Integrand size = 27, antiderivative size = 223 \[ \int \frac {\left (3-x+2 x^2\right )^{3/2}}{\left (2+3 x+5 x^2\right )^3} \, dx=\frac {(3+10 x) \left (3-x+2 x^2\right )^{3/2}}{62 \left (2+3 x+5 x^2\right )^2}+\frac {3 (277+696 x) \sqrt {3-x+2 x^2}}{3844 \left (2+3 x+5 x^2\right )}+\frac {3 \sqrt {\frac {1}{682} \left (366990269+259509026 \sqrt {2}\right )} \arctan \left (\frac {\sqrt {\frac {11}{31 \left (366990269+259509026 \sqrt {2}\right )}} \left (29367+20575 \sqrt {2}+\left (70517+49942 \sqrt {2}\right ) x\right )}{\sqrt {3-x+2 x^2}}\right )}{7688}-\frac {3 \sqrt {\frac {1}{682} \left (-366990269+259509026 \sqrt {2}\right )} \text {arctanh}\left (\frac {\sqrt {\frac {11}{31 \left (-366990269+259509026 \sqrt {2}\right )}} \left (29367-20575 \sqrt {2}+\left (70517-49942 \sqrt {2}\right ) x\right )}{\sqrt {3-x+2 x^2}}\right )}{7688} \] Output:
1/62*(3+10*x)*(2*x^2-x+3)^(3/2)/(5*x^2+3*x+2)^2+3*(277+696*x)*(2*x^2-x+3)^ (1/2)/(19220*x^2+11532*x+7688)+3/5243216*(250287363458+176985155732*2^(1/2 ))^(1/2)*arctan(11^(1/2)/(11376698339+8044779806*2^(1/2))^(1/2)*(29367+205 75*2^(1/2)+(70517+49942*2^(1/2))*x)/(2*x^2-x+3)^(1/2))-3/5243216*(-2502873 63458+176985155732*2^(1/2))^(1/2)*arctanh(11^(1/2)/(-11376698339+804477980 6*2^(1/2))^(1/2)*(29367-20575*2^(1/2)+(70517-49942*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 = 1.24 (sec) , antiderivative size = 572, normalized size of antiderivative = 2.57 \[ \int \frac {\left (3-x+2 x^2\right )^{3/2}}{\left (2+3 x+5 x^2\right )^3} \, dx=\frac {\frac {3306250 \sqrt {3-x+2 x^2} \left (2220+8343 x+10171 x^2+11680 x^3\right )}{\left (2+3 x+5 x^2\right )^2}-42578694225 \text {RootSum}\left [-56-26 \sqrt {2} \text {$\#$1}+17 \text {$\#$1}^2+6 \sqrt {2} \text {$\#$1}^3-5 \text {$\#$1}^4\&,\frac {\log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right )}{-13 \sqrt {2}+17 \text {$\#$1}+9 \sqrt {2} \text {$\#$1}^2-10 \text {$\#$1}^3}\&\right ]+406695200 \text {RootSum}\left [-56-26 \sqrt {2} \text {$\#$1}+17 \text {$\#$1}^2+6 \sqrt {2} \text {$\#$1}^3-5 \text {$\#$1}^4\&,\frac {93 \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right )+10 \sqrt {2} \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}}{-13 \sqrt {2}+17 \text {$\#$1}+9 \sqrt {2} \text {$\#$1}^2-10 \text {$\#$1}^3}\&\right ]+14 \text {RootSum}\left [-56-26 \sqrt {2} \text {$\#$1}+17 \text {$\#$1}^2+6 \sqrt {2} \text {$\#$1}^3-5 \text {$\#$1}^4\&,\frac {4926449381 \sqrt {2} \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}-2660991465 \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{-13 \sqrt {2}+17 \text {$\#$1}+9 \sqrt {2} \text {$\#$1}^2-10 \text {$\#$1}^3}\&\right ]-186 \text {RootSum}\left [-56-26 \sqrt {2} \text {$\#$1}+17 \text {$\#$1}^2+6 \sqrt {2} \text {$\#$1}^3-5 \text {$\#$1}^4\&,\frac {155209944 \sqrt {2} \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}-248390285 \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{-13 \sqrt {2}+17 \text {$\#$1}+9 \sqrt {2} \text {$\#$1}^2-10 \text {$\#$1}^3}\&\right ]}{12709225000} \] Input:
