Integrand size = 30, antiderivative size = 205 \[ \int \frac {A+B x+C x^2}{\sqrt {70+67 x-53 x^2+6 x^3}} \, dx=\frac {C (5-2 x) (7-x) (2+3 x)}{9 \sqrt {70+67 x-53 x^2+6 x^3}}-\frac {\sqrt {19} (9 B+53 C) \sqrt {5-2 x} \sqrt {7-x} \sqrt {2+3 x} E\left (\arcsin \left (\frac {\sqrt {2+3 x}}{\sqrt {23}}\right )|\frac {46}{19}\right )}{27 \sqrt {70+67 x-53 x^2+6 x^3}}-\frac {(2 A+5 B+22 C) \sqrt {-7+x} \sqrt {-5+2 x} \sqrt {2+3 x} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {19}{2}}}{\sqrt {2+3 x}}\right ),\frac {46}{19}\right )}{\sqrt {19} \sqrt {70+67 x-53 x^2+6 x^3}} \] Output:
1/9*C*(5-2*x)*(7-x)*(2+3*x)/(6*x^3-53*x^2+67*x+70)^(1/2)-1/27*19^(1/2)*(9* B+53*C)*(5-2*x)^(1/2)*(7-x)^(1/2)*(2+3*x)^(1/2)*EllipticE(1/23*(2+3*x)^(1/ 2)*23^(1/2),1/19*874^(1/2))/(6*x^3-53*x^2+67*x+70)^(1/2)-1/19*(2*A+5*B+22* C)*(-7+x)^(1/2)*(-5+2*x)^(1/2)*(2+3*x)^(1/2)*EllipticF(1/2*38^(1/2)/(2+3*x )^(1/2),1/19*874^(1/2))*19^(1/2)/(6*x^3-53*x^2+67*x+70)^(1/2)
Time = 10.35 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.81 \[ \int \frac {A+B x+C x^2}{\sqrt {70+67 x-53 x^2+6 x^3}} \, dx=-\frac {\sqrt {5-2 x} \left (138 C \sqrt {5-2 x} \left (-14-19 x+3 x^2\right )+46 \sqrt {46} (9 B+53 C) \sqrt {7-x} \sqrt {2+3 x} E\left (\arcsin \left (\sqrt {\frac {2}{19}} \sqrt {2+3 x}\right )|\frac {19}{46}\right )-27 \sqrt {46} (2 A+14 B+75 C) \sqrt {7-x} \sqrt {2+3 x} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {2}{19}} \sqrt {2+3 x}\right ),\frac {19}{46}\right )\right )}{1242 \sqrt {70+67 x-53 x^2+6 x^3}} \] Input:
Integrate[(A + B*x + C*x^2)/Sqrt[70 + 67*x - 53*x^2 + 6*x^3],x]
Output:
-1/1242*(Sqrt[5 - 2*x]*(138*C*Sqrt[5 - 2*x]*(-14 - 19*x + 3*x^2) + 46*Sqrt [46]*(9*B + 53*C)*Sqrt[7 - x]*Sqrt[2 + 3*x]*EllipticE[ArcSin[Sqrt[2/19]*Sq rt[2 + 3*x]], 19/46] - 27*Sqrt[46]*(2*A + 14*B + 75*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 = 3.22 (sec) , antiderivative size = 1338, normalized size of antiderivative = 6.53, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {2526, 2490, 2486, 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.
