Integrand size = 27, antiderivative size = 222 \[ \int \frac {\left (3-x+2 x^2\right )^{5/2}}{2+3 x+5 x^2} \, dx=-\frac {(226249-99620 x) \sqrt {3-x+2 x^2}}{80000}-\frac {1}{600} (103-60 x) \left (3-x+2 x^2\right )^{3/2}-\frac {7216203 \text {arcsinh}\left (\frac {1-4 x}{\sqrt {23}}\right )}{800000 \sqrt {2}}-\frac {121 \sqrt {\frac {11}{31} \left (-15457+25000 \sqrt {2}\right )} \arctan \left (\frac {\sqrt {\frac {11}{62 \left (-15457+25000 \sqrt {2}\right )}} \left (196-443 \sqrt {2}-\left (690+247 \sqrt {2}\right ) x\right )}{\sqrt {3-x+2 x^2}}\right )}{3125}+\frac {121 \sqrt {\frac {11}{31} \left (15457+25000 \sqrt {2}\right )} \text {arctanh}\left (\frac {\sqrt {\frac {11}{62 \left (15457+25000 \sqrt {2}\right )}} \left (196+443 \sqrt {2}-\left (690-247 \sqrt {2}\right ) x\right )}{\sqrt {3-x+2 x^2}}\right )}{3125} \]
-1/600*(103-60*x)*(2*x^2-x+3)^(3/2)-7216203/1600000*arcsinh(1/23*(1-4*x)*2 3^(1/2))*2^(1/2)-1/80000*(226249-99620*x)*(2*x^2-x+3)^(1/2)-121/96875*arct an(1/62*(196-443*2^(1/2)-x*(690+247*2^(1/2)))*682^(1/2)/(-15457+25000*2^(1 /2))^(1/2)/(2*x^2-x+3)^(1/2))*(-5270837+8525000*2^(1/2))^(1/2)+121/96875*a rctanh(1/62*(196-x*(690-247*2^(1/2))+443*2^(1/2))*682^(1/2)/(15457+25000*2 ^(1/2))^(1/2)/(2*x^2-x+3)^(1/2))*(5270837+8525000*2^(1/2))^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.54 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.07 \[ \int \frac {\left (3-x+2 x^2\right )^{5/2}}{2+3 x+5 x^2} \, dx=\frac {20 \sqrt {3-x+2 x^2} \left (-802347+412060 x-106400 x^2+48000 x^3\right )-21648609 \sqrt {2} \log \left (1-4 x+2 \sqrt {6-2 x+4 x^2}\right )-2044416 \text {RootSum}\left [-56-26 \sqrt {2} \text {$\#$1}+17 \text {$\#$1}^2+6 \sqrt {2} \text {$\#$1}^3-5 \text {$\#$1}^4\&,\frac {368 \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right )+22 \sqrt {2} \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}-119 \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{-13 \sqrt {2}+17 \text {$\#$1}+9 \sqrt {2} \text {$\#$1}^2-10 \text {$\#$1}^3}\&\right ]}{4800000} \]
(20*Sqrt[3 - x + 2*x^2]*(-802347 + 412060*x - 106400*x^2 + 48000*x^3) - 21 648609*Sqrt[2]*Log[1 - 4*x + 2*Sqrt[6 - 2*x + 4*x^2]] - 2044416*RootSum[-5 6 - 26*Sqrt[2]*#1 + 17*#1^2 + 6*Sqrt[2]*#1^3 - 5*#1^4 & , (368*Log[-(Sqrt[ 2]*x) + Sqrt[3 - x + 2*x^2] - #1] + 22*Sqrt[2]*Log[-(Sqrt[2]*x) + Sqrt[3 - x + 2*x^2] - #1]*#1 - 119*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) & ])/4800000
Time = 0.73 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.08, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.481, Rules used = {1308, 27, 2138, 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.
