Integrand size = 27, antiderivative size = 197 \[ \int \frac {\left (3-x+2 x^2\right )^{3/2}}{2+3 x+5 x^2} \, dx=-\frac {1}{100} (49-20 x) \sqrt {3-x+2 x^2}-\frac {2203 \text {arcsinh}\left (\frac {1-4 x}{\sqrt {23}}\right )}{1000 \sqrt {2}}+\frac {11}{125} \sqrt {\frac {11}{31} \left (247+500 \sqrt {2}\right )} \arctan \left (\frac {\sqrt {\frac {11}{62 \left (247+500 \sqrt {2}\right )}} \left (8+61 \sqrt {2}+\left (130+69 \sqrt {2}\right ) x\right )}{\sqrt {3-x+2 x^2}}\right )-\frac {11}{125} \sqrt {\frac {11}{31} \left (-247+500 \sqrt {2}\right )} \text {arctanh}\left (\frac {\sqrt {\frac {11}{62 \left (-247+500 \sqrt {2}\right )}} \left (8-61 \sqrt {2}+\left (130-69 \sqrt {2}\right ) x\right )}{\sqrt {3-x+2 x^2}}\right ) \]
-2203/2000*arcsinh(1/23*(1-4*x)*23^(1/2))*2^(1/2)-1/100*(49-20*x)*(2*x^2-x +3)^(1/2)-11/3875*arctanh(1/62*(8+x*(130-69*2^(1/2))-61*2^(1/2))*682^(1/2) /(-247+500*2^(1/2))^(1/2)/(2*x^2-x+3)^(1/2))*(-84227+170500*2^(1/2))^(1/2) +11/3875*arctan(1/62*(8+61*2^(1/2)+x*(130+69*2^(1/2)))*682^(1/2)/(247+500* 2^(1/2))^(1/2)/(2*x^2-x+3)^(1/2))*(84227+170500*2^(1/2))^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.34 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.16 \[ \int \frac {\left (3-x+2 x^2\right )^{3/2}}{2+3 x+5 x^2} \, dx=\frac {20 (-49+20 x) \sqrt {3-x+2 x^2}-2203 \sqrt {2} \log \left (1-4 x+2 \sqrt {6-2 x+4 x^2}\right )+1936 \text {RootSum}\left [-56-26 \sqrt {2} \text {$\#$1}+17 \text {$\#$1}^2+6 \sqrt {2} \text {$\#$1}^3-5 \text {$\#$1}^4\&,\frac {-36 \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right )+6 \sqrt {2} \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}+13 \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{-13 \sqrt {2}+17 \text {$\#$1}+9 \sqrt {2} \text {$\#$1}^2-10 \text {$\#$1}^3}\&\right ]}{2000} \]
(20*(-49 + 20*x)*Sqrt[3 - x + 2*x^2] - 2203*Sqrt[2]*Log[1 - 4*x + 2*Sqrt[6 - 2*x + 4*x^2]] + 1936*RootSum[-56 - 26*Sqrt[2]*#1 + 17*#1^2 + 6*Sqrt[2]* #1^3 - 5*#1^4 & , (-36*Log[-(Sqrt[2]*x) + Sqrt[3 - x + 2*x^2] - #1] + 6*Sq rt[2]*Log[-(Sqrt[2]*x) + Sqrt[3 - x + 2*x^2] - #1]*#1 + 13*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) & ])/2000
Time = 0.63 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.06, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.407, Rules used = {1308, 27, 2143, 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.