| \(\displaystyle \int \frac {A+B x+C x^2}{\sqrt {6 x^3-53 x^2+67 x+70}} \, dx\) |
| \(\Big \downarrow \) 2526 |
| \(\displaystyle \frac {1}{18} \int \frac {18 A-67 C+2 (9 B+53 C) x}{\sqrt {6 x^3-53 x^2+67 x+70}}dx+\frac {1}{9} C \sqrt {6 x^3-53 x^2+67 x+70}\) |
| \(\Big \downarrow \) 2490 |
| \(\displaystyle \frac {1}{18} \int \frac {\frac {1}{18} (18 (18 A-67 C)+106 (9 B+53 C))+2 (9 B+53 C) \left (x-\frac {53}{18}\right )}{\sqrt {6 \left (x-\frac {53}{18}\right )^3-\frac {1603}{18} \left (x-\frac {53}{18}\right )-\frac {9490}{243}}}d\left (x-\frac {53}{18}\right )+\frac {1}{9} C \sqrt {6 x^3-53 x^2+67 x+70}\) |
| \(\Big \downarrow \) 2486 |
| \(\displaystyle \frac {1}{9} C \sqrt {6 x^3-53 x^2+67 x+70}+\frac {\sqrt {18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}} \sqrt {324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603} \int \frac {162 A+477 B+2206 C+18 (9 B+53 C) \left (x-\frac {53}{18}\right )}{\sqrt {3} \sqrt {18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}} \sqrt {324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603}}d\left (x-\frac {53}{18}\right )}{3 \sqrt {2} \sqrt {2916 \left (x-\frac {53}{18}\right )^3-43281 \left (x-\frac {53}{18}\right )-18980}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} C \sqrt {6 x^3-53 x^2+67 x+70}+\frac {\sqrt {18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}} \sqrt {324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603} \int \frac {162 A+477 B+2206 C+18 (9 B+53 C) \left (x-\frac {53}{18}\right )}{\sqrt {18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}} \sqrt {324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603}}d\left (x-\frac {53}{18}\right )}{3 \sqrt {6} \sqrt {2916 \left (x-\frac {53}{18}\right )^3-43281 \left (x-\frac {53}{18}\right )-18980}}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {1}{9} C \sqrt {6 x^3-53 x^2+67 x+70}+\frac {\sqrt {18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}} \sqrt {324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603} \left (\left (162 A+\frac {\left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) (9 B+53 C)}{\sqrt [3]{18980+35397 i \sqrt {3}}}+477 B+2206 C\right ) \int \frac {1}{\sqrt {18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}} \sqrt {324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603}}d\left (x-\frac {53}{18}\right )+(9 B+53 C) \int \frac {\sqrt {18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}}}{\sqrt {324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603}}d\left (x-\frac {53}{18}\right )\right )}{3 \sqrt {6} \sqrt {2916 \left (x-\frac {53}{18}\right )^3-43281 \left (x-\frac {53}{18}\right )-18980}}\) |
| \(\Big \downarrow \) 1172 |
| \(\displaystyle \frac {1}{9} \sqrt {6 x^3-53 x^2+67 x+70} C+\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} \left (-\frac {\sqrt {\frac {2}{3}} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (162 A+477 B+2206 C+\frac {\left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) (9 B+53 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 ) (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}}}} \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 )}{3 \sqrt {6} \sqrt {2916 \left (x-\frac {53}{18}\right )^3-43281 \left (x-\frac {53}{18}\right )-18980}}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {1}{9} \sqrt {6 x^3-53 x^2+67 x+70} C+\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} \left (\frac {\sqrt {\frac {2}{3}} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (162 A+477 B+2206 C+\frac {\left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) (9 B+53 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 ) (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}}}} \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 )}{3 \sqrt {6} \sqrt {2916 \left (x-\frac {53}{18}\right )^3-43281 \left (x-\frac {53}{18}\right )-18980}}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {1}{9} \sqrt {6 x^3-53 x^2+67 x+70} C+\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} \left (\frac {\left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) (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}}}} \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 (162 A+477 B+2206 C+\frac {\left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) (9 B+53 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 )}{3 \sqrt {6} \sqrt {2916 \left (x-\frac {53}{18}\right )^3-43281 \left (x-\frac {53}{18}\right )-18980}}\) |
Input:
Int[(A + B*x + C*x^2)/Sqrt[70 + 67*x - 53*x^2 + 6*x^3],x]
Output:
(C*Sqrt[70 + 67*x - 53*x^2 + 6*x^3])/9 + (Sqrt[-((1603 + (18980 + (35397*I )*Sqrt[3])^(2/3))/(18980 + (35397*I)*Sqrt[3])^(1/3)) + 18*(-53/18 + x)]*Sq rt[-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]*(((1603 - (189 80 + (35397*I)*Sqrt[3])^(2/3))*(9*B + 53*C)*Sqrt[-((1603 + (18980 + (35397 *I)*Sqrt[3])^(2/3))/(18980 + (35397*I)*Sqrt[3])^(1/3)) + 18*(-53/18 + x)]* Sqrt[-(((18980 + (35397*I)*Sqrt[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])^(2/3))*(-53/18 + x))/(18980 + (35397*I)*Sqrt[3])^(1/ 3) - 324*(-53/18 + x)^2))/(1603 - (18980 + (35397*I)*Sqrt[3])^(2/3))^2)]*E llipticE[ArcSin[53/18 - x], (2*(18980 + (35397*I)*Sqrt[3] - 1603*(18980 + (35397*I)*Sqrt[3])^(1/3)))/(18980 + (35397*I)*Sqrt[3] - 1603*(18980 + (353 97*I)*Sqrt[3])^(1/3) + Sqrt[-3*(18980 + (35397*I)*Sqrt[3])^(2/3)]*(1603 + (18980 + (35397*I)*Sqrt[3])^(2/3)))])/(3*Sqrt[6]*Sqrt[-(18980 + (35397*I)* Sqrt[3])^(2/3)]*Sqrt[-(((1603 + (18980 + (35397*I)*Sqrt[3])^(2/3))/(18980 + (35397*I)*Sqrt[3])^(1/3) - 18*(-53/18 + x))/((Sqrt[-1/3*(18980 + (35397* I)*Sqrt[3])^(2/3)]*(324 - 519372/(18980 + (35397*I)*Sqrt[3])^(2/3)))/324 - (1603 + (18980 + (35397*I)*Sqrt[3])^(2/3))/(18980 + (35397*I)*Sqrt[3])^(1 /3)))]*Sqrt[-1603 + 2569609/(18980 + (35397*I)*Sqrt[3])^(2/3) + (18980 ...