| \(\displaystyle \int \frac {\left (2 x^2-x+3\right )^{5/2}}{5 x^2+3 x+2} \, dx\) |
| \(\Big \downarrow \) 1308 |
| \(\displaystyle -\frac {1}{300} \int -\frac {3 \sqrt {2 x^2-x+3} \left (4981 x^2-2045 x+3154\right )}{4 \left (5 x^2+3 x+2\right )}dx-\frac {1}{600} (103-60 x) \left (2 x^2-x+3\right )^{3/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{400} \int \frac {\sqrt {2 x^2-x+3} \left (4981 x^2-2045 x+3154\right )}{5 x^2+3 x+2}dx-\frac {1}{600} (103-60 x) \left (2 x^2-x+3\right )^{3/2}\) |
| \(\Big \downarrow \) 2138 |
| \(\displaystyle \frac {1}{400} \left (-\frac {1}{100} \int -\frac {7216203 x^2-3779795 x+2136862}{4 \sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx-\frac {1}{200} \sqrt {2 x^2-x+3} (226249-99620 x)\right )-\frac {1}{600} (103-60 x) \left (2 x^2-x+3\right )^{3/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{400} \left (\frac {1}{400} \int \frac {7216203 x^2-3779795 x+2136862}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx-\frac {1}{200} (226249-99620 x) \sqrt {2 x^2-x+3}\right )-\frac {1}{600} (103-60 x) \left (2 x^2-x+3\right )^{3/2}\) |
| \(\Big \downarrow \) 2143 |
| \(\displaystyle \frac {1}{400} \left (\frac {1}{400} \left (\frac {7216203}{5} \int \frac {1}{\sqrt {2 x^2-x+3}}dx+\frac {1}{5} \int -\frac {340736 (119 x+11)}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx\right )-\frac {1}{200} (226249-99620 x) \sqrt {2 x^2-x+3}\right )-\frac {1}{600} (103-60 x) \left (2 x^2-x+3\right )^{3/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{400} \left (\frac {1}{400} \left (\frac {7216203}{5} \int \frac {1}{\sqrt {2 x^2-x+3}}dx-\frac {340736}{5} \int \frac {119 x+11}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx\right )-\frac {1}{200} (226249-99620 x) \sqrt {2 x^2-x+3}\right )-\frac {1}{600} (103-60 x) \left (2 x^2-x+3\right )^{3/2}\) |
| \(\Big \downarrow \) 1090 |
| \(\displaystyle \frac {1}{400} \left (\frac {1}{400} \left (\frac {7216203 \int \frac {1}{\sqrt {\frac {1}{23} (4 x-1)^2+1}}d(4 x-1)}{5 \sqrt {46}}-\frac {340736}{5} \int \frac {119 x+11}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx\right )-\frac {1}{200} (226249-99620 x) \sqrt {2 x^2-x+3}\right )-\frac {1}{600} (103-60 x) \left (2 x^2-x+3\right )^{3/2}\) |
| \(\Big \downarrow \) 222 |
| \(\displaystyle \frac {1}{400} \left (\frac {1}{400} \left (\frac {7216203 \text {arcsinh}\left (\frac {4 x-1}{\sqrt {23}}\right )}{5 \sqrt {2}}-\frac {340736}{5} \int \frac {119 x+11}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx\right )-\frac {1}{200} (226249-99620 x) \sqrt {2 x^2-x+3}\right )-\frac {1}{600} (103-60 x) \left (2 x^2-x+3\right )^{3/2}\) |
| \(\Big \downarrow \) 1368 |
