Integrand size = 25, antiderivative size = 1056 \[ \int \frac {A+B x+C x^2}{\left (2-4 x^2+3 x^3\right )^2} \, dx =\text {Too large to display} \] Output:
9*(179-9*345^(1/2))^(1/3)*(16*A+27*B+24*C)/(3680+230*(179-9*345^(1/2))^(2/ 3)-230*(179-9*345^(1/2))^(1/3)*(4-9*x))-9/230*(-9315+537*345^(1/2))^(1/3)* (288*A+486*B+432*C+(9*(179-9*345^(1/2)+16*(179-9*345^(1/2))^(1/3))*A+(27-3 45^(1/2))*(4+(179-9*345^(1/2))^(1/3))*(9*B+8*C))*(4-9*x)/(179-9*345^(1/2)) ^(2/3))*115^(1/3)/(16-(179-9*345^(1/2))^(2/3))/(16-256/(179-9*345^(1/2))^( 2/3)-(179-9*345^(1/2))^(2/3)-(16+(179-9*345^(1/2))^(2/3))*(4-9*x)/(179-9*3 45^(1/2))^(1/3)-(4-9*x)^2)/(4-16/(179-9*345^(1/2))^(1/3)-(179-9*345^(1/2)) ^(1/3)-9*x)-C/(27*x^3-36*x^2+18)-17496*6^(1/2)/(29993-1611*345^(1/2)+128*( 179-9*345^(1/2))^(2/3)-16*(179-9*345^(1/2))^(4/3))^(1/2)*((9736952672-5241 97920*345^(1/2)+(319636187-17206017*345^(1/2))*(179-9*345^(1/2))^(2/3))*A+ (1229351761-66184515*345^(1/2)+4*(5665339-304929*345^(1/2))*(179-9*345^(1/ 2))^(2/3))*(9*B+8*C))*arctan((179-9*345^(1/2)+16*(179-9*345^(1/2))^(1/3)+2 *(179-9*345^(1/2))^(2/3)*(4-9*x))/(179958-9666*345^(1/2)+768*(179-9*345^(1 /2))^(2/3)-96*(179-9*345^(1/2))^(4/3))^(1/2))/(16-(179-9*345^(1/2))^(2/3)) ^2/(256+16*(179-9*345^(1/2))^(2/3)+(179-9*345^(1/2))^(4/3))^3-2916*(179-9* 345^(1/2))*((479888-25776*345^(1/2)+128*(179-9*345^(1/2))^(4/3)-179*(179-9 *345^(1/2))^(5/3))*A+3*(29993-1611*345^(1/2)+8*(179-9*345^(1/2))^(4/3)-4*( 179-9*345^(1/2))^(5/3))*(9*B+8*C))*ln(16+(179-9*345^(1/2))^(2/3)-(179-9*34 5^(1/2))^(1/3)*(4-9*x))/(16-(179-9*345^(1/2))^(2/3))^2/(256+16*(179-9*345^ (1/2))^(2/3)+(179-9*345^(1/2))^(4/3))^3+2916*((82967216-4466448*345^(1/...
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.03 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.15 \[ \int \frac {A+B x+C x^2}{\left (2-4 x^2+3 x^3\right )^2} \, dx=\frac {1}{230} \left (\frac {2 C \left (-27-16 x+36 x^2\right )+A \left (-36+17 x+48 x^2\right )+B \left (-32-36 x+81 x^2\right )}{2-4 x^2+3 x^3}+\text {RootSum}\left [2-4 \text {$\#$1}^2+3 \text {$\#$1}^3\&,\frac {98 A \log (x-\text {$\#$1})+36 B \log (x-\text {$\#$1})+32 C \log (x-\text {$\#$1})+48 A \log (x-\text {$\#$1}) \text {$\#$1}+81 B \log (x-\text {$\#$1}) \text {$\#$1}+72 C \log (x-\text {$\#$1}) \text {$\#$1}}{-8 \text {$\#$1}+9 \text {$\#$1}^2}\&\right ]\right ) \] Input:
Integrate[(A + B*x + C*x^2)/(2 - 4*x^2 + 3*x^3)^2,x]
Output:
((2*C*(-27 - 16*x + 36*x^2) + A*(-36 + 17*x + 48*x^2) + B*(-32 - 36*x + 81 *x^2))/(2 - 4*x^2 + 3*x^3) + RootSum[2 - 4*#1^2 + 3*#1^3 & , (98*A*Log[x - #1] + 36*B*Log[x - #1] + 32*C*Log[x - #1] + 48*A*Log[x - #1]*#1 + 81*B*Lo g[x - #1]*#1 + 72*C*Log[x - #1]*#1)/(-8*#1 + 9*#1^2) & ])/230
Time = 9.95 (sec) , antiderivative size = 1110, normalized size of antiderivative = 1.05, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {2526, 2490, 2485, 27, 1235, 27, 1200, 2009}
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.
