Integrand size = 27, antiderivative size = 197 \[ \int \frac {\left (3-x+2 x^2\right )^{3/2}}{2+3 x+5 x^2} \, dx=-\frac {1}{100} (49-20 x) \sqrt {3-x+2 x^2}-\frac {2203 \text {arcsinh}\left (\frac {1-4 x}{\sqrt {23}}\right )}{1000 \sqrt {2}}+\frac {11}{125} \sqrt {\frac {11}{31} \left (247+500 \sqrt {2}\right )} \arctan \left (\frac {\sqrt {\frac {11}{62 \left (247+500 \sqrt {2}\right )}} \left (8+61 \sqrt {2}+\left (130+69 \sqrt {2}\right ) x\right )}{\sqrt {3-x+2 x^2}}\right )-\frac {11}{125} \sqrt {\frac {11}{31} \left (-247+500 \sqrt {2}\right )} \text {arctanh}\left (\frac {\sqrt {\frac {11}{62 \left (-247+500 \sqrt {2}\right )}} \left (8-61 \sqrt {2}+\left (130-69 \sqrt {2}\right ) x\right )}{\sqrt {3-x+2 x^2}}\right ) \] Output:
-1/100*(49-20*x)*(2*x^2-x+3)^(1/2)-2203/2000*arcsinh(1/23*(1-4*x)*23^(1/2) )*2^(1/2)+11/3875*(84227+170500*2^(1/2))^(1/2)*arctan(11^(1/2)/(15314+3100 0*2^(1/2))^(1/2)*(8+61*2^(1/2)+(130+69*2^(1/2))*x)/(2*x^2-x+3)^(1/2))-11/3 875*(-84227+170500*2^(1/2))^(1/2)*arctanh(11^(1/2)/(-15314+31000*2^(1/2))^ (1/2)*(8-61*2^(1/2)+(130-69*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 = 0.55 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.16 \[ \int \frac {\left (3-x+2 x^2\right )^{3/2}}{2+3 x+5 x^2} \, dx=\frac {20 (-49+20 x) \sqrt {3-x+2 x^2}-2203 \sqrt {2} \log \left (1-4 x+2 \sqrt {6-2 x+4 x^2}\right )+1936 \text {RootSum}\left [-56-26 \sqrt {2} \text {$\#$1}+17 \text {$\#$1}^2+6 \sqrt {2} \text {$\#$1}^3-5 \text {$\#$1}^4\&,\frac {-36 \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right )+6 \sqrt {2} \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}+13 \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{-13 \sqrt {2}+17 \text {$\#$1}+9 \sqrt {2} \text {$\#$1}^2-10 \text {$\#$1}^3}\&\right ]}{2000} \] Input:
Integrate[(3 - x + 2*x^2)^(3/2)/(2 + 3*x + 5*x^2),x]
Output:
(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 = 0.65 (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 |
| \(\displaystyle \frac {1}{200} \left (\frac {2203 \text {arcsinh}\left (\frac {4 x-1}{\sqrt {23}}\right )}{5 \sqrt {2}}+\frac {968}{5} \left (\sqrt {\frac {1}{341} \left (247+500 \sqrt {2}\right )} \arctan \left (\frac {\sqrt {\frac {11}{62 \left (247+500 \sqrt {2}\right )}} \left (\left (130+69 \sqrt {2}\right ) x+61 \sqrt {2}+8\right )}{\sqrt {2 x^2-x+3}}\right )+\frac {\left (247-500 \sqrt {2}\right ) \text {arctanh}\left (\frac {\sqrt {\frac {11}{62 \left (500 \sqrt {2}-247\right )}} \left (\left (130-69 \sqrt {2}\right ) x-61 \sqrt {2}+8\right )}{\sqrt {2 x^2-x+3}}\right )}{\sqrt {341 \left (500 \sqrt {2}-247\right )}}\right )\right )-\frac {1}{100} (49-20 x) \sqrt {2 x^2-x+3}\) |
Input:
Int[(3 - x + 2*x^2)^(3/2)/(2 + 3*x + 5*x^2),x]
Output:
-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