| \(\displaystyle \frac {1}{400} \left (\frac {1}{400} \left (\frac {7216203 \text {arcsinh}\left (\frac {4 x-1}{\sqrt {23}}\right )}{5 \sqrt {2}}-\frac {340736}{5} \left (\frac {\int \frac {11 \left (\left (130+119 \sqrt {2}\right ) x+11 \sqrt {2}+108\right )}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx}{22 \sqrt {2}}-\frac {\int \frac {11 \left (\left (130-119 \sqrt {2}\right ) x-11 \sqrt {2}+108\right )}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx}{22 \sqrt {2}}\right )\right )-\frac {1}{200} (226249-99620 x) \sqrt {2 x^2-x+3}\right )-\frac {1}{600} (103-60 x) \left (2 x^2-x+3\right )^{3/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{400} \left (\frac {1}{400} \left (\frac {7216203 \text {arcsinh}\left (\frac {4 x-1}{\sqrt {23}}\right )}{5 \sqrt {2}}-\frac {340736}{5} \left (\frac {\int \frac {\left (130+119 \sqrt {2}\right ) x+11 \sqrt {2}+108}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx}{2 \sqrt {2}}-\frac {\int \frac {\left (130-119 \sqrt {2}\right ) x-11 \sqrt {2}+108}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx}{2 \sqrt {2}}\right )\right )-\frac {1}{200} (226249-99620 x) \sqrt {2 x^2-x+3}\right )-\frac {1}{600} (103-60 x) \left (2 x^2-x+3\right )^{3/2}\) |
| \(\Big \downarrow \) 1362 |
| \(\displaystyle \frac {1}{400} \left (\frac {1}{400} \left (\frac {7216203 \text {arcsinh}\left (\frac {4 x-1}{\sqrt {23}}\right )}{5 \sqrt {2}}-\frac {340736}{5} \left (\sqrt {2} \left (15457+25000 \sqrt {2}\right ) \int \frac {1}{62 \left (15457+25000 \sqrt {2}\right )-\frac {11 \left (-\left (\left (690-247 \sqrt {2}\right ) x\right )+443 \sqrt {2}+196\right )^2}{2 x^2-x+3}}d\left (-\frac {-\left (\left (690-247 \sqrt {2}\right ) x\right )+443 \sqrt {2}+196}{\sqrt {2 x^2-x+3}}\right )-\sqrt {2} \left (15457-25000 \sqrt {2}\right ) \int \frac {1}{62 \left (15457-25000 \sqrt {2}\right )-\frac {11 \left (-\left (\left (690+247 \sqrt {2}\right ) x\right )-443 \sqrt {2}+196\right )^2}{2 x^2-x+3}}d\left (-\frac {-\left (\left (690+247 \sqrt {2}\right ) x\right )-443 \sqrt {2}+196}{\sqrt {2 x^2-x+3}}\right )\right )\right )-\frac {1}{200} (226249-99620 x) \sqrt {2 x^2-x+3}\right )-\frac {1}{600} (103-60 x) \left (2 x^2-x+3\right )^{3/2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{400} \left (\frac {1}{400} \left (\frac {7216203 \text {arcsinh}\left (\frac {4 x-1}{\sqrt {23}}\right )}{5 \sqrt {2}}-\frac {340736}{5} \left (\sqrt {2} \left (15457+25000 \sqrt {2}\right ) \int \frac {1}{62 \left (15457+25000 \sqrt {2}\right )-\frac {11 \left (-\left (\left (690-247 \sqrt {2}\right ) x\right )+443 \sqrt {2}+196\right )^2}{2 x^2-x+3}}d\left (-\frac {-\left (\left (690-247 \sqrt {2}\right ) x\right )+443 \sqrt {2}+196}{\sqrt {2 x^2-x+3}}\right )-\frac {\left (15457-25000 \sqrt {2}\right ) \arctan \left (\frac {\sqrt {\frac {11}{62 \left (25000 \sqrt {2}-15457\right )}} \left (-\left (\left (690+247 \sqrt {2}\right ) x\right )-443 \sqrt {2}+196\right )}{\sqrt {2 x^2-x+3}}\right )}{\sqrt {341 \left (25000 \sqrt {2}-15457\right )}}\right )\right )-\frac {1}{200} (226249-99620 x) \sqrt {2 x^2-x+3}\right )-\frac {1}{600} (103-60 x) \left (2 x^2-x+3\right )^{3/2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{400} \left (\frac {1}{400} \left (\frac {7216203 \text {arcsinh}\left (\frac {4 x-1}{\sqrt {23}}\right )}{5 \sqrt {2}}-\frac {340736}{5} \left (-\frac {\left (15457-25000 \sqrt {2}\right ) \arctan \left (\frac {\sqrt {\frac {11}{62 \left (25000 \sqrt {2}-15457\right )}} \left (-\left (\left (690+247 \sqrt {2}\right ) x\right )-443 \sqrt {2}+196\right )}{\sqrt {2 x^2-x+3}}\right )}{\sqrt {341 \left (25000 \sqrt {2}-15457\right )}}-\sqrt {\frac {1}{341} \left (15457+25000 \sqrt {2}\right )} \text {arctanh}\left (\frac {\sqrt {\frac {11}{62 \left (15457+25000 \sqrt {2}\right )}} \left (-\left (\left (690-247 \sqrt {2}\right ) x\right )+443 \sqrt {2}+196\right )}{\sqrt {2 x^2-x+3}}\right )\right )\right )-\frac {1}{200} (226249-99620 x) \sqrt {2 x^2-x+3}\right )-\frac {1}{600} (103-60 x) \left (2 x^2-x+3\right )^{3/2}\) |
-1/600*((103 - 60*x)*(3 - x + 2*x^2)^(3/2)) + (-1/200*((226249 - 99620*x)* Sqrt[3 - x + 2*x^2]) + ((7216203*ArcSinh[(-1 + 4*x)/Sqrt[23]])/(5*Sqrt[2]) - (340736*(-(((15457 - 25000*Sqrt[2])*ArcTan[(Sqrt[11/(62*(-15457 + 25000 *Sqrt[2]))]*(196 - 443*Sqrt[2] - (690 + 247*Sqrt[2])*x))/Sqrt[3 - x + 2*x^ 2]])/Sqrt[341*(-15457 + 25000*Sqrt[2])]) - Sqrt[(15457 + 25000*Sqrt[2])/34 1]*ArcTanh[(Sqrt[11/(62*(15457 + 25000*Sqrt[2]))]*(196 + 443*Sqrt[2] - (69 0 - 247*Sqrt[2])*x))/Sqrt[3 - x + 2*x^2]]))/5)/400)/400
3.1.76.3.1 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)^(p_)*((d_) + (e_.)*(x_) + (f_. )*(x_)^2)^(q_), x_Symbol] :> With[{A = Coeff[Px, x, 0], B = Coeff[Px, x, 1] , C = Coeff[Px, x, 2]}, Simp[(B*c*f*(2*p + 2*q + 3) + C*(b*f*p - c*e*(2*p + q + 2)) + 2*c*C*f*(p + q + 1)*x)*(a + b*x + c*x^2)^p*((d + e*x + f*x^2)^(q + 1)/(2*c*f^2*(p + q + 1)*(2*p + 2*q + 3))), x] - Simp[1/(2*c*f^2*(p + q + 1)*(2*p + 2*q + 3)) Int[(a + b*x + c*x^2)^(p - 1)*(d + e*x + f*x^2)^q*Si mp[p*(b*d - a*e)*(C*(c*e - b*f)*(q + 1) - c*(C*e - B*f)*(2*p + 2*q + 3)) + (p + q + 1)*(b^2*C*d*f*p + a*c*(C*(2*d*f - e^2*(2*p + q + 2)) + f*(B*e - 2* A*f)*(2*p + 2*q + 3))) + (2*p*(c*d - a*f)*(C*(c*e - b*f)*(q + 1) - c*(C*e - B*f)*(2*p + 2*q + 3)) + (p + q + 1)*(C*e*f*p*(b^2 - 4*a*c) - b*c*(C*(e^2 - 4*d*f)*(2*p + q + 2) + f*(2*C*d - B*e + 2*A*f)*(2*p + 2*q + 3))))*x + (p*( c*e - b*f)*(C*(c*e - b*f)*(q + 1) - c*(C*e - B*f)*(2*p + 2*q + 3)) + (p + q + 1)*(C*f^2*p*(b^2 - 4*a*c) - c^2*(C*(e^2 - 4*d*f)*(2*p + q + 2) + f*(2*C* d - B*e + 2*A*f)*(2*p + 2*q + 3))))*x^2, x], x], x]] /; FreeQ[{a, b, c, d, e, f, q}, x] && PolyQ[Px, x, 2] && GtQ[p, 0] && NeQ[p + q + 1, 0] && NeQ[2* p + 2*q + 3, 0] && !IGtQ[p, 0] && !IGtQ[q, 0]
