Integrand size = 27, antiderivative size = 269 \[ \int \frac {1}{\left (3-x+2 x^2\right )^{5/2} \left (2+3 x+5 x^2\right )^3} \, dx=-\frac {12280939-19536786 x}{2824232928 \left (3-x+2 x^2\right )^{3/2}}-\frac {1134826571-1504660754 x}{476353953856 \sqrt {3-x+2 x^2}}+\frac {4+65 x}{1364 \left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right )^2}+\frac {46386+86885 x}{1860496 \left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right )}+\frac {35 \sqrt {\frac {1}{682} \left (2243059557247+2011748500000 \sqrt {2}\right )} \arctan \left (\frac {\sqrt {\frac {11}{31 \left (2243059557247+2011748500000 \sqrt {2}\right )}} \left (1432939+2428746 \sqrt {2}+\left (6290431+3861685 \sqrt {2}\right ) x\right )}{\sqrt {3-x+2 x^2}}\right )}{1800960128}-\frac {35 \sqrt {\frac {1}{682} \left (-2243059557247+2011748500000 \sqrt {2}\right )} \text {arctanh}\left (\frac {\sqrt {\frac {11}{31 \left (-2243059557247+2011748500000 \sqrt {2}\right )}} \left (1432939-2428746 \sqrt {2}+\left (6290431-3861685 \sqrt {2}\right ) x\right )}{\sqrt {3-x+2 x^2}}\right )}{1800960128} \] Output:
-1/2824232928*(12280939-19536786*x)/(2*x^2-x+3)^(3/2)-1/476353953856*(1134 826571-1504660754*x)/(2*x^2-x+3)^(1/2)+1/1364*(4+65*x)/(2*x^2-x+3)^(3/2)/( 5*x^2+3*x+2)^2+1/1860496*(46386+86885*x)/(2*x^2-x+3)^(3/2)/(5*x^2+3*x+2)+3 5/1228254807296*(1529766618042454+1372012477000000*2^(1/2))^(1/2)*arctan(1 1^(1/2)/(69534846274657+62364203500000*2^(1/2))^(1/2)*(1432939+2428746*2^( 1/2)+(6290431+3861685*2^(1/2))*x)/(2*x^2-x+3)^(1/2))-35/1228254807296*(-15 29766618042454+1372012477000000*2^(1/2))^(1/2)*arctanh(11^(1/2)/(-69534846 274657+62364203500000*2^(1/2))^(1/2)*(1432939-2428746*2^(1/2)+(6290431-386 1685*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 = 1.88 (sec) , antiderivative size = 605, normalized size of antiderivative = 2.25 \[ \int \frac {1}{\left (3-x+2 x^2\right )^{5/2} \left (2+3 x+5 x^2\right )^3} \, dx=\frac {\frac {4 \sqrt {3-x+2 x^2} \left (9739335532+218659985088 x+178650961091 x^2+519223213785 x^3+174241614961 x^4+592923725931 x^5-12234606480 x^6+225699113100 x^7\right )}{\left (6+7 x+16 x^2+x^3+10 x^4\right )^2}-2976 \text {RootSum}\left [-56-26 \sqrt {2} \text {$\#$1}+17 \text {$\#$1}^2+6 \sqrt {2} \text {$\#$1}^3-5 \text {$\#$1}^4\&,\frac {-26154346 \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right )+37230166 \sqrt {2} \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}-1193705 \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{-13 \sqrt {2}+17 \text {$\#$1}+9 \sqrt {2} \text {$\#$1}^2-10 \text {$\#$1}^3}\&\right ]-24401712 \text {RootSum}\left [-56-26 \sqrt {2} \text {$\#$1}+17 \text {$\#$1}^2+6 \sqrt {2} \text {$\#$1}^3-5 \text {$\#$1}^4\&,\frac {-3647 \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right )+3172 \sqrt {2} \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}-485 \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{-13 \sqrt {2}+17 \text {$\#$1}+9 \sqrt {2} \text {$\#$1}^2-10 \text {$\#$1}^3}\&\right ]+15 \sqrt {2} \text {RootSum}\left [-56-26 \sqrt {2} \text {$\#$1}+17 \text {$\#$1}^2+6 \sqrt {2} \text {$\#$1}^3-5 \text {$\#$1}^4\&,\frac {-9138129081 \sqrt {2} \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right )+16445754136 \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}+1004412885 \sqrt {2} \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{-13 \sqrt {2}+17 \text {$\#$1}+9 \sqrt {2} \text {$\#$1}^2-10 \text {$\#$1}^3}\&\right ]}{5716247446272} \] Input:
Integrate[1/((3 - x + 2*x^2)^(5/2)*(2 + 3*x + 5*x^2)^3),x]
Output:
((4*Sqrt[3 - x + 2*x^2]*(9739335532 + 218659985088*x + 178650961091*x^2 + 519223213785*x^3 + 174241614961*x^4 + 592923725931*x^5 - 12234606480*x^6 + 225699113100*x^7))/(6 + 7*x + 16*x^2 + x^3 + 10*x^4)^2 - 2976*RootSum[-56 - 26*Sqrt[2]*#1 + 17*#1^2 + 6*Sqrt[2]*#1^3 - 5*#1^4 & , (-26154346*Log[-( Sqrt[2]*x) + Sqrt[3 - x + 2*x^2] - #1] + 37230166*Sqrt[2]*Log[-(Sqrt[2]*x) + Sqrt[3 - x + 2*x^2] - #1]*#1 - 1193705*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) & ] - 24401712*RootSum[-56 - 26*Sqrt[2]*#1 + 17*#1^2 + 6*Sqrt[2]*#1^3 - 5*#1^4 & , (-3647*Log[-(Sqrt[2]*x) + Sqrt[3 - x + 2*x^2] - #1] + 3172*Sqrt[2]*Log[ -(Sqrt[2]*x) + Sqrt[3 - x + 2*x^2] - #1]*#1 - 485*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 ) & ] + 15*Sqrt[2]*RootSum[-56 - 26*Sqrt[2]*#1 + 17*#1^2 + 6*Sqrt[2]*#1^3 - 5*#1^4 & , (-9138129081*Sqrt[2]*Log[-(Sqrt[2]*x) + Sqrt[3 - x + 2*x^2] - #1] + 16445754136*Log[-(Sqrt[2]*x) + Sqrt[3 - x + 2*x^2] - #1]*#1 + 10044 12885*Sqrt[2]*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) & ])/5716247446272
Time = 0.88 (sec) , antiderivative size = 290, 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 = {1305, 27, 2135, 27, 2135, 27, 2135, 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.
| \(\displaystyle \int \frac {1}{\left (2 x^2-x+3\right )^{5/2} \left (5 x^2+3 x+2\right )^3} \, dx\) |
| \(\Big \downarrow \) 1305 |
| \(\displaystyle \frac {65 x+4}{1364 \left (2 x^2-x+3\right )^{3/2} \left (5 x^2+3 x+2\right )^2}-\frac {\int -\frac {11 \left (1560 x^2-785 x+1034\right )}{2 \left (2 x^2-x+3\right )^{5/2} \left (5 x^2+3 x+2\right )^2}dx}{15004}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {1560 x^2-785 x+1034}{\left (2 x^2-x+3\right )^{5/2} \left (5 x^2+3 x+2\right )^2}dx}{2728}+\frac {65 x+4}{1364 \left (2 x^2-x+3\right )^{3/2} \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 2135 |
| \(\displaystyle \frac {\frac {\int \frac {11 \left (1390160 x^2+284771 x+462194\right )}{2 \left (2 x^2-x+3\right )^{5/2} \left (5 x^2+3 x+2\right )}dx}{7502}+\frac {86885 x+46386}{682 \left (2 x^2-x+3\right )^{3/2} \left (5 x^2+3 x+2\right )}}{2728}+\frac {65 x+4}{1364 \left (2 x^2-x+3\right )^{3/2} \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {1390160 x^2+284771 x+462194}{\left (2 x^2-x+3\right )^{5/2} \left (5 x^2+3 x+2\right )}dx}{1364}+\frac {86885 x+46386}{682 \left (2 x^2-x+3\right )^{3/2} \left (5 x^2+3 x+2\right )}}{2728}+\frac {65 x+4}{1364 \left (2 x^2-x+3\right )^{3/2} \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 2135 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {33 \left (130245240 x^2+4179719 x+60094966\right )}{2 \left (2 x^2-x+3\right )^{3/2} \left (5 x^2+3 x+2\right )}dx}{8349}-\frac {12280939-19536786 x}{759 \left (2 x^2-x+3\right )^{3/2}}}{1364}+\frac {86885 x+46386}{682 \left (2 x^2-x+3\right )^{3/2} \left (5 x^2+3 x+2\right )}}{2728}+\frac {65 x+4}{1364 \left (2 x^2-x+3\right )^{3/2} \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {1}{506} \int \frac {130245240 x^2+4179719 x+60094966}{\left (2 x^2-x+3\right )^{3/2} \left (5 x^2+3 x+2\right )}dx-\frac {12280939-19536786 x}{759 \left (2 x^2-x+3\right )^{3/2}}}{1364}+\frac {86885 x+46386}{682 \left (2 x^2-x+3\right )^{3/2} \left (5 x^2+3 x+2\right )}}{2728}+\frac {65 x+4}{1364 \left (2 x^2-x+3\right )^{3/2} \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 2135 |
| \(\displaystyle \frac {\frac {\frac {1}{506} \left (\frac {\int \frac {203665 (263242-409755 x)}{2 \sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx}{2783}-\frac {1134826571-1504660754 x}{253 \sqrt {2 x^2-x+3}}\right )-\frac {12280939-19536786 x}{759 \left (2 x^2-x+3\right )^{3/2}}}{1364}+\frac {86885 x+46386}{682 \left (2 x^2-x+3\right )^{3/2} \left (5 x^2+3 x+2\right )}}{2728}+\frac {65 x+4}{1364 \left (2 x^2-x+3\right )^{3/2} \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {1}{506} \left (\frac {805}{22} \int \frac {263242-409755 x}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx-\frac {1134826571-1504660754 x}{253 \sqrt {2 x^2-x+3}}\right )-\frac {12280939-19536786 x}{759 \left (2 x^2-x+3\right )^{3/2}}}{1364}+\frac {86885 x+46386}{682 \left (2 x^2-x+3\right )^{3/2} \left (5 x^2+3 x+2\right )}}{2728}+\frac {65 x+4}{1364 \left (2 x^2-x+3\right )^{3/2} \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 1368 |
| \(\displaystyle \frac {\frac {\frac {1}{506} \left (\frac {805}{22} \left (\frac {\int -\frac {11 \left (\left (146513+409755 \sqrt {2}\right ) x-263242 \sqrt {2}+672997\right )}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx}{22 \sqrt {2}}-\frac {\int -\frac {11 \left (\left (146513-409755 \sqrt {2}\right ) x+263242 \sqrt {2}+672997\right )}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx}{22 \sqrt {2}}\right )-\frac {1134826571-1504660754 x}{253 \sqrt {2 x^2-x+3}}\right )-\frac {12280939-19536786 x}{759 \left (2 x^2-x+3\right )^{3/2}}}{1364}+\frac {86885 x+46386}{682 \left (2 x^2-x+3\right )^{3/2} \left (5 x^2+3 x+2\right )}}{2728}+\frac {65 x+4}{1364 \left (2 x^2-x+3\right )^{3/2} \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {1}{506} \left (\frac {805}{22} \left (\frac {\int \frac {\left (146513-409755 \sqrt {2}\right ) x+263242 \sqrt {2}+672997}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx}{2 \sqrt {2}}-\frac {\int \frac {\left (146513+409755 \sqrt {2}\right ) x-263242 \sqrt {2}+672997}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx}{2 \sqrt {2}}\right )-\frac {1134826571-1504660754 x}{253 \sqrt {2 x^2-x+3}}\right )-\frac {12280939-19536786 x}{759 \left (2 x^2-x+3\right )^{3/2}}}{1364}+\frac {86885 x+46386}{682 \left (2 x^2-x+3\right )^{3/2} \left (5 x^2+3 x+2\right )}}{2728}+\frac {65 x+4}{1364 \left (2 x^2-x+3\right )^{3/2} \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 1362 |
