Integrand size = 27, antiderivative size = 281 \[ \int \frac {\left (3-x+2 x^2\right )^{5/2}}{\left (2+3 x+5 x^2\right )^3} \, dx=\frac {(11359-12920 x) \sqrt {3-x+2 x^2}}{48050}+\frac {(3+10 x) \left (3-x+2 x^2\right )^{5/2}}{62 \left (2+3 x+5 x^2\right )^2}+\frac {(769+2336 x) \left (3-x+2 x^2\right )^{3/2}}{3844 \left (2+3 x+5 x^2\right )}-\frac {4}{125} \sqrt {2} \text {arcsinh}\left (\frac {1-4 x}{\sqrt {23}}\right )+\frac {\sqrt {11 \left (1+4 \sqrt {2}\right )} \left (2937349+1978861 \sqrt {2}\right ) \arctan \left (\frac {\sqrt {\frac {11}{62 \left (3531015707557+2498852071250 \sqrt {2}\right )}} \left (3957722+2937349 \sqrt {2}+\left (9832420+6895071 \sqrt {2}\right ) x\right )}{\sqrt {3-x+2 x^2}}\right )}{29791000}-\frac {\left (2937349-1978861 \sqrt {2}\right ) \sqrt {11 \left (-1+4 \sqrt {2}\right )} \text {arctanh}\left (\frac {\sqrt {\frac {11}{62 \left (-3531015707557+2498852071250 \sqrt {2}\right )}} \left (3957722-2937349 \sqrt {2}+\left (9832420-6895071 \sqrt {2}\right ) x\right )}{\sqrt {3-x+2 x^2}}\right )}{29791000} \]
1/62*(3+10*x)*(2*x^2-x+3)^(5/2)/(5*x^2+3*x+2)^2+1/3844*(769+2336*x)*(2*x^2 -x+3)^(3/2)/(5*x^2+3*x+2)-4/125*arcsinh(1/23*(1-4*x)*23^(1/2))*2^(1/2)+1/4 8050*(11359-12920*x)*(2*x^2-x+3)^(1/2)-1/29791000*arctanh(1/62*(3957722+x* (9832420-6895071*2^(1/2))-2937349*2^(1/2))*682^(1/2)/(-3531015707557+24988 52071250*2^(1/2))^(1/2)/(2*x^2-x+3)^(1/2))*(2937349-1978861*2^(1/2))*(-11+ 44*2^(1/2))^(1/2)+1/29791000*arctan(1/62*(3957722+2937349*2^(1/2)+x*(98324 20+6895071*2^(1/2)))*682^(1/2)/(3531015707557+2498852071250*2^(1/2))^(1/2) /(2*x^2-x+3)^(1/2))*(2937349+1978861*2^(1/2))*(11+44*2^(1/2))^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.89 (sec) , antiderivative size = 616, normalized size of antiderivative = 2.19 \[ \int \frac {\left (3-x+2 x^2\right )^{5/2}}{\left (2+3 x+5 x^2\right )^3} \, dx=\frac {\frac {15812500 \sqrt {3-x+2 x^2} \left (22552+69621 x+93872 x^2+97155 x^3\right )}{\left (2+3 x+5 x^2\right )^2}-4420600000 \sqrt {2} \log \left (1-4 x+2 \sqrt {6-2 x+4 x^2}\right )+972532000 \text {RootSum}\left [-56-26 \sqrt {2} \text {$\#$1}+17 \text {$\#$1}^2+6 \sqrt {2} \text {$\#$1}^3-5 \text {$\#$1}^4\&,\frac {3781 \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right )+630 \sqrt {2} \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}+150 \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{-13 \sqrt {2}+17 \text {$\#$1}+9 \sqrt {2} \text {$\#$1}^2-10 \text {$\#$1}^3}\&\right ]+682 \sqrt {2} \text {RootSum}\left [-56-26 \sqrt {2} \text {$\#$1}+17 \text {$\#$1}^2+6 \sqrt {2} \text {$\#$1}^3-5 \text {$\#$1}^4\&,\frac {4978708507 \sqrt {2} \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right )-165870920 \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}+1110955025 \sqrt {2} \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{-13 \sqrt {2}+17 \text {$\#$1}+9 \sqrt {2} \text {$\#$1}^2-10 \text {$\#$1}^3}\&\right ]-11 \sqrt {2} \text {RootSum}\left [-56-26 \sqrt {2} \text {$\#$1}+17 \text {$\#$1}^2+6 \sqrt {2} \text {$\#$1}^3-5 \text {$\#$1}^4\&,\frac {492740319684 \sqrt {2} \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right )-128644699540 \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}+55365920925 \sqrt {2} \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{-13 \sqrt {2}+17 \text {$\#$1}+9 \sqrt {2} \text {$\#$1}^2-10 \text {$\#$1}^3}\&\right ]}{138143750000} \]
