Integrand size = 27, antiderivative size = 232 \[ \int \frac {\left (3-x+2 x^2\right )^{3/2}}{\left (2+3 x+5 x^2\right )^2} \, dx=\frac {4}{155} (4-5 x) \sqrt {3-x+2 x^2}+\frac {(3+10 x) \left (3-x+2 x^2\right )^{3/2}}{31 \left (2+3 x+5 x^2\right )}-\frac {2}{25} \sqrt {2} \text {arcsinh}\left (\frac {1-4 x}{\sqrt {23}}\right )+\frac {\sqrt {\frac {11}{31} \left (3169333+2265350 \sqrt {2}\right )} \arctan \left (\frac {\sqrt {\frac {11}{62 \left (3169333+2265350 \sqrt {2}\right )}} \left (3514+2963 \sqrt {2}+\left (9440+6477 \sqrt {2}\right ) x\right )}{\sqrt {3-x+2 x^2}}\right )}{1550}-\frac {\sqrt {\frac {11}{31} \left (-3169333+2265350 \sqrt {2}\right )} \text {arctanh}\left (\frac {\sqrt {\frac {11}{62 \left (-3169333+2265350 \sqrt {2}\right )}} \left (3514-2963 \sqrt {2}+\left (9440-6477 \sqrt {2}\right ) x\right )}{\sqrt {3-x+2 x^2}}\right )}{1550} \]
1/31*(3+10*x)*(2*x^2-x+3)^(3/2)/(5*x^2+3*x+2)-2/25*arcsinh(1/23*(1-4*x)*23 ^(1/2))*2^(1/2)+4/155*(4-5*x)*(2*x^2-x+3)^(1/2)-1/48050*arctanh(1/62*(3514 +x*(9440-6477*2^(1/2))-2963*2^(1/2))*682^(1/2)/(-3169333+2265350*2^(1/2))^ (1/2)/(2*x^2-x+3)^(1/2))*(-1080742553+772484350*2^(1/2))^(1/2)+1/48050*arc tan(1/62*(3514+2963*2^(1/2)+x*(9440+6477*2^(1/2)))*682^(1/2)/(3169333+2265 350*2^(1/2))^(1/2)/(2*x^2-x+3)^(1/2))*(1080742553+772484350*2^(1/2))^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.51 (sec) , antiderivative size = 416, normalized size of antiderivative = 1.79 \[ \int \frac {\left (3-x+2 x^2\right )^{3/2}}{\left (2+3 x+5 x^2\right )^2} \, dx=\frac {50 \left (\frac {55 (7+13 x) \sqrt {3-x+2 x^2}}{2+3 x+5 x^2}-62 \sqrt {2} \log \left (1-4 x+2 \sqrt {6-2 x+4 x^2}\right )\right )+682 \text {RootSum}\left [-56-26 \sqrt {2} \text {$\#$1}+17 \text {$\#$1}^2+6 \sqrt {2} \text {$\#$1}^3-5 \text {$\#$1}^4\&,\frac {999 \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right )+310 \sqrt {2} \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}+100 \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{-13 \sqrt {2}+17 \text {$\#$1}+9 \sqrt {2} \text {$\#$1}^2-10 \text {$\#$1}^3}\&\right ]+11 \text {RootSum}\left [-56-26 \sqrt {2} \text {$\#$1}+17 \text {$\#$1}^2+6 \sqrt {2} \text {$\#$1}^3-5 \text {$\#$1}^4\&,\frac {-72888 \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right )+8230 \sqrt {2} \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}+2025 \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{-13 \sqrt {2}+17 \text {$\#$1}+9 \sqrt {2} \text {$\#$1}^2-10 \text {$\#$1}^3}\&\right ]}{38750} \]
(50*((55*(7 + 13*x)*Sqrt[3 - x + 2*x^2])/(2 + 3*x + 5*x^2) - 62*Sqrt[2]*Lo g[1 - 4*x + 2*Sqrt[6 - 2*x + 4*x^2]]) + 682*RootSum[-56 - 26*Sqrt[2]*#1 + 17*#1^2 + 6*Sqrt[2]*#1^3 - 5*#1^4 & , (999*Log[-(Sqrt[2]*x) + Sqrt[3 - x + 2*x^2] - #1] + 310*Sqrt[2]*Log[-(Sqrt[2]*x) + Sqrt[3 - x + 2*x^2] - #1]*# 1 + 100*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) & ] + 11*RootSum[-56 - 26*Sqrt[2]*#1 + 1 7*#1^2 + 6*Sqrt[2]*#1^3 - 5*#1^4 & , (-72888*Log[-(Sqrt[2]*x) + Sqrt[3 - x + 2*x^2] - #1] + 8230*Sqrt[2]*Log[-(Sqrt[2]*x) + Sqrt[3 - x + 2*x^2] - #1 ]*#1 + 2025*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) & ])/38750
Time = 0.78 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.07, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.481, Rules used = {1302, 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.