| \(\displaystyle \int \frac {A+B x+C x^2}{\left (3 x^3-4 x^2+2\right )^2} \, dx\) |
| \(\Big \downarrow \) 2526 |
| \(\displaystyle \frac {1}{9} \int \frac {9 A+(9 B+8 C) x}{\left (3 x^3-4 x^2+2\right )^2}dx-\frac {C}{9 \left (3 x^3-4 x^2+2\right )}\) |
| \(\Big \downarrow \) 2490 |
| \(\displaystyle \frac {1}{9} \int \frac {\frac {1}{9} (81 A+4 (9 B+8 C))+(9 B+8 C) \left (x-\frac {4}{9}\right )}{\left (3 \left (x-\frac {4}{9}\right )^3-\frac {16}{9} \left (x-\frac {4}{9}\right )+\frac {358}{243}\right )^2}d\left (x-\frac {4}{9}\right )-\frac {C}{9 \left (3 x^3-4 x^2+2\right )}\) |
| \(\Big \downarrow \) 2485 |
| \(\displaystyle 9 \int \frac {81 \left (81 A+36 B+32 C+9 (9 B+8 C) \left (x-\frac {4}{9}\right )\right )}{\left (9 \left (x-\frac {4}{9}\right )+\frac {16+\left (179-9 \sqrt {345}\right )^{2/3}}{\sqrt [3]{179-9 \sqrt {345}}}\right )^2 \left (-81 \left (x-\frac {4}{9}\right )^2+\frac {9 \left (16+\left (179-9 \sqrt {345}\right )^{2/3}\right ) \left (x-\frac {4}{9}\right )}{\sqrt [3]{179-9 \sqrt {345}}}-\left (179-9 \sqrt {345}\right )^{2/3}-\frac {256}{\left (179-9 \sqrt {345}\right )^{2/3}}+16\right )^2}d\left (x-\frac {4}{9}\right )-\frac {C}{9 \left (3 x^3-4 x^2+2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 729 \int \frac {81 A+36 B+32 C+9 (9 B+8 C) \left (x-\frac {4}{9}\right )}{\left (9 \left (x-\frac {4}{9}\right )+\frac {16+\left (179-9 \sqrt {345}\right )^{2/3}}{\sqrt [3]{179-9 \sqrt {345}}}\right )^2 \left (-81 \left (x-\frac {4}{9}\right )^2+\frac {9 \left (16+\left (179-9 \sqrt {345}\right )^{2/3}\right ) \left (x-\frac {4}{9}\right )}{\sqrt [3]{179-9 \sqrt {345}}}-\left (179-9 \sqrt {345}\right )^{2/3}-\frac {256}{\left (179-9 \sqrt {345}\right )^{2/3}}+16\right )^2}d\left (x-\frac {4}{9}\right )-\frac {C}{9 \left (3 x^3-4 x^2+2\right )}\) |
| \(\Big \downarrow \) 1235 |
| \(\displaystyle 729 \left (\frac {\sqrt [3]{\frac {179}{345 \sqrt {345}}-\frac {3}{115}} \left (2 (16 A+27 B+24 C)-\frac {\left (x-\frac {4}{9}\right ) \left (9 \left (179-9 \sqrt {345}+16 \sqrt [3]{179-9 \sqrt {345}}\right ) A+\left (27-\sqrt {345}\right ) \left (4+\sqrt [3]{179-9 \sqrt {345}}\right ) (9 B+8 C)\right )}{\left (179-9 \sqrt {345}\right )^{2/3}}\right )}{6 \left (16-\left (179-9 \sqrt {345}\right )^{2/3}\right ) \left (9 \left (x-\frac {4}{9}\right )+\frac {16+\left (179-9 \sqrt {345}\right )^{2/3}}{\sqrt [3]{179-9 \sqrt {345}}}\right ) \left (-81 \left (x-\frac {4}{9}\right )^2+\frac {9 \left (16+\left (179-9 \sqrt {345}\right )^{2/3}\right ) \left (x-\frac {4}{9}\right )}{\sqrt [3]{179-9 \sqrt {345}}}-\left (179-9 \sqrt {345}\right )^{2/3}-\frac {256}{\left (179-9 \sqrt {345}\right )^{2/3}}+16\right )}-\frac {\sqrt [3]{\frac {179}{345 \sqrt {345}}-\frac {3}{115}} \int -\frac {354294 \left (\frac {9 \left (2864-144 \sqrt {345}-256 \sqrt [3]{179-9 \sqrt {345}}-\left (179-9 \sqrt {345}\right )^{5/3}\right ) A+\left (4833-243 \sqrt {345}-16 \sqrt [3]{179-9 \sqrt {345}} \left (27-\sqrt {345}\right )-4 \left (179-9 \sqrt {345}\right )^{2/3} \left (27-\sqrt {345}\right )\right ) (9 B+8 C)}{179-9 \sqrt {345}}-\frac {9 \left (9 \left (179-9 \sqrt {345}+16 \sqrt [3]{179-9 \sqrt {345}}\right ) A+\left (27-\sqrt {345}\right ) \left (4+\sqrt [3]{179-9 \sqrt {345}}\right ) (9 B+8 C)\right ) \left (x-\frac {4}{9}\right )}{\left (179-9 \sqrt {345}\right )^{2/3}}\right )}{\left (9 \left (x-\frac {4}{9}\right )+\frac {16+\left (179-9 \sqrt {345}\right )^{2/3}}{\sqrt [3]{179-9 \sqrt {345}}}\right )^2 \left (-81 \left (x-\frac {4}{9}\right )^2+\frac {9 \left (16+\left (179-9 \sqrt {345}\right )^{2/3}\right ) \left (x-\frac {4}{9}\right )}{\sqrt [3]{179-9 \sqrt {345}}}-\left (179-9 \sqrt {345}\right )^{2/3}-\frac {256}{\left (179-9 \sqrt {345}\right )^{2/3}}+16\right )}d\left (x-\frac {4}{9}\right )}{1062882 \left (16-\left (179-9 \sqrt {345}\right )^{2/3}\right )}\right )-\frac {C}{9 \left (3 x^3-4 x^2+2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 729 \left (\frac {\sqrt [3]{\frac {179}{345 \sqrt {345}}-\frac {3}{115}} \int \frac {\frac {9 \left (2864-144 \sqrt {345}-256 \sqrt [3]{179-9 \sqrt {345}}-\left (179-9 \sqrt {345}\right )^{5/3}\right ) A+\left (4833-243 \sqrt {345}-16 \sqrt [3]{179-9 \sqrt {345}} \left (27-\sqrt {345}\right )-4 \left (179-9 \sqrt {345}\right )^{2/3} \left (27-\sqrt {345}\right )\right ) (9 B+8 C)}{179-9 \sqrt {345}}-\frac {9 \left (9 \left (179-9 \sqrt {345}+16 \sqrt [3]{179-9 \sqrt {345}}\right ) A+\left (27-\sqrt {345}\right ) \left (4+\sqrt [3]{179-9 \sqrt {345}}\right ) (9 B+8 C)\right ) \left (x-\frac {4}{9}\right )}{\left (179-9 \sqrt {345}\right )^{2/3}}}{\left (9 \left (x-\frac {4}{9}\right )+\frac {16+\left (179-9 \sqrt {345}\right )^{2/3}}{\sqrt [3]{179-9 \sqrt {345}}}\right )^2 \left (-81 \left (x-\frac {4}{9}\right )^2+\frac {9 \left (16+\left (179-9 \sqrt {345}\right )^{2/3}\right ) \left (x-\frac {4}{9}\right )}{\sqrt [3]{179-9 \sqrt {345}}}-\left (179-9 \sqrt {345}\right )^{2/3}-\frac {256}{\left (179-9 \sqrt {345}\right )^{2/3}}+16\right )}d\left (x-\frac {4}{9}\right )}{3 \left (16-\left (179-9 \sqrt {345}\right )^{2/3}\right )}+\frac {\sqrt [3]{\frac {179}{345 \sqrt {345}}-\frac {3}{115}} \left (2 (16 A+27 B+24 C)-\frac {\left (x-\frac {4}{9}\right ) \left (9 \left (179-9 \sqrt {345}+16 \sqrt [3]{179-9 \sqrt {345}}\right ) A+\left (27-\sqrt {345}\right ) \left (4+\sqrt [3]{179-9 \sqrt {345}}\right ) (9 B+8 C)\right )}{\left (179-9 \sqrt {345}\right )^{2/3}}\right )}{6 \left (16-\left (179-9 \sqrt {345}\right )^{2/3}\right ) \left (9 \left (x-\frac {4}{9}\right )+\frac {16+\left (179-9 \sqrt {345}\right )^{2/3}}{\sqrt [3]{179-9 \sqrt {345}}}\right ) \left (-81 \left (x-\frac {4}{9}\right )^2+\frac {9 \left (16+\left (179-9 \sqrt {345}\right )^{2/3}\right ) \left (x-\frac {4}{9}\right )}{\sqrt [3]{179-9 \sqrt {345}}}-\left (179-9 \sqrt {345}\right )^{2/3}-\frac {256}{\left (179-9 \sqrt {345}\right )^{2/3}}+16\right )}\right )-\frac {C}{9 \left (3 x^3-4 x^2+2\right )}\) |
| \(\Big \downarrow \) 1200 |
| \(\displaystyle 729 \left (\frac {\sqrt [3]{-\frac {3}{115}+\frac {179}{345 \sqrt {345}}} \left (2 (16 A+27 B+24 C)-\frac {\left (9 \left (179-9 \sqrt {345}+16 \sqrt [3]{179-9 \sqrt {345}}\right ) A+\left (27-\sqrt {345}\right ) \left (4+\sqrt [3]{179-9 \sqrt {345}}\right ) (9 B+8 C)\right ) \left (x-\frac {4}{9}\right )}{\left (179-9 \sqrt {345}\right )^{2/3}}\right )}{6 \left (16-\left (179-9 \sqrt {345}\right )^{2/3}\right ) \left (9 \left (x-\frac {4}{9}\right )+\frac {16+\left (179-9 \sqrt {345}\right )^{2/3}}{\sqrt [3]{179-9 \sqrt {345}}}\right ) \left (-81 \left (x-\frac {4}{9}\right )^2+\frac {9 \left (16+\left (179-9 \sqrt {345}\right )^{2/3}\right ) \left (x-\frac {4}{9}\right )}{\sqrt [3]{179-9 \sqrt {345}}}-\left (179-9 \sqrt {345}\right )^{2/3}-\frac {256}{\left (179-9 \sqrt {345}\right )^{2/3}}+16\right )}+\frac {\sqrt [3]{-\frac {3}{115}+\frac {179}{345 \sqrt {345}}} \int \left (\frac {18 \left (29993-1611 \sqrt {345}\right ) (-16 A-27 B-24 C)}{\left (179-9 \sqrt {345}\right )^{2/3} \left (256+16 \left (179-9 \sqrt {345}\right )^{2/3}+\left (179-9 \sqrt {345}\right )^{4/3}\right ) \left (9 \sqrt [3]{179-9 \sqrt {345}} \left (x-\frac {4}{9}\right )+\left (179-9 \sqrt {345}\right )^{2/3}+16\right )^2}+\frac {6 \left (-\left (\left (479888-25776 \sqrt {345}+128 \left (179-9 \sqrt {345}\right )^{4/3}-179 \left (179-9 \sqrt {345}\right )^{5/3}\right ) A\right )-3 \left (29993-1611 \sqrt {345}+8 \left (179-9 \sqrt {345}\right )^{4/3}-4 \left (179-9 \sqrt {345}\right )^{5/3}\right ) (9 B+8 C)\right )}{\left (256+16 \left (179-9 \sqrt {345}\right )^{2/3}+\left (179-9 \sqrt {345}\right )^{4/3}\right )^2 \left (9 \sqrt [3]{179-9 \sqrt {345}} \left (x-\frac {4}{9}\right )+\left (179-9 \sqrt {345}\right )^{2/3}+16\right )}+\frac {12 \left (4 \left (943552145-50793219 \sqrt {345}+\left (63691780-3423564 \sqrt {345}\right ) \sqrt [3]{179-9 \sqrt {345}}-32 \left (29993-1611 \sqrt {345}\right ) \left (179-9 \sqrt {345}\right )^{2/3}\right ) A+3 \left (86806320-4672656 \sqrt {345}+\left (9024555-485361 \sqrt {345}\right ) \sqrt [3]{179-9 \sqrt {345}}-8 \left (29993-1611 \sqrt {345}\right ) \left (179-9 \sqrt {345}\right )^{2/3}\right ) (9 B+8 C)+9 \left (\left (82967216-4466448 \sqrt {345}+128 \left (29993-1611 \sqrt {345}\right ) \sqrt [3]{179-9 \sqrt {345}}-179 \left (29993-1611 \sqrt {345}\right ) \left (179-9 \sqrt {345}\right )^{2/3}\right ) A+3 \left (5185451-279153 \sqrt {345}+8 \left (29993-1611 \sqrt {345}\right ) \sqrt [3]{179-9 \sqrt {345}}-4 \left (29993-1611 \sqrt {345}\right ) \left (179-9 \sqrt {345}\right )^{2/3}\right ) (9 B+8 C)\right ) \left (x-\frac {4}{9}\right )\right )}{\left (179-9 \sqrt {345}\right )^{2/3} \left (256+16 \left (179-9 \sqrt {345}\right )^{2/3}+\left (179-9 \sqrt {345}\right )^{4/3}\right )^2 \left (81 \left (179-9 \sqrt {345}\right )^{2/3} \left (x-\frac {4}{9}\right )^2-9 \left (179-9 \sqrt {345}+16 \sqrt [3]{179-9 \sqrt {345}}\right ) \left (x-\frac {4}{9}\right )+\left (179-9 \sqrt {345}\right )^{4/3}-16 \left (179-9 \sqrt {345}\right )^{2/3}+256\right )}\right )d\left (x-\frac {4}{9}\right )}{3 \left (16-\left (179-9 \sqrt {345}\right )^{2/3}\right )}\right )-\frac {C}{9 \left (3 x^3-4 x^2+2\right )}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 729 \left (\frac {\sqrt [3]{-\frac {3}{115}+\frac {179}{345 \sqrt {345}}} \left (2 (16 A+27 B+24 C)-\frac {\left (9 \left (179-9 \sqrt {345}+16 \sqrt [3]{179-9 \sqrt {345}}\right ) A+\left (27-\sqrt {345}\right ) \left (4+\sqrt [3]{179-9 \sqrt {345}}\right ) (9 B+8 C)\right ) \left (x-\frac {4}{9}\right )}{\left (179-9 \sqrt {345}\right )^{2/3}}\right )}{6 \left (16-\left (179-9 \sqrt {345}\right )^{2/3}\right ) \left (9 \left (x-\frac {4}{9}\right )+\frac {16+\left (179-9 \sqrt {345}\right )^{2/3}}{\sqrt [3]{179-9 \sqrt {345}}}\right ) \left (-81 \left (x-\frac {4}{9}\right )^2+\frac {9 \left (16+\left (179-9 \sqrt {345}\right )^{2/3}\right ) \left (x-\frac {4}{9}\right )}{\sqrt [3]{179-9 \sqrt {345}}}-\left (179-9 \sqrt {345}\right )^{2/3}-\frac {256}{\left (179-9 \sqrt {345}\right )^{2/3}}+16\right )}+\frac {\sqrt [3]{-\frac {3}{115}+\frac {179}{345 \sqrt {345}}} \left (\frac {2 \left (29993-1611 \sqrt {345}\right ) (16 A+27 B+24 C)}{\left (179-9 \sqrt {345}\right ) \left (256+16 \left (179-9 \sqrt {345}\right )^{2/3}+\left (179-9 \sqrt {345}\right )^{4/3}\right ) \left (9 \sqrt [3]{179-9 \sqrt {345}} \left (x-\frac {4}{9}\right )+\left (179-9 \sqrt {345}\right )^{2/3}+16\right )}-\frac {4 \sqrt {\frac {6}{29993-1611 \sqrt {345}+128 \left (179-9 \sqrt {345}\right )^{2/3}-16 \left (179-9 \sqrt {345}\right )^{4/3}}} \left (\left (9736952672-524197920 \sqrt {345}+\left (319636187-17206017 \sqrt {345}\right ) \left (179-9 \sqrt {345}\right )^{2/3}\right ) A+\left (1229351761-66184515 \sqrt {345}+4 \left (5665339-304929 \sqrt {345}\right ) \left (179-9 \sqrt {345}\right )^{2/3}\right ) (9 B+8 C)\right ) \arctan \left (\frac {-18 \left (179-9 \sqrt {345}\right )^{2/3} \left (x-\frac {4}{9}\right )+16 \sqrt [3]{179-9 \sqrt {345}}-9 \sqrt {345}+179}{\sqrt {6 \left (29993-1611 \sqrt {345}+128 \left (179-9 \sqrt {345}\right )^{2/3}-16 \left (179-9 \sqrt {345}\right )^{4/3}\right )}}\right )}{\left (179-9 \sqrt {345}\right )^{4/3} \left (256+16 \left (179-9 \sqrt {345}\right )^{2/3}+\left (179-9 \sqrt {345}\right )^{4/3}\right )^2}-\frac {2 \left (\left (479888-25776 \sqrt {345}+128 \left (179-9 \sqrt {345}\right )^{4/3}-179 \left (179-9 \sqrt {345}\right )^{5/3}\right ) A+3 \left (29993-1611 \sqrt {345}+8 \left (179-9 \sqrt {345}\right )^{4/3}-4 \left (179-9 \sqrt {345}\right )^{5/3}\right ) (9 B+8 C)\right ) \log \left (9 \sqrt [3]{179-9 \sqrt {345}} \left (x-\frac {4}{9}\right )+\left (179-9 \sqrt {345}\right )^{2/3}+16\right )}{3 \sqrt [3]{179-9 \sqrt {345}} \left (256+16 \left (179-9 \sqrt {345}\right )^{2/3}+\left (179-9 \sqrt {345}\right )^{4/3}\right )^2}+\frac {2 \left (\left (82967216-4466448 \sqrt {345}+128 \left (29993-1611 \sqrt {345}\right ) \sqrt [3]{179-9 \sqrt {345}}-179 \left (29993-1611 \sqrt {345}\right ) \left (179-9 \sqrt {345}\right )^{2/3}\right ) A+3 \left (5185451-279153 \sqrt {345}+8 \left (29993-1611 \sqrt {345}\right ) \sqrt [3]{179-9 \sqrt {345}}-4 \left (29993-1611 \sqrt {345}\right ) \left (179-9 \sqrt {345}\right )^{2/3}\right ) (9 B+8 C)\right ) \log \left (81 \left (179-9 \sqrt {345}\right )^{2/3} \left (x-\frac {4}{9}\right )^2-9 \left (179-9 \sqrt {345}+16 \sqrt [3]{179-9 \sqrt {345}}\right ) \left (x-\frac {4}{9}\right )+\left (179-9 \sqrt {345}\right )^{4/3}-16 \left (179-9 \sqrt {345}\right )^{2/3}+256\right )}{3 \left (179-9 \sqrt {345}\right )^{4/3} \left (256+16 \left (179-9 \sqrt {345}\right )^{2/3}+\left (179-9 \sqrt {345}\right )^{4/3}\right )^2}\right )}{3 \left (16-\left (179-9 \sqrt {345}\right )^{2/3}\right )}\right )-\frac {C}{9 \left (3 x^3-4 x^2+2\right )}\) |
Input:
Int[(A + B*x + C*x^2)/(2 - 4*x^2 + 3*x^3)^2,x]
Output:
-1/9*C/(2 - 4*x^2 + 3*x^3) + 729*(((-3/115 + 179/(345*Sqrt[345]))^(1/3)*(2 *(16*A + 27*B + 24*C) - ((9*(179 - 9*Sqrt[345] + 16*(179 - 9*Sqrt[345])^(1 /3))*A + (27 - Sqrt[345])*(4 + (179 - 9*Sqrt[345])^(1/3))*(9*B + 8*C))*(-4 /9 + x))/(179 - 9*Sqrt[345])^(2/3)))/(6*(16 - (179 - 9*Sqrt[345])^(2/3))*( (16 + (179 - 9*Sqrt[345])^(2/3))/(179 - 9*Sqrt[345])^(1/3) + 9*(-4/9 + x)) *(16 - 256/(179 - 9*Sqrt[345])^(2/3) - (179 - 9*Sqrt[345])^(2/3) + (9*(16 + (179 - 9*Sqrt[345])^(2/3))*(-4/9 + x))/(179 - 9*Sqrt[345])^(1/3) - 81*(- 4/9 + x)^2)) + ((-3/115 + 179/(345*Sqrt[345]))^(1/3)*((2*(29993 - 1611*Sqr t[345])*(16*A + 27*B + 24*C))/((179 - 9*Sqrt[345])*(256 + 16*(179 - 9*Sqrt [345])^(2/3) + (179 - 9*Sqrt[345])^(4/3))*(16 + (179 - 9*Sqrt[345])^(2/3) + 9*(179 - 9*Sqrt[345])^(1/3)*(-4/9 + x))) - (4*Sqrt[6/(29993 - 1611*Sqrt[ 345] + 128*(179 - 9*Sqrt[345])^(2/3) - 16*(179 - 9*Sqrt[345])^(4/3))]*((97 36952672 - 524197920*Sqrt[345] + (319636187 - 17206017*Sqrt[345])*(179 - 9 *Sqrt[345])^(2/3))*A + (1229351761 - 66184515*Sqrt[345] + 4*(5665339 - 304 929*Sqrt[345])*(179 - 9*Sqrt[345])^(2/3))*(9*B + 8*C))*ArcTan[(179 - 9*Sqr t[345] + 16*(179 - 9*Sqrt[345])^(1/3) - 18*(179 - 9*Sqrt[345])^(2/3)*(-4/9 + x))/Sqrt[6*(29993 - 1611*Sqrt[345] + 128*(179 - 9*Sqrt[345])^(2/3) - 16 *(179 - 9*Sqrt[345])^(4/3))]])/((179 - 9*Sqrt[345])^(4/3)*(256 + 16*(179 - 9*Sqrt[345])^(2/3) + (179 - 9*Sqrt[345])^(4/3))^2) - (2*((479888 - 25776* Sqrt[345] + 128*(179 - 9*Sqrt[345])^(4/3) - 179*(179 - 9*Sqrt[345])^(5/...
