Integrand size = 26, antiderivative size = 520 \[ \int \left (A+B x+C x^2\right ) \left (2+3 x-5 x^2+x^3\right )^p \, dx =\text {Too large to display} \] Output:
C*(x^3-5*x^2+3*x+2)^(p+1)/(3*p+3)+3^(-3+1/2*p)*(3*B+10*C)*(x^3-5*x^2+3*x+2 )^p*AppellF1(2+p,-p,-p,3+p,(5-3*x+8*sin(1/3*Pi+1/3*arcsin(61/128)))/(8*sin (1/3*arcsin(61/128))+8*sin(1/3*Pi+1/3*arcsin(61/128))),1/24*sec(1/3*arcsin (61/128))*(5-3*x+8*sin(1/3*Pi+1/3*arcsin(61/128)))*3^(1/2))*(32*cos(1/3*ar csin(61/128))*(3^(1/2)*cos(1/3*arcsin(61/128))+3*sin(1/3*arcsin(61/128)))) ^p*(5-3*x+8*sin(1/3*Pi+1/3*arcsin(61/128)))^2/(2+p)/((-5+3*x+8*cos(1/6*Pi+ 1/3*arcsin(61/128)))^p)/((-5+3*x+8*sin(1/3*arcsin(61/128)))^p)-3^(-3+1/2*p )*(x^3-5*x^2+3*x+2)^p*AppellF1(p+1,-p,-p,2+p,(5-3*x+8*sin(1/3*Pi+1/3*arcsi n(61/128)))/(8*sin(1/3*arcsin(61/128))+8*sin(1/3*Pi+1/3*arcsin(61/128))),1 /24*sec(1/3*arcsin(61/128))*(5-3*x+8*sin(1/3*Pi+1/3*arcsin(61/128)))*3^(1/ 2))*(32*cos(1/3*arcsin(61/128))*(3^(1/2)*cos(1/3*arcsin(61/128))+3*sin(1/3 *arcsin(61/128))))^p*(5-3*x+8*sin(1/3*Pi+1/3*arcsin(61/128)))*(9*A+15*B+41 *C+8*(3*B+10*C)*sin(1/3*Pi+1/3*arcsin(61/128)))/(p+1)/((-5+3*x+8*cos(1/6*P i+1/3*arcsin(61/128)))^p)/((-5+3*x+8*sin(1/3*arcsin(61/128)))^p)
\[ \int \left (A+B x+C x^2\right ) \left (2+3 x-5 x^2+x^3\right )^p \, dx=\int \left (A+B x+C x^2\right ) \left (2+3 x-5 x^2+x^3\right )^p \, dx \] Input:
Integrate[(A + B*x + C*x^2)*(2 + 3*x - 5*x^2 + x^3)^p,x]
Output:
Integrate[(A + B*x + C*x^2)*(2 + 3*x - 5*x^2 + x^3)^p, x]
Result contains complex when optimal does not.
Time = 4.15 (sec) , antiderivative size = 1899, normalized size of antiderivative = 3.65, 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.
\(\displaystyle \int \left (x^3-5 x^2+3 x+2\right )^p \left (A+B x+C x^2\right ) \, dx\) |
\(\Big \downarrow \) 2526 |
\(\displaystyle \frac {1}{3} \int (3 (A-C)+(3 B+10 C) x) \left (x^3-5 x^2+3 x+2\right )^pdx+\frac {C \left (x^3-5 x^2+3 x+2\right )^{p+1}}{3 (p+1)}\) |
\(\Big \downarrow \) 2490 |
\(\displaystyle \frac {1}{3} \int \left (\frac {1}{3} (9 (A-C)+5 (3 B+10 C))+(3 B+10 C) \left (x-\frac {5}{3}\right )\right ) \left (\left (x-\frac {5}{3}\right )^3-\frac {16}{3} \left (x-\frac {5}{3}\right )-\frac {61}{27}\right )^pd\left (x-\frac {5}{3}\right )+\frac {C \left (x^3-5 x^2+3 x+2\right )^{p+1}}{3 (p+1)}\) |
\(\Big \downarrow \) 2486 |
\(\displaystyle \frac {C \left (x^3-5 x^2+3 x+2\right )^{p+1}}{3 (p+1)}+\frac {1}{3} \left (\left (x-\frac {5}{3}\right )^2+\frac {1}{3} \left (\frac {16}{\sqrt [3]{\frac {1}{2} \left (61+3 i \sqrt {1407}\right )}}+\sqrt [3]{\frac {1}{2} \left (61+3 i \sqrt {1407}\right )}\right ) \left (x-\frac {5}{3}\right )+\frac {1}{18} \left (-32+\frac {512}{\left (\frac {1}{2} \left (61+3 i \sqrt {1407}\right )\right )^{2/3}}+\sqrt [3]{2} \left (61+3 i \sqrt {1407}\right )^{2/3}\right )\right )^{-p} \left (\left (x-\frac {5}{3}\right )^3-\frac {16}{3} \left (x-\frac {5}{3}\right )-\frac {61}{27}\right )^p \left (x-\frac {32 \sqrt [3]{2}+\left (122+6 i \sqrt {1407}\right )^{2/3}}{6 \sqrt [3]{61+3 i \sqrt {1407}}}-\frac {5}{3}\right )^{-p} \int \frac {1}{3} \left (9 A+15 B+41 C+3 (3 B+10 C) \left (x-\frac {5}{3}\right )\right ) \left (\left (x-\frac {5}{3}\right )^2+\frac {1}{3} \left (\frac {16}{\sqrt [3]{\frac {1}{2} \left (61+3 i \sqrt {1407}\right )}}+\sqrt [3]{\frac {1}{2} \left (61+3 i \sqrt {1407}\right )}\right ) \left (x-\frac {5}{3}\right )+\frac {1}{18} \left (-32+\frac {512}{\left (\frac {1}{2} \left (61+3 i \sqrt {1407}\right )\right )^{2/3}}+\sqrt [3]{2} \left (61+3 i \sqrt {1407}\right )^{2/3}\right )\right )^p \left (x-\frac {32 \sqrt [3]{2}+\left (122+6 i \sqrt {1407}\right )^{2/3}}{6 \sqrt [3]{61+3 i \sqrt {1407}}}-\frac {5}{3}\right )^pd\left (x-\frac {5}{3}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {C \left (x^3-5 x^2+3 x+2\right )^{p+1}}{3 (p+1)}+\frac {1}{9} \left (\left (x-\frac {5}{3}\right )^2+\frac {1}{3} \left (\frac {16}{\sqrt [3]{\frac {1}{2} \left (61+3 i \sqrt {1407}\right )}}+\sqrt [3]{\frac {1}{2} \left (61+3 i \sqrt {1407}\right )}\right ) \left (x-\frac {5}{3}\right )+\frac {1}{18} \left (-32+\frac {512}{\left (\frac {1}{2} \left (61+3 i \sqrt {1407}\right )\right )^{2/3}}+\sqrt [3]{2} \left (61+3 i \sqrt {1407}\right )^{2/3}\right )\right )^{-p} \left (\left (x-\frac {5}{3}\right )^3-\frac {16}{3} \left (x-\frac {5}{3}\right )-\frac {61}{27}\right )^p \left (x-\frac {32 \sqrt [3]{2}+\left (122+6 i \sqrt {1407}\right )^{2/3}}{6 \sqrt [3]{61+3 i \sqrt {1407}}}-\frac {5}{3}\right )^{-p} \int \left (9 A+15 B+41 C+3 (3 B+10 C) \left (x-\frac {5}{3}\right )\right ) \left (\left (x-\frac {5}{3}\right )^2+\frac {1}{3} \left (\frac {16}{\sqrt [3]{\frac {1}{2} \left (61+3 i \sqrt {1407}\right )}}+\sqrt [3]{\frac {1}{2} \left (61+3 i \sqrt {1407}\right )}\right ) \left (x-\frac {5}{3}\right )+\frac {1}{18} \left (-32+\frac {512}{\left (\frac {1}{2} \left (61+3 i \sqrt {1407}\right )\right )^{2/3}}+\sqrt [3]{2} \left (61+3 i \sqrt {1407}\right )^{2/3}\right )\right )^p \left (x-\frac {32 \sqrt [3]{2}+\left (122+6 i \sqrt {1407}\right )^{2/3}}{6 \sqrt [3]{61+3 i \sqrt {1407}}}-\frac {5}{3}\right )^pd\left (x-\frac {5}{3}\right )\) |
\(\Big \downarrow \) 1269 |
\(\displaystyle \frac {C \left (x^3-5 x^2+3 x+2\right )^{p+1}}{3 (p+1)}+\frac {1}{9} \left (\left (x-\frac {5}{3}\right )^2+\frac {1}{3} \left (\frac {16}{\sqrt [3]{\frac {1}{2} \left (61+3 i \sqrt {1407}\right )}}+\sqrt [3]{\frac {1}{2} \left (61+3 i \sqrt {1407}\right )}\right ) \left (x-\frac {5}{3}\right )+\frac {1}{18} \left (-32+\frac {512}{\left (\frac {1}{2} \left (61+3 i \sqrt {1407}\right )\right )^{2/3}}+\sqrt [3]{2} \left (61+3 i \sqrt {1407}\right )^{2/3}\right )\right )^{-p} \left (\left (x-\frac {5}{3}\right )^3-\frac {16}{3} \left (x-\frac {5}{3}\right )-\frac {61}{27}\right )^p \left (x-\frac {32 \sqrt [3]{2}+\left (122+6 i \sqrt {1407}\right )^{2/3}}{6 \sqrt [3]{61+3 i \sqrt {1407}}}-\frac {5}{3}\right )^{-p} \left (\frac {1}{2} \left (18 A+\frac {\left (32 \sqrt [3]{2}+\left (122+6 i \sqrt {1407}\right )^{2/3}\right ) (3 B+10 C)}{\sqrt [3]{61+3 i \sqrt {1407}}}+30 B+82 C\right ) \int \left (\left (x-\frac {5}{3}\right )^2+\frac {1}{3} \left (\frac {16}{\sqrt [3]{\frac {1}{2} \left (61+3 i \sqrt {1407}\right )}}+\sqrt [3]{\frac {1}{2} \left (61+3 i \sqrt {1407}\right )}\right ) \left (x-\frac {5}{3}\right )+\frac {1}{18} \left (-32+\frac {512}{\left (\frac {1}{2} \left (61+3 i \sqrt {1407}\right )\right )^{2/3}}+\sqrt [3]{2} \left (61+3 i \sqrt {1407}\right )^{2/3}\right )\right )^p \left (x-\frac {32 \sqrt [3]{2}+\left (122+6 i \sqrt {1407}\right )^{2/3}}{6 \sqrt [3]{61+3 i \sqrt {1407}}}-\frac {5}{3}\right )^pd\left (x-\frac {5}{3}\right )+3 (3 B+10 C) \int \left (\left (x-\frac {5}{3}\right )^2+\frac {1}{3} \left (\frac {16}{\sqrt [3]{\frac {1}{2} \left (61+3 i \sqrt {1407}\right )}}+\sqrt [3]{\frac {1}{2} \left (61+3 i \sqrt {1407}\right )}\right ) \left (x-\frac {5}{3}\right )+\frac {1}{18} \left (-32+\frac {512}{\left (\frac {1}{2} \left (61+3 i \sqrt {1407}\right )\right )^{2/3}}+\sqrt [3]{2} \left (61+3 i \sqrt {1407}\right )^{2/3}\right )\right )^p \left (x-\frac {32 \sqrt [3]{2}+\left (122+6 i \sqrt {1407}\right )^{2/3}}{6 \sqrt [3]{61+3 i \sqrt {1407}}}-\frac {5}{3}\right )^{p+1}d\left (x-\frac {5}{3}\right )\right )\) |
\(\Big \downarrow \) 1179 |
\(\displaystyle \frac {1}{9} \left (\left (x-\frac {5}{3}\right )^2+\frac {1}{3} \left (\frac {16}{\sqrt [3]{\frac {1}{2} \left (61+3 i \sqrt {1407}\right )}}+\sqrt [3]{\frac {1}{2} \left (61+3 i \sqrt {1407}\right )}\right ) \left (x-\frac {5}{3}\right )+\frac {1}{18} \left (-32+\frac {512}{\left (\frac {1}{2} \left (61+3 i \sqrt {1407}\right )\right )^{2/3}}+\sqrt [3]{2} \left (61+3 i \sqrt {1407}\right )^{2/3}\right )\right )^{-p} \left (\left (x-\frac {5}{3}\right )^3-\frac {16}{3} \left (x-\frac {5}{3}\right )-\frac {61}{27}\right )^p \left (\frac {1}{2} \left (18 A+30 B+82 C+\frac {\left (32 \sqrt [3]{2}+\left (122+6 i \sqrt {1407}\right )^{2/3}\right ) (3 B+10 C)}{\sqrt [3]{61+3 i \sqrt {1407}}}\right ) \left (1-\frac {2^{2/3} \sqrt [3]{61+3 i \sqrt {1407}} \left (\frac {32 \sqrt [3]{2}+\left (122+6 i \sqrt {1407}\right )^{2/3}}{\sqrt [3]{61+3 i \sqrt {1407}}}-6 \left (x-\frac {5}{3}\right )\right )}{96+3 \sqrt [3]{2} \left (61+3 i \sqrt {1407}\right )^{2/3}-\sqrt [6]{2} \sqrt {3 \left (-512 2^{2/3}+64 \left (61+3 i \sqrt {1407}\right )^{2/3}-\sqrt [3]{2} \left (61+3 i \sqrt {1407}\right )^{4/3}\right )}}\right )^{-p} \left (\left (x-\frac {5}{3}\right )^2+\frac {1}{3} \left (\frac {16}{\sqrt [3]{\frac {1}{2} \left (61+3 