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 = 5.01 (sec) , antiderivative size = 501, normalized size of antiderivative = 2.54
| method | result | size |
| trager | \(\left (-\frac {49}{100}+\frac {x}{5}\right ) \sqrt {2 x^{2}-x +3}+\frac {2203 \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 )}{2000}-\frac {\operatorname {RootOf}\left (24025 \textit {\_Z}^{4}+163063472 \textit {\_Z}^{2}+2267598080000\right ) \ln \left (\frac {-5429049375 x \operatorname {RootOf}\left (24025 \textit {\_Z}^{4}+163063472 \textit {\_Z}^{2}+2267598080000\right )^{5}-40888264630400 \operatorname {RootOf}\left (24025 \textit {\_Z}^{4}+163063472 \textit {\_Z}^{2}+2267598080000\right )^{3} x +13640929511440000 \sqrt {2 x^{2}-x +3}\, \operatorname {RootOf}\left (24025 \textit {\_Z}^{4}+163063472 \textit {\_Z}^{2}+2267598080000\right )^{2}-154372254960000 \operatorname {RootOf}\left (24025 \textit {\_Z}^{4}+163063472 \textit {\_Z}^{2}+2267598080000\right )^{3}+592661349855657984 \operatorname {RootOf}\left (24025 \textit {\_Z}^{4}+163063472 \textit {\_Z}^{2}+2267598080000\right ) x +52586627694873395200 \sqrt {2 x^{2}-x +3}-2295791036224716800 \operatorname {RootOf}\left (24025 \textit {\_Z}^{4}+163063472 \textit {\_Z}^{2}+2267598080000\right )}{775 x \operatorname {RootOf}\left (24025 \textit {\_Z}^{4}+163063472 \textit {\_Z}^{2}+2267598080000\right )^{2}+6431392 x +5068448}\right )}{100}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+24025 \operatorname {RootOf}\left (24025 \textit {\_Z}^{4}+163063472 \textit {\_Z}^{2}+2267598080000\right )^{2}+163063472\right ) \ln \left (-\frac {8686479 \operatorname {RootOf}\left (\textit {\_Z}^{2}+24025 \operatorname {RootOf}\left (24025 \textit {\_Z}^{4}+163063472 \textit {\_Z}^{2}+2267598080000\right )^{2}+163063472\right ) \operatorname {RootOf}\left (24025 \textit {\_Z}^{4}+163063472 \textit {\_Z}^{2}+2267598080000\right )^{4} x +52493234464 \operatorname {RootOf}\left (24025 \textit {\_Z}^{4}+163063472 \textit {\_Z}^{2}+2267598080000\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+24025 \operatorname {RootOf}\left (24025 \textit {\_Z}^{4}+163063472 \textit {\_Z}^{2}+2267598080000\right )^{2}+163063472\right ) x -246995607936 \operatorname {RootOf}\left (24025 \textit {\_Z}^{4}+163063472 \textit {\_Z}^{2}+2267598080000\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+24025 \operatorname {RootOf}\left (24025 \textit {\_Z}^{4}+163063472 \textit {\_Z}^{2}+2267598080000\right )^{2}+163063472\right )+3382950518837120 \sqrt {2 x^{2}-x +3}\, \operatorname {RootOf}\left (24025 \textit {\_Z}^{4}+163063472 \textit {\_Z}^{2}+2267598080000\right )^{2}-992130849952000 \operatorname {RootOf}\left (\textit {\_Z}^{2}+24025 \operatorname {RootOf}\left (24025 \textit {\_Z}^{4}+163063472 \textit {\_Z}^{2}+2267598080000\right )^{2}+163063472\right ) x +1996846869248000 \operatorname {RootOf}\left (\textit {\_Z}^{2}+24025 \operatorname {RootOf}\left (24025 \textit {\_Z}^{4}+163063472 \textit {\_Z}^{2}+2267598080000\right )^{2}+163063472\right )+9919417776240896000 \sqrt {2 x^{2}-x +3}}{775 x \operatorname {RootOf}\left (24025 \textit {\_Z}^{4}+163063472 \textit {\_Z}^{2}+2267598080000\right )^{2}-1171280 x -5068448}\right )}{15500}\) | \(501\) |
| 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 (-1+\sqrt {2}+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+\frac {3 \sqrt {2}\, \left (-1+\sqrt {2}+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+8-3 \sqrt {2}}\, \sqrt {2}\, \left (1535 \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 (-1+\sqrt {2}+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+24 \sqrt {2}-41\right )}\, \left (\frac {6485 \sqrt {2}\, \left (-1+\sqrt {2}+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+\frac {10368 \left (-1+\sqrt {2}+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+22379 \sqrt {2}+32016\right ) \left (-1+\sqrt {2}+x \right ) \left (8+3 \sqrt {2}\right )}{11692487 \left (\frac {23 \left (-1+\sqrt {2}+x \right )^{4}}{\left (\sqrt {2}+1-x \right )^{4}}+\frac {82 \left (-1+\sqrt {2}+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+23\right ) \left (\sqrt {2}+1-x \right )}\right ) \sqrt {-775687+549362 \sqrt {2}}+2197 \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 (-1+\sqrt {2}+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+24 \sqrt {2}-41\right )}\, \left (\frac {6485 \sqrt {2}\, \left (-1+\sqrt {2}+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+\frac {10368 \left (-1+\sqrt {2}+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+22379 \sqrt {2}+32016\right ) \left (-1+\sqrt {2}+x \right ) \left (8+3 \sqrt {2}\right )}{11692487 \left (\frac {23 \left (-1+\sqrt {2}+x \right )^{4}}{\left (\sqrt {2}+1-x \right )^{4}}+\frac {82 \left (-1+\sqrt {2}+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+23\right ) \left (\sqrt {2}+1-x \right )}\right ) \sqrt {-775687+549362 \sqrt {2}}+3308723 \,\operatorname {arctanh}\left (\frac {31 \sqrt {\frac {8 \left (-1+\sqrt {2}+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+\frac {3 \sqrt {2}\, \left (-1+\sqrt {2}+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 (-1+\sqrt {2}+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+\frac {3 \sqrt {2}\, \left (-1+\sqrt {2}+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 (-1+\sqrt {2}+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+\frac {3 \sqrt {2}\, \left (-1+\sqrt {2}+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+8-3 \sqrt {2}}{\left (1+\frac {-1+\sqrt {2}+x}{\sqrt {2}+1-x}\right )^{2}}}\, \left (1+\frac {-1+\sqrt {2}+x}{\sqrt {2}+1-x}\right ) \left (8+3 \sqrt {2}\right ) \sqrt {-8866+6820 \sqrt {2}}}\) | \(718\) |