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.))/((a_.) + (b_.)* (x_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*((f + g* x)^n/(a + b*x + c*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && In tegersQ[n]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2 *a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^m *(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2*(p + m + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d* m + b*e*m) - b*d*(3*c*d - b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p] )
Int[((e_.) + (f_.)*(x_))^(m_.)*((a_) + (b_.)*(x_) + (d_.)*(x_)^3)^(p_), x_S ymbol] :> With[{r = Rt[-9*a*d^2 + Sqrt[3]*d*Sqrt[4*b^3*d + 27*a^2*d^2], 3]} , Simp[1/d^(2*p) Int[(e + f*x)^m*Simp[18^(1/3)*b*(d/(3*r)) - r/18^(1/3) + d*x, x]^p*Simp[b*(d/3) + 12^(1/3)*b^2*(d^2/(3*r^2)) + r^2/(3*12^(1/3)) - d *(2^(1/3)*b*(d/(3^(1/3)*r)) - r/18^(1/3))*x + d^2*x^2, x]^p, x], x]] /; Fre eQ[{a, b, d, e, f, m}, x] && NeQ[4*b^3 + 27*a^2*d, 0] && ILtQ[p, 0]
Int[(P3_)^(p_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> With[{a = Coeff[P3 , x, 0], b = Coeff[P3, x, 1], c = Coeff[P3, x, 2], d = Coeff[P3, x, 3]}, Su bst[Int[((3*d*e - c*f)/(3*d) + f*x)^m*Simp[(2*c^3 - 9*b*c*d + 27*a*d^2)/(27 *d^2) - (c^2 - 3*b*d)*(x/(3*d)) + d*x^3, x]^p, x], x, x + c/(3*d)] /; NeQ[c , 0]] /; FreeQ[{e, f, m, p}, x] && PolyQ[P3, x, 3]
Int[(Pm_)*(Qn_)^(p_), x_Symbol] :> With[{m = Expon[Pm, x], n = Expon[Qn, x] }, Simp[Coeff[Pm, x, m]*(Qn^(p + 1)/(n*(p + 1)*Coeff[Qn, x, n])), x] + Simp [1/(n*Coeff[Qn, x, n]) Int[ExpandToSum[n*Coeff[Qn, x, n]*Pm - Coeff[Pm, x , m]*D[Qn, x], x]*Qn^p, x], x] /; EqQ[m, n - 1]] /; FreeQ[p, x] && PolyQ[Pm , x] && PolyQ[Qn, x] && NeQ[p, -1]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.09 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.10
| method | result | size |
| default | \(\frac {\left (\frac {27 B}{230}+\frac {8 A}{115}+\frac {12 C}{115}\right ) x^{2}+\left (-\frac {6 B}{115}+\frac {17 A}{690}-\frac {16 C}{345}\right ) x -\frac {16 B}{345}-\frac {6 A}{115}-\frac {9 C}{115}}{x^{3}-\frac {4}{3} x^{2}+\frac {2}{3}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (3 \textit {\_Z}^{3}-4 \textit {\_Z}^{2}+2\right )}{\sum }\frac {\left (48 A \textit {\_R} +81 B \textit {\_R} +72 C \textit {\_R} +98 A +36 B +32 C \right ) \ln \left (x -\textit {\_R} \right )}{9 \textit {\_R}^{2}-8 \textit {\_R}}\right )}{230}\) | \(109\) |
| risch | \(\frac {\left (\frac {27 B}{230}+\frac {8 A}{115}+\frac {12 C}{115}\right ) x^{2}+\left (-\frac {6 B}{115}+\frac {17 A}{690}-\frac {16 C}{345}\right ) x -\frac {16 B}{345}-\frac {6 A}{115}-\frac {9 C}{115}}{x^{3}-\frac {4}{3} x^{2}+\frac {2}{3}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (3 \textit {\_Z}^{3}-4 \textit {\_Z}^{2}+2\right )}{\sum }\frac {\left (3 \left (16 A +27 B +24 C \right ) \textit {\_R} +36 B +98 A +32 C \right ) \ln \left (x -\textit {\_R} \right )}{9 \textit {\_R}^{2}-8 \textit {\_R}}\right )}{230}\) | \(110\) |
Input:
int((C*x^2+B*x+A)/(3*x^3-4*x^2+2)^2,x,method=_RETURNVERBOSE)
Output:
((27/230*B+8/115*A+12/115*C)*x^2+(-6/115*B+17/690*A-16/345*C)*x-16/345*B-6 /115*A-9/115*C)/(x^3-4/3*x^2+2/3)+1/230*sum((48*A*_R+81*B*_R+72*C*_R+98*A+ 36*B+32*C)/(9*_R^2-8*_R)*ln(x-_R),_R=RootOf(3*_Z^3-4*_Z^2+2))
Result contains complex when optimal does not.
Time = 1.20 (sec) , antiderivative size = 5341, normalized size of antiderivative = 5.06 \[ \int \frac {A+B x+C x^2}{\left (2-4 x^2+3 x^3\right )^2} \, dx=\text {Too large to display} \] Input:
integrate((C*x^2+B*x+A)/(3*x^3-4*x^2+2)^2,x, algorithm="fricas")
Output:
Too large to include