| \(\displaystyle \frac {\frac {\frac {1}{506} \left (\frac {805}{22} \left (\frac {\left (2243059557247-2011748500000 \sqrt {2}\right ) \int \frac {1}{-\frac {11 \left (\left (6290431-3861685 \sqrt {2}\right ) x-2428746 \sqrt {2}+1432939\right )^2}{2 x^2-x+3}-31 \left (2243059557247-2011748500000 \sqrt {2}\right )}d\frac {\left (6290431-3861685 \sqrt {2}\right ) x-2428746 \sqrt {2}+1432939}{\sqrt {2 x^2-x+3}}}{\sqrt {2}}-\frac {\left (2243059557247+2011748500000 \sqrt {2}\right ) \int \frac {1}{-\frac {11 \left (\left (6290431+3861685 \sqrt {2}\right ) x+2428746 \sqrt {2}+1432939\right )^2}{2 x^2-x+3}-31 \left (2243059557247+2011748500000 \sqrt {2}\right )}d\frac {\left (6290431+3861685 \sqrt {2}\right ) x+2428746 \sqrt {2}+1432939}{\sqrt {2 x^2-x+3}}}{\sqrt {2}}\right )-\frac {1134826571-1504660754 x}{253 \sqrt {2 x^2-x+3}}\right )-\frac {12280939-19536786 x}{759 \left (2 x^2-x+3\right )^{3/2}}}{1364}+\frac {86885 x+46386}{682 \left (2 x^2-x+3\right )^{3/2} \left (5 x^2+3 x+2\right )}}{2728}+\frac {65 x+4}{1364 \left (2 x^2-x+3\right )^{3/2} \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {\frac {1}{506} \left (\frac {805}{22} \left (\frac {\left (2243059557247-2011748500000 \sqrt {2}\right ) \int \frac {1}{-\frac {11 \left (\left (6290431-3861685 \sqrt {2}\right ) x-2428746 \sqrt {2}+1432939\right )^2}{2 x^2-x+3}-31 \left (2243059557247-2011748500000 \sqrt {2}\right )}d\frac {\left (6290431-3861685 \sqrt {2}\right ) x-2428746 \sqrt {2}+1432939}{\sqrt {2 x^2-x+3}}}{\sqrt {2}}+\sqrt {\frac {1}{682} \left (2243059557247+2011748500000 \sqrt {2}\right )} \arctan \left (\frac {\sqrt {\frac {11}{31 \left (2243059557247+2011748500000 \sqrt {2}\right )}} \left (\left (6290431+3861685 \sqrt {2}\right ) x+2428746 \sqrt {2}+1432939\right )}{\sqrt {2 x^2-x+3}}\right )\right )-\frac {1134826571-1504660754 x}{253 \sqrt {2 x^2-x+3}}\right )-\frac {12280939-19536786 x}{759 \left (2 x^2-x+3\right )^{3/2}}}{1364}+\frac {86885 x+46386}{682 \left (2 x^2-x+3\right )^{3/2} \left (5 x^2+3 x+2\right )}}{2728}+\frac {65 x+4}{1364 \left (2 x^2-x+3\right )^{3/2} \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {\frac {1}{506} \left (\frac {805}{22} \left (\sqrt {\frac {1}{682} \left (2243059557247+2011748500000 \sqrt {2}\right )} \arctan \left (\frac {\sqrt {\frac {11}{31 \left (2243059557247+2011748500000 \sqrt {2}\right )}} \left (\left (6290431+3861685 \sqrt {2}\right ) x+2428746 \sqrt {2}+1432939\right )}{\sqrt {2 x^2-x+3}}\right )+\frac {\left (2243059557247-2011748500000 \sqrt {2}\right ) \text {arctanh}\left (\frac {\sqrt {\frac {11}{31 \left (2011748500000 \sqrt {2}-2243059557247\right )}} \left (\left (6290431-3861685 \sqrt {2}\right ) x-2428746 \sqrt {2}+1432939\right )}{\sqrt {2 x^2-x+3}}\right )}{\sqrt {682 \left (2011748500000 \sqrt {2}-2243059557247\right )}}\right )-\frac {1134826571-1504660754 x}{253 \sqrt {2 x^2-x+3}}\right )-\frac {12280939-19536786 x}{759 \left (2 x^2-x+3\right )^{3/2}}}{1364}+\frac {86885 x+46386}{682 \left (2 x^2-x+3\right )^{3/2} \left (5 x^2+3 x+2\right )}}{2728}+\frac {65 x+4}{1364 \left (2 x^2-x+3\right )^{3/2} \left (5 x^2+3 x+2\right )^2}\) |
