Integrand size = 30, antiderivative size = 321 \[ \int \left (A+B x+C x^2\right ) \sqrt {70+67 x-53 x^2+6 x^3} \, dx=-\frac {(4914 A+11412 B+48913 C) \sqrt {70+67 x-53 x^2+6 x^3}}{3402}+\frac {(378 A+18 B-1301 C) (2+3 x) \sqrt {70+67 x-53 x^2+6 x^3}}{2835}-\frac {1}{189} (9 B+53 C) (5-2 x) (2+3 x) \sqrt {70+67 x-53 x^2+6 x^3}+\frac {1}{27} C (5-2 x) (7-x) (2+3 x) \sqrt {70+67 x-53 x^2+6 x^3}+\frac {\sqrt {19} (605934 A+2068839 B+9927742 C) \sqrt {70+67 x-53 x^2+6 x^3} E\left (\arcsin \left (\frac {\sqrt {2+3 x}}{\sqrt {23}}\right )|\frac {46}{19}\right )}{51030 \sqrt {5-2 x} \sqrt {7-x} \sqrt {2+3 x}}+\frac {\sqrt {19} (1512 A+28017 B+159361 C) \sqrt {70+67 x-53 x^2+6 x^3} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {19}{2}}}{\sqrt {2+3 x}}\right ),\frac {46}{19}\right )}{1890 \sqrt {-7+x} \sqrt {-5+2 x} \sqrt {2+3 x}} \] Output:
-1/3402*(4914*A+11412*B+48913*C)*(6*x^3-53*x^2+67*x+70)^(1/2)+1/2835*(378* A+18*B-1301*C)*(2+3*x)*(6*x^3-53*x^2+67*x+70)^(1/2)-1/189*(9*B+53*C)*(5-2* x)*(2+3*x)*(6*x^3-53*x^2+67*x+70)^(1/2)+1/27*C*(5-2*x)*(7-x)*(2+3*x)*(6*x^ 3-53*x^2+67*x+70)^(1/2)+1/51030*19^(1/2)*(605934*A+2068839*B+9927742*C)*(6 *x^3-53*x^2+67*x+70)^(1/2)*EllipticE(1/23*(2+3*x)^(1/2)*23^(1/2),1/19*874^ (1/2))/(5-2*x)^(1/2)/(7-x)^(1/2)/(2+3*x)^(1/2)+1/1890*19^(1/2)*(1512*A+280 17*B+159361*C)*(6*x^3-53*x^2+67*x+70)^(1/2)*EllipticF(1/2*38^(1/2)/(2+3*x) ^(1/2),1/19*874^(1/2))/(-7+x)^(1/2)/(-5+2*x)^(1/2)/(2+3*x)^(1/2)
Time = 8.77 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.64 \[ \int \left (A+B x+C x^2\right ) \sqrt {70+67 x-53 x^2+6 x^3} \, dx=-\frac {\sqrt {5-2 x} \left (6 \sqrt {5-2 x} \left (-14-19 x+3 x^2\right ) \left (378 A (-53+18 x)+18 B \left (-3608-477 x+270 x^2\right )+C \left (-263777-33678 x-4770 x^2+3780 x^3\right )\right )-2 \sqrt {46} (605934 A+2068839 B+9927742 C) \sqrt {7-x} \sqrt {2+3 x} E\left (\arcsin \left (\sqrt {\frac {2}{19}} \sqrt {2+3 x}\right )|\frac {19}{46}\right )+27 \sqrt {46} (27594 A+113094 B+563287 C) \sqrt {7-x} \sqrt {2+3 x} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {2}{19}} \sqrt {2+3 x}\right ),\frac {19}{46}\right )\right )}{102060 \sqrt {70+67 x-53 x^2+6 x^3}} \] Input:
Integrate[(A + B*x + C*x^2)*Sqrt[70 + 67*x - 53*x^2 + 6*x^3],x]
Output:
-1/102060*(Sqrt[5 - 2*x]*(6*Sqrt[5 - 2*x]*(-14 - 19*x + 3*x^2)*(378*A*(-53 + 18*x) + 18*B*(-3608 - 477*x + 270*x^2) + C*(-263777 - 33678*x - 4770*x^ 2 + 3780*x^3)) - 2*Sqrt[46]*(605934*A + 2068839*B + 9927742*C)*Sqrt[7 - x] *Sqrt[2 + 3*x]*EllipticE[ArcSin[Sqrt[2/19]*Sqrt[2 + 3*x]], 19/46] + 27*Sqr t[46]*(27594*A + 113094*B + 563287*C)*Sqrt[7 - x]*Sqrt[2 + 3*x]*EllipticF[ ArcSin[Sqrt[2/19]*Sqrt[2 + 3*x]], 19/46]))/Sqrt[70 + 67*x - 53*x^2 + 6*x^3 ]
Result contains complex when optimal does not.
Time = 6.87 (sec) , antiderivative size = 1783, normalized size of antiderivative = 5.55, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {2526, 2490, 2486, 27, 1236, 27, 1231, 27, 1269, 1172, 321, 327}
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.