Time = 9.17 (sec) , antiderivative size = 338, normalized size of antiderivative = 0.32 \[ \int \frac {A+B x+C x^2}{\left (2-4 x^2+3 x^3\right )^2} \, dx=\operatorname {RootSum} {\left (24334000 t^{3} + t \left (462976 A^{2} + 821484 A B + 730208 A C + 335664 B^{2} + 596736 B C + 265216 C^{2}\right ) - 19332 A^{3} - 28080 A^{2} B - 24960 A^{2} C - 7776 A B^{2} - 13824 A B C - 6144 A C^{2} + 2187 B^{3} + 5832 B^{2} C + 5184 B C^{2} + 1536 C^{3}, \left ( t \mapsto t \log {\left (x + \frac {34846288000 t^{2} A + 33617421000 t^{2} B + 29882152000 t^{2} C + 3281598600 t A^{2} + 3725006400 t A B + 3311116800 t A C + 925538400 t B^{2} + 1645401600 t B C + 731289600 t C^{2} + 174801904 A^{3} + 622975104 A^{2} B + 553755648 A^{2} C + 615440268 A B^{2} + 1094116032 A B C + 486273792 A C^{2} + 178158852 B^{3} + 475090272 B^{2} C + 422302464 B C^{2} + 125126656 C^{3}}{601168164 A^{3} + 1322255664 A^{2} B + 1175338368 A^{2} C + 1038585888 A B^{2} + 1846374912 A B C + 820611072 A C^{2} + 294722307 B^{3} + 785926152 B^{2} C + 698601024 B C^{2} + 206992896 C^{3}} \right )} \right )\right )} + \frac {- 36 A - 32 B - 54 C + x^{2} \cdot \left (48 A + 81 B + 72 C\right ) + x \left (17 A - 36 B - 32 C\right )}{690 x^{3} - 920 x^{2} + 460} \] Input:
integrate((C*x**2+B*x+A)/(3*x**3-4*x**2+2)**2,x)
Output:
RootSum(24334000*_t**3 + _t*(462976*A**2 + 821484*A*B + 730208*A*C + 33566 4*B**2 + 596736*B*C + 265216*C**2) - 19332*A**3 - 28080*A**2*B - 24960*A** 2*C - 7776*A*B**2 - 13824*A*B*C - 6144*A*C**2 + 2187*B**3 + 5832*B**2*C + 5184*B*C**2 + 1536*C**3, Lambda(_t, _t*log(x + (34846288000*_t**2*A + 3361 7421000*_t**2*B + 29882152000*_t**2*C + 3281598600*_t*A**2 + 3725006400*_t *A*B + 3311116800*_t*A*C + 925538400*_t*B**2 + 1645401600*_t*B*C + 7312896 00*_t*C**2 + 174801904*A**3 + 622975104*A**2*B + 553755648*A**2*C + 615440 268*A*B**2 + 1094116032*A*B*C + 486273792*A*C**2 + 178158852*B**3 + 475090 272*B**2*C + 422302464*B*C**2 + 125126656*C**3)/(601168164*A**3 + 13222556 64*A**2*B + 1175338368*A**2*C + 1038585888*A*B**2 + 1846374912*A*B*C + 820 611072*A*C**2 + 294722307*B**3 + 785926152*B**2*C + 698601024*B*C**2 + 206 992896*C**3)))) + (-36*A - 32*B - 54*C + x**2*(48*A + 81*B + 72*C) + x*(17 *A - 36*B - 32*C))/(690*x**3 - 920*x**2 + 460)
\[ \int \frac {A+B x+C x^2}{\left (2-4 x^2+3 x^3\right )^2} \, dx=\int { \frac {C x^{2} + B x + A}{{\left (3 \, x^{3} - 4 \, x^{2} + 2\right )}^{2}} \,d x } \] Input:
integrate((C*x^2+B*x+A)/(3*x^3-4*x^2+2)^2,x, algorithm="maxima")
Output:
1/230*(3*(16*A + 27*B + 24*C)*x^2 + (17*A - 36*B - 32*C)*x - 36*A - 32*B - 54*C)/(3*x^3 - 4*x^2 + 2) + 1/230*integrate((3*(16*A + 27*B + 24*C)*x + 9 8*A + 36*B + 32*C)/(3*x^3 - 4*x^2 + 2), x)
Exception generated. \[ \int \frac {A+B x+C x^2}{\left (2-4 x^2+3 x^3\right )^2} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((C*x^2+B*x+A)/(3*x^3-4*x^2+2)^2,x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Time = 12.47 (sec) , antiderivative size = 988, normalized size of antiderivative = 0.94 \[ \int \frac {A+B x+C x^2}{\left (2-4 x^2+3 x^3\right )^2} \, dx=\text {Too large to display} \] Input:
int((A + B*x + C*x^2)/(3*x^3 - 4*x^2 + 2)^2,x)
Output:
symsum(log((1728*A^2*x)/13225 + (19683*B^2*x)/52900 + (3888*C^2*x)/13225 - 96*root(z^3 + z*((28936*A^2)/1520875 + (20979*B^2)/1520875 + (16576*C^2)/ 1520875 + (205371*A*B)/6083500 + (45638*A*C)/1520875 + (37296*B*C)/1520875 ) - (864*A*B*C)/1520875 + (729*B^2*C)/3041750 + (324*B*C^2)/1520875 - (384 *A*C^2)/1520875 - (312*A^2*C)/304175 - (486*A*B^2)/1520875 - (351*A^2*B)/3 04175 + (2187*B^3)/24334000 + (96*C^3)/1520875 - (4833*A^3)/6083500, z, k) ^2*x + (3528*A^2)/13225 + (2187*B^2)/13225 + (1728*C^2)/13225 + 162*root(z ^3 + z*((28936*A^2)/1520875 + (20979*B^2)/1520875 + (16576*C^2)/1520875 + (205371*A*B)/6083500 + (45638*A*C)/1520875 + (37296*B*C)/1520875) - (864*A *B*C)/1520875 + (729*B^2*C)/3041750 + (324*B*C^2)/1520875 - (384*A*C^2)/15 20875 - (312*A^2*C)/304175 - (486*A*B^2)/1520875 - (351*A^2*B)/304175 + (2 187*B^3)/24334000 + (96*C^3)/1520875 - (4833*A^3)/6083500, z, k)^2 + (1449 9*A*B)/26450 + (6444*A*C)/13225 + (3888*B*C)/13225 - (588*A*root(z^3 + z*( (28936*A^2)/1520875 + (20979*B^2)/1520875 + (16576*C^2)/1520875 + (205371* A*B)/6083500 + (45638*A*C)/1520875 + (37296*B*C)/1520875) - (864*A*B*C)/15 20875 + (729*B^2*C)/3041750 + (324*B*C^2)/1520875 - (384*A*C^2)/1520875 - (312*A^2*C)/304175 - (486*A*B^2)/1520875 - (351*A^2*B)/304175 + (2187*B^3) /24334000 + (96*C^3)/1520875 - (4833*A^3)/6083500, z, k))/115 - (216*B*roo t(z^3 + z*((28936*A^2)/1520875 + (20979*B^2)/1520875 + (16576*C^2)/1520875 + (205371*A*B)/6083500 + (45638*A*C)/1520875 + (37296*B*C)/1520875) - ...