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 = 3.64 (sec) , antiderivative size = 599, normalized size of antiderivative = 2.70
| method | result | size |
| trager | \(\text {Expression too large to display}\) | \(599\) |
| risch | \(\frac {\left (48000 x^{3}-106400 x^{2}+412060 x -802347\right ) \sqrt {2 x^{2}-x +3}}{240000}+\frac {7216203 \sqrt {2}\, \operatorname {arcsinh}\left (\frac {4 \sqrt {23}\, \left (x -\frac {1}{4}\right )}{23}\right )}{1600000}+\frac {121 \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 (6955 \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 )+10111 \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 )+21342849 \,\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}+993674 \,\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 )}{3003125 \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}}}\) | \(728\) |
| default | \(\text {Expression too large to display}\) | \(4860\) |
(1/5*x^3-133/300*x^2+20603/12000*x-267449/80000)*(2*x^2-x+3)^(1/2)+1/12400 000*RootOf(_Z^2+24025*RootOf(24025*_Z^4-1264358596886528*_Z^2+870315550029 38163200000000)^2-1264358596886528)*ln((43476601*RootOf(_Z^2+24025*RootOf( 24025*_Z^4-1264358596886528*_Z^2+87031555002938163200000000)^2-12643585968 86528)*RootOf(24025*_Z^4-1264358596886528*_Z^2+87031555002938163200000000) ^4*x-785358833218781184*RootOf(24025*_Z^4-1264358596886528*_Z^2+8703155500 2938163200000000)^2*RootOf(_Z^2+24025*RootOf(24025*_Z^4-1264358596886528*_ Z^2+87031555002938163200000000)^2-1264358596886528)*x-7351087056682614784* RootOf(24025*_Z^4-1264358596886528*_Z^2+87031555002938163200000000)^2*Root Of(_Z^2+24025*RootOf(24025*_Z^4-1264358596886528*_Z^2+87031555002938163200 000000)^2-1264358596886528)-177022652311416884166656000*RootOf(24025*_Z^4- 1264358596886528*_Z^2+87031555002938163200000000)^2*(2*x^2-x+3)^(1/2)-9065 0044207406572476825600000*RootOf(_Z^2+24025*RootOf(24025*_Z^4-126435859688 6528*_Z^2+87031555002938163200000000)^2-1264358596886528)*x+40856458713110 3036728934400000*RootOf(_Z^2+24025*RootOf(24025*_Z^4-1264358596886528*_Z^2 +87031555002938163200000000)^2-1264358596886528)+5143797820781846248923715 547955200000*(2*x^2-x+3)^(1/2))/(775*x*RootOf(24025*_Z^4-1264358596886528* _Z^2+87031555002938163200000000)^2-42996957921280*x-30138769768448))-1/800 00*RootOf(24025*_Z^4-1264358596886528*_Z^2+87031555002938163200000000)*ln( (-27172875625*x*RootOf(24025*_Z^4-1264358596886528*_Z^2+870315550029381...
Result contains complex when optimal does not.
