Integrand size = 23, antiderivative size = 747 \[ \int \left (A+B x+C x^2\right ) \left (2-6 x+3 x^3\right )^p \, dx =\text {Too large to display} \] Output:
C*(3*x^3-6*x+2)^(p+1)/(9*p+9)+1/3*2^(3/2+2*p)*B*(3*x^3-6*x+2)^p*AppellF1(p +1,-p,-1-p,2+p,-1/6*(3*x-2*6^(1/2)*cos(1/6*Pi+1/3*arcsin(1/4*6^(1/2))))*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))) ),-1/12*(3*x-2*6^(1/2)*cos(1/6*Pi+1/3*arcsin(1/4*6^(1/2))))*sec(1/3*arcsin (1/4*6^(1/2)))*2^(1/2))*cos(1/3*arcsin(1/4*6^(1/2)))^(p+1)*(3*x-2*6^(1/2)* cos(1/6*Pi+1/3*arcsin(1/4*6^(1/2))))*(cos(1/3*arcsin(1/4*6^(1/2)))-3^(1/2) *sin(1/3*arcsin(1/4*6^(1/2))))^p/(p+1)/((x-2/3*6^(1/2)*sin(1/3*arcsin(1/4* 6^(1/2))))^p)/((x+2/3*6^(1/2)*sin(1/3*Pi+1/3*arcsin(1/4*6^(1/2))))^p)+1/9* (3*x^3-6*x+2)^p*AppellF1(p+1,-p,-p,2+p,-1/6*(3*x-2*6^(1/2)*cos(1/6*Pi+1/3* arcsin(1/4*6^(1/2))))*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)))),-1/12*(3*x-2*6^(1/2)*cos(1/6*Pi+1/3*arcsin(1/4*6 ^(1/2))))*sec(1/3*arcsin(1/4*6^(1/2)))*2^(1/2))*(3*x-2*6^(1/2)*cos(1/6*Pi+ 1/3*arcsin(1/4*6^(1/2))))*(4*cos(1/3*arcsin(1/4*6^(1/2)))*(cos(1/3*arcsin( 1/4*6^(1/2)))-3^(1/2)*sin(1/3*arcsin(1/4*6^(1/2)))))^p*(3*A+2*C-2*6^(1/2)* B*sin(1/3*Pi+1/3*arcsin(1/4*6^(1/2))))/(p+1)/((x-2/3*6^(1/2)*sin(1/3*arcsi n(1/4*6^(1/2))))^p)/((x+2/3*6^(1/2)*sin(1/3*Pi+1/3*arcsin(1/4*6^(1/2))))^p )
\[ \int \left (A+B x+C x^2\right ) \left (2-6 x+3 x^3\right )^p \, dx=\int \left (A+B x+C x^2\right ) \left (2-6 x+3 x^3\right )^p \, dx \] Input:
Integrate[(A + B*x + C*x^2)*(2 - 6*x + 3*x^3)^p,x]
Output:
Integrate[(A + B*x + C*x^2)*(2 - 6*x + 3*x^3)^p, x]
Result contains complex when optimal does not.
Time = 4.48 (sec) , antiderivative size = 1821, normalized size of antiderivative = 2.44, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {2526, 27, 2486, 1269, 1179, 150}
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.
\(\displaystyle \int \left (3 x^3-6 x+2\right )^p \left (A+B x+C x^2\right ) \, dx\) |
\(\Big \downarrow \) 2526 |
\(\displaystyle \frac {1}{9} \int 3 (3 A+2 C+3 B x) \left (3 x^3-6 x+2\right )^pdx+\frac {C \left (3 x^3-6 x+2\right )^{p+1}}{9 (p+1)}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{3} \int (3 A+2 C+3 B x) \left (3 x^3-6 x+2\right )^pdx+\frac {C \left (3 x^3-6 x+2\right )^{p+1}}{9 (p+1)}\) |
\(\Big \downarrow \) 2486 |
\(\displaystyle \frac {C \left (3 x^3-6 x+2\right )^{p+1}}{9 (p+1)}+\frac {1}{3} \left (3 x+\sqrt [3]{9-3 i \sqrt {15}}+\frac {2\ 3^{2/3}}{\sqrt [3]{3-i \sqrt {15}}}\right )^{-p} \left (3 x^3-6 x+2\right )^p \left (9 x^2-3 \left (\frac {2\ 3^{2/3}}{\sqrt [3]{3-i \sqrt {15}}}+\sqrt [3]{9-3 i \sqrt {15}}\right ) x+\left (9-3 i \sqrt {15}\right )^{2/3}+\frac {12 \sqrt [3]{3}}{\left (3-i \sqrt {15}\right )^{2/3}}-6\right )^{-p} \int \left (3 x+\sqrt [3]{9-3 i \sqrt {15}}+\frac {2\ 3^{2/3}}{\sqrt [3]{3-i \sqrt {15}}}\right )^p (3 A+2 C+3 B x) \left (9 x^2-3 \left (\frac {2\ 3^{2/3}}{\sqrt [3]{3-i \sqrt {15}}}+\sqrt [3]{9-3 i \sqrt {15}}\right ) x+\left (9-3 i \sqrt {15}\right )^{2/3}+\frac {12 \sqrt [3]{3}}{\left (3-i \sqrt {15}\right )^{2/3}}-6\right )^pdx\) |
\(\Big \downarrow \) 1269 |
