Integrand size = 26, antiderivative size = 960 \[ \int \left (A+B x+C x^2\right ) \left (2+3 x+4 x^2+x^3\right )^p \, dx =\text {Too large to display} \] Output:
C*(x^3+4*x^2+3*x+2)^(p+1)/(3*p+3)+1/27*(9*A-12*B-(7+(37-3*114^(1/2))^(2/3) )*(3*B-8*C)/(37-3*114^(1/2))^(1/3)+23*C)*(4+7/(37-3*114^(1/2))^(1/3)+(37-3 *114^(1/2))^(1/3)+3*x)*(x^3+4*x^2+3*x+2)^p*AppellF1(p+1,-p,-p,2+p,2*I*(37- 3*114^(1/2))^(1/3)*(4+7/(37-3*114^(1/2))^(1/3)+(37-3*114^(1/2))^(1/3)+3*x) /(21*I-7*3^(1/2)+(3*I+3^(1/2))*(37-3*114^(1/2))^(2/3)),2*I*(37-3*114^(1/2) )^(1/3)*(4+7/(37-3*114^(1/2))^(1/3)+(37-3*114^(1/2))^(1/3)+3*x)/(21*I+7*3^ (1/2)+(3*I-3^(1/2))*(37-3*114^(1/2))^(2/3)))/(p+1)/((1-2*I*(37-3*114^(1/2) )^(1/3)*(4+7/(37-3*114^(1/2))^(1/3)+(37-3*114^(1/2))^(1/3)+3*x)/(21*I+7*3^ (1/2)+(3*I-3^(1/2))*(37-3*114^(1/2))^(2/3)))^p)/((1-2*I*(37-3*114^(1/2))^( 1/3)*(4+7/(37-3*114^(1/2))^(1/3)+(37-3*114^(1/2))^(1/3)+3*x)/(21*I-7*3^(1/ 2)+(3*I+3^(1/2))*(37-3*114^(1/2))^(2/3)))^p)+1/27*(3*B-8*C)*(4+7/(37-3*114 ^(1/2))^(1/3)+(37-3*114^(1/2))^(1/3)+3*x)^2*(x^3+4*x^2+3*x+2)^p*AppellF1(2 +p,-p,-p,3+p,2*I*(37-3*114^(1/2))^(1/3)*(4+7/(37-3*114^(1/2))^(1/3)+(37-3* 114^(1/2))^(1/3)+3*x)/(21*I-7*3^(1/2)+(3*I+3^(1/2))*(37-3*114^(1/2))^(2/3) ),2*I*(37-3*114^(1/2))^(1/3)*(4+7/(37-3*114^(1/2))^(1/3)+(37-3*114^(1/2))^ (1/3)+3*x)/(21*I+7*3^(1/2)+(3*I-3^(1/2))*(37-3*114^(1/2))^(2/3)))/(2+p)/(( 1-2*I*(37-3*114^(1/2))^(1/3)*(4+7/(37-3*114^(1/2))^(1/3)+(37-3*114^(1/2))^ (1/3)+3*x)/(21*I+7*3^(1/2)+(3*I-3^(1/2))*(37-3*114^(1/2))^(2/3)))^p)/((1-2 *I*(37-3*114^(1/2))^(1/3)*(4+7/(37-3*114^(1/2))^(1/3)+(37-3*114^(1/2))^(1/ 3)+3*x)/(21*I-7*3^(1/2)+(3*I+3^(1/2))*(37-3*114^(1/2))^(2/3)))^p)
\[ \int \left (A+B x+C x^2\right ) \left (2+3 x+4 x^2+x^3\right )^p \, dx=\int \left (A+B x+C x^2\right ) \left (2+3 x+4 x^2+x^3\right )^p \, dx \] Input:
Integrate[(A + B*x + C*x^2)*(2 + 3*x + 4*x^2 + x^3)^p,x]
Output:
Integrate[(A + B*x + C*x^2)*(2 + 3*x + 4*x^2 + x^3)^p, x]
Time = 1.89 (sec) , antiderivative size = 1284, normalized size of antiderivative = 1.34, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {2526, 2490, 2486, 27, 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.
