Integrand size = 30, antiderivative size = 319 \[ \int \frac {A+B x+C x^2}{\left (70+67 x-53 x^2+6 x^3\right )^{3/2}} \, dx=-\frac {2 (A+7 B+49 C) (5-2 x) (7-x) (2+3 x)}{207 \left (70+67 x-53 x^2+6 x^3\right )^{3/2}}+\frac {2 (130 A+496 B+2437 C) (5-2 x) (7-x)^2 (2+3 x)}{35397 \left (70+67 x-53 x^2+6 x^3\right )^{3/2}}-\frac {2 (3206 A+7331 B+31238 C) (5-2 x)^2 (7-x)^2 (2+3 x)}{5156163 \left (70+67 x-53 x^2+6 x^3\right )^{3/2}}-\frac {2 (3206 A+7331 B+31238 C) (5-2 x)^{3/2} (7-x)^{3/2} (2+3 x)^{3/2} E\left (\arcsin \left (\frac {\sqrt {2+3 x}}{\sqrt {23}}\right )|\frac {46}{19}\right )}{814131 \sqrt {19} \left (70+67 x-53 x^2+6 x^3\right )^{3/2}}-\frac {2 (8 A-151 B-919 C) (-7+x)^{3/2} (-5+2 x)^{3/2} (2+3 x)^{3/2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {19}{2}}}{\sqrt {2+3 x}}\right ),\frac {46}{19}\right )}{30153 \sqrt {19} \left (70+67 x-53 x^2+6 x^3\right )^{3/2}} \] Output:
-2/207*(A+7*B+49*C)*(5-2*x)*(7-x)*(2+3*x)/(6*x^3-53*x^2+67*x+70)^(3/2)+2/3 5397*(130*A+496*B+2437*C)*(5-2*x)*(7-x)^2*(2+3*x)/(6*x^3-53*x^2+67*x+70)^( 3/2)-2/5156163*(3206*A+7331*B+31238*C)*(5-2*x)^2*(7-x)^2*(2+3*x)/(6*x^3-53 *x^2+67*x+70)^(3/2)-2/15468489*(3206*A+7331*B+31238*C)*(5-2*x)^(3/2)*(7-x) ^(3/2)*(2+3*x)^(3/2)*EllipticE(1/23*(2+3*x)^(1/2)*23^(1/2),1/19*874^(1/2)) *19^(1/2)/(6*x^3-53*x^2+67*x+70)^(3/2)-2/572907*(8*A-151*B-919*C)*(-7+x)^( 3/2)*(-5+2*x)^(3/2)*(2+3*x)^(3/2)*EllipticF(1/2*38^(1/2)/(2+3*x)^(1/2),1/1 9*874^(1/2))*19^(1/2)/(6*x^3-53*x^2+67*x+70)^(3/2)
Time = 10.40 (sec) , antiderivative size = 205, normalized size of antiderivative = 0.64 \[ \int \frac {A+B x+C x^2}{\left (70+67 x-53 x^2+6 x^3\right )^{3/2}} \, dx=\frac {2 \sqrt {5-2 x} \left (C \left (513170+715597 x-187428 x^2\right )+B \left (224420+201115 x-43986 x^2\right )+A \left (-13687+125932 x-19236 x^2\right )\right )+2 \sqrt {46} (3206 A+7331 B+31238 C) \sqrt {7-x} (-5+2 x) \sqrt {2+3 x} E\left (\arcsin \left (\sqrt {\frac {2}{19}} \sqrt {2+3 x}\right )|\frac {19}{46}\right )-27 \sqrt {46} (146 A+194 B+599 C) \sqrt {7-x} (-5+2 x) \sqrt {2+3 x} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {2}{19}} \sqrt {2+3 x}\right ),\frac {19}{46}\right )}{15468489 \sqrt {5-2 x} \sqrt {70+67 x-53 x^2+6 x^3}} \] Input:
Integrate[(A + B*x + C*x^2)/(70 + 67*x - 53*x^2 + 6*x^3)^(3/2),x]
Output:
(2*Sqrt[5 - 2*x]*(C*(513170 + 715597*x - 187428*x^2) + B*(224420 + 201115* x - 43986*x^2) + A*(-13687 + 125932*x - 19236*x^2)) + 2*Sqrt[46]*(3206*A + 7331*B + 31238*C)*Sqrt[7 - x]*(-5 + 2*x)*Sqrt[2 + 3*x]*EllipticE[ArcSin[S qrt[2/19]*Sqrt[2 + 3*x]], 19/46] - 27*Sqrt[46]*(146*A + 194*B + 599*C)*Sqr t[7 - x]*(-5 + 2*x)*Sqrt[2 + 3*x]*EllipticF[ArcSin[Sqrt[2/19]*Sqrt[2 + 3*x ]], 19/46])/(15468489*Sqrt[5 - 2*x]*Sqrt[70 + 67*x - 53*x^2 + 6*x^3])
Result contains complex when optimal does not.
