Integrand size = 25, antiderivative size = 749 \[ \int \frac {A+B x+C x^2}{\sqrt {2-6 x+3 x^3}} \, dx =\text {Too large to display} \] Output:
2/9*C*(3*x^3-6*x+2)^(1/2)-1/9*2^(3/4)*(3*x-2*6^(1/2)*cos(1/6*Pi+1/3*arcsin (1/4*6^(1/2))))^(1/2)*(3*A+2*C+2*6^(1/2)*B*cos(1/6*Pi+1/3*arcsin(1/4*6^(1/ 2))))*EllipticF(2^(1/4)*(3*cos(1/3*arcsin(1/4*6^(1/2)))+3*3^(1/2)*sin(1/3* arcsin(1/4*6^(1/2))))^(1/2)/(3*x+2*6^(1/2)*sin(1/3*Pi+1/3*arcsin(1/4*6^(1/ 2))))^(1/2),(3^(1/2)*cos(1/3*arcsin(1/4*6^(1/2)))/(sin(1/3*arcsin(1/4*6^(1 /2)))+sin(1/3*Pi+1/3*arcsin(1/4*6^(1/2)))))^(1/2))*(3*x-2*6^(1/2)*sin(1/3* arcsin(1/4*6^(1/2))))^(1/2)*(3*x+2*6^(1/2)*sin(1/3*Pi+1/3*arcsin(1/4*6^(1/ 2))))^(1/2)/(3*x^3-6*x+2)^(1/2)/(3*cos(1/3*arcsin(1/4*6^(1/2)))+3*3^(1/2)* sin(1/3*arcsin(1/4*6^(1/2))))^(1/2)+2/9*2^(1/4)*3^(3/4)*B*(3*x-2*6^(1/2)*c os(1/6*Pi+1/3*arcsin(1/4*6^(1/2))))*EllipticE(1/2*(3*x+2*6^(1/2)*sin(1/3*P i+1/3*arcsin(1/4*6^(1/2))))^(1/2)*2^(3/4)/(3*cos(1/3*arcsin(1/4*6^(1/2)))+ 3*3^(1/2)*sin(1/3*arcsin(1/4*6^(1/2))))^(1/2),1/2*(2+2*3^(1/2)*tan(1/3*arc sin(1/4*6^(1/2))))^(1/2))*(-3*x+2*6^(1/2)*sin(1/3*arcsin(1/4*6^(1/2))))^(1 /2)*((1+cos(2/3*arcsin(1/4*6^(1/2)))+3^(1/2)*sin(2/3*arcsin(1/4*6^(1/2)))) /(3^(1/2)*cos(1/3*arcsin(1/4*6^(1/2)))+3*sin(1/3*arcsin(1/4*6^(1/2)))))^(1 /2)*(3*x+2*6^(1/2)*sin(1/3*Pi+1/3*arcsin(1/4*6^(1/2))))^(1/2)/(3*x^3-6*x+2 )^(1/2)/(-3*x+2*6^(1/2)*cos(1/6*Pi+1/3*arcsin(1/4*6^(1/2))))^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 4 in optimal.
Time = 10.41 (sec) , antiderivative size = 589, normalized size of antiderivative = 0.79 \[ \int \frac {A+B x+C x^2}{\sqrt {2-6 x+3 x^3}} \, dx =\text {Too large to display} \] Input:
Integrate[(A + B*x + C*x^2)/Sqrt[2 - 6*x + 3*x^3],x]
Output:
(2*(-3*EllipticF[ArcSin[Sqrt[(x - Root[2 - 6*#1 + 3*#1^3 & , 3, 0])/(Root[ 2 - 6*#1 + 3*#1^3 & , 2, 0] - Root[2 - 6*#1 + 3*#1^3 & , 3, 0])]], (Root[2 - 6*#1 + 3*#1^3 & , 2, 0] - Root[2 - 6*#1 + 3*#1^3 & , 3, 0])/(Root[2 - 6 *#1 + 3*#1^3 & , 1, 0] - Root[2 - 6*#1 + 3*#1^3 & , 3, 0])]*Sqrt[x - Root[ 2 - 6*#1 + 3*#1^3 & , 1, 0]]*(3*A + 2*C + 3*B*Root[2 - 6*#1 + 3*#1^3 & , 1 , 0])*Sqrt[-((x - Root[2 - 6*#1 + 3*#1^3 & , 2, 0])*(x - Root[2 - 6*#1 + 3 *#1^3 & , 3, 0]))] + 9*B*EllipticE[ArcSin[Sqrt[(x - Root[2 - 6*#1 + 3*#1^3 & , 3, 0])/(Root[2 - 6*#1 + 3*#1^3 & , 2, 0] - Root[2 - 6*#1 + 3*#1^3 & , 3, 0])]], (Root[2 - 6*#1 + 3*#1^3 & , 2, 0] - Root[2 - 6*#1 + 3*#1^3 & , 3, 0])/(Root[2 - 6*#1 + 3*#1^3 & , 1, 0] - Root[2 - 6*#1 + 3*#1^3 & , 3, 0 ])]*Sqrt[x - Root[2 - 6*#1 + 3*#1^3 & , 1, 0]]*Sqrt[-((x - Root[2 - 6*#1 + 3*#1^3 & , 2, 0])*(x - Root[2 - 6*#1 + 3*#1^3 & , 3, 0]))]*(Root[2 - 6*#1 + 3*#1^3 & , 1, 0] - Root[2 - 6*#1 + 3*#1^3 & , 3, 0]) + C*(2 - 6*x + 3*x ^3)*Sqrt[-Root[2 - 6*#1 + 3*#1^3 & , 1, 0] + Root[2 - 6*#1 + 3*#1^3 & , 3, 0]]))/(9*Sqrt[2 - 6*x + 3*x^3]*Sqrt[-Root[2 - 6*#1 + 3*#1^3 & , 1, 0] + R oot[2 - 6*#1 + 3*#1^3 & , 3, 0]])
Result contains complex when optimal does not.