| \(\displaystyle \int \frac {\left (2 x^2-x+3\right )^{3/2}}{5 x^2+3 x+2} \, dx\) |
| \(\Big \downarrow \) 1308 |
| \(\displaystyle -\frac {1}{50} \int -\frac {2203 x^2-1195 x+1462}{4 \sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx-\frac {1}{100} \sqrt {2 x^2-x+3} (49-20 x)\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{200} \int \frac {2203 x^2-1195 x+1462}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx-\frac {1}{100} (49-20 x) \sqrt {2 x^2-x+3}\) |
| \(\Big \downarrow \) 2143 |
| \(\displaystyle \frac {1}{200} \left (\frac {2203}{5} \int \frac {1}{\sqrt {2 x^2-x+3}}dx+\frac {1}{5} \int \frac {968 (3-13 x)}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx\right )-\frac {1}{100} (49-20 x) \sqrt {2 x^2-x+3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{200} \left (\frac {2203}{5} \int \frac {1}{\sqrt {2 x^2-x+3}}dx+\frac {968}{5} \int \frac {3-13 x}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx\right )-\frac {1}{100} (49-20 x) \sqrt {2 x^2-x+3}\) |
| \(\Big \downarrow \) 1090 |
| \(\displaystyle \frac {1}{200} \left (\frac {968}{5} \int \frac {3-13 x}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx+\frac {2203 \int \frac {1}{\sqrt {\frac {1}{23} (4 x-1)^2+1}}d(4 x-1)}{5 \sqrt {46}}\right )-\frac {1}{100} (49-20 x) \sqrt {2 x^2-x+3}\) |
| \(\Big \downarrow \) 222 |
| \(\displaystyle \frac {1}{200} \left (\frac {968}{5} \int \frac {3-13 x}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx+\frac {2203 \text {arcsinh}\left (\frac {4 x-1}{\sqrt {23}}\right )}{5 \sqrt {2}}\right )-\frac {1}{100} (49-20 x) \sqrt {2 x^2-x+3}\) |
| \(\Big \downarrow \) 1368 |
| \(\displaystyle \frac {1}{200} \left (\frac {968}{5} \left (\frac {\int -\frac {11 \left (\left (10+13 \sqrt {2}\right ) x-3 \sqrt {2}+16\right )}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx}{22 \sqrt {2}}-\frac {\int -\frac {11 \left (\left (10-13 \sqrt {2}\right ) x+3 \sqrt {2}+16\right )}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx}{22 \sqrt {2}}\right )+\frac {2203 \text {arcsinh}\left (\frac {4 x-1}{\sqrt {23}}\right )}{5 \sqrt {2}}\right )-\frac {1}{100} (49-20 x) \sqrt {2 x^2-x+3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{200} \left (\frac {968}{5} \left (\frac {\int \frac {\left (10-13 \sqrt {2}\right ) x+3 \sqrt {2}+16}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx}{2 \sqrt {2}}-\frac {\int \frac {\left (10+13 \sqrt {2}\right ) x-3 \sqrt {2}+16}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx}{2 \sqrt {2}}\right )+\frac {2203 \text {arcsinh}\left (\frac {4 x-1}{\sqrt {23}}\right )}{5 \sqrt {2}}\right )-\frac {1}{100} (49-20 x) \sqrt {2 x^2-x+3}\) |
| \(\Big \downarrow \) 1362 |