Input:
Int[1/((3 - x + 2*x^2)^(5/2)*(2 + 3*x + 5*x^2)^3),x]
Output:
(4 + 65*x)/(1364*(3 - x + 2*x^2)^(3/2)*(2 + 3*x + 5*x^2)^2) + ((46386 + 86 885*x)/(682*(3 - x + 2*x^2)^(3/2)*(2 + 3*x + 5*x^2)) + (-1/759*(12280939 - 19536786*x)/(3 - x + 2*x^2)^(3/2) + (-1/253*(1134826571 - 1504660754*x)/S qrt[3 - x + 2*x^2] + (805*(Sqrt[(2243059557247 + 2011748500000*Sqrt[2])/68 2]*ArcTan[(Sqrt[11/(31*(2243059557247 + 2011748500000*Sqrt[2]))]*(1432939 + 2428746*Sqrt[2] + (6290431 + 3861685*Sqrt[2])*x))/Sqrt[3 - x + 2*x^2]] + ((2243059557247 - 2011748500000*Sqrt[2])*ArcTanh[(Sqrt[11/(31*(-224305955 7247 + 2011748500000*Sqrt[2]))]*(1432939 - 2428746*Sqrt[2] + (6290431 - 38 61685*Sqrt[2])*x))/Sqrt[3 - x + 2*x^2]])/Sqrt[682*(-2243059557247 + 201174 8500000*Sqrt[2])]))/22)/506)/1364)/2728
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[(2*a*c^2*e - b^2*c*e + b^3*f + b*c*(c*d - 3*a *f) + c*(2*c^2*d + b^2*f - c*(b*e + 2*a*f))*x)*(a + b*x + c*x^2)^(p + 1)*(( d + e*x + f*x^2)^(q + 1)/((b^2 - 4*a*c)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1))), x] - Simp[1/((b^2 - 4*a*c)*((c*d - a*f)^2 - (b*d - a*e)*( c*e - b*f))*(p + 1)) Int[(a + b*x + c*x^2)^(p + 1)*(d + e*x + f*x^2)^q*Si mp[2*c*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1) - (2*c^2*d + b^2*f - c*(b*e + 2*a*f))*(a*f*(p + 1) - c*d*(p + 2)) - e*(b^2*c*e - 2*a*c^2*e - b^3*f - b*c*(c*d - 3*a*f))*(p + q + 2) + (2*f*(2*a*c^2*e - b^2*c*e + b^3*f + b*c*(c*d - 3*a*f))*(p + q + 2) - (2*c^2*d + b^2*f - c*(b*e + 2*a*f))*(b*f *(p + 1) - c*e*(2*p + q + 4)))*x + c*f*(2*c^2*d + b^2*f - c*(b*e + 2*a*f))* (2*p + 2*q + 5)*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] && LtQ[p, -1] && NeQ[(c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f), 0] && !( !IntegerQ[p] && ILtQ[q, -1]) && !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[(a + b*x + c*x^2)^(p + 1)*((d + e*x + f*x^2)^( q + 1)/((b^2 - 4*a*c)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1)))*( (A*c - a*C)*(2*a*c*e - b*(c*d + a*f)) + (A*b - a*B)*(2*c^2*d + b^2*f - c*(b *e + 2*a*f)) + c*(A*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)) - B*(b*c*d - 2*a*c* e + a*b*f) + C*(b^2*d - a*b*e - 2*a*(c*d - a*f)))*x), x] + Simp[1/((b^2 - 4 *a*c)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1)) Int[(a + b*x + c *x^2)^(p + 1)*(d + e*x + f*x^2)^q*Simp[(b*B - 2*A*c - 2*a*C)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1) + (b^2*(C*d + A*f) - b*(B*c*d + A*c*e + a*C*e + a*B*f) + 2*(A*c*(c*d - a*f) - a*(c*C*d - B*c*e - a*C*f)))*(a*f*(p + 1) - c*d*(p + 2)) - e*((A*c - a*C)*(2*a*c*e - b*(c*d + a*f)) + (A*b - a*B )*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)))*(p + q + 2) - (2*f*((A*c - a*C)*(2*a *c*e - b*(c*d + a*f)) + (A*b - a*B)*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)))*(p + q + 2) - (b^2*(C*d + A*f) - b*(B*c*d + A*c*e + a*C*e + a*B*f) + 2*(A*c*( c*d - a*f) - a*(c*C*d - B*c*e - a*C*f)))*(b*f*(p + 1) - c*e*(2*p + q + 4))) *x - c*f*(b^2*(C*d + A*f) - b*(B*c*d + A*c*e + a*C*e + a*B*f) + 2*(A*c*(c*d - a*f) - a*(c*C*d - B*c*e - a*C*f)))*(2*p + 2*q + 5)*x^2, x], x], x]] /; F reeQ[{a, b, c, d, e, f, q}, x] && PolyQ[Px, x, 2] && LtQ[p, -1] && NeQ[(c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f), 0] && !