Time = 5.53 (sec) , antiderivative size = 2085, normalized size of antiderivative = 2.78, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {2526, 27, 2486, 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.
| \(\displaystyle \int \frac {A+B x+C x^2}{\sqrt {3 x^3-6 x+2}} \, dx\) |
| \(\Big \downarrow \) 2526 |
| \(\displaystyle \frac {1}{9} \int \frac {3 (3 A+2 C+3 B x)}{\sqrt {3 x^3-6 x+2}}dx+\frac {2}{9} C \sqrt {3 x^3-6 x+2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \int \frac {3 A+2 C+3 B x}{\sqrt {3 x^3-6 x+2}}dx+\frac {2}{9} C \sqrt {3 x^3-6 x+2}\) |
| \(\Big \downarrow \) 2486 |
| \(\displaystyle \frac {2}{9} C \sqrt {3 x^3-6 x+2}+\frac {\sqrt {3 x+\sqrt [3]{9-3 i \sqrt {15}}+\frac {2\ 3^{2/3}}{\sqrt [3]{3-i \sqrt {15}}}} \sqrt {9 x^2-3 \left (\frac {2\ 3^{2/3}}{\sqrt [3]{3-i \sqrt {15}}}+\sqrt [3]{9-3 i \sqrt {15}}\right ) x+\left (9-3 i \sqrt {15}\right )^{2/3}+\frac {12 \sqrt [3]{3}}{\left (3-i \sqrt {15}\right )^{2/3}}-6} \int \frac {3 A+2 C+3 B x}{\sqrt {3 x+\sqrt [3]{9-3 i \sqrt {15}}+\frac {2\ 3^{2/3}}{\sqrt [3]{3-i \sqrt {15}}}} \sqrt {9 x^2-3 \left (\frac {2\ 3^{2/3}}{\sqrt [3]{3-i \sqrt {15}}}+\sqrt [3]{9-3 i \sqrt {15}}\right ) x+\left (9-3 i \sqrt {15}\right )^{2/3}+\frac {12 \sqrt [3]{3}}{\left (3-i \sqrt {15}\right )^{2/3}}-6}}dx}{3 \sqrt {3 x^3-6 x+2}}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {2}{9} C \sqrt {3 x^3-6 x+2}+\frac {\sqrt {3 x+\sqrt [3]{9-3 i \sqrt {15}}+\frac {2\ 3^{2/3}}{\sqrt [3]{3-i \sqrt {15}}}} \sqrt {9 x^2-3 \left (\frac {2\ 3^{2/3}}{\sqrt [3]{3-i \sqrt {15}}}+\sqrt [3]{9-3 i \sqrt {15}}\right ) x+\left (9-3 i \sqrt {15}\right )^{2/3}+\frac {12 \sqrt [3]{3}}{\left (3-i \sqrt {15}\right )^{2/3}}-6} \left (\left (3 A-\left (\frac {2\ 3^{2/3}}{\sqrt [3]{3-i \sqrt {15}}}+\sqrt [3]{9-3 i \sqrt {15}}\right ) B+2 C\right ) \int \frac {1}{\sqrt {3 x+\sqrt [3]{9-3 i \sqrt {15}}+\frac {2\ 3^{2/3}}{\sqrt [3]{3-i \sqrt {15}}}} \sqrt {9 x^2-3 \left (\frac {2\ 3^{2/3}}{\sqrt [3]{3-i \sqrt {15}}}+\sqrt [3]{9-3 i \sqrt {15}}\right ) x+\left (9-3 i \sqrt {15}\right )^{2/3}+\frac {12 \sqrt [3]{3}}{\left (3-i \sqrt {15}\right )^{2/3}}-6}}dx+B \int \frac {\sqrt {3 x+\sqrt [3]{9-3 i \sqrt {15}}+\frac {2\ 3^{2/3}}{\sqrt [3]{3-i \sqrt {15}}}}}{\sqrt {9 x^2-3 \left (\frac {2\ 3^{2/3}}{\sqrt [3]{3-i \sqrt {15}}}+\sqrt [3]{9-3 i \sqrt {15}}\right ) x+\left (9-3 i \sqrt {15}\right )^{2/3}+\frac {12 \sqrt [3]{3}}{\left (3-i \sqrt {15}\right )^{2/3}}-6}}dx\right )}{3 \sqrt {3 x^3-6 x+2}}\) |
| \(\Big \downarrow \) 1172 |
| \(\displaystyle \frac {2}{9} \sqrt {3 x^3-6 x+2} C+\frac {\sqrt {3 x+\sqrt [3]{9-3 i \sqrt {15}}+\frac {2\ 3^{2/3}}{\sqrt [3]{3-i \sqrt {15}}}} \sqrt {9 x^2-3 \left (\frac {2\ 3^{2/3}}{\sqrt [3]{3-i \sqrt {15}}}+\sqrt [3]{9-3 i \sqrt {15}}\right ) x+\left (9-3 i \sqrt {15}\right )^{2/3}+\frac {12 \sqrt [3]{3}}{\left (3-i \sqrt {15}\right )^{2/3}}-6} \left (\frac {2 \sqrt {\frac {2}{3} \left (-4 \sqrt [3]{3}-\left (3^{2/3}-i \sqrt [6]{3} \sqrt {5}\right ) \sqrt [3]{3-i \sqrt {15}}+4 \left (3-i \sqrt {15}\right )^{2/3}\right )} \left (3 A-\left (\frac {2\ 3^{2/3}}{\sqrt [3]{3-i \sqrt {15}}}+\sqrt [3]{9-3 i \sqrt {15}}\right ) B+2 C\right ) \sqrt {\frac {\sqrt [3]{3-i \sqrt {15}} \left (3 x+\sqrt [3]{9-3 i \sqrt {15}}+\frac {2\ 3^{2/3}}{\sqrt [3]{3-i \sqrt {15}}}\right )}{2\ 3^{2/3}+\sqrt [3]{3} \left (3-i \sqrt {15}\right )^{2/3}-\sqrt {-4 \sqrt [3]{3}-\left (3^{2/3}-i \sqrt [6]{3} \sqrt {5}\right ) \sqrt [3]{3-i \sqrt {15}}+4 \left (3-i \sqrt {15}\right )^{2/3}}}} \sqrt {-\frac {\left (3-i \sqrt {15}\right )^{2/3} \left (-9 x^2+3 \left (\frac {2\ 3^{2/3}}{\sqrt [3]{3-i \sqrt {15}}}+\sqrt [3]{9-3 i \sqrt {15}}\right ) x-\left (9-3 i \sqrt {15}\right )^{2/3}-\frac {12 \sqrt [3]{3}}{\left (3-i \sqrt {15}\right )^{2/3}}+6\right )}{4 \sqrt [3]{3}+\left (3^{2/3}-i \sqrt [6]{3} \sqrt {5}\right ) \sqrt [3]{3-i \sqrt {15}}-4 \left (3-i \sqrt {15}\right )^{2/3}}} \int \frac {1}{\sqrt {\frac {\sqrt [3]{3-i \sqrt {15}} \left (\frac {2\ 3^{2/3}+\sqrt [3]{3} \left (3-i \sqrt {15}\right )^{2/3}-3 \sqrt {-4 \sqrt [3]{3}-\left (3^{2/3}-i \sqrt [6]{3} \sqrt {5}\right ) \sqrt [3]{3-i \sqrt {15}}+4 \left (3-i \sqrt {15}\right )^{2/3}}}{\sqrt [3]{3-i \sqrt {15}}}-6 x\right )}{6 \sqrt {-4 \sqrt [3]{3}-\left (3^{2/3}-i \sqrt [6]{3} \sqrt {5}\right ) \sqrt [3]{3-i \sqrt {15}}+4 \left (3-i \sqrt {15}\right )^{2/3}}}+1} \sqrt {1-\frac {\sqrt [3]{3-i \sqrt {15}} \left (\frac {2\ 3^{2/3}+\sqrt [3]{3} \left (3-i \sqrt {15}\right )^{2/3}-3 \sqrt {-4 \sqrt [3]{3}-\left (3^{2/3}-i \sqrt [6]{3} \sqrt {5}\right ) \sqrt [3]{3-i \sqrt {15}}+4 \left (3-i \sqrt {15}\right )^{2/3}}}{\sqrt [3]{3-i \sqrt {15}}}-6 x\right )}{3 \left (2\ 3^{2/3}+\sqrt [3]{3} \left (3-i \sqrt {15}\right )^{2/3}-\sqrt {-4 \sqrt [3]{3}-\left (3^{2/3}-i \sqrt [6]{3} \sqrt {5}\right ) \sqrt [3]{3-i \sqrt {15}}+4 \left (3-i \sqrt {15}\right )^{2/3}}\right )}}}d\frac {\sqrt {-\frac {\sqrt [3]{3-i \sqrt {15}} \left (\frac {2\ 3^{2/3}+\sqrt [3]{3} \left (3-i \sqrt {15}\right )^{2/3}-3 \sqrt {-4 \sqrt [3]{3}-\left (3^{2/3}-i \sqrt [6]{3} \sqrt {5}\right ) \sqrt [3]{3-i \sqrt {15}}+4 \left (3-i \sqrt {15}\right )^{2/3}}}{\sqrt [3]{3-i \sqrt {15}}}-6 x\right )}{\sqrt {-4 \sqrt [3]{3}-\left (3^{2/3}-i \sqrt [6]{3} \sqrt {5}\right ) \sqrt [3]{3-i \sqrt {15}}+4 \left (3-i \sqrt {15}\right )^{2/3}}}}}{\sqrt {6}}}{3 \sqrt [3]{3-i \sqrt {15}} \sqrt {3 x+\sqrt [3]{9-3 i \sqrt {15}}+\frac {2\ 3^{2/3}}{\sqrt [3]{3-i \sqrt {15}}}} \sqrt {9 x^2-3 \left (\frac {2\ 3^{2/3}}{\sqrt [3]{3-i \sqrt {15}}}+\sqrt [3]{9-3 i \sqrt {15}}\right ) x+\left (9-3 i \sqrt {15}\right )^{2/3}+\frac {12 \sqrt [3]{3}}{\left (3-i \sqrt {15}\right )^{2/3}}-6}}+\frac {\sqrt {\frac {2}{3} \left (-4 \sqrt [3]{3}-\left (3^{2/3}-i \sqrt [6]{3} \sqrt {5}\right ) \sqrt [3]{3-i \sqrt {15}}+4 \left (3-i \sqrt {15}\right )^{2/3}\right )} B \sqrt {3 x+\sqrt [3]{9-3 i \sqrt {15}}+\frac {2\ 3^{2/3}}{\sqrt [3]{3-i \sqrt {15}}}} \sqrt {-\frac {\left (3-i \sqrt {15}\right )^{2/3} \left (-9 x^2+3 \left (\frac {2\ 3^{2/3}}{\sqrt [3]{3-i \sqrt {15}}}+\sqrt [3]{9-3 i \sqrt {15}}\right ) x-\left (9-3 i \sqrt {15}\right )^{2/3}-\frac {12 \sqrt [3]{3}}{\left (3-i \sqrt {15}\right )^{2/3}}+6\right )}{4 \sqrt [3]{3}+\left (3^{2/3}-i \sqrt [6]{3} \sqrt {5}\right ) \sqrt [3]{3-i \sqrt {15}}-4 \left (3-i \sqrt {15}\right )^{2/3}}} \int \frac {\sqrt {1-\frac {\sqrt [3]{3-i \sqrt {15}} \left (\frac {2\ 3^{2/3}+\sqrt [3]{3} \left (3-i \sqrt {15}\right )^{2/3}-3 \sqrt {-4 \sqrt [3]{3}-\left (3^{2/3}-i \sqrt [6]{3} \sqrt {5}\right ) \sqrt [3]{3-i \sqrt {15}}+4 \left (3-i \sqrt {15}\right )^{2/3}}}{\sqrt [3]{3-i \sqrt {15}}}-6 x\right )}{3 \left (2\ 3^{2/3}+\sqrt [3]{3} \left (3-i \sqrt {15}\right )^{2/3}-\sqrt {-4 \sqrt [3]{3}-\left (3^{2/3}-i \sqrt [6]{3} \sqrt {5}\right ) \sqrt [3]{3-i \sqrt {15}}+4 \left (3-i \sqrt {15}\right )^{2/3}}\right )}}}{\sqrt {\frac {\sqrt [3]{3-i \sqrt {15}} \left (\frac {2\ 3^{2/3}+\sqrt [3]{3} \left (3-i \sqrt {15}\right )^{2/3}-3 \sqrt {-4 \sqrt [3]{3}-\left (3^{2/3}-i \sqrt [6]{3} \sqrt {5}\right ) \sqrt [3]{3-i \sqrt {15}}+4 \left (3-i \sqrt {15}\right )^{2/3}}}{\sqrt [3]{3-i \sqrt {15}}}-6 x\right )}{6 \sqrt {-4 \sqrt [3]{3}-\left (3^{2/3}-i \sqrt [6]{3} \sqrt {5}\right ) \sqrt [3]{3-i \sqrt {15}}+4 \left (3-i \sqrt {15}\right )^{2/3}}}+1}}d\frac {\sqrt {-\frac {\sqrt [3]{3-i \sqrt {15}} \left (\frac {2\ 3^{2/3}+\sqrt [3]{3} \left (3-i \sqrt {15}\right )^{2/3}-3 \sqrt {-4 \sqrt [3]{3}-\left (3^{2/3}-i \sqrt [6]{3} \sqrt {5}\right ) \sqrt [3]{3-i \sqrt {15}}+4 \left (3-i \sqrt {15}\right )^{2/3}}}{\sqrt [3]{3-i \sqrt {15}}}-6 x\right )}{\sqrt {-4 \sqrt [3]{3}-\left (3^{2/3}-i \sqrt [6]{3} \sqrt {5}\right ) \sqrt [3]{3-i \sqrt {15}}+4 \left (3-i \sqrt {15}\right )^{2/3}}}}}{\sqrt {6}}}{\sqrt [3]{3-i \sqrt {15}} \sqrt {\frac {\sqrt [3]{3-i \sqrt {15}} \left (3 x+\sqrt [3]{9-3 i \sqrt {15}}+\frac {2\ 3^{2/3}}{\sqrt [3]{3-i \sqrt {15}}}\right )}{2\ 3^{2/3}+\sqrt [3]{3} \left (3-i \sqrt {15}\right )^{2/3}-\sqrt {-4 \sqrt [3]{3}-\left (3^{2/3}-i \sqrt [6]{3} \sqrt {5}\right ) \sqrt [3]{3-i \sqrt {15}}+4 \left (3-i \sqrt {15}\right )^{2/3}}}} \sqrt {9 x^2-3 \left (\frac {2\ 3^{2/3}}{\sqrt [3]{3-i \sqrt {15}}}+\sqrt [3]{9-3 i \sqrt {15}}\right ) x+\left (9-3 i \sqrt {15}\right )^{2/3}+\frac {12 \sqrt [3]{3}}{\left (3-i \sqrt {15}\right )^{2/3}}-6}}\right )}{3 \sqrt {3 x^3-6 x+2}}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {2}{9} \sqrt {3 x^3-6 x+2} C+\frac {\sqrt {3 x+\sqrt [3]{9-3 i \sqrt {15}}+\frac {2\ 3^{2/3}}{\sqrt [3]{3-i \sqrt {15}}}} \sqrt {9 x^2-3 \left (\frac {2\ 3^{2/3}}{\sqrt [3]{3-i \sqrt {15}}}+\sqrt [3]{9-3 i \sqrt {15}}\right ) x+\left (9-3 i \sqrt {15}\right )^{2/3}+\frac {12 \sqrt [3]{3}}{\left (3-i \sqrt {15}\right )^{2/3}}-6} \left (\frac {2 \sqrt {\frac {2}{3} \left (-4 \sqrt [3]{3}-\left (3^{2/3}-i \sqrt [6]{3} \sqrt {5}\right ) \sqrt [3]{3-i \sqrt {15}}+4 \left (3-i \sqrt {15}\right )^{2/3}\right )} \left (3 A-\left (\frac {2\ 3^{2/3}}{\sqrt [3]{3-i \sqrt {15}}}+\sqrt [3]{9-3 i \sqrt {15}}\right ) B+2 C\right ) \sqrt {\frac {\sqrt [3]{3-i \sqrt {15}} \left (3 x+\sqrt [3]{9-3 i \sqrt {15}}+\frac {2\ 3^{2/3}}{\sqrt [3]{3-i \sqrt {15}}}\right )}{2\ 3^{2/3}+\sqrt [3]{3} \left (3-i \sqrt {15}\right )^{2/3}-\sqrt {-4 \sqrt [3]{3}-\left (3^{2/3}-i \sqrt [6]{3} \sqrt {5}\right ) \sqrt [3]{3-i \sqrt {15}}+4 \left (3-i \sqrt {15}\right )^{2/3}}}} \sqrt {-\frac {\left (3-i \sqrt {15}\right )^{2/3} \left (-9 x^2+3 \left (\frac {2\ 3^{2/3}}{\sqrt [3]{3-i \sqrt {15}}}+\sqrt [3]{9-3 i \sqrt {15}}\right ) x-\left (9-3 i \sqrt {15}\right )^{2/3}-\frac {12 \sqrt [3]{3}}{\left (3-i \sqrt {15}\right )^{2/3}}+6\right )}{4 \sqrt [3]{3}+\left (3^{2/3}-i \sqrt [6]{3} \sqrt {5}\right ) \sqrt [3]{3-i \sqrt {15}}-4 \left (3-i \sqrt {15}\right )^{2/3}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-\frac {\sqrt [3]{3-i \sqrt {15}} \left (\frac {2\ 3^{2/3}+\sqrt [3]{3} \left (3-i \sqrt {15}\right )^{2/3}-3 \sqrt {-4 \sqrt [3]{3}-\left (3^{2/3}-i \sqrt [6]{3} \sqrt {5}\right ) \sqrt [3]{3-i \sqrt {15}}+4 \left (3-i \sqrt {15}\right )^{2/3}}}{\sqrt [3]{3-i \sqrt {15}}}-6 x\right )}{\sqrt {-4 \sqrt [3]{3}-\left (3^{2/3}-i \sqrt [6]{3} \sqrt {5}\right ) \sqrt [3]{3-i \sqrt {15}}+4 \left (3-i \sqrt {15}\right )^{2/3}}}}}{\sqrt {6}}\right ),-\frac {2 \sqrt {-4 \sqrt [3]{3}-\left (3^{2/3}-i \sqrt [6]{3} \sqrt {5}\right ) \sqrt [3]{3-i \sqrt {15}}+4 \left (3-i \sqrt {15}\right )^{2/3}}}{2\ 3^{2/3}+\sqrt [3]{3} \left (3-i \sqrt {15}\right )^{2/3}-\sqrt {-4 \sqrt [3]{3}-\left (3^{2/3}-i \sqrt [6]{3} \sqrt {5}\right ) \sqrt [3]{3-i \sqrt {15}}+4 \left (3-i \sqrt {15}\right )^{2/3}}}\right )}{3 \sqrt [3]{3-i \sqrt {15}} \sqrt {3 x+\sqrt [3]{9-3 i \sqrt {15}}+\frac {2\ 3^{2/3}}{\sqrt [3]{3-i \sqrt {15}}}} \sqrt {9 x^2-3 \left (\frac {2\ 3^{2/3}}{\sqrt [3]{3-i \sqrt {15}}}+\sqrt [3]{9-3 i \sqrt {15}}\right ) x+\left (9-3 i \sqrt {15}\right )^{2/3}+\frac {12 \sqrt [3]{3}}{\left (3-i \sqrt {15}\right )^{2/3}}-6}}+\frac {\sqrt {\frac {2}{3} \left (-4 \sqrt [3]{3}-\left (3^{2/3}-i \sqrt [6]{3} \sqrt {5}\right ) \sqrt [3]{3-i \sqrt {15}}+4 \left (3-i \sqrt {15}\right )^{2/3}\right )} B \sqrt {3 x+\sqrt [3]{9-3 i \sqrt {15}}+\frac {2\ 3^{2/3}}{\sqrt [3]{3-i \sqrt {15}}}} \sqrt {-\frac {\left (3-i \sqrt {15}\right )^{2/3} \left (-9 x^2+3 \left (\frac {2\ 3^{2/3}}{\sqrt [3]{3-i \sqrt {15}}}+\sqrt [3]{9-3 i \sqrt {15}}\right ) x-\left (9-3 i \sqrt {15}\right )^{2/3}-\frac {12 \sqrt [3]{3}}{\left (3-i \sqrt {15}\right )^{2/3}}+6\right )}{4 \sqrt [3]{3}+\left (3^{2/3}-i \sqrt [6]{3} \sqrt {5}\right ) \sqrt [3]{3-i \sqrt {15}}-4 \left (3-i \sqrt {15}\right )^{2/3}}} \int \frac {\sqrt {1-\frac {\sqrt [3]{3-i \sqrt {15}} \left (\frac {2\ 3^{2/3}+\sqrt [3]{3} \left (3-i \sqrt {15}\right )^{2/3}-3 \sqrt {-4 \sqrt [3]{3}-\left (3^{2/3}-i \sqrt [6]{3} \sqrt {5}\right ) \sqrt [3]{3-i \sqrt {15}}+4 \left (3-i \sqrt {15}\right )^{2/3}}}{\sqrt [3]{3-i \sqrt {15}}}-6 x\right )}{3 \left (2\ 3^{2/3}+\sqrt [3]{3} \left (3-i \sqrt {15}\right )^{2/3}-\sqrt {-4 \sqrt [3]{3}-\left (3^{2/3}-i \sqrt [6]{3} \sqrt {5}\right ) \sqrt [3]{3-i \sqrt {15}}+4 \left (3-i \sqrt {15}\right )^{2/3}}\right )}}}{\sqrt {\frac {\sqrt [3]{3-i \sqrt {15}} \left (\frac {2\ 3^{2/3}+\sqrt [3]{3} \left (3-i \sqrt {15}\right )^{2/3}-3 \sqrt {-4 \sqrt [3]{3}-\left (3^{2/3}-i \sqrt [6]{3} \sqrt {5}\right ) \sqrt [3]{3-i \sqrt {15}}+4 \left (3-i \sqrt {15}\right )^{2/3}}}{\sqrt [3]{3-i \sqrt {15}}}-6 x\right )}{6 \sqrt {-4 \sqrt [3]{3}-\left (3^{2/3}-i \sqrt [6]{3} \sqrt {5}\right ) \sqrt [3]{3-i \sqrt {15}}+4 \left (3-i \sqrt {15}\right )^{2/3}}}+1}}d\frac {\sqrt {-\frac {\sqrt [3]{3-i \sqrt {15}} \left (\frac {2\ 3^{2/3}+\sqrt [3]{3} \left (3-i \sqrt {15}\right )^{2/3}-3 \sqrt {-4 \sqrt [3]{3}-\left (3^{2/3}-i \sqrt [6]{3} \sqrt {5}\right ) \sqrt [3]{3-i \sqrt {15}}+4 \left (3-i \sqrt {15}\right )^{2/3}}}{\sqrt [3]{3-i \sqrt {15}}}-6 x\right )}{\sqrt {-4 \sqrt [3]{3}-\left (3^{2/3}-i \sqrt [6]{3} \sqrt {5}\right ) \sqrt [3]{3-i \sqrt {15}}+4 \left (3-i \sqrt {15}\right )^{2/3}}}}}{\sqrt {6}}}{\sqrt [3]{3-i \sqrt {15}} \sqrt {\frac {\sqrt [3]{3-i \sqrt {15}} \left (3 x+\sqrt [3]{9-3 i \sqrt {15}}+\frac {2\ 3^{2/3}}{\sqrt [3]{3-i \sqrt {15}}}\right )}{2\ 3^{2/3}+\sqrt [3]{3} \left (3-i \sqrt {15}\right )^{2/3}-\sqrt {-4 \sqrt [3]{3}-\left (3^{2/3}-i \sqrt [6]{3} \sqrt {5}\right ) \sqrt [3]{3-i \sqrt {15}}+4 \left (3-i \sqrt {15}\right )^{2/3}}}} \sqrt {9 x^2-3 \left (\frac {2\ 3^{2/3}}{\sqrt [3]{3-i \sqrt {15}}}+\sqrt [3]{9-3 i \sqrt {15}}\right ) x+\left (9-3 i \sqrt {15}\right )^{2/3}+\frac {12 \sqrt [3]{3}}{\left (3-i \sqrt {15}\right )^{2/3}}-6}}\right )}{3 \sqrt {3 x^3-6 x+2}}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {2}{9} \sqrt {3 x^3-6 x+2} C+\frac {\sqrt {3 x+\sqrt [3]{9-3 i \sqrt {15}}+\frac {2\ 3^{2/3}}{\sqrt [3]{3-i \sqrt {15}}}} \sqrt {9 x^2-3 \left (\frac {2\ 3^{2/3}}{\sqrt [3]{3-i \sqrt {15}}}+\sqrt [3]{9-3 i \sqrt {15}}\right ) x+\left (9-3 i \sqrt {15}\right )^{2/3}+\frac {12 \sqrt [3]{3}}{\left (3-i \sqrt {15}\right )^{2/3}}-6} \left (\frac {\sqrt {\frac {2}{3} \left (-4 \sqrt [3]{3}-\left (3^{2/3}-i \sqrt [6]{3} \sqrt {5}\right ) \sqrt [3]{3-i \sqrt {15}}+4 \left (3-i \sqrt {15}\right )^{2/3}\right )} B \sqrt {3 x+\sqrt [3]{9-3 i \sqrt {15}}+\frac {2\ 3^{2/3}}{\sqrt [3]{3-i \sqrt {15}}}} \sqrt {-\frac {\left (3-i \sqrt {15}\right )^{2/3} \left (-9 x^2+3 \left (\frac {2\ 3^{2/3}}{\sqrt [3]{3-i \sqrt {15}}}+\sqrt [3]{9-3 i \sqrt {15}}\right ) x-\left (9-3 i \sqrt {15}\right )^{2/3}-\frac {12 \sqrt [3]{3}}{\left (3-i \sqrt {15}\right )^{2/3}}+6\right )}{4 \sqrt [3]{3}+\left (3^{2/3}-i \sqrt [6]{3} \sqrt {5}\right ) \sqrt [3]{3-i \sqrt {15}}-4 \left (3-i \sqrt {15}\right )^{2/3}}} E\left (\arcsin \left (\frac {\sqrt {-\frac {\sqrt [3]{3-i \sqrt {15}} \left (\frac {2\ 3^{2/3}+\sqrt [3]{3} \left (3-i \sqrt {15}\right )^{2/3}-3 \sqrt {-4 \sqrt [3]{3}-\left (3^{2/3}-i \sqrt [6]{3} \sqrt {5}\right ) \sqrt [3]{3-i \sqrt {15}}+4 \left (3-i \sqrt {15}\right )^{2/3}}}{\sqrt [3]{3-i \sqrt {15}}}-6 x\right )}{\sqrt {-4 \sqrt [3]{3}-\left (3^{2/3}-i \sqrt [6]{3} \sqrt {5}\right ) \sqrt [3]{3-i \sqrt {15}}+4 \left (3-i \sqrt {15}\right )^{2/3}}}}}{\sqrt {6}}\right )|-\frac {2 \sqrt {-4 \sqrt [3]{3}-\left (3^{2/3}-i \sqrt [6]{3} \sqrt {5}\right ) \sqrt [3]{3-i \sqrt {15}}+4 \left (3-i \sqrt {15}\right )^{2/3}}}{2\ 3^{2/3}+\sqrt [3]{3} \left (3-i \sqrt {15}\right )^{2/3}-\sqrt {-4 \sqrt [3]{3}-\left (3^{2/3}-i \sqrt [6]{3} \sqrt {5}\right ) \sqrt [3]{3-i \sqrt {15}}+4 \left (3-i \sqrt {15}\right )^{2/3}}}\right )}{\sqrt [3]{3-i \sqrt {15}} \sqrt {\frac {\sqrt [3]{3-i \sqrt {15}} \left (3 x+\sqrt [3]{9-3 i \sqrt {15}}+\frac {2\ 3^{2/3}}{\sqrt [3]{3-i \sqrt {15}}}\right )}{2\ 3^{2/3}+\sqrt [3]{3} \left (3-i \sqrt {15}\right )^{2/3}-\sqrt {-4 \sqrt [3]{3}-\left (3^{2/3}-i \sqrt [6]{3} \sqrt {5}\right ) \sqrt [3]{3-i \sqrt {15}}+4 \left (3-i \sqrt {15}\right )^{2/3}}}} \sqrt {9 x^2-3 \left (\frac {2\ 3^{2/3}}{\sqrt [3]{3-i \sqrt {15}}}+\sqrt [3]{9-3 i \sqrt {15}}\right ) x+\left (9-3 i \sqrt {15}\right )^{2/3}+\frac {12 \sqrt [3]{3}}{\left (3-i \sqrt {15}\right )^{2/3}}-6}}+\frac {2 \sqrt {\frac {2}{3} \left (-4 \sqrt [3]{3}-\left (3^{2/3}-i \sqrt [6]{3} \sqrt {5}\right ) \sqrt [3]{3-i \sqrt {15}}+4 \left (3-i \sqrt {15}\right )^{2/3}\right )} \left (3 A-\left (\frac {2\ 3^{2/3}}{\sqrt [3]{3-i \sqrt {15}}}+\sqrt [3]{9-3 i \sqrt {15}}\right ) B+2 C\right ) \sqrt {\frac {\sqrt [3]{3-i \sqrt {15}} \left (3 x+\sqrt [3]{9-3 i \sqrt {15}}+\frac {2\ 3^{2/3}}{\sqrt [3]{3-i \sqrt {15}}}\right )}{2\ 3^{2/3}+\sqrt [3]{3} \left (3-i \sqrt {15}\right )^{2/3}-\sqrt {-4 \sqrt [3]{3}-\left (3^{2/3}-i \sqrt [6]{3} \sqrt {5}\right ) \sqrt [3]{3-i \sqrt {15}}+4 \left (3-i \sqrt {15}\right )^{2/3}}}} \sqrt {-\frac {\left (3-i \sqrt {15}\right )^{2/3} \left (-9 x^2+3 \left (\frac {2\ 3^{2/3}}{\sqrt [3]{3-i \sqrt {15}}}+\sqrt [3]{9-3 i \sqrt {15}}\right ) x-\left (9-3 i \sqrt {15}\right )^{2/3}-\frac {12 \sqrt [3]{3}}{\left (3-i \sqrt {15}\right )^{2/3}}+6\right )}{4 \sqrt [3]{3}+\left (3^{2/3}-i \sqrt [6]{3} \sqrt {5}\right ) \sqrt [3]{3-i \sqrt {15}}-4 \left (3-i \sqrt {15}\right )^{2/3}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-\frac {\sqrt [3]{3-i \sqrt {15}} \left (\frac {2\ 3^{2/3}+\sqrt [3]{3} \left (3-i \sqrt {15}\right )^{2/3}-3 \sqrt {-4 \sqrt [3]{3}-\left (3^{2/3}-i \sqrt [6]{3} \sqrt {5}\right ) \sqrt [3]{3-i \sqrt {15}}+4 \left (3-i \sqrt {15}\right )^{2/3}}}{\sqrt [3]{3-i \sqrt {15}}}-6 x\right )}{\sqrt {-4 \sqrt [3]{3}-\left (3^{2/3}-i \sqrt [6]{3} \sqrt {5}\right ) \sqrt [3]{3-i \sqrt {15}}+4 \left (3-i \sqrt {15}\right )^{2/3}}}}}{\sqrt {6}}\right ),-\frac {2 \sqrt {-4 \sqrt [3]{3}-\left (3^{2/3}-i \sqrt [6]{3} \sqrt {5}\right ) \sqrt [3]{3-i \sqrt {15}}+4 \left (3-i \sqrt {15}\right )^{2/3}}}{2\ 3^{2/3}+\sqrt [3]{3} \left (3-i \sqrt {15}\right )^{2/3}-\sqrt {-4 \sqrt [3]{3}-\left (3^{2/3}-i \sqrt [6]{3} \sqrt {5}\right ) \sqrt [3]{3-i \sqrt {15}}+4 \left (3-i \sqrt {15}\right )^{2/3}}}\right )}{3 \sqrt [3]{3-i \sqrt {15}} \sqrt {3 x+\sqrt [3]{9-3 i \sqrt {15}}+\frac {2\ 3^{2/3}}{\sqrt [3]{3-i \sqrt {15}}}} \sqrt {9 x^2-3 \left (\frac {2\ 3^{2/3}}{\sqrt [3]{3-i \sqrt {15}}}+\sqrt [3]{9-3 i \sqrt {15}}\right ) x+\left (9-3 i \sqrt {15}\right )^{2/3}+\frac {12 \sqrt [3]{3}}{\left (3-i \sqrt {15}\right )^{2/3}}-6}}\right )}{3 \sqrt {3 x^3-6 x+2}}\) |
Input:
Int[(A + B*x + C*x^2)/Sqrt[2 - 6*x + 3*x^3],x]
Output:
(2*C*Sqrt[2 - 6*x + 3*x^3])/9 + (Sqrt[(2*3^(2/3))/(3 - I*Sqrt[15])^(1/3) + (9 - (3*I)*Sqrt[15])^(1/3) + 3*x]*Sqrt[-6 + (12*3^(1/3))/(3 - I*Sqrt[15]) ^(2/3) + (9 - (3*I)*Sqrt[15])^(2/3) - 3*((2*3^(2/3))/(3 - I*Sqrt[15])^(1/3 ) + (9 - (3*I)*Sqrt[15])^(1/3))*x + 9*x^2]*((Sqrt[(2*(-4*3^(1/3) - (3^(2/3 ) - I*3^(1/6)*Sqrt[5])*(3 - I*Sqrt[15])^(1/3) + 4*(3 - I*Sqrt[15])^(2/3))) /3]*B*Sqrt[(2*3^(2/3))/(3 - I*Sqrt[15])^(1/3) + (9 - (3*I)*Sqrt[15])^(1/3) + 3*x]*Sqrt[-(((3 - I*Sqrt[15])^(2/3)*(6 - (12*3^(1/3))/(3 - I*Sqrt[15])^ (2/3) - (9 - (3*I)*Sqrt[15])^(2/3) + 3*((2*3^(2/3))/(3 - I*Sqrt[15])^(1/3) + (9 - (3*I)*Sqrt[15])^(1/3))*x - 9*x^2))/(4*3^(1/3) + (3^(2/3) - I*3^(1/ 6)*Sqrt[5])*(3 - I*Sqrt[15])^(1/3) - 4*(3 - I*Sqrt[15])^(2/3)))]*EllipticE [ArcSin[Sqrt[-(((3 - I*Sqrt[15])^(1/3)*((2*3^(2/3) + 3^(1/3)*(3 - I*Sqrt[1 5])^(2/3) - 3*Sqrt[-4*3^(1/3) - (3^(2/3) - I*3^(1/6)*Sqrt[5])*(3 - I*Sqrt[ 15])^(1/3) + 4*(3 - I*Sqrt[15])^(2/3)])/(3 - I*Sqrt[15])^(1/3) - 6*x))/Sqr t[-4*3^(1/3) - (3^(2/3) - I*3^(1/6)*Sqrt[5])*(3 - I*Sqrt[15])^(1/3) + 4*(3 - I*Sqrt[15])^(2/3)])]/Sqrt[6]], (-2*Sqrt[-4*3^(1/3) - (3^(2/3) - I*3^(1/ 6)*Sqrt[5])*(3 - I*Sqrt[15])^(1/3) + 4*(3 - I*Sqrt[15])^(2/3)])/(2*3^(2/3) + 3^(1/3)*(3 - I*Sqrt[15])^(2/3) - Sqrt[-4*3^(1/3) - (3^(2/3) - I*3^(1/6) *Sqrt[5])*(3 - I*Sqrt[15])^(1/3) + 4*(3 - I*Sqrt[15])^(2/3)])])/((3 - I*Sq rt[15])^(1/3)*Sqrt[((3 - I*Sqrt[15])^(1/3)*((2*3^(2/3))/(3 - I*Sqrt[15])^( 1/3) + (9 - (3*I)*Sqrt[15])^(1/3) + 3*x))/(2*3^(2/3) + 3^(1/3)*(3 - I*S...
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[(Pm_)*(Qn_)^(p_), x_Symbol] :> With[{m = Expon[Pm, x], n = Expon[Qn, x] }, Simp[Coeff[Pm, x, m]*(Qn^(p + 1)/(n*(p + 1)*Coeff[Qn, x, n])), x] + Simp [1/(n*Coeff[Qn, x, n]) Int[ExpandToSum[n*Coeff[Qn, x, n]*Pm - Coeff[Pm, x , m]*D[Qn, x], x]*Qn^p, x], x] /; EqQ[m, n - 1]] /; FreeQ[p, x] && PolyQ[Pm , x] && PolyQ[Qn, x] && NeQ[p, -1]
Result contains complex when optimal does not.
Time = 1.13 (sec) , antiderivative size = 788, normalized size of antiderivative = 1.05
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(788\) |
| risch | \(\text {Expression too large to display}\) | \(1086\) |
| default | \(\text {Expression too large to display}\) | \(1087\) |
Input:
int((C*x^2+B*x+A)/(3*x^3-6*x+2)^(1/2),x,method=_RETURNVERBOSE)
Output:
2/9*C*(3*x^3-6*x+2)^(1/2)+1/9*(A+2/3*C)*6^(1/2)*sin(1/3*arctan(1/3*15^(1/2 )))*((x+1/3*6^(1/2)*sin(1/3*arctan(1/3*15^(1/2)))*3^(1/2)-1/3*6^(1/2)*cos( 1/3*arctan(1/3*15^(1/2))))*6^(1/2)/sin(1/3*arctan(1/3*15^(1/2)))*3^(1/2))^ (1/2)*((x+2/3*6^(1/2)*cos(1/3*arctan(1/3*15^(1/2))))/(-1/3*6^(1/2)*sin(1/3 *arctan(1/3*15^(1/2)))*3^(1/2)+6^(1/2)*cos(1/3*arctan(1/3*15^(1/2)))))^(1/ 2)*(-3*(x-1/3*6^(1/2)*sin(1/3*arctan(1/3*15^(1/2)))*3^(1/2)-1/3*6^(1/2)*co s(1/3*arctan(1/3*15^(1/2))))*6^(1/2)/sin(1/3*arctan(1/3*15^(1/2)))*3^(1/2) )^(1/2)/(3*x^3-6*x+2)^(1/2)*EllipticF(1/6*3^(1/2)*((x+1/3*6^(1/2)*sin(1/3* arctan(1/3*15^(1/2)))*3^(1/2)-1/3*6^(1/2)*cos(1/3*arctan(1/3*15^(1/2))))*6 ^(1/2)/sin(1/3*arctan(1/3*15^(1/2)))*3^(1/2))^(1/2),1/3*I*6^(1/2)*(6^(1/2) *sin(1/3*arctan(1/3*15^(1/2)))*3^(1/2)/(-1/3*6^(1/2)*sin(1/3*arctan(1/3*15 ^(1/2)))*3^(1/2)+6^(1/2)*cos(1/3*arctan(1/3*15^(1/2)))))^(1/2))+1/9*B*6^(1 /2)*sin(1/3*arctan(1/3*15^(1/2)))*((x+1/3*6^(1/2)*sin(1/3*arctan(1/3*15^(1 /2)))*3^(1/2)-1/3*6^(1/2)*cos(1/3*arctan(1/3*15^(1/2))))*6^(1/2)/sin(1/3*a rctan(1/3*15^(1/2)))*3^(1/2))^(1/2)*((x+2/3*6^(1/2)*cos(1/3*arctan(1/3*15^ (1/2))))/(-1/3*6^(1/2)*sin(1/3*arctan(1/3*15^(1/2)))*3^(1/2)+6^(1/2)*cos(1 /3*arctan(1/3*15^(1/2)))))^(1/2)*(-3*(x-1/3*6^(1/2)*sin(1/3*arctan(1/3*15^ (1/2)))*3^(1/2)-1/3*6^(1/2)*cos(1/3*arctan(1/3*15^(1/2))))*6^(1/2)/sin(1/3 *arctan(1/3*15^(1/2)))*3^(1/2))^(1/2)/(3*x^3-6*x+2)^(1/2)*((-1/3*6^(1/2)*s in(1/3*arctan(1/3*15^(1/2)))*3^(1/2)+6^(1/2)*cos(1/3*arctan(1/3*15^(1/2...