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[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 = 1.04 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.79
| method | result | size |
| elliptic | \(\frac {\sqrt {6 x^{3}-53 x^{2}+67 x +70}\, C}{9}+\frac {\left (A -\frac {67 C}{18}\right ) \sqrt {76+114 x}\, \sqrt {483-69 x}\, \sqrt {285-114 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {76+114 x}}{19}, \frac {\sqrt {874}}{46}\right )}{1311 \sqrt {6 x^{3}-53 x^{2}+67 x +70}}+\frac {\left (B +\frac {53 C}{9}\right ) \sqrt {76+114 x}\, \sqrt {483-69 x}\, \sqrt {285-114 x}\, \left (-\frac {23 \operatorname {EllipticE}\left (\frac {\sqrt {76+114 x}}{19}, \frac {\sqrt {874}}{46}\right )}{3}+7 \operatorname {EllipticF}\left (\frac {\sqrt {76+114 x}}{19}, \frac {\sqrt {874}}{46}\right )\right )}{1311 \sqrt {6 x^{3}-53 x^{2}+67 x +70}}\) | \(162\) |
| risch | \(\frac {\sqrt {6 x^{3}-53 x^{2}+67 x +70}\, C}{9}+\frac {\left (18 B +106 C \right ) \sqrt {76+114 x}\, \sqrt {483-69 x}\, \sqrt {285-114 x}\, \left (-\frac {23 \operatorname {EllipticE}\left (\frac {\sqrt {76+114 x}}{19}, \frac {\sqrt {874}}{46}\right )}{3}+7 \operatorname {EllipticF}\left (\frac {\sqrt {76+114 x}}{19}, \frac {\sqrt {874}}{46}\right )\right )}{23598 \sqrt {6 x^{3}-53 x^{2}+67 x +70}}+\frac {A \sqrt {76+114 x}\, \sqrt {483-69 x}\, \sqrt {285-114 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {76+114 x}}{19}, \frac {\sqrt {874}}{46}\right )}{1311 \sqrt {6 x^{3}-53 x^{2}+67 x +70}}-\frac {67 C \sqrt {76+114 x}\, \sqrt {483-69 x}\, \sqrt {285-114 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {76+114 x}}{19}, \frac {\sqrt {874}}{46}\right )}{23598 \sqrt {6 x^{3}-53 x^{2}+67 x +70}}\) | \(216\) |
| default | \(\frac {A \sqrt {76+114 x}\, \sqrt {483-69 x}\, \sqrt {285-114 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {76+114 x}}{19}, \frac {\sqrt {874}}{46}\right )}{1311 \sqrt {6 x^{3}-53 x^{2}+67 x +70}}+\frac {B \sqrt {76+114 x}\, \sqrt {483-69 x}\, \sqrt {285-114 x}\, \left (-\frac {23 \operatorname {EllipticE}\left (\frac {\sqrt {76+114 x}}{19}, \frac {\sqrt {874}}{46}\right )}{3}+7 \operatorname {EllipticF}\left (\frac {\sqrt {76+114 x}}{19}, \frac {\sqrt {874}}{46}\right )\right )}{1311 \sqrt {6 x^{3}-53 x^{2}+67 x +70}}+C \left (\frac {\sqrt {6 x^{3}-53 x^{2}+67 x +70}}{9}-\frac {67 \sqrt {76+114 x}\, \sqrt {483-69 x}\, \sqrt {285-114 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {76+114 x}}{19}, \frac {\sqrt {874}}{46}\right )}{23598 \sqrt {6 x^{3}-53 x^{2}+67 x +70}}+\frac {53 \sqrt {76+114 x}\, \sqrt {483-69 x}\, \sqrt {285-114 x}\, \left (-\frac {23 \operatorname {EllipticE}\left (\frac {\sqrt {76+114 x}}{19}, \frac {\sqrt {874}}{46}\right )}{3}+7 \operatorname {EllipticF}\left (\frac {\sqrt {76+114 x}}{19}, \frac {\sqrt {874}}{46}\right )\right )}{11799 \sqrt {6 x^{3}-53 x^{2}+67 x +70}}\right )\) | \(286\) |