( !IntegerQ[p] && ILtQ[q, -1]) && !IGtQ[q, 0]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 5.10 (sec) , antiderivative size = 511, normalized size of antiderivative = 1.90
| method | result | size |
| trager | \(\text {Expression too large to display}\) | \(511\) |
| risch | \(\frac {225699113100 x^{7}-12234606480 x^{6}+592923725931 x^{5}+174241614961 x^{4}+519223213785 x^{3}+178650961091 x^{2}+218659985088 x +9739335532}{1429061861568 \left (2 x^{2}-x +3\right )^{\frac {3}{2}} \left (5 x^{2}+3 x +2\right )^{2}}+\frac {35 \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 (159009303 \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}}+226212430 \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}}+287037935998 \,\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}-254173077554 \,\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 )}{38075899026176 \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}}}\) | \(746\) |
| default | \(\text {Expression too large to display}\) | \(19014\) |
Input:
int(1/(2*x^2-x+3)^(5/2)/(5*x^2+3*x+2)^3,x,method=_RETURNVERBOSE)
Output:
1/1429061861568*(225699113100*x^7-12234606480*x^6+592923725931*x^5+1742416 14961*x^4+519223213785*x^3+178650961091*x^2+218659985088*x+9739335532)/(10 *x^4+x^3+16*x^2+7*x+6)^2*(2*x^2-x+3)^(1/2)-35/1228254807296*RootOf(_Z^2+21 700825344*RootOf(7909950837888*_Z^4+557599932276474483*_Z^2+15809109481454 101562500)^2+1529766618042454)*ln(-(3609214479402775056*RootOf(_Z^2+217008 25344*RootOf(7909950837888*_Z^4+557599932276474483*_Z^2+158091094814541015 62500)^2+1529766618042454)*RootOf(7909950837888*_Z^4+557599932276474483*_Z ^2+15809109481454101562500)^4*x+133267585012980205221621*RootOf(7909950837 888*_Z^4+557599932276474483*_Z^2+15809109481454101562500)^2*RootOf(_Z^2+21 700825344*RootOf(7909950837888*_Z^4+557599932276474483*_Z^2+15809109481454 101562500)^2+1529766618042454)*x+13985826628845767554155586905000*(2*x^2-x +3)^(1/2)*RootOf(7909950837888*_Z^4+557599932276474483*_Z^2+15809109481454 101562500)^2-310035860689884712026075*RootOf(7909950837888*_Z^4+5575999322 76474483*_Z^2+15809109481454101562500)^2*RootOf(_Z^2+21700825344*RootOf(79 09950837888*_Z^4+557599932276474483*_Z^2+15809109481454101562500)^2+152976 6618042454)-7510779581096536575629672500*RootOf(_Z^2+21700825344*RootOf(79 09950837888*_Z^4+557599932276474483*_Z^2+15809109481454101562500)^2+152976 6618042454)*x+473458193348490384034058466836980000*(2*x^2-x+3)^(1/2)+95336 58115508780780226687500*RootOf(_Z^2+21700825344*RootOf(7909950837888*_Z^4+ 557599932276474483*_Z^2+15809109481454101562500)^2+1529766618042454))/(...
Leaf count of result is larger than twice the leaf count of optimal. 738 vs. \(2 (208) = 416\).