((15812500*Sqrt[3 - x + 2*x^2]*(22552 + 69621*x + 93872*x^2 + 97155*x^3))/ (2 + 3*x + 5*x^2)^2 - 4420600000*Sqrt[2]*Log[1 - 4*x + 2*Sqrt[6 - 2*x + 4* x^2]] + 972532000*RootSum[-56 - 26*Sqrt[2]*#1 + 17*#1^2 + 6*Sqrt[2]*#1^3 - 5*#1^4 & , (3781*Log[-(Sqrt[2]*x) + Sqrt[3 - x + 2*x^2] - #1] + 630*Sqrt[ 2]*Log[-(Sqrt[2]*x) + Sqrt[3 - x + 2*x^2] - #1]*#1 + 150*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) & ] + 682*Sqrt[2]*RootSum[-56 - 26*Sqrt[2]*#1 + 17*#1^2 + 6*Sqrt[ 2]*#1^3 - 5*#1^4 & , (4978708507*Sqrt[2]*Log[-(Sqrt[2]*x) + Sqrt[3 - x + 2 *x^2] - #1] - 165870920*Log[-(Sqrt[2]*x) + Sqrt[3 - x + 2*x^2] - #1]*#1 + 1110955025*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) & ] - 11*Sqrt[2]*RootSum[-56 - 26*Sqrt[2]*#1 + 17*#1^2 + 6*Sqrt[2]*#1^3 - 5*#1^4 & , (492740319684*Sqrt [2]*Log[-(Sqrt[2]*x) + Sqrt[3 - x + 2*x^2] - #1] - 128644699540*Log[-(Sqrt [2]*x) + Sqrt[3 - x + 2*x^2] - #1]*#1 + 55365920925*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) & ])/138143750000
Time = 0.89 (sec) , antiderivative size = 288, normalized size of antiderivative = 1.02, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {1302, 27, 2132, 27, 2138, 27, 2143, 27, 1090, 222, 1368, 27, 1362, 217, 219}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\left (2 x^2-x+3\right )^{5/2}}{\left (5 x^2+3 x+2\right )^3} \, dx\) |
| \(\Big \downarrow \) 1302 |
| \(\displaystyle \frac {(10 x+3) \left (2 x^2-x+3\right )^{5/2}}{62 \left (5 x^2+3 x+2\right )^2}-\frac {1}{62} \int -\frac {5 \left (-16 x^2-14 x+39\right ) \left (2 x^2-x+3\right )^{3/2}}{2 \left (5 x^2+3 x+2\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{124} \int \frac {\left (-16 x^2-14 x+39\right ) \left (2 x^2-x+3\right )^{3/2}}{\left (5 x^2+3 x+2\right )^2}dx+\frac {(10 x+3) \left (2 x^2-x+3\right )^{5/2}}{62 \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 2132 |
| \(\displaystyle \frac {5}{124} \left (\frac {(2336 x+769) \left (2 x^2-x+3\right )^{3/2}}{155 \left (5 x^2+3 x+2\right )}-\frac {1}{155} \int -\frac {\left (-20672 x^2-5900 x+13347\right ) \sqrt {2 x^2-x+3}}{2 \left (5 x^2+3 x+2\right )}dx\right )+\frac {(10 x+3) \left (2 x^2-x+3\right )^{5/2}}{62 \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{124} \left (\frac {1}{310} \int \frac {\left (-20672 x^2-5900 x+13347\right ) \sqrt {2 x^2-x+3}}{5 x^2+3 x+2}dx+\frac {(2336 x+769) \left (2 x^2-x+3\right )^{3/2}}{155 \left (5 x^2+3 x+2\right )}\right )+\frac {(10 x+3) \left (2 x^2-x+3\right )^{5/2}}{62 \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 2138 |