Time = 7.55 (sec) , antiderivative size = 1923, normalized size of antiderivative = 6.03, 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, 1235, 27, 1237, 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.
| \(\displaystyle \int \frac {A+B x+C x^2}{\left (6 x^3-53 x^2+67 x+70\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 2526 |
| \(\displaystyle \frac {1}{18} \int \frac {18 A-67 C+2 (9 B+53 C) x}{\left (6 x^3-53 x^2+67 x+70\right )^{3/2}}dx-\frac {C}{9 \sqrt {6 x^3-53 x^2+67 x+70}}\) |
| \(\Big \downarrow \) 2490 |
| \(\displaystyle \frac {1}{18} \int \frac {\frac {1}{18} (18 (18 A-67 C)+106 (9 B+53 C))+2 (9 B+53 C) \left (x-\frac {53}{18}\right )}{\left (6 \left (x-\frac {53}{18}\right )^3-\frac {1603}{18} \left (x-\frac {53}{18}\right )-\frac {9490}{243}\right )^{3/2}}d\left (x-\frac {53}{18}\right )-\frac {C}{9 \sqrt {6 x^3-53 x^2+67 x+70}}\) |
| \(\Big \downarrow \) 2486 |
| \(\displaystyle -\frac {C}{9 \sqrt {6 x^3-53 x^2+67 x+70}}+\frac {3 \sqrt {2} \left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{3/2} \left (324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603\right )^{3/2} \int \frac {9 \sqrt {3} \left (162 A+477 B+2206 C+18 (9 B+53 C) \left (x-\frac {53}{18}\right )\right )}{\left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{3/2} \left (324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603\right )^{3/2}}d\left (x-\frac {53}{18}\right )}{\left (2916 \left (x-\frac {53}{18}\right )^3-43281 \left (x-\frac {53}{18}\right )-18980\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {C}{9 \sqrt {6 x^3-53 x^2+67 x+70}}+\frac {27 \sqrt {6} \left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{3/2} \left (324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603\right )^{3/2} \int \frac {162 A+477 B+2206 C+18 (9 B+53 C) \left (x-\frac {53}{18}\right )}{\left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{3/2} \left (324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603\right )^{3/2}}d\left (x-\frac {53}{18}\right )}{\left (2916 \left (x-\frac {53}{18}\right )^3-43281 \left (x-\frac {53}{18}\right )-18980\right )^{3/2}}\) |
| \(\Big \downarrow \) 1235 |
| \(\displaystyle \frac {27 \sqrt {6} \left (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}}}\right )^{3/2} \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} \left (\frac {i \sqrt [3]{18980+35397 i \sqrt {3}} \int \frac {157464 \left (\frac {\left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (5139218-4809 \left (18980+35397 i \sqrt {3}\right )^{2/3}+2 \left (18980+35397 i \sqrt {3}\right )^{4/3}\right ) (9 B+53 C)+\sqrt [3]{18980+35397 i \sqrt {3}} \left (2569609-6412 \left (18980+35397 i \sqrt {3}\right )^{2/3}+\left (18980+35397 i \sqrt {3}\right )^{4/3}\right ) (162 A+477 B+2206 C)}{18980+35397 i \sqrt {3}}-\frac {18 \left (i \left (2569609 i+\left (18980 i-35397 \sqrt {3}\right ) \sqrt [3]{18980+35397 i \sqrt {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)\right ) \left (x-\frac {53}{18}\right )}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}\right )}{\left (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}}}\right )^{3/2} \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 )}{33442519248 \sqrt {3} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right )}-\frac {i \sqrt [3]{18980+35397 i \sqrt {3}} \left (9 (3206 A+7331 B+31238 C)-\frac {\left (\left (2569609+\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)\right ) \left (x-\frac {53}{18}\right )}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}\right )}{106191 \sqrt {3} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\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}}\right )}{\left (2916 \left (x-\frac {53}{18}\right )^3-43281 \left (x-\frac {53}{18}\right )-18980\right )^{3/2}}-\frac {C}{9 \sqrt {6 x^3-53 x^2+67 x+70}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {27 \sqrt {6} \left (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}}}\right )^{3/2} \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} \left (\frac {i \sqrt [3]{18980+35397 i \sqrt {3}} \int \frac {\frac {\left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (5139218-4809 \left (18980+35397 i \sqrt {3}\right )^{2/3}+2 \left (18980+35397 i \sqrt {3}\right )^{4/3}\right ) (9 B+53 C)+\sqrt [3]{18980+35397 i \sqrt {3}} \left (2569609-6412 \left (18980+35397 i \sqrt {3}\right )^{2/3}+\left (18980+35397 i \sqrt {3}\right )^{4/3}\right ) (162 A+477 B+2206 C)}{18980+35397 i \sqrt {3}}-\frac {18 \left (i \left (2569609 i+\left (18980 i-35397 \sqrt {3}\right ) \sqrt [3]{18980+35397 i \sqrt {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)\right ) \left (x-\frac {53}{18}\right )}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}}{\left (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}}}\right )^{3/2} \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 )}{212382 \sqrt {3} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right )}-\frac {i \sqrt [3]{18980+35397 i \sqrt {3}} \left (9 (3206 A+7331 B+31238 C)-\frac {\left (\left (2569609+\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)\right ) \left (x-\frac {53}{18}\right )}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}\right )}{106191 \sqrt {3} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\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}}\right )}{\left (2916 \left (x-\frac {53}{18}\right )^3-43281 \left (x-\frac {53}{18}\right )-18980\right )^{3/2}}-\frac {C}{9 \sqrt {6 x^3-53 x^2+67 x+70}}\) |
| \(\Big \downarrow \) 1237 |
| \(\displaystyle \frac {27 \sqrt {6} \left (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}}}\right )^{3/2} \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} \left (\frac {i \sqrt [3]{18980+35397 i \sqrt {3}} \left (\frac {18 \left (18980+35397 i \sqrt {3}\right )^{2/3} (3206 A+7331 B+31238 C) \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 (2569609+1603 \left (18980+35397 i \sqrt {3}\right )^{2/3}+\left (18980+35397 i \sqrt {3}\right )^{4/3}\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}}}}}-\frac {\left (18980+35397 i \sqrt {3}\right )^{2/3} \int \frac {78732 \left (37960 A-173741 B-1164437 C+18 (3206 A+7331 B+31238 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 )}{486 \left (2569609+1603 \left (18980+35397 i \sqrt {3}\right )^{2/3}+\left (18980+35397 i \sqrt {3}\right )^{4/3}\right )}\right )}{212382 \sqrt {3} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right )}-\frac {i \sqrt [3]{18980+35397 i \sqrt {3}} \left (9 (3206 A+7331 B+31238 C)-\frac {\left (\left (2569609+\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)\right ) \left (x-\frac {53}{18}\right )}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}\right )}{106191 \sqrt {3} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\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}}\right )}{\left (2916 \left (x-\frac {53}{18}\right )^3-43281 \left (x-\frac {53}{18}\right )-18980\right )^{3/2}}-\frac {C}{9 \sqrt {6 x^3-53 x^2+67 x+70}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {27 \sqrt {6} \left (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}}}\right )^{3/2} \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} \left (\frac {i \sqrt [3]{18980+35397 i \sqrt {3}} \left (\frac {18 \left (18980+35397 i \sqrt {3}\right )^{2/3} (3206 A+7331 B+31238 C) \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 (2569609+1603 \left (18980+35397 i \sqrt {3}\right )^{2/3}+\left (18980+35397 i \sqrt {3}\right )^{4/3}\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}}}}}-\frac {162 \left (18980+35397 i \sqrt {3}\right )^{2/3} \int \frac {37960 A-173741 B-1164437 C+18 (3206 A+7331 B+31238 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 )}{2569609+1603 \left (18980+35397 i \sqrt {3}\right )^{2/3}+\left (18980+35397 i \sqrt {3}\right )^{4/3}}\right )}{212382 \sqrt {3} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right )}-\frac {i \sqrt [3]{18980+35397 i \sqrt {3}} \left (9 (3206 A+7331 B+31238 C)-\frac {\left (\left (2569609+\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)\right ) \left (x-\frac {53}{18}\right )}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}\right )}{106191 \sqrt {3} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\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}}\right )}{\left (2916 \left (x-\frac {53}{18}\right )^3-43281 \left (x-\frac {53}{18}\right )-18980\right )^{3/2}}-\frac {C}{9 \sqrt {6 x^3-53 x^2+67 x+70}}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {27 \sqrt {6} \left (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}}}\right )^{3/2} \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} \left (\frac {i \sqrt [3]{18980+35397 i \sqrt {3}} \left (\frac {18 \left (18980+35397 i \sqrt {3}\right )^{2/3} (3206 A+7331 B+31238 C) \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 (2569609+1603 \left (18980+35397 i \sqrt {3}\right )^{2/3}+\left (18980+35397 i \sqrt {3}\right )^{4/3}\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}}}}}-\frac {162 \left (18980+35397 i \sqrt {3}\right )^{2/3} \left (\left (37960 A-173741 B-1164437 C+\frac {\left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) (3206 A+7331 B+31238 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 )+(3206 A+7331 B+31238 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 )}{2569609+1603 \left (18980+35397 i \sqrt {3}\right )^{2/3}+\left (18980+35397 i \sqrt {3}\right )^{4/3}}\right )}{212382 \sqrt {3} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right )}-\frac {i \sqrt [3]{18980+35397 i \sqrt {3}} \left (9 (3206 A+7331 B+31238 C)-\frac {\left (\left (2569609+\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)\right ) \left (x-\frac {53}{18}\right )}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}\right )}{106191 \sqrt {3} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\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}}\right )}{\left (2916 \left (x-\frac {53}{18}\right )^3-43281 \left (x-\frac {53}{18}\right )-18980\right )^{3/2}}-\frac {C}{9 \sqrt {6 x^3-53 x^2+67 x+70}}\) |
| \(\Big \downarrow \) 1172 |
| \(\displaystyle \frac {27 \sqrt {6} \left (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}}}\right )^{3/2} \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} \left (\frac {i \sqrt [3]{18980+35397 i \sqrt {3}} \left (\frac {18 \left (18980+35397 i \sqrt {3}\right )^{2/3} (3206 A+7331 B+31238 C) \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 (2569609+1603 \left (18980+35397 i \sqrt {3}\right )^{2/3}+\left (18980+35397 i \sqrt {3}\right )^{4/3}\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}}}}}-\frac {162 \left (18980+35397 i \sqrt {3}\right )^{2/3} \left (-\frac {\sqrt {\frac {2}{3}} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (37960 A-173741 B-1164437 C+\frac {\left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) (3206 A+7331 B+31238 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 ) (3206 A+7331 B+31238 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 )}{2569609+1603 \left (18980+35397 i \sqrt {3}\right )^{2/3}+\left (18980+35397 i \sqrt {3}\right )^{4/3}}\right )}{212382 \sqrt {3} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right )}-\frac {i \sqrt [3]{18980+35397 i \sqrt {3}} \left (9 (3206 A+7331 B+31238 C)-\frac {\left (\left (2569609+\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)\right ) \left (x-\frac {53}{18}\right )}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}\right )}{106191 \sqrt {3} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\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}}\right )}{\left (2916 \left (x-\frac {53}{18}\right )^3-43281 \left (x-\frac {53}{18}\right )-18980\right )^{3/2}}-\frac {C}{9 \sqrt {6 x^3-53 x^2+67 x+70}}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {27 \sqrt {6} \left (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}}}\right )^{3/2} \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} \left (\frac {i \sqrt [3]{18980+35397 i \sqrt {3}} \left (\frac {18 \left (18980+35397 i \sqrt {3}\right )^{2/3} (3206 A+7331 B+31238 C) \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 (2569609+1603 \left (18980+35397 i \sqrt {3}\right )^{2/3}+\left (18980+35397 i \sqrt {3}\right )^{4/3}\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}}}}}-\frac {162 \left (18980+35397 i \sqrt {3}\right )^{2/3} \left (\frac {\sqrt {\frac {2}{3}} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (37960 A-173741 B-1164437 C+\frac {\left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) (3206 A+7331 B+31238 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 ) (3206 A+7331 B+31238 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 )}{2569609+1603 \left (18980+35397 i \sqrt {3}\right )^{2/3}+\left (18980+35397 i \sqrt {3}\right )^{4/3}}\right )}{212382 \sqrt {3} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right )}-\frac {i \sqrt [3]{18980+35397 i \sqrt {3}} \left (9 (3206 A+7331 B+31238 C)-\frac {\left (\left (2569609+\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)\right ) \left (x-\frac {53}{18}\right )}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}\right )}{106191 \sqrt {3} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\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}}\right )}{\left (2916 \left (x-\frac {53}{18}\right )^3-43281 \left (x-\frac {53}{18}\right )-18980\right )^{3/2}}-\frac {C}{9 \sqrt {6 x^3-53 x^2+67 x+70}}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {27 \sqrt {6} \left (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}}}\right )^{3/2} \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} \left (\frac {i \sqrt [3]{18980+35397 i \sqrt {3}} \left (\frac {18 \left (18980+35397 i \sqrt {3}\right )^{2/3} (3206 A+7331 B+31238 C) \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 (2569609+1603 \left (18980+35397 i \sqrt {3}\right )^{2/3}+\left (18980+35397 i \sqrt {3}\right )^{4/3}\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}}}}}-\frac {162 \left (18980+35397 i \sqrt {3}\right )^{2/3} \left (\frac {\left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) (3206 A+7331 B+31238 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 (37960 A-173741 B-1164437 C+\frac {\left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) (3206 A+7331 B+31238 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 )}{2569609+1603 \left (18980+35397 i \sqrt {3}\right )^{2/3}+\left (18980+35397 i \sqrt {3}\right )^{4/3}}\right )}{212382 \sqrt {3} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right )}-\frac {i \sqrt [3]{18980+35397 i \sqrt {3}} \left (9 (3206 A+7331 B+31238 C)-\frac {\left (\left (2569609+\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)\right ) \left (x-\frac {53}{18}\right )}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}\right )}{106191 \sqrt {3} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\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}}\right )}{\left (2916 \left (x-\frac {53}{18}\right )^3-43281 \left (x-\frac {53}{18}\right )-18980\right )^{3/2}}-\frac {C}{9 \sqrt {6 x^3-53 x^2+67 x+70}}\) |
Input:
Int[(A + B*x + C*x^2)/(70 + 67*x - 53*x^2 + 6*x^3)^(3/2),x]
Output:
-1/9*C/Sqrt[70 + 67*x - 53*x^2 + 6*x^3] + (27*Sqrt[6]*(-((1603 + (18980 + (35397*I)*Sqrt[3])^(2/3))/(18980 + (35397*I)*Sqrt[3])^(1/3)) + 18*(-53/18 + x))^(3/2)*(-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 )*(((-1/106191*I)*(18980 + (35397*I)*Sqrt[3])^(1/3)*(9*(3206*A + 7331*B + 31238*C) - (((2569609 + (18980 + (35397*I)*Sqrt[3])^(4/3))*(9*B + 53*C) - (18980 + (35397*I)*Sqrt[3])^(1/3)*(1603 + (18980 + (35397*I)*Sqrt[3])^(2/3 ))*(162*A + 477*B + 2206*C))*(-53/18 + x))/(18980 + (35397*I)*Sqrt[3])^(2/ 3)))/(Sqrt[3]*(1603 - (18980 + (35397*I)*Sqrt[3])^(2/3))*Sqrt[-((1603 + (1 8980 + (35397*I)*Sqrt[3])^(2/3))/(18980 + (35397*I)*Sqrt[3])^(1/3)) + 18*( -53/18 + x)]*Sqrt[-1603 + 2569609/(18980 + (35397*I)*Sqrt[3])^(2/3) + (189 80 + (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] ) + ((I/212382)*(18980 + (35397*I)*Sqrt[3])^(1/3)*((18*(18980 + (35397*I)* Sqrt[3])^(2/3)*(3206*A + 7331*B + 31238*C)*Sqrt[-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])/((2569609 + 1603*(18980 + (35397*I)*Sqrt[3 ])^(2/3) + (18980 + (35397*I)*Sqrt[3])^(4/3))*Sqrt[-((1603 + (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[(d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2 *a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^m *(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2*(p + m + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d* m + b*e*m) - b*d*(3*c*d - b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p] )