| default | \(\text {Expression too large to display}\) | \(3460\) |
Input:
int((2*x^2-x+3)^(3/2)/(5*x^2+3*x+2),x,method=_RETURNVERBOSE)
Output:
(-49/100+1/5*x)*(2*x^2-x+3)^(1/2)+2203/2000*RootOf(_Z^2-2)*ln(4*RootOf(_Z^ 2-2)*x+4*(2*x^2-x+3)^(1/2)-RootOf(_Z^2-2))-1/100*RootOf(24025*_Z^4+1630634 72*_Z^2+2267598080000)*ln((-5429049375*x*RootOf(24025*_Z^4+163063472*_Z^2+ 2267598080000)^5-40888264630400*RootOf(24025*_Z^4+163063472*_Z^2+226759808 0000)^3*x+13640929511440000*(2*x^2-x+3)^(1/2)*RootOf(24025*_Z^4+163063472* _Z^2+2267598080000)^2-154372254960000*RootOf(24025*_Z^4+163063472*_Z^2+226 7598080000)^3+592661349855657984*RootOf(24025*_Z^4+163063472*_Z^2+22675980 80000)*x+52586627694873395200*(2*x^2-x+3)^(1/2)-2295791036224716800*RootOf (24025*_Z^4+163063472*_Z^2+2267598080000))/(775*x*RootOf(24025*_Z^4+163063 472*_Z^2+2267598080000)^2+6431392*x+5068448))-1/15500*RootOf(_Z^2+24025*Ro otOf(24025*_Z^4+163063472*_Z^2+2267598080000)^2+163063472)*ln(-(8686479*Ro otOf(_Z^2+24025*RootOf(24025*_Z^4+163063472*_Z^2+2267598080000)^2+16306347 2)*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+163063472*_Z^2+2267598080000)^2+163063472)*x-246995607936*RootOf(2402 5*_Z^4+163063472*_Z^2+2267598080000)^2*RootOf(_Z^2+24025*RootOf(24025*_Z^4 +163063472*_Z^2+2267598080000)^2+163063472)+3382950518837120*(2*x^2-x+3)^( 1/2)*RootOf(24025*_Z^4+163063472*_Z^2+2267598080000)^2-992130849952000*Roo tOf(_Z^2+24025*RootOf(24025*_Z^4+163063472*_Z^2+2267598080000)^2+163063472 )*x+1996846869248000*RootOf(_Z^2+24025*RootOf(24025*_Z^4+163063472*_Z^2...
Leaf count of result is larger than twice the leaf count of optimal. 542 vs. \(2 (143) = 286\).
Time = 0.10 (sec) , antiderivative size = 542, normalized size of antiderivative = 2.75 \[ \int \frac {\left (3-x+2 x^2\right )^{3/2}}{2+3 x+5 x^2} \, dx =\text {Too large to display} \] Input:
integrate((2*x^2-x+3)^(3/2)/(5*x^2+3*x+2),x, algorithm="fricas")
Output:
-11/250*sqrt(11/31)*sqrt(500*sqrt(2) + 247)*arctan(-1/1309*sqrt(11/31)*(sq rt(11/31)*(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(500*sqrt(2) - 247) + 88*(105*x^3 - 323*x^2 - sq rt(2)*(161*x^3 - 306*x^2 + 56*x + 192) - 336*x + 72)*sqrt(2*x^2 - x + 3))* sqrt(500*sqrt(2) + 247)/(343*x^4 - 400*x^3 + 1136*x^2 + 384*x - 576)) + 11 /250*sqrt(11/31)*sqrt(500*sqrt(2) + 247)*arctan(-1/1309*sqrt(11/31)*(sqrt( 11/31)*(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(500*sqrt(2) - 247) - 88*(105*x^3 - 323*x^2 - sqrt( 2)*(161*x^3 - 306*x^2 + 56*x + 192) - 336*x + 72)*sqrt(2*x^2 - x + 3))*sqr t(500*sqrt(2) + 247)/(343*x^4 - 400*x^3 + 1136*x^2 + 384*x - 576)) - 11/50 0*sqrt(11/31)*sqrt(500*sqrt(2) - 247)*log(11*(2*sqrt(11/31)*sqrt(2*x^2 - x + 3)*(sqrt(2)*(81*x - 394) + 313*x - 475)*sqrt(500*sqrt(2) - 247) + 5831* x^2 + 5236*sqrt(2)*(2*x^2 - x + 3) - 17969*x + 23800)/x^2) + 11/500*sqrt(1 1/31)*sqrt(500*sqrt(2) - 247)*log(-11*(2*sqrt(11/31)*sqrt(2*x^2 - x + 3)*( sqrt(2)*(81*x - 394) + 313*x - 475)*sqrt(500*sqrt(2) - 247) - 5831*x^2 - 5 236*sqrt(2)*(2*x^2 - x + 3) + 17969*x - 23800)/x^2) + 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)