| \(\displaystyle \frac {5}{124} \left (\frac {1}{310} \left (\frac {4}{25} (11359-12920 x) \sqrt {2 x^2-x+3}-\frac {1}{100} \int -\frac {4 \left (61504 x^2-579685 x+1356541\right )}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx\right )+\frac {(2336 x+769) \left (2 x^2-x+3\right )^{3/2}}{155 \left (5 x^2+3 x+2\right )}\right )+\frac {(10 x+3) \left (2 x^2-x+3\right )^{5/2}}{62 \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{124} \left (\frac {1}{310} \left (\frac {1}{25} \int \frac {61504 x^2-579685 x+1356541}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx+\frac {4}{25} \sqrt {2 x^2-x+3} (11359-12920 x)\right )+\frac {(2336 x+769) \left (2 x^2-x+3\right )^{3/2}}{155 \left (5 x^2+3 x+2\right )}\right )+\frac {(10 x+3) \left (2 x^2-x+3\right )^{5/2}}{62 \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 2143 |
| \(\displaystyle \frac {5}{124} \left (\frac {1}{310} \left (\frac {1}{25} \left (\frac {61504}{5} \int \frac {1}{\sqrt {2 x^2-x+3}}dx+\frac {1}{5} \int \frac {11 (605427-280267 x)}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx\right )+\frac {4}{25} \sqrt {2 x^2-x+3} (11359-12920 x)\right )+\frac {(2336 x+769) \left (2 x^2-x+3\right )^{3/2}}{155 \left (5 x^2+3 x+2\right )}\right )+\frac {(10 x+3) \left (2 x^2-x+3\right )^{5/2}}{62 \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{124} \left (\frac {1}{310} \left (\frac {1}{25} \left (\frac {61504}{5} \int \frac {1}{\sqrt {2 x^2-x+3}}dx+\frac {11}{5} \int \frac {605427-280267 x}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx\right )+\frac {4}{25} \sqrt {2 x^2-x+3} (11359-12920 x)\right )+\frac {(2336 x+769) \left (2 x^2-x+3\right )^{3/2}}{155 \left (5 x^2+3 x+2\right )}\right )+\frac {(10 x+3) \left (2 x^2-x+3\right )^{5/2}}{62 \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 1090 |
| \(\displaystyle \frac {5}{124} \left (\frac {1}{310} \left (\frac {1}{25} \left (\frac {11}{5} \int \frac {605427-280267 x}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx+\frac {30752}{5} \sqrt {\frac {2}{23}} \int \frac {1}{\sqrt {\frac {1}{23} (4 x-1)^2+1}}d(4 x-1)\right )+\frac {4}{25} \sqrt {2 x^2-x+3} (11359-12920 x)\right )+\frac {(2336 x+769) \left (2 x^2-x+3\right )^{3/2}}{155 \left (5 x^2+3 x+2\right )}\right )+\frac {(10 x+3) \left (2 x^2-x+3\right )^{5/2}}{62 \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 222 |
| \(\displaystyle \frac {5}{124} \left (\frac {1}{310} \left (\frac {1}{25} \left (\frac {11}{5} \int \frac {605427-280267 x}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx+\frac {30752}{5} \sqrt {2} \text {arcsinh}\left (\frac {4 x-1}{\sqrt {23}}\right )\right )+\frac {4}{25} \sqrt {2 x^2-x+3} (11359-12920 x)\right )+\frac {(2336 x+769) \left (2 x^2-x+3\right )^{3/2}}{155 \left (5 x^2+3 x+2\right )}\right )+\frac {(10 x+3) \left (2 x^2-x+3\right )^{5/2}}{62 \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 1368 |
| \(\displaystyle \frac {5}{124} \left (\frac {1}{310} \left (\frac {1}{25} \left (\frac {11}{5} \left (\frac {\int -\frac {11 \left (-\left (\left (325160-280267 \sqrt {2}\right ) x\right )-605427 \sqrt {2}+885694\right )}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx}{22 \sqrt {2}}-\frac {\int -\frac {11 \left (-\left (\left (325160+280267 \sqrt {2}\right ) x\right )+605427 \sqrt {2}+885694\right )}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx}{22 \sqrt {2}}\right )+\frac {30752}{5} \sqrt {2} \text {arcsinh}\left (\frac {4 x-1}{\sqrt {23}}\right )\right )+\frac {4}{25} \sqrt {2 