Integrate[(3 - x + 2*x^2)^(3/2)/(2 + 3*x + 5*x^2)^3,x]
Output:
((3306250*Sqrt[3 - x + 2*x^2]*(2220 + 8343*x + 10171*x^2 + 11680*x^3))/(2 + 3*x + 5*x^2)^2 - 42578694225*RootSum[-56 - 26*Sqrt[2]*#1 + 17*#1^2 + 6*S qrt[2]*#1^3 - 5*#1^4 & , Log[-(Sqrt[2]*x) + Sqrt[3 - x + 2*x^2] - #1]/(-13 *Sqrt[2] + 17*#1 + 9*Sqrt[2]*#1^2 - 10*#1^3) & ] + 406695200*RootSum[-56 - 26*Sqrt[2]*#1 + 17*#1^2 + 6*Sqrt[2]*#1^3 - 5*#1^4 & , (93*Log[-(Sqrt[2]*x ) + Sqrt[3 - x + 2*x^2] - #1] + 10*Sqrt[2]*Log[-(Sqrt[2]*x) + Sqrt[3 - x + 2*x^2] - #1]*#1)/(-13*Sqrt[2] + 17*#1 + 9*Sqrt[2]*#1^2 - 10*#1^3) & ] + 1 4*RootSum[-56 - 26*Sqrt[2]*#1 + 17*#1^2 + 6*Sqrt[2]*#1^3 - 5*#1^4 & , (492 6449381*Sqrt[2]*Log[-(Sqrt[2]*x) + Sqrt[3 - x + 2*x^2] - #1]*#1 - 26609914 65*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) & ] - 186*RootSum[-56 - 26*Sqrt[2]*#1 + 17*#1 ^2 + 6*Sqrt[2]*#1^3 - 5*#1^4 & , (155209944*Sqrt[2]*Log[-(Sqrt[2]*x) + Sqr t[3 - x + 2*x^2] - #1]*#1 - 248390285*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) & ])/12709 225000
Time = 0.59 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.05, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1302, 27, 1346, 27, 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.
| \(\displaystyle \int \frac {\left (2 x^2-x+3\right )^{3/2}}{\left (5 x^2+3 x+2\right )^3} \, dx\) |
| \(\Big \downarrow \) 1302 |
| \(\displaystyle \frac {(10 x+3) \left (2 x^2-x+3\right )^{3/2}}{62 \left (5 x^2+3 x+2\right )^2}-\frac {1}{62} \int -\frac {3 (63-22 x) \sqrt {2 x^2-x+3}}{2 \left (5 x^2+3 x+2\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{124} \int \frac {(63-22 x) \sqrt {2 x^2-x+3}}{\left (5 x^2+3 x+2\right )^2}dx+\frac {(10 x+3) \left (2 x^2-x+3\right )^{3/2}}{62 \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 1346 |
| \(\displaystyle \frac {3}{124} \left (\frac {(696 x+277) \sqrt {2 x^2-x+3}}{31 \left (5 x^2+3 x+2\right )}-\frac {1}{31} \int -\frac {4453-1804 x}{2 \sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx\right )+\frac {(10 x+3) \left (2 x^2-x+3\right )^{3/2}}{62 \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{124} \left (\frac {1}{62} \int \frac {4453-1804 x}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx+\frac {\sqrt {2 x^2-x+3} (696 x+277)}{31 \left (5 x^2+3 x+2\right )}\right )+\frac {(10 x+3) \left (2 x^2-x+3\right )^{3/2}}{62 \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 1368 |
