*sqrt(2) + 143/31)/(343* x^4 - 400*x^3 + 1136*x^2 + 384*x - 576)) + 1/10*sqrt(110/31*sqrt(2) + 143/ 31)*arctan(1/11*(88*(25*x^3 - 61*x^2 - sqrt(2)*(22*x^3 - 47*x^2 - 8*x + 24 ) - 32*x + 24)*sqrt(2*x^2 - x + 3) - (171*x^4 + 1212*x^3 - 1640*x^2 - 176* sqrt(2)*(6*x^4 + 5*x^3 + 5*x^2 + 12*x) - 3936*x)*sqrt(110/31*sqrt(2) - 143 /31))*sqrt(110/31*sqrt(2) + 143/31)/(343*x^4 - 400*x^3 + 1136*x^2 + 384*x - 576)) + 1/10*sqrt(2)*log(-4*sqrt(2)*sqrt(2*x^2 - x + 3)*(4*x - 1) - 32*x ^2 + 16*x - 25) - 1/20*sqrt(110/31*sqrt(2) - 143/31)*log((49*x^2 + 2*sqrt( 2*x^2 - x + 3)*(sqrt(2)*(22*x - 63) + 41*x - 85)*sqrt(110/31*sqrt(2) - 143 /31) + 44*sqrt(2)*(2*x^2 - x + 3) - 151*x + 200)/x^2) + 1/20*sqrt(110/31*s qrt(2) - 143/31)*log((49*x^2 - 2*sqrt(2*x^2 - x + 3)*(sqrt(2)*(22*x - 63) + 41*x - 85)*sqrt(110/31*sqrt(2) - 143/31) + 44*sqrt(2)*(2*x^2 - x + 3) - 151*x + 200)/x^2)
\[ \int \frac {\sqrt {3-x+2 x^2}}{2+3 x+5 x^2} \, dx=\int \frac {\sqrt {2 x^{2} - x + 3}}{5 x^{2} + 3 x + 2}\, dx \] Input:
integrate((2*x**2-x+3)**(1/2)/(5*x**2+3*x+2),x)
Output:
Integral(sqrt(2*x**2 - x + 3)/(5*x**2 + 3*x + 2), x)
\[ \int \frac {\sqrt {3-x+2 x^2}}{2+3 x+5 x^2} \, dx=\int { \frac {\sqrt {2 \, x^{2} - x + 3}}{5 \, x^{2} + 3 \, x + 2} \,d x } \] Input:
integrate((2*x^2-x+3)^(1/2)/(5*x^2+3*x+2),x, algorithm="maxima")
Output:
integrate(sqrt(2*x^2 - x + 3)/(5*x^2 + 3*x + 2), x)
Exception generated. \[ \int \frac {\sqrt {3-x+2 x^2}}{2+3 x+5 x^2} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((2*x^2-x+3)^(1/2)/(5*x^2+3*x+2),x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Francis algorithm failure for[-1.0, infinity,infinity,infinity,infinity]proot error [1.0,infinity,infinity,inf inity,inf
Timed out. \[ \int \frac {\sqrt {3-x+2 x^2}}{2+3 x+5 x^2} \, dx=\int \frac {\sqrt {2\,x^2-x+3}}{5\,x^2+3\,x+2} \,d x \] Input:
int((2*x^2 - x + 3)^(1/2)/(3*x + 5*x^2 + 2),x)
Output:
int((2*x^2 - x + 3)^(1/2)/(3*x + 5*x^2 + 2), x)
\[ \int \frac {\sqrt {3-x+2 x^2}}{2+3 x+5 x^2} \, dx=\frac {\sqrt {2}\, \mathrm {log}\left (-2 \sqrt {2 x^{2}-x +3}\, \sqrt {2}-4 x +1\right )}{5}+\frac {11 \left (\int \frac {\sqrt {2 x^{2}-x +3}}{10 x^{4}+x^{3}+16 x^{2}+7 x +6}d x \right )}{5}-\frac {11 \left (\int \frac {\sqrt {2 x^{2}-x +3}\, x}{10 x^{4}+x^{3}+16 x^{2}+7 x +6}d x \right )}{5} \] Input:
int((2*x^2-x+3)^(1/2)/(5*x^2+3*x+2),x)
Output:
(sqrt(2)*log( - 2*sqrt(2*x**2 - x + 3)*sqrt(2) - 4*x + 1) + 11*int(sqrt(2* x**2 - x + 3)/(10*x**4 + x**3 + 16*x**2 + 7*x + 6),x) - 11*int((sqrt(2*x** 2 - x + 3)*x)/(10*x**4 + x**3 + 16*x**2 + 7*x + 6),x))/5