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(e*f - d*g)*(d + e*x)^(m + 1)*((a + b* x + c*x^2)^(p + 1)/((m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[1/((m + 1) *(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*Simp[ (c*d*f - f*b*e + a*e*g)*(m + 1) + b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && LtQ[m, -1 ] && (IntegerQ[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/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 = 1.25 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.65
| method | result | size |
| elliptic | \(-\frac {12 \left (\left (\frac {7331 B}{15468489}+\frac {31238 C}{15468489}+\frac {3206 A}{15468489}\right ) x^{2}+\left (-\frac {10585 B}{4884786}-\frac {37663 C}{4884786}-\frac {3314 A}{2442393}\right ) x -\frac {112210 B}{46405467}-\frac {256585 C}{46405467}+\frac {13687 A}{92810934}\right )}{\sqrt {6 x^{3}-53 x^{2}+67 x +70}}+\frac {\left (-\frac {62476 B}{5156163}-\frac {313339 C}{5156163}-\frac {14662 A}{5156163}\right ) \sqrt {76+114 x}\, \sqrt {483-69 x}\, \sqrt {285-114 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {76+114 x}}{19}, \frac {\sqrt {874}}{46}\right )}{1311 \sqrt {6 x^{3}-53 x^{2}+67 x +70}}+\frac {\left (\frac {14662 B}{5156163}+\frac {62476 C}{5156163}+\frac {6412 A}{5156163}\right ) \sqrt {76+114 x}\, \sqrt {483-69 x}\, \sqrt {285-114 x}\, \left (-\frac {23 \operatorname {EllipticE}\left (\frac {\sqrt {76+114 x}}{19}, \frac {\sqrt {874}}{46}\right )}{3}+7 \operatorname {EllipticF}\left (\frac {\sqrt {76+114 x}}{19}, \frac {\sqrt {874}}{46}\right )\right )}{1311 \sqrt {6 x^{3}-53 x^{2}+67 x +70}}\) | \(207\) |
| risch | \(-\frac {2 \left (19236 A \,x^{2}+43986 B \,x^{2}+187428 C \,x^{2}-125932 A x -201115 B x -715597 C x +13687 A -224420 B -513170 C \right )}{15468489 \sqrt {6 x^{3}-53 x^{2}+67 x +70}}+\frac {\left (6412 A +14662 B +62476 C \right ) \sqrt {76+114 x}\, \sqrt {483-69 x}\, \sqrt {285-114 x}\, \left (-\frac {23 \operatorname {EllipticE}\left (\frac {\sqrt {76+114 x}}{19}, \frac {\sqrt {874}}{46}\right )}{3}+7 \operatorname {EllipticF}\left (\frac {\sqrt {76+114 x}}{19}, \frac {\sqrt {874}}{46}\right )\right )}{6759729693 \sqrt {6 x^{3}-53 x^{2}+67 x +70}}-\frac {14662 A \sqrt {76+114 x}\, \sqrt {483-69 x}\, \sqrt {285-114 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {76+114 x}}{19}, \frac {\sqrt {874}}{46}\right )}{6759729693 \sqrt {6 x^{3}-53 x^{2}+67 x +70}}-\frac {62476 B \sqrt {76+114 x}\, \sqrt {483-69 x}\, \sqrt {285-114 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {76+114 x}}{19}, \frac {\sqrt {874}}{46}\right )}{6759729693 \sqrt {6 x^{3}-53 x^{2}+67 x +70}}-\frac {313339 C \sqrt {76+114 x}\, \sqrt {483-69 x}\, \sqrt {285-114 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {76+114 x}}{19}, \frac {\sqrt {874}}{46}\right )}{6759729693 \sqrt {6 x^{3}-53 x^{2}+67 x +70}}\) | \(314\) |
| default | \(A \left (-\frac {12 \left (\frac {3206}{15468489} x^{2}-\frac {3314}{2442393} x +\frac {13687}{92810934}\right )}{\sqrt {6 x^{3}-53 x^{2}+67 x +70}}-\frac {14662 \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 )}{6759729693 \sqrt {6 x^{3}-53 x^{2}+67 x +70}}+\frac {6412 \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 )}{6759729693 \sqrt {6 x^{3}-53 x^{2}+67 x +70}}\right )+B \left (-\frac {12 \left (-\frac {112210}{46405467}+\frac {7331}{15468489} x^{2}-\frac {10585}{4884786} x \right )}{\sqrt {6 x^{3}-53 x^{2}+67 x +70}}-\frac {62476 \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 )}{6759729693 \sqrt {6 x^{3}-53 x^{2}+67 x +70}}+\frac {14662 \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 )}{6759729693 \sqrt {6 x^{3}-53 x^{2}+67 x +70}}\right )+C \left (-\frac {12 \left (-\frac {256585}{46405467}+\frac {31238}{15468489} x^{2}-\frac {37663}{4884786} x \right )}{\sqrt {6 x^{3}-53 x^{2}+67 x +70}}-\frac {313339 \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 )}{6759729693 \sqrt {6 x^{3}-53 x^{2}+67 x +70}}+\frac {62476 \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 )}{6759729693 \sqrt {6 x^{3}-53 x^{2}+67 x +70}}\right )\) | \(488\) |