| \(\displaystyle \frac {3}{124} \left (\frac {1}{62} \left (\frac {\int -\frac {11 \left (-\left (\left (2649-1804 \sqrt {2}\right ) x\right )-4453 \sqrt {2}+6257\right )}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx}{22 \sqrt {2}}-\frac {\int -\frac {11 \left (-\left (\left (2649+1804 \sqrt {2}\right ) x\right )+4453 \sqrt {2}+6257\right )}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx}{22 \sqrt {2}}\right )+\frac {\sqrt {2 x^2-x+3} (696 x+277)}{31 \left (5 x^2+3 x+2\right )}\right )+\frac {(10 x+3) \left (2 x^2-x+3\right )^{3/2}}{62 \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{124} \left (\frac {1}{62} \left (\frac {\int \frac {-\left (\left (2649+1804 \sqrt {2}\right ) x\right )+4453 \sqrt {2}+6257}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx}{2 \sqrt {2}}-\frac {\int \frac {-\left (\left (2649-1804 \sqrt {2}\right ) x\right )-4453 \sqrt {2}+6257}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx}{2 \sqrt {2}}\right )+\frac {\sqrt {2 x^2-x+3} (696 x+277)}{31 \left (5 x^2+3 x+2\right )}\right )+\frac {(10 x+3) \left (2 x^2-x+3\right )^{3/2}}{62 \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 1362 |
| \(\displaystyle \frac {3}{124} \left (\frac {1}{62} \left (\frac {\left (366990269-259509026 \sqrt {2}\right ) \int \frac {1}{-\frac {11 \left (\left (70517-49942 \sqrt {2}\right ) x-20575 \sqrt {2}+29367\right )^2}{2 x^2-x+3}-31 \left (366990269-259509026 \sqrt {2}\right )}d\frac {\left (70517-49942 \sqrt {2}\right ) x-20575 \sqrt {2}+29367}{\sqrt {2 x^2-x+3}}}{\sqrt {2}}-\frac {\left (366990269+259509026 \sqrt {2}\right ) \int \frac {1}{-\frac {11 \left (\left (70517+49942 \sqrt {2}\right ) x+20575 \sqrt {2}+29367\right )^2}{2 x^2-x+3}-31 \left (366990269+259509026 \sqrt {2}\right )}d\frac {\left (70517+49942 \sqrt {2}\right ) x+20575 \sqrt {2}+29367}{\sqrt {2 x^2-x+3}}}{\sqrt {2}}\right )+\frac {\sqrt {2 x^2-x+3} (696 x+277)}{31 \left (5 x^2+3 x+2\right )}\right )+\frac {(10 x+3) \left (2 x^2-x+3\right )^{3/2}}{62 \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {3}{124} \left (\frac {1}{62} \left (\frac {\left (366990269-259509026 \sqrt {2}\right ) \int \frac {1}{-\frac {11 \left (\left (70517-49942 \sqrt {2}\right ) x-20575 \sqrt {2}+29367\right )^2}{2 x^2-x+3}-31 \left (366990269-259509026 \sqrt {2}\right )}d\frac {\left (70517-49942 \sqrt {2}\right ) x-20575 \sqrt {2}+29367}{\sqrt {2 x^2-x+3}}}{\sqrt {2}}+\sqrt {\frac {1}{682} \left (366990269+259509026 \sqrt {2}\right )} \arctan \left (\frac {\sqrt {\frac {11}{31 \left (366990269+259509026 \sqrt {2}\right )}} \left (\left (70517+49942 \sqrt {2}\right ) x+20575 \sqrt {2}+29367\right )}{\sqrt {2 x^2-x+3}}\right )\right )+\frac {\sqrt {2 x^2-x+3} (696 x+277)}{31 \left (5 x^2+3 x+2\right )}\right )+\frac {(10 x+3) \left (2 x^2-x+3\right )^{3/2}}{62 \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {3}{124} \left (\frac {1}{62} \left (\sqrt {\frac {1}{682} \left (366990269+259509026 \sqrt {2}\right )} \arctan \left (\frac {\sqrt {\frac {11}{31 \left (366990269+259509026 \sqrt {2}\right )}} \left (\left (70517+49942 \sqrt {2}\right ) x+20575 \sqrt {2}+29367\right )}{\sqrt {2 x^2-x+3}}\right )+\frac {\left (366990269-259509026 \sqrt {2}\right ) \text {arctanh}\left (\frac {\sqrt {\frac {11}{31 \left (259509026 \sqrt {2}-366990269\right )}} \left (\left (70517-49942 \sqrt {2}\right ) x-20575 \sqrt {2}+29367\right )}{\sqrt {2 x^2-x+3}}\right )}{\sqrt {682 \left (259509026 \sqrt {2}-366990269\right )}}\right )+\frac {\sqrt {2 x^2-x+3} (696 x+277)}{31 \left (5 x^2+3 x+2\right )}\right )+\frac {(10 x+3) \left (2 x^2-x+3\right )^{3/2}}{62 \left (5 x^2+3 x+2\right )^2}\) |