| \(\displaystyle \frac {1}{200} \left (\frac {968}{5} \left (\sqrt {2} \left (247-500 \sqrt {2}\right ) \int \frac {1}{-\frac {11 \left (\left (130-69 \sqrt {2}\right ) x-61 \sqrt {2}+8\right )^2}{2 x^2-x+3}-62 \left (247-500 \sqrt {2}\right )}d\frac {\left (130-69 \sqrt {2}\right ) x-61 \sqrt {2}+8}{\sqrt {2 x^2-x+3}}-\sqrt {2} \left (247+500 \sqrt {2}\right ) \int \frac {1}{-\frac {11 \left (\left (130+69 \sqrt {2}\right ) x+61 \sqrt {2}+8\right )^2}{2 x^2-x+3}-62 \left (247+500 \sqrt {2}\right )}d\frac {\left (130+69 \sqrt {2}\right ) x+61 \sqrt {2}+8}{\sqrt {2 x^2-x+3}}\right )+\frac {2203 \text {arcsinh}\left (\frac {4 x-1}{\sqrt {23}}\right )}{5 \sqrt {2}}\right )-\frac {1}{100} (49-20 x) \sqrt {2 x^2-x+3}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{200} \left (\frac {968}{5} \left (\sqrt {2} \left (247-500 \sqrt {2}\right ) \int \frac {1}{-\frac {11 \left (\left (130-69 \sqrt {2}\right ) x-61 \sqrt {2}+8\right )^2}{2 x^2-x+3}-62 \left (247-500 \sqrt {2}\right )}d\frac {\left (130-69 \sqrt {2}\right ) x-61 \sqrt {2}+8}{\sqrt {2 x^2-x+3}}+\sqrt {\frac {1}{341} \left (247+500 \sqrt {2}\right )} \arctan \left (\frac {\sqrt {\frac {11}{62 \left (247+500 \sqrt {2}\right )}} \left (\left (130+69 \sqrt {2}\right ) x+61 \sqrt {2}+8\right )}{\sqrt {2 x^2-x+3}}\right )\right )+\frac {2203 \text {arcsinh}\left (\frac {4 x-1}{\sqrt {23}}\right )}{5 \sqrt {2}}\right )-\frac {1}{100} (49-20 x) \sqrt {2 x^2-x+3}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{200} \left (\frac {2203 \text {arcsinh}\left (\frac {4 x-1}{\sqrt {23}}\right )}{5 \sqrt {2}}+\frac {968}{5} \left (\sqrt {\frac {1}{341} \left (247+500 \sqrt {2}\right )} \arctan \left (\frac {\sqrt {\frac {11}{62 \left (247+500 \sqrt {2}\right )}} \left (\left (130+69 \sqrt {2}\right ) x+61 \sqrt {2}+8\right )}{\sqrt {2 x^2-x+3}}\right )+\frac {\left (247-500 \sqrt {2}\right ) \text {arctanh}\left (\frac {\sqrt {\frac {11}{62 \left (500 \sqrt {2}-247\right )}} \left (\left (130-69 \sqrt {2}\right ) x-61 \sqrt {2}+8\right )}{\sqrt {2 x^2-x+3}}\right )}{\sqrt {341 \left (500 \sqrt {2}-247\right )}}\right )\right )-\frac {1}{100} (49-20 x) \sqrt {2 x^2-x+3}\) |
-1/100*((49 - 20*x)*Sqrt[3 - x + 2*x^2]) + ((2203*ArcSinh[(-1 + 4*x)/Sqrt[ 23]])/(5*Sqrt[2]) + (968*(Sqrt[(247 + 500*Sqrt[2])/341]*ArcTan[(Sqrt[11/(6 2*(247 + 500*Sqrt[2]))]*(8 + 61*Sqrt[2] + (130 + 69*Sqrt[2])*x))/Sqrt[3 - x + 2*x^2]] + ((247 - 500*Sqrt[2])*ArcTanh[(Sqrt[11/(62*(-247 + 500*Sqrt[2 ]))]*(8 - 61*Sqrt[2] + (130 - 69*Sqrt[2])*x))/Sqrt[3 - x + 2*x^2]])/Sqrt[3 41*(-247 + 500*Sqrt[2])]))/5)/200
3.1.69.3.1 Defintions of rubi rules used