Input:
int((C*x^2+B*x+A)/(6*x^3-53*x^2+67*x+70)^(3/2),x,method=_RETURNVERBOSE)
Output:
-12*((7331/15468489*B+31238/15468489*C+3206/15468489*A)*x^2+(-10585/488478 6*B-37663/4884786*C-3314/2442393*A)*x-112210/46405467*B-256585/46405467*C+ 13687/92810934*A)/(6*x^3-53*x^2+67*x+70)^(1/2)+1/1311*(-62476/5156163*B-31 3339/5156163*C-14662/5156163*A)*(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),1/4 6*874^(1/2))+1/1311*(14662/5156163*B+62476/5156163*C+6412/5156163*A)*(76+1 14*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 = 0.09 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.65 \[ \int \frac {A+B x+C x^2}{\left (70+67 x-53 x^2+6 x^3\right )^{3/2}} \, dx=\frac {\sqrt {6} {\left (6 \, {\left (37960 \, A - 173741 \, B - 1164437 \, C\right )} x^{3} - 53 \, {\left (37960 \, A - 173741 \, B - 1164437 \, C\right )} x^{2} + 67 \, {\left (37960 \, A - 173741 \, B - 1164437 \, C\right )} x + 2657200 \, A - 12161870 \, B - 81510590 \, C\right )} {\rm weierstrassPInverse}\left (\frac {1603}{27}, \frac {18980}{729}, x - \frac {53}{18}\right ) - 18 \, \sqrt {6} {\left (6 \, {\left (3206 \, A + 7331 \, B + 31238 \, C\right )} x^{3} - 53 \, {\left (3206 \, A + 7331 \, B + 31238 \, C\right )} x^{2} + 67 \, {\left (3206 \, A + 7331 \, B + 31238 \, C\right )} x + 224420 \, A + 513170 \, B + 2186660 \, C\right )} {\rm weierstrassZeta}\left (\frac {1603}{27}, \frac {18980}{729}, {\rm weierstrassPInverse}\left (\frac {1603}{27}, \frac {18980}{729}, x - \frac {53}{18}\right )\right ) - 18 \, {\left (6 \, {\left (3206 \, A + 7331 \, B + 31238 \, C\right )} x^{2} - 19 \, {\left (6628 \, A + 10585 \, B + 37663 \, C\right )} x + 13687 \, A - 224420 \, B - 513170 \, C\right )} \sqrt {6 \, x^{3} - 53 \, x^{2} + 67 \, x + 70}}{139216401 \, {\left (6 \, x^{3} - 53 \, x^{2} + 67 \, x + 70\right )}} \] Input:
integrate((C*x^2+B*x+A)/(6*x^3-53*x^2+67*x+70)^(3/2),x, algorithm="fricas" )
Output:
1/139216401*(sqrt(6)*(6*(37960*A - 173741*B - 1164437*C)*x^3 - 53*(37960*A - 173741*B - 1164437*C)*x^2 + 67*(37960*A - 173741*B - 1164437*C)*x + 265 7200*A - 12161870*B - 81510590*C)*weierstrassPInverse(1603/27, 18980/729, x - 53/18) - 18*sqrt(6)*(6*(3206*A + 7331*B + 31238*C)*x^3 - 53*(3206*A + 7331*B + 31238*C)*x^2 + 67*(3206*A + 7331*B + 31238*C)*x + 224420*A + 5131 70*B + 2186660*C)*weierstrassZeta(1603/27, 18980/729, weierstrassPInverse( 1603/27, 18980/729, x - 53/18)) - 18*(6*(3206*A + 7331*B + 31238*C)*x^2 - 19*(6628*A + 10585*B + 37663*C)*x + 13687*A - 224420*B - 513170*C)*sqrt(6* x^3 - 53*x^2 + 67*x + 70))/(6*x^3 - 53*x^2 + 67*x + 70)
\[ \int \frac {A+B x+C x^2}{\left (70+67 x-53 x^2+6 x^3\right )^{3/2}} \, dx=\int \frac {A + B x + C x^{2}}{\left (\left (x - 7\right ) \left (2 x - 5\right ) \left (3 x + 2\right )\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((C*x**2+B*x+A)/(6*x**3-53*x**2+67*x+70)**(3/2),x)
Output:
Integral((A + B*x + C*x**2)/((x - 7)*(2*x - 5)*(3*x + 2))**(3/2), x)
\[ \int \frac {A+B x+C x^2}{\left (70+67 x-53 x^2+6 x^3\right )^{3/2}} \, dx=\int { \frac {C x^{2} + B x + A}{{\left (6 \, x^{3} - 53 \, x^{2} + 67 \, x + 70\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((C*x^2+B*x+A)/(6*x^3-53*x^2+67*x+70)^(3/2),x, algorithm="maxima" )
Output:
integrate((C*x^2 + B*x + A)/(6*x^3 - 53*x^2 + 67*x + 70)^(3/2), x)
\[ \int \frac {A+B x+C x^2}{\left (70+67 x-53 x^2+6 x^3\right )^{3/2}} \, dx=\int { \frac {C x^{2} + B x + A}{{\left (6 \, x^{3} - 53 \, x^{2} + 67 \, x + 70\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((C*x^2+B*x+A)/(6*x^3-53*x^2+67*x+70)^(3/2),x, algorithm="giac")
Output:
integrate((C*x^2 + B*x + A)/(6*x^3 - 53*x^2 + 67*x + 70)^(3/2), x)