| \(\displaystyle \int \left (x^3+4 x^2+3 x+2\right )^p \left (A+B x+C x^2\right ) \, dx\) |
| \(\Big \downarrow \) 2526 |
| \(\displaystyle \frac {1}{3} \int (3 (A-C)+(3 B-8 C) x) \left (x^3+4 x^2+3 x+2\right )^pdx+\frac {C \left (x^3+4 x^2+3 x+2\right )^{p+1}}{3 (p+1)}\) |
| \(\Big \downarrow \) 2490 |
| \(\displaystyle \frac {1}{3} \int \left (\frac {1}{3} (9 (A-C)-4 (3 B-8 C))+(3 B-8 C) \left (x+\frac {4}{3}\right )\right ) \left (\left (x+\frac {4}{3}\right )^3-\frac {7}{3} \left (x+\frac {4}{3}\right )+\frac {74}{27}\right )^pd\left (x+\frac {4}{3}\right )+\frac {C \left (x^3+4 x^2+3 x+2\right )^{p+1}}{3 (p+1)}\) |
| \(\Big \downarrow \) 2486 |
| \(\displaystyle \frac {1}{3} \left (x+\frac {7+\left (37-3 \sqrt {114}\right )^{2/3}}{3 \sqrt [3]{37-3 \sqrt {114}}}+\frac {4}{3}\right )^{-p} \left (\left (x+\frac {4}{3}\right )^3-\frac {7}{3} \left (x+\frac {4}{3}\right )+\frac {74}{27}\right )^p \left (\left (x+\frac {4}{3}\right )^2-\frac {\left (7+\left (37-3 \sqrt {114}\right )^{2/3}\right ) \left (x+\frac {4}{3}\right )}{3 \sqrt [3]{37-3 \sqrt {114}}}+\frac {1}{9} \left (-7+\frac {49}{\left (37-3 \sqrt {114}\right )^{2/3}}+\left (37-3 \sqrt {114}\right )^{2/3}\right )\right )^{-p} \int \frac {1}{3} \left (x+\frac {7+\left (37-3 \sqrt {114}\right )^{2/3}}{3 \sqrt [3]{37-3 \sqrt {114}}}+\frac {4}{3}\right )^p \left (9 A-12 B+23 C+3 (3 B-8 C) \left (x+\frac {4}{3}\right )\right ) \left (\left (x+\frac {4}{3}\right )^2-\frac {\left (7+\left (37-3 \sqrt {114}\right )^{2/3}\right ) \left (x+\frac {4}{3}\right )}{3 \sqrt [3]{37-3 \sqrt {114}}}+\frac {1}{9} \left (-7+\frac {49}{\left (37-3 \sqrt {114}\right )^{2/3}}+\left (37-3 \sqrt {114}\right )^{2/3}\right )\right )^pd\left (x+\frac {4}{3}\right )+\frac {C \left (x^3+4 x^2+3 x+2\right )^{p+1}}{3 (p+1)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \left (x+\frac {7+\left (37-3 \sqrt {114}\right )^{2/3}}{3 \sqrt [3]{37-3 \sqrt {114}}}+\frac {4}{3}\right )^{-p} \left (\left (x+\frac {4}{3}\right )^3-\frac {7}{3} \left (x+\frac {4}{3}\right )+\frac {74}{27}\right )^p \left (\left (x+\frac {4}{3}\right )^2-\frac {\left (7+\left (37-3 \sqrt {114}\right )^{2/3}\right ) \left (x+\frac {4}{3}\right )}{3 \sqrt [3]{37-3 \sqrt {114}}}+\frac {1}{9} \left (-7+\frac {49}{\left (37-3 \sqrt {114}\right )^{2/3}}+\left (37-3 \sqrt {114}\right )^{2/3}\right )\right )^{-p} \int \left (x+\frac {7+\left (37-3 \sqrt {114}\right )^{2/3}}{3 \sqrt [3]{37-3 \sqrt {114}}}+\frac {4}{3}\right )^p \left (9 A-12 B+23 C+3 (3 B-8 C) \left (x+\frac {4}{3}\right )\right ) \left (\left (x+\frac {4}{3}\right )^2-\frac {\left (7+\left (37-3 \sqrt {114}\right )^{2/3}\right ) \left (x+\frac {4}{3}\right )}{3 \sqrt [3]{37-3 \sqrt {114}}}+\frac {1}{9} \left (-7+\frac {49}{\left (37-3 \sqrt {114}\right )^{2/3}}+\left (37-3 \sqrt {114}\right )^{2/3}\right )\right )^pd\left (x+\frac {4}{3}\right )+\frac {C \left (x^3+4 x^2+3 x+2\right )^{p+1}}{3 (p+1)}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {1}{9} \left (x+\frac {7+\left (37-3 \sqrt {114}\right )^{2/3}}{3 \sqrt [3]{37-3 \sqrt {114}}}+\frac {4}{3}\right )^{-p} \left (\left (x+\frac {4}{3}\right )^3-\frac {7}{3} \left (x+\frac {4}{3}\right )+\frac {74}{27}\right )^p \left (\left (x+\frac {4}{3}\right )^2-\frac {\left (7+\left (37-3 \sqrt {114}\right )^{2/3}\right ) \left (x+\frac {4}{3}\right )}{3 \sqrt [3]{37-3 \sqrt {114}}}+\frac {1}{9} \left (-7+\frac {49}{\left (37-3 \sqrt {114}\right )^{2/3}}+\left (37-3 \sqrt {114}\right )^{2/3}\right )\right )^{-p} \left (\left (9 A-\frac {\left (7+\left (37-3 \sqrt {114}\right )^{2/3}\right ) (3 B-8 C)}{\sqrt [3]{37-3 \sqrt {114}}}-12 B+23 C\right ) \int \left (x+\frac {7+\left (37-3 \sqrt {114}\right )^{2/3}}{3 \sqrt [3]{37-3 \sqrt {114}}}+\frac {4}{3}\right )^p \left (\left (x+\frac {4}{3}\right )^2-\frac {\left (7+\left (37-3 \sqrt {114}\right )^{2/3}\right ) \left (x+\frac {4}{3}\right )}{3 \sqrt [3]{37-3 \sqrt {114}}}+\frac {1}{9} \left (-7+\frac {49}{\left (37-3 \sqrt {114}\right )^{2/3}}+\left (37-3 \sqrt {114}\right )^{2/3}\right )\right )^pd\left (x+\frac {4}{3}\right )+3 (3 B-8 C) \int \left (x+\frac {7+\left (37-3 \sqrt {114}\right )^{2/3}}{3 \sqrt [3]{37-3 \sqrt {114}}}+\frac {4}{3}\right )^{p+1} \left (\left (x+\frac {4}{3}\right )^2-\frac {\left (7+\left (37-3 \sqrt {114}\right )^{2/3}\right ) \left (x+\frac {4}{3}\right )}{3 \sqrt [3]{37-3 \sqrt {114}}}+\frac {1}{9} \left (-7+\frac {49}{\left (37-3 \sqrt {114}\right )^{2/3}}+\left (37-3 \sqrt {114}\right )^{2/3}\right )\right )^pd\left (x+\frac {4}{3}\right )\right )+\frac {C \left (x^3+4 x^2+3 x+2\right )^{p+1}}{3 (p+1)}\) |
| \(\Big \downarrow \) 1179 |
| \(\displaystyle \frac {1}{9} \left (x+\frac {7+\left (37-3 \sqrt {114}\right )^{2/3}}{3 \sqrt [3]{37-3 \sqrt {114}}}+\frac {4}{3}\right )^{-p} \left (\left (x+\frac {4}{3}\right )^3-\frac {7}{3} \left (x+\frac {4}{3}\right )+\frac {74}{27}\right )^p \left (\left (9 A-12 B-\frac {\left (7+\left (37-3 \sqrt {114}\right )^{2/3}\right ) (3 B-8 C)}{\sqrt [3]{37-3 \sqrt {114}}}+23 C\right ) \left (\left (x+\frac {4}{3}\right )^2-\frac {\left (7+\left (37-3 \sqrt {114}\right )^{2/3}\right ) \left (x+\frac {4}{3}\right )}{3 \sqrt [3]{37-3 \sqrt {114}}}+\frac {1}{9} \left (-7+\frac {49}{\left (37-3 \sqrt {114}\right )^{2/3}}+\left (37-3 \sqrt {114}\right )^{2/3}\right )\right )^p \left (1-\frac {2 i \sqrt [3]{37-3 \sqrt {114}} \left (3 \left (x+\frac {4}{3}\right )+\frac {7+\left (37-3 \sqrt {114}\right )^{2/3}}{\sqrt [3]{37-3 \sqrt {114}}}\right )}{7 \left (3 i+\sqrt {3}\right )+\left (3 i-\sqrt {3}\right ) \left (37-3 \sqrt {114}\right )^{2/3}}\right )^{-p} \int \left (x+\frac {7+\left (37-3 \sqrt {114}\right )^{2/3}}{3 \sqrt [3]{37-3 \sqrt {114}}}+\frac {4}{3}\right )^p \left (1-\frac {6 i \sqrt [3]{37-3 \sqrt {114}} \left (x+\frac {7+\left (37-3 \sqrt {114}\right )^{2/3}}{3 \sqrt [3]{37-3 \sqrt {114}}}+\frac {4}{3}\right )}{7 \left (3 i+\sqrt {3}\right )+\left (3 i-\sqrt {3}\right ) \left (37-3 \sqrt {114}\right )^{2/3}}\right )^p \left (1-\frac {6 i \sqrt [3]{37-3 \sqrt {114}} \left (x+\frac {7+\left (37-3 \sqrt {114}\right )^{2/3}}{3 \sqrt [3]{37-3 \sqrt {114}}}+\frac {4}{3}\right )}{7 \left (3 i-\sqrt {3}\right )+\left (3 i+\sqrt {3}\right ) \left (37-3 \sqrt {114}\right )^{2/3}}\right )^pd\left (x+\frac {7+\left (37-3 \sqrt {114}\right )^{2/3}}{3 \sqrt [3]{37-3 \sqrt {114}}}+\frac {4}{3}\right ) \left (1-\frac {2 i \sqrt [3]{37-3 \sqrt {114}} \left (3 \left (x+\frac {4}{3}\right )+\frac {7+\left (37-3 \sqrt {114}\right )^{2/3}}{\sqrt [3]{37-3 \sqrt {114}}}\right )}{7 \left (3 i-\sqrt {3}\right )+\left (3 i+\sqrt {3}\right ) \left (37-3 \sqrt {114}\right )^{2/3}}\right )^{-p}+3 (3 B-8 C) \left (\left (x+\frac {4}{3}\right )^2-\frac {\left (7+\left (37-3 \sqrt {114}\right )^{2/3}\right ) \left (x+\frac {4}{3}\right )}{3 \sqrt [3]{37-3 \sqrt {114}}}+\frac {1}{9} \left (-7+\frac {49}{\left (37-3 \sqrt {114}\right )^{2/3}}+\left (37-3 \sqrt {114}\right )^{2/3}\right )\right )^p \left (1-\frac {2 i \sqrt [3]{37-3 \sqrt {114}} \left (3 \left (x+\frac {4}{3}\right )+\frac {7+\left (37-3 \sqrt {114}\right )^{2/3}}{\sqrt [3]{37-3 \sqrt {114}}}\right )}{7 \left (3 i+\sqrt {3}\right )+\left (3 i-\sqrt {3}\right ) \left (37-3 \sqrt {114}\right )^{2/3}}\right )^{-p} \int \left (x+\frac {7+\left (37-3 \sqrt {114}\right )^{2/3}}{3 \sqrt [3]{37-3 \sqrt {114}}}+\frac {4}{3}\right )^{p+1} \left (1-\frac {6 i \sqrt [3]{37-3 \sqrt {114}} \left (x+\frac {7+\left (37-3 \sqrt {114}\right )^{2/3}}{3 \sqrt [3]{37-3 \sqrt {114}}}+\frac {4}{3}\right )}{7 \left (3 i+\sqrt {3}\right )+\left (3 i-\sqrt {3}\right ) \left (37-3 \sqrt {114}\right )^{2/3}}\right )^p \left (1-\frac {6 i \sqrt [3]{37-3 \sqrt {114}} \left (x+\frac {7+\left (37-3 \sqrt {114}\right )^{2/3}}{3 \sqrt [3]{37-3 \sqrt {114}}}+\frac {4}{3}\right )}{7 \left (3 i-\sqrt {3}\right )+\left (3 i+\sqrt {3}\right ) \left (37-3 \sqrt {114}\right )^{2/3}}\right )^pd\left (x+\frac {7+\left (37-3 \sqrt {114}\right )^{2/3}}{3 \sqrt [3]{37-3 \sqrt {114}}}+\frac {4}{3}\right ) \left (1-\frac {2 i \sqrt [3]{37-3 \sqrt {114}} \left (3 \left (x+\frac {4}{3}\right )+\frac {7+\left (37-3 \sqrt {114}\right )^{2/3}}{\sqrt [3]{37-3 \sqrt {114}}}\right )}{7 \left (3 i-\sqrt {3}\right )+\left (3 i+\sqrt {3}\right ) \left (37-3 \sqrt {114}\right )^{2/3}}\right )^{-p}\right ) \left (\left (x+\frac {4}{3}\right )^2-\frac {\left (7+\left (37-3 \sqrt {114}\right )^{2/3}\right ) \left (x+\frac {4}{3}\right )}{3 \sqrt [3]{37-3 \sqrt {114}}}+\frac {1}{9} \left (-7+\frac {49}{\left (37-3 \sqrt {114}\right )^{2/3}}+\left (37-3 \sqrt {114}\right )^{2/3}\right )\right )^{-p}+\frac {C \left (x^3+4 x^2+3 x+2\right )^{p+1}}{3 (p+1)}\) |
| \(\Big \downarrow \) 150 |
| \(\displaystyle \frac {1}{9} \left (x+\frac {7+\left (37-3 \sqrt {114}\right )^{2/3}}{3 \sqrt [3]{37-3 \sqrt {114}}}+\frac {4}{3}\right )^{-p} \left (\left (x+\frac {4}{3}\right )^3-\frac {7}{3} \left (x+\frac {4}{3}\right )+\frac {74}{27}\right )^p \left (\frac {\left (9 A-12 B-\frac {\left (7+\left (37-3 \sqrt {114}\right )^{2/3}\right ) (3 B-8 C)}{\sqrt [3]{37-3 \sqrt {114}}}+23 C\right ) \left (x+\frac {7+\left (37-3 \sqrt {114}\right )^{2/3}}{3 \sqrt [3]{37-3 \sqrt {114}}}+\frac {4}{3}\right )^{p+1} \left (\left (x+\frac {4}{3}\right )^2-\frac {\left (7+\left (37-3 \sqrt {114}\right )^{2/3}\right ) \left (x+\frac {4}{3}\right )}{3 \sqrt [3]{37-3 \sqrt {114}}}+\frac {1}{9} \left (-7+\frac {49}{\left (37-3 \sqrt {114}\right )^{2/3}}+\left (37-3 \sqrt {114}\right )^{2/3}\right )\right )^p \left (1-\frac {2 i \sqrt [3]{37-3 \sqrt {114}} \left (3 \left (x+\frac {4}{3}\right )+\frac {7+\left (37-3 \sqrt {114}\right )^{2/3}}{\sqrt [3]{37-3 \sqrt {114}}}\right )}{7 \left (3 i+\sqrt {3}\right )+\left (3 i-\sqrt {3}\right ) \left (37-3 \sqrt {114}\right )^{2/3}}\right )^{-p} \operatorname {AppellF1}\left (p+1,-p,-p,p+2,\frac {6 i \sqrt [3]{37-3 \sqrt {114}} \left (x+\frac {7+\left (37-3 \sqrt {114}\right )^{2/3}}{3 \sqrt [3]{37-3 \sqrt {114}}}+\frac {4}{3}\right )}{7 \left (3 i+\sqrt {3}\right )+\left (3 i-\sqrt {3}\right ) \left (37-3 \sqrt {114}\right )^{2/3}},\frac {6 i \sqrt [3]{37-3 \sqrt {114}} \left (x+\frac {7+\left (37-3 \sqrt {114}\right )^{2/3}}{3 \sqrt [3]{37-3 \sqrt {114}}}+\frac {4}{3}\right )}{7 \left (3 i-\sqrt {3}\right )+\left (3 i+\sqrt {3}\right ) \left (37-3 \sqrt {114}\right )^{2/3}}\right ) \left (1-\frac {2 i \sqrt [3]{37-3 \sqrt {114}} \left (3 \left (x+\frac {4}{3}\right )+\frac {7+\left (37-3 \sqrt {114}\right )^{2/3}}{\sqrt [3]{37-3 \sqrt {114}}}\right )}{7 \left (3 i-\sqrt {3}\right )+\left (3 i+\sqrt {3}\right ) \left (37-3 \sqrt {114}\right )^{2/3}}\right )^{-p}}{p+1}+\frac {3 (3 B-8 C) \left (x+\frac {7+\left (37-3 \sqrt {114}\right )^{2/3}}{3 \sqrt [3]{37-3 \sqrt {114}}}+\frac {4}{3}\right )^{p+2} \left (\left (x+\frac {4}{3}\right )^2-\frac {\left (7+\left (37-3 \sqrt {114}\right )^{2/3}\right ) \left (x+\frac {4}{3}\right )}{3 \sqrt [3]{37-3 \sqrt {114}}}+\frac {1}{9} \left (-7+\frac {49}{\left (37-3 \sqrt {114}\right )^{2/3}}+\left (37-3 \sqrt {114}\right )^{2/3}\right )\right )^p \left (1-\frac {2 i \sqrt [3]{37-3 \sqrt {114}} \left (3 \left (x+\frac {4}{3}\right )+\frac {7+\left (37-3 \sqrt {114}\right )^{2/3}}{\sqrt [3]{37-3 \sqrt {114}}}\right )}{7 \left (3 i+\sqrt {3}\right )+\left (3 i-\sqrt {3}\right ) \left (37-3 \sqrt {114}\right )^{2/3}}\right )^{-p} \operatorname {AppellF1}\left (p+2,-p,-p,p+3,\frac {6 i \sqrt [3]{37-3 \sqrt {114}} \left (x+\frac {7+\left (37-3 \sqrt {114}\right )^{2/3}}{3 \sqrt [3]{37-3 \sqrt {114}}}+\frac {4}{3}\right )}{7 \left (3 i+\sqrt {3}\right )+\left (3 i-\sqrt {3}\right ) \left (37-3 \sqrt {114}\right )^{2/3}},\frac {6 i \sqrt [3]{37-3 \sqrt {114}} \left (x+\frac {7+\left (37-3 \sqrt {114}\right )^{2/3}}{3 \sqrt [3]{37-3 \sqrt {114}}}+\frac {4}{3}\right )}{7 \left (3 i-\sqrt {3}\right )+\left (3 i+\sqrt {3}\right ) \left (37-3 \sqrt {114}\right )^{2/3}}\right ) \left (1-\frac {2 i \sqrt [3]{37-3 \sqrt {114}} \left (3 \left (x+\frac {4}{3}\right )+\frac {7+\left (37-3 \sqrt {114}\right )^{2/3}}{\sqrt [3]{37-3 \sqrt {114}}}\right )}{7 \left (3 i-\sqrt {3}\right )+\left (3 i+\sqrt {3}\right ) \left (37-3 \sqrt {114}\right )^{2/3}}\right )^{-p}}{p+2}\right ) \left (\left (x+\frac {4}{3}\right )^2-\frac {\left (7+\left (37-3 \sqrt {114}\right )^{2/3}\right ) \left (x+\frac {4}{3}\right )}{3 \sqrt [3]{37-3 \sqrt {114}}}+\frac {1}{9} \left (-7+\frac {49}{\left (37-3 \sqrt {114}\right )^{2/3}}+\left (37-3 \sqrt {114}\right )^{2/3}\right )\right )^{-p}+\frac {C \left (x^3+4 x^2+3 x+2\right )^{p+1}}{3 (p+1)}\) |
Input:
Int[(A + B*x + C*x^2)*(2 + 3*x + 4*x^2 + x^3)^p,x]
Output:
(C*(2 + 3*x + 4*x^2 + x^3)^(1 + p))/(3*(1 + p)) + ((74/27 - (7*(4/3 + x))/ 3 + (4/3 + x)^3)^p*(((9*A - 12*B - ((7 + (37 - 3*Sqrt[114])^(2/3))*(3*B - 8*C))/(37 - 3*Sqrt[114])^(1/3) + 23*C)*(4/3 + (7 + (37 - 3*Sqrt[114])^(2/3 ))/(3*(37 - 3*Sqrt[114])^(1/3)) + x)^(1 + p)*((-7 + 49/(37 - 3*Sqrt[114])^ (2/3) + (37 - 3*Sqrt[114])^(2/3))/9 - ((7 + (37 - 3*Sqrt[114])^(2/3))*(4/3 + x))/(3*(37 - 3*Sqrt[114])^(1/3)) + (4/3 + x)^2)^p*AppellF1[1 + p, -p, - p, 2 + p, ((6*I)*(37 - 3*Sqrt[114])^(1/3)*(4/3 + (7 + (37 - 3*Sqrt[114])^( 2/3))/(3*(37 - 3*Sqrt[114])^(1/3)) + x))/(7*(3*I + Sqrt[3]) + (3*I - Sqrt[ 3])*(37 - 3*Sqrt[114])^(2/3)), ((6*I)*(37 - 3*Sqrt[114])^(1/3)*(4/3 + (7 + (37 - 3*Sqrt[114])^(2/3))/(3*(37 - 3*Sqrt[114])^(1/3)) + x))/(7*(3*I - Sq rt[3]) + (3*I + Sqrt[3])*(37 - 3*Sqrt[114])^(2/3))])/((1 + p)*(1 - ((2*I)* (37 - 3*Sqrt[114])^(1/3)*((7 + (37 - 3*Sqrt[114])^(2/3))/(37 - 3*Sqrt[114] )^(1/3) + 3*(4/3 + x)))/(7*(3*I + Sqrt[3]) + (3*I - Sqrt[3])*(37 - 3*Sqrt[ 114])^(2/3)))^p*(1 - ((2*I)*(37 - 3*Sqrt[114])^(1/3)*((7 + (37 - 3*Sqrt[11 4])^(2/3))/(37 - 3*Sqrt[114])^(1/3) + 3*(4/3 + x)))/(7*(3*I - Sqrt[3]) + ( 3*I + Sqrt[3])*(37 - 3*Sqrt[114])^(2/3)))^p) + (3*(3*B - 8*C)*(4/3 + (7 + (37 - 3*Sqrt[114])^(2/3))/(3*(37 - 3*Sqrt[114])^(1/3)) + x)^(2 + p)*((-7 + 49/(37 - 3*Sqrt[114])^(2/3) + (37 - 3*Sqrt[114])^(2/3))/9 - ((7 + (37 - 3 *Sqrt[114])^(2/3))*(4/3 + x))/(3*(37 - 3*Sqrt[114])^(1/3)) + (4/3 + x)^2)^ p*AppellF1[2 + p, -p, -p, 3 + p, ((6*I)*(37 - 3*Sqrt[114])^(1/3)*(4/3 +...
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[(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]
\[\int \left (C \,x^{2}+B x +A \right ) \left (x^{3}+4 x^{2}+3 x +2\right )^{p}d x\]
Input:
int((C*x^2+B*x+A)*(x^3+4*x^2+3*x+2)^p,x)
Output:
int((C*x^2+B*x+A)*(x^3+4*x^2+3*x+2)^p,x)
\[ \int \left (A+B x+C x^2\right ) \left (2+3 x+4 x^2+x^3\right )^p \, dx=\int { {\left (C x^{2} + B x + A\right )} {\left (x^{3} + 4 \, x^{2} + 3 \, x + 2\right )}^{p} \,d x } \] Input:
integrate((C*x^2+B*x+A)*(x^3+4*x^2+3*x+2)^p,x, algorithm="fricas")
Output:
integral((C*x^2 + B*x + A)*(x^3 + 4*x^2 + 3*x + 2)^p, x)
\[ \int \left (A+B x+C x^2\right ) \left (2+3 x+4 x^2+x^3\right )^p \, dx=\int \left (A + B x + C x^{2}\right ) \left (x^{3} + 4 x^{2} + 3 x + 2\right )^{p}\, dx \] Input:
integrate((C*x**2+B*x+A)*(x**3+4*x**2+3*x+2)**p,x)
Output:
Integral((A + B*x + C*x**2)*(x**3 + 4*x**2 + 3*x + 2)**p, x)