Input:
Int[(3 - x + 2*x^2)^(3/2)/(2 + 3*x + 5*x^2)^3,x]
Output:
((3 + 10*x)*(3 - x + 2*x^2)^(3/2))/(62*(2 + 3*x + 5*x^2)^2) + (3*(((277 + 696*x)*Sqrt[3 - x + 2*x^2])/(31*(2 + 3*x + 5*x^2)) + (Sqrt[(366990269 + 25 9509026*Sqrt[2])/682]*ArcTan[(Sqrt[11/(31*(366990269 + 259509026*Sqrt[2])) ]*(29367 + 20575*Sqrt[2] + (70517 + 49942*Sqrt[2])*x))/Sqrt[3 - x + 2*x^2] ] + ((366990269 - 259509026*Sqrt[2])*ArcTanh[(Sqrt[11/(31*(-366990269 + 25 9509026*Sqrt[2]))]*(29367 - 20575*Sqrt[2] + (70517 - 49942*Sqrt[2])*x))/Sq rt[3 - x + 2*x^2]])/Sqrt[682*(-366990269 + 259509026*Sqrt[2])])/62))/124
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[((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)^(p_)*((d_) + (e _.)*(x_) + (f_.)*(x_)^2)^(q_), x_Symbol] :> Simp[(g*b - 2*a*h - (b*h - 2*g* 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[e*q*(g*b - 2*a*h) - d*(b*h - 2*g*c)*(2*p + 3) + (2*f*q*(g*b - 2*a*h) - e*(b*h - 2*g*c)*(2*p + q + 3))*x - f*(b*h - 2*g*c)* (2*p + 2*q + 3)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && Ne Q[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && LtQ[p, -1] && GtQ[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]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 6.50 (sec) , antiderivative size = 483, normalized size of antiderivative = 2.17
| method | result | size |
| trager | \(\text {Expression too large to display}\) | \(483\) |
| risch | \(\frac {\left (11680 x^{3}+10171 x^{2}+8343 x +2220\right ) \sqrt {2 x^{2}-x +3}}{3844 \left (5 x^{2}+3 x +2\right )^{2}}+\frac {3 \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 (1915561 \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}}+2708832 \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}}+2795860364 \,\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}-3974378870 \,\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 )}{162539696 \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}}}\) | \(726\) |
| default | \(\text {Expression too large to display}\) | \(81552\) |
Input:
int((2*x^2-x+3)^(3/2)/(5*x^2+3*x+2)^3,x,method=_RETURNVERBOSE)
Output:
1/3844*(11680*x^3+10171*x^2+8343*x+2220)/(5*x^2+3*x+2)^2*(2*x^2-x+3)^(1/2) -3/5243216*RootOf(_Z^2+29767936*RootOf(952573952*_Z^4+8009195630656*_Z^2+1 6836233643867169)^2+250287363458)*ln(-(36607893262336*RootOf(_Z^2+29767936 *RootOf(952573952*_Z^4+8009195630656*_Z^2+16836233643867169)^2+25028736345 8)*RootOf(952573952*_Z^4+8009195630656*_Z^2+16836233643867169)^4*x+3262645 