\[ \int \frac {\left (3-x+2 x^2\right )^{3/2}}{2+3 x+5 x^2} \, dx=\int \frac {\left (2 x^{2} - x + 3\right )^{\frac {3}{2}}}{5 x^{2} + 3 x + 2}\, dx \] Input:
integrate((2*x**2-x+3)**(3/2)/(5*x**2+3*x+2),x)
Output:
Integral((2*x**2 - x + 3)**(3/2)/(5*x**2 + 3*x + 2), x)
\[ \int \frac {\left (3-x+2 x^2\right )^{3/2}}{2+3 x+5 x^2} \, dx=\int { \frac {{\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{5 \, x^{2} + 3 \, x + 2} \,d x } \] Input:
integrate((2*x^2-x+3)^(3/2)/(5*x^2+3*x+2),x, algorithm="maxima")
Output:
integrate((2*x^2 - x + 3)^(3/2)/(5*x^2 + 3*x + 2), x)
Exception generated. \[ \int \frac {\left (3-x+2 x^2\right )^{3/2}}{2+3 x+5 x^2} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((2*x^2-x+3)^(3/2)/(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. \[ \int \frac {\left (3-x+2 x^2\right )^{3/2}}{2+3 x+5 x^2} \, dx=\int \frac {{\left (2\,x^2-x+3\right )}^{3/2}}{5\,x^2+3\,x+2} \,d x \] Input:
int((2*x^2 - x + 3)^(3/2)/(3*x + 5*x^2 + 2),x)
Output:
int((2*x^2 - x + 3)^(3/2)/(3*x + 5*x^2 + 2), x)
\[ \int \frac {\left (3-x+2 x^2\right )^{3/2}}{2+3 x+5 x^2} \, dx=\frac {\sqrt {2 x^{2}-x +3}\, x}{5}-\frac {80961 \sqrt {2 x^{2}-x +3}}{62500}+\frac {143163 \sqrt {2}\, \mathrm {log}\left (-2 \sqrt {2 x^{2}-x +3}\, \sqrt {2}-4 x +1\right )}{50000}+\frac {11011 \sqrt {2}\, \mathrm {log}\left (2 \sqrt {2 x^{2}-x +3}\, \sqrt {2}-4 x +1\right )}{6250}+\frac {32791 \left (\int \frac {\sqrt {2 x^{2}-x +3}}{10 x^{4}+x^{3}+16 x^{2}+7 x +6}d x \right )}{15625}+\frac {25168 \left (\int \frac {\sqrt {2 x^{2}-x +3}\, x^{3}}{10 x^{4}+x^{3}+16 x^{2}+7 x +6}d x \right )}{3125}+\frac {44044 \left (\int \frac {\sqrt {2 x^{2}-x +3}\, x^{2}}{10 x^{4}+x^{3}+16 x^{2}+7 x +6}d x \right )}{15625}-\frac {33033 \left (\int \frac {\sqrt {2 x^{2}-x +3}\, x}{10 x^{4}+x^{3}+16 x^{2}+7 x +6}d x \right )}{3125} \] Input:
int((2*x^2-x+3)^(3/2)/(5*x^2+3*x+2),x)
Output:
(50000*sqrt(2*x**2 - x + 3)*x - 323844*sqrt(2*x**2 - x + 3) + 715815*sqrt( 2)*log( - 2*sqrt(2*x**2 - x + 3)*sqrt(2) - 4*x + 1) + 440440*sqrt(2)*log(2 *sqrt(2*x**2 - x + 3)*sqrt(2) - 4*x + 1) + 524656*int(sqrt(2*x**2 - x + 3) /(10*x**4 + x**3 + 16*x**2 + 7*x + 6),x) + 2013440*int((sqrt(2*x**2 - x + 3)*x**3)/(10*x**4 + x**3 + 16*x**2 + 7*x + 6),x) + 704704*int((sqrt(2*x**2 - x + 3)*x**2)/(10*x**4 + x**3 + 16*x**2 + 7*x + 6),x) - 2642640*int((sqr t(2*x**2 - x + 3)*x)/(10*x**4 + x**3 + 16*x**2 + 7*x + 6),x))/250000