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[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_.) + (e_.)*(x_) + (f_.)*(x _)^2)^(q_), x_Symbol] :> Simp[(b*f*(3*p + 2*q) - c*e*(2*p + q) + 2*c*f*(p + q)*x)*(a + b*x + c*x^2)^(p - 1)*((d + e*x + f*x^2)^(q + 1)/(2*f^2*(p + q)* (2*p + 2*q + 1))), x] - Simp[1/(2*f^2*(p + q)*(2*p + 2*q + 1)) Int[(a + b *x + c*x^2)^(p - 2)*(d + e*x + f*x^2)^q*Simp[(b*d - a*e)*(c*e - b*f)*(1 - p )*(2*p + q) - (p + q)*(b^2*d*f*(1 - p) - a*(f*(b*e - 2*a*f)*(2*p + 2*q + 1) + c*(2*d*f - e^2*(2*p + q)))) + (2*(c*d - a*f)*(c*e - b*f)*(1 - p)*(2*p + q) - (p + q)*((b^2 - 4*a*c)*e*f*(1 - p) + b*(c*(e^2 - 4*d*f)*(2*p + q) + f* (2*c*d - b*e + 2*a*f)*(2*p + 2*q + 1))))*x + ((c*e - b*f)^2*(1 - p)*p + c*( p + q)*(f*(b*e - 2*a*f)*(4*p + 2*q - 1) - c*(2*d*f*(1 - 2*p) + e^2*(3*p + q - 1))))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, q}, x] && NeQ[b^2 - 4* a*c, 0] && NeQ[e^2 - 4*d*f, 0] && GtQ[p, 1] && NeQ[p + q, 0] && NeQ[2*p + 2 *q + 1, 0] && !IGtQ[p, 0] && !IGtQ[q, 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]
Int[(Px_)/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_ .)*(x_)^2]), x_Symbol] :> With[{A = Coeff[Px, x, 0], B = Coeff[Px, x, 1], C = Coeff[Px, x, 2]}, Simp[C/c Int[1/Sqrt[d + e*x + f*x^2], x], x] + Simp[ 1/c Int[(A*c - a*C + (B*c - b*C)*x)/((a + b*x + c*x^2)*Sqrt[d + e*x + f*x ^2]), x], x]] /; FreeQ[{a, b, c, d, e, f}, x] && PolyQ[Px, x, 2]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 2.50 (sec) , antiderivative size = 590, normalized size of antiderivative = 2.99
| method | result | size |
| trager | \(\text {Expression too large to display}\) | \(590\) |
| risch | \(\frac {\left (-49+20 x \right ) \sqrt {2 x^{2}-x +3}}{100}+\frac {2203 \sqrt {2}\, \operatorname {arcsinh}\left (\frac {4 \sqrt {23}\, \left (x -\frac {1}{4}\right )}{23}\right )}{2000}+\frac {11 \sqrt {\frac {8 \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+\frac {3 \sqrt {2}\, \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+8-3 \sqrt {2}}\, \sqrt {2}\, \left (1535 \sqrt {-775687+549362 \sqrt {2}}\, \sqrt {2}\, \sqrt {-8866+6820 \sqrt {2}}\, \arctan \left (\frac {\sqrt {-775687+549362 \sqrt {2}}\, \sqrt {-23 \left (8+3 \sqrt {2}\right ) \left (-\frac {23 \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+24 \sqrt {2}-41\right )}\, \left (\frac {6485 \sqrt {2}\, \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+\frac {10368 \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+22379 \sqrt {2}+32016\right ) \left (\sqrt {2}-1+x \right ) \left (8+3 \sqrt {2}\right )}{11692487 \left (\frac {23 \left (\sqrt {2}-1+x \right )^{4}}{\left (\sqrt {2}+1-x \right )^{4}}+\frac {82 \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+23\right ) \left (\sqrt {2}+1-x \right )}\right )+2197 \sqrt {-775687+549362 \sqrt {2}}\, \sqrt {-8866+6820 \sqrt {2}}\, \arctan \left (\frac {\sqrt {-775687+549362 \sqrt {2}}\, \sqrt {-23 \left (8+3 \sqrt {2}\right ) \left (-\frac {23 \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+24 \sqrt {2}-41\right )}\, \left (\frac {6485 \sqrt {2}\, \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+\frac {10368 \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+22379 \sqrt {2}+32016\right ) \left (\sqrt {2}-1+x \right ) \left (8+3 \sqrt {2}\right )}{11692487 \left (\frac {23 \left (\sqrt {2}-1+x \right )^{4}}{\left (\sqrt {2}+1-x \right )^{4}}+\frac {82 \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+23\right ) \left (\sqrt {2}+1-x \right )}\right )+3308723 \,\operatorname {arctanh}\left (\frac {31 \sqrt {\frac {8 \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+\frac {3 \sqrt {2}\, \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+8-3 \sqrt {2}}}{2 \sqrt {-8866+6820 \sqrt {2}}}\right ) \sqrt {2}-1712502 \,\operatorname {arctanh}\left (\frac {31 \sqrt {\frac {8 \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+\frac {3 \sqrt {2}\, \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+8-3 \sqrt {2}}}{2 \sqrt {-8866+6820 \sqrt {2}}}\right )\right )}{120125 \sqrt {\frac {\frac {8 \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+\frac {3 \sqrt {2}\, \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+8-3 \sqrt {2}}{\left (1+\frac {\sqrt {2}-1+x}{\sqrt {2}+1-x}\right )^{2}}}\, \left (1+\frac {\sqrt {2}-1+x}{\sqrt {2}+1-x}\right ) \left (8+3 \sqrt {2}\right ) \sqrt {-8866+6820 \sqrt {2}}}\) | \(718\) |
| default | \(\text {Expression too large to display}\) | \(3460\) |
(-49/100+1/5*x)*(2*x^2-x+3)^(1/2)+1/100*RootOf(24025*_Z^4+163063472*_Z^2+2 267598080000)*ln((5429049375*x*RootOf(24025*_Z^4+163063472*_Z^2+2267598080 000)^5+40888264630400*RootOf(24025*_Z^4+163063472*_Z^2+2267598080000)^3*x+ 13640929511440000*RootOf(24025*_Z^4+163063472*_Z^2+2267598080000)^2*(2*x^2 -x+3)^(1/2)+154372254960000*RootOf(24025*_Z^4+163063472*_Z^2+2267598080000 )^3-592661349855657984*RootOf(24025*_Z^4+163063472*_Z^2+2267598080000)*x+5 2586627694873395200*(2*x^2-x+3)^(1/2)+2295791036224716800*RootOf(24025*_Z^ 4+163063472*_Z^2+2267598080000))/(775*x*RootOf(24025*_Z^4+163063472*_Z^2+2 267598080000)^2+6431392*x+5068448))+1/15500*RootOf(_Z^2+24025*RootOf(24025 *_Z^4+163063472*_Z^2+2267598080000)^2+163063472)*ln(-(-8686479*RootOf(_Z^2 +24025*RootOf(24025*_Z^4+163063472*_Z^2+2267598080000)^2+163063472)*RootOf (24025*_Z^4+163063472*_Z^2+2267598080000)^4*x-52493234464*RootOf(24025*_Z^ 4+163063472*_Z^2+2267598080000)^2*RootOf(_Z^2+24025*RootOf(24025*_Z^4+1630 63472*_Z^2+2267598080000)^2+163063472)*x+3382950518837120*RootOf(24025*_Z^ 4+163063472*_Z^2+2267598080000)^2*(2*x^2-x+3)^(1/2)+246995607936*RootOf(24 025*_Z^4+163063472*_Z^2+2267598080000)^2*RootOf(_Z^2+24025*RootOf(24025*_Z ^4+163063472*_Z^2+2267598080000)^2+163063472)+992130849952000*RootOf(_Z^2+ 24025*RootOf(24025*_Z^4+163063472*_Z^2+2267598080000)^2+163063472)*x+99194 17776240896000*(2*x^2-x+3)^(1/2)-1996846869248000*RootOf(_Z^2+24025*RootOf (24025*_Z^4+163063472*_Z^2+2267598080000)^2+163063472))/(775*x*RootOf(2...
Result contains complex when optimal does not.