Time = 0.34 (sec) , antiderivative size = 322, normalized size of antiderivative = 1.45 \[ \int \frac {\left (3-x+2 x^2\right )^{5/2}}{2+3 x+5 x^2} \, dx=\frac {1}{387500} \, \sqrt {31} \sqrt {1839524522 i \, \sqrt {31} + 4978730614} \log \left (-\frac {\sqrt {2 \, x^{2} - x + 3} {\left (27 \, \sqrt {31} + 49 i\right )} \sqrt {1839524522 i \, \sqrt {31} + 4978730614} + 756250 \, \sqrt {31} {\left (-i \, x + 6 i\right )} - 14368750 \, x + 16637500}{x}\right ) - \frac {1}{387500} \, \sqrt {31} \sqrt {1839524522 i \, \sqrt {31} + 4978730614} \log \left (\frac {\sqrt {2 \, x^{2} - x + 3} {\left (27 \, \sqrt {31} + 49 i\right )} \sqrt {1839524522 i \, \sqrt {31} + 4978730614} - 756250 \, \sqrt {31} {\left (-i \, x + 6 i\right )} + 14368750 \, x - 16637500}{x}\right ) + \frac {1}{387500} \, \sqrt {31} \sqrt {-1839524522 i \, \sqrt {31} + 4978730614} \log \left (-\frac {\sqrt {2 \, x^{2} - x + 3} {\left (27 \, \sqrt {31} - 49 i\right )} \sqrt {-1839524522 i \, \sqrt {31} + 4978730614} + 756250 \, \sqrt {31} {\left (i \, x - 6 i\right )} - 14368750 \, x + 16637500}{x}\right ) - \frac {1}{387500} \, \sqrt {31} \sqrt {-1839524522 i \, \sqrt {31} + 4978730614} \log \left (\frac {\sqrt {2 \, x^{2} - x + 3} {\left (27 \, \sqrt {31} - 49 i\right )} \sqrt {-1839524522 i \, \sqrt {31} + 4978730614} - 756250 \, \sqrt {31} {\left (i \, x - 6 i\right )} + 14368750 \, x - 16637500}{x}\right ) + \frac {1}{240000} \, {\left (48000 \, x^{3} - 106400 \, x^{2} + 412060 \, x - 802347\right )} \sqrt {2 \, x^{2} - x + 3} + \frac {7216203}{3200000} \, \sqrt {2} \log \left (-4 \, \sqrt {2} \sqrt {2 \, x^{2} - x + 3} {\left (4 \, x - 1\right )} - 32 \, x^{2} + 16 \, x - 25\right ) \]
1/387500*sqrt(31)*sqrt(1839524522*I*sqrt(31) + 4978730614)*log(-(sqrt(2*x^ 2 - x + 3)*(27*sqrt(31) + 49*I)*sqrt(1839524522*I*sqrt(31) + 4978730614) + 756250*sqrt(31)*(-I*x + 6*I) - 14368750*x + 16637500)/x) - 1/387500*sqrt( 31)*sqrt(1839524522*I*sqrt(31) + 4978730614)*log((sqrt(2*x^2 - x + 3)*(27* sqrt(31) + 49*I)*sqrt(1839524522*I*sqrt(31) + 4978730614) - 756250*sqrt(31 )*(-I*x + 6*I) + 14368750*x - 16637500)/x) + 1/387500*sqrt(31)*sqrt(-18395 24522*I*sqrt(31) + 4978730614)*log(-(sqrt(2*x^2 - x + 3)*(27*sqrt(31) - 49 *I)*sqrt(-1839524522*I*sqrt(31) + 4978730614) + 756250*sqrt(31)*(I*x - 6*I ) - 14368750*x + 16637500)/x) - 1/387500*sqrt(31)*sqrt(-1839524522*I*sqrt( 31) + 4978730614)*log((sqrt(2*x^2 - x + 3)*(27*sqrt(31) - 49*I)*sqrt(-1839 524522*I*sqrt(31) + 4978730614) - 756250*sqrt(31)*(I*x - 6*I) + 14368750*x - 16637500)/x) + 1/240000*(48000*x^3 - 106400*x^2 + 412060*x - 802347)*sq rt(2*x^2 - x + 3) + 7216203/3200000*sqrt(2)*log(-4*sqrt(2)*sqrt(2*x^2 - x + 3)*(4*x - 1) - 32*x^2 + 16*x - 25)
\[ \int \frac {\left (3-x+2 x^2\right )^{5/2}}{2+3 x+5 x^2} \, dx=\int \frac {\left (2 x^{2} - x + 3\right )^{\frac {5}{2}}}{5 x^{2} + 3 x + 2}\, dx \]
\[ \int \frac {\left (3-x+2 x^2\right )^{5/2}}{2+3 x+5 x^2} \, dx=\int { \frac {{\left (2 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{5 \, x^{2} + 3 \, x + 2} \,d x } \]
Exception generated. \[ \int \frac {\left (3-x+2 x^2\right )^{5/2}}{2+3 x+5 x^2} \, dx=\text {Exception raised: TypeError} \]
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. \[ \int \frac {\left (3-x+2 x^2\right )^{5/2}}{2+3 x+5 x^2} \, dx=\int \frac {{\left (2\,x^2-x+3\right )}^{5/2}}{5\,x^2+3\,x+2} \,d x \]