\(\displaystyle \frac {C \left (3 x^3-6 x+2\right )^{p+1}}{9 (p+1)}+\frac {1}{3} \left (3 x+\sqrt [3]{9-3 i \sqrt {15}}+\frac {2\ 3^{2/3}}{\sqrt [3]{3-i \sqrt {15}}}\right )^{-p} \left (3 x^3-6 x+2\right )^p \left (9 x^2-3 \left (\frac {2\ 3^{2/3}}{\sqrt [3]{3-i \sqrt {15}}}+\sqrt [3]{9-3 i \sqrt {15}}\right ) x+\left (9-3 i \sqrt {15}\right )^{2/3}+\frac {12 \sqrt [3]{3}}{\left (3-i \sqrt {15}\right )^{2/3}}-6\right )^{-p} \left (\left (3 A-\left (\frac {2\ 3^{2/3}}{\sqrt [3]{3-i \sqrt {15}}}+\sqrt [3]{9-3 i \sqrt {15}}\right ) B+2 C\right ) \int \left (3 x+\sqrt [3]{9-3 i \sqrt {15}}+\frac {2\ 3^{2/3}}{\sqrt [3]{3-i \sqrt {15}}}\right )^p \left (9 x^2-3 \left (\frac {2\ 3^{2/3}}{\sqrt [3]{3-i \sqrt {15}}}+\sqrt [3]{9-3 i \sqrt {15}}\right ) x+\left (9-3 i \sqrt {15}\right )^{2/3}+\frac {12 \sqrt [3]{3}}{\left (3-i \sqrt {15}\right )^{2/3}}-6\right )^pdx+B \int \left (3 x+\sqrt [3]{9-3 i \sqrt {15}}+\frac {2\ 3^{2/3}}{\sqrt [3]{3-i \sqrt {15}}}\right )^{p+1} \left (9 x^2-3 \left (\frac {2\ 3^{2/3}}{\sqrt [3]{3-i \sqrt {15}}}+\sqrt [3]{9-3 i \sqrt {15}}\right ) x+\left (9-3 i \sqrt {15}\right )^{2/3}+\frac {12 \sqrt [3]{3}}{\left (3-i \sqrt {15}\right )^{2/3}}-6\right )^pdx\right )\) |
\(\Big \downarrow \) 1179 |
\(\displaystyle \frac {1}{3} \left (3 x+\sqrt [3]{9-3 i \sqrt {15}}+\frac {2\ 3^{2/3}}{\sqrt [3]{3-i \sqrt {15}}}\right )^{-p} \left (3 x^3-6 x+2\right )^p \left (\frac {1}{3} \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 ) \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 )^p \left (1-\frac {2 \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 )}{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 )}\right )^{-p} \int \left (3 x+\sqrt [3]{9-3 i \sqrt {15}}+\frac {2\ 3^{2/3}}{\sqrt [3]{3-i \sqrt {15}}}\right )^p \left (1-\frac {2 \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 )}{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 )}\right )^p \left (1-\frac {2 \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 )}{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 )}\right )^pd\left (3 x+\sqrt [3]{9-3 i \sqrt {15}}+\frac {2\ 3^{2/3}}{\sqrt [3]{3-i \sqrt {15}}}\right ) \left (1-\frac {2 \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 )}{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 )}\right )^{-p}+\frac {1}{3} B \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 )^p \left (1-\frac {2 \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 )}{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 )}\right )^{-p} \int \left (3 x+\sqrt [3]{9-3 i \sqrt {15}}+\frac {2\ 3^{2/3}}{\sqrt [3]{3-i \sqrt {15}}}\right )^{p+1} \left (1-\frac {2 \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 )}{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 )}\right )^p \left (1-\frac {2 \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 )}{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 )}\right )^pd\left (3 x+\sqrt [3]{9-3 i \sqrt {15}}+\frac {2\ 3^{2/3}}{\sqrt [3]{3-i \sqrt {15}}}\right ) \left (1-\frac {2 \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 )}{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 )}\right )^{-p}\right ) \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 )^{-p}+\frac {C \left (3 x^3-6 x+2\right )^{p+1}}{9 (p+1)}\) |
\(\Big \downarrow \) 150 |