| \(\displaystyle \int \sqrt {6 x^3-53 x^2+67 x+70} \left (A+B x+C x^2\right ) \, dx\) |
| \(\Big \downarrow \) 2526 |
| \(\displaystyle \frac {1}{18} \int (18 A-67 C+2 (9 B+53 C) x) \sqrt {6 x^3-53 x^2+67 x+70}dx+\frac {1}{27} C \left (6 x^3-53 x^2+67 x+70\right )^{3/2}\) |
| \(\Big \downarrow \) 2490 |
| \(\displaystyle \frac {1}{18} \int \left (\frac {1}{18} (18 (18 A-67 C)+106 (9 B+53 C))+2 (9 B+53 C) \left (x-\frac {53}{18}\right )\right ) \sqrt {6 \left (x-\frac {53}{18}\right )^3-\frac {1603}{18} \left (x-\frac {53}{18}\right )-\frac {9490}{243}}d\left (x-\frac {53}{18}\right )+\frac {1}{27} C \left (6 x^3-53 x^2+67 x+70\right )^{3/2}\) |
| \(\Big \downarrow \) 2486 |
| \(\displaystyle \frac {1}{27} C \left (6 x^3-53 x^2+67 x+70\right )^{3/2}+\frac {\sqrt {2916 \left (x-\frac {53}{18}\right )^3-43281 \left (x-\frac {53}{18}\right )-18980} \int \frac {\sqrt {18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}} \left (162 A+477 B+2206 C+18 (9 B+53 C) \left (x-\frac {53}{18}\right )\right ) \sqrt {324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603}}{27 \sqrt {3}}d\left (x-\frac {53}{18}\right )}{54 \sqrt {2} \sqrt {18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}} \sqrt {324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{27} C \left (6 x^3-53 x^2+67 x+70\right )^{3/2}+\frac {\sqrt {2916 \left (x-\frac {53}{18}\right )^3-43281 \left (x-\frac {53}{18}\right )-18980} \int \sqrt {18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}} \left (162 A+477 B+2206 C+18 (9 B+53 C) \left (x-\frac {53}{18}\right )\right ) \sqrt {324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603}d\left (x-\frac {53}{18}\right )}{1458 \sqrt {6} \sqrt {18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}} \sqrt {324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603}}\) |
| \(\Big \downarrow \) 1236 |
| \(\displaystyle \frac {1}{27} C \left (6 x^3-53 x^2+67 x+70\right )^{3/2}+\frac {\sqrt {2916 \left (x-\frac {53}{18}\right )^3-43281 \left (x-\frac {53}{18}\right )-18980} \left (\frac {\int -\frac {162 \left (\frac {\left (2569609-1603 \left (18980+35397 i \sqrt {3}\right )^{2/3}+\left (18980+35397 i \sqrt {3}\right )^{4/3}\right ) (9 B+53 C)+\sqrt [3]{18980+35397 i \sqrt {3}} \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) (162 A+477 B+2206 C)-3 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (\left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) (9 B+53 C)-2 \sqrt [3]{18980+35397 i \sqrt {3}} (162 A+477 B+2206 C)\right )}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-18 \left (1134 A+9 \left (371-\frac {8015}{\sqrt [3]{18980+35397 i \sqrt {3}}}-5 \sqrt [3]{18980+35397 i \sqrt {3}}\right ) B+\left (15442-\frac {424795}{\sqrt [3]{18980+35397 i \sqrt {3}}}-265 \sqrt [3]{18980+35397 i \sqrt {3}}\right ) C\right ) \left (x-\frac {53}{18}\right )\right ) \sqrt {324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603}}{\sqrt {18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}}}d\left (x-\frac {53}{18}\right )}{1134}+\frac {1}{63} \sqrt {18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}} \left (324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603\right )^{3/2} (9 B+53 C)\right )}{1458 \sqrt {6} \sqrt {18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}} \sqrt {324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{27} C \left (6 x^3-53 x^2+67 x+70\right )^{3/2}+\frac {\sqrt {2916 \left (x-\frac {53}{18}\right )^3-43281 \left (x-\frac {53}{18}\right )-18980} \left (\frac {1}{63} \sqrt {18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}} \left (324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603\right )^{3/2} (9 B+53 C)-\frac {1}{7} \int \frac {\left (\frac {\left (2569609-1603 \left (18980+35397 i \sqrt {3}\right )^{2/3}+\left (18980+35397 i \sqrt {3}\right )^{4/3}\right ) (9 B+53 C)+\sqrt [3]{18980+35397 i \sqrt {3}} \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) (162 A+477 B+2206 C)-3 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (\left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) (9 B+53 C)-2 \sqrt [3]{18980+35397 i \sqrt {3}} (162 A+477 B+2206 C)\right )}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-18 \left (1134 A+9 \left (371-\frac {8015}{\sqrt [3]{18980+35397 i \sqrt {3}}}-5 \sqrt [3]{18980+35397 i \sqrt {3}}\right ) B+\left (15442-\frac {424795}{\sqrt [3]{18980+35397 i \sqrt {3}}}-265 \sqrt [3]{18980+35397 i \sqrt {3}}\right ) C\right ) \left (x-\frac {53}{18}\right )\right ) \sqrt {324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603}}{\sqrt {18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}}}d\left (x-\frac {53}{18}\right )\right )}{1458 \sqrt {6} \sqrt {18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}} \sqrt {324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603}}\) |
| \(\Big \downarrow \) 1231 |