x^2-x+3} (11359-12920 x)\right )+\frac {(2336 x+769) \left (2 x^2-x+3\right )^{3/2}}{155 \left (5 x^2+3 x+2\right )}\right )+\frac {(10 x+3) \left (2 x^2-x+3\right )^{5/2}}{62 \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{124} \left (\frac {1}{310} \left (\frac {1}{25} \left (\frac {11}{5} \left (\frac {\int \frac {-\left (\left (325160+280267 \sqrt {2}\right ) x\right )+605427 \sqrt {2}+885694}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx}{2 \sqrt {2}}-\frac {\int \frac {-\left (\left (325160-280267 \sqrt {2}\right ) x\right )-605427 \sqrt {2}+885694}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx}{2 \sqrt {2}}\right )+\frac {30752}{5} \sqrt {2} \text {arcsinh}\left (\frac {4 x-1}{\sqrt {23}}\right )\right )+\frac {4}{25} \sqrt {2 x^2-x+3} (11359-12920 x)\right )+\frac {(2336 x+769) \left (2 x^2-x+3\right )^{3/2}}{155 \left (5 x^2+3 x+2\right )}\right )+\frac {(10 x+3) \left (2 x^2-x+3\right )^{5/2}}{62 \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 1362 |
| \(\displaystyle \frac {5}{124} \left (\frac {1}{310} \left (\frac {1}{25} \left (\frac {11}{5} \left (\sqrt {2} \left (3531015707557-2498852071250 \sqrt {2}\right ) \int \frac {1}{-\frac {11 \left (\left (9832420-6895071 \sqrt {2}\right ) x-2937349 \sqrt {2}+3957722\right )^2}{2 x^2-x+3}-62 \left (3531015707557-2498852071250 \sqrt {2}\right )}d\frac {\left (9832420-6895071 \sqrt {2}\right ) x-2937349 \sqrt {2}+3957722}{\sqrt {2 x^2-x+3}}-\sqrt {2} \left (3531015707557+2498852071250 \sqrt {2}\right ) \int \frac {1}{-\frac {11 \left (\left (9832420+6895071 \sqrt {2}\right ) x+2937349 \sqrt {2}+3957722\right )^2}{2 x^2-x+3}-62 \left (3531015707557+2498852071250 \sqrt {2}\right )}d\frac {\left (9832420+6895071 \sqrt {2}\right ) x+2937349 \sqrt {2}+3957722}{\sqrt {2 x^2-x+3}}\right )+\frac {30752}{5} \sqrt {2} \text {arcsinh}\left (\frac {4 x-1}{\sqrt {23}}\right )\right )+\frac {4}{25} \sqrt {2 x^2-x+3} (11359-12920 x)\right )+\frac {(2336 x+769) \left (2 x^2-x+3\right )^{3/2}}{155 \left (5 x^2+3 x+2\right )}\right )+\frac {(10 x+3) \left (2 x^2-x+3\right )^{5/2}}{62 \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {5}{124} \left (\frac {1}{310} \left (\frac {1}{25} \left (\frac {11}{5} \left (\sqrt {2} \left (3531015707557-2498852071250 \sqrt {2}\right ) \int \frac {1}{-\frac {11 \left (\left (9832420-6895071 \sqrt {2}\right ) x-2937349 \sqrt {2}+3957722\right )^2}{2 x^2-x+3}-62 \left (3531015707557-2498852071250 \sqrt {2}\right )}d\frac {\left (9832420-6895071 \sqrt {2}\right ) x-2937349 \sqrt {2}+3957722}{\sqrt {2 x^2-x+3}}+\sqrt {\frac {1}{341} \left (3531015707557+2498852071250 \sqrt {2}\right )} \arctan \left (\frac {\sqrt {\frac {11}{62 \left (3531015707557+2498852071250 \sqrt {2}\right )}} \left (\left (9832420+6895071 \sqrt {2}\right ) x+2937349 \sqrt {2}+3957722\right )}{\sqrt {2 x^2-x+3}}\right )\right )+\frac {30752}{5} \sqrt {2} \text {arcsinh}\left (\frac {4 x-1}{\sqrt {23}}\right )\right )+\frac {4}{25} \sqrt {2 x^2-x+3} (11359-12920 x)\right )+\frac {(2336 x+769) \left (2 x^2-x+3\right )^{3/2}}{155 \left (5 x^2+3 x+2\right )}\right )+\frac {(10 x+3) \left (2 x^2-x+3\right )^{5/2}}{62 \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {5}{124} \left (\frac {1}{310} \left (\frac {1}{25} \left (\frac {30752}{5} \sqrt {2} \text {arcsinh}\left (\frac {4 x-1}{\sqrt {23}}\right )+\frac {11}{5} \left (\sqrt {\frac {1}{341} \left (3531015707557+2498852071250 \sqrt {2}\right )} \arctan \left (\frac {\sqrt {\frac {11}{62 \left (3531015707557+2498852071250 \sqrt {2}\right )}} \left (\left (9832420+6895071 \sqrt {2}\right ) x+2937349 \sqrt {2}+3957722\right )}{\sqrt {2 x^2-x+3}}\right )+\frac {\left (3531015707557-2498852071250 \sqrt {2}\right ) \text {arctanh}\left (\frac {\sqrt {\frac {11}{62 \left (2498852071250 \sqrt {2}-3531015707557\right )}} \left (\left (9832420-6895071 \sqrt {2}\right ) x-2937349 \sqrt {2}+3957722\right )}{\sqrt {2 x^2-x+3}}\right )}{\sqrt {341 \left (2498852071250 \sqrt {2}-3531015707557\right )}}\right )\right )+\frac {4}{25} \sqrt {2 x^2-x+3} (11359-12920 x)\right )+\frac {(2336 x+769) \left (2 x^2-x+3\right )^{3/2}}{155 \left (5 x^2+3 x+2\right )}\right )+\frac {(10 x+3) \left (2 x^2-x+3\right )^{5/2}}{62 \left (5 x^2+3 x+2\right )^2}\) |
((3 + 10*x)*(3 - x + 2*x^2)^(5/2))/(62*(2 + 3*x + 5*x^2)^2) + (5*(((769 + 2336*x)*(3 - x + 2*x^2)^(3/2))/(155*(2 + 3*x + 5*x^2)) + ((4*(11359 - 1292 0*x)*Sqrt[3 - x + 2*x^2])/25 + ((30752*Sqrt[2]*ArcSinh[(-1 + 4*x)/Sqrt[23] ])/5 + (11*(Sqrt[(3531015707557 + 2498852071250*Sqrt[2])/341]*ArcTan[(Sqrt [11/(62*(3531015707557 + 2498852071250*Sqrt[2]))]*(3957722 + 2937349*Sqrt[ 2] + (9832420 + 6895071*Sqrt[2])*x))/Sqrt[3 - x + 2*x^2]] + ((353101570755 7 - 2498852071250*Sqrt[2])*ArcTanh[(Sqrt[11/(62*(-3531015707557 + 24988520 71250*Sqrt[2]))]*(3957722 - 2937349*Sqrt[2] + (9832420 - 6895071*Sqrt[2])* x))/Sqrt[3 - x + 2*x^2]])/Sqrt[341*(-3531015707557 + 2498852071250*Sqrt[2] )]))/5)/25)/310))/124
3.1.78.3.1 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[(A*b*c - 2*a*B*c + a*b*C - (c*(b*B - 2*A*c) - C*(b^2 - 2*a*c))*x)*(a + b*x + c*x^2)^(p + 1)*((d + e*x + f*x^2)^q/(c*(b^2 - 4*a*c)*(p + 1))), x] - Simp[1/(c*(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*(A*b*c - 2*a*B*c + a*b*C) - d*(c*(b*B - 2*A*c)*(2*p + 3) + C*(2*a*c - b^2*(p + 2))) + (2*f*q*(A*b*c - 2*a*B*c + a*b*C) - e*(c*(b*B - 2*A*c)*(2*p + q + 3) + C*(2*a*c*(q + 1) - b^2*(p + q + 2))))*x - f*(c*(b*B - 2*A*c)*(2*p + 2*q + 3) + C*(2*a*c*(2*q + 1) - b^2*(p + 2*q + 2)))*x^2, x], x], x]] /; FreeQ[{a, b, c, d, e, f}, x] && PolyQ[Px, x, 2] && LtQ[p, -1] && GtQ[q, 0] && !IGtQ[q, 0]
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 = 5.00 (sec) , antiderivative size = 613, normalized size of antiderivative = 2.18
| method | result | size |
| trager | \(\text {Expression too large to display}\) | \(613\) |
| risch | \(\frac {11 \left (97155 x^{3}+93872 x^{2}+69621 x +22552\right ) \sqrt {2 x^{2}-x +3}}{96100 \left (5 x^{2}+3 x +2\right )^{2}}+\frac {4 \sqrt {2}\, \operatorname {arcsinh}\left (\frac {4 \sqrt {23}\, \left (x -\frac {1}{4}\right )}{23}\right )}{125}+\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 (132861440 \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 )+187960123 \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 )+197090660657 \,\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}-271286828868 \,\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 )}{923521000 \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}}}\) | \(740\) |