\(\displaystyle \frac {1}{3} \left (3 x+\sqrt [3]{9-3 i \sqrt {15}}+\frac {2\ 3^{2/3}}{\sqrt [3]{3-i \sqrt {15}}}\right )^{-p} \left (3 x^3-6 x+2\right )^p \left (\frac {\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 ) \left (3 x+\sqrt [3]{9-3 i \sqrt {15}}+\frac {2\ 3^{2/3}}{\sqrt [3]{3-i \sqrt {15}}}\right )^{p+1} \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 )^p \left (1-\frac {2 \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 )}{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 )}\right )^{-p} \operatorname {AppellF1}\left (p+1,-p,-p,p+2,\frac {2 \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 )}{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 )},\frac {2 \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 )}{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 )}\right ) \left (1-\frac {2 \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 )}{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 )}\right )^{-p}}{3 (p+1)}+\frac {B \left (3 x+\sqrt [3]{9-3 i \sqrt {15}}+\frac {2\ 3^{2/3}}{\sqrt [3]{3-i \sqrt {15}}}\right )^{p+2} \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 )^p \left (1-\frac {2 \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 )}{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 )}\right )^{-p} \operatorname {AppellF1}\left (p+2,-p,-p,p+3,\frac {2 \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 )}{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 )},\frac {2 \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 )}{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 )}\right ) \left (1-\frac {2 \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 )}{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 )}\right )^{-p}}{3 (p+2)}\right ) \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 )^{-p}+\frac {C \left (3 x^3-6 x+2\right )^{p+1}}{9 (p+1)}\) |
Input:
Int[(A + B*x + C*x^2)*(2 - 6*x + 3*x^3)^p,x]
Output:
(C*(2 - 6*x + 3*x^3)^(1 + p))/(9*(1 + p)) + ((2 - 6*x + 3*x^3)^p*(((3*A - ((2*3^(2/3))/(3 - I*Sqrt[15])^(1/3) + (9 - (3*I)*Sqrt[15])^(1/3))*B + 2*C) *((2*3^(2/3))/(3 - I*Sqrt[15])^(1/3) + (9 - (3*I)*Sqrt[15])^(1/3) + 3*x)^( 1 + p)*(-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)^p*AppellF1[1 + p, -p, -p, 2 + p, (2*(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))/(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)])), (2 *(3 - I*Sqrt[15])^(1/3)*((2*3^(2/3))/(3 - I*Sqrt[15])^(1/3) + (9 - (3*I)*S qrt[15])^(1/3) + 3*x))/(3*(2*3^(2/3) + 3^(1/3)*(3 - I*Sqrt[15])^(2/3) + Sq rt[-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*(1 + p)*(1 - (2*(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))/(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)])))^p*( 1 - (2*(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))/(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)])))^p) + (B*((2*3^(2/3))/(3 - I*Sqrt[15])^(...
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_ ] :> Simp[c^n*e^p*((b*x)^(m + 1)/(b*(m + 1)))*AppellF1[m + 1, -n, -p, m + 2 , (-d)*(x/c), (-f)*(x/e)], x] /; FreeQ[{b, c, d, e, f, m, n, p}, x] && !In tegerQ[m] && !IntegerQ[n] && GtQ[c, 0] && (IntegerQ[p] || GtQ[e, 0])
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(a + b*x + c*x^2)^p/(e*(1 - ( d + e*x)/(d - e*((b - q)/(2*c))))^p*(1 - (d + e*x)/(d - e*((b + q)/(2*c)))) ^p) Subst[Int[x^m*Simp[1 - x/(d - e*((b - q)/(2*c))), x]^p*Simp[1 - x/(d - e*((b + q)/(2*c))), x]^p, x], x, d + e*x], x]] /; FreeQ[{a, b, c, d, e, m , p}, x]