23201744512*RootOf(952573952*_Z^4+8009195630656*_Z^2+16836233643867169)^2* RootOf(_Z^2+29767936*RootOf(952573952*_Z^4+8009195630656*_Z^2+168362336438 67169)^2+250287363458)*x+3707589189779200*RootOf(952573952*_Z^4+8009195630 656*_Z^2+16836233643867169)^2*RootOf(_Z^2+29767936*RootOf(952573952*_Z^4+8 009195630656*_Z^2+16836233643867169)^2+250287363458)-122510784267145793066 24*RootOf(952573952*_Z^4+8009195630656*_Z^2+16836233643867169)^2*(2*x^2-x+ 3)^(1/2)+723862202733749385201*RootOf(_Z^2+29767936*RootOf(952573952*_Z^4+ 8009195630656*_Z^2+16836233643867169)^2+250287363458)*x+175986874983483557 00*RootOf(_Z^2+29767936*RootOf(952573952*_Z^4+8009195630656*_Z^2+168362336 43867169)^2+250287363458)-51523372375740505057718054*(2*x^2-x+3)^(1/2))/(2 1824*x*RootOf(952573952*_Z^4+8009195630656*_Z^2+16836233643867169)^2+92128 844*x+508369))+3/961*RootOf(952573952*_Z^4+8009195630656*_Z^2+168362336438 67169)*ln((585726292197376*x*RootOf(952573952*_Z^4+8009195630656*_Z^2+1683 6233643867169)^5+4629284194657700864*RootOf(952573952*_Z^4+8009195630656*_ Z^2+16836233643867169)^3*x-59321427036467200*RootOf(952573952*_Z^4+8009...
Leaf count of result is larger than twice the leaf count of optimal. 618 vs. \(2 (170) = 340\).
Time = 0.09 (sec) , antiderivative size = 618, normalized size of antiderivative = 2.77 \[ \int \frac {\left (3-x+2 x^2\right )^{3/2}}{\left (2+3 x+5 x^2\right )^3} \, dx =\text {Too large to display} \] Input:
integrate((2*x^2-x+3)^(3/2)/(5*x^2+3*x+2)^3,x, algorithm="fricas")
Output:
1/30752*(6*sqrt(1/682)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)*sqrt(25950902 6*sqrt(2) + 366990269)*arctan(-22/508369*sqrt(1/682)*(sqrt(1/682)*(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(259509026*sqrt(2) - 366990269) + 4*(206756*x^3 - 469094*x^2 - sqr t(2)*(145601*x^3 - 331165*x^2 - 113632*x + 150168) - 157840*x + 213744)*sq rt(2*x^2 - x + 3))*sqrt(259509026*sqrt(2) + 366990269)/(343*x^4 - 400*x^3 + 1136*x^2 + 384*x - 576)) - 6*sqrt(1/682)*(25*x^4 + 30*x^3 + 29*x^2 + 12* x + 4)*sqrt(259509026*sqrt(2) + 366990269)*arctan(-22/508369*sqrt(1/682)*( sqrt(1/682)*(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(259509026*sqrt(2) - 366990269) - 4*(206756*x^ 3 - 469094*x^2 - sqrt(2)*(145601*x^3 - 331165*x^2 - 113632*x + 150168) - 1 57840*x + 213744)*sqrt(2*x^2 - x + 3))*sqrt(259509026*sqrt(2) + 366990269) /(343*x^4 - 400*x^3 + 1136*x^2 + 384*x - 576)) + 3*sqrt(1/682)*(25*x^4 + 3 0*x^3 + 29*x^2 + 12*x + 4)*sqrt(259509026*sqrt(2) - 366990269)*log(3*(22*s qrt(1/682)*sqrt(2*x^2 - x + 3)*(sqrt(2)*(187985*x - 452469) + 264484*x - 6 40454)*sqrt(259509026*sqrt(2) - 366990269) + 24910081*x^2 + 22368236*sqrt( 2)*(2*x^2 - x + 3) - 76763719*x + 101673800)/x^2) - 3*sqrt(1/682)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)*sqrt(259509026*sqrt(2) - 366990269)*log(-3*( 22*sqrt(1/682)*sqrt(2*x^2 - x + 3)*(sqrt(2)*(187985*x - 452469) + 264484*x - 640454)*sqrt(259509026*sqrt(2) - 366990269) - 24910081*x^2 - 2236823...