Time = 0.07 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.06 \[ \int \frac {A+B x+C x^2}{\sqrt {2-6 x+3 x^3}} \, dx=\frac {2}{9} \, \sqrt {3} {\left (3 \, A + 2 \, C\right )} {\rm weierstrassPInverse}\left (8, -\frac {8}{3}, x\right ) - \frac {2}{3} \, \sqrt {3} B {\rm weierstrassZeta}\left (8, -\frac {8}{3}, {\rm weierstrassPInverse}\left (8, -\frac {8}{3}, x\right )\right ) + \frac {2}{9} \, \sqrt {3 \, x^{3} - 6 \, x + 2} C \] Input:
integrate((C*x^2+B*x+A)/(3*x^3-6*x+2)^(1/2),x, algorithm="fricas")
Output:
2/9*sqrt(3)*(3*A + 2*C)*weierstrassPInverse(8, -8/3, x) - 2/3*sqrt(3)*B*we ierstrassZeta(8, -8/3, weierstrassPInverse(8, -8/3, x)) + 2/9*sqrt(3*x^3 - 6*x + 2)*C
\[ \int \frac {A+B x+C x^2}{\sqrt {2-6 x+3 x^3}} \, dx=\int \frac {A + B x + C x^{2}}{\sqrt {3 x^{3} - 6 x + 2}}\, dx \] Input:
integrate((C*x**2+B*x+A)/(3*x**3-6*x+2)**(1/2),x)
Output:
Integral((A + B*x + C*x**2)/sqrt(3*x**3 - 6*x + 2), x)
\[ \int \frac {A+B x+C x^2}{\sqrt {2-6 x+3 x^3}} \, dx=\int { \frac {C x^{2} + B x + A}{\sqrt {3 \, x^{3} - 6 \, x + 2}} \,d x } \] Input:
integrate((C*x^2+B*x+A)/(3*x^3-6*x+2)^(1/2),x, algorithm="maxima")
Output:
integrate((C*x^2 + B*x + A)/sqrt(3*x^3 - 6*x + 2), x)
\[ \int \frac {A+B x+C x^2}{\sqrt {2-6 x+3 x^3}} \, dx=\int { \frac {C x^{2} + B x + A}{\sqrt {3 \, x^{3} - 6 \, x + 2}} \,d x } \] Input:
integrate((C*x^2+B*x+A)/(3*x^3-6*x+2)^(1/2),x, algorithm="giac")
Output:
integrate((C*x^2 + B*x + A)/sqrt(3*x^3 - 6*x + 2), x)
Time = 12.64 (sec) , antiderivative size = 3317, normalized size of antiderivative = 4.43 \[ \int \frac {A+B x+C x^2}{\sqrt {2-6 x+3 x^3}} \, dx=\text {Too large to display} \] Input:
int((A + B*x + C*x^2)/(3*x^3 - 6*x + 2)^(1/2),x)
Output:
(C*((2*(x^3 - 2*x + 2/3)^(1/2))/3 + (4*ellipticF(asin(((x + 1/(3*((15^(1/2 )*1i)/9 - 1/3)^(1/3)) + ((15^(1/2)*1i)/9 - 1/3)^(1/3)/2 - 3^(1/2)*(1/(3*(( 15^(1/2)*1i)/9 - 1/3)^(1/3)) - ((15^(1/2)*1i)/9 - 1/3)^(1/3)/2)*1i)/(1/((1 5^(1/2)*1i)/9 - 1/3)^(1/3) + (3*((15^(1/2)*1i)/9 - 1/3)^(1/3))/2 - 3^(1/2) *(1/(3*((15^(1/2)*1i)/9 - 1/3)^(1/3)) - ((15^(1/2)*1i)/9 - 1/3)^(1/3)/2)*1 i))^(1/2)), (3^(1/2)*(1/(3*((15^(1/2)*1i)/9 - 1/3)^(1/3)) + ((15^(1/2)*1i) /9 - 1/3)^(1/3)/2 - (3^(1/2)*(1/(3*((15^(1/2)*1i)/9 - 1/3)^(1/3)) - ((15^( 1/2)*1i)/9 - 1/3)^(1/3)/2)*1i)/3)*1i)/(2/(3*((15^(1/2)*1i)/9 - 1/3)^(1/3)) - ((15^(1/2)*1i)/9 - 1/3)^(1/3)))*((x + 1/(3*((15^(1/2)*1i)/9 - 1/3)^(1/3 )) + ((15^(1/2)*1i)/9 - 1/3)^(1/3)/2 - 3^(1/2)*(1/(3*((15^(1/2)*1i)/9 - 1/ 3)^(1/3)) - ((15^(1/2)*1i)/9 - 1/3)^(1/3)/2)*1i)/(1/((15^(1/2)*1i)/9 - 1/3 )^(1/3) + (3*((15^(1/2)*1i)/9 - 1/3)^(1/3))/2 - 3^(1/2)*(1/(3*((15^(1/2)*1 i)/9 - 1/3)^(1/3)) - ((15^(1/2)*1i)/9 - 1/3)^(1/3)/2)*1i))^(1/2)*((2/(3*(( 15^(1/2)*1i)/9 - 1/3)^(1/3)) - x + ((15^(1/2)*1i)/9 - 1/3)^(1/3))/(1/((15^ (1/2)*1i)/9 - 1/3)^(1/3) + (3*((15^(1/2)*1i)/9 - 1/3)^(1/3))/2 - 3^(1/2)*( 1/(3*((15^(1/2)*1i)/9 - 1/3)^(1/3)) - ((15^(1/2)*1i)/9 - 1/3)^(1/3)/2)*1i) )^(1/2)*(1/((15^(1/2)*1i)/9 - 1/3)^(1/3) + (3*((15^(1/2)*1i)/9 - 1/3)^(1/3 ))/2 - 3^(1/2)*(1/(3*((15^(1/2)*1i)/9 - 1/3)^(1/3)) - ((15^(1/2)*1i)/9 - 1 /3)^(1/3)/2)*1i)*(-(3^(1/2)*(x/3 + 1/(9*((15^(1/2)*1i)/9 - 1/3)^(1/3)) + ( (15^(1/2)*1i)/9 - 1/3)^(1/3)/6 + (3^(1/2)*(1/(3*((15^(1/2)*1i)/9 - 1/3)...
\[ \int \frac {A+B x+C x^2}{\sqrt {2-6 x+3 x^3}} \, dx=\left (\int \frac {\sqrt {3 x^{3}-6 x +2}}{3 x^{3}-6 x +2}d x \right ) a +\left (\int \frac {\sqrt {3 x^{3}-6 x +2}\, x^{2}}{3 x^{3}-6 x +2}d x \right ) c +\left (\int \frac {\sqrt {3 x^{3}-6 x +2}\, x}{3 x^{3}-6 x +2}d x \right ) b \] Input:
int((C*x^2+B*x+A)/(3*x^3-6*x+2)^(1/2),x)
Output:
int(sqrt(3*x**3 - 6*x + 2)/(3*x**3 - 6*x + 2),x)*a + int((sqrt(3*x**3 - 6* x + 2)*x**2)/(3*x**3 - 6*x + 2),x)*c + int((sqrt(3*x**3 - 6*x + 2)*x)/(3*x **3 - 6*x + 2),x)*b