| \(\displaystyle \frac {1}{27} C \left (6 x^3-53 x^2+67 x+70\right )^{3/2}+\frac {\sqrt {2916 \left (x-\frac {53}{18}\right )^3-43281 \left (x-\frac {53}{18}\right )-18980} \left (\frac {1}{63} (9 B+53 C) \sqrt {18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}} \left (324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603\right )^{3/2}+\frac {1}{7} \left (\frac {\int -\frac {2834352 \left (35 (204984 A+1704825 B+9276529 C)+18 (605934 A+2068839 B+9927742 C) \left (x-\frac {53}{18}\right )\right )}{\sqrt {18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}} \sqrt {324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603}}d\left (x-\frac {53}{18}\right )}{787320}-\frac {1}{45} \sqrt {18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}} \left (\frac {5 \left (2569609+1603 \left (18980+35397 i \sqrt {3}\right )^{2/3}+\left (18980+35397 i \sqrt {3}\right )^{4/3}\right ) (9 B+53 C)}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-18 \left (1134 A+9 \left (371-\frac {8015}{\sqrt [3]{18980+35397 i \sqrt {3}}}-5 \sqrt [3]{18980+35397 i \sqrt {3}}\right ) B+\left (15442-\frac {424795}{\sqrt [3]{18980+35397 i \sqrt {3}}}-265 \sqrt [3]{18980+35397 i \sqrt {3}}\right ) C\right ) \left (x-\frac {53}{18}\right )\right ) \sqrt {324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603}\right )\right )}{1458 \sqrt {6} \sqrt {18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}} \sqrt {324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{27} C \left (6 x^3-53 x^2+67 x+70\right )^{3/2}+\frac {\sqrt {2916 \left (x-\frac {53}{18}\right )^3-43281 \left (x-\frac {53}{18}\right )-18980} \left (\frac {1}{7} \left (-\frac {18}{5} \int \frac {35 (204984 A+1704825 B+9276529 C)+18 (605934 A+2068839 B+9927742 C) \left (x-\frac {53}{18}\right )}{\sqrt {18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}} \sqrt {324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603}}d\left (x-\frac {53}{18}\right )-\frac {1}{45} \sqrt {18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}} \sqrt {324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603} \left (\frac {5 \left (2569609+1603 \left (18980+35397 i \sqrt {3}\right )^{2/3}+\left (18980+35397 i \sqrt {3}\right )^{4/3}\right ) (9 B+53 C)}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-18 \left (x-\frac {53}{18}\right ) \left (1134 A+9 \left (371-\frac {8015}{\sqrt [3]{18980+35397 i \sqrt {3}}}-5 \sqrt [3]{18980+35397 i \sqrt {3}}\right ) B+\left (15442-\frac {424795}{\sqrt [3]{18980+35397 i \sqrt {3}}}-265 \sqrt [3]{18980+35397 i \sqrt {3}}\right ) C\right )\right )\right )+\frac {1}{63} \sqrt {18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}} \left (324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603\right )^{3/2} (9 B+53 C)\right )}{1458 \sqrt {6} \sqrt {18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}} \sqrt {324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603}}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {1}{27} C \left (6 x^3-53 x^2+67 x+70\right )^{3/2}+\frac {\sqrt {2916 \left (x-\frac {53}{18}\right )^3-43281 \left (x-\frac {53}{18}\right )-18980} \left (\frac {1}{63} (9 B+53 C) \sqrt {18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}} \left (324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603\right )^{3/2}+\frac {1}{7} \left (-\frac {1}{45} \sqrt {18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}} \sqrt {324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603} \left (\frac {5 \left (2569609+1603 \left (18980+35397 i \sqrt {3}\right )^{2/3}+\left (18980+35397 i \sqrt {3}\right )^{4/3}\right ) (9 B+53 C)}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-18 \left (1134 A+9 \left (371-\frac {8015}{\sqrt [3]{18980+35397 i \sqrt {3}}}-5 \sqrt [3]{18980+35397 i \sqrt {3}}\right ) B+\left (15442-\frac {424795}{\sqrt [3]{18980+35397 i \sqrt {3}}}-265 \sqrt [3]{18980+35397 i \sqrt {3}}\right ) C\right ) \left (x-\frac {53}{18}\right )\right )-\frac {18}{5} \left (\left (35 (204984 A+1704825 B+9276529 C)+\frac {\left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) (605934 A+2068839 B+9927742 C)}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right ) \int \frac {1}{\sqrt {18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}} \sqrt {324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603}}d\left (x-\frac {53}{18}\right )+(605934 A+2068839 B+9927742 C) \int \frac {\sqrt {18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}}}{\sqrt {324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603}}d\left (x-\frac {53}{18}\right )\right )\right )\right )}{1458 \sqrt {6} \sqrt {18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}} \sqrt {324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603}}\) |
| \(\Big \downarrow \) 1172 |
| \(\displaystyle \frac {1}{27} C \left (6 x^3-53 x^2+67 x+70\right )^{3/2}+\frac {\sqrt {2916 \left (x-\frac {53}{18}\right )^3-43281 \left (x-\frac {53}{18}\right )-18980} \left (\frac {1}{63} (9 B+53 C) \sqrt {18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}} \left (324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603\right )^{3/2}+\frac {1}{7} \left (-\frac {1}{45} \sqrt {18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}} \sqrt {324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603} \left (\frac {5 \left (2569609+1603 \left (18980+35397 i \sqrt {3}\right )^{2/3}+\left (18980+35397 i \sqrt {3}\right )^{4/3}\right ) (9 B+53 C)}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-18 \left (1134 A+9 \left (371-\frac {8015}{\sqrt [3]{18980+35397 i \sqrt {3}}}-5 \sqrt [3]{18980+35397 i \sqrt {3}}\right ) B+\left (15442-\frac {424795}{\sqrt [3]{18980+35397 i \sqrt {3}}}-265 \sqrt [3]{18980+35397 i \sqrt {3}}\right ) C\right ) \left (x-\frac {53}{18}\right )\right )-\frac {18}{5} \left (-\frac {\sqrt {\frac {2}{3}} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (35 (204984 A+1704825 B+9276529 C)+\frac {\left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) (605934 A+2068839 B+9927742 C)}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right ) \sqrt {-\frac {\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}-18 \left (x-\frac {53}{18}\right )}{\frac {1}{324} \sqrt {-\frac {1}{3} \left (18980+35397 i \sqrt {3}\right )^{2/3}} \left (324-\frac {519372}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}}} \sqrt {-\frac {\left (18980+35397 i \sqrt {3}\right )^{2/3} \left (-324 \left (x-\frac {53}{18}\right )^2-\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}-\left (18980+35397 i \sqrt {3}\right )^{2/3}-\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}+1603\right )}{\left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right )^2}} \int \frac {1}{\sqrt {1-\frac {\sqrt {-\frac {1}{3} \left (18980+35397 i \sqrt {3}\right )^{2/3}} \left (-36 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\sqrt {-3 \left (18980+35397 i \sqrt {3}\right )^{2/3}} \left (1-\frac {1603}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}\right )\right )}{2 \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right )}} \sqrt {\frac {\sqrt {-\left (18980+35397 i \sqrt {3}\right )^{2/3}} \sqrt [3]{18980+35397 i \sqrt {3}} \left (-36 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\sqrt {-3 \left (18980+35397 i \sqrt {3}\right )^{2/3}} \left (1-\frac {1603}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}\right )\right )}{\sqrt {3} \left (18980+35397 i \sqrt {3}-1603 \sqrt [3]{18980+35397 i \sqrt {3}}+\sqrt {-3 \left (18980+35397 i \sqrt {3}\right )^{2/3}} \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right )\right )}+1}}d\frac {\sqrt {\frac {\sqrt {-3 \left (18980+35397 i \sqrt {3}\right )^{2/3}} \left (-36 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\sqrt {-3 \left (18980+35397 i \sqrt {3}\right )^{2/3}} \left (1-\frac {1603}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}\right )\right )}{1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}}}}{\sqrt {6}}}{9 \sqrt {-\left (18980+35397 i \sqrt {3}\right )^{2/3}} \sqrt {18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}} \sqrt {324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603}}-\frac {\left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) (605934 A+2068839 B+9927742 C) \sqrt {18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}} \sqrt {-\frac {\left (18980+35397 i \sqrt {3}\right )^{2/3} \left (-324 \left (x-\frac {53}{18}\right )^2-\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}-\left (18980+35397 i \sqrt {3}\right )^{2/3}-\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}+1603\right )}{\left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right )^2}} \int \frac {\sqrt {\frac {\sqrt {-\left (18980+35397 i \sqrt {3}\right )^{2/3}} \sqrt [3]{18980+35397 i \sqrt {3}} \left (-36 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\sqrt {-3 \left (18980+35397 i \sqrt {3}\right )^{2/3}} \left (1-\frac {1603}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}\right )\right )}{\sqrt {3} \left (18980+35397 i \sqrt {3}-1603 \sqrt [3]{18980+35397 i \sqrt {3}}+\sqrt {-3 \left (18980+35397 i \sqrt {3}\right )^{2/3}} \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right )\right )}+1}}{\sqrt {1-\frac {\sqrt {-\frac {1}{3} \left (18980+35397 i \sqrt {3}\right )^{2/3}} \left (-36 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\sqrt {-3 \left (18980+35397 i \sqrt {3}\right )^{2/3}} \left (1-\frac {1603}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}\right )\right )}{2 \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right )}}}d\frac {\sqrt {\frac {\sqrt {-3 \left (18980+35397 i \sqrt {3}\right )^{2/3}} \left (-36 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\sqrt {-3 \left (18980+35397 i \sqrt {3}\right )^{2/3}} \left (1-\frac {1603}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}\right )\right )}{1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}}}}{\sqrt {6}}}{3 \sqrt {6} \sqrt {-\left (18980+35397 i \sqrt {3}\right )^{2/3}} \sqrt {-\frac {\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}-18 \left (x-\frac {53}{18}\right )}{\frac {1}{324} \sqrt {-\frac {1}{3} \left (18980+35397 i \sqrt {3}\right )^{2/3}} \left (324-\frac {519372}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}}} \sqrt {324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603}}\right )\right )\right )}{1458 \sqrt {6} \sqrt {18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}} \sqrt {324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603}}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {1}{27} C \left (6 x^3-53 x^2+67 x+70\right )^{3/2}+\frac {\sqrt {2916 \left (x-\frac {53}{18}\right )^3-43281 \left (x-\frac {53}{18}\right )-18980} \left (\frac {1}{63} (9 B+53 C) \sqrt {18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}} \left (324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603\right )^{3/2}+\frac {1}{7} \left (-\frac {1}{45} \sqrt {18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}} \sqrt {324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603} \left (\frac {5 \left (2569609+1603 \left (18980+35397 i \sqrt {3}\right )^{2/3}+\left (18980+35397 i \sqrt {3}\right )^{4/3}\right ) (9 B+53 C)}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-18 \left (1134 A+9 \left (371-\frac {8015}{\sqrt [3]{18980+35397 i \sqrt {3}}}-5 \sqrt [3]{18980+35397 i \sqrt {3}}\right ) B+\left (15442-\frac {424795}{\sqrt [3]{18980+35397 i \sqrt {3}}}-265 \sqrt [3]{18980+35397 i \sqrt {3}}\right ) C\right ) \left (x-\frac {53}{18}\right )\right )-\frac {18}{5} \left (\frac {\sqrt {\frac {2}{3}} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (35 (204984 A+1704825 B+9276529 C)+\frac {\left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) (605934 A+2068839 B+9927742 C)}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right ) \sqrt {-\frac {\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}-18 \left (x-\frac {53}{18}\right )}{\frac {1}{324} \sqrt {-\frac {1}{3} \left (18980+35397 i \sqrt {3}\right )^{2/3}} \left (324-\frac {519372}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}}} \sqrt {-\frac {\left (18980+35397 i \sqrt {3}\right )^{2/3} \left (-324 \left (x-\frac {53}{18}\right )^2-\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}-\left (18980+35397 i \sqrt {3}\right )^{2/3}-\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}+1603\right )}{\left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {53}{18}-x\right ),\frac {2 \left (18980+35397 i \sqrt {3}-1603 \sqrt [3]{18980+35397 i \sqrt {3}}\right )}{18980+35397 i \sqrt {3}-1603 \sqrt [3]{18980+35397 i \sqrt {3}}+\sqrt {-3 \left (18980+35397 i \sqrt {3}\right )^{2/3}} \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right )}\right )}{9 \sqrt {-\left (18980+35397 i \sqrt {3}\right )^{2/3}} \sqrt {18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}} \sqrt {324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603}}-\frac {\left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) (605934 A+2068839 B+9927742 C) \sqrt {18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}} \sqrt {-\frac {\left (18980+35397 i \sqrt {3}\right )^{2/3} \left (-324 \left (x-\frac {53}{18}\right )^2-\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}-\left (18980+35397 i \sqrt {3}\right )^{2/3}-\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}+1603\right )}{\left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right )^2}} \int \frac {\sqrt {\frac {\sqrt {-\left (18980+35397 i \sqrt {3}\right )^{2/3}} \sqrt [3]{18980+35397 i \sqrt {3}} \left (-36 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\sqrt {-3 \left (18980+35397 i \sqrt {3}\right )^{2/3}} \left (1-\frac {1603}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}\right )\right )}{\sqrt {3} \left (18980+35397 i \sqrt {3}-1603 \sqrt [3]{18980+35397 i \sqrt {3}}+\sqrt {-3 \left (18980+35397 i \sqrt {3}\right )^{2/3}} \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right )\right )}+1}}{\sqrt {1-\frac {\sqrt {-\frac {1}{3} \left (18980+35397 i \sqrt {3}\right )^{2/3}} \left (-36 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\sqrt {-3 \left (18980+35397 i \sqrt {3}\right )^{2/3}} \left (1-\frac {1603}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}\right )\right )}{2 \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right )}}}d\frac {\sqrt {\frac {\sqrt {-3 \left (18980+35397 i \sqrt {3}\right )^{2/3}} \left (-36 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\sqrt {-3 \left (18980+35397 i \sqrt {3}\right )^{2/3}} \left (1-\frac {1603}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}\right )\right )}{1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}}}}{\sqrt {6}}}{3 \sqrt {6} \sqrt {-\left (18980+35397 i \sqrt {3}\right )^{2/3}} \sqrt {-\frac {\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}-18 \left (x-\frac {53}{18}\right )}{\frac {1}{324} \sqrt {-\frac {1}{3} \left (18980+35397 i \sqrt {3}\right )^{2/3}} \left (324-\frac {519372}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}}} \sqrt {324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603}}\right )\right )\right )}{1458 \sqrt {6} \sqrt {18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}} \sqrt {324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603}}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {1}{27} C \left (6 x^3-53 x^2+67 x+70\right )^{3/2}+\frac {\sqrt {2916 \left (x-\frac {53}{18}\right )^3-43281 \left (x-\frac {53}{18}\right )-18980} \left (\frac {1}{63} (9 B+53 C) \sqrt {18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}} \left (324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603\right )^{3/2}+\frac {1}{7} \left (-\frac {1}{45} \sqrt {18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}} \sqrt {324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603} \left (\frac {5 \left (2569609+1603 \left (18980+35397 i \sqrt {3}\right )^{2/3}+\left (18980+35397 i \sqrt {3}\right )^{4/3}\right ) (9 B+53 C)}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-18 \left (1134 A+9 \left (371-\frac {8015}{\sqrt [3]{18980+35397 i \sqrt {3}}}-5 \sqrt [3]{18980+35397 i \sqrt {3}}\right ) B+\left (15442-\frac {424795}{\sqrt [3]{18980+35397 i \sqrt {3}}}-265 \sqrt [3]{18980+35397 i \sqrt {3}}\right ) C\right ) \left (x-\frac {53}{18}\right )\right )-\frac {18}{5} \left (\frac {\left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) (605934 A+2068839 B+9927742 C) \sqrt {18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}} \sqrt {-\frac {\left (18980+35397 i \sqrt {3}\right )^{2/3} \left (-324 \left (x-\frac {53}{18}\right )^2-\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}-\left (18980+35397 i \sqrt {3}\right )^{2/3}-\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}+1603\right )}{\left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right )^2}} E\left (\arcsin \left (\frac {53}{18}-x\right )|\frac {2 \left (18980+35397 i \sqrt {3}-1603 \sqrt [3]{18980+35397 i \sqrt {3}}\right )}{18980+35397 i \sqrt {3}-1603 \sqrt [3]{18980+35397 i \sqrt {3}}+\sqrt {-3 \left (18980+35397 i \sqrt {3}\right )^{2/3}} \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right )}\right )}{3 \sqrt {6} \sqrt {-\left (18980+35397 i \sqrt {3}\right )^{2/3}} \sqrt {-\frac {\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}-18 \left (x-\frac {53}{18}\right )}{\frac {1}{324} \sqrt {-\frac {1}{3} \left (18980+35397 i \sqrt {3}\right )^{2/3}} \left (324-\frac {519372}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}}} \sqrt {324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603}}+\frac {\sqrt {\frac {2}{3}} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (35 (204984 A+1704825 B+9276529 C)+\frac {\left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) (605934 A+2068839 B+9927742 C)}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right ) \sqrt {-\frac {\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}-18 \left (x-\frac {53}{18}\right )}{\frac {1}{324} \sqrt {-\frac {1}{3} \left (18980+35397 i \sqrt {3}\right )^{2/3}} \left (324-\frac {519372}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}}} \sqrt {-\frac {\left (18980+35397 i \sqrt {3}\right )^{2/3} \left (-324 \left (x-\frac {53}{18}\right )^2-\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}-\left (18980+35397 i \sqrt {3}\right )^{2/3}-\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}+1603\right )}{\left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {53}{18}-x\right ),\frac {2 \left (18980+35397 i \sqrt {3}-1603 \sqrt [3]{18980+35397 i \sqrt {3}}\right )}{18980+35397 i \sqrt {3}-1603 \sqrt [3]{18980+35397 i \sqrt {3}}+\sqrt {-3 \left (18980+35397 i \sqrt {3}\right )^{2/3}} \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right )}\right )}{9 \sqrt {-\left (18980+35397 i \sqrt {3}\right )^{2/3}} \sqrt {18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}} \sqrt {324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603}}\right )\right )\right )}{1458 \sqrt {6} \sqrt {18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}} \sqrt {324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603}}\) |
Input:
Int[(A + B*x + C*x^2)*Sqrt[70 + 67*x - 53*x^2 + 6*x^3],x]
Output:
(C*(70 + 67*x - 53*x^2 + 6*x^3)^(3/2))/27 + (Sqrt[-18980 - 43281*(-53/18 + x) + 2916*(-53/18 + x)^3]*(((9*B + 53*C)*Sqrt[-((1603 + (18980 + (35397*I )*Sqrt[3])^(2/3))/(18980 + (35397*I)*Sqrt[3])^(1/3)) + 18*(-53/18 + x)]*(- 1603 + 2569609/(18980 + (35397*I)*Sqrt[3])^(2/3) + (18980 + (35397*I)*Sqrt [3])^(2/3) + (18*(1603 + (18980 + (35397*I)*Sqrt[3])^(2/3))*(-53/18 + x))/ (18980 + (35397*I)*Sqrt[3])^(1/3) + 324*(-53/18 + x)^2)^(3/2))/63 + (-1/45 *(Sqrt[-((1603 + (18980 + (35397*I)*Sqrt[3])^(2/3))/(18980 + (35397*I)*Sqr t[3])^(1/3)) + 18*(-53/18 + x)]*((5*(2569609 + 1603*(18980 + (35397*I)*Sqr t[3])^(2/3) + (18980 + (35397*I)*Sqrt[3])^(4/3))*(9*B + 53*C))/(18980 + (3 5397*I)*Sqrt[3])^(2/3) - 18*(1134*A + 9*(371 - 8015/(18980 + (35397*I)*Sqr t[3])^(1/3) - 5*(18980 + (35397*I)*Sqrt[3])^(1/3))*B + (15442 - 424795/(18 980 + (35397*I)*Sqrt[3])^(1/3) - 265*(18980 + (35397*I)*Sqrt[3])^(1/3))*C) *(-53/18 + x))*Sqrt[-1603 + 2569609/(18980 + (35397*I)*Sqrt[3])^(2/3) + (1 8980 + (35397*I)*Sqrt[3])^(2/3) + (18*(1603 + (18980 + (35397*I)*Sqrt[3])^ (2/3))*(-53/18 + x))/(18980 + (35397*I)*Sqrt[3])^(1/3) + 324*(-53/18 + x)^ 2]) - (18*(((1603 - (18980 + (35397*I)*Sqrt[3])^(2/3))*(605934*A + 2068839 *B + 9927742*C)*Sqrt[-((1603 + (18980 + (35397*I)*Sqrt[3])^(2/3))/(18980 + (35397*I)*Sqrt[3])^(1/3)) + 18*(-53/18 + x)]*Sqrt[-(((18980 + (35397*I)*S qrt[3])^(2/3)*(1603 - 2569609/(18980 + (35397*I)*Sqrt[3])^(2/3) - (18980 + (35397*I)*Sqrt[3])^(2/3) - (18*(1603 + (18980 + (35397*I)*Sqrt[3])^(2/...
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c /(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0] && !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ (Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) )], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
Int[((d_.) + (e_.)*(x_))^(m_)/Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Sy mbol] :> Simp[2*Rt[b^2 - 4*a*c, 2]*(d + e*x)^m*(Sqrt[(-c)*((a + b*x + c*x^2 )/(b^2 - 4*a*c))]/(c*Sqrt[a + b*x + c*x^2]*(2*c*((d + e*x)/(2*c*d - b*e - e *Rt[b^2 - 4*a*c, 2])))^m)) Subst[Int[(1 + 2*e*Rt[b^2 - 4*a*c, 2]*(x^2/(2* c*d - b*e - e*Rt[b^2 - 4*a*c, 2])))^m/Sqrt[1 - x^2], x], x, Sqrt[(b + Rt[b^ 2 - 4*a*c, 2] + 2*c*x)/(2*Rt[b^2 - 4*a*c, 2])]], x] /; FreeQ[{a, b, c, d, e }, x] && EqQ[m^2, 1/4]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*(c*d + 2*c*d*p - b*e*p) + g*c*e*(m + 2*p + 1)*x)*((a + b*x + c*x^2)^p/ (c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), x] - Simp[p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)) Int[(d + e*x)^m*(a + b*x + c*x^2)^(p - 1)*Simp[c*e*f*(b*d - 2* a*e)*(m + 2*p + 2) + g*(a*e*(b*e - 2*c*d*m + b*e*m) + b*d*(b*e*p - c*d - 2* c*d*p)) + (c*e*f*(2*c*d - b*e)*(m + 2*p + 2) + g*(b^2*e^2*(p + m + 1) - 2*c ^2*d^2*(1 + 2*p) - c*e*(b*d*(m - 2*p) + 2*a*e*(m + 2*p + 1))))*x, x], x], x ] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && GtQ[p, 0] && (IntegerQ[p] || !R ationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) && !ILtQ[m + 2*p, 0] && (Integer Q[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 2))), x] + Simp[1/(c*(m + 2*p + 2)) Int[(d + e*x)^(m - 1 )*(a + b*x + c*x^2)^p*Simp[m*(c*d*f - a*e*g) + d*(2*c*f - b*g)*(p + 1) + (m *(c*e*f + c*d*g - b*e*g) + e*(p + 1)*(2*c*f - b*g))*x, x], x], x] /; FreeQ[ {a, b, c, d, e, f, g, p}, x] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (Intege rQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p]) && !(IGtQ[m, 0] && EqQ[f, 0])
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g/e Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Simp[(e*f - d*g)/e Int[(d + e*x)^m*(a + b*x + c*x^2)^ p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && !IGtQ[m, 0]
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[(a + b*x + d*x^3)^p/(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) Int[(e + f*x)^m*Sim p[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]] /; FreeQ[{a, b, d, e, f, m, p}, x] && NeQ[4* b^3 + 27*a^2*d, 0] && !IntegerQ[p]
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]
Time = 1.26 (sec) , antiderivative size = 260, normalized size of antiderivative = 0.81
| method | result | size |