Input:
int((C*x^2+B*x+A)/(6*x^3-53*x^2+67*x+70)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/9*(6*x^3-53*x^2+67*x+70)^(1/2)*C+1/1311*(A-67/18*C)*(76+114*x)^(1/2)*(48 3-69*x)^(1/2)*(285-114*x)^(1/2)/(6*x^3-53*x^2+67*x+70)^(1/2)*EllipticF(1/1 9*(76+114*x)^(1/2),1/46*874^(1/2))+1/1311*(B+53/9*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)*(-23/3*Ellipti cE(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 = 63, normalized size of antiderivative = 0.31 \[ \int \frac {A+B x+C x^2}{\sqrt {70+67 x-53 x^2+6 x^3}} \, dx=\frac {1}{486} \, \sqrt {6} {\left (162 \, A + 477 \, B + 2206 \, C\right )} {\rm weierstrassPInverse}\left (\frac {1603}{27}, \frac {18980}{729}, x - \frac {53}{18}\right ) - \frac {1}{27} \, \sqrt {6} {\left (9 \, B + 53 \, C\right )} {\rm weierstrassZeta}\left (\frac {1603}{27}, \frac {18980}{729}, {\rm weierstrassPInverse}\left (\frac {1603}{27}, \frac {18980}{729}, x - \frac {53}{18}\right )\right ) + \frac {1}{9} \, \sqrt {6 \, x^{3} - 53 \, x^{2} + 67 \, x + 70} C \] Input:
integrate((C*x^2+B*x+A)/(6*x^3-53*x^2+67*x+70)^(1/2),x, algorithm="fricas" )
Output:
1/486*sqrt(6)*(162*A + 477*B + 2206*C)*weierstrassPInverse(1603/27, 18980/ 729, x - 53/18) - 1/27*sqrt(6)*(9*B + 53*C)*weierstrassZeta(1603/27, 18980 /729, weierstrassPInverse(1603/27, 18980/729, x - 53/18)) + 1/9*sqrt(6*x^3 - 53*x^2 + 67*x + 70)*C
\[ \int \frac {A+B x+C x^2}{\sqrt {70+67 x-53 x^2+6 x^3}} \, dx=\int \frac {A + B x + C x^{2}}{\sqrt {\left (x - 7\right ) \left (2 x - 5\right ) \left (3 x + 2\right )}}\, dx \] Input:
integrate((C*x**2+B*x+A)/(6*x**3-53*x**2+67*x+70)**(1/2),x)
Output:
Integral((A + B*x + C*x**2)/sqrt((x - 7)*(2*x - 5)*(3*x + 2)), x)
\[ \int \frac {A+B x+C x^2}{\sqrt {70+67 x-53 x^2+6 x^3}} \, dx=\int { \frac {C x^{2} + B x + A}{\sqrt {6 \, x^{3} - 53 \, x^{2} + 67 \, x + 70}} \,d x } \] 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)
\[ \int \frac {A+B x+C x^2}{\sqrt {70+67 x-53 x^2+6 x^3}} \, dx=\int { \frac {C x^{2} + B x + A}{\sqrt {6 \, x^{3} - 53 \, x^{2} + 67 \, x + 70}} \,d x } \] 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 = 0.06 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.00 \[ \int \frac {A+B x+C x^2}{\sqrt {70+67 x-53 x^2+6 x^3}} \, dx=-\frac {46\,A\,\sqrt {\frac {2\,x}{9}-\frac {5}{9}}\,\sqrt {\frac {3\,x}{23}+\frac {2}{23}}\,\sqrt {\frac {21}{23}-\frac {3\,x}{23}}\,\mathrm {F}\left (\mathrm {asin}\left (\sqrt {\frac {21}{23}-\frac {3\,x}{23}}\right )\middle |\frac {46}{27}\right )-2\,C\,\left (x^3-\frac {53\,x^2}{6}+\frac {67\,x}{6}+\frac {35}{3}\right )+207\,B\,\sqrt {\frac {2\,x}{9}-\frac {5}{9}}\,\sqrt {\frac {3\,x}{23}+\frac {2}{23}}\,\sqrt {\frac {21}{23}-\frac {3\,x}{23}}\,\mathrm {E}\left (\mathrm {asin}\left (\sqrt {\frac {21}{23}-\frac {3\,x}{23}}\right )\middle |\frac {46}{27}\right )+115\,B\,\sqrt {\frac {2\,x}{9}-\frac {5}{9}}\,\sqrt {\frac {3\,x}{23}+\frac {2}{23}}\,\sqrt {\frac {21}{23}-\frac {3\,x}{23}}\,\mathrm {F}\left (\mathrm {asin}\left (\sqrt {\frac {21}{23}-\frac {3\,x}{23}}\right )\middle |\frac {46}{27}\right )+1219\,C\,\sqrt {\frac {2\,x}{9}-\frac {5}{9}}\,\sqrt {\frac {3\,x}{23}+\frac {2}{23}}\,\sqrt {\frac {21}{23}-\frac {3\,x}{23}}\,\mathrm {E}\left (\mathrm {asin}\left (\sqrt {\frac {21}{23}-\frac {3\,x}{23}}\right )\middle |\frac {46}{27}\right )+506\,C\,\sqrt {\frac {2\,x}{9}-\frac {5}{9}}\,\sqrt {\frac {3\,x}{23}+\frac {2}{23}}\,\sqrt {\frac {21}{23}-\frac {3\,x}{23}}\,\mathrm {F}\left (\mathrm {asin}\left (\sqrt {\frac {21}{23}-\frac {3\,x}{23}}\right )\middle |\frac {46}{27}\right )}{3\,\sqrt {6\,x^3-53\,x^2+67\,x+70}} \] Input:
int((A + B*x + C*x^2)/(67*x - 53*x^2 + 6*x^3 + 70)^(1/2),x)
Output:
-(46*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) - 2*C*((67*x)/6 - (53 *x^2)/6 + x^3 + 35/3) + 207*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 ) + 115*B*((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) + 1219*C*((2*x)/9 - 5/9)^(1/2)*((3*x)/23 + 2/23)^(1/2)*(21/23 - (3*x)/23)^(1/2)*ellipticE(as in((21/23 - (3*x)/23)^(1/2)), 46/27) + 506*C*((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))/(3*(67*x - 53*x^2 + 6*x^3 + 70)^(1/2))
\[ \int \frac {A+B x+C x^2}{\sqrt {70+67 x-53 x^2+6 x^3}} \, dx=-\frac {\sqrt {6 x^{3}-53 x^{2}+67 x +70}\, b}{53}+\left (\int \frac {\sqrt {6 x^{3}-53 x^{2}+67 x +70}}{6 x^{3}-53 x^{2}+67 x +70}d x \right ) a +\frac {67 \left (\int \frac {\sqrt {6 x^{3}-53 x^{2}+67 x +70}}{6 x^{3}-53 x^{2}+67 x +70}d x \right ) b}{106}+\frac {9 \left (\int \frac {\sqrt {6 x^{3}-53 x^{2}+67 x +70}\, x^{2}}{6 x^{3}-53 x^{2}+67 x +70}d x \right ) b}{53}+\left (\int \frac {\sqrt {6 x^{3}-53 x^{2}+67 x +70}\, x^{2}}{6 x^{3}-53 x^{2}+67 x +70}d x \right ) c \] Input:
int((C*x^2+B*x+A)/(6*x^3-53*x^2+67*x+70)^(1/2),x)
Output:
( - 2*sqrt(6*x**3 - 53*x**2 + 67*x + 70)*b + 106*int(sqrt(6*x**3 - 53*x**2 + 67*x + 70)/(6*x**3 - 53*x**2 + 67*x + 70),x)*a + 67*int(sqrt(6*x**3 - 5 3*x**2 + 67*x + 70)/(6*x**3 - 53*x**2 + 67*x + 70),x)*b + 18*int((sqrt(6*x **3 - 53*x**2 + 67*x + 70)*x**2)/(6*x**3 - 53*x**2 + 67*x + 70),x)*b + 106 *int((sqrt(6*x**3 - 53*x**2 + 67*x + 70)*x**2)/(6*x**3 - 53*x**2 + 67*x + 70),x)*c)/106