\[ \int \frac {\left (3-x+2 x^2\right )^{3/2}}{\left (2+3 x+5 x^2\right )^3} \, dx=\int \frac {\left (2 x^{2} - x + 3\right )^{\frac {3}{2}}}{\left (5 x^{2} + 3 x + 2\right )^{3}}\, dx \] Input:
integrate((2*x**2-x+3)**(3/2)/(5*x**2+3*x+2)**3,x)
Output:
Integral((2*x**2 - x + 3)**(3/2)/(5*x**2 + 3*x + 2)**3, x)
\[ \int \frac {\left (3-x+2 x^2\right )^{3/2}}{\left (2+3 x+5 x^2\right )^3} \, dx=\int { \frac {{\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{{\left (5 \, x^{2} + 3 \, x + 2\right )}^{3}} \,d x } \] Input:
integrate((2*x^2-x+3)^(3/2)/(5*x^2+3*x+2)^3,x, algorithm="maxima")
Output:
integrate((2*x^2 - x + 3)^(3/2)/(5*x^2 + 3*x + 2)^3, x)
Exception generated. \[ \int \frac {\left (3-x+2 x^2\right )^{3/2}}{\left (2+3 x+5 x^2\right )^3} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((2*x^2-x+3)^(3/2)/(5*x^2+3*x+2)^3,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. \[ \int \frac {\left (3-x+2 x^2\right )^{3/2}}{\left (2+3 x+5 x^2\right )^3} \, dx=\int \frac {{\left (2\,x^2-x+3\right )}^{3/2}}{{\left (5\,x^2+3\,x+2\right )}^3} \,d x \] Input:
int((2*x^2 - x + 3)^(3/2)/(3*x + 5*x^2 + 2)^3,x)
Output:
int((2*x^2 - x + 3)^(3/2)/(3*x + 5*x^2 + 2)^3, x)
\[ \int \frac {\left (3-x+2 x^2\right )^{3/2}}{\left (2+3 x+5 x^2\right )^3} \, dx=\frac {-186400 \sqrt {2 x^{2}-x +3}\, x^{3}-270280 \sqrt {2 x^{2}-x +3}\, x^{2}-377744 \sqrt {2 x^{2}-x +3}\, x +157812 \sqrt {2 x^{2}-x +3}+388037925 \left (\int \frac {\sqrt {2 x^{2}-x +3}}{250 x^{8}+325 x^{7}+720 x^{6}+804 x^{5}+876 x^{4}+579 x^{3}+322 x^{2}+100 x +24}d x \right ) x^{4}+465645510 \left (\int \frac {\sqrt {2 x^{2}-x +3}}{250 x^{8}+325 x^{7}+720 x^{6}+804 x^{5}+876 x^{4}+579 x^{3}+322 x^{2}+100 x +24}d x \right ) x^{3}+450123993 \left (\int \frac {\sqrt {2 x^{2}-x +3}}{250 x^{8}+325 x^{7}+720 x^{6}+804 x^{5}+876 x^{4}+579 x^{3}+322 x^{2}+100 x +24}d x \right ) x^{2}+186258204 \left (\int \frac {\sqrt {2 x^{2}-x +3}}{250 x^{8}+325 x^{7}+720 x^{6}+804 x^{5}+876 x^{4}+579 x^{3}+322 x^{2}+100 x +24}d x \right ) x +62086068 \left (\int \frac {\sqrt {2 x^{2}-x +3}}{250 x^{8}+325 x^{7}+720 x^{6}+804 x^{5}+876 x^{4}+579 x^{3}+322 x^{2}+100 x +24}d x \right )+39854375 \left (\int \frac {\sqrt {2 x^{2}-x +3}\, x^{2}}{250 x^{8}+325 x^{7}+720 x^{6}+804 x^{5}+876 x^{4}+579 x^{3}+322 