Time = 0.32 (sec) , antiderivative size = 316, normalized size of antiderivative = 1.60 \[ \int \frac {\left (3-x+2 x^2\right )^{3/2}}{2+3 x+5 x^2} \, dx=\frac {1}{15500} \, \sqrt {31} \sqrt {316778 i \, \sqrt {31} - 657514} \log \left (-\frac {2 \, \sqrt {2 \, x^{2} - x + 3} {\left (2 \, \sqrt {31} - i\right )} \sqrt {316778 i \, \sqrt {31} - 657514} + 1375 \, \sqrt {31} {\left (-i \, x + 6 i\right )} - 26125 \, x + 30250}{x}\right ) - \frac {1}{15500} \, \sqrt {31} \sqrt {316778 i \, \sqrt {31} - 657514} \log \left (\frac {2 \, \sqrt {2 \, x^{2} - x + 3} {\left (2 \, \sqrt {31} - i\right )} \sqrt {316778 i \, \sqrt {31} - 657514} - 1375 \, \sqrt {31} {\left (-i \, x + 6 i\right )} + 26125 \, x - 30250}{x}\right ) + \frac {1}{15500} \, \sqrt {31} \sqrt {-316778 i \, \sqrt {31} - 657514} \log \left (-\frac {2 \, \sqrt {2 \, x^{2} - x + 3} {\left (2 \, \sqrt {31} + i\right )} \sqrt {-316778 i \, \sqrt {31} - 657514} + 1375 \, \sqrt {31} {\left (i \, x - 6 i\right )} - 26125 \, x + 30250}{x}\right ) - \frac {1}{15500} \, \sqrt {31} \sqrt {-316778 i \, \sqrt {31} - 657514} \log \left (\frac {2 \, \sqrt {2 \, x^{2} - x + 3} {\left (2 \, \sqrt {31} + i\right )} \sqrt {-316778 i \, \sqrt {31} - 657514} - 1375 \, \sqrt {31} {\left (i \, x - 6 i\right )} + 26125 \, x - 30250}{x}\right ) + \frac {1}{100} \, \sqrt {2 \, x^{2} - x + 3} {\left (20 \, x - 49\right )} + \frac {2203}{4000} \, \sqrt {2} \log \left (-4 \, \sqrt {2} \sqrt {2 \, x^{2} - x + 3} {\left (4 \, x - 1\right )} - 32 \, x^{2} + 16 \, x - 25\right ) \]
1/15500*sqrt(31)*sqrt(316778*I*sqrt(31) - 657514)*log(-(2*sqrt(2*x^2 - x + 3)*(2*sqrt(31) - I)*sqrt(316778*I*sqrt(31) - 657514) + 1375*sqrt(31)*(-I* x + 6*I) - 26125*x + 30250)/x) - 1/15500*sqrt(31)*sqrt(316778*I*sqrt(31) - 657514)*log((2*sqrt(2*x^2 - x + 3)*(2*sqrt(31) - I)*sqrt(316778*I*sqrt(31 ) - 657514) - 1375*sqrt(31)*(-I*x + 6*I) + 26125*x - 30250)/x) + 1/15500*s qrt(31)*sqrt(-316778*I*sqrt(31) - 657514)*log(-(2*sqrt(2*x^2 - x + 3)*(2*s qrt(31) + I)*sqrt(-316778*I*sqrt(31) - 657514) + 1375*sqrt(31)*(I*x - 6*I) - 26125*x + 30250)/x) - 1/15500*sqrt(31)*sqrt(-316778*I*sqrt(31) - 657514 )*log((2*sqrt(2*x^2 - x + 3)*(2*sqrt(31) + I)*sqrt(-316778*I*sqrt(31) - 65 7514) - 1375*sqrt(31)*(I*x - 6*I) + 26125*x - 30250)/x) + 1/100*sqrt(2*x^2 - x + 3)*(20*x - 49) + 2203/4000*sqrt(2)*log(-4*sqrt(2)*sqrt(2*x^2 - x + 3)*(4*x - 1) - 32*x^2 + 16*x - 25)
\[ \int \frac {\left (3-x+2 x^2\right )^{3/2}}{2+3 x+5 x^2} \, dx=\int \frac {\left (2 x^{2} - x + 3\right )^{\frac {3}{2}}}{5 x^{2} + 3 x + 2}\, dx \]
\[ \int \frac {\left (3-x+2 x^2\right )^{3/2}}{2+3 x+5 x^2} \, dx=\int { \frac {{\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{5 \, x^{2} + 3 \, x + 2} \,d x } \]
Exception generated. \[ \int \frac {\left (3-x+2 x^2\right )^{3/2}}{2+3 x+5 x^2} \, dx=\text {Exception raised: TypeError} \]
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. \[ \int \frac {\left (3-x+2 x^2\right )^{3/2}}{2+3 x+5 x^2} \, dx=\int \frac {{\left (2\,x^2-x+3\right )}^{3/2}}{5\,x^2+3\,x+2} \,d x \]