i \sqrt {1407}\right )}}+\sqrt [3]{\frac {1}{2} \left (61+3 i \sqrt {1407}\right )}\right ) \left (x-\frac {5}{3}\right )+\frac {1}{18} \left (-32+\frac {512}{\left (\frac {1}{2} \left (61+3 i \sqrt {1407}\right )\right )^{2/3}}+\sqrt [3]{2} \left (61+3 i \sqrt {1407}\right )^{2/3}\right )\right )^p \int \left (x-\frac {32 \sqrt [3]{2}+\left (122+6 i \sqrt {1407}\right )^{2/3}}{6 \sqrt [3]{61+3 i \sqrt {1407}}}-\frac {5}{3}\right )^p \left (\frac {6\ 2^{2/3} \sqrt [3]{61+3 i \sqrt {1407}} \left (x-\frac {32 \sqrt [3]{2}+\left (122+6 i \sqrt {1407}\right )^{2/3}}{6 \sqrt [3]{61+3 i \sqrt {1407}}}-\frac {5}{3}\right )}{96+3 \sqrt [3]{2} \left (61+3 i \sqrt {1407}\right )^{2/3}-\sqrt [6]{2} \sqrt {3 \left (-512 2^{2/3}+64 \left (61+3 i \sqrt {1407}\right )^{2/3}-\sqrt [3]{2} \left (61+3 i \sqrt {1407}\right )^{4/3}\right )}}+1\right )^p \left (\frac {6\ 2^{2/3} \sqrt [3]{61+3 i \sqrt {1407}} \left (x-\frac {32 \sqrt [3]{2}+\left (122+6 i \sqrt {1407}\right )^{2/3}}{6 \sqrt [3]{61+3 i \sqrt {1407}}}-\frac {5}{3}\right )}{96+3 \sqrt [3]{2} \left (61+3 i \sqrt {1407}\right )^{2/3}+\sqrt [6]{2} \sqrt {3 \left (-512 2^{2/3}+64 \left (61+3 i \sqrt {1407}\right )^{2/3}-\sqrt [3]{2} \left (61+3 i \sqrt {1407}\right )^{4/3}\right )}}+1\right )^pd\left (x-\frac {32 \sqrt [3]{2}+\left (122+6 i \sqrt {1407}\right )^{2/3}}{6 \sqrt [3]{61+3 i \sqrt {1407}}}-\frac {5}{3}\right ) \left (1-\frac {2^{2/3} \sqrt [3]{61+3 i \sqrt {1407}} \left (\frac {32 \sqrt [3]{2}+\left (122+6 i \sqrt {1407}\right )^{2/3}}{\sqrt [3]{61+3 i \sqrt {1407}}}-6 \left (x-\frac {5}{3}\right )\right )}{96+3 \sqrt [3]{2} \left (61+3 i \sqrt {1407}\right )^{2/3}+\sqrt [6]{2} \sqrt {3 \left (-512 2^{2/3}+64 \left (61+3 i \sqrt {1407}\right )^{2/3}-\sqrt [3]{2} \left (61+3 i \sqrt {1407}\right )^{4/3}\right )}}\right )^{-p}+3 (3 B+10 C) \left (1-\frac {2^{2/3} \sqrt [3]{61+3 i \sqrt {1407}} \left (\frac {32 \sqrt [3]{2}+\left (122+6 i \sqrt {1407}\right )^{2/3}}{\sqrt [3]{61+3 i \sqrt {1407}}}-6 \left (x-\frac {5}{3}\right )\right )}{96+3 \sqrt [3]{2} \left (61+3 i \sqrt {1407}\right )^{2/3}-\sqrt [6]{2} \sqrt {3 \left (-512 2^{2/3}+64 \left (61+3 i \sqrt {1407}\right )^{2/3}-\sqrt [3]{2} \left (61+3 i \sqrt {1407}\right )^{4/3}\right )}}\right )^{-p} \left (\left (x-\frac {5}{3}\right )^2+\frac {1}{3} \left (\frac {16}{\sqrt [3]{\frac {1}{2} \left (61+3 i \sqrt {1407}\right )}}+\sqrt [3]{\frac {1}{2} \left (61+3 i \sqrt {1407}\right )}\right ) \left (x-\frac {5}{3}\right )+\frac {1}{18} \left (-32+\frac {512}{\left (\frac {1}{2} \left (61+3 i \sqrt {1407}\right )\right )^{2/3}}+\sqrt [3]{2} \left (61+3 i \sqrt {1407}\right )^{2/3}\right )\right )^p \int \left (x-\frac {32 \sqrt [3]{2}+\left (122+6 i \sqrt {1407}\right )^{2/3}}{6 \sqrt [3]{61+3 i \sqrt {1407}}}-\frac {5}{3}\right )^{p+1} \left (\frac {6\ 2^{2/3} \sqrt [3]{61+3 i \sqrt {1407}} \left (x-\frac {32 \sqrt [3]{2}+\left (122+6 i \sqrt {1407}\right )^{2/3}}{6 \sqrt [3]{61+3 i \sqrt {1407}}}-\frac {5}{3}\right )}{96+3 \sqrt [3]{2} \left (61+3 i \sqrt {1407}\right )^{2/3}-\sqrt [6]{2} \sqrt {3 \left (-512 2^{2/3}+64 \left (61+3 i \sqrt {1407}\right )^{2/3}-\sqrt [3]{2} \left (61+3 i \sqrt {1407}\right )^{4/3}\right )}}+1\right )^p \left (\frac {6\ 2^{2/3} \sqrt [3]{61+3 i \sqrt {1407}} \left (x-\frac {32 \sqrt [3]{2}+\left (122+6 i \sqrt {1407}\right )^{2/3}}{6 \sqrt [3]{61+3 i \sqrt {1407}}}-\frac {5}{3}\right )}{96+3 \sqrt [3]{2} \left (61+3 i \sqrt {1407}\right )^{2/3}+\sqrt [6]{2} \sqrt {3 \left (-512 2^{2/3}+64 \left (61+3 i \sqrt {1407}\right )^{2/3}-\sqrt [3]{2} \left (61+3 i \sqrt {1407}\right )^{4/3}\right )}}+1\right )^pd\left (x-\frac {32 \sqrt [3]{2}+\left (122+6 i \sqrt {1407}\right )^{2/3}}{6 \sqrt [3]{61+3 i \sqrt {1407}}}-\frac {5}{3}\right ) \left (1-\frac {2^{2/3} \sqrt [3]{61+3 i \sqrt {1407}} \left (\frac {32 \sqrt [3]{2}+\left (122+6 i \sqrt {1407}\right )^{2/3}}{\sqrt [3]{61+3 i \sqrt {1407}}}-6 \left (x-\frac {5}{3}\right )\right )}{96+3 \sqrt [3]{2} \left (61+3 i \sqrt {1407}\right )^{2/3}+\sqrt [6]{2} \sqrt {3 \left (-512 2^{2/3}+64 \left (61+3 i \sqrt {1407}\right )^{2/3}-\sqrt [3]{2} \left (61+3 i \sqrt {1407}\right )^{4/3}\right )}}\right )^{-p}\right ) \left (x-\frac {32 \sqrt [3]{2}+\left (122+6 i \sqrt {1407}\right )^{2/3}}{6 \sqrt [3]{61+3 i \sqrt {1407}}}-\frac {5}{3}\right )^{-p}+\frac {C \left (x^3-5 x^2+3 x+2\right )^{p+1}}{3 (p+1)}\) |
\(\Big \downarrow \) 150 |
\(\displaystyle \frac {1}{9} \left (\left (x-\frac {5}{3}\right )^2+\frac {1}{3} \left (\frac {16}{\sqrt [3]{\frac {1}{2} \left (61+3 i \sqrt {1407}\right )}}+\sqrt [3]{\frac {1}{2} \left (61+3 i \sqrt {1407}\right )}\right ) \left (x-\frac {5}{3}\right )+\frac {1}{18} \left (-32+\frac {512}{\left (\frac {1}{2} \left (61+3 i \sqrt {1407}\right )\right )^{2/3}}+\sqrt [3]{2} \left (61+3 i \sqrt {1407}\right )^{2/3}\right )\right )^{-p} \left (\left (x-\frac {5}{3}\right )^3-\frac {16}{3} \left (x-\frac {5}{3}\right )-\frac {61}{27}\right )^p \left (\frac {\left (18 A+30 B+82 C+\frac {\left (32 \sqrt [3]{2}+\left (122+6 i \sqrt {1407}\right )^{2/3}\right ) (3 B+10 C)}{\sqrt [3]{61+3 i \sqrt {1407}}}\right ) \left (1-\frac {2^{2/3} \sqrt [3]{61+3 i \sqrt {1407}} \left (\frac {32 \sqrt [3]{2}+\left (122+6 i \sqrt {1407}\right )^{2/3}}{\sqrt [3]{61+3 i \sqrt {1407}}}-6 \left (x-\frac {5}{3}\right )\right )}{96+3 \sqrt [3]{2} \left (61+3 i \sqrt {1407}\right )^{2/3}-\sqrt [6]{2} \sqrt {3 \left (-512 2^{2/3}+64 \left (61+3 i \sqrt {1407}\right )^{2/3}-\sqrt [3]{2} \left (61+3 i \sqrt {1407}\right )^{4/3}\right )}}\right )^{-p} \left (\left (x-\frac {5}{3}\right )^2+\frac {1}{3} \left (\frac {16}{\sqrt [3]{\frac {1}{2} \left (61+3 i \sqrt {1407}\right )}}+\sqrt [3]{\frac {1}{2} \left (61+3 i \sqrt {1407}\right )}\right ) \left (x-\frac {5}{3}\right )+\frac {1}{18} \left (-32+\frac {512}{\left (\frac {1}{2} \left (61+3 i \sqrt {1407}\right )\right )^{2/3}}+\sqrt [3]{2} \left (61+3 i \sqrt {1407}\right )^{2/3}\right )\right )^p \left (x-\frac {32 \sqrt [3]{2}+\left (122+6 i \sqrt {1407}\right )^{2/3}}{6 \sqrt [3]{61+3 i \sqrt {1407}}}-\frac {5}{3}\right )^{p+1} \operatorname {AppellF1}\left (p+1,-p,-p,p+2,-\frac {6\ 2^{2/3} \sqrt [3]{61+3 i \sqrt {1407}} \left (x-\frac {32 \sqrt [3]{2}+\left (122+6 i \sqrt {1407}\right )^{2/3}}{6 \sqrt [3]{61+3 i \sqrt {1407}}}-\frac {5}{3}\right )}{96+3 \sqrt [3]{2} \left (61+3 i \sqrt {1407}\right )^{2/3}-\sqrt [6]{2} \sqrt {3 \left (-512 2^{2/3}+64 \left (61+3 i \sqrt {1407}\right )^{2/3}-\sqrt [3]{2} \left (61+3 i \sqrt {1407}\right )^{4/3}\right )}},-\frac {6\ 2^{2/3} \sqrt [3]{61+3 i \sqrt {1407}} \left (x-\frac {32 \sqrt [3]{2}+\left (122+6 i \sqrt {1407}\right )^{2/3}}{6 \sqrt [3]{61+3 i \sqrt {1407}}}-\frac {5}{3}\right )}{96+3 \sqrt [3]{2} \left (61+3 i \sqrt {1407}\right )^{2/3}+\sqrt [6]{2} \sqrt {3 \left (-512 2^{2/3}+64 \left (61+3 i \sqrt {1407}\right )^{2/3}-\sqrt [3]{2} \left (61+3 i \sqrt {1407}\right )^{4/3}\right )}}\right ) \left (1-\frac {2^{2/3} \sqrt [3]{61+3 i \sqrt {1407}} \left (\frac {32 \sqrt [3]{2}+\left (122+6 i \sqrt {1407}\right )^{2/3}}{\sqrt [3]{61+3 i \sqrt {1407}}}-6 \left (x-\frac {5}{3}\right )\right )}{96+3 \sqrt [3]{2} \left (61+3 i \sqrt {1407}\right )^{2/3}+\sqrt [6]{2} \sqrt {3 \left (-512 2^{2/3}+64 \left (61+3 i \sqrt {1407}\right )^{2/3}-\sqrt [3]{2} \left (61+3 i \sqrt {1407}\right )^{4/3}\right )}}\right )^{-p}}{2 (p+1)}+\frac {3 (3 B+10 C) \left (1-\frac {2^{2/3} \sqrt [3]{61+3 i \sqrt {1407}} \left (\frac {32 \sqrt [3]{2}+\left (122+6 i \sqrt {1407}\right )^{2/3}}{\sqrt [3]{61+3 i \sqrt {1407}}}-6 \left (x-\frac {5}{3}\right )\right )}{96+3 \sqrt [3]{2} \left (61+3 i \sqrt {1407}\right )^{2/3}-\sqrt [6]{2} \sqrt {3 \left (-512 2^{2/3}+64 \left (61+3 i \sqrt {1407}\right )^{2/3}-\sqrt [3]{2} \left (61+3 i \sqrt {1407}\right )^{4/3}\right )}}\right )^{-p} \left (\left (x-\frac {5}{3}\right )^2+\frac {1}{3} \left (\frac {16}{\sqrt [3]{\frac {1}{2} \left (61+3 i \sqrt {1407}\right )}}+\sqrt [3]{\frac {1}{2} \left (61+3 i \sqrt {1407}\right )}\right ) \left (x-\frac {5}{3}\right )+\frac {1}{18} \left (-32+\frac {512}{\left (\frac {1}{2} \left (61+3 i \sqrt {1407}\right )\right )^{2/3}}+\sqrt [3]{2} \left (61+3 i \sqrt {1407}\right )^{2/3}\right )\right )^p \left (x-\frac {32 \sqrt [3]{2}+\left (122+6 i \sqrt {1407}\right )^{2/3}}{6 \sqrt [3]{61+3 i \sqrt {1407}}}-\frac {5}{3}\right )^{p+2} \operatorname {AppellF1}\left (p+2,-p,-p,p+3,-\frac {6\ 2^{2/3} \sqrt [3]{61+3 i \sqrt {1407}} \left (x-\frac {32 \sqrt [3]{2}+\left (122+6 i \sqrt {1407}\right )^{2/3}}{6 \sqrt [3]{61+3 i \sqrt {1407}}}-\frac {5}{3}\right )}{96+3 \sqrt [3]{2} \left (61+3 i \sqrt {1407}\right )^{2/3}-\sqrt [6]{2} \sqrt {3 \left (-512 2^{2/3}+64 \left (61+3 i \sqrt {1407}\right )^{2/3}-\sqrt [3]{2} \left (61+3 i \sqrt {1407}\right )^{4/3}\right )}},-\frac {6\ 2^{2/3} \sqrt [3]{61+3 i \sqrt {1407}} \left (x-\frac {32 \sqrt [3]{2}+\left (122+6 i \sqrt {1407}\right )^{2/3}}{6 \sqrt [3]{61+3 i \sqrt {1407}}}-\frac {5}{3}\right )}{96+3 \sqrt [3]{2} \left (61+3 i \sqrt {1407}\right )^{2/3}+\sqrt [6]{2} \sqrt {3 \left (-512 2^{2/3}+64 \left (61+3 i \sqrt {1407}\right )^{2/3}-\sqrt [3]{2} \left (61+3 i \sqrt {1407}\right )^{4/3}\right )}}\right ) \left (1-\frac {2^{2/3} \sqrt [3]{61+3 i \sqrt {1407}} \left (\frac {32 \sqrt [3]{2}+\left (122+6 i \sqrt {1407}\right )^{2/3}}{\sqrt [3]{61+3 i \sqrt {1407}}}-6 \left (x-\frac {5}{3}\right )\right )}{96+3 \sqrt [3]{2} \left (61+3 i \sqrt {1407}\right )^{2/3}+\sqrt [6]{2} \sqrt {3 \left (-512 2^{2/3}+64 \left (61+3 i \sqrt {1407}\right )^{2/3}-\sqrt [3]{2} \left (61+3 i \sqrt {1407}\right )^{4/3}\right )}}\right )^{-p}}{p+2}\right ) \left (x-\frac {32 \sqrt [3]{2}+\left (122+6 i \sqrt {1407}\right )^{2/3}}{6 \sqrt [3]{61+3 i \sqrt {1407}}}-\frac {5}{3}\right )^{-p}+\frac {C \left (x^3-5 x^2+3 x+2\right )^{p+1}}{3 (p+1)}\) |
Input:
Int[(A + B*x + C*x^2)*(2 + 3*x - 5*x^2 + x^3)^p,x]
Output:
(C*(2 + 3*x - 5*x^2 + x^3)^(1 + p))/(3*(1 + p)) + ((-61/27 - (16*(-5/3 + x ))/3 + (-5/3 + x)^3)^p*(((18*A + 30*B + 82*C + ((32*2^(1/3) + (122 + (6*I) *Sqrt[1407])^(2/3))*(3*B + 10*C))/(61 + (3*I)*Sqrt[1407])^(1/3))*((-32 + 5 12/((61 + (3*I)*Sqrt[1407])/2)^(2/3) + 2^(1/3)*(61 + (3*I)*Sqrt[1407])^(2/ 3))/18 + ((16/((61 + (3*I)*Sqrt[1407])/2)^(1/3) + ((61 + (3*I)*Sqrt[1407]) /2)^(1/3))*(-5/3 + x))/3 + (-5/3 + x)^2)^p*(-5/3 - (32*2^(1/3) + (122 + (6 *I)*Sqrt[1407])^(2/3))/(6*(61 + (3*I)*Sqrt[1407])^(1/3)) + x)^(1 + p)*Appe llF1[1 + p, -p, -p, 2 + p, (-6*2^(2/3)*(61 + (3*I)*Sqrt[1407])^(1/3)*(-5/3 - (32*2^(1/3) + (122 + (6*I)*Sqrt[1407])^(2/3))/(6*(61 + (3*I)*Sqrt[1407] )^(1/3)) + x))/(96 + 3*2^(1/3)*(61 + (3*I)*Sqrt[1407])^(2/3) - 2^(1/6)*Sqr t[3*(-512*2^(2/3) + 64*(61 + (3*I)*Sqrt[1407])^(2/3) - 2^(1/3)*(61 + (3*I) *Sqrt[1407])^(4/3))]), (-6*2^(2/3)*(61 + (3*I)*Sqrt[1407])^(1/3)*(-5/3 - ( 32*2^(1/3) + (122 + (6*I)*Sqrt[1407])^(2/3))/(6*(61 + (3*I)*Sqrt[1407])^(1 /3)) + x))/(96 + 3*2^(1/3)*(61 + (3*I)*Sqrt[1407])^(2/3) + 2^(1/6)*Sqrt[3* (-512*2^(2/3) + 64*(61 + (3*I)*Sqrt[1407])^(2/3) - 2^(1/3)*(61 + (3*I)*Sqr t[1407])^(4/3))])])/(2*(1 + p)*(1 - (2^(2/3)*(61 + (3*I)*Sqrt[1407])^(1/3) *((32*2^(1/3) + (122 + (6*I)*Sqrt[1407])^(2/3))/(61 + (3*I)*Sqrt[1407])^(1 /3) - 6*(-5/3 + x)))/(96 + 3*2^(1/3)*(61 + (3*I)*Sqrt[1407])^(2/3) - 2^(1/ 6)*Sqrt[3*(-512*2^(2/3) + 64*(61 + (3*I)*Sqrt[1407])^(2/3) - 2^(1/3)*(61 + (3*I)*Sqrt[1407])^(4/3))]))^p*(1 - (2^(2/3)*(61 + (3*I)*Sqrt[1407])^(1...