| \(\displaystyle \int \frac {\left (2 x^2-x+3\right )^{3/2}}{\left (5 x^2+3 x+2\right )^2} \, dx\) |
| \(\Big \downarrow \) 1302 |
| \(\displaystyle \frac {(10 x+3) \left (2 x^2-x+3\right )^{3/2}}{31 \left (5 x^2+3 x+2\right )}-\frac {1}{31} \int -\frac {\left (-80 x^2-26 x+69\right ) \sqrt {2 x^2-x+3}}{2 \left (5 x^2+3 x+2\right )}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{62} \int \frac {\left (-80 x^2-26 x+69\right ) \sqrt {2 x^2-x+3}}{5 x^2+3 x+2}dx+\frac {(10 x+3) \left (2 x^2-x+3\right )^{3/2}}{31 \left (5 x^2+3 x+2\right )}\) |
| \(\Big \downarrow \) 2138 |
| \(\displaystyle \frac {1}{62} \left (\frac {8}{5} (4-5 x) \sqrt {2 x^2-x+3}-\frac {1}{100} \int -\frac {20 \left (248 x^2-575 x+1307\right )}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx\right )+\frac {(10 x+3) \left (2 x^2-x+3\right )^{3/2}}{31 \left (5 x^2+3 x+2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{62} \left (\frac {1}{5} \int \frac {248 x^2-575 x+1307}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx+\frac {8}{5} \sqrt {2 x^2-x+3} (4-5 x)\right )+\frac {(10 x+3) \left (2 x^2-x+3\right )^{3/2}}{31 \left (5 x^2+3 x+2\right )}\) |
| \(\Big \downarrow \) 2143 |
| \(\displaystyle \frac {1}{62} \left (\frac {1}{5} \left (\frac {248}{5} \int \frac {1}{\sqrt {2 x^2-x+3}}dx+\frac {1}{5} \int \frac {11 (549-329 x)}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx\right )+\frac {8}{5} \sqrt {2 x^2-x+3} (4-5 x)\right )+\frac {(10 x+3) \left (2 x^2-x+3\right )^{3/2}}{31 \left (5 x^2+3 x+2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{62} \left (\frac {1}{5} \left (\frac {248}{5} \int \frac {1}{\sqrt {2 x^2-x+3}}dx+\frac {11}{5} \int \frac {549-329 x}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx\right )+\frac {8}{5} \sqrt {2 x^2-x+3} (4-5 x)\right )+\frac {(10 x+3) \left (2 x^2-x+3\right )^{3/2}}{31 \left (5 x^2+3 x+2\right )}\) |
| \(\Big \downarrow \) 1090 |
| \(\displaystyle \frac {1}{62} \left (\frac {1}{5} \left (\frac {11}{5} \int \frac {549-329 x}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx+\frac {124}{5} \sqrt {\frac {2}{23}} \int \frac {1}{\sqrt {\frac {1}{23} (4 x-1)^2+1}}d(4 x-1)\right )+\frac {8}{5} \sqrt {2 x^2-x+3} (4-5 x)\right )+\frac {(10 x+3) \left (2 x^2-x+3\right )^{3/2}}{31 \left (5 x^2+3 x+2\right )}\) |
| \(\Big \downarrow \) 222 |
| \(\displaystyle \frac {1}{62} \left (\frac {1}{5} \left (\frac {11}{5} \int \frac {549-329 x}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx+\frac {124}{5} \sqrt {2} \text {arcsinh}\left (\frac {4 x-1}{\sqrt {23}}\right )\right )+\frac {8}{5} \sqrt {2 x^2-x+3} (4-5 x)\right )+\frac {(10 x+3) \left (2 x^2-x+3\right )^{3/2}}{31 \left (5 x^2+3 x+2\right )}\) |
| \(\Big \downarrow \) 1368 |