\[ \int \frac {A+B x+C x^2}{\left (2-4 x^2+3 x^3\right )^2} \, dx=\frac {1458 \left (\int \frac {x^{3}}{9 x^{6}-24 x^{5}+16 x^{4}+12 x^{3}-16 x^{2}+4}d x \right ) b \,x^{3}-1944 \left (\int \frac {x^{3}}{9 x^{6}-24 x^{5}+16 x^{4}+12 x^{3}-16 x^{2}+4}d x \right ) b \,x^{2}+972 \left (\int \frac {x^{3}}{9 x^{6}-24 x^{5}+16 x^{4}+12 x^{3}-16 x^{2}+4}d x \right ) b +1296 \left (\int \frac {x^{3}}{9 x^{6}-24 x^{5}+16 x^{4}+12 x^{3}-16 x^{2}+4}d x \right ) c \,x^{3}-1728 \left (\int \frac {x^{3}}{9 x^{6}-24 x^{5}+16 x^{4}+12 x^{3}-16 x^{2}+4}d x \right ) c \,x^{2}+864 \left (\int \frac {x^{3}}{9 x^{6}-24 x^{5}+16 x^{4}+12 x^{3}-16 x^{2}+4}d x \right ) c +864 \left (\int \frac {1}{9 x^{6}-24 x^{5}+16 x^{4}+12 x^{3}-16 x^{2}+4}d x \right ) a \,x^{3}-1152 \left (\int \frac {1}{9 x^{6}-24 x^{5}+16 x^{4}+12 x^{3}-16 x^{2}+4}d x \right ) a \,x^{2}+576 \left (\int \frac {1}{9 x^{6}-24 x^{5}+16 x^{4}+12 x^{3}-16 x^{2}+4}d x \right ) a -486 \left (\int \frac {1}{9 x^{6}-24 x^{5}+16 x^{4}+12 x^{3}-16 x^{2}+4}d x \right ) b \,x^{3}+648 \left (\int \frac {1}{9 x^{6}-24 x^{5}+16 x^{4}+12 x^{3}-16 x^{2}+4}d x \right ) b \,x^{2}-324 \left (\int \frac {1}{9 x^{6}-24 x^{5}+16 x^{4}+12 x^{3}-16 x^{2}+4}d x \right ) b -432 \left (\int \frac {1}{9 x^{6}-24 x^{5}+16 x^{4}+12 x^{3}-16 x^{2}+4}d x \right ) c \,x^{3}+576 \left (\int \frac {1}{9 x^{6}-24 x^{5}+16 x^{4}+12 x^{3}-16 x^{2}+4}d x \right ) c \,x^{2}-288 \left (\int \frac {1}{9 x^{6}-24 x^{5}+16 x^{4}+12 x^{3}-16 x^{2}+4}d x \right ) c +81 b x +36 b +48 c \,x^{3}-64 c \,x^{2}+72 c x +32 c}{864 x^{3}-1152 x^{2}+576} \] Input:
int((C*x^2+B*x+A)/(3*x^3-4*x^2+2)^2,x)
Output:
(1458*int(x**3/(9*x**6 - 24*x**5 + 16*x**4 + 12*x**3 - 16*x**2 + 4),x)*b*x **3 - 1944*int(x**3/(9*x**6 - 24*x**5 + 16*x**4 + 12*x**3 - 16*x**2 + 4),x )*b*x**2 + 972*int(x**3/(9*x**6 - 24*x**5 + 16*x**4 + 12*x**3 - 16*x**2 + 4),x)*b + 1296*int(x**3/(9*x**6 - 24*x**5 + 16*x**4 + 12*x**3 - 16*x**2 + 4),x)*c*x**3 - 1728*int(x**3/(9*x**6 - 24*x**5 + 16*x**4 + 12*x**3 - 16*x* *2 + 4),x)*c*x**2 + 864*int(x**3/(9*x**6 - 24*x**5 + 16*x**4 + 12*x**3 - 1 6*x**2 + 4),x)*c + 864*int(1/(9*x**6 - 24*x**5 + 16*x**4 + 12*x**3 - 16*x* *2 + 4),x)*a*x**3 - 1152*int(1/(9*x**6 - 24*x**5 + 16*x**4 + 12*x**3 - 16* x**2 + 4),x)*a*x**2 + 576*int(1/(9*x**6 - 24*x**5 + 16*x**4 + 12*x**3 - 16 *x**2 + 4),x)*a - 486*int(1/(9*x**6 - 24*x**5 + 16*x**4 + 12*x**3 - 16*x** 2 + 4),x)*b*x**3 + 648*int(1/(9*x**6 - 24*x**5 + 16*x**4 + 12*x**3 - 16*x* *2 + 4),x)*b*x**2 - 324*int(1/(9*x**6 - 24*x**5 + 16*x**4 + 12*x**3 - 16*x **2 + 4),x)*b - 432*int(1/(9*x**6 - 24*x**5 + 16*x**4 + 12*x**3 - 16*x**2 + 4),x)*c*x**3 + 576*int(1/(9*x**6 - 24*x**5 + 16*x**4 + 12*x**3 - 16*x**2 + 4),x)*c*x**2 - 288*int(1/(9*x**6 - 24*x**5 + 16*x**4 + 12*x**3 - 16*x** 2 + 4),x)*c + 81*b*x + 36*b + 48*c*x**3 - 64*c*x**2 + 72*c*x + 32*c)/(288* (3*x**3 - 4*x**2 + 2))