| default | \(\text {Expression too large to display}\) | \(119458\) |
11/96100*(97155*x^3+93872*x^2+69621*x+22552)/(5*x^2+3*x+2)^2*(2*x^2-x+3)^( 1/2)+1/24025*RootOf(49203200*_Z^4+38530443400861984*_Z^2+75555566255284022 42640625)*ln((36815059399680000*x*RootOf(49203200*_Z^4+38530443400861984*_ Z^2+7555556625528402242640625)^5+35093779815505808148083200*RootOf(4920320 0*_Z^4+38530443400861984*_Z^2+7555556625528402242640625)^3*x+2403325288603 23273780028700000*(2*x^2-x+3)^(1/2)*RootOf(49203200*_Z^4+38530443400861984 *_Z^2+7555556625528402242640625)^2+1823557958071135735680000*RootOf(492032 00*_Z^4+38530443400861984*_Z^2+7555556625528402242640625)^3+79667570132996 79497362622070599592*RootOf(49203200*_Z^4+38530443400861984*_Z^2+755555662 5528402242640625)*x+94293965883068184162712639837646059625*(2*x^2-x+3)^(1/ 2)+1058392736159831951768700463021600*RootOf(49203200*_Z^4+385304434008619 84*_Z^2+7555556625528402242640625))/(24800*x*RootOf(49203200*_Z^4+38530443 400861984*_Z^2+7555556625528402242640625)^2+9922195093316*x+282535863379)) +1/29791000*RootOf(_Z^2+1537600*RootOf(49203200*_Z^4+38530443400861984*_Z^ 2+7555556625528402242640625)^2+1204076356276937)*ln((29452047519744*RootOf (_Z^2+1537600*RootOf(49203200*_Z^4+38530443400861984*_Z^2+7555556625528402 242640625)^2+1204076356276937)*RootOf(49203200*_Z^4+38530443400861984*_Z^2 +7555556625528402242640625)^4*x+18052075604500342109056*RootOf(49203200*_Z ^4+38530443400861984*_Z^2+7555556625528402242640625)^2*RootOf(_Z^2+1537600 *RootOf(49203200*_Z^4+38530443400861984*_Z^2+7555556625528402242640625)...
Result contains complex when optimal does not.
Time = 0.38 (sec) , antiderivative size = 454, normalized size of antiderivative = 1.62 \[ \int \frac {\left (3-x+2 x^2\right )^{5/2}}{\left (2+3 x+5 x^2\right )^3} \, dx=\frac {\sqrt {62} {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} \sqrt {282535863379 i \, \sqrt {31} - 38841172783127} \log \left (\frac {\sqrt {62} \sqrt {2 \, x^{2} - x + 3} \sqrt {282535863379 i \, \sqrt {31} - 38841172783127} {\left (1978861 i \, \sqrt {31} - 13728257\right )} - 38732207104375 \, \sqrt {31} {\left (-i \, x + 6 i\right )} + 735911934983125 \, x - 852108556296250}{x}\right ) - \sqrt {62} {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} \sqrt {282535863379 i \, \sqrt {31} - 38841172783127} \log \left (\frac {\sqrt {62} \sqrt {2 \, x^{2} - x + 3} \sqrt {282535863379 i \, \sqrt {31} - 38841172783127} {\left (-1978861 i \, \sqrt {31} + 13728257\right )} - 38732207104375 \, \sqrt {31} {\left (-i \, x + 6 i\right )} + 735911934983125 \, x - 852108556296250}{x}\right ) - \sqrt {62} {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} \sqrt {-282535863379 i \, \sqrt {31} - 38841172783127} \log \left (\frac {\sqrt {62} \sqrt {2 \, x^{2} - x + 3} {\left (1978861 i \, \sqrt {31} + 13728257\right )} \sqrt {-282535863379 i \, \sqrt {31} - 38841172783127} - 38732207104375 \, \sqrt {31} {\left (i \, x - 6 i\right )} + 735911934983125 \, x - 852108556296250}{x}\right ) + \sqrt {62} {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} \sqrt {-282535863379 i \, \sqrt {31} - 38841172783127} \log \left (\frac {\sqrt {62} \sqrt {2 \, x^{2} - x + 3} {\left (-1978861 i \, \sqrt {31} - 13728257\right )} \sqrt {-282535863379 i \, \sqrt {31} - 38841172783127} - 38732207104375 \, \sqrt {31} {\left (i \, x - 6 i\right )} + 735911934983125 \, x - 852108556296250}{x}\right ) + 1906624 \, \sqrt {2} {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\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 \, {\left (97155 \, x^{3} + 93872 \, x^{2} + 69621 \, x + 22552\right )} \sqrt {2 \, x^{2} - x + 3}}{119164000 \, {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )}} \]
1/119164000*(sqrt(62)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)*sqrt(282535863 379*I*sqrt(31) - 38841172783127)*log((sqrt(62)*sqrt(2*x^2 - x + 3)*sqrt(28 2535863379*I*sqrt(31) - 38841172783127)*(1978861*I*sqrt(31) - 13728257) - 38732207104375*sqrt(31)*(-I*x + 6*I) + 735911934983125*x - 852108556296250 )/x) - sqrt(62)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)*sqrt(282535863379*I* sqrt(31) - 38841172783127)*log((sqrt(62)*sqrt(2*x^2 - x + 3)*sqrt(28253586 3379*I*sqrt(31) - 38841172783127)*(-1978861*I*sqrt(31) + 13728257) - 38732 207104375*sqrt(31)*(-I*x + 6*I) + 735911934983125*x - 852108556296250)/x) - sqrt(62)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)*sqrt(-282535863379*I*sqrt (31) - 38841172783127)*log((sqrt(62)*sqrt(2*x^2 - x + 3)*(1978861*I*sqrt(3 1) + 13728257)*sqrt(-282535863379*I*sqrt(31) - 38841172783127) - 387322071 04375*sqrt(31)*(I*x - 6*I) + 735911934983125*x - 852108556296250)/x) + sqr t(62)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)*sqrt(-282535863379*I*sqrt(31) - 38841172783127)*log((sqrt(62)*sqrt(2*x^2 - x + 3)*(-1978861*I*sqrt(31) - 13728257)*sqrt(-282535863379*I*sqrt(31) - 38841172783127) - 3873220710437 5*sqrt(31)*(I*x - 6*I) + 735911934983125*x - 852108556296250)/x) + 1906624 *sqrt(2)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)*log(-4*sqrt(2)*sqrt(2*x^2 - x + 3)*(4*x - 1) - 32*x^2 + 16*x - 25) + 13640*(97155*x^3 + 93872*x^2 + 6 9621*x + 22552)*sqrt(2*x^2 - x + 3))/(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)
\[ \int \frac {\left (3-x+2 x^2\right )^{5/2}}{\left (2+3 x+5 x^2\right )^3} \, dx=\int \frac {\left (2 x^{2} - x + 3\right )^{\frac {5}{2}}}{\left (5 x^{2} + 3 x + 2\right )^{3}}\, dx \]
\[ \int \frac {\left (3-x+2 x^2\right )^{5/2}}{\left (2+3 x+5 x^2\right )^3} \, dx=\int { \frac {{\left (2 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{{\left (5 \, x^{2} + 3 \, x + 2\right )}^{3}} \,d x } \]
Exception generated. \[ \int \frac {\left (3-x+2 x^2\right )^{5/2}}{\left (2+3 x+5 x^2\right )^3} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Francis algorithm failure for[-1.0, infinity,infinity,infinity,infinity]proot error [1.0,infinity,infinity,inf inity,inf
Timed out. \[ \int \frac {\left (3-x+2 x^2\right )^{5/2}}{\left (2+3 x+5 x^2\right )^3} \, dx=\int \frac {{\left (2\,x^2-x+3\right )}^{5/2}}{{\left (5\,x^2+3\,x+2\right )}^3} \,d x \]