| \(\displaystyle \frac {1}{62} \left (\frac {1}{5} \left (\frac {11}{5} \left (\frac {\int -\frac {11 \left (-\left (\left (220-329 \sqrt {2}\right ) x\right )-549 \sqrt {2}+878\right )}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx}{22 \sqrt {2}}-\frac {\int -\frac {11 \left (-\left (\left (220+329 \sqrt {2}\right ) x\right )+549 \sqrt {2}+878\right )}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx}{22 \sqrt {2}}\right )+\frac {124}{5} \sqrt {2} \text {arcsinh}\left (\frac {4 x-1}{\sqrt {23}}\right )\right )+\frac {8}{5} \sqrt {2 x^2-x+3} (4-5 x)\right )+\frac {(10 x+3) \left (2 x^2-x+3\right )^{3/2}}{31 \left (5 x^2+3 x+2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{62} \left (\frac {1}{5} \left (\frac {11}{5} \left (\frac {\int \frac {-\left (\left (220+329 \sqrt {2}\right ) x\right )+549 \sqrt {2}+878}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx}{2 \sqrt {2}}-\frac {\int \frac {-\left (\left (220-329 \sqrt {2}\right ) x\right )-549 \sqrt {2}+878}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx}{2 \sqrt {2}}\right )+\frac {124}{5} \sqrt {2} \text {arcsinh}\left (\frac {4 x-1}{\sqrt {23}}\right )\right )+\frac {8}{5} \sqrt {2 x^2-x+3} (4-5 x)\right )+\frac {(10 x+3) \left (2 x^2-x+3\right )^{3/2}}{31 \left (5 x^2+3 x+2\right )}\) |
| \(\Big \downarrow \) 1362 |
| \(\displaystyle \frac {1}{62} \left (\frac {1}{5} \left (\frac {11}{5} \left (\sqrt {2} \left (3169333-2265350 \sqrt {2}\right ) \int \frac {1}{-\frac {11 \left (\left (9440-6477 \sqrt {2}\right ) x-2963 \sqrt {2}+3514\right )^2}{2 x^2-x+3}-62 \left (3169333-2265350 \sqrt {2}\right )}d\frac {\left (9440-6477 \sqrt {2}\right ) x-2963 \sqrt {2}+3514}{\sqrt {2 x^2-x+3}}-\sqrt {2} \left (3169333+2265350 \sqrt {2}\right ) \int \frac {1}{-\frac {11 \left (\left (9440+6477 \sqrt {2}\right ) x+2963 \sqrt {2}+3514\right )^2}{2 x^2-x+3}-62 \left (3169333+2265350 \sqrt {2}\right )}d\frac {\left (9440+6477 \sqrt {2}\right ) x+2963 \sqrt {2}+3514}{\sqrt {2 x^2-x+3}}\right )+\frac {124}{5} \sqrt {2} \text {arcsinh}\left (\frac {4 x-1}{\sqrt {23}}\right )\right )+\frac {8}{5} \sqrt {2 x^2-x+3} (4-5 x)\right )+\frac {(10 x+3) \left (2 x^2-x+3\right )^{3/2}}{31 \left (5 x^2+3 x+2\right )}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{62} \left (\frac {1}{5} \left (\frac {11}{5} \left (\sqrt {2} \left (3169333-2265350 \sqrt {2}\right ) \int \frac {1}{-\frac {11 \left (\left (9440-6477 \sqrt {2}\right ) x-2963 \sqrt {2}+3514\right )^2}{2 x^2-x+3}-62 \left (3169333-2265350 \sqrt {2}\right )}d\frac {\left (9440-6477 \sqrt {2}\right ) x-2963 \sqrt {2}+3514}{\sqrt {2 x^2-x+3}}+\sqrt {\frac {1}{341} \left (3169333+2265350 \sqrt {2}\right )} \arctan \left (\frac {\sqrt {\frac {11}{62 \left (3169333+2265350 \sqrt {2}\right )}} \left (\left (9440+6477 \sqrt {2}\right ) x+2963 \sqrt {2}+3514\right )}{\sqrt {2 x^2-x+3}}\right )\right )+\frac {124}{5} \sqrt {2} \text {arcsinh}\left (\frac {4 x-1}{\sqrt {23}}\right )\right )+\frac {8}{5} \sqrt {2 x^2-x+3} (4-5 x)\right )+\frac {(10 x+3) \left (2 x^2-x+3\right )^{3/2}}{31 \left (5 x^2+3 x+2\right )}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{62} \left (\frac {1}{5} \left (\frac {124}{5} \sqrt {2} \text {arcsinh}\left (\frac {4 x-1}{\sqrt {23}}\right )+\frac {11}{5} \left (\sqrt {\frac {1}{341} \left (3169333+2265350 \sqrt {2}\right )} \arctan \left (\frac {\sqrt {\frac {11}{62 \left (3169333+2265350 \sqrt {2}\right )}} \left (\left (9440+6477 \sqrt {2}\right ) x+2963 \sqrt {2}+3514\right )}{\sqrt {2 x^2-x+3}}\right )+\frac {\left (3169333-2265350 \sqrt {2}\right ) \text {arctanh}\left (\frac {\sqrt {\frac {11}{62 \left (2265350 \sqrt {2}-3169333\right )}} \left (\left (9440-6477 \sqrt {2}\right ) x-2963 \sqrt {2}+3514\right )}{\sqrt {2 x^2-x+3}}\right )}{\sqrt {341 \left (2265350 \sqrt {2}-3169333\right )}}\right )\right )+\frac {8}{5} \sqrt {2 x^2-x+3} (4-5 x)\right )+\frac {(10 x+3) \left (2 x^2-x+3\right )^{3/2}}{31 \left (5 x^2+3 x+2\right )}\) |
((3 + 10*x)*(3 - x + 2*x^2)^(3/2))/(31*(2 + 3*x + 5*x^2)) + ((8*(4 - 5*x)* Sqrt[3 - x + 2*x^2])/5 + ((124*Sqrt[2]*ArcSinh[(-1 + 4*x)/Sqrt[23]])/5 + ( 11*(Sqrt[(3169333 + 2265350*Sqrt[2])/341]*ArcTan[(Sqrt[11/(62*(3169333 + 2 265350*Sqrt[2]))]*(3514 + 2963*Sqrt[2] + (9440 + 6477*Sqrt[2])*x))/Sqrt[3 - x + 2*x^2]] + ((3169333 - 2265350*Sqrt[2])*ArcTanh[(Sqrt[11/(62*(-316933 3 + 2265350*Sqrt[2]))]*(3514 - 2963*Sqrt[2] + (9440 - 6477*Sqrt[2])*x))/Sq rt[3 - x + 2*x^2]])/Sqrt[341*(-3169333 + 2265350*Sqrt[2])]))/5)/5)/62
3.1.70.3.1 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 + 2*c*x)*(a + b*x + c*x^2)^(p + 1)*((d + e *x + f*x^2)^q/((b^2 - 4*a*c)*(p + 1))), x] - Simp[1/((b^2 - 4*a*c)*(p + 1)) Int[(a + b*x + c*x^2)^(p + 1)*(d + e*x + f*x^2)^(q - 1)*Simp[2*c*d*(2*p + 3) + b*e*q + (2*b*f*q + 2*c*e*(2*p + q + 3))*x + 2*c*f*(2*p + 2*q + 3)*x^ 2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ [e^2 - 4*d*f, 0] && LtQ[p, -1] && GtQ[q, 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 = 2.40 (sec) , antiderivative size = 511, normalized size of antiderivative = 2.20
| method | result | size |
| trager | \(\text {Expression too large to display}\) | \(511\) |
| risch | \(\frac {11 \left (7+13 x \right ) \sqrt {2 x^{2}-x +3}}{155 \left (5 x^{2}+3 x +2\right )}+\frac {2 \sqrt {2}\, \operatorname {arcsinh}\left (\frac {4 \sqrt {23}\, \left (x -\frac {1}{4}\right )}{23}\right )}{25}+\frac {\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 (126130 \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 )+178601 \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 )+193755859 \,\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}-248376216 \,\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 )}{1489550 \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}}}\) | \(730\) |