| elliptic | \(\frac {2 x^{3} \sqrt {6 x^{3}-53 x^{2}+67 x +70}\, C}{9}+\left (\frac {2 B}{7}-\frac {53 C}{189}\right ) x^{2} \sqrt {6 x^{3}-53 x^{2}+67 x +70}+\left (-\frac {53 B}{105}+\frac {2 A}{5}-\frac {1871 C}{945}\right ) x \sqrt {6 x^{3}-53 x^{2}+67 x +70}+\left (-\frac {3608 B}{945}-\frac {53 A}{45}-\frac {263777 C}{17010}\right ) \sqrt {6 x^{3}-53 x^{2}+67 x +70}+\frac {\left (\frac {7331 A}{90}+\frac {154258 B}{945}+\frac {22387979 C}{34020}\right ) \sqrt {76+114 x}\, \sqrt {483-69 x}\, \sqrt {285-114 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {76+114 x}}{19}, \frac {\sqrt {874}}{46}\right )}{1311 \sqrt {6 x^{3}-53 x^{2}+67 x +70}}+\frac {\left (-\frac {1603 A}{45}-\frac {229871 B}{1890}-\frac {4963871 C}{8505}\right ) \sqrt {76+114 x}\, \sqrt {483-69 x}\, \sqrt {285-114 x}\, \left (-\frac {23 \operatorname {EllipticE}\left (\frac {\sqrt {76+114 x}}{19}, \frac {\sqrt {874}}{46}\right )}{3}+7 \operatorname {EllipticF}\left (\frac {\sqrt {76+114 x}}{19}, \frac {\sqrt {874}}{46}\right )\right )}{1311 \sqrt {6 x^{3}-53 x^{2}+67 x +70}}\) | \(260\) |
| risch | \(\frac {\left (3780 C \,x^{3}+4860 B \,x^{2}-4770 C \,x^{2}+6804 A x -8586 B x -33678 C x -20034 A -64944 B -263777 C \right ) \sqrt {6 x^{3}-53 x^{2}+67 x +70}}{17010}-\frac {\left (1211868 A +4137678 B +19855484 C \right ) \sqrt {76+114 x}\, \sqrt {483-69 x}\, \sqrt {285-114 x}\, \left (-\frac {23 \operatorname {EllipticE}\left (\frac {\sqrt {76+114 x}}{19}, \frac {\sqrt {874}}{46}\right )}{3}+7 \operatorname {EllipticF}\left (\frac {\sqrt {76+114 x}}{19}, \frac {\sqrt {874}}{46}\right )\right )}{44600220 \sqrt {6 x^{3}-53 x^{2}+67 x +70}}+\frac {7331 A \sqrt {76+114 x}\, \sqrt {483-69 x}\, \sqrt {285-114 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {76+114 x}}{19}, \frac {\sqrt {874}}{46}\right )}{117990 \sqrt {6 x^{3}-53 x^{2}+67 x +70}}+\frac {154258 B \sqrt {76+114 x}\, \sqrt {483-69 x}\, \sqrt {285-114 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {76+114 x}}{19}, \frac {\sqrt {874}}{46}\right )}{1238895 \sqrt {6 x^{3}-53 x^{2}+67 x +70}}+\frac {22387979 C \sqrt {76+114 x}\, \sqrt {483-69 x}\, \sqrt {285-114 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {76+114 x}}{19}, \frac {\sqrt {874}}{46}\right )}{44600220 \sqrt {6 x^{3}-53 x^{2}+67 x +70}}\) | \(314\) |
| default | \(A \left (\frac {2 x \sqrt {6 x^{3}-53 x^{2}+67 x +70}}{5}-\frac {53 \sqrt {6 x^{3}-53 x^{2}+67 x +70}}{45}+\frac {7331 \sqrt {76+114 x}\, \sqrt {483-69 x}\, \sqrt {285-114 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {76+114 x}}{19}, \frac {\sqrt {874}}{46}\right )}{117990 \sqrt {6 x^{3}-53 x^{2}+67 x +70}}-\frac {1603 \sqrt {76+114 x}\, \sqrt {483-69 x}\, \sqrt {285-114 x}\, \left (-\frac {23 \operatorname {EllipticE}\left (\frac {\sqrt {76+114 x}}{19}, \frac {\sqrt {874}}{46}\right )}{3}+7 \operatorname {EllipticF}\left (\frac {\sqrt {76+114 x}}{19}, \frac {\sqrt {874}}{46}\right )\right )}{58995 \sqrt {6 x^{3}-53 x^{2}+67 x +70}}\right )+B \left (\frac {2 x^{2} \sqrt {6 x^{3}-53 x^{2}+67 x +70}}{7}-\frac {53 x \sqrt {6 x^{3}-53 x^{2}+67 x +70}}{105}-\frac {3608 \sqrt {6 x^{3}-53 x^{2}+67 x +70}}{945}+\frac {154258 \sqrt {76+114 x}\, \sqrt {483-69 x}\, \sqrt {285-114 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {76+114 x}}{19}, \frac {\sqrt {874}}{46}\right )}{1238895 \sqrt {6 x^{3}-53 x^{2}+67 x +70}}-\frac {229871 \sqrt {76+114 x}\, \sqrt {483-69 x}\, \sqrt {285-114 x}\, \left (-\frac {23 \operatorname {EllipticE}\left (\frac {\sqrt {76+114 x}}{19}, \frac {\sqrt {874}}{46}\right )}{3}+7 \operatorname {EllipticF}\left (\frac {\sqrt {76+114 x}}{19}, \frac {\sqrt {874}}{46}\right )\right )}{2477790 \sqrt {6 x^{3}-53 x^{2}+67 x +70}}\right )+C \left (\frac {2 x^{3} \sqrt {6 x^{3}-53 x^{2}+67 x +70}}{9}-\frac {53 x^{2} \sqrt {6 x^{3}-53 x^{2}+67 x +70}}{189}-\frac {1871 x \sqrt {6 x^{3}-53 x^{2}+67 x +70}}{945}-\frac {263777 \sqrt {6 x^{3}-53 x^{2}+67 x +70}}{17010}+\frac {22387979 \sqrt {76+114 x}\, \sqrt {483-69 x}\, \sqrt {285-114 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {76+114 x}}{19}, \frac {\sqrt {874}}{46}\right )}{44600220 \sqrt {6 x^{3}-53 x^{2}+67 x +70}}-\frac {4963871 \sqrt {76+114 x}\, \sqrt {483-69 x}\, \sqrt {285-114 x}\, \left (-\frac {23 \operatorname {EllipticE}\left (\frac {\sqrt {76+114 x}}{19}, \frac {\sqrt {874}}{46}\right )}{3}+7 \operatorname {EllipticF}\left (\frac {\sqrt {76+114 x}}{19}, \frac {\sqrt {874}}{46}\right )\right )}{11150055 \sqrt {6 x^{3}-53 x^{2}+67 x +70}}\right )\) | \(584\) |
Input:
int((C*x^2+B*x+A)*(6*x^3-53*x^2+67*x+70)^(1/2),x,method=_RETURNVERBOSE)
Output:
2/9*x^3*(6*x^3-53*x^2+67*x+70)^(1/2)*C+(2/7*B-53/189*C)*x^2*(6*x^3-53*x^2+ 67*x+70)^(1/2)+(-53/105*B+2/5*A-1871/945*C)*x*(6*x^3-53*x^2+67*x+70)^(1/2) +(-3608/945*B-53/45*A-263777/17010*C)*(6*x^3-53*x^2+67*x+70)^(1/2)+1/1311* (7331/90*A+154258/945*B+22387979/34020*C)*(76+114*x)^(1/2)*(483-69*x)^(1/2 )*(285-114*x)^(1/2)/(6*x^3-53*x^2+67*x+70)^(1/2)*EllipticF(1/19*(76+114*x) ^(1/2),1/46*874^(1/2))+1/1311*(-1603/45*A-229871/1890*B-4963871/8505*C)*(7 6+114*x)^(1/2)*(483-69*x)^(1/2)*(285-114*x)^(1/2)/(6*x^3-53*x^2+67*x+70)^( 1/2)*(-23/3*EllipticE(1/19*(76+114*x)^(1/2),1/46*874^(1/2))+7*EllipticF(1/ 19*(76+114*x)^(1/2),1/46*874^(1/2)))
Time = 0.08 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.33 \[ \int \left (A+B x+C x^2\right ) \sqrt {70+67 x-53 x^2+6 x^3} \, dx=-\frac {1}{26244} \, \sqrt {6} {\left (204984 \, A + 1704825 \, B + 9276529 \, C\right )} {\rm weierstrassPInverse}\left (\frac {1603}{27}, \frac {18980}{729}, x - \frac {53}{18}\right ) + \frac {1}{51030} \, \sqrt {6} {\left (605934 \, A + 2068839 \, B + 9927742 \, C\right )} {\rm weierstrassZeta}\left (\frac {1603}{27}, \frac {18980}{729}, {\rm weierstrassPInverse}\left (\frac {1603}{27}, \frac {18980}{729}, x - \frac {53}{18}\right )\right ) + \frac {1}{17010} \, {\left (3780 \, C x^{3} + 90 \, {\left (54 \, B - 53 \, C\right )} x^{2} + 18 \, {\left (378 \, A - 477 \, B - 1871 \, C\right )} x - 20034 \, A - 64944 \, B - 263777 \, C\right )} \sqrt {6 \, x^{3} - 53 \, x^{2} + 67 \, x + 70} \] Input:
integrate((C*x^2+B*x+A)*(6*x^3-53*x^2+67*x+70)^(1/2),x, algorithm="fricas" )
Output:
-1/26244*sqrt(6)*(204984*A + 1704825*B + 9276529*C)*weierstrassPInverse(16 03/27, 18980/729, x - 53/18) + 1/51030*sqrt(6)*(605934*A + 2068839*B + 992 7742*C)*weierstrassZeta(1603/27, 18980/729, weierstrassPInverse(1603/27, 1 8980/729, x - 53/18)) + 1/17010*(3780*C*x^3 + 90*(54*B - 53*C)*x^2 + 18*(3 78*A - 477*B - 1871*C)*x - 20034*A - 64944*B - 263777*C)*sqrt(6*x^3 - 53*x ^2 + 67*x + 70)
\[ \int \left (A+B x+C x^2\right ) \sqrt {70+67 x-53 x^2+6 x^3} \, dx=\int \sqrt {\left (x - 7\right ) \left (2 x - 5\right ) \left (3 x + 2\right )} \left (A + B x + C x^{2}\right )\, dx \] Input:
integrate((C*x**2+B*x+A)*(6*x**3-53*x**2+67*x+70)**(1/2),x)
Output:
Integral(sqrt((x - 7)*(2*x - 5)*(3*x + 2))*(A + B*x + C*x**2), x)
\[ \int \left (A+B x+C x^2\right ) \sqrt {70+67 x-53 x^2+6 x^3} \, dx=\int { {\left (C x^{2} + B x + A\right )} \sqrt {6 \, x^{3} - 53 \, x^{2} + 67 \, x + 70} \,d x } \] Input:
integrate((C*x^2+B*x+A)*(6*x^3-53*x^2+67*x+70)^(1/2),x, algorithm="maxima" )
Output:
integrate((C*x^2 + B*x + A)*sqrt(6*x^3 - 53*x^2 + 67*x + 70), x)
\[ \int \left (A+B x+C x^2\right ) \sqrt {70+67 x-53 x^2+6 x^3} \, dx=\int { {\left (C x^{2} + B x + A\right )} \sqrt {6 \, x^{3} - 53 \, x^{2} + 67 \, x + 70} \,d x } \] Input:
integrate((C*x^2+B*x+A)*(6*x^3-53*x^2+67*x+70)^(1/2),x, algorithm="giac")
Output:
integrate((C*x^2 + B*x + A)*sqrt(6*x^3 - 53*x^2 + 67*x + 70), x)
Time = 12.88 (sec) , antiderivative size = 706, normalized size of antiderivative = 2.20 \[ \int \left (A+B x+C x^2\right ) \sqrt {70+67 x-53 x^2+6 x^3} \, dx =\text {Too large to display} \] Input:
int((A + B*x + C*x^2)*(67*x - 53*x^2 + 6*x^3 + 70)^(1/2),x)
Output:
(803*A*x^2)/(9*(67*x - 53*x^2 + 6*x^3 + 70)^(1/2)) - (7216*B)/(27*(67*x - 53*x^2 + 6*x^3 + 70)^(1/2)) - (263777*C)/(243*(67*x - 53*x^2 + 6*x^3 + 70) ^(1/2)) - (2291*A*x)/(45*(67*x - 53*x^2 + 6*x^3 + 70)^(1/2)) - (275126*B*x )/(945*(67*x - 53*x^2 + 6*x^3 + 70)^(1/2)) - (20030519*C*x)/(17010*(67*x - 53*x^2 + 6*x^3 + 70)^(1/2)) - (742*A)/(9*(67*x - 53*x^2 + 6*x^3 + 70)^(1/ 2)) - (424*A*x^3)/(15*(67*x - 53*x^2 + 6*x^3 + 70)^(1/2)) + (12*A*x^4)/(5* (67*x - 53*x^2 + 6*x^3 + 70)^(1/2)) + (35633*B*x^2)/(189*(67*x - 53*x^2 + 6*x^3 + 70)^(1/2)) + (7241*B*x^3)/(315*(67*x - 53*x^2 + 6*x^3 + 70)^(1/2)) - (636*B*x^4)/(35*(67*x - 53*x^2 + 6*x^3 + 70)^(1/2)) + (12*B*x^5)/(7*(67 *x - 53*x^2 + 6*x^3 + 70)^(1/2)) + (2277971*C*x^2)/(3402*(67*x - 53*x^2 + 6*x^3 + 70)^(1/2)) + (24547*C*x^3)/(2835*(67*x - 53*x^2 + 6*x^3 + 70)^(1/2 )) + (16889*C*x^4)/(945*(67*x - 53*x^2 + 6*x^3 + 70)^(1/2)) - (848*C*x^5)/ (63*(67*x - 53*x^2 + 6*x^3 + 70)^(1/2)) + (4*C*x^6)/(3*(67*x - 53*x^2 + 6* x^3 + 70)^(1/2)) + (36869*A*((2*x)/9 - 5/9)^(1/2)*((3*x)/23 + 2/23)^(1/2)* (21/23 - (3*x)/23)^(1/2)*ellipticE(asin((21/23 - (3*x)/23)^(1/2)), 46/27)) /(15*(67*x - 53*x^2 + 6*x^3 + 70)^(1/2)) + (1748*A*((2*x)/9 - 5/9)^(1/2)*( (3*x)/23 + 2/23)^(1/2)*(21/23 - (3*x)/23)^(1/2)*ellipticF(asin((21/23 - (3 *x)/23)^(1/2)), 46/27))/(15*(67*x - 53*x^2 + 6*x^3 + 70)^(1/2)) + (5287033 *B*((2*x)/9 - 5/9)^(1/2)*((3*x)/23 + 2/23)^(1/2)*(21/23 - (3*x)/23)^(1/2)* ellipticE(asin((21/23 - (3*x)/23)^(1/2)), 46/27))/(630*(67*x - 53*x^2 +...
\[ \int \left (A+B x+C x^2\right ) \sqrt {70+67 x-53 x^2+6 x^3} \, dx=\frac {2 \sqrt {6 x^{3}-53 x^{2}+67 x +70}\, a x}{5}-\frac {134 \sqrt {6 x^{3}-53 x^{2}+67 x +70}\, a}{265}+\frac {2 \sqrt {6 x^{3}-53 x^{2}+67 x +70}\, b \,x^{2}}{7}-\frac {53 \sqrt {6 x^{3}-53 x^{2}+67 x +70}\, b x}{105}-\frac {5651 \sqrt {6 x^{3}-53 x^{2}+67 x +70}\, b}{3710}+\frac {2 \sqrt {6 x^{3}-53 x^{2}+67 x +70}\, c \,x^{3}}{9}-\frac {53 \sqrt {6 x^{3}-53 x^{2}+67 x +70}\, c \,x^{2}}{189}-\frac {1871 \sqrt {6 x^{3}-53 x^{2}+67 x +70}\, c x}{945}-\frac {450271 \sqrt {6 x^{3}-53 x^{2}+67 x +70}\, c}{100170}+\frac {15619 \left (\int \frac {\sqrt {6 x^{3}-53 x^{2}+67 x +70}}{6 x^{3}-53 x^{2}+67 x +70}d x \right ) a}{265}+\frac {1922371 \left (\int \frac {\sqrt {6 x^{3}-53 x^{2}+67 x +70}}{6 x^{3}-53 x^{2}+67 x +70}d x \right ) b}{22260}+\frac {57933797 \left (\int \frac {\sqrt {6 x^{3}-53 x^{2}+67 x +70}}{6 x^{3}-53 x^{2}+67 x +70}d x \right ) c}{200340}-\frac {1603 \left (\int \frac {\sqrt {6 x^{3}-53 x^{2}+67 x +70}\, x^{2}}{6 x^{3}-53 x^{2}+67 x +70}d x \right ) a}{265}-\frac {229871 \left (\int \frac {\sqrt {6 x^{3}-53 x^{2}+67 x +70}\, x^{2}}{6 x^{3}-53 x^{2}+67 x +70}d x \right ) b}{11130}-\frac {4963871 \left (\int \frac {\sqrt {6 x^{3}-53 x^{2}+67 x +70}\, x^{2}}{6 x^{3}-53 x^{2}+67 x +70}d x \right ) c}{50085} \] Input:
int((C*x^2+B*x+A)*(6*x^3-53*x^2+67*x+70)^(1/2),x)
Output:
(80136*sqrt(6*x**3 - 53*x**2 + 67*x + 70)*a*x - 101304*sqrt(6*x**3 - 53*x* *2 + 67*x + 70)*a + 57240*sqrt(6*x**3 - 53*x**2 + 67*x + 70)*b*x**2 - 1011 24*sqrt(6*x**3 - 53*x**2 + 67*x + 70)*b*x - 305154*sqrt(6*x**3 - 53*x**2 + 67*x + 70)*b + 44520*sqrt(6*x**3 - 53*x**2 + 67*x + 70)*c*x**3 - 56180*sq rt(6*x**3 - 53*x**2 + 67*x + 70)*c*x**2 - 396652*sqrt(6*x**3 - 53*x**2 + 6 7*x + 70)*c*x - 900542*sqrt(6*x**3 - 53*x**2 + 67*x + 70)*c + 11807964*int (sqrt(6*x**3 - 53*x**2 + 67*x + 70)/(6*x**3 - 53*x**2 + 67*x + 70),x)*a + 17301339*int(sqrt(6*x**3 - 53*x**2 + 67*x + 70)/(6*x**3 - 53*x**2 + 67*x + 70),x)*b + 57933797*int(sqrt(6*x**3 - 53*x**2 + 67*x + 70)/(6*x**3 - 53*x **2 + 67*x + 70),x)*c - 1211868*int((sqrt(6*x**3 - 53*x**2 + 67*x + 70)*x* *2)/(6*x**3 - 53*x**2 + 67*x + 70),x)*a - 4137678*int((sqrt(6*x**3 - 53*x* *2 + 67*x + 70)*x**2)/(6*x**3 - 53*x**2 + 67*x + 70),x)*b - 19855484*int(( sqrt(6*x**3 - 53*x**2 + 67*x + 70)*x**2)/(6*x**3 - 53*x**2 + 67*x + 70),x) *c)/200340