\[ \int \left (A+B x+C x^2\right ) \left (2+3 x+4 x^2+x^3\right )^p \, dx=\int { {\left (C x^{2} + B x + A\right )} {\left (x^{3} + 4 \, x^{2} + 3 \, x + 2\right )}^{p} \,d x } \] Input:
integrate((C*x^2+B*x+A)*(x^3+4*x^2+3*x+2)^p,x, algorithm="maxima")
Output:
integrate((C*x^2 + B*x + A)*(x^3 + 4*x^2 + 3*x + 2)^p, x)
\[ \int \left (A+B x+C x^2\right ) \left (2+3 x+4 x^2+x^3\right )^p \, dx=\int { {\left (C x^{2} + B x + A\right )} {\left (x^{3} + 4 \, x^{2} + 3 \, x + 2\right )}^{p} \,d x } \] Input:
integrate((C*x^2+B*x+A)*(x^3+4*x^2+3*x+2)^p,x, algorithm="giac")
Output:
integrate((C*x^2 + B*x + A)*(x^3 + 4*x^2 + 3*x + 2)^p, x)
Timed out. \[ \int \left (A+B x+C x^2\right ) \left (2+3 x+4 x^2+x^3\right )^p \, dx=\int \left (C\,x^2+B\,x+A\right )\,{\left (x^3+4\,x^2+3\,x+2\right )}^p \,d x \] Input:
int((A + B*x + C*x^2)*(3*x + 4*x^2 + x^3 + 2)^p,x)
Output:
int((A + B*x + C*x^2)*(3*x + 4*x^2 + x^3 + 2)^p, x)
\[ \int \left (A+B x+C x^2\right ) \left (2+3 x+4 x^2+x^3\right )^p \, dx=\text {too large to display} \] Input:
int((C*x^2+B*x+A)*(x^3+4*x^2+3*x+2)^p,x)
Output:
(36*(x**3 + 4*x**2 + 3*x + 2)**p*a*p**2*x + 27*(x**3 + 4*x**2 + 3*x + 2)** p*a*p**2 + 60*(x**3 + 4*x**2 + 3*x + 2)**p*a*p*x + 45*(x**3 + 4*x**2 + 3*x + 2)**p*a*p + 24*(x**3 + 4*x**2 + 3*x + 2)**p*a*x + 18*(x**3 + 4*x**2 + 3 *x + 2)**p*a + 36*(x**3 + 4*x**2 + 3*x + 2)**p*b*p**2*x**2 + 48*(x**3 + 4* x**2 + 3*x + 2)**p*b*p**2*x + 9*(x**3 + 4*x**2 + 3*x + 2)**p*b*p**2 + 48*( x**3 + 4*x**2 + 3*x + 2)**p*b*p*x**2 + 48*(x**3 + 4*x**2 + 3*x + 2)**p*b*p *x + 12*(x**3 + 4*x**2 + 3*x + 2)**p*b*x**2 - 9*(x**3 + 4*x**2 + 3*x + 2)* *p*b + 36*(x**3 + 4*x**2 + 3*x + 2)**p*c*p**2*x**3 + 48*(x**3 + 4*x**2 + 3 *x + 2)**p*c*p**2*x**2 - 56*(x**3 + 4*x**2 + 3*x + 2)**p*c*p**2*x + 21*(x* *3 + 4*x**2 + 3*x + 2)**p*c*p**2 + 36*(x**3 + 4*x**2 + 3*x + 2)**p*c*p*x** 3 + 16*(x**3 + 4*x**2 + 3*x + 2)**p*c*p*x**2 - 80*(x**3 + 4*x**2 + 3*x + 2 )**p*c*p*x + 27*(x**3 + 4*x**2 + 3*x + 2)**p*c*p + 8*(x**3 + 4*x**2 + 3*x + 2)**p*c*x**3 + 22*(x**3 + 4*x**2 + 3*x + 2)**p*c + 1215*int((x**3 + 4*x* *2 + 3*x + 2)**p/(9*p**2*x**3 + 36*p**2*x**2 + 27*p**2*x + 18*p**2 + 9*p*x **3 + 36*p*x**2 + 27*p*x + 18*p + 2*x**3 + 8*x**2 + 6*x + 4),x)*a*p**5 + 3 240*int((x**3 + 4*x**2 + 3*x + 2)**p/(9*p**2*x**3 + 36*p**2*x**2 + 27*p**2 *x + 18*p**2 + 9*p*x**3 + 36*p*x**2 + 27*p*x + 18*p + 2*x**3 + 8*x**2 + 6* x + 4),x)*a*p**4 + 3105*int((x**3 + 4*x**2 + 3*x + 2)**p/(9*p**2*x**3 + 36 *p**2*x**2 + 27*p**2*x + 18*p**2 + 9*p*x**3 + 36*p*x**2 + 27*p*x + 18*p + 2*x**3 + 8*x**2 + 6*x + 4),x)*a*p**3 + 1260*int((x**3 + 4*x**2 + 3*x + ...