Time = 0.04 (sec) , antiderivative size = 263, normalized size of antiderivative = 0.82 \[ \int \frac {A+B x+C x^2}{\left (70+67 x-53 x^2+6 x^3\right )^{3/2}} \, dx=-\frac {2\,\sqrt {\frac {2\,x}{9}-\frac {5}{9}}\,\sqrt {\frac {3\,x}{23}+\frac {2}{23}}\,\sqrt {\frac {21}{23}-\frac {3\,x}{23}}\,\left (\frac {3\,A}{437}-\frac {2\,B}{437}+\frac {4\,C}{1311}\right )\,\left (\frac {27\,\mathrm {E}\left (\mathrm {asin}\left (\sqrt {\frac {21}{23}-\frac {3\,x}{23}}\right )\middle |\frac {46}{27}\right )}{19}+\mathrm {F}\left (\mathrm {asin}\left (\sqrt {\frac {21}{23}-\frac {3\,x}{23}}\right )\middle |\frac {46}{27}\right )-\frac {27\,\sqrt {\frac {2\,x}{9}-\frac {5}{9}}\,\sqrt {\frac {21}{23}-\frac {3\,x}{23}}}{19\,\sqrt {\frac {3\,x}{23}+\frac {2}{23}}}\right )}{\sqrt {6\,x^3-53\,x^2+67\,x+70}}-\frac {92\,\sqrt {\frac {2\,x}{9}-\frac {5}{9}}\,\sqrt {\frac {3\,x}{23}+\frac {2}{23}}\,\sqrt {\frac {21}{23}-\frac {3\,x}{23}}\,\left (\frac {27\,\mathrm {E}\left (\mathrm {asin}\left (\sqrt {\frac {21}{23}-\frac {3\,x}{23}}\right )\middle |\frac {46}{27}\right )}{19}-\frac {23\,\sin \left (2\,\mathrm {asin}\left (\sqrt {\frac {21}{23}-\frac {3\,x}{23}}\right )\right )}{19\,\sqrt {\frac {2\,x}{9}-\frac {5}{9}}}\right )\,\left (\frac {2\,A}{171}+\frac {5\,B}{171}+\frac {25\,C}{342}\right )}{27\,\sqrt {6\,x^3-53\,x^2+67\,x+70}}-\frac {19\,\sqrt {\frac {6\,x}{19}+\frac {4}{19}}\,\sqrt {\frac {15}{19}-\frac {6\,x}{19}}\,\sqrt {\frac {21}{23}-\frac {3\,x}{23}}\,\left (\frac {46\,\mathrm {E}\left (\mathrm {asin}\left (\sqrt {\frac {6\,x}{19}+\frac {4}{19}}\right )\middle |\frac {19}{46}\right )}{27}-\frac {19\,\sin \left (2\,\mathrm {asin}\left (\sqrt {\frac {6\,x}{19}+\frac {4}{19}}\right )\right )}{54\,\sqrt {\frac {21}{23}-\frac {3\,x}{23}}}\right )\,\left (\frac {A}{207}+\frac {7\,B}{207}+\frac {49\,C}{207}\right )}{23\,\sqrt {6\,x^3-53\,x^2+67\,x+70}} \] Input:
int((A + B*x + C*x^2)/(67*x - 53*x^2 + 6*x^3 + 70)^(3/2),x)
Output:
- (2*((2*x)/9 - 5/9)^(1/2)*((3*x)/23 + 2/23)^(1/2)*(21/23 - (3*x)/23)^(1/2 )*((3*A)/437 - (2*B)/437 + (4*C)/1311)*((27*ellipticE(asin((21/23 - (3*x)/ 23)^(1/2)), 46/27))/19 + ellipticF(asin((21/23 - (3*x)/23)^(1/2)), 46/27) - (27*((2*x)/9 - 5/9)^(1/2)*(21/23 - (3*x)/23)^(1/2))/(19*((3*x)/23 + 2/23 )^(1/2))))/(67*x - 53*x^2 + 6*x^3 + 70)^(1/2) - (92*((2*x)/9 - 5/9)^(1/2)* ((3*x)/23 + 2/23)^(1/2)*(21/23 - (3*x)/23)^(1/2)*((27*ellipticE(asin((21/2 3 - (3*x)/23)^(1/2)), 46/27))/19 - (23*sin(2*asin((21/23 - (3*x)/23)^(1/2) )))/(19*((2*x)/9 - 5/9)^(1/2)))*((2*A)/171 + (5*B)/171 + (25*C)/342))/(27* (67*x - 53*x^2 + 6*x^3 + 70)^(1/2)) - (19*((6*x)/19 + 4/19)^(1/2)*(15/19 - (6*x)/19)^(1/2)*(21/23 - (3*x)/23)^(1/2)*((46*ellipticE(asin(((6*x)/19 + 4/19)^(1/2)), 19/46))/27 - (19*sin(2*asin(((6*x)/19 + 4/19)^(1/2))))/(54*( 21/23 - (3*x)/23)^(1/2)))*(A/207 + (7*B)/207 + (49*C)/207))/(23*(67*x - 53 *x^2 + 6*x^3 + 70)^(1/2))
\[ \int \frac {A+B x+C x^2}{\left (70+67 x-53 x^2+6 x^3\right )^{3/2}} \, dx=\text {too large to display} \] Input:
int((C*x^2+B*x+A)/(6*x^3-53*x^2+67*x+70)^(3/2),x)
Output:
(212*sqrt(6*x**3 - 53*x**2 + 67*x + 70)*a*x + 134*sqrt(6*x**3 - 53*x**2 + 67*x + 70)*b*x + 280*sqrt(6*x**3 - 53*x**2 + 67*x + 70)*b + 3816*int((sqrt (6*x**3 - 53*x**2 + 67*x + 70)*x**3)/(36*x**6 - 636*x**5 + 3613*x**4 - 626 2*x**3 - 2931*x**2 + 9380*x + 4900),x)*a*x**3 - 33708*int((sqrt(6*x**3 - 5 3*x**2 + 67*x + 70)*x**3)/(36*x**6 - 636*x**5 + 3613*x**4 - 6262*x**3 - 29 31*x**2 + 9380*x + 4900),x)*a*x**2 + 42612*int((sqrt(6*x**3 - 53*x**2 + 67 *x + 70)*x**3)/(36*x**6 - 636*x**5 + 3613*x**4 - 6262*x**3 - 2931*x**2 + 9 380*x + 4900),x)*a*x + 44520*int((sqrt(6*x**3 - 53*x**2 + 67*x + 70)*x**3) /(36*x**6 - 636*x**5 + 3613*x**4 - 6262*x**3 - 2931*x**2 + 9380*x + 4900), x)*a + 2412*int((sqrt(6*x**3 - 53*x**2 + 67*x + 70)*x**3)/(36*x**6 - 636*x **5 + 3613*x**4 - 6262*x**3 - 2931*x**2 + 9380*x + 4900),x)*b*x**3 - 21306 *int((sqrt(6*x**3 - 53*x**2 + 67*x + 70)*x**3)/(36*x**6 - 636*x**5 + 3613* x**4 - 6262*x**3 - 2931*x**2 + 9380*x + 4900),x)*b*x**2 + 26934*int((sqrt( 6*x**3 - 53*x**2 + 67*x + 70)*x**3)/(36*x**6 - 636*x**5 + 3613*x**4 - 6262 *x**3 - 2931*x**2 + 9380*x + 4900),x)*b*x + 28140*int((sqrt(6*x**3 - 53*x* *2 + 67*x + 70)*x**3)/(36*x**6 - 636*x**5 + 3613*x**4 - 6262*x**3 - 2931*x **2 + 9380*x + 4900),x)*b + 15120*int((sqrt(6*x**3 - 53*x**2 + 67*x + 70)* x**2)/(36*x**6 - 636*x**5 + 3613*x**4 - 6262*x**3 - 2931*x**2 + 9380*x + 4 900),x)*b*x**3 - 133560*int((sqrt(6*x**3 - 53*x**2 + 67*x + 70)*x**2)/(36* x**6 - 636*x**5 + 3613*x**4 - 6262*x**3 - 2931*x**2 + 9380*x + 4900),x)...