x^{2}+100 x +24}d x \right ) x^{4}+47825250 \left (\int \frac {\sqrt {2 x^{2}-x +3}\, x^{2}}{250 x^{8}+325 x^{7}+720 x^{6}+804 x^{5}+876 x^{4}+579 x^{3}+322 x^{2}+100 x +24}d x \right ) x^{3}+46231075 \left (\int \frac {\sqrt {2 x^{2}-x +3}\, x^{2}}{250 x^{8}+325 x^{7}+720 x^{6}+804 x^{5}+876 x^{4}+579 x^{3}+322 x^{2}+100 x +24}d x \right ) x^{2}+19130100 \left (\int \frac {\sqrt {2 x^{2}-x +3}\, x^{2}}{250 x^{8}+325 x^{7}+720 x^{6}+804 x^{5}+876 x^{4}+579 x^{3}+322 x^{2}+100 x +24}d x \right ) x +6376700 \left (\int \frac {\sqrt {2 x^{2}-x +3}\, x^{2}}{250 x^{8}+325 x^{7}+720 x^{6}+804 x^{5}+876 x^{4}+579 x^{3}+322 x^{2}+100 x +24}d x \right )}{28490625 x^{4}+34188750 x^{3}+33049125 x^{2}+13675500 x +4558500} \] Input:
int((2*x^2-x+3)^(3/2)/(5*x^2+3*x+2)^3,x)
Output:
( - 186400*sqrt(2*x**2 - x + 3)*x**3 - 270280*sqrt(2*x**2 - x + 3)*x**2 - 377744*sqrt(2*x**2 - x + 3)*x + 157812*sqrt(2*x**2 - x + 3) + 388037925*in t(sqrt(2*x**2 - x + 3)/(250*x**8 + 325*x**7 + 720*x**6 + 804*x**5 + 876*x* *4 + 579*x**3 + 322*x**2 + 100*x + 24),x)*x**4 + 465645510*int(sqrt(2*x**2 - x + 3)/(250*x**8 + 325*x**7 + 720*x**6 + 804*x**5 + 876*x**4 + 579*x**3 + 322*x**2 + 100*x + 24),x)*x**3 + 450123993*int(sqrt(2*x**2 - x + 3)/(25 0*x**8 + 325*x**7 + 720*x**6 + 804*x**5 + 876*x**4 + 579*x**3 + 322*x**2 + 100*x + 24),x)*x**2 + 186258204*int(sqrt(2*x**2 - x + 3)/(250*x**8 + 325* x**7 + 720*x**6 + 804*x**5 + 876*x**4 + 579*x**3 + 322*x**2 + 100*x + 24), x)*x + 62086068*int(sqrt(2*x**2 - x + 3)/(250*x**8 + 325*x**7 + 720*x**6 + 804*x**5 + 876*x**4 + 579*x**3 + 322*x**2 + 100*x + 24),x) + 39854375*int ((sqrt(2*x**2 - x + 3)*x**2)/(250*x**8 + 325*x**7 + 720*x**6 + 804*x**5 + 876*x**4 + 579*x**3 + 322*x**2 + 100*x + 24),x)*x**4 + 47825250*int((sqrt( 2*x**2 - x + 3)*x**2)/(250*x**8 + 325*x**7 + 720*x**6 + 804*x**5 + 876*x** 4 + 579*x**3 + 322*x**2 + 100*x + 24),x)*x**3 + 46231075*int((sqrt(2*x**2 - x + 3)*x**2)/(250*x**8 + 325*x**7 + 720*x**6 + 804*x**5 + 876*x**4 + 579 *x**3 + 322*x**2 + 100*x + 24),x)*x**2 + 19130100*int((sqrt(2*x**2 - x + 3 )*x**2)/(250*x**8 + 325*x**7 + 720*x**6 + 804*x**5 + 876*x**4 + 579*x**3 + 322*x**2 + 100*x + 24),x)*x + 6376700*int((sqrt(2*x**2 - x + 3)*x**2)/(25 0*x**8 + 325*x**7 + 720*x**6 + 804*x**5 + 876*x**4 + 579*x**3 + 322*x**...