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]
\[\int \left (C \,x^{2}+B x +A \right ) \left (3 x^{3}-6 x +2\right )^{p}d x\]
Input:
int((C*x^2+B*x+A)*(3*x^3-6*x+2)^p,x)
Output:
int((C*x^2+B*x+A)*(3*x^3-6*x+2)^p,x)
\[ \int \left (A+B x+C x^2\right ) \left (2-6 x+3 x^3\right )^p \, dx=\int { {\left (C x^{2} + B x + A\right )} {\left (3 \, x^{3} - 6 \, x + 2\right )}^{p} \,d x } \] Input:
integrate((C*x^2+B*x+A)*(3*x^3-6*x+2)^p,x, algorithm="fricas")
Output:
integral((C*x^2 + B*x + A)*(3*x^3 - 6*x + 2)^p, x)
Timed out. \[ \int \left (A+B x+C x^2\right ) \left (2-6 x+3 x^3\right )^p \, dx=\text {Timed out} \] Input:
integrate((C*x**2+B*x+A)*(3*x**3-6*x+2)**p,x)
Output:
Timed out
\[ \int \left (A+B x+C x^2\right ) \left (2-6 x+3 x^3\right )^p \, dx=\int { {\left (C x^{2} + B x + A\right )} {\left (3 \, x^{3} - 6 \, x + 2\right )}^{p} \,d x } \] Input:
integrate((C*x^2+B*x+A)*(3*x^3-6*x+2)^p,x, algorithm="maxima")
Output:
integrate((C*x^2 + B*x + A)*(3*x^3 - 6*x + 2)^p, x)
\[ \int \left (A+B x+C x^2\right ) \left (2-6 x+3 x^3\right )^p \, dx=\int { {\left (C x^{2} + B x + A\right )} {\left (3 \, x^{3} - 6 \, x + 2\right )}^{p} \,d x } \] Input:
integrate((C*x^2+B*x+A)*(3*x^3-6*x+2)^p,x, algorithm="giac")
Output:
integrate((C*x^2 + B*x + A)*(3*x^3 - 6*x + 2)^p, x)
Timed out. \[ \int \left (A+B x+C x^2\right ) \left (2-6 x+3 x^3\right )^p \, dx=\int \left (C\,x^2+B\,x+A\right )\,{\left (3\,x^3-6\,x+2\right )}^p \,d x \] Input:
int((A + B*x + C*x^2)*(3*x^3 - 6*x + 2)^p,x)
Output:
int((A + B*x + C*x^2)*(3*x^3 - 6*x + 2)^p, x)
\[ \int \left (A+B x+C x^2\right ) \left (2-6 x+3 x^3\right )^p \, dx=\text {too large to display} \] Input:
int((C*x^2+B*x+A)*(3*x^3-6*x+2)^p,x)
Output:
(27*(3*x**3 - 6*x + 2)**p*a*p**2*x + 45*(3*x**3 - 6*x + 2)**p*a*p*x + 18*( 3*x**3 - 6*x + 2)**p*a*x + 27*(3*x**3 - 6*x + 2)**p*b*p**2*x**2 - 36*(3*x* *3 - 6*x + 2)**p*b*p**2 + 36*(3*x**3 - 6*x + 2)**p*b*p*x**2 - 48*(3*x**3 - 6*x + 2)**p*b*p + 9*(3*x**3 - 6*x + 2)**p*b*x**2 - 12*(3*x**3 - 6*x + 2)* *p*b + 27*(3*x**3 - 6*x + 2)**p*c*p**2*x**3 - 36*(3*x**3 - 6*x + 2)**p*c*p **2*x + 18*(3*x**3 - 6*x + 2)**p*c*p**2 + 27*(3*x**3 - 6*x + 2)**p*c*p*x** 3 - 24*(3*x**3 - 6*x + 2)**p*c*p*x + 18*(3*x**3 - 6*x + 2)**p*c*p + 6*(3*x **3 - 6*x + 2)**p*c*x**3 + 4*(3*x**3 - 6*x + 2)**p*c + 1458*int((3*x**3 - 6*x + 2)**p/(27*p**2*x**3 - 54*p**2*x + 18*p**2 + 27*p*x**3 - 54*p*x + 18* p + 6*x**3 - 12*x + 4),x)*a*p**5 + 3888*int((3*x**3 - 6*x + 2)**p/(27*p**2 *x**3 - 54*p**2*x + 18*p**2 + 27*p*x**3 - 54*p*x + 18*p + 6*x**3 - 12*x + 4),x)*a*p**4 + 3726*int((3*x**3 - 6*x + 2)**p/(27*p**2*x**3 - 54*p**2*x + 18*p**2 + 27*p*x**3 - 54*p*x + 18*p + 6*x**3 - 12*x + 4),x)*a*p**3 + 1512* int((3*x**3 - 6*x + 2)**p/(27*p**2*x**3 - 54*p**2*x + 18*p**2 + 27*p*x**3 - 54*p*x + 18*p + 6*x**3 - 12*x + 4),x)*a*p**2 + 216*int((3*x**3 - 6*x + 2 )**p/(27*p**2*x**3 - 54*p**2*x + 18*p**2 + 27*p*x**3 - 54*p*x + 18*p + 6*x **3 - 12*x + 4),x)*a*p - 1944*int((3*x**3 - 6*x + 2)**p/(27*p**2*x**3 - 54 *p**2*x + 18*p**2 + 27*p*x**3 - 54*p*x + 18*p + 6*x**3 - 12*x + 4),x)*b*p* *5 - 4536*int((3*x**3 - 6*x + 2)**p/(27*p**2*x**3 - 54*p**2*x + 18*p**2 + 27*p*x**3 - 54*p*x + 18*p + 6*x**3 - 12*x + 4),x)*b*p**4 - 3672*int((3*...