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]
\[\int \left (C \,x^{2}+B x +A \right ) \left (x^{3}-5 x^{2}+3 x +2\right )^{p}d x\]
Input:
int((C*x^2+B*x+A)*(x^3-5*x^2+3*x+2)^p,x)
Output:
int((C*x^2+B*x+A)*(x^3-5*x^2+3*x+2)^p,x)
\[ \int \left (A+B x+C x^2\right ) \left (2+3 x-5 x^2+x^3\right )^p \, dx=\int { {\left (C x^{2} + B x + A\right )} {\left (x^{3} - 5 \, x^{2} + 3 \, x + 2\right )}^{p} \,d x } \] Input:
integrate((C*x^2+B*x+A)*(x^3-5*x^2+3*x+2)^p,x, algorithm="fricas")
Output:
integral((C*x^2 + B*x + A)*(x^3 - 5*x^2 + 3*x + 2)^p, x)
Timed out. \[ \int \left (A+B x+C x^2\right ) \left (2+3 x-5 x^2+x^3\right )^p \, dx=\text {Timed out} \] Input:
integrate((C*x**2+B*x+A)*(x**3-5*x**2+3*x+2)**p,x)
Output:
Timed out
\[ \int \left (A+B x+C x^2\right ) \left (2+3 x-5 x^2+x^3\right )^p \, dx=\int { {\left (C x^{2} + B x + A\right )} {\left (x^{3} - 5 \, x^{2} + 3 \, x + 2\right )}^{p} \,d x } \] Input:
integrate((C*x^2+B*x+A)*(x^3-5*x^2+3*x+2)^p,x, algorithm="maxima")
Output:
integrate((C*x^2 + B*x + A)*(x^3 - 5*x^2 + 3*x + 2)^p, x)
\[ \int \left (A+B x+C x^2\right ) \left (2+3 x-5 x^2+x^3\right )^p \, dx=\int { {\left (C x^{2} + B x + A\right )} {\left (x^{3} - 5 \, x^{2} + 3 \, x + 2\right )}^{p} \,d x } \] Input:
integrate((C*x^2+B*x+A)*(x^3-5*x^2+3*x+2)^p,x, algorithm="giac")
Output:
integrate((C*x^2 + B*x + A)*(x^3 - 5*x^2 + 3*x + 2)^p, x)
Timed out. \[ \int \left (A+B x+C x^2\right ) \left (2+3 x-5 x^2+x^3\right )^p \, dx=\int \left (C\,x^2+B\,x+A\right )\,{\left (x^3-5\,x^2+3\,x+2\right )}^p \,d x \] Input:
int((A + B*x + C*x^2)*(3*x - 5*x^2 + x^3 + 2)^p,x)
Output:
int((A + B*x + C*x^2)*(3*x - 5*x^2 + x^3 + 2)^p, x)
\[ \int \left (A+B x+C x^2\right ) \left (2+3 x-5 x^2+x^3\right )^p \, dx=\text {too large to display} \] Input:
int((C*x^2+B*x+A)*(x^3-5*x^2+3*x+2)^p,x)
Output:
(90*(x**3 - 5*x**2 + 3*x + 2)**p*a*p**2*x - 54*(x**3 - 5*x**2 + 3*x + 2)** p*a*p**2 + 150*(x**3 - 5*x**2 + 3*x + 2)**p*a*p*x - 90*(x**3 - 5*x**2 + 3* x + 2)**p*a*p + 60*(x**3 - 5*x**2 + 3*x + 2)**p*a*x - 36*(x**3 - 5*x**2 + 3*x + 2)**p*a + 90*(x**3 - 5*x**2 + 3*x + 2)**p*b*p**2*x**2 - 150*(x**3 - 5*x**2 + 3*x + 2)**p*b*p**2*x - 99*(x**3 - 5*x**2 + 3*x + 2)**p*b*p**2 + 1 20*(x**3 - 5*x**2 + 3*x + 2)**p*b*p*x**2 - 150*(x**3 - 5*x**2 + 3*x + 2)** p*b*p*x - 162*(x**3 - 5*x**2 + 3*x + 2)**p*b*p + 30*(x**3 - 5*x**2 + 3*x + 2)**p*b*x**2 - 63*(x**3 - 5*x**2 + 3*x + 2)**p*b + 90*(x**3 - 5*x**2 + 3* x + 2)**p*c*p**2*x**3 - 150*(x**3 - 5*x**2 + 3*x + 2)**p*c*p**2*x**2 - 320 *(x**3 - 5*x**2 + 3*x + 2)**p*c*p**2*x - 96*(x**3 - 5*x**2 + 3*x + 2)**p*c *p**2 + 90*(x**3 - 5*x**2 + 3*x + 2)**p*c*p*x**3 - 50*(x**3 - 5*x**2 + 3*x + 2)**p*c*p*x**2 - 380*(x**3 - 5*x**2 + 3*x + 2)**p*c*p*x - 270*(x**3 - 5 *x**2 + 3*x + 2)**p*c*p + 20*(x**3 - 5*x**2 + 3*x + 2)**p*c*x**3 - 134*(x* *3 - 5*x**2 + 3*x + 2)**p*c + 6318*int((x**3 - 5*x**2 + 3*x + 2)**p/(9*p** 2*x**3 - 45*p**2*x**2 + 27*p**2*x + 18*p**2 + 9*p*x**3 - 45*p*x**2 + 27*p* x + 18*p + 2*x**3 - 10*x**2 + 6*x + 4),x)*a*p**5 + 16848*int((x**3 - 5*x** 2 + 3*x + 2)**p/(9*p**2*x**3 - 45*p**2*x**2 + 27*p**2*x + 18*p**2 + 9*p*x* *3 - 45*p*x**2 + 27*p*x + 18*p + 2*x**3 - 10*x**2 + 6*x + 4),x)*a*p**4 + 1 6146*int((x**3 - 5*x**2 + 3*x + 2)**p/(9*p**2*x**3 - 45*p**2*x**2 + 27*p** 2*x + 18*p**2 + 9*p*x**3 - 45*p*x**2 + 27*p*x + 18*p + 2*x**3 - 10*x**2...