Time = 0.10 (sec) , antiderivative size = 738, normalized size of antiderivative = 2.74 \[ \int \frac {1}{\left (3-x+2 x^2\right )^{5/2} \left (2+3 x+5 x^2\right )^3} \, dx=\text {Too large to display} \] Input:
integrate(1/(2*x^2-x+3)^(5/2)/(5*x^2+3*x+2)^3,x, algorithm="fricas")
Output:
-1/11432494892544*(111090*sqrt(1/682)*(100*x^8 + 20*x^7 + 321*x^6 + 172*x^ 5 + 390*x^4 + 236*x^3 + 241*x^2 + 84*x + 36)*sqrt(2011748500000*sqrt(2) + 2243059557247)*arctan(-22/314332260881*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(2011748500000*sqrt(2) - 2243059557247) + 4*(14041178*x^3 - 36217888 *x^2 - sqrt(2)*(14366395*x^3 - 29579677*x^2 - 1867664*x + 16151928) - 2388 0112*x + 12635616)*sqrt(2*x^2 - x + 3))*sqrt(2011748500000*sqrt(2) + 22430 59557247)/(343*x^4 - 400*x^3 + 1136*x^2 + 384*x - 576)) - 111090*sqrt(1/68 2)*(100*x^8 + 20*x^7 + 321*x^6 + 172*x^5 + 390*x^4 + 236*x^3 + 241*x^2 + 8 4*x + 36)*sqrt(2011748500000*sqrt(2) + 2243059557247)*arctan(-22/314332260 881*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(2011748500000*sqrt(2) - 2243 059557247) - 4*(14041178*x^3 - 36217888*x^2 - sqrt(2)*(14366395*x^3 - 2957 9677*x^2 - 1867664*x + 16151928) - 23880112*x + 12635616)*sqrt(2*x^2 - x + 3))*sqrt(2011748500000*sqrt(2) + 2243059557247)/(343*x^4 - 400*x^3 + 1136 *x^2 + 384*x - 576)) + 55545*sqrt(1/682)*(100*x^8 + 20*x^7 + 321*x^6 + 172 *x^5 + 390*x^4 + 236*x^3 + 241*x^2 + 84*x + 36)*sqrt(2011748500000*sqrt(2) - 2243059557247)*log(35*(22*sqrt(1/682)*sqrt(2*x^2 - x + 3)*(sqrt(2)*(120 22187*x - 39175525) + 27153338*x - 51197712)*sqrt(2011748500000*sqrt(2) - 2243059557247) + 15402280783169*x^2 + 13830619478764*sqrt(2)*(2*x^2 - x...
\[ \int \frac {1}{\left (3-x+2 x^2\right )^{5/2} \left (2+3 x+5 x^2\right )^3} \, dx=\int \frac {1}{\left (2 x^{2} - x + 3\right )^{\frac {5}{2}} \left (5 x^{2} + 3 x + 2\right )^{3}}\, dx \] Input:
integrate(1/(2*x**2-x+3)**(5/2)/(5*x**2+3*x+2)**3,x)
Output:
Integral(1/((2*x**2 - x + 3)**(5/2)*(5*x**2 + 3*x + 2)**3), x)
\[ \int \frac {1}{\left (3-x+2 x^2\right )^{5/2} \left (2+3 x+5 x^2\right )^3} \, dx=\int { \frac {1}{{\left (5 \, x^{2} + 3 \, x + 2\right )}^{3} {\left (2 \, x^{2} - x + 3\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate(1/(2*x^2-x+3)^(5/2)/(5*x^2+3*x+2)^3,x, algorithm="maxima")
Output:
integrate(1/((5*x^2 + 3*x + 2)^3*(2*x^2 - x + 3)^(5/2)), x)
Exception generated. \[ \int \frac {1}{\left (3-x+2 x^2\right )^{5/2} \left (2+3 x+5 x^2\right )^3} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(1/(2*x^2-x+3)^(5/2)/(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. \[ \int \frac {1}{\left (3-x+2 x^2\right )^{5/2} \left (2+3 x+5 x^2\right )^3} \, dx=\int \frac {1}{{\left (2\,x^2-x+3\right )}^{5/2}\,{\left (5\,x^2+3\,x+2\right )}^3} \,d x \] Input:
int(1/((2*x^2 - x + 3)^(5/2)*(3*x + 5*x^2 + 2)^3),x)
Output:
int(1/((2*x^2 - x + 3)^(5/2)*(3*x + 5*x^2 + 2)^3), x)
\[ \int \frac {1}{\left (3-x+2 x^2\right )^{5/2} \left (2+3 x+5 x^2\right )^3} \, dx=\int \frac {\sqrt {2 x^{2}-x +3}}{1000 x^{12}+300 x^{11}+4830 x^{10}+3061 x^{9}+9948 x^{8}+7869 x^{7}+12016 x^{6}+8619 x^{5}+8292 x^{4}+4483 x^{3}+2610 x^{2}+756 x +216}d x \] Input:
int(1/(2*x^2-x+3)^(5/2)/(5*x^2+3*x+2)^3,x)
Output:
int(sqrt(2*x**2 - x + 3)/(1000*x**12 + 300*x**11 + 4830*x**10 + 3061*x**9 + 9948*x**8 + 7869*x**7 + 12016*x**6 + 8619*x**5 + 8292*x**4 + 4483*x**3 + 2610*x**2 + 756*x + 216),x)