| default | \(\text {Expression too large to display}\) | \(28185\) |
11/155*(7+13*x)/(5*x^2+3*x+2)*(2*x^2-x+3)^(1/2)-2/25*RootOf(_Z^2-2)*ln(-4* RootOf(_Z^2-2)*x+4*(2*x^2-x+3)^(1/2)+RootOf(_Z^2-2))+2/155*RootOf(3075200* _Z^4+8645940424*_Z^2+6209490853225)*ln((41069142240000*x*RootOf(3075200*_Z ^4+8645940424*_Z^2+6209490853225)^5+167323715982737600*RootOf(3075200*_Z^4 +8645940424*_Z^2+6209490853225)^3*x+26624410668240000*RootOf(3075200*_Z^4+ 8645940424*_Z^2+6209490853225)^3+4099071357292219000*RootOf(3075200*_Z^4+8 645940424*_Z^2+6209490853225)^2*(2*x^2-x+3)^(1/2)+138906735546068157756*Ro otOf(3075200*_Z^4+8645940424*_Z^2+6209490853225)*x+77561425919598618800*Ro otOf(3075200*_Z^4+8645940424*_Z^2+6209490853225)+5805404150143488427405*(2 *x^2-x+3)^(1/2))/(6200*x*RootOf(3075200*_Z^4+8645940424*_Z^2+6209490853225 )^2+9409004*x+924451))+1/48050*RootOf(_Z^2+384400*RootOf(3075200*_Z^4+8645 940424*_Z^2+6209490853225)^2+1080742553)*ln((65710627584*RootOf(_Z^2+38440 0*RootOf(3075200*_Z^4+8645940424*_Z^2+6209490853225)^2+1080742553)*RootOf( 3075200*_Z^4+8645940424*_Z^2+6209490853225)^4*x+101773581037216*RootOf(307 5200*_Z^4+8645940424*_Z^2+6209490853225)^2*RootOf(_Z^2+384400*RootOf(30752 00*_Z^4+8645940424*_Z^2+6209490853225)^2+1080742553)*x-4066278786433881248 *RootOf(3075200*_Z^4+8645940424*_Z^2+6209490853225)^2*(2*x^2-x+3)^(1/2)-42 599057069184*RootOf(3075200*_Z^4+8645940424*_Z^2+6209490853225)^2*RootOf(_ Z^2+384400*RootOf(3075200*_Z^4+8645940424*_Z^2+6209490853225)^2+1080742553 )-11025935123814325*RootOf(_Z^2+384400*RootOf(3075200*_Z^4+8645940424*_...
Result contains complex when optimal does not.
Time = 0.30 (sec) , antiderivative size = 384, normalized size of antiderivative = 1.66 \[ \int \frac {\left (3-x+2 x^2\right )^{3/2}}{\left (2+3 x+5 x^2\right )^2} \, dx=\frac {\sqrt {62} {\left (5 \, x^{2} + 3 \, x + 2\right )} \sqrt {924451 i \, \sqrt {31} - 34862663} \log \left (\frac {\sqrt {62} \sqrt {2 \, x^{2} - x + 3} \sqrt {924451 i \, \sqrt {31} - 34862663} {\left (1757 i \, \sqrt {31} - 13609\right )} - 35112925 \, \sqrt {31} {\left (-i \, x + 6 i\right )} + 667145575 \, x - 772484350}{x}\right ) - \sqrt {62} {\left (5 \, x^{2} + 3 \, x + 2\right )} \sqrt {924451 i \, \sqrt {31} - 34862663} \log \left (\frac {\sqrt {62} \sqrt {2 \, x^{2} - x + 3} \sqrt {924451 i \, \sqrt {31} - 34862663} {\left (-1757 i \, \sqrt {31} + 13609\right )} - 35112925 \, \sqrt {31} {\left (-i \, x + 6 i\right )} + 667145575 \, x - 772484350}{x}\right ) - \sqrt {62} {\left (5 \, x^{2} + 3 \, x + 2\right )} \sqrt {-924451 i \, \sqrt {31} - 34862663} \log \left (\frac {\sqrt {62} \sqrt {2 \, x^{2} - x + 3} {\left (1757 i \, \sqrt {31} + 13609\right )} \sqrt {-924451 i \, \sqrt {31} - 34862663} - 35112925 \, \sqrt {31} {\left (i \, x - 6 i\right )} + 667145575 \, x - 772484350}{x}\right ) + \sqrt {62} {\left (5 \, x^{2} + 3 \, x + 2\right )} \sqrt {-924451 i \, \sqrt {31} - 34862663} \log \left (\frac {\sqrt {62} \sqrt {2 \, x^{2} - x + 3} {\left (-1757 i \, \sqrt {31} - 13609\right )} \sqrt {-924451 i \, \sqrt {31} - 34862663} - 35112925 \, \sqrt {31} {\left (i \, x - 6 i\right )} + 667145575 \, x - 772484350}{x}\right ) + 7688 \, \sqrt {2} {\left (5 \, x^{2} + 3 \, x + 2\right )} \log \left (-4 \, \sqrt {2} \sqrt {2 \, x^{2} - x + 3} {\left (4 \, x - 1\right )} - 32 \, x^{2} + 16 \, x - 25\right ) + 13640 \, \sqrt {2 \, x^{2} - x + 3} {\left (13 \, x + 7\right )}}{192200 \, {\left (5 \, x^{2} + 3 \, x + 2\right )}} \]
1/192200*(sqrt(62)*(5*x^2 + 3*x + 2)*sqrt(924451*I*sqrt(31) - 34862663)*lo g((sqrt(62)*sqrt(2*x^2 - x + 3)*sqrt(924451*I*sqrt(31) - 34862663)*(1757*I *sqrt(31) - 13609) - 35112925*sqrt(31)*(-I*x + 6*I) + 667145575*x - 772484 350)/x) - sqrt(62)*(5*x^2 + 3*x + 2)*sqrt(924451*I*sqrt(31) - 34862663)*lo g((sqrt(62)*sqrt(2*x^2 - x + 3)*sqrt(924451*I*sqrt(31) - 34862663)*(-1757* I*sqrt(31) + 13609) - 35112925*sqrt(31)*(-I*x + 6*I) + 667145575*x - 77248 4350)/x) - sqrt(62)*(5*x^2 + 3*x + 2)*sqrt(-924451*I*sqrt(31) - 34862663)* log((sqrt(62)*sqrt(2*x^2 - x + 3)*(1757*I*sqrt(31) + 13609)*sqrt(-924451*I *sqrt(31) - 34862663) - 35112925*sqrt(31)*(I*x - 6*I) + 667145575*x - 7724 84350)/x) + sqrt(62)*(5*x^2 + 3*x + 2)*sqrt(-924451*I*sqrt(31) - 34862663) *log((sqrt(62)*sqrt(2*x^2 - x + 3)*(-1757*I*sqrt(31) - 13609)*sqrt(-924451 *I*sqrt(31) - 34862663) - 35112925*sqrt(31)*(I*x - 6*I) + 667145575*x - 77 2484350)/x) + 7688*sqrt(2)*(5*x^2 + 3*x + 2)*log(-4*sqrt(2)*sqrt(2*x^2 - x + 3)*(4*x - 1) - 32*x^2 + 16*x - 25) + 13640*sqrt(2*x^2 - x + 3)*(13*x + 7))/(5*x^2 + 3*x + 2)
\[ \int \frac {\left (3-x+2 x^2\right )^{3/2}}{\left (2+3 x+5 x^2\right )^2} \, dx=\int \frac {\left (2 x^{2} - x + 3\right )^{\frac {3}{2}}}{\left (5 x^{2} + 3 x + 2\right )^{2}}\, dx \]
\[ \int \frac {\left (3-x+2 x^2\right )^{3/2}}{\left (2+3 x+5 x^2\right )^2} \, dx=\int { \frac {{\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{{\left (5 \, x^{2} + 3 \, x + 2\right )}^{2}} \,d x } \]
Exception generated. \[ \int \frac {\left (3-x+2 x^2\right )^{3/2}}{\left (2+3 x+5 x^2\right )^2} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro unding error%%%{174900625,[8]%%%}+%%%{%%{[-419761500,0]:[1,0,-2]%%},[7]%%% }+%%%{-68
Timed out. \[ \int \frac {\left (3-x+2 x^2\right )^{3/2}}{\left (2+3 x+5 x^2\right )^2} \, dx=\int \frac {{\left (2\,x^2-x+3\right )}^{3/2}}{{\left (5\,x^2+3\,x+2\right )}^2} \,d x \]